Harmonic analysis of a mass subject to hysteretic friction ... · [11 ,12 ]. This paper aims to validate these theoretical results on three experimental set-ups. T w o dedicated set-

Harmonic analysis of a mass subject to hysteretic friction:experimental validation

W. Symens, F. Al-Bender, J. Swevers, H. Van BrusselKULeuven, Department of Mechanical Engineering,Celestijnenlaan 300 B, B-3001, Heverlee, Belgiume-mail: [email protected]

AbstractThe industrial demand for machine tools with increasing accuracy obliges us to take a thorough look at thephysical phenomena that are present at small movements of those machine tool slides. One of these phenomena,and probably the most dominant one, is the dependence of the friction force on displacement. This friction forcedependency on the displacement can be described by a static hysteresis function. The influence of this highlynon-linear effect on the dynamics of the system has been theoretically analysed in former communications[11, 12]. This paper aims to validate these theoretical results on three experimental set-ups. Two dedicated set-ups have been built, consisting of a linearly driven rolling element slideway, to specifically study the hystereticfriction behaviour. The experiments performed on these dedicated set-ups are then repeated on one axis of anindustrial pick-and-place device, driven by a linear motor and guided by commercial slideways. The results ofthe experiments on all the set-ups agree qualitatively well with the theoretically predicted ones and point to thedifficulty of accurate quantitative identification, showing that the hysteretic friction behaviour in machine toolsshould be more carefully studied and incorporated in the dynamic identification process of these systems.

1 Introduction

When performing very accurate and small move-ments with machine tools, the system cannot be ef-fectively described by a linear model. Several non-linear phenomena become important, which have tobe considered when controlling the movement of themachine tool axes. One of those nonlinear phenom-ena, probably the most important, is the ”friction” be-haviour between parts that move relative to one an-other. Classical friction models are quasi-static andonly a velocity-force relation is considered, for exam-ple Coulomb friction and the Stribeck effect. Experi-ments however show that the velocity-force relationis not static but depends on the change of velocity[1] and that friction is also displacement-dependent[1, 9]. The friction force dependency on the displace-ment can be described by a static hysteresis function.Although this displacement dependency is only im-portant for small displacements, up to several tens ofmicrons after a velocity reversal, this motion situationis very important in accurate motion systems [9, 8].The region where the influence of the hysteretic be-haviour of the friction force is important is called thepre-sliding/pre-rolling region [1, 9]. All the hithertoobserved friction phenomena, including the hysteretic

position dependency, are combined in a mathematicalfriction model called the ”Leuven model” [6, 7].

From a dynamical point of view, the Stribeck be-haviour of the friction curve gives rise to stick-slipand limit cycle phenomena. This behaviour is exten-sively described and investigated in literature [1, 2,4]. Also the dynamic properties of other velocity-dependent friction phenomena are investigated. Theinfluence of hysteresis phenomena on the dynamicbehaviour of the system is however not thoroughlyexamined. A simplified analysis is carried out in [2]and [4] where the difference in the dynamic behaviourfor very small and very large displacements is consid-ered separately. A thorough analysis of intermediatedisplacement levels or the transition from one modelto the other is however not done. In previous work,the present authors carried out a theoretical analysisof this type of dynamical systems, both for the au-tonomous case as under harmonic excitation. Har-monic excitation is one possible technique to anal-yse non-linear systems that cannot be analysed ana-lytically in the general case. As the analysis showed,interesting and instructive results can be obtained byusing this excitation method. The results of this anal-ysis can be found in [11] and [12] respectively.

1229

The present paper endeavours to validate the ob-tained theoretical results on several experimental set-ups; as a single set-up is not instructive enough todeeply investigate the complexity of the dynamicsarising from the hysteretic behaviour. In first instancea dedicated set-up, consisting of a linearly drivenrolling element slideway, has been built to specificallystudy the hysteretic friction behaviour. Then, in orderto study the dynamics, the experiments on this dedi-cated set-up have been repeated first on a commercialslideway and finally on one axis of an industrial pick-and-place device which is driven by a linear motorand guided by four commercial slideways.

