JOURNAL OF THEORETICAL AND APPLIED MECHANICS 44, 2, pp. 255-277, Warsaw 2006 PIECEWISE LINEAR MODELING OF FRICTION AND STICK-SLIP PHENOMENON IN DISCRETE DYNAMICAL SYSTEMS Dariusz Żardecki Automotive Industry Institute (PIMOT), Warsaw e-mail: zardecki@pimot.org.pl The paper presents an idea and application of a new method of the modeling of mechanical systems with friction. This method is based on the piecewise linear luz(. . .) and tar(. . .) projections and their original mathematical apparatus. The presentedmodels of systems with friction describe the stick-slip phenomenon in detail. Keywords:piecewise linearprojections,mathematicalmodeling, friction, stick-slip 1. Introduction The modeling of dynamic systems with friction is one of the most impor- tant and difficult problems of mechanical science and engineering. Two main categories of problemsarenoticed: thefirst one concerns ”microscopic” friction models and is representative for the tribology and general contact theory. The second one concerns ”macroscopic” descriptions of friction actions (stick-slip phenomena) in discrete systems and is representative for theMBS (multibody systems) as well as the theory of mechanisms. Even though the macrosco- pic description of systems has a simplified character (piecewise linear friction force characteristic is preferred), the synthesis and analysis of such MBSmo- dels is usually very sophisticated and must be supported by a mathematical theory distinctive for non-smooth systemswith constraints (variable-structure differential-algebraic equations and inclusions). Such an approach was discus- sed eg. by Grzesikiewicz (1990) and Brogliatto et al. (2002). Problems of the modeling of friction systems are presented in many pa- pers. They are discussed in several surveys, eg. by Amstrong-Helouvry et al. 256 D. Żardecki (1994), Feeny et al. (1998), Ferri (1995), Gaul and Nitche (2001), Ibrachim (1994a,b), Martins et al. (1990), Oden and Martins (1985), Tworzydlo et al. (1992). Exploring the bibliography, we notice that themost recommended re- ference concerning the friction and stick-slip ”macroscopic”modeling seems to be the article byKarnopp (1985). It contains derivations of variable-structure piecewise linear models for elementary single-mass and two-mass systems. So, Karnopp’smodels can be treated as the referencemodels formodeling of other friction system. In this paper, we focus on the ”macroscopic” piecewise linear approach too. The aim of the paper is a detailed presentation of the original method of modeling of discrete dynamical friction systems basing on the piecewi- se linear luz(. . .) and tar(. . .) projections and their mathematical appara- tus. Using this method, the models have compact analytical forms enabling parametrically-made operations (eg. reductions). In comparison with other methods, just this feature seems to be the main advantage. The single- and two-mass stick-slip models discussed here are strictly compatible with Kar- nop’s models. The derivations include also their degenerate versions when masses go to zero or infinity. 2. Idea of piecewise linear friction and stick-slip description Even so the friction force characteristics canhavedifferentnonlinear forms, the simplest piecewise linear representations shown in Fig.1 express the essence of kinetic and static friction action. Fig. 1. Typical piecewise linear friction force characteristics: FT(V ) – Coulomb-like kinetic friction characteristic, FT(FW) – saturation static friction characteristics. Notation: FT – friction force, V – relative velocity of abrading elements, FW – acting force, FT0 –maximum dry friction force (here the same for kinetic and static friction), C – damping factor) Piecewise linear modeling of friction... 