Advanced Design of Mechanical Systems: From Analysis to Optimization
Multibody systems are used extensively in the investigation of mechanical systems including structural and non-structural applications. It can be argued that among all the areas in solid mechanics the methodologies and applications associated to multibody
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Independent Variable Formulations Wojciech Blajer Institute of Applied Mechanics, Faculty of Mechanical Engineering Technical University of Radom, Radom, Poland
6.1. Introduction In Chapter 5 the dependent variable formulations for simulation of multibody systems were described. The derivation of equations of motion in terms of dependent states, though conceptually simple and easy to handle, leads to large-dimension governing DAEs, which results in computationally inefficient algorithms, burdened further with the constraint violation problem. An old and legitimate approach to the dynamics formulation is therefore to use a minimal number of independent state variables for a unique representation of motion by means of pure reduced-dimension ODEs. The numerical integration of the ODEs is usually by far more efficient compared to the integration of the governing DAEs, and the numerical solution is released from the problem of constraint violation. On the other hand, the minimal-form ODE formulations require often more modeling effort and skill. Effective and computer-oriented modeling procedures of this type are thus highly desirable. In this chapter some basic reduced-dimension ODE formulations used for simulation of multibody systems are reviewed. Firstly, the joint coordinate method for open-loop systems is described using the geometrical concepts of the projection method. The provided matrix formulation of the arising motion equations is supplemented with a compact scheme for the determination of joint reactions. A velocity partitioning method for obtaining the minimal-dimension dynamic equations is then reported, followed by a general projective scheme for independent variable formulations. A relevance of the projective scheme to Gibbs’-Appell’s equations and Kane’s equations is shown. The reduced-dimension ODE formulations for closed-loop systems are finally discussed and illustrated through examples.
6.2. Joint Coordinate Formulation for Open-Loop Systems 6.2.1. Joint Coordinates For an open-loop multibody system, that is a system with tree graph structure, the joint coordinates q [ q1 " qk ]T are most natural independent coordinates that describe the system position on its configuration manifold K (see Section 4.3). An individual coordinate qi is related to a particular joint, and describes the relative
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configurations of the adjacent bodies. There are thus one or more joint coordinates in each joint, which can be either rotational or translational, equal in number to the number of relative degrees of freedom. The vector q for a multibody system contains all the joint coordinates (and the absolute coordinates of the base body if it is not the ground), and its dimension is equal to the number k of degrees of freedom of the system.
a)
b)
Figure 6.1. Joint coordinates: a) relative, b) absolute.
Joint coordinates describe, by definition, relative motions of the adjacent bodies in the joints, often referenced to as relative joint coordinates, illustrated in Figure 6.1a. For planar systems, and
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