In the rest of this paper, section 2 gives a com-prehensive summary of the results of the theoreticalanalysis of the influence of hysteretic friction on thedynamic behaviour of mechanical systems. Section3 hereafter describes the different experimental set-ups of which the dynamic behaviour is investigated.First the hysteretic properties of the set-ups are char-acterised, which are then used to validate the resultsof different stepped-sine experiments. Finally someconclusions and directions for further research arehighlighted in section 4.

2 Dynamic behaviour of a masssubject to hysteretic friction:Theoretical analysis

It is not the intention of this section to give an ex-tensive description of the theoretical analysis of theinfluence of hysteretic friction on the dynamics of amechanical system, but only to state the observationsand results from this analysis. It is those results thatare looked for in the experimental set-ups in Section3. Interested readers can find the details of the anal-ysis in references [11, 12] on which this section ismainly based.

The system under investigation is a mass subjectto hysteretic friction (see Figure 1.A). With hystereticfriction is meant a friction force that is described bya displacement dependent hysteresis function. Fig-ure 1.B shows the time trajectory of the friction forceresulting from an imposed displacement trajectory.A detailed description of the hysteretic friction be-haviour is out of the scope of this paper but can befound in [11].

The main characteristics of this friction force are:

� The force is displacement dependent and not ve-locity dependent.

Figure 1: Mass subject to ”hysteretic friction”

� There is no unique relation between the frictionforce and the displacement: the force also de-pends on the traversed trajectory.

� The states of the system (position and velocity)are not sufficient to describe the friction force.The velocity reversal points have to be remem-bered too.

� If certain memory points (velocity reversals) arerevisited they can be wiped out of the memory.This is called the closing of an internal loop [5].

� The first derivative of the friction force is notcontinuous at velocity reversals.

The mathematical description of the hysteretic fric-tion is based on two rules. The second rule is appliedmost of the time since the first rule (Equation 1)onlyapplies if the friction object enters a region where ithas never been before. The parameters

��and � �

in (Equation 2) change value at each velocity reversal

1230 PROCEEDINGS OFISMA2002 - VOLUME III

point, or when an internal loop in closed [11].

�� with

� �� (1)

�� "!$#&%')(� �� *�,+ � � calculated with the formula

for -/.1032 4 before the reversal

(2)

Both rules are completely characterised by the vir-gin curve � � �� of the hysteretic friction. This curvestarts in general at the origin, has a positive first and anegative second derivative for all � , and saturates forlarge � .

The dynamic equation of the system shown in Fig-ure 1.A can be written as:576� ��89� �;:=</>@? � � �� 1� � �A�CB � (3)

with89� �;:=</>@? � � �� 1� representing the hysteretic fric-

tion force. Within this formula </>@? � � �� represents allthe history of the traversed movement that is relevantfor the future movement and thus has to be remem-bered.

This highly nonlinear dynamic equation can besolved analytically only for the autonomous case,where

�A�CB � � � . Figure 2(b) shows the time evolu-tion of a unit mass with hysteretic friction describedby the virgin curve shown in Figure 2(a) and an initialdisplacement of �ED F . The virgin curve obeys the prop-erties mentioned above an has the following from:

� � �� <HGJI1K �MLONN � �P '=Q (4)

where <HG � �ED R represents the saturation value andN � �ED S is a measure for the curvature of the curve.Figure 2.B clearly shows that the response is non-linear since the mass does not return to its initial equi-librium state being the zero position. The damping ofthe response also is non-linear since there in no ex-ponential decay but this is hard to see on the figure.

In the general case, where�A�CB �UT� � , it is not

possible to derive an analytical solution. To investi-gate this equation some approximating methods canbe used. One of these is the describing functionmethod where the non-linear element

89� �;:=</>@? � � �� 1�is replaced by an element that gives as output the fun-damental component, in Fourier terms, of the out-put of the nonlinear element for a sinusoidal input.Doing this, the system is ’linearised’ in a harmonic

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Figure 2: Autonomous response

way. The manner to further analyse this harmoniclinearisation is then to apply sinusoidal force excita-tions with different amplitude- and frequency levels(�A�CB � � � G*?*>3V�W B ) to the mass and investigate the re-

sulting amplitude level and the phase shift of the dis-placement. In [12] it is shown that with this methodEquation 3 can be rewritten as:576� �YX�Z[�,+ � �&\� ��]^Z"�,+ � � � � � G�_a`cb � W Bd�fe � (5)

whereX�Z W and

]^Zrepresent an equivalent damping

and stiffness term respectively. Figure 3 shows theFrequency Response Functions (FRFs) resulting fromthis equation for different values of

