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All content in this area was uploaded by Ehsan Abdi on Apr 29, 2021
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B(
θ
,t)=
π
2B1cos( p1
θ
+
ω
1t+
φ
1)+B2cos( p2
θ
+
ω
2t+
φ
2)
⎡
⎣⎤
⎦
1
()
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v(
θ
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v
[1
((2 p
1
)
2
−1)
2
B
1
2
cos(2 p
1
θ
+2
ω
1
t+2
φ
1
)
+1
((2 p
2
)
2
−1)
2
B
2
2
cos(2 p
2
θ
+2
ω
2
t+2
φ
2
)
+2
(( p
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−p
2
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2
−1)
2
B
1
B
2
cos(( p
1
−p
2
)
θ
+(
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−
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2
)t+(
φ
1
−
φ
2
))
+2
(( p
1
+p
2
)
2
−1)
2
B
1
B
2
cos(( p
1
+p
2
)
θ
+(
ω
1
+
ω
2
)t+(
φ
1
+
φ
2
))]
Kv
=3D
a
D
c
3
π
2
64E
ym
μ
0
y
c
3
3
()
B(
θ
,t)=
μ
0
g
−1
(
θ
,t)D
a
2J
m
(
θ
,t)d
θ
∫
g
−1
(
θ
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0
−1
+g
s1
−1
cos(
θ
+
φ
s1
)+g
d1
−1
cos(
θ
+
ω
r
t+
φ
d1
)+...
=g
0
−1
1+d
s1
cos(
θ
+
φ
s1
)+d
d1
cos(
θ
+
ω
r
t+
φ
d1
)+...
⎡
⎣⎤
⎦
J
m
(
θ
,t)=ˆ
J
m1
cos( p
1
θ
+
ω
1
t+
φ
1
)+ˆ
J
m2
cos( p
2
θ
+
ω
2
t+
φ
2
)
B
i
=D
a
π
g
0
−1
μ
0
p
i
ˆ
J
mi
B
1
B
2
B
1
d
s
1
/2
B
1
d
s
1
/2
B
2
d
s
1/2
B
2
d
s
1/2
B
1
d
d
1/2
B
1
d
d
1/2
B
2
d
d
1/2
B
2
d
d
1/2
Authorized licensed use limited to: UNIVERSITY OF SOUTHAMPTON. Downloaded on April 29,2021 at 13:22:03 UTC from IEEE Xplore. Restrictions apply.
%%
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p
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1
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2/(2p1)4
2
p
2
2
ω
2
KvB2
2/(2p2)4
p
1
−
p
2
ω
1
−
ω
2
K
v
2B
1
B
2
/(p
1
−p
2
)
4
p
1
+
p
2
ω
1
+
ω
2
K
v
2B
1
B
2
/(p
1
+p
2
)
4
K
v
2d
s1
B
1
2
K
v
2d
s1
B
2
2
2
p
1
−1
2
ω
1
K
v
d
s1
B
1
2
/(2p
1
−1)
4
2
p
1
+1
2
ω
1
K
v
d
s1
B
1
2
/(2p
1
+1)
4
2
p
2
−1
2
ω
2
Kvds1B2
2/(2p2−1)4
2
p
2
+1
2
ω
2
Kvds1B2
2/(2p2+1)4
p
1
−p
2
−1
ω
1
−
ω
2
K
v
2d
s1
B
1
B
2
/(p
1
−p
2
−1)
4
p
1
−p
2
+1
ω
1
−
ω
2
K
v
2d
s1
B
1
B
2
/(p
1
−p
2
+1)
4
p
1
+p
2
−1
ω
1
+
ω
2
K
v
2d
s1
B
1
B
2
/(p
1
+p
2
−1)
4
p
1
+
p
2
+1
ω
1
+
ω
2
K
v
2d
s1
B
1
B
2
/(p
1
+p
2
+1)
4
ω
r
K
v
2d
d1
B
1
2
ω
r
K
v
2d
d1
B
2
2
2
p
1
−1
2
ω
1
−
ω
r
Kvdd1B1
2/(2p1−1)4
2
p
1
+1
2
ω
1
+
ω
r
Kvdd1B1
2/(2p1+1)4
2
p
2
−1
2
ω
2
−
ω
r
Kvdd1B2
2/(2p2−1)4
2
p
2
+1
2
ω
2
+
ω
r
Kvdd1B2
2/(2p2+1)4
p
1
−p
2
−1
ω
1
−
ω
2
−
ω
r
Kv2dd1B1B2/(p1−p2−1)4
p
1
−p
2
+1
ω
1
−
ω
2
+
ω
r
Kv2dd1B1B2/(p1−p2+1)4
p
1
+p
2
−1
ω
1
+
ω
2
−
ω
r
Kv2dd1B1B2/(p1+p2−1)4
p
1
+
p
2
+1
ω
1
+
ω
2
+
ω
r
K
v
2
d
d
1
B
1
B
2/(
p
1+
p
2+1)4
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300 350 400 450 500 550 600 650 700
0
2
4
6
8
10
12
14
Speed (rpm)
Vibration velocity (mm/s rms)
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0 50 100 150 200 250 30
0
Frequency (Hz)
-110
-100
-90
-80
-70
-60
-50
-40
-30
Vibration Amplitude (dB)
48.7 Hz
#2
63.3 Hz
#2
56 Hz
#1
72 Hz
#3
112 Hz
#4
162 Hz
#5
174 Hz
#5
214 Hz
#5
226 Hz
#5 288 Hz
#4
0 50 100 150 200 250 300
Fre
q
uenc
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(
Hz
)
-100
-80
-60
-40
Vibration Amplitude (dB)
5.3 Hz
#3
68 Hz
#1
63 Hz
#2
136 Hz
#4 142 Hz
#5 186 Hz
#5
206 Hz
#5
210 Hz
#4
250 Hz
#6
73.3 Hz
#2
21.3 Hz
#3
300 350 400 450 500 550 600 650 700
Shaft Speed (rpm)
0
50
100
150
200
250
Frequency (Hz)
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