Transactions A.S.M.E.

696 TRANSACTIONS OF THE A.S.M.E. NOVEMBER, 1940
In order to design blades of relatively high frequency, a large
circumferential pitch must be used. In the case of the 50,000-kw
Northwest turbine blades, this pitch is 1.6 in. on the mean diameter.
The Northwest impulse blades are calculated to have a
running frequency of 7100 vibrations per sec. The nozzles have
been spaced on a pitch of 3.484 in., which gives a frequency of the
impressed force of 1837 per sec. The curve, Fig. 10, shows that
the theoretical variation of force on the blade is about 0.69 of the
variation of the equivalent nozzle steam force. There is, however,
a considerable margin between the 7100 per sec frequency of the
blade and the 1837 per sec frequency of the impressed force. The
magnification of motion from secondary resonance is relatively
small under these conditions.
In concluding this section on the stresses in partial-admission
blading, it can be stated that the investigations briefly reviewed
have shown that it is possible to calculate the effects of rapidly
applied loads on partial-admission blades. The only safe ground
for progress in matters of this kind is the establishment of a correct
background of fundamentals. It is believed that this has
been done and that the material constants are sufficiently well
known to permit the prediction of safe proportions for this type
of blading.
The importance of having the natural frequency of the blade
remote from the frequency of the variable impressed force from
the nozzle should be further stressed. Theoretically, if the action
of the steam on the moving blades occurs at the same efficiency
for all relative positions of blades and nozzles, then blades and
nozzles of the same circumferential pitch can be used to avoid
a periodic pulsation in the steam driving force. Actually, some
variation in the efficiency must be expected, as each blade moves
in and out of the flows of the various nozzles; hence, there will
be some periodic variation in the steam driving force on the blades.
If unsuitable design proportions are selected so that the frequency
of the impressed force on the blades is equal to their natural frequency,
then serious and destructive resonance may take place.
The best procedure is to design the blades so that the natural
frequency is substantially higher than the frequency of the impressed
force. The higher the frequency of the blades, the greater
the amount of energy absorbed by damping in each revolution.
The total damping will depend, to some extent, on the variation
of the damping constant with stress.
The general procedure for determining the stresses in partialadmission
blades can be outlined as follows:
a Determination of natural frequency of blade system including
correction for rotation and stiffening effect of shrouding.
b Determination of the frequency of the impressed force exerted
by the steam flow from the nozzle. An equivalent-steamforce
curve is assumed of simple harmonic form, the amplitude of
which must be based on research data.
c Design of blade to place natural frequency of assembled
structure substantially above the frequency of the impressed
force from the nozzles. Secondary resonance is thus avoided.
d The motion of the blade is calculated and plotted in curve
form for the condition of energy stability, in which the energy
imparted to a blade during one passage through the nozzle group
is dissipated by the total damping of the system in the course
of one revolution.
e Care must be taken to determine the most unfavorable phase
relations between the free motion of the blade and the time of
application and release of the impressed force. The phase relations
that will give the maximum amplitude of vibration must be
determined. In other words, the blades are designed to withstand
primary resonance.
f The correct procedure must be adopted properly to take
into account the actual rate of application of the steam load and
the corresponding rate of unloading.
g There must be reliable data available on the damping characteristics
of the blade material for the operating conditions under
consideration.
h The ratio of the maximum amplitude for the condition of
energy stability to the static deflection of the blade under the
action of the steam driving force taken as a steady value gives
the amplification factor to be applied to the bending stress calculated
on the static basis.
i The centrifugal stress must be added to the amplified bending
stress to determine the maximum resultant stress to which
the blade is subjected. All stress concentration factors must
be included.
C e n t r if u g a l S t a b il it y
There is good reason to believe that the initial tightness of
high-temperature blading in the spindle grooves is not maintained
for a long period of time at normal speed and full operating temperature.
This comes about partly due to initial creep in which
the minute irregularities of the machined surfaces are crushed
until a good bearing surface is secured and partly due to elastic
deformation and to the reduction in elastic modulus at high temperature.
It is believed that all manufacturers produce excellent
initial fits of the blade roots in the spindle grooves. At full speed
and temperature, however, the close fit is believed to be lost for
the reasons cited. If it be assumed that the blade root is only
slightly loose at full speed and full operating temperature, conditions
may arise, in which, in partial-admission wheels, the overturning
moment set up by the suddenly applied steam load may
exceed the stabilizing moment of the centrifugal force on the
blade. If this is true, a single blade, or even a group of two or
three blades which are joined together by a single shroud, may
jump in the groove every time the admission arc is passed. If
the blades jump or chatter in their grooves every time the steam
forces are imposed on them, local stresses of sufficient magnitude
to cause failure may be set up in the blade roots. It is, therefore,
necessary to establish a safe ratio between the stabilizing moment
created by the centrifugal force and the overturning moment set
up by the suddenly applied steam load. Experience indicates
that the ratio of the centrifugal moment to the steam driving
moment about the base of the blade should, in most cases, be
greater than 3 for a single blade, or greater than 6 for a pair of
blades tightly shrouded together. In general, it is believed necessary
that the stability factor shall be defined as the quotient of
the centrifugal stabilizing moment divided by the amplified steam
bending moment and that this ratio shall exceed 1.5.
Fig. 11 illustrates the principle of centrifugal stability. The
centrifugal moment is the product of the centrifugal force Fc and
the moment arm KL. The steam moment is the product of the