Dynamic stability of a flexible spring-mounted shaft driven through a universal joint.

Abstract

Dynamic stability of a rotating shaft driven through a universal joint is investigated. The system is composed of a driving shaft and a driven shaft connected by a universal joint. Due to the characteristics of the joint, even if the driving shaft experiences constant torque and rotational speed, the driven shaft experiences fluctuating rotational speed, bending moments and torque. These are sources of potential parametric, forced and flutter type instabilities. The focus of this work is on the lateral instabilities of the driven shaft. Two distinct models are developed for the shaft, namely, a rigid body model and a flexible model. The driven shaft is taken to be pinned at the joint end and to be resting on a compliant bearing at the other end. Both models lead to sets of differential equations with time dependent coefficients. In the flexible case Galerkin's method is used to solve for the spatial dependence of the equations. Stability charts are derived by using the monodromy matrix technique. The results show that for lightly damped shafts, flutter instabilities occur outside the range of practical operational conditions. The transmitted bending moments caused strong parametric instabilities (instability zones of significant size) in the system. By comparing the results from the two models, it is shown that the inclusion of flexibility leads to new zones of instability, not predicted by previous models. These zones, depending on the physical parameters of the system, can occur for practical conditions of operation. It is also shown that outside the parametric instability zones (due to the homogeneous part of the equations) forced instabilities are possible, whereas in certain portions of the zones these instabilities were found to exist. Two procedures are considered to minimize or eliminate the lateral instabilities in the system, namely, an increase in bearing damping and an increase in bearing stiffness.