which gives relative amplitudes of two discs as
 |
(6.13) |
The second mode shape from equation (6.13) represents the case when both masses oscillate in anti-phase with each other (i.e., the direction of rotation of one disc will always be opposite to the other). Both discs will reach their extreme angular positions simultaneously, and both will reach the static equilibrium (untwisted) position also simultaneously. It should be noted that both the discs have same frequency of oscillation (i.e., the time period) but different angular amplitude. A large disc will have lesser amplitude of oscillations as compared to the smaller disc. Figure 6.7 shows this mode shape of the two-rotor system. From two similar triangles in Figure 6.7(a), we have
 |
(6.14) |
where
are node position from discs 1 and 2, respectively (Fig. 6.7a). Since both the masses are always vibrating in the opposite direction, there must be a point on the shaft where the torsional vibration is not taking place, i.e. where the angular displacement is zero. This point is called a node and its location is at present unknown. The location of the node may be established by treating each end of the real system as a separate single-disc cantilever system as shown in Figure 6.7(b and c). The node is treated as the point, where the shaft is rigidly fixed. Hence, basically we will have two single-DOF cantilever rotor systems in torsion (Fig. 6.7b and c) instead of one two-DOF free-free rotor system (Fig. 6.4). Since value of the torsional natural frequency,
is known from equation ((6.11)the frequency of oscillation of each of the single-disc cantilever system must be same and is equal to 
), hence we write
 |
(6.15) |
where is defined by equation (6.11), and 
are the torsional stiffness of two single-DOF cantilever rotor systems, which can be obtained from equation (6.15), as
Lengths 
of these two single-DOF rotor system can be obtained as
 |
(6.16) |
which will give the node position. It should be noted that the shear stress would be maximum at the node point being a fixed end of cantilever rotor systems.