1 Introduction

Because of manufacturing tolerance, wear, plastic deformation, and so on, practical joints in multibody mechanical systems always include clearances. Clearances may lead to friction and impact between the components of joints, which have significant influence on the dynamic responses of the systems.

In recent years, a great number of researchers have studied problems of modeling and analyzing multibody systems including clearance joints^[1-7]. Erkaya and Uzmay^[8] and Erkaya and Doǧan^[9] studied kinematic and dynamic characteristics of planar multibody systems with link flexibility and joint clearance. They also carried out both experimental and numerical investigations to analyze the effects of joint clearance on conventional articulated and partly compliant mechanisms^[10-11]. Flores and his research group have valuable contributions about joint clearance in thematic literature. Dry contact including friction and lubrication effects between bearing and journal parts^[12-15], and different joint types and clearance sizes in two-dimensional and three-dimensional mechanism motions^[16-19] were investigated by them. Rahmanian and Ghazavi^[20] and Farahan et al.^[21-22] studied nonlinear dynamic behaviors and bifurcation in multibody systems with revolute clearance joints. Bai et al.^[23-24] and Bai and Sun^[25] investigated the effects of different normal contact force models and body flexibility on dynamic characteristics of mechanical systems with clearance joints. They also presented the wear prediction for revolute clearance joint in multibody systems using the Archard' s wear model^[26-27]. Qi et al.^[28] and Wang et al.^[29] proposed recursive formulations for multibody systems including frictional joints with tiny clearances using the interactions between bodies, and presented a constraint-based approach for modeling revolute clearance joints of planar multi-rigid-body systems. Based on the absolute nodal coordinate formulation (ANCF), Tian et al.^[30-32] and Wang et al.^[33] investigated planar/spatial flexible multibody systems with dry and lubricated clearance joints. Yan and Guo^[34] and Xiang et al.^[35] presented a general method of kinematic accuracy analysis for the flexible planar mechanisms with uncertain link lengths and joint clearances, and discussed the coupled iterative analyses between system dynamic response and joint wear prediction. However, all these studies mentioned above were based on the modified Coulomb friction model. Although it can avoid numerical difficulties when the relative tangential velocity is in the vicinity of zero, the modified Coulomb friction model does not produce any force at zero relative velocity, and exhibits poor capability to simulate stiction and stick-slip motion^[36-37].

Different from the aforementioned studies, some researchers investigated multibody systems with clearance joints using the LuGre friction model. Muvengei et al.^[38-39] studied the dynamic behaviors of planar multibody systems with LuGre friction at different located revolute clearance joints, and found that the effect of stick-slip friction on the overall dynamic behavior at different speeds varies from one clearance joint to another. Zhao et al.^[40] presented a method for modeling and analyzing planar multibody systems with mixed lubricated revolute joints based on the finite element method and the Lagrange equations. Zheng and Zhou^[41] and Zheng et al.^[42] investigated the dynamics of flexible multibody systems including joints with clearance and lubrication for press systems via the software ADAMS. However, the LuGre friction model introduces extra degree of freedom (DOF), and requires the determination of a large number of parameters, which can hardly be gained even by experiments in most cases. Therefore, it cannot be adopted easily^[36-37].

The purpose of this paper is to compare and analyze the influence of using the Coulomb dry friction model and the modified Coulomb friction model on the dynamic response of the slider-crank mechanism with a revolute clearance joint. The advantages of the Coulomb dry friction model include that it can capture stiction and stick-slip motion and requires a small number of selected parameters. The main difficulty in solving contact problems with dry friction results from instantaneous changes of the frictional forces at transitions from sliding to sticking or reversed sliding. Fortunately, these problems can be solved by the linear complementarity problem (LCP) algorithm^[43-46] or trial-and-error algorithm^[47]. The trial-and-error algorithm can be employed expediently when there is only one pair of contact points. However, the LCP algorithm is much more efficient dealing with multipoint contact problems with friction^[48], and the results can be tested by the trail-and-error algorithm. Flores et al.^[49] proposed an LCP algorithm for dynamic modeling and analysis of rigid multibody systems with translational clearance joints. Zhuang and Wang^[50-51] and Wang et al.^[52] presented a constraint-stabilized method for the planar multi-rigid-body system possessing translational joints with tiny clearances and friction based on the horizontal linear complementarity problem (HLCP) algorithm. Krinner and Thümmel^[53] investigated a planar 6-bar linkage mechanism with revolute clearance joints using the methods of unilateral contacts. Akhadkar et al.^[54-55] used nonsmooth contact dynamics (NSCD) approach to analyze the influence of the joint clearances in a mechanism of a circuit breaker, and validated their studies by comparing with experimental data.

