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Dynamic behaviour of the Laval rotor with a transverse crack
ARTICLE IN PRESS Mechanical Systems and Signal Processing Mechanical Systems and Signal Processing 22 (2008) 790–804 www.elsevier.com/locate/jnlabr/ymssp
Dynamic behaviour of the Laval rotor with a transverse crack Robert Gasch Institut f. Luft-und Raumfahrt, Technical University Berlin, D-10587 Berlin, Germany
Abstract This paper presents an introduction to the dynamic behaviour of a one disc rotor (Laval rotor) having a transverse crack in the elastic shaft. With the aid of a simple crack model the non-linear equations of motion are derived. Due to the weight dominance in the elastic deflection of the horizontal shaft the equations can be simplified to linear but time-variant equations. The stability analysis is carried out with Floquet’s method and the results are presented in an overview diagram. The forced vibrations due to crack and unbalance are discussed in much detail. The orbit decomposition into forward and backward whirls (two-sided spectral order analysis) turns out to be a helpful tool for the understanding the complicated dynamic phenomena. Some hints for crack detection are given. r 2007 Elsevier Ltd. All rights reserved. Keywords: Cracked rotor; Crack model; Floquet’s stability analysis; Crack response; Early crack detection
1. Introduction In the beginning 1970th transverse cracks occurred in the heavy horizontal rotors of some turbosets. They were recognized more or less by chance during the revision after a 2 years run. The early papers presented at the IMECHE Conference on Rotating Machinery in Cambridge, UK, in 1976 [1–3,4], revealed already many peculiarities of the dynamics of a cracked rotor. Since that time plenty of papers were published dealing with crack models, the coupled transversal, torsional and longitudinal vibrations of cracked rotors and—of course—early crack detection to avoid catastrophic failure. Here, however, for a basic understanding we confine ourselves considering only the lateral vibrations of the Laval rotor with a transverse crack.
2. A simple crack model The deflection line of a shaft with a crack in the tension zone (Fig. 1) is superimposed from two parts: the deflection line of the uncracked shaft and an additional deflection from the local flexibility of the crack. This additional part cannot be found from the bending theory. Because for the beam theory a crack is a weakening Tel.: +49 30 314 21 337; fax: +49 30 314 22866.
E-mail address: [email protected] 0888-3270/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ymssp.2007.11.023
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Without crack
plus
Crack contribution Fig. 1. Deflection line; contributions of the uncracked shaft and the local crack flexibility without crack contribution.
1.5
^ H 22 ^ H
33
1.0
2R a ^ H 22
0.5
^ H 33 0.5
a/R
Crack depth a /R
0
^ H 22
0
0.028 0.144 0.374 0.734 1.248 1.947
^ H 33
0
0.001 0.009 0.038 0.107 0.245 0.493
0.1
0.2
0.2
0.4
0.5
0.6
Fig. 2. Main- and cross-flexibility of the open crack from [9].
of the bending stiffness EI on a length zero, only a three-dimensional consideration (or a two-dimensional approximation) is able to yield this additional weakening of the stiffness [9,13,14]. In fact, due to the loss of symmetry a cracked round shaft produces a coupling of lateral vibration, axial and torsional vibration. However for the sake of simplicity we ignore these effects and focus our interest only on the lateral vibration. In [9] we find the crack flexibilities for a general cracked beam element with 6 degrees of freedom at each end. We simplify and pick up only the main flexibility H22 and the cross-flexibility H33 (Fig. 2). These flexibilities are presented in a dimensionless form. To get the physical flexibilities they have to be divided by ER3, where E is the Young’s modulus an R the radius of the shaft, H 22 ¼ H^ 22 =ER3
and
H 33 ¼ H^ 33 =ER3 .
Fig. 2 shows these flexibilities versus crack depth a/R. For small cracks (a/Ro0.5) the cross-flexibility H22 is much smaller than the main flexibility H33, so we go ahead and ignore the cross-flexibility completely. This has the advantage that the crack flexibility is represented by only one parameter. So the understanding of our analytical results will be eased. Thus the deflection at the position of the disc with an open crack nearby can be written as #( ) ( ) " fB wB h0 þ DhB ¼ , (1) fZ vZ h0 þ 0
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fζ l/2
l/2
wζ Fig 3. Crack model (hinge model) for the ‘‘breathing’’ crack.
where h0 is the flexibility of the round uncracked shaft and DhB ¼ H^ 22 ‘2 =ER3 16
(2)
is the (translatoric) additional flexibility of the crack. Hereby the span of the shaft is l. Turning the shaft with the crack into the pressure zone the crack will be closed due to the pre-stressing of the weight and Dhz will become zero; the shaft is round again. This behaviour is represented by the hinge model shown in Fig. 3. The hinge opens in the tension zone and closes in the pressure zone. Thus the crack mechanism is non-linear, respectively, bi-linear due to our assumptions. So for our further calculations we re-write Eq. (1) in the following form: ( ) " # !( f ) wB h0 DhB B ¼ þ f ðtÞ , (3) fZ vZ h0 0 where f(t) is the steering function being f(t) ¼ 0, when the crack is in the pressure zone and f(t) ¼ 1, when the crack is in the tension zone. For very deep cracks a modification of the steering function is advisable [2,11,13]. We now invert Eq. (3), we change from the flexibility presentation to the more practical stiffness formulation for our equations of motion ( ) " # " #!( ) fB wB s0 DsB ¼ f ðtÞ , fZ vZ s0 0 f rot ¼ S0;rot þ DS rot ðurot ; tÞ urot , (4) whereby 1=ðh0 þ DhB Þ ¼ s0 DsB .
