Study On Modeling Methods For Vibration Of Rectangular Plates And Their Coupling Structures

The rectangular plates and their coupling structures are widely used in various engineering fields.Their vibrational characteristics are of practical importance to the performances of the equipment.In the current solution framework,vibration analyses of rectangular plates and their coupled structures are mostly based on Kirchhoff theory.The effect of shear deformation which will result in the over-estimation of vibration frequencies is neglected.The deviation will increase with the increase of plate thickness.The accurately modeling for the plates and their coupled structures can be carried out using the Mindlin plate theory,which considers the shear deformation and rotary inertia.As an important factor,that can reasonably affect the vibration behavior,boundary conditions play a key role in vibration control of structures.Therefore,the investigation on vibration modeling for the rectangular plates and their coupled structures subjected to arbitrary elastic boundary conditions is the key to deeply carry out the theoretical study.The Mindlin plate theory is adopted to formulate the theoretical model.For the factors mentioned above,the research in this dissertation will be of great theoretical significance and practical application value.According to the vibration problem for rectangular plates and their coupled structures,the following research works have been carried out in this dissertation:The traditional series method used in the literature for solving the vibration problems of rectangular plates are usually customized to specific boundary conditions,such as simply supported.An improved Fourier series method is presented for the vibration analysis of Mindlin rectangular plates with general elastically restrained edges.The vibration displacements and the cross-sectional rotations of the mid-plane along the x and y axes are expressed as the superposition of a double Fourier cosine series and four one-dimensional Fourier series.The matrix eigenvalue equation which is equivalent to differential equations of the plate can be derived through using the boundary conditions and the governing equations based on Mindlin plate theory.The natural frequencies and mode shapes can be obtained through solving the matrix equation.Finally the numerical results are presented to validate the accuracy and fast convergence of the presented method.On this basis,the potential and kinetic energy functions of the composite laminated plates are determined.Using the Hamilton principle,the vibration equation of ply laminated plates with arbitrary boundary conditions under the external loads is obtained.The response in time domain under arbitrary load is calculated using the Newmark method.Based on the intensity of vibration theory,the energy flow characteristics of the laminated plates under single-point or multi-point excitation are studied.Due to the couple effect between the longitudinal displacement and the shear displacement,the traditional series method can't solve the in-plane problem effectively.To solve this problem,an improved Fourier series method is presented for the in-plane vibration analysis of rectangular plates.The in-plane displacement fields are expressed the linear combination of a double Fourier series and auxiliary series functions.The general boundary conditions can be represented by two sets of linear springs along each edge,and the matrix eigenvalue equation of the plate with arbitrary boundary conditions can be derived using Rayleigh-Ritz method.Subsequently,the model of the in-plane vibration with point supports is established by Delta function.Modal analysis and harmonic response analysis are carried out to validate the accuracy and convergence characteristic of the current approach.The effects of boundary conditions on the vibration behavior are investigated through mobility analysis.To overcome the limitations of the solution framework for the coupled plates with arbitrary boundary conditions and coupling conditions,the vibration analysis model of coupled plates is constructed using improved Fourier series method and Mindlin plate theory,in which the flexural vibration and in-plane vibration in each plate are both taken into account.To model the arbitrary coupling conditions and boundary supports,six types of springs and five kinds of springs are uniformly distributed along coupling edge and boundary edge,respectively.The displacement functions of the flexural and in-plane vibration are expressed with the improved Fourier series method.Rayleigh-Ritz method is utilized to establish the unified analysis model for coupled plates with arbitrary boundary conditions and coupling conditions.Unlike the most existing solution framework,the current analysis model can deal with the coupled plates with arbitrary couple angles.Numerical examples are presented to demonstrate the correctness of the current method.In the meantime,the effects of boundary conditions and coupling conditions on the vibration behavior are studied.At last,experimental setups of rectangular plates,composite laminated plates and coupling plates are designed and built up.Then various experimental measurement works are carried out.The natural frequencies and mode shapes for experimental setups are obtained.By comparing the results with these theoretical results,the correctness of the method proposed in this dissertation is verified from the perspective of the experiment.