Statistics of complex eigenvalues in friction-induced vibration

Nobari, Amir Hossein; Ouyang, Huajiang; Bannister, P.

doi:10.1016/j.jsv.2014.10.017

Cited by 40 publications

(21 citation statements)

References 36 publications

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“…By definition, the variances of the DE3 complex amplitudes may be presented as the following equation [ Fisher , ; Nobari et al ., ]

σ_{A}^{2} = (⟨⟩, ((), A - (⟨⟩, A)) \cdot {(A - ⟨A⟩)}^{*}),

where ⟨ ⋯ ⟩ is the average value and (⋯)* denotes the conjugate value. We decomposed equation into two parts, i.e., the variance of the absolute amplitudes noted as

σ_{|A|}^{2}

and the contribution of the wave phases as

σ_{φ}^{2}

.…”

Section: Discussionmentioning

confidence: 99%

The variability of nonmigrating tides detected from TIMED/SABER observations

Wan

Ren

et al. 2015

JGR Space Physics

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This paper deals with the variability of the nonmigrating tides detected from the observation of the SABER instrument on board the TIMED satellite during the 11 year solar period from 2002 to 2012. The longitudinal wave number spectra with 1 day resolution were first estimated from the temperature data measured at the MLT altitudes (70–110 km) and at the lower midlatitudes and low latitudes (between ±45°). Then we used the wave number 4 component to obtain the nonmigrating tides in which the dominant component DE3 was further analyzed in detail. We found that the properties of the spatial distribution and large time scale variation of the DE3 component are similar to those of the previous works, which used the interpolated data with 2 month resolution. These properties are that the DE3 component occurs mainly at the low latitudes within ±30° and at the altitudes from 90 to 110 km; the tidal amplitude is larger during boreal summer and early autumn, smaller in spring and almost tends to disappear in winter; the component is slightly stronger during the eastward wind QBO phase than the westward phase. Practically, the higher‐resolution data were used to reveal the day‐to‐day variability of the DE3 component. It is found that (1) the variability occurs mainly at the altitudes from 100 to 110 km with a peak at 106 km; (2) it is strong at the low latitudes and peaks around the equator, as well, slightly stronger in the Southern Hemisphere than in northern one; (3) it is considerably larger around solstitial months than equinoctial months; and (4) it would not experience an obvious interannual variation. The day‐to‐day variability of the DE3 component may be explained by the variance of the absolute amplitudes and the contribution of the wave phases, and the later seems to play more important role.

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“…By definition, the variances of the DE3 complex amplitudes may be presented as the following equation [ Fisher , ; Nobari et al ., ]

σ_{A}^{2} = (⟨⟩, ((), A - (⟨⟩, A)) \cdot {(A - ⟨A⟩)}^{*}),

where ⟨ ⋯ ⟩ is the average value and (⋯)* denotes the conjugate value. We decomposed equation into two parts, i.e., the variance of the absolute amplitudes noted as

σ_{|A|}^{2}

and the contribution of the wave phases as

σ_{φ}^{2}

.…”

Section: Discussionmentioning

confidence: 99%

The variability of nonmigrating tides detected from TIMED/SABER observations

Wan

Ren

et al. 2015

JGR Space Physics

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“…For example, the firstorder perturbation method has been proposed by Butlin and Woodhouse [24] to quantify sensitivity of frictioninduced instabilities to the design parameters. Moreover the recent study of Nobari et al [25] proposes a second-order Taylor expansion to estimate statistical properties of eigenvalues characterizing mode coupling instabilities. Despite its efficiency, this approach has limitations, especially when standard deviations of parameters are important.…”

Section: Introductionmentioning

confidence: 99%

Kriging Surrogate Models for Predicting the Complex Eigenvalues of Mechanical Systems Subjected to Friction-Induced Vibration

Denimal

Nechak

Sinou

et al. 2016

Shock and Vibration

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This study focuses on the kriging based metamodeling for the prediction of parameter-dependent mode coupling instabilities. The high cost of the currently used parameter-dependent Complex Eigenvalue Analysis (CEA) has induced a growing need for alternative methods. Hence, this study investigates capabilities of kriging metamodels to be a suitable alternative. For this aim, kriging metamodels are proposed to predict the stability behavior of a four-degree-of-freedom mechanical system submitted to friction-induced vibrations. This system is considered under two configurations defining two stability behaviors with coalescence patterns of different complexities. Efficiency of kriging is then assessed on both configurations. In this framework, the proposed kriging surrogate approach includes a mode tracking method based on the Modal Assurance Criterion (MAC) in order to follow the physical modes of the mechanical system. Based on the numerical simulations, it is demonstrated by a comparison with the reference parameter-dependent CEA that the proposed kriging surrogate model can provide efficient and reliable predictions of mode coupling instabilities with different complex patterns.

