Whirling of Shaft Experiment — Critical Speed Determination Procedure and Calculations

The whirling of shaft (or critical speed of shaft) experiment is performed in the Theory of Machines laboratory to understand the phenomenon of resonance in rotating shafts. When a shaft rotates at its natural frequency, the amplitude of lateral vibration becomes very large — this speed is called the critical speed or whirling speed. Engineers must design rotating shafts to operate well away from critical speeds to avoid catastrophic failure.

Aim of the Experiment

To determine the critical speed (whirling speed) of a rotating shaft experimentally and compare with the theoretical value using the Dunkerley method and Rayleigh-Ritz method.

Theory

What is Critical Speed?

Every rotating shaft has a transverse natural frequency. When the rotational speed of the shaft equals this natural frequency, resonance occurs and the shaft deflects laterally to very large amplitudes — this is called whirling. The speed at which this occurs is the critical speed (Nc).

Critical Speed for a Simply Supported Shaft (no load)

For a uniform simply supported shaft:

Nc = (π/2L²)√(EI/m) rev/s

where:
E = Young’s modulus of shaft material (Pa)
I = Second moment of area of shaft cross-section (m⁴) = πd⁴/64
m = mass per unit length of shaft (kg/m)
L = length of shaft between supports (m)

Critical Speed for Shaft with Disc Load — Dunkerley’s Method

When one or more disc loads are mounted on the shaft, Dunkerley’s empirical formula gives the combined critical speed:

1/Nc² = 1/N₁² + 1/N₂² + 1/N₃² + … (for each disc)

where N₁, N₂, N₃ are the individual critical speeds considering each disc alone.

For a simply supported shaft with a central disc of mass m at distance a from one end and b from the other:

Static deflection under disc: δ = (m × g × a² × b²)/(3EIL)

Individual critical speed: N = (1/2π)√(g/δ) rev/s

Rayleigh-Ritz Method

The Rayleigh-Ritz method gives the fundamental frequency by equating maximum strain energy to maximum kinetic energy:

ωn² = (ΣW × y) / (ΣW × y²/g)

This gives a slightly better (higher) estimate than Dunkerley’s method.

Apparatus Required

• Whirling of shaft apparatus (two adjustable bearing supports on a rigid frame, variable speed motor)
• Shafts of different diameters (typically 6 mm, 8 mm, 10 mm diameter steel shafts)
• Disc loads (brass or steel discs of known mass)
• Tachometer (non-contact optical for safety)
• Speed controller unit
• Vernier caliper (for shaft diameter measurement)
• Steel rule (for shaft length measurement)

Procedure

1. Measure the shaft diameter (d), length between supports (L), and mass per unit length (m = ρ × A = ρπd²/4).
2. Mount the shaft between the bearings (simply supported end conditions for standard experiment).
3. If using disc loads, mount the disc(s) at their specified positions along the shaft and record position(s).
4. Start the motor at the lowest speed setting. Gradually increase speed using the speed controller.
5. Observe the shaft through a protective shield. As speed increases, the shaft will begin to deflect laterally.
6. Note the speed at which maximum lateral deflection (whirling) occurs — this is the first critical speed (Nc₁).
7. Record the speed using the tachometer. Do not continue at this speed — immediately increase beyond critical speed. The shaft will straighten as speed exceeds Nc₁.
8. Continue increasing speed to observe the second critical speed if the apparatus permits.
9. Reduce speed gradually and stop the motor.
10. Repeat with different shaft diameters and/or disc positions.

Safety Note

Critical: The whirling speed experiment involves rapidly rotating flexible shafts. Always use the protective safety guard provided. Never reach into the apparatus during operation. The shaft can strike the guard during whirling — this is normal but requires the guard to be in place.

Observation Table

Sample Calculation

Given: d = 8 mm, L = 700 mm, E = 200 GPa (steel), ρ = 7850 kg/m³, central disc mass M = 0.5 kg

I = πd⁴/64 = π(0.008)⁴/64 = 2.01 × 10⁻¹⁰ m⁴

m = ρ × π × d²/4 = 7850 × π × (0.008)²/4 = 0.395 kg/m

Critical speed (no load): Nc₀ = (π/2L²)√(EI/m) = …

Static deflection (central disc): δ = MgL³/(48EI) = …

Nc_disc = (1/2π)√(g/δ) = …

Dunkerley: 1/Nc² = 1/Nc₀² + 1/Nc_disc² → Nc = …

Result

• Experimental critical speed = _____ rpm
• Theoretical critical speed (Dunkerley) = _____ rpm
• Percentage error = _____ %
• The shaft was observed to whirl at the critical speed and run straight above it, confirming the theory.

Viva Questions

1. What is the critical speed of a shaft? Why is it important in machine design?
2. Why does a shaft straighten again after passing the critical speed?
3. What is Dunkerley’s method and what is its limitation?
4. How does increasing shaft diameter affect the critical speed?
5. How does adding a disc load to the shaft affect the critical speed?
6. A turbine shaft operates at 3000 rpm. How would you ensure it does not suffer whirling?
7. What is the difference between the first and second critical speed?

Lab Equipment Featured in This Guide

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