Experimental and numerical analysis of self-excited friction oscillator

Chaos, Solitons and Fractals 12 (2001) 1691±1704

www.elsevier.nl/locate/chaos

Experimental and numerical analysis of self-excited friction oscillator ski a,*, Jerzy Wojewoda a, Kazimierz Furmanik b Andrzej Stefan a

Technical University of è odz, Division of Dynamics, Stefanowskiego 1/15, 90-924 è odz, Poland b University of Mining and Metallurgy, Mickiewicza 30, 30-059 Krak ow, Poland Accepted 6 June 2000 Communicated by T. Kapitaniak

Abstract A self-excited friction oscillator has been designed and manufactured to carry out experimental analysis of dry friction phenomenon. A mathematical model of this oscillator has been formulated. The in¯uence of the di€erent types of classical friction characteristics on the dynamical behaviour of the model is investigated by way of numerical analysis. A comparison with dynamics of real oscillator is presented and some reasons of observed di€erences are explained. A particular analysis of experimental data leads to the con®rmation of non-reversible friction characteristics and allows to formulate a hypothesis that a course of such characteristics also depends on value (not only on the sign) of acceleration. Ó 2001 Elsevier Science Ltd. All rights reserved.

1. Introduction Dry friction appears in many, if not all, mechanical systems met in practice of most engineers. Usually, this is an unnecessary e€ect and a most unwanted one. Such self-excited vibrations cause an early wear of the contacting parts of machines and if not under control could cause too much greater consequences. It has been tried to identify the causes of unwanted behaviour of the machines, like: the scream of car tires, brakes and similar sounds. The problems with dry friction appear also in design and exploitation of railvehicles and machines [9]. Stick±slip type behaviour is closely connected with the self-excited vibrations. Such behaviour implicates appearance of the discontinuous di€erential equations in the mathematical model of the discussed system. Detailed character of this discontinuity depends on the friction character assumed. These types of mathematical models arise from description of processes, which are partially continuous and contain periods of stick phases with relatively high friction coecient and slip where this coecient is much smaller. These are the reasons why systems with dry friction possess many di€erent types of dynamical behaviour, like periodical changes, non-periodic, chaotic and sometimes even static [2,8,11,14]. Traditional engineering treatment, due to Coulomb, usually simpli®es friction force in the form of constant value vector directed opposite to the relative velocity of the contacting bodies. Such a vector can take two values with identical level but opposite sign only. Then, it depends on the value of the friction coecient multiplied by the normal force. Newer experiments show non-linear dependence on the contact

*

Corresponding author. E-mail addresses: [email protected] (A. Stefa nski), [email protected] (J. Wojewoda), [email protected] (K. Furmanik). 0960-0779/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 0 - 0 7 7 9 ( 0 0 ) 0 0 1 3 6 - 3

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velocity rather than the constant one. That was why most e€orts were directed to build non-linear friction models and to determine di€erences in maximal values of the static and dynamic friction forces (see [5]). Another attempt to determine di€erent types of friction characteristics ± showing dependencies on the relative acceleration on the contacting surface ± the so-called non-reversible friction characteristics are shown in [1,7,12]. An Important part of the above investigation is the mechanism of transition from and into self-excited vibration. This is usually done by numerical simulations based on non-linear dynamics and bifurcation theory [3,6,10,13,14]. In this paper, some of our results are derived from the experimental rig which is able to generate self-excited vibration. Analysis of the experimentally obtained data allow to state thesis of positive veri®cation of the existence of non-reversible friction characteristics, and, additionally state another thesis ± its dependence not only on the sign of the acceleration but also on its value as well. 2. Experimental rig The rig which is a generator of self-excited vibration is designed and run by Dr K. Furmanik at Cracow Mine and Foundry Academy. General view of the rig is shown in Fig. 1. It consists of two main, steel, coaxially positioned parts: rotating disc with external disc ring (1); friction segment (2), positions of displacement (3), velocity (4) and acceleration (5) sensors are also marked. Scheme of the oscillator is presented in Fig. 2. Friction segment is connected to the ®xed base of the rig by circumferential spring of linear characteristics and the sti€ness k. A sensor ®xed nearby measures the force in the spring and additionally, the displacement of the whole assembly. Friction element is pressed against external disc ring by radial spring of sti€ness k1 with a sensor measuring the pressure force. The source of energy in the entire system is a rotating disc (1) driven by an electrical motor, whose speed can be continuously controlled. The contact point where dry friction appears states a junction and its presence allows self-excited vibration of the segment (2) with friction element (3) to occur during the disc rotation. Disc rotation speed is measured with another sensor and all signals were connected to the PC based Data Acquisition System Microstar DAP 3200A. This system also allows on- and o€-line analysis of recorded data. Detailed parameter values are given in Appendix A.

