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根据古典阴阳互补和现代对偶互补的基本思想,首次建立了线性阻尼情况下Reissner夹层梁动力学的相空间非传统Hamilton型变分原理。这种变分原理不仅能反映此类动力学初值—边值问题的全部特征,而且它的欧拉方程具有辛结构的特征。基于该变分原理,提出一种称之为辛空间有限元—时间子域法的辛算法。这种新方法是由空间域采用有限元法与时间子域采用Lagrange插值多项式插值的时间子域法相结合而成。文中用这种辛算法进行了四种支承条件下夹层梁的动力响应分析。算例的计算结果表明,这种新方法的稳定性、收敛性、计算精度和效率都明显高于国际上常用的Wilson-θ法和Newmark-β法。
According to the basic ideas of classical yin-yang complementarity and modern duality complementarity, the non-traditional Hamilton-type variational principle of phase space for Reissner sandwich beam with linear damping is established for the first time. This variational principle not only reflects all the features of the initial value-boundary value problem of this kind of dynamics, but also its Eulerian equation has the characteristics of symplectic structure. Based on this variational principle, a symplectic algorithm called symplectic space finite element-time sub-domain method is proposed. This new method is a combination of the finite element method in the space domain and the time sub-domain method in which the time sub-domain is interpolated by Lagrange interpolation polynomials. In this paper, the dynamic response of the sandwich beam under four kinds of support conditions is analyzed by using this symplectic algorithm. The calculation results of the example show that the stability, convergence, accuracy and efficiency of this new method are obviously higher than the commonly used Wilson-θ method and Newmark-β method.