Ekaterina Pavlovskaia

Piecewise linear approach to an archetypal oscillator for smooth and discontinuous dynamics.

In a recent paper we examined a model of an arch bridge with viscous damping subjected to a sinusoidally varying central load. We showed how this yields a useful archetypal oscillator which can be used to study the transition from smooth to discontinuous dynamics as a parameter, α, tends to zero. Decreasing this smoothness parameter (a non-dimensional measure of the span of the arch) changes the smooth load–deflection curve associated with snap-buckling into a discontinuous sawtooth. The smooth snap-buckling curve is not amenable to closed-form theoretical analysis, so we here introduce a piecewise linearization that correctly fits the sawtooth in the limit at α=0. Using a Hamiltonian formulation of this linearization, we derive an analytical expression for the unperturbed homoclinic orbit, and make a Melnikov analysis to detect the homoclinic tangling under the perturbation of damping and driving. Finally, a semi-analytical method is used to examine the full nonlinear dynamics of the perturbed piecewise linear system. A chaotic attractor located at α=0.2 compares extremely well with that exhibited by the original arch model: the topological structures are the same, and Lyapunov exponents (and dimensions) are in good agreement.

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.

Piecewise approximate analytical solutions for a Jeffcott rotor with a snubber ring

In the paper two approximate analytical methods for calculating nonlinear dynamic responses of an idealised model of a rotor system are devised in order to obtain robust analytical solutions, and consequently speed up the computations maintaining high computational accuracy. The physical model, which is similar to a Jeffcott rotor, assumes a situation where gyroscopic forces can be neglected and concentrates on the dynamic responses caused by interactions between a whirling rotor and a massless snubber ring, which has much higher stiffness than the rotor. The system is modelled by two second-order differential equations, which are linear for non-contact and strongly nonlinear for contact scenarios. The first and the simpler method has been named one point approximation (1PA) and uses only one point in the first-order Taylor expansion of the nonlinear term. It is suitable for soft impacts and gives a reasonable prediction of responses ranging from period one to period four motion. The second and more accurate method of multiple point approximation (MPA) expands the nonlinear term many times when the rotor and the snubber ring are in contact and it can even be used for calculating chaotic responses. The methods are evaluated by a comparison with direct numerical integration showing an excellent level of accuracy.

Modelling of Ground Moling Dynamics by an Impact Oscillator with a Frictional Slider

This paper describes current research into the mathematical modelling of a vibro-impact ground moling system. Due to the structural complexity of such systems, in the first instance the dynamic response of an idealised impact oscillator is investigated. The model is comprised of an harmonically excited mass simulating the penetrating part of the mole and a visco-elastic slider, which represents the soil resistance. The model has been mathematically formulated and the equations of motion have been developed. A typical nonlinear dynamic analysis reveals a complex behaviour ranging from periodic to chaotic motion. It was found out that the maximum progression coincides with the end of the periodic regime.

Analytical drift reconstruction for visco-elastic impact oscillators operating in periodic and chaotic regimes

An impact oscillator with a drift, which is important in many applications, is considered. The model accounts for the visco-elastic impacts and is capable to mimic the dynamics of a bounded progressive motion. To simplify the dynamic analysis a simple transformation decoupling the original co-ordinates is proposed. As the result the bounded oscillations can be studied separately from the drift as the drift does not influence the dynamics of the bounded system. On the contrary the drift depends on the bounded dynamics and can be reconstructed once the bounded oscillatory motion is determined. The accuracy of the analytical reconstruction allows to calculate even strange chaotic attractors. Evolutions of co-existing periodic and strange attractors were studied.

Bifurcation analysis of a preloaded Jeffcott rotor

Abstract A model of two-degrees-of-freedom Jeffcott rotor system with bearing clearance subjected of an out-of-balance excitation is considered. The influence of preloading and viscous damping of the snubber ring is introduced in the mathematical description. A programme of numerical simulations is conducted to show how the preloading and viscous damping change the dynamics of the rotor system. Bifurcation diagrams and Lyapunov exponents are constructed to explore stability. It is shown that dynamics of the rotor system can be effectively controlled by varying the preloading and the damping both of the rotor and the snubber ring. In the most considered cases preloading stabilises the dynamic responses.

BIFURCATION CONTROL OF A PARAMETRIC PENDULUM

In this paper, we apply chaos control methods to modify bifurcations in a parametric pendulum-shaker system. Specifically, the extended time-delayed feedback control method is employed to maintain stable rotational solutions of the system avoiding period doubling bifurcation and bifurcation to chaos. First, the classical chaos control is realized, where some unstable periodic orbits embedded in chaotic attractor are stabilized. Then period doubling bifurcation is prevented in order to extend the frequency range where a period-1 rotating orbit is observed. Finally, bifurcation to chaos is avoided and a stable rotating solution is obtained. In all cases, the continuous method is used for successive control. The bifurcation control method proposed here allows the system to maintain the desired rotational solutions over an extended range of excitation frequency and amplitude.

Modelling of vibro-impact system driven by beat frequency

Abstract A mathematical model of vibro-impact system accounting for oscillatory and progressive motion, and capable of transferring a high-frequency low-amplitude excitation into low-frequency high-amplitude response is developed. A special beat frequency kinematic excitation was used, which has two distinctive features: (i) the low-frequency modulated excitation is tuned to the natural frequency of the oscillating system, and (ii) the excitation is asymmetric. The model considers also visco-elastic properties of the media. It is demonstrated that this mechanism allows to overcome the resistance force of the media and to move forward. Several different ways to achieve a steady progression without supplying additional energy are explored, however, in all these cases progression rates are relatively low. A significant increase of progression rates is only possible by controlling the motion of the system. A simple control strategy enhancing progression rates substantially is proposed and implemented.

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.