JOURNAL OF THEORETICAL AND APPLIED MECHANICS 44, 1, pp. 185-202, Warsaw 2006 PIECEWISE LINEAR luz(. . .) AND tar(. . .) PROJECTIONS. PART 2 – APPLICATION IN MODELLING OF DYNAMIC SYSTEMS WITH FREEPLAY AND FRICTION Dariusz Żardecki Automotive Industry Institute (PIMOT), Warsaw e-mail: zardecki@pimot.org.pl Thepaperpresents the ideaandexamplesof applicationof anewmethod to the modelling of mechanical systems with freeplay and friction. This method bases on the piecewise linear luz(. . .) and tar(. . .) projections and their originalmathematical apparatus. It is very useful for synthesis of simulation models and description of the stick-slip phenomenon in multi-body systems. Key words: modelling, MBS, freeplay, friction, stick-slip, luz(. . .) and tar(. . .) projection 1. Introduction Basicmodels of spring elements with freeplay (backlash, clereance) as well as dissipative elements with dry friction are based on piecewise linear cha- racteristics (see Grzesikiewicz, 1990 and state-of-the-art papers: Armstrong- Helouvry et al., 1994; Brogliatto et al., 2002; Ibrahim, 1994; Nordin andGut- man, 2002). Suchcharacteristics cohere (Fig.1)with the luz(. . .) and tar(. . .) projections introduced by the author. A simple but very efficient mathematical apparatus has been elaborated for the luz(. . .) and tar(. . .) projections in the first part of the paper (Żar- decki, 2006). Therefore, the proposal of a newmethod formodelling piecewise linear systems using the luz(. . .) and tar(. . .) projections was a natural con- sequence. This paper presents detailed rules of use of the luz(. . .) and tar(. . .) projections in synthesis of mathematical models of systems with freeplay or friction. The method of modelling is described on a representative example 186 D.Żardecki Fig. 1. Idea of description of characteristics by luz(. . .) and tar(. . .) projections of a multi-element gear system (Fig.10) installed by bearings in a fixed stiff casing. In the beginning, necessary mathematical models of elementary sub- systems with single freeplay or friction will be presented. Then, an original method of synthesis of models of complex systems will be described and ap- plied to the modelling of some exemplary multi-element system. 2. Modelling of systems with single freeplay 2.1. Elementary model of elasticity with freeplay Notation: z1,z2 – dispalcements of elements ∆z0 – freeplay parameter (0.5 of total freeplay), ∆z0 =(z1−z2)0 FS12,FS21 – spring force affecting elements 1 and 2, respectively k12 – stiffness coefficient An elementary model of elasticity with freeplay concerns the relationship between the strain force and relative displacement of two coactive toothed elements (Fig.2). The relation between the interacting force FS and relative displacement ∆z of the elements expresses piecewise linear characteristics with the ”dead zone” (Fig.3). Piecewise linear luz(. . .) and tar(. . .) projections. Part 2... 187 Fig. 2. Elastic toothed elements with freeplay Fig. 3. Strain characteristics with freeplay Analytical expressions are FS12 = k12 luz(z1−z2,∆0) FS21 =−k12 luz(z1−z2,∆0) The elementary model of elasticity with freeplay refers to all discrete sys- tems inwhicha toothedmechanismof rigid solids is givenbyweightless springs with freeplay.They canbe simple sliding or rotation elements (rack andpinion elements, bars, shafts, gears, etc.). Analytical descriptions of rotation systems with freeplay concern angular characteristics (strain torque versus angular displacement). 