In chapter 5 gyroscopic couple effect of disc alone was described and the gyroscopic matrix is given as (please note that the disc is at node 2)

It could be seen that the gyroscopic effect leads to coupling of motions in y-z and z-x planes. It can be expanded in the following form to accommodate both the nodes of the shaft element

In equations(10.1)and(10.2)now gyroscopic couple effect (equation(10.4)) would be added and equations of motion of the element, in both planes, can be written as

The above formulation could alternatively be obtained by a more formal approach by using the additional kinetic energy term of the disc developed in Chapter 5 due to gyroscopic moments in finite element formulations of Euler-Bernoulli beams as described in Chapter 9, which will lead to the similar results. This will be illustrated while developing the finite element formulation with the Timoshenko beam model (in which the shear deformation, rotary inertia and gyroscopic effects of shafts will also be considered) in subsequent sections. For the present illustration of an overhang rotor, boundary conditions are given as

After application of boundary conditions(10.6)in equation (10.5), on elimination of the first four and columns the equations of motion reduces to