damping effects induced by a mass moving along a pendulum

Research ArticleDamping Effects Induced by a Mass Moving along a Pendulum

E Gandino S Marchesiello A Bellino A Fasana and L Garibaldi

Department of Mechanical and Aerospace Engineering Politecnico di Torino Cso Duca degli Abruzzi 24 10129 Torino Italy

Correspondence should be addressed to S Marchesiello stefanomarchesiellopolitoit

Received 9 July 2013 Accepted 19 December 2013 Published 1 June 2014

Academic Editor Nuno Maia

Copyright copy 2014 E Gandino et al This 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

The experimental study of damping in a time-varying inertia pendulum is presentedThe system consists of a disk travelling along anoscillating pendulum large swinging angles are reached so that its equation of motion is not only time-varying but also nonlinearSignals are acquired from a rotary sensor but some remarks are also proposed as regards signals measured by piezoelectric orcapacitive accelerometers Time-varying inertia due to the relative motion of the mass is associated with the Coriolis-type effectsappearing in the systemwhich can reduce and also amplify the oscillationsThe analyticalmodel of the pendulum is introduced andan equivalent damping ratio is estimated by applying energy considerations An accurate model is obtained by updating the viscousdamping coefficient in accordance with the experimental data The system is analysed through the application of a subspace-basedtechnique devoted to the identification of linear time-varying systems the so-called short-time stochastic subspace identification(ST-SSI) This is a very simple method recently adopted for estimating the instantaneous frequencies of a system In this paper theST-SSI method is demonstrated to be capable of accurately estimating damping ratios even in the challenging cases when dampingmay turn to negative due to the Coriolis-type effects thus causing amplifications of the system response

1 Introduction

The analysis and simulation [1] of mechanical systems withimposed relative motion of components are challengingtime-varying inertia created by a mass sliding along arotating member is associated with Coriolis-type effectsThe relative movement can excite or reduce the structurevibration providing new means or techniques for activeamplification or attenuation of vibrations A variable lengthmathematical pendulum was used in [2] to examine theconcept of controlling the motion of a system through massreconfiguration that is by sliding a mass towards and awayfrom the pivot A variable length pendulum has also beenconsidered in [3] where a rigorous qualitative investigationof its equation is carried out without any assumption onsmall swinging amplitudes In [4] a physical pendulum wasconsidered to present a technique in which a radially movingmass is treated as a controller to attenuate the pendulumswings A moving mass is a proper characteristic of a time-varying system which is in general one of the sources ofnonstationary signals Another source can be in case of apendulum nonlinearity due to its large swinging amplitudesThe oscillations are also associatedwith the effective damping

ratio which is explicitly determined in [4] from energyconsiderations in terms of the mass motion pattern andthe pendulum parameters This is only one of the severaltechniques that can be adopted to estimate the damping ratiosof a system and in fact it can be seen as part of the largerproblem of dynamic identification as it is actually proposedin this paper

During the last years many efforts have been spent instudying nonstationary signals Among the first works onthe identification of time-varying systems [5 6] introducedthe concept of pseudonatural frequencies that are obtainedby the time-varying state transition matrix The work in [7]proposed a recursive algorithm based on subspace methodsto identify the state matrices and consequently to determinethemodal parameters Other important approaches are thosebased on the Kalman Filter [8] or the parametricmethods asfor example the FS-TARMA [9] which is an extension of theclassical ARMA techniques In [10] a Short-Time StochasticSubspace Identification (ST-SSI) approach has been definedbased on the ldquofrozenrdquo technique where the classical subspaceidentification [11] is applied to a sequence of windowed partsof the signal

Hindawi Publishing CorporationShock and VibrationVolume 2014 Article ID 314527 9 pageshttpdxdoiorg1011552014314527

2 Shock and Vibration

The ST-SSI method can be applied to different kinds ofnonstationary systems in order to estimate the instantaneousfrequencies and for example it has been used to estimatethe frequency in practical systems showing nonlinear effects[12] However instead of extracting a series of time-varyinglinear models the identification of a whole parametric non-linear model is an important instrument for many purposesAmong the past and recent developments [13] the nonlinearsubspace identification (NSI) method has been developedin [14] and improved in [15] for identifying large systemswith lumped nonlinearities Both the ST-SSI and the NSImethods have been applied in [16] for estimating the swingingfrequency of an experimental time-varying inertia pendu-lum whose dynamics is governed by a nonlinear equation ofmotion due to large swinging amplitudes

The same pendulum is considered in this paper forinvestigating a suitable model of damping The paper startswith the description of the experimental set-up Signals areacquired from a rotary sensor but some remarks are alsoproposed for signals measured by piezoelectric or capacitiveaccelerometers After the introduction of the analytical equa-tion of motion the energy approach of [4] and the ST-SSImethod are briefly described and applied for estimating theequivalent damping ratio When the mass is fixed the energyapproach is employed for updating the preliminary modelin order to fit the experimental data Three moving-masscases are finally analysed to demonstrate that the estimatesof the damping factors obtained by means of ST-SSI arevery accurate when compared with those obtained by ananalytical model and by the energy approach

2 Experimental Set-Up

The structure under test is a pendulum with time-varyinginertia a disk on a cart can slide along a runner while thependulum is swinging This structure cannot be consideredsimply as a linear time-variant system the equation ofmotion of the pendulum being nonlinear for large swingingamplitudes

21 Description An overview of the design of the structureis presented in this section together with a descriptionof the instrumentation used for data acquisition Furtherdetails about the experimental set up and the measuredcharacteristics of the considered elements can be found in[16] The pendulum is formed by a thin aluminium runnerallowing the sliding of a cart which can host an addedmass the motion of the mass varies the pendulum inertiaMoreover in order to avoid a nonoptimal clamp betweenthe runner and the shaft due to the large deformability ofaluminium a small metallic plate has also been added atthe root of the aluminium beam near the hinge to limit itstransverse vibrations

The travelling mass is a steel disk of mass 119898119898

=

05025 kg whose motion is regulated by a hand-drivencounterbalancingmassThe latter is connected to themovingmass through a system of pulleys and a cable that canbe considered as nonextendible The complete structure is

shown in Figure 1(a) the main supports plates pulleysbearings and precision shaft are visible

The sensors can also be seen in Figure 1(a) A triaxial andfour monoaxial accelerometers have been mounted along thebeamThe triaxial accelerometer a PCB 356B18 piezoelectricsensor (ICP) is used to express some practical considera-tions in Section 22 To show typical measurement errorssome data sets have been acquired by adding a capacitiveaccelerometer to the system (not shown in the figure)in the same position of the ICP sensor Each monoaxialaccelerometer is a Bruel amp Kjaeligr 4507 B 004 piezoelectricsensor used to measure the transversal vibrations (along the119911 axis in Figure 1(b)) of the pendulum [16]Their signals havenot been analysed in this paper

