In-plane Free Vibration Of Functionally Graded Piezoelectric Material Plate

The vibration of the plate includes the out-of-plane vibration(or lateral vibration)and the in-plane vibration.The frequency band of the out-of-plane vibration is well coupled with the sound field and dominates the energy transfer.Therefore,a lot of researches about the out-of-plane vibration have been solved by different methods and discussed from different angles.In contrast,there are fewer studies on the in-plane vibration because of the higher frequency and inconspicuous performances in engineering applications.Recently,with the advancement of measurement techniques,more and more experiments have proved that the in-plane vibration,as a high-frequency component in the multi-component structure,not only plays a role in energy propagation,but also affects the low-frequency vibration.Furthermore,the high frequency vibration has wide applications in the engineering practical problems,such as the elimination of noise coming from the operative disc brakes and hard discs,and the control of the vibration brought out of the running rail vehicle wheelsets.Therefore,the problem of in-plane free vibration has attracted more and more attentions.Firstly,based on the two-dimension linear elastic theory,the in-plane free vibration differential equation for in-plane Functionally Graded Material rectangular plates were derived.Using the Differential Quadrature Method,the differential equations and boundary conditions are discretized and the characteristic equation of the frequency is obtained.Assuming that the physical parameters of the material vary in the form of an exponential function,the dimensionless natural frequency of in-plane free vibration of Functionally Graded Material rectangular plates were solved numerically.The influence of the gradient exponentα and β,the aspect ratio s and the stiffness of elastic restraint k on the dimensionless frequency is considered under the elastic boundary.And,the modal characteristics of in-plane FGM rectangular plate are studied.Secondly,based on plane stress hypothesis,the in-plane free vibration characteristic of a radial Functionally Graded Piezoelectric Material axisymmetric circular/annular plate is investigated with elastic support on inner and outer boundary.The material properties of Functionally Graded Piezoelectric Material axisymmetric circular/annular plates are assumed to be graded in power-law distribution from along the radial direction.Governing equations of motion are derived based on the linear elastic theory.Analytical solution for giving boundary conditions of the Functionally Graded Piezoelectric Material axisymmetric circular/annular plates are obtained by employing Bessel function.And,the influence of inner radius to outer one ratio η and the gradient exponent β on in-plane free vibration dimensionless frequency of the Functionally Graded Piezoelectric Material axisymmetric circular/annular plates is discussed further.Finally,based on the two-dimension linear elastic theory,the in-plane free vibration differential equation for radial Functionally Graded Piezoelectric Material annular plates were derived.Using the Differential Quadrature Method,the differential equations and boundary conditions are discretized and the characteristic equation of the frequency is obtained.Assuming that the physical parameters of the material vary in the form of a power function,the dimensionless natural frequency of in-plane free vibration of Functionally Graded Piezoelectric Material annular plates were solved numerically.The influence of the gradient exponent p,the inner to outer diameter ratio η,the stiffness of elastic boundaries k and piezoelectric effect on the in-plane free vibration dimensionless frequency is considered under the combination of elastic boundary and the open electrical boundary.And,the modal characteristics of radial Functionally Graded Piezoelectric Material annular plate are studied.