scholarly journals Subharmonic Response of Linear Vibroimpact System with Fractional Derivative Damping to a Randomly Disrobed Periodic Excitation

2015 ◽  
Vol 2015 ◽  
pp. 1-10 ◽  
Author(s):  
Haiwu Rong ◽  
Xiangdong Wang
Keyword(s):  
Fractional Derivative ◽  
Excitation Frequency ◽  
Random Excitation ◽  
Response Amplitude ◽  
System Response ◽  
System Parameters ◽  
Fractional Derivative Damping ◽  
Derivative Term ◽  
Subharmonic Response ◽  
Random Disorder

The subharmonic response of single-degree-of-freedom vibroimpact oscillator with fractional derivative damping and one-sided barrier under narrow-band random excitation is investigated. With the help of a special Zhuravlev transformation, the system is reduced to one without impacts, thereby permitting the applications of asymptotic averaging over the period for slowly varying random process. The analytical expression of the response amplitude is obtained in the case without random disorder, while only the approximate analytical expressions for the steady-state moments of the response amplitude are obtained in the case with random disorder. The effects of the fractional order derivative term, damping term, random disorder, and the coefficient of restitution and other system parameters on the system response are discussed. Theoretical analyses and numerical simulations show that fractional derivative makes both the system damping and stiffness coefficients increase, such that it changes the system parameters region at which the response amplitude reaches the maximum. The system energy loss in collision is equivalent to increasing the damping coefficient of the system. System response amplitude will increase when the excitation frequency is close to the resonant frequency and will decay rapidly when the excitation frequency gradually deviates from the resonance frequency.

Efficacy of a Sliding Beam as a Nonlinear Vibration Isolator

2007 ◽  
Author(s):  
R. A. Ibrahim ◽  
R. J. Somnay
Keyword(s):  
Friction Coefficient ◽  
Initial Conditions ◽  
Excitation Frequency ◽  
Random Excitation ◽  
Excitation Power ◽  
Basins Of Attraction ◽  
System Response ◽  
Density Level ◽  
Vibration Isolator ◽  
Power Spectral

The influence of friction due to beam sliding at its supports on its dynamic behavior and its efficacy as a nonlinear isolator is studied numerically under sinusoidal and random excitation excitations. Under sinusoidal excitation, the equation of motion of the system is solved numerically and the solution is utilized to estimate the system transmissibility. It is found that when the excitation frequency is increased beyond resonance, the friction at the sliding supports serves to improve the transmissibility. The dependence of the response on initial conditions establishes the basins of attraction for different values of friction coefficient and excitation frequency and amplitude. Under random excitation, the system response statistics are estimated from Monte Carlo simulation results for different values of friction coefficient and excitation power spectral density level. The friction is found to result in a significant reduction of the system response mean square.

Finite Difference Method on Non-Uniform Meshes for Time Fractional Diffusion Problem

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Haili Qiao ◽  
Aijie Cheng
Keyword(s):  
Theoretical Analysis ◽  
Fractional Derivative ◽  
Time Discretization ◽  
Summation Formula ◽  
Fractional Diffusion ◽  
Diffusion Problem ◽  
Initial Moment ◽  
Central Difference ◽  
Ideal Convergence ◽  
Derivative Term

AbstractIn this paper, we consider the time fractional diffusion equation with Caputo fractional derivative. Due to the singularity of the solution at the initial moment, it is difficult to achieve an ideal convergence rate when the time discretization is performed on uniform meshes. Therefore, in order to improve the convergence order, the Caputo time fractional derivative term is discretized by the {L2-1_{\sigma}} format on non-uniform meshes, with {\sigma=1-\frac{\alpha}{2}}, while the spatial derivative term is approximated by the classical central difference scheme on uniform meshes. According to the summation formula of positive integer k power, and considering {k=3,4,5}, we propose three non-uniform meshes for time discretization. Through theoretical analysis, different time convergence orders {O(N^{-\min\{k\alpha,2\}})} can be obtained, where N denotes the number of time splits. Finally, the theoretical analysis is verified by several numerical examples.

scholarly journals A CONDITIONAL STOCHASTIC PROJECTION METHOD APPLIED TO A PARAMETRIC VIBRATIONS PROBLEM

2014 ◽  
Vol 20 (6) ◽  
pp. 810-818 ◽  
Author(s):  
Wlodzimierz Brzakala ◽  
Aneta Herbut
Keyword(s):  
Monte Carlo Simulation ◽  
Monte Carlo ◽  
Polynomial Chaos ◽  
Excitation Frequency ◽  
Finite Sequence ◽  
Excitation Amplitude ◽  
Random Excitation ◽  
Parametric Vibrations ◽  
Representative Points ◽  
Cable Stayed Bridges

Parametric vibrations can be observed in cable-stayed bridges due to periodic excitations caused by a deck or a pylon. The vibrations are described by an ordinary differential equation with periodic coefficients. The paper focuses on random excitations, i.e. on the excitation amplitude and the excitation frequency which are two random variables. The excitation frequency ωL is discretized to a finite sequence of representative points, ωL,i Therefore, the problem is (conditionally) formulated and solved as a one-dimensional polynomial chaos expansion generated by the random excitation amplitude. The presented numerical analysis is focused on a real situation for which the problem of parametric resonance was observed (a cable of the Ben-Ahin bridge). The results obtained by the use of the conditional polynomial chaos approximations are compared with the ones based on the Monte Carlo simulation (truly two-dimensional, not conditional one). The convergence of both methods is discussed. It is found that the conditional polynomial chaos can yield a better convergence then the Monte Carlo simulation, especially if resonant vibrations are probable.

