FULL SPECTRA AS A TOOL FOR ANALYSIS OF A SHAFT ROTATING IN

JOURNAL BEARINGS

Ji Tma & Jan Bilo17. listopadu 15, Ostrava Poruba

Czech Republicjiri.tuma@vsb.cz & bilos@volny.cz

VB TU Ostrava Faculty of Mechanical Engineering

Outline

Bently Nevada Rotorkit Instrumentation arrangement Orbit plot, one-side versus two-side spectrum RPM profile & displacement time history RMS full multispectrum Fluid induced instability Bently and Muszynska model Equation of motion Shaft/fluid wedge bearing/system as a servomechanism Vibration modes

Bently Nevada Rotorkit

Bently Nevada Rotorkit - detail

Instrumentation arrangementProximity probes

y(t) imaginary partx(t) real part

Journal bearing

Journal

Shaft

Fluid lubrication

Displacement x(t)

Displacementy(t)

Complex coordinate of the shaft centre position:

X

Y

Orbit plot

+ X

Y

-

A

B = A* (Re)

(Im)

Both the vectors A and B are rotating in opposite direction at the same frequency .

+ X

Y

-

A

B (Re)

(Im)

Real harmonic function of time

Complex harmonic function of time

Ellipse

One-side versus two-side spectrum

Orbit plot

-1,5

0,0

1,5

-1,5 0,0 1,5

X (Re)

Y (Im

)

Time history

-1,5

0,0

1,5

0,0 0,5 1,0

Time [s]

X, Y

(Re,

Im)

Autospectrum

0,00,20,40,60,81,01,2

-5 -4 -3 -2 -1 0 1 2 3 4 5

Frequency [Hz]R

MS

X +

j Y

Time history

-1,5-1,0-0,50,00,51,01,5

0,0 0,5 1,0Time [s]

X (R

e)

Autospectrum

0,0

0,2

0,4

0,6

0,8

-5 -4 -3 -2 -1 0 1 2 3 4 5Frequency [Hz]

RM

S X

Autospectrum

0,0

0,2

0,4

0,6

0,8

0 1 2 3 4 5Frequency [Hz]

RM

S X

Two-side spectrum One-side spectrum

Two-side spectrum

symmetry

non-symmetry

Fourier Transform

FT

FT

RPM profile & displacement time historyTachometer

1000

1500

2000

2500

0 5 10 15Time [s]

RPM

Time History

-200-100

0100200300400500

0 5 10 15Time [s]

mic

ron X

Y;;

Steady-state vibration

RMS full multispectrum of signal x(t) + j y(t)

-100 -9

0 -80 -70 -60 -50 -40 -30 -20 -10 0 10 2

0 30 40 50 60 70 80 90 100

16931935

21832378

24062212

19751727

010203040506070

RMS m

Frequency [Hz]

RPM

Autospectrum : X + jY

60-7050-6040-5030-4020-3010-200-10

0.475 ord 1.0 ord 2.0 ord

Fluid induced instability

Time : X (Resampled) ; Y (Resampled)

-150-100-50

050

100150

0 1 2 3 4 5 6 7 8 9 10Revolution [-]

m

X (Resampled)Y (Resampled)

Orbit plot

-150-100

-500

50100150

-150 0 150

X [m]

Y [

m]

Autospectrum : X (Resampled) + jY (Resampled)

0,475 ord

0

20

40

60

80

-3,0 -2,5 -2,0 -1,5 -1,0 -0,5 0,0 0,5 1,0 1,5 2,0 2,5 3,0

Order [-]

RM

S [

m]

Self- excited whirl vibration

Low rotational speed vibration

Full order-multispectrum of signalx(t) + j y(t)

-2,0

-1,8

-1,6

-1,4

-1,2

-1,0

-0,8

-0,6

-0,4

-0,2 0,0

0,2

0,4

0,6

0,8

1,0

1,2

1,4

1,6

1,8

2,0

174619072057219423252384240123922299218120451896173415551350

RMS dB / ref 1 m

Order [-]

RPMAutospectrum : X (Resampled) + Y (Resampled)

35-4030-3525-3020-2515-2010-155-100-5

Proportionality of the whirl vibrationfrequency to the shaft rotational speed

0.475 ord 1.0 ord

-0.475 ord

Bently and Muszynska model

avgv BearingShaft

= avg

v

Average fluid angular velocity

Fluid circumferential velocity ratio

rotrotrot DK rrF &+=

Fluid forces acting on the rotor

Spring and damper systemrotating at the angular frequency

Fluid wedge

Fluid forces in stationary coordinate

Fluid forcesTransform to stationary coordinates

r Rotating

Stationary

t

( )( ) ( )tjj

tj

rotrot

rot

==

expexp

rrrrr&&

rrrF += jDDK &

Direct Quadrature

Tangentionalforce

Direct force

Gravity force

An example of force balance

Equation of motionAngular velocity is completely independent of the rotor angular velocity

Equation of motion

Perturbation balance force( )( )+= tjmruonPerturbati exp2F

( ) ( )( )+=++ tjmrjDKDM u exp2rrr &&&

( ) ( )( ) ( )+

=jDMK

jmrj u22 expexpA

Amplitude A and phase of the rotor shaft centre-line rotating at the angular velocity

( )( )+= tjexpArSolution takes a form

Shaft/fluid wedge bearing/system as a servomechanism

Direct stiffness

Quadrature stiffness

( )2+=

=

MjDKjKDirect

( ) DjjKQuadrature =

( ) ( )20 +

=MKjD

DjG

Open-loop frequency transfer function

( )jKDirect1

( )jKQuadrature

Rotorload

Fluid wedgesupport

+

-

Rotor shaft eccentricity position

Positive feedback

Equation of motion

Closed-loop stability margin

=> Bently and Muszynskathreshold

( ) 10 =CritjG

According to the Nyquist stability criterion, a margin of stability is resulting from

=> Mechanical resonanceMKCrit =

2

CritCrit =

=

MKCrit

=> Fluid resonance

( )jG0

Real

(-1,0)

stable

margin

unstable0=

0=

0=

Crit=

Crit

Imag

Nyquist plot ofin complex plane

Crit>

Crit

Vibration modes

-120-80-40

04080

120

0 5 10 15 Time [s]

mic

ron

X;;

Crit=

Crit

Unbalance effect Unbalance effect

0,1

1

10

100

0 5 10 15 20

Time [s]

Mag

nitu

de

x2 x1 x2 x3 x

( ) 10 =jG

0

10

20

30

40

50

60

70

1000 1300 1600 1900 2200 2500

RPM

Mag

nitu

de

xcoast down

run up

( ) 2=MtyeccentriciK=>

Magnitude self-control

Crit=

Harmonic envelopes

Fluid Induced vibrationEccentricity

Wal

l

Cen

tre

Stiffness K

Whirl & WhipFluid Induced Instabilities

Whirl vibration

Whip vibration

Bently & Muszynskathreshold

Measurement range for this paper

An example Subharmonic

Harmonics of rotational frequency

Conclusion

This presentation describes using full spectra for rotor system diagnostics

The full spectrum is a good tool for studying rotor instability in journal bearings

The presentation demonstrates whirl vibration and the independence of the ratio relating the precession speed to the shaft rotational speed with respect to the shaft absolute rotational speed

Bently and Muszynska model gives explanation of the rotor instability