Research On Free Vibration Of Arbitrarily Shaped Plates Based On Orthogonal Polynomial Ritz Method

As a structural component,thin plate is widely used in many engineering fields such as mechanical equipment,civil engineering vehicles,ships,aerospace,electronic and optical components.Different working environments,working spaces and loading conditions have different requirements for the geometry of thin plate structure.In addition,some scientific research and engineering fields further require that the thickness and material properties(such as density and elastic modulus,etc.)of plate structure vary according to different function formulations.As a work with important engineering guiding significance,the prediction of vibration characteristics of arbitrarily shaped plate structures mainly uses numerical methods represented by finite element method.However,in engineering design,revealing the analytical mathematical relationship between plate geometry and vibration process will help deepen people’s understanding of the influence of plate geometry on vibration characteristics,and provide theoretical guidance for reasonable and efficient design of reliability and safety of plate structure.Therefore,it is of great theoretical and practical significance to study the semi analytical vibration modeling method and analyze the vibration characteristics of thin plate structures with arbitrary shapes under arbitrary elastic boundary conditions.Based on this goal,this paper carries out the following main work:In view of the difficulties of analytical or semi-analytical mathematical description of complex curve contour and its enclosed plate domain in vibration modeling,and the problems of narrow selection range of admissible functions and high computational complexity in the Ritz method which takes classical polynomial as admissible functions,a semi-analytical method named orthogonal polynomial Ritz method for vibration analysis of plates with arbitrary shapes under general boundary conditions is proposed.Firstly,the arbitrary shape plate domain is divided into several curved edge trapezoid domains,and the energy integral expression is rewritten as the sum of the integrals on all curved edge trapezoid domains.The boundary conditions of the plate are treated by penalty function method,and the Gram-Schmidt orthogonalization method is introduced to generate displacement admissible function,which avoids the complexity of coordinate normalization by existing methods.Then the energy expression is analytically transformed into definite integral by polynomial admissible function and solved analytically or semi-analytically according to the form of contour curve equation.A semi-analytical method for free vibration analysis of arbitrary plates is established by combining Ritz method.And based on the orthogonal polynomial Ritz method,the analytical or semi-analytical results of vibration energy integral of arbitrarily shaped non-homogeneous orthotropic plates of variable thickness under elastic boundary constraints and two kinds of elastic foundation supports are derived and the vibration analysis model is established.The parameter values are discussed through the model parameter analysis,and the convergence,accuracy and universality of the model are proved through numerical and experimental verification.In view of the lack of analytical or semi-analytical analysis methods for in-plane vibration of plates with arbitrary shapes,especially composite plates whose material and geometric properties change with coordinate variables in a function law,under arbitrary elastic boundary conditions,tangential and normal displacement equations of any point on the edge of curve profile of plates were derived,and based on the orthogonal polynomial Ritz method,an analytical model for the in-plane vibration of arbitrarily shaped isotropic plates is established and extended to arbitrarily shaped non-homogeneous orthotropic plates of variable thickness.The values of penalty spring stiffness,orthogonal interval and weight function are discussed.On this basis,the convergence research and numerical verification are carried out to prove the effectiveness and applicability of the method.Taking rectangular non-homogeneous orthotropic plates of variable thickness as an example,the effects of the material inhomogeneity parameters and the plate thickness parameters on the in-plane vibration characteristics of the plate are studied.In view of the limitations that the existing analytical or semi-analytical methods are only applicable to a few through cracked plates with regular shapes and difficult to analyze the problem of crack plates with irregular shapes under complex boundary conditions,a solution model for the free vibration of cracked plates with arbitrary shapes under arbitrary elastic boundary conditions is established.In this model,the arbitrary shape plate with cracks is decomposed into several arbitrary shape subdomains by domain decomposition technique,and the energy of each subdomain is solved according to the orthogonal polynomial Ritz method;arbitrary elastic boundary conditions and subdomain coupling interface continuity conditions of cracked plates are simulated by using linear spring to constrain the plate boundary and the connected plate subdomain respectively;The crack is simulated by taking the stiffness value of subdomain connecting spring as 0.The value of the connection stiffness is discussed.On this basis,the correctness and applicability of the model are verified by the convergence study,the comparison of numerical results and the modal experiment.The variation trend of natural frequencies and mode shapes of cracked regular hexagonal plates with different crack forms,crack lengths and crack angles under several boundary conditions is studied.In view of the current situation that existing researches mainly focus on a few perforated plates with simple contours,and the problem of perforated plates with complex inner and outer contours and complex boundary constraints lacks analytical or semi-analytical analysis methods,the vibration energy of perforated plates is expressed by domain decomposition technique as the sum of the energy of arbitrary subdomains containing the inner boundary potential energy and the potential energy of connected springs.Based on the orthogonal polynomial Ritz method,a semi-analytical model for free vibration analysis of an arbitrarily shaped plate with arbitrarily shaped cutouts under arbitrary elastic boundary conditions is established.The convergence,accuracy and universality of the method are verified by the published results and the finite element software analysis results.Taking the elliptical plate with rhombic cutouts as examples,the effects of cutouts size and position on the vibration characteristics of the plate are studied.Aiming at the problem that the existing research is mainly limited to the coupled structures of rectangular plates and lacks analytical or semi-analytical methods for vibration analysis of coupled structures with complex contour plates under general boundary conditions,a free vibration analysis model of coupled structures composed of arbitrarily shaped plates is established.On the basis of the orthogonal polynomial Ritz method,four types of virtual coupling springs are set on the coupling boundary of the coupling structure,then the bending vibration and in-plane vibration of the sub-plate are all included in the model.The accuracy and applicability of this method are verified by studying the convergence and comparing the results with modal experiments and finite element software.Taking the coupled structure composed of square plate and semicircular plate as an example,the influence of coupling angle on the vibration characteristics of the coupled structure is studied.