Simulating Nonlinear Oscillations of Viscoelastically Damped Mechanical Systems


  • M. D. Monsia Department of Physics/FAST, University of Abomey-Calavi, Abomey-Calavi, Benin
  • Y. J. F. Kpomahou Department of Physics/FAST, University of Abomey-Calavi, Abomey-Calavi, Benin
Volume: 4 | Issue: 6 | Pages: 714-723 | December 2014 |


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.


mathematical modeling, nonlinear oscillations, viscoelastic oscillator, Painlevé equation, exact solution, numerical simulation


Download data is not yet available.


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 DOI:

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 DOI:

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 DOI:

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 DOI:

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 DOI:

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 DOI:

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 DOI:

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 DOI:

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 DOI:

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

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 DOI:

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:

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

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

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 DOI:

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 DOI:

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 DOI:

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

R. Jauregni, F. Silva, “Numerical validation methods” in Numerical Analysis-Theory and Application , IntechOpen, 2011, available at: DOI:


How to Cite

M. D. Monsia and Y. J. F. Kpomahou, “Simulating Nonlinear Oscillations of Viscoelastically Damped Mechanical Systems”, Eng. Technol. Appl. Sci. Res., vol. 4, no. 6, pp. 714–723, Dec. 2014.


Abstract Views: 1113
PDF Downloads: 340

Metrics Information