Numerical methods for calculating the response of a deterministic and stochastically excited Duffing oscillator

When compared to independent harmonic or stochastic excitation, there exist relatively few methods to model the response of non-linear systems to a combination of deterministic and stochastic vibration despite the likelihood of harmonic oscillations containing noise in realistic applications. This paper uses the Duffing oscillator to illustrate how the joint probability density function (JPDF) of the displacement and velocity responds to this form of excitation. Monte Carlo simulations were performed to generate the JPDF which was observed, in general, to spread around the trajectory that would be observed if only deterministic excitation was present. In the deterministic chaotic case, the JPDF is known to be a diffuse chaotic attractor when noise is present. This paper assesses the ability of a useful class of methods, global weighted residual methods, to produce the geometrically complex JPDF responses produced from harmonic and white noise excitation. A technique using a JPDF in the form of a Gram–Charlier type C series was found to produce accurate results, although the method fails due to ill-conditioning as the shape of the JPDF required by the dynamics becomes too complex. The authors would like to thank the EPSRC Doctoral Training Award for funding this research.This is the author accepted manuscript. The final version is available from SAGE via http://dx.doi.org/10.1177/095440621560754

Harmonic Wavelets Procedures and Wiener Path and Integral Methods for Response Determination and Reliability Assessment of Nonlinear Systems/Structures

In this thesis a novel approximate/analytical approach based on the concepts of stochastic averaging and of statistical linearization is developed for the response determination of nonlinear/hysteretic multi-degree-of-freedom (MDOF) systems subject to evolutionary stochastic excitation. The significant advantage of the approach relates to the fact that it is readily applicable for excitations possessing even non-separable evolutionary power spectra (EPS) circumventing ad hoc pre-filtering and pre-processing excitation treatments associated with existing alternative schemes of linearization. Further, the approach can be used, in a rather straightforward manner, in conjunction with recently developed design spectrum based analyses for obtaining peak response estimates without resorting to numerical integration of the nonlinear equations of motion. Furthermore, a novel approximate/analytical Wiener path integral based solution (PIS) is developed and a numerical PIS approach is extended to determine the response and first-passage probability density functions (PDFs) of nonlinear/hysteretic systems subject to evolutionary stochastic excitation. Applications include the versatile Preisach hysteretic model, recently applied in modeling systems equipped with smart material (shape memory alloys) devices used for seismic hazard risk mitigation. The approach is also applied to determine the capsizing probability of a ship, whose rolling dynamics is captured by a softening Duffing oscillator. Finally, novel harmonic wavelets based joint time-frequency response analysis and identification approaches are developed capable of determining the time-varying frequency content of non-stationary complex stochastic phenomena encountered in engineering applications. Specifically, a harmonic wavelets based statistical linearization approach is developed to determine the EPS of the response of nonlinear/hysteretic systems subject to stochastic excitation. In a similar context, an identification approach for nonlinear time-variant systems based on the localization properties of the harmonic wavelet transform is also developed. It can be construed as a generalization of the well established reverse multiple-input/single-output (MISO) spectral identification approach to account for non-stationary inputs and time-varying system parameters. Several linear and nonlinear time-variant systems are used to demonstrate the reliability of the approach

Response and first-passage statistics of nonlinear structural models under evolutionary stochastic loads

In the first half of the thesis, a novel approach is developed for determining the response of a lightly damped nonlinear single-degree of freedom system to a random excitation with an evolutionary broad-band power spectrum. The new approach is based on the coupling of the concepts of stochastic averaging and equivalent linearization. The nonlinearities can be either of the hysteretic or of the 'zero-memory' kind. Moreover, approximate analytical relationships for evaluating the response variance are derived for a number of oscillators. The efficiency and accuracy of the approach is demonstrated by pertinent digital simulations. In the second half of the thesis, an approximate analytical approach is presented for examining the first-passage problem in context with the response of a class of lightly damped nonlinear oscillators to broad-band random excitations. A Markovian approximation both of the response amplitude envelope and of the response energy envelope is considered. This modeling leads to a backward Kolmogorov equation which governs the evolution of the survival probability of the oscillator. The Kolmogorov equation is solved approximately by employing a Galerkin approach. A set of confluent hypergeometric functions is used as an orthogonal basis for the expansions which are involved in the application of the Galerkin approach. The reliability of the derived analytical solution is demonstrated by comparisons of digital data derived by Monte Carlo simulation