2.2. Model of angular elasticity with freeplay for gear elements Notation: δ,γ – angular displacements of gear wheels (the applied sign convention facilitates themodelling of dynamic systems with gears); δ = α1, γ =−α2 MSδγ,MSγδ – spring torques acting on wheel 1 and 2 Fδγ,Fγδ – action/reaction spring forces (Fδγ =−Fγδ) rδ,rγ – radii of wheel 1 and 2 188 D.Żardecki p – gear ratio (whenwithout freeplay δrδ = γrγ so δ = pγ); p = rγ/rδ l – perimetric translocation of wheels; l = rδδ−rγγ l0 – perimetric freeplay (0.5 of total freeplay between teeth) (δ−pγ)0 – angular freeplay parameter (0.5 of total freeplay inwhe- el 1); (δ −pγ)0 = l0/rδ ((δ/p)−γ)0 – angular freeplay parameter (0.5 of total freeplay inwhe- el 2); (δ/p)−γ)0 = l0/rγ K – stiffness coefficient of a pair of teeth kδγ,kγδ – angular stiffness coefficient of a pair of teeth measured fromwheel 1 and2, respectively; kδγ = r 2 δK, kγδ = r 2 γK Wediscuss a simplifiedmodel of the gear that consists of twowell-coacting toothed wheels characterised by effective radii of wheels (radii determine the gear ratio), the perimetric freeplay and the stiffness coefficient between their teeth. This model concerns rather small disturbances. Fig. 4. Gear elements with tooth freeplay Because of freeplay, the spring force can be described by the following formula Fδγ = K luz(l, l0)= K luz(rδδ−rγγ,l0)= rδK luz ( δ− rγ rδ γ, l0 rδ ) Fγδ =−K luz(l, l0)=−K luz(rδδ−rγγ,l0)=−rγK luz (rδ rγ δ−γ, l0 rγ ) The spring torques of wheels are MSδγ = rδFδγ, MSγδ = rγFγδ, hence MSδγ = r 2 δK luz ( δ− rγ rδ γ, l0 rδ ) MSγδ =−r 2 γK luz (rδ rγ δ−γ, l0 rγ ) Piecewise linear luz(. . .) and tar(. . .) projections. Part 2... 189 The relations between torques and relative angular displacements are expres- sed by characteristics of the same type as given in Fig.3, and described by formulas MSδγ = kδγ luz(δ−pγ,(δ −pγ)0)= kγδ p luz (δ p −γ, (δ p −γ ) 0 ) MSγδ =−pkδγ luz(δ −pγ,(δ−pγ)0)=−kγδ luz (δ p −γ, (δ p −γ ) 0 ) =−pMγδ 2.3. Model of angular elasticity with freeplay for elements twisted by elastic shaft with freeplay in its mounting Notation: ψ,δ – angular dispalcements of elements; ψ = α1, δ = α2 MSψδ,MSδψ – spring torque acting on elements 1 and 2, respectively (ψ− δ)0 – angular freeplay parameter (0.5 of total freeplay) kψδ – angular stiffness Fig. 5. Elastic shaft with freeplay in its mounting The relation between torques and relative angular displacements is expres- sed by the same characteristics as those given in Fig.3. It can be written as MSψδ = kψδ luz(ψ− δ,(ψ − δ)0) MSδψ =−kψδ luz(ψ− δ,(ψ − δ)0) 3. Modelling of systems with single friction An elementary dissipation model with dry friction can be formulated for two cases: the first one –when friction exists between amoving element and a 190 D.Żardecki fixed base, the second case – when the friction force acts between twomoving bodies. We discuss here the first model. It will be applied to the description of themodel of a bearing element. The second one (more complicated and not indispensable for modelling of the exemplary system) will be presented in a next special publication. 