Chains of rigid bodies and their numerical simulation by local frame methods

Nicolai Sætran; Antonella Zanna

doi:10.3934/jcd.2019021

Article Contents

2019, Volume 6, Issue 2: 409-427. Doi: 10.3934/jcd.2019021

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Chains of rigid bodies and their numerical simulation by local frame methods

Nicolai Sætran^1, and
Antonella Zanna^2, ,

1.
Nordre Krabbedalen 17, 5178 Loddefjord, Norway
2.
Matematisk Institutt, Universitetet i Bergen, 5020 Bergen, Norway

^* Corresponding author: Antonella Zanna
^* Corresponding author: Antonella Zanna

Received: February 28, 2019

Revised: August 31, 2019

Published: November 2019

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Abstract

We consider the dynamics and numerical simulation of systems of linked rigid bodies (chains). We describe the system using the moving frame method approach of [18]. In this framework, the dynamics of the $ j $th body is described in a frame relative to the $ (j-1) $th one. Starting from the Lagrangian formulation of the system on $ {{\rm{SO}}}(3)^{N} $, the final dynamic formulation is obtained by variational calculus on Lie groups. The obtained system is solved by using unit quaternions to represent rotations and numerical methods preserving quadratic integrals.

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Mathematics Subject Classification: Primary: 70F10, 70E55, 65P99; Secondary: 65K99, 65L05.

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Figure 1. System of chained rigid bodies

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Figure 2. Single solid pendulum, consisting of a rectangular prism. Left: classical planar motion, described by the angle $ \theta $. Right: planar motion, described in 3D by a quaternion and the MFM. At rest (vertical position), the inertial and frame axes are aligned

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Figure 3. Numerical solution for the angle $ \theta $ for the solid pendulum. The MFM is solved with IMR and stepsize $ h = 0.01 $ and the exact solution is computed with RK45 imposing machine precision on the tolerances

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Figure 4. A simulation of the spinning top, idealized as a rod with a disk (left). Right: position of the center of mass in $ [0,30] $ integrated with step size $ h = 0.01 $. See text for the values of the remaining parameters. The formulation of the equations is identical to that of the single pendulum

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Figure 5. 4-body pendulum with torsional springs

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Figure 6. 16-body

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Figure 7. Conservation of quaternion-unit length (left) and relative energy error (right) for the 16-body pendulum solved with GLRK scheme with a step length $ h = 0.01 $

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Figure 8. Spring loaded/damped chain. The red vector represents $ n $, the unit vector in the direction of the displacement of the spring

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Figure 9. 64-body pendulum

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Table 1. Comparison of the error in unit length (for quaternions), relative energy error and function evaulations at $ T = 10 $ for the RK45 and 2-stages GLRK method for a 3D pendulum with 4 links. See text for more details

	Link 1	Link 2	Link 3	Link 4	Relative energy error	Function evaluations
RK45	5.55e-15	2.47e-14	2.24e-14	2.50e-14	7.22e-14	15794
GL2	2.22e-15	5.55e-16	3.00e-15	1.66e-15	3.00e-13	11334

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Table 2. Computational times for systems with $ N $ links. The computational cost grows quadratically with the number of links. See text for more details

$ N $	4	8	16	32	48	64
Time	719ms$ \pm $4.31ms	2.27s$ \pm $15.8ms	10.7s$ \pm $122ms	45.6$ \pm $1.55s	1m43s$ \pm $1.94s	3m24s$ \pm $5.52s

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