Nonlinear vibrations produced by unbalanced motors

Abstract

The present thesis is concerned with the nonlinear dynamics of vibrating systems excited by unbalanced motors. The main focus is the reciprocal (nonideal) interaction which in general exists between the dynamics of the exciter –the unbalanced motor– and that of the vibrating system. Two models were analytically and numerically studied. First, a 2DoF model of a general structure with a cubic nonlinearity, excited by a nonideal motor, was analysed in detail. The second model is a 3DoF simplified representation of the process of vibrocompaction of quartz agglomerates. The first model requires different treatments depending on the order of magnitude of the slope of the motor characteristics. Then, two cases were considered separately: large and small slope. For the first scenario, a new analytical approach was developed, which combines two well – known perturbation techniques: the Averaging Method and the Singular Perturbation Theory. This scheme allows uncovering the system dynamics as composed of three consecutive stages of time. The first two ones occur in a short time scale and can be considered as a fast transient regime. During the third stage, the system dynamics was shown to be well – represented by a reduced 2D system. A detailed analysis of this reduced system allowed obtaining its fixed points and their stability. As a very relevant outcome of the stability analysis, conditions were found for the existence of a Hopf bifurcation, which had not been addressed before in the literature, to the author’s knowledge. This result is particularly significant, for it shows that the stability region of a stationary motion of the system can be smaller than predicted by usual theories. Thus, not taking the Hopf bifurcation into account may lead to unexpected instabilities in real applications. The Hopf bifurcations were analytically investigated and very simple conditions were derived to characterize them as subcritical and supercritical. Moreover, by using the Poincaré – Béndixson theorem, conditions were found under which all trajectories of the reduced system are attracted towards a limit cycle. This kind of motion in the reduced system corresponds to a quasiperiodic oscillation in the original one. The global bifurcations whereby the found limit cycles disappear were numerically analysed, finding homoclinic and saddle – node homoclinic bifurcations. All these results were validated by comparing numerical solutions of the original and reduced systems, which exhibited a remarkable accordance. The case of small slope was also analytically studied in detail. Having found the existence of a resonance manifold in the phase space, the regions far (outer) and close (inner) to the resonance manifold were separately investigated through averaging techniques. Under certain conditions, the inner region contains two fixed points, whose stability was analysed. As an apparent limitation of the procedure, it was addressed that the time of attraction of one the fixed points was much longer than the time of validity of the averaged system. Consequently, it is not obvious whether or not the stability of that fixed point in the averaged system is necessarily the same as in the original system. The main contribution of this part of the thesis consists in having proved, by using attraction arguments, that solutions of the averaged system near the fixed point of interest are actually valid for all time, thereby solving the above difficulty. The existence of a stable fixed point in the resonance region justifies the possibility of resonance capture. As in the case of large slope, numerical simulations were conducted in order to compare solutions of the original and averaged systems. A good agreement was also found in all the considered scenarios. The final part of the thesis considered a real industrial process, where a mixture of granulated quartz and polyester resin is compacted by using a piston with unbalanced electric motors. A simplified 3DoF model of the process was built, including the nonideal coupling between the motor and the vibrating system, impacts and separation between the piston and the quartz slab and a nonlinear constitutive law for the mixture which models the compaction itself. Although the model is not complex enough to give reliable quantitative results, it is a first step towards the construction of more sophisticated models which are able to predict the behaviour of actual compacting machines. Interestingly, it has been shown that the torque – speed curves obtained in the first chapters of the thesis can also be applied, in an approximate way, to the vibrocompaction model. These curves allow predicting whether a particular set of parameters for the process will give an efficient compaction of the mixture. Several simulations have been conducted, illustrating how such a model could be used to understand the effect of each parameter in the final result of the process.