3.1. Elementary frictionmodel for ”moving element–fixed base” system Notation: FT – friction force ∆ż – slip velocity (here ∆ż = ż) F – external force C – damping coefficient FT0 – maximum value of dry friction M – mass of block Fig. 6. Moving element – fixed base system The friction force can be expressed bymodified Coulomb’s characteristics (which take into consideration the possibility of action of static dry friction at zero velocity). Fig. 7. Modified Coulomb’s friction characteristics The analytical description is given by the formula FT = C tar ( ∆ż, FT0 C ) This formula needs only some linear damping and the samemaximum ab- solute values of kinetic aswell as static dry friction. Suchconditions areusually Piecewise linear luz(. . .) and tar(. . .) projections. Part 2... 191 accomplished, especially when the so called Stribeck effect does not appear. Although its conciseness, this formula contains an ample friction description. It express the friction force as a sum of viscous and dry (kinetic and static) friction for every velocity states FT = C tar ( ż, FT0 C ) =     Cż−FT0 if ż < 0 FT0s ∗ if ż =0 Cż+FT0 if ż > 0 where s∗ ∈ [−1,1] so FT = Cż︸︷︷︸ Viscous friction (damping) + FT0 sgn(ż) ︸ ︷︷ ︸ Kinetic dry friction + FT0s ∗ ︸ ︷︷ ︸ Static dry friction ︸ ︷︷ ︸ Dry friction ︸ ︷︷ ︸ Kinetic friction (Coulomb′s) ︸ ︷︷ ︸ Static friction (also stiction) At zero velocity state, the friction force is not calculated on the basis of the modifiedCoulomb’s characteristics. The calculation of FT0s∗ needs discussion of the dynamic model. Themodel of motion dynamics is determined by the differential inclusion Mz̈(t)∈ F(t)−C tar ( ż(t), FT0 C ) where s∗(t)∈ [−1,1] Note, that for ż(t) = 0, Mz̈(t) ∈ F(t)− FT0s∗(t). The replacement of ambiguity inclusion by an explicit relation demands calculation of s∗(t). This will be shown, firstly – using a heuristic rule, then – using general physic principles. As it is well known, the description of dry friction can be done by the following heuristic rule: when the slide velocity goes to zero, the static fric- tion force FTS(t) starts, and the stiction state (when ż(t) = 0, z̈(t) = 0) may exist until FTS(t) ∈ [−FT0,FT0]. If ż(t) = 0, while the condition FTS(t) ∈ [−FT0,FT0], z̈(t) = 0 is impossible, this means only a temporary static friction without stiction. In such a state, z̈(t) 6=0 and FTS(t)=±FT0, where the sign of the friction force asserts its opposite action. Thus FTS(t)= FT0s ∗(t)=      FT0 if F(t)­ FT0 (then z̈(t) 6=0) F(t) if −FT0 < F(t) < FT0 (then z̈(t)= 0) −FT0 if F(t)¬−FT0 (then z̈(t) 6=0) in other words FTS(t)= FT0s ∗(t)= F(t)− luz(F(t),FT0) 192 D.Żardecki The heuristic description of dry friction with stiction corresponds to the S-S mathematical procedure (see its definition in Żardecki (2006)). Marking by s∗∗(t) the singularity variable, which balances the equation ofmotionwhen z̈(t) = 0 (stiction), we have 0 = F(t)−FT0s∗∗(t). Hence, on the basis of the S-S procedure, we obtain FT0s ∗(t)= F(t)− luz(F(t),FT0) Themathematical description of dry friction action ensuing from the heu- ristic rule (or from the S-S procedure) is equivalent to application of some general variation principle (Jourdain’s or Gauss’ principle). On the basis of the Jourdain principle (with extensions) δż variation has to be minimized in relation to