James Ing

Experimental study of impact oscillator with one-sided elastic constraint

In this paper, extensive experimental investigations of an impact oscillator with a one-sided elastic constraint are presented. Different bifurcation scenarios under varying the excitation frequency near grazing are shown for a number of values of the excitation amplitude. The mass acceleration signal is used to effectively detect contacts with the secondary spring. The most typical recorded scenario is when a non-impacting periodic orbit bifurcates into an impacting one via grazing mechanism. The resulting orbit can be stable, but in many cases it loses stability through grazing. Following such an event, the evolution of the attractor is governed by a complex interplay between smooth and non-smooth bifurcations. In some cases, the occurrence of coexisting attractors is manifested through discontinuous transition from one orbit to another through boundary crisis. The stability of non-impacting and impacting period-1 orbits is then studied using a newly proposed experimental procedure. The results are compared with the predictions obtained from standard theoretical stability analysis and a good correspondence between them is shown for different stiffness ratios. A mathematical model of a damped impact oscillator with one-sided elastic constraint is used in the theoretical studies.

Intermittent control of coexisting attractors

This paper proposes a new control method applicable for a class of non-autonomous dynamical systems that naturally exhibit coexisting attractors. The central idea is based on knowledge of a systems basins of attraction, with control actions being applied intermittently in the time domain when the actual trajectory satisfies a proximity constraint with regards to the desired trajectory. This intermittent control uses an impulsive force to perturb one of the system attractors in order to switch the system response onto another attractor. This is carried out by bringing the perturbed state into the desired basin of attraction. The method has been applied to control both smooth and non-smooth systems, with the Duffing and impact oscillators used as examples. The strength of the intermittent control force is also considered, and a constrained intermittent control law is introduced to investigate the effect of limited control force on the efficiency of the controller. It is shown that increasing the duration of the control action and/or the number of control actuations allows one to successfully switch between the stable attractors using a lower control force. Numerical and experimental results are presented to demonstrate the effectiveness of the proposed method.

EXPERIMENTAL BIFURCATIONS OF AN IMPACT OSCILLATOR WITH SMA CONSTRAINT

In this paper we study bifurcations of an impact oscillator with one sided SMA motion constraint. The excitation frequency is used as a bifurcation parameter and two different values of the excitation amplitude are considered. It is shown that as frequency varies, the system exhibits highly nonlinear behavior. A typical bifurcation diagram has a number of resonance regions separated by chaotic motions with additional windows of periodic responses. The evolution of chaotic attractors is recorded experimentally, and changes in the structure of the attractors are shown. A mathematical model is developed and the results of the simulations are compared with the experimental findings. It is shown that the model is capable of accurately predicting not only the resonance structure but also the shape of the periodic and chaotic attractors. Numerical investigations also reveal a number of coexisting attractors at some frequency values. In particular, three attractors are found numerically for A = 0.2 mm and f = 29.474 Hz and their basins of attraction are presented. For A = 0.2 mm and f = 33.463 Hz, four coexisting attractors are found. For both parameter sets, one of the numerically detected attractors was validated experimentally. The undertaken analysis has shown that the hysteretic behavior of the restraint affected the dynamic responses only at the resonances, when the displacements are sufficiently large to trigger phase transformations in the SMA restraint. In nonresonant frequency ranges the restoring force in the SMA constraint is elastic. These findings are consistent with the numerical analysis carried out in [Sitnikova et al., 2008] for a similar system, which showed that the hysteretic behavior of the SMA affects resonant responses and provides a substantial vibration reduction in those regions.

Suppressing nonlinear resonances in an impact oscillator using SMAs

In this paper, we study the resonant responses of an impact oscillator with a one sided SMA motion constraint operating in the pseudoelastic regime. The effectiveness of the SMA restraint in suppressing nonlinear resonances of the impact oscillator is assessed by comparing the dynamic responses of the impact oscillator with SMA and elastic restraints. It is shown that the hysteretic behaviour of the SMA restraint provides an overall vibration reduction in the resonant frequency ranges. Due to the softening behaviour of the SMA element, the resonant frequencies for the SMA oscillator were found to be lower than for the oscillator with an elastic restraint. At each resonance, a single periodic response for the oscillator with the elastic restraint corresponds to two co-existing periodic responses of the SMA oscillator. While at the first resonance peak the emergence of one of the co-existing responses is associated with the hardening effect of the SMA restraint when the pseudoelastic force varies over a complete transformation cycle, at higher frequency resonances incomplete phase transformations in the SMA were detected for both responses. The experimental study undertaken verified the response-modification effects predicted by the numerical analysis conducted under the isothermal approximation. The experimental results showed a good quantitative correspondence with the mathematical modelling.

