Large amplitude non-linear oscillations of a general conservative system

This paper presents new, approximate analytical solutions to large-amplitude oscillations of a general, inclusive of odd and non-odd non-linearity, conservative single-degree-of-freedom system. Based on the original general non-linear oscillating system, two new systems with odd non-linearity are to be addressed. Building on the approximate analytical solutions of odd non-linear systems developed by the authors earlier, we construct the new approximate analytical solutions to the original general non-linear system by combinatory piecing of the approximate solutions corresponding to, respectively, the two new systems introduced. These approximate solutions are valid for small as well as large amplitudes of oscillation for which the perturbation method either provides inaccurate solutions or is inapplicable. Two examples with excellent approximate analytical solutions are presented to illustrate the great accuracy and simplicity of the new formulation.     

Modified Parker-Sochacki method for solving nonlinear oscillators

A 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.     

Construction of approximate analytical solutions to strongly nonlinear damped oscillators

An analytical approximate method for strongly nonlinear damped oscillators is proposed. By introducing phase and amplitude of oscillation as well as a bookkeeping parameter, we rewrite the governing equation into a partial differential equation with solution being a periodic function of the phase. Based on combination of the Newton’s method with the harmonic balance method, the partial differential equation is transformed into a set of linear ordinary differential equations in terms of harmonic coefficients, which can further be converted into systems of linear algebraic equations by using the bookkeeping parameter expansion. Only a few iterations can provide very accurate approximate analytical solutions even if the nonlinearity and damping are significant. The method can be applied to general oscillators with odd nonlinearities as well as even ones even without linear restoring force. Three examples are presented to illustrate the usefulness and effectiveness of the proposed method.     

An analytical approximate technique for a class of strongly non-linear oscillators

An analytical approximate technique for large amplitude oscillations of a class of conservative single degree-of-freedom systems with odd non-linearity is proposed. The method incorporates salient features of both Newton's method and the harmonic balance method. Unlike the classical harmonic balance method, accurate analytical approximate solutions are possible because linearization of the governing differential equation by Newton's method is conducted prior to harmonic balancing. The approach yields simple linear algebraic equations instead of non-linear algebraic equations without analytical solution. With carefully constructed iterations, only a few iterations can provide very accurate analytical approximate solutions for the whole range of oscillation amplitude beyond the domain of possible solution by the conventional perturbation methods or harmonic balance method. Three examples including cubic-quintic Duffing oscillators are presented to illustrate the usefulness and effectiveness of the proposed technique.     

Residue-regulating homotopy method for strongly nonlinear oscillators

Nonlinear Dynamics - It is highly desired yet challenging to obtain analytical approximate solutions to strongly nonlinear oscillators accurately and efficiently. Here we propose a new approach,...     

Application of a modified Lindstedt–Poincaré method in coupled TDOF systems with quadratic nonlinearity and a constant external excitation

An analytical approach is developed for the nonlinear oscillation of a conservative, two-degree-of-freedom (TDOF) mass-spring system with serial combined linear–nonlinear stiffness excited by a constant external force. The main idea of the proposed approach lies in two categories, the first one is the transformation of two nonlinear differential equations of a two-mass system using suitable intermediate variables into a single nonlinear differential equation. Another is the treatment a quadratic nonlinear oscillator (QNO) by the modified Lindstedt–Poincaré (L-P) method presented recently by the authors. The first-order and second-order analytical approximations for the modified L-P method are established for the QNOs with satisfactory results. After solving the nonlinear differential equation, the displacements of two-mass system can be obtained directly from the governing linear second-order differential equation. Unlike the common perturbation method, the modified L-P method is valid for weak as well as strong nonlinear oscillation systems. On the other hand, the new approach yields simple approximate analytical expressions valid for small as well as large amplitudes of oscillation. In short, this new approach yields extended scope of applicability, simplicity, flexibility in application, and avoidance of complicated numerical integration as compared to the previous approaches such as the perturbation and classical harmonic balance methods. Two examples of nonlinear TDOF mass-spring systems excited by a constant external force are selected and the approximate solutions are verified with the exact solutions derived from the Jacobi elliptic function and also the numerical fourth-order Runge–Kutta solutions.     