the s∗ singularity in continuity of the ż = 0 state. This means minimization of |z̈|. The task is following s∗ : min s∗ |z̈| ∧ s∗ ∈ [−1,1] or s∗ : min s∗ ∣ ∣ ∣ Mz̈ FT0 ∣ ∣ ∣=min s∗ ∣ ∣ ∣ F FT0 −s∗ ∣ ∣ ∣ ∧ s∗ ∈ [−1,1] The solution is s∗ =      1 if s∗∗ > 1 s∗∗ if −1¬ s∗∗ ¬ 1 −1 if s∗∗ < −1 where s∗∗ = F FT0 so s∗ = F FT0 − luz ( F FT0 ,1 ) Whenwe apply the Gauss principle, the so called „acceleration energy” is minimized s∗ : min s∗ (Mz̈2) ∧ s∗ ∈ [−1,1] so s∗ : ∂ ∂s∗ (Mz̈2)= 0 ∧ s∗ ∈ [−1,1] s∗ : ∂ ∂s∗ ((F −FT0s∗)2 M ) =0 ∧ s∗ ∈ [−1,1] s∗ : F −FT0s ∗ =0 ∧ s∗ ∈ [−1,1] so s∗ = F FT0 − luz ( F FT0 ,1 ) Piecewise linear luz(. . .) and tar(. . .) projections. Part 2... 193 Therefore, applying the Jourdain or Gauss rules FT0s ∗(t)= F(t)− luz(F(t),FT0) As we see, all methods have given the same result. They mean that the static dry friction FTS = FT0s∗ can be here described by the characteristics shown in Fig.8. Fig. 8. Characteristics of static dry friction Applying the FT0s∗(t) formula, for ż(t)= 0, the inclusion description can be replaced by the differential equation Mz̈(t)= luz(F(t),FT0) which perfectly expresses the essence of the ”stick-slip” phenomenon. Note, that when ż(t) = 0 and −FT0 < F(t) < FT0, then also z̈(t) = 0 (stiction state). When for ż(t) = 0, F(t) < −FT0 or F(t) > FT0, then z̈(t) 6= 0 (no stiction state). In such a case, the state of ż(t) = 0 is only a tempo- rary crossing. When the block is sticky and the excitation F(t) < −FT0 or F(t) > FT0, then the state of slip begins. Summingup, for every ż(t), the dynamicmodel of theblock canbewritten by a differential motion equation with the singularity s∗(t) Mz̈(t)= F(t)−C tar ( ż(t), FT0 C ) where FT0s∗(t)= F(t)− luz(F(t),FT0). This model may be expressed without the singularity using a variable- structural form Mz̈(t)=    F(t)−C tar ( ż(t), FT0 C ) if ż(t) 6=0 luz(F(t),FT0) if ż(t)= 0 Both formsof themodel are equivalent andboth canbeused in simulation. 194 D.Żardecki The presented elementary model of the stick-slip phenomenon is equiva- lent with other models (eg. the well known Karnop model (Karnopp, 1985)). Those models have rather more complicate forms, difficult for analytical ope- rations. A compact, well coherent to parametric operations form of ourmodel is its important feature. Thanks to the luz(. . .) and tar(. . .) mathematical apparatus, formal parametric simplification of the friction model is possible and efficient. This is very important for automation of models synthesis and, generally, for a more efficient, the so-called, MBS software. The easiness of parametric operation on the frictionmodel is shownbelow. Consider a problem of simplification of themodel when themass of the block is assumed to be negligible. In this case, when M =0, the inclusionmodel is degenerated to 0∈−C tar ( ż(t), FT0 C ) +F(t) On the basis of tar(. . .) projection (Theorem 5.3 in the first part (Żardecki, 2006)) such an inclusion passes to the equation form ż(t)= 1 C luz(F(t),FT0) This equation has no s∗(t) and is well determined also for ż(t) = 0. Because of the luz(. . .) description, the stick-slip problem is solved ”automatically”: –when −FT0 < F(t) < FT0, luz(F(t),FT0)= 0and ż(t)= 0 is continued, – when F(t) < −FT0 or F(t) > FT0, the stiction state is terminated and ż(t) 6=0. Analytical descriptions of rotation systems with friction concern angular characteristics of torque and constitutive equations. Here we present a model of a bearing element. 