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.i二 ei.... i二 1. .... (17-6).二1
This equation defines the compliance matrix.ei..， and the elements of the matrixare called the influence coefficients. The compliance matrixis obtained by making
..二 .i. (17-7)
where .i.is the .ronecker delta， and measuring the deflections .i.As . is varied from 1 to， each column of the compliance matrixis obtained. Oncethe compliance matrixis obtained， knowing the initial vibration level in each plane .i， the system of equations
ei. .二 .1. i二 1. .... (17-.).二1
is solved for the correction forces， .. The correction weights can be com-puted from the correction forces.
In general， 2 N sets of amplitude and phase are all that is required by the exact-point speed-balancing method. In balancing with the influence coef-ficient method: (1) initial unbalance amplitudes and phases arerecorded，
(2) trial weights are inserted sequentially at selected locations along therotor， (3) resultant amplitudes and phases are measured at convenientlocations， and (4) required corrective weights are computed and added to the system. Balance planes are obviously where the trial weights are inserted. The influence coefficients (or system parameters) can be stored for future trim balance. The method requires no foreknowledge of the system dynamic response characteristics (although such knowledge is help-ful in selecting the most effectivebalanceplanes， readout locations， and trial weights).
The influence coefficient method examines relative displacements rather than absolute displacements. No assumptions about perfect balancing conditions are made. Its effectiveness is notinfluenced by damping， bymotions of the locations at which readings are taken， or by initially bent rotors. The least-square technique for data processing is applied to find an optimum set of correction weights for a rotor that has a range of operating speeds.

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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