The remainder of this paper is organized as follows. In Section 2, the modeling of the slider-crank mechanism with a revolute clearance joint is presented, and the contact force models utilized in this work are briefly introduced. In Section 3, the algorithms for solving the frictional force are given. In Section 4, three numerical examples are illustrated. Finally, in the last section, the main conclusions of this paper are drawn.

2 Modeling the slider-crank mechanism with a revolute clearance joint 2.1 Modeling the slider-crank mechanism

Consider a planar slider-crank mechanism moving in a vertical plane, as illustrated in Fig. 1. The mechanism consists of four rigid bodies: the ground, an uniform disk (the crank), the rod AB (the connecting rod), and a slider. Joint A connecting the disk with the rod AB is a revolute clearance joint. The journal of joint A is part of the crank, and the bearing of joint A is part of the rod AB. The revolute joint O connecting the ground with the disk, the revolute joint B connecting rod AB with the slider, and the translational joint are ideal joints. The inertial frame of reference Oxy is fixed on the ground, and the relevant coordinates are illustrated in Fig. 1. The gravitational acceleration is taken as acting in the positive x-direction. The moments of inertia of the crank and rod AB are J₁ (about the joint O) and J₂ (about the mass center C₂), respectively. The masses of rod AB and the slider are m₂ and m₃, respectively. The radius of the disk is L₁. The distance between the mass center of rod AB and the bearing center of joint A is a₂, while the distance between the mass center of rod AB and the center of joint B is b₂.

The crank, which is the driving part, rotates with a given driving torque of

(1)

or with a given rheonomic constraint of

(2)

In either case, the crank is subject to a viscous damping torque , while the slider is subject to a viscous damping force c₃ẋ₃.

Remove the constraints generated by joint B and the given rheonomic constraint for the crank, the vector of generalized coordinates of the mechanism q can then be expressed as

(3)

The vectors of the geometric centers of the journal and the bearing of joint A can be easily gained by using the geometrical parameters and generalized coordinates of the mechanism,

(4)

(5)

For convenience of expression, the following notations are used in this paper:

(6)

(7)

where A_{m × n} represents an m × n matrix, and t is time.

2.2 Normal contact force model

The simulation needs to develop a mathematical model for the revolute clearance joint A in the slider-crank mechanism, as shown in Fig. 2. The eccentricity vector d connecting the centers of the bearing and the journal of joint A is defined as

(8)

The unit eccentricity vector n which is normal to the surfaces of collision between the bearing and the journal can then be expressed as

(9)

The unit vector has the same direction as the line of centers of the bearing and the journal. Rotating the vector n in the counter clockwise direction by π /2 gives the unit tangential vector τ,

(10)

When the journal contacts with the bearing, appropriate normal and tangential contact force models are employed and the resultant forces are included as generalized forces in the dynamic equations of the mechanism. In the present study, the normal contact force is calculated by a Hertzian-based contact force model given by Hunt and Crossley^[56] as

(11)

where F_n represents the normal contact force. δ and are the penetration depth and penetration velocity between the journal and the bearing, respectively. K is the stiffness coefficient, while χ is the hysteresis factor. For the case where there is a parabolic distribution of contact stresses, the value of the exponent α is equal to 1.5^[57]. α can be either higher or lower for materials such as polymer or glass^[58]. The coefficient of stiffness K can be obtained by using

(12)

where R_j and R_b are the radii of the journal and the bearing, respectively. σ₁ and σ₂ denote the material parameters for the crank and rod AB, respectively, and they are given by

(13)

in which E_ι and ν_ι are the material Young's modulus and Poisson's ratio associated with the corresponding body, respectively. The hysteresis factor χ can be obtained as^[59]

(14)

where ε_r is the restitution coefficient, and is the initial impact velocity.