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“…Disc Pad Young's modulus E (N/m 2 ) 125 × 10 9 In implementing the contact algorithm at the interface, the mortar method [18] with elementbased integration was adopted. Let us define the pad and disc interface as a slave (nonmortar) and master (mortar) segment.…”

Section: Parametersmentioning

confidence: 99%

“…However, since a large number of samples must be drawn to obtain sufficient accuracy, the computational cost becomes prohibitive. The sensitivity-based method [9] is a low-order Taylor expansion method that approximates the solution near the input parameters. Although this method is computationally efficient, the obtained solution will be inaccurate under large parameter variabilities.…”

Section: Introductionmentioning

confidence: 99%

A Combined Nonstationary Kriging and Support Vector Machine Method for Stochastic Eigenvalue Analysis of Brake Systems

Lee

Park

2019

Applied Sciences

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This paper presents a new metamodel approach based on nonstationary kriging and a support vector machine to efficiently predict the stochastic eigenvalue of brake systems. One of the difficulties in the mode-coupling instability induced by friction is that stochastic eigenvalues represent heterogeneous behavior due to the bifurcation phenomenon. Therefore, the stationarity assumption in kriging, where the response is correlated over the entire random input space, may not remain valid. In this paper, to address this issue, Gibb's nonstationary kernel with step-wise hyperparameters was adopted to reflect the heterogeneity of the stochastic eigenvalues. In predicting the response for unsampled input, the support vector machine-based classification is utilized. To validate the performance, a simplified finite element model of the brake system is considered. Under various types of uncertainties, including different friction coefficients and material properties, stochastic eigenvalue problems are investigated. Through numerical studies, it is seen that the proposed method improves accuracy and robustness compared to conventional stationary kriging.Carlo simulation (MCS), a sensitivity-based approach, polynomial chaos expansion (PCE), and kriging (Gaussian process). MCS [8] is a sampling-based technique which is considered the most robust method for uncertainty quantifications. However, since a large number of samples must be drawn to obtain sufficient accuracy, the computational cost becomes prohibitive. The sensitivity-based method [9] is a low-order Taylor expansion method that approximates the solution near the input parameters. Although this method is computationally efficient, the obtained solution will be inaccurate under large parameter variabilities.PCE and kriging are metamodel (surrogate model) -based techniques that replace the original model with an easy-to-evaluate function. PCE [10], also known as a spectral approach, is a regression-based methods that approximates the model by multivariate polynomials. Kriging [11] is an interpolation-based approach in which the model is assumed to be a realization of a Gaussian process. Since metamodel-based methods can be applied to various levels of uncertainty, numerous studies have utilized them to solve stochastic eigenvalue problems in brake systems [12][13][14][15].However, despite previous studies, it remains challenging to apply PCE and kriging to brake systems with the mode-coupling phenomenon. Since eigenvalues exhibit nonsmooth behavior around the bifurcation point, PCE is not adequate for capturing local solution accuracy. Several studies have tackled this issue by applying multi-element PCE [16], a wavelet basis [17]. Although kriging interpolates each sample and reflects the local characteristics, to the authors' knowledge, previous studies have all been based on the stationarity assumption, which means that the smoothness of response is entirely correlated throughout the random input space. Therefore, for mode-coupling problems where the stationarity a...

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Statistics of complex eigenvalues in friction-induced vibration

Cited by 40 publications

References 36 publications

The variability of nonmigrating tides detected from TIMED/SABER observations

The variability of nonmigrating tides detected from TIMED/SABER observations

Kriging Surrogate Models for Predicting the Complex Eigenvalues of Mechanical Systems Subjected to Friction-Induced Vibration

A Combined Nonstationary Kriging and Support Vector Machine Method for Stochastic Eigenvalue Analysis of Brake Systems

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