Solving The Dynamic Problems Of Structure Of Bar System Using The Spectral Element Method

The finite element method, finite difference method, spectral element method and so on, are used to solve problems of the dynamic response of structures today. Among these numerical methods, the spectral element method applies the interpolation functions which depend on the frequencies, combines with the finite element method, possesses extensive adaptability for the complex boundaries, and has the nature of fast convergence. So this method has important value of depth study.The spectral element method can be considered as the combination of the key features of the conventional finite element method, dynamic stiffness method and spectral analysis method. The exact dynamic stiffness matrix is formulated in the frequency domain by using exact dynamic shape functions that are derived from the wave solutions to the governing equations. So in theory, the spectral element method provides exact frequency-domain solutions. In this paper, the advantages and disadvantages of the spectral element method are summarized, the dynamic stiffness matrices of the bar and beam elements are derived, and the natural frequencies and time domain responses of the bar and beam structures are calculated.Then, in this paper the spectral element method is applied to analyze and calculate the dynamic response of structure of bar system. The structure of bar system is divided into the frame structure and truss structure because of the different ways of connection. The dynamic stiffness matrix for the overall frame structure is assembled. The equations of motion of the overall structure are presented. The natural frequencies and time domain responses are calculated. The present method is compared with the finite element method. From the results we can see that the spectral element method has its unique advantages in the numerical simulation.The connecting rod and truss structures with joints are considered, respectively. The influences of joints on the dynamic behavior of the integral structure are researched using the spectral element method. The joint is modeled as a piecewise linear model, and the spectral element method is extended to solve the piecewise linear problems, which expands the application fields of the spectral element method. The joint structure is considered as a spectral element in the frequency domain. The dynamics stiffness matrix of the whole structure is obtained by integrating the dynamic stiffness matrices of the joint and rod elements, and the dynamics equation of the whole structure is further gotten. The time response curve of the structure is obtained by the calculation of the dynamic equation of the whole structure. The dynamic behaviors of the connecting rod and truss structures with joints are analyzed.