Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 986584 12 pageshttpdxdoiorg1011552013986584

Research ArticleA Frequency Compensation Algorithm of Four-WheelCoherence Random Road

Jun Feng1 Xinjie Zhang123 Konghui Guo12 Fangwu Ma3 and Hamid Reza Karimi4

1 State Key Laboratory of Automotive Simulation and Control Jilin University Changchun Jilin 130022 China2 State Key Laboratory of Advanced Design and Manufacture for Vehicle Body Hunan University Changsha Hunan 410082 China3 Zhejiang Geely Automobile Research Institute Co Ltd Hangzhou 311228 China4Department of Engineering Faculty of Engineering and Science University of Agder 4898 Grimstad Norway

Correspondence should be addressed to Xinjie Zhang xjzhang5885gmailcom

Received 28 July 2013 Revised 15 August 2013 Accepted 15 August 2013

Academic Editor Hui Zhang

Copyright copy 2013 Jun Feng et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The road surface roughness is the main source of kinematic excitation of a moving vehicle which has an important influence onvehicle performance In recent decades random roadmodels have beenwidely studied and a four-wheel random road time domainmodel is usually generated based on the correlation of the four-wheel input in which a coherence function is used to describe thecoherence of the road input between the left and right wheels usually However during our research there are some conditionsthat show that the road PSD (power spectral density) of one wheel is smaller than the other one on the same axle Actually it iscaused by the uncorrelation between the left- and right-wheel road surface roughness Hence a frequency compensation algorithmis proposed to correct the deviation of the PSD of the road input between two wheels on the same axle and it is installed in a 7-DOFvehicle dynamic study The simulation result demonstrates the applicability of the proposed algorithm such that two-wheel roadinput deviation compensation has an important influence on vehicle performances and it can be used for a control system installedin the vehicle to compensate road roughness for damper tuning in the future

1 Introduction

The road surface roughness is the main source of kinematicexcitation of a moving vehicle which has an important influ-ence on ride comfort ride safety vehicle maneuverabilitydriverrsquos and occupantsrsquo comfort and vehicle dynamic load[1ndash3] Measurements of road surface roughness data arepreferably done with laserinertial high-speed profilometersand modern laserinertial road profilometers can record sev-eral parallel profiles simultaneously typically [4] Howeverthey are very expensive inconvenient to use susceptible toexternal factors and so forth And at the same period theroad surface roughness models have been widely investigatedand applied in vehicle dynamics researchAn accurate drivingdynamics simulation of a vehicle on a road section is onlypossible if these input models are accurate themselves [5]and scholars have been focusing on the random road surfaceexcitation simulation and reconstruction The road surfaceroughness as a stochastic process is generally described

by a PSD (power spectral density) function in a frequencydomain and in engineering practice it is regarded as azero-mean stationary (or isotropic) and ergodic Gaussianprocess In 1970s the PSD function was used by Whitehouse[6] and Shinozuka [7] to investigate the road roughnessIn 1993 Cebon [8] proposed a method based on the IFFT(Inverse Fast Fourier Transform) to discretize PSD whichis a simple fast and convenient tool for generating roadsurfaces [2 9] In 1993 Grigoriu [10] proposed a methodbased on the harmony superposition to simulate stationaryrandom processes The harmony superposition method hasan exact theory foundation and is suitable for arbitrarilystated spectral features But it is rather time consuming andhas an increasing demand on computer storage as a large setof trigonometric terms has to be calculated in a simulationprocedure In 1995 the contemporary international standardISO 8608 [11] dealt with road roughness assumption due toclassification of roads into different classes according to theirunevenness equal intensity of road unevenness in the whole

2 Mathematical Problems in Engineering

range ofwavelengths and a general formof the fitted PSDwasgivenAmethod based on linear filtering (auto-regressive andmoving averagemethods orARMAmodelling) was proposedby Yoshimura in 1998 [12] which has a smaller calculationand a faster simulation speed but its precision is not verywell Bogsjo [13] in 2007 summarized different stochasticmodels of parallel road tracks and evaluated their accuracyby comparing the difference of the measured parallel tracksand the synthetic parallel tracks In the same year Ambrozproposed a system for measuring road section parametersthat can be used in driving dynamics simulations In 2012Pazooki et al [14] developed a comprehensive offroad vehicleride dynamicsmodel considering a random roughnessmodelof the two parallel tracks Hassan and Evans [15] collectedthe road longitudinal profile data of a car and truck wheeltracks and then investigated the differences of their surfaceroughness characteristics which showed that the HATI(heavy articulated truck index) lane average values of a truckwheel tracks are higher than car wheel tracksThree new roadprofile models [16] are proposed by Bogsjo et al in 2012which were compared with the classical road surface roadmodel and five different models were fitted to eight mea-sured road surfaces and their accuracy and efficiency werestudied

Generally for some researches and publications aboutrandom road surface reconstruction a single-wheel roadtime domain model is established via the inverse Fouriertransform and a four-wheel road time domain model isgenerated based on the correlation of the four-wheel inputin which there is a time delay between the front and rearaxle and the space is correlated between the left and rightwheel is [17] However during our research there are someconditions show that the road roughness amplitude and theroad PSD of one wheel is smaller than the other one on thesame axle in other words there is a deviation between twotracks on the same axle Also in 1980s Heath [18 19] pointedout that the method used for the calculation of the cross-spectrum from experimental sampled data is inaccurateat high wavenumbers Pazooki et al [14] considered theuncorrelated component between the left and right wheelswhen working on modeling and validation of an off-roadvehicle ride dynamics In fact the deviation between twotracks on the same axle has an important influence onvehicle performances Hence the deviation for two-wheelroad input and its influence on a vehicle dynamics behavior isundertaken in this paper and a PSD frequency compensationalgorithm is proposed to improve the accuracy of the roadinput reconstruction

In this paper single-wheel and a four-wheel road timedomain model is established via the Inverse Fourier Trans-form and the deviation between the two wheels road PSDson the same axle is studied firstly Secondly a road frequencycompensation algorithm is proposed to correct the deviationof the two-wheel roadPSDs on the same axle and it is installedin a road time domain model and then it is studied viasimulation and compared with ISO 8608 Finally the roadfrequency compensation algorithm is applied on a 7-DOFvehicle dynamic study

2 Power Spectral Density of the RoadSurface Roughness

The road surface roughness is usually described in terms ofthe PSDof the road displacement amplitudes [20ndash22] and theroad spectral density depends on the road itself (roughnessand wave numberdistribution) According to ISO 8608 roadroughness level is classified from A to H and a general formof the fitted PSD is given as follows

119866119902 (119899) = 119866119902 (1198990) sdot (119899

1198990

)

minus119882

(1)

where 119899 is the spatial frequency (mminus1) 1198990 is the referencespatial frequency (1198990 = 01mminus1) 119866119902(119899) is the spatial PSD(m2mminus1) 119866119902(1198990) is the spatial PSD at the reference spatialfrequency (m2mminus1) 119882 is undulation exponent and theundulation exponents are given with values in the range from18 to 33

The general expression for the relationship between thevehicle speed 119881 (ms) the spatial frequency 119891 (Hz) and thetemporal frequency 119866119902(119891) (m

2Hz) are presented in

119891 = 119881 sdot 119899

119866119902 (119891) =1

119881sdot 119866119902 (119899) = 119866119902 (1198990) sdot 119899

119882

0sdot119881119882minus1

119891119882

(2)

Taking statistical analysis of the road surface into accountthe spatial frequency 119899 ranges from 0011 to 283mminus1 at thecommonly speed (10ndash30ms) it can guarantee the temporalfrequency 119891 ranges from 033 to 283Hz which considers thenatural frequencies of the sprung mass and unsprung masseffectively

3 Single-Wheel Road Time Domain Model

Discrete PSD in temporal frequency domain is required tosimulate road surface in time domain which can be obtainedby

119866119902 (119891119896)

=

0 119896 = 1 2 119873119897

119866119902 (1198990) 119899119882

0

119881119882minus1

119891119882

119896

119896 = 119873119897 + 1119873119897 + 2 119873119906 + 1

0 119896 = 119873119906 + 2119873119906 + 3 119873

(3)

where 119866119902(119891119896) is the discrete PSD in temporal frequencydomain and119873119897 is the lower limit of the spatial frequency119873119906is the upper limit of the spatial frequency

According to the Fourier transform the relationshipbetween the discrete PSD in temporal frequency domain

Mathematical Problems in Engineering 3

0 2 4 6 8 10 12 14 16 18minus50

0

50

Time (s)

Road

roug

hnes

s (m

m)

Figure 1 Single-wheel road surface time domain signal (119862-levelroad 70Kmh)

and the corresponding spectral amplitude can be obtained asfollows

|119876 (119896)| = 119873radicΔ119891

2119866119902 (119891119896)

Δ119891 =1

119879

(4)

119876 (119896) = |119876 (119896)| sdot 119890119895120601119896 (5)

where 119876(119896) is the road random excitation spectrum 119879is the sampling time |119876(119896)| is the corresponding spectralamplitude 119895 is the imaginary unit and 120601119896 is a uniformlydistributed random variable ranged from 0 to 2120587 119896 =

1 2 119873Then road roughness random excitation signal can be

obtained by the inverse Fourier transform of the complexsequences 119876(119896) (119896 = 1 2 3 119873)

119909 (119899) =1

119873

119873

sum

119896=1

119876 (119896) 119890119895(2120587119873)119899119896

119896 = 1 2 119873 (6)

A single-wheel 119862-level road surface time domain signal isshown in Figure 1 in which the vehicle speed is 70 kmh Acomparison of the road PSD simulated with the ISO 8608standard is shown in Figure 2 which shows that the roadPSD simulated can match the ISO 119862-level road very well Ina word the inverse Fourier transform method for a single-wheel road time domain model is accurate

4 Four-Wheel Road Time Domain Model

A four-wheel vehicle is subjected to excitations due to roadroughness on the left and right wheel paths Hence todescribe the excitations we need a stochasticmodel of parallelroad tracks The model should describe the variation withineach track and the covariation between the tracks [23] Afour-wheel road time domain model is usually generatedbased on the correlation of the four-wheel road input inwhich there is a time delay between the front and rear axleand the space is correlated between the left and right wheels

10minus1 100 101

10minus4

10minus6

10minus8

Frequency (Hz)

PSD

(m2(1

m))

102

ISO standard PSDSimulated PSD

Figure 2 Comparison of the road PSD simulated with the ISO 8608standard

And a four-wheel road surface domain model is establishedbased on the following two assumptions

(1) Thewheel tracks of the vehicle front axle and rear axleare equal and the vehicle keeps in a straight line witha constant speed

(2) The statistical properties of the road profiles at leftand right tires are the same in other words the auto-power spectral density of the road profiles at left andright tires is the same and equal to the standard

While the left and right tracks are usually statistically equiv-alent the actual profiles are not identical Generally thedifference between the left and right tracks brings a rolldisturbance However the information regarding this rolldisturbance is not included in the PSD of the individualwheel paths Hence in addition to the PSD it is appropriateto study the coherence function The statistical propertiesof road roughness between the left and right wheels areusually described by a cross-power spectral density functionor a coherence function [24ndash26] Surprisingly the coherencefunctions of all roads (smoothmotorways main roads pavedcountry roads gravel roads etc) are very similar [27] Andthe coherence function is described by

coh (119899) =10038161003816100381610038161003816119866119909119910 (119899)

10038161003816100381610038161003816

radic119866119909119909 (119899) sdot 119866119910119910 (119899)

(7)

where 119866119909119909(119899) and 119866119910119910(119899) are the auto-spectral densities ofthe individual tracks (m2mminus1) and 119866119909119910(119899) is the associatedcross-spectral density (m2mminus1)119861 is the wheel track (m) and119899 is the spatial frequency (mminus1)

The measurement data yield a coherence function whichdrops from a high value at 119899 = 0 to a very small valueat high-frequency band [28] and the range is from 0 to 1When coh(119899) = 0 the left- and right-wheel road roughnessis completely uncorrelated while the left- and right-wheelroad roughness is perfectly correlated if coh(119899) = 1 Robson[29] pointed out that the coherence function depends on the

4 Mathematical Problems in Engineering

0

02

04

06

08

1

Coh

eren

ce fu

nctio

n

10minus1 100 101 102

Time (s)

Figure 3 The coherence function of the left and right track under119862-level road

2 1

4 3

V

L

B

Figure 4 A top view of four-wheel vehicle

wheel track and spatial frequency The coherence functionwas described by Ammon [30] as

coh (119899 119861) = [1 + (119899119861120576

119899119902

)

120597

]

minus119902

(8)

where the inflection point and the slope of the inflection pointare determined by the parameters 119899119902 and 119902 which can befitted by experimental data 120576 is the density of the coherencefunction between different tracks and 120597 is the frequencyindex The coherence function of the left and right tracksunder 119862-level road is shown in Figure 3

Previously the single-wheel road time domain model andcoherence function have been introduced then a four-wheelroad time domain model will be discussed as follows A topview of four-wheel vehicle is shown in Figure 4 and thespectrum matrix of four-wheel road input is described in

[119866119902 (119891)] =[[[

[

11986611 11986612 11986613 11986614

11986621 11986622 11986623 11986624

11986631 11986632 11986633 11986634

11986641 11986642 11986643 11986644

]]]

]

=

[[[[[[[[[

[

11986611 11986611119890minus1198952120587119891120591

11986613 11986613119890minus1198952120587119891120591

119866111198901198952120587119891120591

11986611 119866131198901198952120587119891120591

11986613

119866lowast

13119866lowast

13119890minus1198952120587119891120591

11986633 11986633119890minus1198952120587119891120591

119866lowast

131198901198952120587119891120591

119866lowast

13119866331198901198952120587119891120591

11986633

]]]]]]]]]

]

(9)

where 119866119894119895 (119894 = 119895) (119894 119895 = 1 2 3 4) is the cross-power spectraldensity 119866119894119895 (119894 = 119895) is the auto-power spectral density 119866lowast

119894119895is

the complex conjugate matrix of 119866119894119895 and 120591 is the time delayof front- and rear-wheel road input which can be expressedas

120591 =119871

119881 (10)

where 119871 is the wheelbase (m) and119881 is the vehicle speed (ms)11986613(119891) can be obtained by

11986613 (119891) =100381610038161003816100381611986613 (119891)

1003816100381610038161003816 11989011989512060113(119891)

= radic11986611 (119891) sdot 11986633 (119891) sdot coh (119891) sdot 11989011989512060113(119891)

(11)

where 12060113(119891) is the phase difference of road PSD of the left-and right-front wheel According to measured pavementsthe phase differences of road PSD of the left- and right-front wheel are approximately 0 at this point the impactson vehicle performance are negligible thus it is assumedthat 12060113(119891) = 0 in this paper Therefore (11) and (9) can berewritten as

11986613 (119891) =100381610038161003816100381611986613 (119891)

1003816100381610038161003816 11989011989512060113(119891) = radic11986611 (119891) sdot 11986633 (119891) sdot coh (119891)

[119866119902 (119891)] = 119866119902 (119891)

[[[[

[

1 119890minus1198952120587119891120591 coh (119891) coh (119891) 119890minus1198952120587119891120591

1198901198952120587119891120591

1 coh (119891) 1198901198952120587119891120591 coh (119891)coh (119891) coh (119891) 119890minus1198952120587119891120591 1 119890

minus1198952120587119891120591

coh (119891) 1198901198952120587119891120591 coh (119891) 1198901198952120587119891120591

1

]]]]

]

(12)

Then the road spectrum relationship between the left andright wheels can be obtained as follows

1198763 (119891) = 1198761 (119891) coh (119891) (13)

And the road spectrum relationship between the front andrear wheels can be obtained as follows

1198762 (119891) = 1198761 (119891) 119890minus1198952120587119891120591

(14)

Mathematical Problems in Engineering 5

LFLR

0 2 4 6 8 10 12 14 16 18minus50

0

50

Time (s)

Road

roug

hnes

s (m

m)

Figure 5 Comparison of left-front wheel (LF) with left-rear wheel(LR) time domain signal

So far a four-wheel time domain model can be obtained viathe following steps

(1) Road roughness level 119866119902(1198990) the vehicle speed 119881the coherence function coh(119891) the lower limit of thespatial frequency119873119897 and the upper limit of the spatialfrequency119873119906 are known as initial conditions

(2) The spectrum of left-front wheel road roughness1198761(119896) can be obtained analogous to the methodof generating single-wheel road random excitationspectrum

(3) The road random excitation spectrum of other wheels1198762(119896) and 1198764(119896) can be obtained from (15)ndash(17) andthere is a time delay between the left-front road input1198761(119896) and the left-rear road input 1198763(119896) The 1198762(119896)and 1198764(119896) are obtained via a coherence functionbetween the left and right wheels

1198762 (119896) = 1198761 (119896) 119890minus1198952120587119891119896120591 (15)

1198763 (119896) = 1198761 (119896) coh (119896) (16)

1198764 (119896) = 1198763 (119896) 119890minus1198952120587119891119896120591 = 1198761 (119896) coh (119896) 119890

minus1198952120587119891119896120591 (17)

(4) The corresponding road roughness random excita-tion signals can be obtained by the inverse Fouriertransform of the complex sequences 119876119894(119896) (119894 =

1 2 3 4)

119909119894 (119899) =1

119873

119873

sum

119896=1

119876119894 (119896) 119890119895(2120587119873)119899119896

119899 = 1 2 119873 (18)

5 Simulation Results

The results indicated in this section include the simulationof the road surfaces discussed in the previous section Anda four-wheel road surface time domain signal for a class 119862random road at the speed of 70Kmh is shown in Figures 5ndash7

RFRR

0 2 4 6 8 10 12 14 16 18minus50

0

50

Time (s)

Road

roug

hnes

s (m

m)

Figure 6 Comparison of right-front wheel (RF) with right-rearwheel (RR) time domain signal

LFRF

0 2 4 6 8 10 12 14 16 18minus50

0

50

Time (s)

Road

roug

hnes

s (m

m)

Figure 7 Comparison of left-front wheel (LF) with right-frontwheel (RF) time domain signal

Figure 5 shows a comparison of the road time domainsignal between the left-front wheel (LF) and the left-rearwheel (LR) and the comparison of the road time domainsignal between the right-front wheel (RF) and the right-rear wheel (RR) is shown in Figure 6 It can be foundobviously that there is time delay between front-wheel andrear-wheel time domain signals which is exactly equal to the119871119881 (see (10)) Figure 8 is a comparison of the road PSDssimulated with the ISO 8608 standards Figures 8(a) and 8(c)demonstrate that the road PSD of the front and rear wheels isconsistent with the ISO 8608 standard values which provesthat the Inverse Fourier Transform is accurate to generatethe front- and rear-wheel time domain signals Howeverfrom Figure 7 we can see that the amplitude of right-frontwheel (RF) is smaller than left-front wheel (LF) time domainsignal simultaneously there is a deviation Δ119866119902(119891) betweenthe road PSD of the left and right wheels from the beginningof midfrequency band in Figures 8(b) and 8(d) and thedeviation increases with the frequency increasing In otherwords the road PSD of the right wheel is smaller than the

6 Mathematical Problems in Engineering

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

(a) Left-front wheel

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

(b) Right-front wheel

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

ISO standard PSDLR simulated PSD

(c) Left-rear wheel

10minus1 100 101

10minus5

Frequency (Hz)PS

D (m

2(1

m))

102

ISO standard PSDRR simulated PSD

(d) Right-rear wheel

Figure 8 Comparison of the road PSDs simulated with the ISO 8608 standards

0 2 4 6 8 10 12 14 16 18minus50

0

50

Time (s)

Road

roug

hnes

s (m

m)

Figure 9 The smaller time domain signal generated based on thedeviation Δ119866

119902(119891)

road PSD of the left wheel on the same axle that is consistentwith the ISO 8608 standard value which can be described by

Δ119866119902 (119891) = 119866119902 (119891) minus 119866119902simu (119891) (19)

where 119866119902(119891) is the standard PSD values 119866119902simu(119891) is thesimulated PSD values and Δ119866119902(119891) is the deviation betweenthe road PSD of the left wheel (consistent with the standard)and right wheel

In addition the right-wheel time domain signal cannotbe obtained when coh(119891) is 0 which only considers thecorrelation of the left- and right-wheel road roughness whileignoring the uncorrelation of the left- and right-wheel roadroughness Since the value of coherence function is between

0 and 1 and it is near 0 at high-frequency band accordingto (16) and (17) it can be found that the reasons for thedeviation aremainly from the coherence functionHence thecompensation of the right-wheel time domain signal will bepresented in the sequel

6 Correction of Four-Wheel Road TimeDomain Model

From the above we know that there is a deviation Δ119866119902(119891)between the road PSD of left and right wheels since theuncorrelation of the left- and right-wheel road roughness isignored Hence considering the uncorrelation of the left andright wheel road roughness a new road spectrum relation-ship between the left and right wheels can be described by

1198761015840

3(119891) = 1198761 (119891) coh (119891) + Δ119876 (119891) (20)

whereΔ119876(119891) reflects the uncorrelation of the left- and right-wheel road roughness which can be obtained by Δ119866119902(119891)based on (4) and (5) A compensation using time domainsignal is generated via the inverse Fourier transformofΔ119876(119891)by using new random sequence numbers which is shown inFigure 9

A new right-wheel time domain signal can be obtained bycompensating the time domain deviation signal A compar-ison of left-front wheel with right-front wheel time domain-compensated signal is shown in Figure 10 and it can be foundthat the amplitude of right-front wheel (RF) and left-frontwheel (LR) time domain signals are the same approximately

Mathematical Problems in Engineering 7

LFRF

0 2 4 6 8 10 12 14 16 18minus50

0

50

Time (s)

Road

roug

hnes

s (m

m)

Figure 10 Comparison of left-front wheel (LF) with right-frontwheel (RF) time domain signal after compensating

Then the new road PSDs of the four wheels can beobtained via Fourier transformwhich is shown in Figure 11 Itshows that the new road PSDs simulated canmatch ISO 8608standards very well In a word this frequency compensationalgorithm for a four-wheel road time domain model isaccurate

In addition for the limit case the method is also applica-ble When coh(119891) is 0 the left- and right-wheel time domainsignals are completely uncorrelated and a comparison of left-front wheel with left-rear wheel time domain signals is shownin Figure 12 It illustrates that the right-front wheel (RF)and left-front wheel (LR) time domain signals are completelyuncorrelated which correspond with the definition of thecorrelation function Figure 13 shows the comparison of theroad PSDs simulatedwith the ISO 8608 standards Obviouslythe new road PSDs simulated can match ISO 8608 standardsvery well

When coh(119891) is 1 the left- and right-wheel road rough-ness are perfectly correlated and a comparison of the left-front wheel with the left-rear wheel time domain signals isshown in Figure 14 It illustrates that the right-front wheel(RF) and left-front wheel (LR) time domain signals arecompletely the same which correspond with the definitionof the correlation function Figure 15 shows the comparisonof the road PSDs simulated with the ISO 8608 standardsObviously the new road PSDs simulated can also matchISO 8608 standards very well In summary this frequencycompensation algorithm for a four-wheel road time domainmodel is accurate

The frequency compensation algorithm can also be usedto correct the four-wheel surface time domain models gener-ated by other methods when there is a deviation between theroad PSDs simulated and the ISO 8608 standards

7 Analysis on Vehicle Dynamic Behavior

Since the road surface irregularities are the main sourceof excitation to the vehicle system to analyze the vehicle

dynamic behavior more accurately a suitable representationof these irregularities is required which is shown in Figure 16

A comparison of vehicle dynamic behavior wo (with orwithout) compensation will be presented as follows Firstly a7-DOF vertical vehicle model is shown in Figure 17 whichconsiders the vertical motion pitch and roll of the sprungmass and the four vertical motions of four unsprung masses

Each corner of the vehicle is identified with 119894 119895 indexwhere 119894 = 119897 119903holds for leftright and 119895 = 119891 119903 for frontrearThese corners (ie positions and velocities of the dynamicalpart) are described as

119911119891119897 = 119911 minus 120572119897119891 + 120573119908 minus 119911119905119891119897

119911119891119903 = 119911 minus 120572119897119891 minus 120573119908 minus 119911119905119891119903

119911119903119897 = 119911 + 120572119897119903 + 120573119908 minus 119911119905119903119897

119911119903119903 = 119911 + 120572119897119903 minus 120573119908 minus 119911119905119903119903

119891119897 = minus 119897119891 +120573119908 minus 119905119891119897

119891119903 = minus 119897119891 minus120573119908 minus 119905119891119903

119903119897 = + 119897119903 +120573119908 minus 119905119903119897

119903119903 = + 119897119903 minus120573119908 minus 119905119903119903

(21)

where 119911 is the center of the sprung mass 120572 (resp 120573) is thepitch (resp roll) angle of the sprungmass around119910-axis (resp119909-axis) at the center of sprung mass 119897119891 119897119903 and 119908 define thevehicle geometrical properties (as shown in Figure 17)

The full vehicle model as depicted on Figure 17 is givenby the following motion equations

119898119905119891119897119905119891119897 + 119896119905119891119897 (119911119905119891119897 minus 119911119891119897) minus 119865119904119891119897 minus 119865119889119891119897 = 0

119898119905119891119903119905119891119903 + 119896119905119891119903 (119911119905119891119903 minus 119911119891119903) minus 119865119904119891119903 minus 119865119889119891119903 = 0

119898119905119903119897119905119903119897 + 119896119905119903119897 (119911119905119903119897 minus 119911119903119897) minus 119865119904119903119897 minus 119865119889119903119897 = 0

119898119905119903119903119905119903119903 + 119896119905119903119903 (119911119905119903119903 minus 119911119903119903) minus 119865119904119903119903 minus 119865119889119903119903 = 0

sum119898 + 119865119904119894119895 + 119865119889119894119895 = 0

sum119868120572 minus 119865119904119891119895119897119891 + 119865119904119903119895119897119903 minus 119865119889119891119895119897119891 + 119865119889119903119895119897119903 = 0

sum119868120573120573 + 119908119865119904119894119897 minus 119908119865119904119894119903 + 119908119865119889119894119897 minus 119908119865119889119894119903 = 0

(22)

where 119898 and 119898119905119894119895 denote the sprung and unsprung massrespectivelyThemoment of inertia about the 119909-axis (resp 119910-axis) of the vehicle sprung mass is denoted as 119868120573 (resp 119868120572)119865119904119894119895 and 119865119889119894119895 are spring force and damper force respectivelywhich can be described by

119865119904119894119895 = 119896119904119894119895119911119894119895 119865119889119894119895 = 119888119894119895119894119895 (23)

where 119896119904119894119895 and 119888119894119895 are spring stiffness and damping coefficientrespectively

Then the full vehicle model can be established by Mat-labSimulink and the outputs include the weighted RMS

8 Mathematical Problems in Engineering

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

(a) Left-front wheel

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

(b) Right-front wheel

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

ISO standard PSDLR simulated PSD

(c) Left-rear wheel

10minus1 100 101

10minus5

Frequency (Hz)PS

D (m

2(1

m))

102

ISO standard PSDRR simulated PSD

(d) Right-rear wheel

Figure 11 Comparison of the new road PSDs of the four wheels simulated with the ISO 8608 standards

LFRF

0 2 4 6 8 10 12 14 16 18minus50

0

50

Time (s)

Road

roug

hnes

s (m

m)

Figure 12 Comparison of left-front wheel (LF) with left-rear wheel(LR) time domain signal when coh(119891) is 0

(root mean square) of acceleration (Ac rms ms2) the rollangle RMS (120573 rms) the pitch angle RMS (120572 rms) and thetire vertical force RMS (Ft FL rms Ft FR rms Ft RL rmsFt RR rms N)

In this paper the frequency compensation algorithm isinstalled in the road input for a sedan model to illustratethe sedanrsquos behavior The parameters of a 119862-class sedan areshown in Table 1 Simulation results run over a 119862-level roadsurface with the speed of 70Kmh tabulated in Table 2 Andthe frequency compensation algorithm can also be installedin the other vehicle runs on a random road

Table 1 Parameters of the 119862-class sedan

Symbol Description Unit Value119898 Vehicle sprung mass kg 2110119898119891119897

Frontleft unsprung mass kg 583119898119891119903

Frontright unsprung mass kg 583119898119903119897

Rearleft unsprung mass kg 633119898119903119903

Rearright unsprung mass kg 633119896119905119891119897

Frontleft vertical tyre stiffness Nm 235000119896119905119891119903

Frontright vertical tyre stiffness Nm 235000119896119905119903119897

Rearleft vertical tyre stiffness Nm 235000119896119905119903119903

Rearright vertical tyre stiffness Nm 235000119896119904119891119897

Frontleft spring stiffness Nm 27500119896119904119891119903

Frontright spring stiffness Nm 27500119896119904119903119897

Rearleft spring stiffness Nm 31300119896119904119903119903

Rearright spring stiffness Nm 31300119897119891

Distance between front-axlecog G m 1455119897119903

Distance between rear-axlecog G m 1515119908 Wheel track m 1610119868120573

Sprung mass roll moment of inertia kgsdotm2 7101119868120572

Sprung mass pitch moment of inertia kgsdotm2 36685

In this analysis the road input of left-front tire is set asa baseline then the right-front road input is obtained via acoherence function wo a compensation and the rear road

Mathematical Problems in Engineering 9

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

(a) Left-front wheel

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

(b) Right-front wheel

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

ISO standard PSDLR simulated PSD

(c) Left-rear wheel

10minus1 100 101

10minus5

Frequency (Hz)PS

D (m

2(1

m))

102

ISO standard PSDRR simulated PSD

(d) Right-rear wheel

Figure 13 Comparison of the road PSDs simulated with the ISO 8608 standards when coh(119891) is 0

LFRF

0 2 4 6 8 10 12 14 16 18minus50

0

50

Time (s)

