Use of penalty formulations in dynamic simulation and analysis of redundantly constrained multibody systems

Abstract

The determination of particular reaction forces in the analysis of redundantly constrained multibody systems requires the consideration of the stiffness distribution in the system. This can be achieved by modelling the components of the mechanical system as flexible bodies. An alternative to this, which we will discuss in this paper, is the use of penalty factors already present in augmented Lagrangian formulations as a way of introducing the structural properties of the physical system into the model. Natural coordinates and the kinematic constraints required to ensure rigid body behaviour are particularly convenient for this. In this paper, scaled penalty factors in an index-3 augmented Lagrangian formulation are employed, together with modelling in natural coordinates, to represent the structural properties of redundantly constrained multibody systems. Forward dynamic simulations for two examples are used to illustrate the material. Results showed that scaled penalty factors can be used as a simple and efficient way to accurately determine the constraint forces in the presence of redundant constraints.

Solvability of reactions in rigid multibody systems with redundant nonholonomic constraints

“…(13), it is sufficient to premultiply the first equation by A −T , to obtain the original form of Eq. (13). This indicates that the solution forq does not depend on the choice of matrix V, thus it is a unique solution.…”

“…Obviously, it is possible to find another matrix V * = V, spanning the null space of the constraint matrix C, and to obtain a different version of Eq. (13). In such a case, the following holds:…”

“…So far, no general method of factors selection is proposed. Similarly, in the case of penalty methods, the authors of [13,14,30] formulated an idea that a flexibility aspect can be incorporated by relating the penalty factors to the real mechanism stiffness, which allows us to solve the indeterminacy problem without adding any major complication to equations of motion. However, so far no general rules on how to relate the penalty parameters with flexibility properties of multibody system parts and joints are provided.…”

A generalized constraint reduction method for reduced order MBS models

“…To ensure correct constraint forces is still an open field of research and several authors (see [8,12,30,31]) deal with the problem of redundant constraint equations, whether or not they result from singular configuration, overconstrained modeling (as in the case of a Fig. 1 One mass oscillator door being modeled using both hinges), etc.…”

“…To overcome the issue of redundant constraint equations in terms of reduced order models, we propose a generalization of the constraint reduction method introduced in [27], based on a singular value decomposition of the reduced constraint Jacobian of the system under consideration. The use of the singular value decomposition method in terms of the elimination of Lagrange multipliers or to handle redundant constraint equations was discussed in the past by various authors; see, for instance, [8,12,15,22,30,31].…”