Stabilization of Parametric Vibrations of a Nonlinear Continuous System

The purpose of the present paper is to solve an active control problem of nonlinear continuous system parametric vibrations excited by the fluctuating force. The problem is solved using the concept of distributed piezoelectric sensors and actuators with a sufficiently large value of velocity feedback. The direct Liapunov method is proposed to establish criteria for the almost sure stochastic stability of the unperturbed (trivial) solution of the shell with closed-loop control. The distributed control is realized by the piezoelectric sensor and actuator, with the changing widths, glued to the upper and lower shell surface. The relation between the stabilization of nonlinear problem and a linearized one is examined. The fluctuating axial force is modeled by the physically realizable ergodic process. The rate velocity feedback is applied to stabilize the shell parametric vibrations.     

Feedback minimization of first-passage failure of quasi non-integrable Hamiltonian systems

The non-linear stochastic optimal control of quasi non-integrable Hamiltonian systems for minimizing their first-passage failure is investigated. A controlled quasi non-integrable Hamiltonian system is reduced to an one-dimensional controlled diffusion process of averaged Hamiltonian by using the stochastic averaging method for quasi non-integrable Hamiltonian systems. The dynamical programming equations and their associated boundary and final time conditions for the problems of maximization of reliability and of maximization of mean first-passage time are formulated. The optimal control law is derived from the dynamical programming equations and the control constraints. The dynamical programming equations for maximum reliability problem and for maximum mean first-passage time problem are finalized and their relationships to the backward Kolmogorov equation for the reliability function and the Pontryagin equation for mean first-passage time, respectively, are pointed out. The boundary condition at zero Hamiltonian is discussed. Two examples are worked out to illustrate the application and effectiveness of the proposed procedure.     

Delay-independent exponential stability of stochastic Cohen–Grossberg neural networks with time-varying delays and reaction–diffusion terms

Different from the approaches used in the earlier papers, in this paper, the Halanay inequality technique, in combination with the Lyapunov method, is exploited to establish a delay-independent sufficient condition for the exponential stability of stochastic Cohen–Grossberg neural networks with time-varying delays and reaction–diffusion terms. Moreover, for the deterministic delayed Cohen–Grossberg neural networks, with or without reaction–diffusion terms, sufficient criteria for their global exponential stability are also obtained. The proposed results improve and extend those in the earlier literature and are easier to verify. An example is also given to illustrate the correctness of our results.     

A two-body continuous-discrete filter

In the theory of classical mechanics, the two-body central forcing problem is formulated as a system of the coupled nonlinear second-order deterministic differential equations. The uncertainty introduced by the small, unmodeled stochastic acceleration is not assumed in the particle dynamics. The small, unmodeled stochastic acceleration produces an additional random force on a particle. Estimation algorithms for a two-body dynamics, without introducing the stochastic perturbation, may cause inaccurate estimation of a particle trajectory. Specifically, this paper examines the effect of the stochastic acceleration on the motion of the orbiting particle, and subsequently, the stochastic estimation algorithm is developed by deriving the evolutions of conditional means and conditional variances for estimating the states of the particle-earth system. The theory of the nonlinear filter of this paper is developed using the Kolmogorov forward equation “between the observations" and a functional difference equation for the conditional probability density “at the observation." The effectiveness of the nonlinear filter is examined on the basis of its ability to preserve perturbation effect felt by the orbiting particle and the signal-to-noise ratio. The Kolmogorov forward equation, however, is not appropriate for the numerical simulations, since it is the equation for the evolution of “the conditional probability density." Instead of the Kolmogorov equation, one derives the evolutions for the moments of the state vector, which in our case consists of positions and velocities of the orbiting body. Even these equations are not appropriate for the numerical implementations, since they are not closed in the sense that computing the evolution of a given moment involves the knowledge of higher order moments. Hence, we consider the approximations to these moment evolution equations. This paper makes a connection between classical mechanics, statistical mechanics and the theory of the nonlinear stochastic filtering. The results of this paper will be of use to astrophysicists, engineers and applied mathematicians, who are interested in applications of the nonlinear filtering theory to the problems of celestial and satellite mechanics. Simulation results are introduced to demonstrate the usefulness of an analytic theory developed, in this paper.     

