Resolving Controversies in the Application of the Method of Multiple Scales and the Generalized Method of Averaging

I compare application of the method of multiple scales with reconstitution and the generalized method of averaging for determining higher-order approximations of three single-degree-of-freedom systems and a two-degree-of-freedom system. Three implementations of the method of multiple scales are considered, namely, application of the method to the system equations expressed as second-order equations, as first-order equations, and in complex-variable form. I show that all of these methods produce the same modulation equations.I address the problem of determining higher-order approximate solutions of the Duffing equation in the case of primary resonance. I show that the conclusions of Rahman and Burton that the method of multiple scales, the generalized method of averaging, and Lie series and transforms might lead to incorrect results, in that spurious solutions occur and the obtained frequency–response curves bear little resemblance to the actual response, is the result of their using parameter values for which the neglected terms are the same order as the retained terms. I show also that spurious solutions cannot be avoided, in general, in any consistent expansion and their presence does not constitute a limitation of the methods. In particular, I show that, for the Duffing equation, the second-order frequency–response equation does not possess spurious solutions for the case of hardening nonlinearity, but possesses spurious solutions for the case of softening nonlinearity. For sufficiently small nonlinearity, the spurious solutions are far removed from the actual response. But as the strength of the nonlinearity increases, these solutions move closer to the backbone and eventually distort it. This is not a drawback of the perturbation methods but an indication of an application of the analysis for parameter values outside the range of validity of the expansion.Also, I address the problem of obtaining non-Hamiltonian modulation equations in the application of the method of multiple scales to multi-degree-of-freedom Hamiltonian systems written as second-order equations in time and how this problem can be overcome by attacking the state-space form of the governing equations. Moreover, I show that application of a variation of the method of Rahman and Burton to multi-degree-of-freedom systems leads to results that do not agree with those obtained with the generalized method of averaging.Contributed by Prof. R.A. Ibrahim.     

范式理论与平均法的等价性分析

分析了范式理论与平均法的等价性。得到的结论是：对含有一对纯虚根的二维非线性系统，使用两种方法得到的结果是等价的，并提供了两个算例来证实其结论的正确性。虽然分析是针对一类二维非线性系统，但其结论同样适合于高维非线性系统。     

Periodic solutions of strongly non-linear Mathieu oscillators

The purpose of this paper is to continue our investigation into periodic solutions of strongly non-linear Mathieu oscillators. The modified version of the generalized averaging method which we developed recently is applied to derive highly accurate analytical expressions for these periodic solutions. These analytical results are used, together with the perturbation methods of multiple time scaling, to obtain second order expressions for the stability regions of these periodic solutions. The analytical research results are verified with numerical computations. Very good agreement is found, which shows the applicability of the modified version of the generalized averaging method to this kind of strongly non-linear oscillators. These oscillators may be used to model the beam-beam interaction in particle accelerators.     

Fourier averaging: a phase-averaging method for periodic flow

A new phase-averaging method, denoted as Fourier averaging, is presented for the investigation of periodic flows. In such flows, the moments of velocity, as estimated from a small number of samples, show fluctuations in their phasewise development. In previous methods these fluctuations are reduced by calculating moments from large phase intervals. Fourier averaging, in contrast, neglects high-frequency fluctuations and assumes that they are of no physical relevance. This method supplies additional information on amplitudes and phase angles of discrete frequencies, which may then be used for visualizations of flow fields at any desired phase increment. The Fourier averaging method was verified empirically by LDA measurements and compared to other methods. It is shown that the results obtained by Fourier averaging are more accurate than for previously known methods. Received: 15 June 1998/Accepted: 15 April 1999     

含分数阶Bingham模型的阻尼减振系统时滞半主动控制

针对基于磁流变液阻尼器的半主动控制系统中存在的时滞问题, 采用了一种将可控的时滞变量引入半主动控制切换条件的控制策略, 研究了考虑时滞的天棚阻尼控制切换条件对半主动阻尼减振系统的影响, 分析了含有分数阶Bingham模型的线性刚度系统在基础激励下的振动特性. 利用平均法得到了系统在含时滞半主动控制策略下主共振响应的近似解析解, 根据Lyapunov理论分析了系统的稳定性. 通过数值解验证了近似解析解的准确性, 二者具有较好的一致性. 利用近似解析解分析了固定激励频率下时滞对系统幅频响应特性的影响, 以及主共振峰值响应和共振频率随时滞变化的特性规律. 结果表明, 含时滞的半主动控制系统存在一个小时滞区间, 使得系统的振幅在主共振峰对应的频率附近低于不考虑时滞时系统的振幅, 且存在最优时滞使得系统的振幅大幅度降低; 而大时滞的引入会加剧系统的振动, 导致系统的颤振. 确定了基于分数阶Bingham模型的线性刚度系统在天棚阻尼半主动控制下的时滞选取原则, 为振动系统半主动阻尼控制中的时滞选取提供了参考.      