A direct measure of the angular position of the pendulumis given by a Penny+Giles SRS280 sealed rotary sensor withan accuracy of plusmn1 over 100∘ connected to the precisionshaft A Celesco PT1A linear potentiometer with amaximumextension of 12m has been connected to the counterweight(see Figure 1(a))The position of the travellingmass along therunner can be simply obtained from this measure

All signals have been acquired and recorded with asampling frequency of 256Hz by using an OROS acquisitionsystem with 32 channels and antialiasing filter

22 Remarks about the Accelerometers In this section someremarks about the signals measured by the accelerometersare proposed together with a comparison with the rotarysensor recordings In the following 120579(119905) is the output ofthe rotary potentiometer which is very accurate at theselow frequencies thus the use of supplementary sensors isnot needed to describe the dynamics of the SDOF systemHowever accelerometers are mounted to give some usefulguidelines in case a potentiometer is not available A piezo-electric sensor (ICP) is not suited to measure the radialacceleration of the pendulumunder exam because it removesthe DC component of the output which is nonnull This iswhy a capacitive accelerometer was chosen for comparison

To show the difference the two accelerometers have beenmounted on the beam in the same position 119904

119886= 093m with

radial direction (the 119909 axis in Figure 1(b)) The signal 120579(119905)of the rotary sensor has been numerically differentiated inorder to obtain

120579(119905) which is used for computing the ldquoactualrdquovalue of the radial acceleration 119886

119903(119905) = 119904

119886

120579(119905)2 shown in

Figure 2(a) The signals acquired by the accelerometers arerepresented in Figure 2(b) the ICPmeasurement 119886

119903ICP(119905) haszero mean and its value is zero for 120579 = 0 and

120579 = 0 (at the endof time history) while the capacitive sensor output 119886

119903cap(119905)

is asymmetric and its value tends to 119892 for 120579 = 0 and 120579 = 0

Clearly none of the two behaviours can be associatedwith theactual value of radial acceleration

Another remark arises from Figure 2(b) the effect of thegravitational acceleration 119892 on the measured signals mustbe taken into account and removed in order to get thecorrect value of the radial acceleration This is due to thefact that the measurement axes of the accelerometer on thependulum have an orientation that largely changes over timewhile most of dynamics applications do not show such a

Shock and Vibration 3

Metallicplate

(a)

s(t)

yx

z

rarrg

120579(t)

p

b

m∙

∙

∙

(b)

Figure 1 (a) Complete structure At the top the reinforcement plate is highlighted The travelling mass and the counterweight are visible onthe left and on the right respectively (b) Pendulum with the travelling mass

0

1

2

3

4

5

6

7

8

ar

(ms2)

0 10 20 30 40 50 60

Time (s)

(a)

0

0

5

10

10

15

20

20 30 40 50 60minus10

minus5

ar

(ms2)

Time (s)

CapacitiveICP

(b)

Figure 2 (a) Actual value of the radial acceleration (b) Measured radial accelerations by the capacitive and the ICP sensor

behaviour A ldquocleaningrdquo operation can be thought of as beingtrivial but it cannot be performed on signals from classicalICP accelerometers because of the continuous componentremoval described above When the capacitive sensor signalis considered as ldquocorruptedrdquo by the presence of 119892 an estimate(indicated by and) of the actual radial acceleration is obtainedafter cleaning

119886119903cap (119905) = 119886119903 (119905) + 119892 cos 120579 (119905) (1a)

CLEANING997888997888997888997888997888997888997888997888rarr 119886

119903cap (119905) = 119886119903cap (119905) minus 119892 cos 120579 (119905) (1b)

Figure 3(a) shows a comparison between the actual radialacceleration 119886

119903(119905) and the estimate (1b) an almost perfect

correspondence is now obtained A similar approach can beadopted if the tangential direction (the 119910 axis in Figure 1(b))is considered By differentiating again the signal

120579(119905) in orderto obtain

120579(119905) the ldquoactualrdquo value of the tangential acceleration


ActualCorrected capacitive

0

1

2

3

4

5

6

7

8ar

(ms2)

0 10 20 30 40 50 60

Time (s)

(a) Radial direction

2

4

6

8

10

at

(ms2)

0 10 20 30 40 50 60 70

Time (s)

0

minus10

minus8

minus6

minus4

minus2


(b) Tangential direction

Figure 3 Comparisons between actual accelerations and corrected capacitive estimates

119886119905(119905) = 119904

119886

120579(119905) has been computed The capacitive estimate

119886119905cap(119905) can be obtained from the measured signal 119886

119905cap(119905)as 119886119905cap(119905) = 119886

119905cap(119905) minus 119892 sin 120579(119905) The comparison betweenthe actual tangential acceleration and the capacitive estimateis shown in Figure 3(b) again a perfect agreement can beobserved In conclusion an angle measurement is necessaryin this kind of problems

23 Equation of Motion The system shown in Figure 1(b)consists of a rigid bar with no flexural effects the pendulumis simply a nonlinear SDOF system

From the rotational equilibrium of the system the equa-tion of the swinging motion is

(119868up + 119898119898119904(119905)2

)120579 (119905)

+ (119888V + 2119898119898119904 (119905) 119904 (119905))120579 (119905) + (119875up + 119892119898119898119904 (119905)) sin 120579 (119905)

= 119868tot (119905)120579 (119905) + 119862tot (119905)

120579 (119905) + 119875tot (119905) sin 120579 (119905) = 0

(2)

in which the subscript ldquouprdquo refers to the quantities notdepending on the mass position 119904(119905) These have beenestimated by means of the Nonlinear Subspace Identification(NSI) method in [16] in which an updating procedure wasperformed to build an accuratemodel tuned on themeasuredresults The angle swept by the pendulum is indicated by 120579(119905)whilst 119904(119905) is the distance of the disk from the hinge Otherterms appearing in (2) are the gravitational acceleration 119892 =

981ms2 and a viscous damping coefficient 119888V Given thevalue119898

119898= 05025 kg of the travellingmass model (2) can be

completely defined by exploiting the results obtained in [16]119868up = 01292 kgm2 119875up = 18380 kgm2 sminus2 moreover anoverall estimate of 119888V = 0035 kgm

2 sminus1 is used for preliminarycomparisons

The analytical form of the time-variant viscous dampingratio is then defined as

120577 (119905) =

119862tot (119905)

2radic119868tot (119905) 119875tot (119905) (3)

3 Methodologies

Two methodologies are adopted in this paper for obtainingestimates of the equivalent viscous damping ratio The firstbased on energy considerations is evaluated as a baseline forupdating the model and is used for comparisons with thesecond technique based on subspace identification

31 Energy Considerations A method for deriving an esti-mate of the damping ratio which is briefly described in thissection and used for successive comparisons can be foundin [4] The normalised total energy of the pendulum withmoving mass can be written as

