Statistical energy analysis of nonlinear vibrating systems

Spelman, G; Langley, R.S.

doi:10.1098/rsta.2014.0403

Cited by 19 publications

(15 citation statements)

References 16 publications

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“…Of course, to date we have only considered only a few structures that have low modal density. Certainly the findings in [24] suggest that multi-mode effects can become very important for modally dense structures. Further research should be performed to better understand when this type of simplification can be relied on and when multi-mode models should be considered.…”

Section: Discussionmentioning

confidence: 99%

Relationships between nonlinear normal modes and response to random inputs

Schoneman

Allen

Kuether

2017

Mechanical Systems and Signal Processing

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The ability to model nonlinear structures subject to random excitation is of key importance in designing hypersonic aircraft and other advanced aerospace vehicles. When a structure is linear, superposition can be used to construct its response to a known spectrum in terms of its linear modes. Superposition does not hold for a nonlinear system, but several works have shown that a system's dynamics can still be understood qualitatively in terms of its nonlinear normal modes (NNMs). This work investigates the connection between a structure's undamped nonlinear normal modes and the spectrum of its response to high amplitude random forcing. Two examples are investigated: a spring-mass system and a clamped-clamped beam modeled within a geometrically nonlinear finite element package. In both cases, an intimate connection is observed between the smeared peaks in the response spectrum and the frequency-energy dependence of the nonlinear normal modes. In order to understand the role of coupling between the underlying linear modes, reduced order models with and without modal coupling terms are used to separate the effect of each NNM's backbone from the nonlinear couplings that give rise to internal resonances. In the cases shown here, uncoupled, single-degree-of-freedom nonlinear models are found to predict major features in the response with reasonable accuracy; a highly inexpensive approximation such as this could be useful in design and optimization studies. More importantly, the results show that a reduced order model can be expected to give accurate results only if it is also capable of accurately predicting the frequency-energy dependence of the nonlinear modes that are in the frequency band of the excitation.

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Section: Discussionmentioning

confidence: 99%

Relationships between nonlinear normal modes and response to random inputs

Schoneman

Allen

Kuether

2017

Mechanical Systems and Signal Processing

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“…The classical SEA approach is valid under a number of limiting assumptions [1,[8][9][10][11]. Efforts to examine SEA for nonlinear systems have been limited in scope [12]. To the best of our knowledge, there are not many papers devoted to nonlinear SEA.…”

Section: Introductionmentioning

confidence: 99%

“…Ref. [12] studies the energy cascade phenomenon due to large deformations in coupled vibrating plates. Ref.…”

Section: Introductionmentioning

confidence: 99%

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Entropy and Mixing Entropy for Weakly Nonlinear Mechanical Vibrating Systems

Sotoudeh

2019

Entropy

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In this paper, we examine Khinchin’s entropy for two weakly nonlinear systems of oscillators. We study a system of coupled Duffing oscillators and a set of Henon–Heiles oscillators. It is shown that the general method of deriving the Khinchin’s entropy for linear systems can be modified to account for weak nonlinearities. Nonlinearities are modeled as nonlinear springs. To calculate the Khinchin’s entropy, one needs to obtain an analytical expression of the system’s phase volume. We use a perturbation method to do so, and verify the results against the numerical calculation of the phase volume. It is shown that such an approach is valid for weakly nonlinear systems. In an extension of the author’s previous work for linear systems, a mixing entropy is defined for these two oscillators. The mixing entropy is the result of the generation of entropy when two systems are combined to create a complex system. It is illustrated that mixing entropy is always non-negative. The mixing entropy provides insight into the energy behavior of each system. The limitation of statistical energy analysis motivates this study. Using the thermodynamic relationship of temperature and entropy, and Khinchin’s entropy, one can define a vibrational temperature. Vibrational temperature can be used to derive the power flow proportionality, which is the backbone of the statistical energy analysis. Although this paper is motivated by statistical energy analysis application, it is not devoted to the statistical energy analysis of nonlinear systems.

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“…Each of these methods began by analyzing different prototypical systems, overcoming obstacles such as different coordinate systems [14], spatial parametric or geometric distributions [15], and property uncertainty [16]. Reinforcements, discontinuities [17], and even nonlinear systems have been investigated [18,19]. After a few iterations and further refinement, these techniques are being applied to product-related problems [20,21].…”

Section: Introductionmentioning

confidence: 99%

High Frequency Analysis of a Point-Coupled Parallel Plate System

Culver

Dowell

2017

Journal of Vibration and Acoustics

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The root-mean-square (RMS) response of various points in a system comprised of two parallel plates coupled at a point undergoing high frequency, broadband transverse point excitation of one component is considered. Through this prototypical example, asymptotic modal analysis (AMA) is extended to two coupled continuous dynamical systems. It is shown that different points on the plates respond with different RMS magnitudes depending on their spatial relationship to the excitation or coupling points in the system. The ability of AMA to accurately compute the RMS response of these points (namely, the excitation point, the coupling points, and the hot lines through the excitation or coupling points) in the system is shown. The behavior of three representative prototypical configurations of the parallel plate system considered is: two similar plates (in both geometry and modal density), two plates with similar modal density but different geometry, and two plates with similar geometry but different modal density. After examining the error between reduced modal methods (such as AMA) to classical modal analysis (CMA), it is determined that these several methods are valid for each of these scenarios. The data from the various methods will also be useful in evaluating the accuracy of other methods including statistical energy analysis (SEA).

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Statistical energy analysis of nonlinear vibrating systems

Cited by 19 publications

References 16 publications

Relationships between nonlinear normal modes and response to random inputs

Relationships between nonlinear normal modes and response to random inputs

Entropy and Mixing Entropy for Weakly Nonlinear Mechanical Vibrating Systems

High Frequency Analysis of a Point-Coupled Parallel Plate System

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