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

3

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:

4

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

5

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

6

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

7

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

8

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!

9

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

10

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

11

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

12

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

13

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

14

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

15

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

13

F = 0.0266 lb 90°

Step 4: Calculate Applied Force

16

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

17

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

18

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

19

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

20

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

21

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

22

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

23

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

24

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

25

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

26

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

27

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

32

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

28

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

29

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