338 W. J. Morrell et al. by applying balance weights at weight planes, which requires knowledge of influence coefficients to select the appropriate weight. Influence coefficients simply provide information on how a rotor will respond to weight applied at different locations. Modal balancing has proven very effective for balancing rotors when the rotor has been accelerated through the critical speed being balanced, but more efficient ways of balancing rotors are desired to reduce the cost and time needed. Thus, DSF balancing is introduced as alternative to modal balancing for investigation. Modal and deflection shape balancing are governed by the same equations of motion but arrive at their solutions differently. Equation (35.1) shows the equations of motion in matrix form for a rotor at steady state with symmetric stiffness and damping. M 0 0 M ¨x ¨y + DS 0 0 DS ˙x ˙y + K 0 0 K x y = Fcos (ωt +δ) Fsin(ωt +δ) (35.1) In the above equation M is the rotor or modal mass, DS is the lateral external modal damping, and K is lateral isotropic stiffness. Modal balancing takes deflection data gathered during a balance test run, uses that data to identify the mode being addressed, and individual modes are balanced. DSF balancing takes the same deflection data and treats the rotor’s shape as a combination of deflections needed to create the measured rotor shape. Correct placement and magnitude of the corrective weights for both modal and DSF balancing requires weight planes and influence coefficients, and the weight planes do not need to be the same for the two methods. Modal balancing weight planes are in close proximity to antinodes of the mode being balanced while DSF balancing simply uses some number of weight planes along the length of the rotor. Influence coefficients for the two approaches consist of one influence coefficient for each mode in modal balancing and would correspond to the number of weight planes in use for DSF balancing. For instance, in the case of the 28-in. steel beam spinning above the third critical speed the rotor has gone through three critical speeds, corresponding to three mode shapes, and would require three sets of influence coefficients for the modal balancing approach. The number of influence coefficients for the DSF balancing approach would depend on the number of weight planes being used. 35.3 Analysis DSF and modal balancing approaches are compared through simulation in MATLAB. A 28-in. long 0.8 in. outside diameter steel beam with a 0.025-in. wall thickness is used as a simulated rotor. A randomly distributed imbalance is applied to the beam and modal magnitudes are determined at the first three critical speeds: 113, 454, and 1003 Hz. The modal magnitudes are then balanced using the modal and the DSF approaches. The two approaches are compared by • The reduction in modal magnitude at 113 Hz when balancing at 113 Hz • The reduction in modal magnitude at 454 Hz when balancing at 454 Hz • The reduction in modal magnitude at 1003 Hz when balancing at 113 and 454 Hz together These comparisons are performed for 5, 10, 15, 20, and 28 DSF weight planes equally spaced along the length of the rotor. The MSF approach always uses three weight planes corresponding to the antinodes of the first and second flexural modes. A total of 20,000 random imbalances were simulated and the final modal magnitude, normalized against the initial modal magnitude from the applied imbalance, was logged to create a distribution of the balancing results. Each balancing approach, Modal or DSF, is applied only once to each imbalance. The probability of each result is also plotted. Additionally, the effect of the number of weight planes used for the DSF approach is shown. 35.3.1 5 Weight Planes Plots comparing the reduction in modal magnitudes using 5 DSF weight planes are presented below. Results from balancing at 113 Hz seen in Fig. 35.1 show very slightly better performance from the DSF approach when compared to the modal approach and Fig. 35.2 shows very similar performance when balancing at 454 Hz, but in both cases the magnitudes are so small that both approaches might be considered functionally equivalent. The biggest difference between approaches is shown in Fig. 35.3, where the impact on the modal amplitude of the third mode when balancing occurs only at the first and second critical speeds. The DSF produces clearly lower modal amplitudes than the modal approach for this condition.
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