Modal Analysis of Multi-Degrees-of-Freedom Systems with Singular Matrices: Analytical Dynamics Approach
Publication: Journal of Engineering Mechanics
Volume 143, Issue 6
Abstract
Complex mechanical (e.g., multibody) systems with different types of constraints are generally performed through analytical dynamics methods. In some cases, however, it is possible that the (augmented) mass and/or stiffness matrices may derive to be singular; consequently, modal analysis, which is used extensively in the classical dynamics literature, fails. In this paper, if the uniqueness condition is satisfied by the constraints, a properly modified modal analysis is elucidated into analytical dynamics leading to the evaluation of the natural frequencies in a simple and straightforward way. Under that framework, advances of both classical and analytical dynamics are taken into consideration for evaluating the structural response.
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Acknowledgments
The authors are extremely grateful to the anonymous two reviewers and the associate editor who have afforded them considerable assistance in enhancing both the quality of the findings and the clarity of their presentation. They would like to gratefully acknowledge the support of this work by the EPSRC and ESRC Centre for Doctoral Training on Quantification and Management of Risk and Uncertainty in Complex Systems and Environments (EP/L015927/1).
References
Antoniou, E. N., Pantelous, A. A., Kougioumtzoglou, I. A., and Pirrotta, A. (2017). “Response determination of linear dynamical systems with singular matrices: A polynomial matrix theory approach.” Appl. Math. Model., 42, 423–440.
Ardema, M. D. (2005). Analytical dynamics: Theory and applications, Springer, New York.
Bae, D. S., and Haug, E. J. (1987). “A recursive formulation for constraint mechanical system dynamics. Part II: Closed loop systems.” Mech. Struct. Mach., 15(4), 481–506.
Ben-Israel, A., and Greville, T. N. E. (2003). Generalized inverses: Theory and applications, 2nd Ed., Springer, New York.
Campbell, S. L., and Meyer, C. D., Jr. (1979). Generalized inverses of linear transformations, Dover Publications, New York.
Critchley, J. H., and Anderson, K. S. (2003). “A generalized recursive coordinate reduction method for multi-body dynamic systems.” Int. J. Multiscale Comp. Eng., 1(2–3), 20–200.
De Falco, D., Pennestrì, E., and Vita, L. (2009). “Investigation of the influence of pseudoinverse matrix calculations on multibody dynamics simulations by means of the Udwadia-Kalaba formulation.” J. Aerosp. Eng., 365–372.
Di Matteo, A., Lo Iacono, F., Navarra, G., and Pirrotta, A. (2014a). “Direct evaluation of the equivalent linear damping for TLCD systems in random vibration for pre-design purposes.” Int. J. Non Linear Mech., 63, 19–30.
Di Matteo, A., Lo Iacono, F., Navarra, G., and Pirrotta, A. (2014b). “Experimental validation of a direct pre-design formula for TLCD.” Eng. Struct., 75, 528–538.
Featherstone, R. (1987). Robot dynamics algorithms, Kluwer, New York.
Fragkoulis, V. C., Kougioumtzoglou, I. A., and Pantelous, A. A. (2016). “Linear random vibration of structural systems with singular matrices.” J. Eng. Mech., .
Fragkoulis, V. C., Kougioumtzoglou, I. A., Pantelous, A. A., and Pirrotta, A. (2015). “Higher order matrix differential equations with singular coefficient matrices.” Proc., Int. Conf. on Numerical Analysis and Applied Mathematics (ICNAAM-2014), AIP Publishing, Melville, NY.
Garcia de Jalón, J., Guetiérrez-López, M. D. (2013). “Multibody dynamics with redundant constraints and singular mass matrix: Existence, uniqueness, and determination of solutions for accelerations and constraint forces.” Multibody Syst. Dyn., 30(3), 311–341.
Gashi, B., and Pantelous, A. A. (2013). “Linear backward stochastic differential equations of descriptor type: Regular systems.” Stoch. Anal. Appl., 31(1), 142–166.
Gashi, B., and Pantelous, A. A. (2015). “Linear backward stochastic differential systems of descriptor type with structure and applications to engineering.” Prob. Eng. Mech., 40, 1–11.
Gauss, C. F. (1829). “Uber ein neues allgemeines Grundgsetz der Mechanik.” J Reine. Angew. Math., 1829(4), 232–235 (in German).
