scholarly journals Stochastic P-Bifurcation of a Bistable Viscoelastic Beam with Fractional Constitutive Relation under Gaussian White Noise

2018 ◽  
Vol 2018 ◽  
pp. 1-10
Author(s):  
Yajie Li ◽  
Zhiqiang Wu ◽  
Guoqi Zhang ◽  
Feng Wang
Keyword(s):  
White Noise ◽  
Minimum Mean Square Error ◽  
Gaussian White Noise ◽  
Singularity Theory ◽  
Original System ◽  
Viscoelastic Beam ◽  
Damping Force ◽  
Restoring Force ◽  
Transition Set ◽  
Axially Moving

In this paper, we study the stochastic P-bifurcation problem for axially moving of a bistable viscoelastic beam with fractional derivatives of high order nonlinear terms under Gaussian white noise excitation. First, using the principle for minimum mean square error, we show that the fractional derivative term is equivalent to a linear combination of the damping force and restoring force, so that the original system can be simplified to an equivalent system. Second, we obtain the stationary Probability Density Function (PDF) of the system’s amplitude by stochastic averaging, and using singularity theory, we find the critical parametric condition for stochastic P-bifurcation of amplitude of the system. Finally, we analyze the types of the stationary PDF curves of the system qualitatively by choosing parameters corresponding to each region within the transition set curve. We verify the theoretical analysis and calculation of the transition set by showing the consistency of the numerical results obtained by Monte Carlo simulation with the analytical results. The method used in this paper directly guides the design of the fractional order viscoelastic material model to adjust the response of the system.

scholarly journals Stochastic P-bifurcation in a bistable Van der Pol oscillator with fractional time-delay feedback under Gaussian white noise excitation

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Yajie Li ◽  
Zhiqiang Wu ◽  
Guoqi Zhang ◽  
Feng Wang ◽  
Yuancen Wang
Keyword(s):  
Time Delay ◽  
White Noise ◽  
Gaussian White Noise ◽  
Singularity Theory ◽  
Van Der Pol Oscillator ◽  
Order System ◽  
Damping Force ◽  
Van Der Pol ◽  
Restoring Force ◽  
White Noise Excitation

Abstract The stochastic P-bifurcation behavior of a bistable Van der Pol system with fractional time-delay feedback under Gaussian white noise excitation is studied. Firstly, based on the minimal mean square error principle, the fractional derivative term is found to be equivalent to the linear combination of damping force and restoring force, and the original system is further simplified to an equivalent integer order system. Secondly, the stationary Probability Density Function (PDF) of system amplitude is obtained by stochastic averaging, and the critical parametric conditions for stochastic P-bifurcation of system amplitude are determined according to the singularity theory. Finally, the types of stationary PDF curves of system amplitude are qualitatively analyzed by choosing the corresponding parameters in each area divided by the transition set curves. The consistency between the analytical solutions and Monte Carlo simulation results verifies the theoretical analysis in this paper.

scholarly journals Stochastic transition behaviors in a generalized Van der Pol oscillator with fractional damping under Gaussian white noise

2021 ◽  
pp. 125-125
Author(s):  
Yajie Li ◽  
Zhiqiang Wu ◽  
Qixun Lan ◽  
Yujie Cai ◽  
Huafeng Xu ◽  
...  
Keyword(s):  
White Noise ◽  
Gaussian White Noise ◽  
Singularity Theory ◽  
Original System ◽  
Stochastic Averaging ◽  
Van Der Pol Oscillator ◽  
Van Der Pol ◽  
Fractional Damping ◽  
Transition Set ◽  
Fractional Order Controller

The stochastic P-bifurcation behavior of bi-stability in a generalized Van der Pol oscillator with a fractional damping under multiplicative Gaussian white noise excitation is investigated. Firstly, using the principle of minimal mean square error, the nonlinear stiffness terms can be equivalent to a linear stiffness which is a function of the system amplitude, and the original system is simplified to an equivalent integer order Van der Pol system. Secondly, the system amplitude?s stationary Probability Density Function (PDF) is obtained by stochastic averaging. And then according to the singularity theory, the critical parametric conditions for the system amplitude?s stochastic P-bifurcation are found. Finally, the types of the system?s stationary PDF curves of amplitude are qualitatively analyzed by choosing the corresponding parameters in each area divided by the transition set curves. The consistency between the analytical results and the numerical results obtained from Monte Carlo simulation verifies the theoretical analysis in this paper and the method used in this paper can directly guide the design of the fractional order controller to adjust the response of the system.

