Wide-band Random Excitation Research Articles

Study of improved fatigue damage calculation of spacecraft equipment under wideband random vibration excitation by applying Rice distribution

Study of improved fatigue damage calculation of spacecraft equipment under wideband random vibration excitation by applying Rice distribution

Response and reliability analysis of random time-delay controlled systems undergo wide-band random excitations

This paper proposed a new procedure for response and reliability analysis of a multi-degree-of-freedom (MDOF) random time-delay controlled nonlinear system subjected to wide-band random excitations. A two-steps method is presented to deal with random time delay: Firstly, by modeling the random jump time-delay as a Markov jump process, the random delayed control is formulated as the one with discrete Markov jump varying delay; secondly, based on the randomly periodic property of the system, the Markov jump delay is taken out of the control as a parameter. A new scheme based on limit averaging and stochastic averaging principle is developed, which suppresses the rapidly varying processes to generate a less-dimensional probability-weighted system. The Fokker–Planck–Kolmogorov (FPK) equation and the backward Kolmogorov (BK) equation are then deduced and solved, from which the quantitative characteristics of the response and reliability of the system are obtained. Finally, an example is worked out in detail, from which the application and effectiveness of the proposed method are demonstrated.

Random Response of Wake-Oscillator Excited by Fluctuating Wind

Abstract Many wake-oscillator models applied to study vortex-induced vibration (VIV) are assumed to be excited by ideal wind that is assumed to be uniform flow with constant velocity. However, in the field of wind engineering, the real wind is generally described as being composed of mean wind and fluctuating wind. A wake-oscillator excited by fluctuating wind should be treated as a randomly excited and dissipated multi-degree-of-freedom (DOF) nonlinear system. The study of such a system is challenging and so far there is no exact solution available. The present paper aims to carry out the study on the stochastic dynamics of VIV. The stochastic averaging method of quasi-integrable Hamiltonian systems under wideband random excitation is applied to study both the original Hartlen–Currie wake-oscillator model and a modified version excited by fluctuating wind. The probability and statistics of the random response of wake-oscillator in resonant (lock-in) case and in non-resonant case are analytically obtained, and the analytical results are confirmed using numerical simulation of the original system.

Nonlinear Stochastic Optimal Control of MDOF Partially Observable Linear Systems Excited by Combined Harmonic and Wide-Band Noises

In this paper, nonlinear stochastic optimal control of multi-degree-of-freedom (MDOF) partially observable linear systems subjected to combined harmonic and wide-band random excitations is investigated. Based on the separation principle, the control problem of a partially observable system is converted into a completely observable one. The dynamic programming equation for the completely observable control problem is then set up based on the stochastic averaging method and stochastic dynamic programming principle, from which the nonlinear optimal control law is derived. To illustrate the feasibility and efficiency of the proposed control strategy, the responses of the uncontrolled and optimal controlled systems are respectively obtained by solving the associated Fokker–Planck–Kolmogorov (FPK) equation. Numerical results show the proposed control strategy can dramatically reduce the response of stochastic systems subjected to both harmonic and wide-band random excitations.

A quasi-Gaussian approximation method for the Duffing oscillator with colored additive random excitation

Statistical analysis of stochastic dynamical systems is of considerable importance for engineers as well as scientists. Engineering applications require approximate statistical methods with a trade-off between accuracy and simplicity. Most exact and approximate methods available in the literature to study stochastic differential equations (SDEs) are best suited for linear or lightly nonlinear systems. When a system is highly nonlinear, e.g., a system with multiple equilibria, the accuracy of conventional methods degrades. This problem is addressed in this article, and a novel method is introduced for statistical analysis of special types of essentially nonlinear SDEs. In particular, second-order dynamical systems with nonlinear stiffness and additive random excitations are considered. The proposed approximate method can estimate second-order moments of the state vector (namely position and velocity), not only in the case of white noise excitation, but also when the excitation is a correlated noise. To illustrate the efficiency, a second-order dynamical system with bistable Duffing-type nonlinearity is considered as the case study. Results of the proposed method are compared with the Gaussian moment closure approximation for two types of colored noise excitations, one with first-order dynamics and the other with second-order dynamics. In the absence of exact closed-form solutions, Monte Carlo simulations are considered as the reference ideal solution. Results indicate that the proposed method gives proper approximations for the mean square value of position, for which the Gaussian moment closure method cannot provide good estimations. On the other hand, both methods provide acceptable estimations for the mean square value of velocity in terms of accuracy. Such nonlinear SDEs especially arise in energy-harvesting applications, when the ambient vibration can be modeled as a wideband random excitation. In such conditions, linear energy harvesters are no longer optimal designs, but nonlinear broadband harvesting techniques are hoped to show much better performance.

