Cubic-quintic Duffing Oscillator Research Articles

Generic numerical and analytical methods for solving nonlinear oscillators

Nonlinear oscillations are a challenging problem due to the high complexity of the underlying differential equations. This paper focuses on the nonlinear oscillator with cubic and harmonic restoring force as it represents a very general case. In particular, many well-known oscillators (e.g., the Duffing equation, the capillary oscillator, the cubic-quintic Duffing oscillator, the simple pendulum, etc) are subcases of it. Accurate solutions in terms of Fourier series functions for different oscillation amplitudes were derived using an energy-based numerical method and fitting processes. In addition, an analytical approach was developed by finding the derivative of the restoring force function at the point x=c0A, where A is the amplitude of the oscillation, and c0 is a constant. Using this approach, the error was negligible (∼0.02–0.22%) even for A → ∞. The methods and solutions for the nonlinear oscillator with cubic and harmonic restoring force can be equally applied to all its subcases.

Nonlinear oscillations for beginners. Calculating period using an elementary computational approach

Determining the period of a nonlinear oscillation is a challenging task that requires a strong mathematical background in solving nonlinear differential equations. However, the procedure can be significantly simplified using the area under the 1/|υ|= f(x) graph, where υ is the velocity of the oscillating object and x is its displacement from its equilibrium position. The proposed method requires elementary computational tools and is appropriate for determining the period of any nonlinear undamped oscillation. Characteristic examples are presented, such as the simple pendulum, the oscillation with a power-law restoring force, and the cubic-quintic Duffing oscillator. The proposed approach provides accurate results and is appropriate for introductory physics and mechanics courses.

Analytical Approximant to a Quadratically Damped Forced Cubic-Quintic Duffing Oscillator

The cubic-quintic Duffing oscillator of a system with strong quadratic damping and forcing is considered. We give elementary approximate analytical solution to this oscillator in terms of exponential and trigonometric functions. We compare the analytical approximant with the Runge–Kutta numerical solution. The approximant allows us to estimate the points at which the solution crosses the horizontal axis.

SIMPLIFIED HAMILTONIAN-BASED FREQUENCY-AMPLITUDE FORMULATION FOR NONLINEAR VIBRATION SYSTEMS

The Hamiltonian-based frequency formulation has been hailed as an unprecedented success for it gives a straightforward insight into a complex nonlinear vibration system with simple calculation. This paper gives a systematical analysis of the formulation, and two simplified formulations are suggested. The cubic-quintic Duffing oscillator is used as an example to show extremely simple calculation and remarkable accuracy. It can be used as a paradigm for many other applications, and the one-step solving process has cleaned up the road of the nonlinear vibration theory.

A short remark on He’s frequency formulation

Recently Shen suggested a modification of He’s frequency formulation using Lagrange interpolation with great success for the cubic-quintic Duffing oscillator. This paper proposed an alternative modification of He’s frequency formulation by dividing the oscillators into two extreme conditions when the amplitude is either extremely small or remarkably large, and He’s frequency formulation is used for each case, and a final frequency formulation is obtained by matching the two extreme conditions. Comparison of the approximate frequency with the exact one for various amplitudes shows good agreement.

Global residue harmonic balance method for strongly nonlinear oscillator with cubic and harmonic restoring force

Abstract This paper focuses on the numerical investigation of a strongly nonlinear oscillator with cubic and harmonic restoring force. We transform this oscillator as a free damped cubic-quintic Duffing oscillator equation by Taylor approximation. The approximated solutions with high accuracy are provided by using the global residue harmonic balance method (GRHBM) without any discretization or restrict assumptions. The sensitive analysis of the approximation or the frequency with respect to the amplitude is considered in detail. Numerical comparisons with Runge–Kutta method and harmonic balance method are given to show the efficiency and stability of GRHBM.