Road

roug

hnes

s (m

m)

Figure 14 Comparison of left-front wheel (LF) with left-rear wheel(LR) time domain signal when coh(119891) is 1

input has a time delay dependent on the wheel base andvehicle speed Table 2 shows that the 120572 rms Ft FL rms andFt RL rms without compensation are almost equal to thevalues with compensation since the PSDs of the left-side roadinput are the same while the Ac rms 120573 rms Ft FR rms andFt RR rmswithout compensation are significant smaller thanthe values with compensation which is precisely caused bythe previously mentioned deviation since there is a deviationbetween the road PSDs of the two wheels on the same axlewhich is the main reason that the acceleration RMS the rollangle RMS the RMS of tire vertical force of the right-front

Table 2 Simulation results

Results Without compensation With compensation DifferenceAc rms 03638 04081 108120572 rms 00062 00062 0120573 rms 910119890 minus 4 00017 464Ft FL rms 10115119890 + 3 10162119890 + 3 05Ft FR rms 06824119890 + 3 10100119890 + 3 327Ft RL rms 10514119890 + 3 10566119890 + 3 05Ft RR rms 07190119890 + 3 10534119890 + 3 314

and right-rear wheels without compensation are smaller thanthe values with compensation Hence the deviation of thePSD of the road input between two wheels on the same axlehas an important influence on vehicle performances whichillustrates the importance of the frequency compensationalgorithm proposed which can be used for a control systeminstalled in the vehicle to compensate road roughness fordamper tuning in the future

8 Conclusions

The road surface roughness has an important influence onride comfort ride safety vehicle maneuverability driverrsquosand occupantsrsquo comfort and vehicle dynamic load In recentdecades random road models have been widely studied anda four-wheel random road time domain model is usuallygenerated based on the correlation of the four-wheel input in

10 Mathematical Problems in Engineering

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

(a) Left-front wheel

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

(b) Right-front wheel

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

ISO standard PSDLR simulated PSD

(c) Left-rear wheel

10minus1 100 101

10minus5

Frequency (Hz)PS

D (m

2(1

m))

102

ISO standard PSDRR simulated PSD

(d) Right-rear wheel

Figure 15 Comparison of the road PSDs simulated with the ISO 8608 standards when coh(119891) is 1

Vehicledynamicbehavior

VehiclemodelRoad model

Figure 16 Analysis on the vehicle dynamic behavior

which a coherence function is used to describe the coherenceof the road input between the left and right wheels usuallyGenerally the value of coherence function is between 0 and 1and it is near 0 at high-frequency band As a result the roadroughness amplitude and the road PSD of one track will besmaller than the other one on the same axle actually it iscaused by the uncorrelation between the left- and right-wheelroad surface roughness Hence a frequency compensationalgorithm is proposed to correct the deviation of the PSDof the road input between two wheels on the same axle andit is installed in a 7-DOF vehicle dynamic study The mainconclusions can be drawn from the current study as follows

(1) It is simple and accurate to generate a single-wheeltime signal model by using the inverse Fourier trans-form method

(2) With the Inverse Fourier Transform method viacoherence function the road PSDs of the front andrear wheels are consistent with the standard valueshowever there is a deviation between the two-wheelroad PSDs on the same axle The coherence function

value is between 0 and 1 and it is near 0 at high-frequency band As a result the road roughnessamplitude and the road PSD of one track will besmaller than the other one on the same axle

(3) A road frequency compensation algorithm is pro-posed to correct the deviation of the two wheels roadPSDs on the same axle and it is installed in a roadtime domain model The simulation results showedthat the road PSDs with frequency compensation areconsistent with the ISO 8608 standard values whichdemonstrates that the proposed algorithm is correctespecially for some limit case such as the left andright wheel road input are completely uncorrelated orperfectly correlated

(4) The vehicle dynamic behavior with road frequencycompensation show that the deviation compensationof two wheels on the same axle has an importantinfluence on vehicle performances which should beconsidered in the vehicle dynamic studies and theroad frequency compensation can be used for acontrol system installed in the vehicle to compensateroad roughness for damper tuning in the future

Acknowledgments

Special thanks are due to the Open Research Fund Pro-gram of the State Key Laboratory of Advanced Design andManufacturing for Vehicle Body (31115028) the China Post-doctoral Science Foundation (2012M520028) the NationalNatural Science Foundation of China (51205155) and the

Mathematical Problems in Engineering 11

lf

lr

w

I120572 120572 I120573 120573

xGy

z

z

ztrl

zrl

crl

mrl

ksrl

ktrl

ksfr

mfr

ksfl

zfl

cfl

mfl

ktfl

ztfl

cfrztfr

zfr ktfr

Figure 17 7 DOF full vehicle model

National Basic Research Program of China (973 Program)(2011CB711201) for supporting authorsrsquo research

References

[1] P Mucka ldquoRoad waviness and the dynamic tyre forcerdquo Inter-national Journal of Vehicle Design vol 36 no 2-3 pp 216ndash2322004

[2] Z Yonglin and Z Jiafan ldquoNumerical simulation of stochasticroad process using white noise filtrationrdquo Mechanical Systemsand Signal Processing vol 20 no 2 pp 363ndash372 2006

[3] K Ahlin J Granlund and F Lindstrom ldquoComparing roadprofiles with vehicle perceived roughnessrdquo International Journalof Vehicle Design vol 36 no 2-3 pp 270ndash286 2004

[4] S M Karamihas and M W Sayers The Little Book of ProfilingUniversity of Michigan Ann Arbor Mich USA 1998

[5] MAmbroz G Sustersic and I Prebil ldquoCreatingmodels of roadsections and their use in driving dynamics simulationsrdquo VehicleSystem Dynamics vol 45 no 10 pp 911ndash924 2007

[6] D J Whitehouse and J F Archard ldquoThe properties of randomsurfaces of significance in their contactrdquoProceedings of the RoyalSociety of London A vol 316 pp 97ndash121 1970

[7] M Shinozuka and C-M Jan ldquoDigital simulation of randomprocesses and its applicationsrdquo Journal of Sound and Vibrationvol 25 no 1 pp 111ndash128 1972

[8] D Cebon Interaction between Heavy Vehicles and Roads vol951 Society of Automotive Engineers 1993

[9] J-J Wu ldquoSimulation of rough surfaces with FFTrdquo TribologyInternational vol 33 no 1 pp 47ndash58 2000

[10] M Grigoriu ldquoOn the spectral representation method in sim-ulationrdquo Probabilistic Engineering Mechanics vol 8 no 2 pp75ndash90 1993

[11] ISO Mechanical Vibration-Road Surface Profiles-Reporting ofMeasured Data International Organization for Standardiza-tion 1995

[12] T Yoshimura ldquoA semi-active suspension of passenger cars usingfuzzy reasoning and the field testingrdquo International Journal ofVehicle Design vol 19 no 2 pp 150ndash166 1998

[13] K Bogsjo ldquoEvaluation of stochastic models of parallel roadtracksrdquo Probabilistic Engineering Mechanics vol 22 no 4 pp362ndash370 2007

[14] A Pazooki S Rakheja andD Cao ldquoModeling and validation ofoff-road vehicle ride dynamicsrdquoMechanical Systems and SignalProcessing vol 28 pp 679ndash695 2012

[15] R Hassan and R Evans ldquoRoad roughness characteristics incar and truck wheel tracksrdquo International Journal of PavementEngineering 2012

[16] K Bogsjo K Podgorski and I Rychlik ldquoModels for roadsurface roughnessrdquo Vehicle System Dynamics vol 50 no 5 pp725ndash747 2012

[17] S A Hassan and O S Lashine ldquoIdentification of road surfacequalities for linear and non-linear vehicle modelingrdquo in Pro-ceedings of the 20th International Modal Analysis Conference(IMAC rsquo02) pp 318ndash323 Society for Experimental Mechanics2002

[18] A N Heath ldquoApplication of the isotropic road roughnessassumptionrdquo Journal of Sound and Vibration vol 115 no 1 pp131ndash144 1987

[19] A N Heath ldquoEvaluation of the isotropic road roughnessassumptionrdquoVehicle SystemDynamics vol 17 supplement 1 pp157ndash160 1988

[20] D Cebon and D E Newland ldquoThe artificial generation of roadsurface topography by the inverse FFT methodrdquo Vehicle SystemDynamics vol 12 no 1ndash3 pp 160ndash165 1983

[21] C J Dodds ldquoThe laboratory simulation of vehicle servicestressrdquo Journal of Engineering for Industry vol 96 no 2 pp 391ndash398 1974

[22] D Cebon and D E Newlan ldquoThe artificial generation of roadsurface topography by the inverse FFT methodrdquo in Proceedingsof the 8th IAVSD-Symposium on the Dynamics of Vehicles onRoads and Tracks pp 29ndash42 Swets and Zeitlinger CambridgeMass USA 1983

[23] C J Dodds and J D Robson ldquoThe description of roadroughnessrdquo Journal of Sound and Vibration vol 31 no 2 pp175ndash183 1973

[24] K M A Kamash and J D Robson ldquoImplications of isotropy inrandom surfacesrdquo Journal of Sound and Vibration vol 54 no 1pp 131ndash145 1977

[25] K M A Kamash and J D Robson ldquoThe application of isotropyin road surface modellingrdquo Journal of Sound and Vibration vol57 no 1 pp 89ndash100 1978

12 Mathematical Problems in Engineering

[26] G C Carter ldquoCoherence and time delay estimationrdquo Proceed-ings of the IEEE vol 75 no 2 pp 236ndash255 1987

[27] K Bogsjo ldquoCoherence of road roughness in left and rightwheel-pathrdquo Vehicle System Dynamics vol 46 supplement 1 pp 599ndash609 2008

[28] R Hongbin C Sizhong andW Zhicheng ldquoModel of excitationof random road profile in time domain for a vehicle withfour wheelsrdquo in Proceedings of the International Conference onMechatronic Science Electric Engineering and Computer (MECrsquo11) pp 2332ndash2335 Jilin China August 2011

[29] J D Robson ldquoRoad surface description and vehicle responserdquoInternational Journal of Vehicle Design vol 1 no 1 pp 25ndash351979

[30] D Ammon ldquoProblems in road surface modellingrdquo VehicleSystem Dynamics vol 20 pp 28ndash41 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

range ofwavelengths and a general formof the fitted PSDwasgivenAmethod based on linear filtering (auto-regressive andmoving averagemethods orARMAmodelling) was proposedby Yoshimura in 1998 [12] which has a smaller calculationand a faster simulation speed but its precision is not verywell Bogsjo [13] in 2007 summarized different stochasticmodels of parallel road tracks and evaluated their accuracyby comparing the difference of the measured parallel tracksand the synthetic parallel tracks In the same year Ambrozproposed a system for measuring road section parametersthat can be used in driving dynamics simulations In 2012Pazooki et al [14] developed a comprehensive offroad vehicleride dynamicsmodel considering a random roughnessmodelof the two parallel tracks Hassan and Evans [15] collectedthe road longitudinal profile data of a car and truck wheeltracks and then investigated the differences of their surfaceroughness characteristics which showed that the HATI(heavy articulated truck index) lane average values of a truckwheel tracks are higher than car wheel tracksThree new roadprofile models [16] are proposed by Bogsjo et al in 2012which were compared with the classical road surface roadmodel and five different models were fitted to eight mea-sured road surfaces and their accuracy and efficiency werestudied

Generally for some researches and publications aboutrandom road surface reconstruction a single-wheel roadtime domain model is established via the inverse Fouriertransform and a four-wheel road time domain model isgenerated based on the correlation of the four-wheel inputin which there is a time delay between the front and rearaxle and the space is correlated between the left and rightwheel is [17] However during our research there are someconditions show that the road roughness amplitude and theroad PSD of one wheel is smaller than the other one on thesame axle in other words there is a deviation between twotracks on the same axle Also in 1980s Heath [18 19] pointedout that the method used for the calculation of the cross-spectrum from experimental sampled data is inaccurateat high wavenumbers Pazooki et al [14] considered theuncorrelated component between the left and right wheelswhen working on modeling and validation of an off-roadvehicle ride dynamics In fact the deviation between twotracks on the same axle has an important influence onvehicle performances Hence the deviation for two-wheelroad input and its influence on a vehicle dynamics behavior isundertaken in this paper and a PSD frequency compensationalgorithm is proposed to improve the accuracy of the roadinput reconstruction

In this paper single-wheel and a four-wheel road timedomain model is established via the Inverse Fourier Trans-form and the deviation between the two wheels road PSDson the same axle is studied firstly Secondly a road frequencycompensation algorithm is proposed to correct the deviationof the two-wheel roadPSDs on the same axle and it is installedin a road time domain model and then it is studied viasimulation and compared with ISO 8608 Finally the roadfrequency compensation algorithm is applied on a 7-DOFvehicle dynamic study

2 Power Spectral Density of the RoadSurface Roughness

The road surface roughness is usually described in terms ofthe PSDof the road displacement amplitudes [20ndash22] and theroad spectral density depends on the road itself (roughnessand wave numberdistribution) According to ISO 8608 roadroughness level is classified from A to H and a general formof the fitted PSD is given as follows

119866119902 (119899) = 119866119902 (1198990) sdot (119899

1198990

)

minus119882

(1)

where 119899 is the spatial frequency (mminus1) 1198990 is the referencespatial frequency (1198990 = 01mminus1) 119866119902(119899) is the spatial PSD(m2mminus1) 119866119902(1198990) is the spatial PSD at the reference spatialfrequency (m2mminus1) 119882 is undulation exponent and theundulation exponents are given with values in the range from18 to 33

The general expression for the relationship between thevehicle speed 119881 (ms) the spatial frequency 119891 (Hz) and thetemporal frequency 119866119902(119891) (m

2Hz) are presented in

119891 = 119881 sdot 119899

119866119902 (119891) =1

119881sdot 119866119902 (119899) = 119866119902 (1198990) sdot 119899

119882

0sdot119881119882minus1

119891119882

(2)

Taking statistical analysis of the road surface into accountthe spatial frequency 119899 ranges from 0011 to 283mminus1 at thecommonly speed (10ndash30ms) it can guarantee the temporalfrequency 119891 ranges from 033 to 283Hz which considers thenatural frequencies of the sprung mass and unsprung masseffectively

3 Single-Wheel Road Time Domain Model

Discrete PSD in temporal frequency domain is required tosimulate road surface in time domain which can be obtainedby

119866119902 (119891119896)

=

0 119896 = 1 2 119873119897

119866119902 (1198990) 119899119882

0

119881119882minus1

119891119882

119896

119896 = 119873119897 + 1119873119897 + 2 119873119906 + 1

0 119896 = 119873119906 + 2119873119906 + 3 119873

(3)

where 119866119902(119891119896) is the discrete PSD in temporal frequencydomain and119873119897 is the lower limit of the spatial frequency119873119906is the upper limit of the spatial frequency

According to the Fourier transform the relationshipbetween the discrete PSD in temporal frequency domain

Mathematical Problems in Engineering 3

0 2 4 6 8 10 12 14 16 18minus50

0

50

Time (s)

Road

roug

hnes

s (m

m)

Figure 1 Single-wheel road surface time domain signal (119862-levelroad 70Kmh)

and the corresponding spectral amplitude can be obtained asfollows

|119876 (119896)| = 119873radicΔ119891

2119866119902 (119891119896)

Δ119891 =1

119879

(4)

119876 (119896) = |119876 (119896)| sdot 119890119895120601119896 (5)

where 119876(119896) is the road random excitation spectrum 119879is the sampling time |119876(119896)| is the corresponding spectralamplitude 119895 is the imaginary unit and 120601119896 is a uniformlydistributed random variable ranged from 0 to 2120587 119896 =

1 2 119873Then road roughness random excitation signal can be

obtained by the inverse Fourier transform of the complexsequences 119876(119896) (119896 = 1 2 3 119873)

119909 (119899) =1

119873

119873

sum

119896=1

119876 (119896) 119890119895(2120587119873)119899119896

119896 = 1 2 119873 (6)

A single-wheel 119862-level road surface time domain signal isshown in Figure 1 in which the vehicle speed is 70 kmh Acomparison of the road PSD simulated with the ISO 8608standard is shown in Figure 2 which shows that the roadPSD simulated can match the ISO 119862-level road very well Ina word the inverse Fourier transform method for a single-wheel road time domain model is accurate

4 Four-Wheel Road Time Domain Model

A four-wheel vehicle is subjected to excitations due to roadroughness on the left and right wheel paths Hence todescribe the excitations we need a stochasticmodel of parallelroad tracks The model should describe the variation withineach track and the covariation between the tracks [23] Afour-wheel road time domain model is usually generatedbased on the correlation of the four-wheel road input inwhich there is a time delay between the front and rear axleand the space is correlated between the left and right wheels

10minus1 100 101

10minus4

10minus6

10minus8

Frequency (Hz)

PSD

(m2(1

m))

102

ISO standard PSDSimulated PSD

Figure 2 Comparison of the road PSD simulated with the ISO 8608standard

And a four-wheel road surface domain model is establishedbased on the following two assumptions

(1) Thewheel tracks of the vehicle front axle and rear axleare equal and the vehicle keeps in a straight line witha constant speed

(2) The statistical properties of the road profiles at leftand right tires are the same in other words the auto-power spectral density of the road profiles at left andright tires is the same and equal to the standard

While the left and right tracks are usually statistically equiv-alent the actual profiles are not identical Generally thedifference between the left and right tracks brings a rolldisturbance However the information regarding this rolldisturbance is not included in the PSD of the individualwheel paths Hence in addition to the PSD it is appropriateto study the coherence function The statistical propertiesof road roughness between the left and right wheels areusually described by a cross-power spectral density functionor a coherence function [24ndash26] Surprisingly the coherencefunctions of all roads (smoothmotorways main roads pavedcountry roads gravel roads etc) are very similar [27] Andthe coherence function is described by

coh (119899) =10038161003816100381610038161003816119866119909119910 (119899)

10038161003816100381610038161003816

radic119866119909119909 (119899) sdot 119866119910119910 (119899)

(7)

where 119866119909119909(119899) and 119866119910119910(119899) are the auto-spectral densities ofthe individual tracks (m2mminus1) and 119866119909119910(119899) is the associatedcross-spectral density (m2mminus1)119861 is the wheel track (m) and119899 is the spatial frequency (mminus1)

The measurement data yield a coherence function whichdrops from a high value at 119899 = 0 to a very small valueat high-frequency band [28] and the range is from 0 to 1When coh(119899) = 0 the left- and right-wheel road roughnessis completely uncorrelated while the left- and right-wheelroad roughness is perfectly correlated if coh(119899) = 1 Robson[29] pointed out that the coherence function depends on the

4 Mathematical Problems in Engineering

0

02

04

06

08

1

Coh

eren

ce fu

nctio

n

10minus1 100 101 102

Time (s)

Figure 3 The coherence function of the left and right track under119862-level road

2 1

4 3

V

L

B

Figure 4 A top view of four-wheel vehicle

wheel track and spatial frequency The coherence functionwas described by Ammon [30] as

coh (119899 119861) = [1 + (119899119861120576

119899119902

)

120597

]

minus119902

(8)

where the inflection point and the slope of the inflection pointare determined by the parameters 119899119902 and 119902 which can befitted by experimental data 120576 is the density of the coherencefunction between different tracks and 120597 is the frequencyindex The coherence function of the left and right tracksunder 119862-level road is shown in Figure 3

Previously the single-wheel road time domain model andcoherence function have been introduced then a four-wheelroad time domain model will be discussed as follows A topview of four-wheel vehicle is shown in Figure 4 and thespectrum matrix of four-wheel road input is described in

[119866119902 (119891)] =[[[

[

11986611 11986612 11986613 11986614

11986621 11986622 11986623 11986624

11986631 11986632 11986633 11986634

11986641 11986642 11986643 11986644

]]]

]

=

[[[[[[[[[

[

11986611 11986611119890minus1198952120587119891120591

11986613 11986613119890minus1198952120587119891120591

119866111198901198952120587119891120591

11986611 119866131198901198952120587119891120591

11986613

119866lowast

13119866lowast

13119890minus1198952120587119891120591

11986633 11986633119890minus1198952120587119891120591

119866lowast

131198901198952120587119891120591

119866lowast

13119866331198901198952120587119891120591

11986633

]]]]]]]]]

]

(9)

where 119866119894119895 (119894 = 119895) (119894 119895 = 1 2 3 4) is the cross-power spectraldensity 119866119894119895 (119894 = 119895) is the auto-power spectral density 119866lowast

119894119895is

the complex conjugate matrix of 119866119894119895 and 120591 is the time delayof front- and rear-wheel road input which can be expressedas

120591 =119871

119881 (10)

where 119871 is the wheelbase (m) and119881 is the vehicle speed (ms)11986613(119891) can be obtained by

11986613 (119891) =100381610038161003816100381611986613 (119891)

1003816100381610038161003816 11989011989512060113(119891)

= radic11986611 (119891) sdot 11986633 (119891) sdot coh (119891) sdot 11989011989512060113(119891)

(11)

where 12060113(119891) is the phase difference of road PSD of the left-and right-front wheel According to measured pavementsthe phase differences of road PSD of the left- and right-front wheel are approximately 0 at this point the impactson vehicle performance are negligible thus it is assumedthat 12060113(119891) = 0 in this paper Therefore (11) and (9) can berewritten as

11986613 (119891) =100381610038161003816100381611986613 (119891)

1003816100381610038161003816 11989011989512060113(119891) = radic11986611 (119891) sdot 11986633 (119891) sdot coh (119891)

[119866119902 (119891)] = 119866119902 (119891)

[[[[

[

1 119890minus1198952120587119891120591 coh (119891) coh (119891) 119890minus1198952120587119891120591

1198901198952120587119891120591

1 coh (119891) 1198901198952120587119891120591 coh (119891)coh (119891) coh (119891) 119890minus1198952120587119891120591 1 119890

minus1198952120587119891120591

coh (119891) 1198901198952120587119891120591 coh (119891) 1198901198952120587119891120591

1

]]]]

]

(12)

Then the road spectrum relationship between the left andright wheels can be obtained as follows

1198763 (119891) = 1198761 (119891) coh (119891) (13)

And the road spectrum relationship between the front andrear wheels can be obtained as follows

1198762 (119891) = 1198761 (119891) 119890minus1198952120587119891120591

(14)

Mathematical Problems in Engineering 5

LFLR

0 2 4 6 8 10 12 14 16 18minus50

0

50

Time (s)

Road

roug

hnes

s (m

m)

Figure 5 Comparison of left-front wheel (LF) with left-rear wheel(LR) time domain signal

So far a four-wheel time domain model can be obtained viathe following steps

(1) Road roughness level 119866119902(1198990) the vehicle speed 119881the coherence function coh(119891) the lower limit of thespatial frequency119873119897 and the upper limit of the spatialfrequency119873119906 are known as initial conditions

(2) The spectrum of left-front wheel road roughness1198761(119896) can be obtained analogous to the methodof generating single-wheel road random excitationspectrum

(3) The road random excitation spectrum of other wheels1198762(119896) and 1198764(119896) can be obtained from (15)ndash(17) andthere is a time delay between the left-front road input1198761(119896) and the left-rear road input 1198763(119896) The 1198762(119896)and 1198764(119896) are obtained via a coherence functionbetween the left and right wheels

1198762 (119896) = 1198761 (119896) 119890minus1198952120587119891119896120591 (15)

1198763 (119896) = 1198761 (119896) coh (119896) (16)

1198764 (119896) = 1198763 (119896) 119890minus1198952120587119891119896120591 = 1198761 (119896) coh (119896) 119890

minus1198952120587119891119896120591 (17)

(4) The corresponding road roughness random excita-tion signals can be obtained by the inverse Fouriertransform of the complex sequences 119876119894(119896) (119894 =

1 2 3 4)

119909119894 (119899) =1

119873

119873

sum

119896=1

119876119894 (119896) 119890119895(2120587119873)119899119896

119899 = 1 2 119873 (18)

5 Simulation Results

The results indicated in this section include the simulationof the road surfaces discussed in the previous section Anda four-wheel road surface time domain signal for a class 119862random road at the speed of 70Kmh is shown in Figures 5ndash7

RFRR

0 2 4 6 8 10 12 14 16 18minus50

0

50

Time (s)

Road

roug

hnes

s (m

m)

Figure 6 Comparison of right-front wheel (RF) with right-rearwheel (RR) time domain signal

LFRF

0 2 4 6 8 10 12 14 16 18minus50

0

50

Time (s)

Road

roug

hnes

s (m

m)

Figure 7 Comparison of left-front wheel (LF) with right-frontwheel (RF) time domain signal

Figure 5 shows a comparison of the road time domainsignal between the left-front wheel (LF) and the left-rearwheel (LR) and the comparison of the road time domainsignal between the right-front wheel (RF) and the right-rear wheel (RR) is shown in Figure 6 It can be foundobviously that there is time delay between front-wheel andrear-wheel time domain signals which is exactly equal to the119871119881 (see (10)) Figure 8 is a comparison of the road PSDssimulated with the ISO 8608 standards Figures 8(a) and 8(c)demonstrate that the road PSD of the front and rear wheels isconsistent with the ISO 8608 standard values which provesthat the Inverse Fourier Transform is accurate to generatethe front- and rear-wheel time domain signals Howeverfrom Figure 7 we can see that the amplitude of right-frontwheel (RF) is smaller than left-front wheel (LF) time domainsignal simultaneously there is a deviation Δ119866119902(119891) betweenthe road PSD of the left and right wheels from the beginningof midfrequency band in Figures 8(b) and 8(d) and thedeviation increases with the frequency increasing In otherwords the road PSD of the right wheel is smaller than the

6 Mathematical Problems in Engineering

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

(a) Left-front wheel

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

(b) Right-front wheel

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

ISO standard PSDLR simulated PSD

(c) Left-rear wheel

10minus1 100 101

10minus5

Frequency (Hz)PS

D (m

2(1

m))

102

ISO standard PSDRR simulated PSD

(d) Right-rear wheel

Figure 8 Comparison of the road PSDs simulated with the ISO 8608 standards

0 2 4 6 8 10 12 14 16 18minus50

0

50

Time (s)

Road

roug

hnes

s (m

m)

Figure 9 The smaller time domain signal generated based on thedeviation Δ119866

119902(119891)

road PSD of the left wheel on the same axle that is consistentwith the ISO 8608 standard value which can be described by

Δ119866119902 (119891) = 119866119902 (119891) minus 119866119902simu (119891) (19)

where 119866119902(119891) is the standard PSD values 119866119902simu(119891) is thesimulated PSD values and Δ119866119902(119891) is the deviation betweenthe road PSD of the left wheel (consistent with the standard)and right wheel

In addition the right-wheel time domain signal cannotbe obtained when coh(119891) is 0 which only considers thecorrelation of the left- and right-wheel road roughness whileignoring the uncorrelation of the left- and right-wheel roadroughness Since the value of coherence function is between

0 and 1 and it is near 0 at high-frequency band accordingto (16) and (17) it can be found that the reasons for thedeviation aremainly from the coherence functionHence thecompensation of the right-wheel time domain signal will bepresented in the sequel

6 Correction of Four-Wheel Road TimeDomain Model

From the above we know that there is a deviation Δ119866119902(119891)between the road PSD of left and right wheels since theuncorrelation of the left- and right-wheel road roughness isignored Hence considering the uncorrelation of the left andright wheel road roughness a new road spectrum relation-ship between the left and right wheels can be described by

1198761015840

3(119891) = 1198761 (119891) coh (119891) + Δ119876 (119891) (20)

whereΔ119876(119891) reflects the uncorrelation of the left- and right-wheel road roughness which can be obtained by Δ119866119902(119891)based on (4) and (5) A compensation using time domainsignal is generated via the inverse Fourier transformofΔ119876(119891)by using new random sequence numbers which is shown inFigure 9

A new right-wheel time domain signal can be obtained bycompensating the time domain deviation signal A compar-ison of left-front wheel with right-front wheel time domain-compensated signal is shown in Figure 10 and it can be foundthat the amplitude of right-front wheel (RF) and left-frontwheel (LR) time domain signals are the same approximately

Mathematical Problems in Engineering 7

LFRF

0 2 4 6 8 10 12 14 16 18minus50

0

50

Time (s)

Road

roug

hnes

s (m

m)

Figure 10 Comparison of left-front wheel (LF) with right-frontwheel (RF) time domain signal after compensating

Then the new road PSDs of the four wheels can beobtained via Fourier transformwhich is shown in Figure 11 Itshows that the new road PSDs simulated canmatch ISO 8608standards very well In a word this frequency compensationalgorithm for a four-wheel road time domain model isaccurate

In addition for the limit case the method is also applica-ble When coh(119891) is 0 the left- and right-wheel time domainsignals are completely uncorrelated and a comparison of left-front wheel with left-rear wheel time domain signals is shownin Figure 12 It illustrates that the right-front wheel (RF)and left-front wheel (LR) time domain signals are completelyuncorrelated which correspond with the definition of thecorrelation function Figure 13 shows the comparison of theroad PSDs simulatedwith the ISO 8608 standards Obviouslythe new road PSDs simulated can match ISO 8608 standardsvery well

When coh(119891) is 1 the left- and right-wheel road rough-ness are perfectly correlated and a comparison of the left-front wheel with the left-rear wheel time domain signals isshown in Figure 14 It illustrates that the right-front wheel(RF) and left-front wheel (LR) time domain signals arecompletely the same which correspond with the definitionof the correlation function Figure 15 shows the comparisonof the road PSDs simulated with the ISO 8608 standardsObviously the new road PSDs simulated can also matchISO 8608 standards very well In summary this frequencycompensation algorithm for a four-wheel road time domainmodel is accurate

The frequency compensation algorithm can also be usedto correct the four-wheel surface time domain models gener-ated by other methods when there is a deviation between theroad PSDs simulated and the ISO 8608 standards

