JOURNAL OF THEORETICAL AND APPLIED MECHANICS 43, 4, pp. 875-892, Warsaw 2005 DAMPING OF VIBRATIONS IN A POWER TRANSMISSION SYSTEM CONTAINING A FRICTION CLUTCH Zbigniew Skup Institute of Machine Design Fundamentals, Warsaw University of Technology e-mail: zskup@ipbm.simr.pw.edu.pl This paper presents a theoretical study of the process of damping of non- linear vibrations in a three-mass model of a power transmission system with a multi-disc flexible friction clutch switched on and off electroma- gnetically. Steady-statemotionof the system is subject toharmonic exci- tation. The problem is considered on the assumption of a uniform unit pressure distribution between the contacting surfaces of the cooperating friction discs. Structural friction, small relative sliding of the clutch discs and linear viscotic damping have also been taken into account. In the case of sliding, the friction coefficient is not constant but depends on the relative angular velocity of slowly slidingdiscs.The aimof the analysis is to assess the influence of geometric parameters of the system, its exter- nal load, unit pressure, viscotic damping on resonance curves and phase shift angle of steady-state vibrations. The equations of motion of the examined system are solved bymeans of the slowly varying parameters (Van der Pol) method and digital simulation. Keywords:non-linearvibrations,viscoticdamping, frictionclutch, struc- tural friction, hysteresis loop 1. Introduction Friction clutches of usual design, including single and multi-disc systems, have an important property of damping torsional vibrations as a result of mi- cro andmacro slip between torsionally flexible discs. The sliding effect in the elastic range of the material of cooperating elements is called the structural friction. This phenomenon iswell knowand referred to as a structural hystere- sis loop (see Godman andKlamp, 1956; Pian, 1957 orCaughey, 1960 for early studies). In the Polish literature, an overview of structural friction problems 876 Z.Skup with applications can be found in the works byOsiński (1986, 1998), Giergiel (1990), Gałkowski (1981), Kosior and Wróbel (1986). Structural friction is a natural source of dampingpresent in every real device. In friction clutches, the magnitudeof dissipation canbecontrolled in suchaway that thebestdynamic properties of the entire transmission system are obtained. Nominal driving or resistance torques of such systems are usually disturbed by additional forces of a periodical or random nature. From the point of view of clutch design, it is important to establish a rela- tion between the external driving load and corresponding torsional motion of the transmission system. Therefore, a dynamic analysis based onmore advan- ced models is necessary. During the past two decades, attention was mainly focused on dynamical analysis of systems with structural friction, using rela- tively simplemodels of both the stick-slip process and themechanical system. More advanced stick-slip models were developed based mainly on finite ele- ments (see Buczkowski and Kleiber, 1997; Buczkowski, 1999; Grudziński et al., 1992; Pietrzakowski, 1986; Zboiński and Ostachowicz, 1997). A number of papers devoted to various dynamical