Page 217 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

204 Properties of Vibrating Systems Chap. 6

6-30 In a manner similar to Prob. 6-2:9, show that
= 0, 1, 2, .
6-31 Evaluate the numerical coefficients for the equations of motion for the second and
third modes of Example 6.10-E
6-32 If the acceleration ¿i(t) of the ground in Example 6.10-1 is a single sine pulse of
amplitude and duration /|, as shown in Fig. P6-32, determine the maximum q for
each mode and the value of given in Sec. 6.10.

Figure P6-32.

6-33 The normal modes of the double pendulum of Prob. 5-9 are given as
o>| = , ft»2 = 1-850-y/j

0.707 \
4> i 1.00 / ’
(1)
- 0.707 \
1.00 /
If the lower mass is given an impulse determine the response in terms of the
normal modes.
6-34 The normal modes of the three-mass torsional system of Fig. P6-6 are given for
/, = ^2 = J3 and K^=K^ = K^.
(0.328 \ J(i)\
= 0.591 , ~k~ 0.198,
(0.737)

1 0.737
4 > 2 = 0.328 1.555,
i -0.591
( 0.591
= -0.737 3.247
0.328
Determine the equations of motion if a torque M(t) is applied to the free end. If
M(t) = M(^u(t), where w(r) is a unit step function, determine the time solution and the
maximum response of the end mass from the shock spectrum.
6-35 Using two normal modes, set up the equations of motion for the five-story building
,
whose foundation stiffness in translation and rotation are A:, and = 00 respectively
(see Fig. P6-35).