Gear-pair Element Equations : Equations (7.109) and (7.110) can be combined in the stiffness matrix form as

(7.111)

On combining equations (7.106) and (7.111), we get

which is the equation of motion of the gear-pair element. It should be worth mentioning that φ_zg2 is the actual rotational displacement of right hand end of the shaft that is attached to gears 2. Hence, no scaling is required of the mode shapes obtained from the above formulation. Following examples will illustrate the use of the gear-pair element for the geared and branched systems by using FEM.

Example 7.8 For a simple-gear rotor system as shown in Figure 7.23, find torsional natural frequencies. 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 G equals to 0.8 × 10¹¹ N/m² , the polar mass moment of inertia of discs and gears are I_pA = 24 Nm² , I_pB = 10 Nm² , I_pgA = 5 Nm² , and I_pgB = 3 Nm² .

Figure 7.23 Two-discs with a geared system

Solution : Now this problem will be solved by using the FEM for illustration of the method to geared system. The pinion and the gear have appreciable amount of the polar mass moment of inertia. Let us divide the geared system in two elements (Fig. 7.23). Denote the node number of the disc on branch A as 1, the gear as 2, the gear as 3 and the node number of the disc on branch B as 4 (Fig. 7.23).

The following data could be obtained for the present rotor system:

Now elemental equations will be written one by one for each element, and the same would be then assembled to get global equation of motion of the geared system.