Bounds on the vibrational energy that can be harvested from random base motion

This paper is concerned with the development of upper bounds on the energy harvesting performance of a general multi-degree-of-freedom nonlinear electromechanical system that is subjected to random base motion and secondary applied periodic forces. The secondary forces are applied with the aim of enhancing the energy harvested from the base motion, and they may constitute direct excitation, or they may produce parametric terms in the equations of motion. It is shown that when the base motion has white noise acceleration then the power input by the base is always πS0M/2πS0M/2 where S0 is the single sided spectral density of the acceleration, and M is the mass of the system. This implies that although the secondary forces may enhance the energy harvested by causing a larger fraction of the power input from the base to be harvested rather than dissipated, there is an upper limit on the power that can be harvested. Attention is then turned to narrow band excitation, and it is found that in the absence of secondary forces a bound can be derived for a single degree of freedom system with linear damping and arbitrary nonlinear stiffness. The upper bound on the power input by the base is πMmax[S(ω)]/2, where S(ω)S(ω) is the single sided base acceleration spectrum. The validity of this result for more general systems is found to be related to the properties of the first Wiener kernel, and this issue is explored analytically and by numerical simulation.This work was funded in part through the EPSRC Programme Grant “Engineering Nonlinearity” EP/K003836/1.This is the final published version. It originally appeared at http://www.sciencedirect.com/science/article/pii/S0022460X14009092#

The level crossing rates and associated statistical properties of a random frequency response function

This work is concerned with the statistical properties of the frequency response function of the energy of a random system. Earlier studies have considered the statistical distribution of the function at a single frequency, or alternatively the statistics of a band-average of the function. In contrast the present analysis considers the statistical fluctuations over a frequency band, and results are obtained for the mean rate at which the function crosses a specified level (or equivalently, the average number of times the level is crossed within the band). Results are also obtained for the probability of crossing a specified level at least once, the mean rate of occurrence of peaks, and the mean trough-to-peak height. The analysis is based on the assumption that the natural frequencies and mode shapes of the system have statistical properties that are governed by the Gaussian Orthogonal Ensemble (GOE), and the validity of this assumption is demonstrated by comparison with numerical simulations for a random plate. The work has application to the assessment of the performance of dynamic systems that are sensitive to random imperfections

The crossing rates, exceedance probabilities, and related statistical properties of the energy frequency response functions of a random built-up system

A method is derived for computing a number of key statistical properties of the vibrational energy of each component of a built-up system that has random properties. The energy is considered to be a random function of frequency, and the derived statistical properties include: the mean rate at which the energy crosses a specified level, the probability that the energy will exceed a specified level within a given frequency band, the mean trough-to-peak height, the rate of occurrence of peaks, and the mean quefrency (a measure of the rate of fluctuation of the energy). The analysis is based on combining statistical energy analysis with a non-parametric model of uncertainty based on the Gaussian orthogonal ensemble and avoids the use of Monte Carlo simulations or large computational models. By way of example, the method is applied to a number of coupled plate systems. </jats:p

On the statistical mechanics of structural vibration

The analysis of the structural dynamics of a complex engineering structure has much in common with the subject of statistical mechanics. Both are concerned with the analysis of large systems in the presence of various sources of randomness, and both are concerned with the possibility of emergent laws that might be used to provide a simplified approach to the analysis of the system. The aim of the present work is to apply a number of the concepts of statistical mechanics to structural dynamic systems in order to provide new insights into the system behaviour under various conditions. The work is foundational, in that it is based on employing the fundamental equations of motion of the system in conjunction with various definitions of entropy, and no recourse is made to emergent laws that are accepted in thermodynamics. The analysis covers closed (undamped and unforced) and open (forced and damped) systems, linear and nonlinear systems, and both single systems and coupled systems. The fact that the system itself can be random leads to a number of results that differ from those found in classical statistical mechanics, where the initial conditions might be considered to be random but the Hamiltonian is taken to be well defined. For example, the occurrence of a stationary state in a closed system normally requires nonlinearity and coarse-graining of the statistical distribution, but neither condition is required for a random system. For coupled systems it is shown that under certain conditions both Statistical Energy Analysis (SEA) and Transient Statistical Energy Analysis (TSEA) are emergent laws, and insights are gained as to the validity of these laws. The analysis is supported by a number of numerical examples to illustrate key points

An efficient model of drillstring dynamics

This is the author accepted manuscript. The final version is available from Elsevier via http://dx.doi.org/10.1016/j.jsv.2015.06.033High amplitude vibration regimes can cause significant damage to oilwell drillstrings: torsional stick-slip oscillation, forward whirl and backward whirl are each associated with different kinds of damage. There is a need for models of drillstring dynamics that can predict this variety of phenomena that are: efficient enough to carry out parametric studies; simple enough to provide insight into the underlying physics, and which retain sufficient detail to correlate to real drillstrings. The modelling strategy presented in this paper attempts to balance these requirements. It includes the dynamics of the full length of the drillstring over a wide bandwidth but assumes that the main nonlinear effects are due to spatially localised regions of strong nonlinearity, for example at the drillbit cutting interface and at stabilisers where the borehole wall clearance is smallest. The equations of motion can be formed in terms of this reduced set of degrees of freedom, coupled to the nonlinear contact laws and solved by time-domain integration. Two implementations of this approach are presented, using (1) digital filters and (2) a Finite Element model to describe the linear dynamics. Choosing a sampling period that is less than the group delay between nonlinear degrees of freedom results in a decoupled set of equations that can be solved very efficiently. Several cases are presented which demonstrate a variety of phenomena, including stick-slip oscillation; forward whirl and backward whirl. Parametric studies are shown which reveal the conditions which lead to high amplitude vibration regimes, and an analytic regime boundary is derived for torsional stick-slip oscillation. The digital filter and Finite Element models are shown to be in good agreement and are similarly computationally efficient. The digital filter approach has the advantage of more intuitive interpretation, while the Finite Element model is more readily implemented using existing software packages.The authors are grateful to industrial support and permission to publish this work, and would like to thank Prof Jim Woodhouse for technical contributions, and Dr Louis Kovalevski for assistance with the Finite Element model

Statistical energy analysis of nonlinear vibrating systems.