Gohberg, I., Lancaster, P., and Rodman, L. (1986). Invariant subspaces of matrices with applications: classics in applied mathematics, 2nd Ed., Vol. 51, Society for Industrial and Applied Mathematics, New York.
Greville, T. N. E. (1960). “Some applications of the pseudoinverse of a matrix.” SIAM Rev., 2(1), 15–22.
Kalogeropoulos, G. I., Karageorgos, A. D., and Pantelous, A. A. (2014). “On the solution of higher order linear homogeneous complex descriptor matrix differential systems of Apostol-Kolodner type.” J. Frank. Inst., 351(3), 1756–1777.
Kalogeropoulos, G. I., and Pantelous, A. A. (2008). “On generalized regular stochastic differential delay systems with time invariant coefficients.” Stoch. Anal. Appl., 26(5), 1076–1094.
Lagrange, J. L. (1787). Mecanique analytique, Mme Ve Courcier, Paris (in French).
Laulusa, A., and Bauchau, O. A. (2007). “Review of classical approaches for constraint enforcement in multibody systems ASME.” J. Comput. Nonlinear Dynam., 3(1), 011004–011004-8.
Mariti, L., Belfiore, N. P., Pennestrì, E., and Valentini, P. P. (2011). “Comparison of solution strategies for multibody dynamics equations.” Int. J. Numer. Meth. Eng., 88(7), 637–656.
Pars, L. A. (1979). A treatise on analytical dynamics, Ox Bow Press, Woodridge, IL.
Pradhan, S., Modi, V. J., and Misra, A. K. (1997). “Order N formulation for flexible multi-body systems in tree topology: Lagrangian approach.” J. Guid. Control Dyn., 20(4), 665–672.
Roberts, J. B., and Spanos, P. D. (2003). Random vibration and statistical linearization, Dover Publications, New York.
Schiehlen, W. O. (1984). “Dynamics of complex multibody systems.” SM Arch., 9(2), 159–195.
Schutte, A. D., and Udwadia, F. E. (2011). “New approach to the modeling of complex multi-body dynamical systems.” ASME J. Appl. Mech., 78(2), 021018.
Udwadia, F. E., and Di Massa, G. (2011). “Sphere rolling on a moving surface: Application of the fundamental equation of constrained motion.” Simul. Modell. Pract. Theory, 19(4), 1118–1138.
Udwadia, F. E., and Kalaba, R. E. (1992). “A new perspective on constrained motion.” Proc. R. Soc. A, 439(1906), 407–410.
Udwadia, F. E., and Kalaba, R. E. (2001). “Explicit equations of motion for mechanical systems with nonideal constraints.” ASME J. Appl. Mech., 68(3), 462–467.
Udwadia, F. E., and Kalaba, R. E. (2002). “On the foundations of analytical dynamics.” Int. J. Nonlin. Mech., 37(6), 1079–1090.
Udwadia, F. E., and Kalaba, R. E. (2007). Analytical dynamics: A new approach, Cambridge University Press, New York.
Udwadia, F. E., Kalaba, R. E., and Hee-Chang, E. (1997). “Equation of motion for constrained mechanical systems and the extended D’Alembert’s principle.” Q. Appl. Math. LV, 55(2), 321–331.
Udwadia, F. E., and Phohomsiri, P. (2006). “Explicit equations of motion for constrained mechanical systems with singular mass matrices and applications to multi-body dynamics.” Proc. R. Soc. A, 462(2071), 2097–2117.
Udwadia, F. E., and Schutte, A. D. (2010). “Equations of motion for general constrained systems in Lagrangian mechanics.” Acta Mech., 213(1), 111–129.
Udwadia, F. E., and Wanichanon, T. (2012). Explicit equations of motion of constrained systems with applications to multi-body dynamics, Springer, New York, 315–348.
Udwadia, F. E., and Wanichanon, T. (2013). “On general nonlinear constrained mechanical systems.” Numer. Algebra, Contr. Optim., 3(3), 425–443.
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Published In
Journal of Engineering Mechanics
Volume 143 • Issue 6 • June 2017
Copyright
©2017 American Society of Civil Engineers.
History
Received: Jan 14, 2016
Accepted: Nov 9, 2016
Published online: Feb 16, 2017
Published in print: Jun 1, 2017
Discussion open until: Jul 16, 2017
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