7 Analysis on Vehicle Dynamic Behavior

Since the road surface irregularities are the main sourceof excitation to the vehicle system to analyze the vehicle

dynamic behavior more accurately a suitable representationof these irregularities is required which is shown in Figure 16

A comparison of vehicle dynamic behavior wo (with orwithout) compensation will be presented as follows Firstly a7-DOF vertical vehicle model is shown in Figure 17 whichconsiders the vertical motion pitch and roll of the sprungmass and the four vertical motions of four unsprung masses

Each corner of the vehicle is identified with 119894 119895 indexwhere 119894 = 119897 119903holds for leftright and 119895 = 119891 119903 for frontrearThese corners (ie positions and velocities of the dynamicalpart) are described as

119911119891119897 = 119911 minus 120572119897119891 + 120573119908 minus 119911119905119891119897

119911119891119903 = 119911 minus 120572119897119891 minus 120573119908 minus 119911119905119891119903

119911119903119897 = 119911 + 120572119897119903 + 120573119908 minus 119911119905119903119897

119911119903119903 = 119911 + 120572119897119903 minus 120573119908 minus 119911119905119903119903

119891119897 = minus 119897119891 +120573119908 minus 119905119891119897

119891119903 = minus 119897119891 minus120573119908 minus 119905119891119903

119903119897 = + 119897119903 +120573119908 minus 119905119903119897

119903119903 = + 119897119903 minus120573119908 minus 119905119903119903

(21)

where 119911 is the center of the sprung mass 120572 (resp 120573) is thepitch (resp roll) angle of the sprungmass around119910-axis (resp119909-axis) at the center of sprung mass 119897119891 119897119903 and 119908 define thevehicle geometrical properties (as shown in Figure 17)

The full vehicle model as depicted on Figure 17 is givenby the following motion equations

119898119905119891119897119905119891119897 + 119896119905119891119897 (119911119905119891119897 minus 119911119891119897) minus 119865119904119891119897 minus 119865119889119891119897 = 0

119898119905119891119903119905119891119903 + 119896119905119891119903 (119911119905119891119903 minus 119911119891119903) minus 119865119904119891119903 minus 119865119889119891119903 = 0

119898119905119903119897119905119903119897 + 119896119905119903119897 (119911119905119903119897 minus 119911119903119897) minus 119865119904119903119897 minus 119865119889119903119897 = 0

119898119905119903119903119905119903119903 + 119896119905119903119903 (119911119905119903119903 minus 119911119903119903) minus 119865119904119903119903 minus 119865119889119903119903 = 0

sum119898 + 119865119904119894119895 + 119865119889119894119895 = 0

sum119868120572 minus 119865119904119891119895119897119891 + 119865119904119903119895119897119903 minus 119865119889119891119895119897119891 + 119865119889119903119895119897119903 = 0

sum119868120573120573 + 119908119865119904119894119897 minus 119908119865119904119894119903 + 119908119865119889119894119897 minus 119908119865119889119894119903 = 0

(22)

where 119898 and 119898119905119894119895 denote the sprung and unsprung massrespectivelyThemoment of inertia about the 119909-axis (resp 119910-axis) of the vehicle sprung mass is denoted as 119868120573 (resp 119868120572)119865119904119894119895 and 119865119889119894119895 are spring force and damper force respectivelywhich can be described by

119865119904119894119895 = 119896119904119894119895119911119894119895 119865119889119894119895 = 119888119894119895119894119895 (23)

where 119896119904119894119895 and 119888119894119895 are spring stiffness and damping coefficientrespectively

Then the full vehicle model can be established by Mat-labSimulink and the outputs include the weighted RMS

8 Mathematical Problems in Engineering

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

(a) Left-front wheel

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

(b) Right-front wheel

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

ISO standard PSDLR simulated PSD

(c) Left-rear wheel

10minus1 100 101

10minus5

Frequency (Hz)PS

D (m

2(1

m))

102

ISO standard PSDRR simulated PSD

(d) Right-rear wheel

Figure 11 Comparison of the new road PSDs of the four wheels simulated with the ISO 8608 standards

LFRF

0 2 4 6 8 10 12 14 16 18minus50

0

50

Time (s)

Road

roug

hnes

s (m

m)

Figure 12 Comparison of left-front wheel (LF) with left-rear wheel(LR) time domain signal when coh(119891) is 0

(root mean square) of acceleration (Ac rms ms2) the rollangle RMS (120573 rms) the pitch angle RMS (120572 rms) and thetire vertical force RMS (Ft FL rms Ft FR rms Ft RL rmsFt RR rms N)

In this paper the frequency compensation algorithm isinstalled in the road input for a sedan model to illustratethe sedanrsquos behavior The parameters of a 119862-class sedan areshown in Table 1 Simulation results run over a 119862-level roadsurface with the speed of 70Kmh tabulated in Table 2 Andthe frequency compensation algorithm can also be installedin the other vehicle runs on a random road

Table 1 Parameters of the 119862-class sedan

Symbol Description Unit Value119898 Vehicle sprung mass kg 2110119898119891119897

Frontleft unsprung mass kg 583119898119891119903

Frontright unsprung mass kg 583119898119903119897

Rearleft unsprung mass kg 633119898119903119903

Rearright unsprung mass kg 633119896119905119891119897

Frontleft vertical tyre stiffness Nm 235000119896119905119891119903

Frontright vertical tyre stiffness Nm 235000119896119905119903119897

Rearleft vertical tyre stiffness Nm 235000119896119905119903119903

Rearright vertical tyre stiffness Nm 235000119896119904119891119897

Frontleft spring stiffness Nm 27500119896119904119891119903

Frontright spring stiffness Nm 27500119896119904119903119897

Rearleft spring stiffness Nm 31300119896119904119903119903

Rearright spring stiffness Nm 31300119897119891

Distance between front-axlecog G m 1455119897119903

Distance between rear-axlecog G m 1515119908 Wheel track m 1610119868120573

Sprung mass roll moment of inertia kgsdotm2 7101119868120572

Sprung mass pitch moment of inertia kgsdotm2 36685

In this analysis the road input of left-front tire is set asa baseline then the right-front road input is obtained via acoherence function wo a compensation and the rear road

Mathematical Problems in Engineering 9

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

(a) Left-front wheel

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

(b) Right-front wheel

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

ISO standard PSDLR simulated PSD

(c) Left-rear wheel

10minus1 100 101

10minus5

Frequency (Hz)PS

D (m

2(1

m))

102

ISO standard PSDRR simulated PSD

(d) Right-rear wheel

Figure 13 Comparison of the road PSDs simulated with the ISO 8608 standards when coh(119891) is 0

LFRF

0 2 4 6 8 10 12 14 16 18minus50

0

50

Time (s)

Road

roug

hnes

s (m

m)

Figure 14 Comparison of left-front wheel (LF) with left-rear wheel(LR) time domain signal when coh(119891) is 1

input has a time delay dependent on the wheel base andvehicle speed Table 2 shows that the 120572 rms Ft FL rms andFt RL rms without compensation are almost equal to thevalues with compensation since the PSDs of the left-side roadinput are the same while the Ac rms 120573 rms Ft FR rms andFt RR rmswithout compensation are significant smaller thanthe values with compensation which is precisely caused bythe previously mentioned deviation since there is a deviationbetween the road PSDs of the two wheels on the same axlewhich is the main reason that the acceleration RMS the rollangle RMS the RMS of tire vertical force of the right-front

Table 2 Simulation results

Results Without compensation With compensation DifferenceAc rms 03638 04081 108120572 rms 00062 00062 0120573 rms 910119890 minus 4 00017 464Ft FL rms 10115119890 + 3 10162119890 + 3 05Ft FR rms 06824119890 + 3 10100119890 + 3 327Ft RL rms 10514119890 + 3 10566119890 + 3 05Ft RR rms 07190119890 + 3 10534119890 + 3 314

and right-rear wheels without compensation are smaller thanthe values with compensation Hence the deviation of thePSD of the road input between two wheels on the same axlehas an important influence on vehicle performances whichillustrates the importance of the frequency compensationalgorithm proposed which can be used for a control systeminstalled in the vehicle to compensate road roughness fordamper tuning in the future

8 Conclusions

The road surface roughness has an important influence onride comfort ride safety vehicle maneuverability driverrsquosand occupantsrsquo comfort and vehicle dynamic load In recentdecades random road models have been widely studied anda four-wheel random road time domain model is usuallygenerated based on the correlation of the four-wheel input in

10 Mathematical Problems in Engineering

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

(a) Left-front wheel

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

(b) Right-front wheel

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

ISO standard PSDLR simulated PSD

(c) Left-rear wheel

10minus1 100 101

10minus5

Frequency (Hz)PS

D (m

2(1

m))

102

ISO standard PSDRR simulated PSD

(d) Right-rear wheel

Figure 15 Comparison of the road PSDs simulated with the ISO 8608 standards when coh(119891) is 1

Vehicledynamicbehavior

VehiclemodelRoad model

Figure 16 Analysis on the vehicle dynamic behavior

which a coherence function is used to describe the coherenceof the road input between the left and right wheels usuallyGenerally the value of coherence function is between 0 and 1and it is near 0 at high-frequency band As a result the roadroughness amplitude and the road PSD of one track will besmaller than the other one on the same axle actually it iscaused by the uncorrelation between the left- and right-wheelroad surface roughness Hence a frequency compensationalgorithm is proposed to correct the deviation of the PSDof the road input between two wheels on the same axle andit is installed in a 7-DOF vehicle dynamic study The mainconclusions can be drawn from the current study as follows

(1) It is simple and accurate to generate a single-wheeltime signal model by using the inverse Fourier trans-form method

(2) With the Inverse Fourier Transform method viacoherence function the road PSDs of the front andrear wheels are consistent with the standard valueshowever there is a deviation between the two-wheelroad PSDs on the same axle The coherence function

value is between 0 and 1 and it is near 0 at high-frequency band As a result the road roughnessamplitude and the road PSD of one track will besmaller than the other one on the same axle

(3) A road frequency compensation algorithm is pro-posed to correct the deviation of the two wheels roadPSDs on the same axle and it is installed in a roadtime domain model The simulation results showedthat the road PSDs with frequency compensation areconsistent with the ISO 8608 standard values whichdemonstrates that the proposed algorithm is correctespecially for some limit case such as the left andright wheel road input are completely uncorrelated orperfectly correlated

(4) The vehicle dynamic behavior with road frequencycompensation show that the deviation compensationof two wheels on the same axle has an importantinfluence on vehicle performances which should beconsidered in the vehicle dynamic studies and theroad frequency compensation can be used for acontrol system installed in the vehicle to compensateroad roughness for damper tuning in the future

Acknowledgments

Special thanks are due to the Open Research Fund Pro-gram of the State Key Laboratory of Advanced Design andManufacturing for Vehicle Body (31115028) the China Post-doctoral Science Foundation (2012M520028) the NationalNatural Science Foundation of China (51205155) and the

Mathematical Problems in Engineering 11

lf

lr

w

I120572 120572 I120573 120573

xGy

z

z

ztrl

zrl

crl

mrl

ksrl

ktrl

ksfr

mfr

ksfl

zfl

cfl

mfl

ktfl

ztfl

cfrztfr

zfr ktfr

Figure 17 7 DOF full vehicle model

National Basic Research Program of China (973 Program)(2011CB711201) for supporting authorsrsquo research

References

[1] P Mucka ldquoRoad waviness and the dynamic tyre forcerdquo Inter-national Journal of Vehicle Design vol 36 no 2-3 pp 216ndash2322004

[2] Z Yonglin and Z Jiafan ldquoNumerical simulation of stochasticroad process using white noise filtrationrdquo Mechanical Systemsand Signal Processing vol 20 no 2 pp 363ndash372 2006

[3] K Ahlin J Granlund and F Lindstrom ldquoComparing roadprofiles with vehicle perceived roughnessrdquo International Journalof Vehicle Design vol 36 no 2-3 pp 270ndash286 2004

[4] S M Karamihas and M W Sayers The Little Book of ProfilingUniversity of Michigan Ann Arbor Mich USA 1998

[5] MAmbroz G Sustersic and I Prebil ldquoCreatingmodels of roadsections and their use in driving dynamics simulationsrdquo VehicleSystem Dynamics vol 45 no 10 pp 911ndash924 2007

[6] D J Whitehouse and J F Archard ldquoThe properties of randomsurfaces of significance in their contactrdquoProceedings of the RoyalSociety of London A vol 316 pp 97ndash121 1970

[7] M Shinozuka and C-M Jan ldquoDigital simulation of randomprocesses and its applicationsrdquo Journal of Sound and Vibrationvol 25 no 1 pp 111ndash128 1972

[8] D Cebon Interaction between Heavy Vehicles and Roads vol951 Society of Automotive Engineers 1993

[9] J-J Wu ldquoSimulation of rough surfaces with FFTrdquo TribologyInternational vol 33 no 1 pp 47ndash58 2000

[10] M Grigoriu ldquoOn the spectral representation method in sim-ulationrdquo Probabilistic Engineering Mechanics vol 8 no 2 pp75ndash90 1993

[11] ISO Mechanical Vibration-Road Surface Profiles-Reporting ofMeasured Data International Organization for Standardiza-tion 1995

[12] T Yoshimura ldquoA semi-active suspension of passenger cars usingfuzzy reasoning and the field testingrdquo International Journal ofVehicle Design vol 19 no 2 pp 150ndash166 1998

[13] K Bogsjo ldquoEvaluation of stochastic models of parallel roadtracksrdquo Probabilistic Engineering Mechanics vol 22 no 4 pp362ndash370 2007

[14] A Pazooki S Rakheja andD Cao ldquoModeling and validation ofoff-road vehicle ride dynamicsrdquoMechanical Systems and SignalProcessing vol 28 pp 679ndash695 2012

[15] R Hassan and R Evans ldquoRoad roughness characteristics incar and truck wheel tracksrdquo International Journal of PavementEngineering 2012

[16] K Bogsjo K Podgorski and I Rychlik ldquoModels for roadsurface roughnessrdquo Vehicle System Dynamics vol 50 no 5 pp725ndash747 2012

[17] S A Hassan and O S Lashine ldquoIdentification of road surfacequalities for linear and non-linear vehicle modelingrdquo in Pro-ceedings of the 20th International Modal Analysis Conference(IMAC rsquo02) pp 318ndash323 Society for Experimental Mechanics2002

[18] A N Heath ldquoApplication of the isotropic road roughnessassumptionrdquo Journal of Sound and Vibration vol 115 no 1 pp131ndash144 1987

[19] A N Heath ldquoEvaluation of the isotropic road roughnessassumptionrdquoVehicle SystemDynamics vol 17 supplement 1 pp157ndash160 1988

[20] D Cebon and D E Newland ldquoThe artificial generation of roadsurface topography by the inverse FFT methodrdquo Vehicle SystemDynamics vol 12 no 1ndash3 pp 160ndash165 1983

[21] C J Dodds ldquoThe laboratory simulation of vehicle servicestressrdquo Journal of Engineering for Industry vol 96 no 2 pp 391ndash398 1974

[22] D Cebon and D E Newlan ldquoThe artificial generation of roadsurface topography by the inverse FFT methodrdquo in Proceedingsof the 8th IAVSD-Symposium on the Dynamics of Vehicles onRoads and Tracks pp 29ndash42 Swets and Zeitlinger CambridgeMass USA 1983

[23] C J Dodds and J D Robson ldquoThe description of roadroughnessrdquo Journal of Sound and Vibration vol 31 no 2 pp175ndash183 1973

[24] K M A Kamash and J D Robson ldquoImplications of isotropy inrandom surfacesrdquo Journal of Sound and Vibration vol 54 no 1pp 131ndash145 1977

[25] K M A Kamash and J D Robson ldquoThe application of isotropyin road surface modellingrdquo Journal of Sound and Vibration vol57 no 1 pp 89ndash100 1978

12 Mathematical Problems in Engineering

[26] G C Carter ldquoCoherence and time delay estimationrdquo Proceed-ings of the IEEE vol 75 no 2 pp 236ndash255 1987

[27] K Bogsjo ldquoCoherence of road roughness in left and rightwheel-pathrdquo Vehicle System Dynamics vol 46 supplement 1 pp 599ndash609 2008

[28] R Hongbin C Sizhong andW Zhicheng ldquoModel of excitationof random road profile in time domain for a vehicle withfour wheelsrdquo in Proceedings of the International Conference onMechatronic Science Electric Engineering and Computer (MECrsquo11) pp 2332ndash2335 Jilin China August 2011

[29] J D Robson ldquoRoad surface description and vehicle responserdquoInternational Journal of Vehicle Design vol 1 no 1 pp 25ndash351979

[30] D Ammon ldquoProblems in road surface modellingrdquo VehicleSystem Dynamics vol 20 pp 28ndash41 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

0 2 4 6 8 10 12 14 16 18minus50

0

50

Time (s)

Road

roug

hnes

s (m

m)

Figure 1 Single-wheel road surface time domain signal (119862-levelroad 70Kmh)

and the corresponding spectral amplitude can be obtained asfollows

|119876 (119896)| = 119873radicΔ119891

2119866119902 (119891119896)

Δ119891 =1

119879

(4)

119876 (119896) = |119876 (119896)| sdot 119890119895120601119896 (5)

where 119876(119896) is the road random excitation spectrum 119879is the sampling time |119876(119896)| is the corresponding spectralamplitude 119895 is the imaginary unit and 120601119896 is a uniformlydistributed random variable ranged from 0 to 2120587 119896 =

1 2 119873Then road roughness random excitation signal can be

obtained by the inverse Fourier transform of the complexsequences 119876(119896) (119896 = 1 2 3 119873)

119909 (119899) =1

119873

119873

sum

119896=1

119876 (119896) 119890119895(2120587119873)119899119896

119896 = 1 2 119873 (6)

A single-wheel 119862-level road surface time domain signal isshown in Figure 1 in which the vehicle speed is 70 kmh Acomparison of the road PSD simulated with the ISO 8608standard is shown in Figure 2 which shows that the roadPSD simulated can match the ISO 119862-level road very well Ina word the inverse Fourier transform method for a single-wheel road time domain model is accurate

4 Four-Wheel Road Time Domain Model

A four-wheel vehicle is subjected to excitations due to roadroughness on the left and right wheel paths Hence todescribe the excitations we need a stochasticmodel of parallelroad tracks The model should describe the variation withineach track and the covariation between the tracks [23] Afour-wheel road time domain model is usually generatedbased on the correlation of the four-wheel road input inwhich there is a time delay between the front and rear axleand the space is correlated between the left and right wheels

10minus1 100 101

10minus4

10minus6

10minus8

Frequency (Hz)

PSD

(m2(1

m))

102

ISO standard PSDSimulated PSD

Figure 2 Comparison of the road PSD simulated with the ISO 8608standard

And a four-wheel road surface domain model is establishedbased on the following two assumptions

(1) Thewheel tracks of the vehicle front axle and rear axleare equal and the vehicle keeps in a straight line witha constant speed

(2) The statistical properties of the road profiles at leftand right tires are the same in other words the auto-power spectral density of the road profiles at left andright tires is the same and equal to the standard

While the left and right tracks are usually statistically equiv-alent the actual profiles are not identical Generally thedifference between the left and right tracks brings a rolldisturbance However the information regarding this rolldisturbance is not included in the PSD of the individualwheel paths Hence in addition to the PSD it is appropriateto study the coherence function The statistical propertiesof road roughness between the left and right wheels areusually described by a cross-power spectral density functionor a coherence function [24ndash26] Surprisingly the coherencefunctions of all roads (smoothmotorways main roads pavedcountry roads gravel roads etc) are very similar [27] Andthe coherence function is described by

coh (119899) =10038161003816100381610038161003816119866119909119910 (119899)

10038161003816100381610038161003816

radic119866119909119909 (119899) sdot 119866119910119910 (119899)

(7)

where 119866119909119909(119899) and 119866119910119910(119899) are the auto-spectral densities ofthe individual tracks (m2mminus1) and 119866119909119910(119899) is the associatedcross-spectral density (m2mminus1)119861 is the wheel track (m) and119899 is the spatial frequency (mminus1)

The measurement data yield a coherence function whichdrops from a high value at 119899 = 0 to a very small valueat high-frequency band [28] and the range is from 0 to 1When coh(119899) = 0 the left- and right-wheel road roughnessis completely uncorrelated while the left- and right-wheelroad roughness is perfectly correlated if coh(119899) = 1 Robson[29] pointed out that the coherence function depends on the

4 Mathematical Problems in Engineering

0

02

04

06

08

1

Coh

eren

ce fu

nctio

n

10minus1 100 101 102

Time (s)

Figure 3 The coherence function of the left and right track under119862-level road

2 1

4 3

V

L

B

Figure 4 A top view of four-wheel vehicle

wheel track and spatial frequency The coherence functionwas described by Ammon [30] as

coh (119899 119861) = [1 + (119899119861120576

119899119902

)

120597

]

minus119902

(8)

where the inflection point and the slope of the inflection pointare determined by the parameters 119899119902 and 119902 which can befitted by experimental data 120576 is the density of the coherencefunction between different tracks and 120597 is the frequencyindex The coherence function of the left and right tracksunder 119862-level road is shown in Figure 3

Previously the single-wheel road time domain model andcoherence function have been introduced then a four-wheelroad time domain model will be discussed as follows A topview of four-wheel vehicle is shown in Figure 4 and thespectrum matrix of four-wheel road input is described in

[119866119902 (119891)] =[[[

[

11986611 11986612 11986613 11986614

11986621 11986622 11986623 11986624

11986631 11986632 11986633 11986634

11986641 11986642 11986643 11986644

]]]

]

=

[[[[[[[[[

[

11986611 11986611119890minus1198952120587119891120591

11986613 11986613119890minus1198952120587119891120591

119866111198901198952120587119891120591

11986611 119866131198901198952120587119891120591

11986613

119866lowast

13119866lowast

13119890minus1198952120587119891120591

11986633 11986633119890minus1198952120587119891120591

119866lowast

131198901198952120587119891120591

119866lowast

13119866331198901198952120587119891120591

11986633

]]]]]]]]]

]

(9)

where 119866119894119895 (119894 = 119895) (119894 119895 = 1 2 3 4) is the cross-power spectraldensity 119866119894119895 (119894 = 119895) is the auto-power spectral density 119866lowast

119894119895is

the complex conjugate matrix of 119866119894119895 and 120591 is the time delayof front- and rear-wheel road input which can be expressedas

120591 =119871

119881 (10)

where 119871 is the wheelbase (m) and119881 is the vehicle speed (ms)11986613(119891) can be obtained by

11986613 (119891) =100381610038161003816100381611986613 (119891)

1003816100381610038161003816 11989011989512060113(119891)

= radic11986611 (119891) sdot 11986633 (119891) sdot coh (119891) sdot 11989011989512060113(119891)

(11)

where 12060113(119891) is the phase difference of road PSD of the left-and right-front wheel According to measured pavementsthe phase differences of road PSD of the left- and right-front wheel are approximately 0 at this point the impactson vehicle performance are negligible thus it is assumedthat 12060113(119891) = 0 in this paper Therefore (11) and (9) can berewritten as

11986613 (119891) =100381610038161003816100381611986613 (119891)

1003816100381610038161003816 11989011989512060113(119891) = radic11986611 (119891) sdot 11986633 (119891) sdot coh (119891)

[119866119902 (119891)] = 119866119902 (119891)

[[[[

[

1 119890minus1198952120587119891120591 coh (119891) coh (119891) 119890minus1198952120587119891120591

1198901198952120587119891120591

1 coh (119891) 1198901198952120587119891120591 coh (119891)coh (119891) coh (119891) 119890minus1198952120587119891120591 1 119890

minus1198952120587119891120591

coh (119891) 1198901198952120587119891120591 coh (119891) 1198901198952120587119891120591

1

]]]]

]

(12)

Then the road spectrum relationship between the left andright wheels can be obtained as follows

1198763 (119891) = 1198761 (119891) coh (119891) (13)

And the road spectrum relationship between the front andrear wheels can be obtained as follows

1198762 (119891) = 1198761 (119891) 119890minus1198952120587119891120591

(14)

Mathematical Problems in Engineering 5

LFLR

0 2 4 6 8 10 12 14 16 18minus50

0

50

Time (s)

Road

roug

hnes

s (m

m)

Figure 5 Comparison of left-front wheel (LF) with left-rear wheel(LR) time domain signal

So far a four-wheel time domain model can be obtained viathe following steps

(1) Road roughness level 119866119902(1198990) the vehicle speed 119881the coherence function coh(119891) the lower limit of thespatial frequency119873119897 and the upper limit of the spatialfrequency119873119906 are known as initial conditions

(2) The spectrum of left-front wheel road roughness1198761(119896) can be obtained analogous to the methodof generating single-wheel road random excitationspectrum

(3) The road random excitation spectrum of other wheels1198762(119896) and 1198764(119896) can be obtained from (15)ndash(17) andthere is a time delay between the left-front road input1198761(119896) and the left-rear road input 1198763(119896) The 1198762(119896)and 1198764(119896) are obtained via a coherence functionbetween the left and right wheels

1198762 (119896) = 1198761 (119896) 119890minus1198952120587119891119896120591 (15)

1198763 (119896) = 1198761 (119896) coh (119896) (16)

1198764 (119896) = 1198763 (119896) 119890minus1198952120587119891119896120591 = 1198761 (119896) coh (119896) 119890

minus1198952120587119891119896120591 (17)

(4) The corresponding road roughness random excita-tion signals can be obtained by the inverse Fouriertransform of the complex sequences 119876119894(119896) (119894 =

1 2 3 4)

119909119894 (119899) =1

119873

119873

sum

119896=1

119876119894 (119896) 119890119895(2120587119873)119899119896

119899 = 1 2 119873 (18)

5 Simulation Results

The results indicated in this section include the simulationof the road surfaces discussed in the previous section Anda four-wheel road surface time domain signal for a class 119862random road at the speed of 70Kmh is shown in Figures 5ndash7

RFRR

0 2 4 6 8 10 12 14 16 18minus50

0

50

Time (s)

Road

roug

hnes

s (m

m)

Figure 6 Comparison of right-front wheel (RF) with right-rearwheel (RR) time domain signal

LFRF

0 2 4 6 8 10 12 14 16 18minus50

0

50

Time (s)

Road

roug

hnes

s (m

m)

Figure 7 Comparison of left-front wheel (LF) with right-frontwheel (RF) time domain signal

Figure 5 shows a comparison of the road time domainsignal between the left-front wheel (LF) and the left-rearwheel (LR) and the comparison of the road time domainsignal between the right-front wheel (RF) and the right-rear wheel (RR) is shown in Figure 6 It can be foundobviously that there is time delay between front-wheel andrear-wheel time domain signals which is exactly equal to the119871119881 (see (10)) Figure 8 is a comparison of the road PSDssimulated with the ISO 8608 standards Figures 8(a) and 8(c)demonstrate that the road PSD of the front and rear wheels isconsistent with the ISO 8608 standard values which provesthat the Inverse Fourier Transform is accurate to generatethe front- and rear-wheel time domain signals Howeverfrom Figure 7 we can see that the amplitude of right-frontwheel (RF) is smaller than left-front wheel (LF) time domainsignal simultaneously there is a deviation Δ119866119902(119891) betweenthe road PSD of the left and right wheels from the beginningof midfrequency band in Figures 8(b) and 8(d) and thedeviation increases with the frequency increasing In otherwords the road PSD of the right wheel is smaller than the

6 Mathematical Problems in Engineering

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

(a) Left-front wheel

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

(b) Right-front wheel

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

ISO standard PSDLR simulated PSD

(c) Left-rear wheel

10minus1 100 101

10minus5

Frequency (Hz)PS

D (m

2(1

m))

102

ISO standard PSDRR simulated PSD

(d) Right-rear wheel

Figure 8 Comparison of the road PSDs simulated with the ISO 8608 standards

0 2 4 6 8 10 12 14 16 18minus50

0

50

Time (s)

Road

roug

hnes

s (m

m)

Figure 9 The smaller time domain signal generated based on thedeviation Δ119866

119902(119891)

road PSD of the left wheel on the same axle that is consistentwith the ISO 8608 standard value which can be described by

Δ119866119902 (119891) = 119866119902 (119891) minus 119866119902simu (119891) (19)

where 119866119902(119891) is the standard PSD values 119866119902simu(119891) is thesimulated PSD values and Δ119866119902(119891) is the deviation betweenthe road PSD of the left wheel (consistent with the standard)and right wheel

In addition the right-wheel time domain signal cannotbe obtained when coh(119891) is 0 which only considers thecorrelation of the left- and right-wheel road roughness whileignoring the uncorrelation of the left- and right-wheel roadroughness Since the value of coherence function is between

0 and 1 and it is near 0 at high-frequency band accordingto (16) and (17) it can be found that the reasons for thedeviation aremainly from the coherence functionHence thecompensation of the right-wheel time domain signal will bepresented in the sequel

6 Correction of Four-Wheel Road TimeDomain Model

From the above we know that there is a deviation Δ119866119902(119891)between the road PSD of left and right wheels since theuncorrelation of the left- and right-wheel road roughness isignored Hence considering the uncorrelation of the left andright wheel road roughness a new road spectrum relation-ship between the left and right wheels can be described by

1198761015840

3(119891) = 1198761 (119891) coh (119891) + Δ119876 (119891) (20)

whereΔ119876(119891) reflects the uncorrelation of the left- and right-wheel road roughness which can be obtained by Δ119866119902(119891)based on (4) and (5) A compensation using time domainsignal is generated via the inverse Fourier transformofΔ119876(119891)by using new random sequence numbers which is shown inFigure 9

