Stability Analysis of an Inverted Pendulum Subjected to Combined High Frequency Harmonics and Stochastic Excitations

Stability of vertical upright position of an inverted pendulum with its suspension point subjected to high frequency harmonics and stochastic excitations is investigated. Two classes of excitations, i.e., combined high frequency harmonic excitation and Gaussian white noise excitation, and high frequency bounded noise excitation, respectively, are considered. Firstly, the terms of high frequency harmonic excitations in the equation of motion of the system can be set equivalent to nonlinear stiffness terms by using the method of direct separation of motions. Then the stochastic averaging method of energy envelope is used to derive the averaged Ito stochastic differential equation for system energy. Finally, the stability with probability 1 of the system is studied by using the largest Lyapunov exponent obtained from the averaged Ito stochastic differential equation. The effects of system parameters on the stability of the system are discussed, and some examples are given to illustrate the efficiency of the proposed procedure.     

Higher-order stochastic averaging to study stability of a fractional viscoelastic column

Stochastic stability of a fractional viscoelastic column axially loaded by a wideband random force is investigated by using the method of higher-order stochastic averaging. By modelling the wideband random excitation as Gaussian white noise and real noise and assuming the viscoelastic material to follow the fractional Kelvin–Voigt constitutive relation, the motion of the column is governed by a fractional stochastic differential equation, which is justifiably and uniformly approximated by an averaged system of Itô stochastic differential equations. Analytical expressions are obtained for the moment Lyapunov exponent and the Lyapunov exponent of the fractional system with small damping and weak random fluctuation. The effects of various parameters on the stochastic stability of the system are discussed.     

A STOCHASTIC OPTIMAL SEMI-ACTIVE CONTROL STRATEGY FOR ER/MR DAMPERS

A stochastic optimal semi-active control strategy for randomly excited systems using electrorheological/magnetorheological (ER/MR) dampers is proposed. A system excited by random loading and controlled by using ER/MR dampers is modelled as a controlled, stochastically excited and dissipated Hamiltonian system with n degrees of freedom. The control forces produced by ER/MR dampers are split into a passive part and an active part. The passive control force is further split into a conservative part and a dissipative part, which are combined with the conservative force and dissipative force of the uncontrolled system, respectively, to form a new Hamiltonian and an overall passive dissipative force. The stochastic averaging method for quasi-Hamiltonian systems is applied to the modified system to obtain partially completed averaged Itô stochastic differential equations. Then, the stochastic dynamical programming principle is applied to the partially averaged Itô equations to establish a dynamical programming equation. The optimal control law is obtained from minimizing the dynamical programming equation subject to the constraints of ER/MR damping forces, and the fully completed averaged Itô equations are obtained from the partially completed averaged Itô equations by replacing the control forces with the optimal control forces and by averaging the terms involving the control forces. Finally, the response of semi-actively controlled system is obtained from solving the final dynamical programming equation and the Fokker-Planck-Kolmogorov equation associated with the fully completed averaged Itô equations of the system. Two examples are given to illustrate the application and effectiveness of the proposed stochastic optimal semi-active control strategy.     

STOCHASTIC AVERAGING OF DUHEM HYSTERETIC SYSTEMS

The response of Duhem hysteretic system to externally and/or parametrically non-white random excitations is investigated by using the stochastic averaging method. A class of integrable Duhem hysteresis models covering many existing hysteresis models is identified and the potential energy and dissipated energy of Duhem hysteretic component are determined. The Duhem hysteretic system under random excitations is replaced equivalently by a non-hysteretic non-linear random system. The averaged Ito's stochastic differential equation for the total energy is derived and the Fokker-Planck-Kolmogorov equation associated with the averaged Ito's equation is solved to yield stationary probability density of total energy, from which the statistics of system response can be evaluated. It is observed that the numerical results by using the stochastic averaging method is in good agreement with that from digital simulation.     

Stationary response of stochastic viscoelastic system with the right unilateral nonzero offset barrier impacts

The stationary response of viscoelastic dynamical system with the right unilateral nonzero offset barrier impacts subjected to stochastic excitations is investigated. First, the viscoelastic force is approximately treated as equivalent terms associated with effects. Then, the free vibro-impact(VI) system is absorbed to describe the periodic motion without impacts and quasi-periodic motion with impacts based upon the level of system energy. The stochastic averaging of energy envelope(SAEE) is adopted to seek the stationary probability density functions(PDFs). The detailed theoretical results for Van der Pol viscoelastic VI system with the right unilateral nonzero offset barrier are solved to demonstrate the important effects of the viscoelastic damping and nonzero rigid barrier impacts condition. Monte Carlo(MC) simulation is also performed to verify the reliability of the suggested approach. The stochastic P-bifurcation caused by certain system parameters is further explored. The variation of elastic modulus from negative to zero and then to positive witnesses the evolution process of stochastic P-bifurcation. From the vicinity of the common value to a wider range, the relaxation time induces the stochastic P-bifurcation in the two interval schemes.     

