A number of investigations have concerned themselves with the optimum selections of the number of balancing planes necessary to balance a flexiblerotor. To perform an ideal balance on a flexiblerotor， as many balancing planes as unbalances are needed. The perfect balance is either impractical or uneconomical. Two substitute approaches for deciding the number of bal-ancing planes have been proposed.
One is the so-called N-plane approach. This approach states that only N-planes are necessary for a rotor system running over N critical speeds. Theother technique， called the ( N +2)-planeapproach， requires two additional planes. These two additional planes are for the two-bearing system and are necessary in this school of balancing.
The N-plane is based on the concepts of the modal technique. FromEquation (17-5)， there are N principal modes that need to be .ero forthe perfect balance of arotor， which runs through Nth critical speed.Thus，
N-planes located at the peaks of the principal modes will be enough for cancelling these modes. From the point of view of residual forces andmoments at the support bearings， ( N + 2)-planes are better than N-planes.
If one can balance at designspeed， that point is ideal， but there may beproblems while trying to go through the various criticals. Thus， it is best to balance the unit through the entire operation range. The number of speeds to be selected is also very important. Tests conducted show that when the points were taken at the critical speed and at a point .ust after the criticalspeed， the best balance results throughout the operating range wereobtained， as seen in Figure 17-9.
.pplication of Balancing .echni.ues
.sing the influence coefficient technique for multiplane balancing is sim-ply an extension of the logic， which is ..hardwired". into the standard balan-cing machine. This extension has been made possible by the availability of better electronics and easier access to computers.
Practical balancing may now be performed in any reasonable number ofplanes at virtually any reasonable number of speeds. The one-plane， low-speed balancing operation is perhaps the simplest application of themethod， where a known weight at a known radial location (often in the form of wax added by hand) is used to determine balance sensitivity of the part to be balanced in a spin-up fixture. This procedure can effectively remove an unbalance force from a component. Two-plane balancing is simply an extension to permit unbalance moments as well as forces to be removed. Inseveral instances， the sensitivities associated with these types of machines can be predetermined (the machine may be calibrated) and the values stored to permit one-start balancing. Balancing a fully assembled rotor operating inits running environment， whether rigid or flexible innature， represents the ultimate application.
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