Application Of Improved Rayleigh-ritz Method To Vibration Characteristics Analysis Of Complex Beam,plate And Frame Structures

The beam,plate elements,and the framework structure composed of them play a crucial role in engineering,ranging from high-rise buildings,bridges,aerospace fields to microelectronic component domains.The dynamic problems involved in these structures are of particular importance and have always been a research focus for scholars.Currently,the main computational methods include analytical methods,finite element methods,meshless methods,and other numerical algorithms.Among these methods,analytical methods can only solve very ideal and simple cases,while numerical algorithms have not yet formed a unified solving pattern.For the vibration problems of beams,plates,or framework structures under complex conditions,there is no analytical solution,and finite element software is primarily used for calculation at present.This article introduces an improved Rayleigh-Ritz method(IRRM)for computing the vibration characteristics of beams,plates,and framework structures under complex conditions.The main work carried out in this article is briefly summarized as follows:(1)In order to demonstrate the applicability of the computational method in this article,the vibration characteristics of rectangular thick plates with different boundary conditions and thicknesses were compared and analyzed using seven different computational methods,including IRRM,classical 2D Fourier series analytical method,Discrete-Layer method(D-L),Generalized Quadrilateral Coordinated Finite Element Method(ATF-PQ4a),Galerkin MeshFree Method(EFG),and ABAQUS-2D/3D calculation.The results show that the results of IRRM,ATF-PQ4 a,EFG,and ABAQUS-2D have a very high degree of consistency and accuracy.IRRM has the advantages of unified calculation mode,good convergence,high accuracy,and high efficiency.(2)Based on the Euler-Bernoulli beam theory,this method presented in this paper is used to analyze the natural frequencies of continuous multi-span beams with arbitrary boundary conditions and arbitrary concentrated masses.Various factors affecting the calculation accuracy,convergence,and efficiency are summarized.Furthermore,the applicability of this method is demonstrated through examples in the aerospace and bridge engineering fields.Additionally,the improved Rayleigh-Ritz method(IRRM)is applied to derive computational methods based on both the classical Timoshenko beam theory(OTB)and the modified Timoshenko beam theory(MTB).A comparison of the calculation results for OTB and MTB beams is conducted,and the two theories are compared from the perspective of energy methods.(3)Regarding the vibration characteristics of plates,four constraints are considered:different tensile and compressive moduli,medium-thick plates,arbitrary position openings,and arbitrary boundary conditions.Firstly,the stress-strain relationship for medium-thick plates with bi-modulus is analyzed based on the elasticity theory and Mindlin’s theory for mediumthick plates.Then,the geometric discontinuity problem of the plate is solved,and the improved Rayleigh-Ritz method(IRRM)for plates is established.Subsequently,the frequency and mode shape variations of bi-modulus rectangular medium-thick plates with single or coupled factors,such as different plate thicknesses,modulus ratios,opening sizes,and opening positions,are analyzed in detail.The results are compared with finite element calculations to validate the effectiveness and accuracy of the computational method.(4)A novel method is proposed for analyzing the vibration characteristics of frame structures with arbitrary planar shapes,cross-sectional forms,and combined sections.The analysis process utilizes the finite element method to analyze the straight members in the structure.Furthermore,a coupling approach that combines the improved Rayleigh-Ritz method(IRRM)with the finite element method(FEM)is innovatively employed to analyze the nodes with curved connections.This method establishes a completely new approach for analyzing the vibration characteristics of planar frame structures with arbitrary shapes.By leveraging the advantages of IRRM and FEM,the method achieves a balance between efficiency and accuracy.It demonstrates broad applicability and strong suitability,closely resembling the workflow of finite element analysis.This approach represents a new avenue that surpasses previous methods used to analyze planar frame structures.Finally,the improved Rayleigh-Ritz method exhibits advantages of high computational accuracy,good convergence,simple principle,and ease of programming implementation in the analysis of the vibration characteristics of beam,plate elements,and frame structures under complex conditions.It provides a feasible computational method and theoretical basis for practical engineering applications.