problems of friction clutches was pre- sented by Skup (1991a,b, 1998, 2001, 2003, 2004), who developed an analytic description of thedynamic friction torque in amulti-disc clutchwith torsional- ly flexible discs and shafts, and applied this result to solve vibration problems in transmission systems related to various excitation loads. The relation between an external load and relative angular displacements of discs is of the fundamental importance for the design of friction clutches and their proper selection for particular engine-machine systems. The degree of energy dissipation in a power transmission system can be controlled in order to obtain the best dynamical properties of the whole sys- tem. The traditional professional literature treats frictional torsion dampers, frictional clutches and brakes as joints of rigid bodies. Therefore, the effect of natural damping has been neglected. The author of this paper takes into consideration the elasticity of thema- terial of cooperating elements in a friction clutch. The problem is investigated on the assumption of a uniform distribution pressures, non-uniform friction coefficient and linear viscous damping. The problem of deriving a precise ma- thematical description of the structural friction is very complicated because of the complexity of the friction phenomenon as well as difficulties in descri- bing the stress and strain states present in the sliding zone. Therefore, the mathematical description is based onmany simplifications. The assumptions concerning the properties of the material and friction forces are the sameas in the classical theoryof elasticity and structural friction Damping of vibrations in a power transmission system... 877 theory. In the case when friction forces are smaller than applied loadings, there is amacro-slide (a kinetic coefficient of friction) between the cooperating elements. Such a phenomenon is accompanied by the occurrence of friction forces. Studieswhichhavebeenconducted so farby theauthor in thedomainof mechanical systemswith structural frictionwere based on the assumption that there was no sliding between the cooperating elements (in the case of a static coefficient of friction, formicro-sliding). The phenomena of structural friction and macro-slip appear simultaneously during engagement or overloading the damper. Characteristics of friction are based also on the experimental research pre- sented by Grudziński et al. (1992), Kołacin (1971), Popp and Stelter (1990), Skup (1998). Most of the work has been restricted to the analysis of a one- degree-of-freedom system. 2. Equations of motion of the mechanical system We assume a three-mass model of a mechanical system which consists of an engine (E), friction clutch (C), reduced mass (RM) and a working machi- ne (WM), as shown in Fig.1. Structural friction occurs between the coopera- ting surfaces of discs of a friction clutch (C). Fig. 1. Physical model of the considered power transmission system Therefore, equations of motion of the considered system may be written down as follows I1ϕ̈11+Mz =M(t)+Mm I2ϕ̈12−Mz +k1(ϕ12−ϕ13)+ c1(ϕ̇12− ϕ̇13)=0 (2.1) I3ϕ̈13−k1(ϕ12−ϕ13)− c1(ϕ̇12− ϕ̇13)+Mr =0 where 878 Z.Skup Fig. 2. Mechanical systemwith a non-linear hysteresis