Periodic Motions And Bifurcations Of Multi-degree-of-freedom Vibro-impact Systems

By view of engineering, using the modern theory and the method of nonlinear dynamics, this dissertation studies periodic motions and bifurcations of some multi-degree-of-freedom vibro-impact systems in the two aspects of theory and numerical simulations. The main research work involves:First, some multi-degree-of-freedom vibro-impact systems, such as the three-degree-of-freedom vibro-impact system with an unconstrained rigid body, the three-degree-of-freedom vibro-impact system with a rigid constrain and the three-degree-of-freedom vibro-impact system having symmetrically placed rigid stops with a rigid constrain, are considered. The systems are uncoupled by using modal matrix approach. Single-impact periodic motion and the Poincarémapping of the systems are derived analytically. The existence condition and stability criterion of period n single-impact motions are derived. So stability and local bifurcations of one-impact periodic motion are analyzed theoretically by using Jacobian matrix of the Poincarémapping.Second, Neimark-Sacker bifurcation of periodic motion of some multi-degree-of-freedom vibro-impact systems in non-resonance and weak resonance cases are analyzed. Local dynamical behaviors of the systems are studied by center manifold theory and normal form method of high dimensional mapping. Routes from Neimark-Sacker bifurcation of periodic motions via phase locking or torus doubling to chaos are discussed by numerical simulations.Third, subharmonic bifurcation or Neimark-Sacker bifurcation of periodic motions of some multi-degree-of-freedom vibro-impact systems in 1:4, 1:3, 1:2 resonance cases are analyzed. Local dynamical behaviors of the systems are studied by center manifold theory and normal form method of high dimensional mapping. Routes from subharmonic bifurcation or Neimark-Sacker bifurcation of periodic motions of the systems in 1:4, 1:3, 1:2 resonance cases to chaos are discussed by numerical simulations. Here the processes from subharmonic bifurcations of periodic motions of some systems nearby the bifurcation value of 1:4 resonance cases to q=4/4 periodic motions, which occur via T_?, T_on, or T_in tangent bifurcation, are analyzed in detail. And the following routes from q= 4/4 periodic motions to chaos are discussed:â… . q=4/4 fixed points→an attracting invariant circle(quasi-periodic motion)→phase locking→chaos.â…¡. q=4/4 fixed points→four attracting invariant circles corresponding to q=4/4 fixed points→phase locking→chaos.â…¢. q=4/4 fixed points→q=8/8 fixed points→eight attracting invariant circles corresponding to q=8/8 fixed points→phase locking→chaos.â…£. q=4/4 fixed points→q=8/8 fixed points→q=16/16 fixed points→……→chaos(route from q=4/4 periodic motion of the system via Feigenhaum series to chaos).Fourth, routes from period doubling bifurcation of periodic motions of the systems via Feigenhaum series, Neimark-Sacker bifurcation or "grazing" to chaos are discussed by numerical simulations.Fifth, routes from pitchfork bifurcation of periodic motions of the systems via Neimark-Sacker bifurcation or period doubling bifurcation to chaos are discussed by numerical simulations.Sixth, using the center manifold-normal form method of maps and simulations, codimension two bifurcation of multi-degree-of-freedom vibro-impact systems under the three conditions which a real eigenvalueλ₁=+1 and a pair of complex conjugate eigenvalues |λ_{2, 3}|=1, a real eigenvalueλ₁=-1 and a pair of complex conjugate eigenvalues |λ_{2, 3}|=1 or two pairs of complex conjugate eigenvalues |λ_{2, 3}|=1 escape the unit circle simultaneously, are deeply analyzed. Nearby the bifurcation value of codimension two bifurcation, local dynamical behaviors of the systems are studied.Seventh, in order to design impact tools which need larger impact velocity and larger regions of one-impact periodic motions in practicing engineer, global bifurcations are investigated and some designing methods which can be used to optimize the system parameters are obtained.