Heavy top constrained-rigid-body example

We provide here a simple application of the Kynema formulation for the heavy-top problem, which is a rotating body fixed to the ground by a spherical joint. It is a common benchmark problem for constrained-rigid-body dynamics and for testing Lie-group time integrators like that used in Kynema. We follow the problem description found in [@Bruls-etal:2012], but with the key difference that we formulate the problem in inertial coordinates rather than material coordinates. We assume the heavy top is a thin disk with mass \(m=15\) kg. The \(6\times6\) mass matrix in material coordinates is

\[\begin{split}\underline{\underline{M}}^* = \begin{bmatrix} 15 \mathrm{~kg}& 0 & 0 & 0 & 0 & 0\\ 0 & 15 \mathrm{~kg} & 0 & 0 & 0 & 0\\ 0 & 0 & 15 \mathrm{~kg} & 0 & 0 & 0\\ 0 & 0 & 0 & 0.234375 \mathrm{~kg~m}^2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0.234375 \mathrm{~kg~m}^2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0.46875 \mathrm{~kg~m}^2 \\ \end{bmatrix}\end{split}\]

The heavy-top center of mass reference position and orientation (see Eq. (5)) are given by

\[\begin{split}\underline{x}^\mathrm{r} = ( 0, 0 , -1 )^T\,, \quad \underline{\underline{R}}^\mathrm{r} = \underline{\underline{I}} \,, \\\end{split}\]

respectively. The only component of external force (see Eq. (6)) is gravity:

\[\underline{f} = [0,0,-g,0,0,0]^T\]

where \(g=9.81\) m/s\(^2\). The problem is constrained such that the center of mass is located 1 m from the origin, which can be written as three constraint equations as

\[\underline{\Phi} = \underline{\underline{R}}\, \underline{x}^\mathrm{r} - \underline{x}^c \in \mathbb{R}^3\]

where \(\underline{\Phi} \in \mathbb{R}^3\), \(\underline{x}^c\) is the current center-of-mass position, and for which the constraint gradient matrix is

\[\underline{\underline{B}} = \begin{bmatrix} -\underline{\underline{I}} & \widetilde{- \underline{\underline{R}}\, \underline{x}^\mathrm{r}} \end{bmatrix}\]

\(\underline{\underline{B}} \in \mathbb{R}^{3 \times 6}\). The stiffness matrix associated with linearization of the constraint forces (see Eq. (3)) is

\[\begin{split}\underline{\underline{K}}^\Phi = \begin{bmatrix} \underline{\underline{0}} & \underline{\underline{0}}\\ \underline{\underline{0}} & \widetilde{\lambda} \, \widetilde{\underline{\underline{R}} \underline{x}^\mathrm{r}} \end{bmatrix}\end{split}\]

where \(\underline{\lambda} \in \mathbb{R}^3\) are the Lagrange multipliers. The Kynema regression test suite includes the spinning, heavy top problem with the following initial conditions:

\[\begin{split}\begin{aligned} \underline{u}^\mathrm{init} &= \left[ 0, 1, 1 \right]^T \, \mathrm{m}\\ \underline{\underline{R}}^\mathrm{init} &= \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos(\theta) & - \sin(\theta) \\ 0 & \sin(\theta) & \cos(\theta) \end{bmatrix}\,, \end{aligned}\end{split}\]

where \(\theta = \pi/2\),

\[\begin{split}\begin{aligned} \omega^\mathrm{init} &= (-4.61538,-150,0)^T \, \mathrm{rad/s}\\ \dot{\underline{u}}^\mathrm{init} &= \widetilde{\omega^\mathrm{init}}\left(\underline{x}^\mathrm{r}+\underline{u}^\mathrm{init}\right)\, \mathrm{m/s} \end{aligned}\end{split}\]

Brüls, O., A. Cardona, and M. Arnold. 2012. “Lie Group Generalized-\(\alpha\) Time Integration For Constrained Flexible Multibody Systems.” Mechanism and Machine Theory, 121–37.