Should a Flexible Rotor Be Balanced in N or (N + 2) Planes?

Abstract

The problem of balancing flexible rotors consists mainly of eliminating rotating bearing forces. Analytical expressions are derived for the deformation and the rotating bearing forces of a rotor, using orthogonal functions. With this kind of representation it is possible to set up simple conditions for the vanishing rotating bearing forces. They lead to a linear system of equations giving the compensating unbalances in each of a set number of balancing planes. Two methods used in practice are theoretically explained and compared. The “N” method employs N planes for balancing a speed range up to, and including, the Nth critical speed and can be characterized by the condition A = 0, see equation (13). The “(N + 2)” method requires two more planes for the same speed range and is characterized by A = 0 and B = 0. It is proved that lim N→∞ B = 0, so that in the limiting case of an infinite number of balancing planes (speed range from zero to infinity) both methods are of equal value. The two methods differ for finite N in their accuracy and the amount of calculation. Considering simple examples with known unbalance distribution it will be shown that the main error of the N method is the result of treating B as equal to 0, which it is not, thus accounting for the greater accuracy of the N + 2 method. The additional effort needed for the latter method is justified in those cases where greater accuracy is demanded.