Short Notes: Critical Speeds of Shafts | Short Notes for Mechanical Engineering PDF Download

 Page 1


Critical Speeds of Shafts 
Critical Speed
• 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) u)2
v
e
when u)n = u j
Rotor
Shaft with vibrate
Critical speed,
Page 2


Critical Speeds of Shafts 
Critical Speed
• 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) u)2
v
e
when u)n = u j
Rotor
Shaft with vibrate
Critical speed,
where, c j = 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,
kv = mealy
Critical speed for Simple Shaft
• Bending Critical Speed: We can also write function as total displacement
r,,, = Re'(w t'^
where,
R =
m c o ~ a
(k - m co2)+ (cco)2 
Hence, dynamic magnifier and phase angle.
H (a ;) = - =
a ^ /(l-r : ): + (2£r):
9 — tan'
ill
1 - r 2
For an undamped rotor resonance occurs, 
When u j = ujn
Also at resonance, e p = 90° 
mu)2a = cojR
R m c o 
a c
1
or = —
tTJX ^ >
Distance of the centre of gravity from the bearing axis or whirl amplitude
Rl = R '+ a1 + 2Ra cos < f >
and R0 = a
1+ (2 ;r)*
(1+ /•*)*+U n ­
critical Speed for Multi-mas System
• Bending Critical speed
° The synchronous whirl frequency increases with the rotational speed 
linearly and can be represented 1 x rev excitation frequency, whenever, 
this excitation line intersects the natural frequencies, critical speeds
occur.
Page 3


Critical Speeds of Shafts 
Critical Speed
• 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) u)2
v
e
when u)n = u j
Rotor
Shaft with vibrate
Critical speed,
where, c j = 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,
kv = mealy
Critical speed for Simple Shaft
• Bending Critical Speed: We can also write function as total displacement
r,,, = Re'(w t'^
where,
R =
m c o ~ a
(k - m co2)+ (cco)2 
Hence, dynamic magnifier and phase angle.
H (a ;) = - =
a ^ /(l-r : ): + (2£r):
9 — tan'
ill
1 - r 2
For an undamped rotor resonance occurs, 
When u j = ujn
Also at resonance, e p = 90° 
mu)2a = cojR
R m c o 
a c
1
or = —
tTJX ^ >
Distance of the centre of gravity from the bearing axis or whirl amplitude
Rl = R '+ a1 + 2Ra cos < f >
and R0 = a
1+ (2 ;r)*
(1+ /•*)*+U n ­
critical Speed for Multi-mas System
• Bending Critical speed
° The synchronous whirl frequency increases with the rotational speed 
linearly and can be represented 1 x rev excitation frequency, whenever, 
this excitation line intersects the natural frequencies, critical speeds
occur.
Campbell diagram
Durkerley's lower bound approximation,
Durkertey’s lower bound approximation
Considering n degree of freedom,
1 1 1 1 ^
Where influence coefficient
a ,
Rayleigh’s upper bound approximation,