# Efficient Solving Methods Exploiting Sparsity of Matrix in Real-Time Multibody Dynamic Simulation with Relative Coordinate Formulation

• Choi, Gyoojae (Korea Automotive Technology Institute(KATECH)) ;
• Yoo, Yungmyun (Korea Automotive Technology Institute(KATECH)) ;
• Im, Jongsoon (Korea Automotive Technology Institute(KATECH))
• Published : 2001.08.01

#### Abstract

In this paper, new methods for efficiently solving linear acceleration equations of multibody dynamic simulation exploiting sparsity for real-time simulation are presented. The coefficient matrix of the equations tends to have a large number of zero entries according to the relative joint coordinate numbering. By adequate joint coordinate numbering, the matrix has minimum off-diagonal terms and a block pattern of non-zero entries and can be solved efficiently. The proposed methods, using sparse Cholesky method and recursive block mass matrix method, take advantages of both the special structure and the sparsity of the coefficient matrix to reduce computation time. The first method solves the η$\times$η sparse coefficient matrix for the accelerations, where η denotes the number of relative coordinates. In the second method, for vehicle dynamic simulation, simple manipulations bring the original problem of dimension η$\times$η to an equivalent problem of dimension 6$\times$6 to be solved for the accelerations of a vehicle chassis. For vehicle dynamic simulation, the proposed solution methods are proved to be more efficient than the classical approaches using reduced Lagrangian multiplier method. With the methods computation time for real-time vehicle dynamic simulation can be reduced up to 14 per cent compared to the classical approach.

#### References

1. Amirouche, F. M. L, 1992, Computational Methods in Multibody Dynamics, Prenticwe-Hall International, Inc
2. Bae, D. S., Lee, J. K. Cho, H. J. and Yae, H., 2000, 'An Explicit Integration Method for Realtime Simulation of Multibody Vehicle Models,' Computational Methods in Applied Mechanics and Engineering, Vol. 187, pp. 337-350 https://doi.org/10.1016/S0045-7825(99)00138-3
3. Besinger, F. H., Cebon, D. and Cole, D. J., 1995, 'Force Control of a Semi-Active Damper,' Vehicle System Dynamics, Voll. 24, pp. 695-723 https://doi.org/10.1080/00423119508969115
4. Choi, G. J., Yoo, Y. M., Lee, K. P., Yoon, Y. S., 2000, 'A Real-Time Multibody Vehicle Dynamic Analysis Method Using Suspension Composite Joints,' International Journal of Vehicle Design, Vol. 24, pp. 259-273 https://doi.org/10.1504/IJVD.2000.005184
5. Cuadrado, J., Cardenal, J. and Bayo, E., 1997, 'Modeling and Solution Methods for Efficient Real-Time Simulation of Multibody Dynamics,' Multibody System Dynamics, Vol. 1, pp. 259-280 https://doi.org/10.1023/A:1009754006096
6. Jennings, A., 1977, Matrix Computation for Engineers and Scientists, John Wiley & Sons
7. Lee, K. P. and Yoon, Y. S., 1998, 'Efficiency Comparision of Multibody Dynamic Analysis Algorithms for Real-Time Simulation,' JSME International Journal, Series C, Vol. 41, pp. 813-821
8. Serban, R., Negrut, D., Haug, E. J. and Potra, F. A. 1997, 'A Topology-Based Approach for Exploiting Sparsity in Multibody Dynamics in Cartesian Formulation,' Mechanics of Structures and Machines, Vol. 25, pp. 379-396 https://doi.org/10.1080/08905459708905295
9. Tewarson, R. P., 1972, Sparse Matrices,NewYork : Academic Press
10. Wang, J. T. and Huston, R. L., 1987, 'kane's Equations with Undetermined Multipliers-Ap-Proach to Constrained Multibody Systems,' ASME Journal of Applied Mechanics, Vol. 54, pp. 424-429
11. Wehage, R. A. and Haug, E. J., 1982, 'Generalized Coordinate Partitioning of Dimension Reduction in Analysis of Constrained Dynamic Systems,' ASME Journal of Mechanical Design, Vol. 104, pp. 247-255