Codimension two bifurcation of periodic vibro-impact and chaos of a dual component system
Introduction
Impact oscillators arise whenever the components of a vibrating system collide with rigid obstacles or with each other. Such system with impacts exist in a wide variety of engineering applications, particularly in mechanisms and machines with clearances or gaps. Examples of these types of machines and equipment include vibration hammer, impact dampers, machinery for compacting, milling and forming, shakers, offshore structures, gears, piping systems, wheel-rail interaction of high speed railway coaches, etc. In the past several years, dynamics of mechanical systems with impacts have been the subject of several investigations, and many new problems of theory have been advanced in researches into the vibro-impact problems. Some researchers, including global bifurcation [1], [2] and singularities [3], [4], [5] etc., were unfolded for the vibro-impact systems. Recently, a few researchers began to focus to attention on the phenomena of Hopf bifurcation of the vibro-impact systems [6], [7], [8], [9]. The purpose of the present Letter is to focus attention on codimension two bifurcation of period motion with one impact. The vibro-impact systems, under the condition of codimension two bifurcation, can exhibit richer and more complicated quasi-period impact motions. It is found that there exist not only supercritical and subcritical Hopf bifurcations of period 1 single-impact motions near by the point of codimension two bifurcation, but also Hopf bifurcation of period 2 two-impact motion occurs. Transition of different forms of 1/1 impact points, near by the bifurcation point, is demonstrated, and different routes from period 1 single-impact motions to chaos are also observed by numerical simulations.
Section snippets
The mechanical model and Poincaré map
The mechanical model for a two-degree-of-freedom vibro-impact system with masses M1 and M2 is shown in Fig. 1. The vibro-impact system is similar to the model of the impact-forming machine with double masses. The masses are connected to linear springs with stiffness K1,K2 and K3, and linear viscous dashpots with damping constants C1,C2 and C3. The excitations on both masses are harmonic with amplitudes P1 and P2. The excitation frequency and the phase τ are the same for both masses. The mass M
The center manifold and normal form
Taking the forcing frequency ω and the gap δ as the control parameters, we continue to consider the Poincaré map . is a fixed point for the map for ν in some neighborhood of a critical value ν=νc at which Df(ν,0) satisfies the following assumption:
H.1 Df(ν,0) has double eigenvalues λ1(νc) and λ2(νc) on the unit circle, and λ1(νc)=λ2(νc)=−1.
H.2 The other eigenvalues λ3(νc) and λ4(νc) stay inside the unit circle, i.e., |λi(νc)|<1, i=3,4.
For all ν in some neighborhood of νc, the map (5),
Local bifurcation of the simplified map
In view of the formula (11), the period two points Z satisfies the equation Ignoring the terms of high order, the solution of Eq. (12) become If a2<0, there exist the period two points Z=(Z1,Z2)T, and they are symmetrical at the origin. The linearized maps of Φ(z;ε) at the fixed point and Φ2(z;ε) at the period two points, respectively, are given by In view
Numerical analysis
The vibro-impact system, with parameters: μm=3.976211, k1=4.996932, k3=3.0, δ=0.04935, ζ=0.1, and R=0.8, has been chosen to be analyzed. The forcing frequency and gap are taken as the control parameters, i.e., ν=(ω,δ)T. The eigenvalues of Df(ν,0) are computed with ω∈[4.378,4.45] and δ∈[0.047,0.052]. All eigenvalues of Df(ν,0) stay inside the unit circle for ν=(4.378,0.052)T. By gradually increasing ω and decreasing δ from the point ν=(4.378,0.052)T to change the control parameter, we found
Conclusions
The vibro-impact system, under the condition of codimension two bifurcation, can exhibit more complicated quasi-periodic impact motions. Near by the point of codimension two bifurcation there exist not only supercritical and subcritical Hopf bifurcations of q=1/1 impact motion, but also Hopf bifurcation of q=2/2 impact motion occurs.
By choosing different value of gap δ (δ>δc) and varying the forcing frequency ω near by the bifurcation point νc, we can found that the vibro-impact system can
Acknowledgements
The authors gratefully acknowledge the support of the National Science Foundation of China (10172042, 10072051).
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