� G . It can beseen that for low excitation amplitudes a mass-spring-damper characteristic is obtained. When the ampli-tude increases, the eigenfrequency decreases and thedamping increases. For large excitation amplitudes,the FRF shows a kind of massline characteristic. Inbetween, there is a region in which the FRF is verysensitive to small changes in the excitation amplitudeand therefore we call this region the ’sensitive’ or ’in-termediate’ region. For small excitation amplitudes

the resonance frequency

tends to G �hg'�i�j k=l� as

� G tends to zero, with] G � d

dx � � ��nmmm G , the derivativeof the virgin curve at zero.

To validate the results, a Simulink model has beenbuilt. Figure 4 shows the results of these simulations1 .These FRFs are obtained by applying a stepped-sineforce with increasing frequency on the model andmeasuring the phase and amplitude of the fundamen-tal component of the resulting steady-state displace-ment. It can be seen that again a mass-spring-dampercharacteristic is obtained for low excitation ampli-

1The excitation levels of the simulation and the describingfunction analysis are not the same.

NON-LINEARITIES: IDENTIFICATION AND MODELLING 1231

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Figure 3: FRFs with describing function approxima-tion

tudes and the massline characteristic for high exci-tation amplitudes. The curves in the intermediateregion are however less smooth than those obtainedwith the describing function approximation. This canbe explained by the fact that the conditions to applythis approximation are less satisfied when the exci-tation amplitude closely approximates the saturationlevel of the virgin curve of the hysteretic friction, andthese excitation amplitudes are indeed the ones of theintermediate region.

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Figure 4: FRFs with Simulink model

Based on the theoretical analysis, it can thus beconcluded that, for the autonomous case, an oscilla-tory behaviour is obtained with non-linear dampingcharacteristics. For the general excitation case no an-alytical solution can be obtained. A harmonic anal-

ysis however shows that for small force excitationamplitudes a mass-spring-damper characteristic is ob-tained and for large excitation amplitudes a kind ofmassline is obtained. The distinction between ’small’and ’large’ amplitudes is related to the saturation levelof the virgin curve. Higher(lower) amplitudes thanthis saturation level are referred to as ’large’(’small’)amplitudes. The next section will investigate if thesecharacteristics also can be obtained on practical ex-perimental set-ups. In particular, the behaviour ofthe system for excitation amplitudes around the sat-uration level of the hysteretic friction will be thor-oughly examined, as for this region the describingfunction approximation is less accurate and the be-haviour is very sensitive to the different system pa-rameters. This high sensitivity enlarges the influenceof measurement noise on the obtained results in thisregion, as is shown in [12] for some simulation cases.

3 Dynamic behaviour of a masssubject to hysteretic friction:Experimental validation

This paragraph describes different experimental set-ups that are used to validate the theoretical results de-scribed in the previous section. Since the hysteresisbehaviour can show position sensitivity, our intentionis to observe the different results qualitatively ratherthan quantitatively. The exact identification of thedifferent parameters that influence the behaviour isthus not of our main concern, only global character-istics are looked for. The influence and sensitivity ofcertain parameters in the identification process willhowever be quoted. These remarks can be used asa starting point in looking for appropriate dedicatedidentification schemes for the different phenomena.

Three experimental set-ups are successively de-scribed. The first one is a dedicated set-up to test thefriction behaviour of roller bearings an identify thehysteresis loops. The second set-up also is a dedicatedone but now an industrial roller slideway is used in-stead of a self assembled roller bearing. The third set-up consists of one axis of an industrial pick-and-placemachine. All the set-ups are first briefly described.Thereafter the hysteretic friction is characterised andsome stepped-sine identification experiments are per-formed. At the end of this section the results of thedifferent set-ups are compared and discussed.


