Question

Derive the expression for the frequency of free torsional vibrations for a shaft fixed at one end and carrying a load on the free end. 

Answer 1

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.