Abstract
This paper presents a formulation that expresses kinematic pairs in form of holonomic constraints as functions of a measure of the relative position and orientation of the connected bodies expressed in vector form. While formulating the relative position measure is straightforward, expressing a suitable measure of the relative orientation requires some care. The problem is addressed by computing the Euler vector of the product of the actual and prescribed relative rotation matrices. By arbitrarily combining the error measures in up to six independent equations, a general family of holonomic rheonomic constraints can be formulated. The relative motion between the bodies can be constrained or specified component-wise, respectively, resulting in scleronomic or rheonomic constraints. The proposed formulation is implemented in a free, general-purpose multibody solver; numerical applications to generic mechanical and aerospace problems are presented.
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Notes
Also known as “Cardan joint,” or “Hooke’s joint” in the Anglo-Saxon literature, since after Girolamo Cardano’s early description in 1557, Robert Hooke, probably in 1667, realized that the transmitted angular velocity was not uniform.
Although parameter c can be loosely interpreted as a compliance, its value should be as small as possible, with a lower limit only based on matrix conditioning considerations.
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Acknowledgements
The implementation in MBDyn was partially sponsored by Hutchinson CdR. Mr. Daniel Benoualid’s support of free multibody software is gratefully acknowledged. The support of Dr. Alessandro Fumagalli in the implementation of the code is also acknowledged.
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Appendix: Perturbation of angular error
Appendix: Perturbation of angular error
The perturbation of the angular error of Eq. (8) needs to be computed at the orientation matrix level, namely
As a consequence, \(\boldsymbol {\epsilon}_{\boldsymbol {\theta}\delta} = \mathbf {R}_{a}^{T}(\boldsymbol {\theta}_{b\delta} - \boldsymbol {\theta}_{a\delta})\). Furthermore, since ϵ θ δ =Γ(ϵ θ )δ ϵ θ ,
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Masarati, P. A formulation of kinematic constraints imposed by kinematic pairs using relative pose in vector form. Multibody Syst Dyn 29, 119–137 (2013). https://doi.org/10.1007/s11044-012-9320-0
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DOI: https://doi.org/10.1007/s11044-012-9320-0