Modeling and analysis of rigid multibody systems with driving constraints and frictional translation joints

Transcription

1 Acta Mechanica Sinica (2014) 30(3): DOI /s RESEARCH PAPER Modeling and analysis of rigid multibody systems with driving constraints and frictional translation joints Fang-Fang Zhuang Qi Wang Received: 24 May 2013 / Revised: 9 July 2013 / Accepted: 17 December 2013 The Chinese Society of Theoretical and Applied Mechanics and Springer-Verlag Berlin Heidelberg 2014 Abstract An approach is proposed for modeling and analyses of rigid multibody systems with frictional translation joints and driving constraints. The geometric constraints of translational joints with small clearance are treated as bilateral constraints by neglecting the impact between sliders and guides. Firstly, the normal forces acting on sliders, the driving constraint forces (or moments) and the constraint forces of smooth revolute joints are all described by complementary conditions. The frictional contacts are characterized by a setvalued force law of Coulomb s dry friction. Combined with the theory of the horizontal linear complementarity problem (HLCP), an event-driven scheme is used to detect the transitions of the contact situation between sliders and guides, and the stick-slip transitions of sliders, respectively. And then, all constraint forces in the system can be computed easily. Secondly, the dynamic euations of multibody systems are written at the acceleration-force level by the Lagrange multiplier techniue, and the Baumgarte stabilization method is used to reduce the constraint drift. Finally, a numerical example is given to show some non-smooth dynamical behaviors of the studied system. The obtained results validate the feasibility of algorithm and the effect of constraint stabilization. Keywords Multibody systems Lagrange multipliers Driving constraints Coulomb s friction Horizontal linear complementarity problem (HLCP) 1 Introduction In the recent years, many works have been conducted to study the dynamic effect of revolute joints with clearance in The project is supported by the National Natural Science Foundation of China ( and ). F.-F. Zhuang Q. Wang ( fi ) Beihang University, Being, China bhwangi@sina.com multibody mechanical systems. However, the study of translational joints with clearance has not yet received enough attention [1 8]. In multibody mechanical systems with frictional translational joints, there are several different contact configurations of sliders and guides, including: (1) there is no contact between elements; (2) one corner of slider is in contact with the guide surface; (3) two corners of slider are in contact with the guide surface [9 11]. In addition, each contact point may be in stick or slip phase, and this situation greatly enlarges the number of contact configurations. The non-smooth dynamical method is valid and adopted for modeling this type of multibody system, since it is simple and able to deal with all possible different configurations in a unified manner. The stick-slip and contact-detachment transitions are called as non-smooth events. The linear complementarity problem (LCP) is one of the most popular techniues to detect non-smooth events of multibody systems with unilateral constraints [12 14]. Glocker and Pfeiffer [15] analyzed the contacts under dry friction, and formulated the conditions of stick-slip transitions and the contact problem as LCP. Flores et al. [16] applied this method to model the planar rigid multibody system with translational clearance joints and simulate the dynamic response of a planar slidercrank mechanism with slider clearance. They pointed out that some numerical difficulties can arise when the