Abstract
Based on the differential cubature (DC) principle, a dynamic procedure for simultaneous discretization of time and space is developed. A spatial–temporal differential cubature analysis method for dynamic problems is established with the Timoshenko shear beam; the reliability of analysis results obtained by which is verified, and the stability of the numerical scheme is studied. This method is extended to the two-dimensional structure, and the forced vibration analysis is carried out with the thin plate as an example. The research shows that the method can acquire highly accurate numerical results, and the calculated time-history numerical solution of beam displacement is extremely consistent with the analytical solution, which can adopt to the changes in beam properties and load parameters. With fewer nodes and longer time step than the finite element method (FEM), the method in this paper can still obtain stable and accurate results when solving displacement responses of plate under forced vibration. The numerical stability of this method is closely related to the grid form and the size of time step, and the increase in the number of nodes in the time domain is conducive to increasing the stability range.
Highlights
Traditional dynamic analysis methods can be divided into two categories: one is based on coordinate transformation [1,2], such as mode superposition method, frequency domain method, etc; and another kind of direct solution method for dynamic differential equations can continue to be divided into display difference method and implicit progressive integration method [3–5]
Let the time step length be 0.16 s, the displacement obtained by differential cubature method (DCM) at each time step is compared with the analytical solution and the relevant reference solution
A spatio–temporal dynamic analysis method based on multi-dimensional differential quadrature principle is proposed, and one-dimensional and two-dimensional transient problems are analyzed, respectively
Summary
Structural dynamic analysis generally refers to calculating the response of the structure according to the known structure and dynamic load, so as to determine the bearing capacity and dynamic characteristics of the structure, and provide a reasonable basis for structural design. Traditional dynamic analysis methods can be divided into two categories: one is based on coordinate transformation [1,2], such as mode superposition method, frequency domain method, etc; and another kind of direct solution method for dynamic differential equations can continue to be divided into display difference method and implicit progressive integration method [3–5] These analysis methods have certain application scope. The DQM is only suitable for the regular region formed by orthogonal superposition of multiple one-dimensional directions To cope with this limitation, Civan proposed the differential cubature method (DCM) [17,18] based on the theory of multivariate interpolation approximation, and later researchers [19–22] mainly applied it to the boundary value problems in two-dimensional space domain, such as the natural frequency, bending, buckling analysis of plate and shell, and the transfer of steady-state space and convection diffusion. In order to expand the spatial dimension of the spatio–temporal dynamic analysis method, the response analysis of the forced vibration of the thin plate is carried out based on the three-dimensional DCM, and evaluated its accuracy and computational efficiency by comparing the finite element solutions
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