As shown in Fig. 2, the penetration depth between the journal and the bearing can be calculated by

(15)

2.3 Tangential contact force models

The Coulomb dry friction model is a fundamental and simple friction model for dry contacting surfaces. It has good capability to simulate stiction and to capture the stick-slip motion, and it requires a smaller number of selected parameters and less computational time than the bristle-based friction models^[36-37]. Besides, the coefficients of Coulomb friction can be easily obtained by consulting handbooks on friction or by doing experiments. Therefore, the Coulomb dry friction model is employed as the tangential contact model for the components of joint A in this paper.

The Coulomb dry friction model is expressed as

(16)

where F_τ is the frictional force acting on the contact points between the journal and the bearing^[60]. v_τ is the relative tangential velocity of the contact points, and a_τ = dv_τ /dt. μ_s and μ are the coefficients of static and kinetic friction, respectively. μ_s is generally larger than μ. sgn(x), which is the sign function, is defined by

(17)

while Sgn(x), which is the multivalued function, is defined as^[61]

(18)

In order to make comparison, a modified Coulomb friction model which many researchers have adopted is given here,

(19)

where c_d is a dynamic coefficient of correction, which is expressed as

(20)

in which v₀ and v₁ are given tolerances for the relative tangential velocity^{[12-19, 30-33]}. These two Coulomb friction models are illustrated in Fig. 3.

2.4 Dynamic equations for the slider-crank mechanism

For a kinematically constrained rigid multibody system, the Lagrange equations of the first kind are given by

(21)

where T is the kinetic energy of the mechanism. is the vector of generalized velocities. Q is the vector of generalized forces. Φ is the kinematic independent holonomic constraints vector^[62], while Φ_q is the Jacobian matrix of the holonomic constraints . λ is the vector of Lagrange multipliers.

The kinetic energy of the mechanism is expressed as

(22)

where M is the positive definite mass matrix

(23)

The vector of generalized forces is expressed as

(24)

where

and

in which F is the vector of generalized externally applied forces,

(25)

or

(26)

and

The constraint equations can be expressed as

(27)

or

(28)

To keep the constraint violations under control, the method of Baumgarte stabilization^[63] can be used,

(29)

where α_B>0, and β_B>0, and they can be chosen following the instruction of Ref. [64].

From Eq. (29), the following equation can be obtained:

(30)

where . Substituting Eqs. (22) and (24) into Eq. (21) results in

(31)

Equation (31) is inserted into Eq. (30), which gives the vector of Lagrange multipliers,

(32)

where

Substituting Eq. (32) into Eq. (31) leads to the differential dynamic equations for the system,

(33)

where

3 Calculating the frictional force and solving the ordinary differential equations 3.1 Complementary conditions between accelerations and friction saturations

In order to solve the dynamic equation (33), the frictional force between the journal and the bearing of the clearance joint A and F_τ must be determined. When v_τ ≠ 0, the journal is sliding relative to the bearing. Therefore, the frictional force can be calculated by the first equation of Eq. (16). When v_τ = 0, there are the following three cases:

(ⅰ) remains sticking.

(ⅱ) commences negative sliding.

(ⅲ) commences positive sliding.

Therefore, when v_τ = 0, the determination of the stick-slip transition and the calculation of the static frictional force between the journal and bearing will be difficult. Using the non-smooth dynamic approach^[48], one can transform this problem to an LCP and solve it.

An LCP is a set of linear equations that can be expressed as

(34)

subject to inequality complementarity conditions

(35)

for which the vectors u and v can be calculated for given A and b^[65]. That is to say, the LCP is the problem of finding solutions u ∈ ℝⁿ and v ∈ ℝⁿ for conditions (35) and Eq. (34), in which b is a given n-dimensional vector, and A is a given n × n matrix.

When v_τ = 0, the positive and negative parts of the acceleration a_τ are defined as^[66]

(36)

The saturations of friction are defined as^[66]

(37)

Then, the accelerations a_τ⁺ and a_τ^- are complementary to the friction saturations F_τ⁺ and F_τ^-, respectively. From Eqs. (36) and (37), one can obtain the following three equations:

(38)

The above equations will be used in the next subsection.

3.2 LCP algorithm for the static frictional force

Taking the derivative of Eq. (8) versus time yields

(39)

where

Taking the derivative of Eq. (10) with respect to time leads to

(40)

where

Substituting Eq. (39) into Eq. (40) results in

(41)

The relative tangential velocity of the contact points between the journal and bearing is

(42)

where

Therefore, a_τ can be expressed as

(43)

where

Inserting Eq. (33) into Eq. (43) yields

(44)

Combining Eq. (44) with Eq. (38) results in

(45)

where

(46)

Some numerical methods for the LCP, such as Lemke's algorithm, can be adopted to solve Eq. (45) and conditions (46) to obtain the solution F_τ^-. Substituting F_τ^- into the second equation of Eq. (38), the static frictional force F_τ can be calculated.

