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Research ArticleVibrationControlPerformanceAnalysis andShake-TableTest of aPounding Tuned Rotary Mass Damper under the Earthquake

Shujin Li Lei Sun and Fan Kong

School of Civil Engineering and Architecture Wuhan University of Technology Luoshi Road No 122 Wuhan 430070 China

Correspondence should be addressed to Fan Kong kongfanwhuteducn

Received 28 April 2019 Accepted 11 June 2019 Published 1 August 2019

Academic Editor Marco Tarabini

Copyright copy 2019 Shujin Li et al -is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

-e voided biaxial concrete slab has been widely used in the engineering field -e slab has become a popular choice for designersand architects looking to reduce slab thickness and overall structure weight recently Utilizing the empty space in the voided slaband introducing the structural control technology of mass damper into it a new pounding tuned rotary mass damper (PTRMD) isproposed in this paper -is damper is designed to locate in the prefabricated hollow module to mitigate response of structuresubject to disastrous excitations -e damper combines the characteristics of pounding mechanisms (PMDs) and tuned rotarymass dampers (TRMDs)-is is achieved by a ball rolling on a curved orbit and a fixed stroke-limiting plate-e structural controlperformance of the PTRMD is studied numerically and verified experimentally Specifically first the motion equations for asingle-degree-of-freedom (SDOF) and multiple-degree-of-freedom (MDOF) system with PTRMDs are derived Furthermorenumerical results show that the PTRMD provides significant energy dissipation and thus is quite effective in reducing thestructure response Besides the PTRMD generally exhibits better control performance and robustness in terms of vibrationsuppression compared with the TRMD proposed by the authors before Finally a shake-table test is conducted to verify thedamping effect of a PTRMD-controlled SDOF system Pertinent results confirm the effectiveness and robustness of PTRMDs forstructural control

1 Introduction

Structural systems with voided biaxial slabs have been widelyused in engineering applications in Europe [1 2] and ChinaSchnellenbach-Held and Pfeffer [3] for instance in-vestigated the structural behavior of a type of biaxial voidedslab called BubbleDeck used in Europe -is voided slabconsists of a hollow ball made of recycled industrial plastic[3] BubbleDeck is employed to act as a normal monolithictwo-way spanning concrete slab Other examples of similarhollow modules include U-boot [2] -e main advantages ofbiaxial voided slabs are that they reduce dead weight andextend the span of structural floors [4] -e most importantfeature of these hollow modules is their large voided interiorspace

In a previous study [5] the authors proposed a new typeof damper namely the tuned rolling mass damper (TRMD)that takes advantage of the large space in voided modules to

avoid the excessive occupation of building space often en-countered in pendulum-like tuned mass damper (TMD)applications However the study shows the TRMDs have aproblem as for TMDs ie they have limited response re-duction capacity for seismic control [6ndash8] -e reason mayattribute to that the frequency response function of the linearTMDs is quite narrow compared to the frequency band ofseismic response of MDOF systems -erefore an in-troduction of nonlinear mechanism into the linear TMDsmay benefit to the controlling effect In this regard one mayconsider the use of the pounding tuned mass damper(PTMD) that combines a traditional TMD and a collisionmechanism proposed in a previous study [9] Specificallybased on a traditional TMD a PTMD adds buffer made ofviscoelastic materials to limit the stroke of the mass [10]-esystem acts as a traditional linear TMD without poundingwhen the stroke is inside of the buffers whereas it acts as anonlinear TMD when a larger stroke occurs -is high

HindawiShock and VibrationVolume 2019 Article ID 4038657 14 pageshttpsdoiorg10115520194038657

degree of nonlinearity is due to the impact between theoscillator and the buffers In the impact process mechanicalenergy is dissipated as heat and noise

Zhang et al [9] utilized PTMD for response control of atransmission tower -e performance of the PTMD underearthquake conditions was studied and the numerical resultsconfirmed that the PTMD was more effective than a TMD-e influence of parameters such as the mass ratio clearanceseismic intensity and structural damping ratio was analyzedLi et al [11] investigated the performance of a PTMD on atraffic signal pole -e pounding mechanism of the PTMDwas verified experimentally with its performance underconditions of free vibration and resonant excitation exceedingthat of a TMD [12ndash14] -e acceleration response of thecontrol signal under conditions of sine-wave excitation wasreduced by 55 which is an important and significant resultLi et al [12] applied a PTMD to a subsea jumper to study therobustness of the damper when confronted with detuningeffects It was found that [12] the PTMD performed best whenthe excitation frequency was slightly lower than the funda-mental frequency which is the optimal frequency Xue et al[13] took an offshore platform as an example and reached thesame conclusion ie PTMD is more effective and robust thanTMD Previous studies demonstrate that viscoelastic collisionmechanisms significantly improve the damping performanceand robustness of the traditional TMDs However moststudies [14ndash16] investigated impact dampers be they theo-retical or experimental focusing on the performance of SDOFsystems under simple excitation conditions such as sinusoidalloading Most civil structures for instance multistorybuildings experiencing situations such as strong winds orearthquakes cannot reasonably be approximated as an SDOFsystem and complex external loading is likely to induce morethan just the fundamental mode [16]

In this paper a novel pounding tuned rotary massdamper (PTRMD) that combines a buffered poundingmechanism to the design of a TRMD is presented Specif-ically first the equations of motion relating to SDOFMDOF structure controlled by a PTRMD are derived Nextthe control performance of three different loading condi-tions for an SDOF system including free vibration har-monic excitations and seismic excitation is investigatedFurthermore a 6-story structure is used as an illustrativeexample for control performance of an MDOF systemsequipped with PTRMD PTRMDs of varying frequencies areintroduced into this numerical simulation to study the ro-bustness of the system Finally experiments involving freevibration and seismic response of PTRMD-controlled SDOFstructure are conducted to verify the effectiveness of theproposed control device

2 Pounding Tuned Rotary MassDamper System

21 Primary Structure A structure with a voided biaxialreinforced concrete slab is the primary structure to becontrolled -e main elements of this slab are prefabricatedhollow box-like modules (Figure 1) located between the

reinforcement grids of the main beams and the ribbedbeams -e modules are used as the side formwork site-casting the concrete beams In this way only the bottomformworks of the concrete slab are needed thus saving aconsiderable mass of the structure Figure 2 shows a sim-plified diagram of a hollow floor with a pounding tunedrotary mass damper -e damper consists of a single rect-angular hollow box a ball rolling along an arch path andtwo buffering stop plates Compared to the other TMDs theproposed PTRMD system does not alter the architecturalinterior space excessively Previous studies [13] indicatedthat an enhanced control effect could be achieved by in-troducing a flexible buffer zone between a moving dampermass and its boundaries Stop plates covered with visco-elastic materials are therefore installed in the hollow-floorcavity

22 Models of Mass Dampers Figures 3(a)ndash3(c) illustrate aTRMD pounding mass damper (PMD) and PTRMDmodelused for a hollow-ribbed floor -e PTRMD consists of twoparts ie the TRMD part and the pounding part -eTRMD part as shown in Figure 3(a) is a hollowmodule withan arced path supporting a rolling ball -is part absorbsstructural mechanical energy by tuning the damper fre-quency to the fundamental structure frequency and throughthe friction generated on the mass-path interface -epounding part is two stroke-limiting plates covered withviscoelastic materials located on both sides of the massequilibrium position A poundingimpact mass damperwithout tunable frequency is shown in Figure 3(b) Once themass stroke exceeds the allowed clearance the mass impactsthe inner side of the plates causing mechanical energy todissipate in the form of heat and noise In brief the PTRMDhas two energy dissipation mechanisms one derived fromthe pounding mechanism and another originated from thetuned mass damper

23 Governing Equation for an SDOF PTRMD Structure-e structure equipped with a PTRMD can be considered asa model with two degrees of freedom for numerical analysisas shown in Figure 3(c) In this context one may useLagrangersquos equation to derive the equations of the motion ofthe controlled system in which the angular motion of therolling mass is considered to be a small quantity -at is

d

dt

zT

z _qi

1113888 1113889minuszT

zqi

+zV

zqi

Qnci i 1 2 (1)

Figure 1 Prefabricated box of hollow floor

2 Shock and Vibration

where T is the kinetic energy of the controlled structure V isits potential energy qi is the ith generalized coordinates _qi isthe generalized velocity of the ith coordinate and Qnc

i is thenonconservative force with respect to qi

First consider the situation when no collision occurs Inthis case the angular displacement of the mass θ satisfiesminusθm le θ le θm where θm is the angular displacement of theoscillator with respect to the center of the arced path when acollision occurs In this case the kinetic and potential energyof the controlled structure can be written as follows

T 12

M _x2

+12

m( _x + ρ _θ cos θ)2

+12

m(ρ _θ sin θ)2

+15

m(ρ _θ)2

(2)

V 12

Kx2

+ mgρ(1minus cos θ) (3)

where x _x and eurox are the displacement velocity and ac-celeration of the main structure respectively M is the massof the main structure K is its stiffness θ _θ and euroθ are theangular displacement angular velocity and angular accel-eration of the oscillator respectively m is the mass of theoscillatorball ρ is the radius difference between the arcedpath and the oscillator and g is the acceleration of gravity

-e nonconservative force Qnci including the external

force and the damping force of the structure can be writtenas follows

Qnc1 f(t)minusC _x

Qnc2 0

(4)

where C is the damping constant of the main structure andf(t) is the external force Assuming the angular motion ofthe oscillator θ is very small [5 6] and combingequations (2)ndash(4) with equation (1) yield the equation ofmotion

(M + m) eurox + mρeuroθ + C _x + Kx f(t) (5a)

euroθ +5eurox

7ρ+5gθ7ρ

0 (5b)

Equations (5a) and (5b) can be rewritten in a morecompact form

M + m mρ57ρ

1⎡⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎦eurox

euroθ

⎧⎨

⎩

⎫⎬

⎭ +C 00 01113890 1113891

_x

_θ1113896 1113897

+

K 0

05g

7ρ

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

x

θ1113896 1113897 f(t)

01113896 1113897

(6)

-e left side of equation (5b) shows that the undampednatural frequency of the oscillator depends on the radiusdifference between the arced path and the oscillator whichcan be written as ωd

5g(7ρ)

1113968 To obtain an optimum

controlled system the natural frequency of the TRMD needsto match the first modal frequency of the main structure

Next consider the situation when a collision occurs Inthis case θ leminusθm or θ ge θm the ball collides with one of thestroke-limiting plates -e potential energy of the controlledstructure can be written as follows