loop and linear viscotic damping reduced to two-degrees-of-freedom I1,I2,I3 – mass moments of inertia of the driving and driven part, respectively ϕ11,ϕ12,ϕ13 – angular displacements Mz – clutch friction torque in a cycle represented by the structural hysteresis loop (Fig.2), dependent on the relative angular displacement, its vibration amplitude and its sign of velocity, respectively c1 – coefficient of viscous damping (Fig.2) Mr – resistance torque M(t)+Mm – variable engine torque described by the constant ave- rage value of the nominal driving torque Mm and discrete torque M(t) in the form of a harmonic exci- tation, i.e. M(t)=M0cosωt (2.2) and M0 – amplitude of the excitation torque ω – angular velocity of the excitation torque t – time and Mz = { M(ϕ1,A1, ϕ̇1) for ρ<ρ1 MT(ϕ̇) for R­ ρ>ρ1 (2.3) Having used the results presented by Skup (2001), a limited radius sliding zone ρ1, shown in Fig.3, and M = M(ϕ1,A1, ϕ̇1), shown in Fig.2, were determined. Thus, M(ϕ1,A1, ϕ̇1)= 1 √ η3 ( √ A1 2 + √ A1+ϕsgnϕ̇sgnϕ̇− √ 2A1 2 − √ 2A1 2 sgnϕ̇ ) (2.4) Damping of vibrations in a power transmission system... 879 Fig. 3. Load distribution in the frictional pair and ν = 3 2πµpR3 ρ1 = r 3 √ 1+ναM 0¬α¬ 1 η3 = κ1ν 2 6 κ1 = 2δ(k1 +k2) k1k2 δ= µpR 6 k1 =Gh1 k2 =Gh2 (2.5) where η3,α – nondimensional parameters k1,k2 – stiffness of discs h1,h2 – their thickness µ – friction coefficient p – pressure per unit area r,R – internal and external radius of the discs G – shear modulus MT(ϕ̇) – friction torque dependent on the sign of relative angular velocity. Themoment of friction in the friction clutch is described as below MT(ϕ̇)= 2π R ∫ ρ1 p(ρ)ρ2µ(ϕ̇) dρ (2.6) where ρ is the radius (r¬ ρ¬R), µ(ϕ̇) – variable value of the friction coeffi- cient dependent on the relative angular velocity. The hysteresis loop described by (2.4) and (2.6) is shown in Fig.4. In the case of macro-slide of the collaborating discs and plunger (ρ1 = r), we obtain MT(ϕ̇)= 2 3 πp(R3−r3)µ(ϕ̇) (2.7) 880 Z.Skup Fig. 4. Hysteresis loops: MT(ϕ̇),M(ϕ1,A1, ϕ̇1) –moments of friction for kinetic and static friction Fig. 5. Variation of the friction coefficient in function of relative speed of sliding discs In the papers by Grudziński et al. (1992) Kołacin (1991), Pop and Stelter (1990), Skup (1998), theoretical studies were confirmed by experimental rese- arch. Therefore, variation of the friction coefficient µ(ϕ̇) shown in Fig.5 can be determined in the following form: µ(ϕ̇)= (a1− c1ϕ̇2)sgnϕ̇− b1ϕ̇+d1ϕ̇3 (2.8) where a2, b2, c2, d2 are constant parameters. The numerical results were ob- tained for the following set of data for dry friction a2 =0.25 b2 =0.03 c2 =0.02 d2 =0.002 ϕ̇=ω=1rad/s (2.9) Damping of vibrations in a power transmission system... 881 3. The solution to equations of motion Since we are interested in steady motion of the considered system, we assume Mm = Mr, which provides a uniform rotation of the undisturbed system. Introducing new variables: ϕ1 =ϕ11−ϕ12 and ϕ2 =ϕ12−ϕ13 in forms of relative angles of torsion, we can reduce equations (2.1) to two second-order non-linear differential equations describing relative torsional vibration ϕ̈1− cϕ̇2+f1(ϕ1,A1, ϕ̇1)−mϕ2−B= zcosωt (3.1) ϕ̈2+wϕ̇2+nϕ2−βf1(ϕ1,A1, ϕ̇1)−A=0 where c= c1 I2 f1(ϕ1,A1, ϕ̇1)= Mz