119864 (119905) =

1

2

120579(119905)2

+ 120596119904(119905)2

(1 minus cos 120579 (119905)) (4)

where120596119904(119905)2

= (119875up+119892119898119898119904(119905))(119868up+119898119898119904(119905)2

) is related to theldquoinstantaneousrdquo frequency (whenmass119898

119898is at location 119904(119905))

The two terms of 119864(119905) can be interpreted as the normalisedkinetic and potential energies of the pendulum respectivelyIn [4] a mathematically equivalent form of (2) is rewritteninvolving 119864(119905) and its rate of change then an integral formis considered over one swinging period 119879 and the followingapproximation of the damping ratio for the pendulum withmoving mass is defined

120577 cong

1

4120587

119864 (0) minus 119864 (119879)

119864 (0)

(5)


Thus the damping ratio is defined in (5) by the nor-malised energies that can always be calculated for a given setof conditions at the beginning (represented by 119864(0)) and atthe end (119864(119879)) of each cycle In this way a single value for 120577can be obtained at each cycle in order to get an approxima-tion of the ldquoinstantaneousrdquo damping ratio expressed by (3)This method has two main limitations (i) every estimate of 120577from (5) can be obtained only if exactly a cycle is considered(ii) equation (5) involves the application of (4) which impliesfull knowledge of the pendulum parameters and the massmotion pattern contained in the definition of 120596

119904(119905)

32 Subspace Identification The procedure for the identifi-cation of linear time-varying systems is called Short-TimeStochastic Subspace Identification (ST-SSI) [10 17]The basicidea of the method consists of windowing the signal intomany parts and considering the system as time-invariant ineach time window the process is called frozen technique

If the output data are measured at discrete times witha sampling interval Δ119905 and the input is a discrete signalcharacterised by a zero-order hold between consecutivesample points the discrete-time state-space model of ageneral linear time-varying system at a time instant 119905 = 119903Δ119905

can be obtained The frozen technique considers constantstate matrices during each time step so that the followingrepresentation can be adopted

119909 (119903 + 1) = 119860 (119903) 119909 (119903) + 119861 (119903) 119906 (119903) + 119908 (119903)

119910 (119903) = 119862 (119903) 119909 (119903) + 119863 (119903) 119906 (119903) + V (119903) (6a)

FROZEN TECHNIQUE997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888997888rarr

119909 (119903 + 1) = 119860119909 (119903) + 119861119906 (119903) + 119908 (119903)

119910 (119903) = 119862119909 (119903) + 119863119906 (119903) + V (119903) (6b)

where 119860(119903) and 119861(119903) are not constant and in general theirclosed forms are unknown [6] 119909(119903) is the state vector 119906(119903)the input vector and 119910(119903) the output vector119908(119903) and V(119903) areprocess and measurement error respectively The completetime record is split into time windows (frozen system) whoselength corresponds to a period 119879 (about 400 samples) forcomparing the ST-SSI results with those obtained throughthe energy approach of previous Section 31 such a restrictionto a cycle however is not a limitation of the present ST-SSImethod Usually the windows are almost completely over-lapped except for a sampling period Δ119905 (or a multiple) andtheir length can be arbitrarily chosen If window lengths areshort the data-driven subspace method [11] is preferred withrespect to the covariance-driven version [18] which needsmore samples to obtain accurate results Subspace methodsdo not need any a priori knowledge of the system parametersas they identify the state-space matrices of (6b) startingfrom the measured system responses Natural frequenciesand damping ratios are then extracted by computing theeigenvalues of the identified matrix 119860 in every window Anextensive study about the time-varying swinging frequencyof the pendulum has been performed in [16] in which thecontribution of nonlinearity was analysed by means of theNonlinear Subspace Identification (NSI) method [14] Thispaper is focused on the more challenging topic of damping

4 Results

In this section the results are presented At first the massis fixed and preliminary considerations about the model aredrawn the overall estimate of the viscous damping coefficientis updated to fit the experimental results and an accuratemodel is thus obtained Then three moving-mass cases areconsidered and the results obtained by the ST-SSI methodare validated by comparisons with the method in [4] and theupdated analytical model

41 Fixed Mass By considering the mass fixed in 119904(119905) =

119904 = 49 cm (and obviously 119904(119905) equiv 0) (5) can be appliedand the obtained damping factors can be used to computeestimates for the viscous damping coefficient at each cycleThese values can be then compared with the overall estimate119888V = 0035 kgm2 sminus1 defined by a preliminary nonlinearidentification [16]The comparison is shown in Figure 4(a) itis clear that a constant 119888V value is not proper for representingthe experimental evidence An updating procedure is thencarried out for obtaining a new model in which the viscousdamping coefficient is no more constant indeed it is afunction of the rootmean square (RMS) of angular velocity ineach cycle named 120579 The updating simply consists of a poly-nomial fitting of the viscous coefficients obtained throughthe energy considerations of Section 31 In particular thevariable 119862tot is updated to 119862tot = 119888V(

120579) + 2119898

119898119904(119905) 119904(119905) and

consequently the analytical form (3) of the viscous dampingratio changes into the following ldquoupdatedrdquo version

120577up (119905) =119862tot (119905)

2radic119868tot (119905) 119875tot (119905) (7)

Figure 4(a) shows the behaviour of 119888V during a swingingdecay as expected it accurately fits the energy approachestimates In Section 42 the new damping model (7) will bethe baseline for comparisons in more complicated situationsinvolving the mass moving along the beam

In order to have a better visualisation of the experimentaldamping characteristic the damping force 119865V = 119888V(

120579)120579 is

represented in Figure 4(b) as a function of 120579 Two zonescan be distinguished for small angular velocities (120579 lt

005 rad sminus1) a Coulomb friction is prevalent while for largerangular velocities the contribution of the viscous damping ismore evident

42 Moving Mass In this section three moving-mass casesare considered in order to show the results obtained bymeans of the ST-SSI method In Figure 5 the measured masspositions along the beam and the swinging amplitudes overtime are shown for each case

421 Case 1 The mass is moving downwards from thetop to the middle of the beam (Figure 5(a)) No evidenceof increasing swinging amplitudes due to the Coriolis-typeeffect can be observed in Figure 5(b) since the velocity 119904 ispositive


0 5 10 15 20 25 30 35 40002

004

006

008

01

012

014

Overall estimateApproach of ref [4]Polynomial fitting

Time (s)

Visc

ous c

oeffi

cien

tc

(kg m

2sminus

1)

(a)

0 005 01 015 02 025 03 035 04 045 050

0005

001

0015

Dam

ping

forc

e (N

)

120579 (rads)

(b)

Figure 4 Fixed-mass case (a) Viscous coefficient as a function of time during a swinging decay (b) Damping force as a function of 120579 at eachcycle