Stochastic dynamics and wavelets techniques for system response analysis and diagnostics: Diverse applications in structural and biomedical engineering

In the first part of the dissertation, a novel stochastic averaging technique based on a Hilbert transform definition of the oscillator response displacement amplitude is developed. In comparison to standard stochastic averaging, the requirement of “a priori” determination of an equivalent natural frequency is bypassed, yielding flexibility in the ensuing analysis and potentially higher accuracy. Further, the herein proposed Hilbert transform based stochastic averaging is adapted for determining the time-dependent survival probability and first-passage time probability density function of stochastically excited nonlinear oscillators, even endowed with fractional derivative terms. To this aim, a Galerkin scheme is utilized to solve approximately the backward Kolmogorov partial differential equation governing the survival probability of the oscillator response. Next, the potential of the stochastic averaging technique to be used in conjunction with performance-based engineering design applications is demonstrated by proposing a stochastic version of the widely used incremental dynamic analysis (IDA). Specifically, modeling the excitation as a non-stationary stochastic process possessing an evolutionary power spectrum (EPS), an approximate closed-form expression is derived for the parameterized oscillator response amplitude probability density function (PDF). In this regard, IDA surfaces are determined providing the conditional PDF of the engineering demand parameter (EDP) for a given intensity measure (IM) value. In contrast to the computationally expensive Monte Carlo simulation, the methodology developed herein determines the IDA surfaces at minimal computational cost. In the second part of the dissertation, a novel multiple-input/single-output (MISO) system identification technique is developed for parameter identification of nonlinear and time-variant 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 sub-systems. Next, a recently developed L1-norm minimization procedure based on compressive sensing theory is applied for determining the wavelet coefficients of the available incomplete non-stationary input-output (excitation-response) data. Several numerical examples are considered for assessing the reliability of the technique, even in the presence of incomplete and corrupted data. These include a 2-DOF time-variant Duffing oscillator endowed with fractional derivative terms, as well as a 2-DOF system subject to flow-induced forces where the non-stationary sea state possesses a recently proposed evolutionary version of the JONSWAP spectrum. In the third part of this dissertation, a joint time-frequency analysis technique based on generalized harmonic wavelets (GHWs) is developed for dynamic cerebral autoregulation (DCA) performance quantification. DCA is the continuous counter-regulation of the cerebral blood flow by the active response of cerebral blood vessels to the spontaneous or induced blood pressure fluctuations. Specifically, various metrics of the phase shift and magnitude of appropriately defined GHW-based transfer functions are determined based on data points over the joint time-frequency domain. The potential of these metrics to be used as a diagnostics tool for indicating healthy versus impaired DCA function is assessed by considering both healthy individuals and patients with unilateral carotid artery stenosis. Next, another application in biomedical engineering is pursued related to the Pulse Wave Imaging (PWI) technique. This relies on ultrasonic signals for capturing the propagation of pressure pulses along the carotid artery, and eventually for prognosis of focal vascular diseases (e.g., atherosclerosis and abdominal aortic aneurysm). However, to obtain a high spatio-temporal resolution the data are acquired at a high rate, in the order of kilohertz, yielding large datasets. To address this challenge, an efficient data compression technique is developed based on the multiresolution wavelet decomposition scheme, which exploits the high correlation of adjacent RF-frames generated by the PWI technique. Further, a sparse matrix decomposition is proposed as an efficient way to identify the boundaries of the arterial wall in the PWI technique

Random vibration response statistics for fatigue analysis of nonlinear structures