Iz1 m= k1 I2 k1 = πGd41 32l1 B= Mm I1 z= M0 I1 w= c1 Iz2 n= k1 Iz2 A= Mm I3 Iz1 = I1I2 I1+ I2 Iz2 = I2I3 I2+ I3 β= I1 I1+ I2 (3.2) Let the solution to the system of equations (3.1) be approximated by ϕ1 =A1cosθ1 ϕ2 =A2cosθ2 (3.3) where θ1 =ωt−φ1 θ2 = θ1−φ2 (3.4) and A1,A2, φ1, φ1 are all slowly varying functions of time t. Then ϕ̇1 = Ȧ1 cosθ1+A1φ̇1 sinθ1−A1ω sinθ1 (3.5) ϕ̇2 = Ȧ2 cosθ2+A2φ̇2 sinθ2−A2ω sinθ2 By analogy to Lagrange’smethod of variation of a parameter, it is permissible to propose the following Ȧ1 cosθ1+A1φ̇1 sinθ1 =0 (3.6) Ȧ2 cosθ2+A2φ̇2 sinθ2 =0 882 Z.Skup Thus ϕ̈1 =ωA1φ̇1cosθ1−A1ω2 cosθ1−ωȦ1 sinθ1 (3.7) ϕ̈2 =ωA2φ̇2cosθ2−A2ω2 cosθ2−ωȦ2 sinθ2 Substituting equations (3.7) and (3.5)2 into equations of motion (3.1) and using formulas (3.3), (3.4), (3.6), are arrives at ωA1φ̇1 cosθ1−A1ω2cosθ1−ωȦ1 sinθ1+ cA2ω sinθ2+ + f1(A1,θ1)−mA2 cosθ2−B= z cos(θ1+φ1) (3.8) ωA2φ̇2 cosθ2−A2ω2cosθ2−ωȦ2 sinθ2−wA2ω sinθ2+ +nA2cosθ2−βf1(A1,θ1)−A=0 Multiplying equation (3.6)1 by ωcosθ1, equation (3.8)1 by sinθ1, then subtracting the sides and using formula (3.4), we obtain −A1ω2 sinθ1 cosθ1−ωȦ1+f1(A1,θ1)sinθ1−B sinθ1+ (3.9) −mA2cosθ2 sin(θ2+φ2)+ cA2ω sinθ2 sin(θ2+φ2)= z sinθ1cos(θ1+φ1) Since the variables A1, A2, φ1 and φ2 are assumed to be slowly varying, they remain essentially constant over one cycle of θ1. Thus equation (3.9) may be averaged over one cycle of θ1, which gives −ωȦ1+ 1 2 cA2ωcosφ2+ 1 2π 2π ∫ 0 f1(A1,θ1)sinθ1 dθ1− 1 2 mA2 sinφ2 = (3.10) =− 1 2 z sinφ1 Multiplying equation (3.6)1 by ω sinθ1, equation (3.8)1 by cosθ1, adding the sides, and using formula (3.4), gives ωA1φ̇1−A1ω2cos2θ1+ cA2ω sinθ2 cos(θ2+φ2)+f1(A1,θ1)cosθ1+ (3.11) −mA2cosθ2cos(θ2+φ2)−Bcosθ1 = z cosθ1cos(θ1+φ1) Averaging equation (3.11) over one cycle of θ1, gives ωA1φ̇1− 1 2 cA2ω sinφ2− 1 2 A1ω 2+ 1 2π 2π ∫ 0 f1(A1,θ1)cosθ1 dθ1+ (3.12) − 1 2 mA2cosφ2 = 1 2 z cosφ1 Damping of vibrations in a power transmission system... 883 Similarly,multiplying equation (3.6)2 by ωcosθ2, equation (3.8)2 by sinθ2, subtracting the sides, and using formula (3.4), yields ωȦ2+A2ω 2 sinθ2cosθ2+wA2ω sin 2θ2−nA2 sinθ2cosθ2+ (3.13) +βf1(A1,θ1)sin(θ1−φ2)+Asinθ2 =0 With equation (3.13) averaged over one cycle of θ1, we obtain ωȦ2+ 1 2 wA2ω+ β cosφ2 2π 2π ∫ 0 f1(A1,θ1)sin1 dθ1+ (3.14) − β sinφ2 2π 2π ∫ 0 f1(A1,θ1)cosθ1 dθ1 =0 Finally, multiplying equation (3.6)2 by ω sinθ2, equation (3.8)2 by cosθ2, adding the sides, and using formula (3.4), gives ωA2φ̇2−A2ω2cos2θ2−wA2ω sinθ2 cosθ2+nA2cos2θ2+ (3.15) −βf1(A1,θ1)cos(θ1−φ2)−Acosθ2 =0 After averaging over one cycle of θ1, equation (3.15) takes the following form ωA2φ̇2− 1 2 A2ω 2+ 1 2 nA2− β cosφ2 2π 2π ∫ 0 f1(A1,θ1)cosθ1 dθ1+ (3.16) −β sinφ2 2π 2π ∫ 0 f1(A1,θ1)sinθ1 dθ1 =0 Steady-state equations (3.10), (3.12), (3.14) and (3.16) can be obtained when Ȧ1 = Ȧ2 = φ̇1 = φ̇2 =0 When the following notations are introduced S1 = 1 π 2π ∫ 0 f1(A1,θ1)sinθ1 dθ1 (3.17) C1 = 1 π 2π ∫ 0 f1(A1,θ1)cosθ1 dθ1 884 Z.Skup equations (3.10), (3.12), (3.14) and (3.16) assume the following form −S1+mA2 sinφ2− cA2ωcosφ2 =−z sinφ1 −A1ω2+C1−mA2cosφ2− cA2ω sinφ2 = z cosφ1 (3.18) wA2ω+βS1 cosφ2−βC1 sinφ2 =0 (n−ω2)A2−βC1cosφ2−βS1 sinφ2 =0 The variable φ1 may be eliminated from the foregoing equations by squ- aring and