Fig. 1. Experimental rig.

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3 Fig. 2. Scheme of the oscillator (see description in Section 1). Fig. 3. Tangent forces acting at the vibrating segment with friction element.

3. Mathematical model Forces acting in the tangent direction at the segment with friction element during disc rotation are shown in Fig. 3. Fig. 4(a) shows radial components of forces acting at the segment and Fig. 4(b) shows radial components of forces at friction element. Detailed list of nomenclature used is presented in Appendix A. From the scheme in Fig. 3, a dynamical equation of moments acting at the segment with friction element can be written as _ ˆ Mf …/_ †; …I ‡ I1 †/ ‡ Ms …/† ‡ Md …/; /† r

…1†

where Ms is the moment of the force in circumferential spring, Md the moment of the force in the bearing and Mf the moment of the friction force between friction element and disc ring. In the ®rst of mathematical models considered here (model I described later in the paper), let us assume all system parameters being constant in time, friction surface (external ring of the disc) is ideally perfect, no eccentricity in disc ®xing point at its rotation axis, zero value of the reaction force deviation in the bearing d, (this angle is to be omitted with respect to the friction moment in the bearing and its mathematical description can be highly complicated). Schemes of radial loads of the segment (Fig. 4(a)) and the friction element (Fig. 4(b)) allow to formulate radial pressure forces as follows: _ ˆ P ‡ m1 Rc1 /_ 2 ; N …/†

Fig. 4. Radial forces acting at the vibrating segment (a) and friction element (b).

…2†

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_ ˆ P ‡ kRs sin2 / ‡ m1 Rc /_ 2 : Z…/; /†

…3†

The force in the circumferential spring is S…/† ˆ kRs sin /:

…4†

Using Eqs. (2)±(4) in Eq. (1) we obtain: 1 Ms …/† ˆ SRs cos / ˆ R2s sin 2/; 2

…5†

h i _ ˆ Z…/; /†r _ 0 f1 sign/_ ˆ P ‡ kRs sin2 / ‡ m1 Rc …/† _ 2 r0 f1 sign/; _ Md …/; /†

…6†

h i _ …vr †R sign/_ ˆ P ‡ m1 Rc1 …/† _ 2 f …vr †R sign/_ : Mf …/_ r † ˆ N …/†f r r

…7†

Then after rearranging the full form of the equation of motion the following formula for the friction force is obtained: h i _ 2 r0 f1 sign/_ …I ‡ I1 †/ ‡ 12 kR2s sin 2/ ‡ P ‡ kRs sin2 / ‡ m1 Rc …/† h i f …vr † ˆ ; …8† _ 2 R sign/_ P ‡ m1 Rc1 …/† r where vr ˆ …x0

_ /†R:

…9†

After substituting real parameter values in Eq. (8), we are able to get real dependence of the friction coecient on the disc-friction element relative velocity during slip phase ± vr 6ˆ 0. Then, after approximating this real characteristics with chosen model of the friction, we can use it in numerical model of the system. Observation and identi®cation of the di€erences between dynamics of the real system and dynamics of its model allows to correct the model or assumed friction function force aiming for maximum similarity of the simulation and experimental data. For simulation needs of Eq. (1) we can denote it in the form of a system of the ®rst-order di€erential equations. After dividing Eq. (1) by …I ‡ I1 † and substituting / ˆ /1 and /_ ˆ /2 , we get another form of Eq. (1): /_ 1 ˆ /2 ; 1 /_ 2 ˆ ‰Mf …vr ; /2 † I ‡ I1

Ms …/1 †

Md …/1 ; /2 †Š

…10†

for the slip phase of the contacting surfaces …vr 6ˆ 0† and /_ 1 ˆ x0 ; Mn …x0 ; fs † ˆ Ms …/1 † ‡ Md …/1 ; x0 †

…11†

for the stick phase of the contacting surfaces …vr ˆ 0†, where Mn …x0 ; fs † is the moment of undeveloped friction force. The boundary between stick and slip phases is described by inequality resulting from Eq. (11): 1 2     kR sin 2/ ‡ P ‡ kRs sin2 / ‡ m1 Rc x2 r0 f1 6 fs P ‡ m1 Rc1 x2 R: …12† 0 0 2 s During stick phase the moment of undeveloped friction force is statically balanced with the circumferential spring moment and much smaller moment of friction in the bearing. When the sum of the above moments exceeds maximum value of static friction moment (right-hand side of Eq. (12)) slip phase begins, during which dynamical moment of inertia force acting on the segment appears, and friction moment depends on

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momentary angular and relative velocities (Eq. (7)). Stick phase periodically returns when relative velocity approaches zero (Eq. (9)), and, when at the same time Eq. (12) is ful®lled. These oscillations take place around static equilibrium position, whose angular co-ordinate is described by   1 2PRf …vr ˆ x0 R† /s ˆ arcsin : …13† 2 kR2s Value of the static co-ordinate of the equilibrium position (Eq. (13)) is obtained from equality of the friction moment (Eq. (7)), and, circumferential spring force moment (Eq. (5)), with ®xed segment …/_ r ˆ w0 ; /_ ˆ 0†. In the second mathematical model analysed here (model II described later here) some non-zero value of the disc eccentricity e in its rotation axis is assumed. This aims for simpli®cation in modelling instabilities of the real system motion. When e 6ˆ 0, then friction element with mass m1 has additional motion possibility in radial direction. Scheme of load applied to this element is shown in Fig. 5. Equation of its motion in such case in radial direction is _ 2 Rc …/ † ˆ N …r; / †; m1r ‡ k1 r ‡ m1 …/† r r

…14†

where radial displacement of the friction element is r ˆ rk ‰ 1

e sin /r Š:

…15†

Base value of the rk displacement in Eq. (15) results from initially introduced radial spring pressure rk ˆ P =k1 , see Fig. 5, and relative angular displacement of the disc and friction element /r can be obtained by integrating relative angular velocity Z t2 _ dt /r ˆ …x0 /† …16† t1

After double di€erentiation of Eq. (15) and substitution in Eq. (14) formula describing momentary pressure between friction element and disc ring in function of relative displacement is obtained. Using this in segment with friction element equation of motion and after variables separation …/ ˆ /1 ; /_ ˆ /2 ; /r ˆ /3 † the following set of di€erential equations modeling the oscillator during slip phase (with eccentricity) is obtained: /_ 1 ˆ /2 ; Mf …vr ; /2 ; /3 † Ms …/1 † Md …/1 ; /2 ; /3 † /_ 2 ˆ ; I ‡ I1 …/3 † ‡ a…/3 † /_ ˆ x0 / ; 3

2

Fig. 5. Scheme of load applied to the friction element if non-zero eccentricity e is assumed.