K

CM

TRMD

X

m

θ

(a)

K

C

M

PMD

XM

kbXm

mcbkbcb

(b)

K

CM

PTRMD

X

kb kbcb cbm

θ

(c)

Figure 3 Schematic diagram of the physical model of a (a) TRMD (b) PMD and (c) PTRMD

Stop plate

Viscoelastic materials

Rolling ball

Hollow moduleArch path

Figure 2 Schematic diagram of PTRMD

Shock and Vibration 3

V 12

Kx2

+ mgρ cos θminus cos θm( 1113857 +12

kb ρ θminus θm( 11138571113858 11138592

(7)

where kb is the equivalent contact stiffness for the PTRMDand cb is the equivalent contact damping constant for thePTRMD [17 18] In this case the nonconservative force Qnc

i

can be expressed as follows

Qnc1 f(t)minus c _x

Qnc2 minuscb _θρ2

(8)

Once again as the angular motion of the oscillator θ isvery small combining equation (2) with equations (7) and(8) gives


euroα +5eurox

7ρ+5cb _α7m

+5kbα7m

0 (9b)

where α θ minus θm Equations (9a) and (9b) can be cast in acompact form

M + m mρ57ρ

1⎡⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎦eurox

αeuro1113896 1113897 +

C 0

05cb

7m

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦_x

_α1113896 1113897

+K 0

05kb

7m

⎡⎢⎢⎣ ⎤⎥⎥⎦x

α1113896 1113897

f(t)

01113896 1113897

(10)

24 Governing Equation of an MDOF PTRMD Structure-e equation of motion of aMDOF structure with a PTRMDlocated at the top floor can be written as follows [12]

Meurox(t) + C _x(t) + Kx(t) F(t) + Hfc(t) (11)

where eurox(t) _x(t) and x(t) are the acceleration velocity anddisplacement vectors of the MDOF structure respectivelyEach vector for example x(t) [x1 x2 xn θ]T con-tains n entries for the n-DOF structure and one entry for thetuned mass MC and K are the (n + 1) times (n + 1) massdamping and stiffness matrices respectively F(t) denotesthe vector of the external excitation fc(t) is the interactionforce between the main structure and the ball whichcan be calculated using equation (12) [12 19ndash21]H [0 0 0 11113980radicradicradicradicradic11139791113978radicradicradicradicradic1113981

n times]T denotes the location of fc(t) which can

be written as

fc(t)

minus5g

7ρθ |θ|le θm(no collision)

minus5cb7m

_θminus5kb

7mθ minus θm( 1113857 |θ|gt θm(collision)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(12)

In equation (11) MC and K are defined as

M

Ms + ΓΓTm Γmρ

ΓT57ρ

1⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C Cs 00 0

1113890 1113891

K Ks 00 0

1113890 1113891

(13)

where MsCs and Ks are the n times n mass damping andstiffness matrices of the controlled structure respectivelyΓ [0 0 0 11113980radicradicradicradicradic11139791113978radicradicradicradicradic1113981

nminus1 times]T denotes the location of the damper

3 Control Performance of an SDOFStructure with PTRMD

A single-degree-of-freedom structure with a PTRMD sub-jected to three different loading scenarios free vibrationharmonic excitation and different intensities of earthquakeexcitation is considered-e parameters are listed in Table 1and θm was chosen to be 013 rad [6 13] A PMD shown inFigure 3(b) with the same parameters is employed forcomparison where the stroke length equals to the projectionlength of the arced path in the PTRMD

31 Free Vibration In the case of free vibration an initialdisplacement of 0015m is applied to the structure Figure 4shows the dynamic responses of the structure with PMDwith PTRMD and without control for comparison -efigure shows that the PTRMD mitigates the vibration ef-fectively For the first 87 s the damping rate with thePTRMD is quite rapid whereas after 87 s a beat oscillationcan be observed in the dynamic response of the structureOne can conclude that the PTRMD is no longer functioningafter 87 s indicating that the rolling ball does not impactwith the plates anymore From then on the system works asa TRMD and dissipates mechanical energy by means ofstructural damping and the interface friction Further in-spection illustrates that the dynamic response of a structurewith a TRMD diminishes with time

In the case of a PMD (again see Figure 4) the vibrationreduction is quite effective However it took 5 s for the PMDto reduce the displacement of the main structure to 1mmwhile the PTRMD only needed 38 s -is implies that thevibration reduction performance of a PTRMD is better thana PMD Figure 5 shows a comparison of the velocity of theball rolling in the hollow with the PMD and PTRMD It isnoticeable that the ballrsquos velocity in the PTRMD system isfaster than the one in the PMD system from 55 s to 87 sBesides for a PMD one may conclude that the velocity ofthe ball at the moment it collides with the stroke-limitingplates depends entirely on the linear velocity of the ball at theend of the previous collision By contrast the ball in aPTRMD system located in the curved track can absorb thestructural energy convert it into its own kinetic energy and


finally dissipate the energy out -e preceding argumentindicates that the energy-consuming capacity of a PTRMD ismore efficient especially at the early phase of the response

32HarmonicExcitation For the forced vibrations subject toharmonic excitation consider first the resonant case ie theexcitation frequency equals to the fundamental frequency ofthe structure In this case choose f(t) 100 middot sinωt andω 84 rads Figure 4 shows a comparison between theresponses of the same structure without control controlled bya PMD and controlled by the PTRMD Similar to the situ-ation in Section 31 parameters of the stroke-limiting plate(cb kb) in the PMD equal the ones in the PTRMD Excitationwith nonresonant frequency is also investigated from 0 to15 rads [13]

For the resonant case Figure 6 shows a comparison ofthe responses to harmonic excitation for the three differentscenarios It can be seen that the PTRMD has the bestcontrol performance although the PMD also reduced thestationary amplitude significantly Figure 7 shows the re-sponse amplitudes of the main structure versus excitationfrequencies From Figure 7 one can conclude that thePTRMDs are more effective than PMDs in the resonant caseWhen the frequency of the excitation is lower than 8 rads orhigher than 9 rads both the PTRMD and the PMD areunable to control the vibration effectively

33 Seismic Control Performance Considering the samestructures uncontrolled or controlled by the PMD andPTRMD subjected to El Centro and Kobe excitation with a

peak value of 110 cms2 the EL Centro wave (May 181940) has a north-south acceleration peak of 3417 cms2and the Kobe wave (January 16 1995 Kobe Japan) has anorth-south acceleration peak of 821 cms2 Figures 8(a)and 8(b) show response comparisons between the threedifferent control scenarios under two earthquake exci-tations respectively It is seen that although the effec-tiveness of the PMD and PTRMD systems on peakmitigation is not significant the responses after the peakare well controlled

To further investigate the effectiveness of the PTRMDcontrol efficiency from an energy perspective is examinedFigures 9(a) 9(b) 10(a) and 10(b) show the total amount ofenergy input from the earthquake excitation (input energy)and the amount of energy dissipated by structural dampingand by the PTRMD or PMD For the uncontrolled structure

0 5 10 15 20 25Time (s)

ndash04

ndash03

ndash02

ndash01

0

01

02

03

04

Vel

ocity

of b

all (

ms

)

PMD controlPTRMD control

Collision ends at 87s in PTRMD

Figure 5 Velocity of the rolling ball in PMD and PTRMD whenthe structure subject to initial displacement

Table 1 Parameters of the PTRMD model

M K C m ρ kb cb

840 kg 58700Nm

280Nmiddotsm 168 kg 01002m 3000Nm 50Nmiddotsm

0 5 10 15 20 25Time (s)

ndash0015

ndash001

ndash0005

0

0005

001

0015

Disp

str

uct

(m)

Without controlPMD controlPTRMD control


Figure 4 Free decaying displacement of the uncontrolled structureand the controlled structure with PMD and PTRMD

0 5 10 15 20 25 30 35 40Time (s)

ndash005

ndash004

ndash003

ndash002

ndash001

0

001

002

003

004

005D

isp s

truc

t (m

)


Figure 6 Response of PMD- and PTRMD-controlled structuresubject to harmonic excitation


the sum of the structural strain energy kinetic energy andstructural viscous damped energy equals the seismic inputenergy For the controlled structures the sum of the three isless than the input energy of the earthquake -e differencebetween them is the energy dissipated by the external controldevice From Figures 10(a) and 10(b) one may conclude thatthe reason for the PTRMD failing to reduce the peaks can beattributed to the following at the early stage of the seismicresponse neither the angular displacement nor the angularacceleration of the ball oscillator is large enough for pro-ducing a collision With the increment of the structuralresponse and the angular response of the oscillator effectivecollisions are finally produced therefore achieving thedesigned purpose of energy dissipation

-e control effectiveness of the PTRMD and PMD arecompared in Table 2 -e energy reduction ratio is 530

and 418 for the two dampers under the Kobe excitationrespectively For El Centro excitation case the energy re-duction ratio for the PMD system is 457 and increases to468 for the PTRMD system

4 Seismic Control of an MDOFStructure with PTRMD

Consider a six-floor structure with a lumped-mass model-e lumped masses are assumed to be the same at each floorie 16315 t -e stiffness and damping coefficients of eachstory reduce from the top floor to the ground floor Insteadof assuming a value for the pounding stiffness and thepounding damping they can be obtained from a linearviscoelastic pounding model [22ndash25]

0 5 10 15 20 25 30 35 40Time (s)

ndash005

ndash004

ndash003

ndash002

ndash001

0

001

002

003

004

005

Disp

str

uct

(m)


(a)

0 5 10 15 20 25 30 35 40Time (s)

ndash004

ndash003

ndash002

ndash001

0

001

002

003

004D

isp s

truc

t (m

)

Without damperPMDPTRMD

(b)

Figure 8 Displacement of the controlled structure under (a) Kobe earthquake and (b) El Centro earthquake

2 4 6 8 10 12 14Frequency (rads)

0

0005

001

0015

002

0025

003

0035

004

0045

Disp

lace

men

t (m

)


f(t) = 100sin(ωt)

Figure 7 Comparison of the excitation frequency versus response amplitude curves


ξ minusln e

π2 +(ln e)21113969 (14)

kb m

(1 + λ)T2c

π2 +(ln e)2

1113960 1113961

cb 2m ln e

(1 + λ)Tc

(15)

where ξ is the damping ratio of the pounding boundary λ isthe mass ratio defined as the oscillator mass to the floor masswhere the damper installed and e is the restitutive coefficient

0 5 10 15 20 25 30 35 40Time (s)