257 Such characteristics can be analytically written using the luz(. . .) and tar(. . .) piecewise linear projections (compare Fig.1 and Fig.2) luz(x,a)=x+ |x−a|− |x+a| 2 tar(x,a) = luz(x,a)−1 =x+asgh(x) where sgh(x)=        −1 if x< 0 s∗ ∈ [−1,1] if x=0 1 if x> 0 Fig. 2. Geometric interpretations of piecewise linear projections The luz(. . .) and tar(. . .) projections have a surprisingly simple ma- thematical apparatus which was formulated with proofs by Żardecki (2001, 2006b). Here wewill explore only particular formulas. Theywill be usedwhen necessary. Applying the tar(. . .) projection in description of Coulomb’s-like charac- teristics, we can express in a compact formula both friction forces, the kinetic force – for a non-zero relative velocity as well as the static one – for the zero velocity. Such ”extended Coulomb characteristics” have form FT =C tar ( V, FT0 C ) = CV +FT0 sgnV ︸ ︷︷ ︸ Kinetic friction (V 6=0) + FT0s ∗ ︸ ︷︷ ︸ Static friction (V=0) The kinetic friction force is expressed one-to-one, while the static friction FTS demands an additional description in form of a function of the acting force. Having the formof saturation characteristics, the static friction characteristics can be can be analytically written with using the luz(. . .) projection FTS =FT0s ∗ =Fw− luz(Fw,FT0) Obviously, the acting force FW depends on the system configuration. Only in the simplest single-mass system (Fig.3), the force FW is identical with the 258 D. Żardecki external input force F . In multibody systems with many frictions sources, where the structure of an object depends on many stick-slip processes, the forces FW mayhave a complicate character andmust bederived for a concrete but temporary structure of the system. Fig. 3. Single-mass systemwith friction; M –mass, F – external acting force (notation of friction parameters in accordance with Fig.1) Havinganalytical formsofkinetic andstatic fiction forces,wecan formulate the stick-slip systemmodel.For the single-mass system, themodel is (Żardecki, 2006b) Mz̈(t)+C tar ( ż(t), FT0 C ) =F(t) where FT0s ∗(t)=F(t)− luz(F(t),FT0) Themodel in a compact formcanbe rewritten as a variable structure equation Mz̈(t)=    F(t)−C tar ( ż(t), FT0 C ) if ż(t) 6=0 luz(F(t),FT0) if ż(t)= 0 Thisvariable-structure formula strictly correspondsto the single-massKar- nopp model (1985) and clearly describes the stick-slip phenomenon in the system. When ż(t) = 0 and |F(t)| ¬ FT0, we obtain luz(F(t),FT0) = 0, hence also z̈(t) = 0. It means the stiction. When |F(t)| > FT0, we have luz(F(t),FTo) 6=0 and z̈(t) 6=0 – the slip state. Advantages ofusing the luz(. . .) and tar(. . .)projections concernnotonly brief analytical forms of friction characteristics and clear stick-slip description. Using their mathematical apparatus, we can transform the stick-slip models by parametric operations, which is the main benefit. For example: • We can reduce analytically the order of the model. For the single-mass system, when M =0, the compact model simplifies to the form C tar ( ż(t), FT0 C ) =F(t) Piecewise linear modeling of friction... 259 Using the theorem on degeneration and the formula k tar(x,a) = = tar(kx,ka) (Żardecki, 2006a), we obtain after inversion the final un- coupled form Cż(t)= luz(F(t),FT0) which effectively explains the stick-slip phenomenon in the mass less system (nomotion for −FT0 ¬F(t)¬FT0) • We can simplify non–homonegous frictionmodels using substitutive pa- rameters. For such a single-mass but multi-friction system, its compact model has form Mz̈(t)+ i=n ∑ i=1 Ci tar ( ż(t), FT0i Ci ) =F(t) By the formula k1 tar(x,a1)+k2 tar(x,a2)= (k1+k2)tar ( x, k1a1+k2a2 k1+k2 ) (Żardecki, 2006a), the model is reduced to the form with substitutive parameters Mz̈(t)+C tar ( ż(t), FT0 C ) =F(t) where C = n ∑ i Ci FT0 = n ∑ i FT0i The luz(. . .) and tar(. . .) projections are especially useful in the synthesis andanalysis of non-trivial friction systemmodels.This is presented in thenext sections. 3. Modeling of dynamics of a two-mass system with single friction The simplest example of a two-mass systemwith friction (Fig.4) concerns twomoving blocks placed one on another. The blocks are subject to two exter- nal forces and the friction force between them. The friction force is described according to the extended Coulomb cha- racteristics. Because the action of static friction is not given as an explicit 260 D. Żardecki Fig. 4. Two-mass systemwith single friction; M1,M2 –masses of blocks, F1, F2 – external forces, FT12 – friction force, FT012 –maximum dry friction force, C12 – damping factor dependence yet, the primary dynamical system model must be expressed by inclusion forms. Here M1z̈1(t)+C12 tar ( ż1(t)− ż2(t), FT012 C12 ) ∈F1(t) M2z̈2(t)−C12 tar ( ż1(t)− ż2(t), FT012 C12 ) ∈F2(t) where s∗12(t)∈ [−1,1]. If ż1(t)− ż2(t)= 