3.1 Dedicated set-up with roller slide-way

Figures 5 and 6 show the set-up considered in thissubsection. It consists of two V-grooved slidewaysbetween which two balls roll. One slideway is fixed,while the other is actuated with a linear actuator(Bruel & Kjær 4810). The force on the slideway ismeasured with a force sensor (Kistler-GEPA 9031).The relative displacement between the two slidewaysis measured with a Bently displacement sensor (type3300). As Figure 5 shows, there is also a possibil-ity to vary the preload on the slideways. Also differ-ent slideways can be mounted and different balls canbe inserted in the set-up. This way, the influence ofthese variables on the hysteretic friction can be mea-sured and classified. These influences are howeverout of the scope of this paper, as it is only the inten-tion to validate the dynamical behaviour described inthe previous section, but can be found in [13]. Theslideways used in these experiments have a V grooveangle of 90 � and the diameter of the balls is 1,5mm2.

Figure 5: Dedicated set-up with roller slideway

Several sinusoidal force excitations with differentamplitudes are applied to the slideways and the re-sulting displacements are measured. Some of these

2It is not this configuration that is shown in the figures. Theconfiguration in the figures more clearly visualises the differentcomponents of the set-up and is therefore chosen to describe theset-up.

Figure 6: Detail of the set-up with roller slideway

curves are shown in Figure 7. The curves are ob-tained by averaging out several measurement loopsto reduce the influence of noise on the measurements.All the measurements are done at a frequency of 1Hzto minimise the influence of the inertial forces in themeasurement. The hysteretic character of the frictioncan clearly be seen in the figure and therefore thesecurves are called ’hysteresis loops’.

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Figure 7: Hysteresis loops on dedicated set-up withroller slideway

The hysteresis loops can be used to identify thevirgin curve. This identification can be done by fittinga curve to the measurements that obeys the propertiesmentioned in Section 2, for example Equation 4. Theparameters <$G and N are then optimised by minimis-ing the error between the fit and the experimental datain a least square sense. For the sake of brevity suchan identification is only presented for the second ded-icated test set-up.

Figure 8 shows the measured FRFs on the set-upfor some excitation levels. These measurements areobtained by a stepped-sine identification with increas-ing frequency. Figure 9 shows the experimental re-sults for a stepped-sine with decreasing frequency.

Investigation of Figures 8 and 9 shows that theresults qualitatively agree with the predicted be-


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Figure 8: Experimental FRF using stepped-sine exci-tation with increasing frequency

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Figure 9: Experimental FRF using stepped-sine exci-tation with decreasing frequency

haviour and thus support the theoretical analysis ofthe previous section. For this configuration

G �g'�i�j �� g G�� g�� g� �� R�� where the mass is mea-sured by weighing the components and

] G is ob-tained by approximating the derivative of the hys-teretic friction at zero displacement based on the mea-sured hysteresis loops of Figure 7. This

G �� R��is even much higher than the one predicted by theFRF measurement with the lowest excitation ampli-tude ( ��^�ED F�� ). Careful investigation of Figure 3however shows that

G can deviate substantially fromthe resonance frequency

for curves equivalent with

the ones experimentally measured. The results arethus not contradictory, but it would be interestingto do experiments with lower excitation amplitudes.This is however restricted by the accuracy of the mea-

surement system. It needs also be stressed that theidentification of

] G is hard to do since the derivativeof a measurement is needed, which enlarges the un-certainty region on

G . Since it is not the intentionof this paper to carefully identify the system param-eters, but only to investigate the qualitative dynamicbehaviour, the obtained result is considered adequate.

Some quantitative deviations which show up canbe explained as follows. The phase shift at high fre-quencies result from the different filters that are usedto reduce the noise of the position and force measure-ments. At low frequencies the measurements are lesssmooth than the results from the simulations. Thiscan be explained by the fact that most measurementsare done in the ’sensitive region’ and therefore smallmodelling errors or small deviations of the propertiesof the friction with respect to the position have a largeinfluence on the FRF measurements. This is prob-ably also the reason for the differences between theFRFs for the increasing and decreasing stepped-sines.Such deviations between increasing and decreasingstepped-sines are not observed in the Simulink model,and therefore it can be concluded that they do notresult from the hysteretic friction itself, but from anunmodelled phenomenon. Position dependency ofthe hysteretic friction parameters can be such a phe-nomenon. This observation also demonstrates that itis hard to identify these parameters accurately.

