### Simulating Nonlinear Oscillations of Viscoelastically Damped Mechanical Systems

#### Abstract

The aim of this work is to propose a mathematical model in terms of an exact analytical solution that may be used in numerical simulation and prediction of oscillatory dynamics of a one-dimensional viscoelastic system experiencing large deformations response. The model is represented with the use of a mechanical oscillator consisting of an inertial body attached to a nonlinear viscoelastic spring. As a result, a second-order first-degree Painlevé equation has been obtained as a law, governing the nonlinear oscillatory dynamics of the viscoelastic system. Analytical resolution of the evolution equation predicts the existence of three solutions and hence three damping modes of free vibration well known in dynamics of viscoelastically damped oscillating systems. Following the specific values of damping strength, over-damped, critically-damped and under-damped solutions have been obtained. It is observed that the rate of decay is not only governed by the damping degree but, also by the magnitude of the stiffness nonlinearity controlling parameter. Computational simulations demonstrated that numerical solutions match analytical results very well. It is found that the developed mathematical model includes a nonlinear extension of the classical damped linear harmonic oscillator and incorporates the Lambert nonlinear oscillatory equation with well-known solutions as special case. Finally, the three damped responses of the current mathematical model devoted for representing mechanical systems undergoing large deformations and viscoelastic behavior are found to be asymptotically stable.

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E. I. Rivin, “Use of Stiffness /Damping/Natural Frequency Criteria inVibration Control”, IFToMM World Congress, Besanson, June 2-6, 2007

F. J. Lockett, Nonlinear Viscoelastic Solids, Academic Press, 1972

Y. M. de Haan, G. M. Sluimer, “Standard linear solid model for dynamic and time dependent behaviour of building materials”, Heron , Vol. 46, No. 1, pp.49-76, 2001

H. H. Hilton, S. Yi, “Generalized viscoelastic 1-dof deterministic nonlinear oscillators”, Nonlinear Dynamics, Vol. 36, No. 2-4, pp. 281-298, 2004

S. Laroze, Mécanique des Structures. Tome 3, Masson, 1992

M. D. Monsia, “A mathematical model for predicting the relaxation of creep strains in materials”, Physical Review & Research International, Vol.2, No. 3, pp.107-124, 2012

R. D. Bauer, “Rheological approaches of arteries”, Biorheology Suppl. I, Vol. 1, pp.159-167, 1984

G. Kerschen, K. Worden, A. F. Vakakis, J-.C. Golinval, “Past, present and future of nonlinear system identification in structural dynamics”, Mechanical Systems and Signal Processing, Vol. 20, No. 3, pp.505-592, 2006

H. Jrad, J. L. Dion, F. Renaud, I. Tawfiq, M. Haddar, “Experimental characterization, modeling and parametric identification of the non linear dynamic behavior of viscoelastic components”, European Journal of Mechanics-A/Solids, Vol. 42 pp. 176-187, 2013

H. Jrad, J. L. Dion, F. Renaud, I. Tawfiq, M. Haddar, “A new approach for nonlinear generalized Maxwell model for depicting dynamic behaviour of viscoelastic elements-parameters identification and validation”, 18th Symposium of Vibrations, Shocks and Noise, France, July 3-5, 2012

M. D. Monsia, “A Simplified nonlinear generalized Maxwell model for predicting the time dependent behavior of viscoelastic materials”, World Journal of Mechanics, Vol.1, No. 3, pp. 158-167, 2011

M. D. Monsia, “Modeling the nonlinear rheological behavior of materials with a hyper-exponential type function”, Mechanical Engineering Research, Vol.1, No. 1, pp. 103-109, 2011

M. D. Monsia, “A nonlinear mechanical model for predicting the dynamics response of materials under a constant loading”, Journal of Materials Science Research, Vol.1, No. 1, pp. 90-100, 2012

M. D. Monsia, Y. J. F. Kpomahou, “Predicting the dynamic behavior of materials with a nonlinear modified Voigt model”, Journal of Materials Science Research, Vol. 1, No. 2, pp. 166-173, 2012

M. D. Monsia, Y. J. F. Kpomahou, “A theoretical characterization of time dependent materials by using a hyperlogistic-type model'”, Mechanical Engineering Research, Vol. 2, No. 1, pp. 36-43, 2012

L. Cveticanin, “Oscillator with strong quadratic damping force”, Publications de l'Institut Mathématique, Nouvelle série, Vol. 85, No. 99, pp. 119-130, 2009

M. M. Rashidi, A. Shooshtari, O. Anwar Bég, “Homotopy Perturbation Study of Nonlinear Vibration of Von Karman Rectangular Plates”, Computer & Structures, Vol. 106-107, pp. 46-55, 2012

J. D. Keckić, “Additions to Kamke's treatise ,VI: A nonlinear second order equation”, Publications de l'Institut Mathématique, nouvelle série, Vol. 19, No. 33, pp.81-82, 1975

J. D. Keckić, “Additions to Kamke's treatise ,VII: Variation of parameters for nonlinear second order differential equations”, Ser. Mat. Fiz., Vol. 544-576 , pp. 31-36, 1946, available at: http://pefmath2.etf.bg.ac.rs/files/100/549.pdf

P. Painlevé, “Mémoire sur les équations différentielles dont l'intégrale générale est uniforme”, Bull. Soc. Math. France, Vol. 28, pp. 201-261, 1900

L. Roth, “On the solutions of certain differential equations of the second order”, Phil. Mag. Vol. 32, No. 211, pp.155-164, 1941

M. D. Monsia, “A mathematical model for predicting the nonlinear deformation response of viscoelastic materials”, International Journal of Applied Mathematics and Mechanics, Vol. 8, No. 16, pp. 22-30, 2012

R. Conte, “Partial integrability of the anharmonic oscillator”, Journal of Nonlinear Mathematical Physics , Vol. 14, No. 3, pp. 462-473, 2007

R. Conte, M. Musette, The Painlevé Handbook, Springer, 2008

N. A. Kudryashov, “Nonlinear differential equations with exact solutions”, arXiv:nlin/0311058v1[nlin. SI]

D. W. Jordan, P. Smith, Nonlinear Ordinary Differential Equations, Oxford University Press, 2007

J.-H. He, “Some asymptotic methods for strongly nonlinear equations”, International Journal of Modern Physics B, Vol. 20, No. 10, pp.1141-1199, 2006

M. Pellicer, J. Solà-Morales, “Analysis of a viscoelastic spring-mass model Journal of Mathematical Analysis and Applications, Vol. 294, No. 2, pp. 687-698, 2004

L. Cveticanin, “Vibrations of the nonlinear oscillator with quadratic nonlinearity”, Physica A: Statistical Mechanics and its Applications, Vol. 341, pp. 123-135, 2004

R. Jauregni, F. Silva, “Numerical validation methods” in Numerical Analysis-Theory and Application , IntechOpen, 2011, available at: www.intechopen.com/download/pdf/19916

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