A torsional system consists of two discs mounted on a shaft, as shown in Fig. 1.

Fig 1. Two DOF Torsional System

The shaft can be divided into three segments, having rotational spring constants as k_t1, k_t2, and k_t3, as shown in Figure 1.
* Mass moment of inertia of the discs are J₁ and J₂, and M_t1 and M_t2 are the applied torque and the rotational degree of freedoms are θ₁ and θ₂.

The differential equations of rotational motion for the discs can be written as,

Consider a harmonic oscillation where θ_i = Θ_i sin ωt. The frequencies can be obtained by equating the determinant to zero,

And,

Which is the characteristic equation.
The roots of the equations are,

where ω₁ and ω₂ are the natural frequencies of the system.