A methodology for modeling and simulating frictional translational clearance joint in multibody systems including a flexible slider part

Mechanism and Machine Theory 0 0 0 (2019) 103603

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Research paper

A methodology for modeling and simulating frictional translational clearance joint in multibody systems including a flexible slider part Xudong Zheng a,b, Jing Li c, Qi Wang b,∗, Qingmin Liao a a

Tsinghua Shenzhen International Graduate School, Tsinghua University, Shenzhen 518055, China School of Aeronautic Science and Engineering, Beihang University, Beijing 100083, China c China Aerospace Academy of Systems Science and Engineering, Beijing 100048, China b

a r t i c l e

i n f o

Article history: Received 7 August 2019 Revised 20 August 2019 Accepted 22 August 2019 Available online xxx MSC: 70E55 74M10 74M15 Keywords: Multibody systems Translational clearance joint Flexible slider Finite element method Stick-Slip

a b s t r a c t The objective of this paper is to present a simple and effective methodology for modeling and simulating frictional translational clearance joint in planar multibody systems including a flexible slider part. By using the finite element method (FEM), the slider is divided into finite elements, and the distributed body forces and boundary stresses (contact forces) on the slider are equivalent to the nodal forces. The normal contact forces are described by a nonlinear viscoelastic contact force model, while the frictional forces are evaluated by the Coulomb dry friction model. By lumping the mass matrix and presenting it in diagonal form, the equations of motion are then inertially decoupled, and the frictional forces are solved independently via trial-and-error algorithm. A penalty method for the revolute joint connecting the slider and other bodies is applied, and equations of motion are integrated numerically. Finally, two numerical examples are given to test the feasibility and effectiveness of the methodology in this paper. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction Over the last decades, researchers have conducted a lot of studies considering multibody mechanical systems with imperfect joints [1–19], and reviews were made in this field [20,21]. As a branch of this area, modeling and simulating translational joints with clearances in multibody systems have attracted the attention of many authors. Wilson and Fawcett [22], Farahanchi and Shaw [23], and Klepp [24] were the pioneers in studying mechanism dynamics with imperfect translational joints. In recent years, Qi et al. [25] presented a frictional contact analysis for multibody systems with spatial prismatic joints on the basis of tiny clearances. Flores et al. [26,27] proposed a methodology for dynamic modeling and analysis of rigid multibody systems with translational clearance joints using the nonsmooth dynamics approach. In their studies, the resulting contact-impact problem was formulated and solved as a linear complementarity problem (LCP) [28,29], which was embedded in the Moreau time-stepping method. Flores [27] demonstrated that some numerical difficulties arose when the clearance size was very small, which led to drift problem. Acary et al. [30,31] investigated the numerical time-integration algorithms for nonsmooth mechanical systems subjected to unilateral contacts, impacts and Coulomb’s friction, and verified their algorithms by comparing with Flores’ study [27]. To overcome the drift problem, Zhuang and wang [32,33] combined ∗

Corresponding author. E-mail address: [email protected] (Q. Wang).

https://doi.org/10.1016/j.mechmachtheory.2019.103603 0094-114X/© 2019 Elsevier Ltd. All rights reserved.

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Fig. 1. FEM discretization of the slider.

