Consider a shaft of negligible mass whose upper end is fixed and the lower end carries a heavy disc. If the disc is given a twist about its vertical axis and then released, it will start oscillating about the axis, which are known as Torsional vibrations.
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.
At any instant, the torque acting on the disc are:
(1) Inertia torque and
(2) Restoring torque or restoring force (or spring torque)
The inertia torque is equal to accelerating torque but opposite in direction.
Equating equations (i) and (ii), the equation of motion is
The fundamental equation of the simple harmonic motion is
Comparing equations (iii) and (iv),
Note :
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