Page 237 - Steam Turbines Design, Applications, and Rerating

216 Chapter Eleven

Figure 11.31 Blade natural frequency vs. phase angle β over the speed
range of 0–7500 r/min (blade with integral tip shroud). (Provenzale and
Skok, ASME Paper 73-Pet-17)

from flow distortion due to nozzles, the modes and frequency of concern
have been marked by large shaded circles (Fig. 11.29).
Results from Weaver and Prohl’s paper are reproduced alongside the
SAFE diagram (Fig. 11.30). The shaded circles on the SAFE diagram
point to the mode(s) of concern as predicted by Weaver and Prohl (after
numerous response calculations).
The data provided in the paper by Provenzale and Skok (ASME
Paper 73-Pet-17) were obtained by a response calculation of the blades
under the action of aerodynamic forces. Forces were defined by a fre-
quency and a shape defined by a parameter β. The relationship
between β and nodal diameters is β= 360°= N nodal diameter (N is the
number of blades).
The total number of blades based on the data provided in the paper
is estimated to be 264. On the basis of the theory of bladed disk vibra-
tion, there should be 264 frequencies; each of these will be excited by a
force shape matching the mode shapes if frequencies coincide. Figures
2 and 3 of the paper by Provenzale and Skok are reproduced here as