Theoretical modelling, analysis and validation of the shaft motion and dynamic forces during rotor–stator contact

Abstract

This paper deals with the theoretical study of a horizontal shaft, partially levitated by a passive magnetic bearing, impacting its stator. Rigid body dynamics are utilised in order to describe the governing nonlinear equations of motion of the shaft interacting with a passive magnetic bearing and stator. Expressions for the restoring magnetic forces are derived using Biot Savart law for uniformed magnetised bar magnets and the contact forces are derived by use of a compliant contact force model. The theoretical mathematical model is verified with experimental results, and shows good agreements. However, the simulated contact forces are higher in magnitude compared to the experimental results. The cause of this disagreement is addressed and shows that the formulation of the theoretical contact force model slightly overestimates the forces acting during a full annular backward whirl motion.

Introduction

Impact is a complex physical phenomenon, which occurs when two or more bodies collide with each other. The contact implies a continuous process which takes place over a finite time. Gilardi and Sharf [1] give a comprehensive literature survey on contact dynamics and discuss various contact models. Two different approaches for impact and contact analysis exist. The first approach is referred to as impulse-momentum method. The method is confined primarily to impact between rigid bodies. Hence, assumes that the interaction between the objects happens in a short period of time and that the configuration of the impacting bodies does not change significantly. For such a model, the dynamical analysis is divided into two intervals, before and after impacts. In order to assess the process of energy transfer and dissipation between the two bodies, various coefficients are employed. The primary coefficients are the coefficient of restitution and the impulse ratio. However, the application of the impulse-momentum method leads to several problems. The main problem is that energy conservation principles may be violated during frictional impacts, and the discrete approach is not extendible to generic multibody systems. The second method for impact analysis is by use of compliant, continuous, contact force models. These models overcome the difficulties associated with the impulse-momentum method. In the continuous contact model the forces and deformations vary and act in a continuous manner during impacts. Different continuous contact models have been proposed to describe the interaction force at the surfaces of two contacting bodies. The first model was developed by Hertz [2] and based on the theory of elasticity for frictionless contact, to calculate indention without the use of damping. The local impact stiffness in this model between the two colliding bodies depends on the material properties, geometric properties and computed by using elastostatic theory. This model is only applicable for contact between rigid bodies where no energy dissipation takes place. To include damping in the contact modelling a simple spring-damper model was proposed, where the contact force is represented by a linear spring-damper element such as given in the Kelvin–Voigt model. The impact is schematically represented with a linear spring in conjunction with linear damping, accounting for the energy dissipation. However, this model has some weaknesses. The contact force at the beginning of impact is discontinuous due to the damping term. At separation the local indention tends to zero where the relative velocity between the two bodies becomes zero. This induces a negative force keeping the bodies together. Furthermore, the coefficient of restitution is independent on the impact velocities. To overcome these problems of the spring-damper model and to retain the advantages of the Hertz's model, Hunt and Crossley [3] proposed a contact model. This model includes a nonlinear damping term defined in terms of local penetration and the corresponding rate. This model satisfies the force boundary conditions during impact and separation and gives a correct description of the contact force behaviour. The energy dissipation in this model is related to the ingoing and outgoing velocities. Another important aspect is that the damping depends on the local indention $\delta$. This has its advantages since a contact area increases with deformation and a plastic region is more likely to develop for large indention. A further development of this model was proposed by Lankarani and Nikravesh [4]. In this model a hysteresis damping function is incorporated to represent the energy dissipation during impacts.

The different contact force approaches outlined above have been employed by many researchers on the study of rotor to stator contact. Yanabe et al. [5] numerically studied the collision of a rotor with an annular guard during the passage through a critical speed. Their results were based upon two different approaches; the collision theory where the law of conservation of momentum and the coefficient of restitution were used, and the compliant contact force approach where they assumed the contact force to behave as a linear spring with no damping. Popprath and Ecker [6] numerically studied the dynamics of a Jeffcott-rotor having intermittent contact and interacting with its stator. The stator was implemented as an additional vibratory system and the contact forces were modelled by use of a spring-damper model accounting for the energy dissipation during impact. Their results were studied through Poincaré maps and bifurcation diagrams, where they discovered that the damping parameter of the stator suspension has a significant influence on the type of motion obtained for the combined rotor–stator system. In the end of the nineties the application of the Hunt and Crossley's compliant contact force model started to become employed in the rotor to stator contact force modelling. Both Fumagalli [7] and Bartha [8] made use of this compliant contact force model in their theoretical studies. This contact model has also been employed in recent years in the research work of Childs and Bhattacharya [9], Wilkes et al. [10] and lately in the research work of Childs and Kumar [11]. The extension of the Hunt and Crossley contact force model proposed by Lankarani and Nikravesh is employed in this paper. This contact force model has been utilised extensively in the work of Flores et al. [12], [13], [14], [15] for the study of mechanical systems with revolute joints and in the work of Lahriri et al. [16] for the rotor to stator contact modelling. Yet, the correct description of the contact stiffness and damping parameter is a difficult task since these are functions of the material properties, wear and damages and surface temperature. These parameters have an effect on the contact force behaviour and magnitude. Different researchers have experimentally investigated the rotor motion and behaviour during contact with its stator. The investigations revealed that the rotor can exhibit complicated vibration phenomena and the rotating system showed a rich class of nonlinear related dynamics. Among these, Chu and Lu [17] experimentally demonstrated the existence of fractional sub-harmonic vibrations, multiple super-harmonic vibrations and even chaotic motions during rotor rub and impacts.

The main original contribution of this work relies on experimental validation of a nonlinear contact force model, taking into account compliance, dry friction, penetration rate and penetration velocity during a contact event. Using (1) a special test rig completely monitored by force transducers, displacement sensors and accelerometers and (2) a theoretical model derived by multiphysics, it possible to compare theoretical and experimental values of the dynamic contact forces during a full annular contact state. In light of these results, the theoretical employed compliance contact force model is validated.