0

10

20

30

40

50

60

70

80En

ergy

(J)

StrucdampedStrucdamped + kineticStrucdamped + kinetic + strainInput

(a)

0 5 10 15 20 25 30 35 40Time (s)

0

10

20

30

40

50

60

70

Ener

gy (J

)


(b)

Figure 9 Input and dissipated energy of the controlled structure under Kobe earthquake with a (a) PTRMD and (b) PMD

0 5 10 15 20 25 30 35 40Time (s)

0

10

20

30

40

50

60

70

Ener

gy (J

)


(a)

0 5 10 15 20 25 30 35 40Time (s)

0

10

20

30

40

50

60

Ener

gy (J

)


(b)

Figure 10 Input and dissipated energy of the controlled structure under El Centro earthquake with a (a) PTRMD and (b) PMD

Table 2 Comparison of control performance between PTRMDand PMD under earthquake excitation

Earthquakewave Damper D0 (m) Dctrl (m) EI (J) EP (J) ηE ()

Kobe wave PTRMD 00404 00378 735 390 530PMD 00373 600 251 418

El centrowave

PTRMD 00307 00297 389 182 468PMD 00298 386 176 457

Dctrl and D0 displacement responses of the main structure with andwithout damper ηE EPEI ηE energy reduction ratios EI input en-ergy EP energy dissipation of damper


defined as the ratio of the prepounding velocity to thepostpounding velocity Generally the restitutive coefficientshould be measured to determine kb and cb Alternatively inthis numerical example one may adopt a value of poundingdamping coefficient ξ suggested by other studies and obtainthe restitutive coefficient e by using equation (14) Assumingξ 01 leads to kb 427 kNm2 and cb 808Nmiddotsm [26]Other parameters of the PTRMD-controlled MDOF struc-ture are listed in Table 3 In this case the mass ratio of theoscillator to the first model mass is 05

-e following cases illustrate the control performance ofthe PTRMD in reducing the MDOF structural response

Case 1 free vibration -is case investigates the freedecaying response of the controlled MDOF structurewith initial nonzero displacement -e frequency of thePTRMD is tuned to the first modal frequency to attainan optimal effectCase 2 forced vibration-is case investigates the forcedharmonic response of the controlled MDOF structuresubject to a sinusoidal excitation F 2000 sin(ω1 middot t) atthe top floor and ω1 1HzCase 3 robustness analysis -is case investigates therobustness of the PTRMD with three different detunedfrequencies (083 098 and 113Hz)

41 Free Vibration -e stroke-limiting angle imposed bythe pounding mechanism may have a significant influenceon the control effect Figure 11 shows the displacement rootmean square (RMS) of the top story with an initial dis-placement 0015m One can observe that the RMS decreasesrapidly as increasing θm until θm 010 rad after this theRMS increases rapidly as increasing θm and finally reaches astationary value -e preceding results could be anticipatedsince an excessively small clearance (2θm) between thepounding boundaries may lead to ineffective collisions in-creasing whereas a large clearance may deactivate thepounding mechanism -erefore the optimal poundingangle θm 010 rad is adopted in this case

In Figures 12(a) and 13 the dynamic responses of thestructure without control with a TRMD and with a PTRMDare compared It can be seen that a PTRMD provides distinctadvantages regarding the displacement and accelerationreduction when compared to a TRMD with the same pa-rameters Specifically both the TRMD and PTRMD havelimited control effect at the beginning phase of the responseAfter that the displacement and acceleration of the structurewith the PTRMD are significantly reduced On the contrarybecause oscillator-path friction is assumed to be zero eventhough a TRMD can absorb energy from the structure quitefast it cannot dissipate the absorbed energy through aneffective damping mechanism Again the TRMD absorbedenergy transfers to the structure and finally makes thestructure response exhibiting the so-called beat behaviorAccording to the authorsrsquo previous studies the amplitude ofthe beats may be smaller but still exists if oscillator-pathfriction is considered As for the PTRMD although theresponse mitigation rate is slower than the TRMD no beatbehavior is observed in the response -e rotation angle of

the oscillator shown in Figure 12(b) does not exceed thelimitation of the small quantity assumption which is oftenregarded as 03 rad (or about ca 20deg) -e rotation angle ofthe oscillator shows that it is reasonable to use the smallquantity assumptions for a linearized equation of motion

429e ForcedVibrationCase Using a similar method as inSection 31 one can determine the value of the poundingangle θm to be 021 rad Figures 14 and 15 display the topstory responses of the structure with and without a PTRMDwhen the excitation is a sinusoidal wave Compared to thedisplacement of the TRMD whose peak value at the topfloor decreased from 008m to 006m the displacement forthe PTRMD is much smaller at 002m A similar conclusionis visible in the acceleration response In this situation thePTRMDperformed better because the poundingmechanismprovides an additional control mechanism

Figure 16 shows the control effect of the TRMD andPTRMD in the frequency domain where the horizontal axisdenotes the frequencies of harmonic excitations and thevertical stands for the response amplitude at the stationarystage It is seen that the PTRMD performed better perfor-mance over a wide frequency range Although at a specificfrequency interval for example at 098Hz the displacementof the structure with the TRMD is smaller than it is with thePTRMD and the PTRMD displayed better control for theranges 09 to 096Hz and 099 to 11Hz

43 9e PTRMD with Detuned Frequencies -e robustnessof the PTRMD and TRMD can be investigated by detuningtheir natural frequencies Both free vibration and forcedvibration are considered

Figures 17 and 18 show the free vibration of the structurewith differently tuned TRMDs and PTRMDs -e optimallytuned PTRMD (ie at 096Hz) decreases the RMS of thedisplacement by 442 However detuned PTRMDs exhibitdifferent performance Specifically the response RMS re-duction ratio of the structure with a 110Hz PTRMD is419 and with a 082Hz PTRMD it is only 93 Contrarilythe optimally tuned nonfriction TRMD reduced the RMS ofthe displacement response by 317 Figure 19 further il-lustrates the relationship between the vibration reductionratio and the detuning ratio-e preceding results imply thata PTRMD outperforms a nonfriction or low-friction TRMDfor vibration control

Figures 20 and 21 show the response of the structurewhen subjected to sinusoidal excitation with the same fre-quency as in Section 42 Consider PTRMDs with threefrequencies including optimally tuned at frequency 10Hzand two detuned frequencies at 115Hz (+15) and 085Hz(minus15) For the optimally tuned frequency the peak dis-placement is reduced by 713 whereas for plusmn15 detunedfrequencies it is reduced by 575 and 671 respectively Itcan be concluded that the detuned frequencies have a limitedimpact on the performance of a PTRMDwhen the excitationis a sinusoidal wave It can also be seen from Figure 22 thatthe vibration control performance of the PTRMD de-teriorated less than that of the TRMD when the dampers aredetuned


5 Experimental Verification

51 Experimental Setup Experiments were conducted toinvestigate the effectiveness of the PTRMD and verify thenumerical results An experimental structure illustrated inFigure 23 was built to simulate an SDOF linear oscillator-e test model consisted of two sets of 400 times 100 times 1mmflexible columnsmade of steel strips and a 300 times 100 times 10mmbeammade of aluminum alloy In order to fix the model withthe shaking table a 300 times 100 times 10mm aluminum alloybottom plate was made -e columns were bolted rigidly tothe beam and the base such that the ends were rotationallyfixed-e base was fixed to a unidirectional shake table (Shake

Table 2 Quanser) that was driven by an electrodynamicshaker to produce base excitation Dynamic tests show thatthe frequency of the primary structure isfn 227Hz and thedamping ratio is ξ 013

Figure 24 is a photo of the PTRMDdevice Two steel stopplates were fixed on the curved orbit to act as stroke-limitingstops for the freely moving impact mass -e curved trackwas made of an aluminum alloy plate -e side baffles weremade from acrylic-based resins that allowed the ball to roll inthe direction of the structural vibration -e parameters ofthe PTRMD are listed in Table 4

Experiments were performed to cover two cases First ofall a free-vibration experiment was carried out by applying

Table 3 Model parameters of a PTRMD-controlled MDOF structure

First mode massMs (kg)

First mode stiffnessKs (kNm)

First mode frequencyf (Hz)

Ball massm (kg) Mass ratio (mM6) λ

Radius of ballr (m) Radius of arc track R (m)

79 times 104 312 times 106 1 3953 00242 023 0407

0 01 02 03 04 05 06 07Pounding angle θm (rad)

25

3

35

4

RMS

disp

(m

m)

Figure 11 Relation of the structural displacement RMS and pounding angle

Without controlTRMD controlPTRMD control

10 20 30 40 50 60 70 800Time (s)

ndash0015

ndash001

ndash0005

0

0005

001

0015

Disp

str

uct

(m)

(a)

Ball

Pounding angle

ndash03

ndash02

ndash01

0

01

02

03

Disp

bal

l (ra

d)

10 20 30 40 50 60 70 800Time (s)

(b)

Figure 12 Displacement response of (a) the uncontrolled or TRMD-PTRMD-controlled structure and (b) the oscillator when the structuresubject to an initial displacement


an initial displacement to the structure and then releasing it-en the shaking table tests with different earthquake ex-citations were conducted

52 Experimental Results -e responses of the PTRMD-controlled structure are compared to the responses of theuncontrolled structure to verify the effectiveness of thePTRMD model Figure 25 relates to the free vibrationscenario As can be seen at the beginning of the responsethe displacement mitigation performance of the damper israther limited As time goes on the damper exhibits sig-nificant performance improvement -e preceding obser-vation is reasonable since effective collision cannot beestablished until several cycles of oscillation Specifically thedisplacement amplitude of the controlled structure onlyreached 3mm whereas for the uncontrolled structure itreached 126mm after 10 s Note also that the wave shape of

the controlled response obtained by the experiment isconsistent with the one shown in Figure 12(a) qualitativelydemonstrating the effectiveness of the proposed damper

To quantify the vibration control performance one candefine the response reduction ratio as

βt Dun minusDc

Duntimes 100 (16)

where Dc and Dun are the displacement envelope of the

structure calculated by D(t)

x2 + ( _xωd )21113969

at a certaintime instant with and without the PTRMD where ωd is thedamped natural frequency In this numerical simulationbecause no flexible viscoelastic material is attached on thestroke-limiting plate the contact stiffness kb is set to be 1 times

109 and the constitutive coefficient is 001Table 5 shows the response reduction ratio of experi-

mental test and of the numerical simulation for controlled

Acce

l str

uct

(ms

2 )


