Page 176 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Chap. 5 Problems 163

5-21 Set up the matrix equation of motion for the system shown in Fig. P5-21 using
coordinates Xj and X2 at m and 2m. Determine the equation for the normal mode
frequencies and describe the mode shapes.

Figure P5-21.

5-22 In Prob. 5-21, if the coordinates x at m and 0 are used, what form of coupling will
result?
5-23 Compare Probs. 5-9 and 5-10 in matrix form and indicate the type of coupling present
in each coordinate system.
5-24 The following information is given for the automobile shown in Fig. P5-24.
W = 3500 1b = 2000 1b/ft
/, = 4.4 ft ^2 = 2400 1b/ft
I2 = 5.6 ft
r = 4 ft = radius of gyration about c.g.
Determine the normal modes of vibration and locate the node for each mode.

Figure P5-24.

5-25 Referring to Problem 5-24 prove in general that the uncoupled natural frequencies are
always between the coupled natural frequencies.
5-26 For Problem 5-24, if we include the mass of the wheels and the stiffness of the tires,
the problem becomes that of 4 DOF. Draw the spring-mass model and show that its
equation of motion is
' m 1
1 ^ X
J \ e
-1
1^0 I

1 ^ 0- 1 ^
(*, + kj) (^2^2 ^ 1^1) 1 -ki 0\
(kili-kA) -h k2l\^ 1 k^l^ 0
=
- *1 /C,/, 1 {kf^ + k^) 0 ■^1 0
-k2 -k^lj ' 0 (*o + ^ 2) < ^2} 10;