…17†

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where 2

I1 …/3 † ˆ m1 ‰Rc1 …1 ‡ e sin /3 †Š ; a…vr ; /3 † ˆ m1 R…1 ‡ e sin /3 †

P e cos /3 f …vr †; k1

 Md …/1 ; /2 ; /3 † ˆ f1 r0 P …1

e sin /3 † ‡ kRs sin2 /1

 mRc /22 sign/2 ;

 Mf …vr ; /1 ; /2 † ˆ P …1

e sin /3 † ‡ mRc1 /22 …1 ‡ e sin /3 †  ‡ mr0 e sin /3 …x0 /2 † R…1 ‡ e sin /3 †f …vr †sign…x0

/2 †;

circumferential force moment Ms …/1 † remains the same after introducing eccentricity into the mathematical model and is described by Eq. (5). During stick phase the model is identically described as simpli®ed case (Eq. (11)). Non-zero value of the eccentricity e results in the third dimension of the model phase space (Eq. (17)). This means possibility of quasi-periodic and chaotic solutions of the system, contrary to the model with zero eccentricity (Eq. (11)), where only periodic solutions are possible.

4. Experimental results versus numerical simulations Numerical modeling of the dynamical system with dry friction needs to assume some friction characteristics. Its optimal shape requires analysis of experimental data. Then, after approximation of these data with assumed friction model and applying in simulation process, comparison is available. Such characteristics can be obtained as function of the kinetic friction fk versus relative velocity vr with parameters and variables values assumed on experimental data used in Eqs. (8) and (9). Experimental data recorded from sensors were visualised, recorded and analysed in the DAP system. Five signals were chosen for digital records: displacement x; velocity v and tangent acceleration a at the measurement radius; force in the radial spring P; disc rotation speed n0 . Typical data record interval was 50 s (digitised every 0.001 s), tangent velocity measurement radius equal to 0.075 m, displacement and acceleration measurement point radius was 0.22 m (see Fig. 1). Usually data were recorded for some chosen disc rotation speed n0 values. Then, _ /;  P ; x0 . results were analysed in software way (Excel, Origin), where real values were obtained as: /; /; After substituting them in Eqs. (8) and (9), experimental picture of investigated friction function is obtained. In Fig. 6, the approximations of this real friction function which has been done using three di€erent friction models: (I) Classical Coulomb model (Fig. 6(a)), which assumes constant static friction coecient fs and smaller, also of constant value kinetic friction coecient fk .  vr ˆ 0; fs ; f …vr † ˆ …18† fk signvr ; vr 6ˆ 0: (II) Stelter±Popp model [5] (Fig. 6(b)), which has non-linear friction characteristics described as:   fs fk 2 f …vr † ˆ …19† ‡ fk ‡ g2 vr sign…vr †: 1 ‡ g1 jvr j (III) Non-reversible friction characteristics [12], which shape depends on the sign of relative acceleration ar in the form: ( f …vr † ˆ

fk ‰1 ‡ fs fkfk exp… ajvr j†Šsign…vr †; fk ‰1 exp… bjvr j†Šsign…vr †;

signar > 0; signar < 0:

…20†

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Fig. 6. The approximations of experimental friction function (shown in grey) using 3 friction models: Coulomb model (a); Popp± Stelter model (b); non-reversible friction characteristics (c); fs ˆ 0:15, fk ˆ 0:10, g1 ˆ 500, g2 ˆ 0:1, a ˆ 100, b ˆ 1000.

