Abstract
This paper presents a recursive direct differentiation method for sensitivity analysis of flexible multibody systems. Large rotations and translations in the system are modeled as rigid body degrees of freedom while the deformation field within each body is approximated by superposition of modal shape functions. The equations of motion for the flexible members are differentiated at body level and the sensitivity information is generated via a recursive divide and conquer scheme. The number of differentiations required in this method is minimal. The method works concurrently with the forward dynamics simulation of the system and requires minimum data storage. The use of divide and conquer framework makes the method linear and logarithmic in complexity for serial and parallel implementation, respectively, and ideally suited for general topologies. The method is applied to a flexible two arm robotic manipulator to calculate sensitivity information and the results are compared with the finite difference approach.
Article PDF
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.Avoid common mistakes on your manuscript.
References
Anderson, K.S., Hsu, Y.H.: Analytic full-recursive sensitivity analysis for multibody systems design. In: Proceedings of the International Congress of Theoretical and Applied Mechanics (ICTAM 2000), Paper-1874. Chicago, IL (2000)
Bestle, D., Eberhard, P.: Analysing and optimizing multibody systems. Struct. Mach. 20, 67–92 (1992)
Haftka, R.T., Guerdal, Z.: Elements of Structural Optimization. Kluwer Academic, Dordrecht (1992)
Anderson, K.S., Hsu, Y.H.: Low operational order analytic sensitivity analysis for tree-type multibody dynamic systems. J. Guid. Control Dyn. 24(6), 1133–1143 (2001)
Haug, E., Ehle, P.H.: Second-order design sensitivity analysis of mechanical system dynamics. Int. J. Numer. Methods Eng. 18, 1699–1717 (1982)
Bestle, D., Seybold, J.: Sensitivity analysis of constrained optimization in dynamic systems. Arch. Appl. Mech. 62, 181–190 (1992)
Eberhard, P.: Analysis and optimization of complex multibody systems using advanced sensitivity methods. Math. Mech. 76, 40–43 (1996)
Haug, E., Wehage, R.A., Mani, N.K.: Design sensitivity analysis of large-scaled constrained dynamic mechanical systems. Trans. ASME 106, 156–162 (1984)
Ding, J.Y., Pan, Z.K., Chen, L.Q.: Second order adjoint sensitivity analysis of multibody systems described by differential-algebraic equations. Multibody Syst. Dyn. 18, 599–617 (2007)
Chang, C.O., Nikravesh, P.E.: Optimal design of mechanical systems with constaint violation stabilization method. J. Mech. Transm. Autom. Des. 107, 493–498 (1985)
Pagalday, J., Aranburu, I., Avello, A., Jalon, J.D.: Multibody dynamics optimization by direct differentiation methods using object oriented programming. In: Proceedings of the IUTAM Symposium on Optimization of Mechanical Systems, pp. 213–220. Stuttgart, Germany (1995)
Etman, L.: Optimization of multibody systems using approximation concepts. Ph.D. thesis, Technische Universiteit Eindhoven (1997)
Bischof, C.H.: On the automatic differentiation of computer programs and an application to multibody systems. In: Proceedings of the IUTAM Symposium on Optimization of Mechanical Systems, pp. 41–48 (1996)
Barthelemy, J.F., Hall, L.: Automatic differentiation as a tool in engineering design. Struct. Optim. 9, 76–82 (1995)
Greene, W., Haftka, R.: Computational aspects of sensitivity calculations in transient structural analysis. Comput. Struct. 32(2), 433–443 (1989)
Tak, T.: A recursive approach to design sensitivity analysis of multibody systems using direct differentiation. Ph.D. thesis, University of Iowa, Iowa City (1990)
Dias, J.M.P., Pereira, M.: Sensitivity analysis of rigid-flexible multibody systems. Multibody Syst. Dyn. 1, 303–322 (1997)
Serban, R., Haug, E.J.: Kinematic and kinetics derivatives for multibody system analyses. Mech. Struct. Mach. 26(2), 145–173 (1998)
Jain, A., Rodrigues, G.: Sensitivity analysis of multibody systems using spatial operators. In: Proceedings of the International Conference on Method and Models in Automation and Robotics (MMAR 2000). Miedzyzdroje, Poland (2000)
Hsu, Y.H., Anderson, K.S.: Efficient direct differentiation sensitivity analysis for general multi-rigid-body systems. In: Proceedings of the Third Symposium on Multibody Dynamics and Vibrations, ASME Design Engineering Technical Conference 2001 (DETC01), DETC2001/VIB-21335. Pittsburgh, PA (2001).
