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Efficient and accurate modeling of rigid rods


Ten years ago, an original semi-recursive formulation for the dynamic simulation of large-scale multibody systems was presented by García de Jalón et al. (Advances in Computational Multibody Systems, pp. 1–23, 2005). By taking advantage of the cut-joint and rod-removal techniques through a double-step velocity transformation, this formulation proved to be remarkably efficient. The rod-removal technique was employed, primarily, to reduce the number of differential and constraint equations. As a result, inertia and external forces were applied to neighboring bodies. Those inertia forces depended on unknown accelerations, a fact that contributed to the complexity of the system inertia matrix. In search of performance improvement, this paper presents an approximation of rod-related inertia forces by using accelerations from previous time-steps. Additionally, a mass matrix partition is carried out to preserve the accuracy of the original formulation. Three extrapolation methods, namely, point, linear Lagrange and quadratic Lagrange extrapolation methods, are introduced to evaluate the unknown rod-related inertia forces. In order to assess the computational efficiency and solution accuracy of the presented approach, a general-purpose MATLAB/C/C++ simulation code is implemented. A 15-DOF, 12-rod sedan vehicle model with MacPherson strut and multi-link suspension systems is modeled, simulated and analyzed.

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  1. Cartesian coordinates of points and Cartesian components of unit vectors [9, 11, 13].


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Financial support of the first author from the CSC research fellowship is acknowledged, as well as funding from the Ministry of Science and Innovation of Spain under Research Projects OPTIVIRTEST (TRA2009-14513-C02-01) and DOPTCARR (TRA2012-38826-C02-01).

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Correspondence to Yongjun Pan.

Appendix: Vehicle model details

Appendix: Vehicle model details

Some additional details about the vehicle model depicted in Figs. 1 and 8 are gathered here. The number of bodies, joints, constraints and coordinates is shown in Table 7. The DOF count is detailed in Table 8. Finally, detailed descriptions of rod elements and cut joints are presented in Tables 9 and 10, respectively.

Table 7 Element count
Table 8 Degree of freedom count
Table 9 Rods
Table 10 Cut joints

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Pan, Y., Callejo, A., Bueno, J.L. et al. Efficient and accurate modeling of rigid rods. Multibody Syst Dyn 40, 23–42 (2017).

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  • Multibody dynamics
  • Semi-recursive Maggi’s formulation
  • Rod-removal technique
  • Polynomial extrapolation
  • Vehicle dynamics