Accurate approximate closed-form solutions for the cubic-quintic Duffing oscillator are obtained in terms of elementary functions. To do this, we use the previous results obtained using a cubication method in which the restoring force is expanded in Chebyshev polynomials and the original nonlinear differential equation is approximated by a cubic Duffing equation. Explicit approximate solutions are then expressed as a function of the complete elliptic integral of the first kind and the Jacobi elliptic function cn. Then we obtain other approximate expressions for these solutions, which are expressed in terms of elementary functions. To do this, the relationship between the complete elliptic integral of the first kind and the arithmetic-geometric mean is used and the rational harmonic balance method is applied to obtain the periodic solution of the original nonlinear oscillator.

We introduce a novel analytical approximate technique, called Optimal Variational Method (OVM), to investigate the nonlinear behaviour of a rotating electrical machine modelled as an oscillator with cubic elastic restoring force and time variable coefficients. The proposed procedure involves the presence of some initially unknown convergence-control parameters whose values are later optimally determined. Comparisons between the obtained results and exact ones reveal that OVM is very effective and convenient providing highly accurate results.

A review on mathematical and mechanical models of the vocal cords is given. The basic model is a two-mass nonlinear oscillator system which is accepted to be the basic one for mechanical description in voice production. The model is not only extended into three, five, and more mass systems, systems with time variable parameters and three-dimensional systems, but also simplified into one-mass system with coupled two-direction deflection and damping functions. The corresponding mathematical models are the systems of coupled second-order differential equations which describe the vibrations of the symmetric and asymmetric vocal folds. The models give the conditions for the regular and irregular motions like bifurcation and deterministic chaos in vocal folds. The obtained results are of special interest for detecting the pathology of vocal cords, when there are no visual effects of disease. Based on the results given in the paper, the objectives for future investigation in this matter are given.

A multi-degree-of-freedom dynamical system with local cubic nonlinearities subjected to super/subharmonic excitation is considered in this paper. The purpose of this paper is to approximate the nonlinear response of system at super/sub harmonic resonance. For many situations, single resonance mode is often observed to be leading as system enters into super/sub harmonic resonance. In this case, the single modal natural resonance theory can be applied to reduce the system model and a simplified model with only a single DOF is always obtained. Thus, an approximate solution and the analytical expression of frequency response relation are then derived using classical perturbation analysis. While the system is controlled by multiple modes, modal analysis for linearized system is used to decide dominant modes. The reduced model governed by these relevant modes is found and results in an approximate numerical solutions. An illustrative example of the discrete mass-spring-damper nonlinear vibration system with ten DOFs is examined. The approximation results are validated by comparing them with the calculations from direct numerical integration of the equation of motion of the original nonlinear system. Comparably good agreements are obtained.

The industrial applications use instruments and machines operating at high speeds, developing high forces, low temperatures, corrosive environments, extreme pressures, and so forth. Under these conditions, the elasticity of elements such a machine is built of cannot be ignored anymore, and models are needed to more accurately “grasp” the mechanical phenomena accompanying the operation. The vibrations and the loss of stability are the main effects occurring under these conditions. For the study on this kind of systems with rigid motion and elastic elements, numerous models have been elaborated, the main idea being the discretization of the elements and the use of finite element method. Finally, second-order differential equations with variable coefficients are obtained; these equations are strong nonlinear ones due to the time-dependent values of angular speed and acceleration, and they can be linearized considering a very short period of time, in which the motion is considered to be “frozen.” The aim of this paper is to present some characteristic properties of these systems.

We study the dynamic mathematical model for an infinitely long cylinder composed of an isotropic incompressible Ogden material with a microvoid at its center, where the outer surface of the cylinder is subjected to a uniform radial tensile load. Using the incompressibility condition and the boundary conditions, we obtain a second-order nonlinear ordinary differential equation that describes the motion of the microvoid with time. Qualitatively, we find that this equation has two types of solutions. One is a classical nonlinear periodic solution which describes that the motion of the microvoid is a nonlinear periodic oscillation; the other is a blow-up solution. Significantly, for the isotropic incompressible Ogden material, there exist some special values of material parameters, the phase diagrams of the motion equation have homoclinic orbits, which means that the amplitude of a nonlinear periodic oscillation increases discontinuously with the increasing load.

The dynamic characteristics and parametric instability of a rotating electrorheological (ER) sandwich beam with rotary oscillation are numerically analyzed. Assuming that the angular velocity of an ER sandwich beam varies harmonically, the dynamic equation of the rotating beam is first derived based on Hamilton's principle. Then the coupling and nonlinear equation is discretized and solved by the finite element method. The multiple scales method is employed to determine the parametric instability of the structures. The effects of electric field on the natural frequencies, loss factor, and regions of parametric instability are presented. The results obtained indicate that the ER material layer has a significant effect on the vibration characteristics and parametric instability regions, and the ER material can be used to adjust the dynamic characteristics and stability of the rotating flexible beams.

Energy balance method (EBM) is extended for high-order nonlinear oscillators. To illustrate the effectiveness of the method, a cubic-quintic Duffing oscillator was chosen. The maximum relative errors of the frequencies of the oscillator read 1.25% and 0.6% for the first- and second-order approximation, respectively. The third-order approximation has an accuracy as high as 0.008%. Excellent agreement of the approximated frequencies and periodic solutions with the exact ones is demonstrated for several values of parameters of the oscillator.