有界噪声激励下带有时滞反馈的随机Mathieu-Duffing系统的响应

研究了参数激励下带有时滞反馈的随机Mathieu-Duffing方程的主参数共振响应问题.运用多尺度方法分离了系统的快慢变量.分析了系统的分岔性质,发现调谐参数、时滞、时滞项的系数以及非线性项的强度等都可以影响系统的分岔行为,适当选择这些参数可以改变系统的分岔响应.同时,还讨论了非零解的稳定性,得到了非零解稳定的充要条件,而且发现在随机激励的带宽较小时,系统的多解现象仍然存在,分岔和跳跃现象仍会发生,数值模拟验证了理论推导的有效性. 关键词： 随机Mathieu-Duffing系统 多尺度 稳定性 分岔     

Frequency domain response of a parametrically excited riser under random wave forces

Floating Production, Drilling, Storage and Offloading units represent a new technology with a promising future in the offshore oil industry. An important role is played by risers, which are installed between the subsea wellhead and the Tension Leg Deck located in the middle of the moon-pool in the hull. The inevitable heave motion of the floating hull causes a time-varying axial tension in the riser. This time dependent tension may have an undesirable influence on the lateral deflection response of the riser, with random wave forces in the frequency domain. To investigate this effect, a riser is modeled as a Bernoulli–Euler beam. The axial tension is expressed as a static part, along with a harmonic dynamic part. By linearizing the wave drag force, the riser's lateral deflection is obtained through a partial differential equation containing a time-dependent coefficient. Applying the Galerkin method, the equation is reduced to an ordinary differential equation that can be solved using the pseudo-excitation method in the frequency domain. Moreover, the Floquet–Liapunov theorem is used to estimate the stability of the vibration system in the space of parametric excitation. Finally, stability charts are obtained for some numerical examples, the correctness of the proposed method is verified by comparing with Monte-Carlo simulation and the influence of the parametric excitation on the frequency domain responses of the riser is discussed.     

Dynamic stability of a slender beam under horizontal–vertical excitations

The dynamic stability of a vertically standing cantilevered beam simultaneously excited in both horizontal and vertical directions at its base is studied theoretically. The beam is assumed to be an inextensible Euler–Bernoulli beam. The governing equation of motion is derived using Hamilton's principle and has a nonlinear elastic term and a nonlinear inertia term. A forced horizontal external term is added to the parametrically excited system. Applying Galerkin's method for the first bending mode, the forced Mathieu–Duffing equation is derived. The frequency response is obtained by the harmonic balance method, and its stability is investigated using the phase plane method. Excitation frequencies in the horizontal and vertical directions are taken as 1:2, from which we can investigate the influence of the forced response under horizontal excitation on the parametric instability region under vertical excitation. Three criteria for the instability boundary are proposed. The influences of nonlinearities and damping of the beam on the frequency response and parametric instability region are also investigated. The present analytical results for instability boundaries are compared with those of experiments carried out by one of the authors.     

Chatter suppression by parametric excitation: Model and experiments

Chatter vibration leads to challenges in precise machining due to its harmful effect on productivity and surface quality. In this study, a chatter suppression method based on parametric excitation was developed. The effect of parametric excitation on self-excited vibration was investigated based on a model of a van der Pol-Mathieu-Duffing oscillator with a time delay. It reveals that there can be a zero solution for the oscillator under the effect of parametric excitation, while it is impossible to have a stable zero equation without parametric excitation. The stability of a parametrically excited vibration system regarding the regenerative effect in the cutting processes was studied by the averaging method. The stability analysis shows that parametric excitation with an appropriate frequency and large amplitude has a chatter suppression effect no matter whether the waveform is a sinusoidal wave, square wave or triangular wave. To validate the effect of parametric excitation for chatter suppression, experiments were conducted based on a magnetorheological (MR) fluid-controlled boring bar, which can generate high-frequency parametric excitation based on the quick response of the MR fluid. Cutting experiments with an excitation current of different waveforms and diverse frequencies show that chatter can be significantly suppressed by the effect of parametric excitation.     