Baumgarte’s stabilization method with horizontal linear complementarity problem (HLCP) approach for dynamic modeling and simulating of frictional translational joints with tiny clearances in rigid multibody systems. Using Ting’s rotatability laws, Ting et al. [34,35] presented a novel kinematic model to quantify the effects of the joint clearance on the output position uncertainty of any single loop linkage containing revolute or prismatic joints. Zhao et al. [36] proposed a numerical model for piston dynamics and lubrication analysis by coupling the lubrication model of piston skirt-liner system and the dynamics model of crank-piston mechanism. Zhan et al. [37] analyzed the general planar parallel manipulators with interval clearance variables of revolute and translational joints. All these above-mentioned studies about imperfect translational joints were based on rigid slider models, and the influence of the slider deformation was not studied. In view of this, Zhang and Wang [38] analyzed the deformation of the slider and the clearance size using KED technique and penalty method. However, Zhang’s study mainly has three shortcomings: First, tedious judgments of the contact states between the slider and its guide were necessary in his work. Second, the coupling between rigid body and flexible body motion was ignored [39], and a very large Young’s modulus was applied in his study. Third, his method was time consuming. In our previous study, we found that under the effect of Coulomb dry friction model, the multibody systems with imperfect revolute joints may exhibit stick-slip motions [40]. The present study further extends the research object to multibody systems with imperfect translational joints. The revolute joints are treated as ideal in this paper. This paper proposes a modeling and simulation method for frictional translational joint with a flexible slider and clearance in multibody systems. The main difficulties in solving mechanical systems with impacts and friction result from instantaneous changes of contact forces at transitions from contact to separation, and slipping to sticking or reversed slipping. Two types of methods were proposed to solve these problems: trial-and-error algorithm [41] and complementary-based algorithm [28,29,42–49]. Generally speaking, the trial-and-error algorithm can be employed expediently when there is only one pair of contact points, while the complementary-based algorithm can be adopted to deal with multipoint contact problems with friction [50]. Therefore, the previous researchers often applied the complementary-based algorithm to address flexible mechanical systems involving unilateral contact and friction [44–48]. However, the complementary-based algorithm needs to calculate the linear (or nonlinear) complementarity equations, and sometimes the calculation is time consuming. As described later, when the corresponding body mainly undergoes translational motions, by lumping the mass matrix, the equations of motion are then inertially decoupled, thus, the frictional forces acting at the surface nodes can be solved independently of each other via trial-and-error algorithm. The proposed method is not only different from complementary-based algorithm, but also from KED based approach. The method is simple and effective, and can overcome the shortcomings in Ref. [38]. Additionally, the proposed method may be extended to other flexible mechanical systems with impacts and friction. The reminder of this paper is organized as follows: In Section 2, the FEM discretization of the slider, normal and tangential contact forces, as well as penalty method for the revolute joint are introduced. In Section 3, Equations of motion of the system are established, and computer procedures for dynamic analysis are given. In Section 4, two demonstrative examples are presented, and the results are compared with previous studies. Finally, in the last section, the main conclusions from this study are drawn. 2. FEM discretization of the slider and contact forces 2.1. FEM discretization of the slider As shown in Fig. 1, a translational joint often consists of the guide. For simplicity in this study, the guide is fixed on the ground, and an inertial frame of reference Oxy is established on the guide [27]. The clearance size in the translational

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Fig. 2. Normal contact forces at the nodes.

joints is generally very small, thus, the rotational motion of the slider can be neglected, i.e., the slider mainly undergoes translational motions. To describe the flexibility of the slider, the FEM technique is used, and the slider is divided into a finite number of elements. The distributed body forces and boundary stresses (normal and tangential contact forces) on the slider are equivalent to the nodal forces. The nodes are labeled as illustrated in Fig. 1, and each node has two degrees of freedom. The revolute joint is considered as perfect, that is to say, there is no clearance and friction. The revolute joint connects the slider with other bodies via the node nR . 2.2. Normal contact forces As shown in Fig. 2, the normal contact forces acting at the surface nodes are described by a nonlinear viscoelastic contact force model [51–53]. The contact force model work only if the surface nodes are in contact with or penetrate the guide surfaces, and the normal contact forces can never be negative [51]. The normal contact force acting at node i is expressed as



FiN =

max{K δiα + χ δiα δ˙ i , 0}, δi ≥ 0 0, δi < 0

(1)