Case 1 is useful for demonstrating once again the needfor the updating procedure performed in Section 41 Esti-mates of the damping factors obtained through the energyapproach of Section 31 and the ST-SSImethod of Section 32are compared with the overall estimate of (3) in Figure 6(a)the energy-approach and the ST-SSI estimates are similarbut there is no correspondence with the overall estimate Thesame estimates can be compared with the updated value of(7) in Figure 6(b) a good level of agreement can be observedconfirming the reliability of the damping model obtained inSection 41 for a fixed mass As a general remark emergingfromFigure 6 the ST-SSImethod gives slightly overestimatedvalues of damping with respect to those obtained by theenergy approachThe reason is themodel order 119899 selected forrepresenting the state-space model (6b) in this paper it hasbeen fixed to 2 for simplicity but more accurate estimates ofdamping can be obtained by increasing 119899 and by investigatingthe results by means of stabilisation diagrams

422 Case 2 The mass is moving downwards and upwardsin an almost regular way (Figure 5(c)) An excitation of theswinging amplitudes due to the Coriolis-type effect can beobserved in Figure 5(d) at about 10 seconds since the velocity119904 is negative and large enough to change the sign of 120577upin (7) A less clear excitation can also be observed after 20seconds These qualitative considerations are confirmed bythe damping factors represented in Figure 7(a) The energy-approach and the ST-SSI estimates are compared with theupdated value of (7) an excellent agreement can be observedfor the ST-SSI estimates

423 Case 3 This is the more challenging case the mass ismoving along the beam in a more irregular way especiallyfrom 5 to 12 seconds (Figure 5(e)) From a detailed inspection

of Figure 5(f) four zones are recognised in which an ampli-fication of the swinging amplitudes due to the Coriolis-typeeffect can be observed they are marked by letters A B Cand D In particular zones A and B are short and could bedifficult to distinguish but the representation of the dampingfactors in Figure 7(b) can be useful in this sense In factFigure 7(b) reveals the four zones in which the equivalentdamping factor is negative and these are found to correspondto those of Figure 5(f) Moreover in Figure 7(b) the energy-approach and the ST-SSI estimates are compared with theupdated value of (7) the ST-SSI estimates are very accurate

As a final consideration observe that Figures 7(a) and7(b) are represented on the same scale Case 3 appears tobe more challenging also because larger and fast-varyingvalues of damping factors are involved However resultsare satisfactory even in such a complicated case in whichhigh mass velocities cause substantial changes in the systemdynamics

5 Conclusions

The experimental study of damping in a time-varying iner-tia pendulum is presented The system consists of a disktravelling along a pendulum this relative motion which isassociated with Coriolis-type effects can be exploited forattenuation or amplification of the pendulum oscillationsAt first signals measured by piezoelectric or capacitiveaccelerometers are compared with the output of a rotarysensor for swinging systems like the pendulum under examthe piezoelectric sensor is not suited because it removes theDC component of the output The effect of gravitationalacceleration can be taken into account and correctly removedfrom the measured signal of a capacitive acceleration butangle measurement is needed


0 5 10 15 20

01

015

02

025

03

035

04

045

05

Time (s)

Mas

s pos

ition

s(m

)Ca

se 1

(a)

Ang

le120579

(∘)

0 5 10 15 20

0

10

20

30

40

minus10

minus20

minus30

minus40

Time (s)

(b)

Case

2

0 5 10 15 20 25

01

02

03

04

05

06

07

08

09

Time (s)

Mas

s pos

ition

s(m

)

(c)

Ang

le120579

(∘)

0 5 10 15 20 25

Time (s)

0

10

20

30

minus10

minus20

minus30

(d)

Case

3

0 5 10 15 20 25

01

02

03

04

05

06

07

08

09

Time (s)

Mas

s pos

ition

s(m

)

(e)

Ang

le120579

(∘)

0 5 10 15 20 25

Time (s)

0

10

20

30

minus10

minus20

minus30

A

B

C

D

(f)

Figure 5 Moving mass (a b) Case 1 (c d) Case 2 (e f) Case 3 (a c e) Mass position along the beam (b d f) Swinging amplitude overtime


0 5 10 15 20001

002

003

004

005

006

007

008

009

Dam

ping

fact

or 120577

Overall estimateApproach of ref [4]

Time (s)

ST-SSI

(a)

Updated

0 5 10 15 20001

002

003

004

005

006

007

008

009

Dam

ping

fact

or 120577

Approach of ref [4]

Time (s)

ST-SSI

(b)

Figure 6 Moving-mass Case 1 Estimates of the damping factors compared with the overall estimate of (3) (a) and with the updated valueof (7) (b)

0 5 10 15 20 25

Time (s)

Dam

ping

fact

or 120577

0

005

01

015

minus01

minus005

UpdatedApproach of ref [4]ST-SSI

(a)

A B

C

D

0 5 10 15 20 25

Time (s)

Dam

ping

fact

or 120577

0

005

01

015

minus01

minus005


(b)

Figure 7 Estimates of the damping factors compared with the updated value of (7) Moving-mass Case 2 (a) and Case 3 (b)

The analytical equation of motion is then introducedand the theoretical viscous damping coefficient is updated inorder to obtain an accurate damping model The updatingprocedure investigates the approach of a published workin which the damping ratios are derived from energy con-siderations provided that the system parameters and themass motion pattern are known The system is analysed

through the application of the Short-Time Stochastic Sub-space Identification (ST-SSI) which is very simple and doesnot require any a priori knowledge about the systemThe ST-SSI estimates of the damping factors are compared with thoseobtained by an analytical model and by the energy approachThree moving-mass cases are presented to demonstrate thatthe ST-SSI estimates are very accurate even in the challenging


cases in which damping may turn to negative due to theCoriolis-type effects

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] W Szyszkowski and E Sharbati ldquoOn the FEM modeling ofmechanical systems controlled by relative motion of a membera pendulum-mass interaction test caserdquo Finite Elements inAnalysis and Design vol 45 no 10 pp 730ndash742 2009

[2] W Szyszkowski and D S D Stilling ldquoControlling angularoscillations through mass reconfiguration a variable lengthpendulum caserdquo International Journal of Non-LinearMechanicsvol 37 no 1 pp 89ndash99 2002

[3] A A Zevin and L A Filonenko ldquoA qualitative investigationof the oscillations of a pendulum with a periodically varyinglength and amathematical model of a swingrdquo Journal of AppliedMathematics and Mechanics vol 71 no 6 pp 892ndash904 2007

[4] W Szyszkowski and D S D Stilling ldquoOn damping propertiesof a frictionless physical pendulum with a moving massrdquoInternational Journal of Non-LinearMechanics vol 40 no 5 pp669ndash681 2005

[5] K Liu ldquoIdentification of linear time-varying systemsrdquo Journalof Sound and Vibration vol 206 no 4 pp 487ndash505 1997