Stationary response of nonlinear Markovian jump system under wide-band random excitation

An approximate procedure for predicting the stationary response of a stochastically excited nonlinear Markovian jump system under wide-band random excitation is proposed. Firstly, a weighted-averag...

Elastic perfectly plastic oscillator under random loads: Linearization and response power spectral density

The randomly-excited elastic-perfectly-plastic oscillator—hysteretic bilinear oscillator with zero secondary stiffness—has been extensively researched. The vast majority of the work on that system has investigated the time-domaine statistics of the response. No studies have focused on the power spectral densities. This study specifically examines the system's velocity power spectral density—the system's displacement is nonstationary—under wide-band random excitations by means of a statistical linearization/stochastic averaging technique that is developed in one existing and two new procedures, one with constant, the other with amplitude-dependent parameters. The three procedures are evaluated against Monte Carlo simulation and classical Gaussian linearization in terms of the velocity average power, the peak frequency of the power spectral density, its peak value, its bandwidth, and its overall shape around the main frequency. The best predictions were yielded by the new procedure with amplitude-dependent parameters, which combines an extended amplitude-phase transformation, a linearization with random parameters into a Maxwell system, a correlation-function-based criterion, the conditional power spectral density concept, and a power conservation correction step. The new procedures can be adapted to apply to other hysteretic systems.

Soft and Hard Piezoelectric Ceramics and Single Crystals for Random Vibration Energy Harvesting

AbstractVibration‐based energy harvesting for enabling next‐generation self‐powered devices is a rapidly growing research area. In real‐world applications, the ambient vibrational energy is often available in non‐deterministic forms rather than the extensively studied deterministic scenarios, such as simple harmonic excitation. It is of interest to choose the best piezoelectric material for a given random excitation. Here, performance comparisons of various soft and hard piezoelectric ceramics and single crystals are presented for electrical power generation under band‐limited off‐resonance and wideband random vibration energy‐harvesting scenarios. For low‐frequency off‐resonance excitation, it is found that soft piezoelectric ceramics based upon lead zirconate titanate (e.g., PZT‐5H and PZT‐5A) outperform their hard counterparts (e.g., PZT‐4 and PZT‐8), and likewise soft single crystals based upon lead magnesium niobate and lead titanate as well as PZT (e.g., PMN‐PT and PMN‐PZT) outperform the relatively hard ones (e.g., manganese‐doped PMN‐PZT‐Mn). Overall, for such off‐resonance random vibrations, PMN‐PT is the most suitable choice among the materials studied. For wideband random excitation with a bandwidth covering the fundamental resonance of the harvester, hard piezoelectric ceramics offer larger power output compared to soft ceramics, and likewise hard single crystals produce larger power compared to their soft counterparts. Remarkably, a hard piezoelectric ceramic (e.g., PZT‐8) can outperform a soft single crystal (e.g., PMN‐PT) for wideband random vibration energy harvesting.

Statistical Linearization Analysis of a Hydropneumatic Suspension System With Nonlinearity

Hydropneumatic suspension has the characteristics of nonlinear stiffness, nonlinear friction damping, and nonlinear hydraulic damping, and therefore, it has been widely applied to heavy vehicles. A hydropneumatic suspension with a special structure was proposed and studied in this paper. The nonlinear quarter-vehicle model was built by using Newton’s laws according to its configuration. Then, this nonlinear mathematic model was linearized through the statistical linearization method on the basis of random vibration theory. Next, the transfer functions of the suspension system subjected to random road excitation were built according to the statistical linearization model and the power density spectrum approach. In addition, the responses of vehicle body acceleration, tire relative dynamic load, and suspension deflection with respect to the wideband random excitation due to road roughness were obtained according to the James formula. Next, the influences of the equivalent damping ratio and the equivalent frequency ratio on vehicle riding comfort, riding safety, handing stability, and suspension reliability were analyzed through simulations. These results provided the basic principles for selecting the reasonable hydropneumatic suspension parameters.

