Application of euler midpoint symplectic integration method for the solution of dynamic equilibrium equations

The dynamic equilibrium equations \pmb M \ddot \pmb x +\pmb C\dot \pmb x + \pmb K \pmb x = \pmb R are solved by the Euler midpoint implicit integration method. The properties ofJacobi matrix of the algorithm are discussed in detail, and it is shown that Jacobi matrix independent of the external load vector\pmb R is symplectic if \pmb C = 0, and the amplitude of alleigenvalues of symplectic matrix are equal to unity.It is proved that the Newmark method with \delta = 0.5 and \alpha =0.25 is just the Euler midpoint implicit integration method; and for aconservative system, it is a structure-preserving algorithm, which meansthat the energy of the system is preserved through the solution process.Numerical analyses are carried out to illustrate the advantages ofthe symplectic algorithm in the solution ofnon-conservative systems.The accuracy of structure-preserving algorithm is not sensitive to theratio of the frequency of the external force to that of the system,while the accuracy of Newmark algorithm with \delta \ne 0.5 issensitive to that ratio.