Investigation of properties of some coupled nonlinear oscillators

Interaction between nonlinear oscillators are investigated in this thesis. Subject covers vast area from chemical reactions,the movements of eyes of some reptiles to moving styles of horses and the effect of earthquakes to the buildings. Especially the abrupt changing of behaviour of nonlinear coupled oscillators as a function of some parameters and appearance of unstability of some modes of motion and stability of others (bifurcation) are subject of great concern. For example the way motion of horses change as a function of their velocity. As their velocity increase , at some critical velocity , normal walking horses abruptly start trotting. With further increase in velocity horses stop trotting and start galloping. At this moment it is said that trotting loses stability and galloping are stabilized. This motion can be analysed as a coupling of four nonlinear oscillator (Jacobus, 2001).Interaction between only two nonlinear oscillators are investigated in this thesis. Transfer of energy from one coupled oscillators to other and increasing of amplitude of other oscillator due to this energy tranfer and becoming unstable is investigated. Energy transfer is calculated by Energy-Rate Method (Jazar, 2008).In order to make one way only energy transfer possible , nonlinear coupled asymmetric oscillators are taken into consideration. Effect of parametric and normal resonances and their influence on energy transfer various different nonlinear couplings are investigated by chosing different coupling mechanism and change of amplitude and velocity of each oscillator in time are calculated by numerical integration. This amplitudes and velocities are used to calculate energy transfer in some parts of (w ,b) frequency-amplitude space. Positive values of energy transfer correspondes to unstable and the rest is stable part of this space. For every coupled oscillator taken into consideration the effect of nonlinearity , forcing and parametric resonance are interpreted.In the last example of nonlinear coupled oscillator , an oscillator of constant frequency coupled to an oscillator of soft nonlinearity are investigated as a model ofnonlinear interaction between eartquake and a building. Transfer of energy from one oscillator (eartquake) to the other (building) are calculated. In the model frequency of building are chosen a lithe higher then eartquake?s and frequency of building is assummed to decrease as its amplitude increases.For this assumptions for forcing frequency ( w ) and amplitude ( b ) stable and unstable areas in ( w ,b) space are determined by calculating energy transfer integral. Interaction between oscillators show dominant parametric resonance features. This is also worked out from analitical analysis of differential equation of the system under consideration.To investigate the nature of motion of oscillator (building) as one crosses in to unstable region from stable one , an arbitrary ( w = 0.5 ) forcing frequency are chosen and change of amplitude in time of forced oscillator for different forcing amplitudes ( b = 0.133 ve b = 1.225 ) are calculated. From analysis of these results it is found that , as it is moved nearer to unstable region amplitude increases and frequency of oscillation decreases and at the border it becomes unstable. It is a reminecent of period doubling of parametric resonans.In further work other cases of earthquake-building interaction and development of more general model is planned.