Approximate Periodic Solutions Research Articles

Application of Adaptation HAM for Nonlinear Oscillator Typified as A Mass Attached to A Stretched Elastic Wire

This paper applies the adaptation of homotopy analysis method (AHAM) for the first time to obtained the periodic solutions for the oscillation of a mass attached to a stretched elastic wire. The AHAM approach can be applied directly to the governing equation without rewrite it in a form that does not contain the square-root expression. More precisely, with the help of the homotopy polynomials procedure the nonlinear term of the problem can be decompose as a series of polynomials to overcomes the difficulty arising in calculating complicated integrals. A comparative study between AHAM and other existing solutions obtained by several authors is conducted to demonstrate the simplicity and the efficiency of AHAM. The approximate frequency and periodic solution for both small and large amplitude of oscillations show a good agreement with the numerical solution.

Analytical and numerical construction of vertical periodic orbits about triangular libration points based on polynomial expansion relations among directions

Innovated by the nonlinear modes concept in the vibrational dynamics, the vertical periodic orbits around the triangular libration points are revisited for the Circular Restricted Three-body Problem. The $\zeta $ -component motion is treated as the dominant motion and the $\xi$ and $\eta $ -component motions are treated as the slave motions. The slave motions are in nature related to the dominant motion through the approximate nonlinear polynomial expansions with respect to the $\zeta $ -position and $\zeta $ -velocity during the one of the periodic orbital motions. By employing the relations among the three directions, the three-dimensional system can be transferred into one-dimensional problem. Then the approximate three-dimensional vertical periodic solution can be analytically obtained by solving the dominant motion only on $\zeta $ -direction. To demonstrate the effectiveness of the proposed method, an accuracy study was carried out to validate the polynomial expansion (PE) method. As one of the applications, the invariant nonlinear relations in polynomial expansion form are used as constraints to obtain numerical solutions by differential correction. The nonlinear relations among the directions provide an alternative point of view to explore the overall dynamics of periodic orbits around libration points with general rules.

Higher-order approximate analytical solutions to nonlinear oscillatory systems arising in engineering problems

In the current paper, a powerful approximate analytical approach namely the global residue harmonic balance method (GRHBM) is proposed for obtaining higher-order approximate frequency and periodic solution of nonlinear conservative oscillatory systems arising in engineering problems. The proposed method has a main difference with other traditional harmonic balance methods such that the residual errors obtained in pervious order approximation are used in the present one. Comparison of the obtained results with the exact and numerical solution as well as well-known analytical methods such as Hamiltonian approach, Max–Min approach, variational approach, and He’s amplitude–frequency formulation reveals the correctness and usefulness of the GRHBM. It is shown that the results are valid for different values of system parameters and both small and large amplitudes. Hence, the method can be easily applied to other strongly nonlinear conservative oscillatory systems. Furthermore, using the obtained analytical expressions, the effect of amplitude and system parameters on nonlinear frequency is studied.

An Effective Iteration Method for a Class of Nonlinear Singular Perturbed Problems

In this paper we investigate a class of singular perturbed differential equations with initial conditions. Based on the equations cannot solved easily by the traditional methods, we give a simple but effective method for finding their approximate periodic solutions firstly. That is, we introduce an iterative technique to the singular perturbed nonlinear system and then adopt the classical perturbed technique to give the approximate solutions. Secondly, some typical examples given to illustrate the methods is a useful tool in solving the nonlinear singular perturbed systems. At last, we present a windowing technique for the system. The convergence processes can be speeded up when we take the windowing technique. These methods can be easily extended to other nonlinear systems and found widely applicable in engineering.

On the Harmonic Oscillations for the Motion of a Dynamical System

This work touches two important cases for the motion of a pendulum called Sub and Ultra-harmonic cases. The small parameter method is used to obtain the approximate analytic periodic solutions of the equation of motion when the pivot point of the pendulum moves in an elliptic path. Moreover, the fourth order Runge-Kutta method is used to investigate the numerical solutions of the considered model. The comparison between both the analytical solution and the numerical ones shows high consistency between them.

Higher accuracy analytical approximations to nonlinear oscillators with discontinuity by energy balance method

A modified energy balance method (EBM) has been presented to obtain higher-order approximate periodic solutions from the integral expression (i.e., first order differential equation) of some nonlinear oscillators. It has been shown that the method is more effective than the classical EBM as well as some recently developed approaches such as homotopy perturbation method, harmonic balance method and iterative method.

Approximate analytical solutions to a conservative oscillator using global residue harmonic balance method

In the current research paper, a conservative system comprising of a mass grounded by linear and nonlinear springs in series connection is studied. The equation of motion for the aforementioned system has been derived as a nonlinear ordinary differential equation with inertia and static–type cubic nonlinearities. The global residue harmonic balance method is applied to obtain an approximate analytical frequency and periodic solution of the problem. Using the obtained analytical expressions, the influences of the hardening and softening nonlinear spring on the non–dimensional frequency are investigated. The results show that developing the system nonlinearity leads the displacement of the mass and the deflection of linear spring to approach each other. Moreover, comparison of the results obtained using the proposed procedure with those achieved by other methods such as numerical method, variational iteration method and harmonic balance approach demonstrates the accuracy and advantages of the current approach.

