Application of the Bezier integration technique with enhanced stability in forward dynamics of constrained multibody systems with Baumgarte stabilization method

Abstract

This paper deals with the study and application of a numerical integration scheme based on the Bézier curves to obtain stable and accurate solutions of the governing differential–algebraic equations (DAEs) of constrained multibody systems. For this purpose, the standard Lagrange multiplier method is utilized to derive the equations of motion for constrained mechanical systems. It is well known that constraints violation and instability are amongst the main computational difficulties of various numerical integration algorithms to provide accurate solutions for DAEs. In the present study, the Baumgarte stabilization technique is employed, as well as a criterion to select the Baumgarte parameters based on the stability domain analysis. Stability regions based on dimensionless Baumgarte parameters are obtained for different orders of Bézier curves together with other classic numerical integration algorithms, namely the Adams–Bashforth and Runge–Kutta techniques. It is shown that the proper identification and selection of Baumgarte parameters is highly dependent on the integration scheme utilized. A comparative analysis performed within the present work reveals that the Bézier method provides substantially wider region of stability for the same level of computational effort. A planar slider-crank mechanism is considered as an example of application to demonstrate and implement the Bézier technique, which permits to examine the benefits of the presented stability analysis. It is demonstrated that for certain values of Baumgarte parameters, the Bézier method remains stable, while other methods become unstable. Due to the good performance of the applied method based on the Bézier approach, both in terms of efficiency and stability, it is expected that the introduced integration method to be used for various benchmark studies performed under the umbrella of multibody dynamics.

Appendix

For the slider-crank mechanism mentioned in Sect. 5, the mass matrix can be defined as

$$\mathbf{M}=\mathrm{diag}\left(\begin{array}{ccccc}{J}_{1}& {m}_{2}& {m}_{2}& {J}_{2}& {m}_{3}\end{array}\right)$$

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where ${J}_{1}$ and ${J}_{2}$ represent the mass moment of inertia for bodies 1 and 2, respectively, and ${m}_{2}$ and ${m}_{3}$ indicate masses of bodies 2 and 3. The corresponding constraint Jacobian matrix, vector ${\varvec{\upgamma}}$ and vector $\mathbf{g}$ are expressed as

$${\mathbf{C}}_{\mathbf{q}}=\left[\begin{array}{ccccc}-{l}_{1}\mathrm{sin}{\theta }_{1}& -1& 0& -\frac{{l}_{2}}{2}\mathrm{sin}{\theta }_{2}& 0\\ {l}_{1}\mathrm{cos}{\theta }_{1}& 0& -1& \frac{{l}_{2}}{2}\mathrm{cos}{\theta }_{2}& 0\\ 0& 1& 0& -\frac{{l}_{2}}{2}\mathrm{sin}{\theta }_{2}& -1\\ 0& 0& 1& \frac{{l}_{2}}{2}\mathrm{cos}{\theta }_{2}& 0\end{array}\right]$$

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$${\varvec{\upgamma}}=\left[\begin{array}{c}{l}_{1}\mathrm{cos}{\theta }_{1}{\dot{{\theta }_{1}}}^{2}+\frac{{l}_{2}}{2}\mathrm{cos}{\theta }_{2}{\dot{{\theta }_{2}}}^{2}\\ {l}_{1}\mathrm{sin}{\theta }_{1}{\dot{{\theta }_{1}}}^{2}+\frac{{l}_{2}}{2}\mathrm{sin}{\theta }_{2}{\dot{{\theta }_{2}}}^{2}\\ \frac{{l}_{2}}{2}\mathrm{cos}{\theta }_{2}{\dot{{\theta }_{2}}}^{2}\\ \frac{{l}_{2}}{2}\mathrm{sin}{\theta }_{2}{\dot{{\theta }_{2}}}^{2}\end{array}\right]$$

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$$\mathbf{g}={\left[\begin{array}{ccccc}0& 0& {m}_{2}g& 0& 0\end{array}\right]}^{\mathrm{T}}$$

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