A Direct Eigenanalysis of Multibody System in Equilibrium

This paper presents a direct eigenanalysis procedure for multibody system in equilibrium. The first kind Lagrange’s equation of the dynamics of multibody system is a set of differential algebraic equations, and the equations can be used to solve the equilibrium of the system. The vibration of the system about the equilibrium can be described by the linearization of the governing equation with the generalized coordinates and the multipliers as the perturbed variables. But the multiplier variables and the generalize coordinates are not in the same dimension. As a result, the system matrices in the perturbed vibration equations are badly conditioned, and a direct application of the mature eigensolvers does not guarantee a correct solution to the corresponding eigenvalue problem. This paper discusses the condition number of the problem and proposes a method for preconditioning the system matrices, then the corresponding eigenvalue problem of the multibody system about equilibrium can be smoothly solved with standard eigensolver such as ARPACK. In addition, a necessary frequency shift technology is also presented in the paper. The importance of matrix conditioning and the effectiveness of the presented method for preconditioning are demonstrated with numerical examples.

[1]  O. Schenk,et al.  ON FAST FACTORIZATION PIVOTING METHODS FOR SPARSE SYMMETRI C INDEFINITE SYSTEMS , 2006 .

[2]  Michael Beitelschmidt,et al.  Comparison of model reduction techniques for large mechanical systems , 2008 .

[3]  D. Sorensen.  Numerical methods for large eigenvalue problems , 2002, Acta Numerica.

[4]  Pierangelo Masarati.  Direct eigenanalysis of constrained system dynamics , 2009 .

[5]  Ahmed A. Shabana,et al.  Dynamics of Multibody Systems , 2020 .

[6]  Olivier A. Bauchau,et al.  Scaling of Constraints and Augmented Lagrangian Formulations in Multibody Dynamics Simulations , 2009 .

[7]  Olaf Schenk,et al.  Solving unsymmetric sparse systems of linear equations with PARDISO , 2004, Future Gener. Comput. Syst..

[8]  Ahmed A. Shabana,et al.  Flexible Multibody Dynamics: Review of Past and Recent Developments , 1997 .

[9]  Danny C. Sorensen,et al.  Deflation Techniques for an Implicitly Restarted Arnoldi Iteration , 1996, SIAM J. Matrix Anal. Appl..

[10]  M. Géradin,et al.  Flexible Multibody Dynamics: A Finite Element Approach , 2001 .

[11]  Pierangelo Masarati,et al.  Reduced Order Models Using Generalized Eigenanalysis , 2011 .

[12]  V. Cossalter,et al.  The Modal Analysis of a Motorcycle in Straight Running and on a Curve , 2004 .

[13]  D. C. Lewis.  Review: Cornelius Lanczos, The variational principles of mechanics , 1951 .

[14]  C. Lanczos.  The variational principles of mechanics , 1949 .

[15]  Søren Nielsen,et al.  A component mode synthesis algorithm for multibody dynamics of wind turbines , 2009 .

[16]  Daniel Dopico,et al.  On the optimal scaling of index three DAEs in multibody dynamics , 2008 .

[17]  H. Saberi Najafi,et al.  A new restarting method in the Arnoldi algorithm for computing the eigenvalues of a nonsymmetric matrix , 2004, Appl. Math. Comput..

[18]  Rosario Chamorro,et al.  Stability analysis of vehicles on circular motions using multibody dynamics , 2008 .

[19]  Giuseppe Quaranta,et al.  Seismic Attenuation System Synthesis by Reduced Order Models from Multibody Analysis , 2007 .

[20]  Yimin Zhang,et al.  Eigenvalue problem of constrained flexible multibody systems , 1997 .