Symplectic Method For Free Vibration Analysis Of Elastic Cylindrical Shell Structures

With the developing of space flight,shipbuilding and automobile,the cylindrical shells structures are used for large industrial equipment,such as spacecraft shell,automobile energy absorption device,submarine ship body,plane sank system resistance and so on.Cylindrical shell structures have a series of advantages,such as simple form,easy maching,superiority mechanical performance and so forth.With the progress of material science,more and more new materials are used in the maching of cylindrical shells,for example,functionally graded materials,nano-materials etc.Because of the developing of the new materials,the existing theorys and methods may not applicable,it has theoretical and practical significance for studying the vibration behaviors of the cylindrical shells with new materials.Based on Donnell and Reissner shell theories,this paper establishes the Hamiltonian solving models of macroscopic cylindrical shells and microcosmic cylindrical shells with size effect,analyses the vibration of the structures,and obtains the analytical solution of the problem.In the Hamiltonian system,the questions are attributed to the solution of the symplectic eigenvalues and eigensolutions.The basic unknown vectors are expressed by the linear combination of the eigensolutions,the unknown coefficients are determined by the conjugate symplectic orthogonal relations among the eigensolutions vectors and the boundary conditons,thus the expression of the basic unknown vectors and the frequency equations are obtained.The symplectic method breaks through the limitation of the traditonal semi-inverse method,gives the analytical solution directly.The main points of the paper contains:(1)An analytical method is used to establish the Hamilton system of static behavior of cylindrical shells.Based on the simple shell theory,dispacement and equivalent shear,rotation angle and equivalent moment are dual variables for each other in the Hamiltonian system.The above variables are seen as the basic unknown variables,the Hamiltonian function is gained by the Hamiltonian variational principle.In symplectic space,the variables separation method is applied to stactic analysis of cylindrical shells,the question is then ascribed to symplectic eigenproblem.Thereafter,combining boundary conditions and conjugate symplectic orthogonal relations,the analytical expression of the problem of the static behaviour of cylindrical shells is obtained.According to the study,zero eigensolution is the solution corresponding to Saint-Venant principle,the nonzero eigensolution corresponds to the boundary effect of Saint-Venant principle.(2)An analytical method is used to establish the Hamilton system of free vibration of cylindrical shells.Take the Donnell shell and Reissner shell as research objects,take axial displcement,circumferential displacement,radiual displacement,rotation angle,membrance force,surface force,shear force and equivalent moment as the basic unknown vectors,the Hamiltonian canonical equation of the free vibration of ealstic cylindrical shells is established by the aid of Hamilton variables principle.The basic unknown vectors can be expressed by the linear combination of the symplectic eigensolution.The frequency equations and vibration modes are obtained by the symplectic eigensolution and boundary conditions of cylindrical shells.The roots of the frequency equations equivalent to the frequency of the free vibration of the cylindrical shells.According to the numerical examples,the symplectic method is applied to the analyze of the free vibration of cylindrical shells,high accuracy frequencies and smoothness vibration modes are obtained.(3)An analytical method is used to establish the Hamiltonian system for the free vibration of functionally graded cylindrical shells.Based on the functionally graded cylindrical shells models of the Donnell and Reissner shell theories,the Hamiltonian canonical equation of the free vibration of functionally graded cylindrical shells is established.Take a full consideration of the material properties and volume fraction,the Hamiltonian system is built.Meanwhile,the analysis solution and the frequency equations of free vibration of functionally graded cylindrical shells can be obtained.According to the research of analysis solution,it shows that volume fractions are inversely proportional to frenquency of free vibration of functionally graded cylindrical shells.Moreover,the paremeter study points out that,the boundary conditions,size parameters,circumferential wave numbers and the axial half-wave numbers are sensitive to the frenquency,especially the different functional graded cylidrical shell models.(4)An analytical method is used to establish the Hamilton system of radial free vibration of microcosmic cylindrical shells with size effect.Based on the Eringen's nonlocal elastic theory and Reissner shell theory,the Hamiltonian analysis model is established.In Hamitonian system,axial displcement and local surface force,circumferential displacement and local membrance force,radiual displacement and local equivalent shear force,rotation angle and local moment are the dual variables for each other.The problem of breathing vibration of microcosmic cylindrical shells contributes to the solution of the symplectic eigenvalue and the eigensolution.The numerical examples show that the nonlocal paremeter has an obvious effect on the frequency of the microcosmic cylindrical shells,as the nonlocal paremeter increases,the frequency decreases.In addition,the model can be promoted to the radial free vibration of cylindrical shells with nanometer structure,like Single-Walled Carbon Nanotubes.