Mukherjee, R.M., Bhalerao, K.D., Anderson, K.S.: A divide-and-conquer direct differentiation approach for multibody system sensitivity analysis. Struct. Multidiscipl. Optim. 35, 413–429 (2007)
Neto, M.A., Ambrósio, J.A.C., Leal, R.P.: Sensitivity analysis of flexible multibody systems using composite materials components. Int. J. Numer. Methods Eng. 77(3), 386–413 (2009)
Mukherjee, R., Anderson, K.S.: A logarithmic complexity divide-and-conquer algorithm for multi-flexible articulated body systems. Comput. Nonlinear Dyn. 2(1), 10–21 (2007)
Shabana, A., Wehage, R.: A coordinate reduction technique for transient analysis of spatial structures with large angular rotations. J. Struct. Mech. 11, 401–431 (1989)
Craig, R.R., Bampton, M.C.: Coupling of substructures for dynamic analyses. AIAA J. 6, 1313–1319 (1968)
Yoo, W., Haug, E.: Dynamics of flexible mechanical systems using vibration and static correction modes. J. Mech. Transm. Autom. Des. 108, 315–322 (1986)
Pereira, M., Proença, P.: Dynamic analysis of spatial flexible multibody systems using joint coordinates. Int. J. Numer. Methods Eng. 32, 1799–1812 (1991)
Nikravesh, P., Lin, Y.S.: Use of principal axes as the floating reference frame for a moving deformable body. Multibody Syst. Dyn. 13(2), 211–231 (2005)
Lehner, M., Eberhard, P.: On the use of moment-matching to build reduced order models in flexible multibody dynamics. Multibody Syst. Dyn. 16(2), 191–211 (2006)
Featherstone, R.: A divide-and-conquer articulated body algorithm for parallel O(log (n)) calculation of rigid body dynamics. Part 1: Basic algorithm. Int. J. Robot. Res. 18(9), 867–875 (1999)
Featherstone, R.: A divide-and-conquer articulated body algorithm for parallel O(log (n)) calculation of rigid body dynamics. Part 2: Trees, loops, and accuracy. Int. J. Robot. Res. 18(9), 876–892 (1999)
Mukherjee, R., Anderson, K.S.: An orthogonal complement based divide-and-conquer algorithm for constrained multibody systems. Nonlinear Dyn. 48(1-2), 199–215 (2007)
Schwertassek, R., Wallrapp, O., Shabana, A.: Flexible multibody simulation and choice of shape functions. Nonlinear Dyn. 20(4), 361–380 (1999)
Berzeri, M., Shabana, A.: Development of simple models for the elastic forces in the absolute nodal co-ordinate formulation. J. Sound Vib. 235(4), 539–565 (2000)
Kane, T.R., Levinson, D.A.: Dynamics: Theory and Application. McGraw-Hill, New York (1985)
Botz, M., Hagedorn, P.: Dynamic simulation of multibody systems including planar elastic beams using autolev. Eng. Comput. 14(4), 456–479 (1997)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bhalerao, K.D., Poursina, M. & Anderson, K.S. An efficient direct differentiation approach for sensitivity analysis of flexible multibody systems. Multibody Syst Dyn 23, 121–140 (2010). https://doi.org/10.1007/s11044-009-9176-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11044-009-9176-0