Stochastic stability of a fractional viscoelastic column under bounded noise excitation

The stability of a viscoelastic column under the excitation of stochastic axial compressive load is investigated in this paper. The material of the column is modeled using a fractional Kelvin–Voigt constitutive relation, which leads to that the equation of motion is governed by a stochastic fractional equation with parametric excitation. The excitation is modeled as a bounded noise, which is a realistic model of stochastic fluctuation in engineering applications. The method of stochastic averaging is used to approximate the responses of the original dynamical system by a new set of averaged variables which are diffusive Markov vector. An eigenvalue problem is formulated from the averaged equations, from which the moment Lyapunov exponent is determined for the column system with small damping and weak excitation. The effects of various parameters on the stochastic stability and significant parametric resonance are discussed and confirmed by simulation results.     

参数激励与晶体摆动场辐射的稳定性

寻找新光源, 特别是短波长相干光源备受关注. 本文讨论了晶体摆动场辐射作为短波长激光的可能性和必须满足的基本条件; 指出了至今尚未获得可利用的短波长激光可能不只是技术原因, 而且还有物理原因. 利用参数激励方法对这个问题进行了分析. 在经典力学框架内和偶极近似下, 引入正弦平方势, 把粒子运动方程化为具有阻尼项和参数激励项的摆方程. 利用Melnikov方法讨论了系统的稳定性, 并对系统的临界条件进行了分析. 结果表明: 系统的稳定性与其参数有关, 只需适当调节这些参数, 系统的稳定性就可以原则上得到保证. 关键词： 晶体摆动场辐射 沟道辐射 参数激励 稳定性     

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.     

LARGEST LYAPUNOV EXPONENT AND ALMOST CERTAIN STABILITY ANALYSIS OF SLENDER BEAMS UNDER A LARGE LINEAR MOTION OF BASEMENT SUBJECT TO NARROWBAND PARAMETRIC EXCITATION

The first order approximate solutions of a set of non-liner differential equations, which is established by using Kane's method and governs the planar motion of beams under a large linear motion of basement, are systematically derived via the method of multiple scales. The non-linear dynamic behaviors of a simply supported beam subject to narrowband random parametric excitation, in which either the principal parametric resonance of its first mode or a combination parametric resonance of the additive type of its first two modes with or without 3:1 internal resonance between the first two modes is taken into consideration, are analyzed in detail. The largest Lyapunov exponent is numerically obtained to determine the almost certain stability or instability of the trivial response of the system and the validity of the stability is verified by direct numerical integration of the equation of motion of the system.     

声悬浮液滴的表面毛细波及八阶扇谐振荡

采用声悬浮方法研究了自由液滴表面的毛细波形成机理,并利用主动调制声场技术激发了液滴的八阶扇谐振荡.实验结果表明,当声场调制频率接近液滴本征频率的两倍时,液滴将由轴对称受迫振荡向非轴对称扇谐振荡模态转变.实验与理论分析证实,参数共振是毛细波与扇谐振荡的形成原因.扇谐振荡的本征频率随液滴赤道半径的增大而减小,可通过修正的Rayleigh方程来描述. 关键词： 声悬浮 液滴 毛细波 扇谐振荡     

一类准周期参激非线性相对转动动力系统的稳定性与时滞反馈控制

建立了一类含准周期参数激励和时滞反馈的相对转动非线性系统的动力学方程. 采用多尺度法求解1/2亚谐波主参数共振下的分岔响应方程,并分析了系统的稳定性. 在求解非受控系统的定常解的基础上,通过讨论系统的动力学特性,研究了准周期参数激励对系统响应的影响. 采用时滞反馈控制的方法对系统分岔和极限环(域)进行控制,数值模拟的结果表明通过改变时滞参数可以实现对系统分岔的控制,并能有效地控制极限环(域)的幅值和稳定性. 关键词： 相对转动 准周期参激 时滞反馈 极限环     

Parametric resonance and cooling on an atom chip

This paper observes the parametric excitation on atom chip by measuring the trap loss when applying a parametric modulation. By modulating the current in chip wires, it modulates not only the trap frequency but also the trap position. It shows that the strongest resonance occurs when the modulation frequency equals to the trap frequency. The resonance amplitude increases exponentially with modulation depth. Because the Z-trap is an anharmonic trap, there exists energy selective excitation which would cause parametric cooling. We confirm this effect by observing the temperature of atom cloud dropping.     

