Closed-Form Exact Solutions for the Unforced Quintic Nonlinear Oscillator

Closed-form exact solutions for the periodic motion of the one-dimensional, undamped, quintic oscillator are derived from the first integral of the nonlinear differential equation which governs the behaviour of this oscillator. Two parameters characterize this oscillator: one is the coefficient of the linear term and the other is the coefficient of the quintic term. Not only the common case in which both coefficients are positive but also all possible combinations of positive and negative values of these coefficients which provide periodic motions are considered. The set of possible combinations of signs of these coefficients provides four different cases but only three different pairs of period-solution. The periods are given in terms of the complete elliptic integral of the first kind and the solutions involve Jacobi elliptic function. Some particular cases obtained varying the parameters that characterize this oscillator are presented and discussed. The behaviour of the periods as a function of the initial amplitude is analysed and the exact solutions for several values of the parameters involved are plotted. An interesting feature is that oscillatory motions around the equilibrium point that is not at x = 0 are also considered.This work was supported by the “Generalitat Valenciana” of Spain, under Project PROMETEOII/2015/015 and by the Universidad de Alicante, Spain, under Project GITE-09006-UA

High-order approximate solutions of strongly nonlinear cubic-quintic Duffing oscillator based on the harmonic balance method

In this paper, a new reliable analytical technique has been introduced based on the Harmonic Balance Method (HBM) to determine higher-order approximate solutions of the strongly nonlinear cubicquintic Duffing oscillator. The application of the HBM leads to very complicated sets of nonlinear algebraic equations. In this technique, the high-order nonlinear algebraic equations are approximated in the form of a power series solution, and this solution produces desired results even for small as well as large amplitudes of oscillation. Moreover, a suitable truncation formula is found in which the solution measures better results than existing results and it saves a lot of calculation. It is highly noteworthy that using the proposed technique, the third-order approximate solutions gives an excellent agreement as compared with the numerical solutions (considered to be exact). The proposed technique is applied to the strongly nonlinear cubic-quintic Duffing oscillator to reveals its novelty, reliability and wider applicability

Exact solution for the unforced Duffing oscillator with cubic and quintic nonlinearities

The nonlinear differential equation governing the periodic motion of the one-dimensional, undamped, unforced cubic–quintic Duffing oscillator is solved exactly, providing exact expressions for the period and the solution. The period is given in terms of the complete elliptic integral of the first kind and the solution involves Jacobi elliptic functions. Some particular cases obtained varying the parameters that characterize this oscillator are presented and discussed. The behaviour of the period as a function of the initial amplitude is analysed, the exact solutions and velocities for several values of the initial amplitude are plotted, and the Fourier series expansions for the exact solutions are also obtained. All this allows us to conclude that the quintic term appearing in the cubic–quintic Duffing equation makes this nonlinear oscillator not only more complex but also more interesting to study.This work was supported by the “Generalitat Valenciana” of Spain, under project PROMETEOII/2015/015, and by the “Vicerrectorado de Tecnologías de la Información” of the University of Alicante, Spain, under project GITE-09006-UA

Analytical approximate solutions for the cubic-quintic Duffing oscillator in terms of elementary functions

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.This work was supported by the “Generalitat Valenciana” of Spain, under Project PROMETEO/2011/021, and by the “Vicerrectorado de Tecnología e Innovación Educativa” of the University of Alicante, Spain, under Project GITE-09006-UA

Direct normal form analysis of oscillators with different combinations of geometric nonlinear stiffness terms

Nonlinear oscillators with geometric stiffness terms can be used to model a range of structural elements such as cables, beams and plates. In particular, single-degree-of-freedom (SDOF) systems are commonly studied in the literature by means of different approximate analytical methods. In this work, an analytical study of nonlinear oscillators with different combinations of geometric polynomial stiffness nonlinearities is presented. To do this, the method of direct normal forms (DNF) is applied symbolically using Maple software. Closed form (approximate) expressions of the corresponding frequency-amplitude relationships (or backbone curves) are obtained for both ε and ε2expansions, and a general pattern for ε truncation is presented in the case of odd nonlinear terms. This is extended to a system of two degrees-of-freedom, where linear and nonlinear cubic and quintic coupling terms exist. Considering the non-resonant case, an example is shown to demonstrate how the single mode backbone curves of the two degree-of-freedom system can be computed in an analogous manner to the approach used for the SDOF analysis. Numerical verifications are also presented using COCO numerical continuation toolbox in Matlab for the SDOF examples

Energy Method to Obtain Approximate Solutions of Strongly Nonlinear Oscillators

We introduce a nonlinearization procedure that replaces the system potential energy by an equivalent representation form that is used to derive analytical solutions of strongly nonlinear conservative oscillators. We illustrate the applicability of this method by finding the approximate solutions of two strongly nonlinear oscillators and show that this procedure provides solutions that follow well the numerical integration solutions of the corresponding equations of motion

Generalization of hyperbolic perturbation solution for heteroclinic orbits of strongly nonlinear self-excited oscillator

A generalized hyperbolic perturbation method for heteroclinic solutions is presented for strongly nonlinear self-excited oscillators in the more general form of x⋅⋅+g(x)=ɛf(μ,x,x⋅)x··+g(x)=ɛf(μ,x,x·). The advantage of this work is that heteroclinic solutions for more complicated and strong nonlinearities can be analytically derived, and the previous hyperbolic perturbation solutions for Duffing type oscillator can be just regarded as a special case of the present method. The applications to cases with quadratic-cubic nonlinearities and with quintic-septic nonlinearities are presented. Comparisons with other methods are performed to assess the effectiveness of the present method.postprin

Equivalent Representation Form of Oscillators with Elastic and Damping Nonlinear Terms

In this work we consider the nonlinear equivalent representation form of oscillators that exhibit nonlinearities in both the elastic and the damping terms. The nonlinear damping effects are considered to be described by fractional power velocity terms which provide better predictions of the dissipative effects observed in some physical systems. It is shown that their effects on the system dynamics response are equivalent to a shift in the coefficient of the linear damping term of a Duffing oscillator. Then, its numerical integration predictions, based on its equivalent representation form given by the well-known forced, damped Duffing equation, are compared to the numerical integration values of its original equations of motion. The applicability of the proposed procedure is evaluated by studying the dynamics response of four nonlinear oscillators that arise in some engineering applications such as nanoresonators, microresonators, human wrist movements, structural engineering design, and chain dynamics of polymeric materials at high extensibility, among others

Analytical study on the non-linear vibration of Euler-Bernoulli beams

In this study, He’s Variational Approach Method (VAM) is used to obtain an accurate analytical solution for the nonlinear vibrations of Euler-Bernoulli beams subjected to axial loads. It is demonstrated that the method works very well for the whole range of initial amplitudes and does not need small perturbation. It is sufficiently accurate in the case of both linear and nonlinear physics and engineering problems. Finally, the accuracy of the solution obtained with the approximate VAM method is shown graphically and compared with that of the numerical solution