where FiN is the normal contact force. δ i and δ˙ i are the penetration depth and penetration velocity of node i, respectively. δ i ≥ 0 indicates that node i is in contact with the guide, while δ i < 0 means that node i is out of touch with the guide. The penetration depths and penetration velocities of the surface nodes can be obtained expediently by the state variables of these nodes: displacements and velocities. K is the generalized stiffness parameter, while χ is the hysteresis factor. The exponent α defines the degree of nonlinearity of the contact, and it is equal to 1.5 for metallic contacts [52]. 2.3. Tangential contact forces There are mainly three types of friction models that have been adopted widely in the field of modeling and simulating multibody systems with imperfect joints. The most popular type is the modified Coulomb friction model [1–15,43], since it can avoid numerical difficulties when the relative tangential velocity is in the vicinity of zero. However, this friction model does not produce any force at zero relative velocity and exhibits poor capability to capture stiction and stick-slip motion [54]. The LuGre friction model has attracted the attention of some researchers [13,16,17], since it can simulate stickslip motion and pre-sliding displacement observed by experiments. The shortcoming is that it needs to determine many parameters, which are difficult to obtain even by experiments. Therefore, the application of this model is limited. The third type is the Coulomb dry friction model [27–29,32,33,38]. This model has the superiority to simulate stiction and to capture stick-slip motion. It has been demonstrated that the Coulomb dry friction model requires less computational time and fewer selected parameters than the LuGre model [55,56]. What’s more, the Coulomb friction coefficients can be easily obtained by experiments or by consulting handbooks on friction. The disadvantage of the Coulomb dry friction model is that when the relative tangential velocity of two contacting objects is equal to zero, it is difficult to detect the stick-slip motion and to calculate the static frictional force. The Coulomb dry friction model is adopted to evaluate the tangential contact forces in this work, and this paper exactly aims at dealing with the difficulties yielded by the friction model.

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Fig. 3. Coulomb dry friction model.

Fig. 4. Penalty method for the revolute joint.

The Coulomb dry friction model can be expressed as



Fiτ =

−μFiN sgn  ( v iτ ) ,  − min |Fix |, μ FiN sgn(Fix ),

v iτ  = 0 v iτ = 0

(2)

where Fiτ is the frictional force acting at node i. μ is the coefficient of kinetic friction, while μ is that of static friction, and μ is usually smaller than μ . viτ is the tangential velocity (component of velocity in x direction) of node i. Fix represents the resultant of forces acting at node i in x direction, and sgn( ) is the sign function. Since the tangential velocities of the surface nodes can hardly be equal to the exact value of zero in computational process, when |viτ | is not larger than a tiny velocity tolerance vɛ , viτ is treated as zero [57,58]. Thus, the Coulomb dry friction model employed in this paper is written as follows:



Fiτ =

−μFiN sgn  ( v iτ ) ,  − min |Fix |, μ FiN sgn(Fix ),

|viτ | > vε |viτ | ≤ vε

(3)

The Coulomb dry friction model is illustrated in Fig. 3. 2.4. Penalty method for the revolute joint There are two common techniques dealing with the perfect revolute joint. One is the famous Lagrange multipliers method, the other is the penalty method [59], which is equivalent to two springs acting at the pin of the joint, as illustrated in Fig. 4. The penalty method is easy to implement as compared to the Lagrange multipliers method, which introduces the differential-algebraic equations that are difficult to solve. Thus, the penalty method is applied for the perfect revolute joint in this study. The forces from the revolute joint acting at the node nR are evaluated as



FRX = kRX (xR − xnR ) FRY = kRY (yR − ynR )

(4)

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where FRX and FRY are two forces from the revolute joint. xR and yR are the components of the position vector of the revolute joint, while xnR and ynR are that of the node nR . kRX and kRY are the penalty factors. 3. Equations of motion of the system and computer procedures 3.1. Equations of motion of the system According to the theory of FEM, the equations of motion of the slider can be written as

M u¨ + C u˙ + Ku = Q˜ + Q N + Q τ + Q FR

(5)

where M, C, and K are mass, damping, and stiffness matrices of the slider, respectively. u, u˙ , and u¨ are the vectors of displacements, velocities, and accelerations of the nodes, respectively. QN is the equivalent nodal forces vector of normal contact forces, while Qτ is that of tangential contact forces. Q FR contains the forces from the revolute joint, and Q˜ is the equivalent nodal forces vector of other forces such as gravity and externally applied forces. The Rayleigh damping matrix is applied:

C = ξ M + ηK

(6)

where the parameters ξ and η are determined experimentally [60]. Other bodies of the system are treated as rigid, thus, their equations of motion are



q¨ = f t, q, q˙ , F R



(7)

where q, q˙ , and q¨ are the vectors of generalized coordinates, velocities, and accelerations of other bodies, respectively. FR contains the forces from the revolute joint. Combining Eqs. (5) and (7) yields the equations of motion of the system:



