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Bifurcation in a planar four-bar mechanism with revolute clearance joint

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Abstract

Generally, multi-body mechanical systems in which the impact phenomena occur exhibit chaotic and quasi-periodic behavior. Mechanisms with clearance joint are categorized in such systems due to contact between the connecting bodies. In this paper, we would investigate the nonlinear dynamic behavior of a four-bar mechanism with joint and clearance at the connection between the coupler and rocker. Motion equations are derived based on the Lagrangian approach. The nonlinear continuous contact force model proposed by Lankarani and Nikravesh is utilized to evaluate the normal contact force developed in the journal–bearing system. Furthermore, the friction effect on the clearance joint is considered using a modified Coulomb friction law. The dynamical behaviors are numerically identified in discrete state space, based on the Poincaré portraits. Numerical simulations display both periodic and chaotic motions in the system behavior. Therefore, bifurcation analysis has been performed with a change in the size of clearance corresponding to the various values of crank rotational velocities. Then, we compare some clearance ranges, in which the system response becomes stable for these cases. Fast Fourier transformation is also applied to analyze the frequency spectrum of the system response.

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Abbreviations

\(c_\mathrm{{r}} \) :

Restitution coefficient

\(c_\mathrm{{f}} \) :

Friction coefficient

\(c_\mathrm{{d}} \) :

Dynamic correction coefficient

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

Normal contact force component

\(F_\mathrm{{t}}\,{(}\mathrm {N}{)}\) :

Tangential contact force component

\(K\,{(}{\mathrm{{N/m}}}{)}\) :

Stiffness coefficient

\(M_2\,{(}\mathrm{{N\,m}}{)}\) :

External moment exerted on the crank

\({{{\varvec{Q}}_\mathbf{c }}}\,{(}\mathrm {N}{)}\) :

Resultant contact force

\(r\,{(}\mathrm {m}{)}\) :

Magnitude of clearance vector

\({{{\varvec{r}}_{{\varvec{i}}}^{{\varvec{O}}}}}\,{(}\mathrm {m}{)}\) :

Position vector of bearing center

\({{\varvec{r}}}_{{\varvec{i}}} ^{{\varvec{Q}}}\,{(}\mathrm {m}{)}\) :

Position vector of contact point on the bearing

\({{\varvec{r}}}_{{\varvec{j}}} ^{{\varvec{O}}}\,{(}\mathrm {m}{)}\) :

Position vector of journal center

\({{\varvec{r}}}_{{\varvec{j}}} ^{{\varvec{Q}}}\,{(}\mathrm {m}{)}\) :

Position vector of contact point on the journal

\(\dot{r}\,{(}\mathrm {m/s}{)}\) :

Normal component of relative velocity

\(R_i\,{(}\mathrm {m}{)}\) :

Bearing radius

\(R_j\,{(}\mathrm {m}{)}\) :

Journal radius

\(v_\mathrm{{t}}\,{(}\mathrm {{m/s}} {)}\) :

Relative tangential velocity

\(X_{\mathrm{{rel}}}\,{(}\mathrm {m}{)}\) :

Journal center displacement relative to bearing center in X-direction

\(\dot{X}_{\mathrm{{rel}}}\,{(}\mathrm{{m/s}}{)}\) :

Journal center velocity relative to bearing center in X-direction

\(Y_{\mathrm{{rel}}}\,{(}\mathrm {m}{)}\) :

Journal center displacement relative to bearing center in Y-direction

\(\dot{Y}_{\mathrm{{rel}}}\,{(}\mathrm{{m/s}}{)}\) :

Journal center velocity relative to bearing center in Y-direction

\(\alpha \,{(}{\mathrm{{rad}}}{)}\) :

Orientation of clearance vector

\(\gamma \,{(}{\mathrm{{rad}}}{)}\) :

Orientation of resultant contact force

\(\omega _2\,{(}\mathrm{{rad/s}}{)}\) :

Crank rotational velocity

\(\delta \,{(}\mathrm {m}{)}\) :

Relative penetration depth

\(\dot{\delta }^{(-)}\,{(}\mathrm{{m/s}}{)}\) :

Initial impact velocity

\(\chi \,{(}\mathrm{{N\,s/m^{2}}}{)}\) :

Hysteresis damping coefficient

\(\phi \,{(}\mathrm{{rad}}{)}\) :

Angle between the normal and tangential contact force component

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Authors and Affiliations

  1. Mechanical Engineering Department, Engineering Faculty, Tarbiat Modares University, Tehran, Iran

    Sajjad Boorghan Farahan, Mohammad Reza Ghazavi & Sasan Rahmanian

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  1. Sajjad Boorghan Farahan
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  2. Mohammad Reza Ghazavi
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  3. Sasan Rahmanian
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Correspondence to Mohammad Reza Ghazavi.

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Farahan, S.B., Ghazavi, M.R. & Rahmanian, S. Bifurcation in a planar four-bar mechanism with revolute clearance joint. Nonlinear Dyn 87, 955–973 (2017). https://doi.org/10.1007/s11071-016-3091-8

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  • DOI: https://doi.org/10.1007/s11071-016-3091-8

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