The response of two-degree-of-freedom systems with quadratic and cubic non-linearities to multifrequency parametric excitations

The response of two d.o.f. systems with quadratic and cubic non-linearities to multi-frequency parametric excitations is determined by using the method of multiple scales. Four first-order ordinary differential equations are derived to describe the modulation of the amplitudes and the phases when principal parametric resonances of both modes and combination resonances of the additive and difference type occur simultaneously. In all cases the steady state solutions and their stability are determined. Numerical results depicting the various resonances are presented.     

The response of single degree of freedom systems with quadratic and cubic non-linearities to a subharmonic excitation

The response of single degree of freedom systems with quadratic and cubic nonlinearities to a subharmonic excitation is investigated. The method of multiple scales is used to derive two first order ordinary differential equations that govern the evolution of the amplitude and phase of the subharmonic. These equations are used to obtain the steady state solutions and their stability. The results identify two critical values ζ₁ and ζ₂, where ζ₂>ζ₁, for the excitation amplitude f. The value ζ₂ is the threshold for the stability of the trivial solution. When f>ζ₂, subharmonic oscillations of finite amplitude are always excited. When f<ζ₁, subharmonic oscillations cannot be excited. But when ζ₁<f<ζ₂, subharmonic oscillations may or may not be excited, depending on the initial conditions. Also, the method of harmonic balance is applied to a special case of the problem considered. It is shown that, although the method seems straightforward, it can lead to erroneous results if extreme care is not taken in the ordering of the different terms.     

Combination tones in the response of single degree of freedom systems with quadratic and cubic non-linearities

The method of multiple scales is used to analyze the response of a single-degree-of-freedom system to either the combination resonance of the additive type $\text{Ω}{}_2\text{+ Ω}{}_1\text{≈ ω}{}_0$ or the combination resonance of the difference type $\text{Ω}{}_2\text{? Ω}{}_1\text{≈ ω}{}_0$, where $\text{Ω}{}_1$ and $\text{Ω}{}_2$ are the frequencies of the excitation and ω₀ is the linear undamped natural frequency of the system. To the second approximation, the combination resonance of the additive type has three effects on the steady state response. First, it produces terms having the frequencies $\text{Ω}{}_1\text{, Ω}{}_2$ and $\text{Ω}{}_2\text{+ Ω}{}_1$ at first order and terms having the frequencies 0, $\text{2Ω}{}_1\text{, 2Ω}{}_2\text{, Ω}{}_2\text{? Ω}{}_1$, $\text{2(Ω}{}_2\text{+ Ω}{}_1\text{), Ω}{}_2\text{+ 2Ω}{}_1$ and $\text{2Ω}{}_2\text{+ Ω}{}_1$ at second order. Second, it produces a shift in the natural frequency of the system. Third, it produces a virtual primary-resonant excitation having the frequency $\text{Ω}{}_2\text{+ Ω}{}_1\text{≈ ω}{}_0$ that makes the component having the frequency $\text{Ω}{}_2\text{+ Ω}{}_1$ be of first rather than second order. Similar effects are produced by a combination resonance of the difference type or a superharmonic resonance of order two.     

The response of two-degree-of-freedom systems with quadratic non-linearities to a parametric excitation

The method of multiple scales is used to analyze the response of two-degree-of-freedom systems with quadratic non-linearities to a parametric harmonic excitation having the frequency Ω. Four ordinary differential equations are derived to describe the modulation of the amplitudes and the phases when ω₂ ≈ 2ω₁ and either $\text{Ω ≈ 2ω}{}_1$ or $\text{Ω ≈ 2ω}{}_2$, where ω₁ and ω₂ are the linear undamped natural frequencies of the system. Two critical values ζ₁ and ζ₂ of the amplitude F of the excitation are identified in the analysis. When F >ζ₂, the amplitude of the directly excited mode grows exponentially with time according to the linear analysis, whereas the amplitudes of both modes achieve steady state constant values, irrespective of the initial amplitudes, according to the non-linear analysis. When F < ζ₁, the motion decays to zero according to both the linear and non-linear analyses. When ζ₁ ? F ? ζ₂, the motion decays to zero according to the linear analysis, whereas it achieves a periodic steady state or decays to zero depending on the initial amplitudes according to the non-linear analysis. This is an example of subcritical instability. When $\text{Ω ≈ 2ω}{}_2$, the steady state value of the higher mode, which is directly excited, is a constant that is independent of the excitation of amplitude F, whereas the amplitude of the lower mode, which is indirectly excited through internal resonance, grows with the excitation amplitude F. This is another example of saturation.     

Normal modes of a model radiating system

In order to gain insight into normal modes of realistic radiating systems, we study the simple model problem of a finite string and a semi-infinite string coupled by a spring. As expected there is a family of modes which are basically the modes of the finite string slowly damped by the radiation of energy to infinity on the semi-infinite string. But we also study another family of modes, found by Dyson in a different model problem, which are strongly damped modes of the semi-infinite string itself. These may be analogous to the modes of black holes, and they are likely to be present in relativistic stars as well. The question of whether the instability in these modes which Dyson found is present in realistic stars remains open.     

