Nonlinear Stochastic Dynamics, Chaos and Reliability Analysis for Single Degree Freedom Model of a Rotor Blade

Kumar, Pankaj; Narayanan, Sriharini

doi:10.1115/gt2008-50736

Cited by 3 publications

(3 citation statements)

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“…Some common examples of harmonically forced nonlinear systems include rotor systems with bearing nonlinearities [1][2][3], turbomachinery [4,5], wind turbines [6], and nonlinear electrical circuits [7][8][9]. The response of the human musculoskeletal system is also highly nonlinear [10][11][12][13], and researchers often study neural network control systems as a form of oscillatory control force [14][15][16][17].…”

Section: Introductionmentioning

confidence: 99%

Numerical Continuation of Periodic Orbits for Harmonically Forced Nonlinear Systems

Sracic

Allen

2011

Civil Engineering Topics, Volume 4

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A nonlinear structure will often respond periodically when it is excited with a sinusoidal force. Several methods are available that can compute the periodic response for various drive frequencies, which is analogous to the frequency response function for a linear system. The simplest approach would be to compute a sequence of simulations where the equations of motion are integrated until damping drives the system to steady state, but that approach suffers from a number of drawbacks. Recently, numerical methods have been proposed that use a solution branch continuation technique to find the free response of unforced, undamped nonlinear systems for different values of a control parameter. These are attractive because they are built around broadly applicable time-integration routines, so they are applicable to a wide range of systems. However, the continuation approach is not typically used to calculate the periodic response of a structural dynamic system to a harmonic force. This work adapts the numerical continuation approach to find the periodic, forced steady-state response of a nonlinear system. The method uses an adaptive procedure with a prediction step and a mode switching correction step based on Newton-Raphson methods. Once a branch of solutions has been computed, it explains how a full spectrum of harmonic forcing conditions affect the dynamic response of the nonlinear system. The approach is developed and applied to calculate nonlinear frequency response curves for a Duffing oscillator and a low order nonlinear cantilever beam.

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Section: Introductionmentioning

confidence: 99%

Numerical Continuation of Periodic Orbits for Harmonically Forced Nonlinear Systems

Sracic

Allen

2011

Civil Engineering Topics, Volume 4

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“…Random analysis of vibrating structures has received much more attention in recent years [1][2][3][4]. Mechanical structures always contain unavoidable uncertain factors such as geometrical errors, uncertainty of material properties, and manufacturing errors, which constitute the basic random parameters of the vibrating system.…”

Section: Introductionmentioning

confidence: 99%

A New Method for Reliability-Based Sensitivity Analysis of Dynamic Random Systems

Wang

2019

Mathematical Problems in Engineering

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A novel numerical method for investigating time-dependent reliability and sensitivity issues of dynamic systems is proposed, which involves random structure parameters and is subjected to stochastic process excitation simultaneously. The Karhunen–Loève (K-L) random process expansion method is used to express the excitation process in the form of a series of deterministic functions of time multiplied by independent zero-mean standard random quantities, and the discrete points are made to be the same as Legendre integration points. Then, the precise Gauss–Legendre integration is used to solve the oscillation differential functions. Considering the independent relationship of the structural random parameters and the parameters of random process, the time-varying moments of the response are evaluated by the point estimate method. Combining with the fourth-moment method theory of reliability analysis, the dynamic reliability response can be evaluated. The dynamic reliability curve is useful for getting the weakness time so as to avoid breakage. Reliability-based sensitivity analysis gives the importance sort of the distribution parameters, which is useful for increasing system reliability. The result obtained by the proposed method is accurate enough compared with that obtained by the Monte Carlo simulation (MCS) method.

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“…Analytical solutions are only known for special classes of nonlinear stochastic systems [9]. Therefore, numerical techniques like the path integral (PI) method [10][11][12][13][14][15], global Galerkin method [16], finite element (FE) [17][18][19][20][21], finite difference (FD) [22,23], and multi scale finite elements [24] have been developed to obtain the solution of the FP equation. There are inherent difficulties associated with the FE/FD methods in the solution of the FP equation especially for high dimensional systems [25].…”

Section: Introductionmentioning

confidence: 99%

Efficient path integral solution of Fokker–Planck equation: response, bifurcation and periodicity of nonlinear systems

Kumar

Narayanan

2011

Int J Adv Eng Sci Appl Math

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Response of nonlinear systems subjected to harmonic, parametric, and random excitation is of importance in the field of structural dynamics. The transitional probability density function (PDF) of the random response of nonlinear systems under white or coloured noise excitation (delta-correlated) is governed by both the forward Fokker-Planck (FP) and the backward Kolmogorov equations. This article presents a new approach for efficient numerical implementation of the path integral (PI) method in the solution of the FP equation for some nonlinear systems subjected to white noise, parametric and combined harmonic and white noise excitations. The modified PI method is based on a non-Gaussian transition PDF and the Gauss-Legendre integration scheme. The steady state PDF, jump phenomenon, noise induced state changes of PDF are studied by the modified PI method. Effects of white noise intensity, amplitude and frequency of harmonic excitation and the level of nonlinearity on stochastic jump and bifurcation behaviours of a hardening Duffing oscillator are also investigated.

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Nonlinear Stochastic Dynamics, Chaos and Reliability Analysis for Single Degree Freedom Model of a Rotor Blade

Cited by 3 publications

References 0 publications

Numerical Continuation of Periodic Orbits for Harmonically Forced Nonlinear Systems

Numerical Continuation of Periodic Orbits for Harmonically Forced Nonlinear Systems

A New Method for Reliability-Based Sensitivity Analysis of Dynamic Random Systems

Efficient path integral solution of Fokker–Planck equation: response, bifurcation and periodicity of nonlinear systems

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