GATE ME 2019 Set 2 — Question 51
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Theory of Machines & Vibrations → Balancing, Flywheels & Gyroscopes → Gyroscope & Gyroscopic Couple
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Question
A uniform disc with radius and a mass of kg is mounted centrally on a horizontal axle of negligible mass and length of . The disc spins counter-clockwise about the axle with angular speed , when viewed from the right-hand side bearing, . The axle precesses about a vertical axis at in the clockwise direction when viewed from above. Let and (positive upwards) be the resultant reaction forces due to the mass and the gyroscopic effect, at bearings and , respectively. Assuming and , the ratio of the larger to the smaller bearing reaction force (considering appropriate signs) is ______.
Correct answer
-3
Solution
1.Identify given parameters:
- Mass of disc =
- Radius of disc =
- Length of axle
- Distance from center to each bearing and is .
- Spin speed (CCW from right vector points right, ).
- Precession speed (CW from top vector points down, ).
The weight acts downwards at the center. The static reactions at and are equal and upward:
(positive upwards).
3.Calculate Gyroscopic Couple:
- Mass moment of inertia of the disc .
- Active gyroscopic couple magnitude .
- Direction of active couple: (out of page).
- Reactive gyroscopic couple: (into page).
- A reactive couple vector into the page (clockwise in the plane) creates an upward force at and a downward force at .
- Magnitude of reaction due to couple: .
- (upwards)
- (downwards)
- The larger force is and the smaller force is .
- Ratio .
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