MBD_demo

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Closed-loop Pendulum smaple code by 3D Multi-Body Dynamics

Overview

The Closed-loop Pendulum demo which is modeled by Quaternion based Multi-Body Dynamics.

The following dynamics for multi-bodies are solved ¹.

$$ \left[ \begin{array}{cccc} {\bf M_G} & {\bf 0} & \partial_{{\bf r_G}} {\bf C}^T & {\bf 0} \\ {\bf 0} & 4{\bf L_G}^T {\bf J_G}' {\bf L_G} & \partial_{\bf \varepsilon} {\bf C}^T & \partial_{\bf \varepsilon} {\bf C}^T_E \\ \partial_{{\bf r_G}} {\bf C} & \partial_{\bf \varepsilon} {\bf C} & {\bf 0} & {\bf 0} \\ {\bf 0} & \partial_{\bf \varepsilon} {\bf C}_E & {\bf 0} & {\bf 0} \\ \end{array} \right] \left[ \begin{array}{c} \ddot{\bf r}_G \\ \ddot{\bf \varepsilon} \\ {\bf \lambda} \\ {\bf \lambda}_E \\ \end{array} \right] =\left[ \begin{array}{c} {\bf F}_G \\ 8d_T {\bf L}^T_G {\bf J}'_G d_T {\bf L}_G {\bf \varepsilon} + 2 {\bf L}^T_G {\bf n}'_G \\ {\bf \gamma} \\ {\bf \gamma}_E \\ \end{array} \right], $$

where

$$ {\bf C} ({\bf r_G}, {\bf \varepsilon}) = {\bf 0}, $$

is the geometrical constraint condition, and

$$ {\bf C}_E ({\bf \varepsilon}) := \left[ \begin{array}{c} {\bf \varepsilon}^T_1 {\bf \varepsilon}_1 - 1 \\ \vdots \\ {\bf \varepsilon}^T_N {\bf \varepsilon}_N - 1 \\ \end{array} \right] = {\bf 0}, $$

is constraint condition for Euler parameter.

Baumgarte constraint stabilization method ² is employed to above Differential-algebraic system of equations (DAE).

Preparation before analysis

[Step 1] Install the ToolBoxes

The following ToolBoxes in “./ ToolBoxes/” are required,

“mmwrite” by Micah Richert: https://jp.mathworks.com/matlabcentral/fileexchange/15881-mmwrite

[Step 1.2] Add path to installed ToolBoxes

Modify "add_pathes.m".

[Step 2] Start GUI form

Open the “GUI.fig” from MATLAB.

[Step 2.1] Pre-setting

Push the "Parameters" buttun and edit parameters.

[Step 3] Start analysis

Push the “exe” button and wait until the finish of the analysis.

[Step 4] Plot results

Push the “plot” button.

Image

Time series of energy (conserved)

Error of constraint conditions

Movie

My.Movie.mp4

Chaotic behevior is generated because the masses of the left and right rigid bodies are different.

References

日本機械学会編，マルチボディダイナミクス〈1〉基礎理論，コロナ社． ↩
J. Baumgarte，“Stabilization of Constraints and Integrals of Motion in Dynamical Systems”，Computer Methods in Applied Mechanics and Engineering，pp.1–16，1972. ↩