Page 78 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 3.4 Whirling of Rotating Shafts 65

Example 3.4-1
Turbines operating above the critical speed must run through dangerous speed at
resonance each time they are started or stopped. Assuming the critical speed to be
reached with amplitude determine the equation for the amplitude buildup with
time. Assume zero damping.
Solution: We will assume synchronous whirl as before, which makes 6 = (o = constant and
^ = 0. However, r and r terms must be retained unless shown to be zero. With c = 0
for the undamped case, the general equations of motion reduce to
.• k A ^
r + ------r = ew^ cos 4>
I ^ I (a)
2rco = Cio" sin cf)

The solution of the second equation with initial deflection equal to is
ea>
r -- sm (p + r,) (b)
DifTerentiating this equation twice, we find that r = 0; so the first equation with the
above solution for r becomes

(c)

Because the right side of this equation is constant, it is satisfied only if the coefficient
of t is zero:

-o.2)sin<^ = 0 (d)
which leaves the remaining terms:

I ------0) \f eoj^ cos (f)
2 \
k k
\m I (e)

With CO= yjk/m , the first equation is satisfied, but the second equation is satisfied
only if cos (/) = 0, or (/> = 7t/ 2. Thus, we have shown that at a> = or at

resonance, the phase angle is tt/2 as before for the damped case, and the amplitude
builds up linearly according to the equation shown in Fig. 3.4-4.

Figure 3.4-4. Amplitude and phase
relationship of synchronous whirl
with viscous damping.