Page 218 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Chap. 6 Problems 205

Figure P6-35.
6-36 The lateral and torsional oscillations of the system shown in Fig. P6-36 will have equal
natural frequencies for a specific value of a/L. Determine this value, and assuming
that there is an eccentricity e of mass equal to me, determine the equations of motion.

Figure P6-36.

6-37 Assume that a three-story building with rigid floor girders has Rayleigh damping. If
the modal dampings for the first and second modes are 0.05% and 0.13%, respectively,
determine the modal damping for the third mode.
6-38 The normal modes of a 3-DOF system with = m2 = m^ and /c, = k2 = arc
given as
/ 0.737) /-0.591 ) I 0.328^
(/>! = 0.591 , (f>2= { 0-328 , { -0.737
i 0.328) i 0.737) i 0.591
Verify the orthogonal properties of these modes.
6-39 The system of Prob. 6-38 is given an initial displacement of
( 0.520)
- 0.100
( 0.205 )
and released. Determine how much of each mode will be present in the free vibration.
6-40 In general, the free vibration of an undamped system can be represented by the modal
sum
j
X{t) = A¿(f)I sin CO¡t + cos (Ot