On the stability properties of a damped oscillator with a periodically time-varying mass

In this paper the vibrations of a damped, linear, single degree of freedom oscillator (sdofo) with a time-varying mass will be considered. Both the free and forced vibrations of the oscillator will be studied. For the free vibrations the minimal damping rates will be computed, for which the oscillator is always stable. The forced vibrations are partly due to small masses, which are periodically hitting and leaving the oscillator with different velocities. Since these small masses stay for some time on the oscillator surface the effective mass of the oscillator will periodically vary in time. Additionally, an external harmonic force will be applied to the oscillator. Not only solutions of the oscillator equations will be constructed, but also stability properties for the free, and for the forced vibrations will be presented for various parameter values. For the external, harmonic forcing case an interesting resonance condition will be derived.     

An Analytic Approach to Torus Bifurcation in a Quasiperiodically Forced Duffing Oscillator

Abstract

A weakly nonlinear quasiconservative Duffing oscillator under quasiperiodic forcing is studied with the help of an analytic expression for the complex Poincare mapping. This mapping is then used to analyze the quasiperiodic response of the oscillator and the different zones of various periodicity. This map plays the same significant role as the averages equations in the theory of a periodically forced Duffing oscillator.     

水平往复运动的半圆形轨道上弹簧振子的混沌行为

设计一个简单的受迫弹簧振子实验装置 ,建立该系统的动力学方程 ,用数值计算并结合多种分析方法求解和判断非线性方程解的各种性质 ,即系统运动的各种状态 ,结果表明 :在系统参数的某个范围内 ,弹簧振子的运动状态呈现十分丰富的分岔、混沌等现象     

Forced periodic oscillations and the Jones polynomial

We show that forced periodic oscillations in a nonlinear damped oscillator can be classified by means of the Jones (knot) polynomial; this is done by associating to any periodic oscillation a braid. We also discuss the relation of this approach to (Lie-point) symmetry analysis of the associated differential equations.     

太阳强迫厄尔尼诺/南方涛动充电振子模型的Hopf分岔与混沌

本文研究了一类太阳强迫的厄尔尼诺/南方涛动(ENSO)充电振子数理模型,通过数学变换将此ENSO振子方程组变换为有周期强迫项的van der Pol-Duffing方程,利用谐波平衡法定性分析得到此ENSO系统发生Hopf分岔的条件并做简单数值模拟,结果发现随着强迫作用增大,11年周期太阳循环强迫的ENSO系统经历准周期、倍频锁相到混沌的过程.     

Suppression of the primary resonance vibrations of a forced nonlinear system using a dynamic vibration absorber

In a single degree-of-freedom weakly nonlinear oscillator subjected to periodic external excitation, a small-amplitude excitation may produce a relatively large-amplitude response under primary resonance conditions. Jump and hysteresis phenomena that result from saddle-node bifurcations may occur in the steady-state response of the forced nonlinear oscillator. A simple mass-spring-damper vibration absorber is thus employed to suppress the nonlinear vibrations of the forced nonlinear oscillator for the primary resonance conditions. The values of the spring stiffness and mass of the vibration absorber are significantly lower than their counterpart of the forced nonlinear oscillator. Vibrational energy of the forced nonlinear oscillator is transferred to the attached light mass through linked spring and damper. As a result, the nonlinear vibrations of the forced oscillator are greatly reduced and the vibrations of the absorber are significant. The method of multiple scales is used to obtain the averaged equations that determine the amplitude and phases of the first-order approximate solutions to primary resonance vibrations of the forced nonlinear oscillator. Illustrative examples are given to show the effectiveness of the dynamic vibration absorber for suppressing primary resonance vibrations. The effects of the linked spring and damper and the attached mass on the reduction of nonlinear vibrations are studied with the help of frequency response curves, the attenuation ratio of response amplitude and the desensitisation ratio of the critical amplitude of excitation.     