clearance size is very small, this will lead to the well-known drift problem. Many different approaches have been proposed to reduce the constraint violations, as presented in Refs. [17 21]. In previous works [22 24], the translational joints were treated as bilateral constraints when the small clearance sizes and impacts between the sliders and guides were ignored. In this case, the absolute values of normal contact forces acting on the sliders can appear in the dynamic euations of the system [23, 25]. Hence it is needed to detect the signs of normal contact forces before solving the dynamic euations, which may lead to numerical difficulties. The traditional numerical techniue to solve this problem is the trial-and-error

2 438 F.-F. Zhuang, Q. Wang method [26]. Although the non-smooth multibody systems with unilateral or bilateral constraints, as well as those with driving constraints are very common in engineering design, there are only few papers which focus on the smooth rigid body dynamics with driving constraints [27 30]. In this paper, an approach is proposed for modeling and analyses of the rigid multibody systems with frictional translational joints and driving constraints. The theory of nonsmooth multibody dynamics is applied to the multibody system with bilateral constraints. The geometric constraints of translational joints are treated as bilateral constraints, and the normal forces acting on sliders are described by the complementarity conditions. The driving forces (or moments) and constraint forces of revolute joints, together with the transitions of contact situations between sliders and guides, and the stick-slip transitions of sliders in the system are formulated as a horizontal linear complementarity problem (HLCP). The Baumgarte stabilization method is introduced to the constraint euations to reduce the constraint violation. This paper is organized as follows. The mathematical model of translational joints with Coulomb s friction and small clearance is presented in Sect. 2. The dynamical euation is given in Sect. 3 for the non-smooth planar rigid multibody system with driving constraints and frictional translation joints. The numerical algorithm based on HLCP is provided in Sect. 4. In Sect. 5, a numerical example is analyzed to validate the algorithm and highlight the main dynamical phenomenon. The paper ends with conclusions in Sect Mathematical models of translational joints with Coulomb s friction and small clearance 2.1 Normal contact law In this subsection, the normal contact laws of the planar translational joints in a multibody system with n frictional translation joints are presented by means of complementary conditions. Figure 1 shows a planar translational joint with clearance. For the i-th translational joint of the system, the length and width of slider are a i, b i,andd i is the total distance among the guide surfaces, respectively. δ i is the clearance, which can be expressed as δ i = 0.5(d i b i ). P i is the smooth pin connecting the slider i to other body of the system. The surfaces AB and CD of the slider are defined as surface (+) and surface ( ), respectively. It is shown that, when the slider moves inside its guide, there are several possible configurations for the relative position between the i-th slider and its guide, as illustrated in Fig. 1 to Fig. 3, where the distributed normal forces acting on the slider can be indicated by, F( ), F(+) and F ( ), respectively. These different configurations are given as follows: (1) There is no contact between the two elements, i.e., the slider moves inside the guide without any contact, and conseuently there is no reaction force at the i-th slider. Then = 0, ( j = 1, 2), as shown in Fig. 