If the undamped torsional natural frequency analysis in- dicates an interference between an undamped torsional nat- ural frequency and a shaft rotative speed or other potential excitation mechanism, and the train design cannot be altered sufficiently to remove the resonant interference, then a damped torsional response and vibratory stress analysis must be performed to ensure that rotor shafts and couplings are not overstressed. Potential areas of vibratory stress con- centrations are couplings and shaft ends. Results generated from this type of analysis may indicate that shaft ends must be re-sized to safely accommodate the high levels of vibra- tory stress resulting from close operation to a torsional nat- ural frequency. Figure 2-24 displays a typical plot of calculated oscillatory stresses versus the reference fre- quency (low speed shaft). The two peaks present in this plot indicate excitation of the first (1st) and second (2nd) train torsional natural frequencies by 1×and 2×operating speed torque pulsations, respectively.
Calculated stresses will be governed by assumptions re- garding the level of available torsional damping as well as overall expected torque excitation levels at given frequen- cies. These key parameters are normally set by mutual con- sent of both the purchaser and the vendor and are based on experience, measurement, and/or available literature.
ysis. The Campbell diagram in Figure 2-20 cross-plots the frequencies of the modes with the shaft running speed, and Figure 2-19, Views a–e present the first five modeshapes.
The Campbell diagram indicates a potential interference, within 10 percent, between the fundamental torsional mode and 1×speed. The plot also indicates that no interference ex- ists between the 1×and 2×electrical line frequencies and any undamped torsional natural frequency. Examination of the corresponding torsional modeshape indicates that shaft twisting of the three individual units is minimal; torsional twisting is principally confined to the couplings for this mode. This implies that the torsional stiffness of the cou- plings is significantly lower than the torsional stiffnesses of the surrounding shafts. Hence, the frequency of this mode is governed by the torsional stiffness of the couplings. Altering the torsional stiffness of one or both couplings will allow the design engineer to shift the frequency of the potentially problematic mode. Figure 2-21 displays the result of tuning the coupling torsional stiffnesses so the potentially problem- atic mode has been shifted clear of the operating speed of the unit. Although motor drive trains typically possess more sources of torsional excitation than a turbine driven train, they are also usually limited to a single operating speed so torsional detuning is often not difficult to accomplish. Most coupling vendors will readily adjust the torsional stiffness of a coupling within a range of at least ± 25 percent. Note, how- ever, that tuning the coupling stiffness may adversely impact the service factor of the coupling.
Careful examination of the first and second torsional modes for the motor-driven-compressor train indicates that most of the twisting occurs in the vicinity of the couplings.
As just mentioned, this situation implies that the couplings are the torsionally soft elements in the train and that their tor- sional stiffnesses will govern the locations of the fundamen- tal two modes. In general, machinery trains will have the same number of coupling controlled modes as couplings.
These modes are torsionally significant and require de-tun- ing if they interfere with potential excitation frequencies. In motor-gear-compressor trains, the third mode is almost al- ways associated with the torsional characteristics of the mo- tor. The third mode calculated for this example is typical of motor controlled modes: calculated angular deflections are predominantly found in the low-speed shafting with the largest change in angular deflection occurring through the motor. Note that the node point of the motion is located nearly at motor midspan so that the two ends of the core vi- brate out of phase. This motion is analogous to the out-of- phase free vibration observed in a system composed of two masses connected by a single spring. Higher order modes contain single unit out-of-phase motions similar to the motor controlled mode. In Figure 2-19, View e displays the com- pressor controlled torsional mode.
Copyright American Petroleum Institute Reproduced by IHS under license with API
Figure 2-22—Sample Train Torsional Campbell Diagram for a Typical Turbine-Compressor Train
First mode =1841 CPM Second mode = 12514 CPM Third mode = 13630 CPM
1 x reference speed
180.0120.080.040.0140.0100.060.020.00.0 90% speed = 4140 RPM 110% speed = 5080 RPMNormal speed = 4800
TRAIN TORSIONAL CAMPBELL DIAGRAM
0.0 10.0 20.0 30.0 40.0 50.0 60.0
Reference speed x 102 (RPM) Torsional natural frequency x 102 (CPM)
TUTORIAL ON THEAPI STANDARDPARAGRAPHSCOVERINGROTORDYNAMICS ANDBALANCING 103
Figure 2-23—Torsional Modeshapes for a Typical Turbine-Compressor Train
1.00.50.0-0.5-1.0
Normalized amplitude (NDIM)
Compressor rotor mode
Steam turbine rotor mode Fundamental
mode
3rd mode = 13630 CPM 2nd mode = 12514 CPM
1st mode = 1841 CPM 1.00.50.0-0.5-1.0
Normalized amplitude (NDIM) 1.00.50.0
0.0 -0.5-1.0
Normalized amplitude (NDIM) Coupling Compressor EndTurbine
40.0 80.0 120.0 160.0 200.0 240.0 280.0
40.0 80.0 120.0 160.0 200.0 240.0 280.0
40.0 80.0 120.0 160.0 200.0 240.0 280.0
Copyright American Petroleum Institute Reproduced by IHS under license with API
1/2 x 2nd Torsional natural frequency
1 x 1st torsional natural frequency
1250 RPM 1587 RPM
Torsional shaft stress (psi p-p)
Operating speed (RPM) 0
0 500 1000 1500 2000 2500
2000 4000 6000 8000
Figure 2-24—A Typical Plot of Calculated Oscillatory Stresses Versus the Reference Frequency (Low Shaft Speed)
TUTORIAL ON THEAPI STANDARDPARAGRAPHSCOVERINGROTORDYNAMICS ANDBALANCING 105
c. A variable speed motor drive.
Variable speed motor drives require a transient torsional analysis because such motor designs result in high-level transient torque pulsations at the beginning of the unit start.
Machinery trains with synchronous motor drivers that undergo asynchronous unit starts require a transient tor- sional analysis to determine train response during unit start to the transient torque pulsations resulting from the oscil- lating air gap torque of the motor. This time-transient anal- ysis, for the full period of train acceleration to normal running speed (synchronizing speed), is normally calcu- lated for both full and reduced synchronous motor terminal voltage. Figure 2-25 presents a typical plot from a transient torsional analysis.
Since the pulsating component of synchronous motor torque changes linearly in frequency from 2×electric line frequency at 0 percent speed to 0 Hertz at 100 percent speed, and achieves a maximum amplitude at approximately