3.2. Friction model of bearing element Notation: α̇ – angular velocity Mα – external torque MT – friction torque µ – damping coefficient MT0 – maximum friction torque I – moment of inertia A stiff solid (inertial or non-inertial) element having a bearing in a fixed base is analysed here. Piecewise linear luz(. . .) and tar(. . .) projections. Part 2... 195 Fig. 9. One-mass element with friction in bearing In this case, the Coulomb characteristics describing friction in the bearing relates the friction torquewith the angularvelocity.Assuming that the element may be inertial or non-inertial, one presents two variants of themathematical model. • Themodel of inertial elementwith friction in the bearing (two variants): – equation with singularity Iαα̈i(t)= Mα(t)−µα tar ( α̇(t), MT0α µα ) where for α̇(t)= 0 MT0αs ∗ α(t)= Mα(t)− luz ( Mα(t),MT0α ) – variable-structure equation (without singularity) Iαα̈(t)=    Mα(t)−µα tar ( α̇(t), MT0α µ ) if α̇(t) 6=0 luz(Mα(t),MT0α) if α̇(t)= 0 • Themodel of non-inertial element with friction in the bearing µαα̇(t)= luz(Mα(t),MT0α) Themodel of the bearing element will be useful for synthesis of themodel of the consideredmulti-body system. 4. Modelling of multi-body systems with freeplay and friction An efficient method of modelling some class of MBS systems with fre- eplay and friction has been elaborated on the base of simple piecewise linear 196 D.Żardecki models of elementary subsystems and the luz(. . .) and tar(. . .) projections. This class of systems concerns especially mechanisms which can be treated as rotational systems with fixed axles of rotation. They have invariable mecha- nical structures, but because of the stick-slip phenomenon theirmathematical description have variable structural forms. Themethod is following: • Firstly, a complementary discrete physical model is created. It can be built with: – stiff solid (inertial or non-inertial) elements – spring elements with freeplay – dissipative elements with dry friction. • In the primary stage ofmodelling, all friction sub-systems are treated as sub-systemshavingnon-zeroviscous friction, andall freeplay connections are treated as sub-systems having non-zero elasticity. Solid elements are treated as inertial bodies as well. Therefore, the primary physical model has a redundant form. • Then equations of motion for the primarymodel are built. The Lagran- ge or other well known method can be used. All equations are created by balancing the inertial forces or torques with external excitation, dis- sipation as well as elasticity ones which are described by the luz(. . .) and tar(. . .) projections. As a result, we obtain a redundantmathema- tical model. Coordinates of bodies and their derivatives are the model variables. Themass and geometric parameters of the solid elements and parameters of piecewise linear characteristics are the parameters of the model. • The applicable model is obtained from the redundant model. This ope- ration is done by formal parametric and assymptotic reduction. This means that wemust determine analytical forms of limitations – eg. very small masses or moments of inertia tend to zero, very