The response of one-degree-of-freedom systems with cubic and quartic non-linearities to a harmonic excitation

The method of multiple scales is used to analyze the response of single-degree-of freedom systems with cubic and quartic non-linearities to a harmonic excitation. Two first-order ordinary differential equations describing the evolution of the amplitude and the phase are derived for superharmonic resonances of order two and four, subharmonic resonances of order one-half and one-fourth, and the supersubharmonic resonances of order $\text{3}\text{2}$ and $\text{2}\text{3}$. In all cases, the steady state solutions and their stability are determined and representative numerical results are included.     

Two-degree-of-freedom systems with quadratic non-linearities subjected to parametric and self excitation

The method of multiple scales is used to study the response of two-degree-of-freedom systems with quadratic non-linearities under the simultaneous effects of a harmonic parametric excitation and self excitation. The principal parametric resonance of the first mode and a three-to-one internal resonance is considered, followed by the case of internal and parametric resonance of the second mode. In both cases the stability of the system is also studied. Amplitude and frequency response curves are presented for both cases. The character of stability and the mode in which the system loses its stability is also discussed.     

The response of two-degree-of-freedom systems with quadratic non-linearities to a combination parametric resonance

The response of two-degree-of-freedom systems with quadratic non-linearities to a combination parametric resonance in the presence of two-to-one internal resonances is investigated. The method of multiple scales is used to construct a first order uniform expansion yielding four first order non-linear ordinary differential equations governing the modulation of the amplitudes and the phases of the two modes. Steady state responses and their stability are computed for selected values of the system parameters. The effects of detuning the internal resonance, detuning the parametric resonance, the phase and magnitude of the second mode parametric excitation, and the initial conditions are investigated. The first order perturbation solution predicts qualitatively the trivial and non-trivial stable steady state solutions and illustrates both the quenching and saturation phenomena. In addition to the steady state solutions, other periodic solutions are predicted by the perturbation amplitude and phase modulation equations. These equations predict a transition from constant steady state non-trivial responses to limit cycle responses (Hopf bifurcation). Some limit cycles are also shown to experience period doubling bifurcations. The perturbation solutions are verified by numerically integrating the governing differential equations.     

The response of multidegree-of-freedom systems with quadratic non-linearities to a harmonic parametric resonance

An analysis is presented of the response of multidegree-of-freedom systems with quadratic non-linearities to a harmonic parametric excitation in the presence of an internal resonance of the combination type ω₃ ≈ ω₂ + ω₁, where the ω_n are the linear natural frequencies of the systems. In the case of a fundamental resonance of the third mode (i.e., $\text{Ω ≈ω}{}_3$, where Ω is the frequency of the excitation), one can identify two critical values _{ζ1 and ζ2, where ζ2 ? ζ1}, of the amplitude F of the excitation. The value F = ζ₂ corresponds to the transition from stable to unstable solutions. When F < ζ₁, the motion decays to zero according to both linear and non-linear theories. When F >ζ₂, the motion grows exponentially with time according to the linear theory but the non-linearity limits the motion to a finite amplitude steady state. The amplitude of the third mode, which is directly excited, is independent of F, whereas the amplitudes of the first and second modes, which are indirectly excited through the internal resonance, are functions of F. When ζ₁ ? F ? ζ₂, the motion decays or achieves a finite amplitude steady state depending on the initial conditions according to the non-linear theory, whereas it decays to zero according to the linear theory. This is an example of subcritical instability. In the case of a fundamental resonance of either the first or second mode, the trivial response is the only possible steady state. When F ? ζ₂, the motion decays to zero according to both linear and non-linear theories. When F >ζ₂, the motion grows exponentially with time according to the linear theory but it is aperiodic according to the non-linear theory. Experiments are being planned to check these theoretical results.     

Three-to-one internal resonances in a general cubic non-linear continuous system

A general continuous system with an arbitrary cubic non-linearity is considered. The non-linearity is expressed in terms of an arbitrary cubic operator. Three-to-one internal resonance case is considered. A general approximate solution is presented for the system. Amplitude and phase modulation equations are derived. Steady state solutions and their stability are discussed in the general sense. The sufficiency condition for such resonances to occur is derived. Finally the algorithm is applied to a beam resting on a non-linear elastic foundation.     

Convection in chemical fronts with quadratic and cubic autocatalysis

Convection in chemical fronts enhances the speed and determines the curvature of the front. Convection is due to density gradients across the front. Fronts propagating in narrow vertical tubes do not exhibit convection, while convection develops in tubes of larger diameter. The transition to convection is determined not only by the tube diameter, but also by the type of chemical reaction. We determine the transition to convection for chemical fronts with quadratic and cubic autocatalysis. We show that quadratic fronts are more stable to convection than cubic fronts. We compare these results to a thin front approximation based on an eikonal relation. In contrast to the thin front approximation, reaction-diffusion models show a transition to convection that depends on the ratio between the kinematic viscosity and the molecular diffusivity. (c) 2002 American Institute of Physics.     

