Torsional and lateral vibrations in flexible rotor systems with friction

Self-sustained vibrations which can appear in mechanical systems often limit the performance of such systems or can even cause failure or damage to such systems. Moreover, different types of vibration can appear in dynamical systems. In order to gain an improved understanding and to predict different types of vibrations which appear in mechanical systems, it is important to understand the causes for such vibrations and the interaction between those vibrations. In this thesis, we address on the one hand friction-induced vibrations in flex- ible mechanical systems, and on the other hand lateral vibrations caused by mass- unbalance in rotor systems, and the interaction between those two types of vibrations. Although a lot of theoretical research has been done on vibrations in flexible rotor systems, a limited number of papers is available which include experimental results on the friction-induced vibrations and on the interaction between different types of vibrations. For this purpose, we have designed and constructed an experimental drill-string set-up which exhibits both types of vibration. The set-up consists of a DC-motor, two rotating (upper and lower) discs, a low-stiffness string, which connects the two discs, and an additional brake at the lower disc. The lower disc can rotate around its geometric center and is also free to move in lateral direction. The configuration of the experimental set-up is representative for many other mechanical systems, in which friction or unbalance can deteriorate the system performance by the appearance of vibrations. For example, when the lower disc is fixed in lateral direction (i.e. when the lower disc only rotates), the system forms a configuration of two masses, coupled by a flexibility, of which one is subject to friction and the other is driven by an actuator. In this context, one can think of set-ups such as printers, pick and place machines, industrial and domestic robots, braking mechanisms and many others. Moreover, when mass-unbalance is present at the lower disc and the disc can move in lateral direction, this configuration can be recognized in drilling systems which are used for exploration of oil and gas, in electrical shavers, in various turbines, pumps, fans and so on. The drill-string set-up is modelled and the parameters of the model are estimated. The comparison between responses of the experimental set-up and estimated model indicates a high accuracy of the obtained parameter estimates. The steady-state behaviour of the drill-string system has been analyzed when various constant input voltages are applied to the DC motor; first, when only torsional and no lateral vi- brations occur and, second, when both torsional and lateral vibrations appear in the system. When analyzing the friction-induced vibrations a discontinuous static friction model is used. We have chosen such a model and not a more complicated dynamical friction model since it accounts for the friction characteristics which are crucial for the global dynamics of the system but avoids unnecessary complexity. A discontin- uous friction model leads to a discontinuous model of the system dynamics which exhibits both friction-induced vibrations and the interaction between friction-induced vibrations and vibrations due to mass-unbalance. As a result of the analysis on a theoretical, numerical and experimental level the following conclusions are drawn. When analyzing the set-up with only friction-induced torsional vibrations and no lateral vibrations, the main conclusion is that a subtle interplay of negative damping characteristics at low velocities and viscous friction at higher velocities determines the occurrence and nature of friction-induced limit cycling. It also determines the range of parameters for which these limit cycles sustain. Furthermore, the level of posi- tive damping at very low velocities relative to the negative damping level at slightly higher velocities determines whether torsional vibrations with or without stick-slip can occur. Then, both model-based and experimental bifurcation analysis confirm that discontinuous bifurcations play a crucial role in the creation and disappearance of these limit cycles. Also, the way in which such friction characteristics are influ- enced by physical conditions such as temperature and normal forces on the frictional contact is experimentally studied. An important observation is that the normal force in the frictional contact influences the friction force in a rather complex way and can induce a higher negative damping level (for larger normal forces), which in turn can give rise to limit cycles of a larger amplitude for a larger range of constant input voltages to the DC motor. The analysis of the set-up, when both torsional and lateral vibrations are present, leads to the main conclusion that two types of torsional vibrations can appear. Firstly, friction-induced torsional vibrations and, secondly, torsional vibrations due to cou- pling between torsional and lateral dynamics may appear. Furthermore, if mass- unbalance is present at the lower disc, the amplitude of friction-induced vibrations and the region in which these vibrations occur, both decrease compared to the situ- ation without mass-unbalance. Moreover, it is shown that if the mass-unbalance is large enough then torsional vibrations can disappear entirely. Next, on a simulation level it is shown that torsional vibrations due to coupling between torsional and lateral modes appear for input voltages to the DC motor which are higher than the so-called critical voltage, which is related to the critical angular velocity inducing resonance in lateral direction. Due to limitations in the available DC motor, those vibrations are studied only at a simulation level. Finally, the knowledge obtained in this thesis provides a better understanding of the causes for torsional and lateral vibrations. Moreover, based on this knowledge, various control strategies may be designed and tested on the designed set-up in or- der to eliminate torsional and lateral vibrations. Furthermore, the results presented here can support the design of various braking mechanisms, pumps and fans in pre- venting the occurrence of or in decreasing the amplitude of friction-induced torsional vibrations and lateral vibrations due to mass-unbalance.