M u¨ +C u˙ + Ku = Q˜ + Q N + Q τ + Q FR q¨ = f t, q, q˙ , F R

(8)

3.2. Computer procedures for dynamic analysis of the system In order to integrate the differential equations of motion (8) numerically, the vectors of contact forces QN and Qτ must be calculated first. Since QN relates to the state variables of the slider u and u˙ , it can be easily obtained. However, the calculation of Qτ is much more difficult. As for the surface node i, when |viτ | > vɛ , node i is slipping relative to the guide, thus, the frictional force Fiτ can be evaluated by the first equation of Eq. (3). On the contrary, when |viτ | ≤ vɛ , the frictional force Fiτ is multivalued and it depends on the resultant of forces acting at node i in x direction, Fix . And in most cases, Fix cannot be obtained obviously. By lumping the mass matrix M and presenting it in diagonal form, Eq. (5) is then inertially decoupled [60]. Therefore, the frictional forces acting at the surface nodes can be solved independently of each other via trial-and-error algorithm. On some occasions such lumping is physically obvious, in others this is not the case and a rational procedure is required. As Eq. (5) is inertially decoupled, the equations of motion of the surface node i can be written as

⎧   F ⎨m2i−1 u¨ 2i−1 + c2i−1, j u˙ j + k2i−1, j u j = Q˜2i−1 + Q2Ni−1 + Q2τi−1 + Q2Ri−1 j

⎩

m2i u¨ 2i +

 j

j

c2i, j u˙ j +

 j

k2i, j u j = Q˜2i + Q2Ni + Q2τi + Q2FRi

(9)

which are the rows 2i − 1 and 2i of Eq. (5). Thus,



Q2Ni−1



Q2Ni or



Q2Ni−1



Q2Ni and



Q2τi−1 Q2τi





0 = ( for the upper surface ) −FiN





0 = ( for the lower surface ) FiN



(10)

(11)



=

Fiτ 0

What’s more, viτ = u˙ 2i−1 .

(12)

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Fig. 5. Computer procedures for multibody systems including frictional translational joint with a flexible slider and clearance.

Therefore, the trial-and-error algorithm for the frictional force Fiτ acting at the surface node i is expressed as

if (FiN = 0 ) Fiτ = 0 else if (FiN > 0 & |viτ | > vε ) Fiτ = −μFiN sgn(viτ ) else if ( FiN > 0 & |viτ | ≤ vε ) Fiτ = c2i−1, j u˙ j + k2i−1, j u j − Q˜2i−1 − Q2FRi−1 j

(13)

j

if (Fiτ > μ FiN ) Fiτ := μ FiN if (Fiτ < −μ FiN ) Fiτ := −μ FiN end if After obtaining the vector Qτ , the equations of motion (8) can be integrated by a numerical method for ordinary differential equations (ODEs). In this paper, the fourth-order Runge–Kutta method is adopted to solve the ODEs, since it has high calculation accuracy and convergence. Other integration algorithms such as Newmark method and Generalized-α method could also be applied to solve the ODEs. The computer procedures for multibody systems including frictional translational joint with a flexible slider and clearance is illustrated in Fig. 5. 4. Numerical examples In order to test the applicability and correctness of the methodology proposed by this paper, two numerical examples are given, and the results are compared with previous studies. The first is a stick-slip oscillator [38], while the second is a planar slider-crank mechanism [32]. 4.1. Example I: A stick-slip oscillator The FEM discretization of a stick-slip oscillator is illustrated in Fig. 6. The slider is subjected to the applied force F = Fmax sin t and is attached to a spring of stiffness k. The uniform slider is divided into 6 four-node quadrilateral elements, and the parameters and initial conditions necessary to perform the dynamic analysis of the stick-slip oscillator are listed in Table 1. Fig. 7 (b) shows the velocities of four corner nodes (nodes 1, 4, 9, and 12) in x direction, and it indicates that the slider moves with stick-slip phenomena. The results agree with Ref. [38] closely. Fig. 8(b) depicts the sum of frictional forces acting on the slider. To test the convergence of the proposed method, the results are further compared with a 9 elements model, as shown by the dashed lines in Fig. 6. Generally speaking, the results consist with Ref. [38], however, it may be noticed that there are some unconformities. This is because Young’s modulus and clearance size applied in present study are not identical with theirs. Based on KED technique, a very large Young’s modulus was adopted in their study, but this is not the case for ours. Fig. 9(a) further compares the sum of frictional forces acting on the slider with different Young’s modulus.