Modified Parker-Sochacki method for solving nonlinear oscillators

ABSTRACTA new approach is presented for solving nonlinear oscillatory systems. Parker-Sochacki method (PSM) is combined with Laplace-Padé resummation method to obtain approximate periodic solutions for three nonlinear oscillators. The first one is Duffing oscillator with quintic nonlinearity which has odd nonlinearity. The second one is Helmholtz oscillator which has even nonlinearity. The last one is a strongly nonlinear oscillator, namely; relativistic harmonic oscillator which has a fractional order nonlinearity. Solutions are also obtained using Runge-Kutta numerical method (RKM) and Lindstedt-Poincare method (LPM). However, the LPM could not be used to solve the relativistic harmonic oscillator since it is a strongly nonlinear oscillator. The comparison between these solutions shows that the convergence zone for the Parker-Sochacki with Laplace-Padé method (PSLPM) is remarkably increased compared to PSM method. It also shows that the PSLPM solutions are in excellent agreement with LPM solutions for Duffing oscillator and are superior to LPM solutions in case of Helmholtz oscillator. The PSLPM succeeded to give an accurate periodic solution for the relativistic harmonic oscillator. For a wide range of solution domain, comparing PSLPM with RKM prove the correctness of the PSLPM method. Hence, the PSLPM method can be used with satisfied confidence to solve a broad class of nonlinear oscillators.

A new analytical approach to investigate the strongly nonlinear oscillators

In this paper, a modified harmonic balance method is used to investigate the strongly nonlinear oscillators. The approximate frequency and periodic solution for both small and large amplitude of oscillations show a good agreement with the numerical solution (considered to be exact). One of the most advantages of the presented method is its simplicity. Results found in this paper are compared with other existing solutions (obtained by several authors) and those results are more accurate than other existing solutions. The method is illustrated by examples.

A novel analytical approximation technique for highly nonlinear oscillators based on the energy balance method

In the present paper, a novel analytical approximation technique has been proposed based on the energy balance method (EBM) to obtain approximate periodic solutions for the focus generalized highly nonlinear oscillators. The expressions of the natural frequency-amplitude relationship are obtained using a novel analytical way. The accuracy of the proposed method is investigated on three benchmark oscillatory problems, namely, the simple relativistic oscillator, the stretched elastic wire oscillator (with a mass attached to its midpoint) and the Duffing-relativistic oscillator. For an initial oscillation amplitude A0=100, the maximal relative errors of natural frequency found in three oscillators are 2.1637%, 0.0001% and 1.201%, respectively, which are much lower than the errors found using the existing methods. It is highly remarkable that an excellent accuracy of the approximate natural frequency has been found which is valid for the whole range of large values of oscillation amplitude as compared with the exact ones. Very simple solution procedure and high accuracy that is found in three benchmark problems reveal the novelty, reliability and wider applicability of the proposed analytical approximation technique.

Linear and nonlinear free vibrations of electrostatically actuated micro-/nanomechanical resonators

This paper is concerned with linear and nonlinear free vibrations of an axially stressed clamped---clamped micro-/nanobeam actuated by one-sided electrode configuration. Galerkin procedure and a proper shape function are used to establish a dynamic reduced-order model, where the integral of the electromechanical forcing term is obtained in an explicit and closed form. This model makes the analysis of dynamic behaviors convenient for the micro-/nanobeam. For linear free vibration, the fundamental natural frequency formulas of the system are validated by comparing with published experiment data while the Newton method coupled with the harmonic balance method is developed to obtain analytical approximate period and periodic solution for the nonlinear system. These approximate solutions have explicit expressions and they can readily be used to derive the effects of various physical parameters on dynamic behavior of micro-/nanobeams. Exact solution in an integral form is also presented. For small as well as large amplitudes of nonlinear vibration, the new analytical approximate solutions show excellent agreement with respect to the numerically integrated solutions obtained from the integral-form exact solutions.

A simple harmonic balance method for solving strongly nonlinear oscillators

In this paper, a simple harmonic balance method (HBM) is proposed to obtain higher-order approximate periodic solutions of strongly nonlinear oscillator systems having a rational and an irrational force. With the proposed procedure, the approximate frequencies and the corresponding periodic solutions can be easily determined. It gives high accuracy for both small and large amplitudes of oscillations and better result than those obtained by other existing results. The main advantage of the present method is that its simplicity and the second-order approximate solutions almost coincide with the corresponding numerical solutions (considered to be exact). The method is illustrated by examples. The present method is very effective and convenient method for solving strongly nonlinear oscillator systems arising in nonlinear science and engineering.

Free Oscillations of a Nonlinear Oscillator with an Exponential Non-Viscous Damping

This paper deals with the nonlinear oscillations of an exponential non-viscous damping oscillator. An analytic technique, namely Optimal Homotopy Perturbation Method (OHPM) is employed to propose an analytic approach to solve nonlinear oscillations. Our procedure proved to very effective and accurate and did not require a small or large parameters in the nonlinear equation or in the initial conditions. An excellent agreement of the approximate frequencies and periodic solutions with the numerical ones has been demonstrated.