10 20 30 40 50 60 70 800Time (s)

ndash06

ndash04

ndash02

0

02

04

06

Figure 13 Acceleration response of the structure under freevibration


10 20 30 40 50 60 70 800Time (s)

ndash01

ndash0075

ndash005

ndash0025

0

0025

005

0075

01

Disp

str

uct

(m)

Figure 14 Displacement response of the structure under sinu-soidal excitation

Acce

l str

uct

(ms

2 )


10 20 30 40 50 60 70 800Time (s)

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4

Figure 15 Acceleration response of the structure under sinusoidalexcitation


05 1 15 2 25 30Frequency (Hz)

0

002

004

006

008

01D

ispla

cem

ent (

m)

Figure 16 Response of the structure subjected to excitation ofvaried frequencies


Without control085Hz PTRMD

100Hz PTRMD115Hz PTRMD

10 20 30 40 50 60 70 800Time (s)

ndash01

ndash0075

ndash005

ndash0025

0

0025

005

0075

01D

isp s

truc

t (m

)

Figure 21 Response of structure controlled by PTRMD withdifferent frequencies for forced vibration

Without control078Hz TRMD

092Hz TRMD106Hz TRMD

10 20 30 40 50 60 70 800Time (s)

ndash0015

ndash001

ndash0005

0

0005

001

0015

Disp

str

uct

(m)

Figure 17 Response of the structure controlled by TRMD withdifferent frequencies for free vibration



10 20 30 40 50 60 70 800Time (s)

ndash0015

ndash001

ndash0005

0

0005

001

0015

Disp

str

uct

(m)

Figure 18 Response of the structure controlled by PTRMD withdifferent frequencies for free vibration

TRMDPTRMD

ndash15 ndash10 ndash5 0 5 10 15 20ndash20Detuning ratio ()

0

5

10

15

20

25

30

35

40

45

Vibr

atio

n re

duct

ion

()

Figure 19 Reduction ratio of the TRMD and PTRMD with dif-ferent frequencies



10 20 30 40 50 60 70 800Time (s)

ndash01

ndash0075

ndash005

ndash0025

0

0025

005

0075

01

Disp

str

uct

(m)

Figure 20 Response of structure controlled by TRMD with dif-ferent frequencies for forced vibration

TRMDPTRMD


0

10

20

30

40

50

60

70

80

Vibr

atio

n re

duct

ion

()

Figure 22 Reduction ratio of TRMD and PTRMD


structure at a different time instant -e experiment datashows that the PTRMD provided a satisfactory responsecontrol effect Besides the results given by the numericalsimulations show reasonable agreement compared to theexperimental data -e difference between the experimentaland numerical results may attribute to the error in modelfabrication excitation generating of the shaking table andparameters (for example kb cb and e) identification

Figures 26(a) and 26(b) show a comparison between theresponse of the primary structure with and without thePTRMD for a Cape Mendocino (April 25 1992 and north-south) earthquake as an input It can be seen that the dis-placement and acceleration of the controlled structure aregreatly mitigated especially after the 5th sec -e controlledresponse of the model structure subject to El Centroearthquake excitation exhibits similar behavior which is notshown here-e results measured from the experimental testfor Cape Mendocino and El Centro waves are listed inTables 6 and 7 demonstrating the vibration control effec-tiveness of the PTRMD

Figure 24 Experiment model of PTRMD

Figure 23 Photograph of the test structure

Table 4 Parameters of PTRMD in test

Radius of arc track R (mm) Radius of ball r (mm) Ball mass m (g) Radius difference ρ (mm) D (mm)50 15 120 345 68where D is the arc clearance between two stops

Disp

str

uct

(mm

)

Without controlWith PTRMD

ndash25

ndash20

ndash15

ndash10

ndash5

0

5

10

15

20

25

5 10 15 20 25 300Time (s)

Figure 25 Displacement time history of dynamic response in freevibration


6 Concluding Remarks

A novel pounding tuned rotary mass damper (PTRMD)exclusively used for the voided biaxial slabs has been in-troduced in this paper -e proposed damper has beendeveloped by introducing the nonlinear pounding mecha-nism to the tuned rotary mass damper (TRMD) proposed bythe authors in previous studies Numerical analysis has beenused to investigate the control performance of the proposedPTRMD in reducing structural response Both single-de-gree-of-freedom (SDOF) and multiple-degree-of-freedom(MDOF) lumped-mass models have been used for thispurpose In the numerical analysis specifically the equationof motion of the controlled structure has been establishedusing Lagrangersquos equation while the pounding mechanismhas been described using a parallel connection of a lineardashpot and a spring (Kelvin model) -e PTRMD controlperformance has been studied quantitatively in different

cases including free vibration and forced vibration withsinusoidal excitation and seismic excitation An experi-mental study has been carried out to validate PTRMDcontrol performance obtained by numerical analysisAccording to the numerical simulation and experimentalstudy one may draw the following conclusions

(1) PTRMD outperforms the pounding mass damper(PMD) in reducing the response of an SDOF systemin the case of free vibration and sinusoidal excitationPTRMD exhibits comparably better performance inmitigating response of an SDOF system subject toearthquake excitations

(2) PTRMD cannot effectively reduce the responseamplitude at the early stage when the structure issubject to earthquake excitation -e control per-formance becomes significant after the poundingmechanism is completely activated

Table 5 Damping effect of the structure with PTRMD at 10 s 20 s and 30 s

D|t10 (mm) β|t10 D|t20 (mm) β|t20 D|t30 (mm) β|t30

Experimental results Uncontrolled 1262 735 856 806 526 871Controlled 334 163 068

Numerical results Uncontrolled 1141 5706 701 6166 490 7023Controlled 490 269 146

Disp

str

uct

(mm

)


ndash15

ndash10

ndash5

0

5

10

15

5 10 15 200Time (s)

(a)

Acce

l str

uct

(ms

2 )Without controlWith PTRMD

ndash4

ndash2

0

2

4

5 10 15 200Time (s)

(b)

Figure 26 Time history of dynamic response under Cape Mendocino excitation (a) Displacement (b) Acceleration

Table 6 Dynamic responses of the main structure under Cape Mendocino excitation

Cases Max displ ampl (mm) Displ RMS (mm) Max acc ampl (ms2) Acc RMS (ms2)Uncontrolled 1352 644 349 125Controlled 1041 295 294 053Redn ratio () 2300 5419 1576 5760

Table 7 Dynamic responses of the main structure under El Centro excitation



(3) PTRMD with optimal pounding angle performsbetter than the TRMD in the case of reducingMDOFsystem response in the cases of free vibration andforced vibration with sinusoidal excitation -ePTRMD has exhibited improved robustness com-pared with the TRMD when the frequency of thedamper is detuned

(4) -e shaking table experiment results agree reason-ably with the numerical ones validating the effec-tiveness of the proposed PTRMD

Data Availability

-e numerical simulation and other data used to support thefindings of this study are available from the correspondingauthor upon request

Conflicts of Interest

-e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

-is work was supported financially by the National NaturalScience Foundation of China (NSFC) (Grant no 51678464)

References

[1] T Lai Structural Behavior of BubbleDeckreg Slabs and 9eirApplication to Lightweight Bridge Decks Massachusetts In-stitute of Technology Cambridge MA USA 2010

[2] A Churakov ldquoBiaxial hollow slab with innovative types ofvoidsrdquo Construction of Unique Buildings ad Structures vol 6no 21 pp 70ndash88 2014

[3] M Schnellenbach-Held and K Pfeffer ldquoPunching behavior ofbiaxial hollow slabsrdquo Cement and Concrete Compositesvol 24 no 6 pp 551ndash556 2002

[4] W B Ali and G S Urgessa ldquoStructural capacities ofspherically voided biaxial slab (SVBS)rdquo in Proceedings of theStructures Congress 2014 pp 785ndash796 Boston MA USAApril 2014

[5] S Li L Fu and F Kong ldquoSeismic response reduction ofstructures equipped with a voided biaxial slab-based tunedrolling mass damperrdquo Shock and Vibration vol 2015 ArticleID 760394 15 pages 2015

[6] S J Li J X Wang L Sun and F Kong ldquoParameter opti-mization of new energy dissipation device based on hollowfloor slabrdquo Journal of Architecture amp Civil Engineering vol 34no 2 pp 10ndash17 2017 in Chinese

[7] C-M Chang S Shia and Y-A Lai ldquoSeismic design of passivetuned mass damper parameters using active control algo-rithmrdquo Journal of Sound and Vibration vol 426 pp 150ndash1652018

[8] G Li L Li and P Zhu ldquoGalloping control for iced conductorsusing tuned mass dampers with fixed time-delayed feedbackrdquoShock and Vibration vol 2019 Article ID 4823457 9 pages2019

[9] P Zhang G Song H-N Li and Y X Lin ldquoSeismic control ofpower transmission tower using pounding TMDrdquo Journal ofEngineering Mechanics vol 139 no 10 pp 1395ndash1406 2013

[10] L Zheng K Li Y Ouyang and J Shan ldquoPerformance-basedoptimal design of tuned impact damper for seismically excited

nonlinear buildingrdquo Engineering Structures vol 160pp 314ndash327 2018

[11] L Li G Song M Singla and Y-L Mo ldquoVibration control of atraffic signal pole using a pounding tuned mass damper withviscoelastic materials (II) experimental verificationrdquo Journalof Vibration and Control vol 21 no 4 pp 670ndash675 2015

[12] H Li P Zhang G Song D Patil and Y Mo ldquoRobustnessstudy of the pounding tuned mass damper for vibrationcontrol of subsea jumpersrdquo Smart Materials and Structuresvol 24 no 9 article 095001 2015

[13] Q Xue J Zhang J He and C Zhang ldquoControl performanceand robustness of pounding tuned mass damper for vibrationreduction in SDOF structurerdquo Shock and Vibration vol 2016Article ID 8021690 15 pages 2016

[14] J Chen and R Yang ldquoVibration control of tuned rolling-balldamper in wind turbinesrdquo Journal of Tongji University vol 41no 8 pp 1145ndash1150 2013 in Chinese

[15] C Bapat and S Sankar ldquoSingle unit impact damper in free andforced vibrationrdquo Journal of Sound and Vibration vol 99no 1 pp 85ndash94 1985

[16] L Zuo and S A Nayfeh ldquoMinimax optimization of multi-degree-of-freedom tuned-mass dampersrdquo Journal of Soundand Vibration vol 272 no 3ndash5 pp 893ndash908 2004

