Abstract
An inertial shaker as a vibratory system with impact is considered. By means of differential equations, periodicity and matching conditions, the theoretical solution of periodic n−1 impacting motion can be obtained and the Poincaré map is established. Dynamics of the system are studied with special attention to interaction of Hopf and period doubling bifurcations corresponding to a codimension-2 one when a pair of complex conjugate eigenvalues crosses the unit circle and the other eigenvalue crosses −1 simultaneously for the Jacobi matrix. The four-dimensional map can be reduced to a three-dimensional normal form by the center manifold theorem and the theory of normal forms. The two-parameter unfoldings of local dynamical behavior are put forward and the singularity is investigated. It is proved that there exist curve doubling bifurcation (a torus doubling bifurcation), Hopf bifurcation of 2–2 fixed points as well as period doubling bifurcation and Hopf bifurcation of 1–1 fixed points near the critical point. Numerical results indicate that the vibro-impact system presents complicated and interesting curve doubling bifurcation and Hopf bifurcation as the two controlling parameters vary.
Highlights
Vibro-impact systems are often encountered in practice, for instance, in the models of hammerlike devices, rotor-casing dynamical systems, collisions of solids, ships moored at dockside, etc
We study one kind of codimension-2 bifurcation of system (1), which is characterized by the so-called Hopf–Flip degeneracy, and satisfy: (H.1) Að0Þ 1⁄4 Df0ð0Þ has a pair of complex conjugate eigenvalues l0; l0 on the unit circle, one real eigenvalues l1 1⁄4 À1; and another real eigenvalue jl2jo1: (H.2) Non-resonant condition ln0a1; n 1⁄4 1; 2; 3; 4; 5; 6 and ln0a À 1; n 1⁄4 4; 5: Let pðlÞ 1⁄4 l4 þ a3l3 þ a2l2 þ a1l þ a0 1⁄4 0
We have studied the interaction dynamics of Hopf and period doubling bifurcations of the two-degree-of-freedom vibro-impact system shown in Fig. 1 by theoretical analysis and numerical simulations
Summary
Vibro-impact systems are often encountered in practice, for instance, in the models of hammerlike devices, rotor-casing dynamical systems, collisions of solids, ships moored at dockside, etc. Map and normal form approach, we investigate the interaction of Hopf and period doubling bifurcation (the so-called Hopf–Flip bifurcation [11]) of inertial impacting shaker, which is a 2-d.o.f. vibro-impact system. To make it possible to analyze this complex problem and make the calculations easier and correct, we take advantage of a symbolic software, like MAPLE. Mx’À þ my’À 1⁄4 Mx’þ þ my’þ; ð3Þ x’þ À y’þ 1⁄4 ÀRðx’À À y’ÀÞ; ð4Þ where x’À and y’À represent, respectively, the approach velocities of M and m at the instant of impact. x’þ and y’þ represent, respectively, the departure velocities of M and m at the instant, which are given by x’ þ
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