The Spectro-geometric Method For Dynamics Modeling And Characteristics Analysis Of Plate And Shell Structures

Accurate grasp of vibration characteristics is the key and premise for implementing vibration isolation and control of a structure and system as well as low noise design of mechanical equipment.This thesis focuses on plate and shell structures which are commonly used in engineering application,to carry out modeling analysis for their dynamic characteristics with elastic boundary and coupling conditions.This work not only has very important instructive significance in engineering field but also provides a theoretic foundation for the research of vibration control and sound radiation of structures.In view of the deficiencies of existing methods in medium frequency range analysis of structural vibration and acoustic problems,a so-called Spectro-Geometric Method(SGM)based on the improved Fourier series method for dynamics modeling and characteristics analysis of structures is presented in this thesis.This Method has more simple displacement admissible function expressions and the solving process become much easier.The geometry of a structure is accurately described in terms of mathematical or design parameters and an unknown solution variable on the structure is expressed in spectral form as a trigonometric series expansion in SGM.The discontinuous problems encountered in differentials of the displacement functions along the structure edges will be overcomed with supplementary functions.It will lead to a substantial reduction of the time consumption in creating computational models.Theoretically,the models created by SGM will introduce no discretization errors.Also,the solution accuracy can be adjusted by changing the terms of the series expansion.The results of free and force vibration analysis for various plate and shell structures and their coupled systems show that SGM has high convergence speed,accurate prediction and high calculation efficiency.Analysis model for an arbitrary shape straight-sided quadrilateral plate with elastically restrained boundary conditions is established by combining SGM with coordinate transformation technology in this thesis,because current vibration analysis of the plate,plate with holes as well as coupled plate structures are mainly focus on classical boundary conditions and regular geometric shapes.The elastic boundary supports are realized by setting four sets of springs along each boundary.The modal frequencies and modal shapes for different kinds of straight-sided quadrilateral plates are calculated through solving an eigenvalue problem after the structural characteristic equations are obtained by using Rayleigh-Ritz method.Since the displacements are constructed sufficiently smooth over thesolution domain,it can be proven mathematically that the weak form solving process based on such energy principle is equivalent to solving structural governing differential equations and boundary conditions directly.The effects of geometric parameters on the modal characteristics for straight-sided quadrilateral plates with different kinds of holes are investigated by combining SGM with numerical method.Different coupling conditions for coupled plate structures are realized by setting four sets of coupling springs along the conjunctions.The analysis model based on SGM breaks through other current research in which the coupled plate structures should be rigid coupling.Free and forced vibrations for three kinds of representative coupled plate structures which are more close to the practical applications are calculated.By using SGM,vibration analytical model for a doubly curved shallow shell with elastically restrained boundary conditions is constructed by considering coupling effects between three direction displacements of the shell.According to different Gaussian curvatures,free vibrations of curved shallow spherical shells,open shallow cylindrical shells and hyperbolic paraboloidal shells are analyzed.The results show that the vibration characteristics of various types of shallow shells can be accurately solved by SGM.Variation trend of the natural frequency with curvature,effects of changing the stiffnesses of boundary springs and the position of exciting force on the frequency responses for the forced vibrations of shallow shells are systemically investigated.Four displacements of a three dimensional curved beam are expressed by one-dimensional trigonometric series respectively and then the modal characteristics of the beam are obtained exactly by using SGM.Free vibrations of doubly curved shallow shells reinforced by any number of beams with arbitrary lengths are also analyzed in this thesis.The shallow shell and the reinforcing beams are coupled by three kinds of linear springs and three kinds of rotational springs.Effects of geometric parameters of stiffeners,coupling conditions and locations of the reinforcing beams on vibration characteristics of stiffened shells are investigated.The unified approach for the dynamics modeling and characteristics analysis of various open and closed shells,which are with greater radius of curvature and higher coupling effect between the displacements of different directions,has not been established up to now.By using SGM,a unified analysis model of dynamic characteristics for three types of shells: open cylindrical shells,open conical shells and open spherical shells are established in this thesis.Then the varying trend of the natural frequencies with circumferential angles,conicities and boundary conditions for different kinds of deep shells are presented.Two straight edges of the open cylindrical shells and open conical shells with 360-degree circumferential angle are rigidcoupled by four sets of springs,and then the vibration problems for closed cylindrical shells and closed conical shells can be solved without changing any modeling process steps.What should be done is to replace the potential energies stored in the boundary restraining springs with the potential energies stored in the coupling springs.Accuracy of the equivalent model is proved by the modal calculation results.Mathematical model of a joined conical-cylindrical shell structure is obtained using the same way as a coupled plate structure.Results of the free and forced vibrations for the joined conical-cylindrical shell structure show the effects of coupling conditions on the modal characteristics as well as strong coupling effect between the displacements of different directions.