A modified method of averaging for solving a class of nonlinear equations

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Free Vibration Analysis of Rotating Nonlinearly Elastic Structures with Symmetry: An Efficient Group-Equivariance Approach

A study is made of the dynamics of oscillating systems with a slowly varying parameter. A slowly varying forcing periodically crosses a critical value corresponding to a pitchfork bifurcation. The instantaneous phase portrait exhibits a centre when the forcing does not exceed the critical value, and a saddle and two centres with an associated double homoclinic loop separatrix beyond this value. The aim of this study is to construct a Poincaré map in order to describe the dynamics of the system as it repeatedly crosses the bifurcation point. For that purpose averaging methods and asymptotic matching techniques connecting local solutions are applied. Given the initial state and the values of the parameters the properties of the Poincaré map can be studied. Both sensitive dependence on initial conditions and (quasi) periodicity are observed. Moreover, Lyapunov exponents are computed. The asymptotic expressions for the Poincaré map are compared with numerical solutions of the full system that includes small damping.     

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.     

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.     

Dynamical responses of airfoil models with harmonic excitation under uncertain disturbance

In this paper, we investigate nonlinear dynamical responses of two-degree-of-freedom airfoil (TDOFA) models driven by harmonic excitation under uncertain disturbance. Firstly, based on the deterministic airfoil models under the harmonic excitation, we introduce stochastic TDOFA models with the uncertain disturbance as Gaussian white noise. Subsequently, we consider the amplitude–frequency characteristic of deterministic airfoil models by the averaging method, and also the stochastic averaging method is applied to obtain the mean-square response of given stochastic TDOFA systems analytically. Then, we carry out numerical simulations to verify the effectiveness of the obtained analytic solution and the influence of harmonic force on the system response is studied. Finally, stochastic jump and bifurcation can be found through the random responses of system, and probability density function and time history diagrams can be obtained via Monte Carlo simulations directly to observe the stochastic jump and bifurcation. The results show that noise can induce the occurrence of stochastic jump and bifurcation, which will have a significant impact on the safety of aircraft.     

Harmonic Balance Based Averaging: Approximate Realizations of an Asymptotic Technique

Averaging is a classical asymptotic technique commonly used to studyweakly nonlinear oscillations via small perturbations of the harmonicoscillator. If the unperturbed oscillator is autonomous and stronglynonlinear, but with a two-parameter family of periodic solutions, thenaveraging is allowed in principle but typically not considered feasibleunless (a) the required family of unperturbed periodic solutions can befound in closed form, and (b) the averaging integrals can be found inclosed form. Often, the foregoing requirements cannot be met. Here, itis shown how both these difficulties can be bypassed using the classicalbut heuristic approximation method of harmonic balance, to obtain approximate realizations of the asymptotic analytical technique. Theadvantages of the present approach are that (a) closed form solutions tothe unperturbed problem are not needed, and (b) the heuristic andasymptotic parts of the calculation are kept conceptually distinct, withscope for refining the former, while preserving the asymptotic nature ofthe latter. Several examples are provided, including oscillators with astrong cubic nonlinearity, velocity dependent nonlinear terms (includinga strongly nonconservative system), a nondifferentiable characteristic,and a strongly nonlinear but homogeneous function of order 1; dynamicphenomena investigated include damped oscillations, limit cycles, forcedoscillations near resonance, and subharmonic entrainment. Goodapproximations are obtained in each case.     

Method of constructing estimates of the solution of equations of filtration of an aerated liquid

The exact solutions of the nonlinear equations of filtration of an aerated liquid have been obtained in [1–3]. In [4] the system of equations of an aerated liquid have been reduced to the heat-conduction equation under certain assumptions. An approximate method of computing the nonsteady flow of an aerated liquid is given in [5], where the real flow pattern is replaced by a computational scheme of successive change of stationary states. In [6] the same problem is solved by the method of averaging. In the present article estimates of the solution of the equations for nonstationary filtration of an aerated liquid in one-dimensional layer are constructed under certain conditions imposed on the desired functions. These estimates can be used as approximate solutions with known error or for the verification of the accuracy of different approximate methods. We note that the use of comparison theorem for the estimate of solutions of equations of nonlinear filtration is discussed in [7–9]. The methods of constructing estimates of solutions of various problems of heat conduction are given in [10, 11]     

随机平均规范形方法

计算随机规范形系数是应用随机规范形方法的关键．提出一种应用随机平均计算随机规范形系数的方法．为了说明方法的有效性,对白噪声激励的Duffing系统,经过变换,对于相应的平均方程,比较了精确解、规范形方法解和能量包络方法解的稳态概率密度,结果表明,当非线性项系数较小时,三者完全一致．当非线性项系数较大时,规范形方法所得解与精确解相差不大．     

An improved approach for random parametric response of dynamic systems with non-linear inertia

A non-Gaussian closure scheme is developed for determining the stationary response of dynamic systems including non-linear inertia and stochastic coefficients. Numerical solutions are obtained and examined for their validity based on the preservation of moments properties. The method predicts the jump phenomenon, for all response statistics at an excitation level very close to the threshold level of the condition of almost sure stability. In view of the increased degree of non-linearity, resulting from the non-Gaussian closure scheme, the mean square of the response displacement is found to be less than those values predicted by other methods such as the Gaussian closure or the first order stochastic averaging.     

