Nonlinear Oscilations And Global Dynamics Of Flexible Cantilever

In order to investigate the nonlinear dynamics of the flexible robotic arm, we can simplify the robotic arm into the model of the flexible cantilever. The dynamic properties of the nonlinear planar cantilever and the nonlinear nonplanar cantilever are investigated, The main contents are as follows.The equation of motion for the planar cantilever is obtained by using the theory of elastic mechanics. Then, the governing partial differential equation is analyzed by two processes. In the first case, we use the Galerkin procedure to get a two degree of freedom nonlinear system under combined forcing and parametric excitations. The method of multiple scales is used to get the averaged equations of the system obtained above. For the second case, the method of multiple scales is used to obtain the partial differential averaged equations. Based on the averaged equations obtained above, the Galerkin procedure is used to obtain four first-order nonlinear ordinary differential equations, governing the modulation of the amplitudes and phases of the two modes. The results obtained above are compared in the chapter 2 to show to have a good agreement qualitatively. For the nonlinear planar cantilever, the two resonant cases are considered. One of the two cases is primary resonance-principal parametric resonance and 1:3 internal resonance. The other is 1/2 subharmonic resonance- principal parametric resonance and 1:3 internal resonance.The equations of motion for the nonlinear nonplanar flexible cantilever are derived by using the generalized Hamilton's principle. Then, the Galerkin procedure and the method of multiple scales are used to give the perturbation analysis of the system and the average equations. The three resonant cases are considered in this dissertation. The first case is primary resonance-principal parametric resonance and 1:1 internal resonance. The second case is 1/2 subharmonic resonance-principal parametric resonance and 1:1 internal resonance. The final case is primary resonance for the second mode-principal parametric resonance and 1:2 internal resonance.The adjoint operator method and the theory of normal form are used to investigated the normal forms in the four dimensional nonlinear systems with cubic nonlinearities. We give two methods to compute the normal form for the four dimensional nonlinear systems. The corresponding Maple programs are also given in the appendix. The results obtained in this dissertation have a completely agreement with the results obtained in other papers.Based on the averaged equations of the nonlinear nonplanar flexible cantilever, thenormal form of the averaged equations is obtaineiby the Maple program given in the appendix. The global bifurcation analysis of the nonlinear nonplanar cantilever is given by a global perturbation method developed by Kovacic and Wiggins. It is found that the nonlinear nonplanar cantilever can undergo the Hopf bifurcation, heteroclinic bifurcations and Silnikov-type homoclinic orbit to saddle focus, which means that the nonlinear nonplanar cantilever can give rise to the chaotic motion in the sense of Smale horseshoes. The chaotic motion of the planar cantilever and the nonplanar cantilever are investigated by the numerical simulation. The numerical results show the existence of chaotic motion in the planar cantilever and the nonplanar cantilever. The period-doubling solution in the nonplanar cantilever is also found. For the planar cantilever, the period-three, period-multiplying, quasi-periodic motions are showed. The transfer from period-three motion to chaotic motion and the transfer from quasi-period motion to chaotic motion are given by the numerical simulation.