Consider ashaft of negligilbe mass whose one end is fixed and the other end carrying a disc as shown in Fig.
Let θ = Angular displacement of the shaft from mean position after time t in radians,
m = Mass of disc in kg,
I = Mass moment of inertia of disc in kg-m2 = m.k2,
k = Radius of gyration in metres,
q = Torsional stiffness of the shaft in n-m.
Restoring force = q.θ ...(i)
and accelerating force = I x d2θ/dt2 ...(ii)
Equating equations (i) and (ii), the equation of motion is
The fundamental equation of the simple harmonic motion is
d2θ/dt2 + ω2. x = 0
Comparing equations (iii) and (iv),
Note : The value of the torsional stiffness q may be obtained from the torsion equation,
where C = Modulus of rigidity for the shaft material,
J = polar moment of inertia of the shaft cross-section;
π/32 d4; d is the diameter of the shaft, and l = Length of the shaft.