adding equations (3.18)1,2. This gives (S1−mA2 sinφ2+cA2ωcosφ2)2+(C1−A1ω2−cA2ω sinφ2−mA2 cosφ2)2 = z2 (3.19) Equations (3.18)3,4 may be rewritten in the following form sinφ2 = A2[S1(n−ω2)+wωC1] β(C21 +S 2 1) (3.20) cosφ2 = A2[C1(n−ω2)−wωS1] β(C21 +S 2 1) In order to determine the amplitude A2, the second equation has to be formulated by means of squaring and adding the sides of equations (3.20). Performing the indicated operations and rearranging the equations, are obtains A22 = β2(S21 +C 2 1) [(n−ω2)2+w2ω2] (3.21) Equations (3.20) can be used to eliminate the variable φ2 from equation (3.19). Therefore, substituting equations (3.20) and (3.21) into equation (3.19) and rearranging them, gives βz2 =(C21 +S 2 1)(α1−α4)+A1(α2S1+α3C1)+α5A21 (3.22) where x= β2 [(n−ω2)2+w2ω2] α1 =β[1+x(m 2+ c2ω2)] α2 =2xω 3[(n−ω2)−wm] α3 =2ω2{x[m(n−ω2)+ω2w]−β} α4 =2x[ω 2cw−m(n−ω2)] α5 =βω4 (3.23) Damping of vibrations in a power transmission system... 885 For ϕ̇=0, there appears a discontinuity yn M(ϕ1,A1, ϕ̇1). To avoid this while integrating Eqs. (3.17), we confine our considerations to a single half- period (motion between two stops). Thus, the integration interval (from 0 to 2π) of the right-hand terms of the above equations is divided into two sub-intervals: from 0 to π for negative sgnϕ̇1 and from π to 2π for positive sgnϕ̇1. Such a procedure, for instance, was adopted by Caughey (1960), Osiński (1998) and Skup (1998). Therefore, after substituting formulas (2.4) and (2.7) into equations (3.17) by using formula (3.2) and subsequent integration, we obtain the following relationships after some transformations C1 = 1 π 2π ∫ 0 f1(A,θ)cosθ dθ= = 1 πIz1 { π ∫ 0 [M(ϕ1,A1, ϕ̇1)+MT(ϕ̇)]cosθ dθ↓sgn ϕ̇<0 } + + 1 πIz1 { 2π ∫ π [M(ϕ1,A1, ϕ̇1)+MT(ϕ̇)]cosθ dθ↓sgn ϕ̇>0 } = 8 √ A1 3πIz1 √ 2η3 (3.24) S1 = 1 π 2π ∫ 0 f1(A,θ)cosθ dθ= = 1 πIz1 { π ∫ 0 [M(ϕ1,A1, ϕ̇1)+MT(ϕ̇)]sinθ dθ↓sgn ϕ̇<0 } + + 1 πIz1 { 2π ∫ π [M(ϕ1,A1, ϕ̇1)+MT(ϕ̇)]sinθ dθ↓sgn ϕ̇>0 } = = 1 πIz1 [ 4F (2 3 c2A 2 1ω 2−a2)− 4 √ A1 3 √ 2η3 ] Finally, substituting equations (3.24) into equation (3.22) and (3.21) and using formulas (3.23), gives T8A 8 1+T7A 7 1+T6A 6 1−T5A51+T4A41+T3A31+T2A21+T1A1+T0 =0 (3.25) A22 = x π2I2z1 {32A1 9η3 + [ 4F (2 3 c2A 2 1ω 2−a2 ) − 4 √ A1 3 √ 2η3 ]} 886 Z.Skup where T0 =β 2z4+α18(α18−2βz2)+α214 T5 =2(α11α12+α10α13) T1 =2[α13(α18−βz2)]+α14α16 T6 =α211+2α10α12 T2 =α 2 18−2α12(βz2−α18)+α216+2α14α15 T7 =2α10α11 T3 =2[α12α13+α11(α18−βz2)]+α15α16 T8 =α210 T4 =α 2 12+α 2 15+2[α11α13+α10(α18−βz2)] (3.26) and α6 = 8Fc2ω 2 3πIz1 α7 = 4Fa2 πIz1 α8 = 16F 3πIz1 √ 2η3 α9 = 8 3πIz1 √ 2η3 α10 =α1α 2 6−α4α26 α11 =α2α6 α12 =2α6α7(α4−α1)+α5 α13 =(α1−α4)(α28+α29)−α2α7 α14 =2α7α8(α1−α4) α15 =2α6α8(α4−α1) α16 =α3α9−α2α8 α18 =α1α7 (3.27) Thus, the formulated steady-state problem has been reduced to a set of two equations, i.e. (3.25), with two unknown amplitudes A1 and A2. Equation (3.25)1 was solved by means of the Newton-Raphson Iterative Technique method. We had to choose one from the eight roots of equation (3.25)1 which would satisfy the physical condition. That root takes a specific value of the deformation amplitude in the exa- mined system. For such a value of A1, the value of A2 was calculated with formula (3.25)2 in function of the forced vibration frequency. 