Numerical simulations of the investigated oscillator showed, that choice of any of the above friction models does not in¯uence to the system dynamics and character of its solution in the phase space. Only after applying more detailed analysis one can observe di€erences in behaviour of models with di€erent friction characteristics. This is the reason why in the summary presented below only friction model (I) is shown in Fig. 6(a). Phase portraits showing dynamics of the investigated oscillator are shown in Figs. 7±9. They were generated from the experiment and numerical simulation from mathematical models (I) and (II) (with zero and non-zero eccentricity values). Visualisation ranges are the same on all of the phase portraits what makes comparison easier. Figs. 7(a), 8(a) and 9(a) show rather chaotic dynamics of the system, but this does not seem to be a typical deterministic chaos caused by non-linearity or discontinuity of equations of motion. Irregular vibrations of the segment with friction element seems to be an e€ect of in¯uence of factors not considered in mathematical model, such as errors on friction surface, irregular cooperation friction element and the disc ring, unbalanced of the disc rotation speed, etc. Irregular motion of the oscillator can be summarised result of the above factors. It presents oscillating decrease or increase of the vibration amplitude, micro-slips and attractors inclination observed during stick phase caused by unbalanced motion of the disc. Experimental friction characteristics (Fig. 6) presents some area shape form inside which all traces pass, instead of regularly linear or non-linear shape. Numerically generated attractors shown in Figs. 7(b), 8(b) and 9(b) show limit cycles with periodically repeated patterns of stick and slip phases. This picture is an ideal form possible in ideal conditions only when no manufacturing or assembling errors occur, rotation speed of the disc is ideally stable and all other parameters and conditions do not introduce any meaningful disturbances. In such model there is no chance to observe chaotic behaviour due to the fact that it is described in two-dimensional phase space, where according to Poincare± Bendixon theorem chaotic motion is impossible [4]. Introduction of non-zero value of the eccentricity e in mathematical model (II) caused disappearing of periodic motion with constant amplitude (Figs. 7(c), 8(c) and 9(c)). Vibration amplitude varies during disc motion. All changes present character of quasi-periodic

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Fig. 7. Phase portraits showing dynamics of the investigated oscillator obtained from: experiment (a); numerical simulations based on the mathematical model (I) with e ˆ 0 (b); based on the model (II) with e ˆ 0:0002 (c); n0 ˆ 0:25 rev/min.

attractor or a limit cycle with very long period. Such period should be equal to the period of disc rotation. Comparing both experimental and numerical phase portraits we can say both ranges of segment vibration are very close. Similar shape and size of attractors in phase space presented in Figs. 7±9. It can also be seen that both systems real and numerical oscillate around similar static equilibrium position /s . Its value calculated from Eq. (13) equals /s ˆ 0:00219 rad (under condition f …vr † ˆ fk ˆ 0:1 obtained from friction characteristics (I)). In the numerical model, only one disturbing factor has been considered ± the eccentricity e. Its in¯uence to the model dynamics is seen after comparison phase portraits in phase space in Figs. 7(b), 8(b) and 9(b) and 7(c), 8(c) and 9(c), respectively. Non-zero eccentricity value has broke ideal periodic dynamics of numerical model and made it more similar to the real one. But, apart from the eccentricity there are few more, mentioned above factors, which disturb motion of discussed oscillator. We assume further modeling of these factors could make simulation results much more like experimental ones.

5. Friction and relative acceleration The basic aim of our investigation was the analysis of the friction characteristics. As it has been said in the previous chapter the choice of discussed model of friction has no meaningful in¯uence on the global dynamics of a system with dry friction in numerical simulations. But identi®cation of the real character of friction characteristics in function of relative velocity is the key point in proper description of the dry friction. Detailed analysis presented here has been done for minimal disc rotation speed equal to 0.25 rev/ min. At such small speeds any disturbance caused by varying of system parameters, mostly by friction condition at disc circumference is smaller in the considered experiment duration.

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Fig. 8. Phase portraits showing dynamics of the investigated oscillator obtained from: experiment (a); numerical simulations based on the mathematical model (I) with e ˆ 0 (b); based on the model (II) with e ˆ 0:0002 (c); n0 ˆ 1 rev/min.

