Question

The disc of a torsional pendulum has a

moment of inertia of 600 kg-cm2 and is

immersed in a

viscous fluid. The brass shaft attached to

it is of 10 cm diameter and 40 cm long.

when the pendulum is vibrating, the

observed amplitudes on the same side of

the rest position for successive cycles are

90, 60, 40. Determine 1) logarithmic

decrement

2) damping torque at unit velocity 3) the

periodic time of vibration

assume for the brass shaft g= 4.4 * 10

Answer 1

Answer:

fₙ = ωn/2π

Explanation:

Step 1:

Determine the Circular Natural frequency

ωₙ= √s/m

Step 2:

Determine the Damping Coefficient C using below method x1 = 9 deg, x2= 6 deg and x3 = 4 deg

Convert x1,x2 and x3 in radians We know that

x₁/x₂=x₂/x₃

Also

[x1/x3] = [x1/x2][x2/x3] = [x1/x2]2 or

[x1/x2] = [x1/x3]1/2

Also we know that

㏒e(x₁/x₂)=ax²/√(ωn)²-a²

Where a = c/2m

From the above expression, Determine the value of ‘c’

‘c’ is the Damping force per unit velocity [in N/m/s]

Step 3:

Determine cc using cc = 2mωn

Determine Logarithmic Decrement using

δ=2πc/√(сc)²-c²

Step 4:

Time period tp = 1/fn

For Damped vibration fₙ = ωd/2π where ωd = Sqrt [ωn2-a2] and a = c/2m

Determine Periodic time of vibration tp from the above expression Step 5:

For Undamped vibration [if disc is removed from viscous fluid]

Determine natural frequency using

fₙ = ωn/2π