First-exit problem of MDOF strongly nonlinear oscillators under wide-band random excitations without internal resonances

The first-exit problem of strongly nonlinear oscillators with multi-degrees-of-freedom (MDOF) is studied theoretically in this paper. The excitations are modelled as wide-band random noises. When there is no internal resonance, the equations of motion of the original system are reduced to a set of averaged Ito stochastic differential equations (SDEs) after stochastic averaging. Two partial differential equations (PDEs), namely the backward Kolmogorov equation governing the conditional reliability function and the Pontryagin equation governing the mean first-exit time, are established under appropriate boundary and (or) initial conditions. The theoretical method is applied to study a 3-DOF strongly nonlinear system. Long expressions of drift and diffusion coefficients of the averaged Ito SDEs are obtained by using software Mathematica. The qualitative boundary conditions of the associated backward Kolmogorov equation and the Pontryagin equation are quantified by examining the behaviour of drift coefficients at the lower boundary. The method of finite difference is used to solve the corresponding high-dimensional PDEs. Monte Carlo simulation is performed to verify the validity of the proposed theoretical method.

Stochastic Optimal Control of Stayed Cable Vibrations with Wide-Band Random Wind Excitation Using Axial Support Motion

This paper proposes an optimal control strategy and corresponding method, to use the advantageous characteristics of the axial support motion to control the stayed cable vibrations in plane by wind excitation. Through a thorough investigation of wide-band random wind excitation and spatial relativity of the wind load along the cable, this study derives realistic and accurate motion equations of the control system. These motion equations are simplified into the form of the first four modes using the Galerkin method. The Itô stochastic differential equations of partial average for the system energy are derived using the stochastic averaging method of the quasi-integrable Hamiltonian systems. Furthermore, the dynamic programming equations with performance index are established by the stochastic dynamic programming principle. The random optimal control law is obtained by solving the dynamic programming equations using the approximate method. The numerical results show that, when standard deviations of the axial support motion are the same, the reduction of the standard deviation of the cable acceleration, and the cable displacement under the stochastic optimal control (SOC) law proposed by this paper, is larger than that under bilinear (BL) control. The SOC method proposed in this paper provides better results than the BL method.

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.

Time-delayed stochastic optimal control of strongly non-linear systems with actuator saturation by using stochastic maximum principle

A time-delayed stochastic optimal bounded control strategy for strongly non-linear systems under wide-band random excitations with actuator saturation is proposed based on the stochastic averaging method and the stochastic maximum principle. First, the partially averaged Itô equation for the system amplitude is derived by using the stochastic averaging method for strongly non-linear systems. The time-delayed feedback control force is approximated by a control force without time delay based on the periodically random behavior of the displacement and velocity of the system. The partially averaged Itô equation for the system energy is derived from that for the system amplitude by using Itô formula and the relation between system amplitude and system energy. Then, the adjoint equation and maximum condition of the partially averaged control problem are derived based on the stochastic maximum principle. The saturated optimal control force is determined from maximum condition and solving the forward–backward stochastic differential equations (FBSDEs). For infinite time-interval ergodic control, the adjoint variable is stationary process and the FBSDE is reduced to a ordinary differential equation. Finally, the stationary probability density of the Hamiltonian and other response statistics of optimally controlled system are obtained from solving the Fokker–Plank–Kolmogorov (FPK) equation associated with the fully averaged Itô equation of the controlled system. For comparison, the optimal control forces obtained from the time-delayed bang–bang control and the control without considering time delay are also presented. An example is worked out to illustrate the proposed procedure and its advantages.

Enhanced vibrational energy harvester based on velocity amplification

This article presents a fundamental investigation in which velocity amplification is employed in non-resonant structures to enhance the power harvested from ambient vibrations. Velocity amplification is achieved utilising sequential collisions between free-moving masses, and the final velocity is proportional to the number of masses and the mass ratios selected. The governing theory is discussed, particularly how the final velocity scales with the number of masses. This article examines n-mass velocity-amplified vibration energy harvesters and examines their performance relative to single-mass harvesters. Electromagnetic energy conversion is chosen as it is fundamental in allowing the free movement of the masses. Experimental results from two- and three-mass prototypes are presented that demonstrate a wider frequency response and a gain in power of 33 times compared to single-mass configurations under wideband random excitation. The volume of the devices was constrained, which resulted in the two-mass system outperforming the triple-mass system counter to expectations. This was caused by the triple-mass device experiencing an increased number of impact due to the volume constraint, leading to high losses in the system. It is recommended that in order to realise the full benefits of the triple-mass system, additional volume for mass actuation is required.