1. Fig. 1 Planar translational joint with clearance constituted by the i-th slider and its guide (2) One corner of the slider is in contact with the guide surface, and then only one normal force acting on the corner is positive, as shown in Fig. 2. For example, > 0, = F ( ) = 0inFig.2a;F ( ) > 0, = = 0 in Fig. 2b; > 0, = 0inFig.2c;and F ( ) > 0, = = 0 in Fig. 2d. (3) Two opposite slider corners are in contact with the guide surfaces, as shown in Figs. 3a and 3b, and then two normal forces acting on the two corners are positive, for example, F ( ) > 0, F ( ) = 0, F ( ) > 0, = 0or = 0, > 0, F ( ) = 0, > 0. (4) Two adjacent slider corners are in contact with the guide surface, as shown in Figs. 3c and 3d. In this case, one surface of the slider is in contact with the guide surface. When corners A and B are in contact with the guide surface, > 0, F ( ) = 0, > 0, F ( ) = 0. When corners C and D are in contact with the guide surface, = 0, F ( ) > 0, = 0, F ( ) > 0. Fig. 2 One corner of the i-th slider in contact with its guide Fig. 3 Two corners of the i-th slider in contact with its guide

3 Modeling and analysis of rigid multibody systems with driving constraints and frictional translation joints 439 For any configuration of the relative position between the i-th slider and its guide, the magnitudes of the normal forces, F( ), F(+),andF( ) are subjected to the following complementarity conditions 0, F ( ) 0, F ( ) = 0, i = 1, 2,, n; j = 1, 2. (1) If the multibody system consists of n frictional translation joints, formula (1) can be written in the matrix form as 0, F ( ) 0, F ( ) = 0, (2) where = [ 11, F(+) 12,, F(+) n1, F(+) n2 ]T R 2n, F ( ) = [F ( ) 11, F( ) 12,, F( ) n1, F( ) n2 ]T R 2n, which will be of use later. 2.2 Tangential contact law When the contacting bodies slide or tend to slide relative to each other, tangential forces are generated. These forces are usually referred to as friction forces. The Coulomb s friction law of sliding friction can represent the most fundamental and simplest model of friction between dry contacting surfaces. When δ i > 0andδ i 0, the motion of the slider in the guide is rectilinear translation. Therefore, the relative tangential velocity of the contact point of slider ġ Ti can be replaced by that of its mass center ġ c Ti when we detect whether this velocity is zero. The symbol * denotes the surface (+) or the surface ( ) of the slider. According to the Coulomb s friction law, the tangential friction force λ Ti actingonthe surface of the i-th slider, which contacts the guide, can be expressed as [31] λ Ti = μ i F Ni sgn(ġc Ti ), ġc Ti 0, μ i F Ni sgn( gc Ti ), ġc Ti = 0, i = 1, 2A#,, n, (3) where FNi is the magnitude of the normal contact force acting on the surface * of the i-th slider, which can be expressed as FNi = F + F, (4) μ i and μ i are the coefficients of kinetic friction and static friction between the i-th slider and its guide, respectively. sgn(x) is a set-valued function defined by +1, x > 0, sgn(x) := [ 1, +1], x = 0, 1, x < 0. The contact set which describes the kinematic state of contact points is defined by H = {i ġ c Ti = 0} with p elements, where the elements of the set H correspond to the potentially active constraints in the tangential direction (sticking). g Hi is defined as the relative acceleration of the contact point in the tangential direction when ġ c Ti = 0. And, it is split into positive and negative parts as follows [32] g + Hi := 1 2 ( g Hi + g Hi ), g Hi := 1 2 ( g Hi g Hi ), i H. (5) Then, the friction saturation is defined by [32] λ + H0i := μ i F Ni + λ Hi, where λ Hi λ H0i := μ i F Ni λ Hi, i H, (6) is the tangential friction force acting on the surface * of the i-th slider whose relative