Stochastic models of damped vibrations

1996 ◽  
Vol 33 (04) ◽  
pp. 1159-1168 ◽  
Author(s):  
Maged Elshamy
Keyword(s):  
Random Fields ◽  
Space Variable ◽  
Displacement Function ◽  
Time Variable ◽  
Stochastic Disturbances ◽  
Large Deviations Principle ◽  
Stochastic Perturbations ◽  
Random Forcing ◽  
Non Gaussian ◽  
Random Fluctuations

In this article we study stochastic perturbations of partial differential equations describing forced-damped vibrations of a string. Two models of such stochastic disturbances are considered; one is triggered by an initial white noise, and the other is in the form of non-Gaussian random forcing. Let uε (t, x) be the displacement at time t of a point x on a string, where the time variable t ≧ 0, and the space variable . The small parameter ε controls the intensity of the random fluctuations. The random fields uε (t, x) are shown to satisfy a large deviations principle, and the random deviations of the unperturbed displacement function are analyzed as the noise parameter ε tends to zero.

Stochastic models of damped vibrations

1996 ◽  
Vol 33 (4) ◽  
pp. 1159-1168 ◽  
Author(s):  
Maged Elshamy
Keyword(s):  
Random Fields ◽  
Space Variable ◽  
Displacement Function ◽  
Time Variable ◽  
Stochastic Disturbances ◽  
Large Deviations Principle ◽  
Stochastic Perturbations ◽  
Random Forcing ◽  
Non Gaussian ◽  
Random Fluctuations

In this article we study stochastic perturbations of partial differential equations describing forced-damped vibrations of a string. Two models of such stochastic disturbances are considered; one is triggered by an initial white noise, and the other is in the form of non-Gaussian random forcing. Let uε (t, x) be the displacement at time t of a point x on a string, where the time variable t ≧ 0, and the space variable . The small parameter ε controls the intensity of the random fluctuations. The random fields uε (t, x) are shown to satisfy a large deviations principle, and the random deviations of the unperturbed displacement function are analyzed as the noise parameter ε tends to zero.

A Stability Property of Stochastic Vibration

2007 ◽  
Vol 44 (02) ◽  
pp. 444-457
Author(s):  
M. Elshamy
Keyword(s):  
Wave Equation ◽  
Random Fields ◽  
Stability Property ◽  
Space Variable ◽  
Time Variable ◽  
Initial Condition ◽  
Brownian Sheet ◽  
Random Forcing ◽  
Continuity Properties ◽  
Stochastic Wave Equation

Let u(t,x) be the displacement at time t of a point x on a string; the time variable t varies in the interval I≔[0,T] and the space variable x varies in the interval J≔[0,L], where T and L are fixed positive constants. The displacement u(t,x) is the solution to a stochastic wave equation. Two forms of random excitations are considered, a white noise in the initial condition and a nonlinear random forcing which involves the formal derivative of a Brownian sheet. In this article, we consider the continuity properties of solutions to this equation. Smoothness characteristics of these random fields, in terms of Hölder continuity, are also investigated.

scholarly journals A Stability Property of Stochastic Vibration

2007 ◽  
Vol 44 (02) ◽  
pp. 444-457
Author(s):  
M. Elshamy
Keyword(s):  
Wave Equation ◽  
Random Fields ◽  
Stability Property ◽  
Space Variable ◽  
Time Variable ◽  
Initial Condition ◽  
Brownian Sheet ◽  
Random Forcing ◽  
Continuity Properties ◽  
Stochastic Wave Equation

Let u(t,x) be the displacement at time t of a point x on a string; the time variable t varies in the interval I≔[0,T] and the space variable x varies in the interval J≔[0,L], where T and L are fixed positive constants. The displacement u(t,x) is the solution to a stochastic wave equation. Two forms of random excitations are considered, a white noise in the initial condition and a nonlinear random forcing which involves the formal derivative of a Brownian sheet. In this article, we consider the continuity properties of solutions to this equation. Smoothness characteristics of these random fields, in terms of Hölder continuity, are also investigated.

scholarly journals A Stability Property of Stochastic Vibration

2007 ◽  
Vol 44 (2) ◽  
pp. 444-457
Author(s):  
M. Elshamy
Keyword(s):  
Wave Equation ◽  
Random Fields ◽  
Stability Property ◽  
Space Variable ◽  
Time Variable ◽  
Initial Condition ◽  
Brownian Sheet ◽  
Random Forcing ◽  
Continuity Properties ◽  
Stochastic Wave Equation

Let u(t,x) be the displacement at time t of a point x on a string; the time variable t varies in the interval I≔[0,T] and the space variable x varies in the interval J≔[0,L], where T and L are fixed positive constants. The displacement u(t,x) is the solution to a stochastic wave equation. Two forms of random excitations are considered, a white noise in the initial condition and a nonlinear random forcing which involves the formal derivative of a Brownian sheet. In this article, we consider the continuity properties of solutions to this equation. Smoothness characteristics of these random fields, in terms of Hölder continuity, are also investigated.

Randomly perturbed vibrations

1995 ◽  
Vol 32 (02) ◽  
pp. 417-428 ◽  
Author(s):  
M. Elshamy
Keyword(s):  
Dirichlet Boundary ◽  
Space Variable ◽  
Initial Conditions ◽  
Linear Wave ◽  
Stochastic Perturbations ◽  
Uniform Topology ◽  
Linear Wave Equation ◽  
Random Forcing ◽  
Continuity Properties ◽  
Positive Time

Let u ε(t, x) be the position at time t of a point x on a string, where the time variable t varies in an interval I: = [0, T], T is a fixed positive time, and the space variable x varies in an interval J. The string is performing forced vibrations and also under the influence of small stochastic perturbations of intensity ε. We consider two kinds of random perturbations, one in the form of initial white noise, and the other is a nonlinear random forcing which involves the formal derivative of a Brownian sheet. When J has finite endpoints, a Dirichlet boundary condition is imposed for the solutions of the resulting non-linear wave equation. Assuming that the initial conditions are of sufficient regularity, we analyze the deviations u ε(t, x) from u 0(t, x), the unperturbed position function, as the intensity of perturbation ε ↓ 0 in the uniform topology. We also discuss some continuity properties of the realization of the solutions u ε(t, x).

Randomly perturbed vibrations

1995 ◽  
Vol 32 (2) ◽  
pp. 417-428 ◽  
Author(s):  
M. Elshamy
Keyword(s):  
Dirichlet Boundary ◽  
Space Variable ◽  
Initial Conditions ◽  
Linear Wave ◽  
Stochastic Perturbations ◽  
Uniform Topology ◽  
Linear Wave Equation ◽  
Random Forcing ◽  
Continuity Properties ◽  
Positive Time

Let uε(t, x) be the position at time t of a point x on a string, where the time variable t varies in an interval I: = [0, T], T is a fixed positive time, and the space variable x varies in an interval J. The string is performing forced vibrations and also under the influence of small stochastic perturbations of intensity ε. We consider two kinds of random perturbations, one in the form of initial white noise, and the other is a nonlinear random forcing which involves the formal derivative of a Brownian sheet. When J has finite endpoints, a Dirichlet boundary condition is imposed for the solutions of the resulting non-linear wave equation. Assuming that the initial conditions are of sufficient regularity, we analyze the deviations uε(t, x) from u0(t, x), the unperturbed position function, as the intensity of perturbation ε ↓ 0 in the uniform topology. We also discuss some continuity properties of the realization of the solutions uε(t, x).

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