New Insight Into Energy Harvesting via Axially-Loaded Beams

2009 ◽  
Author(s):  
Mohammed F. Daqaq
Keyword(s):  
Steady State ◽  
Energy Harvesting ◽  
Axial Load ◽  
Excitation Frequency ◽  
Axial Loading ◽  
Response Amplitude ◽  
Resistive Load ◽  
Steady State Response ◽  
State Response ◽  
Axially Loaded

Driven by the study of Leland and Wright [1], this manuscript delves into the qualitative understanding of energy harvesting using axially-loaded beams. Using a simple nonlinear electromechanical model and the method of multiple scales, we study the general nonlinear physics of energy harvesting from a piezoelectric beam subjected to static axial loading and traversal dynamic excitation. We obtain analytical expressions for the steady-state response amplitude, the voltage drop across a resistive load, and the output power. We utilize these expression to study the effect of the axial loading on the overall nonlinear behavior of the harvester. It is demonstrated that, in addition to the ability of tuning the harvester to the excitation frequency via axial load variations, the axial load aids in i) increasing the electric damping in the system thereby enhancing the energy transfer from the beam to the electric load, ii) amplifying the effect of the external excitation on the structure, and hence, increases the steady-state response amplitude and output voltage, and iii) increasing the bandwidth of the harvester by enhancing the effective nonlinearity of the system.

Beating Effects in a Single Nonlinear Dynamical System in a Neighborhood of the Resonance

2016 ◽  
Vol 849 ◽  
pp. 76-83
Author(s):  
Jiří Náprstek ◽  
Cyril Fischer
Keyword(s):  
Harmonic Balance ◽  
Excitation Frequency ◽  
Nonlinear Dynamical System ◽  
Numerical Continuation ◽  
System Response ◽  
Algebraic Equations ◽  
Single Degree Of Freedom ◽  
Nonlinear Dynamical ◽  
The Difference ◽  
Eigen Frequency

The exact coincidence of external excitation and basic eigen-frequency of a single degree of freedom (SDOF) nonlinear system produces stationary response with constant amplitude and phase shift. When the excitation frequency differs from the system eigen-frequency, various types of quasi-periodic response occur having a character of a beating process. The period of beating changes from infinity in the resonance point until a couple of excitation periods outside the resonance area. Theabove mentioned phenomena have been identified in many papers including authors’ contributions. Nevertheless, investigation of internal structure of a quasi-period and its dependence on the difference of excitation and eigen-frequency is still missing. Combinations of harmonic balance and small parameter methods are used for qualitative analysis of the system in mono- and multi-harmonic versions. They lead to nonlinear differential and algebraic equations serving as a basis for qualitativeanalytic estimation or numerical description of characteristics of the quasi-periodic system response. Zero, first and second level perturbation techniques are used. Appearance, stability and neighborhood of limit cycles is evaluated. Numerical phases are based on simulation processes and numerical continuation tools. Parametric evaluation and illustrating examples are presented.

Asymptotic Approximations for Systems Incorporating Fractional Derivative Damping

1996 ◽  
Vol 118 (3) ◽  
pp. 572-579 ◽  
Author(s):  
B. S. Liebst ◽  
P. J. Torvik
Keyword(s):  
Fractional Derivative ◽  
Natural Frequency ◽  
Fractional Derivatives ◽  
Asymptotic Approximations ◽  
Damping Factor ◽  
Polynomial Equations ◽  
Algebraic Expressions ◽  
Dependent Behavior ◽  
Fractional Derivative Damping ◽  
Damping Materials

Viscoelastic constitutive relationships incorporating fractional derivatives have been previously shown to be extremely useful in describing the frequency dependent behavior of common damping materials. However, the implementation of such models in the analysis of damped mechanical systems is somewhat complicated by the fact that polynomial equations with noninteger order exponents must be solved. This paper develops accurate approximations from which the damping factor and damped natural frequency of such systems may be obtained by evaluating relatively simple algebraic expressions.

Galerkin Scheme-Based Determination of Survival Probability of Oscillators With Fractional Derivative Elements

2016 ◽  
Vol 83 (12) ◽  
Author(s):  
Pol D. Spanos ◽  
Alberto Di Matteo ◽  
Yezeng Cheng ◽  
Antonina Pirrotta ◽  
Jie Li
Keyword(s):  
Fractional Derivative ◽  
Survival Probability ◽  
Stochastic Averaging ◽  
Kolmogorov Equation ◽  
System Response ◽  
Statistical Linearization ◽  
Van Der Pol ◽  
Galerkin Scheme ◽  
Confluent Hypergeometric Functions

In this paper, an approximate semi-analytical approach is developed for determining the first-passage probability of randomly excited linear and lightly nonlinear oscillators endowed with fractional derivative elements. The amplitude of the system response is modeled as one-dimensional Markovian process by employing a combination of the stochastic averaging and the statistical linearization techniques. This leads to a backward Kolmogorov equation which governs the evolution of the survival probability of the oscillator. Next, an approximate solution of this equation is sought by resorting to a Galerkin scheme. Specifically, a convenient set of confluent hypergeometric functions, related to the corresponding linear oscillator with integer-order derivatives, is used as orthogonal basis for this scheme. Applications to the standard viscous linear and to nonlinear (Van der Pol and Duffing) oscillators are presented. Comparisons with pertinent Monte Carlo simulations demonstrate the reliability of the proposed approximate analytical solution.

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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