Soliton approach to the noisy Burgers equation: Steepest descent method

The noisy Burgers equation in one spatial dimension is analyzed by means of the Martin-Siggia-Rose technique in functional form. In a canonical formulation the morphology and scaling behavior are accessed by mean of a principle of least action in the asymptotic non-perturbative weak noise limit. The ensuing coupled saddle point field equations for the local slope and noise fields, replacing the noisy Burgers equation, are solved yielding nonlinear localized soliton solutions and extended linear diffusive mode solutions, describing the morphology of a growing interface. The canonical formalism and the principle of least action also associate momentum, energy, and action with a soliton-diffusive mode configuration and thus provides a selection criterion for the noise-induced fluctuations. In a ``quantum mechanical'' representation of the path integral the noise fluctuations, corresponding to different paths in the path integral, are interpreted as ``quantum fluctuations'' and the growth morphology represented by a Landau-type quasi-particle gas of ``quantum solitons'' with gapless dispersion and ``quantum diffusive modes'' with a gap in the spectrum. Finally, the scaling properties are dicussed from a heuristic point of view in terms of a``quantum spectral representation'' for the slope correlations. The dynamic eponent z=3/2 is given by the gapless soliton dispersion law, whereas the roughness exponent zeta =1/2 follows from a regularity property of the form factor in the spectral representation. A heuristic expression for the scaling function is given by spectral representation and has a form similar to the probability distribution for Levy flights with index $z$.Comment: 30 pages, Revtex file, 14 figures, to be submitted to Phys. Rev.

Understanding how kurtosis is transferred from input acceleration to stress response and it's influence on fatigue life

High cycle fatigue of metals typically occurs through long term exposure to time varying loads which, although modest in amplitude, give rise to microscopic cracks that can ultimately propagate to failure. The fatigue life of a component is primarily dependent on the stress amplitude response at critical failure locations. For most vibration tests, it is common to assume a Gaussian distribution of both the input acceleration and stress response. In real life, however, it is common to experience non-Gaussian acceleration input, and this can cause the response to be non-Gaussian. Examples of non-Gaussian loads include road irregularities such as potholes in the automotive world or turbulent boundary layer pressure fluctuations for the aerospace sector or more generally wind, wave or high amplitude acoustic loads. The paper first reviews some of the methods used to generate non-Gaussian excitation signals with a given power spectral density and kurtosis. The kurtosis of the response is examined once the signal is passed through a linear time invariant system. Finally an algorithm is presented that determines the output kurtosis based upon the input kurtosis, the input power spectral density and the frequency response function of the system. The algorithm is validated using numerical simulations. Direct applications of these results include improved fatigue life estimations and a method to accelerate shaker tests by generating high kurtosis, non-Gaussian drive signals

Identification of nonlinear vibrating structures: Part II -- Applications

A time-domain procedure for the identification of nonlinear vibrating structures, presented in a companion paper, is applied to a "calibration" problem which incorporates realistic test situations and nonlinear structural characteristics widely encountered in the applied mechanics field. The "data" set is analyzed to develop suitable, approximate nonlinear system representations. Subsequently, a "validation" test is conducted to demonstrate the range of validity of the method under discussion. It is shown that the procedure furnishes a convenient means for constructing reduced-order nonlinear nonparametric mathematical models of reasonably high fidelity in regard to reproducing the response of the test article under dynamic loads that differ from the identification test loads

Towards Efficient Maximum Likelihood Estimation of LPV-SS Models

How to efficiently identify multiple-input multiple-output (MIMO) linear parameter-varying (LPV) discrete-time state-space (SS) models with affine dependence on the scheduling variable still remains an open question, as identification methods proposed in the literature suffer heavily from the curse of dimensionality and/or depend on over-restrictive approximations of the measured signal behaviors. However, obtaining an SS model of the targeted system is crucial for many LPV control synthesis methods, as these synthesis tools are almost exclusively formulated for the aforementioned representation of the system dynamics. Therefore, in this paper, we tackle the problem by combining state-of-the-art LPV input-output (IO) identification methods with an LPV-IO to LPV-SS realization scheme and a maximum likelihood refinement step. The resulting modular LPV-SS identification approach achieves statical efficiency with a relatively low computational load. The method contains the following three steps: 1) estimation of the Markov coefficient sequence of the underlying system using correlation analysis or Bayesian impulse response estimation, then 2) LPV-SS realization of the estimated coefficients by using a basis reduced Ho-Kalman method, and 3) refinement of the LPV-SS model estimate from a maximum-likelihood point of view by a gradient-based or an expectation-maximization optimization methodology. The effectiveness of the full identification scheme is demonstrated by a Monte Carlo study where our proposed method is compared to existing schemes for identifying a MIMO LPV system

Stochastic response determination and spectral identification of complex dynamic structural systems

Uncertainty propagation in engineering mechanics and dynamics is a highly challenging problem that requires development of analytical/numerical techniques for determining the stochastic response of complex engineering systems. In this regard, although Monte Carlo simulation (MCS) has been the most versatile technique for addressing the above problem, it can become computationally daunting when faced with high-dimensional systems or with computing very low probability events. Thus, there is a demand for pursuing more computationally efficient methodologies. Further, most structural systems are likely to exhibit nonlinear and time-varying behavior when subjected to extreme events such as severe earthquake, wind and sea wave excitations. In such cases, a reliable identification approach is behavior and for assessing its reliability. Current work addresses two research themes in the field of stochastic engineering dynamics related to the above challenges. In the first part of the dissertation, the recently developedWiener Path Integral (WPI) technique for determining the joint response probability density function (PDF) of nonlinear systems subject to Gaussian white noise excitation is generalized herein to account for non-white, non-Gaussian, and non-stationary excitation processes. Specifically, modeling the excitation process as the output of a filter equation with Gaussian white noise as its input, it is possible to define an augmented response vector process to be considered in the WPI solution technique. A significant advantage relates to the fact that the technique is still applicable even for arbitrary excitation power spectrum forms. In such cases, it is shown that the use of a filter approximation facilitates the implementation of the WPI technique in a straightforward manner, without compromising its accuracy necessarily. Further, in addition to dynamical systems subject to stochastic excitation, the technique can also account for a special class of engineering mechanics problems where the media properties are modeled as non-Gaussian and non-homogeneous stochastic fields. Several numerical examples pertaining to both single- and multi-degree-of freedom systems are considered, including a marine structural system exposed to flow-induced non-white excitation, as well as a beam with a non-Gaussian and non-homogeneous Young’s modulus. Comparisons with MCS data demonstrate the accuracy of the technique. In the second part of the dissertation, a novel multiple-input/single-output (MISO) system identification technique is developed for parameter identification of nonlinear time-variant multi-degree-of-freedom oscillators with fractional derivative terms subject to incomplete non-stationary data. The technique utilizes a representation of the nonlinear restoring forces as a set of parallel linear subsystems. In this regard, the oscillator is transformed into an equivalent MISO system in the wavelet domain. Next, a recently developed L1-norm minimization procedure based on compressive sampling theory is applied for determining the wavelet coefficients of the available incomplete non-stationary input-output (excitation-response) data. Finally, these wavelet coefficients are utilized to determine appropriately defined time- and frequency-dependent wavelet based frequency response functions and related oscillator parameters. A nonlinear time-variant system with fractional derivative elements is used as a numerical example to demonstrate the reliability of the technique even in cases of noise corrupted and incomplete data