Effects of internal viscous damping on the stability of a rotating shaft driven through a universal joint

Abstract

A rotating flexible shaft, with both external and internal viscous damping, driven through a universal joint is considered. The mathematical model consists of a set of coupled, linear partial differential equations with time-dependent coefficients. Use of Galerkin's technique leads to a set of coupled linear differential equations with time-dependent coefficients. Using these differential equations some effects of internal viscous damping on parametric and flutter instability zones are investigated by the monodromy matrix technique. The flutter zones are also obtained on discarding the time-dependent coefficients in the differential equations which leads to an eigenvalue analysis. A one-term Galerkin approximation aided this analysis. Two different shafts (“automotive” and “lab”) were considered. Increasing internal damping is always stabilizing as regards to parametric instabilities. For flutter type instabilities it was found that increasing internal damping is always stabilizing for rotational speeds v below the first critical speed, v 1. For v> v 1, there is a value of the internal viscous damping coefficient, C iv , which depends on the rotational speed and torque, above which destabilization occurs. The value of C iv (“critical value”) at which the unstable zone first enters the practical range of operation was determined. The dependence of C iv critical on the external damping was investigated. It was found for the automotive case that a four-fold increase in external damping led to an increase of about 20% of the critical value. For the lab model an increase of two orders of magnitude of the external damping led to an increase of critical value of only 10%. For the automotive shaft it was found that this critical value also removed the parametric instabilities out of the practical range. For the lab model it is not always possible to completely stabilize the system by increasing the internal damping. For this model using C iv critical, parametric instabilities are still found in the practical range of operation.