[17] K Li and A P Darby ldquoModelling a buffered impact dampersystem using a spring-damper model of impactrdquo StructuralControl amp Health Monitoring vol 16 no 3 pp 287ndash3022009

[18] Q Xue J Zhang J He C Zhang and G Zou ldquoSeismiccontrol performance for pounding tuned massed damperbased on viscoelastic pounding force analytical methodrdquoJournal of Sound and Vibration vol 411 pp 362ndash377 2017

[19] R Jankowski ldquoPounding force response spectrum underearthquake excitationrdquo Engineering Structures vol 28 no 8pp 1149ndash1161 2006

[20] G F D S Rebouccedilas I F Santos and J J -omsen ldquoVali-dation of vibro-impact force models by numerical simulationperturbation methods and experimentsrdquo Journal of Soundand Vibration vol 413 pp 291ndash307 2018

[21] L Gordillo T-P Sun and X Cheng ldquoDynamics of dropimpact on solid surfaces evolution of impact force and self-similar spreadingrdquo Journal of Fluid Mechanics vol 840pp 190ndash214 2018

[22] WWang XWang X Hua G Song and Z Chen ldquoVibrationcontrol of vortex-induced vibrations of a bridge deck by asingle-side pounding tuned mass damperrdquo EngineeringStructures vol 173 pp 61ndash75 2018

[23] Y Zuo G Sun and H Li ldquoResponse analysis of curved bridgewith unseating failure control system under near-fault groundmotionsrdquo IOP Conference Series Earth amp EnvironmentalScience vol 108 article 022065 2018

[24] E A Mavronicola P C Polycarpou and P KomodromosldquoSpatial seismic modeling of base-isolated buildingspounding against moat walls effects of ground motiondirectionality and mass eccentricityrdquo Earthquake Engi-neering amp Structural Dynamics vol 46 no 7 pp 1161ndash1179 2017

[25] A Guo Y Shen J Bai and H Li ldquoApplication of the en-durance time method to the seismic analysis and evaluation ofhighway bridges considering pounding effectsrdquo EngineeringStructures vol 131 pp 220ndash230 2017

[26] WWang X Hua XWang Z Chen and G Song ldquoOptimumdesign of a novel pounding tuned mass damper under har-monic excitationrdquo Smart Material Structures vol 26 no 5article 055024 2017


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degree of nonlinearity is due to the impact between theoscillator and the buffers In the impact process mechanicalenergy is dissipated as heat and noise

Zhang et al [9] utilized PTMD for response control of atransmission tower -e performance of the PTMD underearthquake conditions was studied and the numerical resultsconfirmed that the PTMD was more effective than a TMD-e influence of parameters such as the mass ratio clearanceseismic intensity and structural damping ratio was analyzedLi et al [11] investigated the performance of a PTMD on atraffic signal pole -e pounding mechanism of the PTMDwas verified experimentally with its performance underconditions of free vibration and resonant excitation exceedingthat of a TMD [12ndash14] -e acceleration response of thecontrol signal under conditions of sine-wave excitation wasreduced by 55 which is an important and significant resultLi et al [12] applied a PTMD to a subsea jumper to study therobustness of the damper when confronted with detuningeffects It was found that [12] the PTMD performed best whenthe excitation frequency was slightly lower than the funda-mental frequency which is the optimal frequency Xue et al[13] took an offshore platform as an example and reached thesame conclusion ie PTMD is more effective and robust thanTMD Previous studies demonstrate that viscoelastic collisionmechanisms significantly improve the damping performanceand robustness of the traditional TMDs However moststudies [14ndash16] investigated impact dampers be they theo-retical or experimental focusing on the performance of SDOFsystems under simple excitation conditions such as sinusoidalloading Most civil structures for instance multistorybuildings experiencing situations such as strong winds orearthquakes cannot reasonably be approximated as an SDOFsystem and complex external loading is likely to induce morethan just the fundamental mode [16]

In this paper a novel pounding tuned rotary massdamper (PTRMD) that combines a buffered poundingmechanism to the design of a TRMD is presented Specif-ically first the equations of motion relating to SDOFMDOF structure controlled by a PTRMD are derived Nextthe control performance of three different loading condi-tions for an SDOF system including free vibration har-monic excitations and seismic excitation is investigatedFurthermore a 6-story structure is used as an illustrativeexample for control performance of an MDOF systemsequipped with PTRMD PTRMDs of varying frequencies areintroduced into this numerical simulation to study the ro-bustness of the system Finally experiments involving freevibration and seismic response of PTRMD-controlled SDOFstructure are conducted to verify the effectiveness of theproposed control device

2 Pounding Tuned Rotary MassDamper System

21 Primary Structure A structure with a voided biaxialreinforced concrete slab is the primary structure to becontrolled -e main elements of this slab are prefabricatedhollow box-like modules (Figure 1) located between the

reinforcement grids of the main beams and the ribbedbeams -e modules are used as the side formwork site-casting the concrete beams In this way only the bottomformworks of the concrete slab are needed thus saving aconsiderable mass of the structure Figure 2 shows a sim-plified diagram of a hollow floor with a pounding tunedrotary mass damper -e damper consists of a single rect-angular hollow box a ball rolling along an arch path andtwo buffering stop plates Compared to the other TMDs theproposed PTRMD system does not alter the architecturalinterior space excessively Previous studies [13] indicatedthat an enhanced control effect could be achieved by in-troducing a flexible buffer zone between a moving dampermass and its boundaries Stop plates covered with visco-elastic materials are therefore installed in the hollow-floorcavity

22 Models of Mass Dampers Figures 3(a)ndash3(c) illustrate aTRMD pounding mass damper (PMD) and PTRMDmodelused for a hollow-ribbed floor -e PTRMD consists of twoparts ie the TRMD part and the pounding part -eTRMD part as shown in Figure 3(a) is a hollowmodule withan arced path supporting a rolling ball -is part absorbsstructural mechanical energy by tuning the damper fre-quency to the fundamental structure frequency and throughthe friction generated on the mass-path interface -epounding part is two stroke-limiting plates covered withviscoelastic materials located on both sides of the massequilibrium position A poundingimpact mass damperwithout tunable frequency is shown in Figure 3(b) Once themass stroke exceeds the allowed clearance the mass impactsthe inner side of the plates causing mechanical energy todissipate in the form of heat and noise In brief the PTRMDhas two energy dissipation mechanisms one derived fromthe pounding mechanism and another originated from thetuned mass damper

23 Governing Equation for an SDOF PTRMD Structure-e structure equipped with a PTRMD can be considered asa model with two degrees of freedom for numerical analysisas shown in Figure 3(c) In this context one may useLagrangersquos equation to derive the equations of the motion ofthe controlled system in which the angular motion of therolling mass is considered to be a small quantity -at is

d

dt

zT

z _qi

1113888 1113889minuszT

zqi

+zV

zqi

Qnci i 1 2 (1)

Figure 1 Prefabricated box of hollow floor





T 12

M _x2

+12


+12

m(ρ _θ sin θ)2

+15

m(ρ _θ)2

(2)

V 12

Kx2





Qnc1 f(t)minusC _x

Qnc2 0

(4)



euroθ +5eurox

7ρ+5gθ7ρ

0 (5b)


M + m mρ57ρ

1⎡⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎦eurox

euroθ

⎧⎨

⎩

⎫⎬

⎭ +C 00 01113890 1113891

_x

_θ1113896 1113897

+

K 0

05g

7ρ

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

x

θ1113896 1113897 f(t)

01113896 1113897

(6)


5g(7ρ)




K

CM

TRMD

X

m

θ

(a)

K

C

M

PMD

XM

kbXm

mcbkbcb

(b)

K

CM

PTRMD

X

kb kbcb cbm

θ

(c)


Stop plate


Rolling ball




V 12

Kx2


kb ρ θminus θm( 11138571113858 11138592

(7)


i


Qnc1 f(t)minus c _x

Qnc2 minuscb _θρ2

(8)



euroα +5eurox

7ρ+5cb _α7m

+5kbα7m

0 (9b)


M + m mρ57ρ

1⎡⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎦eurox

αeuro1113896 1113897 +

C 0

05cb

7m

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦_x

_α1113896 1113897

+K 0

05kb

7m

⎡⎢⎢⎣ ⎤⎥⎥⎦x

α1113896 1113897

f(t)

01113896 1113897

(10)





be written as

fc(t)

minus5g


minus5cb7m

_θminus5kb


⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(12)


M

Ms + ΓΓTm Γmρ

ΓT57ρ

1⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C Cs 00 0

1113890 1113891

K Ks 00 0

1113890 1113891

(13)














0 5 10 15 20 25Time (s)

ndash04

ndash03

ndash02

ndash01

0

01

02

03

04

Vel

ocity

of b

all (

ms

)





M K C m ρ kb cb

840 kg 58700Nm


0 5 10 15 20 25Time (s)

ndash0015

ndash001

ndash0005

0

0005

001

0015

Disp

str

uct

(m)




0 5 10 15 20 25 30 35 40Time (s)

ndash005

ndash004

ndash003

ndash002

ndash001

0

001

002

003

004

005D

isp s

truc

t (m

)









0 5 10 15 20 25 30 35 40Time (s)

ndash005

ndash004

ndash003

ndash002

ndash001

0

001

002

003

004

005

Disp

str

uct

(m)


(a)

0 5 10 15 20 25 30 35 40Time (s)

ndash004

ndash003

ndash002

ndash001

0

001

002

003

004D

isp s

truc

t (m

)


(b)



0

0005

001

0015

002

0025

003

0035

004

0045

Disp

lace

men

t (m

)


f(t) = 100sin(ωt)



ξ minusln e

π2 +(ln e)21113969 (14)

kb m

(1 + λ)T2c

π2 +(ln e)2

1113960 1113961

cb 2m ln e

(1 + λ)Tc

(15)


0 5 10 15 20 25 30 35 40Time (s)

0

10

20

30

40

50

60

70

80En

ergy

(J)


(a)

0 5 10 15 20 25 30 35 40Time (s)

0

10

20

30

40

50

60

70

Ener

gy (J

)


(b)


0 5 10 15 20 25 30 35 40Time (s)

0

10

20

30

40

50

60

70

Ener

gy (J

)


(a)

0 5 10 15 20 25 30 35 40Time (s)

0

10

20

30

40

50

60

Ener

gy (J

)


(b)





El centrowave

PTRMD 00307 00297 389 182 468PMD 00298 386 176 457


























79 times 104 312 times 106 1 3953 00242 023 0407


25

3

35

4

RMS

disp

(m

m)