tangential velocity is zero, i.e., ġ c Ti = 0. The friction saturations λ + H0i and λ H0i are complementary to the accelerations g + Hi and g Hi, respectively. The complementarity conditions can be expressed as g + Hi 0, λ + H0i 0, g + g Hi 0, λ H0i 0, g Hi λ + H0i = 0, Hi λ H0i = 0, i H. (7) This set of relations describes the states of continual sticking as well as the transitions to sliding. The relative tangential velocities and the accelerations of the potential contact points of the i-th slider can be written as follows [31] ġ Ti (, ) = W T Ti, g Ti (,, ) = W T Ti + ŴW T Ti, in which W Ti dw Ti dt = ġ Ti = i = 1, 2,, n, (8) [ ġ Ti 1, ġ Ti 2,, ġ Ti k ] T, ŴW Ti =. Euation (8) can be rewritten in the matrix notation of ġg T (, ) = W T, g T (,, ) = W T + ŴW T, (9) where ġg T (, ) = [ġ (+) T1, ġ( ) T1,, ġ(+) Tn, ġ( ) Tn ]T, g T (,, ) = [ g (+) T1, g( ) T1,, g(+) Tn, g( ) Tn ]T, W T = [W (+) T1, W( ) T1,, W(+) Tn, W( ) ŴW T = [ŴW (+) T1,ŴW( ) T1,, ŴW(+) Tn,ŴW( ) Tn ]T. The calculation of the relative tangential velocities of the potential contact points of the i-th slider ġ Ti (, ) andthematrixw Ti are given in Ref. [31]. Euation (4) can be written in the matrix notation of Tn ]T, F N = H + GF ( ), (10) in which F N = [ N1, F( ) N1, F(+) N2, F( ) N2,, F(+) Nn, F( ) Nn ]T 11 R 2n and H = diag[h 1, H 2,, H n ], H i = 00, G = 00 diag[g 1, G 2,, G n ], G i =,(i = 1, 2,, n). 11 The tangential friction force λ T R 2n can be written as λ T = μf N + Vλ H, (11) where μ = diag[ μ 1, μ 2,, μ n ] with μ i = μ (+) i sgn(ġ c Ti ) 0 0 μ ( ) i sgn(ġ c Ti ), (i = 1, 2,, n). λ H = {λ Ti ġc Ti = 0} R 2p (i H; = (+), ( )), V R 2n 2p consists of the nonzero column vectors of matrix U, U = diag[u 1, U 2,, U n ], in which U i = diag[1 sgn(ġ c Ti ) ]

4 440 F.-F. Zhuang, Q. Wang R 2 2 (i = 1, 2,, n). On the right hand side of E. (11), the first term (μf N ) presents the kinetic friction force and the second term (Vλ H ) describes the static friction force. When the relative velocity in tangential direction ġ c Ti = 0(i H), the relative acceleration of the contact point in tangential direction can be expressed in the matrix form as g H = W H + ŴW H, (12) where W H = V T W T, ŴW H = V T ŴW T. Euations (5) (7) can be combined into the matrix form as follows g + H = 1 2 ( g H + g H ), g H = 1 2 ( g H g H ), (13) λ + H0 = μ H F N + λ H, λ H0 = μ H F N λ H, (14) g + H 0, λ+ H0 0, g + H λ+ H0 = 0, g H 0, λ H0 0, g H λ H0 = 0. (15) From Es.(13) and (14), the following functions can be obtained g H = g + H g H, (16) λ + H0 = 2μ H F N λ H0, (17) μ where μ H = V T μ R 2p 2n,andμ μ = diag[ μ 1, μ 2,, μ n] μ (+) with μ i [1 sgn(ġ c Ti i = ) ] 0 0 μ ( ) i [1 sgn(ġ c Ti ) ] (i = 1, 2,, n). 3 Dynamical analysis 3.1 Dynamic euations Let R k be a k-dimensional vector in the generalized coordinates of multibody systems. When the clearance sizes of the translational joints are so small that the sizes of the clearances and the impacts between sliders and guides can be neglected, the constraints of the translational joints and the ideal revolute joints can be treated as bilateral scleronomic holonomic constraints. Hence, the constraint euations of these joints are expressed as Φ CR () = 0, (18) Φ CT () where Φ CR () = 0andΦ CT () = 0 are the vector forms of constraint euations for the revolute and the translational of the system, respectively. The driving constraints in the system are bilateral rheonomic holonomic constraints, whose constraint euations are expressed as Φ D (, t) = 0. (19) Euations (18) and (19) can be written as Φ CR () Φ(, t) = Φ CT () = 0. (20) Φ D (, t) μ The dynamical euation of the multibody system with frictional translational joints and driving constraints can be written as M = Q E + Q