Versal Deformation and Local Bifurcation Analysis of Time-Periodic Nonlinear Systems

In this study a local semi-analytical method of quantitativebifurcation analysis for time-periodic nonlinear systems is presented.In the neighborhood of a local bifurcation point the system equationsare simplified via Lyapunov–Floquet transformation whichtransforms the linear part of the equation into a dynamically equivalenttime-invariant form. Then the time-periodic center manifoldreduction is used to separate the `critical' states and reduce thedimension of the system to a possible minimum. The center manifoldequations can be simplified further via time-dependent normal formtheory. For most codimension one cases these nonlinear normal forms arecompletely time-invariant. Versal deformation of thesetime-invariant normal forms can be found and the bifurcation phenomenoncan be studied in the neighborhood of the critical point. However, ingeneral, it is not a trivial task to find a quantitatively correctversal deformation for time-periodic systems. In order to do so, onemust find a relationship between the bifurcation parameter of theoriginal time-periodic system and the versal deformation parameter ofthe time-invariant normal form. Essentially one needs to find theeigenvalues of the fundamental solution matrix of the time-periodicproblem in terms of the system parameters, which, in general, cannot bedone due to computational difficulties. In this work two ideas areproposed to achieve this goal. The eigenvalues of the fundamentalsolution matrix can be related to the versal deformation parameter bysensitivity analysis and an approximation of any desired order can beobtained. This idea requires a symbolic computational procedure whichcan be very time consuming in some cases. An alternative method issuggested for faster results in which a second or higher order curvefitting technique is used to find the relationship. Once thisrelationship is established, closed form post-bifurcation steady-statesolutions can be obtained for flip, symmetry breaking, transcritical andsecondary Hopf bifurcations. Unlike averaging and perturbation methods,the proposed technique is applicable at any bifurcation point in theparameter space. As physical examples, a simple and a double pendulumsubjected to periodic parametric excitation are considered. A simple twodegrees of freedom model is also studied and the results are comparedwith those obtained from the traditional averaging method. All resultsare verified by numerical integration. It is observed that the proposedtechnique yields results which are very close to the numericalsolutions, unlike the averaging method.     

Approximate analytical solutions for oscillatory and rotational motion of a parametric pendulum

In this paper, the authors have studied dynamic responses of a parametric pendulum by means of analytical methods. The fundamental resonance structure was determined by looking at the undamped case. The two typical responses, oscillations and rotations, were investigated by applying perturbation methods. The primary resonance boundaries for oscillations and pure rotations were computed, and the approximate analytical solutions for small oscillations and period-one rotations were obtained. The solution for the rotations has been derived for the first time. Comparisons between the analytical and numerical results show good agreements.     

Analysis of the periodic solutions of a smooth and discontinuous oscillator

In this paper, the periodic solutions of the smooth and discontinuous (SD) oscillator, which is a strongly irrational nonlinear system are discussed for the system having a viscous damping and an external harmonic excitation. A four dimensional averaging method is employed by using the complete Jacobian elliptic integrals directly to obtain the perturbed primary responses which bifurcate from both the hyperbolic saddle and the non-hyperbolic centres of the unperturbed system. The stability of these periodic solutions is analysed by examining the four dimensional averaged equation using Lyapunov method. The results presented herein this paper are valid for both smooth ( α > 0) and discontinuous ( α = 0) stages providing the answer to the question why the averaging theorem spectacularly fails for the case of medium strength of external forcing in the Duffing system analysed by Holmes. Numerical calculations show a good agreement with the theoretical predictions and an excellent efficiency of the analysis for this particular system, which also suggests the analysis is applicable to strongly nonlinear systems.     

Periodic solutions of rigid body–viscous flow interaction

This paper describes the work on extending the finite element method to cover interactions between a viscous flow and a moving body. The problem configuration of interest is that of an arbitrarily shaped body undergoing a simple harmonic motion in an otherwise undisturbed incompressible fluid. The finite element modelling is based on a primitive variables representation of the Navier-Stokes equations using curved isoparametric elements. The non-linear boundary conditions on the moving body are obtained using Taylor series expansion to approximate the velocities at the fixed finite element grid points. The method of averaging is used to analyse the resulting periodic motion of the fluid. The stability of the periodic solutions is studied by introducing small perturbations and applying Floquet theory. Numerical results are obtained for several example body shapes and compared with published experimental results. Good agreement is obtained for the basic non-linear phenomenon of steady streaming.     

Averaging and reduction

This paper deals with the averaging method which is the most useful technique for perturbation for a differential system. We show that in reality it is a reduction of the differential system, i.e. the initial system is replaced by a reduced system and a global connection between the two systems is emphasized. This connection is absolutely necessary in order to associate the solutions of the two systems in a structural way. In addition, this new approach (averaging as reduction) provides us with new reduced systems and the method of averaging is thus greatly extended.