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Modeling and simulation of planar multibody systems considering multiple revolute clearance joints

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Abstract

This paper presents a general procedure for dynamic modeling and simulation of planar multibody systems considering multiple revolute clearance joints. The normal contact force is evaluated by a hybrid continuous contact force model, established on the base of the Lankarani–Nikravesh (L–N) model and the elastic foundation model. The LuGre friction law is employed to describe the tangential effect. The effectiveness of the presented methodology is demonstrated through the comparisons with the MSC ADAMS software simulation results of a slider–crank mechanism with clearance joints. Then, the system behavior affected by dynamic interaction of three revolute clearance joints is analyzed and some kinds of 27 combination modes are presented. Additionally, a comprehensive analysis of the system responses in a wide range of dynamic simulation parameters is conducted to find out some inner rules existed in the mechanical system with revolute clearance joints. Results show that there exists a strong dynamic interaction between different clearance joints, indicating that all joints should be modeled as imperfect to achieve a further understanding of multibody system behavior. Also, the clearance joint nearer to the input link is found to suffer more serious contact effects, require more input torque and cost longer computational time. Furthermore, the system dynamics relies on many factors, even a small change of which may lead to different system responses, changing from periodic to chaotic and the other way around. In addition, several characteristic values have been captured from the simulation results, which need to be paid more attention in use.

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Abbreviations

\(A_i \) :

Area of ith element

\(\mathbf{A}_{I} ,\mathbf{A}_{\textit{II}} \) :

Transformation matrices

c :

Radial clearance

\(c_\mathrm{e} \) :

Restitution coefficient

\(D_i \) :

Damping coefficient for the ith element

\(D_i^{\bmod } \) :

Modified damping coefficient of the ith element

\(\mathbf{e}\) :

Eccentricity vector

E :

Elastic modulus

\(\mathbf{F}_\mathrm{T} \) :

Tangential friction force

\(\mathbf{F}_\mathrm{N} \) :

Overall contact force in the contact area

\(\mathbf{F}_{\mathrm{N}i} \) :

The ith element total contact force

\(F_{\mathrm{N}ix} ,F_{\mathrm{N}iy} \) :

Components along axes x and y of F \(_{\mathrm{N}i}\)

\(F_{\mathrm{N}x} , F_{\mathrm{N}y} \) :

Components along axes x and y of F \(_\mathrm{N}\)

h :

Length of the pin

\(H_j \) :

Weight coefficient

\(K_i \) :

Contact stiffness coefficient for the ith element

\(L_i \) :

Thickness of the elastic layer

\(\mathbf{M}\) :

Global system mass matrix

\(m_1 \) :

Number of the bearing points

\(m_2 \) :

Number of the elements

\(\mathbf{n}\) :

Unit direction of eccentricity vector

n :

Contact exponent

\(p_i \) :

Contact pressure of spring i

\(\mathbf{q}\) :

Vector of generalized coordinates

\({\ddot{\mathbf{q}}}\) :

Vector of acceleration

\(\mathbf{Q}^{A}\) :

Vector of applied loads

\(R_1,R_2 \) :

The radii of the bushing and pin

\(\mathbf{r}_{I} \), \(\mathbf{r}_{\textit{II}} \), \(\mathbf{r}_{I}^{P_1 } \), \(\mathbf{r}_{I}^B \), \(\mathbf{r}_{\textit{II}}^{P_2} \),\(\mathbf{r}_{\textit{II}}^C \) :

Global vectors

\(\mathbf{s}_{I} \), \(\mathbf{s}_{\textit{II}} \) :

Local vectors

t :

Time

z :

Internal state

\({\varvec{\uptau }} \) :

Tangential velocity direction

\(\mu _\mathrm{k} \) :

Kinetic friction coefficient

\(\mu _\mathrm{s} \) :

Static friction coefficient

\(v_\mathrm{N} \),\(v_\mathrm{T} \) :

Normal and tangential velocities

\(v_\mathrm{S} \) :

Characteristic Stribeck velocity

\(\upsilon \) :

Poisson’s ratio

\(\sigma _0 \) :

Bristle stiffness

\(\sigma _1 \) :

Microscopic damping

\(\sigma _2 \) :

The viscous friction coefficient

\(\delta _{si} \) :

Deformation of the ith spring

\(\delta _i \) :

Penetration of ith element

\(\dot{\delta }_i \) :

Relative penetration velocity of ith element

\(\dot{\delta }_i ^{(-)}\) :

Initial penetration velocity of ith element

\(\delta _{ij} \) :

Penetration of thejth bearing point in the ith element

\({\varvec{\Phi }} \) :

Constraint vector

\({\varvec{\Phi }} _\mathbf{q} \) :

Jacobian matrix

\({\varvec{\lambda }} \) :

Vector of Lagrange multipliers

\({\varvec{\uprho }} \) :

The vector of quadratic velocity terms

\(\theta _i\) :

The ith Element angle from pin center

\(\varphi _{{\textit{I}}} ,\varphi _{{II}} \) :

Local orienting angles relative to the global x-axis

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Acknowledgements

We would like to express our sincere thanks to the editor and anonymous reviewers for their useful comments and suggestions which help substantially improve the manuscript. The warmhearted and valuable advice of Professor Paulo Flores from the University of Minho regarding the structure and overall description of the paper is highly appreciated. The research work presented in this paper was supported by National Natural Science Foundation of China [Grant numbers: 11472137]. The innovation of graduate student training project in Jiangsu province [Grant numbers: KYLX16_0488] is also acknowledged.

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  1. School of Mechanical Engineering, Nanjing University of Science and Technology, No. 200 Xiaolingwei Road, Nanjing, 210094, Jiangsu, China

    Jia Ma & Linfang Qian

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  1. Jia Ma
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Ma, J., Qian, L. Modeling and simulation of planar multibody systems considering multiple revolute clearance joints. Nonlinear Dyn 90, 1907–1940 (2017). https://doi.org/10.1007/s11071-017-3771-z

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