0 they are M1z̈1(t)+FT012s ∗ 12(t)∈F1(t) M2z̈2(t)−FT012s ∗ 12(t)∈F2(t) Formal calculation of s∗12(t) (or static friction force FT012s ∗ 12(t)) can be based on the Gauss rule. This requires minimization of the so-called acceleration energy Q. Thus s∗12 : min s∗ 12 ( Q(s∗12)= M1(z̈1(s∗12)) 2+M2(z̈2(s∗12)) 2 2 ) ∧ s∗12 ∈ [−1,1] For concrete s∗12 M1(z̈1(s ∗ 12)) 2 = (F1−FT012s∗12) 2 M1 M2(z̈2(s ∗ 12)) 2 = (F2+FT012s∗12) 2 M2 So s∗12 : min s∗ 12 ((M1+M2)FT012 2M1M2 ( s∗12− M2F1−M1F2 (M1+M2)FT012 )2 + (F1+F2)2 2M1M2 ) ∧ s∗12 ∈ [−1,1] For s∗12 ∈ [−1,1] optimal solution is s∗12 = M2F1−M1F2 (M1+M2)FT012 Piecewise linear modeling of friction... 261 Forarbitrary F1 and F2, the solution s∗12(F1,F2)mustbe saturated.Therefore finally s∗12(t)= M2F1(t)−M1F2(t) (M1+M2)FT012 − luz (M2F1(t)−M1F2(t) (M1+M2)FT012 ,1 ) FTS12(t)=FT012s ∗ 12(t)= = M2F1(t)−M1F2(t) M1+M2 − luz (M2F1(t)−M1F2(t) M1+M2 ,FT012 ) Thederivationof these formulasmaybedone indifferentways.Because the s∗12 singularity is related to the cross-friction description, when ∆ż= ż1− ż2 is an independent variable, a new inclusion based on ∆ż and ∆z̈ variables is useful. After combination operations, we obtain M1M2 M1+M2 [z̈1(t)− z̈2(t)]+C tar ( ż1(t)− ż2(t), FT012 C ) ∈ M2F1(t)−M1F2(t) M1+M2 Comparing this inclusion with the compact single-mass system model (Sec- tion 2), we notice that they have the same mathematical structure. The sub- stitutive parameters andvariables of themodelmaybe interpreted in the same way. So, in theanalysis of relativemotion,we candirectlyuse themethods that have been applied to the single-mass frictionmodel, also the semi-heuristic S-S or formal variation procedure (Żardecki, (2006a,b). Such calculations give the same results as FTS12 obtained on the basis of the Gauss principle. Taking into account the result concerning the static friction force FTS12, we obtain for ż1(t)= ż2(t) the following equation of relative motion M1M2 M1+M2 [z̈1(t)− z̈2(t)] = luz(F12(t),FT012) where F12(t)= M2F1(t)−M1F2(t) M1+M2 This equation expresses the stick-slip phenomenon in relation to the resultant excitation F12 and the limitations FT012 on the static friction force FTS12. Particularly, when −FT012 ¬ F12(t) ¬ FT012 also z̈1(t) = z̈2(t), thus the stiction state starts. In the stiction state F12(t) = FTS12(t) (linear part of saturation characteristics).When F12(t)¬−FT012 or F12(t)­FT012 we have z̈1(t) 6= z̈2(t), and the slip state happens. In such a situation ż1(t) = ż2(t) entails temporary static friction without stiction. 262 D. Żardecki Anew aspect of the stick-slip description is taken up by the analysis men- tioned below. When ż1(t)= ż2(t), the equations for z1(t), z2(t) are M1z̈1(t)= M1 M1+M2 [F1(t)+F2(t)]+ luz (M2F1(t)−M1F2(t) M1+M2 ,FT012 ) M2z̈2(t)= M2 M1+M2 [F1(t)+F2(t)]− luz (M2F1(t)−M1F2(t) M1+M2 ,FT012 ) Furthermore if −FT012 ¬F12(t)= M2F1(t)−M1F2(t) M1+M2 ¬FT012 then (M1+M2)z̈1(t)=F1(t)+F2(t) or (M1+M2)z̈2(t)=F1(t)+F2(t) These equations have identical forms. It means that z̈1(t) = z̈2(t) (stiction state), andmotion is described by one of these equations only. Summing up, the compact-form model of a two-mass frictional system is M1z̈1(t)+C tar ( ż1(t)− ż2(t), FT012 C ) =F1(t) M2z̈2(t)−C tar ( ż1(t)− ż2(t), FT012 C ) =F2(t) where FT012s ∗ 12(t)= M2F1(t)−M1F2(t) M1+M2 − luz (M2F1(t)−M1F2(t) M1+M2 ,FT012 ) This model can be expressed in a variable-structure formwithout s∗12: — if ż1(t) 6= ż2(t) M1z̈1(t)=F1(t)−C12 tar ( ż1(t)− ż2(t), FT012 C12 ) M2z̈2(t)=F2(t)+C12 tar ( ż1(t)− ż2(t), FT012 C12 ) — if ż1(t)= ż2(t) M1z̈1(t)= M1 M1+M2 [F1(t)+F2(t)]+ luz (M2F1(t)−M1F2(t) M1+M2 ,FT012 ) M2z̈2(t)= M2 M1+M2 [F1(t)+F2(t)]− luz (M2F1(t)−M1F2(t) M1+M2 ,FT012 ) Piecewise linear modeling of friction... 