Numerical Analysis Of Non-linear Dynamics And Chaos Control For MDOF Rotor Systems

By means of an integral expression of non-linear dynamics equation an explicit precise integration algorithm with four order accuracy and self-correct and self-starting to solve this equation is given. This method is adapted to solve the non-conservative system with multi-DOF and strong nonlinear. The non-linear dynamic equations are divided into some blocks or groups. Using the precise integration for each block of the equations the lower order matrix can be used to solve these integral equations. A new step-by-step integral procedure of dynamics equations is presented. The general expression of the solution of dynamics equations is obtained on the basis of the homogenous analytical solutions of dynamics equations and Duhamel integration. The explicit analytical integration algorithm, which is characterized by fourth-order accuracy, self-starting and self-correcting, is employed to discretize the equivalent load terms. A new approach, which is direct integration method with integral model (DIM-IM) to solve dynamic governing equations, is developed. Numerical examples show that the results are highly accurate in comparison with Newmark> Wilson-# -. Houbolt and central difference method.A new numerical method of solving the stability and Hopf bifurcation of the flexible axis system with the Jeffcott rotor supported on the nonlinear unsteady-state oil-film is presented. By means of the Lyapunov stability theory and interval halving method, the angle velocity instability (the point of the Hopf bifurcation) is calculated. At the point of co = G>C , it is the Hopf bifurcation thatthe rotor orbit of shaft centerline is the periodic whirl of small amplitude. The method is suitable for the multi-freedom system.The dynamics behaviors of the flexible Jeffcott rotor system supported by unsteady short dynamic bearing are investigated. Based on nonlinear unsteady-state dynamic n -oil film force model described by three functions the local stability of the periodic solutions with the controlling parameters, rotational speed ratio, imbalance amount, damping ratio and viscidity, are predicted by using the Floquet multiplier. It is found that the period doubling bifurcation is caused by a certain imbalance amount and the Hopf bifurcation is created by the lost stability of the oil-film. By means of the precise integration method with Lagrangian interpolation the trajectory of the shaft center, the Poincare mapping and the bifurcation graphs are numerically given. The results predicted by the Floquet theory are checked and the long-term dynamic behavior of the system is predicted. It is shown that the system has rich nonlinear behaviors at somemcombination of the four parameters, for examples, multi-frequency subharmonic resonance, as well as chaos phenomenon from doubling bifurcation and twice Hopf bifurcation. Using analytical and numerical methods the local stability, the bifurcation and long term behaviors of the imbalance rotor dynamic system are investigated in a wide range of various parametric configurations. It is concluded that the results of both of the two methods are the same for analyzing local stability of the periodic solutions. But after the bifurcation of the periodic solutions the dynamic behaviors of the system with the parameters changed only can be analyzed by numerical method. This is the limitation of the Floquet theory. Combining both methods some suggestions on the design parameters of oil-film bearing-flexible rotor system can be offered. Comparing the dynamic oil-film force model to the steady one, the rationality of the dynamic oil-film force model is noted. The computation results show that the more the imbalance amount, damping ratio and the viscidity, the more the no-synchronization motions of the system will be restrained. The rotational speed of the oil-film lost stability is increased.An approach about the no-linear whirl dynamics behavior of elastic shaft Jeffcott rotor with a transverse crack, which is supported on the unsteady and no-linear oil film, is given. The numeric...