3.3 Trial-and-error algorithm for the static frictional force

As the mechanism has only one pair of contact points, the trial-and-error algorithm can be employed, and the results generated by the trial-and-error algorithm can be compared with those yielded by the LCP algorithm to test the validity of the LCP algorithm. When v_τ = 0, suppose a_τ = 0 firstly. Then, Eq. (44) leads to

(47)

If |F_τ| ≤ μ_sF_n, the static frictional force is then solved by Eq. (47), otherwise the frictional force is determined by

(48)

By using the LCP algorithm or the trial-and-error algorithm, the static frictional force F_τ can be obtained. Inserting F_τ into Eq. (33), the dynamic equations can be solved by numerical methods for ordinary differential equations. In this study, Lemke's algorithm is used to solve the LCP, and the ODE15s in MATLAB is adopted to solve Eq. (33). The simulation flowchart is given in Fig. 4.

4 Numerical examples

As the focus of this study is to investigate the influence of using the two Coulomb friction models on the dynamical response of the slider-crank mechanism with a revolute clearance joint, the journal and the bearing of the clearance joint are in permanent contact in all numerical examples, which is so called "contact or following mode"^[15].

The parameters of the slider-crank mechanism are set as

The coefficients for the Hertzian-based contact force model are^[46-47]

The viscous damping coefficients for the crank and the slider are

The Baumgarte stabilization constants are set as

The gravitational acceleration is set as g = 9.8 m/s². When v_τ ≤ ε = 1 × 10^-5 m/s, the relative tangential velocity of the contact points between the journal and bearing is treated as zero^[68-69].

The velocity tolerances for the modified Coulomb friction model are set as

4.1 Example Ⅰ: the crank rotating with a given driving torque

In this example, the crank is driven by a given sinusoidal driving torque,

The initial conditions of the mechanism are set as

Using the above initial conditions, the dynamic response is simulated with the following four cases in the revolute clearance joint A: (ⅰ) no friction; (ⅱ) using the modified Coulomb friction model with μ = 0.10; (ⅲ) using the Coulomb dry friction model with μ = μ_s = 0.10; (ⅳ) using the Coulomb dry friction model with μ = 0.10 and μ_s = 0.11. The dynamic response of the slider-crank mechanism with ideal joints (ideal slider-crank mechanism) is also simulated to compare with the above results. Figure 5 shows the time histories of angular position and angular velocity of the crank. When the revolute clearance joint has no friction, the motion of the slider-crank mechanism with a revolute clearance joint is identical with ideal slider-crank mechanism, as shown in Fig. 5(a). Because of the friction in the clearance joint A, the amplitude of the crank angular position reduces and the motion lags, as illustrated in Figs. 5(b), 5(c), and 5(d). The modified Coulomb friction model does not capture stick-slip motion, but the Coulomb dry friction model does.

Figure 6 depicts the time histories of the relative tangential velocity, the frictional force, and the normal contact force of the contact points between the journal and the bearing. The contact points show stick-slip motion if the Coulomb dry friction model is applied, as shown in Figs. 6(b), 6(c), and 6(d), where Fig. 6(d) is a magnified view of the blue rectangle in Fig. 6(c). Due to the difference between static and kinetic friction coefficients, the relative tangential velocity and the contact forces oscillate severely when the contact points move from stick to slip, as shown in Figs. 6(c) and 6(d).

Figure 7 shows the comparison of the crank angular position based on two different algorithms: trial-and-error algorithm and LCP algorithm, using the Coulomb dry friction model with μ = 0.10 and μ_s = 0.11. It can be seen that the two different algorithms yield identical results.

In order to investigate the differences between the modified Coulomb friction model and the Coulomb dry friction model, Fig. 8 gives the plots of the ratio F_τ /F_n versus v_τ using the two Coulomb friction models. It can be observed from Fig. 8 that the Coulomb dry friction model exhibits the stick friction characteristics when the relative tangential velocity is in the vicinity of zero.