Response Analysis of van der Pol Vibro-Impact System with Coulomb Friction Under Gaussian White Noise

2018 ◽  
Vol 28 (13) ◽  
pp. 1830043 ◽  
Author(s):  
Meng Su ◽  
Wei Xu ◽  
Guidong Yang
Keyword(s):  
White Noise ◽  
Coulomb Friction ◽  
Gaussian White Noise ◽  
Original System ◽  
Stochastic Averaging ◽  
Response Analysis ◽  
State Variables ◽  
Van Der Pol ◽  
Impact System ◽  
The Impact

In this paper, the stationary response of a van der Pol vibro-impact system with Coulomb friction excited by Gaussian white noise is studied. The Zhuravlev nonsmooth transformation of the state variables is utilized to transform the original system to a new system without the impact term. Then, the stochastic averaging method is applied to the equivalent system to obtain the stationary probability density functions (pdfs). The accuracy of the analytical results obtained from the proposed procedure is verified by those from the Monte Carlo simulation based on the original system. Effects of different damping coefficients, restitution coefficients, amplitudes of friction and noise intensities on the response are discussed. Additionally, stochastic P-bifurcations are explored.

scholarly journals Steady-State Dynamical Response of a Strongly Nonlinear System with Impact and Coulomb Friction Subjected to Gaussian White Noise Excitation

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Guidong Yang ◽  
Dongmei Huang ◽  
Wei Li ◽  
Meng Su ◽  
Francesco Pellicano
Keyword(s):  
Steady State ◽  
White Noise ◽  
Coulomb Friction ◽  
Gaussian White Noise ◽  
Original System ◽  
Probability Density Functions ◽  
Dynamical Response ◽  
Strongly Nonlinear ◽  
White Noise Excitation ◽  
Strongly Nonlinear System

The paper is devoted to the steady-state dynamical response analysis of a strongly nonlinear system with impact and Coulomb friction subjected to Gaussian white noise excitation. The Zhuravlev nonsmooth transformation of the state variables combined with the Dirac delta function is utilized to simplify the original system to one without velocity jump. Then, the steady-state probability density functions of the transformed system are derived in terms of the stochastic averaging method of energy envelope. The effectiveness of the presented analytical procedure is verified by those from the Monte Carlo simulation based on the original system. Effects of different restitution coefficients, amplitudes of friction, and noise intensities on the steady-state dynamical responses are investigated in detail. Results show different intensities of Gaussian white noise can affect the peaks value of the probability density functions, whereas the variations of restitution coefficients and amplitudes of friction can induce the occurrence of stochastic P-bifurcation.

Driver Steering Control and a New Perspective on Car Handling Qualities

2005 ◽  
Vol 219 (10) ◽  
pp. 1041-1051 ◽  
Author(s):  
R S Sharp
Keyword(s):  
White Noise ◽  
Gaussian White Noise ◽  
Design Parameters ◽  
Steering Wheel ◽  
Steering Control ◽  
Pass Filter ◽  
Low Pass Filter ◽  
Handling Qualities ◽  
Gaussian White Noise Process ◽  
Low Pass

The article is about steering control of cars by drivers, concentrating on following the lateral profile of the roadway, which is presumed visible ahead of the car. It builds on previously published work, in which it was shown how the driver's preview of the roadway can be combined with the linear dynamics of a simple car to yield a problem of discrete-time optimal-linear-control-theory form. In that work, it was shown how an optimal ‘driver’ of a linear car can convert the path preview sample values, modelled as deriving from a Gaussian white-noise process, into steering wheel displacement commands to cause the car to follow the previewed path with an attractive compromise between precision and ease. Recognizing that real roadway excitation is not so rich in high frequencies as white-noise, a low-pass filter is added to the system. The white-noise sample values are filtered before being seen by the driver. Numerical results are used to show that the optimal preview control is unaltered by the inclusion of the low-pass filter, whereas the feedback control is affected diminishingly as the preview increases. Then, using the established theoretical basis, new results are generated to show time-invariant optimal preview controls for cars and drivers with different layouts and priorities. Tight and loose controls, representing different balances between tracking accuracy and control effort, are calculated and illustrated through simulation. A new performance criterion with handling qualities implications is set up, involving the minimization of the preview distance required. The sensitivities of this distance to variations in the car design parameters are calculated. The influence of additional rear wheel steering is studied from the viewpoint of the preview distance required and the form of the optimal preview gain sequence. Path-following simulations are used to illustrate relatively high-authority and relatively low-authority control strategies, showing manoeuvring well in advance of a turn under appropriate circumstances. The results yield new insights into driver steering control behaviour and vehicle design optimization. The article concludes with a discussion of research in progress aimed at a further improved understanding of how drivers control their vehicles.

Spectral element analysis for an axially moving viscoelastic beam

2004 ◽  
Vol 18 (7) ◽  
pp. 1159-1168 ◽  
Author(s):  
Hyungmi Oh ◽  
Jooyong Cho ◽  
Usik Lee
Keyword(s):  
Spectral Element ◽  
Viscoelastic Beam ◽  
Element Analysis ◽  
Axially Moving
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