Free and forced random vibration analysis of sandwich plates with thick viscoelastic cores

Free vibrations and the transverse response of sandwich plates with viscoelastic cores under wide-band random excitation is studied with special attention to the so-called pumping, thickness-shear and stretching modes. The quadratic displacement field is adopted for all displacement components of the core to accurately capture the higher modes excited by the wide-band excitation. The Love-Kirchhoff plate theory is used for the face layers. The viscoelastic behavior of the core is modeled by the Golla–Hughes–McTavish method. An analytical solution using the normal mode method is provided for the simply supported boundary conditions by including a different family of modes. The effects of some geometric and material properties on the frequencies, damping ratios and also root mean square responses are explored. The participation of the through-the-thickness deformation in the bending mode vibration of the top layer is also investigated, which is found to be mostly resulting from the second order term of the transverse displacement expansion in symmetric configurations. Moreover, the alteration of the response with the exclusion of a different family of modes from the solution is investigated.

Analysis of a passive control of a chain under wide-band random excitation by nonlinear energy sinks using generalized orthogonal decompositions

The objective of this paper is to show how the generalized orthogonal decomposition named smooth decomposition can be used to analyze the energy pumping phenomenon in the context of vibration reduction under wide-band random excitation.

First-passage failure of a business cycle model under time-delayed feedback control and wide-band random excitation

First-passage failure of a classical business cycle model subject to time-delayed feedback control and wide-band random excitation is investigated in this paper. First, we get the averaged equation by using a stochastic averaging method, and then a backward Kolmogorov equation governing the conditional reliability function and a set of generalized Pontryagin equations governing the conditional moments of first-passage time are established. The conditional reliability function, the conditional probability density and moments of first-passage time are obtained by solving the backward Kolmogorov equation and generalized Pontryagin equations analytically and numerically with suitable initial and boundary conditions. The results show that time delay in the feedback control forces may remarkably reduce the conditional reliability and the mean first-passage time of the controlled economics system.

Stochastic averaging of n-dimensional non-linear dynamical systems subject to non-Gaussian wide-band random excitations

The averaged generalized Fokker–Planck–Kolmogorov (GFPK) equation for response of n-dimensional ( n-d) non-linear dynamical systems to non-Gaussian wide-band stationary random excitation is derived from the standard form of equation of motion. The explicit expressions for coefficients of the fourth-order approximation of the averaged GFPK equation are given in series form. Conditions for convergences of these series are pointed out. The averaged GFPK equation is then reduced to that for 1-d dynamical systems derived by Stratonovich and compared with the closed form of GFPK equation for n-d dynamical systems subject to Poisson white noise derived by Di Paola and Falsone. Finally, this averaged GFPK equation is further reduced to that for quasi linear system subject to non-Gaussian wide-band stationary random excitation. Stationary probability density for quasi linear system subject to filtered Poisson white noise is obtained. Theoretical results for an example are confirmed by using Monte-Carlo simulation for different parameter values.

Stochastic Averaging of Quasi Linear Systems Subject to Multi- time-delayed Feedback Control and Wide-band Random Excitation

A stochastic averaging technique for resonant and non-resonant quasi linear systems subject to multi-time-delayed feedback control and wide-band random excitation is developed. The technique is then applied to predict the response of one and two degree-of-freedom linear oscillators with nonlinear dampings subject to wide-band random excitations and multi-time-delayed feedback control. It is shown through comparison with the result from Monte Carlo simulation that the proposed technique yields quite a good result, even for long delay time, and that the time-delayed feedback control may result in phenomenological bifurcation.

First-passage failure of quasi linear systems subject to multi-time-delayed feedback control and wide-band random excitation

A procedure for studying the first-passage failure of quasi-linear systems subject to multi-time-delayed feedback control and wide-band random excitation is proposed. The stochastic averaging method for quasi-integrable Hamiltonian systems is first introduced. The backward Kolmogorov equation governing the conditional reliability function and a set of generalized Pontryagin equations governing the conditional moments of first-passage time are then established. The conditional reliability function, the conditional probability density and moments of first-passage time are obtained by solving the backward Kolmogorov equation and generalized Pontryagin equations with suitable initial and boundary conditions. An example is given to illustrate the proposed procedure and the results from digital simulation are obtained to verify the effectiveness of the proposed procedure. The effects of time delay in feedback control forces on the conditional reliability function, conditional probability density and moments of first-passage time are analyzed.