4.2 Example Ⅱ: the crank rotating with a given constraint

In this example, the crank rotates with a given rheonomic constraint,

The initial conditions of the mechanism are set as

The dynamic response of the mechanism is simulated for the revolute clearance joint A modeled by the following four cases: (ⅰ) no friction; (ⅱ) the modified Coulomb friction model with μ = 0.050; (ⅲ) the Coulomb dry friction model with μ = μ_s = 0.050; (ⅳ) the Coulomb dry friction model with μ = 0.050 and μ_s = 0.055. The dynamic response of ideal slider-crank mechanism is also simulated to compare with the above results. In this system, the Lagrange multiplier λ₁ corresponding to the rheonomic constraint is the driving torque acting on the crank. The time histories of λ₁ for the clearance joint A modeled by Cases (ⅰ), (ⅱ), (ⅲ), and for the ideal slider-crank mechanism are given in Figs. 9(a) and 9(b), where Fig. 9(b) is a magnified view of the blue rectangle in Fig. 9(a). Due to the transient effect of stick friction, the driving torque acting on the crank are slightly different using the two different Coulomb friction models. The time histories of λ₁ for the clearance joint A modeled by Cases (ⅲ) and (ⅳ) and for the ideal slider-crank mechanism are shown in Figs. 9(c) and 9(d), where Fig. 9(d) is a magnified view of the blue rectangle in Fig. 9(c). Figure 10 illustrates the time histories of relative tangential velocity, frictional force, and normal contact force of the contact points between the journal and the bearing in Case (ⅳ). Because of the difference between the static and kinetic friction coefficients, the relative tangential velocity, the contact forces, and the driving torque acting on the crank oscillate severely when the contact points move from stick to slip, as shown in Figs. 9(c), 9(d), and 10.

Figure 11 shows the time histories of the Lagrange multiplier λ₁ using the Coulomb dry friction model with different friction coefficients. Simulation results show that lager friction coefficients correspond to larger amplitudes of the Lagrange multiplier λ₁. In other words, the increase in the friction coefficient leads to the increase in the amplitude of the driving torque on the crank, which is consistent with our common sense.

4.3 Example Ⅲ: equilibrium of the mechanism

In this example, the driving crank torque is zero. The initial conditions of the mechanism are set as

Using the above initial conditions, the dynamic behavior of the mechanism is simulated with the following three cases in the revolute clearance joint A: (ⅰ) no friction; (ⅱ) using the modified Coulomb friction model with μ = 0.15; (ⅲ) using the Coulomb dry friction model with μ = 0.15 and μ_s = 0.20. The dynamic response of ideal slider-crank mechanism is also simulated to compare with the above results. Figure 12 illustrates the time histories of the crank angular position θ₁ in these cases. If the Coulomb dry friction model is adopted as the tangential contact force model for the revolute clearance joint A, the system can balance at θ₁ = π - 0.01 due to static friction. However, the crank can hardly keep balance in other cases, and it moves from θ₁ = π - 0.01 to θ₁ =0, as shown in Fig. 12.

Figure 13 gives the time histories of relative tangential velocity and frictional force of the contact points between the journal and the bearing in Case (ⅲ). In this case, the relative tangential velocity of the contact points is zero. Then, the contact points are subject to static frictional forces, which are unequal to zero.

5 Conclusions

This paper aims at comparing and analyzing the effects of the Coulomb dry friction model and the modified Coulomb friction model on the dynamic behavior of slider-crank mechanism with a revolute clearance joint. The normal force of the contact points between the journal and the bearing is expressed as a nonlinear function of the penetration depth and penetration velocity. The tangential force is described by the Coulomb dry friction model, which has stick friction characteristics. The dynamic equations of the system are obtained by using the Lagrange equations of the first kind and the Baumgarte stabilization method. The static frictional force is solved via the trial-and-error algorithm and the LCP algorithm to compare the simulation results. The results generated by using the Coulomb dry friction model are compared with those yielded by the modified Coulomb friction model.

The study shows that due to stick friction, the slider-crank mechanism with a revolute clearance joint may exhibit stick-slip motion, and can balance at some special positions, while the mechanism with ideal joints cannot. Because of the difference between the static and kinetic friction coefficients for the Coulomb dry friction model, the relative tangential velocity of the contact points, the contact forces, and the driving crank torque may oscillate severely when the contact points move from stick to slip.