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Fig. 6. FEM discretization of a stick-slip oscillator.

Table 1 Parameters and initial conditions of the stick-slip oscillator. Geometric characteristics

Width of slider Height of slider Clearance size Position of applied force

a = 0.6 m h = 0.5 m c = 2.5 mm s = 0.05 m

Material properties

Density Young’s modulus Poisson’s ratio Rayleigh parameters

ρ = 1/3 kg/m2

Contact parameters

Stiffness parameter Hysteresis factor Exponent Coefficient of kinetic friction Coefficient of static friction Velocity tolerance

K = 1 × 104 N/m1.5 χ = 1 × 105 N · s/m2.5 α = 1.5 μ = 0.15 μ = 0.20 vε = 1 × 10−5 m/s

External forces

Amplitude of applied force Angular frequency of applied force Spring stiffness Gravitational acceleration

Fmax = 1.5 N = π /10 k = 1.0 N/s g = 9.81 m/s2

Initial conditions

Vector of displacements Vector of velocities

u=0 u˙ = 0

E = 10 0 0 kPa υ = 0.3 ξ = 0.010, η = 0.544

Fig. 7. Velocities in x direction: (a) Reference [38]; (b) this paper.

If Young’s modulus increases, the results are more consistent with Ref. [38], which is the same with our analysis. Fig. 9(b) illustrates the sum of frictional forces with different clearance sizes. It can be concluded from Fig. 9 that both Young’s modulus and clearance size make a great influence on the sum of frictional forces, and thus affect the dynamic behavior of the slider. The frictional forces acting at the surface nodes are illustrated in Fig. 10. It shows that the frictional forces acting at nodes 2 and 3 are always zero, which means that these nodes are never in contact with the guide. Additionally, it can be seen that all nodes have the opportunity to depart from the guide. As the perfect convergence of the proposed method, a very small number of elements can be used, which is helpful for saving the calculation time. In this study, to simulate 50 s, the elapsed CPU time for the 6 elements model is only 235 s, which is much smaller 8 h in Zhang’s work [38].

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Fig. 8. Sum of frictional forces on the slider: (a) Ref. [38]; (b) this paper.

Fig. 9. Sum of frictional forces on the slider: (a) with different Young’s modulus; (b) with different clearance size.

Fig. 11 illustrates the normal and tangential contact forces acting at the four corner nodes and average tangential velocity of these nodes v¯ τ , and v¯ τ = 14 (v1τ + v4τ + v9τ + v12τ ). The solid lines in Fig. 11(a) are normal contact forces at the left corner nodes (nodes 1 and 9), which show that these values are subjected to the complementarity conditions [32]:

F1N ≥ 0, F9N ≥ 0, F1N · F9N = 0

(14)

Similarly, as shown in the dashed lines, the normal contact forces at the right corner nodes (nodes 4 and 12) are subjected to

F4N ≥ 0, F12N ≥ 0, F4N · F12N = 0

(15)

It should be highlighted that when two opposite corners of a rigid slider touch the guide with zero tangential velocities, a well-known statically indeterminate problem will arise. In these situations, the values of frictional forces at the corners cannot be calculated exactly for the rigid model. These problems can be solved by considering the deformation of the slider, as depicted in Figs. 10 and 11(b). 4.2. Example II: A planar slider-crank mechanism A planar slider-crank mechanism including frictional translational joint with a flexible slider and clearance is shown in Fig. 12. In this system, the crank and the connecting rod are uniform rigid rods, and all revolute joints are perfect. The driving crank moment is P = P0 sin t. The uniform slider is divided into 4 four-node quadrilateral elements, and the connecting rod is linked with the slider through the centric node (node 5). The parameters and initial conditions used in the simulation are listed in Table 2. The angular position and angular velocity of the crank are illustrated in Fig. 13. It can be seen that the crank shows a periodic stick-slip motion. The results generated in this work agree with Ref. [32] perfectly. The occurrence of stick-slip motion relates to the angular frequency of the applied driving crank moment [32]. Fig. 14 shows the sum of frictional forces acting on the slider. The results are consistent with Ref. [32]. However, due to the flexibility of the slider, the maximum static frictional force yielded in this work is smaller than that generated by a rigid model, as illustrated in the green circles. An 8