Interaction of fundamental parametric resonances with subharmonic resonances of order one-half

The interaction of fundamental parametric resonances with subharmonic resonances of order one-half in a single-degree-of-freedom system with quadratic and cubic nonlinearities is investigated. The method of multiple scales is used to derive two first-order ordinary differential equations that describe the modulation of the amplitude and the phase of the response with the non-linearity and both resonances. These equations are used to determine the steady state solutions and their stability. Conditions are derived for the quenching or enhancement of a parametric resonance by the addition of a subharmonic resonance of order one-half. The degree of quenching or enhancement depends on the relative amplitudes and phases of the excitations. The analytical results are verified by numerically integrating the original governing differential equation.     

[Russian Text Ignored]

The stability of the electromagnetic plasma confinement by powerful external s-polarized pump waves is considered. The parametric excitation of standing electromagnetic waves along the plasma boundary with frequencies close to the frequency of the pump wave leads to a periodic density modulation of the plasma boundary. The density disturbances along the direction of the external wave field are connected to the excitation of transverse p-polarized surface waves while the modulation in the direction perpendicular to the pump field are created by the parametric interaction between the external wave and s-polarized trapped leaking oscillations. Only when the leaking waves are excited the scale length of the modulation is larger than half the free space wave length of the incident radiation.     

Dynamical parametric instability of carbon nanotubes under axial harmonic excitation by nonlocal continuum theory

Structures under parametric load can be induced to the parametric instability in which the excitation frequency is located the instability region. In the present work, the parametric instability of double-walled carbon nanotubes is studied. The axial harmonic excitation is considered and the nonlocal continuum theory is applied. The critical equation is derived as the Mathieu form by the Galerkin's theory and the instability condition is presented with the Bolotin's method. Numerical calculations are performed and it can be seen that the van der Waals interaction can enhance the stability of double-walled nanotubes under the parametric excitation. The parametric instability becomes more obvious with the matrix stiffness decreasing and small scale coefficient increasing. The parametric instability is going to be more significant for higher mode numbers. For the nanosystem with the soft matrix and higher mode number, the small scale coefficient and the ratio of the length to the diameter have obvious influences on the starting point of the instability region.     

Spectral modulation detection as a function of modulation frequency, carrier bandwidth, and carrier frequency region

The present study investigates the nature of spectral envelope perception using a spectral modulation detection task in which sinusoidal spectral modulation is superimposed upon a noise carrier. The principal goal of this study is to characterize spectral envelope perception in terms of the influence of modulation frequency (cycles/octave), carrier bandwidth (octaves), and carrier frequency region (defined by lower and upper cutoff frequencies in Hz). Spectral modulation detection thresholds measured as a function of spectral modulation frequency result in a spectral modulation transfer function (SMTF). The general form of the SMTF is bandpass in nature, with a minimum modulation detection threshold in the region between 2 to 4 cycles/octave. SMTFs are not strongly dependent on carrier bandwidth (ranging from 1 to 6 octaves) or carrier frequency region (ranging from 200 to 12 800 Hz), with the exception of carrier bands restricted to very low audio frequencies (e.g., 200-400 Hz). Spectral modulation detection thresholds do not depend on the presence of random level variations or random modulation phase across intervals. The SMTFs reported here and associated excitation pattern computations are considered in terms of a linear systems approach to spectral envelope perception and potential underlying mechanisms for the perception of spectral features.     

