Stationary response probability density of nonlinear random vibrating systems: a data-driven method

Physical and engineering systems are often subjected to combined harmonic and random excitations. The random excitation is often modeled as Gaussian white noise for mathematical tractability. However, in practice, the random excitation is nonwhite. This paper investigates the stationary response probability density of strongly nonlinear oscillators under combined harmonic and wide-band noise excitations. By using generalized harmonic functions, a new stochastic averaging procedure for estimating stationary response probability density of strongly nonlinear oscillators under combined harmonic and wide-band noise excitations is developed. The damping can be linear and (or) nonlinear and the excitations can be external and (or) parametric. After stochastic averaging, the system state is represented by two-dimensional time-homogeneous diffusive Markov processes. The method of reduced Fokker–Planck–Kolmogorov equation is used to investigate the stationary response of the vibration system. A nonlinearly damped Duffing oscillator is taken as an example to show the application and validity of the method. In the case of primary external resonance, based on the stationary joint probability density of amplitude and phase difference, the stochastic jump of the Duffing oscillator and P-bifurcation as the system parameters change are examined for the first time. The agreement between the analytical results and those from Monte Carlo simulation of original system shows that the proposed procedure works quite well.

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Constructing transient response probability density of non-linear system through complex fractional moments

International Journal of Non-Linear Mechanics ◽

10.1016/j.ijnonlinmec.2014.06.004 ◽

2014 ◽

Vol 65 ◽

pp. 253-259 ◽

Cited By ~ 7

Author(s):

Xiaoling Jin ◽

Yong Wang ◽

Zhilong Huang ◽

Mario Di Paola

Keyword(s):

Linear System ◽

Probability Density ◽

Transient Response ◽

Response Probability ◽

Non Linear ◽

Fractional Moments ◽

Non Linear System

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Path Integral Method for Nonlinear Systems Under Levy White Noise

ASCE-ASME J Risk and Uncert in Engrg Sys Part B Mech Engrg ◽

10.1115/1.4036703 ◽

2017 ◽

Vol 3 (3) ◽

Cited By ~ 1

Author(s):

Alberto Di Matteo ◽

Antonina Pirrotta

Keyword(s):

Probability Density Function ◽

Nonlinear Systems ◽

Probability Density ◽

Density Function ◽

Path Integral ◽

Response Probability ◽

Stability Index ◽

Integral Solution ◽

Stable Random Variables ◽

White Noises

In this paper, the probabilistic response of nonlinear systems driven by alpha-stable Lévy white noises is considered. The path integral solution is adopted for determining the evolution of the probability density function of nonlinear oscillators. Specifically, based on the properties of alpha-stable random variables and processes, the path integral solution is extended to deal with Lévy white noises input with any value of the stability index alpha. It is shown that at the limit when the time increments tend to zero, the Einstein–Smoluchowsky equation, governing the evolution of the response probability density function, is fully restored. Application to linear and nonlinear systems under different values of alpha is reported. Comparisons with pertinent Monte Carlo simulation data and analytical solutions (when available) demonstrate the accuracy of the results.

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Stochastic Response and Stability Analysis of Single Leg Articulated Tower

Volume 2: Safety and Reliability; Pipeline Technology ◽

10.1115/omae2003-37032 ◽

2003 ◽

Cited By ~ 3

Author(s):

A. K. Banik ◽

T. K. Datta

Keyword(s):

Asymptotic Stability ◽

Probability Density ◽

Parametric Excitation ◽

Stochastic Averaging ◽

Stochastic Response ◽

Sea State ◽

Stationary Response ◽

Stability In Probability ◽

Hydrodynamic Damping ◽

Asymptotic Stability In Probability

The stationary response and asymptotic stability in probability of an articulated tower under random wave excitation are investigated. The articulated tower is modelled as a SDOF system having stiffness nonlinearity, damping nonlinearity and parametric excitation. Using a stochastic averaging procedure and Fokker-Plank-Kolomogorov equation (FPK), the probability density function of the stationary solution is obtained for random sea state represented by a P-M sea spectrum. The method involves a Van-Der-Pol transformation of the nonlinear equation of motion to convert it to the Ito’s stochastic differential equation with averaged drift and diffusion coefficients. The asymptotic stability in probability of the system is investigated by obtaining the averaged Ito’s equation for the Hamiltonian of the system. The asymptotic stability is examined approximately by investigating the asymptotic behaviour of the diffusion process Y(t) at its two boundaries Y = 0 and ∞. As an illustrative example, an articulated tower in a sea depth of 150 m is considered. The tower consists of hollow cylinder of varying diameter along the height, providing the required buoyancy of the system. Wave forces on the structure are calculated using Morrison’s equation. The stochastic response and the stability conditions are obtained for a sea state represented by P-M spectrum with 16m significant wave height. The results of the study indicate that the probability density of the stationary response obtained by the stochastic averaging procedure is in very good agreement with that obtained from digital simulation. Further, the articulated tower is found to be asymptotically stable under the parametric excitation arising due to hydrodynamic damping.

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