## Torsionals

Besides the inherent lateral natural frequency characteristic, compressors are also influenced by torsional natural frequencies. All torsionally flexible drive trains are subject to non-steady or oscillatory excitation torques during normal operation of the system. These excitation torques can be an inherent function of either the driver or the driven equipment and, when superimposed on the normal operating torque, may appear to be of negligible concern. However, when combined with the high inertia loads of many turbomachinery trains and a torsional resonant frequency of the system, these diminutive ripples can result in a tidal wave of problems.

The torsional resonant response of a system is an interaction of all the components in the train. Calculation of torsional natural frequencies is based on the entire system and these frequencies are valid only for that given arrangement. If any component of the train is replaced by an item with torsional characteristics different from the original, the system tor sional response must be recalculated and new torsional natural frequencies determined. Occasionally, an original equipment manufacturer is requested to calculate the torsional and lateral critical speeds of the supplied item. Unfortunately, the purchaser is unaware that this request is of limited value since the torsional response of a single item in a train is meaningless. Likewise, a torsional shop test will yield meaningless results if the train is not assembled and tested with every item destined for the field.

The interesting aspect of torsional problems in turbomachinery systems is that the first indication of a problem is usually a ruptured shaft or coupling in the field. Silent and deadly, a torsional response can lurk at synchronous or non-synchronous frequencies, and be steady or transient in nature. Once a torsional problem is found in the field and the excitations are determined to be inherent in the system, the only solution avail able, to put the system back on line quickly, is to decouple the excitation source or to dampen the system response.

Table 9-2 contains some sources of torsional excitations encountered in the operation of turbomachinery systems.

A quick review of system torsional response may help explain why a resilient coupling works. Figure 9-14 is a torsional single degree of freedom system with a disk having a torsional moment of inertia J connected to a massless torsional spring K.

Newton's law for a rotating body states;

Table 9-2

Typical Sources of Torsional Excitation in Turbomachinery Systems

Table 9-2

Typical Sources of Torsional Excitation in Turbomachinery Systems

SOURCE |
FREQUENCY |
MODE | |||||||||||||||||||||||||||||

Gear |
Mesh |
No. of Teeth * Gear RPM |
Steady | ||||||||||||||||||||||||||||

High Spot |
1* Gear RPM |
Steady | |||||||||||||||||||||||||||||

Quality |
2* Gear RPM |
Steady | |||||||||||||||||||||||||||||

Steam Turbine |
Nozzle Passing |
No. of Blades « RPM |
Steady | ||||||||||||||||||||||||||||

Frequency Drive |
No. of Pulses « Motor Line Frequency |
Steady | |||||||||||||||||||||||||||||

Synchronous Start |
2k Slip Frequency |
Transient | |||||||||||||||||||||||||||||

Compressors |
Centrifugal Surge |
Broad Band |
Transient | ||||||||||||||||||||||||||||

Reciprocating |
No. of Cylinders » RPM |
Steady | |||||||||||||||||||||||||||||

Electrical Faults |
Varies |
where T - torque J = mass moment of inertia a = angular acceleration Angular acceleration is the time rate of change of angular velocity which, in turn, is the time rate of change of angular displacement. where 6 = angular displacement, rad (0 = angular velocity, rad/sec Assume that oscillatory excitation torque of T0 sin tot is applied to the system in Figure 9-14. By definition, when the excitation frequency coincides with the torsional natural frequency of the model, all torques will balance and the system will be in a state of resonance. For equilibrium, the following equation of motion must be satisfied where = inertial torque CO = damping torque k0 = stiffness torque In its simplest form, damping is neglected and no external forcing function is applied, resulting in the equation separating the variables, The general solution of this second order differential equation is Assuming the disk is displaced 90 radians and then released, the following initial conditions apply: The first condition yields Differentiating and substituting, the second condition yields resulting in the specific solution The period, P, of this vibration is The reciprocal of the period is where f„ = torsional natural frequency, cps For a complex, multimass system like that shown in Figure 9-15, the equations of motion become quite complex, especially if a forcing function exists and internal damping is included. Inertial damping (damping to ground) is neglected. The equations of motion for this system would take the form: To sin wl CPLG Gear CPLG Compressor CPLG Gear CPLG Compressor To sin wl Figure 9-15. Multimass torsional model. Figure 9-15. Multimass torsional model. Jj9{ + Cj CGi -02) + K3 (0] -02) = T0 sintot (9.18) J202 + c2 (02 - 03) + Q (0y - 02) + K2 (02 - 03) - Kj (0j - 02) = 0 J3e3 + C3 (03 - 04 ) + C2 (02 - 03 ) + K3 (03 - 04 ) - K2 (02 - 03 ) = 0 J404 + C4 (04 - 05) + C3 (03 - 04 ) + K4 (04 - 05) - K3 (03 - 04 ) = 0 J505 + C4 (04 - 05 ) + K4 (04 - 05 ) = 0 The solutions to a problem of this magnitude can be found in references [3, 7] and others. Figures 9-16 and 9-17 are torsional mode shape diagrams of some typical systems. While the rigorous solution to the multimass damped system is not within the scope of this book, several interesting points should be made.
Figure 9-16. Typical torsional interference map. Figure 9-16. Typical torsional interference map. MOTOR j GEAR \ COMPRESSOR COUPUNG COUPLING MOTOR j GEAR \ COMPRESSOR COUPUNG COUPLING Figure 9-17. Typical plot of torsional angular deflection. 1. An nth degree of freedom system will have (n - 1) natural frequen cies, 2. Equation 9.11 indicates that the torsional natural frequencies ot a system are a function of the torsional inertia and stiffness of the system. 3. The natural frequencies of a damped system are essentially the same as the undamped systems for all realistic values of damping. 4. The displacements of the system at resonance will be a function of the magnitude of the driving or excitation source and damping. 5. Damping in the system represents dissipation of vibratory energy that reduces the amplitudes in the system. 6. Damping is a function of the angular velocity change across the damper. As previously mentioned, the torsional response of a system is a function of the stiffness and inertia in the train. While some parameters of the system can be changed, the inertia J is usually fixed by the basic process. For example, the J value for a centrifugal compressor is largely a function of impeller diameters and widths. These, in turn, are set by the required head and flow. Theoretically, hub and shroud thickness could be varied to tune the system. However, any change in impeller hub thickness, shroud thickness, or shaft diameter may significantly alter the lateral response of the unit. These modifications would lead to trade-offs that probably would not be considered acceptable from a process or operational point of view. As a necessary competitive procedure, some machine manufacturing must proceed in parallel to the engineering analysis. Basic changes to impellers and shafts while torsional problems are worked out would definitely slow the process down and impact delivery. The driver cannot be designed until the system inertia is known, yet the torsional analysis cannot be completed until driver parameters are set. So it would appear that an iterative design and analysis procedure would be required and could go on for quite some time. This would complicate things considerably if it were not for the couplings. |

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