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.     

Response and stability of a SDOF strongly nonlinear stochastic system with light damping modeled by a fractional derivative

A stochastic averaging procedure for a single-degree-of-freedom (SDOF) strongly nonlinear system with light damping modeled by a fractional derivative under Gaussian white noise excitations is developed by using the so-called generalized harmonic functions. The approximate stationary probability density and the largest Lyapunov exponent of the system are obtained from the averaged Itô stochastic differential equation of the system. It is shown that the approximate stationary solutions obtained by using the stochastic averaging procedure agree well with those from the numerical simulation of original systems. The effects of system parameters on the approxiamte stationary probability density and the largest Lyapunov exponent of the system are also discussed.     

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.     

窄带随机噪声作用下单自由度非线性干摩擦系统的响应

研究了单自由度非线性干摩擦系统在窄带随机噪声参数激励下的主共振响应问题.用Krylov-Bogoliubov平均法得到了关于慢变量的随机微分方程.在没有随机扰动情形,得到了系统响应幅值满足的代数方程.在有随机扰动情形,用线性化方法和矩方法给出了系统响应稳态矩计算的近似计算公式.讨论了系统阻尼项、非线性项、随机扰动项和干摩擦项等参数对于系统响应的影响.理论计算和数值模拟表明,当非线性强度增大时系统的响应显著变小,系统分岔点滞后;随着激励频率的增大系统响应变大,而当激励频率小于一定的值时,系统响应为零;增加干 关键词： 单自由度非线性干摩擦系统 主共振响应 Krylov-Bogoliubov平均法     

Response of SDOF nonlinear oscillators with lightly fractional derivative damping under real noise excitations

This paper deals with the response of single-degree-of-freedom (SDOF) strongly nonlinear oscillator with lightly fractional derivative damping to external and (or) parametric real noise excitations. First, the state vector of the displacement and the velocity is approximated by one-dimensional time-homogeneous diffusive Markov process of amplitude through using the stochastic averaging method. Then, the stationary probability density of amplitude is obtained by solving the Fokker-Planck-Kolmogorov (FPK) equation associated with the averaged It? equation of amplitude, in which the Fourier series expansions are used to obtain the explicit expressions of the drift and diffusion coefficients. Finally, the response of a Duffing oscillator with lightly fractional derivative damping under external and parametric real noise excitations is evaluated by using the proposed procedure and compared with that from the Monte Carlo simulation of original system.     

Stochastic averaging of quasi-integrable Hamiltonian systems with delayed feedback control

A stochastic averaging method for quasi-integrable Hamiltonian systems with time-delayed feedback control is proposed. First, a quasi-integrable Hamiltonian system with delayed feedback control subjected to Gaussian white noise excitations is formulated and then transformed into Itô stochastic differential equations without time delay. Then, the averaged Itô stochastic differential equations for the system are derived and the stationary solution of the averaged Fokker–Planck–Kolmogorov (FPK) equation associated with the averaged Itô equations is obtained for both non-resonant and resonant cases. Finally, three examples are worked out in detail to illustrate the application and effectiveness of the proposed method and the effect of time delayed feedback control on the response of the systems.     

Energy diffusion controlled reaction rate of reacting particle driven by broad-band noise

The energy diffusion controlled reaction rate of a reacting particle with linear weak damping and broad-band noise excitation is studied by using the stochastic averaging method. First, the stochastic averaging method for strongly nonlinear oscillators under broad-band noise excitation using generalized harmonic functions is briefly introduced. Then, the reaction rate of the classical Kramers' reacting model with linear weak damping and broad-band noise excitation is investigated by using the stochastic averaging method. The averaged It? stochastic differential equation describing the energy diffusion and the Pontryagin equation governing the mean first-passage time (MFPT) are established. The energy diffusion controlled reaction rate is obtained as the inverse of the MFPT by solving the Pontryagin equation. The results of two special cases of broad-band noises, i.e. the harmonic noise and the exponentially corrected noise, are discussed in details. It is demonstrated that the general expression of reaction rate derived by the authors can be reduced to the classical ones via linear approximation and high potential barrier approximation. The good agreement with the results of the Monte Carlo simulation verifies that the reaction rate can be well predicted using the stochastic averaging method.     