CD + Q F, (21) where M = M() R k k is the positive definite mass matrix, Q E = Q + T Ṁ, andq is the vector of generalized forces of active forces, T is the kinetic energy of the system, ṀṀṀ = dm. Q CD and Q F are generalized forces of constraint dt forces of the system. Q CD in E. (21) can be expressed as Q CD = Φ CR () T λ CR + Φ CT () T λ CT + Φ D (, t) T λ D, (22) where λ CR and λ CT represent the reaction forces of revolute joints, and the resultant force or the resultant moment of normal forces acting on the slider, respectively [33]. λ D is the vector of driving forces or driving moments. Q F in E. (21) can be expressed as Q F = W T T λ T, (23) where λ T = [λ T T1,λT T2,,λT Tn ]T, λ Ti = [λ (+) Ti, λ( ) Ti ]T,andλ Ti is the tangential friction force acting on the surface * of the i-th slider (i = 1, 2,, n). W T is defined in Es. (8) and (9). Thus, E. (20) can be rewritten as M = Q E + Φ CR ()T λ CR + Φ CT ()T λ CT +Φ D (, t) T λ D + W T T λ T. (24) The calculation of the vectors λ CR, λ CT,andλ D in E. (24) will be discussed in Sect. 3.2 below. Using the Baumgarte stabilization method, the constraint euations are replaced by Φ + 2α Φ + β 2 Φ = 0, (25) where α and β are prescribed positive constants, respectively, which represent the feedback control parameters for the velocity and the position constraint violations [18, 20, 34, 35]. 3.2 Calculation of the Lagrange multipliers For all sliders of the system, the resultant force or the resultant moment of normal forces acting on the slider can be written in the matrix form as [31] λ CT = S + KF ( ), (26) 1 1 where S = diag[s 1, S 2,, S n ], S i = a a, K = 1 1 diag[k 1, K 2,, K n ], K i =,(i = 1, 2,, n). a a The Lagrange multipliers λ CR and λ D are unknown, and need to be calculated. In order to set up the complementary relations, the Lagrange multipliers λ CR and λ D are split into two parts, which can be represented as

5 Modeling and analysis of rigid multibody systems with driving constraints and frictional translation joints 441 λ CR = λ CR+ λ CR, (27) λ D = λ D+ λ D, (28) where λ CR+ and λ D+ denote the positive part of λ CR and λ D, respectively. λ CR and λ D denote the negative part of λ CR and λ D, respectively. They can be expressed as λ CR+ = 1 2 ( λcr + λ CR ), λ CR = 1 2 ( λcr λ CR ), λ D+ = 1 2 ( λd + λ D ), λ D = 1 2 ( λd λ D ). The positive and negative parts of the Lagrange multipliers λ CR and λ D are subjected to the following complementarity conditions λ CR+ 0, λ CR 0, λ CR+ λ CR = 0, (29) λ D+ 0, λ D 0, λ D+ λ D = 0. (30) In the next section, the above complementary conditions will be used together with the dynamic euations of the multibody system to obtain the HLCP of the fully coupled state transition problem. 4 The event-driven scheme based on HLCP and the Baumgarte stabilization method 4.1 Formulation of HLCP The HLCP [36] formulation to calculate the reaction forces of revolute joints, the driving forces (moments), the normal contact forces and tangential friction forces are given in this section. For this purpose, E. (25) can be rewritten as Φ + Φ + 2αΦ + β 2 Φ = 0, (31) where Φ = Φ, Φ = dφ. Substituting E. (16) into dt E. (12) yields g + H g H = W H + ŴW H. (32) Introducing E. (11) into E. (24) leads to = M 1 Q E + M 1 Φ CR () T λ CR + M 1 Φ CT () T λ CT +M 1 Φ D (, t) T λ D + M 1 W T T μf N + M 1 W T T Vλ H. (33) Substituting Es. (10), (14), (26) (28), and (33) into Es. (31) and (32) results in Φ M 1 Φ CR ()T λ CR+ + Φ M 1 Φ D (, t)t λ D+ +Φ M 1 [Φ CT () T K + W T T (μ Vμ H )G]F ( ) = Φ M 1 Φ CR () T λ CR + Φ M 1 Φ D (, t) T λ D Φ M 1 [Φ CT ()T S + W T T (μ Vμ H )H] Φ M 1 W T T Vλ+ H0 Φ M 1 Q E Φ 2αΦ β 2 Φ, (34) V T W T M 1 Φ CR () T λ CR+ + V T W T M 1 Φ D () T λ D+ +V T W T M 1 [Φ CT () T K + W T T (μ Vμ H )G]F ( ) g + H = V T W T M 1 Φ CR ()T λ CR + V T W T M 1 Φ D ()T λ D V T W T M 1 [Φ CT () T S + W T T (μ Vμ H )H] V T W T M 1 W T T Vλ+ H0 g H V T W T M 1 Q E V T ŴW T.