4. Numerical results The following data has been assumed in the numerical calculations h1 =0.00125m h2 =0.00103m r=0.050m M0 =20Nm µ=0.21 d=0.035m Damping of vibrations in a power transmission system... 887 I1 =0.35 kgm 2 I2 =0.08 kgm 2 I3 =0.420 kgm 2 l=0.25m p=1.2 ·105 N/m2 G=8.1 ·1010 N/m2 c1 =0.50Nms On the basis of results of the numerical analysis, it has been found that all resonance curves do not start from zero but tend asomptically to zero in the superresonance range (Fig.6–Fig.9). The response curves are typical for the ”soft” type of resonance (Fig.6). The influence of the loading amplitude (Fig.6), unit pressures (Fig.7), viscotic damping (Fig.8) and the internal radius (Fig.9) on the process of vibration damping has been examined in the numerical calculations as well. Diagrams in Fig.6 show that the maximal values of the amplitudes A1 and A2 in the first resonance are significantly higher than theirmaximal values in the second resonance. Fig. 6. Resonant curves for various amplitudes of the excitation torque M0 There exists an optimal clamp of the clutch plates, where the resonance amplitudes in the first and second resonance reach the minimum (see Fig.7). The reason for this is the increase of the sliding zone of the cooperating disc surfaces, which maximizes the loss of energy. When the amplitude rises, the difference between the maximal values of the amplitudes A1 in the first and second resonance grows a little, and the difference between the maximal values of the amplitude A2 decreases a little. 888 Z.Skup Fig. 7. Resonant curves for various values of the unit pressure p The examined system has a ”soft” frequency characteristic and damping diagram. An increase in the viscotic damping causes a decrease in the reso- nance amplitudes A1 and A2, particularly in the first resonance (Fig.8). Fig. 8. Resonant curves for various values of the viscotic damping coefficient c1 When the unit pressure increases, the sliding zone decreases, which enta- ils weaker energy dissipation and decreased damping capability of the power transmission system. The less is the internal radius, themore visible becomes the damping vibration effect (Fig.9). Damping of vibrations in a power transmission system... 889 Fig. 9. Resonant curves for various values of the internal radius r of the discs 5. Concluding remarks Structural frictionbetweencontacting surfaces ofdiscs in the frictionclutch causes increased performance of the examined system in terms of vibration damping. The author of the paper has carefully examined the effect of the most important parameters of the vibrating power transmission system with a friction clutch on resonant amplitudes. On the basis of the obtained results, it has been found that all resonance curves start from a non-dimensional resonance amplitude and tend asympto- tically to zero in the post-resonance zone. They also tend to assume a more smooth form in that zone. The damping effect is strongest for the optimal value of the friction force when the area of relative slide between the co- oporating surfaces of discs is largest. The effects of structural friction and viscotic damping can be used in order to improve the design of dynamic systems. Yet, it shouldbenoted that vibrationdampingby friction clutches is consi- derably influencedby the following factors: amplitude of forcing, unit pressure, coefficient of viscous damping and internal radius of discs. The examined sys- tem has a ”soft” frequency characteristic and attenuation diagram. 