Importance sampling for randomly excited dynamical systems

An importance sampling technique for linear and non-linear dynamical systems subjected to random excitations is presented. Applying a transformation of probability measures, controls are introduced in the system of Itô stochastic differential equations such that the sample trajectories can be influenced in a predetermined way. As is shown, there exist controls resulting in unbiased zero-variance estimators. However, these optimal controls are in general not accessible and have to be replaced by sub-optimal ones derived from an optimization procedure analogous to the first order reliability method known from time-invariant problems. The efficiency of the proposed Monte Carlo simulation technique is demonstrated by estimating first-passage probabilities of typical oscillators under external white-noise excitation.     

Non-trivial effect of fast vibration on the dynamics of a class of non-linearly damped mechanical systems

The effect of very high-frequency excitation on the slow dynamics of a class of non-linearly damped mechanical oscillators is considered. Two different models of damping namely, piecewise linear and pth power damping are considered. Fast excitation is modelled as triangular, sinusoidal and random base excitation. The effect of fast excitation is theoretically analyzed using the method of direct partition of motion (MDPM) and direct simulation. The method of numerical averaging is also used, where damping characteristics or excitations are not amenable to analytical techniques. Fast excitation has the non-trivial effect of increasing and decreasing the low-velocity damping of hard and soft dampers, respectively. The effect of fast excitation on the transient and steady state slow dynamics of the system is investigated by direct numerical integration of the equation of motion.     

Lyapunov control of squeezed noise quantum trajectory

The quantum trajectory renders the optimal estimation of quantum state. It is a classical Itô stochastic differential equation. The Lyapunov global stabilization problem is solved for squeezed noise quantum trajectory. Lyapunov control stabilizes the quantum system toward one eigenstate. A two-level bistable quantum system is simulated as an example.     

Josephson磁通孤子振荡器中热噪声研究

研究了可作为微波振荡器的长Sosephson隧道结中的磁通孤子在有限温度下受热噪声影响的运动。当Josephson隧道结被偏置在零场台阶状态,热噪声使孤子的运动速度在平均值附近涨落,孤子的平均速度对应于隧道结的平均电压,也对应于对外辐射电磁波的平均频率,速度的涨落对应于辐射一有限的频带宽度。研究表明,隧道结典型辐射频率为10GHZ,在液氦温区其频带宽约为1KHZ,是很窄的。热噪声同时也激发了一些连续模。用反散射方法的微扰论讨论了每模式的平均能量为k_BT,而孤子在低速极限的平均动能为1 关键词：     

Fresnel Type Path Integral for the Stochastic Schrödinger Equation

A path integral representation is obtained for the stochastic partial differential equation of Schrödinger type arising in the theory of open quantum systems subject to continuous nondemolition measurement and filtering, known as the a posteriori or Belavkin equation. The result is established by means of Fresnel-type integrals over paths in configuration space. This is achieved by modifying the classical action functional in the expression for the amplitude along each path by means of a stochastic Itô integral. This modification can be regarded as an extension of Menski's path integral formula for a quantum system subject to continuous measurement to the case of the stochastic Schrödinger equation.     

The Feynman–Kac formula for Schrödinger operators on vector bundles over complete manifolds

Methods from stochastic analysis are combined with functional analytic methods in order to prove a Feynman–Kac formula för Schrödinger type operators with nonnegative locally square integrable potentials on vector bundles over complete Riemannian manifolds. In particular, we obtain a Feynman–Kac–Itô formula on manifolds for Schrödinger operators with magnetic fields.     

Multiple time scales and the empirical models for stochastic volatility

The most common stochastic volatility models such as the Ornstein–Uhlenbeck (OU), the Heston, the exponential OU (ExpOU) and Hull–White models define volatility as a Markovian process. In this work we check the applicability of the Markovian approximation at separate times scales and will try to answer the question which of the stochastic volatility models indicated above is the most realistic. To this end we consider the volatility at both short (a few days) and long (a few months) time scales as a Markovian process and estimate for it the coefficients of the Kramers–Moyal expansion using the data for Dow-Jones Index. It has been found that the empirical data allow to take only the first two coefficients of expansion to be non-zero that define form of the volatility stochastic differential equation of Itô. It proved to be that for the long time scale the empirical data support the ExpOU model. At the short time scale the empirical model coincides with ExpOU model for the small volatility quantities only.     

Quantum stochastic differential equations for boson and fermion systems—method of non-equilibrium thermo field dynamics

A unified canonical operator formalism for quantum stochastic differential equations, including the quantum stochastic Liouville equation and the quantum Langevin equation both of the Itô and the Stratonovich types, is presented within the framework of non-equilibrium thermo field dynamics (NETFD). It is performed by introducing an appropriate martingale operator in the Schrödinger and the Heisenberg representations with fermionic and bosonic Brownian motions. In order to decide the double tilde conjugation rule and the thermal state conditions for fermions, a generalization of the system consisting of a vector field and Faddeev-Popov ghosts to dissipative open situations is carried out within NETFD.