An experimental study into the bilinear oscillator close to grazing

A linear oscillator undergoing impact with a secondary support is studied experimentally. Smooth as well as nonsmooth bifurcations are observed. The bifurcations are explained with help from simulations based on mapping solutions between locally smooth subspaces. Experimental stability studies are also presented, justifying the normal form maps used to show the response close to grazing for systems with and without prestress of the secondary spring. The high degree of correspondence lends support to the modelling approach, and the highly complicated response justifies continued study of this system.

Severity Analysis of Stick-slip Bifurcation in Drill-string Dynamics under Parameter Variation

Bifurcations are an interesting class of dynamic behavior exhibited by many nonlinear systems. Drill string typically employed in underground oil exploration are highly nonlinear systems whose dynamic behavior has generated huge research interest. Though it is well-known that typical drill-strings behave like piecewise smooth systems and the severity of bifurcations they exhibit has not been studied in depth. In this work, a two degree of freedom (2-DOF) model of the drill-string is constructed and its dynamic behavior. In-depth analysis of the effect of parameter variation on the severity of bifurcations is conducted. This can potentially deliver key insights in the design of control strategies aimed at suppressing problematic stick-slip oscillation.

Delay-Induced Coexistence of Attractors in a Controlled Drill-String

The realistic modeling and effective control of drill-strings has been an ongoing research challenge. This has recently come to focus due to the volatility in the oil industry. Owing to the severely nonlinear nature of the drill-string, evolved nonlinear control techniques have recently been proposed to overcome the inherent stick-slip dynamics which are severely detrimental to the drilling performance as well as structural health of any given drill-string. Yet, most of the controller performance is analysed without including the significant delay intrinsic to the overall system. In this paper, the impact of system delay on the overall performance of a controlled drill-string is studied via extensive simulations. The analysis presents the impact of delay on three recently proposed sliding-mode control schemes. A surprising coexistence of attractors is observed from the delayed system on the third controller. This result will potentially impact the design of implementable control schemes proposed in future.

Through the looking-glass of the grazing bifurcation

It is well-known that for vibro-impact systems an existence of a periodic solution with a low-velocity impact (so-called grazing) may yield a complex behavior of solutions. Slightly changing the parameters of a system with grazing, we may obtain a periodic solution which passes near the delimiter without touching it. In this paper we show that sometimes existence of these solutions can provide chaos.

Complex Nonlinear Response of a Piecewise Linear Oscillator: Experiment and Simulation

In this work an experimental piecewise linear oscillator is presented. This consists of a linear mass-spring-damper undergoing intermittent contact with a slender beam, which can be modelled as providing stiffness support only. Experimental bifurcation diagrams are presented in which a complex response is observed. Smooth bifurcations are recorded, but more typically rapid transitions between attractors were observed. All of the following are shown to occur: coexisting attractors, basin erosion associated with grazing trajectories, pairs of unstable periodic orbits, chaotic response, loss of stability through grazing and saddle node bifurcations. All of these come into play to generate the observed experimental bifurcation scenarios. It is shown that a global analysis is important in understanding the system response, especially close to grazing conditions. Although grazing is known to cause a local change in stability, it was found that more often grazing of one orbit would cause a change in the other orbits which resulted in boundary crisis and annihilation of the attractor.

Dynamics of the archetypal piecewise linear oscillator close to grazing

Abstract A linear oscillator undergoing impact with a secondary support is studied numerically. Smooth as well as nonsmooth bifurcations are observed. The bifurcations are explained with help from simulations based on mapping solutions between locally smooth subspaces. The obtained results are verified by comparison with previous experimental studies. Experimental stability analysis is also presented, justifying the normal form maps used to show the response close to grazing for systems with and without prestress of the secondary spring. The high degree of correspondence lends support to the modelling approach, as does the close match between experiments and simulations.