Page 219 - Steam Turbines Design, Applications, and Rerating

198 Chapter Eleven

for packeted bladed disk assembly by Dresser-Rand, is called a SAFE
diagram.
The use of the SAFE diagram for a bladed disk assembly (not a sin-
gle blade or a packet of blades as usually considered) facilitates the
design evaluation process.

11.4.1 Definition of resonance
Each blade on a rotating turbine disk experiences a dynamic force
when it rotates through the nonuniform flow emanating from station-
ary vanes. The dynamic response (e.g., stress and displacements) levels
experienced by the blades depend on:

1. The natural frequencies of the bladed disk and their associated
mode shapes
2. The frequency, the shape, and the magnitude of the dynamic force
which is a function of the turbine speed, number of stationary vanes,
and their location around the annulus and/or the number of inter-
ruptions in the flow passage, e.g., struts and their location around
the annulus
3. The energy-dissipating properties called damping provided by blade
material, frictional slip between joints, aerodynamic damping from
steam, etc.

A turbine bladed disk may get into a state of vibration where the
energy buildup is a maximum. This is exemplified by maxima in its
response (stress, displacement, etc.) and minima in its resistance to the
exciting force. This condition is called a state of resonance. There are
two simultaneous conditions for the energy built up per cycle of vibra-
tion to be a maximum. These conditions are:
1. The frequency of the exciting force equals the natural frequency of
vibration.
2. The exciting force profile has the same shape as the associated mode
shape of vibration.

These have been demonstrated by theoretical calculations and also by
measured responses of turbine bladed disks. Thus for a resonance to
occur, both of the above conditions must be met.

11.4.2 Mode shape
The deflected shape attained by a vibrating bladed disk at its natural
frequency is called its mode shape. This is shown by plotting the relative