It would have been interesting to excite the sys-tem with higher force amplitudes but doing this thedisplacement amplitude of the slideway exceeds themeasurement range of the displacement sensor; thisis why a second test set-up is built.

3.2 Dedicated set-up with commercialslideway

Figure 10 shows the set-up considered in this subsec-tion. It consists of a commercial linear roller slideway(Schneeberger MR25 G2-A-02-09). In this set-up thecarriage is fixed, while the rail is driven by a linearvoice-coil. The position of this rail is measured withan optical laser encoder (Renishaw ML10).

Just as like for the roller slideways of the formerparagraph, different hysteresis loops are measuredand, based on these loops, the parameters of the vir-gin curve given by Equation 1 are optimised in a leastsquare sense. Figure 11 shows these measurementsand the identified virgin curve with < G � �/D �"F�� andN � �ED �� 5 . The measured mass of the slideway is1.143kg.


Figure 10: Set-up with industrial slideway

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Figure 11: Hysteresis loops for the set-up with indus-trial slideway

As Figure 11 shows, the fit corresponds quantita-tively well to the measurements. Better forms of thevirgin curve, with more parameter flexibility, can cer-tainly be found. However, as the purpose of this initialstudy is to validate the expected behaviour of Section2, and not to exactly identify all the parameters, thepresented fit is considered good enough.

Figure 12 shows the experimental response of thesystem to a force step of 3 N. The figure clearly showsthat a mass-spring-damper characteristic is obtainedfor such small force excitations. The non-linear char-acter of the response is hard to see immediately in thefigure, but is nevertheless present, as shown by morededicated analysis. This analysis is however not pre-sented for reasons of brevity. The small beat that canbe seen in the measurement results from other (lowerfrequency) resonances of the set-up. The presence ofthese resonances is shown next.

Figure 13 shows the simulated FRF response fora stepped-sine excitation with different excitation of1,3,4,5,6,7 and 8 N. Figure 14 shows the measuredFRFs on the set-up for a stepped-sine with increasing

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Figure 12: Response to a force step

frequency3 .

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Figure 13: Simulated FRF for set-up with industrialslideway

For this system G is equal to

g'�i�j g � G�� g G�� g � g�� k �� R�� based on the identified virgin curve. Similarconsiderations as for the previous set-up can be madeconcerning this

G . Also note that this G can be iden-

tified based on a step response as shown in Figure 12.Thus, if a good stepresponse measurement is possible,this method is more appropriate to use. The reason forthis is that, in the method based on the step response,the period of the vibration of a measured signal hasto be identified, whereas in the previous method thederivative of the measurement signal has to be identi-fied. It is a well known fact that numerical differen-tiation is an operation that is very sensitive to noise,making the previously used identification method lessattractive.

Comparison of Figures 13 and 14 shows that they3Note that the excitation levels of the simulation and the ex-

periment are not the same.


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Figure 14: Experimental FRFs for set-up with com-mercial slideway

again qualitatively agree, as the curves in both fig-ures cover approximately the same region. The regionresulting from the simulations is bigger than that ofthe experiments as the practically applicable excita-tion range is smaller than that of the simulations. Themost prominent difference between the figures is thatmany more resonances are present in the FRFs. Theseresonances originate from other structural flexibilitiesthat were not present in the first. One possible way tolower the frequency band of interest, and thus avoid-ing the other resonances in the FRF, is to add mass tothe system since the eigenfrequency is inversely pro-portional to the square root of the mass. Figure 15shows the experimental results when an extra mass of0.845 kg is added to the slideway.

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Figure 15: Experimental FRFs with extra mass added

It is clear to see that the eigenfrequency decreases

in this case but a very large mass has to be added to theslideway to lower the eigenfrequency to a reasonablylow value.

As in the previous set-up, the phase shift is dis-torted by the measurement equipment. The phaseshift goes to -200 � for high frequencies instead of thepredicted -180 � since the sampling rate is a little toosmall. Also, as in the previous set-up the measure-ments are less smooth than the simulations at low fre-quencies which again can be explained by the fact thatmost measurements are done in the ’sensitive region’.For this set-up it was not possible to excite the systemwith higher force amplitudes since the voice coil actu-ator cannot produce a high enough force. Such mea-surements are however possible with the third set-upwhich is described next.

