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Kruger-CriticalSpeeds-Shafts.pdf

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Kruger-CriticalSpeeds-Shafts.pdf
Technical Bulletin
TBN017.0/1998
CRITICAL SPEED OF SHAFTS
All rotating shaft, even in the absence of
external laod, deflect during rotation.
The combined weight of a shaft and
wheel can cause deflection that will
create resonant vibration at certain
speeds, known as Critical Speed.
suggests that the maximum operation
speed should not exceed 75% of the
critical speed.
Criticalspeed,Nc =
The magnitude of deflection depends
upon the followings :-
30
π
g
δst
where :
g = gravity acceleration (9.81 m/s2)
δst = total maximum static deflection
(a) stiffness of the shaft and it’s support
(b) total mass of shaft and attached parts
(c) unbalance of the mass with respect
to the axis of rotation
(d) the amount of damping in the
system
Critical speed depend upon the
magnitude or location of the load or
load carried by the shaft, the length of
the shaft, its diameter and the kind of
bearing support.
Therefore, the calculation of critical
speed for fan shaft is necessary.
Critical Speed Equation (Nc)
Total Maximum Static Deflection (δst)
There are two method used to calculate
critical
speed,
Rayleigh-Ritz
and
Dunkerley Equation. Both the RayleighRitz and Dunkerley equation are an
approximations to the first natural
frequency of vibration, which is assumed
to be nearly equal to the critical speed of
rotation.
The maximum static deflection, δst, is
obtained by adding both the maximum
static deflection of the rotating shaft and
the load.
(1) Maximum static deflection on shaft
(δstI)
In general, the Rayleigh-Ritz equation
overestimates and the Dunkerley
equation underestimates the natural
frequency.
1.1)
The equation illustrated below is the
Rayleigh-Ritz equation, good practice
δstI=
1-5
5wL3
384EI
Technical Bulletin - TBN017.0/1998
1.2)
2.3)
δstI=
wL3
8EI
δst2=
WA(3L2 − 4A 2 )
24EI
δst2=
WL3
3EI
(2) Maximum static deflection on load
only (δst2)
2.4)
2.1 )
δst2=
WL
48EI
3
where :
2.2 )
δst2=
w = weight of shaft, kg
W = weight of wheel, kg
E = modulus of elasticity, kg/m2
for shaft C40=200 x108 kg/m2
I = moment of inertia=πD4/64, m4
L = length of shaft, m
WB(L2 − B2 ) 3 / 2
9 3EIL
Shaft Diameter
D (mm)
20
25
30
35
40
45
50
55
60
70
Moment of inertia
I (m4)
7.85 x 10-9
19.17 x 10-9
39.76 x 10-9
73.66 x 10-9
125.66 x 10-9
201.29 x 10-9
306.79 x 10-9
449.18 x 10-9
636.17 x 10-9
1178.59 x 10-9
Table I
2-5
Weight per metre
(kg/m)
2.47
3.85
5.51
7.99
9.87
13.00
15.40
18.70
22.20
30.20
Technical Bulletin - TBN017.0/1998
Example 1
Given the following specifications, find
the critical speed.
(b) Total maximum static deflection
(δst)
δst = δst1 + δst2
= 0.00018 + 0.000139
= 0.000319 m
Model : KAT 15/15 S2
with 2-bearings
(d) Critical Speed (Nc)
Diameter of shaft, D = 40 mm
Weight of wheel, W = 7.5 kg
= 1.37 m
Shaft length, L
= 0.205 m
Length, A
Moment of inertia, I = 125.66 x 10-9 m4
Modulus of Elasticity, E= 200 x108 kg/m2
Nc =
=
(C40)
Shaft weight, w=1.37 x 9.87
5wL3
---------------- refer to Fig. 1.1
384EI
5(13.52)(1.37 ) 3
384(200x10 8 )(125.66x10 −9 )
=0.00018 m
(b) Deflection from load only (δst2)
δst2=
=
30
9.81
π 0.000319
Safety factor 25%, therefore max.
operation speed = 1675 x 0.75
= 1256 rpm
(a) Deflection from shaft weight only
(δst I)
=
g
δst
=1675 rpm
=13.52 kg ---- refer to Table I
δstI=
30
π
WA(3L2 − 4A 2 )
--refer to Fig. 2.3
24EI
7.5(0.205)[3(1.37 ) 2 − 4(0.205) 2 ]
24(200 x10 8 )(125 .66x10 −9 )
=0.000139 m
3-5
Technical Bulletin - TBN017.0/1998
Example 2
(b) Deflection from load only (δst2)
To check critical speed for KAT 12/12 S3
with 2-bearing, one side of the bearing
overhung.
δst2=
=
WA(3L2 − 4 A 2 )
24EI
5.4(0.197 )[3(1.114 ) 2 − 40(0.197 ) 2
24(200 x10 8 )(73.66x10 −9 )
=0.000107 m
Diameter of shaft, D = 35 mm
Weight of wheel, W = 5.4 kg
Moment of inertia, I = 73.66 x 10-9 m4
Modulus of Elasticity, E= 200 x108 kg/m2
(c) Total maximum static deflection
(δst)
(C40)
δst =δstI+δst2
=0.000109 +0.000107
Check Critical Speed For Long Span
=0.000216m
(d) Critical speed for long span (Nc)
Nc =
Length, A
Length, L
Shaft weight,w
= 0.197 m
= 1.114 m
= 8.9 kg
=
(a) Deflection from shaft weight (δst I)
δstI=
=
30
π
30
π
g
δst
9.81
0.000216
=2035rpm
5wL
384EI
3
Safety Factor 25%, therefore max.
operation speed = 2035 rpm x 0.75
= 1526 rpm
5(8.9)(1.114 ) 3
384(200 x10 8 )(73.66x10 −9 )
=0.000109 m
4-5
Technical Bulletin - TBN017.0/1998
Check Critical Speed For Overhung
(d) Critical Speed at overhung (Nc)
Nc =
=
Length, A = 0.5215
Length, L = 0.534 m
shaft weight,w = 4.27 kg
9.81
0.000228
Conclusion
wL3
8EI
=0.000055 m
Long Span
Critical Speed
Max. operation speed
= 2035 rpm
= 1526 rpm
Overhung
Critical Speed
Max. operation speed
= 1980 rpm
= 1485 rpm
Therefore, the max. operation speed for
this KAT 12/12 S3 should be according
to the overhung, ie. whichever lesser,
which is = 1485 rpm
(b) Deflection from load only (δst2)
=
30
π
Safety factor 25%, max. operation
speed = 1980 x 0.75 = 1485 rpm
4.27(0.534) 3
=
8(200x10 8 )(73.66x10 −9 )
δst2=
g
δst
= 1980rpm
(a) Deflection from shaft weight only
(δstI)
δstI=
30
π
WA 3
3EI
5.4(0.5215) 3
3(200 x10 8 )(73.66x10 −9 )
=0.000173 m
(b) Total maximum static deflection
(δst)
δst =δstI+δst2
=0.000055+0.000173
=0.000228m
5-5
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