A new right-wheel time domain signal can be obtained bycompensating the time domain deviation signal A compar-ison of left-front wheel with right-front wheel time domain-compensated signal is shown in Figure 10 and it can be foundthat the amplitude of right-front wheel (RF) and left-frontwheel (LR) time domain signals are the same approximately

Mathematical Problems in Engineering 7

LFRF

0 2 4 6 8 10 12 14 16 18minus50

0

50

Time (s)

Road

roug

hnes

s (m

m)

Figure 10 Comparison of left-front wheel (LF) with right-frontwheel (RF) time domain signal after compensating

Then the new road PSDs of the four wheels can beobtained via Fourier transformwhich is shown in Figure 11 Itshows that the new road PSDs simulated canmatch ISO 8608standards very well In a word this frequency compensationalgorithm for a four-wheel road time domain model isaccurate

In addition for the limit case the method is also applica-ble When coh(119891) is 0 the left- and right-wheel time domainsignals are completely uncorrelated and a comparison of left-front wheel with left-rear wheel time domain signals is shownin Figure 12 It illustrates that the right-front wheel (RF)and left-front wheel (LR) time domain signals are completelyuncorrelated which correspond with the definition of thecorrelation function Figure 13 shows the comparison of theroad PSDs simulatedwith the ISO 8608 standards Obviouslythe new road PSDs simulated can match ISO 8608 standardsvery well

When coh(119891) is 1 the left- and right-wheel road rough-ness are perfectly correlated and a comparison of the left-front wheel with the left-rear wheel time domain signals isshown in Figure 14 It illustrates that the right-front wheel(RF) and left-front wheel (LR) time domain signals arecompletely the same which correspond with the definitionof the correlation function Figure 15 shows the comparisonof the road PSDs simulated with the ISO 8608 standardsObviously the new road PSDs simulated can also matchISO 8608 standards very well In summary this frequencycompensation algorithm for a four-wheel road time domainmodel is accurate

The frequency compensation algorithm can also be usedto correct the four-wheel surface time domain models gener-ated by other methods when there is a deviation between theroad PSDs simulated and the ISO 8608 standards

7 Analysis on Vehicle Dynamic Behavior

Since the road surface irregularities are the main sourceof excitation to the vehicle system to analyze the vehicle

dynamic behavior more accurately a suitable representationof these irregularities is required which is shown in Figure 16

A comparison of vehicle dynamic behavior wo (with orwithout) compensation will be presented as follows Firstly a7-DOF vertical vehicle model is shown in Figure 17 whichconsiders the vertical motion pitch and roll of the sprungmass and the four vertical motions of four unsprung masses

Each corner of the vehicle is identified with 119894 119895 indexwhere 119894 = 119897 119903holds for leftright and 119895 = 119891 119903 for frontrearThese corners (ie positions and velocities of the dynamicalpart) are described as

119911119891119897 = 119911 minus 120572119897119891 + 120573119908 minus 119911119905119891119897

119911119891119903 = 119911 minus 120572119897119891 minus 120573119908 minus 119911119905119891119903

119911119903119897 = 119911 + 120572119897119903 + 120573119908 minus 119911119905119903119897

119911119903119903 = 119911 + 120572119897119903 minus 120573119908 minus 119911119905119903119903

119891119897 = minus 119897119891 +120573119908 minus 119905119891119897

119891119903 = minus 119897119891 minus120573119908 minus 119905119891119903

119903119897 = + 119897119903 +120573119908 minus 119905119903119897

119903119903 = + 119897119903 minus120573119908 minus 119905119903119903

(21)

where 119911 is the center of the sprung mass 120572 (resp 120573) is thepitch (resp roll) angle of the sprungmass around119910-axis (resp119909-axis) at the center of sprung mass 119897119891 119897119903 and 119908 define thevehicle geometrical properties (as shown in Figure 17)

The full vehicle model as depicted on Figure 17 is givenby the following motion equations

119898119905119891119897119905119891119897 + 119896119905119891119897 (119911119905119891119897 minus 119911119891119897) minus 119865119904119891119897 minus 119865119889119891119897 = 0

119898119905119891119903119905119891119903 + 119896119905119891119903 (119911119905119891119903 minus 119911119891119903) minus 119865119904119891119903 minus 119865119889119891119903 = 0

119898119905119903119897119905119903119897 + 119896119905119903119897 (119911119905119903119897 minus 119911119903119897) minus 119865119904119903119897 minus 119865119889119903119897 = 0

119898119905119903119903119905119903119903 + 119896119905119903119903 (119911119905119903119903 minus 119911119903119903) minus 119865119904119903119903 minus 119865119889119903119903 = 0

sum119898 + 119865119904119894119895 + 119865119889119894119895 = 0

sum119868120572 minus 119865119904119891119895119897119891 + 119865119904119903119895119897119903 minus 119865119889119891119895119897119891 + 119865119889119903119895119897119903 = 0

sum119868120573120573 + 119908119865119904119894119897 minus 119908119865119904119894119903 + 119908119865119889119894119897 minus 119908119865119889119894119903 = 0

(22)

where 119898 and 119898119905119894119895 denote the sprung and unsprung massrespectivelyThemoment of inertia about the 119909-axis (resp 119910-axis) of the vehicle sprung mass is denoted as 119868120573 (resp 119868120572)119865119904119894119895 and 119865119889119894119895 are spring force and damper force respectivelywhich can be described by

119865119904119894119895 = 119896119904119894119895119911119894119895 119865119889119894119895 = 119888119894119895119894119895 (23)

where 119896119904119894119895 and 119888119894119895 are spring stiffness and damping coefficientrespectively

Then the full vehicle model can be established by Mat-labSimulink and the outputs include the weighted RMS

8 Mathematical Problems in Engineering

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

(a) Left-front wheel

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

(b) Right-front wheel

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

ISO standard PSDLR simulated PSD

(c) Left-rear wheel

10minus1 100 101

10minus5

Frequency (Hz)PS

D (m

2(1

m))

102

ISO standard PSDRR simulated PSD

(d) Right-rear wheel

Figure 11 Comparison of the new road PSDs of the four wheels simulated with the ISO 8608 standards

LFRF

0 2 4 6 8 10 12 14 16 18minus50

0

50

Time (s)

Road

roug

hnes

s (m

m)

Figure 12 Comparison of left-front wheel (LF) with left-rear wheel(LR) time domain signal when coh(119891) is 0

(root mean square) of acceleration (Ac rms ms2) the rollangle RMS (120573 rms) the pitch angle RMS (120572 rms) and thetire vertical force RMS (Ft FL rms Ft FR rms Ft RL rmsFt RR rms N)

In this paper the frequency compensation algorithm isinstalled in the road input for a sedan model to illustratethe sedanrsquos behavior The parameters of a 119862-class sedan areshown in Table 1 Simulation results run over a 119862-level roadsurface with the speed of 70Kmh tabulated in Table 2 Andthe frequency compensation algorithm can also be installedin the other vehicle runs on a random road

Table 1 Parameters of the 119862-class sedan

Symbol Description Unit Value119898 Vehicle sprung mass kg 2110119898119891119897

Frontleft unsprung mass kg 583119898119891119903

Frontright unsprung mass kg 583119898119903119897

Rearleft unsprung mass kg 633119898119903119903

Rearright unsprung mass kg 633119896119905119891119897

Frontleft vertical tyre stiffness Nm 235000119896119905119891119903

Frontright vertical tyre stiffness Nm 235000119896119905119903119897

Rearleft vertical tyre stiffness Nm 235000119896119905119903119903

Rearright vertical tyre stiffness Nm 235000119896119904119891119897

Frontleft spring stiffness Nm 27500119896119904119891119903

Frontright spring stiffness Nm 27500119896119904119903119897

Rearleft spring stiffness Nm 31300119896119904119903119903

Rearright spring stiffness Nm 31300119897119891

Distance between front-axlecog G m 1455119897119903

Distance between rear-axlecog G m 1515119908 Wheel track m 1610119868120573

Sprung mass roll moment of inertia kgsdotm2 7101119868120572

Sprung mass pitch moment of inertia kgsdotm2 36685

In this analysis the road input of left-front tire is set asa baseline then the right-front road input is obtained via acoherence function wo a compensation and the rear road

Mathematical Problems in Engineering 9

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

(a) Left-front wheel

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

(b) Right-front wheel

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

ISO standard PSDLR simulated PSD

(c) Left-rear wheel

10minus1 100 101

10minus5

Frequency (Hz)PS

D (m

2(1

m))

102

ISO standard PSDRR simulated PSD

(d) Right-rear wheel

Figure 13 Comparison of the road PSDs simulated with the ISO 8608 standards when coh(119891) is 0

LFRF

0 2 4 6 8 10 12 14 16 18minus50

0

50

Time (s)

Road

roug

hnes

s (m

m)

Figure 14 Comparison of left-front wheel (LF) with left-rear wheel(LR) time domain signal when coh(119891) is 1

input has a time delay dependent on the wheel base andvehicle speed Table 2 shows that the 120572 rms Ft FL rms andFt RL rms without compensation are almost equal to thevalues with compensation since the PSDs of the left-side roadinput are the same while the Ac rms 120573 rms Ft FR rms andFt RR rmswithout compensation are significant smaller thanthe values with compensation which is precisely caused bythe previously mentioned deviation since there is a deviationbetween the road PSDs of the two wheels on the same axlewhich is the main reason that the acceleration RMS the rollangle RMS the RMS of tire vertical force of the right-front

Table 2 Simulation results

Results Without compensation With compensation DifferenceAc rms 03638 04081 108120572 rms 00062 00062 0120573 rms 910119890 minus 4 00017 464Ft FL rms 10115119890 + 3 10162119890 + 3 05Ft FR rms 06824119890 + 3 10100119890 + 3 327Ft RL rms 10514119890 + 3 10566119890 + 3 05Ft RR rms 07190119890 + 3 10534119890 + 3 314

and right-rear wheels without compensation are smaller thanthe values with compensation Hence the deviation of thePSD of the road input between two wheels on the same axlehas an important influence on vehicle performances whichillustrates the importance of the frequency compensationalgorithm proposed which can be used for a control systeminstalled in the vehicle to compensate road roughness fordamper tuning in the future

8 Conclusions

The road surface roughness has an important influence onride comfort ride safety vehicle maneuverability driverrsquosand occupantsrsquo comfort and vehicle dynamic load In recentdecades random road models have been widely studied anda four-wheel random road time domain model is usuallygenerated based on the correlation of the four-wheel input in

10 Mathematical Problems in Engineering

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

(a) Left-front wheel

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

(b) Right-front wheel

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

ISO standard PSDLR simulated PSD

(c) Left-rear wheel

10minus1 100 101

10minus5

Frequency (Hz)PS

D (m

2(1

m))

102

ISO standard PSDRR simulated PSD

(d) Right-rear wheel

Figure 15 Comparison of the road PSDs simulated with the ISO 8608 standards when coh(119891) is 1

Vehicledynamicbehavior

VehiclemodelRoad model

Figure 16 Analysis on the vehicle dynamic behavior

which a coherence function is used to describe the coherenceof the road input between the left and right wheels usuallyGenerally the value of coherence function is between 0 and 1and it is near 0 at high-frequency band As a result the roadroughness amplitude and the road PSD of one track will besmaller than the other one on the same axle actually it iscaused by the uncorrelation between the left- and right-wheelroad surface roughness Hence a frequency compensationalgorithm is proposed to correct the deviation of the PSDof the road input between two wheels on the same axle andit is installed in a 7-DOF vehicle dynamic study The mainconclusions can be drawn from the current study as follows

(1) It is simple and accurate to generate a single-wheeltime signal model by using the inverse Fourier trans-form method

(2) With the Inverse Fourier Transform method viacoherence function the road PSDs of the front andrear wheels are consistent with the standard valueshowever there is a deviation between the two-wheelroad PSDs on the same axle The coherence function

value is between 0 and 1 and it is near 0 at high-frequency band As a result the road roughnessamplitude and the road PSD of one track will besmaller than the other one on the same axle

(3) A road frequency compensation algorithm is pro-posed to correct the deviation of the two wheels roadPSDs on the same axle and it is installed in a roadtime domain model The simulation results showedthat the road PSDs with frequency compensation areconsistent with the ISO 8608 standard values whichdemonstrates that the proposed algorithm is correctespecially for some limit case such as the left andright wheel road input are completely uncorrelated orperfectly correlated

(4) The vehicle dynamic behavior with road frequencycompensation show that the deviation compensationof two wheels on the same axle has an importantinfluence on vehicle performances which should beconsidered in the vehicle dynamic studies and theroad frequency compensation can be used for acontrol system installed in the vehicle to compensateroad roughness for damper tuning in the future

Acknowledgments

Special thanks are due to the Open Research Fund Pro-gram of the State Key Laboratory of Advanced Design andManufacturing for Vehicle Body (31115028) the China Post-doctoral Science Foundation (2012M520028) the NationalNatural Science Foundation of China (51205155) and the

Mathematical Problems in Engineering 11

lf

lr

w

I120572 120572 I120573 120573

xGy

z

z

ztrl

zrl

crl

mrl

ksrl

ktrl

ksfr

mfr

ksfl

zfl

cfl

mfl

ktfl

ztfl

cfrztfr

zfr ktfr

Figure 17 7 DOF full vehicle model

National Basic Research Program of China (973 Program)(2011CB711201) for supporting authorsrsquo research

References

[1] P Mucka ldquoRoad waviness and the dynamic tyre forcerdquo Inter-national Journal of Vehicle Design vol 36 no 2-3 pp 216ndash2322004

[2] Z Yonglin and Z Jiafan ldquoNumerical simulation of stochasticroad process using white noise filtrationrdquo Mechanical Systemsand Signal Processing vol 20 no 2 pp 363ndash372 2006

[3] K Ahlin J Granlund and F Lindstrom ldquoComparing roadprofiles with vehicle perceived roughnessrdquo International Journalof Vehicle Design vol 36 no 2-3 pp 270ndash286 2004

[4] S M Karamihas and M W Sayers The Little Book of ProfilingUniversity of Michigan Ann Arbor Mich USA 1998

[5] MAmbroz G Sustersic and I Prebil ldquoCreatingmodels of roadsections and their use in driving dynamics simulationsrdquo VehicleSystem Dynamics vol 45 no 10 pp 911ndash924 2007

[6] D J Whitehouse and J F Archard ldquoThe properties of randomsurfaces of significance in their contactrdquoProceedings of the RoyalSociety of London A vol 316 pp 97ndash121 1970

[7] M Shinozuka and C-M Jan ldquoDigital simulation of randomprocesses and its applicationsrdquo Journal of Sound and Vibrationvol 25 no 1 pp 111ndash128 1972

[8] D Cebon Interaction between Heavy Vehicles and Roads vol951 Society of Automotive Engineers 1993

[9] J-J Wu ldquoSimulation of rough surfaces with FFTrdquo TribologyInternational vol 33 no 1 pp 47ndash58 2000

[10] M Grigoriu ldquoOn the spectral representation method in sim-ulationrdquo Probabilistic Engineering Mechanics vol 8 no 2 pp75ndash90 1993

[11] ISO Mechanical Vibration-Road Surface Profiles-Reporting ofMeasured Data International Organization for Standardiza-tion 1995

[12] T Yoshimura ldquoA semi-active suspension of passenger cars usingfuzzy reasoning and the field testingrdquo International Journal ofVehicle Design vol 19 no 2 pp 150ndash166 1998

[13] K Bogsjo ldquoEvaluation of stochastic models of parallel roadtracksrdquo Probabilistic Engineering Mechanics vol 22 no 4 pp362ndash370 2007

[14] A Pazooki S Rakheja andD Cao ldquoModeling and validation ofoff-road vehicle ride dynamicsrdquoMechanical Systems and SignalProcessing vol 28 pp 679ndash695 2012

[15] R Hassan and R Evans ldquoRoad roughness characteristics incar and truck wheel tracksrdquo International Journal of PavementEngineering 2012

[16] K Bogsjo K Podgorski and I Rychlik ldquoModels for roadsurface roughnessrdquo Vehicle System Dynamics vol 50 no 5 pp725ndash747 2012

[17] S A Hassan and O S Lashine ldquoIdentification of road surfacequalities for linear and non-linear vehicle modelingrdquo in Pro-ceedings of the 20th International Modal Analysis Conference(IMAC rsquo02) pp 318ndash323 Society for Experimental Mechanics2002

[18] A N Heath ldquoApplication of the isotropic road roughnessassumptionrdquo Journal of Sound and Vibration vol 115 no 1 pp131ndash144 1987

[19] A N Heath ldquoEvaluation of the isotropic road roughnessassumptionrdquoVehicle SystemDynamics vol 17 supplement 1 pp157ndash160 1988

[20] D Cebon and D E Newland ldquoThe artificial generation of roadsurface topography by the inverse FFT methodrdquo Vehicle SystemDynamics vol 12 no 1ndash3 pp 160ndash165 1983

[21] C J Dodds ldquoThe laboratory simulation of vehicle servicestressrdquo Journal of Engineering for Industry vol 96 no 2 pp 391ndash398 1974

[22] D Cebon and D E Newlan ldquoThe artificial generation of roadsurface topography by the inverse FFT methodrdquo in Proceedingsof the 8th IAVSD-Symposium on the Dynamics of Vehicles onRoads and Tracks pp 29ndash42 Swets and Zeitlinger CambridgeMass USA 1983

[23] C J Dodds and J D Robson ldquoThe description of roadroughnessrdquo Journal of Sound and Vibration vol 31 no 2 pp175ndash183 1973

[24] K M A Kamash and J D Robson ldquoImplications of isotropy inrandom surfacesrdquo Journal of Sound and Vibration vol 54 no 1pp 131ndash145 1977

[25] K M A Kamash and J D Robson ldquoThe application of isotropyin road surface modellingrdquo Journal of Sound and Vibration vol57 no 1 pp 89ndash100 1978

12 Mathematical Problems in Engineering

[26] G C Carter ldquoCoherence and time delay estimationrdquo Proceed-ings of the IEEE vol 75 no 2 pp 236ndash255 1987

[27] K Bogsjo ldquoCoherence of road roughness in left and rightwheel-pathrdquo Vehicle System Dynamics vol 46 supplement 1 pp 599ndash609 2008

[28] R Hongbin C Sizhong andW Zhicheng ldquoModel of excitationof random road profile in time domain for a vehicle withfour wheelsrdquo in Proceedings of the International Conference onMechatronic Science Electric Engineering and Computer (MECrsquo11) pp 2332ndash2335 Jilin China August 2011

[29] J D Robson ldquoRoad surface description and vehicle responserdquoInternational Journal of Vehicle Design vol 1 no 1 pp 25ndash351979

[30] D Ammon ldquoProblems in road surface modellingrdquo VehicleSystem Dynamics vol 20 pp 28ndash41 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

0

02

04

06

08

1

Coh

eren

ce fu

nctio

n

10minus1 100 101 102

Time (s)

Figure 3 The coherence function of the left and right track under119862-level road

2 1

4 3

V

L

B

Figure 4 A top view of four-wheel vehicle

wheel track and spatial frequency The coherence functionwas described by Ammon [30] as

coh (119899 119861) = [1 + (119899119861120576

119899119902

)

120597

]

minus119902

(8)

where the inflection point and the slope of the inflection pointare determined by the parameters 119899119902 and 119902 which can befitted by experimental data 120576 is the density of the coherencefunction between different tracks and 120597 is the frequencyindex The coherence function of the left and right tracksunder 119862-level road is shown in Figure 3

Previously the single-wheel road time domain model andcoherence function have been introduced then a four-wheelroad time domain model will be discussed as follows A topview of four-wheel vehicle is shown in Figure 4 and thespectrum matrix of four-wheel road input is described in

[119866119902 (119891)] =[[[

[

11986611 11986612 11986613 11986614

11986621 11986622 11986623 11986624

11986631 11986632 11986633 11986634

11986641 11986642 11986643 11986644

]]]

]

=

[[[[[[[[[

[

11986611 11986611119890minus1198952120587119891120591

11986613 11986613119890minus1198952120587119891120591

119866111198901198952120587119891120591

11986611 119866131198901198952120587119891120591

11986613

119866lowast

13119866lowast

13119890minus1198952120587119891120591

11986633 11986633119890minus1198952120587119891120591

119866lowast

131198901198952120587119891120591

119866lowast

13119866331198901198952120587119891120591

11986633

]]]]]]]]]

]

(9)

where 119866119894119895 (119894 = 119895) (119894 119895 = 1 2 3 4) is the cross-power spectraldensity 119866119894119895 (119894 = 119895) is the auto-power spectral density 119866lowast

119894119895is

the complex conjugate matrix of 119866119894119895 and 120591 is the time delayof front- and rear-wheel road input which can be expressedas

120591 =119871

119881 (10)

where 119871 is the wheelbase (m) and119881 is the vehicle speed (ms)11986613(119891) can be obtained by

11986613 (119891) =100381610038161003816100381611986613 (119891)

1003816100381610038161003816 11989011989512060113(119891)

= radic11986611 (119891) sdot 11986633 (119891) sdot coh (119891) sdot 11989011989512060113(119891)

(11)

where 12060113(119891) is the phase difference of road PSD of the left-and right-front wheel According to measured pavementsthe phase differences of road PSD of the left- and right-front wheel are approximately 0 at this point the impactson vehicle performance are negligible thus it is assumedthat 12060113(119891) = 0 in this paper Therefore (11) and (9) can berewritten as

11986613 (119891) =100381610038161003816100381611986613 (119891)

1003816100381610038161003816 11989011989512060113(119891) = radic11986611 (119891) sdot 11986633 (119891) sdot coh (119891)

[119866119902 (119891)] = 119866119902 (119891)

[[[[

[

1 119890minus1198952120587119891120591 coh (119891) coh (119891) 119890minus1198952120587119891120591

1198901198952120587119891120591

1 coh (119891) 1198901198952120587119891120591 coh (119891)coh (119891) coh (119891) 119890minus1198952120587119891120591 1 119890

minus1198952120587119891120591

coh (119891) 1198901198952120587119891120591 coh (119891) 1198901198952120587119891120591

1

]]]]

]

(12)

Then the road spectrum relationship between the left andright wheels can be obtained as follows

1198763 (119891) = 1198761 (119891) coh (119891) (13)

And the road spectrum relationship between the front andrear wheels can be obtained as follows

1198762 (119891) = 1198761 (119891) 119890minus1198952120587119891120591

(14)

Mathematical Problems in Engineering 5

LFLR

0 2 4 6 8 10 12 14 16 18minus50

0

50

Time (s)

Road

roug

hnes

s (m

m)

Figure 5 Comparison of left-front wheel (LF) with left-rear wheel(LR) time domain signal

So far a four-wheel time domain model can be obtained viathe following steps

(1) Road roughness level 119866119902(1198990) the vehicle speed 119881the coherence function coh(119891) the lower limit of thespatial frequency119873119897 and the upper limit of the spatialfrequency119873119906 are known as initial conditions

(2) The spectrum of left-front wheel road roughness1198761(119896) can be obtained analogous to the methodof generating single-wheel road random excitationspectrum

(3) The road random excitation spectrum of other wheels1198762(119896) and 1198764(119896) can be obtained from (15)ndash(17) andthere is a time delay between the left-front road input1198761(119896) and the left-rear road input 1198763(119896) The 1198762(119896)and 1198764(119896) are obtained via a coherence functionbetween the left and right wheels

1198762 (119896) = 1198761 (119896) 119890minus1198952120587119891119896120591 (15)

1198763 (119896) = 1198761 (119896) coh (119896) (16)

1198764 (119896) = 1198763 (119896) 119890minus1198952120587119891119896120591 = 1198761 (119896) coh (119896) 119890

minus1198952120587119891119896120591 (17)

(4) The corresponding road roughness random excita-tion signals can be obtained by the inverse Fouriertransform of the complex sequences 119876119894(119896) (119894 =

1 2 3 4)

119909119894 (119899) =1

119873

119873

sum

119896=1

119876119894 (119896) 119890119895(2120587119873)119899119896

119899 = 1 2 119873 (18)

5 Simulation Results

The results indicated in this section include the simulationof the road surfaces discussed in the previous section Anda four-wheel road surface time domain signal for a class 119862random road at the speed of 70Kmh is shown in Figures 5ndash7

RFRR

0 2 4 6 8 10 12 14 16 18minus50

0

50

Time (s)

Road

roug

hnes

s (m

m)

Figure 6 Comparison of right-front wheel (RF) with right-rearwheel (RR) time domain signal

LFRF

0 2 4 6 8 10 12 14 16 18minus50

0

50

Time (s)

Road

roug

hnes

s (m

m)

Figure 7 Comparison of left-front wheel (LF) with right-frontwheel (RF) time domain signal

Figure 5 shows a comparison of the road time domainsignal between the left-front wheel (LF) and the left-rearwheel (LR) and the comparison of the road time domainsignal between the right-front wheel (RF) and the right-rear wheel (RR) is shown in Figure 6 It can be foundobviously that there is time delay between front-wheel andrear-wheel time domain signals which is exactly equal to the119871119881 (see (10)) Figure 8 is a comparison of the road PSDssimulated with the ISO 8608 standards Figures 8(a) and 8(c)demonstrate that the road PSD of the front and rear wheels isconsistent with the ISO 8608 standard values which provesthat the Inverse Fourier Transform is accurate to generatethe front- and rear-wheel time domain signals Howeverfrom Figure 7 we can see that the amplitude of right-frontwheel (RF) is smaller than left-front wheel (LF) time domainsignal simultaneously there is a deviation Δ119866119902(119891) betweenthe road PSD of the left and right wheels from the beginningof midfrequency band in Figures 8(b) and 8(d) and thedeviation increases with the frequency increasing In otherwords the road PSD of the right wheel is smaller than the

6 Mathematical Problems in Engineering

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

(a) Left-front wheel

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

(b) Right-front wheel

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

ISO standard PSDLR simulated PSD

(c) Left-rear wheel

10minus1 100 101

10minus5

Frequency (Hz)PS

D (m

2(1

m))

102

ISO standard PSDRR simulated PSD

(d) Right-rear wheel

Figure 8 Comparison of the road PSDs simulated with the ISO 8608 standards

0 2 4 6 8 10 12 14 16 18minus50

0

50

Time (s)

Road

roug

hnes

s (m

m)

Figure 9 The smaller time domain signal generated based on thedeviation Δ119866

119902(119891)

road PSD of the left wheel on the same axle that is consistentwith the ISO 8608 standard value which can be described by

Δ119866119902 (119891) = 119866119902 (119891) minus 119866119902simu (119891) (19)

where 119866119902(119891) is the standard PSD values 119866119902simu(119891) is thesimulated PSD values and Δ119866119902(119891) is the deviation betweenthe road PSD of the left wheel (consistent with the standard)and right wheel

In addition the right-wheel time domain signal cannotbe obtained when coh(119891) is 0 which only considers thecorrelation of the left- and right-wheel road roughness whileignoring the uncorrelation of the left- and right-wheel roadroughness Since the value of coherence function is between

0 and 1 and it is near 0 at high-frequency band accordingto (16) and (17) it can be found that the reasons for thedeviation aremainly from the coherence functionHence thecompensation of the right-wheel time domain signal will bepresented in the sequel

6 Correction of Four-Wheel Road TimeDomain Model

From the above we know that there is a deviation Δ119866119902(119891)between the road PSD of left and right wheels since theuncorrelation of the left- and right-wheel road roughness isignored Hence considering the uncorrelation of the left andright wheel road roughness a new road spectrum relation-ship between the left and right wheels can be described by

1198761015840

3(119891) = 1198761 (119891) coh (119891) + Δ119876 (119891) (20)

whereΔ119876(119891) reflects the uncorrelation of the left- and right-wheel road roughness which can be obtained by Δ119866119902(119891)based on (4) and (5) A compensation using time domainsignal is generated via the inverse Fourier transformofΔ119876(119891)by using new random sequence numbers which is shown inFigure 9

A new right-wheel time domain signal can be obtained bycompensating the time domain deviation signal A compar-ison of left-front wheel with right-front wheel time domain-compensated signal is shown in Figure 10 and it can be foundthat the amplitude of right-front wheel (RF) and left-frontwheel (LR) time domain signals are the same approximately

Mathematical Problems in Engineering 7

LFRF

0 2 4 6 8 10 12 14 16 18minus50

0

50

Time (s)

Road

roug

hnes

s (m

m)

Figure 10 Comparison of left-front wheel (LF) with right-frontwheel (RF) time domain signal after compensating

Then the new road PSDs of the four wheels can beobtained via Fourier transformwhich is shown in Figure 11 Itshows that the new road PSDs simulated canmatch ISO 8608standards very well In a word this frequency compensationalgorithm for a four-wheel road time domain model isaccurate

In addition for the limit case the method is also applica-ble When coh(119891) is 0 the left- and right-wheel time domainsignals are completely uncorrelated and a comparison of left-front wheel with left-rear wheel time domain signals is shownin Figure 12 It illustrates that the right-front wheel (RF)and left-front wheel (LR) time domain signals are completelyuncorrelated which correspond with the definition of thecorrelation function Figure 13 shows the comparison of theroad PSDs simulatedwith the ISO 8608 standards Obviouslythe new road PSDs simulated can match ISO 8608 standardsvery well

When coh(119891) is 1 the left- and right-wheel road rough-ness are perfectly correlated and a comparison of the left-front wheel with the left-rear wheel time domain signals isshown in Figure 14 It illustrates that the right-front wheel(RF) and left-front wheel (LR) time domain signals arecompletely the same which correspond with the definitionof the correlation function Figure 15 shows the comparisonof the road PSDs simulated with the ISO 8608 standardsObviously the new road PSDs simulated can also matchISO 8608 standards very well In summary this frequencycompensation algorithm for a four-wheel road time domainmodel is accurate

The frequency compensation algorithm can also be usedto correct the four-wheel surface time domain models gener-ated by other methods when there is a deviation between theroad PSDs simulated and the ISO 8608 standards