3.3 Industrial pick-and-place machine

Figure 16 shows the set-up considered in this subsec-tion. It is an industrial pick-and-place machine pro-duced by Philips consisting of three linear motors andtwo rotative motors. The linear motor moves the armaround in the plane while one rotative motor rotatesthe arm around its axis and the other moves the arm upand down. In this experiment only the middle linearmotor (LiMMS, [3]) is used. This motor moves alongtwo linear slideways, each with two carriages (THKSR15W). The position of the motor is measured withan optical linear encoder with a resolution of 1 � 5(Heidenhain LIDA 201).

Figure 16: FlexCell

As in the former paragraphs different hysteresisloops are measured. Figure 17 shows these measure-ments. The mass of the middle motor and the armis 35.45kg which results in

G � g'�i�j g � g � g G�� k �� R�� .


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Figure 17: Hysteresis loops for the pick-and-placemachine

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Figure 18: Experimental FRFs on the FlexCell

Figure 18 shows some measured FRFs on the set-up for a stepped-sine with increasing frequency.Thisbehaviour is qualitatively very similar to that of Fig-ure 4. Figure 19 shows four FRFs with the same am-plitude

� G � K � D F�� . Two FRFs are obtained withan increasing stepped-sine and two with a decreas-ing stepped-sine. All the FRFs ’jump’ at a certainfrequency (for one of the decreasing FRFs this jumppoint is lower than 0,1Hz). These ’jump’ frequen-cies however differ for the different FRFs. Since thesejumps seem to occur at arbitrary (unrepeatable) pointswithin the sensitive region, the authors believe thatthese jumps results from a very high sensitivity of theFRF to the exact parameters of the virgin curve forexcitation levels in the ’sensitive’ region. In particularthe saturation level of the hysteresis plays a dominantrole in the ’jumps’. Because the parameters are posi-tion dependent, it is very hard to get exactly the sameresults when repeating the measurements. Note withrespect to this also that frequency scale of Figure 18

is different from that of Figure 19. This is because themeasurements are carried out at different positions ofthe linear motor which have slightly different satura-tion values. The saturation value of Figure 19 is alittle higher than that of Figure 18 and these curvesare thus not situated between the

� G � � D �"F�� andK � D �"F�� curve of Figure 18, but between the K � D �"F��and K*F D �"F�� curve. The differences between the FRFsof Figure 19 for this set-up are similar to those ob-served for the dedicated set-up in Paragraph 3.1.

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Figure 19: FRFs on the FlexCell with� G � K � D F��

for increasing and decreasing frequencies

3.4 Discussion and comparison of thedifferent set-ups

All the set-ups considered in this section validatethe dynamic behaviour predicted by the theoreticalanalysis in [11, 12] qualitatively. For low excita-tion amplitudes a mass-spring-damper characteristicis obtained, for high excitation amplitudes a kind ofmassline characteristic.

The origin of the quantitative deviations differs forthe different set-ups. For the first set-up the non-smoothness of the FRFs and the differences betweenthe results for increasing and decreasing stepped-sines can be explained by the fact that the FRFsare very sensitive to the saturation level of the hys-teretic friction and this level can be position depen-dent. Also, the moving slideway is not perfectly kine-matically guided so that tilting is possible which influ-ences the hysteretic parameters. On the second set-upthe characteristic resonance frequency resulting fromthe hysteretic friction is much higher than that in thetwo other set-ups. As there are other structural res-


onances with lower resonance frequencies, these popup in the FRFs. In the last set-up similar considera-tions can be made as for the first set-up. The motoris kinematically over-determined, which can result ininternal stresses that influence the parameters of thefriction. In this way the hysteretic parameters becomeposition dependent.