large stiffness – to infinity. Calculations are supported by the mathematical apparatus of luz(. . .) and tar(. . .) projections. When the model is provided for simulation investigations, its form should not contain the so-called stiff differential equations as well as any equations of constraints. • As a result of successive reductions, we obtain successive aproximations of the primarymodel. The main advantage of the method is simplicity of the primary model and mathematical formalism of the model reduction. Simplifications of the reducedmodel ensue frommathematical formulas of the luz(. . .) and tar(. . .) Piecewise linear luz(. . .) and tar(. . .) projections. Part 2... 197 projections. Oftentimes, reduction of equations seems to be very complicated or even impossible be realised, while application of the luz(. . .) and tar(. . .) projections makes them surprisingly simple. A representative example of application of this method is shown below. 4.1. Model of exemplary multi-body system with freeplay and friction Notation: Symbols are the same as in Section 2 and Section 3. A multi-body rotation system consists of two inertial solids, two shafts and two gear wheels (Fig.10).We assume that the gear wheels are weightless. There are three freeplays: one between gears teeth and two – in sockets of shafts. The rotation elements have four bearings with friction. This system is driven by two external torques Mψ(t), Mϕ(t). Fig. 10. An example of a multi-body rotation system The primary redundant model is given by Newton’s equations of motion Iψψ̈(t)+µψ tar ( ψ̇(t), MT0ψ µψ ) +kψδ luz(ψ(t)−δ(t),(ψ − δ)0)= Mψ(t) Iδδ̈(t)+µδ tar ( δ̇(t), MT0δ µδ ) −kψδ luz(ψ(t)− δ(t),(ψ − δ)0)+ +kδγ luz(δ(t)−pγ(t),(δ −pγ)0)= 0 198 D.Żardecki Iδγ̇(t)+µγ tar ( γ̇(t), MT0γ µγ ) −pkδγ luz(δ(t)−pγ(t),(δ −pγ)0)+ +kγϕ luz(γ(t)−ϕ(t),(γ −ϕ)0)= 0 Iϕϕ̇(t)+µϕ tar ( ϕ̇(t), MT0ϕ µϕγ ) −kγϕ luz(γ(t)−ϕ(t),(γ −ϕ)0)= Mϕ(t) These equations will be simplified, therefore at this moment we need not to determine their variable-structural forms or equations describing the singula- rities s∗ϕ, s ∗ δ, s ∗ γ, s ∗ ψ, which are necessary for zero velocities. Simplification 1:when Iδ = Iγ =0 (weightless gear) The second and third equation assumes a degenerated form µδ tar ( δ̇(t), MT0δ µδ ) −kψδ luz(ψ(t)− δ(t),(ψ − δ)0)+ +kδγ luz(δ(t)−pγ(t),(δ −pγ)0)= 0 µγ tar ( γ̇(t), MT0γ µγ ) −pkδγ luz(δ(t)−pγ(t),(δ −pγ)0)+ +kγϕ luz(γ(t)−ϕ(t),(γ −ϕ)0)= 0 ApplyingTheorems 2.1, 3.2, see Żardecki (2006), we obtain these equations in disentangled form. The applicable model of the system has a form Iψψ̈(t)+µψ tar ( ψ̇(t), MT0ψ µψ ) +kψδ luz(ψ(t)−δ(t),(ψ − δ)0)= Mψ(t) where for ψ̇(t)= 0 MT0ψs ∗ ψ(t)= Mψψ(t)− luz(Mψψ(t),MT0ψ) and Mψψ(t)= Mψ(t)−kψδ luz(ψ(t)− δ(t),(ψ − δ)0) µδδ̇(t)+ luz(kψδ luz(ψ(t)− δ(t),(ψ − δ)0)+ −kδγ luz(δ(t)−pγ(t),(δ −pγ)0),MT0δ ) =0 µγγ̇(t)+ luz(−pkδγ luz(δ(t)−pγ(t),(δ −pγ)0)+ +kγϕ luz(γ(t)−ϕ(t),(γ −ϕ)0),MT0γ)= 0 Iϕϕ̇(t)+µϕ tar ( ϕ̇(t), MT0ϕ µϕγ ) −kγϕ luz(γ(t)−ϕ(t),(γ −ϕ)0)= Mϕ(t) where for ϕ̇(t)= 0 MT0ϕs ∗ ϕ(t)= Mϕϕ(t)− luz(Mϕϕ(t),MT0ϕ) and Mϕϕ(t)= Mϕ(t)+kγϕ luz(γ(t)−ϕ(t),(γ −ϕ)0) Piecewise linear luz(. . .) and tar(. . .) projections. Part 2... 