On the forced vibrations of an oscillator with a periodically time-varying mass

In this paper the forced vibrations of a linear, single degree of freedom oscillator (sdofo) with a time-varying mass will be studied. The forced vibrations are due to small masses which are periodically hitting and leaving the oscillator with different velocities. Since these small masses stay for some time on the oscillator surface the effective mass of the oscillator will periodically vary in time. Additionally, an external harmonic force will be applied to the oscillator with a time-varying mass. Not only solutions of the oscillator equation will be constructed, but also stability properties for the forced vibrations will be presented for various parameter values.     

Evolution of a velocity-dependent quantum forced anharmonic oscillator

The exact solution to a velocity-dependent quantum forced anharmonic oscillator is derived by using integral operators and an iteration method. The study is carried out in operational form by use of the creation and annihilation operators of the oscillator. The time development of the displacement and momentum operators of the anharmonic oscillator is given. These operators are presented as a Laplace transform and a subsequent inverse Laplace transform of suitable functionals.     

Quantum harmonic oscillator with time-dependent mass and frequency

The quantum harmonic oscillator with time-dependent mass and frequency is analyzed by using the canonical transformation method. The varying mass and frequency of the system are reduced to constant mass and frequency, and the corresponding eigenvalues and eigenvectors are derived. The exact time-dependent coherent state of the harmonic oscillator is constructed and shown to be equivalent to the squeezed state. Damped harmonic oscillators with different frictions and forced time-dependent harmonic oscillators are also discussed.     

依赖于速度的量子受迫非简谐振子的演化

运用积分算子和迭代方法求得了依赖于速度的量子受迫非简谐振子的精确解。给出了非简谐振子坐标和动量算符的时间演化。这些算符被表示成适当泛函的拉普拉斯变换及随后的逆拉普拉斯交换的形式。 关键词：     

Noise-induced critical breakdown of phase lockings in a forced van der Pol oscillator

The effect of noise on the period of the van der Pol oscillator forced by a sinusoidal signal is analyzed in detail. The dynamics of the forced nonlinear oscillator with noise is described by the evolution of probability densities by a Markov operator. The oscillator shows a 1 : 1 phase-locked oscillation whose period is equal to the period of the sinusoidal forcing when the forcing is sufficiently strong. Noise modulates the oscillator's period as expected, but we show that this modulation is not uniform; the period of the oscillator changes drastically at a certain strength of noise. Thus this noise-induced breakdown of the phase locking can be considered as a stochastic bifurcation. The breakdown of stochastic phase-locking or the stochastic bifurcation is also analyzed from the viewpoint of the spectra of the Markov operator.     

Forced Time-Dependent Harmonic Oscillators in Non-Commutative Space

For the time-dependent harmonic oscillator and generalized harmonic oscillator with or without external forces in non-commutative space, wave functions, and geometric phases are derived using the Lewis-Riesenfeld invariant. Coherent states are obtained as the ground state of the forced system. Quantum fluctuations are calculated too. It is seen that geometric phases and quantum fluctuations are greatly affected by the non-commutativity of the space.     

A wake oscillator with frequency dependent coupling for the modeling of vortex-induced vibration

The aim of this paper is to improve the phenomenological modeling of vortex-induced vibration of an elastically mounted cylinder in fluid flow. To this end an attempt is made to introduce a wake oscillator model that conforms to both the free and forced vibration experiments. This approach is new as in the past wake oscillator models have been tuned to the free vibration experiments only. First, an existing wake oscillator model is improved by properly including the effect of stall and that of the relatively large attack angles in the course of vortex-induced vibration. Then, to comply with the forced vibration experiments, the model is enhanced by introducing frequency dependent coupling. Such a coupling allows reproduction of the measured frequency dependence of the fluid force on the cylinder. The time domain representation of this coupling is a convolution integral. It is found in this paper that if the wake oscillator is modeled with a Van der Pol equation, it is impossible to find one set of frequency dependent coefficients that conforms to the forced vibration experiments at all amplitudes of cylinder motion. Moreover, the frequency dependencies identified for each frequency separately do not seem to satisfy the Kramers-Kronig relations. Based on the above findings, it is concluded that the nonlinearities in the wake oscillator model used in this paper cannot accurately model the results of vortex-induced vibration measurements. The idea proposed in this paper, that a consistent wake oscillator model must conform to the forced vibration experiments as well, is expected to be a powerful tool in the search for the correct nonlinearity.     