Oscillators with a quasi-constant restoring force: approximations for motion

In this work approximate solutions to conservative single-degree of freedom oscillators with a restoring force close to the one with a constant magnitude are derived. Approximate solutions are assumed as a truncated Fourier series and harmonic balancing is applied. In addition, the assumption that the response of the oscillators considered is close to the response of the antisymmetric oscillator is introduced. It is suggested in a novel way how to modify the differential equation of motion with the assumed solution so as to derive explicit expressions for the frequency and the amplitudes of harmonics in the first, second and third approximation are presented. The comparison of the results obtained with numerical solutions as well as with some existing approximate analytical results from the literature is also carried out, showing excellent accuracy.     

A further investigation about a simple model of the hunting behavior of some old locomotives

This study is concerned with forced damped purely nonlinear oscillators and their behaviour at different excitation frequencies. First, their dynamics is considered numerically for the response determined in the vicinity of a backbone curve with the aim of detecting coexisting responses that have not been found analytically so far. Both the cases of low and high excitation amplitudes are investigated. Second, the angular excitation frequency is lowered significantly for different powers of nonlinearity, and the system’s behaviour is examined qualitatively, which has not been considered previously related to a general class of purely nonlinear oscillators. It is illustrated that the response at a low-valued angular excitation frequency has a form of bursting oscillations, consisting of fast oscillations around a slow flow. Finally, approximate analytical solutions are presented for the slow and fast flow for a general class of purely nonlinear oscillators.     

Accurate analytical perturbation approach for large amplitude vibration of functionally graded beams

The present work derives the accurate analytical solutions for large amplitude vibration of thin functionally graded beams. In accordance with the Euler–Bernoulli beam theory and the von Kármán type geometric non-linearity, the second-order ordinary differential equation having odd and even non-linearities can be formulated through Hamilton's principle and Galerkin's procedure. This ordinary differential equation governs the non-linear vibration of functionally graded beams with different boundary constraints. Building on the original non-linear equation, two new non-linear equations with odd non-linearity are to be constructed. Employing a generalised Senator–Bapat perturbation technique as an ingenious tool, two newly formulated non-linear equations can be solved analytically. By selecting the appropriate piecewise approximate solutions from such two new non-linear equations, the analytical approximate solutions of the original non-linear problem are established. The present solutions are directly compared to the exact solutions and the available results in the open literature. Besides, some examples are selected to confirm the accuracy and correctness of the current approach. The effects of boundary conditions and vibration amplitudes on the non-linear frequencies are also discussed.     

Accurate Higher-Order Analytical Approximate Solutions to Large-Amplitude Oscillating Systems with a General Non-Rational Restoring Force

A new approximate analytical approach for accurate higher-order nonlinear solutions of oscillations with large amplitude is presented in this paper. The oscillatory system is subjected to a non-rational restoring force. This approach is built upon linearization of the governing dynamic equation associated with the method of harmonic balance. Unlike the classical harmonic balance method, simple linear algebraic equations instead of nonlinear algebraic equations are obtained upon linearization prior to harmonic balancing. This approach also explores large parameter regions beyond the classical perturbation methods which in principle are confined to problems with small parameters. It has significant contribution as there exist many nonlinear problems without small parameters. Through some examples in this paper, we establish the general approximate analytical formulas for the exact period and periodic solution which are valid for small as well as large amplitudes of oscillation.     