199 Simplification 2: when Iδ = Iγ = 0 and µδ = µγ = 0, MT0δ = MT0γ = 0 (weightless gear with perfect bearing, but with teeth feeplay). The second and third equations assume an involved form kψδ luz(ψ(t)− δ(t),(ψ − δ)0)−kδγ luz(δ(t)−pγ(t),(δ −pγ)0)= 0 −pkδγ luz(δ(t)−pγ(t),(δ −pγ)0)+kγϕ luz(γ(t)−ϕ(t),(γ −ϕ)0)= 0 These equations are entangled constraints for the first and forth equation. To reduce the variables δ(t) and γ(t), the second and third equations are transformed to the form luz(ψ(t)− δ(t),(ψ − δ)0)= kδγ kψδ luz(δ(t)−pγ(t),(δ −pγ)0) luz(pγ(t)−pϕ(t),p(γ −ϕ)0)= p2kδγ kγϕ luz(δ(t)−pγ(t),(δ −pγ)0) Applying properties of the luz(. . .), on the basis of Theorem 4.3 (Żardecki, 2006), we find luz(ψ(t)− δ(t),(ψ − δ)0)= kδγ kψδ p2kδγ kγϕ + kδγ kψδ +1 · · luz(ψ(t)−pϕ(t),(ψ − δ)0+(δ −pγ)0+p(γ −ϕ)0) luz(pγ(t)−pϕ(t),p(γ −ϕ)0)= p2kδγ kγϕ p2kδγ kγϕ + kδγ kψδ +1 luz(ψ(t)−pϕ(t),(δ −pγ)0) As the final result, we obtain the model without algebraic constraints (!) Iψψ̈(t)+µψ tar ( ψ̇(t), MT0ψ µψ ) +kψϕ luz(ψ(t)−pϕ(t),(ψ −pϕ)0)= Mψ(t) where for ψ̇(t)= 0 MT0ψs ∗ ψ(t)= Mψψ(t)− luz(Mψψ(t),MT0ψ) and Mψψ(t)= Mψ(t)−kψϕ luz(ψ(t)−pϕ(t),(ψ −pϕ)0) Iϕϕ̇(t)+µϕ tar ( ϕ̇(t), MT0ϕ µϕγ ) −pkψϕ luz(ψ(t)−pϕ(t),(ψ −pϕ)0)= Mϕ(t) where for ϕ̇(t)= 0 MT0ϕs ∗ ϕ(t)= Mϕϕ(t)− luz(Mϕϕ(t),MT0ϕ) and Mϕϕ(t)= Mϕ(t)+pkψϕ luz(ψ(t)−pϕ(t),(ψ −pϕ)0) 200 D.Żardecki where kψϕ is the reduced stiffness coefficient kψϕ = ( 1 kψδ + 1 kδγ + p2 kγϕ ) −1 and (ψ−pϕ)0 is the reduced freeplay parameter (ψ−pϕ)0 =(ψ− δ)0+(δ−pγ)0+p(γ −ϕ)0 Note, when the gear stiffness kδγ →∞ (practically kδγ � kψδ,kγϕ), then kψϕ = ( 1 kψδ + 1 kδγ + p2 kγϕ ) −1 kδγ −→ ( 1 kψδ + p2 kγϕ ) −1 This result confirms the possibility of operation with the piecewise linear model by its reduced parameters. Their theoretical as well as well known in practice mathematical forms are compatible. Simplification 3: where Iδ = Iγ = 0, µδ = µγ =0, MT0δ = MT0γ = 0, and kδγ � kψδ,kγϕ (kδγ →∞), (ψ− δ)0 =(γ −ϕ)0 =(δ−pγ)= 0. (Ideal kinematic gear with stiff teeth in the system without freeplay and dry friction). The reduced linear model is Iψψ̈(t)+µψψ̇(t)+kψϕ(ψ(t)−pϕ(t))= Mψ(t) Iϕϕ̈(t)+µϕϕ̇(t)−pkψϕ(ψ(t)−pϕ(t)) = Mϕ(t) As we can see, all these simplifications could be strictly formal. 5. Final remarks In this paper, the idea and examples of application of a new method to themodelling of mechanical systemswith freeplay and friction have been pre- sented. The method is based on the piecewise linear luz(. . .) and tar(. . .) projections and their originalmathematical apparatus. It is very useful for de- scription of stick-slip processes inmulti-body systems which can be described by piecewise – linear equations. The presentedmethod has been already applied to synthesis of simulation models of steering systems with freeplays and dry friction (see for example Lozia and Żardecki, 2002, 2005; Żardecki, 1998, 2005a,b). Piecewise linear luz(. . .) and tar(. . .) projections. 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Część 2 – Zastosowanie w modelowaniu układów dynamicznych z luzem i tarciem Streszczenie Artykuł przedstawia ideę i przykłady zastosowania nowej metody modelowania układówmechanicznych z luzem i tarciem.Metoda bazuje na przedziałami liniowych odwzorowaniach luz(. . .) i tar(. . .) oraz ich oryginalnym aparaciematematycznym. Metoda jest bardzo użyteczna dla syntezy modeli symulacyjnych i opisu zjawiska stick-slip w układach wielomasowych. Manuscript received May 13, 2005; accepted for print November 10, 2005