scholarly journals Dynamic Analysis of a Plate on the Generalized Foundation with Fractional Damping Subjected to Random Excitation

2018 ◽  
Vol 2018 ◽  
pp. 1-10 ◽  
Author(s):  
Ali Hosseinkhani ◽  
Davood Younesian ◽  
Saman Farhangdoust
Keyword(s):  
Probability Function ◽  
Stochastic Averaging ◽  
Random Excitation ◽  
Stochastic Averaging Method ◽  
Function Technique ◽  
Stationary Probability ◽  
Support Design ◽  
Governing Differential Equation ◽  
The Galerkin Method ◽  
Generalized Harmonic Function

Stochastic response of a plate on the generalized foundation driven by random excitation is solved in this paper. Governing differential equation is obtained by employing the Galerkin method. The generalized harmonic function technique is applied to the governing equation of motion. Using the stochastic averaging method (SAM), the system is approximated by the time homogeneous diffusive Markov process. Corresponding approximate stationary probability function is achieved by solving associated Fokker-Plank-Kolmogorov (FPK). An analytical solution is presented for the stationary probability of the amplitude and velocity. Validity of the stationary probability is verified by Monte-Carlo simulation. Parametric study is carried out to investigate effects of foundation parameters and excitation intensity on the stationary probability function. It is found that the fractional properties act similar to the foundation stiffness and damping and can be employed as a new control parameter for the support design.

Random Response of Wake-Oscillator Excited by Fluctuating Wind

2021 ◽  
pp. 1-33
Author(s):  
Mao Lin Deng ◽  
Genjin Mu ◽  
Weiqiu Zhu
Keyword(s):  
Hamiltonian Systems ◽  
Averaging Method ◽  
Stochastic Averaging ◽  
Random Excitation ◽  
Stochastic Averaging Method ◽  
Random Response ◽  
Integrable Hamiltonian Systems ◽  
Wake Oscillator ◽  
Fluctuating Wind ◽  
Oscillator Models

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. While in the field of wind engineering, the real wind generally is described to be composed of mean wind and fluctuating wind. The wake-oscillator excited by fluctuating wind should be treated as a randomly excited and dissipated multi-degree of freedom (DOF) nonlinear system. The involved studies are very difficult and so far there are no exact solutions available. The present paper aims to carry out some study works on the stochastic dynamics of VIV. The stochastic averaging method of quasi integrable Hamiltonian systems under wideband random excitation is applied to study the Hartlen-Currie wake-oscillator model and its modified model excited by fluctuating wind. The probability and statistics of the random response of wake-oscillator in resonant or lock-in case and in non-resonant case are analytically obtained, and the theoretical results are confirmed by using numerical simulation of original system. Finally, it is pointed out that the stochastic averaging method of quasi integrable Hamiltonian systems under wideband random excitation can also be applied to other wake-oscillator models, such as Skop-Griffin model and Krenk-Nielsen model excited by fluctuating wind.

scholarly journals Applying higher order stochastic averaging method to the Van der Pol system with timedelay under random excitation

2019 ◽  
Vol 2 (2) ◽  
pp. 102-109
Author(s):  
Hao Ngoc Duong ◽  
Anh Dong Nguyen ◽  
Dung Quang Nguyen
Keyword(s):  
Time Delay ◽  
Averaging Method ◽  
Order Approximation ◽  
Original System ◽  
Stochastic Averaging ◽  
Random Excitation ◽  
Stochastic Averaging Method ◽  
Higher Order ◽  
Van Der Pol ◽  
System With Time Delay

The paper investigated the Van der Pol system with time-delay under random excitation by the higher stochastic averaging method. The original system was expressed in terms without time-delay under the assumption that the state variabled of the system were slowly varying processed. Then the higher stochastic averaging method was applied on the approximation system. By this technique, the analytical expression of the stationary probability density function for the Van der Pol system with time-delay under random excitation was showed in higher order approximation for the first time. Effects of the parameter time-delay on the system’s response were investigated. The analytical results were suited well to numerical ones obtained by Monte-Carlo method. It was also showed that the higher order averaging solution was better than the one obtained by the traditional stochastic averaging method.