Fig. 10 shows a fragment of time history of the friction element and disc relative velocity. Stick and slip phases are distinct, and one can observe even during stick phase, a change in relative velocity takes place, what can be connected to some disturbances in disc rotation speed. Additionally, using experimental data collected during stick phase friction characteristics cannot be determined due to fact the relative velocity is zero and friction force is not yet fully developed. This is why in our investigation only the results obtained for values of the relative velocity vr > 0:01 m/s, where real stick phases are not included, are meaningful. Fig. 11 shows real friction characteristics in two phases of motion ± relative acceleration and deceleration of the oscillator friction surfaces. Experimental results are marked in grey and approximated by a power trend line. One can see that the friction characteristics are di€erent in both the above-mentioned phases of motion. Relative velocity trend line shows a decreasing nature when this velocity increases (Fig. 11(a)) (positive exponent of the trend line equation), then after change of the relative acceleration such an inclination still remains (Fig. 11(b)) (negative exponent of the trend line equation), which agrees with results of other authors [1,7,12] and con®rms to the non-reversible dry friction character. For further analysis of experimental series recorded during deceleration phase of the relative motion, from maximum to zero value of relative velocity were analysed. Then, the above data were gathered in two separate ®les, depending on the mean value of the relative acceleration ar , which was calculated as arithmetic mean of all acceleration values in considered ®le. Friction characteristics generated from the above-described ®les are presented in Fig. 12. Approximated characteristics have distinct shapes for any two di€erent values of mean relative acceleration ar . From these results we can formulate the following hypothesis: The shape of non-reversible dry friction characteristics depends not only on the momentary value of relative velocity and sign of the relative acceleration but also on the value of the acceleration.

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Fig. 9. Phase portraits showing dynamics of the investigated oscillator obtained from: experiment (a); numerical simulations based on the mathematical model (I) with e ˆ 0 (b); based on model (II) with e ˆ 0:0002 (c); n0 ˆ 1:5 rev/min.

Fig. 10. The time diagram of the friction element and disk relative velocity.

When such a hypothesis is considered in physics of the dry friction model it leads to appearance of signi®cant similarities between types of solutions of system with dry friction and gives a picture of its friction characteristics. If motion of the investigated system is of regular nature, i.e., has a periodic or multiperiodic attractor, then change of relative velocity and acceleration repeat for every period of system motion. According to the above hypothesis, cycles of changes in friction coecient values as relative velocity function are also repeatable.

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Fig. 11. Relative velocity trend line (its equation is shown in the top part) in: acceleration phase (a); deceleration phase (b).

Fig. 12. The approximate friction characteristics for two di€erent values of mean relative acceleration (given in the top part of the picture).

Then, non-reversible friction characteristics possess a shape predicted in mathematical description of its model (Eq. (20)) and contains two lines representing relative acceleration and deceleration phases only (Fig. 13(a)). In case of motion with multiplicated period, friction characteristics can take shape as shown in Fig. 13(b). Appearance of additional lines in these characteristics is caused by existence of sub-orbits of such limiting cycles at which di€erent traces of relative velocity and acceleration are realised. In case of a chaotic attractor, there exist in®nitely many unstable orbits lying on this attractor. As an e€ect there appear in®nitely many possible traces, which ®ll an area of friction characteristics graph, as phase portrait of chaotic attractor ®lls some area of the phase space (Fig. 13(c)). Our hypothesis has been formulated only using experimental results recorded during motion phase with decreasing relative velocity (Fig. 12(a)). If it is correct, we can assume that a similar detailed e€ect of friction characteristics in phase of increasing relative velocity could appear. This must yet be shown in next experiments. 6. Conclusions This paper presents analysis an of the experimentally recorded data and numerical simulations of the oscillator with self-excited vibrations caused by dry friction. Analysis of the physics of the dry friction behaviour and numerical con®rmation of the results were primary tasks of the investigation. An agreement

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Fig. 13. The dependence between friction characteristics and system motion character arising from presented hypothesis for (see Section 6): simple periodic motion (a); multi-periodic motion (b); chaotic motion (c).