(35) Thus Es. (34) and (35) are combined with E. (17) to yield an HLCP formulation which can be represented as M 1 Φ CR ()T M 1 Φ D (, t)t M 1 M KG 0 0 M 2 M 2 Φ CR ()T M 2 Φ D ()T M 2 M KG E μ H G 0 E λ CR+ λ D+ F ( ) g + H λ H0 M 1 Φ CR () T M 1 Φ D (, t) T M 1 M S H M 1 W T T V 0 = M 2 Φ CR ()T M 2 Φ D ()T M 2 M S H M 2 W T T V E 0 0 2μ H H E 0 λ CR λ D λ + H0 g H M 1 Q E Φ 2αΦ β 2 Φ + M 2 Q E V T ŴW T, (36) 0 where E denotes the identity matrix, M 1 = Φ M 1, M 2 = V T W T M 1, M KG = Φ CT ()T K + W T T G, M S H = Φ CT ()T S + W T T H,and W T T = WT T (μ Vμ H ). Euation (36) together with the complementarity conditions described by Es. (2), (15), (29), and (30) forms the horizontal linear complementarity problem (HLCP). 4.2 Algorithm The frictional contact problem can be formulated and solved as HLCP. Euation (36) can be written in the following form A 0 y = A 1 x + b, (37) subjected to the ineuality complementarity conditions of y 0, x 0, y T x = 0, (38) where A 0 = M 1 Φ CR () T M 1 Φ D (, t) T M 1 M KG 0 0 M 2 M 2 Φ CR () T M 2 Φ D () T M 2 M KG E μ H G 0 E,

6 442 F.-F. Zhuang, Q. Wang M 1 Φ CR ()T M 1 Φ D (, t)t M 1 M S H M 1 W T T V 0 A 1 = M 2 Φ CR ()T M 2 Φ D ()T M 2 M S H M 2 W T T V E 0 0 2μ H H E 0 M 1 Q E Φ 2αΦ β 2 Φ b = M 2 Q E V T ŴW T, 0 y = x = λ CR+ λ D+ F ( ) g + H λ H0 λ CR λ D λ + H0 g H,. If A 0 in E. (37) is a non-singular matrix, the HLCP can be transformed into an LCP. A more efficient algorithm for LCP, such as Lemke s algorithm [37, 38], can be utilized to solve this problem. The changes of the relative tangential velocities of sliders need to be detected using event-driven schemes based on HLCP. Hence, the following steps have to be performed: (1) Give the geometric, inertial parameters of the system, and specify the simulation time t and the initial conditions 0, 0. (2) Compute ġg T to detect the motion states of sliders of the system, and obtain M, Φ, Φ, Φ CR (), Φ CT (), Φ D (, t), Φ, W T, ŴW T, V, μ, μ H. (3) Solve E. (36) by Lemke s algorithm to get the solutions λ CR+, λ CR, λ D+, λ D,, F ( ), g + H, g H, λ+ H, λ H. (4) Substitute these solutions into E. (33) to solve the ordinary differential euation (ODE) by Runge Kutta method. (5) Update the initial conditions 0, 0 with the solutions of ODE and repeat steps (2) (4). 5 Numerical example A slider-bar system with driving constraint is shown in Fig. 4, with a slider of mass m 2, length a and width b. This system is subjected to an applied force of F = F 0 sin(ωt) and is attached to a spring of stiffness k and length L 0.The slider moves in the guide for which the coefficients of kinetic friction and static friction are μ and μ, respectively. The bar AB of mass m 1, length L is driven by an electric motor with a constant angular velocity of ω 0, and turns from the horizontal position. The journal bearing of the motor is treated as an, ideal revolute joint. The general coordinates of the system can be written as = [x 1, y 1,θ 1, x 2, y 2,θ 2 ] T,wherex 1 and y 1 are the coordinates of the mass center C of the bar, θ 1 is the absolute angle of rotation of the bar, x 2 and y 2 are the coordinates of the mass center A of the slider, θ 2 is the absolute angle of rotation of the slider. When the clearance between the slider and its guide is negligibly small, the constraint euations of the system can be written as Φ CR () Φ(, t) = Φ CT () = 0, Φ D (, t) where Φ CR () = Φ CT () = 1 () 2 () = x 1 x 2 0.5L cos θ 1 y 1 y 2 0.5L sin θ, 1 Φ CR Φ CR 1 () 2 () = Φ CT Φ CT y 2 θ 2 Φ D (, t) = θ 1 θ 2 ω 0 t., For