10 20 30 40 50 60 70 800Time (s)

ndash0015

ndash001

ndash0005

0

0005

001

0015

Disp

str

uct

(m)

(a)

Ball

Pounding angle

ndash03

ndash02

ndash01

0

01

02

03

Disp

bal

l (ra

d)

10 20 30 40 50 60 70 800Time (s)

(b)







βt Dun minusDc

Duntimes 100 (16)



x2 + ( _xωd )21113969




Acce

l str

uct

(ms

2 )


10 20 30 40 50 60 70 800Time (s)

ndash06

ndash04

ndash02

0

02

04

06



10 20 30 40 50 60 70 800Time (s)

ndash01

ndash0075

ndash005

ndash0025

0

0025

005

0075

01

Disp

str

uct

(m)


Acce

l str

uct

(ms

2 )


10 20 30 40 50 60 70 800Time (s)

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4



05 1 15 2 25 30Frequency (Hz)

0

002

004

006

008

01D

ispla

cem

ent (

m)





10 20 30 40 50 60 70 800Time (s)

ndash01

ndash0075

ndash005

ndash0025

0

0025

005

0075

01D

isp s

truc

t (m

)




10 20 30 40 50 60 70 800Time (s)

ndash0015

ndash001

ndash0005

0

0005

001

0015

Disp

str

uct

(m)




10 20 30 40 50 60 70 800Time (s)

ndash0015

ndash001

ndash0005

0

0005

001

0015

Disp

str

uct

(m)


TRMDPTRMD


0

5

10

15

20

25

30

35

40

45

Vibr

atio

n re

duct

ion

()




10 20 30 40 50 60 70 800Time (s)

ndash01

ndash0075

ndash005

ndash0025

0

0025

005

0075

01

Disp

str

uct

(m)


TRMDPTRMD


0

10

20

30

40

50

60

70

80

Vibr

atio

n re

duct

ion

()









Disp

str

uct

(mm

)


ndash25

ndash20

ndash15

ndash10

ndash5

0

5

10

15

20

25

5 10 15 20 25 300Time (s)












Disp

str

uct

(mm

)


ndash15

ndash10

ndash5

0

5

10

15

5 10 15 200Time (s)

(a)

Acce

l str

uct

(ms


ndash4

ndash2

0

2

4

5 10 15 200Time (s)

(b)









Data Availability




Acknowledgments


References































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





T 12

M _x2

+12


+12

m(ρ _θ sin θ)2

+15

m(ρ _θ)2

(2)

V 12

Kx2





Qnc1 f(t)minusC _x

Qnc2 0

(4)



euroθ +5eurox

7ρ+5gθ7ρ

0 (5b)


M + m mρ57ρ

1⎡⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎦eurox

euroθ

⎧⎨

⎩

⎫⎬

⎭ +C 00 01113890 1113891

_x

_θ1113896 1113897

+

K 0

05g

7ρ

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

x

θ1113896 1113897 f(t)

01113896 1113897

(6)


5g(7ρ)




K

CM

TRMD

X

m

θ

(a)

K

C

M

PMD

XM

kbXm

mcbkbcb

(b)

K

CM

PTRMD

X

kb kbcb cbm

θ

(c)


Stop plate


Rolling ball




V 12

Kx2


kb ρ θminus θm( 11138571113858 11138592

(7)


i


Qnc1 f(t)minus c _x

Qnc2 minuscb _θρ2

(8)



euroα +5eurox

7ρ+5cb _α7m

+5kbα7m

0 (9b)


M + m mρ57ρ

1⎡⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎦eurox

αeuro1113896 1113897 +

C 0

05cb

7m

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦_x

_α1113896 1113897

+K 0

05kb

7m

⎡⎢⎢⎣ ⎤⎥⎥⎦x

α1113896 1113897

f(t)

01113896 1113897

(10)





be written as

fc(t)

minus5g


minus5cb7m

_θminus5kb


⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(12)


M

Ms + ΓΓTm Γmρ

ΓT57ρ

1⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C Cs 00 0

1113890 1113891

K Ks 00 0

1113890 1113891

(13)














0 5 10 15 20 25Time (s)

ndash04

ndash03

ndash02

ndash01

0

01

02

03

04

Vel

ocity

of b

all (

ms

)





M K C m ρ kb cb

840 kg 58700Nm


0 5 10 15 20 25Time (s)

ndash0015

ndash001

ndash0005

0

0005

001

0015

Disp

str

uct

(m)




0 5 10 15 20 25 30 35 40Time (s)

ndash005

ndash004

ndash003

ndash002

ndash001

0

001

002

003

004

005D

isp s

truc

t (m

)









0 5 10 15 20 25 30 35 40Time (s)

ndash005

ndash004

ndash003

ndash002

ndash001

0

001

002

003

004

005

Disp

str

uct

(m)


(a)

0 5 10 15 20 25 30 35 40Time (s)

ndash004

ndash003

ndash002

ndash001

0

001

002

003

004D

isp s

truc

t (m

)


(b)



0

0005

001

0015

002

0025

003

0035

004

0045

Disp

lace

men

t (m

)


f(t) = 100sin(ωt)



ξ minusln e

π2 +(ln e)21113969 (14)

kb m

(1 + λ)T2c

π2 +(ln e)2

1113960 1113961

cb 2m ln e

(1 + λ)Tc

(15)


0 5 10 15 20 25 30 35 40Time (s)

0

10

20

30

40

50

60

70

80En

ergy

(J)


(a)

0 5 10 15 20 25 30 35 40Time (s)

0

10

20

30

40

50

60

70

Ener

gy (J

)


(b)


0 5 10 15 20 25 30 35 40Time (s)

0

10

20

30

40

50

60

70

Ener

gy (J

)


(a)

0 5 10 15 20 25 30 35 40Time (s)

0

10

20

30

40

50

60

Ener

gy (J

)


(b)





El centrowave

PTRMD 00307 00297 389 182 468PMD 00298 386 176 457


























79 times 104 312 times 106 1 3953 00242 023 0407


25

3

35

4

RMS

disp

(m

m)



10 20 30 40 50 60 70 800Time (s)

ndash0015

ndash001

ndash0005

0

0005

001

0015

Disp

str

uct

(m)

(a)

Ball

Pounding angle

ndash03

ndash02

ndash01

0

01

02

03

Disp

bal

l (ra

d)

10 20 30 40 50 60 70 800Time (s)

(b)







βt Dun minusDc

Duntimes 100 (16)



x2 + ( _xωd )21113969




Acce

l str

uct

(ms

2 )


10 20 30 40 50 60 70 800Time (s)

ndash06

ndash04

ndash02

0

02

04

06



10 20 30 40 50 60 70 800Time (s)

ndash01

ndash0075

ndash005

ndash0025

0

0025

005

0075

01

Disp

str

uct

(m)


Acce

l str

uct

(ms

2 )


10 20 30 40 50 60 70 800Time (s)

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4



05 1 15 2 25 30Frequency (Hz)

0

002

004

006

008

01D

ispla

cem

ent (

m)





10 20 30 40 50 60 70 800Time (s)

ndash01

ndash0075

ndash005

ndash0025

0

0025

005

0075

01D

isp s

truc

t (m

)




10 20 30 40 50 60 70 800Time (s)

ndash0015

ndash001

ndash0005

0

0005

001

0015

Disp

str

uct

(m)




10 20 30 40 50 60 70 800Time (s)

ndash0015

ndash001

ndash0005

0

0005

001

0015

Disp

str

uct

(m)


TRMDPTRMD


0

5

10

15

20

25

30

35

40

45

Vibr

atio

n re

duct

ion

()




10 20 30 40 50 60 70 800Time (s)

ndash01

ndash0075

ndash005

ndash0025

0

0025

005

0075

01

Disp

str

uct

(m)


TRMDPTRMD


0

10

20

30

40

50

60

70

80

Vibr

atio

n re

duct

ion

()









Disp

str

uct

(mm

)


ndash25

ndash20

ndash15

ndash10

ndash5

0

5

10

15

20

25

5 10 15 20 25 300Time (s)












Disp

str

uct

(mm

)


ndash15

ndash10

ndash5

0

5

10

15

5 10 15 200Time (s)

(a)

Acce

l str

uct

(ms


ndash4

ndash2

0

2

4

5 10 15 200Time (s)

(b)









Data Availability




Acknowledgments


References































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


V 12

Kx2


kb ρ θminus θm( 11138571113858 11138592

(7)


i


Qnc1 f(t)minus c _x

Qnc2 minuscb _θρ2

(8)



euroα +5eurox

7ρ+5cb _α7m

+5kbα7m

0 (9b)


M + m mρ57ρ

1⎡⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎦eurox

αeuro1113896 1113897 +

C 0

05cb

7m

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦_x

_α1113896 1113897

+K 0

05kb

7m

⎡⎢⎢⎣ ⎤⎥⎥⎦x

α1113896 1113897

f(t)

01113896 1113897

(10)





be written as

fc(t)

minus5g


minus5cb7m

_θminus5kb


⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(12)


M

Ms + ΓΓTm Γmρ

ΓT57ρ

1⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C Cs 00 0

1113890 1113891

K Ks 00 0

1113890 1113891

(13)














0 5 10 15 20 25Time (s)

ndash04

ndash03

ndash02

ndash01

0

01

02

03

04

Vel

ocity

of b

all (

ms

)





M K C m ρ kb cb

840 kg 58700Nm


0 5 10 15 20 25Time (s)

ndash0015

ndash001

ndash0005

0

0005

001

0015

Disp

str

uct

(m)




0 5 10 15 20 25 30 35 40Time (s)

ndash005

ndash004

ndash003

ndash002

ndash001

0

001

002

003

004

005D

isp s

truc

t (m

)









0 5 10 15 20 25 30 35 40Time (s)

ndash005

ndash004

ndash003

ndash002

ndash001

0

001

002

003

004

005

Disp

str

uct

(m)


(a)

0 5 10 15 20 25 30 35 40Time (s)

ndash004

ndash003

ndash002

ndash001

0

001

002

003

004D

isp s

truc

t (m

)


(b)



0

0005

001

0015

002

0025

003

0035

004

0045

Disp

lace

men

t (m

)


f(t) = 100sin(ωt)



ξ minusln e

π2 +(ln e)21113969 (14)

kb m

(1 + λ)T2c

π2 +(ln e)2

1113960 1113961

cb 2m ln e

(1 + λ)Tc

(15)


0 5 10 15 20 25 30 35 40Time (s)