Statistical analysis methods are developed for determining fatigue time to failure for nonlinear structures when subjected to random loading. The change in the response, as structures progress from a linear regime to a large amplitude nonlinear regime, is studied in both the time and frequency domains. The analyses in the two domains are shown to compliment each other, allowing keen understanding of the physical fundamentals of the problem. Analysis of experimental random vibration data, obtained at Wright Patterson Air Force Base, is included to illustrate the challenge for a real, multi-mode, nonlinear structure. The reverse path frequency response identification method was used with the displacement and strain response to estimate nonlinear frequency response functions. The coherence functions of these estimates provided insight into nonlinear models of the system. Time domain analysis of the nonlinear data showed how the displacement and strain departed from a normal distribution. Inverse distribution function methods were used to develop functions that related the linear to the nonlinear response of the structure. These linear to the nonlinear functions were subsequently used to estimate probability density functions that agreed well with measured histograms. Numerical simulations of a nonlinear single degree of freedom system were produced to study other aspects of the large deflection structural response. Very large sample size data sets of displacement, velocity, acceleration and stress were used to quantify the rate of convergence of several random response statistics. The inverse distribution function method was also applied to the simulation results to estimate normal and peak linear to nonlinear functions. These functions were shown to be useful for probability density function estimates and for estimating rates of response zero crossings. Fatigue analysis was performed on the experimental and simulated linear and nonlinear data. The time to failure estimates for the nonlinear results was shown to increase dramatically when compared to the linear results. The nonlinear stresses have significant positive mean values due to membrane effects, that when used with fatigue equations that account for mean stresses, show reductions in time to failure. Further studies of the nonlinear increase in the frequency of stress response are included in the fatigue analysi

Path integral techniques and Gröbner basis approaches for stochastic response analysis and optimization of diverse nonlinear dynamic systems

This thesis focuses primarily on generalizations and enhancements of the Wiener path integral (WPI) technique for stochastic response analysis and optimization of diverse nonlinear dynamic systems of engineering interest. Concisely, the WPI technique, which has proven to be a potent mathematical tool in theoretical physics, has been recently extended to address problems in stochastic engineering dynamics. Herein, the WPI technique has been significantly enhanced in terms of computational efficiency and versatility; these results are presented in Chapters 2-5. Specifically, in Chapter 2 a brief introduction to the standard WPI solution approach is outlined. In Chapter 3, a novel methodology is presented, which utilizes theoretical results from calculus of variations to extend the WPI for determining marginalized response PDFs of n-degree-of-freedom (n-DOF) nonlinear systems. The associated computational cost relates to the dimension of the PDF and is essentially independent from the dimension n of the system. In several commonly encountered cases, the aforementioned methodology improves the computational efficiency of the WPI by orders of magnitude, and exhibits a significant advantage over the commonly utilized Monte-Carlo-simulation (MCS). Moreover, in Chapter 4, an extension of the WPI technique is presented for addressing the challenge of determining the stochastic response of nonlinear dynamical systems under the presence of singularities in the diffusion matrix. The key idea behind this approach is to partition the original system into an underdetermined system of SDEs corresponding to a nonsingular diffusion matrix and an underdetermined system of homogeneous differential equations; the latter is treated as a dynamic constraint that allows for employing constrained variational/optimization solution methods. In Chapter 5, this approach is applied for the stochastic response analysis and optimization of electromechanical vibratory energy harvesters. Next, in Chapter 6, a technique from computational algebraic geometry has been developed, which is based on the concept of Gröbner basis and is capable of determining the entire solution set of systems of polynomial equations. This technique has been utilized to address diverse challenging problems in engineering mechanics. First, after formulating the WPI as a minimization problem, it is shown in Chapter 7 that the corresponding objective function is convex, and thus, convergence of numerical schemes to the global optimum is guaranteed. Second, in Chapter 8, the computational algebraic geometry technique has been applied to the challenging problem of determining nonlinear normal modes (NNMs) corresponding to multi-degree-of-freedom dynamical systems as defined in [1], and has been shown to yield improvements in accuracy compared to the standard treatment in the literature

