Abstract
We present the new, general, explicit form of the equations of motion for constrained mechanical systems applicable to systems with singular mass matrices. The systems may have holonomic and/or non-holonomic constraints, which may or may not satisfy D'Alembert's principle at each instant of time. The equation provides new insights into the behaviour of constrained motion and opens up new ways of modelling complex multi-body systems. Examples are provided and applications of the equation to such systems are illustrated.
Highlights
One of the central problems in analytical dynamics is the determination of equations of motion for constrained mechanical systems
We present the necessary and sufficient conditions for the equations of motions to uniquely determine the accelerations of the system. We show that these general equations are identical to the familiar, explicit equations previously obtained by Udwadia & Kalaba (2001) when the mass matrices are restricted to being positive definite
The general equation of motion of a constrained mechanical system described by relations (2.1) (2.3), whether or not the matrix M that arises in the description of the unconstrained motion of the system is singular, is given by q€ Z MC
Summary
One of the central problems in analytical dynamics is the determination of equations of motion for constrained mechanical systems. Udwadia & Kalaba (1992) discovered a simple, explicit set of equations of motion for general constrained mechanical systems Their equations can deal with holonomic and/ or non-holonomic constraints that are not necessarily independent. As known (Pars 1979), when the minimum number of coordinates is employed for describing the (unconstrained) motion of mechanical systems, the corresponding set of Lagrange equations usually yield mass matrices that are non-singular; they are symmetric, and positive definite (Pars 1979). Singular mass matrices can, and do, arise in the modelling of complex, multi-body mechanical systems Such occurrences are most frequent when describing mechanical systems with more than the minimum number of required generalized coordinates, so that the coordinates are not independent of one another, and are subjected to constraints.
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