this example, the system parameters are set as follows, m 1 = 2.0 kg, m 2 = 1.0 kg, a = 0.6m, b = 0.5m, k = 1.0N/m, Ω = π/2s 1, L 0 = 0.25 m, μ = 0.03, μ = 0.04, α = 90, β = 10, F 0, L, andω 0 will be given in the following three cases. Fig. 4 Slider-bar system with driving constraint Case 1 In this case, let F 0 = 3.0N, L = 2.0m,ω 0 = π/4s 1.Time evolutions of the velocity and acceleration of the slider are shown in Fig. 5. It is found that the motion of the slider is periodic. When both the velocity and the acceleration of slider remain zero, stick phase takes place in the motion. The slider slips if either the velocity or the acceleration of slider is nonzero. From Fig. 5, the value of slider acceleration jumps when the stick-slip transition occurs or the direction of slider s velocity changes, due to the variation of the value of friction forces acting on the slider.

7 Modeling and analysis of rigid multibody systems with driving constraints and frictional translation joints 443 Fig. 5 Time evolution of the velocity and acceleration of the slider The time evolution of F 1 (t), F 2 (t), F 3 (t), and F 4 (t) are shown in Fig. 6, where F i (t) is the magnitude of normal contact force acting on the corner i of the slider. Two adjacent slider corners and two opposite slider corners are in contact with the guide surfaces alternately in the motion. In particular, initially, corners 2 and 4 of the slider remain in contact with the upper and the lower surfaces of the guide, respectively. Next, corners 1 and 2 of the slider remain in contact with the lower surface of the guide, that is, a surface of the slider is in contact with the guide. And then, corners 1 and 3 of the slider are in contact with the guide surfaces. Finally, the lower surface of the slider is in contact with the guide again. Figure 7 shows the time evolution of the driving moment λ D. It is found that a small abrupt change of the magnitude of driving moment occurs when the magnitude of slider s acceleration jumps. This phenomenon results from the direction change of frictional forces acting on the slider. Fig. 6 Time evolutions of F 1 (t), F 2 (t), F 3 (t), and F 4 (t) the value of slider s acceleration jumps when the direction of slider s velocity changes. Figures 9 and 10 illustrate time evolution of F 1 (t), F 2 (t), F 3 (t), F 4 (t), and the driving moment λ D, respectively. In this case, the contact situations of the slider inside the guide are similar to those in Case 1. Fig. 7 Time evolution of driving moment λ D Case 2 Let F 0 = 5.0N, L = 2.0m, ω 0 = π/4s 1. Figure 8 shows time evolution of the velocity and acceleration of the slider, where the slider s motion is periodic without stick phenomena because F 0 in this case is bigger than that in Case 1, and Fig. 8 Time evolution of the velocity and acceleration of the slider

8 444 F.-F. Zhuang, Q. Wang Fig. 9 Time evolution of F 1 (t), F 2 (t), F 3 (t), and F 4 (t) Fig. 3 appear in this case, and the maximum values of F i (t) and λ D are bigger than those in Case 1 because L and ω are bigger than those in Case 1 and Case 2. Compared Case 1 with Case 2, when the amplitude of the external force acting on the slider is changed, such as F 0 = 3.0N or F 0 = 5.0 N, the stick phenomena of the slider occur in Case 1 whereas it does not in Case 2. If the amplitude of the external force acting on the slider is not large enough, the stick phase would take place in the motion. Fig. 10 Time evolution of driving moment λ D Case 3 Here, let F 0 = 1.0N,L = 4.0m,ω 0 = π s 1. The time evolution of the velocity and acceleration of the slider are shown in Fig. 11. The slider s motion is periodic without stick phase because ω 0 in this case is bigger than that in Case 1. The time evolutions of F 1 (t), F 2 (t), F 3 (t), F 4 (t), and λ D are illustrated in Figs. 12 and 13, respectively. It is shown that all contact configurations of the slider and the guide shown in Fig. 11 Time evolution of the velocity and acceleration of the slider Fig. 12 Time evolution of F 1 (t), F 2 (t), F 3 (t), and F 4 (t)

