Structure-preserving properties of Strmer-Verlet scheme for mathematical pendulum

The structure-preserving property, in both the time domain and the frequency domain, is an important index for evaluating validity of a numerical method. Even in the known structure-preserving methods such as the symplectic method, the inherent conservation law in the frequency domain is hardly conserved. By considering a mathematical pendulum model, a Strmer-Verlet scheme is first constructed in a Hamiltonian framework. The conservation law of the Strmer-Verlet scheme is derived, including the total energy expressed in the time domain and periodicity in the frequency domain. To track the structure-preserving properties of the Strmer-Verlet scheme associated with the conservation law, the motion of the mathematical pendulum is simulated with different time step lengths. The numerical results illustrate that the Strmer-Verlet scheme can preserve the total energy of the model but cannot preserve periodicity at all. A phase correction is performed for the Strmer-Verlet scheme. The results imply that the phase correction can improve the conservative property of periodicity of the Strmer-Verlet scheme.

Structure-preserving Analysis on Folding and Unfolding Process of Undercarriage

The main idea of the structure-preserving method is to preserve the intrinsic geometric properties of the continuous system as much as possible in numerical algorithm design. The geometric constraint in the multi-body systems, one of the difficulties in the numerical methods that are proposed for the multi-body systems, can also be regarded as a geometric property of the multi-body systems. Based on this idea, the symplectic precise integration method is applied in this paper to analyze the kinematics problem of folding and unfolding process of nose undercarriage. The Lagrange governing equation is established for the folding and unfolding process of nose undercarriage with the generalized defined displacements firstly. And then, the constrained Hamiltonian canonical form is derived from the Lagrange governing equation based on the Hamiltonian variational principle. Finally, the symplectic precise integration scheme is used to simulate the kinematics process of nose undercarriage during folding and unfolding described by the constrained Hamiltonian canonical formulation. From the numerical results, it can be concluded that the geometric constraint of the undercarriage system can be preserved well during the numerical simulation on the folding and unfolding process of undercarriage using the symplectic precise integration method.