Mei Symmetry and Conservation Laws for Time-Scale Nonshifted Hamilton Equations

The Mei symmetry and conservation laws for time-scale nonshifted Hamilton equations are explored, and the Mei symmetry theorem is presented and proved. Firstly, the time-scale Hamilton principle is established and extended to the nonconservative case. Based on the Hamilton principles, the dynamic equations of time-scale nonshifted constrained mechanical systems are derived. Secondly, for the time-scale nonshifted Hamilton equations, the definitions of Mei symmetry and their criterion equations are given. Thirdly, Mei symmetry theorems are proved, and the Mei-type conservation laws in time-scale phase space are driven. Two examples show the validity of the results.

The Use of Zero-Mass Particles in Analytical and Multi-body Dynamics: sphere rolling on an arbitrary surface

Zero-mass particles are, as a rule, never used in analytical dynamics, because they lead to singular mass matrices. However, recent advances in the development of the explicit equations of motion of constrained mechanical systems with singular mass matrices permit their use under certain circumstances. This paper shows that the use of such particles can be very efficacious in some problems in analytical dynamics that have resisted easy, general formulations, and in obtaining the equations of motion for complex multi-body systems. We explore the ease and simplicity that suitably used zero-mass particles can provide in formulating and simulating the equations of motion of a rigid, non-homogeneous sphere rolling under gravity, without slipping, on an arbitrarily prescribed surface. Computational results comparing the significant difference in the motion of a homogeneous sphere and a non-homogeneous sphere rolling down an asymmetric arbitrarily prescribed surface are obtained, along with measures of the accuracy of the computations. While the paper shows the usefulness of zero-mass particles applied to the classical problem of a rolling sphere, the development given is described in a general enough manner to be applicable to numerous other problems in analytical and multi-body dynamics that may have much greater complexity.

Application of Gauss principle of least constraint in multibody systems with redundant constraints

Redundancy in the constrained mechanical systems often occurs in complex multibody mechanic systems in the existence of excessive constraints and singular positions due to system motion. In this work, Gauss principle of least constraint (GPLC) is applied to solve the dynamic motion of system with redundant constraints without changing the physics of system. Furthermore, the particle swarm optimization method is used to handle the minimization optimization problem. Eventually, the effectiveness of GPLC is validated through the dynamic modelling and simulation of two numerical examples (a planar four-bar mechanism and a spatial parallelogram mechanism). The simulation results are analyzed and compared with those obtained from Udwaia-Phohomsiri formulation and augmented Lagrangian formulation, in terms of constraint violation, computational efficiency and variation of the mechanical energy. From the viewpoint of computational efficiency and accuracy, GPLC can be regarded as a practical real-time simulation method for multibody systems with redundant constraints.

Backward Differentiation Formula and Newmark-Type Index-2 and Index-1 Integration Schemes for Constrained Mechanical Systems

Various methods for solving systems of differential-algebraic equations (DAE systems) are known from literature. Here, an alternative approach is suggested, which is based on a collocated constraints approach (CCA). The basic idea of the method is to introduce intermediate time points. The approach is rather general and may basically be applied for solving arbitrary DAE systems. Here, the approach is discussed for constrained mechanical systems of index-3. Application of the presented formulations for nonmechanical higher index DAE systems is also possible. We discuss index-2 formulations with one intermediate time point and index-1 implementations with two intermediate time points. The presented technique is principally independent of the time discretization method and may be applied in connection with different time integration schemes. Here, implementations are investigated for backward differentiation formula (BDF) and Newmark-type integrator schemes. A direct application of the presented approach yields a system of discretized equations with larger dimensions. The increased dimension of the discretized system of equations may be considered as the main drawback of the presented technique. The main advantage is that the approach may be used in a very straightforward manner for solving rather arbitrary multiphysical DAE systems with arbitrary index. Hence, the method might, for instance, be attractive for general purpose DAE integrators, since the approach is not tailored for special DAE systems (e.g., constrained mechanical systems). Numerical examples will demonstrate the straightforward application of the approach.