Dynamic responses of axially moving viscoelastic beam under a randomly disordered periodic excitation

We investigate dynamic responses of axially moving viscoelastic beam subject to a randomly disordered periodic excitation. The method of multiple scales is used to derive the analytical expression of first-order uniform expansion of the solution. Based on the largest Lyapunov exponent, the almost sure stability of the trivial steady-state solution is examined. Meanwhile, we obtain the first-order and the second-order steady-state moments for the non-trivial steady-state solutions. Specially, we discuss the first mode theoretically and numerically. Results show that under the same conditions of the parameters, as the intensity of the random excitation increases, non-trivial steady-state solution fluctuation will become strenuous, which will result in the non-trivial steady-state solution lose stability and the trivial steady-state solution can be a possible. In the case of parametric principal resonance, the stochastic jump is observed for the first mode, which indicates that the stationary joint probability density concentrates at the non-trivial solution branch when the random excitation is small, but with the increase of intensity of the random excitation, the probability of the trivial steady-state solution will become larger. This phenomenon of stochastic jump can be defined as a stochastic bifurcation.     

Marangoni instability of a liquid layer with insoluble surfactant under heat flux modulation

We investigate the parametric excitation of Marangoni convection by a periodic flux modulation in a liquid layer with insoluble surfactant absorbed on the nondeformable free surface. The stability analysis of the convective system is performed for arbitrary wave numbers of the disturbances. An interesting feature of the onset of convection is the existence of bifurcating neutral curves with double minima, one of which corresponds to a quasi-periodic solution, and the other one corresponds to a subharmonic solution. The evolution of the subharmonic instability region depending on the amplitude of the external heat flux modulation and the frequency of the modulation is studied. The quasi-periodic neutral curve is close to the oscillatory neutral curve of the nonmodulated problem.     

Resonance response of a single-degree-of-freedom nonlinear vibro-impact system to a narrow-band random parametric excitation

The resonant response of a single-degree-of-freedom nonlinear vibro-impact oscillator with a one-sided barrier to a narrow-band random parametric excitation is investigated. The narrow-band random excitation used here is a bounded random noise. The analysis is based on a special Zhuravlev transformation, which reduces the system to one without impacts, thereby permitting the applications of random averaging over "fast" variables. The averaged equations are solved exactly and an algebraic equation of the amplitude of the response is obtained for the case without random disorder. The methods of linearization and moment are used to obtain the formula of the mean-square amplitude approximately for the case with random disorder. The effects of damping, detuning, restitution factor, nonlinear intensity, frequency and magnitude of random excitations are analysed. The theoretical analyses are verified by numerical results. Theoretical analyses and numerical simulations show that the peak response amplitudes will reduce at large damping or large nonlinear intensity and will increase with large amplitude or frequency of the random excitations. The phenomenon of stochastic jump is observed, that is, the steady-state response of the system will jump from a trivial solution to a large non-trivial one when the amplitude of the random excitation exceeds some threshold value, or will jump from a large non-trivial solution to a trivial one when the intensity of the random disorder of the random excitation exceeds some threshold value.     

Response of nonlinear random business cycle model with time delay state feedback

The steady state response and bifurcation of nonlinear random business cycle model to random narrow-band excitation with time delay state feedback are studied in this paper. The method of multiple scales is used to determine the business cycle model of modulation of amplitude and phase. The effects of delay, detuning, bandwidth and magnitude of random excitation on dynamics of the business cycle system are investigated. The results show that the complex dynamics such as bifurcation, jump domain and so on are induced by time delay and the phenomena that multiple solution or bifurcation is induced by noise.     

Observation of the coherent state in a system of nuclear spin-wave pairs excited by microwave noise pumping

The parametric excitation of nuclear magnons by a microwave noise field was observed in an antiferromagnet. Two critical pumping amplitudes were found to exist. The first one corresponds to the onset of nonlinear microwave absorption. Above the second amplitude, strong phase correlations appear in a system of excited magnon pairs to form a nonequilibrium Bose condensate, which produces intense coherent electromagnetic radiation from the sample and gives rise to the coherent response of parametric magnons to the modulation of their spectrum (modulation response). It was found that, for the noise pumping, the contribution from the processes of elastic magnon relaxation to the threshold pumping amplitudes becomes nonadditive.