7 Analysis on Vehicle Dynamic Behavior

Since the road surface irregularities are the main sourceof excitation to the vehicle system to analyze the vehicle

dynamic behavior more accurately a suitable representationof these irregularities is required which is shown in Figure 16

A comparison of vehicle dynamic behavior wo (with orwithout) compensation will be presented as follows Firstly a7-DOF vertical vehicle model is shown in Figure 17 whichconsiders the vertical motion pitch and roll of the sprungmass and the four vertical motions of four unsprung masses

Each corner of the vehicle is identified with 119894 119895 indexwhere 119894 = 119897 119903holds for leftright and 119895 = 119891 119903 for frontrearThese corners (ie positions and velocities of the dynamicalpart) are described as

119911119891119897 = 119911 minus 120572119897119891 + 120573119908 minus 119911119905119891119897

119911119891119903 = 119911 minus 120572119897119891 minus 120573119908 minus 119911119905119891119903

119911119903119897 = 119911 + 120572119897119903 + 120573119908 minus 119911119905119903119897

119911119903119903 = 119911 + 120572119897119903 minus 120573119908 minus 119911119905119903119903

119891119897 = minus 119897119891 +120573119908 minus 119905119891119897

119891119903 = minus 119897119891 minus120573119908 minus 119905119891119903

119903119897 = + 119897119903 +120573119908 minus 119905119903119897

119903119903 = + 119897119903 minus120573119908 minus 119905119903119903

(21)

where 119911 is the center of the sprung mass 120572 (resp 120573) is thepitch (resp roll) angle of the sprungmass around119910-axis (resp119909-axis) at the center of sprung mass 119897119891 119897119903 and 119908 define thevehicle geometrical properties (as shown in Figure 17)

The full vehicle model as depicted on Figure 17 is givenby the following motion equations

119898119905119891119897119905119891119897 + 119896119905119891119897 (119911119905119891119897 minus 119911119891119897) minus 119865119904119891119897 minus 119865119889119891119897 = 0

119898119905119891119903119905119891119903 + 119896119905119891119903 (119911119905119891119903 minus 119911119891119903) minus 119865119904119891119903 minus 119865119889119891119903 = 0

119898119905119903119897119905119903119897 + 119896119905119903119897 (119911119905119903119897 minus 119911119903119897) minus 119865119904119903119897 minus 119865119889119903119897 = 0

119898119905119903119903119905119903119903 + 119896119905119903119903 (119911119905119903119903 minus 119911119903119903) minus 119865119904119903119903 minus 119865119889119903119903 = 0

sum119898 + 119865119904119894119895 + 119865119889119894119895 = 0

sum119868120572 minus 119865119904119891119895119897119891 + 119865119904119903119895119897119903 minus 119865119889119891119895119897119891 + 119865119889119903119895119897119903 = 0

sum119868120573120573 + 119908119865119904119894119897 minus 119908119865119904119894119903 + 119908119865119889119894119897 minus 119908119865119889119894119903 = 0

(22)

where 119898 and 119898119905119894119895 denote the sprung and unsprung massrespectivelyThemoment of inertia about the 119909-axis (resp 119910-axis) of the vehicle sprung mass is denoted as 119868120573 (resp 119868120572)119865119904119894119895 and 119865119889119894119895 are spring force and damper force respectivelywhich can be described by

119865119904119894119895 = 119896119904119894119895119911119894119895 119865119889119894119895 = 119888119894119895119894119895 (23)

where 119896119904119894119895 and 119888119894119895 are spring stiffness and damping coefficientrespectively

Then the full vehicle model can be established by Mat-labSimulink and the outputs include the weighted RMS

8 Mathematical Problems in Engineering

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

(a) Left-front wheel

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

(b) Right-front wheel

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

ISO standard PSDLR simulated PSD

(c) Left-rear wheel

10minus1 100 101

10minus5

Frequency (Hz)PS

D (m

2(1

m))

102

ISO standard PSDRR simulated PSD

(d) Right-rear wheel

Figure 11 Comparison of the new road PSDs of the four wheels simulated with the ISO 8608 standards

LFRF

0 2 4 6 8 10 12 14 16 18minus50

0

50

Time (s)

Road

roug

hnes

s (m

m)

Figure 12 Comparison of left-front wheel (LF) with left-rear wheel(LR) time domain signal when coh(119891) is 0

(root mean square) of acceleration (Ac rms ms2) the rollangle RMS (120573 rms) the pitch angle RMS (120572 rms) and thetire vertical force RMS (Ft FL rms Ft FR rms Ft RL rmsFt RR rms N)

In this paper the frequency compensation algorithm isinstalled in the road input for a sedan model to illustratethe sedanrsquos behavior The parameters of a 119862-class sedan areshown in Table 1 Simulation results run over a 119862-level roadsurface with the speed of 70Kmh tabulated in Table 2 Andthe frequency compensation algorithm can also be installedin the other vehicle runs on a random road

Table 1 Parameters of the 119862-class sedan

Symbol Description Unit Value119898 Vehicle sprung mass kg 2110119898119891119897

Frontleft unsprung mass kg 583119898119891119903

Frontright unsprung mass kg 583119898119903119897

Rearleft unsprung mass kg 633119898119903119903

Rearright unsprung mass kg 633119896119905119891119897

Frontleft vertical tyre stiffness Nm 235000119896119905119891119903

Frontright vertical tyre stiffness Nm 235000119896119905119903119897

Rearleft vertical tyre stiffness Nm 235000119896119905119903119903

Rearright vertical tyre stiffness Nm 235000119896119904119891119897

Frontleft spring stiffness Nm 27500119896119904119891119903

Frontright spring stiffness Nm 27500119896119904119903119897

Rearleft spring stiffness Nm 31300119896119904119903119903

Rearright spring stiffness Nm 31300119897119891

Distance between front-axlecog G m 1455119897119903

Distance between rear-axlecog G m 1515119908 Wheel track m 1610119868120573

Sprung mass roll moment of inertia kgsdotm2 7101119868120572

Sprung mass pitch moment of inertia kgsdotm2 36685

In this analysis the road input of left-front tire is set asa baseline then the right-front road input is obtained via acoherence function wo a compensation and the rear road

Mathematical Problems in Engineering 9

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

(a) Left-front wheel

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

(b) Right-front wheel

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

ISO standard PSDLR simulated PSD

(c) Left-rear wheel

10minus1 100 101

10minus5

Frequency (Hz)PS

D (m

2(1

m))

102

ISO standard PSDRR simulated PSD

(d) Right-rear wheel

Figure 13 Comparison of the road PSDs simulated with the ISO 8608 standards when coh(119891) is 0

LFRF

0 2 4 6 8 10 12 14 16 18minus50

0

50

Time (s)

Road

roug

hnes

s (m

m)

Figure 14 Comparison of left-front wheel (LF) with left-rear wheel(LR) time domain signal when coh(119891) is 1

input has a time delay dependent on the wheel base andvehicle speed Table 2 shows that the 120572 rms Ft FL rms andFt RL rms without compensation are almost equal to thevalues with compensation since the PSDs of the left-side roadinput are the same while the Ac rms 120573 rms Ft FR rms andFt RR rmswithout compensation are significant smaller thanthe values with compensation which is precisely caused bythe previously mentioned deviation since there is a deviationbetween the road PSDs of the two wheels on the same axlewhich is the main reason that the acceleration RMS the rollangle RMS the RMS of tire vertical force of the right-front

Table 2 Simulation results

Results Without compensation With compensation DifferenceAc rms 03638 04081 108120572 rms 00062 00062 0120573 rms 910119890 minus 4 00017 464Ft FL rms 10115119890 + 3 10162119890 + 3 05Ft FR rms 06824119890 + 3 10100119890 + 3 327Ft RL rms 10514119890 + 3 10566119890 + 3 05Ft RR rms 07190119890 + 3 10534119890 + 3 314

and right-rear wheels without compensation are smaller thanthe values with compensation Hence the deviation of thePSD of the road input between two wheels on the same axlehas an important influence on vehicle performances whichillustrates the importance of the frequency compensationalgorithm proposed which can be used for a control systeminstalled in the vehicle to compensate road roughness fordamper tuning in the future

8 Conclusions

The road surface roughness has an important influence onride comfort ride safety vehicle maneuverability driverrsquosand occupantsrsquo comfort and vehicle dynamic load In recentdecades random road models have been widely studied anda four-wheel random road time domain model is usuallygenerated based on the correlation of the four-wheel input in

10 Mathematical Problems in Engineering

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

(a) Left-front wheel

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

(b) Right-front wheel

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

ISO standard PSDLR simulated PSD

(c) Left-rear wheel

10minus1 100 101

10minus5

Frequency (Hz)PS

D (m

2(1

m))

102

ISO standard PSDRR simulated PSD

(d) Right-rear wheel

Figure 15 Comparison of the road PSDs simulated with the ISO 8608 standards when coh(119891) is 1

Vehicledynamicbehavior

VehiclemodelRoad model

Figure 16 Analysis on the vehicle dynamic behavior

which a coherence function is used to describe the coherenceof the road input between the left and right wheels usuallyGenerally the value of coherence function is between 0 and 1and it is near 0 at high-frequency band As a result the roadroughness amplitude and the road PSD of one track will besmaller than the other one on the same axle actually it iscaused by the uncorrelation between the left- and right-wheelroad surface roughness Hence a frequency compensationalgorithm is proposed to correct the deviation of the PSDof the road input between two wheels on the same axle andit is installed in a 7-DOF vehicle dynamic study The mainconclusions can be drawn from the current study as follows

(1) It is simple and accurate to generate a single-wheeltime signal model by using the inverse Fourier trans-form method

(2) With the Inverse Fourier Transform method viacoherence function the road PSDs of the front andrear wheels are consistent with the standard valueshowever there is a deviation between the two-wheelroad PSDs on the same axle The coherence function

value is between 0 and 1 and it is near 0 at high-frequency band As a result the road roughnessamplitude and the road PSD of one track will besmaller than the other one on the same axle

(3) A road frequency compensation algorithm is pro-posed to correct the deviation of the two wheels roadPSDs on the same axle and it is installed in a roadtime domain model The simulation results showedthat the road PSDs with frequency compensation areconsistent with the ISO 8608 standard values whichdemonstrates that the proposed algorithm is correctespecially for some limit case such as the left andright wheel road input are completely uncorrelated orperfectly correlated

(4) The vehicle dynamic behavior with road frequencycompensation show that the deviation compensationof two wheels on the same axle has an importantinfluence on vehicle performances which should beconsidered in the vehicle dynamic studies and theroad frequency compensation can be used for acontrol system installed in the vehicle to compensateroad roughness for damper tuning in the future

Acknowledgments

Special thanks are due to the Open Research Fund Pro-gram of the State Key Laboratory of Advanced Design andManufacturing for Vehicle Body (31115028) the China Post-doctoral Science Foundation (2012M520028) the NationalNatural Science Foundation of China (51205155) and the

Mathematical Problems in Engineering 11

lf

lr

w

I120572 120572 I120573 120573

xGy

z

z

ztrl

zrl

crl

mrl

ksrl

ktrl

ksfr

mfr

ksfl

zfl

cfl

mfl

ktfl

ztfl

cfrztfr

zfr ktfr

Figure 17 7 DOF full vehicle model

National Basic Research Program of China (973 Program)(2011CB711201) for supporting authorsrsquo research

References

[1] P Mucka ldquoRoad waviness and the dynamic tyre forcerdquo Inter-national Journal of Vehicle Design vol 36 no 2-3 pp 216ndash2322004

[2] Z Yonglin and Z Jiafan ldquoNumerical simulation of stochasticroad process using white noise filtrationrdquo Mechanical Systemsand Signal Processing vol 20 no 2 pp 363ndash372 2006

[3] K Ahlin J Granlund and F Lindstrom ldquoComparing roadprofiles with vehicle perceived roughnessrdquo International Journalof Vehicle Design vol 36 no 2-3 pp 270ndash286 2004

[4] S M Karamihas and M W Sayers The Little Book of ProfilingUniversity of Michigan Ann Arbor Mich USA 1998

[5] MAmbroz G Sustersic and I Prebil ldquoCreatingmodels of roadsections and their use in driving dynamics simulationsrdquo VehicleSystem Dynamics vol 45 no 10 pp 911ndash924 2007

[6] D J Whitehouse and J F Archard ldquoThe properties of randomsurfaces of significance in their contactrdquoProceedings of the RoyalSociety of London A vol 316 pp 97ndash121 1970

[7] M Shinozuka and C-M Jan ldquoDigital simulation of randomprocesses and its applicationsrdquo Journal of Sound and Vibrationvol 25 no 1 pp 111ndash128 1972

[8] D Cebon Interaction between Heavy Vehicles and Roads vol951 Society of Automotive Engineers 1993

[9] J-J Wu ldquoSimulation of rough surfaces with FFTrdquo TribologyInternational vol 33 no 1 pp 47ndash58 2000

[10] M Grigoriu ldquoOn the spectral representation method in sim-ulationrdquo Probabilistic Engineering Mechanics vol 8 no 2 pp75ndash90 1993

[11] ISO Mechanical Vibration-Road Surface Profiles-Reporting ofMeasured Data International Organization for Standardiza-tion 1995

[12] T Yoshimura ldquoA semi-active suspension of passenger cars usingfuzzy reasoning and the field testingrdquo International Journal ofVehicle Design vol 19 no 2 pp 150ndash166 1998

[13] K Bogsjo ldquoEvaluation of stochastic models of parallel roadtracksrdquo Probabilistic Engineering Mechanics vol 22 no 4 pp362ndash370 2007

[14] A Pazooki S Rakheja andD Cao ldquoModeling and validation ofoff-road vehicle ride dynamicsrdquoMechanical Systems and SignalProcessing vol 28 pp 679ndash695 2012

[15] R Hassan and R Evans ldquoRoad roughness characteristics incar and truck wheel tracksrdquo International Journal of PavementEngineering 2012

[16] K Bogsjo K Podgorski and I Rychlik ldquoModels for roadsurface roughnessrdquo Vehicle System Dynamics vol 50 no 5 pp725ndash747 2012

[17] S A Hassan and O S Lashine ldquoIdentification of road surfacequalities for linear and non-linear vehicle modelingrdquo in Pro-ceedings of the 20th International Modal Analysis Conference(IMAC rsquo02) pp 318ndash323 Society for Experimental Mechanics2002

[18] A N Heath ldquoApplication of the isotropic road roughnessassumptionrdquo Journal of Sound and Vibration vol 115 no 1 pp131ndash144 1987

[19] A N Heath ldquoEvaluation of the isotropic road roughnessassumptionrdquoVehicle SystemDynamics vol 17 supplement 1 pp157ndash160 1988

[20] D Cebon and D E Newland ldquoThe artificial generation of roadsurface topography by the inverse FFT methodrdquo Vehicle SystemDynamics vol 12 no 1ndash3 pp 160ndash165 1983

[21] C J Dodds ldquoThe laboratory simulation of vehicle servicestressrdquo Journal of Engineering for Industry vol 96 no 2 pp 391ndash398 1974

[22] D Cebon and D E Newlan ldquoThe artificial generation of roadsurface topography by the inverse FFT methodrdquo in Proceedingsof the 8th IAVSD-Symposium on the Dynamics of Vehicles onRoads and Tracks pp 29ndash42 Swets and Zeitlinger CambridgeMass USA 1983

[23] C J Dodds and J D Robson ldquoThe description of roadroughnessrdquo Journal of Sound and Vibration vol 31 no 2 pp175ndash183 1973

[24] K M A Kamash and J D Robson ldquoImplications of isotropy inrandom surfacesrdquo Journal of Sound and Vibration vol 54 no 1pp 131ndash145 1977

[25] K M A Kamash and J D Robson ldquoThe application of isotropyin road surface modellingrdquo Journal of Sound and Vibration vol57 no 1 pp 89ndash100 1978

12 Mathematical Problems in Engineering

[26] G C Carter ldquoCoherence and time delay estimationrdquo Proceed-ings of the IEEE vol 75 no 2 pp 236ndash255 1987

[27] K Bogsjo ldquoCoherence of road roughness in left and rightwheel-pathrdquo Vehicle System Dynamics vol 46 supplement 1 pp 599ndash609 2008

[28] R Hongbin C Sizhong andW Zhicheng ldquoModel of excitationof random road profile in time domain for a vehicle withfour wheelsrdquo in Proceedings of the International Conference onMechatronic Science Electric Engineering and Computer (MECrsquo11) pp 2332ndash2335 Jilin China August 2011

[29] J D Robson ldquoRoad surface description and vehicle responserdquoInternational Journal of Vehicle Design vol 1 no 1 pp 25ndash351979

[30] D Ammon ldquoProblems in road surface modellingrdquo VehicleSystem Dynamics vol 20 pp 28ndash41 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

LFLR

0 2 4 6 8 10 12 14 16 18minus50

0

50

Time (s)

Road

roug

hnes

s (m

m)

Figure 5 Comparison of left-front wheel (LF) with left-rear wheel(LR) time domain signal

So far a four-wheel time domain model can be obtained viathe following steps

(1) Road roughness level 119866119902(1198990) the vehicle speed 119881the coherence function coh(119891) the lower limit of thespatial frequency119873119897 and the upper limit of the spatialfrequency119873119906 are known as initial conditions

(2) The spectrum of left-front wheel road roughness1198761(119896) can be obtained analogous to the methodof generating single-wheel road random excitationspectrum

(3) The road random excitation spectrum of other wheels1198762(119896) and 1198764(119896) can be obtained from (15)ndash(17) andthere is a time delay between the left-front road input1198761(119896) and the left-rear road input 1198763(119896) The 1198762(119896)and 1198764(119896) are obtained via a coherence functionbetween the left and right wheels

1198762 (119896) = 1198761 (119896) 119890minus1198952120587119891119896120591 (15)

1198763 (119896) = 1198761 (119896) coh (119896) (16)

1198764 (119896) = 1198763 (119896) 119890minus1198952120587119891119896120591 = 1198761 (119896) coh (119896) 119890

minus1198952120587119891119896120591 (17)

(4) The corresponding road roughness random excita-tion signals can be obtained by the inverse Fouriertransform of the complex sequences 119876119894(119896) (119894 =

1 2 3 4)

119909119894 (119899) =1

119873

119873

sum

119896=1

119876119894 (119896) 119890119895(2120587119873)119899119896

119899 = 1 2 119873 (18)

5 Simulation Results

The results indicated in this section include the simulationof the road surfaces discussed in the previous section Anda four-wheel road surface time domain signal for a class 119862random road at the speed of 70Kmh is shown in Figures 5ndash7

RFRR

0 2 4 6 8 10 12 14 16 18minus50

0

50

Time (s)

Road

roug

hnes

s (m

m)

Figure 6 Comparison of right-front wheel (RF) with right-rearwheel (RR) time domain signal

LFRF

0 2 4 6 8 10 12 14 16 18minus50

0

50

Time (s)

Road

roug

hnes

s (m

m)

Figure 7 Comparison of left-front wheel (LF) with right-frontwheel (RF) time domain signal

Figure 5 shows a comparison of the road time domainsignal between the left-front wheel (LF) and the left-rearwheel (LR) and the comparison of the road time domainsignal between the right-front wheel (RF) and the right-rear wheel (RR) is shown in Figure 6 It can be foundobviously that there is time delay between front-wheel andrear-wheel time domain signals which is exactly equal to the119871119881 (see (10)) Figure 8 is a comparison of the road PSDssimulated with the ISO 8608 standards Figures 8(a) and 8(c)demonstrate that the road PSD of the front and rear wheels isconsistent with the ISO 8608 standard values which provesthat the Inverse Fourier Transform is accurate to generatethe front- and rear-wheel time domain signals Howeverfrom Figure 7 we can see that the amplitude of right-frontwheel (RF) is smaller than left-front wheel (LF) time domainsignal simultaneously there is a deviation Δ119866119902(119891) betweenthe road PSD of the left and right wheels from the beginningof midfrequency band in Figures 8(b) and 8(d) and thedeviation increases with the frequency increasing In otherwords the road PSD of the right wheel is smaller than the

6 Mathematical Problems in Engineering

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

(a) Left-front wheel

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

(b) Right-front wheel

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

ISO standard PSDLR simulated PSD

(c) Left-rear wheel

10minus1 100 101

10minus5

Frequency (Hz)PS

D (m

2(1

m))

102

ISO standard PSDRR simulated PSD

(d) Right-rear wheel

Figure 8 Comparison of the road PSDs simulated with the ISO 8608 standards

0 2 4 6 8 10 12 14 16 18minus50

0

50

Time (s)

Road

roug

hnes

s (m

m)

Figure 9 The smaller time domain signal generated based on thedeviation Δ119866

119902(119891)

road PSD of the left wheel on the same axle that is consistentwith the ISO 8608 standard value which can be described by

Δ119866119902 (119891) = 119866119902 (119891) minus 119866119902simu (119891) (19)

where 119866119902(119891) is the standard PSD values 119866119902simu(119891) is thesimulated PSD values and Δ119866119902(119891) is the deviation betweenthe road PSD of the left wheel (consistent with the standard)and right wheel

In addition the right-wheel time domain signal cannotbe obtained when coh(119891) is 0 which only considers thecorrelation of the left- and right-wheel road roughness whileignoring the uncorrelation of the left- and right-wheel roadroughness Since the value of coherence function is between

0 and 1 and it is near 0 at high-frequency band accordingto (16) and (17) it can be found that the reasons for thedeviation aremainly from the coherence functionHence thecompensation of the right-wheel time domain signal will bepresented in the sequel

6 Correction of Four-Wheel Road TimeDomain Model

From the above we know that there is a deviation Δ119866119902(119891)between the road PSD of left and right wheels since theuncorrelation of the left- and right-wheel road roughness isignored Hence considering the uncorrelation of the left andright wheel road roughness a new road spectrum relation-ship between the left and right wheels can be described by

1198761015840

3(119891) = 1198761 (119891) coh (119891) + Δ119876 (119891) (20)

whereΔ119876(119891) reflects the uncorrelation of the left- and right-wheel road roughness which can be obtained by Δ119866119902(119891)based on (4) and (5) A compensation using time domainsignal is generated via the inverse Fourier transformofΔ119876(119891)by using new random sequence numbers which is shown inFigure 9

A new right-wheel time domain signal can be obtained bycompensating the time domain deviation signal A compar-ison of left-front wheel with right-front wheel time domain-compensated signal is shown in Figure 10 and it can be foundthat the amplitude of right-front wheel (RF) and left-frontwheel (LR) time domain signals are the same approximately

Mathematical Problems in Engineering 7

LFRF

0 2 4 6 8 10 12 14 16 18minus50

0

50

Time (s)

Road

roug

hnes

s (m

m)

Figure 10 Comparison of left-front wheel (LF) with right-frontwheel (RF) time domain signal after compensating

Then the new road PSDs of the four wheels can beobtained via Fourier transformwhich is shown in Figure 11 Itshows that the new road PSDs simulated canmatch ISO 8608standards very well In a word this frequency compensationalgorithm for a four-wheel road time domain model isaccurate

In addition for the limit case the method is also applica-ble When coh(119891) is 0 the left- and right-wheel time domainsignals are completely uncorrelated and a comparison of left-front wheel with left-rear wheel time domain signals is shownin Figure 12 It illustrates that the right-front wheel (RF)and left-front wheel (LR) time domain signals are completelyuncorrelated which correspond with the definition of thecorrelation function Figure 13 shows the comparison of theroad PSDs simulatedwith the ISO 8608 standards Obviouslythe new road PSDs simulated can match ISO 8608 standardsvery well

When coh(119891) is 1 the left- and right-wheel road rough-ness are perfectly correlated and a comparison of the left-front wheel with the left-rear wheel time domain signals isshown in Figure 14 It illustrates that the right-front wheel(RF) and left-front wheel (LR) time domain signals arecompletely the same which correspond with the definitionof the correlation function Figure 15 shows the comparisonof the road PSDs simulated with the ISO 8608 standardsObviously the new road PSDs simulated can also matchISO 8608 standards very well In summary this frequencycompensation algorithm for a four-wheel road time domainmodel is accurate

The frequency compensation algorithm can also be usedto correct the four-wheel surface time domain models gener-ated by other methods when there is a deviation between theroad PSDs simulated and the ISO 8608 standards

7 Analysis on Vehicle Dynamic Behavior

Since the road surface irregularities are the main sourceof excitation to the vehicle system to analyze the vehicle

dynamic behavior more accurately a suitable representationof these irregularities is required which is shown in Figure 16

A comparison of vehicle dynamic behavior wo (with orwithout) compensation will be presented as follows Firstly a7-DOF vertical vehicle model is shown in Figure 17 whichconsiders the vertical motion pitch and roll of the sprungmass and the four vertical motions of four unsprung masses

Each corner of the vehicle is identified with 119894 119895 indexwhere 119894 = 119897 119903holds for leftright and 119895 = 119891 119903 for frontrearThese corners (ie positions and velocities of the dynamicalpart) are described as

119911119891119897 = 119911 minus 120572119897119891 + 120573119908 minus 119911119905119891119897

119911119891119903 = 119911 minus 120572119897119891 minus 120573119908 minus 119911119905119891119903

119911119903119897 = 119911 + 120572119897119903 + 120573119908 minus 119911119905119903119897

119911119903119903 = 119911 + 120572119897119903 minus 120573119908 minus 119911119905119903119903

119891119897 = minus 119897119891 +120573119908 minus 119905119891119897

119891119903 = minus 119897119891 minus120573119908 minus 119905119891119903

119903119897 = + 119897119903 +120573119908 minus 119905119903119897

119903119903 = + 119897119903 minus120573119908 minus 119905119903119903

(21)

where 119911 is the center of the sprung mass 120572 (resp 120573) is thepitch (resp roll) angle of the sprungmass around119910-axis (resp119909-axis) at the center of sprung mass 119897119891 119897119903 and 119908 define thevehicle geometrical properties (as shown in Figure 17)

The full vehicle model as depicted on Figure 17 is givenby the following motion equations

119898119905119891119897119905119891119897 + 119896119905119891119897 (119911119905119891119897 minus 119911119891119897) minus 119865119904119891119897 minus 119865119889119891119897 = 0

119898119905119891119903119905119891119903 + 119896119905119891119903 (119911119905119891119903 minus 119911119891119903) minus 119865119904119891119903 minus 119865119889119891119903 = 0

119898119905119903119897119905119903119897 + 119896119905119903119897 (119911119905119903119897 minus 119911119903119897) minus 119865119904119903119897 minus 119865119889119903119897 = 0

119898119905119903119903119905119903119903 + 119896119905119903119903 (119911119905119903119903 minus 119911119903119903) minus 119865119904119903119903 minus 119865119889119903119903 = 0

sum119898 + 119865119904119894119895 + 119865119889119894119895 = 0

sum119868120572 minus 119865119904119891119895119897119891 + 119865119904119903119895119897119903 minus 119865119889119891119895119897119891 + 119865119889119903119895119897119903 = 0

sum119868120573120573 + 119908119865119904119894119897 minus 119908119865119904119894119903 + 119908119865119889119894119897 minus 119908119865119889119894119903 = 0

(22)

where 119898 and 119898119905119894119895 denote the sprung and unsprung massrespectivelyThemoment of inertia about the 119909-axis (resp 119910-axis) of the vehicle sprung mass is denoted as 119868120573 (resp 119868120572)119865119904119894119895 and 119865119889119894119895 are spring force and damper force respectivelywhich can be described by

119865119904119894119895 = 119896119904119894119895119911119894119895 119865119889119894119895 = 119888119894119895119894119895 (23)

where 119896119904119894119895 and 119888119894119895 are spring stiffness and damping coefficientrespectively

Then the full vehicle model can be established by Mat-labSimulink and the outputs include the weighted RMS

8 Mathematical Problems in Engineering

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

(a) Left-front wheel

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

(b) Right-front wheel

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

ISO standard PSDLR simulated PSD

(c) Left-rear wheel

10minus1 100 101

10minus5

Frequency (Hz)PS

D (m

2(1

m))

102

ISO standard PSDRR simulated PSD

(d) Right-rear wheel

Figure 11 Comparison of the new road PSDs of the four wheels simulated with the ISO 8608 standards

LFRF

0 2 4 6 8 10 12 14 16 18minus50

0

50

Time (s)

Road

roug

hnes

s (m

m)

Figure 12 Comparison of left-front wheel (LF) with left-rear wheel(LR) time domain signal when coh(119891) is 0

(root mean square) of acceleration (Ac rms ms2) the rollangle RMS (120573 rms) the pitch angle RMS (120572 rms) and thetire vertical force RMS (Ft FL rms Ft FR rms Ft RL rmsFt RR rms N)

In this paper the frequency compensation algorithm isinstalled in the road input for a sedan model to illustratethe sedanrsquos behavior The parameters of a 119862-class sedan areshown in Table 1 Simulation results run over a 119862-level roadsurface with the speed of 70Kmh tabulated in Table 2 Andthe frequency compensation algorithm can also be installedin the other vehicle runs on a random road

Table 1 Parameters of the 119862-class sedan

Symbol Description Unit Value119898 Vehicle sprung mass kg 2110119898119891119897

Frontleft unsprung mass kg 583119898119891119903

Frontright unsprung mass kg 583119898119903119897

Rearleft unsprung mass kg 633119898119903119903

Rearright unsprung mass kg 633119896119905119891119897

Frontleft vertical tyre stiffness Nm 235000119896119905119891119903

Frontright vertical tyre stiffness Nm 235000119896119905119903119897

Rearleft vertical tyre stiffness Nm 235000119896119905119903119903

Rearright vertical tyre stiffness Nm 235000119896119904119891119897

Frontleft spring stiffness Nm 27500119896119904119891119903

Frontright spring stiffness Nm 27500119896119904119903119897

Rearleft spring stiffness Nm 31300119896119904119903119903

Rearright spring stiffness Nm 31300119897119891

Distance between front-axlecog G m 1455119897119903

Distance between rear-axlecog G m 1515119908 Wheel track m 1610119868120573

Sprung mass roll moment of inertia kgsdotm2 7101119868120572

Sprung mass pitch moment of inertia kgsdotm2 36685

In this analysis the road input of left-front tire is set asa baseline then the right-front road input is obtained via acoherence function wo a compensation and the rear road

