Modelling and analysis of gyroscopic effects in undamped rotating systems

Abstract

The gyroscopic effect has always been an interesting phenomenon in the field of rotor dynamics owing to its unique and complicated results on the dynamics of rotating systems. The finite element method is commonly used in this field to model and analyse this effect. This method is based on developing discrete mathematical models of rotating finite elements and using them to discretise rotating structures. Structures with complex geometries are usually discretised using finite solid or shell elements while simpler systems composing of flexible shaft and rigid discs or isolated blades are discretised using finite beam elements. Despite the accuracy provided by solid/shell elements for complex structures, the requirement of significant modelling and computational effort becomes a drawback for this modelling technique. On the contrary, for simpler structures, beam elements provide satisfactorily accurate results with relatively less effort put into the modelling and computation stages. Using these relatively simpler models, the gyroscopic effects generated by the flexible rotating structures can be modelled and the outcomes of this phenomenon on the dynamics of a rotor can be investigated effectively. In this thesis, it is aimed to include the gyroscopic effects in the models of rotating flexible shaft-disc-blade systems and analyse the results using finite beam elements. Besides the conventional beam element model of rotating shafts and blades, a flexible disc structure in the form of a so-called spiderweb is proposed. By doing so, a fully flexible rotating shaft-disc-blade system's dynamics under the gyroscopic influence is examined with relatively less degrees of freedom. Moreover, the effects of disc and blade flexibility on the dynamics are discussed using the proposed disc model. Considering the lack of capabilities of some popular computer programs for modelling the gyroscopic effects of rotating beam elements, it is also aimed to contribute to this field with the models presented throughout this thesis. A comprehensive literature survey summarises the development of the rotor dynamics field over the years and reveals modern studies' focal points in the dynamics of rotating systems and the methods they employ. Furthermore, the place and the importance of the gyroscopic effects and finite element method in the literature is emphasised. The gyroscopic effect is caused by the gyroscopic moments in stationary reference frame, or the Coriolis forces in rotating reference frame. Although at first glance they seem different, they are directly related to each other over the Coriolis acceleration. It is noticed that the relation between these two cases is not sufficiently explained in the the rotor dynamics literature, therefore, the generation of the gyroscopic moments and the Coriolis forces, as well as the relation between these two is explained and clarified in some detail in the second chapter of the thesis. The gyroscopic effect's distinctive feature is its ability to generate complicated results on the dynamic behaviour of rotating objects. The gyroscopic effect is a speeddependent phenomenon and causes the system's dynamics to change with changing rotational speed. This leads to the natural frequencies of the system to become speeddependent as well. Some mode shapes, can also be highly influenced by the gyroscopic effects. Therefore, the gyroscopic effects necessitate to carry out the analyses at a range of rotational speeds. The gyroscopic effect introduces complexity to the mode shapes of rotating systems, which in return manifests itself as whirling motion of shafts and occurrence of travelling waves on discs. With these radical changes in the dynamics of rotors, the gyroscopic effect is accepted to be crucial to be accounted for in modelling and analysis of rotating structures. In the modelling stage, first, the conventionel model of flexible shaft-rigid disc system is presented. The flexible shaft is discretised using Timoshenko beam elements with 8 degrees of freedom. The dynamics of these elements, as well as the rigid disc's, are expressed in stationary reference frame while taking the gyroscopic effects caused by the gyroscopic moments into account. Later, a more general Timoshenko beam element with 12 degrees of freedom is presented. This element is used to discretise flexible shaft-disc-blade system. Due to the nonaxissymmetric geometry of the system, the modelling is carried out in rotating reference frame. Therefore, the gyroscopic effect is taken into account by modelling the Coriolis forces. Owing to the orientation difference between a shaft and a blade with respect to the axis of rotation, separate beam element models are presented and used for modelling shafts and blades. Although the formulation of structural stiffness and mass matrices are the same, the gyroscopic and spin softening matrices differ between the shaft and the blade elements. In order to account for the disc flexibility, a special disc model in the form of a socalled spiderweb is presented. This structure is composed of radial and connecting elements, both of which use the same mathematical model with the blade elements. Finally, an 8-noded isoparametric hexahedral solid element and the derivation of its matrices are presented. With the models developed, a flexibe shaft-rigid disc system's dynamics is analysed first. The free vibration problem of the system is solved at different rotational speeds, and the natural frequencies and mode shapes are predicted. The gyroscopic effect is observed to vary with the order of the shaft's bending vibrations. As expected, depending on the location of the disc on the shaft, some modes experience strong gyroscopic effect generated by the disc. It is seen that the natural frequencies split almost linearly with increasing rotational speed, and there, naturally, occurs no shaftdisc interactions because of the disc's rigidity. With the intention to investigate the gyroscopic effects in rotors, flexible shaft-flexible disc system, flexible shaft-disc-blade system and flexible bladed disc systems are also analysed. In these models, the so-called spiderweb disc is used to model a flexible disc, and the blades are intentionally oriented perpendicular to the disc in order to enchance the gyroscopic coupling in rotating reference frame. The natural frequencies in rotating reference frame and the mode shapes of the systems are predicted at different rotational speeds. Accordingly, Campbell diagrams are utilised to present the natural frequencies as a function of rotational speed. When the blades are attached, it is seen that the system dynamics gets tremendously complicated. The complex mode shapes of the shaft and the disc are presented by visualising the vibration patterns of the nodes on the bodies in various modes. It is seen that the spiderweb disc is capable of exhibiting the occurance of travelling waves on the disc. Taking the disc's flexibility into account is found to be essential in order to be able to predict shaft-disc interactions and the so-called veering phenomenon in several mode shapes of different systems. Moreover, with the proposed model, a merging type of instability is predicted in a flexible shaft-disc-blade system. Following the analyses of rotor models using beam finite elements, a solid bladed disc model is developed using 8-noded solid elements. The results of this model are used to predict how a solid bladed disc system behaves under the rotational effects. Then, the results are compared with those corresponding to previous systems modelled using beam elements. It is found that both types of models yield quite similar results in terms of the frequency splits caused by the gyroscopic effect, spin softening effect in blade dominated modes and the occurance of travelling waves on the disc. This thesis investigated the complicated and interesing nature of the gyroscopic effects in rotating systems using various models based on the finite element approach. Besides the gyroscopic effect's generation mechanism, the relationship between the gyroscopic moments in stationary reference frame and the Coriolis forces in rotating reference frame are explained in some detail. By proposing a spiderweb disc structure discretised using finite beam elements, a cost-effective model of flexible shaft-disc-blade system is obtained. It is concluded that this system can provide accurate results and insight into the dynamics of rotating simplified shaft-disc-blade systems which can be highly influenced by rotational effects such as the gyroscopic effects.