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Modal Balancing
Modal balancing is based on the fact that a flexible rotor may be balanced by eliminating the effect of the unbalance distribution in a mode-by-modesequence. Typical principal modes of a symmetric， uniform shaft are shown in Figure 17-.. The deflections of a rotor at any speed may be represented by the sum of various modal deflections multiplied by constants dependent on speed
F(..且)二 BFr(且)..r(.)(17-5)r二1
where F(.，且) represents the amplitude of transversevibrations， as a function of the distance along the shaft at a rotational speed且. BFr (且) and .r(.) express， respectively， the complex coefficient at rotating speed且 and the rth principal mode.
Thus， arotor， which has been balanced atall critical speeds， is alsobalanced at any other speed. For end-bearing rotors， the recommended procedure is: (1) balance the shaft asa rigid body， (2) balance for eachcritical speed in the operatingrange， and (3) balance out the remaining noncritical modes as far as possible at the running speed. Balance planes picked are the ones wherein the maximum amplitudes of vibration occur.
Modal balancing is one of the proven methods for flexible rotor balan-cing. Modal balancing has also been applied to problems of dissimilar lateralstiffness， hystereticwhirl， and to complexshaft-bearing problems. In many discussions on modal balancing fluid-film damping is not included. In other instances rolling-element bearing effectsare neglected. In suchcases， the practical usefulness of the modal method is not fully defined.

Several problems hinder the application of the modal technique to morecomplexsystems. To use the technique， calculated information is required on the mode shapes and natural frequencies of the system to be balanced. The accuracy of the computed results depends on the capabilities of the computer program used and onthe input data(dimension， coefficients， system model effectiveness) used in the calculations. In turbomachinerywhere system damping is significant， as with fluid-film bearings， problems arise. The mode shapes and resonant frequencies of heavily damped systems often bear little resemblance to undamped mode shapes and frequencies. The reliance of modal balancing on predicted modes and frequencies is at least aninconvenience and， without proper response programs， can be a significant disadvantage.
At present， no general-purpose modal balancing computer programs exist that are comparable in nature to the programs developed for the influence coefficient (multiplane) method. Such a program would require calculated modal amplitudes and phaseangles， and that the measured amplitudes and phase angles of the rotor bearing system be balanced. The programwould then be run for each separate rotor whirlmode， including the full-speed residual balance correction. At present， no general analysis suitable for programming exists.
Multiplane Balancing (Influence Coefficient Method)
Modal balancing came into being to alleviate the problems of the supercritical rotor unbalance of the steam turbine-generator industry. It combined the then available techniques for calculating response amplitudes for the various rotor vibrational modes with the available instruments formeasuring actual installed vibration levels. In recentyears， more systems have been designed for supercritical operation. Newer types of sensors andinstruments are becoming available， making it feasible to obtain precision in amplitude and phase measurement. Minicomputers for operation on theshop floor or in balancingpits， and time-sharing terminals for in-the-fieldaccess to large computers， are now commonly available. The newest multi-plane balancing techniques owe their success to advancement in these areas.
The influence coefficient method is simple to apply， and data are now easily obtainable. Consider a rotor with discs. The method of influence coefficients provides the means for measuring the compliance characteristics of the rotor.
.et .1， ...， ..， ...， . be the forces acting on the shaft. Then the deflec-tion .i in the i-plane is given by
　
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