9 Modeling and analysis of rigid multibody systems with driving constraints and frictional translation joints 445 Therefore, the amplitude of the external force can affect the motion states of the slider. The obtained results show that the stick phenomenon occurs in slider s motion if 0 < F , and the slider remains slipping without stick phenomena if F 0 > Figure 14 shows the drift of constraint euations. It can be seen that the constraint drift grows indefinitely with time when the Baumgarte stabilization method is not utilized. However, it can remain bounded Φ < by means of the Baumgarte stabilization method. Fig. 13 Time evolution of driving moment λ D Fig. 14 Time evolution of the constrain drift 6 Conclusions The rigid multibody systems with driving constraints and frictional translation joints are investigated in this paper. The geometric constraints of translational joints are treated as bilateral constraints by neglecting the clearance sizes and impacts between sliders and guides. The theory of non-smooth multibody dynamics is applied for modeling and analyses of these multibody systems. In order to compute the driving forces (or moments), constraint forces of the system and to detect the contact situation transitions and stick-slip transitions of sliders, the normal constraint forces acting on the corners of sliders are described by complementary conditions. Next, the driving forces (or moments) and constraint forces of smooth revolute joints are decomposed into the positive and the negative parts, which are also subjected to the complementary conditions, respectively, and the frictional contacts of sliders are characterized by a set-valued force law of Coulomb s dry friction. Finally, the problem of computing the driving forces (moments) and constraint forces as well as detecting the contact situation transitions and stick-slip transitions of sliders is formulated as an HLCP, which is solved then by an event-driven scheme. The dynamic euations of the system are written at the acceleration-force level by the Lagrange multiplier method, and the Baumgarte stabilization method is introduced in the constraint euations to reduce the constraint drift. In the numerical example, a slider-bar system with driving constraint and frictional translational joint is presented to illustrate the validity of the algorithm. From the numerical results of three different cases, in which the slider s motion is periodic with several contact configurations between the slider and the guide, the stick phase may occur when the amplitude of the external force acting on the slider is small. The constraint drift can be restrained by using the Baumgarte stabilization method. References 1 Flores, P., Ambrósio, J.: Revolute joints with clearance in multibody systems. Comput. Struct. 82, (2004) 2 Flores, P., Ambrósio, J., Claro, J.C.P., et al.: Kinematics and dynamics of multibody systems with imperfect joints: Models and case studies. Lecture Notes in Applied and Computational Mechanics, Volumn 34. Springer, Berlin (2008) 3 Tian, Q., Zhang, Y., Chen, L., et al.: Dynamics of spatial flexible multibody systems with clearance and lubricated spherical joints. Comput. Struct. 87, (2009) 4 Liu, C.S., Zhang, K., Yang, R.: The FEM analysis and approximate model for cylindrical joints with clearances. Mech. Mach.

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