[6] K Liu ldquoExtension of modal analysis to linear time-varyingsystemsrdquo Journal of Sound andVibration vol 226 no 1 pp 149ndash167 1999

[7] F Tasker A Bosse and S Fisher ldquoReal-time modal parameterestimation using subspace methods applicationsrdquo MechanicalSystems and Signal Processing vol 12 no 6 pp 809ndash823 1998

[8] S-W Lee J-S Lim S J Baek and K-M Sung ldquoTime-varyingsignal frequency estimation by VFF Kalman filteringrdquo SignalProcessing vol 77 no 3 pp 343ndash347 1999

[9] A G Poulimenos and S D Fassois ldquoOutput-only stochasticidentification of a time-varying structure via functional seriesTARMAmodelsrdquoMechanical Systems and Signal Processing vol23 no 4 pp 1180ndash1204 2009

[10] A Bellino L Garibaldi and S Marchesiello ldquoTime-varyingoutput-only identification of a cracked beamrdquo Key EngineeringMaterials vol 413-414 pp 643ndash650 2009

[11] P van Overschee and B de Moor Subspace Identification forLinear Systems Theory Implementation Applications KluwerAcademic Publishers Boston Mass USA 1996

[12] A Bellino L Garibaldi and S Marchesiello ldquoDetermination ofmoving load characteristics by output-only identification overthe Pescara beamsrdquo Journal of Physics Conference Series vol305 no 1 Article ID 012079 2011

[13] G Kerschen K Worden A F Vakakis and J-C GolinvalldquoPast present and future of nonlinear system identification instructural dynamicsrdquoMechanical Systems and Signal Processingvol 20 no 3 pp 505ndash592 2006

[14] S Marchesiello and L Garibaldi ldquoA time domain approachfor identifying nonlinear vibrating structures by subspacemethodsrdquoMechanical Systems and Signal Processing vol 22 no1 pp 81ndash101 2008

[15] E Gandino and S Marchesiello ldquoIdentification of a duffingoscillator under different types of excitationrdquo Mathematical

Problems in Engineering vol 2010 Article ID 695025 15 pages2010

[16] A Bellino A Fasana E Gandino L Garibaldi and S March-esiello ldquoA time-varying inertia pendulum analytical modellingand experimental identificationrdquoMechanical Systems and SignalProcessing vol 47 no 1-2 pp 120ndash138 2014

[17] S Marchesiello S Bedaoui L Garibaldi and P Argoul ldquoTime-dependent identification of a bridge-like structurewith crossingloadsrdquoMechanical Systems and Signal Processing vol 23 no 6pp 2019ndash2028 2009

[18] E Gandino L Garibaldi and S Marchesiello ldquoCovariance-driven subspace identification a complete input-outputapproachrdquo Journal of Sound and Vibration vol 332 no 26 pp7000ndash7017 2013

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



The ST-SSI method can be applied to different kinds ofnonstationary systems in order to estimate the instantaneousfrequencies and for example it has been used to estimatethe frequency in practical systems showing nonlinear effects[12] However instead of extracting a series of time-varyinglinear models the identification of a whole parametric non-linear model is an important instrument for many purposesAmong the past and recent developments [13] the nonlinearsubspace identification (NSI) method has been developedin [14] and improved in [15] for identifying large systemswith lumped nonlinearities Both the ST-SSI and the NSImethods have been applied in [16] for estimating the swingingfrequency of an experimental time-varying inertia pendu-lum whose dynamics is governed by a nonlinear equation ofmotion due to large swinging amplitudes

The same pendulum is considered in this paper forinvestigating a suitable model of damping The paper startswith the description of the experimental set-up Signals areacquired from a rotary sensor but some remarks are alsoproposed for signals measured by piezoelectric or capacitiveaccelerometers After the introduction of the analytical equa-tion of motion the energy approach of [4] and the ST-SSImethod are briefly described and applied for estimating theequivalent damping ratio When the mass is fixed the energyapproach is employed for updating the preliminary modelin order to fit the experimental data Three moving-masscases are finally analysed to demonstrate that the estimatesof the damping factors obtained by means of ST-SSI arevery accurate when compared with those obtained by ananalytical model and by the energy approach

2 Experimental Set-Up

The structure under test is a pendulum with time-varyinginertia a disk on a cart can slide along a runner while thependulum is swinging This structure cannot be consideredsimply as a linear time-variant system the equation ofmotion of the pendulum being nonlinear for large swingingamplitudes

21 Description An overview of the design of the structureis presented in this section together with a descriptionof the instrumentation used for data acquisition Furtherdetails about the experimental set up and the measuredcharacteristics of the considered elements can be found in[16] The pendulum is formed by a thin aluminium runnerallowing the sliding of a cart which can host an addedmass the motion of the mass varies the pendulum inertiaMoreover in order to avoid a nonoptimal clamp betweenthe runner and the shaft due to the large deformability ofaluminium a small metallic plate has also been added atthe root of the aluminium beam near the hinge to limit itstransverse vibrations

The travelling mass is a steel disk of mass 119898119898

=

05025 kg whose motion is regulated by a hand-drivencounterbalancingmassThe latter is connected to themovingmass through a system of pulleys and a cable that canbe considered as nonextendible The complete structure is

shown in Figure 1(a) the main supports plates pulleysbearings and precision shaft are visible

The sensors can also be seen in Figure 1(a) A triaxial andfour monoaxial accelerometers have been mounted along thebeamThe triaxial accelerometer a PCB 356B18 piezoelectricsensor (ICP) is used to express some practical considera-tions in Section 22 To show typical measurement errorssome data sets have been acquired by adding a capacitiveaccelerometer to the system (not shown in the figure)in the same position of the ICP sensor Each monoaxialaccelerometer is a Bruel amp Kjaeligr 4507 B 004 piezoelectricsensor used to measure the transversal vibrations (along the119911 axis in Figure 1(b)) of the pendulum [16]Their signals havenot been analysed in this paper

A direct measure of the angular position of the pendulumis given by a Penny+Giles SRS280 sealed rotary sensor withan accuracy of plusmn1 over 100∘ connected to the precisionshaft A Celesco PT1A linear potentiometer with amaximumextension of 12m has been connected to the counterweight(see Figure 1(a))The position of the travellingmass along therunner can be simply obtained from this measure

All signals have been acquired and recorded with asampling frequency of 256Hz by using an OROS acquisitionsystem with 32 channels and antialiasing filter

22 Remarks about the Accelerometers In this section someremarks about the signals measured by the accelerometersare proposed together with a comparison with the rotarysensor recordings In the following 120579(119905) is the output ofthe rotary potentiometer which is very accurate at theselow frequencies thus the use of supplementary sensors isnot needed to describe the dynamics of the SDOF systemHowever accelerometers are mounted to give some usefulguidelines in case a potentiometer is not available A piezo-electric sensor (ICP) is not suited to measure the radialacceleration of the pendulumunder exam because it removesthe DC component of the output which is nonnull This iswhy a capacitive accelerometer was chosen for comparison

