Critical Speeds of a Rotor With Unequal Shaft Flexibilities, Mounted in Bearings of Unequal Flexibility—I

Abstract

Abstract This paper contains a study of the motion of a rotor possessing unequal flexibilities in two mutually perpendicular directions and mounted in bearings which likewise possess different stiffnesses in two mutually perpendicular directions, say, the horizontal and vertical directions. A two-pole turbogenerator is an example of such a rotor. As known, the effect of ρ, the fractional inequality in rotor flexibilities, by itself, that is, without unequal bearing flexibilities, consists in giving rise to an unstable range of speed near the critical whose width depends upon ρ; the larger ρ is, the wider is the unstable range. If neither ρ nor σ vanishes (σ = fractional inequality of bearing flexibilities), the investigation becomes more difficult due to the fact that the principal flexibility directions are fixed for the bearings but rotate for the rotor. The differential equations of motion now acquire periodic coefficients whose period is half the rotation period. Their solution shows that near the main critical speed, for small ρ the unstable range splits up into three parts, which coalesce into one range for large ρ. The separation of three unstable ranges increases with σ. A further slight instability also occurs when the rotor speed is near half of the main critical speed. If one attempts to revolve such a rotor at a constant speed inside an unstable range, it may, even if in perfect balance, whirl with an exponentially increasing amplitude. With friction the amount of instability decreases and the size of the unstable regions decreases too, and with sufficient friction complete stability is restored. The friction necessary to restore stability is computed and the effect of unbalance on the steady-state amplitudes is studied. The solution of linear differential equations with periodic coefficients leads to an infinite number of algebraic linear equations in an infinite number of coefficients. Various ways of solving these equations and of speeding up the convergence of the solution are discussed in detail.