Approximate Analytical Solution of Two Coupled Time Fractional Nonlinear Schrödinger Equations

In this article, we consider the well known nonlinear Schrodinger equation (NLS) with fractional time derivative and derive its approximate analytical solution using the homotopy analysis method (HAM). We also applied HAM to two coupled time fractional NLS and constructed its approximate solution. The question of convergence of the obtained solution is discussed. The obtained approximate analytical periodic wave solution, solitary wave solution and the effect of time fractional order $$\alpha $$ are shown graphically.

Higher order explicit solutions for nonlinear dynamic model of column buckling using variational approach and variational iteration algorithm-II

This paper deals with the determination of approximate solutions for a model of column buckling using two efficient and powerful methods called He’s variational approach and variational iteration algorithm-II. These methods are used to find analytical approximate solution of nonlinear dynamic equation of a model for the column buckling. First and second order approximate solutions of the equation of the system are achieved. To validate the solutions, the analytical results have been compared with those resulted from Runge-Kutta 4th order method. A good agreement of the approximate frequencies and periodic solutions with the numerical results and the exact solution shows that the present methods can be easily extended to other nonlinear oscillation problems in engineering. The accuracy and convenience of the proposed methods are also revealed in comparisons with the other solution techniques.

Study of a piecewise linear dynamic system with negative and positive stiffness

The present paper mainly focuses on numerical and analytical study of a piecewise linear dynamic oscillator with negative stiffness followed by positive stiffness which has not been studied to date. The dynamic system of interest stems from a previous analytical and experimental research on adaptive negative stiffness for the purpose of seismic protection. Numerical algorithms meant specifically for simulating piecewise smooth (PWS) systems like this nonlinear system are studied. An appropriate combination of negative stiffness and adequate damping can reduce the peak restoring or transmitted force with a slightly larger peak displacement. Essentially, the negative stiffness system in a dynamic system is very beneficial in reducing the amount of force transmitted. The exact solution is derived for free vibration. A modified Lindstedt–Poincaré method (modified L–P method) is adopted to derive approximate periodic solutions for the forced and damped system and its frequency-response curves are obtained through numerical simulation. The modified L–P solution obtained for the forced and damped case is found to agree well with the numerical results. In the piecewise linear dynamic system with initial negative stiffness followed by positive stiffness, it is found that the response remains bounded in a limit cycle. This system behaves similar to a van der Pol oscillator wherein negative damping is followed by positive damping. Presented herein is a special case as defined by the specified parameter ranges; thus, to make it more general future work is needed.

Global residue harmonic balance method to periodic solutions of a class of strongly nonlinear oscillators

A new approach, namely the global residue harmonic balance method, was advanced to determine the accurate analytical approximate periodic solution of a class of strongly nonlinear oscillators. A class of nonlinear jerk equation containing velocity-cubed and velocity times displacements-squared was taken as a typical example. Unlike other harmonic balance methods, all the former residual errors are introduced in the present approximation to improve the accuracy. Comparison of the result obtained using this approach with the exact one and simplicity and efficiency of the proposed procedure. The method can be easily extended to other strongly nonlinear oscillators.

Approximate periodic solution for nonlinear jerk equation as a third-order nonlinear equation via modified differential transform method

Purpose – The nonlinear jerk equation is a third-order nonlinear equation that describes some physical phenomena and in general form is given by: x = J (x, x, x). The purpose of this paper is to employ the modified (MDTM) differential transform method (DTM) to obtain approximate periodic solutions of two cases of nonlinear jerk equation. Design/methodology/approach – The approach is based on MDTM that is developed by combining DTM, Laplace transform and Padé approximant. Findings – Comparison of results obtained by MDTM with those obtained by numerical solutions indicates the excellent accuracy of solution. Originality/value – The MDTM is extended to determining approximate periodic solution of third-order nonlinear differential equations.

Accurate periodic solution for nonlinear vibration of thick circular sector slab

In this paper we consider a periodic solution for nonlinear free vibration of conservative systems for thick circular sector slabs. In Energy Balance Method (EBM) contrary to the conventional methods, only one iteration leads to high accuracy of the solutions. The excellent agreement of the approximate frequencies and periodic solutions with the exact ones could be established. Some patterns are given to illustrate the effectiveness and convenience of the methodology. Comparing with numerical solutions shows that the energy balance method can converge to the numerical solutions very rapidly which are valid for a wide range of vibration amplitudes as indicated in this paper.

Explicit solution of the large amplitude transverse vibrations of a flexible string under constant tension

This paper presents the analytical simulation of string with large amplitudes using the Variational Iteration Method (VIM) and Hamiltonian Approach (HA). In order to verify the precision of the presented methods, current results were compared with He's Variational Approach and Runge-Kutta 4th order. It has been found that these methods are well suited for a range of parameters and the approximate frequencies and periodic solutions show a good agreement with other techniques. The results show that both methods can be easily extended to other nonlinear oscillations and it can be predicted that both methods can be found widely applicable in engineering.