Research On Torus Motions In Forced Vibration Systems At Resonance Case

Nonlinear forced vibration systems could produce complicated dynamic behaviors, whose behaviors will become more complicated when the ratio between the outside exciting frequency and the inherent frequency is rational number. As parameters vary in some neighborhood of the critical point, behaviors of the system under consideration are periodic or quasi-periodic. Systems forced by outside periodic force which have small amplitude will be considered in this paper, torus of which will be investigated when the ratio between the exciting frequency and the linearization frequency of the original unperturbed systems is 4:1. Computation of resonance region is studied tentatively at 1:5 weak resonance point. What will be studied mainly consist of:In the second chapter, the differential equations of motion of a one-degree-of-freedom systems with dry friction are established according to the Newton's second law. The velocity of the body is smaller than that of the friction surface. Equations of motion are expanded to their Taylor series when the velocity of the body is small. With a change of coordinates in some form and a change of scale of time, the differential equations of motion of the system take their complex form. Based on the theorem that the solutions of differential equation continuously depend on the initial-value and the parameters, we can expand the solutions of the differential equations in complex form to power series according to the initial-value and the parameters. Equating the corresponding coefficients of both sides of the equations, the power series solutions of differential equations can be obtained, through which we could establish Poincare map of the system. Analysising the Poincare map by the bifurcation theory of fixed point, we obtain the torus solution of the system. Theoretical conclusions are verified by numerical simulation.A standard form Van der pol equation is considered in the third chapter, whose simple form make it clear that generalized Hopf bifurcation could take place in it. With a appropriate change of coordinate-parameter, degeneracy of the equation is eliminated. We are able to study its torus solution according to the methods stated in chapter 2. The theoretical conclusions are also verified by numerical simulation.In chapter 4, computation of resonance region is studied tentatively at 1:5 weak resonance point, which is corresponding to periodic orbit of period 5. First of all, procedure of computing the fourth terms of Poincare map is established. Then, coefficients of Poincare normal form in 1:5 resonance case are deduced. Finally, polar coordinate form of plane map in 1:5 resonance case is provided. The method determining Arnold tongue is given.