Critical Speeds of Shafts

Why it matters

Understanding the critical speeds of shafts is essential for mechanical engineers to prevent resonance in rotating machinery, which can lead to excessive vibrations and potential failure. This knowledge is crucial in the design and maintenance of equipment such as turbines, engines, and pumps.

Key ideas

• Critical Speed: The speed at which the natural frequency of a rotating shaft coincides with the frequency of rotation, causing resonance.
• Resonance: A condition where the amplitude of vibration becomes excessively large due to the matching of natural frequency and external frequency.
• Damping: The presence of damping can reduce the amplitude of vibrations at critical speeds.
• Factors Affecting Critical Speed: Shaft material, length, diameter, and support conditions (e.g., simply supported, fixed).

Model behind the static-deflection formula

For a simplified single-disk rotor on a massless elastic shaft, let δ be the static shaft deflection at the disk under that disk’s weight, with the same supports used in the vibration model. Since kδ = mg, ω_n = √(k/m) = √(g/δ). The synchronous undamped critical speed is ω_c = ω_n; divide by 2π for Hz or multiply by 60/(2π) for rpm. Multiple disks, distributed shaft mass, bearing flexibility, and gyroscopic effects require a richer model.

Formulas

• n_c = (1/2π) * √(g/δ)
  • n_c: Critical speed (Hz)
  • g: Acceleration due to gravity (9.81 m/s²)
  • δ: Static deflection of the shaft (m)

Worked example

Given: A single disk on a simply supported massless elastic shaft gives a static deflection of 0.002 m under its own weight. Neglect damping and gyroscopic effects.

1. Identify the formula: Use n_c = (1/2π) * √(g/δ).
2. Substitute the values: n_c = (1/2π) * √(9.81 / 0.002).
3. Calculate: n_c = (1/6.2832) * √(4905).
4. Result: n_c ≈ 11.14 Hz.

Final Answer: 11.14 Hz, approximately 669 rpm

Common mistakes

• Confusing critical speed with operating speed.
• Ignoring the effects of damping.
• Incorrectly calculating static deflection.

For GATE ME

Questions often involve calculating the critical speed of a shaft given its dimensions and material properties. Practice problems involving different support conditions and damping effects.

Quick check

1. What is critical speed?
2. How does damping affect critical speed?
3. What is the formula for calculating critical speed?

Answers: 1. The speed at which resonance occurs. 2. It reduces vibration amplitude. 3. n_c = (1/2π) * √(g/δ).