Lawrence N. Virgin

Resonances of a Harmonically Forced Duffing Oscillator with Time Delay State Feedback

The paper presents analytical and numerical studies of the primary resonance and the 1/3 subharmonic resonance of a harmonically forced Duffing oscillator under state feedback control with a time delay. By using the method of multiple scales, the first order approximations of the resonances are derived and the effect of time delay on the resonances is analyzed. The concept of an equivalent damping related to the delay feedback is proposed and the appropriate choice of the feedback gains and the time delay is discussed from the viewpoint of vibration control. In order to numerically solve the problem of history dependence prior to the start of excitation, the concepts of the Poincaré section and fixed points are generalized. Then, a modified shooting scheme associated with the path following technique is proposed to locate the periodic motion of the delayed system. The numerical results show the efficacy of the first order approximations of the resonances.

THE NONLINEAR ROLLING RESPONSE OF A VESSEL INCLUDING CHAOTIC MOTIONS LEADING TO CAPSIZE IN REGULAR SEAS

Abstract The rolling motion of a ship has been successfully modelled using a semi-empirical nonlinear differential equation by a number of researchers. Experimental data has been used to model nonlinear damping and righting lever characteristics and comparison with observed behaviour has been reasonably good. The present article describes a numerical, phenomenological approach to analyse this type of behaviour. The stability of the periodic motion, and in particular the possibility of capsize, is explored with reference to qualitative prediction techniques. The appearance of chaotic motions in regular beam seas is a new feature which should be of interest to designers. The inability of traditional quantitative methods, such as the perturbation technique, to detect chaos is a further justification for using numerical simulation guided by dynamical systems theory.

Bragg grating-based fibre optic sensors in structural health monitoring

This work first considers a review of the dominant current methods for fibre Bragg grating wavelength interrogation. These methods include WDM interferometry, tunable filter (both Fabry–Perot and acousto-optic) demultiplexing, CCD/prism technique and a newer hybrid method utilizing Fabry–Perot and interferometric techniques. Two applications using these techniques are described: hull loads monitoring on an all-composite fast patrol boat and bolt pre-load loss monitoring in a composite beam in conjunction with a state-space modelling data analysis technique.

Structural health monitoring through chaotic interrogation

The field of vibration based structural health monitoring involves extracting a ‘feature’ which robustly quantifies damage induced changes to the structure in the presence of ambient variation, that is, changes in ambient temperature, varying moisture levels, etc. In this paper, we present an attractor-based feature derived from the field of nonlinear time-series analysis. Emphasis is placed on the use of chaos for the purposes of system interrogation. The structure is excited with the output of a chaotic oscillator providing a deterministic (low-dimensional) input. Use is made of the Kaplan–Yorke conjecture in order to ‘tune’ the Lyapunov exponents of the driving signal so that varying degrees of damage in the structure will alter the state space properties of the response attractor. The average local attractor variance ratio (ALAVR) is suggested as one possible means of quantifying the state space changes. Finite element results are presented for a thin aluminum cantilever beam subject to increasing damage, as specified by weld line separation, at the clamped end. Comparisons of the ALAVR to two modal features are evaluated through the use of a performance metric.

Chaos and Period-Adding; Experimental and Numerical Verification of the Grazing Bifurcation

SummaryExperimental results are presented for a single-degree-of-freedom horizontally excited pendulum that is allowed to impact with a rigid stop at a fixed angle θ to the vertical. By inclining the apparatus, the pendulum is allowed to swing in an effectively reduced gravity, so that for each fixed θ less than a critical value, a forcing frequency is found such that a period-one limit cycle motion just grazes with the stop. Experimental measurements show the immediate onset of chaotic dynamics and a period-adding cascade for slightly higher frequencies. These results are compared with a numerical simulation and continuation of solutions to a mathematical model of the system, which shows the same qualitative effects. From the model, the theory of discontinuity mappings due to Nordmark is applied to derive the coefficients of the square-root normal form map of the grazing bifurcation for this system. The grazing periodic orbit and its linearisation are found using a numerical continuation method for hybrid systems. From this, the normal-form coefficients are computed, which in this case imply that a jump to chaos and period-adding cascade occurs. Excellent quantitative agreement is found between the model simulation and the map, even over wide parameter ranges. Qualitatively, both accurately predict the experimental results, and after a slight change in the effective damping value, a striking quantitative agreement is found too.

Grazing bifurcations and basins of attraction in an impact-friction oscillator

Abstract This paper describes some interesting global dynamic behavior in the response of a double-sided, harmonically-forced, impact oscillator including the influence of Coulomb damping. The system under study is modeled as piecewise linear in both its force-deflection and force-velocity characteristics. Grazing bifurcations caused by this latter effect are a new feature. The paper has two distinct but related foci. First, the study of basins of attraction provides information regarding the complete solution set for the system, given a specific set of parameters. Second, grazing bifurcations represent the primary source of sudden change in qualitative behavior as a system parameter is varied. The numerical technique of cell-to-cell mapping provides a useful insight into the relation between these two. Thus, both local and global issues are addressed – indeed it is the interplay of periodic attractors and their basins of attraction that dominates bifurcational behavior.

Spatial chaos and localization phenomena in nonlinear elasticity

Abstract Classical static-dynamic analogies are invoked to demonstrate spatial chaos and localization of deformations in the elastica of a post-buckled strut. Some conjectures are then made relating homoclinic events in the dynamic analogy of a strut on a nonlinear elastic foundation to the spatial localization of the buckling pattern.

Accurate numerical integration of state-space models for aeroelastic systems with free play

Results show the importance of accurately locating the switching point between linear subdomains when numerically integrating a piecewise linear system of equations. Henons method locates the switching point to a high degree of accuracy in one integration step while eliminating the need for a specified tolerance

Use of data-driven phase space models in assessing the strength of a bolted connection in a composite beam

This work explores the role of empirical dynamical models in deducing the level of preload loss in a bolted connection. Specifically, we examine the functional relationship between data gleaned from locations on either side of the connection using nonlinear predictive models. This relationship, as quantified by a measure of prediction error, changes as a function of bolt loosening, thus allowing both the presence and magnitude of the axial load to be identified. The models are based on a phase space portrayal of the system dynamics and require only that the structures response be low dimensional. The technique is demonstrated experimentally on a composite beam fastened to steel plates with four instrumented bolts. Results are compared to a similar approach using an auto-regressive (AR) modeling technique.

The stability of limit-cycle oscillations in a nonlinear aeroelastic system

The effects of a freeplay structural nonlinearity on an aeroelastic system are studied experimentally. Particular attention is paid to the stability of a periodic nonlinear aeroelastic response, known as limit–cycle oscillations (LCOs). The major thrust of this research lies in the application of relatively recently developed techniques from nonlinear dynamics and signal processing to the realm of experimental aeroelasticity. Innovations from the field of nonlinear dynamics include time–delay embedded coordinates to reconstruct system dynamics, a Poincaré section to assess the periodic nature of a response and to prescribe an operating point about which a linear description of the dynamics can be approximated, stochastic perturbations to assess the stability and robustness of responses, and a basin of attraction measure to assess initial condition dependence. A novel system–identification approach is used to generate a linear approximation of the experimental system dynamics about the LCO. This technique makes use of a rotating slotted cylinder gust generator and incorporates a least–squares fit of the resulting transient dynamics. An extension to this method is then developed based on the outcome of relatively large disturbances to the flow and hence airfoil, to obtain global stability.