Nonlinear random coupled motions of structural elements with quadratic nonlinearities

Serhan, Samir J.; Nayfeh, Ali H.

doi:10.1007/bf00045299

Cited by 3 publications

(2 citation statements)

References 26 publications

(26 reference statements)

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“…17,18 Recently, a number of different studies have proposed alternative or modified perturbation methods, such as parameter splitting approaches 19,20 and the homotopy perturbation method, [21][22][23] that have shown improved performance for highly nonlinear oscillators under harmonic excitation. For combined excitation, the method of multiple scales has been employed in combination with Gaussian closure 24,25 deterministic averaging 26 and stochastic averaging, 27 or on its own in adapted forms for specific combined excitation problems. 28,29 The primary drawback of the majority of methods mentioned in the preceding section is that they can be difficult to generalize and have only been applied to highly specific and simple oscillators.…”

Section: Introductionmentioning

confidence: 99%

A time and ensemble equivalent linearization method for nonlinear systems under combined harmonic and random excitation

Hickey,

Butlin,

Langley

et al. 2023

Proceedings of the Institution of Mechanical Engineers, Part C:

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An Equivalent Linearization technique, termed an Equivalent Linearization Time and Ensemble Expectation (EL-TEE) approach, is used to develop an alternative method for estimating the response of a nonlinear oscillator to a combination of deterministic harmonic and random white noise excitation. The approach is based on applying equivalent linearization and averaging over the time period of one harmonic excitation cycle. This gives a set of coupled nonlinear equations that can be solved for the response averaged over time and across the ensemble. The primary advantages of the proposed method are its computational speed, ability to return physically meaningful linearization matrices and that it can be applied to a wide variety of nonlinearities. The method is applied to three example test systems: the well-known single degree of freedom Duffing oscillator; a single degree of freedom system with a displacement constraint imposing a discontinuous nonlinearity; and a multi degree of freedom oscillator with a localized polynomial nonlinearity that has also been examined experimentally. It is shown that the response predicted matches well with Monte Carlo results from direct time integration at a fraction of the computational cost, and the method is capable of reproducing key results observed experimentally.

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Section: Introductionmentioning

confidence: 99%

A time and ensemble equivalent linearization method for nonlinear systems under combined harmonic and random excitation

Hickey,

Butlin,

Langley

et al. 2023

Proceedings of the Institution of Mechanical Engineers, Part C:

View full text Add to dashboard Cite

show abstract

“…Later, it was extended for analyzing the system subjected to random excitations and was named stochastic averaging method (Stratonovich, 1967; Roberts and Spanos, 1986). Combining the methods of multiple scales and second-order closure, the responses of a Duffing–Rayleigh oscillator (Nayfeh and Serhan, 1990) and a two-degree-of-freedom system (Serhan and Nayfeh, 1991) were analyzed. The stationary joint PDFs of response amplitude and phase difference of a Duffing oscillator were analyzed based on the harmonic balance and stochastic averaging method (Haiwu et al, 2001).…”

Section: Introductionmentioning

confidence: 99%

Investigation on the EPC method in analyzing the nonlinear oscillators under both harmonic and Gaussian white noise excitations

Zhu

Gong

et al. 2022

Journal of Vibration and Control

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The objective of the current paper is to study and extend the exponential-polynomial-closure (EPC) method in analyzing the non-stationary response probabilistic solutions of nonlinear stochastic oscillators excited by combined deterministic harmonic and Gaussian white noise excitations. The probabilistic solution of the non-stationary responses of nonlinear stochastic oscillator is expressed as an exponential function of polynomial with time-variant coefficients and then the Fokker–Planck–Kolmogorov equation is solved approximately. Two validation examples are presented. The results obtained by Monte Carlo simulation (MCS) and EPC method are presented to show their good agreement, which confirms the effectiveness of the EPC method for the probabilistic solutions to nonlinear stochastic oscillators. The advantage of the EPC method for analyzing the oscillators with strong nonlinearities is investigated by comparison with Equivalent Linearization method. Furthermore, compared with MCS, the computational efficiency by using the EPC method is increased by about 56 times in the first example and 44 times in the second example.

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Periodic Response of a Duffing Oscillator Under Combined Harmonic and Random Excitations

Zhu

Guo

2015

Journal of Vibration and Acoustics

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This paper presents a solution procedure to investigate the periodic response of a Duffing oscillator under combined harmonic and random excitations. The solution procedure consists of an implicit harmonic balance method and a Gaussian closure method. The implicit harmonic balance method, previously developed for the case of harmonic excitation, is extended to the present case of combined harmonic and random excitations with the help of the Gaussian closure method. The amplitudes of the periodic response and the steady variances can be automatically found by the proposed solution procedure. First, the response process is separated into the mean part and the random process part. Then the Gaussian closure method is adopted to reformulate the original equation into two coupled differential equations. One is a deterministic equation about the mean part and the other is a stochastic equivalent linear equation. In terms of these two coupled equations, the implicit harmonic balance method is used to obtain a set of nonlinear algebraic equations relating to amplitudes, frequency, and variance. The resulting equations are not explicitly determined and they can be implicitly solved by nonlinear equation routines available in most mathematical libraries. Three illustrative examples are further investigated to show the effectiveness of the proposed solution procedure. Furthermore, the proposed solution procedure is particularly convenient for programming and it can be extended to obtain the periodic solutions of the response of multi degrees-of-freedom systems.

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Nonlinear random coupled motions of structural elements with quadratic nonlinearities

Cited by 3 publications

References 26 publications

A time and ensemble equivalent linearization method for nonlinear systems under combined harmonic and random excitation

A time and ensemble equivalent linearization method for nonlinear systems under combined harmonic and random excitation

Investigation on the EPC method in analyzing the nonlinear oscillators under both harmonic and Gaussian white noise excitations

Periodic Response of a Duffing Oscillator Under Combined Harmonic and Random Excitations

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