Studies on structural safety in stochastically excited Duffing oscillator with double potential wells

The effects of the Gaussian white noise excitation on structural safety due to erosion of safe basin in Duffing oscillator with double potential wells are studied in the present paper. By employing the well-developed stochastic Melnikov condition and Monte–Carlo method, various eroded basins are simulated in deterministic and stochastic cases of the system, and the ratio of safe initial points (RSIP) is presented in some given limited domain defined by the system’s Hamiltonian for various parameters or first-passage times. It is shown that structural safety control becomes more difficult when the noise excitation is imposed on the system, and the fractal basin boundary may also appear when the system is excited by Gaussian white noise only. From the RSIP results in given limited domain, sudden discontinuous descents in RSIP curves may occur when the system is excited by harmonic or stochastic forces, which are different from the customary continuous ones in view of the first-passage problems. In addition, it is interesting to find that RSIP values can even increase with increasing driving amplitude of the external harmonic excitation when the Gaussian white noise is also present in the system. The project supported by the National Natural Science Foundation of China (10302025 and 10672140). The English text was polished by Yunming Chen.     

Stability of noisy nonlinear auto-parametric systems

The purpose of this work is to examine the stationary motion and stability properties of stationary motion of two degree-of-freedom noisy auto-parametric systems We shall use analytical techniques to extend the existing results to examine such multi-dimensional nonlinear systems with noise, and in particular additive white noise. We obtain an approximation for the top Lyapunov exponent, the exponential growth rate, of the response of the so-called single-mode stationary motion. We show analytically that the top Lyapunov exponent is positive, and for small values of noise intensity ɛ and dissipation ɛ² the exponent grows in proportion with ɛ^2/3.     

Non-Linear Self-Excited Random Vibrations and Stability of an SDOF System with Parametric Noises

Abstract. The paper presents the solution to the properties of stochastic response of a system with random parametric noises, which is prone to the loss of aerodynamical stability. The system is described by an equation of van der Pol type with the negative linear, and with the positive cubic dampings. The coefficients of the linear damping and of the stiffness include the multiplicative random perturbations, the external excitation being given as a sum of a deterministic function and of an additive perturbation. All three input random processes are supposed to be Gaussian and centered, with the non-zero mutual stochastic parameters, as it corresponds to the properties of real systems. The solution has been based on the method of stochastic linearisation and of the subsequent solution of the Fokker–Planck–Kolmogorov equation in the sense of the first and second stochastic moments for the transient and stationary states. There have been demonstrated several effects, which are typical for systems with parametric noises, differentiating them from the systems with constant coefficients. The principal attention has been devoted to the properties of the spectral density of the response, the character of which changes abruptly with the degree of non-linearity of the damping and of the level of random perturbations.Sommario. La presente memoria studia le proprietà della risposta stocastica di un sistema con eccitazione casuale parametrica, che tende alla perdita della stabilità aerodinamica. Il sistema è descritto mediante un'equazione del tipo di van der Pole con il termine lineare dello smorzamento negativo e il termine cubico positivo. Poichá l'eccitazione esterna è la somma di una funzione deterministica e di una perturbazione additiva, i coefficienti dello smorzamento lineare e della rigidezza comprendono le perturbazioni casuali moltiplicative. I tre processi stocastici di eccitazione sono assunti gaussiani e a media nulla con parametri stocastici incrociati diversi da zero, come si verifica per le proprietà dei sistemi reali. La soluzione è basata sul metodo della linearizzazione stocastica e della successiva soluzione dell'equazione di Fokker-Planck-Kolmogorov studiando i primi e i secondi momenti statistici per gli stati transitori e stazionari. Vengono mostrati diversi effetti, tipici dei sistemi con eccitazione parametrica, differenziandoli dai sistemi a coefficienti costanti. Particolare attenzione è rivolta alle proprietà della densità spettrale della risposta le cui caratteristiche cambiano bruscamente con il grado di non linearità dello smorzamento e del livello di casualità delle perturbazioni.     

