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ARTICLE IN PRESS JOURNAL OF SOUND AND VIBRATION Journal of Sound and Vibration 269 (2004) 61–89
   JOURNAL OFSOUND AND VIBRATION www.elsevier.com/locate/jsvi Journal of Sound and Vibration 269 (2004) 61–89 Effect of high-frequency excitation on a class of mechanical systems with dynamic friction S. Chatterjee*, T.K. Singha, S.K. Karmakar Department of Mechanical Engineering, Bengal Engineering College, Deemed University,P.O. Botanic Garden, Howrah-711103, West Bengal, India Received 9 August 2002; accepted 19 December 2002 Abstract The effect of high-frequency excitation on a class of systems with friction is considered. Friction isrepresented by the LuGre and the elasto-plastic model of friction. Analytical expressions are obtained forthe effective friction characteristics under two types of fast excitation. Numerical simulation usingMATLAB validates the analytical results. The stability of a velocity tracking system with friction isdiscussed in light of the effective friction characteristics. Numerical simulation of a MATLAB t SIMULINK model is carried out to unfold the basic physical mechanism underneath the mathematicalexpressions. r 2003 Elsevier Ltd. All rights reserved. 1. Introduction Friction between two sliding surfaces plays a central controlling role in the dynamic behaviourof a number of systems. Various complex dynamical features like stick–slip motion, self–excitedand chaotic oscillations are often identified with the presence of friction in joints and contactinterfaces. Research on dynamical systems with friction has a long and rich history [1]. Recentadvances in the precision mechatronic systems have brought a fresh impetus in the researchinterest on this topic. Except in a few cases, the side effects of friction lead to loss of functionalaccuracy of many systems and call for compensatory control arrangements. Though Coulomb’sdry friction model is useful for the majority of rough engineering calculations, it is hardly suitablefor some sophisticated applications because this model disregards the microscopic degrees of freedom of contact, which are important for some applications. For example, in precisemechatronic systems, the velocity and length scale of the motion becomes comparable to the ARTICLE IN PRESS *Corresponding author. E-mail address:  shy@mech.becs.ac.in (S. Chatterjee).0022-460X/03/$-see front matter r 2003 Elsevier Ltd. All rights reserved.doi:10.1016/S0022-460X(03)00004-X  length and time scale involved in the microscopic degrees of freedom of contact. Suchunderstanding has led to the development of new phenomenological models of friction [2–4] andvarious compensation techniques. One of these compensation techniques considers high-frequency oscillation (dither) [5–7] in mitigating some evil effects of friction and shows a greatdeal of promise.Engineers in reducing the effect of friction have used high-frequency oscillation long back [8,9].However, only very recently, a systematic study of this effect has been initiated in a more generalsetup. Research [10–12] has revealed that any excitation operating at a time scale much fastercompared to the natural time scale of a system may bring forth non-trivial changes in thedynamics of non-linear systems. In recent literature, such excitations are termed as fast vibration.Fast vibration have been shown to effectively change certain characteristics of mechanical systemssuch as equilibrium states [13], linear stiffness [14], damping [15] and natural frequencies [16]. Suitably designed fast excitation may also significantly influence certain non-linear features likerestoring and energy dissipation characteristics, frequency response and bifurcation behaviour of non-linear systems. Another very important application of fast vibration is pivoted around thesmoothening effect of the same on discontinuous system characteristics. Dry-friction character-istics are phemenologically described by a discontinuous fiction–velocity relationship, thediscontinuity being at zero velocity, i.e., when velocity reversal takes place. Velocity reversal ofteninvolves sticking of two interacting surfaces, and during sticking phase, friction force assumes avalue that depends on the external load. Thus, there exists a difference in the level of friction forcein sticking and sliding phase. This is responsible for stick–slip motion. During low-velocity sliding,friction is shown to have drooping characteristics with increasing velocity. This is known asStribeck effect, and responsible for self-excited oscillation of various systems. Thomsen [5]considers a similar model of friction to investigate into the effect of fast excitation on stick–slipdynamics. It is shown that a properly chosen fast vibration is capable of suppressing self-excitedoscillation and stick–slip motion. Due to the effect of fast-vibration the zero-velocity discontinuityof friction force is removed by an equivalent viscous damping characteristics, and the low-velocitydrooping characteristic tends to flatten out simultaneously. Feeny and Moon [6] haveexperimentally demonstrated the possibility of using fast vibration in quenching friction-drivenchaos. Tuned dither has also been successfully applied in eliminating friction-induced error inrobotic manipulators [7].A great number of recent phenomenological friction models consider only the macroscopicdegrees of freedom. This implies that all microscopic degrees of freedom are much faster than themacroscopic ones. Such an assumption breaks loose when the velocity of sliding becomescomparable to the ratio of microscopic length scale to macroscopic time scale [17]. Thismicroscopic length scale may be of the same order of magnitude as the size of the asperities of thecontact surface or the correlation length of the surface roughness. Therefore, when velocitybecomes very low, one has to pay attention also in the microscopic degrees of freedom, which arein general faster in dynamics. Moreover, coming to the understanding of the effect of fastvibration, consideration of the interaction of fast microscopic degrees of freedom and the fastexcitation becomes important. Few recent models of friction incorporate the microscopic degreesof freedom, and such models are known as dynamic models of friction [2–4]. In the present