To show the difference the two accelerometers have beenmounted on the beam in the same position 119904

119886= 093m with

radial direction (the 119909 axis in Figure 1(b)) The signal 120579(119905)of the rotary sensor has been numerically differentiated inorder to obtain

120579(119905) which is used for computing the ldquoactualrdquovalue of the radial acceleration 119886

119903(119905) = 119904

119886

120579(119905)2 shown in

Figure 2(a) The signals acquired by the accelerometers arerepresented in Figure 2(b) the ICPmeasurement 119886

119903ICP(119905) haszero mean and its value is zero for 120579 = 0 and

120579 = 0 (at the endof time history) while the capacitive sensor output 119886

119903cap(119905)

is asymmetric and its value tends to 119892 for 120579 = 0 and 120579 = 0

Clearly none of the two behaviours can be associatedwith theactual value of radial acceleration

Another remark arises from Figure 2(b) the effect of thegravitational acceleration 119892 on the measured signals mustbe taken into account and removed in order to get thecorrect value of the radial acceleration This is due to thefact that the measurement axes of the accelerometer on thependulum have an orientation that largely changes over timewhile most of dynamics applications do not show such a


Metallicplate

(a)

s(t)

yx

z

rarrg

120579(t)

p

b

m∙

∙

∙

(b)


0

1

2

3

4

5

6

7

8

ar

(ms2)

0 10 20 30 40 50 60

Time (s)

(a)

0

0

5

10

10

15

20

20 30 40 50 60minus10

minus5

ar

(ms2)

Time (s)

CapacitiveICP

(b)



119886119903cap (119905) = 119886119903 (119905) + 119892 cos 120579 (119905) (1a)

CLEANING997888997888997888997888997888997888997888997888rarr 119886









0

1

2

3

4

5

6

7

8ar

(ms2)

0 10 20 30 40 50 60

Time (s)


2

4

6

8

10

at

(ms2)

0 10 20 30 40 50 60 70

Time (s)

0

minus10

minus8

minus6

minus4

minus2




119886119905(119905) = 119904

119886



119905cap(119905)as 119886119905cap(119905) = 119886




(119868up + 119898119898119904(119905)2

)120579 (119905)

+ (119888V + 2119898119898119904 (119905) 119904 (119905))120579 (119905) + (119875up + 119892119898119898119904 (119905)) sin 120579 (119905)

= 119868tot (119905)120579 (119905) + 119862tot (119905)

120579 (119905) + 119875tot (119905) sin 120579 (119905) = 0

(2)







120577 (119905) =

119862tot (119905)

2radic119868tot (119905) 119875tot (119905) (3)

3 Methodologies



119864 (119905) =

1

2

120579(119905)2

+ 120596119904(119905)2

(1 minus cos 120579 (119905)) (4)

where120596119904(119905)2

= (119875up+119892119898119898119904(119905))(119868up+119898119898119904(119905)2


119898is at location 119904(119905))


120577 cong

1

4120587

119864 (0) minus 119864 (119879)

119864 (0)

(5)



119904(119905)




119909 (119903 + 1) = 119860 (119903) 119909 (119903) + 119861 (119903) 119906 (119903) + 119908 (119903)

119910 (119903) = 119862 (119903) 119909 (119903) + 119863 (119903) 119906 (119903) + V (119903) (6a)


119909 (119903 + 1) = 119860119909 (119903) + 119861119906 (119903) + 119908 (119903)

119910 (119903) = 119862119909 (119903) + 119863119906 (119903) + V (119903) (6b)


4 Results




120579) + 2119898

119898119904(119905) 119904(119905) and


120577up (119905) =119862tot (119905)

2radic119868tot (119905) 119875tot (119905) (7)



120579)120579 is






0 5 10 15 20 25 30 35 40002

004

006

008

01

012

014


Time (s)

Visc

ous c

oeffi

cien

tc

(kg m

2sminus

1)

(a)

0 005 01 015 02 025 03 035 04 045 050

0005

001

0015

Dam

ping

forc

e (N

)

120579 (rads)

(b)







5 Conclusions



0 5 10 15 20

01

015

02

025

03

035

04

045

05

Time (s)

Mas

s pos

ition

s(m

)Ca

se 1

(a)

Ang

le120579

(∘)

0 5 10 15 20

0

10

20

30

40

minus10

minus20

minus30

minus40

Time (s)

(b)

Case

2

0 5 10 15 20 25

01

02

03

04

05

06

07

08

09

Time (s)

Mas

s pos

ition

s(m

)

(c)

Ang

le120579

(∘)

0 5 10 15 20 25

Time (s)

0

10

20

30

minus10

minus20

minus30

(d)

Case

3

0 5 10 15 20 25

01

02

03

04

05

06

07

08

09

Time (s)

Mas

s pos

ition

s(m

)

(e)

Ang

le120579

(∘)

0 5 10 15 20 25

Time (s)

0

10

20

30

minus10

minus20

minus30

A

B

C

D

(f)



0 5 10 15 20001

002

003

004

005

006

007

008

009

Dam

ping

fact

or 120577


Time (s)

ST-SSI

(a)

Updated

0 5 10 15 20001

002

003

004

005

006

007

008

009

Dam

ping

fact

or 120577

Approach of ref [4]

Time (s)

ST-SSI

(b)


0 5 10 15 20 25

Time (s)

Dam

ping

fact

or 120577

0

005

01

015

minus01

minus005


(a)

A B

C

D

0 5 10 15 20 25

Time (s)

Dam

ping

fact

or 120577

0

005

01

015

minus01

minus005


(b)








References






















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Metallicplate

(a)

s(t)

yx

z

rarrg

120579(t)

p

b

m∙

∙

∙

(b)


0

1

2

3

4

5

6

7

8

ar

(ms2)

0 10 20 30 40 50 60

Time (s)

(a)

0

0

5

10

10

15

20

20 30 40 50 60minus10

minus5

ar

(ms2)

Time (s)

CapacitiveICP

(b)



119886119903cap (119905) = 119886119903 (119905) + 119892 cos 120579 (119905) (1a)

CLEANING997888997888997888997888997888997888997888997888rarr 119886









0

1

2

3

4

5

6

7

8ar

(ms2)

0 10 20 30 40 50 60

Time (s)


2

4

6

8

10

at

(ms2)

0 10 20 30 40 50 60 70

Time (s)

0

minus10

minus8

minus6

minus4

minus2




119886119905(119905) = 119904

119886



119905cap(119905)as 119886119905cap(119905) = 119886




(119868up + 119898119898119904(119905)2

)120579 (119905)

+ (119888V + 2119898119898119904 (119905) 119904 (119905))120579 (119905) + (119875up + 119892119898119898119904 (119905)) sin 120579 (119905)