Response Statistics of a Beam-Mass Oscillator Under Combined Harmonic and Random Excitation

1995 ◽  
Vol 1 (2) ◽  
pp. 225-247 ◽  
Author(s):  
Stephen Ekwaro-Osire ◽  
Atila Ertas
Keyword(s):  
Steady State ◽  
Harmonic Component ◽  
Stochastic Averaging ◽  
Random Excitation ◽  
Stochastic Averaging Method ◽  
Time Averaging ◽  
Mean Square ◽  
Mean Square Response ◽  
The Mean ◽  
Non Gaussian

In the present study, the response statistics of a beam-mass oscillator under combined harmonic and random excitation were investigated. The Gaussian and non-Gaussian closure schemes, in conjunction with the stochastic averaging method, were used to solve for the mean square response. The influence of the oscillator parameters on the response statistics was studied. The harmonic component of the excitation was observed to manifest itself, as an oscillation, in the steady-state mean square response. Results obtained showed that the non-Gaussian solution yields higher steady-state mean square responses than those obtained from the Gaussian solution. It was further shown that the harmonic time-varying properties of the oscillator are preserved by omitting the time-averaging in the stochastic averaging procedure.

scholarly journals Response of Duffing Oscillator with Time Delay Subjected to Combined Harmonic and Random Excitations

2017 ◽  
Vol 2017 ◽  
pp. 1-8
Author(s):  
D. N. Hao ◽  
N. D. Anh
Keyword(s):  
Time Delay ◽  
Probability Density ◽  
Duffing Oscillator ◽  
Original System ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
Auxiliary Function ◽  
Equivalent Linearization ◽  
Stationary Probability ◽  
Stationary Probability Density

This paper aims to investigate the stationary probability density functions of the Duffing oscillator with time delay subjected to combined harmonic and white noise excitation by the method of stochastic averaging and equivalent linearization. By the transformation based on the fundamental matrix of the degenerate Duffing system, the paper shows that the displacement and the velocity with time delay in the Duffing oscillator can be computed approximately in non-time delay terms. Hence, the stochastic system with time delay is transformed into the corresponding stochastic non-time delay equation in Ito sense. The approximate stationary probability density function of the original system can be found by combining the stochastic averaging method, the equivalent linearization method, and the technique of auxiliary function. The response of Duffing oscillator is investigated. The analytical results are verified by numerical simulation results.

scholarly journals Vibrations of an Elastic Beam Subjected by Two Kinds of Moving Loads and Positioned on a Foundation having Fractional Order Viscoelastic Physical Properties

2021 ◽  
Author(s):  
Lionel Merveil Anague Tabejieu ◽  
Blaise Roméo Nana Nbendjo ◽  
Giovanni Filatrella
Keyword(s):  
Physical Properties ◽  
Fractional Order ◽  
Elastic Beam ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
Moderate Increase ◽  
Moving Loads ◽  
Resonant Amplitude ◽  
Wind Turbulence ◽  
And Shifts

The present chapter investigates both the effects of moving loads and of stochastic wind on the steady-state vibration of a first mode Rayleigh elastic beam. The beam is assumed to lay on foundations (bearings) that are characterized by fractional-order viscoelastic material. The viscoelastic property of the foundation is modeled using the constitutive equation of Kelvin-Voigt type, which contain fractional derivatives of real order. Based to the stochastic averaging method, an analytical explanation on the effects of the viscoelastic physical properties and number of the bearings, additive and parametric wind turbulence on the beam oscillations is provided. In particular, it is found that as the number of bearings increase, the resonant amplitude of the beam decreases and shifts towards larger frequency values. The results also indicate that as the order of the fractional derivative increases, the amplitude response decreases. We are also demonstrated that a moderate increase of the additive and parametric wind turbulence contributes to decrease the chance for the beam to reach the resonance. The remarkable agreement between the analytical and numerical results is also presented in this chapter.

An improved energy envelope stochastic averaging method and its application to a nonlinear oscillator

2016 ◽  
Vol 23 (1) ◽  
pp. 119-130 ◽  
Author(s):  
Yaping Zhao
Keyword(s):  
Steady State ◽  
Probability Density Function ◽  
Probability Density ◽  
Averaging Method ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
System Response ◽  
Mean Square ◽  
Fpk Equation ◽  
The Mean

An improved stochastic averaging method of the energy envelope is proposed, whose application sphere is extensive and whose implementation is convenient. An oscillating system with both nonlinear damping and stiffness is taken into account. Its averaged Fokker-Planck-Kolmogorov (FPK) equation in respect of the transition probability density function of the energy envelope is deduced by virtue of the method mentioned above. Under the initial and boundary conditions, the joint probability density function as to the displacement and velocity of the system is worked out in closed form after solving the averaged FPK equation by right of a technique based on the integral transformation. With the aid of the special functions, the transient solutions of the probabilistic characteristics of the system response are further derived analytically, including the probability density functions and the mean square values. A simple approach to generate the ideal white noise is drastically ameliorated in order to produce the stationary wide-band stochastic external excitation for the Monte Carlo simulating investigation of the nonlinear system. Both the theoretical solution and the numerical solution of the probabilistic properties of the system response are obtained, which are extremely coincident with each other. The numerical simulation and the theoretical computation all show that the time factor has a certain influence on the probability characteristics of the response. For example, the probabilistic distribution of the displacement tends to be scattered and the mean square displacement trends toward its steady-state value as time goes by. Of course the transient process to reach the steady-state value will obviously be shorter if the damping of the system is greater.