An equivalent polynomial approximation technique in nonlinear structural dynamics

Summary A new technique is proposed to obtain an approximate probability density for the response of a general nonlinear system under Gaussian white noise excitations. In this new technique, the original nonlinear system is replaced by another equivalent nonlinear system, structured by the polynomial formula, for which the exact solution of stationary probability density function is obtainable. Since the equivalent nonlinear system structured in this paper originates directly from certain classes of real nonlinear mechanical systems, the technique is applied to some very challenging nonlinear systems in order to show its power and efficiency. The calculated results show that applying the technique presented here can yield exact stationary solutions for the nonlinear oscillators. This is obtained by using an energy-dependent system, and for a nonlinearity of a more complex type. A more accurate approximate solution is then available, and is compared with the approximation. Application of the technique is illustrated by examples.     

非线性模态的分类和新的求解方法

引入不可分偶数维不变流形的概念来定义非线性模态．在此基础上，揭示出了一种新的模态——耦合非线性模态，并对实际系统中各种可能的模态进行了分类．这种分类可能是新的构筑非线性模态理论的框架．用此方法构造非线性模态，得到的模态振子具有范式的形式，形式最简、却能反映原系统在平衡点附近的主要动力学行为，且易于得到非线性频率及非线性稳定性等方面的信息．不仅适用于分析一般的多自由度系统，还可用于分析奇数维系统；不仅可构造内共振系统的非耦合模态，还可用于构造内共振耦合模态．从掌握的资料看，以前的方法还不能解决上述所有问题     

A New Method for Approximate Analytical Solutions to Nonlinear Oscillations of Nonnatural Systems

This paper deals with nonlinear oscillations of a conservative,nonnatural, single-degree-of-freedom system with odd nonlinearity. Bycombining the linearization of the governing equation with the method ofharmonic balance, we establish approximate analytical solutions for thenonlinear oscillations of the system. Unlike the classical harmonicbalance method, the linearization is performed prior to proceeding withharmonic balancing thus resulting in linear algebraic equations insteadof nonlinear algebraic equations. Hence, we are able to establish theapproximate analytical formulas for the exact period and periodicsolution. These approximate solutions are valid for small as well aslarge amplitudes of oscillation. Two examples are presented toillustrate that the proposed formulas can give excellent approximateresults.     

Oscillators with a power-form restoring force and fractional derivative damping: Application of averaging

In this paper free oscillators with a power-form restoring force and with a fractional derivative damping term are considered. An analytical approach based on the averaging method is adjusted to derive analytical expressions for the amplitude and phase of oscillations. Effects of the fractional-order derivative on the amplitude and frequency of oscillations are discussed in several examples, including a generalized van der Pol oscillator, purely nonlinear oscillators and a linear oscillator.     

Energy localization and white noise-induced enhancement of response in a micro-scale oscillator array

In this work, the authors seek to develop an analytical framework to understand the influence of noise on an array of micro-scale oscillators with special attention to the phenomenon of intrinsic localized modes (ILMs). It was recently shown by one of the authors and co-workers (Dick et al. in Nonlinear Dyn. 54:13, 2008) that ILMs can be realized as nonlinear vibration modes. Building on this work, it is shown here that white noise excitation, by itself, is unable to produce ILMs in an array of coupled nonlinear oscillators. However, in the case of an array subjected to a combined deterministic and random excitation, the obtained numerical results indicate the existence of a threshold noise strength beyond which the ILM at one location in attenuated whilst the localization in strengthened at another location in the array. The numerical results further motivate the formulation of a general analytical framework wherein the Fokker–Planck equation is derived for a typical coupled oscillator cell of the array subjected to a combined white noise and deterministic excitation. With a set of approximations, the moment evolution equations are derived from the Fokker–Planck equation and they are numerically solved. These solutions indicate that once a localization event occurs in the array, a random excitation with noise strength above a threshold value contributes to the sustenance of the event. It is also observed that an excitation with a higher noise strength results in enhanced response amplitudes for oscillators in the center of the array. The efforts presented in this paper, in addition to providing an analytical framework for developing a fundamental understanding of the influence of white noise on the dynamics of coupled oscillator arrays, suggest that noise may be potentially used to manipulate the formation and persistence of ILMs in such arrays. Furthermore, the occurrence of enhanced response amplitudes due to an excitation with a high noise strength indicates that the framework may also be used to investigate stochastic resonance-type phenomena in coupled arrays of nonlinear oscillators including micro-scale oscillator arrays.     