A New Approach for Solving the Complex Cubic-Quintic Duffing Oscillator Equation for Given Arbitrary Initial Conditions

The nonlinear differential equation governing the periodic motion of the one-dimensional, undamped, and unforced cubic-quintic Duffing oscillator is solved exactly, providing exact expressions for the period and the solution. The period as well as the exact analytic solution is given in terms of the famous Weierstrass elliptic function. An integrable case of a damped cubic-quintic equation is presented. Mathematica code for solving both cubic and cubic-quintic Duffing equations is given in Appendix at the end.

An integral of motion for the damped cubic-quintic Duffing oscillator with variable coefficients

In this short communication, integrals of motion for the undamped and damped cubic-quintic Duffing oscillators with time-dependent coefficients are obtained for the first time in the literature under appropriate analytical conditions. The integrals of motion are obtained using Noether’s theorem, and the conditions for their existence are directly related to the well-known Milne–Pinney equation, which is associated to the Ermakov–Lewis systems. We perform here some numerical simulations to illustrate the validity of our analytical results.

Coupling of energy and harmonic balance method for solving a conservative oscillator with strong odd-nonlinearity

In this paper, a new analytical technique, combining the energy balance method (EBM) with harmonic balance method (HBM), is presented to obtain higher-order approximations of a conservative oscillator with strong odd-nonlinearity. To show the accuracy of the present method, one nonlinear oscillator named as cubic-quintic Duffing oscillator is investigated. The results obtained in this paper are compared with those results determined by other methods and exact solutions. The results give high accuracy and also provide better result than other existing results for both small and large amplitudes of oscillation. The main advantage of the present paper is its simplicity, which contains a few harmonic terms with lower order terms and these terms make the solution quickly converges. The present technique can be used to other nonlinear oscillators.

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 cubic-quintic 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.

Application of Variational Iteration Method for Dropping Damage Evaluation of the Suspension Spring Packaging System

The dropping damage evaluation for packaging system is essential for safe transportation and storage. A dynamic model of nonlinear cubic-quintic Duffing oscillator for the suspension spring packaging system was proposed. Then, a first-order approximate solution was obtained by applying He’s variable iteration method. Based on the results, a damage evaluation equation was derived, which reveals the main controlling physical parameters for damage potential of drop to packaged products concretely. Finally, the dropping damage boundary curves and surfaces for the system were discussed. It was found that decreasing the suspension angle can improve the safe region of the system.

Stability Analysis of Damped Cubic-Quintic Duffing Oscillator

This paper presents a comprehensive stability analysis of the dynamics of the damped cubic-quintic Duffing oscillator. We employ the derivative expansion method to investigate the slightly damped cubic-quintic Duffing oscillator obtaining a uniformly valid solution. We obtain a uniformly valid solution of the un-damped cubic-quintic Duffing oscillator as a special case of our solution. A phase plane analysis of the damped cubic-quintic Duffing oscillator is undertaken showing some chaotic dynamics which sends a signal that the oscillator may be useful as model for prediction of earth- quake occurrence.

Exact solution of the cubic-quintic Duffing oscillator

In this paper, we derive the exact solution of the cubic-quintic Duffing oscillator based on the use of Jacobi elliptic functions. We also showed that the exact angular frequency of this cubic-quintic Duffing equation is given in terms of the complete elliptic integral of the first kind.

Solution of nonlinear cubic-quintic Duffing oscillators using He’s Energy Balance Method

In this study, He’s Energy Balance Method (EBM) was applied to solve strong nonlinear Duffing oscillators with cubic-quintic nonlinear restoring force. The complete EBM solution procedure of the cubic-quintic Duffing oscillator equation is presented. For illustration of effectiveness and convenience of the EBM, different cases of cubic-quintic Duffing oscillator with different parameters of α, β and γ were compared with the exact solution. We found that the solutions were valid for small as well as large amplitudes of oscillation. The results show that the EBM is very convenient and precise, so it can be widely applicable in engineering and other sciences.