Explicit dispersion relation for strongly nonlinear flexural waves using the homotopy analysis method

Nonlinear Dynamics - Using the homotopy analysis method, we present an explicit frequency-versus-wavenumber nonlinear dispersion relation for a flexural elastic beam. In our analysis, we employ the...     

Asymptotic analysis of strongly nonlinear oscillators

In this paper, an asymptotic method is presented for the analysis of a class of strongly nonlinear autonomous oscillators. The equations governing the amplitude and phase factor are obtained, and the amplitude and stability of the corresponding limit cycles are determined.     

Separatrices and limit cycles of strongly nonlinear oscillators by the perturbation-incremental method

The perturbation-incremental method is applied to determine the separatrices and limit cycles of strongly nonlinear oscillators. Conditions are derived under which a limit cycle is created or destroyed. The latter case may give rise to a homoclinic orbit or a pair of heteroclinic orbits. The limit cycles and the separatrices can be calculated to any desired degree of accuracy. Stability and bifurcations of limit cycles will also be discussed.     

A non-linear seales method for strongly non-linear oscillators

A non-linear seales method is presented for the analysis of strongly non-linear oseillators of the form % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaamXvP5wqonvsaeHbfv3ySLgzaGqbdiqb-Hha4zaadaGaey4kaSIa% am4zaiaacIcacqWF4baEcaGGPaGae8xpa0JaeqyTduMaamOzaiaacI% cacqWF4baEcqWFSaalcuWF4baEgaGaaiaabMcaaaa!4FEC!\[\ddot x + g(x) = \varepsilon f(x,\dot x{\text{)}}\], where g(x) is an arbitrary non-linear function of the displacement x. We assumed that % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaamXvP5wqonvsaeHbfv3ySLgzaGqbdiab-Hha4jaacIcacqWF0baD% cqWFSaalcqaH1oqzcaGGPaGaeyypa0Jae8hEaG3aaSbaaSqaaiaaic% daaeqaaOGaaiikaiabe67a4jaacYcacqaH3oaAcaGGPaGaey4kaSYa% aabmaeaacqaH1oqzdaahaaWcbeqaaiaad6gaaaaabaGaamOBaiabg2% da9iaaigdaaeaacaWGTbGaeyOeI0IaaGymaaqdcqGHris5aOGae8hE% aG3aaSbaaSqaaiab-5gaUbqabaGccaGGOaGaeqOVdGNaaiykaiabgU% caRiaad+eacaGGOaGaeqyTdu2aaWbaaSqabeaacaWGTbaaaOGaaiyk% aaaa!67B9!\[x(t,\varepsilon ) = x_0 (\xi ,\eta ) + \sum\nolimits_{n = 1}^{m - 1} {\varepsilon ^n } x_n (\xi ) + O(\varepsilon ^m )\], where % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiaabsgacqaH+oaEcaGGVaGaaeizaiaadshacqGH9aqpdaaeWaqa% aiabew7aLnaaCaaaleqabaGaamOBaaaaaeaacaWGUbGaeyypa0JaaG% ymaaqaaiaad2gaa0GaeyyeIuoakiaadkfadaWgaaWcbaGaamOBaaqa% baGccaGGOaGaeqOVdGNaaiykaaaa!4FFC!\[{\text{d}}\xi /{\text{d}}t = \sum\nolimits_{n = 1}^m {\varepsilon ^n } R_n (\xi )\], % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiaabsgacqaH3oaAcaGGVaGaaeizaiaadshacqGH9aqpdaaeWaqa% aiabew7aLnaaCaaaleqabaGaamOBaaaaaeaacaWGUbGaeyypa0JaaG% imaaqaaiaad2gaa0GaeyyeIuoakiaadofadaWgaaWcbaGaamOBaaqa% baGccaGGOaGaeqOVdGNaaiilaiabeE7aOjaacMcaaaa!5241!\[{\text{d}}\eta /{\text{d}}t = \sum\nolimits_{n = 0}^m {\varepsilon ^n } S_n (\xi ,\eta )\], and R _n,S _nare to be determined in the course of the analysis. This method is suitable for the systems with even non-linearities as well as with odd non-linearities. It can be viewed as a generalization of the two-variable expansion procedure. Using the present method we obtained a modified Krylov-Bogoliubov method. Four numerical examples are presented which served to demonstrate the effectiveness of the present method.     