paper,the effect of fast vibration on the dynamics of a class of systems represented by dynamic modelsof friction is investigated. For the present investigation, the LuGre [2] and the single-state ARTICLE IN PRESS S. Chatterjee et al. / Journal of Sound and Vibration 269 (2004) 61–89 62  elasto-plastic models of friction [3] are considered. The effect of fast vibration on the stick–slip,and self-excited oscillation has been discussed in the light of fast vibration-induced effectivefriction characteristics. Detail numerical simulation is carried out to understand the mechanism of the effect of fast vibration on friction-driven dynamics. The organisation of the paper is brieflygiven below.The discussion starts with a brief review of the existing mathematical models of friction inSection 2. Section 3 deals with a simple non-inertial contact model considering only themicroscopic degrees of freedom of the contact, and the effect of high-frequency velocity variationis studied by the method of harmonic balance. In Section 4, an example system consisting of anelastic slider moving on a frictional surface is considered. The slider is modelled as a single-degree-of-freedom system and the frictional contact is modelled according to the LuGre dynamic frictionmodel. In Section 5, the effect of high-frequency excitation on the friction characteristics of theexample system discussed in Section 4 is studied analytically. Two different kinds of fastexcitation, namely sinusoidal force excitation and square wave velocity excitation are consideredfor analytical estimation of the effective friction characteristics. Analytical results are alsocompared with that obtained from the direct numerical simulation of the system. The effect of fast-excitation on stick–slip instability of the system is considered in Section 6. In Section 7, amore rigorous elasto-plastic model of friction is considered for numerical investigation into themechanism of the effect of fast excitation on stick–slip motion. 2. Models of friction Modern literature is rich in significantly large number of mathematical models of friction. Eachof these models is relevant only for one or few phenomenology and operating ranges of its owninterest. For a particular problem, selection of the appropriate friction model is very important.Depending on the length and time scale involved, the existing models of friction in literature canbe classified into two broad categories, namely macroscopic and microscopic models.In macroscopic models, friction is represented as a dissipative function of relative velocity of sliding. Such models are mainly valid in situations where only macroscopic structural degrees of freedom involving relatively slower time scale are of paramount importance. The simplest form of such macroscopic models is the Coulomb’s dry friction model given as follows: F   ¼  F  c  Sgn ð v Þ :  ð 2 : 1 Þ This model only recognises the fact that friction force  F   is constant and depends on the sign of relative sliding velocity. However, Coulomb’s model cannot describe the stick–slip or the Stribeckeffect. There exist a number of extensions of Coulomb’s model, which consider the stick–slip andthe Stribeck effect. Such models are called kinetic friction models (KFM) and may in general begiven by F   ¼ g ð v Þ ;  v a 0 F  e  if   v  ¼  0 and  j F  e j o F  s F  s  Sgn ð F  e Þ  otherwise 8><>:9>=>; ;  ð 2 : 2 Þ ARTICLE IN PRESS S. Chatterjee et al. / Journal of Sound and Vibration 269 (2004) 61–89  63  where g ð v Þ ¼  F  c  þ ð F  s    F  c Þ e j v = v s j d þ  F  v v : In Eq. (2.2),  F  e  is the external force,  F  s  is the maximum level of multi-valued friction force duringsticking and  F  c  is the kinetic friction force. vs is the characteristic Stribeck velocity and  F  v  is theviscous friction coefficient.In recent times, a series of sophisticated models friction have been developed considering themicroscopic degrees of freedoms of friction interface. Such models are known as ‘bristle models’,where asperities of the friction interface are considered as elastic spring-like bristles. When atangential force is applied, the bristles deflect like springs, and the friction force is represented asthe average deflection force of the spring-like bristles. When deflection of the bristles is sufficientlylarge, the bristles start slipping. The velocity of sliding determines the average deflection duringsteady slipping. The LuGre model [2] is the most widely used form of microscopic model of friction that relies on bristles interpretation. Recently, Dupont et al. [3] further generalise theLuGre model to incorporate stiction phenomena in a more rigorous manner. The generalisedLuGre model, known as the ‘single-state elasto-plastic model’ is described as F   ¼  s 0 z  þ  s 1 d z d t  þ  s 2 v ;  s 0 ; s 1 ; s 2  >  0 ; d z d t  ¼  v    s 0 a ð z ; v Þj v j zg ð v Þ  ;  ð 2 : 3 Þ where  v  is the relative velocity between the matting surfaces and  z  is the average deflection of thebristles.  g ð v Þ  models the Stribeck effect and the most common form of   g ð v Þ  is as follows: g ð v Þ ¼  F  c  þ ð F  s    F  c Þ e j v = v s j d :  ð 2 : 4 Þ s 0  and  s 1  represents bristle stiffness and damping, respectively.  s 2  is viscous damping coefficientand  v s  is the characteristic Stribeck velocity. To render the model dissipative, the bristle dampingterm  s 1  is taken to be a function of velocity. The most common form of this function is given by s 1 ð v Þ ¼  #   s 1 e j v = v d  j d ;  ð 2 : 5 Þ where  v d   is a characteristic velocity and  d   is a positive quantity, and these two parameters modelsthe rate of change of bristle damping with sliding velocity. The function  a ð z ; v Þ  (introducedby Dupont et al. [3]) controls different phases of friction process like sticking, elasto-plasticpre-sliding and pure sliding by assuming different values at different states as follows: a ð z ; v Þ ¼ 0 for  j z j o z ba a m ð z ; z ba ; z ss Þ  for  z ba o j z j o z ss ð v Þ 1 for  j z j  >  z ss ð v Þ 8><>:9>=>; when Sgn ð v Þ ¼  Sgn ð z Þ¼  0 f g  when Sgn ð v Þ a Sgn ð z Þ ; where 0 o a m ð : Þ o 1 and  z ss ð v Þ ¼  g ð v Þ s 0 : ð 2 : 6 Þ ARTICLE IN PRESS S. Chatterjee et al. / Journal of Sound and Vibration 269 (2004) 61–89 64
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