Benchmark Solutions Of Linear Random Vibration Responses For Mindlin Plates

Engineering structures are often subjected to strong random excitation such as earthquake,wind and wave loads,which results in complex dynamic behaviors.Therefore,it is fairly important to calculate the random vibration response accurately.In order to develop the numerical method of random vibration analysis,it is necessary to provide exact benchmark solutions for random vibration response of structures.For the linear system,the pseudo excitation method can transform the stationary random excitation into the deterministic harmonic excitation,so as to calculate the accurate random vibration response of the structure efficiently.Based on the pseudo excitation method and semi-analytical method,the exact solutions of the stationary random vibration responses of elastic Mindlin rectangular plate and orthotropic Mindlin rectangular plate under six boundary conditions with simple support on opposite sides are obtained,which can be used as the benchmark solutions for the development of other numerical methods.The present research work has important theoretical significance and engineering application value.For the elastic Mindlin rectangular plate,the exact natural frequencies and analytical modes of transverse free vibration of SSSS,SSSC,SCSC,SFSF,SSSF and SCSF(S stands for simple support,C stands for fixed support and F for free edge)boundary conditions are obtained by solving the closed-form free vibration eigenequations.Subsequently,the forced vibration equation of Mindlin plate with damping is derived,and the frequency response function is decoupled by mode superposition method and modal orthogonality.The formulas of power spectral density function(PSD)and root mean square(RMS)of transverse deflection,velocity and acceleration responses of Mindlin rectangular plate under stationary random excitation such as random point excitation,base acceleration excitation or surface excitation are derived by introducing the pseudo excitation method.In order to improve the calculation efficiency,a semi-analytical method for discretization in frequency domain and mode coordinates is proposed.Finally,the semi analytical solutions of the linear random vibration responses are compared with the analytical solutions,ANSYS calculated results and MATLAB finite element program results,which indicate the high precision of the benchmark solutions.For the elastic orthotropic Mindlin rectangular plates,similarly,the exact natural frequencies and analytical modes of transverse free vibration under six boundary conditions(SSSS,SSSC,SCSC,SFSF,SSSF and SCSF)are obtained by solving the closed-form free vibration eigenequations.Then,the damped forced vibration equation of orthotropic Mindlin plate is derived.The frequency response function is also decoupled by mode superposition method and mode orthogonality.Based on the pseudo excitation method,the formulas of power spectral density function and root mean square of the transverse deflection,velocity and acceleration response of orthotropic Mindlin rectangular plate under the base acceleration excitation are established.Finally,the random excitation and the analytical modes are discretized in frequency domain and space respectively,and the efficient and accurate semianalytical method is advanced.The power spectrum function and root mean square distribution as well as the corresponding benchmark solutions are obtained by semi-analytical method.