Excitation of stable transverse wavepackets with quadratic and cubic susceptibilities

We have identified a family of (2+1)D spatial solitary waves which can stably propagate in bulk media in the presence of coexisting diffraction, self-focusing Kerr and quadratic nonlinearities. In a conspicuous range of excitation conditions close to the stationary solutions, the emerging wavepackets are immune to the detrimental occurrence of filamentation and collapse, typical of pure Kerr media. The presence of a second-order contribution to the cubic nonlinear response is, therefore, able to prevent optical damage in applications relying on self-guidance. We show that the cross-phase modulation plays an important effect on stability. Our estimate shows that the effects of the cubic susceptibility cannot be neglected below a certain beam size in realistic crystals (e.g. KTP or similar).     

Occurrence of stable periodic modes in a pendulum with cubic damping

Dynamical systems with nonlinear damping show interesting behavior in the periodic and chaotic phases. The Froude pendulum with cubical and linear damping is a paradigm for such a system. In this work the driven Froude pendulum is studied by the harmonic balancing method; the resulting nonlinear response curves are studied further for resonance and stability of symmetric oscillations with relatively low damping. The stability analysis is carried out by transforming the system of equations to the linear Mathieu equation.     

具有二阶和三阶非线性一维光子晶体中的耦合模孤子

研究同时具有二阶和三阶非线性的一维光子晶体中的耦合孤子动力学.从Maxwell方程出发,利用多重尺度法,导出了光学整流场与两个基频电场包络的非线性耦合模方程组,给出了耦合模方程组的孤子解.结果表明,由于二阶非线性导致的光学整流场对基频电场有调制作用,使得两个基频电场分量可以呈现为亮孤子亮孤子、暗孤子暗孤子及亮孤子-暗孤子对 当两个基频电场的振动频率趋于光子晶体频带的带边频率时,光学整流场消失     

具有二阶和三阶非线性一维光子晶体中的耦合模孤子

研究同时具有二阶和三阶非线性的一维光子晶体中的耦合孤子动力学. 从Maxwell方程出发,利用多重尺度法,导出了光学整流场与两个基频电场包络的非线性耦合模方程组,给出了耦合模方程组的孤子解. 结果表明,由于二阶非线性导致的光学整流场对基频电场有调制作用,使得两个基频电场分量可以呈现为亮孤子-亮孤子、暗孤子-暗孤子及亮孤子-暗孤子对;当两个基频电场的振动频率趋于光子晶体频带的带边频率时,光学整流场消失.     

Nonlinear vibration of a curved beam under uniform base harmonic excitation with quadratic and cubic nonlinearities

This paper presents nonlinear vibration analysis of a curved beam subject to uniform base harmonic excitation with both quadratic and cubic nonlinearities. The Galerkin method is employed to discretize the governing equations. A high-dimensional model that can take nonlinear model coupling into account is derived, and the incremental harmonic balance (IHB) method is employed to obtain the steady-state response of the curved beam. The cases investigated include softening stiffness, hardening stiffness and modal energy transfer. The stability of the periodic solutions for given parameters is determined by the multi-variable Floquet theory using Hsu's method. Particular attention is paid to the anti-symmetric response with and without excitation, as the excitation frequency is close to the first and third natural frequencies of the system. The results obtained with the IHB method compare very well with those obtained via numerical integration.     

Dynamics of localized waves with large amplitude in a weakly dispersive medium with a quadratic and positive cubic nonlinearity

The dynamics of localized waves is analyzed in the framework of a model described by the Korteweg-de Vries (KdV) equation with account made for the cubic positive nonlinearity (the Gardner equation). In particular, the interaction process of two solitons is considered, and the dynamics of a “breathing” wave packet (a breather) is discussed. It is shown that solitons of the same polarity interact as in the case of the Korteweg-de Vries equation or modified Korteweg-de Vries equation, whereas the interaction of solitons of different polarity is qualitatively different from the classical case. An example of “unpredictable” behavior of the breather of the Gardner equation is discussed.     

Normal modes of a resistive nonuniform plasma

The propagation and dissipation properties of magnetohydrodynamic waves in a nonuniform, highlyconducting plasma, is investigated with a normal mode approach. The interaction between the perturbation and the non-uniform supporting medium is analyzed as the main mechanism able to produce the small scale spatial structure necessary to dissipate efficiently the wave energy. Two fundamental classes of modes are found, characterized by their resistive or ideal asymptotic behavior; the damping rates are shown to be orders of magnitude larger than those obtained when the plasma is perfectly homogeneous, and an application to the problem of solar coronal heating is discussed.Astronomia e Scienza dello spazio, Universitá di Firenze, Italy. Observatoire de Paris-Meudon (DESPA), France. Published in Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 37, No. 5, pp. 563–579, May, 1994.