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Fig. 10. Frictional forces at the nodes: (a) nodes 1 and 2; (b) nodes 3 and 4; (c) nodes 9 and 10; (d) nodes 11 and 12.

Fig. 11. Normal and frictional forces at nodes 1, 4, 9, and 12 and average tangential velocities of the nodes: (a) normal contact forces; (b) frictional forces.

Fig. 12. Slider-crank mechanism with a frictional translational joint.

four-node quadrilateral elements model is also made to verify the convergence of the results, as illustrated in Figs. 12 and 14(b). The 4 elements discretization of the slider can generate acceptable results, as depicted in Figs 13 and 14. In present work, to simulate 200 s, the 4 elements model takes about 91 s computation time, while the 8 elements model spends about 214 s

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Table 2 Parameters and initial conditions of the slider-crank mechanism. Geometric characteristics

Length of crank OA Length of connecting rod AB Width of slider Height of slider Clearance size

L 1 = 1.8 m L 2 = 2.0 m a = 0.5 m h = 0.3 m c = 2.5 mm

Material properties

Mass of crank OA Mass of connecting rod AB Density of slider Young’s modulus of slider Poisson’s ratio of slider Rayleigh parameters of slider

m1 = 2.0 kg m2 = 1.0 kg ρ = 20/3 kg/m2 E = 10 0 0 kPa υ = 0.3 ξ = 0.010, η = 0.544

Contact parameters

Stiffness parameter Hysteresis factor Exponent Coefficient of kinetic friction Coefficient of static friction Velocity tolerance

K = 1 × 104 N/m1.5 χ = 1 × 105 N · s/m2.5 α = 1.5 μ = 0.03 μ = 0.04 vε = 1 × 10−5 m/s kRX = kRY = 1 × 105 N/m

Penalty factors for joint B External forces

Initial conditions

Amplitude of applied moment Angular frequency of applied moment Gravitational acceleration Displacements vector of the slider Velocities vector of the slider Generalized coordinates vector of other bodies Generalized velocities vector of other bodies

P0 = 6.0 Nm

= π /6

g = 9.81m/ s2

u=0 u˙ = 0 T q0 = [θ10 , θ20 ] T = [0 rad, 0 rad]

T q˙ 0 = θ˙ 10 , θ˙ 20 = [0 rad/s, 0 rad/s]

Fig. 13. Angular position and velocity of the crank: (a) and (c) Ref. [32]; (b) and (d) this paper.

T

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Fig. 14. Sum of frictional forces on the slider: (a) Ref. [32]; (b) this paper.

Fig. 15. Normal contact forces at the nodes: (a) nodes 1, 2, and 3; (b) nodes 7, 8, and 9.

Fig. 16. Frictional forces at the nodes: (a) nodes 1, 2, and 3; (b) nodes 7, 8, and 9.

Figs. 15 and 16 illustrate the normal and tangential contact forces acting at the surface nodes for the 4 elements model, respectively. It can be inferred that when the motion is stable, only lower surface of the slider is in contact with the guide, therefore, different clearance sizes will no longer play an important role in affecting the systems’s dynamic behavior. In Fig. 16, the frictional forces acting at the nodes of lower surface have the mutation phenomena during the transitions of the nodes from stick to slip. Fig. 17 depicts the differences between position vector components: xR − x5 and yR − y5 . As described in Section 2.4, xR and yR are the components of the position vector of the revolute joint B (one end of the connecting rod), while x5 and y5 are those of node 5. It can be seen that these differences can remain bounded within 8 × 10−4 m based on the penalty method for the perfect revolute joint.

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Fig. 17. Differences between position vector components: (a) xR − x5 ; (b) yR − y5 .