On the effects of non-linear elements in the reliability-based optimal design of stochastic dynamical systems

The aim of the present paper is to study the effects of non-linear devices on the reliability-based optimal design of structural systems subject to stochastic excitation. One-dimensional hysteretic devices are used for modelling the non-linear system behavior while non-stationary filtered white noise processes are utilized to represent the stochastic excitation. The reliability-based optimization problem is formulated as the minimization of the expected cost of the structure for a specified failure probability. Failure is assumed to occur when any one of the output states of interest exceeds in magnitude some specified threshold level within a given time duration. Failure probabilities are approximated locally in terms of the design variables during the optimization process in a parallel computing environment. The approximations are based on a local interpolation scheme and on an efficient simulation technique. Specifically, a subset simulation scheme is adopted and integrated into the proposed optimization process. The local approximations are then used to define a series of explicit approximate optimization problems. A sensitivity analysis is performed at the final design in order to evaluate its robustness with respect to design and system parameters. Numerical examples are presented in order to illustrate the effects of hysteretic devices on the design of two structural systems subject to earthquake excitation. The obtained results indicate that the non-linear devices have a significant effect on the reliability and global performance of the structural systems.     

Lyapunov exponent and stochastic stability of quasi-non-integrable Hamiltonian systems

An n degree-of-freedom (DOF) non-integrable Hamiltonian system subject to light damping and weak stochastic excitation is called quasi-non-integrable Hamiltonian system. In the present paper, the stochastic averaging of quasi-non-integrable Hamiltonian systems is briefly reviewed. A new norm in terms of the square root of Hamiltonian is introduced in the definitions of stochastic stability and Lyapunov exponent and the formulas for the Lyapunov exponent are derived from the averaged Itô equations of the Hamiltonian and of the square root of Hamiltonian. It is inferred that the Lyapunov exponent so obtained is the first approximation of the largest Lyapunov exponent of the original quasi-non-integrable Hamiltonian systems and the necessary and sufficient condition for the asymptotic stability with probability one of the trivial solution of the original systems can be obtained approximately by letting the Lyapunov exponent to be negative. This inference is confirmed by comparing the stability conditions obtained from negative Lyapunov exponent and by examining the sample behaviors of averaged Hamiltonian or the square root of averaged Hamiltonian at trivial boundary for two examples. It is also verified by the largest Lyapunov exponent obtained using small noise expansion for the second example.     

Non-linear stochastic response of a shallow cable

The paper considers the stochastic response of geometrical non-linear shallow cables. Large rain-wind induced cable oscillations with non-linear interactions have been observed in many large cable stayed bridges during the last decades. The response of the cable is investigated for a reduced two-degrees-of-freedom system with one modal coordinate for the in-plane displacement and one for the out-of-plane displacement. At first harmonic varying chord elongation at excitation frequencies close to the corresponding eigenfrequencies of the cable is considered in order to identify stable modes of vibration. Depending on the initial conditions the system may enter one of two states of vibration in the static equilibrium plane with the out-of-plane displacement equal to zero, or a whirling state with the out-of-plane displacement different from zero. Possible solutions are found both analytically and numerically. Next, the chord elongation is modelled as a narrow-banded Gaussian stochastic process, and it is shown that all the indicated harmonic solutions now become instable with probability one. Instead, the cable jumps randomly back and forth between the two in-plane and the whirling mode of vibration. A theory for determining the probability of occupying either of these modes at a certain time is derived based on a homogeneous, continuous time three states Markov chain model. It is shown that the transitional probability rates can be determined by first-passage crossing rates of the envelope process of the chord wise component of the support point motion relative to a safe domain determined from the harmonic analysis of the problem.