Critical Speed of Shaft- 

• The critical speed essentially depends on
• Critical or whirling or whipping speed is the speed at which the shaft tends to vibrate violently in transverse direction.
  • The eccentricity of the C.G of the rotating masses from the axis of rotation of the shaft.
  • Diameter of the disc
  • Span (length) of the shaft, and
  • Type of supports connections at its ends.

For equilibrium,

ky = m(y+e) ω²

y→∞, when ω_n = ω

Critical speed,

where, ω = Angular velocity of shaft

k = stiffness of shaft

e = Initial eccentricity of centre of mass of rotor.

m = Mass of rotor

y = Additional of rotor due to centrifugal force

Dynamic force on the bearings, 

Critical speed for Simple Shaft

• Bending Critical Speed: We can also write function as total displacement

r_ω = Re^i(ωt-φ)

where, 

Hence, dynamic magnifier and phase angle.

For an undamped rotor resonance occurs,

When ω = ω_n

Also at resonance, φ = 90°

mω²a = cωR

Distance of the centre of gravity from the bearing axis or whirl amplitude

Critical Speed for Multi-mas System

• Bending Critical speed
  • The synchronous whirl frequency increases with the rotational speed linearly and can be represented 1 × rev excitation frequency, whenever, this excitation line intersects the natural frequencies, critical speeds occur.

Durkerley’s lower bound approximation,

Considering n degree of freedom,

Where the influence coefficient 

Rayleigh’s upper bound approximation,