Mathematical Problems in Engineering 9

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

(a) Left-front wheel

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

(b) Right-front wheel

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

ISO standard PSDLR simulated PSD

(c) Left-rear wheel

10minus1 100 101

10minus5

Frequency (Hz)PS

D (m

2(1

m))

102

ISO standard PSDRR simulated PSD

(d) Right-rear wheel

Figure 13 Comparison of the road PSDs simulated with the ISO 8608 standards when coh(119891) is 0

LFRF

0 2 4 6 8 10 12 14 16 18minus50

0

50

Time (s)

Road

roug

hnes

s (m

m)

Figure 14 Comparison of left-front wheel (LF) with left-rear wheel(LR) time domain signal when coh(119891) is 1

input has a time delay dependent on the wheel base andvehicle speed Table 2 shows that the 120572 rms Ft FL rms andFt RL rms without compensation are almost equal to thevalues with compensation since the PSDs of the left-side roadinput are the same while the Ac rms 120573 rms Ft FR rms andFt RR rmswithout compensation are significant smaller thanthe values with compensation which is precisely caused bythe previously mentioned deviation since there is a deviationbetween the road PSDs of the two wheels on the same axlewhich is the main reason that the acceleration RMS the rollangle RMS the RMS of tire vertical force of the right-front

Table 2 Simulation results

Results Without compensation With compensation DifferenceAc rms 03638 04081 108120572 rms 00062 00062 0120573 rms 910119890 minus 4 00017 464Ft FL rms 10115119890 + 3 10162119890 + 3 05Ft FR rms 06824119890 + 3 10100119890 + 3 327Ft RL rms 10514119890 + 3 10566119890 + 3 05Ft RR rms 07190119890 + 3 10534119890 + 3 314

and right-rear wheels without compensation are smaller thanthe values with compensation Hence the deviation of thePSD of the road input between two wheels on the same axlehas an important influence on vehicle performances whichillustrates the importance of the frequency compensationalgorithm proposed which can be used for a control systeminstalled in the vehicle to compensate road roughness fordamper tuning in the future

8 Conclusions

The road surface roughness has an important influence onride comfort ride safety vehicle maneuverability driverrsquosand occupantsrsquo comfort and vehicle dynamic load In recentdecades random road models have been widely studied anda four-wheel random road time domain model is usuallygenerated based on the correlation of the four-wheel input in

10 Mathematical Problems in Engineering

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

(a) Left-front wheel

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

(b) Right-front wheel

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

ISO standard PSDLR simulated PSD

(c) Left-rear wheel

10minus1 100 101

10minus5

Frequency (Hz)PS

D (m

2(1

m))

102

ISO standard PSDRR simulated PSD

(d) Right-rear wheel

Figure 15 Comparison of the road PSDs simulated with the ISO 8608 standards when coh(119891) is 1

Vehicledynamicbehavior

VehiclemodelRoad model

Figure 16 Analysis on the vehicle dynamic behavior

which a coherence function is used to describe the coherenceof the road input between the left and right wheels usuallyGenerally the value of coherence function is between 0 and 1and it is near 0 at high-frequency band As a result the roadroughness amplitude and the road PSD of one track will besmaller than the other one on the same axle actually it iscaused by the uncorrelation between the left- and right-wheelroad surface roughness Hence a frequency compensationalgorithm is proposed to correct the deviation of the PSDof the road input between two wheels on the same axle andit is installed in a 7-DOF vehicle dynamic study The mainconclusions can be drawn from the current study as follows

(1) It is simple and accurate to generate a single-wheeltime signal model by using the inverse Fourier trans-form method

(2) With the Inverse Fourier Transform method viacoherence function the road PSDs of the front andrear wheels are consistent with the standard valueshowever there is a deviation between the two-wheelroad PSDs on the same axle The coherence function

value is between 0 and 1 and it is near 0 at high-frequency band As a result the road roughnessamplitude and the road PSD of one track will besmaller than the other one on the same axle

(3) A road frequency compensation algorithm is pro-posed to correct the deviation of the two wheels roadPSDs on the same axle and it is installed in a roadtime domain model The simulation results showedthat the road PSDs with frequency compensation areconsistent with the ISO 8608 standard values whichdemonstrates that the proposed algorithm is correctespecially for some limit case such as the left andright wheel road input are completely uncorrelated orperfectly correlated

(4) The vehicle dynamic behavior with road frequencycompensation show that the deviation compensationof two wheels on the same axle has an importantinfluence on vehicle performances which should beconsidered in the vehicle dynamic studies and theroad frequency compensation can be used for acontrol system installed in the vehicle to compensateroad roughness for damper tuning in the future

Acknowledgments

Special thanks are due to the Open Research Fund Pro-gram of the State Key Laboratory of Advanced Design andManufacturing for Vehicle Body (31115028) the China Post-doctoral Science Foundation (2012M520028) the NationalNatural Science Foundation of China (51205155) and the

Mathematical Problems in Engineering 11

lf

lr

w

I120572 120572 I120573 120573

xGy

z

z

ztrl

zrl

crl

mrl

ksrl

ktrl

ksfr

mfr

ksfl

zfl

cfl

mfl

ktfl

ztfl

cfrztfr

zfr ktfr

Figure 17 7 DOF full vehicle model

National Basic Research Program of China (973 Program)(2011CB711201) for supporting authorsrsquo research

References

[1] P Mucka ldquoRoad waviness and the dynamic tyre forcerdquo Inter-national Journal of Vehicle Design vol 36 no 2-3 pp 216ndash2322004

[2] Z Yonglin and Z Jiafan ldquoNumerical simulation of stochasticroad process using white noise filtrationrdquo Mechanical Systemsand Signal Processing vol 20 no 2 pp 363ndash372 2006

[3] K Ahlin J Granlund and F Lindstrom ldquoComparing roadprofiles with vehicle perceived roughnessrdquo International Journalof Vehicle Design vol 36 no 2-3 pp 270ndash286 2004

[4] S M Karamihas and M W Sayers The Little Book of ProfilingUniversity of Michigan Ann Arbor Mich USA 1998

[5] MAmbroz G Sustersic and I Prebil ldquoCreatingmodels of roadsections and their use in driving dynamics simulationsrdquo VehicleSystem Dynamics vol 45 no 10 pp 911ndash924 2007

[6] D J Whitehouse and J F Archard ldquoThe properties of randomsurfaces of significance in their contactrdquoProceedings of the RoyalSociety of London A vol 316 pp 97ndash121 1970

[7] M Shinozuka and C-M Jan ldquoDigital simulation of randomprocesses and its applicationsrdquo Journal of Sound and Vibrationvol 25 no 1 pp 111ndash128 1972

[8] D Cebon Interaction between Heavy Vehicles and Roads vol951 Society of Automotive Engineers 1993

[9] J-J Wu ldquoSimulation of rough surfaces with FFTrdquo TribologyInternational vol 33 no 1 pp 47ndash58 2000

[10] M Grigoriu ldquoOn the spectral representation method in sim-ulationrdquo Probabilistic Engineering Mechanics vol 8 no 2 pp75ndash90 1993

[11] ISO Mechanical Vibration-Road Surface Profiles-Reporting ofMeasured Data International Organization for Standardiza-tion 1995

[12] T Yoshimura ldquoA semi-active suspension of passenger cars usingfuzzy reasoning and the field testingrdquo International Journal ofVehicle Design vol 19 no 2 pp 150ndash166 1998

[13] K Bogsjo ldquoEvaluation of stochastic models of parallel roadtracksrdquo Probabilistic Engineering Mechanics vol 22 no 4 pp362ndash370 2007

[14] A Pazooki S Rakheja andD Cao ldquoModeling and validation ofoff-road vehicle ride dynamicsrdquoMechanical Systems and SignalProcessing vol 28 pp 679ndash695 2012

[15] R Hassan and R Evans ldquoRoad roughness characteristics incar and truck wheel tracksrdquo International Journal of PavementEngineering 2012

[16] K Bogsjo K Podgorski and I Rychlik ldquoModels for roadsurface roughnessrdquo Vehicle System Dynamics vol 50 no 5 pp725ndash747 2012

[17] S A Hassan and O S Lashine ldquoIdentification of road surfacequalities for linear and non-linear vehicle modelingrdquo in Pro-ceedings of the 20th International Modal Analysis Conference(IMAC rsquo02) pp 318ndash323 Society for Experimental Mechanics2002

[18] A N Heath ldquoApplication of the isotropic road roughnessassumptionrdquo Journal of Sound and Vibration vol 115 no 1 pp131ndash144 1987

[19] A N Heath ldquoEvaluation of the isotropic road roughnessassumptionrdquoVehicle SystemDynamics vol 17 supplement 1 pp157ndash160 1988

[20] D Cebon and D E Newland ldquoThe artificial generation of roadsurface topography by the inverse FFT methodrdquo Vehicle SystemDynamics vol 12 no 1ndash3 pp 160ndash165 1983

[21] C J Dodds ldquoThe laboratory simulation of vehicle servicestressrdquo Journal of Engineering for Industry vol 96 no 2 pp 391ndash398 1974

[22] D Cebon and D E Newlan ldquoThe artificial generation of roadsurface topography by the inverse FFT methodrdquo in Proceedingsof the 8th IAVSD-Symposium on the Dynamics of Vehicles onRoads and Tracks pp 29ndash42 Swets and Zeitlinger CambridgeMass USA 1983

[23] C J Dodds and J D Robson ldquoThe description of roadroughnessrdquo Journal of Sound and Vibration vol 31 no 2 pp175ndash183 1973

[24] K M A Kamash and J D Robson ldquoImplications of isotropy inrandom surfacesrdquo Journal of Sound and Vibration vol 54 no 1pp 131ndash145 1977

[25] K M A Kamash and J D Robson ldquoThe application of isotropyin road surface modellingrdquo Journal of Sound and Vibration vol57 no 1 pp 89ndash100 1978

12 Mathematical Problems in Engineering

[26] G C Carter ldquoCoherence and time delay estimationrdquo Proceed-ings of the IEEE vol 75 no 2 pp 236ndash255 1987

[27] K Bogsjo ldquoCoherence of road roughness in left and rightwheel-pathrdquo Vehicle System Dynamics vol 46 supplement 1 pp 599ndash609 2008

[28] R Hongbin C Sizhong andW Zhicheng ldquoModel of excitationof random road profile in time domain for a vehicle withfour wheelsrdquo in Proceedings of the International Conference onMechatronic Science Electric Engineering and Computer (MECrsquo11) pp 2332ndash2335 Jilin China August 2011

[29] J D Robson ldquoRoad surface description and vehicle responserdquoInternational Journal of Vehicle Design vol 1 no 1 pp 25ndash351979

[30] D Ammon ldquoProblems in road surface modellingrdquo VehicleSystem Dynamics vol 20 pp 28ndash41 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

(a) Left-front wheel

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

(b) Right-front wheel

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

ISO standard PSDLR simulated PSD

(c) Left-rear wheel

10minus1 100 101

10minus5

Frequency (Hz)PS

D (m

2(1

m))

102

ISO standard PSDRR simulated PSD

(d) Right-rear wheel

Figure 8 Comparison of the road PSDs simulated with the ISO 8608 standards

0 2 4 6 8 10 12 14 16 18minus50

0

50

Time (s)

Road

roug

hnes

s (m

m)

Figure 9 The smaller time domain signal generated based on thedeviation Δ119866

119902(119891)

road PSD of the left wheel on the same axle that is consistentwith the ISO 8608 standard value which can be described by

Δ119866119902 (119891) = 119866119902 (119891) minus 119866119902simu (119891) (19)

where 119866119902(119891) is the standard PSD values 119866119902simu(119891) is thesimulated PSD values and Δ119866119902(119891) is the deviation betweenthe road PSD of the left wheel (consistent with the standard)and right wheel

In addition the right-wheel time domain signal cannotbe obtained when coh(119891) is 0 which only considers thecorrelation of the left- and right-wheel road roughness whileignoring the uncorrelation of the left- and right-wheel roadroughness Since the value of coherence function is between

0 and 1 and it is near 0 at high-frequency band accordingto (16) and (17) it can be found that the reasons for thedeviation aremainly from the coherence functionHence thecompensation of the right-wheel time domain signal will bepresented in the sequel

6 Correction of Four-Wheel Road TimeDomain Model

From the above we know that there is a deviation Δ119866119902(119891)between the road PSD of left and right wheels since theuncorrelation of the left- and right-wheel road roughness isignored Hence considering the uncorrelation of the left andright wheel road roughness a new road spectrum relation-ship between the left and right wheels can be described by

1198761015840

3(119891) = 1198761 (119891) coh (119891) + Δ119876 (119891) (20)

whereΔ119876(119891) reflects the uncorrelation of the left- and right-wheel road roughness which can be obtained by Δ119866119902(119891)based on (4) and (5) A compensation using time domainsignal is generated via the inverse Fourier transformofΔ119876(119891)by using new random sequence numbers which is shown inFigure 9

A new right-wheel time domain signal can be obtained bycompensating the time domain deviation signal A compar-ison of left-front wheel with right-front wheel time domain-compensated signal is shown in Figure 10 and it can be foundthat the amplitude of right-front wheel (RF) and left-frontwheel (LR) time domain signals are the same approximately

Mathematical Problems in Engineering 7

LFRF

0 2 4 6 8 10 12 14 16 18minus50

0

50

Time (s)

Road

roug

hnes

s (m

m)

Figure 10 Comparison of left-front wheel (LF) with right-frontwheel (RF) time domain signal after compensating

Then the new road PSDs of the four wheels can beobtained via Fourier transformwhich is shown in Figure 11 Itshows that the new road PSDs simulated canmatch ISO 8608standards very well In a word this frequency compensationalgorithm for a four-wheel road time domain model isaccurate

In addition for the limit case the method is also applica-ble When coh(119891) is 0 the left- and right-wheel time domainsignals are completely uncorrelated and a comparison of left-front wheel with left-rear wheel time domain signals is shownin Figure 12 It illustrates that the right-front wheel (RF)and left-front wheel (LR) time domain signals are completelyuncorrelated which correspond with the definition of thecorrelation function Figure 13 shows the comparison of theroad PSDs simulatedwith the ISO 8608 standards Obviouslythe new road PSDs simulated can match ISO 8608 standardsvery well

When coh(119891) is 1 the left- and right-wheel road rough-ness are perfectly correlated and a comparison of the left-front wheel with the left-rear wheel time domain signals isshown in Figure 14 It illustrates that the right-front wheel(RF) and left-front wheel (LR) time domain signals arecompletely the same which correspond with the definitionof the correlation function Figure 15 shows the comparisonof the road PSDs simulated with the ISO 8608 standardsObviously the new road PSDs simulated can also matchISO 8608 standards very well In summary this frequencycompensation algorithm for a four-wheel road time domainmodel is accurate

The frequency compensation algorithm can also be usedto correct the four-wheel surface time domain models gener-ated by other methods when there is a deviation between theroad PSDs simulated and the ISO 8608 standards

7 Analysis on Vehicle Dynamic Behavior

Since the road surface irregularities are the main sourceof excitation to the vehicle system to analyze the vehicle

dynamic behavior more accurately a suitable representationof these irregularities is required which is shown in Figure 16

A comparison of vehicle dynamic behavior wo (with orwithout) compensation will be presented as follows Firstly a7-DOF vertical vehicle model is shown in Figure 17 whichconsiders the vertical motion pitch and roll of the sprungmass and the four vertical motions of four unsprung masses

Each corner of the vehicle is identified with 119894 119895 indexwhere 119894 = 119897 119903holds for leftright and 119895 = 119891 119903 for frontrearThese corners (ie positions and velocities of the dynamicalpart) are described as

119911119891119897 = 119911 minus 120572119897119891 + 120573119908 minus 119911119905119891119897

119911119891119903 = 119911 minus 120572119897119891 minus 120573119908 minus 119911119905119891119903

119911119903119897 = 119911 + 120572119897119903 + 120573119908 minus 119911119905119903119897

119911119903119903 = 119911 + 120572119897119903 minus 120573119908 minus 119911119905119903119903

119891119897 = minus 119897119891 +120573119908 minus 119905119891119897

119891119903 = minus 119897119891 minus120573119908 minus 119905119891119903

119903119897 = + 119897119903 +120573119908 minus 119905119903119897

119903119903 = + 119897119903 minus120573119908 minus 119905119903119903

(21)

where 119911 is the center of the sprung mass 120572 (resp 120573) is thepitch (resp roll) angle of the sprungmass around119910-axis (resp119909-axis) at the center of sprung mass 119897119891 119897119903 and 119908 define thevehicle geometrical properties (as shown in Figure 17)

The full vehicle model as depicted on Figure 17 is givenby the following motion equations

119898119905119891119897119905119891119897 + 119896119905119891119897 (119911119905119891119897 minus 119911119891119897) minus 119865119904119891119897 minus 119865119889119891119897 = 0

119898119905119891119903119905119891119903 + 119896119905119891119903 (119911119905119891119903 minus 119911119891119903) minus 119865119904119891119903 minus 119865119889119891119903 = 0

119898119905119903119897119905119903119897 + 119896119905119903119897 (119911119905119903119897 minus 119911119903119897) minus 119865119904119903119897 minus 119865119889119903119897 = 0

119898119905119903119903119905119903119903 + 119896119905119903119903 (119911119905119903119903 minus 119911119903119903) minus 119865119904119903119903 minus 119865119889119903119903 = 0

sum119898 + 119865119904119894119895 + 119865119889119894119895 = 0

sum119868120572 minus 119865119904119891119895119897119891 + 119865119904119903119895119897119903 minus 119865119889119891119895119897119891 + 119865119889119903119895119897119903 = 0

sum119868120573120573 + 119908119865119904119894119897 minus 119908119865119904119894119903 + 119908119865119889119894119897 minus 119908119865119889119894119903 = 0

(22)

where 119898 and 119898119905119894119895 denote the sprung and unsprung massrespectivelyThemoment of inertia about the 119909-axis (resp 119910-axis) of the vehicle sprung mass is denoted as 119868120573 (resp 119868120572)119865119904119894119895 and 119865119889119894119895 are spring force and damper force respectivelywhich can be described by

119865119904119894119895 = 119896119904119894119895119911119894119895 119865119889119894119895 = 119888119894119895119894119895 (23)

where 119896119904119894119895 and 119888119894119895 are spring stiffness and damping coefficientrespectively

Then the full vehicle model can be established by Mat-labSimulink and the outputs include the weighted RMS

8 Mathematical Problems in Engineering

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

(a) Left-front wheel

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

(b) Right-front wheel

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

ISO standard PSDLR simulated PSD

(c) Left-rear wheel

10minus1 100 101

10minus5

Frequency (Hz)PS

D (m

2(1

m))

102

ISO standard PSDRR simulated PSD

(d) Right-rear wheel

Figure 11 Comparison of the new road PSDs of the four wheels simulated with the ISO 8608 standards

LFRF

0 2 4 6 8 10 12 14 16 18minus50

0

50

Time (s)

Road

roug

hnes

s (m

m)

Figure 12 Comparison of left-front wheel (LF) with left-rear wheel(LR) time domain signal when coh(119891) is 0

(root mean square) of acceleration (Ac rms ms2) the rollangle RMS (120573 rms) the pitch angle RMS (120572 rms) and thetire vertical force RMS (Ft FL rms Ft FR rms Ft RL rmsFt RR rms N)

In this paper the frequency compensation algorithm isinstalled in the road input for a sedan model to illustratethe sedanrsquos behavior The parameters of a 119862-class sedan areshown in Table 1 Simulation results run over a 119862-level roadsurface with the speed of 70Kmh tabulated in Table 2 Andthe frequency compensation algorithm can also be installedin the other vehicle runs on a random road

Table 1 Parameters of the 119862-class sedan

Symbol Description Unit Value119898 Vehicle sprung mass kg 2110119898119891119897

Frontleft unsprung mass kg 583119898119891119903

Frontright unsprung mass kg 583119898119903119897

Rearleft unsprung mass kg 633119898119903119903

Rearright unsprung mass kg 633119896119905119891119897

Frontleft vertical tyre stiffness Nm 235000119896119905119891119903

Frontright vertical tyre stiffness Nm 235000119896119905119903119897

Rearleft vertical tyre stiffness Nm 235000119896119905119903119903

Rearright vertical tyre stiffness Nm 235000119896119904119891119897

Frontleft spring stiffness Nm 27500119896119904119891119903

Frontright spring stiffness Nm 27500119896119904119903119897

Rearleft spring stiffness Nm 31300119896119904119903119903

Rearright spring stiffness Nm 31300119897119891

Distance between front-axlecog G m 1455119897119903

Distance between rear-axlecog G m 1515119908 Wheel track m 1610119868120573

Sprung mass roll moment of inertia kgsdotm2 7101119868120572

Sprung mass pitch moment of inertia kgsdotm2 36685

In this analysis the road input of left-front tire is set asa baseline then the right-front road input is obtained via acoherence function wo a compensation and the rear road

Mathematical Problems in Engineering 9

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

(a) Left-front wheel

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

(b) Right-front wheel

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

ISO standard PSDLR simulated PSD

(c) Left-rear wheel

10minus1 100 101

10minus5

Frequency (Hz)PS

D (m

2(1

m))

102

ISO standard PSDRR simulated PSD

(d) Right-rear wheel

Figure 13 Comparison of the road PSDs simulated with the ISO 8608 standards when coh(119891) is 0

LFRF

0 2 4 6 8 10 12 14 16 18minus50

0

50

Time (s)

Road

roug

hnes

s (m

m)

Figure 14 Comparison of left-front wheel (LF) with left-rear wheel(LR) time domain signal when coh(119891) is 1

input has a time delay dependent on the wheel base andvehicle speed Table 2 shows that the 120572 rms Ft FL rms andFt RL rms without compensation are almost equal to thevalues with compensation since the PSDs of the left-side roadinput are the same while the Ac rms 120573 rms Ft FR rms andFt RR rmswithout compensation are significant smaller thanthe values with compensation which is precisely caused bythe previously mentioned deviation since there is a deviationbetween the road PSDs of the two wheels on the same axlewhich is the main reason that the acceleration RMS the rollangle RMS the RMS of tire vertical force of the right-front

Table 2 Simulation results

Results Without compensation With compensation DifferenceAc rms 03638 04081 108120572 rms 00062 00062 0120573 rms 910119890 minus 4 00017 464Ft FL rms 10115119890 + 3 10162119890 + 3 05Ft FR rms 06824119890 + 3 10100119890 + 3 327Ft RL rms 10514119890 + 3 10566119890 + 3 05Ft RR rms 07190119890 + 3 10534119890 + 3 314

and right-rear wheels without compensation are smaller thanthe values with compensation Hence the deviation of thePSD of the road input between two wheels on the same axlehas an important influence on vehicle performances whichillustrates the importance of the frequency compensationalgorithm proposed which can be used for a control systeminstalled in the vehicle to compensate road roughness fordamper tuning in the future

8 Conclusions

The road surface roughness has an important influence onride comfort ride safety vehicle maneuverability driverrsquosand occupantsrsquo comfort and vehicle dynamic load In recentdecades random road models have been widely studied anda four-wheel random road time domain model is usuallygenerated based on the correlation of the four-wheel input in

10 Mathematical Problems in Engineering

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

(a) Left-front wheel

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

(b) Right-front wheel

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

ISO standard PSDLR simulated PSD

(c) Left-rear wheel

10minus1 100 101

10minus5

Frequency (Hz)PS

D (m

2(1

m))

102

ISO standard PSDRR simulated PSD

(d) Right-rear wheel

Figure 15 Comparison of the road PSDs simulated with the ISO 8608 standards when coh(119891) is 1

Vehicledynamicbehavior

VehiclemodelRoad model

Figure 16 Analysis on the vehicle dynamic behavior

which a coherence function is used to describe the coherenceof the road input between the left and right wheels usuallyGenerally the value of coherence function is between 0 and 1and it is near 0 at high-frequency band As a result the roadroughness amplitude and the road PSD of one track will besmaller than the other one on the same axle actually it iscaused by the uncorrelation between the left- and right-wheelroad surface roughness Hence a frequency compensationalgorithm is proposed to correct the deviation of the PSDof the road input between two wheels on the same axle andit is installed in a 7-DOF vehicle dynamic study The mainconclusions can be drawn from the current study as follows

(1) It is simple and accurate to generate a single-wheeltime signal model by using the inverse Fourier trans-form method

(2) With the Inverse Fourier Transform method viacoherence function the road PSDs of the front andrear wheels are consistent with the standard valueshowever there is a deviation between the two-wheelroad PSDs on the same axle The coherence function

value is between 0 and 1 and it is near 0 at high-frequency band As a result the road roughnessamplitude and the road PSD of one track will besmaller than the other one on the same axle

(3) A road frequency compensation algorithm is pro-posed to correct the deviation of the two wheels roadPSDs on the same axle and it is installed in a roadtime domain model The simulation results showedthat the road PSDs with frequency compensation areconsistent with the ISO 8608 standard values whichdemonstrates that the proposed algorithm is correctespecially for some limit case such as the left andright wheel road input are completely uncorrelated orperfectly correlated

(4) The vehicle dynamic behavior with road frequencycompensation show that the deviation compensationof two wheels on the same axle has an importantinfluence on vehicle performances which should beconsidered in the vehicle dynamic studies and theroad frequency compensation can be used for acontrol system installed in the vehicle to compensateroad roughness for damper tuning in the future

Acknowledgments

Special thanks are due to the Open Research Fund Pro-gram of the State Key Laboratory of Advanced Design andManufacturing for Vehicle Body (31115028) the China Post-doctoral Science Foundation (2012M520028) the NationalNatural Science Foundation of China (51205155) and the

Mathematical Problems in Engineering 11

lf

lr

w

I120572 120572 I120573 120573

xGy

z

z

ztrl

zrl

crl

mrl

ksrl

ktrl

ksfr

mfr

ksfl

zfl

cfl

mfl

ktfl

ztfl

cfrztfr

zfr ktfr

Figure 17 7 DOF full vehicle model

National Basic Research Program of China (973 Program)(2011CB711201) for supporting authorsrsquo research

References

[1] P Mucka ldquoRoad waviness and the dynamic tyre forcerdquo Inter-national Journal of Vehicle Design vol 36 no 2-3 pp 216ndash2322004

[2] Z Yonglin and Z Jiafan ldquoNumerical simulation of stochasticroad process using white noise filtrationrdquo Mechanical Systemsand Signal Processing vol 20 no 2 pp 363ndash372 2006

[3] K Ahlin J Granlund and F Lindstrom ldquoComparing roadprofiles with vehicle perceived roughnessrdquo International Journalof Vehicle Design vol 36 no 2-3 pp 270ndash286 2004

[4] S M Karamihas and M W Sayers The Little Book of ProfilingUniversity of Michigan Ann Arbor Mich USA 1998

[5] MAmbroz G Sustersic and I Prebil ldquoCreatingmodels of roadsections and their use in driving dynamics simulationsrdquo VehicleSystem Dynamics vol 45 no 10 pp 911ndash924 2007

[6] D J Whitehouse and J F Archard ldquoThe properties of randomsurfaces of significance in their contactrdquoProceedings of the RoyalSociety of London A vol 316 pp 97ndash121 1970

[7] M Shinozuka and C-M Jan ldquoDigital simulation of randomprocesses and its applicationsrdquo Journal of Sound and Vibrationvol 25 no 1 pp 111ndash128 1972

[8] D Cebon Interaction between Heavy Vehicles and Roads vol951 Society of Automotive Engineers 1993

[9] J-J Wu ldquoSimulation of rough surfaces with FFTrdquo TribologyInternational vol 33 no 1 pp 47ndash58 2000

[10] M Grigoriu ldquoOn the spectral representation method in sim-ulationrdquo Probabilistic Engineering Mechanics vol 8 no 2 pp75ndash90 1993

[11] ISO Mechanical Vibration-Road Surface Profiles-Reporting ofMeasured Data International Organization for Standardiza-tion 1995

[12] T Yoshimura ldquoA semi-active suspension of passenger cars usingfuzzy reasoning and the field testingrdquo International Journal ofVehicle Design vol 19 no 2 pp 150ndash166 1998

[13] K Bogsjo ldquoEvaluation of stochastic models of parallel roadtracksrdquo Probabilistic Engineering Mechanics vol 22 no 4 pp362ndash370 2007

[14] A Pazooki S Rakheja andD Cao ldquoModeling and validation ofoff-road vehicle ride dynamicsrdquoMechanical Systems and SignalProcessing vol 28 pp 679ndash695 2012

[15] R Hassan and R Evans ldquoRoad roughness characteristics incar and truck wheel tracksrdquo International Journal of PavementEngineering 2012

[16] K Bogsjo K Podgorski and I Rychlik ldquoModels for roadsurface roughnessrdquo Vehicle System Dynamics vol 50 no 5 pp725ndash747 2012

[17] S A Hassan and O S Lashine ldquoIdentification of road surfacequalities for linear and non-linear vehicle modelingrdquo in Pro-ceedings of the 20th International Modal Analysis Conference(IMAC rsquo02) pp 318ndash323 Society for Experimental Mechanics2002

[18] A N Heath ldquoApplication of the isotropic road roughnessassumptionrdquo Journal of Sound and Vibration vol 115 no 1 pp131ndash144 1987