263 Thepresentedmodel is strictly compatible withKarnop’smodel (1985). It is clear when ourmodel is rewritten in a traditional formwithout the luz(. . .) and tar(. . .) projections. Now,we cananalyze particular situationswhenoneof themassparameters (for example M2) goes to infinity or goes to zero. • When M2 → ∞, the state z̈2(t) = 0 must be steady. It means perma- nent blockade of this block. Hence also ż2(t) = 0. After an asymptotic transformation, this compact-form model can be given by one equation for the moving mass M1 M1z̈1(t)+C12 tar ( ż1(t), FT012 C12 ) =F1(t) where FT012s ∗ 12(t)=F1(t)− luz(F1(t),FT012) In such a case, the two-mass system with cross-friction becomes the single-mass elementary system presented in Section 2. • When M1 → 0, motion of the massless element has a kinetic character. After inversion, the first inclusion of the primary model transforms to the equation ż1(t)− ż2(t)= 1 C luz(F1(t),FT012) Finally, the second equation can be written as M2z̈2(t)=F1(t)+F2(t). Note: when also −FT012 ¬F1(t)¬FT012, we have ż1(t)− ż2(t) = 0. It means the stiction state. It is independent of action of the secondmass. 4. Modeling of dynamics of a two-mass system with two friction forces The simplest system with simultaneous action of several various friction forces is the two-mass systemwith two friction forces (Fig.5). It is an extension of the two-mass system discussed in Section 3. In this case, the bottom block interacts with friction forces not only with the top block but also with the fixed basis. The primarymathematical inclusionmodel is M1z̈1+C12 tar ( ż1− ż2, FT012 C12 ) ∈F1 M2z̈2−C12 tar ( ż1− ż2, FT012 C12 ) +C20 tar ( ż2, FT020 C20 ) ∈F2 where s∗12 ∈ [−1,1], s ∗ 20 ∈ [−1,1]. 264 D. Żardecki Fig. 5. Two-mass systemwith two friction forces; M1,M2 –masses of blocks, F1, F2 – external forces, FT12, FT20 – friction forces, FT012, FT020 –maximal dry friction forces, C12,C20 – damping factors Thesingularities of dry friction characteristics concern thevelocities ż1−ż2 or ż2 at zero points. Therefore, we must analyze three cases: (1) ż1 = ż2 6=0 (then ż1− ż2 =0 and ż2 6=0 – problem of s∗12) (2) ż1 6=0, ż2 =0 (then ż1− ż2 6=0 and ż2 =0 – problem of s∗20) (3) ż1 = ż2 =0 (then ż1− ż2 =0 and ż2 =0 – problem of s∗12 and s ∗ 20) Analysis (the Gauss rule is applied): — If ż1 = ż2 6=0 theminimization task is s∗12 : min s∗ 12 ( Q(s∗12)= M1(z̈1(s∗12)) 2+M2(z̈2(s∗12)) 2 2 ) ∧ s∗12 ∈ [−1,1] where M1(z̈1(s ∗ 12)) 2 = (F1−FT012s∗12) 2 M1 M2(z̈2(s ∗ 12)) 2 = ( F2−C20 tar ( ż2, FT020 C20 ) +FT012s∗12 )2 M2 For s∗12 ∈ [−1,1] s∗12 = M2F1−M1 ( F2−C20 tar ( ż2, FT020 C20 )) (M1+M2)FT012 For arbitrary excitations F1,F2, ż2, the solution s∗12(F1,F2, ż2) must be satu- rated. Thus finally s∗12(t)= Fw(t) FT012 − luz (Fw(t) FT012 ,1 ) Piecewise linear modeling of friction... 265 or FT012s ∗ 12(t)=Fw(t)− luz(Fw(t),FT012) where Fw(t)= M2F1(t)−M1F2(t) M1+M2 + M1C20 M1+M2 tar ( ż2(t), FT020 C20 ) —If ż1 6=0, ż2 =0 theminimization task has a form s∗20 : min s∗ 12 ( Q(s∗20)= M1(z̈1(s∗20)) 2+M2(z̈2(s∗20)) 2 2 ) ∧ s∗20 ∈ [−1,1] where M1(z̈1(s ∗ 20)) 2 = 1 M1 [ F1−C12 tar ( ż1− ż2), FT012 C12 )]2 =A (independent of s∗20) and M2(z̈2(s ∗ 20)) 2 = 1 M2 [ F2+C12 tar ( ż1− ż2), FT012 C12 ) −FT20s ∗ 20 ]2 Thus, for s∗20 ∈ [−1,1] the solution is s∗20 = 1 FT020 [ F2−C12 tar ( ż1, FT012 C12 )] For arbitrary excitations F2, ż1, the solution s∗20(F2, ż1) must be saturated. So s∗20(t)= 1 FT012 [ F2(t)−C12 tar ( ż2(t), FT012 C12 )] + − luz ( 1 FT012 [ F2(t)−C12 tar ( ż2(t), FT012 C12 )] ,1 ) FT020s ∗ 20(t)=F2(t)−C12 tar ( ż2(t), FT012 C12 ) + − luz ( F2(t)−C12 tar ( ż2(t), FT012 C12 ) ,FT020 ) —If ż1 = ż2 =0 theminimization task is s∗12,s ∗ 20 : min s∗ 12 ( Q(s∗12,s ∗ 20)= M1(z̈1(s∗12,s ∗ 20)) 2+M2(z̈2(s∗12,s ∗ 20)) 2 2 ) ∧ s∗12,s ∗ 20 ∈ [−1,1] 266 D. Żardecki For concrete values s∗12, s ∗ 20, we have M1z̈ 2 1 = (F1−FT012s∗12) 2 M1 M2z̈ 2 2 = (F2+FT012s∗12−FT020s ∗ 20) 2 M2 Q(s∗12,s ∗ 20)= (M1+M2)F2T012 2M1M2 (s∗12) 2+ F2T020 2M2 (s∗20) 2− FT012FT020 M2 s∗12s ∗ 20+ + (M1F2−M2F1)FT012 M1M2 s∗12− F2FT020 M2 s∗20+ M1F 2 2 +M2F 2 1 2M1M2 For s∗12 ∈ [−1,1] and s ∗ 20 ∈ [−1,1] the minimization solution must fulfill (convex functionminimization) ∂Q(s∗12,s ∗ 20) ∂s∗12 = (M1+M2)F2T012 M1M2 s∗12− FT012FT020 M2 s∗20+ + (M1F2−M2F1)FT012 M1M2 =0 ∂Q(s∗12,s ∗ 20) ∂s∗20 = F2T020 M2 s∗20− FT012FT020 M2 s∗12− F2FT020 M2 =0 ∂Q2(s∗12,s ∗ 20) ∂(s∗12) 2 = (M1+M2)F2T012 M1M2 > 0 ∂Q2(s∗12,s ∗ 20) ∂(s∗20) 2 = F2T020 M2 > 0 (fulfilled) Thus, the system     (M1+M2)F2T012 M1M2 − FT012FT020 M2 − FT012FT020 M2 F2T020 M2     [ s∗12 s∗20 ] =     − (M1F2−M2F1)FT012 M1M2 − F2FT020 M2     yields for s∗12 ∈ [−1,1] and s ∗ 20 ∈ [−1,1] the solution s∗12 = F1 FT012 s∗20 = F1+F2 FT020 For arbitrary excitations F1,F2, the solution s∗12(F1,F2), s ∗ 20(F1,F2) must be saturated. Thus finally s∗12(t)= F1(t) FT012 − luz (F1(t) FT012 ,1 ) s∗20(t)= F1(t)+F2(t) FT020 − luz (F1(t)+F2(t) FT020 ,1 ) Piecewise linear modeling of friction... 