Accurate Beam Element And Active Vibration Control Of Piezoelectric Laminated Beams

An accurate and efficient laminated piezoelectric beam element for the static and dynamic numerical analysis of laminated piezoelectric beams is presented and used in the analysis of active vibration control of composite laminates with embedded or surface-bonded piezoelectric layers.The objectives of this thesis are two-folded as below.(1)A refined third-order shear deformable beam theory is used for the kinematics of laminated composite beams.And a layer-wise theory(LWT),which contains the electric parameter at each layer interfaces,is adopted to model the electrical field of the beams.The element formulation is deduced from the mechanical-electro coupled variational principle and Hamiltonian principle.By using the quasi-conforming element technique,a two-noded laminated piezoelectric beam element is derived,in which element stiffness matrix is given explicitly.The performance and excellent features of this laminated piezoelectric beam element are evaluated by the numerical results of static and dynamic analysis.(2)A reduced-order state space model is used for designing the control system.The active vibration control of laminated piezoelectric beams containing distributed piezoelectric sensors and actuators is used to compare the influences of control parameters on the control effects of each control algorithm and the performances of different control algorithms.The genetic algorithm is used to optimize the weighting matrix in the linear quadratic regulator controller.The following conclusions can be draw from the finite element results and analysis of the active vibration control presented in this thesis:(1)The present laminated piezoelectric beam element which has a bending strain in the form of a linear function is defined in terms of the transverse shear strain of beam cross-section.In addition,the coupled electromechanical field is fully considered by using a layer-wise theory.Therefore,the present element has a high computational efficiency.By using the quasi-conforming element technique,the element formulation is derived,in which element stiffness matrix is given explicitly.This present element is free from locking problems as well as free from the time consuming numerical integration.In addition,the electrical freedom is eliminated to improve the computational efficiency.(2)Both classical control laws and optimal control laws are employed in this thesis,which can effectively suppress the excessive vibration in laminated piezoelectric beams.The classical control laws considered are constant gain displacement feedback control,constant gain velocity feedback control and Lyapunov feedback control.The optimal control laws considered are Linear Quadratic Regulator(LQR)and Linear Quadratic Gaussian control(LQG).Compared with classical control laws,the optimal control laws provide a more effective control with smaller actuation voltage and less energy consumption.For a LQR controller,the weighting matrices have a great influence on controlled responses of laminated piezoelectric structures.Furthermore,it is feasible to optimize the weighting matrices in LQR control by genetic algorithm,which enriches the selection strategy of weighting matrices in LQR control.The present two-noded laminated piezoelectric beam element provides an accurate and efficient computational model for static and dynamic analysis and active vibration control analysis of laminated piezoelectric beams with coupled mechanical and electrical properties used in various engineering applications.