Abstract
The modeling of multi-dof systems calls for various concepts that were not required when studying single-dof systems. For example, the concept ofdegree of freedom, to begin with, plays a major role here. Furthermore, the concept ofvector of generalized coordinates, and the correspondingvector of generalized speeds arise only in multi-dof systems. Associated with these vectors, we have the vector of generalized force, while the conceptsof generalized mass matrix,generalized damping matrix andgeneralized stiffness matrix enter in this context as natural extensions of their scalar counterparts encountered in single-dof systems. We shall introduce these concepts via thevector form of the Lagrange governing equations. One more concept that pertains to multi-dof systems is that ofrigid modes, namely, non-trivial motions under which the potential energy of the whole system does not undergo any change.
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Notes
- 1.
In the case at hand, the trace of M has a physical meaning, as the diagonal entries of M bear the same units, those of moment of inertia. Depending on the choice of the generalized coordinates, these entries may bear distinct units, in which case the trace is meaningless. In such cases, the eigenvalues must be computed to prove positive-definiteness.
- 2.
Notice the recursive nature of these relations: the second is based on the first, as the third on the second. Recursion is at the core of computational algorithms in multibody system dynamics.
- 3.
As opposed to differential.
- 4.
Taken, with some modifications, fromCannon [2].
References
Angeles J, Espinosa I (1981) Suspension-system synthesis for mass transport vehicles with prescribed dynamic behavior, ASME Paper 81-DET-44. Proc. 1981 ASME Design Engineering Technical Conference, Hartford, 20–23 Sept 1981
Cannon RH (1967) Dynamics of physical systems, McGraw-Hill Book Co., New York
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Angeles, J. (2011). Modeling of Multi-dof Mechanical Systems. In: Dynamic Response of Linear Mechanical Systems. Mechanical Engineering Series. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-1027-1_4
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DOI: https://doi.org/10.1007/978-1-4419-1027-1_4
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