Analysis of synchronous and asynchronous whirl of jeffcott rotor considering material non-linearity

Abstract

Whirling is defined as the rotation of the plane made by the bent shaft and the line of centers of the bearing. The phenomenon of whirling of shaft is often explained in terms of a “Jeffcott rotor model”. The Jeffcott rotor consists of a simply supported flexible shaft with a rigid thin disc mounted at the mid-span. Whirling results from various causes such as: a) Mass eccentricity (rotating unbalance), b) Lack of initial straightness of the shaft, c) Non-homogenous material, d) Unbalanced magnetic pull in case of electrical machinery e) Lubricant viscosity, f) Shaft material’s initial stiffness and number of supports, g) Unbalanced centrifugal forces, h) Hysteresis characteristics of shaft materials etc.

In this thesis whirling is extensively studied considering eccentric mass centre of the rotor on the shaft. Generally, two modes of whirling (synchronous and asynchronous) can be observed in various rotating machines. In the synchronous motion of the shaft, the whirl (also called orbital) speed and its own spin speed are equal. However, in case of asynchronous whirl motion of the shaft, the orbital speed and its own spin speed are not equal.

Previous studies of Jeffcott rotor mostly dealt with synchronous whirl considering steady-state vibration and involving linearly elastic shaft materials. Therefore, present thesis aims to focus following unexplored but important points concerning whirling of shafts: asynchronous mode of whirl, whirl during transient vibration and, the effect of material non-linearity (that is, the shaft material has a non-linear stress-strain relation and Hook’s law cannot be applied) on the shaft’s response.

To accomplish the goals, at first, a new theoretical model has been developed to predict response of shafts during steady-state whirl (both synchronous mode as well as asynchronous mode) in terms of exact solutions. Thus, differences between the shaft responses are analyzed for synchronous and asynchronous modes of whirl during steady-state vibration.

Secondly, a new mathematical model has been developed to predict response of shafts during unsteady-state (transient) whirl in terms of numerical solutions. Original non-linear second order governing differential equations are solved as an initial value problem.

Next, results from transient solutions and exact solutions are compared.

Thirdly, material non-linearity issue is handled in terms of a mathematical model that has been developed to solve pure bending of a shaft (that is whirling), the material of which does not follow Hooke’s law. It should be noted here that, SMA inherently has highly non-linear stress-strain curves in tension and compression. Thus, bending moment-curvature and reduced modulus-curvature relations are obtained for a superelastic SMA shaft. Results are used to predict effect of material non-linearity on the response of a whirling shaft.

Effect of various other factors like damping ratio, eccentricity ratio, whirl speed ratio and spin ratio on response of whirling shafts are studied and analyzed. Some salient findings are as follows.

The Jeffcott rotor system behaves as a single degree of freedom system. For steady-state and synchronous whirl, increased damping greatly reduces whirling amplitude and its maximum value is at spin ratio of unity, because of resonance. However, for large value of spin ratio, non-dimensional whirling amplitude approaches unity. Another interesting finding is that, all load-spin ratio curves intersect at a spin ratio of 1.414.

Interestingly, for asynchronous whirl, resonance does not occur at a spin ratio of unity. Rather, it is the whirl speed ratio that determines at what speed resonance will occur.

As material non-linearity is taken into account some portion of the shaft is found to experience stresses (different magnitude in tension and compression) beyond proportional limit. Effect of material non-linearity becomes prominent as whirl amplitude starts to become large. Moreover, the maximum deflection of a shaft is found to be much larger when material non-linearity is considered in comparison to the case of linearly elastic shaft material.

When, unsteady-state whirl of Jeffcott rotor is considered,the peak value of whirling amplitude is found to be almost equal to that for steady-state whirl. In turn, it proves soundness of the entire mathematical scheme because steady-state solutions are exact (obtained analytically from the simplified mathematical model) and transient solutions are obtained by numerical method from the original non-linear governing equations. Beyond resonance, whirl amplitude increases with the spin ratio. Also increased eccentricity increases the whirl amplitude. However, peaks at resonance disappear with increasing damping ratio.

Finally, an experimental setup is constructed and distinct whirling is demonstrated at and above resonance. Material non-linearity and inelastic behavior of shaft is demonstrated by permanently bent shaft due to whirling. Experimental observations are explained in terms of mathematical predictions.

Key words: Jeffcott rotor, Whirl, Synchronous whirl, Asynchronous whirl, Material non-linearity, Steady-state whirl, Unsteady-state whirl, Threshold bending moment, Reduced modulus, Transient solution.