obtained is of the quantity level ± experimentally and numerically generated attractors of the considered system both have overlapped regions in which they exist in phase space. This happens independently on friction model assumed in numerical simulations. Detailed analysis of real system vibration show their chaotic character contrary to periodic behaviour of the model. It has been detected that the reason for chaotic behaviour in this case is mainly caused by external disturbances in the system. After including one of the disturbing external factors ± the disc eccentricity, a much better agreement in simulation results was obtained. A general conclusion can be stated as, a probability of closer to real picture prediction of dynamical behaviour of the system rises when one considers a more complicated model, obtained by including external disturbances. In the opposite way, when eliminating the disturbances from the real system, a simpler model can ful®ll simulation requirements. Such a conclusion can be extended to most of the dynamical systems, not being limited to the ones with dry friction only. Detailed analysis of experimental records con®rmed dependence of the dry friction coecient on relative velocity in a non-reversible way. The term of non-reversibility, introduced by Powell and Wiercigroch [7], means an existence of two di€erent traces of friction characteristics in relative acceleration and deceleration

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phases. Analysis of friction characteristics dependence on mean value of the relative velocity showed a relation between this acceleration and friction characteristics shape. On the basis of our results, we have stated the hypothesis that non-reversible friction characteristics in the shape proposed by Powell and Wiercigroch very well approximates real character of friction force only in case of simple, periodic motion of dynamical system, when mean relative velocity calculated for the whole period is constant. In case of motion with multi-periods, our hypothesis predicts the possibility of appearance of additional traces in friction characteristics in function of relative velocity, due to few or more acceleration and deceleration phases (with di€erent values of mean relative acceleration) during the motion. According to the hypothesis, chaotic motion of system with dry friction leads to the presence of in®nitely many friction characteristics ®lling an area in the graph of friction force versus relative velocity. Graphical presentation of such a conclusion is presented in Fig. 13. Here, friction characteristics is a representation of dynamical system motion character. However, this is not a simple mapping which shifts description of system dynamics from one co-ordinate system to another, because friction characteristics are an integral part of dynamical system model and also infuence its motion. We have to stress the following ± the hypothesis presented needs further veri®cation in next series of experiments which are to be done in near future. Acknowledgements This paper has been prepared with support of the Polish Committee of Scienti®c Investigation (KBN) under the Project No. 7T07A01011. Appendix A 1. System parameters and their real values: f1 ˆ 0:01 (friction coecient of the bearing) I ˆ 0:0337 …kg m2 † (mass inertia moment of the segment related to its rotation axis) I1 ˆ 0:00259 …kg m2 † (mass inertia moment of the friction element related to its rotation axis) k ˆ 11632:8 (N/m) (circumferential spring sti€ness) k1 ˆ 49050 (N/m) (radial spring sti€ness) m ˆ 0:84 (kg) (segment mass) m1 ˆ 0:05654 (kg) (friction element mass) r0 ˆ 0:01 (m) (central bearing radius) R ˆ 0:2475 (m) (internal radius of the disc ring) Rs ˆ 0:22 (m) (radius of the ®xing point of the circumferential spring) Rc ˆ 0:1833 (m) (segment mass centre radius) Rc1 ˆ 0:2125 (m) (friction element mass centre). 2. System variables: a …m=s2 † (tangent acceleration) ar …m=s2 † (relative tangent acceleration at contact point) ar …m=s2 † (mean relative acceleration at contact point) e (non-dimensional disc eccentricity) f …vr † (friction force function) fs (static friction coecient) fk (kinetic friction coecient) n0 (rev/min) (rotational disc speed) N (N) (dynamic pressure force between friction element and disc ring) P (N) (static pressure force between friction element and the disc ring (force in radial spring k1 )) r (m) (radial displacement of the friction element) S (N) (force in circumferential spring k) t (s) (time)

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