The Effects of Correlated Noise on Bifurcation in Birhythmicity Driven by Delay

2018 ◽  
Vol 28 (10) ◽  
pp. 1850127 ◽  
Author(s):  
Lijuan Ning ◽  
Zhidan Ma
Keyword(s):  
Averaging Method ◽  
Harmonic Approximation ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
Self Control ◽  
Correlated Noise ◽  
State Variables ◽  
Stationary Probability Density Function ◽  
Fpk Equation ◽  
Control Feedback

We consider bifurcation regulations under the effects of correlated noise and delay self-control feedback excitation in a birhythmic model. Firstly, the term of delay self-control feedback is transferred into state variables without delay by harmonic approximation. Secondly, FPK equation and stationary probability density function (SPDF) for amplitude can be theoretically mapped with stochastic averaging method. Thirdly, the intriguing effects on bifurcation regulations in a birhythmic model induced by delay and correlated noise are observed, which suggest the violent dependence of bifurcation in this model on delay and correlated noise. Particularly, the inner limit cycle (LC) is always standing due to noise. Lastly, the validity of analytical results was confirmed by Monte Carlo simulation for the dynamics.

Asymptotic Stability of Controlled Nonlinear Stochastic Systems Considering the Dynamics of Sensors and Actuators

2021 ◽  
pp. 2150180
Author(s):  
Jiaojiao Sun ◽  
Zuguang Ying ◽  
Ronghua Huan ◽  
Weiqiu Zhu
Keyword(s):  
Asymptotic Stability ◽  
Stochastic Systems ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
Stable Region ◽  
High Dimensional ◽  
Nonlinear Stochastic System ◽  
Dimensional System ◽  
Sensors And Actuators ◽  
Controlled System

A closed-loop controlled system usually consists of the main structure, sensors, and actuators. In this paper, asymptotic stability of trivial solutions of a controlled nonlinear stochastic system considering the dynamics of sensors and actuators is investigated. Considering the inherent and intentional nonlinearities and random loadings, the coupled dynamic equations of the controlled system with sensors and actuators are given, which are further formulated by a controlled, randomly excited, dissipated Hamiltonian system. The Hamiltonian of the controlled system is introduced, and, based on the stochastic averaging method, the original high-dimensional system is reduced to a one-dimensional averaged system. The analytical expression of Lyapunov exponent of the averaged system is derived, which gives the approximately necessary and sufficient condition of the asymptotic stability of trivial solutions of the original high-dimensional system. The validation of the proposed method is demonstrated by a four-degree-of-freedom controlled system under pure stochastically parametric excitations in detail. A comparative analysis, which is related to the stochastic asymptotic stability of the system with and without considering the dynamics of sensors and actuators, is carried out to investigate the effect of their dynamics on the motion of the controlled system. Results show that ignoring the dynamics of sensors and actuators will get a shrink stable region of the controlled system.

STOCHASTIC HOPF BIFURCATION OF QUASI-INTEGRABLE HAMILTONIAN SYSTEMS WITH FRACTIONAL DERIVATIVE DAMPING

2012 ◽  
Vol 22 (04) ◽  
pp. 1250083 ◽  
Author(s):  
F. HU ◽  
W. Q. ZHU ◽  
L. C. CHEN
Keyword(s):  
Hopf Bifurcation ◽  
Fractional Derivative ◽  
Hamiltonian Systems ◽  
Averaging Method ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
Bifurcation Parameter ◽  
Integrable Hamiltonian Systems ◽  
Fractional Derivative Damping ◽  
Stochastic Hopf Bifurcation

The stochastic Hopf bifurcation of multi-degree-of-freedom (MDOF) quasi-integrable Hamiltonian systems with fractional derivative damping is investigated. First, the averaged Itô stochastic differential equations for n motion integrals are obtained by using the stochastic averaging method for quasi-integrable Hamiltonian systems. Then, an expression for the average bifurcation parameter of the averaged system is obtained and a criterion for determining the stochastic Hopf bifurcation of the system by using the average bifurcation parameter is proposed. An example is given to illustrate the proposed procedure in detail and the numerical results show the effect of fractional derivative order on the stochastic Hopf bifurcation.

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