Existence and analytical predictions of periodic motions in a periodically forced,nonlinear friction oscillator

The grazing bifurcation, stick phenomena and periodic motions in a periodically forced, nonlinear friction oscillator are investigated. The nonlinear friction force is approximated by a piecewise linear, kinetic friction model with the static force. The total forces for the input and output flows to the separation boundary are introduced, and the force criteria for the onset and vanishing of stick motions are developed through such input and output flow forces. The periodic motions of such an oscillator are predicted analytically through the corresponding mapping structure. Illustrations of the periodic motions in such a piecewise friction model are given for a better understanding of the stick motion with the static friction. The force responses are presented, which agreed very well with the force criteria. If the fully nonlinear friction force is modeled by several portions of piecewise linear functions, the periodically forced, nonlinear friction oscillator can be predicted more accurately. However, for the fully nonlinear friction force model, only the numerical investigation can be carried out.     

Fermion Coherence Hamiltonians

We have established that the most general form of Hamiltonian that preserves fermionic coherent states stable in time, is that of the nonstationary free fermionic oscillator. This is to be compared with the earlier result of boson coherence Hamiltonian, which is of the more general form of the nonstationary forced bosonic oscillator. If however one admits Grassmann variables as Hamiltonian parameters then the coherence Hamiltonian takes again the form of (Grassmannian fermionic) forced oscillator.     

Two coupled impact oscillators

A forced impact oscillator is coupled to a second freely moving oscillator with no amplitude constraint. The influence of the coupling strength and the external frequency is studied in detail. For intermediate coupling strength many of the long-period or chaotic motions are suppressed and replaced by short-period motions at least for not too high frequencies. Phase space trajectories are discussed for some characteristic examples.     

Forced synchronization of a delayed-feedback oscillator

Dynamics of a delayed-feedback oscillator with cubic nonlinearity driven by an external harmonic signal is considered. Regimes of forced synchronization and their stability conditions are studied analytically and numerically for the cases of single-frequency and multiple-frequency (self-modulation) operation of the free-running oscillator. Special attention is paid to synchronization of the oscillator with bistability of steady states which arises near the boundary of generation zone. Results of numerical simulation of bifurcation transitions to synchronization regime for different driving frequencies are presented. Main differences from the classical picture of synchronization for a system with one degree of freedom are discussed.     

一维受迫谐振子量子运动的经典特性

对经典一维受迫谐振子量子化,求解量子化后体系的时间演化算符.应用相空间准概率分布函数,研究了体系的量子特性.研究结果表明,初始为真空态,经过时间演化,系统波函数是一个二维高斯波包;波包中心的振幅和相位受到作用力的调制,成为调幅、调相波,波包中心的运动与经典受迫谐振子的运动形式相同.     

强迫的Oregonator振子混沌维数与立方映象

本文论证了强迫的Oregonator振子可以用立方映象刻划其行为,其依据为描述此振子的四变量方程有三个不动点,混沌的转换函数呈现三次曲线特点,找到了隶属三点周期的符号BK;C,四点周期的符号BK;LC,B²K;C,BRK;C,并画出了BRK;C的环状线,找到了一条倍周期分岔道路。在混沌域中研究了五个混沌态,其维数介于2.02—2.37之间,混沌域中央维数偏低,贴近四点周期线维数偏高。 关键词：     

Linear relaxation processes governed by fractional symmetric kinetic equations

The fractional symmetric Fokker-Planck and Einstein-Smoluchowski kinetic equations that describe the evolution of systems influenced by stochastic forces distributed with stable probability laws are derived. These equations generalize the known kinetic equations of the Brownian motion theory and involve symmetric fractional derivatives with respect to velocity and space variables. With the help of these equations, the linear relaxation processes in the force-free case and for the linear oscillator is analytically studied. For a weakly damped oscillator, a kinetic equation for the distribution in slow variables is obtained. Linear relaxation processes are also studied numerically by solving the corresponding Langevin equations with the source given by a discrete-time approximation to white Levy noise. Numerical and analytical results agree quantitatively.