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CRITICAL SPEED OF SHAFT

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

03-Critical-speeds-of-shafts (1)

y, when Ï‰n = Ï‰

03-Critical-speeds-of-shafts (2)

Critical speed,

image001

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, 03-Critical-speeds-of-shafts (3)

Critical speed for Simple Shaft

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

rω = Rei(ωt-φ)

where, 03-Critical-speeds-of-shafts (4)

Hence, dynamic magnifier and phase angle.

03-Critical-speeds-of-shafts (5)

For an undamped rotor resonance occurs,

When Ï‰ = ωn

Also at resonance, Ï† = 90°

mω2a = cωR

image002

03-Critical-speeds-of-shafts (6)

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

03-Critical-speeds-of-shafts (7)

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.

03-Critical-speeds-of-shafts (8)

Durkerley’s lower bound approximation,

03-Critical-speeds-of-shafts (9)

Considering n degree of freedom,

03-Critical-speeds-of-shafts (10)

Where the influence coefficient 03-Critical-speeds-of-shafts (11)

Rayleigh’s upper bound approximation,

03-Critical-speeds-of-shafts (12)


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