where k_B is the stiffness of shaft B. At the gear pair, the following conditions hold

(e)

where is the angular displacement. Equation (e) can be combined as

(f)

or

(g)

with the gear ratio transformation matrix is given as

(h)

Noting equation (g), equation (c) can be written as

(i)

Noting equation (a), equation (i) can be expressed as

(j)

with the overall transformation matrix is given as

(k)

The overall transformation matrix can also be written as

(l)

Boundary conditions of the problem are  since both ends are free. Equation (j) can be written in expanded form as

(m)