Global bifurcation analysis and chaos of an arch structure with parametric and forced excitation

doi:10.1016/j.mechrescom.2009.08.007

Mechanics Research Communications

Volume 37, Issue 1, January 2010, Pages 67-71

https://doi.org/10.1016/j.mechrescom.2009.08.007 Get rights and content

Abstract

The global bifurcations and chaotic motions are investigated analytically for an arch structure with parametric and forced excitation. The critical curves separating the chaotic and non-chaotic regions are drawn, which show that the system in the case of 1:1 resonance is more easily chaotically excited than the case of 1:2 resonance. There exist “uncontrollable regions” or “chaotic bands” for the system as the natural frequency varies. There also exists a “controllable frequency” for the system with linear and cubic parametric excitation. The system can be chaotically excited through infinite subharmonic bifurcations of odd/even orders. Numerical results agree with the analytical ones.

Introduction

It is well known that arch structures are widely used in support structures of aircraft, and of mechanical arms of robots. Because of the extreme light weight and flexibility and heavy load imposed at the same time in such systems, abundant dynamical behaviors are exhibited for this class of systems and extensive attention has been paid to these problems by many researchers (Ariaratnam, 1985, Humphery, 1966, Hsu, 1969). By means of a numerical analytical method, a nonlinear vibration equation of the arch structures with parametric and forced excitation was established and some dynamical phenomena were revealed by Liu et al. (2000). Using singularity theory, Chen et al., 2007, Zhou and Chen, 2008 studied the local bifurcations in the case of 1/2 subharmonic resonance and primary resonance of this structure, respectively, and abundant bifurcation phenomena were presented there.

In this paper, the chaotic motions of the arch structures are studied analytically with Melnikov’s method. The critical curves separating the chaotic and non-chaotic regions are drawn. Associated poincaré sections are numerically computed, which verify the analytical results. The subharmonic bifurcations are also investigated.

Section snippets

Formulation of the problem

Consider an arch structure (see Fig. 1). Combining the finite element method with an approximate analytical method, the nonlinear vibration equation of arch structures with parametric and forced excitation was established as follows (Liu et al., 2000, Chen et al., 2007). $\ddot{u} + ω^{2} u + D_{10} \dot{u} + D_{12} u^{2} + D_{13} u^{3} + (D_{14} u + D_{15} u^{2} + D_{16} u^{3}) \sin γ t - D_{17} \sin γ t = 0$ where u is the amplitude caused by the live load, and the coefficients $D_{1 j} (j = 0, 1, \dots, 7)$ are determined by the parameters of the arch structure, initial state data,

Melnikov analysis of chaotic motions

The system with heteroclinic orbits may have chaotic behaviors if both branches of the heteroclinic cycle have transverse intersection points (Wiggins, 1990), or only one branch of the cycle has its tangent intersection perturbed into transverse intersections (Blackmore, 2005). Using the classical Melnikov’s method (Guckenheimer and Holmes, 1983, Wiggins, 1990), the chaotic motions for system (5), which arise from the transverse intersections of both branches of the heteroclinic cycle, are

Subharmonic bifurcations

Subharmonic bifurcation for system (5) is considered in this section. It can be computed that the subharmonic Melnikov function for the periodic orbits (8) satisfying the resonance condition $mT = {nT}_{k}$ is $M^{m / n} (τ_{0}) = {\tilde{D}}_{10} I_{0} (m, n) - ({\tilde{D}}_{14} I_{1} (m, n) + {\tilde{D}}_{16} I_{3} (m, n)) \cos Ω τ_{0} + (D_{17} I_{4} (m, n) - {\tilde{D}}_{15} I_{2} (m, n)) \sin Ω τ_{0}$ where $I_{j} (m, n), (j = 0, 1, 2, 3, 4)$ see the Appendix (A.2).

Numerical simulations

In this section using the fourth-order Runge–Kutta method, the Poincaré sections of system (5) in two cases, that is, the “uncontrollable regions” and “controllable frequency”, are obtained.

Choosing the system parameters $ε = 0.01$ , $δ = 5$ , ${\tilde{D}}_{10} = 0.1$ , ${\tilde{D}}_{14} = {\tilde{D}}_{15} = D_{17} = 1$ , ${\tilde{D}}_{16} = 2$ , $ω = 1$ , $γ = ω + ε δ$ or $γ = 2 ω + ε δ$ , this is the case of “uncontrollable regions”. Taking the initial values $x_{1} (0) = 0.8$ , $x_{2} (0) = 0$ , we get the Poincaré sections of system (5) as in Fig. 6. From Fig. 6 we can see that the systems of both 1:1 and

Conclusions

The global bifurcations and chaotic motions of an arch structure with parametric and forced excitation are studied by means of Melnikov’s method. The critical curves separating the chaotic and non-chaotic regions are plotted. There exist “uncontrollable regions” in which the system is always chaotically excited when the natural frequency is small. There also exists a “controllable frequency” for the system with linear and cubic parametric excitation. The system can be chaotically excited

Acknowledgements

The project supported by National Natural Science Foundation of China (10632040, 10972099), China Postdoctoral Science Foundation (No. 20090450765), and the Natural Science Foundation of Tianjin, China (No. 09JCZDJC26800).

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Stability of shallow arches against snap-through under time wise step loads
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View full text

Global bifurcation analysis and chaos of an arch structure with parametric and forced excitation

Abstract

Introduction

Section snippets

Formulation of the problem

Melnikov analysis of chaotic motions

Subharmonic bifurcations

Numerical simulations

Conclusions

Acknowledgements

Mechanics Research Communication

Dynamic buckling of shallow curved structures under stochastic loads

Nonlinear Dynamics

New models for chaotic dynamics

Regular and Chaotic Dynamics

Non-linear Oscillations, Dynamical Systems and Bifurcation of Vector Fields

Stability of shallow arches against snap-through under time wise step loads

AIAA Journal