A perturbation-incremental method for strongly non-linear non-autonomous oscillators

A perturbation-incremental method is extended for the analysis of strongly non-linear non-autonomous oscillators of the form , where g(x) and are arbitrary non-linear functions of their arguments, and ε can take arbitrary values. Limit cycles of the oscillators can be calculated to any desired degree of accuracy and their stabilities are determined by the Floquet theory. Branch switching at period-doubling bifurcation along a frequency-response curve is made simple by the present method. Subsequent continuation of an emanating branch is also discussed.     

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.     

Homoclinic and heteroclinic solutions of cubic strongly nonlinear autonomous oscillators by the hyperbolic perturbation method

Nonlinear Dynamics - The hyperbolic perturbation method is applied to determining the homoclinic and heteroclinic solutions of cubic strongly nonlinear autonomous oscillators of the form...     

Transient mode localization in coupled strongly nonlinear exactly solvable oscillators

Coupled strongly nonlinear oscillators, whose characteristic is close to linear for low amplitudes but becomes infinitely growing as the amplitude approaches certain limit, are considered in this paper. Such a model may serve for understanding the dynamics of elastic structures within the restricted space bounded by stiff constraints. In particular, this study focuses on the evolution of vibration modes as the energy is gradually pumped into or dissipates out of the system. For instance, based on the two degrees of freedom system, it is shown that the in-phase and out-of-phase motions may follow qualitatively different scenarios as the system’ energy increases. So the in-phase mode appears to absorb the energy with equipartition between the masses. In contrast, the out-of-phase mode provides equal energy distribution only until certain critical energy level. Then, as a result of bifurcation of the 1:1 resonance path, one of the masses becomes a dominant energy receiver in such a way that it takes the energy not only from the main source but also from another mass.     

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.     

Nonstationary probability densities of strongly nonlinear single-degree-of-freedom oscillators with time delay

The approximate nonstationary probability density of a nonlinear single-degree-of-freedom (SDOF) oscillator with time delay subject to Gaussian white noises is studied. First, the time-delayed terms are approximated by those without time delay and the original system can be rewritten as a nonlinear stochastic system without time delay. Then, the stochastic averaging method based on generalized harmonic functions is used to obtain the averaged Itô equation for amplitude of the system response and the associated Fokker–Planck–Kolmogorov (FPK) equation governing the nonstationary probability density of amplitude is deduced. Finally, the approximate solution of the nonstationary probability density of amplitude is obtained by applying the Galerkin method. The approximate solution is expressed as a series expansion in terms of a set of properly selected basis functions with time-dependent coefficients. The proposed method is applied to predict the responses of a Van der Pol oscillator and a Duffing oscillator with time delay subject to Gaussian white noise. It is shown that the results obtained by the proposed procedure agree well with those obtained from Monte Carlo simulation of the original systems.     

Improved L-P method for solving strongly nonlinear problems

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｓｔｒｏｎｇｌｙｎｏｎｌｉｎｅａｒｏｓｃｉｌｌａｔｉｏｎｐｒｏｂｌｅｍ¨ｕ ω20 ｕ =εｆ(ｕ , ｕ ,¨ｕ) , (1 )ｗｈｅｒｅｔｈｅｓｙｍｂｏｌ“·”ｄｅｎｏｔｅｓｔｈｅｄｉｆｆｅｒｅｎｔｉａｔｉｏｎｗｉｔｈｒｅｓｐｅｃｔｔｏｔｈｅｉｎｄｅｐｅｎｄｅｎｔｖａｒｉａｂｌｅｔａｎｄω0ａｎｄεａｒｅａｌｌｔｈｅａｒｂｉｔｒａｒｙｃｏｎｓｔａｎｔｓ,ｈａｖｅｉｎｔｈｅｒｅｃｅｎｔｙｅａｒｓｓｅｖｅｒａｌｉｍｐｒｏｖｅｄＬ＿Ｐｍｅｔｈｏｄｓｗｈｅｎεｉｓｎｏｔｓｍａｌｌ (ｓｔｒｏ…     

An equivalent nonlinearization method for strongly nonlinear oscillations

An equivalent nonlinearization method is proposed for the study of certain kinds of strongly nonlinear oscillators. This method is to express the nonlinear restored force of an oscillatory system by a polynomial of degree two or three such that the asymptotic solutions can be derived in terms of elliptic functions. The least squares method is used to determine the coefficients of approximate polynomials. The advantage of present method is that it is valid for relatively large oscillations. As an application, a strongly nonlinear oscillator with slowly varying parameters resulted from free-electron laser is studied in detail. Comparisons are made with other methods to assess the accuracy of the present method.     