Nonlinearities in practical systems can arise in contacts between components, possibly from friction or impacts. However, it is also known that quadratic and cubic nonlinearity can occur in the stiffness of structural elements undergoing large amplitude vibration, without the need for local contacts. Nonlinearity due purely to large amplitude vibration can then result in significant energy being found in frequency bands other than those being driven by external forces. To analyse this phenomenon, a method is developed here in which the response of the structure in the frequency domain is divided into frequency bands, and the energy flow between the frequency bands is calculated. The frequency bands are assigned an energy variable to describe the mean response and the nonlinear coupling between bands is described in terms of weighted summations of the convolutions of linear modal transfer functions. This represents a nonlinear extension to an established linear theory known as statistical energy analysis (SEA). The nonlinear extension to SEA theory is presented for the case of a plate structure with quadratic and cubic nonlinearity.The authors would like to acknowledge the ‘Engineering Nonlinearity’ Program Grant, funded by EPSRC that has funded the research content of this paper. We have no competing interests.This is the author accepted manuscript. The final version is available from RSC via http://dx.doi.org/10.1098/rsta.2014.040

Scaling of slow-drift motion with platform size and its importance for floating wind turbines

Slow drift is a large, low-frequency motion of a floating platform caused by nonlinear hydrodynamic forces. Although slow drift is a well-known phenomenon for ships and other floating structures, new platforms for floating wind turbines are significantly smaller in scale, and it is yet to be established how important slow drift is for them. In this paper we derive an approximate expression for the scaling of the slow drift motion with platform size, mooring characteristics and wave conditions. This suggests that slow drift may be less important for floating wind turbines than other, larger, floating structures. The accuracy of the approximations is discussed; in the one case where detailed data is available, the approximate result is found to be conservative by a factor of up to 40.Engineering and Physical Sciences Research Council (doctoral training award ID: 1089390), GL Garrad Hassa

Analysis of the power flow in nonlinear oscillators driven by random excitation using the first Wiener kernel

Random excitation of mechanical systems occurs in a wide variety of structures and, in some applications, calculation of the power dissipated by such a system will be of interest. In this paper, using the Wiener series, a general methodology is developed for calculating the power dissipated by a general nonlinear multi-degree-of freedom oscillatory system excited by random Gaussian base motion of any spectrum. The Wiener series method is most commonly applied to systems with white noise inputs, but can be extended to encompass a general non-white input. From the extended series a simple expression for the power dissipated can be derived in terms of the first term, or kernel, of the series and the spectrum of the input. Calculation of the first kernel can be performed either via numerical simulations or from experimental data and a useful property of the kernel, namely that the integral over its frequency domain representation is proportional to the oscillating mass, is derived. The resulting equations offer a simple conceptual analysis of the power flow in nonlinear randomly excited systems and hence assist the design of any system where power dissipation is a consideration. The results are validated both numerically and experimentally using a base-excited cantilever beam with a nonlinear restoring force produced by magnets

Complex but negligible: Non-linearity of the inertial coupling between the platform and blades of floating wind turbines

Approximate linearised models can be important for preliminary design of floating wind turbines, but their value depends on how well they approximate the real-world non-linear behaviour. This paper focuses on the non-linear inertial coupling between motion of the floating platform and the blade dynamics, using a simplified model to demonstrate how the inertial coupling works, and systematically studying the linearity of the dynamic blade response to different directions, amplitudes and frequencies of motion. Simplified equations of motion are derived and approximately solved analytically, showing that the blade response contains harmonics at a range of frequencies, some linear and some non-linear in the amplitude of the platform motion. Comparison to numerical simulations shows that the analytical results were qualitatively useful but inaccurate for large platform motions. Because of the multiple harmonics in the response, there are more combinations of rotor speeds and platform motions leading to large resonant blade responses and non-linear behaviour than might be expected. Overall, for realistically low rotor speeds and platform frequencies (below 20 rpm and 0.2 Hz), non-linear inertial loading due to platform motion should be negligible. The implications of this work for the use of linearised structural models and the relevance of scale model testing are discussed

The effect of generalized force correlations on the response statistics of a harmonically driven random system

If the physical properties of a structural component are sufficiently random then the statistical distribution of the natural frequencies and mode shapes tends to a universal distribution associated with the Gaussian Orthogonal Ensemble (GOE) of random matrices. Previous work has exploited this result to yield expressions for the relative variance of the energy of the response of a random system to harmonic excitation. The derivation of these expressions employed random point process theory, and in the theoretical development it was assumed that the modal generalised forces were uncorrelated. Although this assumption is often valid, there are cases in which correlations between the generalised forces can significantly affect the response variance, and in the present work the existing theory is extended to include correlations of this type. The extended theory is applicable to both single frequency responses and to band average responses, and the developed closed form expressions are validated by comparison with direct simulations for a random plate structure.Elke Deckers contribution was funded through The Research Fund KU Leuve