scholarly journals Closure to “Discussion of ‘Response Envelope Statistics for Nonlinear Oscillators With Random Excitation’” (1978, ASME J. Appl. Mech., 45, p. 964)

1978 ◽  
Vol 45 (4) ◽  
pp. 964-965
Author(s):  
W. D. Iwan ◽  
P.-T. Spanos
Keyword(s):  
Nonlinear Oscillators ◽  
Random Excitation ◽  
Response Envelope

Response Envelope Statistics for Nonlinear Oscillators With Random Excitation

1978 ◽  
Vol 45 (1) ◽  
pp. 170-174 ◽  
Author(s):  
W. D. Iwan ◽  
P.-T. Spanos
Keyword(s):  
Computer Simulation ◽  
Linear System ◽  
Analytical Method ◽  
Nonlinear Oscillator ◽  
Duffing Oscillator ◽  
Gaussian White Noise ◽  
Nonlinear Oscillators ◽  
Random Excitation ◽  
Weakly Nonlinear ◽  
Response Envelope

An approximate analytical method is presented for determining both the stationary and nonstationary amplitude or envelope response statistics of a lightly damped and weakly nonlinear oscillator subject to Gaussian white noise. The method is based on the solution of an equivalent linear system whose parameters are functions of the response itself. The solution derived by the approximate method is compared with that obtained by computer simulation for a Duffing oscillator.

Stochastic Averaging of Strongly Nonlinear Oscillators Under Combined Harmonic and Wide-Band Noise Excitations

2008 ◽  
Vol 130 (5) ◽  
Author(s):  
Y. J. Wu ◽  
W. Q. Zhu
Keyword(s):  
Probability Density ◽  
Wide Band ◽  
Nonlinear Oscillators ◽  
Stochastic Averaging ◽  
Random Excitation ◽  
Stationary Response ◽  
Strongly Nonlinear ◽  
Band Noise ◽  
Wide Band Noise ◽  
Strongly Nonlinear Oscillators

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.

Nonstationary Response Envelope Probability Densities of Nonlinear Oscillators

2006 ◽  
Vol 74 (2) ◽  
pp. 315-324 ◽  
Author(s):  
P. D. Spanos ◽  
A. Sofi ◽  
M. Di Paola
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Planck Equation ◽  
Nonlinear Oscillators ◽  
Basis Functions ◽  
Time Dependent ◽  
Fokker Planck Equation ◽  
Response Envelope ◽  
Fokker Planck

The nonstationary random response of a class of lightly damped nonlinear oscillators subjected to Gaussian white noise is considered. An approximate analytical method for determining the response envelope statistics is presented. Within the framework of stochastic averaging, the procedure relies on the Markovian modeling of the response envelope process through the definition of an equivalent linear system with response-dependent parameters. An approximate solution of the associated Fokker-Planck equation is derived by resorting to a Galerkin scheme. Specifically, the nonstationary probability density function of the response envelope is expressed as the sum of a time-dependent Rayleigh distribution and of a series expansion in terms of a set of properly selected basis functions with time-dependent coefficients. These functions are the eigenfunctions of the boundary-value problem associated with the Fokker-Planck equation governing the evolution of the probability density function of the response envelope of a linear oscillator. The selected basis functions possess some notable properties that yield substantial computational advantages. Applications to the Van der Pol and Duffing oscillators are presented. Appropriate comparisons to the data obtained by digital simulation show that the method, being nonperturbative in nature, yields reliable results even for large values of the nonlinearity parameter.

Response Spectra for Nonlinear Oscillators

1975 ◽  
Vol 97 (4) ◽  
pp. 1223-1226 ◽  
Author(s):  
J. E. Manning
Keyword(s):  
Nonlinear Oscillator ◽  
Response Spectrum ◽  
Duffing Oscillator ◽  
Response Spectra ◽  
Nonlinear Oscillators ◽  
Random Excitation ◽  
Heuristic Approach ◽  
Perturbation Approach ◽  
Spectral Bandwidth ◽  
Nonlinear Spring

Methods are presented for calculating the response spectrum of a nonlinear oscillator with broadband random excitation. A perturbation approach is used for oscillators with slightly nonlinear spring elements to show that the effect of the nonlinearity is a slight shift in the spectrum peak. A new heuristic approach is used for oscillators with large nonlinearities and small damping to show that in addition to a shift in the peak an increase in spectral bandwidth also occurs. The Duffing oscillator is studied as a specific example.

Performance analysis of global local mean square error criterion of stochastic linearization for nonlinear oscillator

2019 ◽  
Author(s):  
Nguyen Cao Thang ◽  
Luu Xuan Hung
Keyword(s):  
Performance Analysis ◽  
Mean Square Error ◽  
Nonlinear Oscillators ◽  
Degree Of Freedom ◽  
Equivalent Linearization ◽  
Mean Square ◽  
Error Criterion ◽  
Stochastic Linearization ◽  
Local Mean ◽  
Mean Square Error Criterion

The paper presents a performance analysis of global-local mean square error criterion of stochastic linearization for some nonlinear oscillators. This criterion of stochastic linearization for nonlinear oscillators bases on dual conception to the local mean square error criterion (LOMSEC). The algorithm is generally built to multi degree of freedom (MDOF) nonlinear oscillators. Then, the performance analysis is carried out for two applications which comprise a rolling ship oscillation and two degree of freedom one. The improvement on accuracy of the proposed criterion has been shown in comparison with the conventional Gaussian equivalent linearization (GEL).

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