Example 6.11 For a simple geared system as shown in Figure 6.32 find torsional natural frequencies and mode shapes. Find also the location of the node point on the shaft (i.e., the location of the point where the angular twist during torsional vibrations is zero). The shaft ‘A’ has the diameter of 5 cm and the length of 0.75 m, and the shaft ‘B’ has the diameter of 4 cm and the length of 1.0 m. Take the modulus of rigidity of the shaft ,the polar mass moment of inertia of discs are . Neglect the inertia of gears. 

Figure 6.32 A two-disc geared system

Solution: On taking shaft B as the input shaft (or the reference shaft) as shown in Figure 6.33, the gear ratio can be defined as

where D is the nominal diameter of the gear.

Figure 6.33 A geared system