APPROXIMATE STOCHASTIC TECHNIQUES FOR DIVERSE ENGINEERING DYNAMICS APPLICATIONS

Generally, deterministic approaches are used in practice to analyze dynamic systems. Variations in loading conditions and material properties are taken into account by either selecting high, low or average values. Consequently, the uncertainty inherent in almost every dynamic analysis is considered just intuitively. To realistically capture the behavior of a dynamic system the intrinsic randomness must be appropriately modeled requiring concepts and methods of mathematical statistics and probability theory, as well as, random vibration theory. Undeniably, stochastic dynamics based approaches provide a more realistic modeling of the dynamic response of engineered systems allowing for enhanced design solutions. The prevailing approach used in the industry is the Monte Carlo simulation method. However, a well-known shortcoming of the method is the extensive computational cost required. Further, the class of problems of nonlinear random vibrations that lend themselves to exact solutions (e.g., via the associated Fokker-Planck-Kolmogorov equation) is extremely limited. Therefore, approximate approaches are desired for solving nonlinear stochastic dynamics problems. The current thesis seeks to exploit approximate stochastic dynamics tools to solve engineering dynamics problems encountered in practice. In particular, the primary focus is directed towards the recently developed Wiener path integral technique, which has been shown to poses certain advantages over alternative well-established solution methodologies, namely, computational efficiency and accuracy. Two applications are investigated: the stochastic response of nonlinear vibratory energy harvesters, and, the depth determination of ice gouging events. The accuracy/reliability of the approximate approaches is demonstrated via comparisons with pertinent Monte Carlo simulation data

Dynamics of optically levitated nanoparticles in high vacuum

Nanotechnology was named one of the key enabling technologies by the European Commission and its tremendous impact was envisioned early by 20th century physicist R.Feynman in his now oft-quoted talk "Plenty of Room at the bottom". Nanotechnology and nanoscience deal with structures barely visible with an optical microscope, yet much bigger than simple molecules. Matter at this mesoscale is often awkward to explore as it contains too many atoms to be easily understood by straightforward application of quantum mechanics (although the fundamental laws still apply). Yet, these systems are not so large as to be completely free of quantum effects; thus, they do not simply obey the classical physics governing the macroworld. It is precisely in this intermediate regime, the mesoworld, that unforeseen properties of collective systems emerge. To fully exploit the potential of nanotechnology, a thorough understanding of these properties is paramount. The objective of the present thesis is to investigate and to control the dynamics of an optically levitated particle in high vacuum, a system which belongs to the broader class of nanomechanical oscillators. Nanomechanical oscillators exhibit high resonance frequencies, diminished active masses, low power consumption and high quality factors - significantly higher than those of electrical circuits. These attributes make them suitable for sensing, transduction and signal processing. Furthermore, nanomechanical systems are expected to open up investigations of the quantum behavior of mesoscopic systems. Testing the predictions of quantum theory on macroscopic scales is one of today's outstanding challenges of modern physics and addresses fundamental questions on our understanding of the world. The state-of-the-art in nanomechanics itself has exploded in recent years, driven by a combination of interesting new systems and vastly improved fabrication capabilities. Despite major break-throughs, including ground state cooling, observation of radiation pressure shot noise, squeezing and demonstrated ultra-high force and mass sensitivity, difficulties in reaching ultra-high mechanical quality (Q) factors still pose a major limitation for many of the envisioned applications and significant improvements in mechanical quality (Q) factors are generally needed to facilitate quantum coherent manipulation. This is difficult given that many mechanical systems are approaching fundamental limits of dissipation. To overcome the limitations set by dissipation, I developed an experiment to trap and cool nanoparticles in high vacuum. The combination of nanoparticles and vacuum trapping results in a very light and ultra-high-Q mechanical oscillator. In fact, the Q-factor achieved with this setup is the highest observed so far in any nano- or micromechanical system. The scope of the thesis ranges from a detailed description of the experimental apparatus and proof-of-principle experiments (parametric feedback cooling) to the first observation of phenomena owing to the unique parameters of this novel optomechanical system (thermal nonlinearities). Aside from optomechanics and optical trapping, the topics covered include the dynamics of complex (nonlinear) systems and the study of fluctuation theorems, the latter playing a pivotal role in statistical physics. Optically trapped nanoparticles are just beginning to emerge as a new class of optomechanical systems. Owing to their unique mechanical properties, there is clearly a vast and untapped potential for further research. Primary examples of how levitated particles in high vacuum can impact other fields and inspire new research avenues have been the first observation of thermal nonlinearities in a mechanical oscillator and the study of fluctuation relations with a high-Q nanomechanical resonator. Based on recent progress in the field, a plethora of fundamental research opportunities and novel applications are expected to emerge as this still young field matures