A Modified Perturbation Technique Depending Upon an Artificial Parameter

In this paper, a modified perturbation method is proposed to search for analytical solutions of nonlinear oscillators without possible small parameters. An artificial perturbation equation is carefully constructed by embedding an artificial parameter, which is used as expanding parameter. It reveals that various traditional perturbation techniques can be powerfully applied in this theory. Some examples, such as the Duffing equation and the van der Pol equation, are given here to illustrate its effectiveness and convenience. The results show that the obtained approximate solutions are uniformly valid on the whole solution domain, and they are suitable not only for weak nonlinear systems, but also for strongly nonlinear systems. In applying the new method, some special techniques have been emphasized for different problems.     

Averaging Oscillations with Small Fractional Damping and Delayed Terms

We demonstrate the method of averaging for conservative oscillators which may be strongly nonlinear, under small perturbations including delayed and/or fractional derivative terms. The unperturbed systems studied here include a harmonic oscillator, a strongly nonlinear oscillator with a cubic nonlinearity, as well as one with a nonanalytic nonlinearity. For the latter two cases, we use an approximate realization of the asymptotic method of averaging, based on harmonic balance. The averaged dynamics closely match the full numerical solutions in all cases, verifying the validity of the averaging procedure as well as the harmonic balance approximations therein. Moreover, interesting dynamics is uncovered in the strongly nonlinear case with small delayed terms, where arbitrarily many stable and unstable limit cycles can coexist, and infinitely many simultaneous saddle-node bifurcations can occur.     

Preliminary report on the energy balance for nonlinear oscillations

In this paper, a reliable technique for calculating angular frequencies of nonlinear oscillators is developed. The new algorithm offers a promising approach by constructing a Hamiltonian for the nonlinear oscillator. Some illustrative examples are given.     

Transitions from strongly to weakly-nonlinear dynamics in a class of exactly solvable oscillators and nonlinear beat phenomena

In this paper, a regular perturbation tool is suggested to bridge the gap between weakly and strongly nonlinear dynamics based on exactly solvable oscillators with trigonometric characteristics considered by Nesterov (Proc. Mosc. Inst. Power Eng. 357:68–70, 1978). It is shown that the corresponding action-angle variables linearize the original oscillators with no special functions involved. As a result, linear and strongly nonlinear areas of the dynamics are described within the same perturbation procedure. The developed tool is applied then to analyzing the nonlinear beat and energy localization phenomena in two linearly coupled Duffing oscillators. It is shown that the principal phase variable describing the beat phenomena is governed by the hardening Nesterov oscillator with some perturbation due to qubic nonlinearity and coupling between the oscillators. As a result, the above class of strongly nonlinear oscillators is given clear physical meaning, whereas a closed form analytical solution is obtained for nonlinear beat and localization dynamics. Based on this solution, necessary and sufficient conditions for onset of energy localization are obtained.     

Weakly coupled parametrically forced oscillator networks: existence,stability, and symmetry of solutions

In this paper, we discuss existence, stability, and symmetry of solutions for networks of parametrically forced oscillators. We consider a nonlinear oscillator model with strong 2:1 resonance via parametric excitation. For uncoupled systems, the 2:1 resonance property results in sets of solutions that we classify using a combinatorial approach. The symmetry properties for solution sets are presented as are the group operators that generate the isotropy subgroups. We then impose weak coupling and prove that solutions from the uncoupled case persist for small coupling by using an appropriate Poincaré map and the Implicit Function Theorem. Solution bifurcations are investigated as a function of coupling strength and forcing frequency using numerical continuation techniques. We find that the characteristics of the single oscillator system are transferred to the network under weak coupling. We explore interesting dynamics that emerge with larger coupling strength, including anti-synchronized chaos and unsynchronized chaos. A classification for the symmetry-breaking that occurs due to weak coupling is presented for a simple example network.