267 and FT012s ∗ 12(t)=F1(t)− luz(F1(t),FT012) FT020s ∗ 20 =F1(t)+F2(t)− luz(F1(t)+F2(t),FT020) Concluding, the model of the two-mass system with two frictions forces can be described in a compact form by differential equations of motion M1z̈1+C12 tar ( ż1− ż2, FT012 C12 ) =F1 M2z̈2−C12 tar ( ż1− ż2, FT012 C12 ) +C20 tar ( ż2, FT020 C20 ) =F2 where s∗12 = Fw12 FT012 − luz (Fw12 FT012 ,1 ) s∗20 = Fw20 FT020 − luz (Fw20 FT020 ,1 ) and Fw12 =    M2F1−M1F2 M1+M2 + M1C20 M1+M2 tar ( ż2, FT020 C20 ) if ż2 6=0 F1 if ż2 =0 Fw20 =    F2+C12 tar ( ż1, FT012 C12 ) if ż1 6=0 F1+F2 if ż1 =0 Displacing s∗12, s ∗ 20, the model can be expressed also by variable-structure equations: — If ż1 6= ż2 6=0 M1z̈1 =F1−C12 tar ( ż1− ż2), FT012 C12 ) M2z̈2 =F2+C12 tar ( ż1− ż2, FT012 C12 ) −C20 tar ( ż2, FT020 C20 ) —If ż1 = ż2 6=0 M1z̈1 = M1(F1+F2) M1+M2 − M1C20 M1+M2 tar ( ż2, FT020 C20 ) + +luz (M2F1−M1F2 M1+M2 + M1C20 M1+M2 tar ( ż2, FT020 C20 ) ,FT012 ) M2z̈2 = M2(F1+F2) M1+M2 − M2C20 M1+M2 tar ( ż2, FT020 C20 ) + − luz (M2F1−M1F2 M1+M2 + M1C20 M1+M2 tar ( ż2, FT020 C20 ) ,FT012 ) 268 D. Żardecki Note, if −FT012 ¬ M2F1−M1F2 M1+M2 + M1C20 M1+M2 tar ( ż2, FT020 C20 ) ¬FT012 then z̈1 = F1+F2 M1+M2 − C20 M1+M2 tar ( ż2, FT020 C20 ) z̈2 = F1+F2 M1+M2 − C20 M1+M2 tar ( ż2, FT020 C20 ) whichmeans that z̈1 = z̈2, and the stiction state appears. — If ż1 6=0, ż2 =0 M1z̈1 =F1−C12 tar ( ż1, FT012 C12 ) M2z̈2 = luz ( F2+C12 tar ( ż1, FT012 C12 ) ,FT020 ) It means that when −FT020 ¬F2+C12 tar ( ż1, FT012 C12 ) ¬FT020 also z̈2 = 0. In other words, the bottom block is stuck to its base, and only the top block is moving according to the equation of motion M1z̈1 =F1−C12 tar ( ż1, FT012 C12 ) —If ż1 = ż2 =0 M1z̈1 = luz(F1,FT012) M2z̈2 =− luz(F1,FT012)+ luz(F1+F2,FT020) Note: - when −FT012 ¬ F1 ¬ FT012 and −FT020 ¬ F1 + F2 ¬ FT020 then z̈1 = z̈2 = 0 thus the blocks are stuck together and the bottom one is stuck to its base - when luz(F1 +F2,FT020) = luz(F1,FT012) 6= 0 the bottom block is stuck, while equalization of both blocks velocities is only temporary, since z̈1 6= z̈2 =0. Piecewise linear modeling of friction... 269 The presented modeling of a two-mass system with two friction forces basing on the luz(. . .) and tar(. . .) mathematical apparatus and the Gauss rule gives results strictly compatible withKarnop’smodel (1985). Ourmodel have amore compact description, better for analytical transformations anddetailed analysis. It is interesting that for the singular state ż1 = ż2 =0, the model of two- element system with two friction forces can not be directly used (by setting FT020 =0) for the description ofmotion of the systemwith cross-friction only! (Note that transformation of the two-mass elementary system’smodel into the single-mass elementarymodelwas regular).Whydowehaveno transformation regularity in this case? We explain this in the following considerations. When FT020 =0 has been set a priori (before minimization) Q(s∗12,s ∗ 20)= (M1+M2)F2T012 2M1M2 (s∗12) 2+ (M1F2−M2F1)FT012 M1M2 s∗12+ + M1F 2 2 +M2F 2 1 2M1M2 Thus we obtain s∗12 : FT012s ∗ 12 = M2F1−M1F2 M1+M2 ∧ s∗12 ∈ [−1,1] and finally, the proper model of the cross-friction system FT012s ∗ 12 = M2F1−M1F2 M1+M2 − luz (M2F1−M1F2 M1+M2 ,FT012 ) When FT020 = 0 has been set after minimization, we obtain generally a dif- ferent and false result FT012s∗12 = F1 − luz(F1,FT012) (good result only for F2 =0). We can draw the same conclusion when analyzing the model with FT012 =0. When FT012 =0 has been set before minimization Q(s∗12,s ∗ 20)= F2T020 2M2 (s∗20) 2− F2FT020 M2 s∗20+ M1F 2 2 +M2F 2 1 2M1M2 Thus s∗20 : F2−FT020s ∗ 20 =0 ∧ s ∗ 20 ∈ [−1,1] and finally FT020s ∗ 20 =F2− luz(F2,FT020) When FT012 = 0 is set a posteriori, a different result FT020s∗20 = F1 +F2 − luz(F1 +F2,FT020) is calculated (the same only for F1 = 0). As we see, the 270 D. Żardecki exactmodel of the two-mass systemwith two friction forcesdoesnot let regular conversion to elementary models. Another question seems to be interesting for better understanding of the model of the two-mass system with two friction forces. The question is if the model of such a complex system can be created on the basis of simpler (here elementary) exact sub-models? Let