[19] A N Heath ldquoEvaluation of the isotropic road roughnessassumptionrdquoVehicle SystemDynamics vol 17 supplement 1 pp157ndash160 1988

[20] D Cebon and D E Newland ldquoThe artificial generation of roadsurface topography by the inverse FFT methodrdquo Vehicle SystemDynamics vol 12 no 1ndash3 pp 160ndash165 1983

[21] C J Dodds ldquoThe laboratory simulation of vehicle servicestressrdquo Journal of Engineering for Industry vol 96 no 2 pp 391ndash398 1974

[22] D Cebon and D E Newlan ldquoThe artificial generation of roadsurface topography by the inverse FFT methodrdquo in Proceedingsof the 8th IAVSD-Symposium on the Dynamics of Vehicles onRoads and Tracks pp 29ndash42 Swets and Zeitlinger CambridgeMass USA 1983

[23] C J Dodds and J D Robson ldquoThe description of roadroughnessrdquo Journal of Sound and Vibration vol 31 no 2 pp175ndash183 1973

[24] K M A Kamash and J D Robson ldquoImplications of isotropy inrandom surfacesrdquo Journal of Sound and Vibration vol 54 no 1pp 131ndash145 1977

[25] K M A Kamash and J D Robson ldquoThe application of isotropyin road surface modellingrdquo Journal of Sound and Vibration vol57 no 1 pp 89ndash100 1978

12 Mathematical Problems in Engineering

[26] G C Carter ldquoCoherence and time delay estimationrdquo Proceed-ings of the IEEE vol 75 no 2 pp 236ndash255 1987

[27] K Bogsjo ldquoCoherence of road roughness in left and rightwheel-pathrdquo Vehicle System Dynamics vol 46 supplement 1 pp 599ndash609 2008

[28] R Hongbin C Sizhong andW Zhicheng ldquoModel of excitationof random road profile in time domain for a vehicle withfour wheelsrdquo in Proceedings of the International Conference onMechatronic Science Electric Engineering and Computer (MECrsquo11) pp 2332ndash2335 Jilin China August 2011

[29] J D Robson ldquoRoad surface description and vehicle responserdquoInternational Journal of Vehicle Design vol 1 no 1 pp 25ndash351979

[30] D Ammon ldquoProblems in road surface modellingrdquo VehicleSystem Dynamics vol 20 pp 28ndash41 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

LFRF

0 2 4 6 8 10 12 14 16 18minus50

0

50

Time (s)

Road

roug

hnes

s (m

m)

Figure 10 Comparison of left-front wheel (LF) with right-frontwheel (RF) time domain signal after compensating

Then the new road PSDs of the four wheels can beobtained via Fourier transformwhich is shown in Figure 11 Itshows that the new road PSDs simulated canmatch ISO 8608standards very well In a word this frequency compensationalgorithm for a four-wheel road time domain model isaccurate

In addition for the limit case the method is also applica-ble When coh(119891) is 0 the left- and right-wheel time domainsignals are completely uncorrelated and a comparison of left-front wheel with left-rear wheel time domain signals is shownin Figure 12 It illustrates that the right-front wheel (RF)and left-front wheel (LR) time domain signals are completelyuncorrelated which correspond with the definition of thecorrelation function Figure 13 shows the comparison of theroad PSDs simulatedwith the ISO 8608 standards Obviouslythe new road PSDs simulated can match ISO 8608 standardsvery well

When coh(119891) is 1 the left- and right-wheel road rough-ness are perfectly correlated and a comparison of the left-front wheel with the left-rear wheel time domain signals isshown in Figure 14 It illustrates that the right-front wheel(RF) and left-front wheel (LR) time domain signals arecompletely the same which correspond with the definitionof the correlation function Figure 15 shows the comparisonof the road PSDs simulated with the ISO 8608 standardsObviously the new road PSDs simulated can also matchISO 8608 standards very well In summary this frequencycompensation algorithm for a four-wheel road time domainmodel is accurate

The frequency compensation algorithm can also be usedto correct the four-wheel surface time domain models gener-ated by other methods when there is a deviation between theroad PSDs simulated and the ISO 8608 standards

7 Analysis on Vehicle Dynamic Behavior

Since the road surface irregularities are the main sourceof excitation to the vehicle system to analyze the vehicle

dynamic behavior more accurately a suitable representationof these irregularities is required which is shown in Figure 16

A comparison of vehicle dynamic behavior wo (with orwithout) compensation will be presented as follows Firstly a7-DOF vertical vehicle model is shown in Figure 17 whichconsiders the vertical motion pitch and roll of the sprungmass and the four vertical motions of four unsprung masses

Each corner of the vehicle is identified with 119894 119895 indexwhere 119894 = 119897 119903holds for leftright and 119895 = 119891 119903 for frontrearThese corners (ie positions and velocities of the dynamicalpart) are described as

119911119891119897 = 119911 minus 120572119897119891 + 120573119908 minus 119911119905119891119897

119911119891119903 = 119911 minus 120572119897119891 minus 120573119908 minus 119911119905119891119903

119911119903119897 = 119911 + 120572119897119903 + 120573119908 minus 119911119905119903119897

119911119903119903 = 119911 + 120572119897119903 minus 120573119908 minus 119911119905119903119903

119891119897 = minus 119897119891 +120573119908 minus 119905119891119897

119891119903 = minus 119897119891 minus120573119908 minus 119905119891119903

119903119897 = + 119897119903 +120573119908 minus 119905119903119897

119903119903 = + 119897119903 minus120573119908 minus 119905119903119903

(21)

where 119911 is the center of the sprung mass 120572 (resp 120573) is thepitch (resp roll) angle of the sprungmass around119910-axis (resp119909-axis) at the center of sprung mass 119897119891 119897119903 and 119908 define thevehicle geometrical properties (as shown in Figure 17)

The full vehicle model as depicted on Figure 17 is givenby the following motion equations

119898119905119891119897119905119891119897 + 119896119905119891119897 (119911119905119891119897 minus 119911119891119897) minus 119865119904119891119897 minus 119865119889119891119897 = 0

119898119905119891119903119905119891119903 + 119896119905119891119903 (119911119905119891119903 minus 119911119891119903) minus 119865119904119891119903 minus 119865119889119891119903 = 0

119898119905119903119897119905119903119897 + 119896119905119903119897 (119911119905119903119897 minus 119911119903119897) minus 119865119904119903119897 minus 119865119889119903119897 = 0

119898119905119903119903119905119903119903 + 119896119905119903119903 (119911119905119903119903 minus 119911119903119903) minus 119865119904119903119903 minus 119865119889119903119903 = 0

sum119898 + 119865119904119894119895 + 119865119889119894119895 = 0

sum119868120572 minus 119865119904119891119895119897119891 + 119865119904119903119895119897119903 minus 119865119889119891119895119897119891 + 119865119889119903119895119897119903 = 0

sum119868120573120573 + 119908119865119904119894119897 minus 119908119865119904119894119903 + 119908119865119889119894119897 minus 119908119865119889119894119903 = 0

(22)

where 119898 and 119898119905119894119895 denote the sprung and unsprung massrespectivelyThemoment of inertia about the 119909-axis (resp 119910-axis) of the vehicle sprung mass is denoted as 119868120573 (resp 119868120572)119865119904119894119895 and 119865119889119894119895 are spring force and damper force respectivelywhich can be described by

119865119904119894119895 = 119896119904119894119895119911119894119895 119865119889119894119895 = 119888119894119895119894119895 (23)

where 119896119904119894119895 and 119888119894119895 are spring stiffness and damping coefficientrespectively

Then the full vehicle model can be established by Mat-labSimulink and the outputs include the weighted RMS

8 Mathematical Problems in Engineering

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

(a) Left-front wheel

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

(b) Right-front wheel

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

ISO standard PSDLR simulated PSD

(c) Left-rear wheel

10minus1 100 101

10minus5

Frequency (Hz)PS

D (m

2(1

m))

102

ISO standard PSDRR simulated PSD

(d) Right-rear wheel

Figure 11 Comparison of the new road PSDs of the four wheels simulated with the ISO 8608 standards

LFRF

0 2 4 6 8 10 12 14 16 18minus50

0

50

Time (s)

Road

roug

hnes

s (m

m)

Figure 12 Comparison of left-front wheel (LF) with left-rear wheel(LR) time domain signal when coh(119891) is 0

(root mean square) of acceleration (Ac rms ms2) the rollangle RMS (120573 rms) the pitch angle RMS (120572 rms) and thetire vertical force RMS (Ft FL rms Ft FR rms Ft RL rmsFt RR rms N)

In this paper the frequency compensation algorithm isinstalled in the road input for a sedan model to illustratethe sedanrsquos behavior The parameters of a 119862-class sedan areshown in Table 1 Simulation results run over a 119862-level roadsurface with the speed of 70Kmh tabulated in Table 2 Andthe frequency compensation algorithm can also be installedin the other vehicle runs on a random road

Table 1 Parameters of the 119862-class sedan

Symbol Description Unit Value119898 Vehicle sprung mass kg 2110119898119891119897

Frontleft unsprung mass kg 583119898119891119903

Frontright unsprung mass kg 583119898119903119897

Rearleft unsprung mass kg 633119898119903119903

Rearright unsprung mass kg 633119896119905119891119897

Frontleft vertical tyre stiffness Nm 235000119896119905119891119903

Frontright vertical tyre stiffness Nm 235000119896119905119903119897

Rearleft vertical tyre stiffness Nm 235000119896119905119903119903

Rearright vertical tyre stiffness Nm 235000119896119904119891119897

Frontleft spring stiffness Nm 27500119896119904119891119903

Frontright spring stiffness Nm 27500119896119904119903119897

Rearleft spring stiffness Nm 31300119896119904119903119903

Rearright spring stiffness Nm 31300119897119891

Distance between front-axlecog G m 1455119897119903

Distance between rear-axlecog G m 1515119908 Wheel track m 1610119868120573

Sprung mass roll moment of inertia kgsdotm2 7101119868120572

Sprung mass pitch moment of inertia kgsdotm2 36685

In this analysis the road input of left-front tire is set asa baseline then the right-front road input is obtained via acoherence function wo a compensation and the rear road

Mathematical Problems in Engineering 9

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

(a) Left-front wheel

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

(b) Right-front wheel

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

ISO standard PSDLR simulated PSD

(c) Left-rear wheel

10minus1 100 101

10minus5

Frequency (Hz)PS

D (m

2(1

m))

102

ISO standard PSDRR simulated PSD

(d) Right-rear wheel

Figure 13 Comparison of the road PSDs simulated with the ISO 8608 standards when coh(119891) is 0

LFRF

0 2 4 6 8 10 12 14 16 18minus50

0

50

Time (s)

Road

roug

hnes

s (m

m)

Figure 14 Comparison of left-front wheel (LF) with left-rear wheel(LR) time domain signal when coh(119891) is 1

input has a time delay dependent on the wheel base andvehicle speed Table 2 shows that the 120572 rms Ft FL rms andFt RL rms without compensation are almost equal to thevalues with compensation since the PSDs of the left-side roadinput are the same while the Ac rms 120573 rms Ft FR rms andFt RR rmswithout compensation are significant smaller thanthe values with compensation which is precisely caused bythe previously mentioned deviation since there is a deviationbetween the road PSDs of the two wheels on the same axlewhich is the main reason that the acceleration RMS the rollangle RMS the RMS of tire vertical force of the right-front

Table 2 Simulation results

Results Without compensation With compensation DifferenceAc rms 03638 04081 108120572 rms 00062 00062 0120573 rms 910119890 minus 4 00017 464Ft FL rms 10115119890 + 3 10162119890 + 3 05Ft FR rms 06824119890 + 3 10100119890 + 3 327Ft RL rms 10514119890 + 3 10566119890 + 3 05Ft RR rms 07190119890 + 3 10534119890 + 3 314

and right-rear wheels without compensation are smaller thanthe values with compensation Hence the deviation of thePSD of the road input between two wheels on the same axlehas an important influence on vehicle performances whichillustrates the importance of the frequency compensationalgorithm proposed which can be used for a control systeminstalled in the vehicle to compensate road roughness fordamper tuning in the future

8 Conclusions

The road surface roughness has an important influence onride comfort ride safety vehicle maneuverability driverrsquosand occupantsrsquo comfort and vehicle dynamic load In recentdecades random road models have been widely studied anda four-wheel random road time domain model is usuallygenerated based on the correlation of the four-wheel input in

10 Mathematical Problems in Engineering

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

(a) Left-front wheel

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

(b) Right-front wheel

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

ISO standard PSDLR simulated PSD

(c) Left-rear wheel

10minus1 100 101

10minus5

Frequency (Hz)PS

D (m

2(1

m))

102

ISO standard PSDRR simulated PSD

(d) Right-rear wheel

Figure 15 Comparison of the road PSDs simulated with the ISO 8608 standards when coh(119891) is 1

Vehicledynamicbehavior

VehiclemodelRoad model

Figure 16 Analysis on the vehicle dynamic behavior

which a coherence function is used to describe the coherenceof the road input between the left and right wheels usuallyGenerally the value of coherence function is between 0 and 1and it is near 0 at high-frequency band As a result the roadroughness amplitude and the road PSD of one track will besmaller than the other one on the same axle actually it iscaused by the uncorrelation between the left- and right-wheelroad surface roughness Hence a frequency compensationalgorithm is proposed to correct the deviation of the PSDof the road input between two wheels on the same axle andit is installed in a 7-DOF vehicle dynamic study The mainconclusions can be drawn from the current study as follows

(1) It is simple and accurate to generate a single-wheeltime signal model by using the inverse Fourier trans-form method

(2) With the Inverse Fourier Transform method viacoherence function the road PSDs of the front andrear wheels are consistent with the standard valueshowever there is a deviation between the two-wheelroad PSDs on the same axle The coherence function

value is between 0 and 1 and it is near 0 at high-frequency band As a result the road roughnessamplitude and the road PSD of one track will besmaller than the other one on the same axle

(3) A road frequency compensation algorithm is pro-posed to correct the deviation of the two wheels roadPSDs on the same axle and it is installed in a roadtime domain model The simulation results showedthat the road PSDs with frequency compensation areconsistent with the ISO 8608 standard values whichdemonstrates that the proposed algorithm is correctespecially for some limit case such as the left andright wheel road input are completely uncorrelated orperfectly correlated

(4) The vehicle dynamic behavior with road frequencycompensation show that the deviation compensationof two wheels on the same axle has an importantinfluence on vehicle performances which should beconsidered in the vehicle dynamic studies and theroad frequency compensation can be used for acontrol system installed in the vehicle to compensateroad roughness for damper tuning in the future

Acknowledgments

Special thanks are due to the Open Research Fund Pro-gram of the State Key Laboratory of Advanced Design andManufacturing for Vehicle Body (31115028) the China Post-doctoral Science Foundation (2012M520028) the NationalNatural Science Foundation of China (51205155) and the

Mathematical Problems in Engineering 11

lf

lr

w

I120572 120572 I120573 120573

xGy

z

z

ztrl

zrl

crl

mrl

ksrl

ktrl

ksfr

mfr

ksfl

zfl

cfl

mfl

ktfl

ztfl

cfrztfr

zfr ktfr

Figure 17 7 DOF full vehicle model

National Basic Research Program of China (973 Program)(2011CB711201) for supporting authorsrsquo research

References

[1] P Mucka ldquoRoad waviness and the dynamic tyre forcerdquo Inter-national Journal of Vehicle Design vol 36 no 2-3 pp 216ndash2322004

[2] Z Yonglin and Z Jiafan ldquoNumerical simulation of stochasticroad process using white noise filtrationrdquo Mechanical Systemsand Signal Processing vol 20 no 2 pp 363ndash372 2006

[3] K Ahlin J Granlund and F Lindstrom ldquoComparing roadprofiles with vehicle perceived roughnessrdquo International Journalof Vehicle Design vol 36 no 2-3 pp 270ndash286 2004

[4] S M Karamihas and M W Sayers The Little Book of ProfilingUniversity of Michigan Ann Arbor Mich USA 1998

[5] MAmbroz G Sustersic and I Prebil ldquoCreatingmodels of roadsections and their use in driving dynamics simulationsrdquo VehicleSystem Dynamics vol 45 no 10 pp 911ndash924 2007

[6] D J Whitehouse and J F Archard ldquoThe properties of randomsurfaces of significance in their contactrdquoProceedings of the RoyalSociety of London A vol 316 pp 97ndash121 1970

[7] M Shinozuka and C-M Jan ldquoDigital simulation of randomprocesses and its applicationsrdquo Journal of Sound and Vibrationvol 25 no 1 pp 111ndash128 1972

[8] D Cebon Interaction between Heavy Vehicles and Roads vol951 Society of Automotive Engineers 1993

[9] J-J Wu ldquoSimulation of rough surfaces with FFTrdquo TribologyInternational vol 33 no 1 pp 47ndash58 2000

[10] M Grigoriu ldquoOn the spectral representation method in sim-ulationrdquo Probabilistic Engineering Mechanics vol 8 no 2 pp75ndash90 1993

[11] ISO Mechanical Vibration-Road Surface Profiles-Reporting ofMeasured Data International Organization for Standardiza-tion 1995

[12] T Yoshimura ldquoA semi-active suspension of passenger cars usingfuzzy reasoning and the field testingrdquo International Journal ofVehicle Design vol 19 no 2 pp 150ndash166 1998

[13] K Bogsjo ldquoEvaluation of stochastic models of parallel roadtracksrdquo Probabilistic Engineering Mechanics vol 22 no 4 pp362ndash370 2007

[14] A Pazooki S Rakheja andD Cao ldquoModeling and validation ofoff-road vehicle ride dynamicsrdquoMechanical Systems and SignalProcessing vol 28 pp 679ndash695 2012

[15] R Hassan and R Evans ldquoRoad roughness characteristics incar and truck wheel tracksrdquo International Journal of PavementEngineering 2012

[16] K Bogsjo K Podgorski and I Rychlik ldquoModels for roadsurface roughnessrdquo Vehicle System Dynamics vol 50 no 5 pp725ndash747 2012

[17] S A Hassan and O S Lashine ldquoIdentification of road surfacequalities for linear and non-linear vehicle modelingrdquo in Pro-ceedings of the 20th International Modal Analysis Conference(IMAC rsquo02) pp 318ndash323 Society for Experimental Mechanics2002

[18] A N Heath ldquoApplication of the isotropic road roughnessassumptionrdquo Journal of Sound and Vibration vol 115 no 1 pp131ndash144 1987

[19] A N Heath ldquoEvaluation of the isotropic road roughnessassumptionrdquoVehicle SystemDynamics vol 17 supplement 1 pp157ndash160 1988

[20] D Cebon and D E Newland ldquoThe artificial generation of roadsurface topography by the inverse FFT methodrdquo Vehicle SystemDynamics vol 12 no 1ndash3 pp 160ndash165 1983

[21] C J Dodds ldquoThe laboratory simulation of vehicle servicestressrdquo Journal of Engineering for Industry vol 96 no 2 pp 391ndash398 1974

[22] D Cebon and D E Newlan ldquoThe artificial generation of roadsurface topography by the inverse FFT methodrdquo in Proceedingsof the 8th IAVSD-Symposium on the Dynamics of Vehicles onRoads and Tracks pp 29ndash42 Swets and Zeitlinger CambridgeMass USA 1983

[23] C J Dodds and J D Robson ldquoThe description of roadroughnessrdquo Journal of Sound and Vibration vol 31 no 2 pp175ndash183 1973

[24] K M A Kamash and J D Robson ldquoImplications of isotropy inrandom surfacesrdquo Journal of Sound and Vibration vol 54 no 1pp 131ndash145 1977

[25] K M A Kamash and J D Robson ldquoThe application of isotropyin road surface modellingrdquo Journal of Sound and Vibration vol57 no 1 pp 89ndash100 1978

12 Mathematical Problems in Engineering

[26] G C Carter ldquoCoherence and time delay estimationrdquo Proceed-ings of the IEEE vol 75 no 2 pp 236ndash255 1987

[27] K Bogsjo ldquoCoherence of road roughness in left and rightwheel-pathrdquo Vehicle System Dynamics vol 46 supplement 1 pp 599ndash609 2008

[28] R Hongbin C Sizhong andW Zhicheng ldquoModel of excitationof random road profile in time domain for a vehicle withfour wheelsrdquo in Proceedings of the International Conference onMechatronic Science Electric Engineering and Computer (MECrsquo11) pp 2332ndash2335 Jilin China August 2011

[29] J D Robson ldquoRoad surface description and vehicle responserdquoInternational Journal of Vehicle Design vol 1 no 1 pp 25ndash351979

[30] D Ammon ldquoProblems in road surface modellingrdquo VehicleSystem Dynamics vol 20 pp 28ndash41 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

(a) Left-front wheel

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

(b) Right-front wheel

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

ISO standard PSDLR simulated PSD

(c) Left-rear wheel

10minus1 100 101

10minus5

Frequency (Hz)PS

D (m

2(1

m))

102

ISO standard PSDRR simulated PSD

(d) Right-rear wheel

Figure 11 Comparison of the new road PSDs of the four wheels simulated with the ISO 8608 standards

LFRF

0 2 4 6 8 10 12 14 16 18minus50

0

50

Time (s)

Road

roug

hnes

s (m

m)

Figure 12 Comparison of left-front wheel (LF) with left-rear wheel(LR) time domain signal when coh(119891) is 0

(root mean square) of acceleration (Ac rms ms2) the rollangle RMS (120573 rms) the pitch angle RMS (120572 rms) and thetire vertical force RMS (Ft FL rms Ft FR rms Ft RL rmsFt RR rms N)

In this paper the frequency compensation algorithm isinstalled in the road input for a sedan model to illustratethe sedanrsquos behavior The parameters of a 119862-class sedan areshown in Table 1 Simulation results run over a 119862-level roadsurface with the speed of 70Kmh tabulated in Table 2 Andthe frequency compensation algorithm can also be installedin the other vehicle runs on a random road

Table 1 Parameters of the 119862-class sedan

Symbol Description Unit Value119898 Vehicle sprung mass kg 2110119898119891119897

Frontleft unsprung mass kg 583119898119891119903

Frontright unsprung mass kg 583119898119903119897

Rearleft unsprung mass kg 633119898119903119903

Rearright unsprung mass kg 633119896119905119891119897

Frontleft vertical tyre stiffness Nm 235000119896119905119891119903

Frontright vertical tyre stiffness Nm 235000119896119905119903119897

Rearleft vertical tyre stiffness Nm 235000119896119905119903119903

Rearright vertical tyre stiffness Nm 235000119896119904119891119897

Frontleft spring stiffness Nm 27500119896119904119891119903

Frontright spring stiffness Nm 27500119896119904119903119897

Rearleft spring stiffness Nm 31300119896119904119903119903

Rearright spring stiffness Nm 31300119897119891

Distance between front-axlecog G m 1455119897119903

Distance between rear-axlecog G m 1515119908 Wheel track m 1610119868120573

Sprung mass roll moment of inertia kgsdotm2 7101119868120572

Sprung mass pitch moment of inertia kgsdotm2 36685

In this analysis the road input of left-front tire is set asa baseline then the right-front road input is obtained via acoherence function wo a compensation and the rear road

Mathematical Problems in Engineering 9

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

(a) Left-front wheel

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

(b) Right-front wheel

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

ISO standard PSDLR simulated PSD

(c) Left-rear wheel

10minus1 100 101

10minus5

Frequency (Hz)PS

D (m

2(1

m))

102

ISO standard PSDRR simulated PSD

(d) Right-rear wheel

Figure 13 Comparison of the road PSDs simulated with the ISO 8608 standards when coh(119891) is 0

LFRF

0 2 4 6 8 10 12 14 16 18minus50

0

50

Time (s)

Road

roug

hnes

s (m

m)

Figure 14 Comparison of left-front wheel (LF) with left-rear wheel(LR) time domain signal when coh(119891) is 1

input has a time delay dependent on the wheel base andvehicle speed Table 2 shows that the 120572 rms Ft FL rms andFt RL rms without compensation are almost equal to thevalues with compensation since the PSDs of the left-side roadinput are the same while the Ac rms 120573 rms Ft FR rms andFt RR rmswithout compensation are significant smaller thanthe values with compensation which is precisely caused bythe previously mentioned deviation since there is a deviationbetween the road PSDs of the two wheels on the same axlewhich is the main reason that the acceleration RMS the rollangle RMS the RMS of tire vertical force of the right-front

Table 2 Simulation results

Results Without compensation With compensation DifferenceAc rms 03638 04081 108120572 rms 00062 00062 0120573 rms 910119890 minus 4 00017 464Ft FL rms 10115119890 + 3 10162119890 + 3 05Ft FR rms 06824119890 + 3 10100119890 + 3 327Ft RL rms 10514119890 + 3 10566119890 + 3 05Ft RR rms 07190119890 + 3 10534119890 + 3 314

and right-rear wheels without compensation are smaller thanthe values with compensation Hence the deviation of thePSD of the road input between two wheels on the same axlehas an important influence on vehicle performances whichillustrates the importance of the frequency compensationalgorithm proposed which can be used for a control systeminstalled in the vehicle to compensate road roughness fordamper tuning in the future

8 Conclusions

The road surface roughness has an important influence onride comfort ride safety vehicle maneuverability driverrsquosand occupantsrsquo comfort and vehicle dynamic load In recentdecades random road models have been widely studied anda four-wheel random road time domain model is usuallygenerated based on the correlation of the four-wheel input in

10 Mathematical Problems in Engineering

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

(a) Left-front wheel

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

(b) Right-front wheel

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

ISO standard PSDLR simulated PSD

(c) Left-rear wheel

10minus1 100 101

10minus5

Frequency (Hz)PS

D (m

2(1

m))

102

ISO standard PSDRR simulated PSD

(d) Right-rear wheel

Figure 15 Comparison of the road PSDs simulated with the ISO 8608 standards when coh(119891) is 1

Vehicledynamicbehavior

VehiclemodelRoad model

Figure 16 Analysis on the vehicle dynamic behavior

which a coherence function is used to describe the coherenceof the road input between the left and right wheels usuallyGenerally the value of coherence function is between 0 and 1and it is near 0 at high-frequency band As a result the roadroughness amplitude and the road PSD of one track will besmaller than the other one on the same axle actually it iscaused by the uncorrelation between the left- and right-wheelroad surface roughness Hence a frequency compensationalgorithm is proposed to correct the deviation of the PSDof the road input between two wheels on the same axle andit is installed in a 7-DOF vehicle dynamic study The mainconclusions can be drawn from the current study as follows

(1) It is simple and accurate to generate a single-wheeltime signal model by using the inverse Fourier trans-form method

(2) With the Inverse Fourier Transform method viacoherence function the road PSDs of the front andrear wheels are consistent with the standard valueshowever there is a deviation between the two-wheelroad PSDs on the same axle The coherence function

value is between 0 and 1 and it is near 0 at high-frequency band As a result the road roughnessamplitude and the road PSD of one track will besmaller than the other one on the same axle

(3) A road frequency compensation algorithm is pro-posed to correct the deviation of the two wheels roadPSDs on the same axle and it is installed in a roadtime domain model The simulation results showedthat the road PSDs with frequency compensation areconsistent with the ISO 8608 standard values whichdemonstrates that the proposed algorithm is correctespecially for some limit case such as the left andright wheel road input are completely uncorrelated orperfectly correlated

(4) The vehicle dynamic behavior with road frequencycompensation show that the deviation compensationof two wheels on the same axle has an importantinfluence on vehicle performances which should beconsidered in the vehicle dynamic studies and theroad frequency compensation can be used for acontrol system installed in the vehicle to compensateroad roughness for damper tuning in the future

Acknowledgments

Special thanks are due to the Open Research Fund Pro-gram of the State Key Laboratory of Advanced Design andManufacturing for Vehicle Body (31115028) the China Post-doctoral Science Foundation (2012M520028) the NationalNatural Science Foundation of China (51205155) and the

Mathematical Problems in Engineering 11

lf

lr

w

I120572 120572 I120573 120573

xGy

z

z

ztrl

zrl

crl

mrl

ksrl

ktrl

ksfr

mfr

ksfl

zfl

cfl

mfl

ktfl

ztfl

cfrztfr

zfr ktfr

Figure 17 7 DOF full vehicle model

National Basic Research Program of China (973 Program)(2011CB711201) for supporting authorsrsquo research

References

[1] P Mucka ldquoRoad waviness and the dynamic tyre forcerdquo Inter-national Journal of Vehicle Design vol 36 no 2-3 pp 216ndash2322004

[2] Z Yonglin and Z Jiafan ldquoNumerical simulation of stochasticroad process using white noise filtrationrdquo Mechanical Systemsand Signal Processing vol 20 no 2 pp 363ndash372 2006

[3] K Ahlin J Granlund and F Lindstrom ldquoComparing roadprofiles with vehicle perceived roughnessrdquo International Journalof Vehicle Design vol 36 no 2-3 pp 270ndash286 2004

[4] S M Karamihas and M W Sayers The Little Book of ProfilingUniversity of Michigan Ann Arbor Mich USA 1998

[5] MAmbroz G Sustersic and I Prebil ldquoCreatingmodels of roadsections and their use in driving dynamics simulationsrdquo VehicleSystem Dynamics vol 45 no 10 pp 911ndash924 2007

[6] D J Whitehouse and J F Archard ldquoThe properties of randomsurfaces of significance in their contactrdquoProceedings of the RoyalSociety of London A vol 316 pp 97ndash121 1970

[7] M Shinozuka and C-M Jan ldquoDigital simulation of randomprocesses and its applicationsrdquo Journal of Sound and Vibrationvol 25 no 1 pp 111ndash128 1972

[8] D Cebon Interaction between Heavy Vehicles and Roads vol951 Society of Automotive Engineers 1993

