Page 146 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 5.1 The Normal Mode Analysis 133

a;? = 0 .6 3 4 ^

Figure 5.1-2. Normal modes of the system shown in Figure 5.1-1.

Each normal mode oscillation can then be written as
( 1)
_ . . 0.731 . . . , , , X

—^ i\ 1 1.00 } sin (w,/ + (//,)
- A , -2.73 sin + ^ 2)
1.00
These normal modes are displayed graphically in Fig. 5.1-2. In the first normal mode,
the two masses move in phase; in the second mode, the two masses move in
opposition, or out of phase with each other.
Example 5.1-2 Rotational System
We now describe the rotational system shown in Fig. 5.1-3 with coordinates 0^ and 62

measured from the inertial reference. From the free-body diagram of two disks, the
torque equations are
7,0, = -K,e, + K2{02 - 0,)
(5.1-8)
7202 = - ^ 2(^2 - ^1) - ^ 3^2
It should be noted that Eqs. (5.1-8) are similar in form to those of Eqs. (5.1-1) and
only the symbols are different. The rotational moment of inertia 7 now replaces
the mass m, and instead of the translational stiffness /c, we have the rotational
stiffness K.

Figure 5.1-3.