applynowaconcept of the so-called ”modeldecomposition-aggregation method”. Such decomposition can be done in two ways. Fig. 6. Two concepts of model decomposition In both cases, the system is replaced by equivalent three-mass systems. Of course, M21+M20 =M2. The coordinates of sub-systems are z1, z21 and z20. Thus our complex system is treated as a series of sub-systems containing indi- vidual frictions. For the description of these sub-systems, elementary (basing on general physical rules, i.e. on exact formulas) friction models can be used. For calculation of the final model of a complex system, we have to apply the operation KA → ∞ or KB → ∞. We will only discuss the most singular velocity state when the total stiction state appears. When the decomposition is made according to methodA, for the singular state ż1 = ż21, ż20 = 0, we have model description based on two elementary subsystemsmodels. Lattering FA =KA(z21−z20) Piecewise linear modeling of friction... 271 M1z̈1 = M1 M1+M21 (F1+F2−FA)+ luz (M21F1−M1(F2−FA) M1+M21 ,FT012 ) M21z̈21 = M21 M1+M21 (F1+F2−FA)−− luz (M21F1−M1(F2−FA) M1+M21 ,FT012 ) and M20z̈20 = luz(FA,FT020). Setting KA → ∞ we have z21 = z20 = z2, ż21 = ż20 = ż2 = ż1 = 0, z̈21 = z̈20 = z̈2. By summing the second and third equation, the complex model passes to M1z̈1 = M1 M1+M21 (F1+F2−FA)+ luz (M21F1−M1(F2−FA) M1+M21 ,FT012 ) M2z̈2 = M21 M1+M21 (F1+F2−FA)− luz (M21F1−M1(F2−FA) M1+M21 ,FT012 ) + +luz(FA,FT020) As yet FA = KA(z21 − z20) is formally indeterminate (indeterminacy of the type ”∞-0”). Admittedly the system must fulfill Gauss’ rule, so the accele- ration energy Q treated as a function of the unknown variable FA should be minimal. Because of strong non-linear form of Q(FA), formal analytical minimization is very complicated. We do this with little heuristic roundabout effort, we use themethod incorporated in the S-Sprocedure (Żardecki, 2006a). We know thatminimum-minimorumof Q(FA) is warranted by the stuck state when the accelerations of masses are zero. In the state z̈12 = z̈21 = z̈20 = 0, our three-mass model is M1z̈1= M1 M1+M21 (F1+F2−FA)+ luz (M21F1−M1(F2−FA) M1+M21 ,FT012 ) =0 M21z̈21= M21 M1+M21 (F1+F2−FA)− luz (M21F1−M1(F2−FA) M1+M21 ,FT012 ) =0 M20z̈20 = luz(FA,FT020)= 0 Because summation of the first and second equation F1 +F2 −FA = 0, one finds M1z̈1 = luz(F1,FT012)= 0 M21z̈21 =− luz(F1,FT012)= 0 M20z̈20 = luz(F1+F2,FT020)= 0 It means that the stuck state (optimal condition for minimization of Q(FA)) is warranted by the function FA(F1,F2) = F1 + F2 with conditions F1 ∈ [−FT012,FT012] and F1+F2 ∈ [−FT020,FT020]. 272 D. Żardecki When F1 /∈ [−FT012,FT012] or F1 +F2 /∈ [−FT020,FT020], the total stuck state vanishes. So for the aggregatedmodel,we finally obtain the proper result M1z̈1 = luz(F1,FT012) M2z̈2 =− luz(F1,FT012)+ luz(F1+F2,FT020) Similar calculation repeated for the model decomposed according to me- thodB (in this case FB(F1,F2)=F1 also gives the same final result. The ”decomposition-aggregation” method and the S-S procedure seem to be very useful for theoretical verification and analysis of multibody models with multiple friction sources. Such models can be very complicated, so affir- mative proofs are desirable. Basing on thederivedmathematicalmodel,we cananalyze situationswhen themass parameters go to infinity or go to zero.We discuss several non-trivial situations: (1) M1 →∞, (2) M2 →∞, (3) M1 → 0, (4) M2 → 0. (1) When M1 → ∞, the top element is motionless and the moving block (mass M2) acts under two friction forces. The state z̈1(t) = 0, ż1(t) = 0 results from parametrical transformation of the first model inclusion. Thus, the secondmodel inclusion is M2z̈2+C12 tar ( ż2, FT012 C12 ) +C20 tar ( ż2, FT020 C20 ) ∈F2 Finally, we obtain M2z̈2(t)=F2(t)− (C12+C20)tar ( ż2(t), FT012+FT020 C12+C20 ) where (FT012+FT020)s ∗ 12(t)=F1(t)− luz(F1(t),FT012+FT020) (2) When M2 → ∞, the state z̈2(t) = 0, ż2(t) = 0 must be steady. After asymptotic transformation, the model is given by one equation for the mass M1 M1z̈1(t)=F1(t)−C12 tar ( ż1(t), FT012 C12 ) where for ż2(t)= 0 FT012s ∗ 12(t)=F1(t)− luz(F1(t),FT012) (3)When M1 → 0 themotion of thismassless element has a kinetic character C12 tar ( ż1(t)− ż2(t), FT012 C12 ) ∈F1(t) Piecewise linear modeling of friction... 