= 119868tot (119905)120579 (119905) + 119862tot (119905)

120579 (119905) + 119875tot (119905) sin 120579 (119905) = 0

(2)







120577 (119905) =

119862tot (119905)

2radic119868tot (119905) 119875tot (119905) (3)

3 Methodologies



119864 (119905) =

1

2

120579(119905)2

+ 120596119904(119905)2

(1 minus cos 120579 (119905)) (4)

where120596119904(119905)2

= (119875up+119892119898119898119904(119905))(119868up+119898119898119904(119905)2


119898is at location 119904(119905))


120577 cong

1

4120587

119864 (0) minus 119864 (119879)

119864 (0)

(5)



119904(119905)




119909 (119903 + 1) = 119860 (119903) 119909 (119903) + 119861 (119903) 119906 (119903) + 119908 (119903)

119910 (119903) = 119862 (119903) 119909 (119903) + 119863 (119903) 119906 (119903) + V (119903) (6a)


119909 (119903 + 1) = 119860119909 (119903) + 119861119906 (119903) + 119908 (119903)

119910 (119903) = 119862119909 (119903) + 119863119906 (119903) + V (119903) (6b)


4 Results




120579) + 2119898

119898119904(119905) 119904(119905) and


120577up (119905) =119862tot (119905)

2radic119868tot (119905) 119875tot (119905) (7)



120579)120579 is






0 5 10 15 20 25 30 35 40002

004

006

008

01

012

014


Time (s)

Visc

ous c

oeffi

cien

tc

(kg m

2sminus

1)

(a)

0 005 01 015 02 025 03 035 04 045 050

0005

001

0015

Dam

ping

forc

e (N

)

120579 (rads)

(b)







5 Conclusions



0 5 10 15 20

01

015

02

025

03

035

04

045

05

Time (s)

Mas

s pos

ition

s(m

)Ca

se 1

(a)

Ang

le120579

(∘)

0 5 10 15 20

0

10

20

30

40

minus10

minus20

minus30

minus40

Time (s)

(b)

Case

2

0 5 10 15 20 25

01

02

03

04

05

06

07

08

09

Time (s)

Mas

s pos

ition

s(m

)

(c)

Ang

le120579

(∘)

0 5 10 15 20 25

Time (s)

0

10

20

30

minus10

minus20

minus30

(d)

Case

3

0 5 10 15 20 25

01

02

03

04

05

06

07

08

09

Time (s)

Mas

s pos

ition

s(m

)

(e)

Ang

le120579

(∘)

0 5 10 15 20 25

Time (s)

0

10

20

30

minus10

minus20

minus30

A

B

C

D

(f)



0 5 10 15 20001

002

003

004

005

006

007

008

009

Dam

ping

fact

or 120577


Time (s)

ST-SSI

(a)

Updated

0 5 10 15 20001

002

003

004

005

006

007

008

009

Dam

ping

fact

or 120577

Approach of ref [4]

Time (s)

ST-SSI

(b)


0 5 10 15 20 25

Time (s)

Dam

ping

fact

or 120577

0

005

01

015

minus01

minus005


(a)

A B

C

D

0 5 10 15 20 25

Time (s)

Dam

ping

fact

or 120577

0

005

01

015

minus01

minus005


(b)








References






















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











0

1

2

3

4

5

6

7

8ar

(ms2)

0 10 20 30 40 50 60

Time (s)


2

4

6

8

10

at

(ms2)

0 10 20 30 40 50 60 70

Time (s)

0

minus10

minus8

minus6

minus4

minus2




119886119905(119905) = 119904

119886



119905cap(119905)as 119886119905cap(119905) = 119886




(119868up + 119898119898119904(119905)2

)120579 (119905)

+ (119888V + 2119898119898119904 (119905) 119904 (119905))120579 (119905) + (119875up + 119892119898119898119904 (119905)) sin 120579 (119905)

= 119868tot (119905)120579 (119905) + 119862tot (119905)

120579 (119905) + 119875tot (119905) sin 120579 (119905) = 0

(2)







120577 (119905) =

119862tot (119905)

2radic119868tot (119905) 119875tot (119905) (3)

3 Methodologies



119864 (119905) =

1

2

120579(119905)2

+ 120596119904(119905)2

(1 minus cos 120579 (119905)) (4)

where120596119904(119905)2

= (119875up+119892119898119898119904(119905))(119868up+119898119898119904(119905)2


119898is at location 119904(119905))


120577 cong

1

4120587

119864 (0) minus 119864 (119879)

119864 (0)

(5)



119904(119905)




119909 (119903 + 1) = 119860 (119903) 119909 (119903) + 119861 (119903) 119906 (119903) + 119908 (119903)

119910 (119903) = 119862 (119903) 119909 (119903) + 119863 (119903) 119906 (119903) + V (119903) (6a)


119909 (119903 + 1) = 119860119909 (119903) + 119861119906 (119903) + 119908 (119903)

119910 (119903) = 119862119909 (119903) + 119863119906 (119903) + V (119903) (6b)


4 Results




120579) + 2119898

119898119904(119905) 119904(119905) and


120577up (119905) =119862tot (119905)

2radic119868tot (119905) 119875tot (119905) (7)



120579)120579 is






0 5 10 15 20 25 30 35 40002

004

006

008

01

012

014


Time (s)

Visc

ous c

oeffi

cien

tc

(kg m

2sminus

1)

(a)

0 005 01 015 02 025 03 035 04 045 050

0005

001

0015

Dam

ping

forc

e (N

)

120579 (rads)

(b)







5 Conclusions



0 5 10 15 20

01

015

02

025

03

035

04

045

05

Time (s)

Mas

s pos

ition

s(m

)Ca

se 1

(a)

Ang

le120579

(∘)

0 5 10 15 20

0

10

20

30

40

minus10

minus20

minus30

minus40

Time (s)

(b)

Case

2

0 5 10 15 20 25

01

02

03

04

05

06

07

08

09

Time (s)

Mas

s pos

ition

s(m

)

(c)

Ang

le120579

(∘)

0 5 10 15 20 25

Time (s)

0

10

20

30

minus10

minus20

minus30

(d)

Case

3

0 5 10 15 20 25

01

02

03

04

05

06

07

08

09

Time (s)

Mas

s pos

ition

s(m

)

(e)

Ang

le120579

(∘)

0 5 10 15 20 25

Time (s)

0

10

20

30

minus10

minus20

minus30

A

B

C

D

(f)



0 5 10 15 20001

002

003

004

005

006

007

008

009

Dam

ping

fact

or 120577


Time (s)

ST-SSI

(a)

Updated

0 5 10 15 20001

002

003

004

005

006

007

008

009

Dam

ping

fact

or 120577

Approach of ref [4]

Time (s)

ST-SSI

(b)


0 5 10 15 20 25

Time (s)

