Post on 17-Jan-2016
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Synchronous Dynamic Stiffness
2
What is Dynamic Stiffness?Dynamic Stiffness is the static spring stiffness of the mechanical system complemented by the dynamic effects of mass and damping.
M
K
D,
Input Output
Force Motion
M MassD DampingK Spring Fluid circumferential average velocity ratio
Machine parameterscharacterizing the system
F r
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Synchronous Dynamic Stiffness
• Synchronous means that the unbalance weight moves together with (is attached to) the rotating rotor.
• Put most simply, the Synchronous Dynamic Stiffness, KDS , is the ratio of the applied force to the rotor system
KDS = rF
response:
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Arrow indicates a VECTOR quantity; it has magnitude (length) and phase (angular orientation
conventionally measured in degrees of lag)
=Synchronous
Response Motion
Synchronous Dynamic Stiffness
Unbalance Force
Synchronous Rotor Response is Controlled by Synchronous Dynamic Stiffness
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r = (mru2)
(M2 +jD(1 ) + K)
Dynamic stiffness: (K M2 ) + jD(1 )Direct term + Quadrature term
Phase Lag ofunbalance
is the Fluid Circumferential Average Velocity Ratio
Synchronous Rotor Response Equation
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Synchronous Dynamic Stiffness = K Mr 2 + jD(1-)D
ire
ct
Sti
ffn
ess
Qu
ad
ratu
re
Sti
ffn
ess
+
0
-
+
0
-
Slope is D(1)
Curve is K M r 2K
Resonance peak: K Mr 2 = 0
(parabola)
K/Mr =
Synchronous Dynamic Stiffness Versus Speed
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Why Identify Dynamic Stiffness?
• Dynamic Stiffness is the relationship between machine parameters and measured vibration response.
• Parameter identification provides information for analytical modeling.
• Trending of Dynamic Stiffness can provide valuable information on changes in machine parameters.
• Dynamic Stiffness can be used to estimate the dynamic forces acting in a machine
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How Do We Measure Synchronous Dynamic Stiffness?
• Obtain reference data (startup or shutdown).
• Attach a known unbalance mass to the rotor in a known location.
• Get the new startup data
• Use data to calculate the Dynamic Stiffness.
• This can be done with every balance shot!
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1) Obtain data from startup or shutdown or single speed.
30
60
90
120
150
180
0
0 500 1000 1500 2000 2500 3000rpm
Ph
ase
lag
(d
eg)
0
5
10
15
20
0 500 1000 1500 2000 2500 3000rpm
Am
plit
ud
e (M
il p
p)
0°330°
300°
270°
240°
210°180°
150°
120°
90°
60°
30°
20 mil pp full scale
Rotn
2) Select a point. Here, the point is
2000 rpm:3.19 mil pp 177°
3) Convert from polar form to rectangular:
3.19 mil pp 177° = 3.186 +j0.167 mil pp
Step 1: Obtain Reference Data
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Original Response Vector3.19 mil pp 177°
90°
0°330°
300°
270°
240°
210°180°
150°
120°
60°
30°
Rotn
4 mil pp full scale
O
Original Response Vector
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1) Add known Calibration Weight at known location.
0.09 grams 90°
2) Obtain new data at same speed.
2000 rpm:3.74 mil pp 206°
45
90
135
180
225
0
0 500 1000 1500 2000 2500 3000rpm
Ph
ase
Lag
(d
eg)
0
5
10
15
20
0 500 1000 1500 2000 2500 3000rpm
Am
plit
ud
e (m
il p
p)
0°330°
300°
270°
240°
210°180°
150°
120°
90°
60°
30°
20 mil pp full scale
ROTN
Knownunbalance
0.09 g 90°
3) Convert from polar form to rectangular:
3.74 mil pp 206° = 3.361 j1.640 mil pp
Step 2: Add Weight and Get Data
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New Response Vector3.74 mil pp 206°
This Vector contains theOriginal response vectorand the response to the
Calibration Weight
90°
0°330°
300°
270°
240°
210°180°
150°
120°
60°
30°
ROTN
4 mil pp full scale
OO + C
New Response Vector
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Polar RectangularRun
3.361 + j1.640 ) mil ppO + C 2
1 (ref) 3.186 j 0.167 )
( 3.74 206° )
( 3.19 177° ) mil ppO
0.176 j 1.8061.81 264° mil ppC
This is the change in rotor system responsedue to the weight that was added.
r
Step 3: Subtract Equivalent Data to See Effect of Weight Only
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90°
0°330°
300°
270°
240°
210°180°
150°
120°
60°
30°
ROTN
4 mil pp full scale
OO + C
r
r
Response Vector due to Calibration Weight only
1.81 mil pp 264°
The r vector is the sameas the C vector used inbalancing.
Effect of Calibration Weight
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Phase Angle ofUnbalance
Mass ofUnbalance
Radius ofUnbalance
RotativeSpeed
2000
2 12
revmin
radrev
min60 sec
90°
Convert toEnglish
Metricresult
F=(0.09 103kg)(30 103m)
F mru2 e
j
F FHIK 118.4 N
lb
4.448 N10 90
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F = 0.0266 lb 90°
Step 4: Calculate Applied Force
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1) Because Dynamic Stiffness is a zero to peak type of parameter (lb/in orN/m), first convert the vibration amplitude from pp to zero-to-peak.While we are at it, convert from mils to inches:
1 81 1. mil pp 2642
in1000 mil
0 905 10 3. in 264
2) Calculate the Dynamic Stiffness:
Phase Lag of Response Vector
KDS = FA
KDS =Fr =
Fe j
Ae j=
FA e j (
Step 5: CalculateSynchronous Dynamic Stiffness
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3) IMPORTANT! Because the phase angles were measured asphase lag, use the negative of the calculated Dynamic Stiffnessphase angle.