273 So ż1(t)− ż2(t)= 1 C luz(F1(t),FT012) The second inclusion can be written as M2z̈2(t)+C20 tar ( ż2(t), FT020 C20 ) ∈F1(t)+F2(t) So M2z̈2(t)+C20 tar ( ż2(t), FT020 C20 ) =F1(t)+F2(t) where FT020s ∗ 20(t)=F1(t)+F2(t)− luz(F1(t)+F2(t),FT020) When also −FT012 ¬F1(t)¬FT012, we have ż1(t)− ż2(t) = 0. It means the stiction state of both elements. It is independent of action of the secondmass. (4) When M2 → 0, motion of this massless element has a kinetic character. Here M1z̈1(t)+C12 tar ( ż1(t)− ż2(t), FT012 C12 ) ∈F1(t) −C12 tar ( ż1(t)− ż2(t), FT012 C12 ) +C20 tar ( ż2(t), FT020 C20 ) ∈F2(t) This system is generally extremly complicated. Using the luz(. . .) and tar(. . .) mathematical apparatus, we can simplify it for the case when F2(t) = 0. In such a case, the bottom element becomes a separation sheet for the top block, so this case is very practical. We use the following theorem (Żardecki, 2006b): for a­ 0, b ­ 0, k ­ 0 when tar(y,b)= k tar(x−y,a) then y(x)=        x− 1 k+1 luz(x,ka− b) if ka­ b k k+1 luz ( x, b−ka k ) if ka¬ b Therefore, because we have here a= FT012 C12 b= FT020 C20 k= C12 C20 ka− b= C12 C20 FT012 C12 − FT020 C20 = FT012−FT020 C20 274 D. Żardecki when tar ( ż2(t), FT020 C20 ) = C12 C20 tar ( ż1(t)− ż2(t), FT012 C12 ) =0 here then ż2(t)=       ż1(t)− C20 C12+C20 luz ( ż1(t), FT012−FT020 C20 ) if FT012 ­FT020 C12 C12+C20 luz ( ż1(t), FT020−FT012 C12 ) if FT012 ¬FT020 Setting this relationship to the first model inclusion, we obtain for F2(t)= 0 M1ż1(t)∈ { B1(t) if FT012 ­FT020 B2(t) if FT012 ¬FT020 where B1(t)=F1(t)−C12 tar ( C20 C12+C20 luz ( ż1(t), FT012−FT020 C20 ) , FT012 C12 ) B2(t)=F1(t)−C12 tar ( ż1(t)− C12 C12+C20 luz ( ż1(t), FT020−FT012 C12 ) , FT012 C12 ) In this formula, the tar(. . .) is undetermined for |ż1(t)| ¬ (FT012−FT020)/C20 if FT012 ­ FT020 and for ż1(t) = 0 if FT012 ¬ FT020. But in both cases M1z̈1(t)∈F1(t)−FT012s∗12(t). Applying the Gauss’ rule and the optimization task, we finally obtain for F2(t) = 0 and arbitrary F1(t) the saturation equ- ation FT012s∗12(t)=F1(t)− luz(F1(t),FT012), which enables one to replace the inclusion form by an equation without implicit form. The variable-structure form of this equation explains the stick-slip process for all cases of model’s conditions. 5. Final remarks In this paper, the idea and examples of application of a new method of modeling mechanical systems with freeplay and friction has been presented. Themethod is based on the piecewise linear luz(. . .) and tar(. . .) projections and their original mathematical apparatus. It is very useful for the descrip- tion of stick-slip processes in multi-body systems which can be described by piecewise-linear equations. Piecewise linear modeling of friction... 275 The derived friction models basing on the luz(. . .) and tar(. . .) projec- tions are strictly compatible with the legitimate Karnoppmodels. Themodels can be directly used in multi-body systems when inclusions contain simple individual tar(. . .) components, for example Miz̈i(t)∈Fi(z1(t),z2(t), . . . ,zn(t), ż1(t), ż2(t), . . . , żn(t), t)+ −Cij tar ( żi(t)− żj(t), FT0ij Cij ) When the structure of a multibody system ismore complicated, the synthesis of a model must bemore sophisticated, as it has been presented in Section 4. But in such cases, the application of the luz(. . .) and tar(. . .) mathematical apparatus yields excellent final results – a ready-to-simulation model without implicit forms. The piecewise-linear approximation based on the luz(. . .) and tar(. . .) projections can also be applied to friction characteristics expressing Stribeck’s effect, for asymmetric characteristics an so on. Even stick-slip models have been derived here for the simplest friction characteristics, their final forms can be easily adapted to othermore complicated characteristics. For example, whenmagnitudes of kinetic and static dry friction forces are not identical, in the variable-structure model two different parameters FT0K and FT0S can be applied. The presentedmethod has been already applied to several simulationmo- dels of systems with friction. Most of them concern car steering mechanisms (models with dry frictions in king-pins and gears) – see papers by Lozia and Żardecki (2002, 2005) as well as by Żardecki (2005a). Acknowledgments The work has been