## Disk Resonance Equations & Interference Diagrams

A 2-diameter mode shape for a covered impeller is shown below. Two areas vibrate in phase while two other areas are simultaneously out-of-phase.

Excitation forces such as from inlet or diffuser vanes can give rise to resonant response of vibratory modes. Potentially responsive diametral modes are equal to one-half the number of rotating blades; up to 7-diameter for 15 blades. These are mainly modes with vibration in the axial direction; however there is relative tangential vibration of the blades that accept potential flow forcing functions, just as they accept the steady state loading.

The equations required to check for disk mode "interaction resonance" and "disk critical speeds" are as follows:

Equation (1)                                           Equation (2)

Not at Disk Critical Speeds:              At Disk Critical Speeds:

(a)  |y × S| ± |z × B| = n                        (a)   For B >1

(b)      y × S = h                                     (b)    y × S = h = n

(c)      fr = y × S × w                              (c)    fr = n × w

where:

B = number of rotating blades

S = number of stationary elements

fr = natural frequency at speed, Hz

h = harmonic of speed

n = number of diameter nodal lines

y & z = integers > 0 (typically only 1 and 2 are important)

w = rotating speed, Hz (rpm / 60)

Note that number of blades should not be equal to number of stationary vanes as then circle and/or torsional modes could be excited.

Campbell diagram is given below for field testing with strain gauges where a five-diameter mode of an impeller had high response to two times number of inlet vanes. Equation 1 is satisfied as |2 x 16 - 27 = 5. Inlet vanes are not as troublesome as diffuser vanes, but the 5-diameter mode was the cause of failures due to increased wake excitation from liquid ingestion into a mixed gas refrigeration compressor.

Campbell diagram below is for the three-diameter mode; however there was no response shown by strain gauges due to first harmonic of 16 vanes because there was phase cancellation. Equation 1 was not satisfied.

In addition for this case the 11-diameter mode was not resonant in the speed range.

Interaction equations 1 and 2 above for worrisome resonance can be shown using an interference diagram as plotted below for a somewhat larger impeller.

The interference diagram plots interference lines for disk critical speeds and for interaction resonance due to blade and vane numbers. For this case the 5-diameter mode is resonant in the speed range; other modes have phase cancellation. Many other impellers with minimal liquid ingestion with the same design did not have failures. That may not have been the case if there had been 16 diffuser vanes near the impeller tip, as interaction is much more of a concern.

Disk Critical Speeds are where the number of nodal diameters is coincident with the same number times speed; i.e. Equation 2. The interference diagram above shows that all diameter mode frequencies are above number of diameters times 6300 rpm maximum speed.

This article describes covered impellers. Future submissions will feature unshrouded impellers and turbine bladed disks with other considerations, especially due to coupled blade modes and mistuning.

Reference: Kushner, F., 2004, "Rotating Component Modal Analysis And Resonance Avoidance Recommendations", Tutorial, Proceedings of the 33rd Turbomachinery Symposium, Turbomachinery Laboratory, Texas A&M University, College Station, TX. Pp. 143-161.