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Correlation dimension and bifurcation analysis for the planar slider-crank mechanism with multiple clearance joints

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Abstract

Joint clearance serves as a crucial element of nonlinearity in multibody systems. The quantization of the system chaos is conducive to not only the understanding of the nonlinear nature but the rationalization of system controlled parameters. In the present work, the system dynamics for the planar slider-crank mechanism with multiple clearance joints is depicted by the correlation dimension and bifurcation actions. Considering the obliqueness of the slider, the general configuration of the planar joint is proposed. Generalized coordinates and the Lagrangian approach are adopted to derive the dynamic motion equations. The effects of clearance size and driving speed on the bifurcations of the dynamic response are investigated. Furthermore, the fractal dimension of the strange attractor is identified by the correlation dimension from time series. Based on the Cao method, the Mutual Information (MI) function, and the Grassberger-Procaccia (G-P) algorithm, the controlled factors in the evaluation of correlation dimension are cautiously determined. Ultimately, the compound effect of translational and revolute clearance joints on the mechanism dynamics is featured. The numerical results testify that the correlation dimension of the slider displacement approximately saturates beyond a specific translational clearance value. Moreover, with the parameters used in this work, the complexity of system response seems to be more sensitive to the variation of translational clearance size than with the revolute joint.

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Abbreviations

\(\mathbf{r}_{HK}\) :

Global position vector of point \(H\) to point \(K\)

\(\mathbf{s}_{k}^{OH}\) :

Position vector of point \(H\) in the body fixed coordinate of body \(k\) respect to the origin \(O_{k}\)

\(\mathbf{R}_{k}\) :

Rotation matrix of body \(k\)

\(\theta _{k}\) :

Revolute angle of the mass center of body \(k\) [rad]

\(\boldsymbol{\delta }\), \(\delta \) :

Vector and value of penetration [m]

\(\mathbf{v}_{T}\) :

Tangential velocity vector [m/s]

\(\mathbf{F}_{N}\), \(\mathbf{F}_{f}\) :

Normal and friction force vectors at the contact points [N]

\(\mathbf{Q}_{c}\) :

Vector of the resultant contact force [N]

\(K\) :

Generalized contact stiffness [N/m]

\(n\) :

Exponent of the force deformation characteristics

\(E_{k}\) :

Elastic module of body \(k\) [Pa]

\(\upsilon _{k}\) :

Poisson’s ratio of body \(k\)

\(\dot{\delta }^{\left ( - \right )}\) :

Initial impact velocity [m/s]

\(c_{e}\) :

Restitution coefficient

\(c_{t}\) :

Normal clearance in the translational joint [m]

\(c_{r}\) :

Normal clearance in the revolute joint [m]

\(S\) :

Area of the contact region [m2]

\(L_{s}\), \(L_{c}\) :

Length and width of the rectangular surface for contact region [m]

\(\mu \) :

Friction coefficient

\(z\) :

Average bristle deflection [m/N]

\(\sigma _{0}\) :

Bristle stiffness [N/m]

\(\sigma _{1}\) :

Microscopic damping [Ns/m]

\(\sigma _{2}\) :

Viscous friction coefficient

\(\mu _{k}\) :

Coefficient of kinetic friction

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

Coefficient of static friction

\(\gamma \) :

Shape factor of Stribeck curve

\(\omega \) :

Angular velocity of the crank [rad/s]

\(\varphi \) :

Orientation angle of resultant contact forces [rad]

\(\mathbf{M}_{c}\) :

Moment vector originated in contact forces [Nm]

\(L\) :

Lagrangian function [Nm]

\(T\), \(U\) :

System kinetic and potential energies [Nm]

\(q_{k}\) :

Generalized coordinate

\(Q_{nc,k}\) :

Generalized force

\(\dot{u}\) :

Time derivative of variable \(u\)

\(\tau \) :

Time delay

\(p(x, y)\) :

Joint probability density for time series \(x\) and \(y\)

\(C\)(\(\varepsilon \)):

Correlation function respect to the radial \(\varepsilon \)

\(m\) :

Embedding dimension

\(D\) :

Correlation dimension

\(E_{1}(m), E_{2}(m)\) :

Indexes determined by the Cao method

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Acknowledgement

The work is supported by Natural Science Foundation of the National Natural Science Foundation of China (No. 52005230) and National Science and Technology Major Project of China (No. 2019ZX04029-001).

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Authors and Affiliations

  1. School of Mechanical Engineering, Nanjing University of Science and Technology, Nanjing, 210094, China

    Xuze Wu, Yu Sun & Yu Wang

  2. School of Mechanical Engineering, Jiangsu University of Technology, Changzhou, 213001, China

    Yu Chen

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Wu, X., Sun, Y., Wang, Y. et al. Correlation dimension and bifurcation analysis for the planar slider-crank mechanism with multiple clearance joints. Multibody Syst Dyn 52, 95–116 (2021). https://doi.org/10.1007/s11044-020-09769-3

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