Page 100 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Chap. 3 Problems 87

flies off, determine the buildup of vibration if the natural frequeney of the system is 18
eps with damping of = 0.10.
3-16 A solid disk weighing 10 lb is keyed to the center of a \-in. steel shaft 2 ft between
bearings. Determine the lowest critical speed. (Assume the shaft to be simply sup­
ported at the bearings.)
3-17 Convert all units in Prob. 3-16 to the SI system and recalculate the lowest critical
speed.
3-18 The rotor of a turbine 13.6 kg in mass is supported at the midspan of a shaft with
bearings 0.4064 m apart, as shown in Fig. P3-18. The rotor is known to have an
unbalance of 0.2879 kg • cm. Determine the forces exerted on the bearings at a speed
of 6000 rpm if the diam eter of the steel shaft is 2.54 cm. Compare this result with that
of the same rotor mounted on a steel shaft of diam eter 1.905 cm. (Assume the shaft to
be simply supported at the bearings.)

Figure P3-18.
3-19 For turbines operating above the critical speed, stops are provided to limit the
amplitude as they run through the critical speed. In the turbine of Prob. 3-18, if the
clearance between the 2.54-cm shaft and the stops is 0.0508 cm, and if the eccentricity
is 0.0212 cm, determine the time required for the shaft to hit the stops. Assume that
the critical speed is reached with zero amplitude.
3-20 Figure P3-20 represents a simplified diagram of a spring-supported vehicle traveling
over a rough road. Determine the equation for the amplitude of IT as a function of the
speed, and determine the most unfavorable speed.
V _
---------E H - - E H -
k{x-y)

Figure P3-20.

3-21 The springs of an automobile trailer are compressed 10.16 cm under its weight. Find
the critical speed when the trailer is traveling over a road with a profile approximated
by a sine wave of amplitude 7.62 cm and wavelength of 14.63 m. W hat will be the
amplitude of vibration at 64.4 km /h ? (Neglect damping.)
3-22 The point of suspension of a simple pendulum is given by a harmonic motion Xq =
A gsincai along a horizontal line, as shown in Fig. P3-22. Write the differential
equation of motion for a small amplitude of oscillation using the coordinates shown.
Determine the solution for x/X q, and show that when ou = yÎ2o)^, the node is found at
the midpoint of /. Show that in general the distance h from the mass to the node is
given by the relation h = where = yjg/l •