Statistical Linearization of Nonlinear Structural Systems with Singular Matrices

AbstractA generalized statistical linearization technique is developed for determining approximately the stochastic response of nonlinear dynamic systems with singular matrices. This system modeling can arise when a greater than the minimum number of coordinates is utilized, and can be advantageous, for instance, in cases of complex multibody systems where the explicit formulation of the equations of motion can be a nontrivial task. In such cases, the introduction of additional/redundant degrees of freedom can facilitate the formulation of the equations of motion in a less labor-intensive manner. Specifically, relying on the generalized matrix inverse theory and on the Moore-Penrose (M-P) matrix inverse, a family of optimal and response-dependent equivalent linear matrices is derived. This set of equations in conjunction with a generalized excitation-response relationship for linear systems leads to an iterative determination of the system response mean vector and covariance matrix. Further, it is proved ...

[1]  P. Spanos,et al.  An Efficient Wiener Path Integral Technique Formulation for Stochastic Response Determination of Nonlinear MDOF Systems , 2015 .

[2]  C. D. Meyer,et al.  Generalized inverses of linear transformations , 1979 .

[3]  Firdaus E. Udwadia,et al.  Equations of motion for general constrained systems in Lagrangian mechanics , 2010 .

[4]  O. Bauchau,et al.  Review of Classical Approaches for Constraint Enforcement in Multibody Systems , 2008 .

[5]  Jianbing Chen,et al.  Stochastic Dynamics of Structures , 2009 .

[6]  Yu-Kweng Michael Lin.  Probabilistic Theory of Structural Dynamics , 1976 .

[7]  Athanasios A. Pantelous,et al.  On the solution of higher order linear homogeneous complex σ-α descriptor matrix differential systems of Apostol-Kolodner type , 2014, J. Frankl. Inst..

[8]  Athanasios A. Pantelous,et al.  Linear Random Vibration of Structural Systems with Singular Matrices , 2016 .

[9]  M. Grigoriu.  Stochastic Calculus: Applications in Science and Engineering , 2002 .

[10]  A. Pirrotta,et al.  Probabilistic response of nonlinear systems under combined normal and Poisson white noise via path integral method , 2011 .

[11]  E. Haug,et al.  A Recursive Formulation for Constrained Mechanical System Dynamics: Part II. Closed Loop Systems , 1987 .

[12]  Roy Featherstone,et al.  Robot Dynamics Algorithms , 1987 .

[13]  T. S. Koko,et al.  Finite element analysis and design of actively controlled piezoelectric smart structures , 2004 .

[14]  Parviz E. Nikravesh,et al.  Euler Parameters in Computational Kinematics and Dynamics. Part 1 , 1985 .

[15]  P. Spanos,et al.  Random vibration and statistical linearization , 1990 .

[16]  R. Kalaba,et al.  Analytical Dynamics: A New Approach , 1996 .

[17]  Firdaus E. Udwadia,et al.  New Approach to the Modeling of Complex Multibody Dynamical Systems , 2011 .

[18]  Firdaus E. Udwadia,et al.  ON GENERAL NONLINEAR CONSTRAINED MECHANICAL SYSTEMS , 2013 .

[19]  Leopold Alexander Pars,et al.  A Treatise on Analytical Dynamics , 1981 .

[20]  R. J. Peppin.  An Introduction to Random Vibrations, Spectral and Wavelet Analysis , 1994 .

[21]  Wilfred D. Iwan,et al.  On the existence and uniqueness of solutions generated by equivalent linearization , 1978 .

[22]  Athanasios A. Pantelous,et al.  Linear backward stochastic differential systems of descriptor type with structure and applications to engineering , 2015 .

[23]  S. Pradhan,et al.  Order N Formulation for Flexible Multibody Systems in Tree Topology: Lagrangian Approach , 1997 .

[24]  A. Naess,et al.  Response statistics of nonlinear, compliant offshore structures by the path integral solution method , 1993 .

[25]  K. Anderson,et al.  A Generalized Recursive Coordinate Reduction Method for Multibody System Dynamics , 2003 .

[26]  Pol D. Spanos,et al.  Nonstationary Stochastic Response Determination of Nonlinear Systems: A Wiener Path Integral Formalism , 2014 .

[27]  Leslaw Socha.  Linearization Methods for Stochastic Dynamic Systems , 2008 .

[28]  Ettore Pennestrì,et al.  Comparison of solution strategies for multibody dynamics equations , 2011 .

[29]  T. Greville,et al.  Some Applications of the Pseudoinverse of a Matrix , 1960 .

[30]  B Karimpoor,et al.  ACTIVE VIBRATION CONTROL OF FRAME SMART STRUCTURES , 2016 .

[31]  W. Schiehlen.  Dynamics of complex multibody systems , 1984 .

[32]  Phailaung Phohomsiri,et al.  Explicit equations of motion for constrained mechanical systems with singular mass matrices and applications to multi-body dynamics , 2006, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[33]  L. Vita,et al.  Investigation of the influence of pseudoinverse matrix calculations on multibody dynamics simulations by means of the udwadia-kalaba formulation , 2009 .

[34]  E. Haug,et al.  A recursive formulation constrained mechanical system dynamics. I: Open loop systems , 1987 .

[35]  Matthias Morzfeld,et al.  The Decoupling of Second-Order Linear Systems with a Singular Mass Matrix , 2013 .