Page 102 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Chap. 3 Problems 89

instrument on a rubber pad with the following properties: k = 2802 N /cm and
f == 0.10. W hat acceleration is transmitted to the instrument?
3-30 If the instrument of Prob. 3-29 can only tolerate an acceleration of 2.03 cm /s^, suggest
a solution assuming that the same rubber pad is the only isolator available. Give
numerical values to substantiate your solution.
3-31 For the system shown in Fig. P3-31, verify that the transmissibility TR = \x /y \ is the
same as that for force. Plot the transmissibility in decibels, 201og|TO| versus (o/a>^
between co/co^ = 1.50 to 10 with ^ = 0.02, 0 .0 4 ,..., 0.10.

X Figure P3-31.

3-32 Show that the energy dissipated per cycle for viscous friction can be expressed by
iV,,

1 - ((0/i0„y] + [2é{(ü/(0„)Ÿ
3-33 Show that for viscous damping, the loss factor rj is independent of the amplitude and
proportional to the frequency.
3-34 Express the equation for the free vibration of a single-DOF system in terms of the loss
factor 7] at resonance.
3-35 Show that t„ / tj plotted against is a quarter circle where tj is the damped natural
period, and is the undamped natural period.
3-36 For small damping, the energy dissipated per cycle divided by the peak potential
energy is equal to 28 and also to 1 /Q . [See Eq. (3.7-6).] For viscous damping, show
that
TTCO, ,
i
5 = —r ^
3-37 In general, the energy loss per cycle is a function of both amplitude and frequency.
State under what condition the logarithmic decrement 8 is independent of the
amplitude.
3-38 Coulomb damping between dry surfaces is a constant D always opposed to the motion.
Determine the equivalent viscous damping.
3-39 Using the result of Prob. 3-38, determine the amplitude of motion of a spring-mass
system with Coulomb damping when excited by a harmonic force sin (ot. U nder
what condition can this motion be maintained?
3-40 Plot the results of Prob. 3-39 in the permissible range.
3-41 The shaft of a torsiograph, shown in Fig. P3-41, undergoes harmonic torsional
oscillation sin cot. Determ ine the expression for the relative amplitude of the outer
wheel with respect to (a) the shaft and (b) a fixed reference.