Verify that blade packet mode resonance is safe

There are many steam turbine rotor stages with blades connected together for added reliability, making them much more difficult to excite from alternating forces such as upstream vane wakes and potential flow effects such as partial admission. Procedures are described to be sure a “safe” design considers variations giving mistuned resonant response factors.

Compressor and turbine cantilevered blade mistuning effects on response are described in numerous publications. Mistuning can help in reducing response from flutter (negative aerodynamic damping), but can increase response by a large amount for forced response at resonance. Figures below from Reference 1 shows how mistuning affects response at a resonant frequency of a responsive blade mode giving increased individual blade response near the same frequency. For the design shown the tuned 3-diameter mode would not respond from forces if resonant at six times speed, but the mistuned blades would respond.

Tuned Blades with Three-Diameter Mode Shape

Mode Shape with Blade Mistuning

For steam turbines the technique of packeting to intentionally have some blades out-of-phase with excitation forces has been used for decades, as explained in Weaver & Prohl, references 2 & 3. By choosing numbers of blades and vanes per circle along with number of blades per packet phase cancellation of forces can be optimized. This is especially important if there is resonance of fundamental modes that are much more excitable than modes at higher frequencies. However blade packet out-of-phase modes can sometimes cause failures, especially if endurance limit is reduced due to corrosion. Resonant response factor calculation is also summarized in the author’s 2004 tutorial, reference 4. When blade packet mode natural frequency calculations are made using finite element programs for a complete blade disk it is assumed that all blades and shrouds are identical. Thus there are system modes that are in diametral patterns. Some thus have claimed fundamental modes of packeted blade assemblies will not respond as long as the excitation harmonic is above a certain value; e.g. see Reference 5.

This review of mistuning will be for a typical short blade in the front end of a steam turbine, with excitation from upstream nozzle passing frequency. Figure shows FEA (finite element analysis) result for 120 identical blades attached to a rigid disk, shrouded so that there are 24, 5-bladed packets.

12-Diameter Mode Shape

This is a beam model with all blades exactly the same; exaggerated motion showing 12-diameter mode – 12 packets in phase, 12 packets out of phase. Theory says that there will not be any diameter modes with the fundamental mode blade shape for excitation above 12 times operating speed. Thus why worry about excitation such as from 36 upstream nozzles? That harmonic would only excite a 36-diameter mode that cannot exist. The table below shows the maximum number is 12 where packets have all blades moving together in phase. There would not be a low number of nozzles such as 12 with 120 blades. Table of modal frequencies has results for tuned packets using two FEA methods, and also a mistuned case with the blades in one packet having increased stiffness. For other designs there can be more separation of frequencies for the various modes.

An actual design such as a turbine stage with locking pieces will be mistuned. Even with a locking blade, there can be some added causes to give difference in frequency with other packets. Many stages have a variation in number of blades per packet around the circumference, dimensional manufacturing differences including root/disk rim tolerances, and there will be variation in deposits, wear and erosion increasing variation between packets. A blade packet just one percent off resonance can have 1/2 the amplitude of a packet that is at exact resonance.

Mistuned Blade Packet Mode Shape – Blades In Phase

The FEA analysis was rerun for one of the packets stiffer than the others. Result shows an individual mistuned fundamental mode at 2167 Hz. The one mistuned packet is representative of a packet with much higher stiffness – an actual case could be closer to the other nodal diameter frequencies. There still are other nodal diameters that may or may not respond, but the mistuned packet resonant response factor depends on phase cancellation with the excitation harmonic.

Table of frequencies given above for tuned packets has values agreeing for two FEA methods, and also a mistuned case with the blades in one packet having increased stiffness. For other designs there can be more separation of frequencies for the various modes. For the mistuned case there is no 12-diameter mode; rather the mistuned blade packet is near 2160 Hz and would be excited at 36 times 3600 rpm speed, nozzle passing frequency.

Using Weaver & Prohl equations (References 1&2) the table below show use of 6 blades instead of five is a better choice for mistuned packets besides the tuned case.

In fact 10 blades per packet is optimum. As each span would then equal 30 degrees the shroud should be able to be installed over the tangs for proper assembly using proven peening methods. If not six blades per packet could be selected reducing response to ½ that for five per packet.

For another design with 108 upstream nozzles giving fundamental mode resonance, response factors would be different. Here again 10 blades per packet would be optimum. With 5 blades per packet resonant response factor could be 0.65 which is much too high and not safe per Weaver & Prohl equations. A future article will discuss mistuning of longer blades assembled into packets, with resonance at low harmonics of speed.

References:

1. Castanier, M. P., Ceccio S. L., Epureanu, B, I., Pierre, C, (2007) “Next-Generation Modeling, Analysis, and Testing of the Vibration of Mistuned Bladed Disks,” AFOSR Grant FA9550-04-1-0099, The University of Michigan, Ann Arbor, MI, USA.

2. Prohl, M. A., 1958,  “A Method for Calculating Vibration Frequency and Stress of Banded Group of Turbine Buckets,” Trans. ASME, Vol. 78, pp. 169-180, 1958.

3. Weaver, F. L., and Prohl, M. A., 1958, “High Frequency Vibration of Steam Turbine Buckets,” Trans. ASME, Vol. 78, pp. 181-189, 1958.

4. 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.

http://turbolab.tamu.edu/proc/turboproc/T33/t33-17.pdf

5. Bloch, Heinz P., Chapter 11: Campbell, Goodman, and Safe Diagrams for Steam Turbine Blades, “A Practical Guide To Steam Turbine Technology”, McGraw-Hill, 1996.