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Document Title
Fundamentals of Stress and Vibration Chapter Title
[A Practical guide for aspiring Designers / Analysts] 2. Engineering Mechanics

The angular momentum along the ‘y-axis’ is much smaller than the angular momentum along the
‘x-axis’. Therefore, we get:

Net angular momentum = I ω i + I Ω j
p
d
= I ω i (neglecting I assuming the disc to be thin)
p
d
Differentiating the net angular momentum, we get:

di
I ω , where i is the radius vector.
p
dt
Since the tangent vector is along the negative ‘z − axis’, we get:

di dθ

I ω = I ω −k Ω = −I ωΩ k - - - - (2.67)
p
p
p
dθ dt
Equation (2.67) can be vectorially represented as follows:
−I ωΩ k = Ω j × I ω

p
p
Since the gyroscopic torque is an inertia torque, the reaction must be considered on the shaft, which

is along the positive ‘z-axis’ k .
This means that, the gyroscopic torque goes to balance the gravity torque on the shaft, as shown in
[Fig 2.80], which is along the negative ‘z-axis’ −k .

[Fig 2.80: Forces on the disc shaft system]

Vectorially, we have:

Gravity torque = r × mg = r i × mg −j = −r mg k
For equilibrium, the bearing also experiences a vertical upward reaction equal to ‘mg’

Page 90 QP No. SSC/Q4401, Version 1.0, NSQF Level 7, Compliant with Aero and Auto Industries,