It can thus be concluded from the performed ex-periments that the theoretically predicted dynamic be-haviour is predominantly present in the set-ups andthe hysteretic friction behaviour is practically veryrelevant to investigate. The influence of other fric-tion phenomena, like the Stribeck effect, has a smallereffect on these measurements than the displacementdependent hysteretic friction. This is so because thedifference between the dynamic and static frictioncoefficient is rather small for roller bearings as hasbeen also found in [10] and [4]. In [10] the perfor-mance of different friction models, when used in thefeed-forward calculation of a controller for mechan-ical systems, is experimentally compared. These ex-periments revealed that the displacement dependentfriction characteristics are far more important than thevelocity dependent ones in such calculations.

The different experiments also show that the mag-nitude of the different parameters can differ signif-icantly. In the second set-up

G is for examplemuch higher than for the other two set-ups. This re-sults from a higher saturation value and a lower slip-distance. It can be interesting for manufacturers ofcommercial slideways to identify and classify thesecharacteristics for their slideways as the can signifi-cantly influence the dynamical behaviour of the sys-tems they are used in, as is shown in this paper.

4 Conclusions and further re-search

In this paper, the theoretical analysis of mass-hysteretic-spring systems as reported in [11, 12] havebeen validated experimentally on three set-ups: onededicated set-up with roller slideway and one with anindustrial slideway and one axis of an industrial pick-and-place machine. Different stepped-sine experi-ments on these set-ups show qualitatively the theoret-ically predicted behaviour and therefore support thetheoretical results and the presence and importance ofhysteretic friction in roller slideways. A mass-spring-damper characteristic is obtained for low excitationamplitudes a kind of mass characteristic for high ex-citation amplitudes. The tests on the different set-ups

also showed that it is hard to accurately identify theparameters of the hysteretic friction in such a way thatall experiments can accurately be simulated. The rea-son for this is that other phenomena, such as otherstructural resonances, can impair the measurementsand that the parameters are position dependent. Basedon all these observations can be stated that there is asignificant region in the amplitude-frequency domainwhere the response of the system is very sensitive tothe system and excitation parameters. As this is aninherent property of systems with hysteretic friction,and does not result from external parameters, the re-sponse of the system in this region is hard to identify,even if dedicated and accurate measurement tools areused. In fact, if one would succeed in identifying thesystem in this region, it would be almost impossibleto use this information to modify the characteristicsof the system, due to the high sensitivity of the pa-rameters, and thus the identification results would beagain practically useless.

Because of the importance of hysteretic frictionon the dynamics of mechanical positioning systemsthis non-linear phenomenon should not be neglectedin the control design of such systems. Even when de-signing linear controllers the presence of an hystereticfriction component should be kept in mind when iden-tifying a linear model of the system. In further re-search, therefore, different ways to deal with hys-teretic friction in the control design will be investi-gated. More specifically, the influence of hystereticfriction on the feedback compensator design will beinvestigated. Also ways to deal with this non-linearbehaviour more rigorously in the feedback controllerdesign will be studied.

Acknowledgements

This research is sponsored by the Belgian programof Interuniversity Poles of Attraction by the BelgianState, Prime Minister’s Office, Science Policy Pro-gramming (IUAP). W. Symens is a Research Assis-tant of the Fund for Scientific Research - Flanders(Belgium) (F.W.O.). The authors wish to thank ErikDe Zeeuw and Diederik Douwen for their help in per-forming some of the experiments presented in this pa-per and Vincent Lampaert to put his set-up at our dis-posal. The scientific responsibility is assumed by itsauthors.


References

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[3] P. Heijmans. LiMMS motor module. PhilipsCentre for manufacturing Technology, CFT,Eindhoven, 1995.

[4] R. Hensen. Controlled Mchanical Systems withFriction. PhD thesis, Technical University Eind-hoven, February 2002.

[5] Mayergoyz I.D. Mathematical models of hys-teresis. Springer-Verlag, New York, 1991.

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[10] Lampaert V., Swevers J., and Al-Bender F.Experimental comparison of different frictionmodels for accurate low-velocity tracking. InMediterranean Conference on Control and Au-tomation. Submitted for publication.

[11] Symens W., Al-Bender F., Swevers J., andVan Brussel H. Dynamic charaterisation ofhysteresis elements in mechanical systems. InAmerican Control Conference, pages 4129–4134, May 2002.

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Harmonic analysis of a mass subject to hysteretic friction ... · [11 ,12 ]. This paper aims to validate these theoretical results on three experimental set-ups. T w o dedicated set-