Dynamic Stability Of Reticulated Shell Under Periodic Load And Its GPU Parallel Computing

The long-span reticulated shells are one of widely used important types of building structures. They have the characteristics of weak stiffness, small damping and low natural frequency, which results in possible dynamic failures under dynamic loads. Previous researches on dynamic stability of this kind of structure are about the most unfavorable critical load judged by the curves of load-response. However, the time-history analysis requires a lot of computation time and only some specific load can be considered, so the essential relationship between structural dynamic characteristics and load properties can not be determined to arrive at more general conclusions. Therefore, the dynamic stability problem for a large span reticulated shell structures is considered in this paper, a finite element method based on the stability theory of Mathieu-Hill is developed, and a parallel computing platform using GPU is stablished. The main works is listed as follows:(1) The Mathieu-Hill equations based on the finite element method for the structure with and without damping are built.The approximated equation of the first order for determining dynamic unstable regions is derived. The main instability boundaries of different modes are obtained by solving the eigenvalue problem of polynomial matrix. Taking a simply supported beam for example, the accuracy of numerical results are verified by comparing with the analytic solution. Meanwhile, the parametric response of parametric points in the instability regions is calculated by using Runge-Kutta to prove the accuracy of dynamic unstable regions.(2) To solve the problem of excessive time-consuming in solving the eigenvalue problem of polynomial matrix, the GPU parallel computing technique is applied, and a Matlab parallel computing platform is established based on GPU. The matrix multiplication is carried out and tested through large matrixes, which verifies the feasibility and effectiveness of the platform.(3) The parallel computing method is derived for polynomial matrix eigenvalues problem. The correctness of GPU results is verified by an example. The parallel computing efficiency of single precision and double precision of GPU is also compared by a big random matrix. It's shown that GPU double precision computing has good computational efficiency and accuracy, which can solve the problem of time-consuming in polynomial matrix eigenvalue calculations.(4) The dynamic instability regions of a double-layer reticulated shell structure under a single-point periodic load is calculated by using GPU double precision computing. The influence of different damping ratios on instability regions is discussed. It's demonstrated that increasing the damping ratio will reduce the range of dynamic unstable regions, increase the critical load of dynamic instability and decrease the risk of parametric vibration. Moreover, the cases of multiple-point excitations show that more points lead to the expansion of dynamic unstable regions, and make instability regions of different modes overlap. This causes a more dangerous region of parametric vibration, which is verified by the structural parametric responses.