0

10

20

30

40

50

60

70

80En

ergy

(J)


(a)

0 5 10 15 20 25 30 35 40Time (s)

0

10

20

30

40

50

60

70

Ener

gy (J

)


(b)


0 5 10 15 20 25 30 35 40Time (s)

0

10

20

30

40

50

60

70

Ener

gy (J

)


(a)

0 5 10 15 20 25 30 35 40Time (s)

0

10

20

30

40

50

60

Ener

gy (J

)


(b)





El centrowave

PTRMD 00307 00297 389 182 468PMD 00298 386 176 457


























79 times 104 312 times 106 1 3953 00242 023 0407


25

3

35

4

RMS

disp

(m

m)



10 20 30 40 50 60 70 800Time (s)

ndash0015

ndash001

ndash0005

0

0005

001

0015

Disp

str

uct

(m)

(a)

Ball

Pounding angle

ndash03

ndash02

ndash01

0

01

02

03

Disp

bal

l (ra

d)

10 20 30 40 50 60 70 800Time (s)

(b)







βt Dun minusDc

Duntimes 100 (16)



x2 + ( _xωd )21113969




Acce

l str

uct

(ms

2 )


10 20 30 40 50 60 70 800Time (s)

ndash06

ndash04

ndash02

0

02

04

06



10 20 30 40 50 60 70 800Time (s)

ndash01

ndash0075

ndash005

ndash0025

0

0025

005

0075

01

Disp

str

uct

(m)


Acce

l str

uct

(ms

2 )


10 20 30 40 50 60 70 800Time (s)

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4



05 1 15 2 25 30Frequency (Hz)

0

002

004

006

008

01D

ispla

cem

ent (

m)





10 20 30 40 50 60 70 800Time (s)

ndash01

ndash0075

ndash005

ndash0025

0

0025

005

0075

01D

isp s

truc

t (m

)




10 20 30 40 50 60 70 800Time (s)

ndash0015

ndash001

ndash0005

0

0005

001

0015

Disp

str

uct

(m)




10 20 30 40 50 60 70 800Time (s)

ndash0015

ndash001

ndash0005

0

0005

001

0015

Disp

str

uct

(m)


TRMDPTRMD


0

5

10

15

20

25

30

35

40

45

Vibr

atio

n re

duct

ion

()




10 20 30 40 50 60 70 800Time (s)

ndash01

ndash0075

ndash005

ndash0025

0

0025

005

0075

01

Disp

str

uct

(m)


TRMDPTRMD


0

10

20

30

40

50

60

70

80

Vibr

atio

n re

duct

ion

()









Disp

str

uct

(mm

)


ndash25

ndash20

ndash15

ndash10

ndash5

0

5

10

15

20

25

5 10 15 20 25 300Time (s)












Disp

str

uct

(mm

)


ndash15

ndash10

ndash5

0

5

10

15

5 10 15 200Time (s)

(a)

Acce

l str

uct

(ms


ndash4

ndash2

0

2

4

5 10 15 200Time (s)

(b)









Data Availability




Acknowledgments


References































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia








0 5 10 15 20 25Time (s)

ndash04

ndash03

ndash02

ndash01

0

01

02

03

04

Vel

ocity

of b

all (

ms

)





M K C m ρ kb cb

840 kg 58700Nm


0 5 10 15 20 25Time (s)

ndash0015

ndash001

ndash0005

0

0005

001

0015

Disp

str

uct

(m)




0 5 10 15 20 25 30 35 40Time (s)

ndash005

ndash004

ndash003

ndash002

ndash001

0

001

002

003

004

005D

isp s

truc

t (m

)









0 5 10 15 20 25 30 35 40Time (s)

ndash005

ndash004

ndash003

ndash002

ndash001

0

001

002

003

004

005

Disp

str

uct

(m)


(a)

0 5 10 15 20 25 30 35 40Time (s)

ndash004

ndash003

ndash002

ndash001

0

001

002

003

004D

isp s

truc

t (m

)


(b)



0

0005

001

0015

002

0025

003

0035

004

0045

Disp

lace

men

t (m

)


f(t) = 100sin(ωt)



ξ minusln e

π2 +(ln e)21113969 (14)

kb m

(1 + λ)T2c

π2 +(ln e)2

1113960 1113961

cb 2m ln e

(1 + λ)Tc

(15)


0 5 10 15 20 25 30 35 40Time (s)

0

10

20

30

40

50

60

70

80En

ergy

(J)


(a)

0 5 10 15 20 25 30 35 40Time (s)

0

10

20

30

40

50

60

70

Ener

gy (J

)


(b)


0 5 10 15 20 25 30 35 40Time (s)

0

10

20

30

40

50

60

70

Ener

gy (J

)


(a)

0 5 10 15 20 25 30 35 40Time (s)

0

10

20

30

40

50

60

Ener

gy (J

)


(b)





El centrowave

PTRMD 00307 00297 389 182 468PMD 00298 386 176 457


























79 times 104 312 times 106 1 3953 00242 023 0407


25

3

35

4

RMS

disp

(m

m)



10 20 30 40 50 60 70 800Time (s)

ndash0015

ndash001

ndash0005

0

0005

001

0015

Disp

str

uct

(m)

(a)

Ball

Pounding angle

ndash03

ndash02

ndash01

0

01

02

03

Disp

bal

l (ra

d)

10 20 30 40 50 60 70 800Time (s)

(b)







βt Dun minusDc

Duntimes 100 (16)



x2 + ( _xωd )21113969




Acce

l str

uct

(ms

2 )


10 20 30 40 50 60 70 800Time (s)

ndash06

ndash04

ndash02

0

02

04

06



10 20 30 40 50 60 70 800Time (s)

ndash01

ndash0075

ndash005

ndash0025

0

0025

005

0075

01

Disp

str

uct

(m)


Acce

l str

uct

(ms

2 )


10 20 30 40 50 60 70 800Time (s)

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4



05 1 15 2 25 30Frequency (Hz)

0

002

004

006

008

01D

ispla

cem

ent (

m)





10 20 30 40 50 60 70 800Time (s)

ndash01

ndash0075

ndash005

ndash0025

0

0025

005

0075

01D

isp s

truc

t (m

)




10 20 30 40 50 60 70 800Time (s)

ndash0015

ndash001

ndash0005

0

0005

001

0015

Disp

str

uct

(m)




10 20 30 40 50 60 70 800Time (s)

ndash0015

ndash001

ndash0005

0

0005

001

0015

Disp

str

uct

(m)


TRMDPTRMD


0

5

10

15

20

25

30

35

40

45

Vibr

atio

n re

duct

ion

()




10 20 30 40 50 60 70 800Time (s)

ndash01

ndash0075

ndash005

ndash0025

0

0025

005

0075

01

Disp

str

uct

(m)


TRMDPTRMD


0

10

20

30

40

50

60

70

80

Vibr

atio

n re

duct

ion

()









Disp

str

uct

(mm

)


ndash25

ndash20

ndash15

ndash10

ndash5

0

5

10

15

20

25

5 10 15 20 25 300Time (s)












Disp

str

uct

(mm

)


ndash15

ndash10

ndash5

0

5

10

15

5 10 15 200Time (s)

(a)

Acce

l str

uct

(ms


ndash4

ndash2

0

2

4

5 10 15 200Time (s)

(b)









Data Availability




Acknowledgments


References































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia







0 5 10 15 20 25 30 35 40Time (s)

ndash005

ndash004

ndash003

ndash002

ndash001

0

001

002

003

004

005

Disp

str

uct

(m)


(a)

0 5 10 15 20 25 30 35 40Time (s)

ndash004

ndash003

ndash002

ndash001

0

001

002

003

004D

isp s

truc

t (m

)


(b)



0

0005

001

0015

002

0025

003

0035

004

0045

Disp

lace

men

t (m

)


f(t) = 100sin(ωt)



ξ minusln e

π2 +(ln e)21113969 (14)

kb m

(1 + λ)T2c

π2 +(ln e)2

1113960 1113961

cb 2m ln e

(1 + λ)Tc

(15)


0 5 10 15 20 25 30 35 40Time (s)

0

10

20

30

40

50

60

70

80En

ergy

(J)


(a)

0 5 10 15 20 25 30 35 40Time (s)

0

10

20

30

40

50

60

70

Ener

gy (J

)


(b)


0 5 10 15 20 25 30 35 40Time (s)

0

10

20

30

40

50

60

70

Ener

gy (J

)


(a)

0 5 10 15 20 25 30 35 40Time (s)

0

10

20

30

40

50

60

Ener

gy (J

)


(b)





El centrowave

PTRMD 00307 00297 389 182 468PMD 00298 386 176 457


























79 times 104 312 times 106 1 3953 00242 023 0407


25

3

35

4

RMS

disp

(m

m)



10 20 30 40 50 60 70 800Time (s)

ndash0015

ndash001

ndash0005

0

0005

001

0015

Disp

str

uct

(m)

(a)

Ball

Pounding angle

ndash03

ndash02

ndash01

0

01

02

03

Disp

bal

l (ra

d)

10 20 30 40 50 60 70 800Time (s)

(b)







βt Dun minusDc

Duntimes 100 (16)



x2 + ( _xωd )21113969




Acce

l str

uct

(ms

2 )


10 20 30 40 50 60 70 800Time (s)

ndash06

ndash04

ndash02

0

02

04

06



10 20 30 40 50 60 70 800Time (s)

ndash01

ndash0075

ndash005

ndash0025

0

0025

005

0075

01

Disp

str

uct

(m)


Acce

l str

uct

(ms

2 )


10 20 30 40 50 60 70 800Time (s)

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4



05 1 15 2 25 30Frequency (Hz)

0

002

004

006

008

01D

ispla

cem

ent (

m)





10 20 30 40 50 60 70 800Time (s)

ndash01

ndash0075

ndash005

ndash0025

0

0025

005

0075

01D

isp s

truc

t (m

)




10 20 30 40 50 60 70 800Time (s)

ndash0015

ndash001

ndash0005

0

0005

001

0015

Disp

str

uct

(m)




10 20 30 40 50 60 70 800Time (s)

ndash0015

ndash001

ndash0005

0

0005

001

0015

Disp

str

uct

(m)


TRMDPTRMD


0

5

10

15

20

25

30

35

40

45

Vibr

atio

n re

duct

ion

()




10 20 30 40 50 60 70 800Time (s)

ndash01

ndash0075

ndash005

ndash0025

0

0025

005

0075

01

Disp

str

uct

(m)


