Abstract
This article discusses one of the solutions proposed in the literature to Gauss’ principle known as principle of ‘least constraint’, proposing a clarifying interpretation that allows significant computational improvements when considering its application to problems with non-diagonal mass matrix. The case of non-symmetric mass matrix is briefly discussed as well.
Highlights
In computational mechanics, the importance of constrained systems modelling is highlighted by its use in many fields of application
When dealing with non-diagonal matrices, the computation of the square root of the matrix can be very expensive, possibly reducing the appeal of the formulation. This may happen, e.g. when modelling rigid and/or flexible multibody systems, using a consistent inertia formulation, or accounting for the rigid-body motion of bodies whose inertia tensor is not isotropic, or whose motion is not referred to the centre of mass
The solution proposed by Udwadia and Kalaba appears interesting because its use of a pseudoinversion allows to transparently handle overconstrained systems, which are characterized by a rank-deficient matrix A
Summary
The importance of constrained systems modelling is highlighted by its use in many fields of application. The problem of formulating the laws that systems of bodies must obey during their motion has been solved many years ago by great scientists and mathematicians like Newton, d’Alembert, Lagrange, and Euler, the very same principles can be viewed from different perspectives and formulated in different, yet equivalent, manners, as pointed out by Gauss himself in reference [1] This is the case, e.g. of Hamilton’s, Gauss’, Jourdain’s, and Gibbs–Appell’s principles, and of Maggi–Kane’s equations, as illustrated in many textbooks and articles on analytical dynamics [2, 3]. When dealing with non-diagonal matrices, the computation of the square root of the matrix can be very expensive, possibly reducing the appeal of the formulation This may happen, e.g. when modelling rigid and/or flexible multibody systems, using a consistent inertia formulation, or accounting for the rigid-body motion of bodies whose inertia tensor is not isotropic, or whose motion is not referred to the centre of mass. Appendix 2 shows how the principle and the proposed solution can be applied to those cases characterized by a non-symmetric inertia matrix, illustrated by practical examples
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