[9] J-J Wu ldquoSimulation of rough surfaces with FFTrdquo TribologyInternational vol 33 no 1 pp 47ndash58 2000

[10] M Grigoriu ldquoOn the spectral representation method in sim-ulationrdquo Probabilistic Engineering Mechanics vol 8 no 2 pp75ndash90 1993

[11] ISO Mechanical Vibration-Road Surface Profiles-Reporting ofMeasured Data International Organization for Standardiza-tion 1995

[12] T Yoshimura ldquoA semi-active suspension of passenger cars usingfuzzy reasoning and the field testingrdquo International Journal ofVehicle Design vol 19 no 2 pp 150ndash166 1998

[13] K Bogsjo ldquoEvaluation of stochastic models of parallel roadtracksrdquo Probabilistic Engineering Mechanics vol 22 no 4 pp362ndash370 2007

[14] A Pazooki S Rakheja andD Cao ldquoModeling and validation ofoff-road vehicle ride dynamicsrdquoMechanical Systems and SignalProcessing vol 28 pp 679ndash695 2012

[15] R Hassan and R Evans ldquoRoad roughness characteristics incar and truck wheel tracksrdquo International Journal of PavementEngineering 2012

[16] K Bogsjo K Podgorski and I Rychlik ldquoModels for roadsurface roughnessrdquo Vehicle System Dynamics vol 50 no 5 pp725ndash747 2012

[17] S A Hassan and O S Lashine ldquoIdentification of road surfacequalities for linear and non-linear vehicle modelingrdquo in Pro-ceedings of the 20th International Modal Analysis Conference(IMAC rsquo02) pp 318ndash323 Society for Experimental Mechanics2002

[18] A N Heath ldquoApplication of the isotropic road roughnessassumptionrdquo Journal of Sound and Vibration vol 115 no 1 pp131ndash144 1987

[19] A N Heath ldquoEvaluation of the isotropic road roughnessassumptionrdquoVehicle SystemDynamics vol 17 supplement 1 pp157ndash160 1988

[20] D Cebon and D E Newland ldquoThe artificial generation of roadsurface topography by the inverse FFT methodrdquo Vehicle SystemDynamics vol 12 no 1ndash3 pp 160ndash165 1983

[21] C J Dodds ldquoThe laboratory simulation of vehicle servicestressrdquo Journal of Engineering for Industry vol 96 no 2 pp 391ndash398 1974

[22] D Cebon and D E Newlan ldquoThe artificial generation of roadsurface topography by the inverse FFT methodrdquo in Proceedingsof the 8th IAVSD-Symposium on the Dynamics of Vehicles onRoads and Tracks pp 29ndash42 Swets and Zeitlinger CambridgeMass USA 1983

[23] C J Dodds and J D Robson ldquoThe description of roadroughnessrdquo Journal of Sound and Vibration vol 31 no 2 pp175ndash183 1973

[24] K M A Kamash and J D Robson ldquoImplications of isotropy inrandom surfacesrdquo Journal of Sound and Vibration vol 54 no 1pp 131ndash145 1977

[25] K M A Kamash and J D Robson ldquoThe application of isotropyin road surface modellingrdquo Journal of Sound and Vibration vol57 no 1 pp 89ndash100 1978

12 Mathematical Problems in Engineering

[26] G C Carter ldquoCoherence and time delay estimationrdquo Proceed-ings of the IEEE vol 75 no 2 pp 236ndash255 1987

[27] K Bogsjo ldquoCoherence of road roughness in left and rightwheel-pathrdquo Vehicle System Dynamics vol 46 supplement 1 pp 599ndash609 2008

[28] R Hongbin C Sizhong andW Zhicheng ldquoModel of excitationof random road profile in time domain for a vehicle withfour wheelsrdquo in Proceedings of the International Conference onMechatronic Science Electric Engineering and Computer (MECrsquo11) pp 2332ndash2335 Jilin China August 2011

[29] J D Robson ldquoRoad surface description and vehicle responserdquoInternational Journal of Vehicle Design vol 1 no 1 pp 25ndash351979

[30] D Ammon ldquoProblems in road surface modellingrdquo VehicleSystem Dynamics vol 20 pp 28ndash41 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

(a) Left-front wheel

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

(b) Right-front wheel

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

ISO standard PSDLR simulated PSD

(c) Left-rear wheel

10minus1 100 101

10minus5

Frequency (Hz)PS

D (m

2(1

m))

102

ISO standard PSDRR simulated PSD

(d) Right-rear wheel

Figure 13 Comparison of the road PSDs simulated with the ISO 8608 standards when coh(119891) is 0

LFRF

0 2 4 6 8 10 12 14 16 18minus50

0

50

Time (s)

Road

roug

hnes

s (m

m)

Figure 14 Comparison of left-front wheel (LF) with left-rear wheel(LR) time domain signal when coh(119891) is 1

input has a time delay dependent on the wheel base andvehicle speed Table 2 shows that the 120572 rms Ft FL rms andFt RL rms without compensation are almost equal to thevalues with compensation since the PSDs of the left-side roadinput are the same while the Ac rms 120573 rms Ft FR rms andFt RR rmswithout compensation are significant smaller thanthe values with compensation which is precisely caused bythe previously mentioned deviation since there is a deviationbetween the road PSDs of the two wheels on the same axlewhich is the main reason that the acceleration RMS the rollangle RMS the RMS of tire vertical force of the right-front

Table 2 Simulation results

Results Without compensation With compensation DifferenceAc rms 03638 04081 108120572 rms 00062 00062 0120573 rms 910119890 minus 4 00017 464Ft FL rms 10115119890 + 3 10162119890 + 3 05Ft FR rms 06824119890 + 3 10100119890 + 3 327Ft RL rms 10514119890 + 3 10566119890 + 3 05Ft RR rms 07190119890 + 3 10534119890 + 3 314

and right-rear wheels without compensation are smaller thanthe values with compensation Hence the deviation of thePSD of the road input between two wheels on the same axlehas an important influence on vehicle performances whichillustrates the importance of the frequency compensationalgorithm proposed which can be used for a control systeminstalled in the vehicle to compensate road roughness fordamper tuning in the future

8 Conclusions

The road surface roughness has an important influence onride comfort ride safety vehicle maneuverability driverrsquosand occupantsrsquo comfort and vehicle dynamic load In recentdecades random road models have been widely studied anda four-wheel random road time domain model is usuallygenerated based on the correlation of the four-wheel input in

10 Mathematical Problems in Engineering

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

(a) Left-front wheel

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

(b) Right-front wheel

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

ISO standard PSDLR simulated PSD

(c) Left-rear wheel

10minus1 100 101

10minus5

Frequency (Hz)PS

D (m

2(1

m))

102

ISO standard PSDRR simulated PSD

(d) Right-rear wheel

Figure 15 Comparison of the road PSDs simulated with the ISO 8608 standards when coh(119891) is 1

Vehicledynamicbehavior

VehiclemodelRoad model

Figure 16 Analysis on the vehicle dynamic behavior

which a coherence function is used to describe the coherenceof the road input between the left and right wheels usuallyGenerally the value of coherence function is between 0 and 1and it is near 0 at high-frequency band As a result the roadroughness amplitude and the road PSD of one track will besmaller than the other one on the same axle actually it iscaused by the uncorrelation between the left- and right-wheelroad surface roughness Hence a frequency compensationalgorithm is proposed to correct the deviation of the PSDof the road input between two wheels on the same axle andit is installed in a 7-DOF vehicle dynamic study The mainconclusions can be drawn from the current study as follows

(1) It is simple and accurate to generate a single-wheeltime signal model by using the inverse Fourier trans-form method

(2) With the Inverse Fourier Transform method viacoherence function the road PSDs of the front andrear wheels are consistent with the standard valueshowever there is a deviation between the two-wheelroad PSDs on the same axle The coherence function

value is between 0 and 1 and it is near 0 at high-frequency band As a result the road roughnessamplitude and the road PSD of one track will besmaller than the other one on the same axle

(3) A road frequency compensation algorithm is pro-posed to correct the deviation of the two wheels roadPSDs on the same axle and it is installed in a roadtime domain model The simulation results showedthat the road PSDs with frequency compensation areconsistent with the ISO 8608 standard values whichdemonstrates that the proposed algorithm is correctespecially for some limit case such as the left andright wheel road input are completely uncorrelated orperfectly correlated

(4) The vehicle dynamic behavior with road frequencycompensation show that the deviation compensationof two wheels on the same axle has an importantinfluence on vehicle performances which should beconsidered in the vehicle dynamic studies and theroad frequency compensation can be used for acontrol system installed in the vehicle to compensateroad roughness for damper tuning in the future

Acknowledgments

Special thanks are due to the Open Research Fund Pro-gram of the State Key Laboratory of Advanced Design andManufacturing for Vehicle Body (31115028) the China Post-doctoral Science Foundation (2012M520028) the NationalNatural Science Foundation of China (51205155) and the

Mathematical Problems in Engineering 11

lf

lr

w

I120572 120572 I120573 120573

xGy

z

z

ztrl

zrl

crl

mrl

ksrl

ktrl

ksfr

mfr

ksfl

zfl

cfl

mfl

ktfl

ztfl

cfrztfr

zfr ktfr

Figure 17 7 DOF full vehicle model

National Basic Research Program of China (973 Program)(2011CB711201) for supporting authorsrsquo research

References

[1] P Mucka ldquoRoad waviness and the dynamic tyre forcerdquo Inter-national Journal of Vehicle Design vol 36 no 2-3 pp 216ndash2322004

[2] Z Yonglin and Z Jiafan ldquoNumerical simulation of stochasticroad process using white noise filtrationrdquo Mechanical Systemsand Signal Processing vol 20 no 2 pp 363ndash372 2006

[3] K Ahlin J Granlund and F Lindstrom ldquoComparing roadprofiles with vehicle perceived roughnessrdquo International Journalof Vehicle Design vol 36 no 2-3 pp 270ndash286 2004

[4] S M Karamihas and M W Sayers The Little Book of ProfilingUniversity of Michigan Ann Arbor Mich USA 1998

[5] MAmbroz G Sustersic and I Prebil ldquoCreatingmodels of roadsections and their use in driving dynamics simulationsrdquo VehicleSystem Dynamics vol 45 no 10 pp 911ndash924 2007

[6] D J Whitehouse and J F Archard ldquoThe properties of randomsurfaces of significance in their contactrdquoProceedings of the RoyalSociety of London A vol 316 pp 97ndash121 1970

[7] M Shinozuka and C-M Jan ldquoDigital simulation of randomprocesses and its applicationsrdquo Journal of Sound and Vibrationvol 25 no 1 pp 111ndash128 1972

[8] D Cebon Interaction between Heavy Vehicles and Roads vol951 Society of Automotive Engineers 1993

[9] J-J Wu ldquoSimulation of rough surfaces with FFTrdquo TribologyInternational vol 33 no 1 pp 47ndash58 2000

[10] M Grigoriu ldquoOn the spectral representation method in sim-ulationrdquo Probabilistic Engineering Mechanics vol 8 no 2 pp75ndash90 1993

[11] ISO Mechanical Vibration-Road Surface Profiles-Reporting ofMeasured Data International Organization for Standardiza-tion 1995

[12] T Yoshimura ldquoA semi-active suspension of passenger cars usingfuzzy reasoning and the field testingrdquo International Journal ofVehicle Design vol 19 no 2 pp 150ndash166 1998

[13] K Bogsjo ldquoEvaluation of stochastic models of parallel roadtracksrdquo Probabilistic Engineering Mechanics vol 22 no 4 pp362ndash370 2007

[14] A Pazooki S Rakheja andD Cao ldquoModeling and validation ofoff-road vehicle ride dynamicsrdquoMechanical Systems and SignalProcessing vol 28 pp 679ndash695 2012

[15] R Hassan and R Evans ldquoRoad roughness characteristics incar and truck wheel tracksrdquo International Journal of PavementEngineering 2012

[16] K Bogsjo K Podgorski and I Rychlik ldquoModels for roadsurface roughnessrdquo Vehicle System Dynamics vol 50 no 5 pp725ndash747 2012

[17] S A Hassan and O S Lashine ldquoIdentification of road surfacequalities for linear and non-linear vehicle modelingrdquo in Pro-ceedings of the 20th International Modal Analysis Conference(IMAC rsquo02) pp 318ndash323 Society for Experimental Mechanics2002

[18] A N Heath ldquoApplication of the isotropic road roughnessassumptionrdquo Journal of Sound and Vibration vol 115 no 1 pp131ndash144 1987

[19] A N Heath ldquoEvaluation of the isotropic road roughnessassumptionrdquoVehicle SystemDynamics vol 17 supplement 1 pp157ndash160 1988

[20] D Cebon and D E Newland ldquoThe artificial generation of roadsurface topography by the inverse FFT methodrdquo Vehicle SystemDynamics vol 12 no 1ndash3 pp 160ndash165 1983

[21] C J Dodds ldquoThe laboratory simulation of vehicle servicestressrdquo Journal of Engineering for Industry vol 96 no 2 pp 391ndash398 1974

[22] D Cebon and D E Newlan ldquoThe artificial generation of roadsurface topography by the inverse FFT methodrdquo in Proceedingsof the 8th IAVSD-Symposium on the Dynamics of Vehicles onRoads and Tracks pp 29ndash42 Swets and Zeitlinger CambridgeMass USA 1983

[23] C J Dodds and J D Robson ldquoThe description of roadroughnessrdquo Journal of Sound and Vibration vol 31 no 2 pp175ndash183 1973

[24] K M A Kamash and J D Robson ldquoImplications of isotropy inrandom surfacesrdquo Journal of Sound and Vibration vol 54 no 1pp 131ndash145 1977

[25] K M A Kamash and J D Robson ldquoThe application of isotropyin road surface modellingrdquo Journal of Sound and Vibration vol57 no 1 pp 89ndash100 1978

12 Mathematical Problems in Engineering

[26] G C Carter ldquoCoherence and time delay estimationrdquo Proceed-ings of the IEEE vol 75 no 2 pp 236ndash255 1987

[27] K Bogsjo ldquoCoherence of road roughness in left and rightwheel-pathrdquo Vehicle System Dynamics vol 46 supplement 1 pp 599ndash609 2008

[28] R Hongbin C Sizhong andW Zhicheng ldquoModel of excitationof random road profile in time domain for a vehicle withfour wheelsrdquo in Proceedings of the International Conference onMechatronic Science Electric Engineering and Computer (MECrsquo11) pp 2332ndash2335 Jilin China August 2011

[29] J D Robson ldquoRoad surface description and vehicle responserdquoInternational Journal of Vehicle Design vol 1 no 1 pp 25ndash351979

[30] D Ammon ldquoProblems in road surface modellingrdquo VehicleSystem Dynamics vol 20 pp 28ndash41 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

(a) Left-front wheel

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

(b) Right-front wheel

10minus1 100 101

10minus5

Frequency (Hz)

PSD

(m2(1

m))

102

ISO standard PSDLR simulated PSD

(c) Left-rear wheel

10minus1 100 101

10minus5

Frequency (Hz)PS

D (m

2(1

m))

102

ISO standard PSDRR simulated PSD

(d) Right-rear wheel

Figure 15 Comparison of the road PSDs simulated with the ISO 8608 standards when coh(119891) is 1

Vehicledynamicbehavior

VehiclemodelRoad model

Figure 16 Analysis on the vehicle dynamic behavior

which a coherence function is used to describe the coherenceof the road input between the left and right wheels usuallyGenerally the value of coherence function is between 0 and 1and it is near 0 at high-frequency band As a result the roadroughness amplitude and the road PSD of one track will besmaller than the other one on the same axle actually it iscaused by the uncorrelation between the left- and right-wheelroad surface roughness Hence a frequency compensationalgorithm is proposed to correct the deviation of the PSDof the road input between two wheels on the same axle andit is installed in a 7-DOF vehicle dynamic study The mainconclusions can be drawn from the current study as follows

(1) It is simple and accurate to generate a single-wheeltime signal model by using the inverse Fourier trans-form method

(2) With the Inverse Fourier Transform method viacoherence function the road PSDs of the front andrear wheels are consistent with the standard valueshowever there is a deviation between the two-wheelroad PSDs on the same axle The coherence function

value is between 0 and 1 and it is near 0 at high-frequency band As a result the road roughnessamplitude and the road PSD of one track will besmaller than the other one on the same axle

(3) A road frequency compensation algorithm is pro-posed to correct the deviation of the two wheels roadPSDs on the same axle and it is installed in a roadtime domain model The simulation results showedthat the road PSDs with frequency compensation areconsistent with the ISO 8608 standard values whichdemonstrates that the proposed algorithm is correctespecially for some limit case such as the left andright wheel road input are completely uncorrelated orperfectly correlated

(4) The vehicle dynamic behavior with road frequencycompensation show that the deviation compensationof two wheels on the same axle has an importantinfluence on vehicle performances which should beconsidered in the vehicle dynamic studies and theroad frequency compensation can be used for acontrol system installed in the vehicle to compensateroad roughness for damper tuning in the future

Acknowledgments

Special thanks are due to the Open Research Fund Pro-gram of the State Key Laboratory of Advanced Design andManufacturing for Vehicle Body (31115028) the China Post-doctoral Science Foundation (2012M520028) the NationalNatural Science Foundation of China (51205155) and the

Mathematical Problems in Engineering 11

lf

lr

w

I120572 120572 I120573 120573

xGy

z

z

ztrl

zrl

crl

mrl

ksrl

ktrl

ksfr

mfr

ksfl

zfl

cfl

mfl

ktfl

ztfl

cfrztfr

zfr ktfr

Figure 17 7 DOF full vehicle model

National Basic Research Program of China (973 Program)(2011CB711201) for supporting authorsrsquo research

References

[1] P Mucka ldquoRoad waviness and the dynamic tyre forcerdquo Inter-national Journal of Vehicle Design vol 36 no 2-3 pp 216ndash2322004

[2] Z Yonglin and Z Jiafan ldquoNumerical simulation of stochasticroad process using white noise filtrationrdquo Mechanical Systemsand Signal Processing vol 20 no 2 pp 363ndash372 2006

[3] K Ahlin J Granlund and F Lindstrom ldquoComparing roadprofiles with vehicle perceived roughnessrdquo International Journalof Vehicle Design vol 36 no 2-3 pp 270ndash286 2004

[4] S M Karamihas and M W Sayers The Little Book of ProfilingUniversity of Michigan Ann Arbor Mich USA 1998

[5] MAmbroz G Sustersic and I Prebil ldquoCreatingmodels of roadsections and their use in driving dynamics simulationsrdquo VehicleSystem Dynamics vol 45 no 10 pp 911ndash924 2007

[6] D J Whitehouse and J F Archard ldquoThe properties of randomsurfaces of significance in their contactrdquoProceedings of the RoyalSociety of London A vol 316 pp 97ndash121 1970

[7] M Shinozuka and C-M Jan ldquoDigital simulation of randomprocesses and its applicationsrdquo Journal of Sound and Vibrationvol 25 no 1 pp 111ndash128 1972

[8] D Cebon Interaction between Heavy Vehicles and Roads vol951 Society of Automotive Engineers 1993

[9] J-J Wu ldquoSimulation of rough surfaces with FFTrdquo TribologyInternational vol 33 no 1 pp 47ndash58 2000

[10] M Grigoriu ldquoOn the spectral representation method in sim-ulationrdquo Probabilistic Engineering Mechanics vol 8 no 2 pp75ndash90 1993

[11] ISO Mechanical Vibration-Road Surface Profiles-Reporting ofMeasured Data International Organization for Standardiza-tion 1995

[12] T Yoshimura ldquoA semi-active suspension of passenger cars usingfuzzy reasoning and the field testingrdquo International Journal ofVehicle Design vol 19 no 2 pp 150ndash166 1998

[13] K Bogsjo ldquoEvaluation of stochastic models of parallel roadtracksrdquo Probabilistic Engineering Mechanics vol 22 no 4 pp362ndash370 2007

[14] A Pazooki S Rakheja andD Cao ldquoModeling and validation ofoff-road vehicle ride dynamicsrdquoMechanical Systems and SignalProcessing vol 28 pp 679ndash695 2012

[15] R Hassan and R Evans ldquoRoad roughness characteristics incar and truck wheel tracksrdquo International Journal of PavementEngineering 2012

[16] K Bogsjo K Podgorski and I Rychlik ldquoModels for roadsurface roughnessrdquo Vehicle System Dynamics vol 50 no 5 pp725ndash747 2012

[17] S A Hassan and O S Lashine ldquoIdentification of road surfacequalities for linear and non-linear vehicle modelingrdquo in Pro-ceedings of the 20th International Modal Analysis Conference(IMAC rsquo02) pp 318ndash323 Society for Experimental Mechanics2002

[18] A N Heath ldquoApplication of the isotropic road roughnessassumptionrdquo Journal of Sound and Vibration vol 115 no 1 pp131ndash144 1987

[19] A N Heath ldquoEvaluation of the isotropic road roughnessassumptionrdquoVehicle SystemDynamics vol 17 supplement 1 pp157ndash160 1988

[20] D Cebon and D E Newland ldquoThe artificial generation of roadsurface topography by the inverse FFT methodrdquo Vehicle SystemDynamics vol 12 no 1ndash3 pp 160ndash165 1983

[21] C J Dodds ldquoThe laboratory simulation of vehicle servicestressrdquo Journal of Engineering for Industry vol 96 no 2 pp 391ndash398 1974

[22] D Cebon and D E Newlan ldquoThe artificial generation of roadsurface topography by the inverse FFT methodrdquo in Proceedingsof the 8th IAVSD-Symposium on the Dynamics of Vehicles onRoads and Tracks pp 29ndash42 Swets and Zeitlinger CambridgeMass USA 1983

[23] C J Dodds and J D Robson ldquoThe description of roadroughnessrdquo Journal of Sound and Vibration vol 31 no 2 pp175ndash183 1973

[24] K M A Kamash and J D Robson ldquoImplications of isotropy inrandom surfacesrdquo Journal of Sound and Vibration vol 54 no 1pp 131ndash145 1977

[25] K M A Kamash and J D Robson ldquoThe application of isotropyin road surface modellingrdquo Journal of Sound and Vibration vol57 no 1 pp 89ndash100 1978

12 Mathematical Problems in Engineering

[26] G C Carter ldquoCoherence and time delay estimationrdquo Proceed-ings of the IEEE vol 75 no 2 pp 236ndash255 1987

[27] K Bogsjo ldquoCoherence of road roughness in left and rightwheel-pathrdquo Vehicle System Dynamics vol 46 supplement 1 pp 599ndash609 2008

[28] R Hongbin C Sizhong andW Zhicheng ldquoModel of excitationof random road profile in time domain for a vehicle withfour wheelsrdquo in Proceedings of the International Conference onMechatronic Science Electric Engineering and Computer (MECrsquo11) pp 2332ndash2335 Jilin China August 2011

[29] J D Robson ldquoRoad surface description and vehicle responserdquoInternational Journal of Vehicle Design vol 1 no 1 pp 25ndash351979

[30] D Ammon ldquoProblems in road surface modellingrdquo VehicleSystem Dynamics vol 20 pp 28ndash41 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

lf

lr

w

I120572 120572 I120573 120573

xGy

z

z

ztrl

zrl

crl

mrl

ksrl

ktrl

ksfr

mfr

ksfl

zfl

cfl

mfl

ktfl

ztfl

cfrztfr

zfr ktfr

Figure 17 7 DOF full vehicle model

National Basic Research Program of China (973 Program)(2011CB711201) for supporting authorsrsquo research

References

[1] P Mucka ldquoRoad waviness and the dynamic tyre forcerdquo Inter-national Journal of Vehicle Design vol 36 no 2-3 pp 216ndash2322004

[2] Z Yonglin and Z Jiafan ldquoNumerical simulation of stochasticroad process using white noise filtrationrdquo Mechanical Systemsand Signal Processing vol 20 no 2 pp 363ndash372 2006

[3] K Ahlin J Granlund and F Lindstrom ldquoComparing roadprofiles with vehicle perceived roughnessrdquo International Journalof Vehicle Design vol 36 no 2-3 pp 270ndash286 2004

[4] S M Karamihas and M W Sayers The Little Book of ProfilingUniversity of Michigan Ann Arbor Mich USA 1998

[5] MAmbroz G Sustersic and I Prebil ldquoCreatingmodels of roadsections and their use in driving dynamics simulationsrdquo VehicleSystem Dynamics vol 45 no 10 pp 911ndash924 2007

[6] D J Whitehouse and J F Archard ldquoThe properties of randomsurfaces of significance in their contactrdquoProceedings of the RoyalSociety of London A vol 316 pp 97ndash121 1970

[7] M Shinozuka and C-M Jan ldquoDigital simulation of randomprocesses and its applicationsrdquo Journal of Sound and Vibrationvol 25 no 1 pp 111ndash128 1972

[8] D Cebon Interaction between Heavy Vehicles and Roads vol951 Society of Automotive Engineers 1993

[9] J-J Wu ldquoSimulation of rough surfaces with FFTrdquo TribologyInternational vol 33 no 1 pp 47ndash58 2000

[10] M Grigoriu ldquoOn the spectral representation method in sim-ulationrdquo Probabilistic Engineering Mechanics vol 8 no 2 pp75ndash90 1993

[11] ISO Mechanical Vibration-Road Surface Profiles-Reporting ofMeasured Data International Organization for Standardiza-tion 1995

[12] T Yoshimura ldquoA semi-active suspension of passenger cars usingfuzzy reasoning and the field testingrdquo International Journal ofVehicle Design vol 19 no 2 pp 150ndash166 1998

[13] K Bogsjo ldquoEvaluation of stochastic models of parallel roadtracksrdquo Probabilistic Engineering Mechanics vol 22 no 4 pp362ndash370 2007

[14] A Pazooki S Rakheja andD Cao ldquoModeling and validation ofoff-road vehicle ride dynamicsrdquoMechanical Systems and SignalProcessing vol 28 pp 679ndash695 2012

[15] R Hassan and R Evans ldquoRoad roughness characteristics incar and truck wheel tracksrdquo International Journal of PavementEngineering 2012

[16] K Bogsjo K Podgorski and I Rychlik ldquoModels for roadsurface roughnessrdquo Vehicle System Dynamics vol 50 no 5 pp725ndash747 2012

[17] S A Hassan and O S Lashine ldquoIdentification of road surfacequalities for linear and non-linear vehicle modelingrdquo in Pro-ceedings of the 20th International Modal Analysis Conference(IMAC rsquo02) pp 318ndash323 Society for Experimental Mechanics2002

[18] A N Heath ldquoApplication of the isotropic road roughnessassumptionrdquo Journal of Sound and Vibration vol 115 no 1 pp131ndash144 1987

[19] A N Heath ldquoEvaluation of the isotropic road roughnessassumptionrdquoVehicle SystemDynamics vol 17 supplement 1 pp157ndash160 1988

[20] D Cebon and D E Newland ldquoThe artificial generation of roadsurface topography by the inverse FFT methodrdquo Vehicle SystemDynamics vol 12 no 1ndash3 pp 160ndash165 1983

[21] C J Dodds ldquoThe laboratory simulation of vehicle servicestressrdquo Journal of Engineering for Industry vol 96 no 2 pp 391ndash398 1974

[22] D Cebon and D E Newlan ldquoThe artificial generation of roadsurface topography by the inverse FFT methodrdquo in Proceedingsof the 8th IAVSD-Symposium on the Dynamics of Vehicles onRoads and Tracks pp 29ndash42 Swets and Zeitlinger CambridgeMass USA 1983

[23] C J Dodds and J D Robson ldquoThe description of roadroughnessrdquo Journal of Sound and Vibration vol 31 no 2 pp175ndash183 1973

[24] K M A Kamash and J D Robson ldquoImplications of isotropy inrandom surfacesrdquo Journal of Sound and Vibration vol 54 no 1pp 131ndash145 1977

[25] K M A Kamash and J D Robson ldquoThe application of isotropyin road surface modellingrdquo Journal of Sound and Vibration vol57 no 1 pp 89ndash100 1978

12 Mathematical Problems in Engineering

[26] G C Carter ldquoCoherence and time delay estimationrdquo Proceed-ings of the IEEE vol 75 no 2 pp 236ndash255 1987

[27] K Bogsjo ldquoCoherence of road roughness in left and rightwheel-pathrdquo Vehicle System Dynamics vol 46 supplement 1 pp 599ndash609 2008

[28] R Hongbin C Sizhong andW Zhicheng ldquoModel of excitationof random road profile in time domain for a vehicle withfour wheelsrdquo in Proceedings of the International Conference onMechatronic Science Electric Engineering and Computer (MECrsquo11) pp 2332ndash2335 Jilin China August 2011

[29] J D Robson ldquoRoad surface description and vehicle responserdquoInternational Journal of Vehicle Design vol 1 no 1 pp 25ndash351979

[30] D Ammon ldquoProblems in road surface modellingrdquo VehicleSystem Dynamics vol 20 pp 28ndash41 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

[26] G C Carter ldquoCoherence and time delay estimationrdquo Proceed-ings of the IEEE vol 75 no 2 pp 236ndash255 1987

[27] K Bogsjo ldquoCoherence of road roughness in left and rightwheel-pathrdquo Vehicle System Dynamics vol 46 supplement 1 pp 599ndash609 2008

[28] R Hongbin C Sizhong andW Zhicheng ldquoModel of excitationof random road profile in time domain for a vehicle withfour wheelsrdquo in Proceedings of the International Conference onMechatronic Science Electric Engineering and Computer (MECrsquo11) pp 2332ndash2335 Jilin China August 2011

[29] J D Robson ldquoRoad surface description and vehicle responserdquoInternational Journal of Vehicle Design vol 1 no 1 pp 25ndash351979

[30] D Ammon ldquoProblems in road surface modellingrdquo VehicleSystem Dynamics vol 20 pp 28ndash41 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of