KDS 29 lb / in 174
FKDS A
0 0266 (90°
. lbKDS
0.905 103 in
Step 5: CalculateSynchronous Dynamic Stiffness
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KDS KDirect jKQuadrature
The Dynamic Stiffnessconsists of two parts:
Direct
KD = (29 lb/in) cos(174°)
KD = 28.8 lb/in
Quadrature
KQ = (29 lb/in) sin(174°)
KQ = 3.0 lb/in
KQ = KDS sin KKD = KDS cos K
Separate into Direct and Quadrature Dynamic Stiffnesses
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Calculation of Synchronous Dynamic Stiffness can be performed over the entire speed range of a startup or shutdown.
Using Synchronous Dynamic Stiffness, Rotor Parameters can be obtained from the Dynamic Stiffness plots or from two samples.
0 500 1000 1500 2000 2500 3000rpm
-300
-200
-100
0
100
200
Dir
ect
DS
(lb
/in)
0 500 1000 1500 2000 2500 3000rpm
-300
-200
-100
0
100
200
Qu
adra
ture
DS
(lb
/in)
KD = 28.8 lb/inKD = 28.8 lb/in
KQ = lb/inKQ = lb/in
K
K
M
Slope = D(1)
Calculate Synchronous Dynamic Stiffness Over Entire Speed Range
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1) Find the balance resonance speed.
res = 1800 rpm
30
60
90
120
150
180
0
0 500 1000 1500 2000 2500 3000rpm 1800
Ph
ase
lag
(d
eg)
0
5
10
15
20
0 500 1000 1500 2000 2500 3000RPM
Am
plit
ud
e (m
il p
p)
0°330°
300°
270°
240°
210°180°
150°
120°
90°
60°
30°
20 Mils PP Full Scale
ROTN
1800
1800
Obtaining Rotor Parameters from Two Points
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Apply this relationship for the balance resonance and our previous sample:
Solve both equations for M:
KD K M2
Resonance = 1800 rpm 2000 rpm
K M res 2 0 K K MD 20002
MK K K
res
D
220002
( )
Use Direct Dynamic Stiffness Equation
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Solve for the rotor system stiffness, K:
Plug in the numbers:
K = 120 lb/in (21,000 N/m)
K
FHGIKJ
LNMM
OQPP
28 8
12000
1800
2
. lb / in
rpm
rpm
KKD
res
1 2000
2
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Substitute known K into previous mass equation:
M = 0.0034 lb·s2/in (0.59 kg)
MK
res
2 2
120 lb / in
188 rad / sa f
Find Rotor System Modal Mass
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1) Use Quadrature Dynamic Stiffness expression:
2) For this machine, = 0. Also, Synchronous Quadrature Dynamic Stiffness at zero speed is zero.
If significant fluid effects are present, must be estimated.
Damping is:
D = 0.014 lb·s/in (2.5 Ns/m)
K DQ 1 a f
DKQ1
3 0 a f
. lb / in
209 rad / s
Find Rotor System Modal Damping
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Solve both equations for the response vector,
Aemru e j
KHme jj
DS
2
Force
Response
Dynamic Stiffness
Weight
Response
Influence Vector
j
ju
DSAe
emrK
2
j
j
me
AeH
Relationship Between Synchronous Dynamic Stiffness and Influence Vector
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H
H
juj
uuDS e
Hr
He
rH
rK
222
Notice!
Influence Vectors are usually calculated using positive phase lag data. The expressions for Dynamic Stiffness require phase lag to be negative. For this reason, the negative sign above has been eliminated in the equations to the left.
Hu
DS Hr
K 2
KDS
u
Kr
H 2
Relationship Between Synchronous Dynamic Stiffness and Influence Vector
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Given: Influence vector H = 4.68 mil pp/gram 46° lag Radius of unbal ru = 30 mm Rotative speed = 2000 rpm
K
30 10 2000 21
4 6812
12 54 10
100046
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2
mrevmin
radrev
min60 s
mil ppgram
in1000 mil
min
gramskg
. .DS
K
r
HDS H 2
Convertto English
MetricResult
K DS 22 100 46,
N
m
1
4 4482 54
lb
N 10
m
in-2
..FH IK FH IK
LNM
OQP
K DS 126 lb / in 46
Example: Calculating the Dynamic Stiffness from the Influence Vector
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Estimating Forces Using DS
Dynamic Stiffness is the ratioof the Force to the Response: r
FK DS
rKF DS
Rearranging the equation allowsus to estimate the force givenmeasured vibration:
Newly measured Response
Original Dynamic Stiffness
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Summary
• The Synchronous Dynamic Stiffness is simply the ratio of the applied force to the rotor system response.
• Rotor parameters can be calculated from the Synchronous Dynamic Stiffness.
• Synchronous Dynamic Stiffness and Influence Vectors are closely related, and one can be calculated from the other.
• Both Dynamic Stiffness and Influence Vectors can be calculated every time a balance shot is performed.
The End