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.

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.

Mean-Square Response of a Second-Order System to Nonstationary, Random Excitation

1970 ◽  
Vol 37 (3) ◽  
pp. 612-616 ◽  
Author(s):  
L. L. Bucciarelli ◽  
C. Kuo
Keyword(s):  
Narrow Band ◽  
Random Excitation ◽  
Second Order ◽  
Envelope Function ◽  
Order System ◽  
Mean Square ◽  
Forcing Function ◽  
Mean Square Response ◽  
Noise Function ◽  
The Mean

The mean-square response of a lightly damped, second-order system to a type of non-stationary random excitation is determined. The forcing function on the system is taken in the form of a product of a well-defined, slowly varying envelope function and a noise function. The latter is assumed to be white or correlated as a narrow band process. Taking advantage of the slow variation of the envelope function and the small damping of the system, relatively simple integrals are obtained which approximate the mean-square response. Upper bounds on the mean-square response are also obtained.

Mean-Square Response of Thermoviscoelastic Medium to Nonstationary Random Excitation

1976 ◽  
Vol 43 (1) ◽  
pp. 150-158 ◽  
Author(s):  
W. Mosberg ◽  
M. Yildiz
Keyword(s):  
Random Excitation ◽  
Irreversible Thermodynamics ◽  
Envelope Function ◽  
Statistical Characteristics ◽  
Mean Square ◽  
Statistical Property ◽  
Step Functions ◽  
Mean Square Response ◽  
The Mean ◽  
Thermoviscoelastic Medium

The mean-square wave response of a lightly damped thermoviscoelastic medium to a special type of nonstationary random excitation is determined. The excitation function on the thermoviscoelastic medium is taken in the form of a product of a well-defined, slowly varying envelope function, and a part which prescribes the statistical characteristics of the excitation. Both the unit step and rectangular step functions are used for the envelope function, and both white noise and noise with an exponentially decaying harmonic correlation function are used to prescribe the statistical property of the excitation. By taking into consideration the slow variation envelope function and the wave characteristics of the lightly damped thermoviscoelastic medium, the mean-square response (as a function of temperature, excitation, and damping parameters with the aid of reversible and irreversible thermodynamics) is evaluated.

Mean-Square Response of Simple Mechanical Systems to Nonstationary Random Excitation

1969 ◽  
Vol 36 (2) ◽  
pp. 221-227 ◽  
Author(s):  
R. L. Barnoski ◽  
J. R. Maurer
Keyword(s):  
White Noise ◽  
Random Noise ◽  
Random Excitation ◽  
Correlated Noise ◽  
Mean Square ◽  
Single Degree Of Freedom ◽  
Step Functions ◽  
Unit Step ◽  
Mean Square Response ◽  
The Mean

This paper concerns the mean-square response of a single-degree-of-freedom system to amplitude modulated random noise. The formulation is developed in terms of the frequency-response function of the system and generalized spectra of the nonstationary random excitation. Both the unit step and rectangular step functions are used for the amplitude modulation, and both white noise and noise with an exponentially decaying harmonic correlation function are considered. The time-varying mean-square response is shown not to exceed its stationary value for white noise. For correlated noise, however, it is shown that the system mean-square response may exceed its stationary value.

Random Vibration of Beams

1962 ◽  
Vol 29 (2) ◽  
pp. 267-275 ◽  
Author(s):  
S. H. Crandall ◽  
Asim Yildiz
Keyword(s):  
Timoshenko Beam ◽  
Dynamic Models ◽  
Random Excitation ◽  
Viscous Damping ◽  
Mean Square ◽  
Euler Beam ◽  
Rayleigh Beam ◽  
Mean Square Response ◽  
The Mean ◽  
Band Limited

The calculated response of a uniform beam to stationary random excitation depends greatly on the dynamical model postulated, on the damping mechanism assumed, and on the nature of the random excitation process. To illustrate this, the mean square deflections, slopes, bending moments, and shear forces have been compared for four different dynamical models, with three different damping mechanisms, subjected to a distributed transverse loading process which is uncorrelated spacewise and which is either ideally “white” timewise or band-limited with an upper cut-off frequency. The dynamic models are the Bernoulli-Euler beam, the Timoshenko beam, and two intermediate models, the Rayleigh beam, and a beam which has the shear flexibility of the Timoshenko beam but not the rotatory inertia. The damping mechanisms are transverse viscous damping, rotatory viscous damping, and Voigt viscoelasticity. It is found that many of the mean-square response quantities are finite when the excitation is ideally white (i.e., when the input has infinite mean square); however, some of the responses are unbounded. For these cases the rate of growth of the response as the cut-off frequency of the excitation is increased is obtained.

Proximity Spectra of Oscillators Under Random Excitation

1980 ◽  
Vol 102 (2) ◽  
pp. 329-337
Author(s):  
S. F. Masri
Keyword(s):  
Cross Correlation ◽  
Random Excitation ◽  
Intensity Function ◽  
Mean Square ◽  
System Parameters ◽  
Spatially Correlated ◽  
Mean Square Response ◽  
The Mean ◽  
Two Degree Of Freedom ◽  
Different Parts

The transient mean-square response of an uncoupled two-degree-of-freedom system under spatially correlated nonstationary random excitation is determined. The excitation is modulated white noise with an intensity function that resembles the envelopes of typical earthquakes. Results of the analysis are used to investigate the effects of (a) cross correlation, (b) shape of the intensity function, (c) delay time associated with the arrival of the excitation at different parts of the system, and (d) dimensionless system parameters on the mean-square proximity spectra.

scholarly journals Reconstruction of Diffusion Coefficients and Power Exponents from Single Lagrangian Trajectories

Fluids ◽  
2021 ◽  
Vol 6 (3) ◽  
pp. 111
Author(s):  
Leonid M. Ivanov ◽  
Collins A. Collins ◽  
Tetyana Margolina
Keyword(s):  
Black Sea ◽  
Diffusion Coefficients ◽  
Current System ◽  
Mean Square Displacement ◽  
California Current ◽  
The Black Sea ◽  
Mean Square ◽  
California Current System ◽  
The Mean ◽  
Non Gaussian

Using discrete wavelets, a novel technique is developed to estimate turbulent diffusion coefficients and power exponents from single Lagrangian particle trajectories. The technique differs from the classical approach (Davis (1991)’s technique) because averaging over a statistical ensemble of the mean square displacement (<X2>) is replaced by averaging along a single Lagrangian trajectory X(t) = {X(t), Y(t)}. Metzler et al. (2014) have demonstrated that for an ergodic (for example, normal diffusion) flow, the mean square displacement is <X2> = limT→∞τX2(T,s), where τX2 (T, s) = 1/(T − s) ∫0T−s(X(t+Δt) − X(t))2 dt, T and s are observational and lag times but for weak non-ergodic (such as super-diffusion and sub-diffusion) flows <X2> = limT→∞≪τX2(T,s)≫, where ≪…≫ is some additional averaging. Numerical calculations for surface drifters in the Black Sea and isobaric RAFOS floats deployed at mid depths in the California Current system demonstrated that the reconstructed diffusion coefficients were smaller than those calculated by Davis (1991)’s technique. This difference is caused by the choice of the Lagrangian mean. The technique proposed here is applied to the analysis of Lagrangian motions in the Black Sea (horizontal diffusion coefficients varied from 105 to 106 cm2/s) and for the sub-diffusion of two RAFOS floats in the California Current system where power exponents varied from 0.65 to 0.72. RAFOS float motions were found to be strongly non-ergodic and non-Gaussian.

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