A power-form method for dynamic systems: investigating the steady-state response of strongly nonlinear oscillators by their equivalent Duffing-type equation

This paper aims to apply a transformation method that replaces the elastic forces of the original equation of motion with a power-form elastic term. The accuracy obtained from the derived equivalent equations of motion is evaluated by studying the finite-amplitude damped, forced vibration of a vertically suspended load body supported by incompressible, homogeneous, and isotropic viscohyperelastic elastomer materials. Numerical integrations of the original equations of two oscillators described by neo-Hookean and Mooney–Rivlin viscohyperelastic elastomer material models, and their equivalent equations of motion, are compared to the frequency–amplitude steady-state solutions obtained from the harmonic balance and the averaging methods. It is shown from numerical integrations and approximate steady-state solutions that the equivalent equations predict well the original system dynamic response despite having higher system nonlinearities.

     

Comparison of multiple scales and KBM methods for strongly nonlinear oscillators with slowly varying parameters

Two versions of multiple scales and Krylov–Bogoliubov–Mitropolsky (KBM) methods are extended to obtain the asymptotic solutions of a class of strongly nonlinear oscillators with slowly varying parameters. A comparison of the two methods is made to show that the multiple scales method is equivalent to the KBM method for the first order approximation. An example of pendulum with slowly varying length is given to illustrate the comparison of the two methods.     

First-passage failure of strongly nonlinear oscillators under combined harmonic and real noise excitations

First-passage failure of strongly nonlinear oscillators under combined harmonic and real noise excitations is studied. The motion equation of the system is reduced to a set of averaged Itô stochastic differential equations by stochastic averaging in the case of resonance. Then, the backward Kolmogorov equation governing the conditional reliability function and a set of generalized Pontryagin equations governing the conditional moments of first-passage time are established. Finally, the conditional reliability function and the conditional probability density and mean first-passage time are obtained by solving the backward Kolmogorov equation and Pontryagin equation with suitable initial and boundary conditions. The procedure is applied to Duffing–van der Pol system in resonant case and the analytical results are verified by Monte Carlo simulation.     

A generalized frequency detuning method for multidegree-of-freedom oscillators with nonlinear stiffness

In this paper, we derive a frequency detuning method for multi-degree-of-freedom oscillators with nonlinear stiffness. This approach includes a matrix of detuning parameters, which are used to model the amplitude dependent variation in resonant frequencies for the system. As a result, we compare three different approximations for modeling the affect of the nonlinear stiffness on the linearized frequency of the system. In each case, the response of the primary resonances can be captured with the same level of accuracy. However, harmonic and subharmonic responses away from the primary response are captured with significant differences in accuracy. The detuning analysis is carried out using a normal form technique, and the analytical results are compared with numerical simulations of the response. Two examples are considered, the second of which is a two degree-of-freedom oscillator with cubic stiffnesses.     

A new method for the periodic solution of strongly nonlinear systems

A new method for the periodic solution of strongly nonlinear system is given. By using this method, the existance and stability of the periodic solution can be decided, and the approximate expression of the periodic solution can also be found. The project supported by the National Natural Science Foundation of China     

Generalized hyperbolic perturbation method for homoclinic solutions of strongly nonlinear autonomous systems

A generalized hyperbolic perturbation method is presented for homoclinic solutions of strongly nonlinear autonomous oscillators,in which the perturbation procedure is improved for those systems whose exact homoclinic generating solutions cannot be explicitly derived.The generalized hyperbolic functions are employed as the basis functions in the present procedure to extend the validity of the hyperbolic perturbation method.Several strongly nonlinear oscillators with quadratic,cubic,and quartic nonlinearity are studied in detail to illustrate the efficiency and accuracy of the present method.     

Existence, number, and stability of limit cycles in weakly dissipative, strongly nonlinear oscillators

Oscillators control many functions of electronic devices, but are subject to uncontrollable perturbations induced by the environment. As a consequence, the influence of perturbations on oscillators is a question of both theoretical and practical importance. In this paper, a method based on Abelian integrals is applied to determine the emergence of limit cycles from centers, in strongly nonlinear oscillators subject to weak dissipative perturbations. It is shown how Abelian integrals can be used to determine which terms of the perturbation are influent. An upper bound to the number of limit cycles is given as a function of the degree of a polynomial perturbation, and the stability of the emerging limit cycles is discussed. Formulas to determine numerically the exact number of limit cycles, their stability, shape and position are given.