Quasi-Lagrangian equations and its energy-conservative numerical integration for nonlinear dynamic systems

In dynamic simulations, it is important to maintain the same properties and structure as the original system. For conservative systems modeled using Hamiltonian equations and solved with symplectic numerical solvers, the energy-conservative property of the original system can be maintained. However, there are also many dynamic systems modeled under body-fixed frames and contain nonlinear dissipation, making it difficult to maintain energy conservation unless the energy loss from dissipation is also considered. Maintaining total energy conservation during numerical simulations is therefore crucial. To address this problem, this research uses quasi-Lagrangian equations to model dynamic systems described in body-fixed frames. A novel energy-constraint Euler integrator is proposed to solve the quasi-Lagrangian equations of the dynamic model. This integrator takes the energy conservation law as an algebraic constraint and forms a differential algebra equation (DAE) system from the original ordinary differential equation (ODE) system. By solving the DAE system implicitly, the numerical results that maintain the total energy conservation of the system are obtained. To compare the proposed method to common symplectic and non-symplectic numerical integrators, initial value problems for the proposed models are solved. The common symplectic solvers cannot maintain the total energy conservation property of the dynamic models described in quasi-coordinates due to the asymmetric structure of the quasi-Lagrangian equations. On the other hand, the proposed energy-constraint Euler integrator strictly maintains the energy conservation property of the system, working well for both conservative and non-conservative dynamic systems.