Robust Partial Quadratic Eigenvalue Assignment Using Receptances And System Matrices In Engineering Vibration Control

Structural vibration control arise in suppressing structure vibrations of large vibra-tion structures when excited by different dynamic loads such as wind load,earthquake load,or mechanical vibration load,etc.Active Vibration Control is one of the most scientific vibration control techniques.The practical control external force is calculated on computer using the measured natural frequencies and model shapes via realtime monitoring and it is then used to change the physical parameters of the large vibration structures(e.g.,mass,damping and stiffness coefficients)through the actuator,and thus reduce the dynamical response of the large vibration structures.In engineering,a vibration structure is often discretized into a second-order dif-ferential equation by some discretization techniques such as the finite element method.Then,some active vibration control technique is used to avoid some dangerous vibra-tions(e.g,resonance)caused by external forces.By using feedback control external force,the partial quadratic eigenvalue assignment problem aims to reassign few un-wanted resonant natural frequencies and keep the remaining natural frequencies and model shapes of the vibration structure unchanged.For practical effectiveness,the vibration control design should possess minimum feedback norms and robustness.That is,the feedback control external force should be as small as possible to reduce the energy consumption and noise,and simultaneously the vibration structure should be insensitive as much as possible to the perturbation of physical parameters.This thesis focuses on the solution of the robust partial quadratic eigenvalue assignment problem based on the measured receptances matrix data,the system matrices(i.e.,the mass,damping and stiffness matrices)and a few unwanted natural frequencies and corresponding model shapes since the receptance matrix data can be measured via experiment.This thesis turns the robust partial quadratic eigenvalue assignment problem into a new nonconvex optimization problem,where the objective function aims to measure both the feedback norm and the sensitivity of a vibration structure to the perturbation of physical parameters,and the parameterized matrix is required to satisfying the addi-tional self-adjointness.This thesis derives the explicit expression of the gradient of the nonconvex objective function in terms of the parameterized matrix and then develops a gradient-based optimization method.To remove the self-adjoint constraint on the parameterized matrix and for convenience of real operation,the proposed nonconvex objective function is regarded as the function of the real form of the parameterized matrix and then the proposed nonconvex optimization problem becomes a real unconstrained nonconvex optimization problem.This thesis derives the explicit expression of the gradient of the nonconvex objective function in terms of the real form of the parameterized matrix and then develops a gradient-based optimization method.Finally,some numerical examples are given to illustrate the validity of the proposed method for solving the robust partial quadratic eigenvalue assignment problem.