Dam

ping

fact

or 120577

0

005

01

015

minus01

minus005


(a)

A B

C

D

0 5 10 15 20 25

Time (s)

Dam

ping

fact

or 120577

0

005

01

015

minus01

minus005


(b)








References






















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119904(119905)




119909 (119903 + 1) = 119860 (119903) 119909 (119903) + 119861 (119903) 119906 (119903) + 119908 (119903)

119910 (119903) = 119862 (119903) 119909 (119903) + 119863 (119903) 119906 (119903) + V (119903) (6a)


119909 (119903 + 1) = 119860119909 (119903) + 119861119906 (119903) + 119908 (119903)

119910 (119903) = 119862119909 (119903) + 119863119906 (119903) + V (119903) (6b)


4 Results




120579) + 2119898

119898119904(119905) 119904(119905) and


120577up (119905) =119862tot (119905)

2radic119868tot (119905) 119875tot (119905) (7)



120579)120579 is






0 5 10 15 20 25 30 35 40002

004

006

008

01

012

014


Time (s)

Visc

ous c

oeffi

cien

tc

(kg m

2sminus

1)

(a)

0 005 01 015 02 025 03 035 04 045 050

0005

001

0015

Dam

ping

forc

e (N

)

120579 (rads)

(b)







5 Conclusions



0 5 10 15 20

01

015

02

025

03

035

04

045

05

Time (s)

Mas

s pos

ition

s(m

)Ca

se 1

(a)

Ang

le120579

(∘)

0 5 10 15 20

0

10

20

30

40

minus10

minus20

minus30

minus40

Time (s)

(b)

Case

2

0 5 10 15 20 25

01

02

03

04

05

06

07

08

09

Time (s)

Mas

s pos

ition

s(m

)

(c)

Ang

le120579

(∘)

0 5 10 15 20 25

Time (s)

0

10

20

30

minus10

minus20

minus30

(d)

Case

3

0 5 10 15 20 25

01

02

03

04

05

06

07

08

09

Time (s)

Mas

s pos

ition

s(m

)

(e)

Ang

le120579

(∘)

0 5 10 15 20 25

Time (s)

0

10

20

30

minus10

minus20

minus30

A

B

C

D

(f)



0 5 10 15 20001

002

003

004

005

006

007

008

009

Dam

ping

fact

or 120577


Time (s)

ST-SSI

(a)

Updated

0 5 10 15 20001

002

003

004

005

006

007

008

009

Dam

ping

fact

or 120577

Approach of ref [4]

Time (s)

ST-SSI

(b)


0 5 10 15 20 25

Time (s)

Dam

ping

fact

or 120577

0

005

01

015

minus01

minus005


(a)

A B

C

D

0 5 10 15 20 25

Time (s)

Dam

ping

fact

or 120577

0

005

01

015

minus01

minus005


(b)








References






















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 5 10 15 20 25 30 35 40002

004

006

008

01

012

014


Time (s)

Visc

ous c

oeffi

cien

tc

(kg m

2sminus

1)

(a)

0 005 01 015 02 025 03 035 04 045 050

0005

001

0015

Dam

ping

forc

e (N

)

120579 (rads)

(b)







5 Conclusions



0 5 10 15 20

01

015

02

025

03

035

04

045

05

Time (s)

Mas

s pos

ition

s(m

)Ca

se 1

(a)

Ang

le120579

(∘)

0 5 10 15 20

0

10

20

30

40

minus10

minus20

minus30

minus40

Time (s)

(b)

Case

2

0 5 10 15 20 25

01

02

03

04

05

06

07

08

09

Time (s)

Mas

s pos

ition

s(m

)

(c)

Ang

le120579

(∘)

0 5 10 15 20 25

Time (s)

0

10

20

30

minus10

minus20

minus30

(d)

Case

3

0 5 10 15 20 25

01

02

03

04

05

06

07

08

09

Time (s)

Mas

s pos

ition

s(m

)

(e)

Ang

le120579

(∘)

0 5 10 15 20 25

Time (s)

0

10

20

30

minus10

minus20

minus30

A

B

C

D

(f)



0 5 10 15 20001

002

003

004

005

006

007

008

009

Dam

ping

fact

or 120577


Time (s)

ST-SSI

(a)

Updated

0 5 10 15 20001

002

003

004

005

006

007

008

009

Dam

ping

fact

or 120577

Approach of ref [4]

Time (s)

ST-SSI

(b)


0 5 10 15 20 25

Time (s)

Dam

ping

fact

or 120577

0

005

01

015

minus01

minus005


(a)

A B

C

D

0 5 10 15 20 25

Time (s)

Dam

ping

fact

or 120577

0

005

01

015

minus01

minus005


(b)








References






















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 5 10 15 20

01

015

02

025

03

035

04

045

05

Time (s)

Mas

s pos

ition

s(m

)Ca

se 1

(a)

Ang

le120579

(∘)

0 5 10 15 20

0

10

20

30

40

minus10

minus20

minus30

minus40

Time (s)

(b)

Case

2

0 5 10 15 20 25

01

02

03

04

05

06

07

08

09

Time (s)

Mas

s pos

ition

s(m

)

(c)

Ang

le120579

(∘)

0 5 10 15 20 25

Time (s)

0

10

20

30

minus10

minus20

minus30

(d)

Case

3

0 5 10 15 20 25

01

02

03

04

05

06

07

08

09

Time (s)

Mas

s pos

ition

s(m

)

(e)

Ang

le120579

(∘)

0 5 10 15 20 25

Time (s)

0

10

20

30

minus10

minus20

minus30

A

B

C

D

(f)



0 5 10 15 20001

002

003

004

005

006

007

008

009

Dam

ping

fact

or 120577


Time (s)

ST-SSI

(a)

Updated

0 5 10 15 20001

002

003

004

005

006

007

008

009

Dam

ping

fact

or 120577

Approach of ref [4]

Time (s)

ST-SSI

(b)


0 5 10 15 20 25

Time (s)

Dam

ping

fact

or 120577

0

005

01

015

minus01

minus005


(a)

A B

C

D

0 5 10 15 20 25

Time (s)

Dam

ping

fact

or 120577

0

005

01

015

minus01

minus005


(b)








References






















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 5 10 15 20001

002

003

004

005

006

007

008

009

Dam

ping

fact

or 120577


Time (s)

ST-SSI

(a)

Updated

0 5 10 15 20001

002

003

004

005

006

007

008

009

Dam

ping

fact

or 120577

Approach of ref [4]

Time (s)

ST-SSI

(b)


0 5 10 15 20 25

Time (s)

Dam

ping

fact

or 120577

0

005

01

015

minus01

minus005


(a)

A B

C

D

0 5 10 15 20 25

Time (s)

Dam

ping

fact

or 120577

0

005

01

015

minus01

minus005


(b)








References






















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













References






















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









damping effects induced by a mass moving along a pendulum

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