supported by grants No. 9T12C07108, 9T12C05819 and 4T07B05928 (project sponsored by the Ministry of Science and Informatics in 2005-2007). References 1. Armstrong-Helouvry B., Dupont P., Canudas de Wit C., 1994, A Survey of models, analysis tools and compensation methods for the control of machines with friction,Automatica, 30, 7, 1083-1138 276 D. Żardecki 2. Brogliatto B., Dam A.A.T., Paoli L., Genot F., Abadie M., 2002, Numerical simulation of finite dimensional multibody nonsmooth mechanical systems,ASME Applied Mechanical Review, 55, 2, 107-150 3. Feeny B., Guran A., Hinrichs N., Popp K., 1998, Historical review on dry friction and stick-slip phenomena,ASME Applied Mechanics Reviews, 51, 5, 321-341 4. Ferri A., 1995, Friction damping and isolation systems,ASME Transactions. Special 50th Anniversary Design Issue, 117B, 196-206 5. Gaul L., Nitche R., 2001, The role of friction in mechanical joints, ASME Applied Mechanical Reviev, 54, 2, 93-105 6. Grzesikiewicz W., 1990, Dynamics of mechanical systems with constraints, Prace Naukowe Politechniki Warszawskiej. Mechanika, 117 [in Polish] 7. Ibrahim R.A., 1994a, Friction-induced vibration, chatter, squeal, and chaos. Part I: Mechanics of contact and friction, ASME Applied Mechanical Review, 47, 7, 209-226 8. Ibrahim R.A., 1994b, Friction-induced vibration, chatter, squeal, and chaos. Part II: Dynamics and modeling, ASME Applied Mechanical Review, 47, 7, 227-253 9. Karnopp D., 1985, Computer simulation of stick-slip friction in mechanical dynamic systems. Transactions of the ASME. Journal of Dynamic Systems, Measurement, and Control, 107, 100-103 10. Lozia Z., Żardecki D., 2002, Vehicle dynamics simulation with inclusion of freeplay and dry friction in steering system, (SAE Paper 2002-01-0619), SAE 2002 Transactions Journal of Passenger Car – Mechanical Systems, 907-923 11. Lozia Z., Żardecki D., 2005,Dynamics of steering systemwith freeplay and dry friction – comparative simulation investigation for 2WS ands 4WS vehic- les, (SAE Paper 2005-01-1261), Steering and Suspension, Tires and Wheels Symposium 2005, SAE Special Publication SP-1915, 1-10 12. Martins J., Oden J., Simoes F., 1990,A study of static and kinetic friction, International Journal of Engineering Science, 28, 1, 29-92 13. Oden J., Martins J., 1985,Models and computational methods for dynamic friction phenomena,ComputerMethods inAppliedMechanics and Engineering, 52, 527-634 14. TworzydloW.,BeckerE.,Oden J., 1992,Numericalmodeling of friction- induced vibration and dynamic instabilities,ASME DE, 49, 13-32 15. Żardecki D., 2001, The luz(. . .) and tar(. . .) projections – a theoretical background and an idea of application in a modeling of discrete mechanical systems with backlashes or frictions,Biuletyn WAT,L, 5, 125-160 [in Polish] Piecewise linear modeling of friction... 277 16. Żardecki D., 2005a, Piecewise-linear modeling of dynamic systems with fre- eplay and friction, Proceedings of 8th DSTA Conference, Łódź, TU of Łódź, 321-332 17. Żardecki D., 2005b, Steering system freeplay and friction in vehicle dynamic models and simulations,The Archives of Transport,XVII, 1, 65-95 18. ŻardeckiD., 2006a,Piecewise linear luz(. . .) and tar(. . .)projections.Part1 – Theoretical background and friction, Journal of Theoretical and Applied Me- chanics, 44, 1, 163-184 19. ŻardeckiD., 2006b,Piecewise linear luz(. . .) and tar(. . .) projection.Part 2 – Application in modelling of dynamic systems with freeplay and friction, Journal of Theoretical and Applied Mechanics, 44, 1, 185-202 Przedziałami liniowe modelowanie tarcia i zjawiska ”stick-slip” w układach dyskretnych Streszczenie W artykule przedstawia się ideę i zastosowanie nowejmetodymodelowania ukła- dówmechanicznych z tarciem.Opracowanametoda bazuje na przedziałami liniowych odwzorowaniach luz(. . .) i tar(. . .) oraz ich oryginalnym aparacie matematycznym. Dzięki zastosowaniu odwzorowań luz(. . .) i tar(. . .) modele układów z tarciemmają analityczne formy doskonale wyrażające zmiennostrukturalny opis zjawiska ”stick- slip”. Za sprawą aparatu matematycznego luz(. . .) i tar(. . .) modele te mogą być przekształcane (np. redukowane) w sposób parametryczny, co stanowi główną zaletę metody. Manuscript received October 7, 2005; accepted for print January 30, 2006