In this work, the classical mechanical Voigt model is modified and extended to finite deformations by using a rational elastic spring force function to describe accurately the nonlinear time-dependent deformation response of some viscoelastic materials. As theoretical results, a hyperlogistic-type function has been found as the deformation versus time relationship. This growth model appeared powerful to reproduce mathematically as shown by numerical works, any S-shaped experimental data. Compared with some previous models, the present one-dimensional formulation gives the advantage to assure or to control via an explicit material parameter, to speak, via the coefficient of inertia, the nonlinearity of the model. The proposed model demonstrated then the importance to consider in the material modeling the inertial coefficient.

In this paper, a one-dimensional nonlinear modified and extended Voigt model with constant material parameters is formulated to represent mathematically the time deformation behavior of a variety of viscoelastic materials. A binomial law is used as a nonlinear elastic force function. Numerical illustrations performed show that the hyperlogistic-type solution obtained is very useful to reproduce any S-shaped experimental curve.

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Authors introduce a generalized singular differential equation of quadratic Liénard type for study of exact classical and quantum mechanical solutions. The equation is shown to exhibit periodic solutions and to include the linear harmonic oscillator equation and the Painlevé-Gambier XVII equation as special cases. It is also shown that the equation may exhibit discrete eigenstates as quantum behavior under Nikiforov-Uvarov approach after several point transformations.

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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In this paper, nonlinear dynamics study of a RLC series circuit modeled by a generalized Van der Pol oscillator is investigated. After establishing a new general class of nonlinear ordinary differential equation, a forced Van der Pol oscillator subjected to an inertial nonlinearity is derived. According to the external excitation strength, harmonic, subharmonic and superharmonic oscillatory states are obtained using the multiple time scales method. Bifurcation diagrams displayed by the model for each system parameter are performed numerically through the fourth-order Runge–Kutta algorithm.