5. Conclusions This paper presented a novel methodology for modeling and simulating frictional translational clearance joint in planar multibody systems including a flexible slider part. Firstly, the slider was divided into a finite number of elements based on FEM, and the distributed body forces and boundary stresses (normal and tangential contact forces) acting on the slider were equivalent to the nodal forces. Secondly, the normal contact forces were obtained by using a nonlinear viscoelastic contact force model, while the frictional forces were described by the Coulomb dry friction model. Thirdly, by lumping the mass matrix and presenting it in diagonal form, the equations of motion were inertially decoupled, and the frictional forces were solved independently via trial-and-error algorithm. After that, the equations of motion were integrated numerically. Finally, two numerical examples were presented, and the results were compared with previous studies. The methodology proposed in this paper can be adopted to investigate the clearance in the translational joint as well as the deformation of the slider, and the statically indeterminate problem can be solved naturally. Compared with the KED based approach [38], the proposed method mainly has three improvements: (1) Instead of judging the contact states between the slider and its guide, the method just judge the contact states of each separate surface node, and the judgments are in a “for loop” for all surface nodes in the program, which is easier to operate; (2) This study found that both Young’s modulus and clearance size sometimes make a great influence on the sum of frictional forces acting on the slider, and thus affect the systems’ dynamic behavior; (3) As the perfect convergence of the proposed method, a very small number of elements can be adopted, which has significantly improved the calculation efficiency. Further studies could be done on rheonomic constraint between the slider and other bodies. The attempt of extending the proposed method to other flexible mechanical systems with impacts and friction will also be conducted in the future. Acknowledgments This work was supported by the National Natural Science Foundation of China [grant number 11772021]. References [1] P. Flores, J. Ambrósio, J.C. Claro, H.M. Lankarani, Translational joints with clearance in rigid multibody systems, J. Comput. Nonlinear Dyn. 3 (1) (2008) 011007. [2] P. Flores, Modeling and simulation of wear in revolute clearance joints in multibody systems, Mech. Mach. Theory 44 (6) (2009) 1211–1222. [3] P. Flores, C.S. Koshy, H.M. Lankarani, J. Ambrósio, J.C.P. Claro, Numerical and experimental investigation on multibody systems with revolute clearance joints, Nonlinear Dyn. 65 (4) (2011) 383–398. [4] C.S. Koshy, P. Flores, H.M. Lankarani, Study of the effect of contact force model on the dynamic response of mechanical systems with dry clearance joints: computational and experimental approaches, Nonlinear Dyn. 73 (1–2) (2013) 325–338. [5] F. Marques, F. Isaac, N. Dourado, A.P. Souto, P. Flores, H.M. Lankarani, A study on the dynamics of spatial mechanisms with frictional spherical clearance joints, J. Comput. Nonlinear Dyn. 12 (5) (2017) 051013. [6] S.c. Erkaya, Clearance-induced vibration responses of mechanical systems: computational and experimental investigations, J. Braz. Soc. Mech. Sci. Eng. 40 (2) (2018) 90:1–12, doi:10.1007/s40430- 018- 1015-x. [7] S.c. Erkaya, Experimental investigation of flexible connection and clearance joint effects on the vibration responses of mechanisms, Mech. Mach. Theory 121 (2018) 515–529. [8] S. Rahmanian, M.R. Ghazavi, Bifurcation in planar slider-crank mechanism with revolute clearance joint, Mech. Mach. Theory 91 (2015) 86–101. [9] S.B. Farahan, M.R. Ghazavi, S. Rahmanian, Nonlinear dynamic analysis of a four-bar mechanism having revolute joint with clearance, J. Theor. Appl. Vib.Acoust. 2 (1) (2016) 91–106. [10] S.B. Farahan, M.R. Ghazavi, S. Rahmanian, Bifurcation in a planar four-bar mechanism with revolute clearance joint, Nonlinear Dyn. 87 (2) (2017) 955–973. [11] Q. Tian, Y. Zhang, L. Chen, J. Yang, Simulation of planar flexible multibody systems with clearance and lubricated revolute joints, Nonlinear Dyn. 60 (4) (2010) 489–511.

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