Modeling and Control of Dynamical Effects due to Impact on Flexible Structures

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In the first part of this dissertation, we consider modeling and approximation of impact dynamics on flexible structures. A nonlinear model is developed through Hertz law of impact in conjunction with the dynamic equation of the flexible structure. We have analyzed this nonlinear model and established the existence and uniqueness of solutions of the nonlinear equation. A numerical method is developed based on the contraction mapping principle. By utilizing the fact that impact interval is very short in general, one may approximate the transfer functions of the systems to which the impacting bodies belong by Taylor polynomials of low order. We have developed the first and second order approximations. The first order approximation yields a special function which can be used for analytical and computational purposes. The second order approximation leads to a two-parameter family of ordinary differential equations of which the solutions exhibit universal features of impact problems. Simulation results of various examples have demonstrated the usefulness of the developed numerical method and approximation methods.

The second part of this dissertation is devoted to control of impact dynamics. We have formulated and studied a control problem where a linear system is subjected to a series of impact forces. The impact forces are treated as disturbances to the system and modeled as finite duration events using the theory developed in part one. A reasonable control objective is to design a feedback controller to minimize the energy transferred from the disturbances to the controlled outputs in the L2 norm sense. Under the assumption that the disturbance information is known a priori, a (sub) optimal control strategy is derived based on dynamic game theory. We shown that, by taking advantage of the fact that the duration of each impact force is very short in general, we can derive a series of approximate solutions of the nonlinear problem. The higher order terms may be negligible for the disturbance attenuation problem in some applications. Hence, the approximation with the leading term renders a linear one. A (sub) Hcontroller is derived and a procedure to compute such a controller is given. The (sub) optimal solution is naturally associated with the existence of a stabilizing periodic solution of coupled Raccati equations. Hamiltonian theory is employed to analyze the coupled Raccati equations. Finally, we investigate the digital implementation of this control algorithm by using a sampled-data controller. We have shown that under a certain sampling condition, the controller structure could become simpler than the continuous version.