The Non-linear Vibration Analysis of the Multi-layered Thin-walled Cylindrical Shell
|Course||General and Fundamental Mechanics|
|Keywords||axial movement multi-layered numerical solution Lindstedt-Poincare multi-scale stability|
In this paper, the studied model is the multi-layered thin-walled cylindrical shell with the boundary of clamped-free. We analyse the nature of the nonlinear vibration of the system under the conditions of different parameters with numerical method and two approximate analytical methods.First of all, in the foundation of layered shell theory, we establish the wave equation of the multi-layered thin-walled cylindrical shell with axial movement. Applying the Galerkin we discretize the wave equation, and then we will obtain four mutual coupling mode equations. We achieve the numerical solutions of the system with the Runge-Kutta method and obtain the amplitude-frequency characteristic curves under the conditions of different parameters. Through the analysis of the curves, we can see that this system has obvious soft characteristic; the two modes are coupling, that is the phenomenon of internal resonance, which is resulted from the two closed natural frequencies and the speed of axial movement; we also can see the influences of the amplitude of excitation, damping and speed on the amplitude-frequency characteristic curves.We can gain the approximate solutions applying the Lindstedt-Poincare method when the primary resonance and 1:1 internal resonance occur in the system, with which we can plot the amplitude-frequency characteristic curves. Then we analyse the parameter vibration of the system and study the influences of excitation amplitude, damping and speed on the vibration characteristic. The study indicates that when the exciting force is greater, the damping is smaller and the speed is slower, the response amplitude is greater, the range of interval resonance is more extensive, the phenomenon of the coupling between the two modes is more obvious and the soft feature of the system is more marked, which is similar with the conclusion of the numerical solutions on the amplitude-frequency characteristic curves. And then we determine the stability of the zero solutions applying the theorem of the first rank differential coefficient.And then we also get the approximate solutions applying multi-scale method when the primary resonance and 1:1 internal resonance occur in the system. Studying the influences of the three parameters on the amplitude-frequency characteristic curves, we can see that the conclusion is similar with the conclusion of Lindstedt-Poincare. With once approximate theory we determine the stability of the zero solutions.The last, comparing the two approximate analytical solutions which are got applying Lindstedt-Poincare method and multi-scale method, we can see that the two solutions are similar which are different on the amplitude and resonance frequency.