890 Z.Skup References 1. BuczkowskiR., 1999,Statisticalmodelingof roughsurfacesandfinite element contact analysis, Rozprawa habilitacyjna,Prace IPPT-PAN, 3 2. Buczkowski R., Kleiber M., 1997, Elasto-plastic interface model for 3D frictional orthotropic contact problems, Int. Numer. Methods Eng., 40, 599-619 3. Caughey T. K., 1960, Sinusoidal excitation of a system with bilinear hyste- resis,Trans. ASME, 82, 640-643 4. Gałkowski Z., 1981,Analysis of non-stationary processes in a dynamical sys- tem with structural friction, Ph.D. thesis, (original title in Polish: Badanie procesów niestacjonarnych układu dynamicznego z uwzględnieniem tarcia kon- strukcyjnego),WarsawUniversity of Technology 5. Giergiel J., 1990,Tłumienie drgań mechanicznych, PWN,Warszawa 6. 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SkupZ., 1998,Wpływ tarcia konstrukcyjnegowwielotarczowymsprzęgle cier- nym na drgania w układzie napędowym,Prace naukowe PW, Mechanika, 167 18. Skup Z., 2001, Analysis of frictional vibration damper with structural friction taken into consideration,Machine Dynamics Problems, 25, 2, 103-114 19. Skup Z., 2003, Active and passive damping of vibrations by torsional damper during steady statemotion of a power transmission system, Journal of Theore- tical and Applied Mechanics, 41, 1, 155-167. 20. Skup Z., 2004, Vibration damping in a power transmission system containing a friction clutch,Mathematik and Mechanik, PAMM, 4, Issue 1, 111-121 21. Szadkowski A.,MorfordR.B., 1992,Clutch engagement simulation: enga- gement without throttle, SAE Technical Paper Series, 920766, 103-116 22. Zboiński G., Ostachowicz A., 1997, A general FE algorithm for 3D incre- mental analysis of frictional contact problems of elastoplasticity, Finite Ele- ments in Analysis and Design, 27, 289-305 Tłumienie drgań w układzie napędowym zawierającym sprzęgło cierne Streszczenie Artykuł przedstawia rozważania teoretyczne procesu tłumienia drgań nielinio- wych w układzie napędowym o trzech stopniach swobody ze sprzęgłem ciernymwłą- czanym elektromagnetycznie. Przedmiotem rozważań jest ruch ustalony układu pod- danego wymuszeniu harmonicznemu. Zagadnienie rozpatrywane jest przy założeniu stałego rozkładu nacisku pomiędzywspółpracującymi powierzchniami tarcz ciernych. Uwzględniane jest tarcie konstrukcyjne, mały względny poślizg tarcz sprzęgła oraz liniowe tłumienie wiskotyczne. W przypadku poślizgu współczynnik tarcia nie jest 892 Z.Skup stały, a zależy od względnej prędkości powoli ślizgających się tarcz. Celem analizy jest zbadanie wpływu geometrycznych parametrówukładu, obciążenia zewnętrznego, nacisku jednostkowego, wiskotycznego współczynnika tłumienia na krzywe rezonan- sowe i kąta przesunięcia fazowego dla drgań ustalonych. Równania ruchu badanego układuzostały rozwiązanemetodąpowoli zmieniających się parametrów(metodaVan der Pola) i metodą symulacji cyfrowej. Manuscript received April 25, 2005; accepted for print July 7, 2005