The Response of an Oscillator With Bilinear Hysteresis to Stationary Random Excitation

1978 ◽  
Vol 45 (4) ◽  
pp. 923-928 ◽  
Author(s):  
J. B. Roberts
Keyword(s):  
Random Process ◽  
Joint Distribution ◽  
Narrow Band ◽  
Digital Simulation ◽  
Stochastic Averaging ◽  
Random Excitation ◽  
Stationary Random Process ◽  
Simulation Results ◽  
Analytical Result ◽  
Theoretical Results

By applying the technique of stochastic averaging, a simple analytical result is obtained for the joint distribution of the displacement and velocity of a bilinear oscillator excited by a stationary random process. A comparison of theoretical results deduced from this distribution with corresponding digital simulation results shows that the theory is accurate in circumstances where the response is narrow-band in nature.

Visco-Elastic Systems Under Narrow-Band Excitation

2001 ◽  
Author(s):  
Wei Xu ◽  
Haiwu Rong ◽  
Tong Fang
Keyword(s):  
Steady State ◽  
Limit Cycle ◽  
Narrow Band ◽  
Stochastic Averaging ◽  
Steady State Solution ◽  
Random Excitation ◽  
Primary System ◽  
Secondary System ◽  
Comparison Results ◽  
Elastic Systems

Abstract The study of the response of nonlinear systems to narrow-band random excitation is importance. For example, the excitation of secondary system would be a narrow-band random process if the primary system could be modeled as a single -degree-of-freedom system with light damping subject to borad-band excitation. In the theory of nonlinear random vibration, most results obtained so far are attributed to the response of nonlinear oscillators to borad-band random excitation. In comparison, results on the effect of narrow-band excitation on non-linear oscillators are quite limited. Furthermore, some results in this area are disputable. For linear viscoelastic systems under both additive and multiplicative borad-band excitation excitations, Ariaratnam studied the stochastic stability of the system by using the method of stochastic averaging. Cai, Lin and Xu determined the condition for asymptotic sample stability of the system by using an improved stochastic averging procedure. In this paper, the response of visco-elastic systems to combined deterministic harmonic and random excitation is investigated. The method of harmonic balance and the method of stochastic averaging are used to determine the response of the system. Theoretical analyses and numerical simulations show that when the intensity of the random excitation increase, the steady state solution may change form a limit cycle to a diffused limit cycle. Under some conditions the system may have two steady state solutions and jumps may exist.

A Note on the Extreme Wave Load and the Associated Probability of Failure

1986 ◽  
Vol 30 (02) ◽  
pp. 123-126
Author(s):  
A. E. Mansour
Keyword(s):  
Probability Distribution ◽  
Gaussian Process ◽  
Random Process ◽  
Narrow Band ◽  
Spectral Width ◽  
Probability Of Failure ◽  
Stationary Random Process ◽  
Wave Load ◽  
Extreme Wave ◽  
Wave Statistics

Introduction and background - The probability distribution of the peak process of a stationary random process with zero mean was first determined by Rice [1]. 2 Following his basic derivation, Longuet-Higgins [2] and Cartwright and Longuet-Higgins [3] evaluated various wave statistics, first for a narrow-band Gaussian process, then extended the results for a Gaussian process of any spectral width.

scholarly journals A Convergent Collocation Approach for Generalized Fractional Integro-Differential Equations Using Jacobi Poly-Fractonomials

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 979
Author(s):  
Sandeep Kumar ◽  
Rajesh K. Pandey ◽  
H. M. Srivastava ◽  
G. N. Singh
Keyword(s):  
Differential Equation ◽  
Fractional Derivative ◽  
Collocation Method ◽  
Caputo Fractional Derivative ◽  
Fractional Derivatives ◽  
Special Cases ◽  
Collocation Approach ◽  
Linear And Nonlinear ◽  
Simulation Results ◽  
Theoretical Results

In this paper, we present a convergent collocation method with which to find the numerical solution of a generalized fractional integro-differential equation (GFIDE). The presented approach is based on the collocation method using Jacobi poly-fractonomials. The GFIDE is defined in terms of the B-operator introduced recently, and it reduces to Caputo fractional derivative and other fractional derivatives in special cases. The convergence and error analysis of the proposed method are also established. Linear and nonlinear cases of the considered GFIDEs are numerically solved and simulation results are presented to validate the theoretical results.

Induced transformations of recurrent a.m.s. dynamical systems

2015 ◽  
Vol 15 (02) ◽  
pp. 1550010
Author(s):  
Sheng Huang ◽  
Mikael Skoglund
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Random Process ◽  
Ergodic Theorem ◽  
Finite Measure ◽  
Stationary Random Process ◽  
Finite State ◽  
Ratio Ergodic Theorem

This note proves that an induced transformation with respect to a finite measure set of a recurrent asymptotically mean stationary dynamical system with a sigma-finite measure is asymptotically mean stationary. Consequently, the Shannon–McMillan–Breiman theorem, as well as the Shannon–McMillan theorem, holds for all reduced processes of any finite-state recurrent asymptotically mean stationary random process. As a by-product, a ratio ergodic theorem for asymptotically mean stationary dynamical systems is presented.

Calculation of Nonlinear Systems Under Narrow Band Excitation Using Equivalent Linearization and Path Continuation

2021 ◽  
Author(s):  
Alwin Förster ◽  
Lars Panning-von Scheidt
Keyword(s):  
Nonlinear Systems ◽  
Narrow Band ◽  
Gaussian White Noise ◽  
Random Excitation ◽  
Mean Value ◽  
Equivalent Linearization ◽  
Efficient Procedure ◽  
Stochastic Excitations ◽  
Wide Range ◽  
The One

Abstract Turbomachines experience a wide range of different types of excitation during operation. On the structural mechanics side, periodic or even harmonic excitations are usually assumed. For this type of excitation there are a variety of methods, both for linear and nonlinear systems. Stochastic excitation, whether in the form of Gaussian white noise or narrow band excitation, is rarely considered. As in the deterministic case, the calculations of the vibrational behavior due to stochastic excitations are even more complicated by nonlinearities, which can either be unintentionally present in the system or can be used intentionally for vibration mitigation. Regardless the origin of the nonlinearity, there are some methods in the literature, which are suitable for the calculation of the vibration response of nonlinear systems under random excitation. In this paper, the method of equivalent linearization is used to determine a linear equivalent system, whose response can be calculated instead of the one of the nonlinear system. The method is applied to different multi-degree of freedom nonlinear systems that experience narrow band random excitation, including an academic turbine blade model. In order to identify multiple and possibly ambiguous solutions, an efficient procedure is shown to integrate the mentioned method into a path continuation scheme. With this approach, it is possible to track jump phenomena or the influence of parameter variations even in case of narrow band excitation. The results of the performed calculations are the stochastic moments, i.e. mean value and variance.

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