TRMDPTRMD


0

10

20

30

40

50

60

70

80

Vibr

atio

n re

duct

ion

()









Disp

str

uct

(mm

)


ndash25

ndash20

ndash15

ndash10

ndash5

0

5

10

15

20

25

5 10 15 20 25 300Time (s)












Disp

str

uct

(mm

)


ndash15

ndash10

ndash5

0

5

10

15

5 10 15 200Time (s)

(a)

Acce

l str

uct

(ms


ndash4

ndash2

0

2

4

5 10 15 200Time (s)

(b)









Data Availability




Acknowledgments


References































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


ξ minusln e

π2 +(ln e)21113969 (14)

kb m

(1 + λ)T2c

π2 +(ln e)2

1113960 1113961

cb 2m ln e

(1 + λ)Tc

(15)


0 5 10 15 20 25 30 35 40Time (s)

0

10

20

30

40

50

60

70

80En

ergy

(J)


(a)

0 5 10 15 20 25 30 35 40Time (s)

0

10

20

30

40

50

60

70

Ener

gy (J

)


(b)


0 5 10 15 20 25 30 35 40Time (s)

0

10

20

30

40

50

60

70

Ener

gy (J

)


(a)

0 5 10 15 20 25 30 35 40Time (s)

0

10

20

30

40

50

60

Ener

gy (J

)


(b)





El centrowave

PTRMD 00307 00297 389 182 468PMD 00298 386 176 457


























79 times 104 312 times 106 1 3953 00242 023 0407


25

3

35

4

RMS

disp

(m

m)



10 20 30 40 50 60 70 800Time (s)

ndash0015

ndash001

ndash0005

0

0005

001

0015

Disp

str

uct

(m)

(a)

Ball

Pounding angle

ndash03

ndash02

ndash01

0

01

02

03

Disp

bal

l (ra

d)

10 20 30 40 50 60 70 800Time (s)

(b)







βt Dun minusDc

Duntimes 100 (16)



x2 + ( _xωd )21113969




Acce

l str

uct

(ms

2 )


10 20 30 40 50 60 70 800Time (s)

ndash06

ndash04

ndash02

0

02

04

06



10 20 30 40 50 60 70 800Time (s)

ndash01

ndash0075

ndash005

ndash0025

0

0025

005

0075

01

Disp

str

uct

(m)


Acce

l str

uct

(ms

2 )


10 20 30 40 50 60 70 800Time (s)

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4



05 1 15 2 25 30Frequency (Hz)

0

002

004

006

008

01D

ispla

cem

ent (

m)





10 20 30 40 50 60 70 800Time (s)

ndash01

ndash0075

ndash005

ndash0025

0

0025

005

0075

01D

isp s

truc

t (m

)




10 20 30 40 50 60 70 800Time (s)

ndash0015

ndash001

ndash0005

0

0005

001

0015

Disp

str

uct

(m)




10 20 30 40 50 60 70 800Time (s)

ndash0015

ndash001

ndash0005

0

0005

001

0015

Disp

str

uct

(m)


TRMDPTRMD


0

5

10

15

20

25

30

35

40

45

Vibr

atio

n re

duct

ion

()




10 20 30 40 50 60 70 800Time (s)

ndash01

ndash0075

ndash005

ndash0025

0

0025

005

0075

01

Disp

str

uct

(m)


TRMDPTRMD


0

10

20

30

40

50

60

70

80

Vibr

atio

n re

duct

ion

()









Disp

str

uct

(mm

)


ndash25

ndash20

ndash15

ndash10

ndash5

0

5

10

15

20

25

5 10 15 20 25 300Time (s)












Disp

str

uct

(mm

)


ndash15

ndash10

ndash5

0

5

10

15

5 10 15 200Time (s)

(a)

Acce

l str

uct

(ms


ndash4

ndash2

0

2

4

5 10 15 200Time (s)

(b)









Data Availability




Acknowledgments


References































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia

























79 times 104 312 times 106 1 3953 00242 023 0407


25

3

35

4

RMS

disp

(m

m)



10 20 30 40 50 60 70 800Time (s)

ndash0015

ndash001

ndash0005

0

0005

001

0015

Disp

str

uct

(m)

(a)

Ball

Pounding angle

ndash03

ndash02

ndash01

0

01

02

03

Disp

bal

l (ra

d)

10 20 30 40 50 60 70 800Time (s)

(b)







βt Dun minusDc

Duntimes 100 (16)



x2 + ( _xωd )21113969




Acce

l str

uct

(ms

2 )


10 20 30 40 50 60 70 800Time (s)

ndash06

ndash04

ndash02

0

02

04

06



10 20 30 40 50 60 70 800Time (s)

ndash01

ndash0075

ndash005

ndash0025

0

0025

005

0075

01

Disp

str

uct

(m)


Acce

l str

uct

(ms

2 )


10 20 30 40 50 60 70 800Time (s)

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4



05 1 15 2 25 30Frequency (Hz)

0

002

004

006

008

01D

ispla

cem

ent (

m)





10 20 30 40 50 60 70 800Time (s)

ndash01

ndash0075

ndash005

ndash0025

0

0025

005

0075

01D

isp s

truc

t (m

)




10 20 30 40 50 60 70 800Time (s)

ndash0015

ndash001

ndash0005

0

0005

001

0015

Disp

str

uct

(m)




10 20 30 40 50 60 70 800Time (s)

ndash0015

ndash001

ndash0005

0

0005

001

0015

Disp

str

uct

(m)


TRMDPTRMD


0

5

10

15

20

25

30

35

40

45

Vibr

atio

n re

duct

ion

()




10 20 30 40 50 60 70 800Time (s)

ndash01

ndash0075

ndash005

ndash0025

0

0025

005

0075

01

Disp

str

uct

(m)


TRMDPTRMD


0

10

20

30

40

50

60

70

80

Vibr

atio

n re

duct

ion

()









Disp

str

uct

(mm

)


ndash25

ndash20

ndash15

ndash10

ndash5

0

5

10

15

20

25

5 10 15 20 25 300Time (s)












Disp

str

uct

(mm

)


ndash15

ndash10

ndash5

0

5

10

15

5 10 15 200Time (s)

(a)

Acce

l str

uct

(ms


ndash4

ndash2

0

2

4

5 10 15 200Time (s)

(b)









Data Availability




Acknowledgments


References































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia






βt Dun minusDc

Duntimes 100 (16)



x2 + ( _xωd )21113969




Acce

l str

uct

(ms

2 )


10 20 30 40 50 60 70 800Time (s)

ndash06

ndash04

ndash02

0

02

04

06



10 20 30 40 50 60 70 800Time (s)

ndash01

ndash0075

ndash005

ndash0025

0

0025

005

0075

01

Disp

str

uct

(m)


Acce

l str

uct

(ms

2 )


10 20 30 40 50 60 70 800Time (s)

ndash4

ndash3

ndash2

ndash1

0

1

2

3

4



05 1 15 2 25 30Frequency (Hz)

0

002

004

006

008

01D

ispla

cem

ent (

m)





10 20 30 40 50 60 70 800Time (s)

ndash01

ndash0075

ndash005

ndash0025

0

0025

005

0075

01D

isp s

truc

t (m

)




10 20 30 40 50 60 70 800Time (s)

ndash0015

ndash001

ndash0005

0

0005

001

0015

Disp

str

uct

(m)




10 20 30 40 50 60 70 800Time (s)

ndash0015

ndash001

ndash0005

0

0005

001

0015

Disp

str

uct

(m)


TRMDPTRMD


0

5

10

15

20

25

30

35

40

45

Vibr

atio

n re

duct

ion

()




10 20 30 40 50 60 70 800Time (s)

ndash01

ndash0075

ndash005

ndash0025

0

0025

005

0075

01

Disp

str

uct

(m)


TRMDPTRMD


0

10

20

30

40

50

60

70

80

Vibr

atio

n re

duct

ion

()









Disp

str

uct

(mm

)


ndash25

ndash20

ndash15

ndash10

ndash5

0

5

10

15

20

25

5 10 15 20 25 300Time (s)












Disp

str

uct

(mm

)


ndash15

ndash10

ndash5

0

5

10

15

5 10 15 200Time (s)

(a)

Acce

l str

uct

(ms


ndash4

ndash2

0

2

4

5 10 15 200Time (s)

(b)









Data Availability




Acknowledgments


References































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




10 20 30 40 50 60 70 800Time (s)

ndash01

ndash0075

ndash005

ndash0025

0

0025

005

0075

01D

isp s

truc

t (m

)




10 20 30 40 50 60 70 800Time (s)

ndash0015

ndash001

ndash0005

0

0005

001

0015

Disp

str

uct

(m)




10 20 30 40 50 60 70 800Time (s)

ndash0015

ndash001

ndash0005

0

0005

001

0015

Disp

str

uct

(m)


TRMDPTRMD


0

5

10

15

20

25

30

35

40

45

Vibr

atio

n re

duct

ion

()




10 20 30 40 50 60 70 800Time (s)

ndash01

ndash0075

ndash005

ndash0025

0

0025

005

0075

01

Disp

str

uct

(m)


TRMDPTRMD


0

10

20

30

40

50

60

70

80

Vibr

atio

n re

duct

ion

()









Disp

str

uct

(mm

)


ndash25

ndash20

ndash15

ndash10

ndash5

0

5

10

15

20

25

5 10 15 20 25 300Time (s)












Disp

str

uct

(mm

)


ndash15

ndash10

ndash5

0

5

10

15

5 10 15 200Time (s)

(a)

Acce

l str

uct

(ms


ndash4

ndash2

0

2

4

5 10 15 200Time (s)

(b)









Data Availability




Acknowledgments


References































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia








Disp

str

uct

(mm

)


ndash25

ndash20

ndash15

ndash10

ndash5

0

5

10

15

20

25

5 10 15 20 25 300Time (s)












Disp

str

uct

(mm

)


ndash15

ndash10

ndash5

0

5

10

15

5 10 15 200Time (s)

(a)

Acce

l str

uct

(ms


ndash4

ndash2

0

2

4

5 10 15 200Time (s)

(b)









Data Availability




Acknowledgments


References































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia











Disp

str

uct

(mm

)


ndash15

ndash10

ndash5

0

5

10

15

5 10 15 200Time (s)

(a)

Acce

l str

uct

(ms


ndash4

ndash2

0

2

4

5 10 15 200Time (s)

(b)









Data Availability




Acknowledgments


References































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




Data Availability




Acknowledgments


References































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


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