(5)
So Dsz is the loss of stiffness due to the opening of the crack and s0 ¼ 1/h0. Having a real rotor Dsz can easily be determined by an experiment, measuring the natural frequencies with open and closed crack (Fig. 4). pffiffiffiffiffiffiffiffiffiffi Taking the natural frequency of the shaft with closed crack o0 ¼ s0 =m as a reference frequency to which we relate the natural frequency oz with an open crack (Fig. 4), we get the parameter 2 oB DsB ¼1 . o0 s0 The square root oz/o0 will be used later as dimensionless crack-depth parameter. Until now the stiffness matrix in Eq. (4) is defined in rotating, rotor fixed coordinates. We now shift to an inertial frame. From Fig. 5 we find the well-known transformation matrix for the deflections ( ) " #( ) wB cos Ot sin Ot w ¼ , vZ sin Ot cos Ot v urot ¼ TðtÞ uinertial
(6)
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Crack (S0 - ΔSζ) / m
ωζ =
ω0 =
S0 / m
Fig. 4. Experimental determination of the stiffness loss DsB due to the crack.
η vη v y
w wζ
Ωt ζ
z
Fig. 5. Rotating and non-rotating coordinates.
and analogously for the forces f rot ¼ TðtÞ f inertial , f inertial ¼ T1 ðtÞ f rot ðtÞ.
(7)
Inserting these transformations of forces and deflections into Eq. (4), yields the stiffness matrix, in the inertial frame SðtÞ ¼ S0 þ DSðtÞ ¼ T1 ðS0;rot þ DSrot ÞT. Written explicitly " S0 þ DSðtÞ ¼
and worked out further " s0 S0 þ DSðtÞ ¼
"
#
s0
# s0
cos2 Ot
sin Ot cos Ot
sin Ot cos Ot
sin2 Ot
1 þ cos 2Ot 1 f ðtÞ DsB 2 sin 2Ot
sin 2Ot
f ðtÞDsB
s0
(8)
1 cos 2Ot
#
.
(9)
This stiffness matrix is twofold time dependent: once due to the transformation from rotor-fixed to inertial coordinates and further due to the steering function f(t) being 1 or 0 depending from the crack position in the compression or the tension zone of the shaft. Since f(t) depends on the shaft deflection u as well, being more
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carefully we should write S ¼ Sðu; tÞ ¼ S0 þ DSðu; tÞ.
3. Equations of motion and its linearization for a horizontal shaft—the breathing crack We now consider the equations of motion for our simple Laval rotor with the disc mass m, the damping coefficient d and the stiffness matrix S(u, t) just derived (Eq. (9)). On the right-hand side (RHS) there are the vectors of weight and unbalance forces p0, respectively, pu ( ) " #( ) " #( ) " #( ) ( ) cos ðb þ OtÞ s11 s12 m d mg w€ w_ w 2 , þ þ ¼ þ mO sin ðb þ OtÞ s21 s22 m d 0 v€ v_ v M€u þ D_u þ Sðu; tÞu ¼ p0 þ pu .
(10)
The weight vector on the RHS is of great importance when linearizing the equations of motion. For with heavy crack endangered horizontal rotors the static deflection u0 is large compared to the additional deflections caused by the dynamics of the rotating shaft. This is confirmed by climps in Figs. 6 and 7, derived from wstat ¼ mg=s0 and o20 ¼ s0 =m0 , where o0 is the natural frequency (critical speed) of the uncracked system (Fig. 4).
η
v
v
y
ε
w
w
β z
Ωt
Ωt
ζ
Fig. 6. Laval rotor with a cracked shaft and the equations of motion (Eq. (21)).
1000
2000
3000 n0 [min-1 ]
wstat [mm]
100
200
300
ω0 [rad / s]
0.5
1.0
1.5 Fig 7. Static deflection and natural frequency o0.
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MG MG
MG
MG
MG
MG
Ωt 0
90 open
180 closed
360 ψ = Ωt
270 closed
open
wz stat wz
f (t) 1 open 0
open
closed 90
180
270
ψ = Ωt 360
Fig. 8. Static deflection and opening and closing of the ‘‘crack’’ versus angle of rotation; steering function f(t).
Having a (first) critical speed of 600–800 rpm like with heavy generators or low-pressure turbines we get static deflections of 1 mm or more (Fig. 7). Whilst dynamic deflections of 0.1 mm are already very serious. So it is obvious to split the total deflection mg=s0 uðtÞ ¼ u0 þ DuðtÞ ¼ þ DuðtÞ, (11) 0 knowing that Du(t) is small, Du(t)5u0. In other words the weight deflection is dominating and will govern the opening and closing of the crack, because the vibration amplitudes Du(t) are small compared to the static deflection. Fig. 8 shows the ‘‘breathing’’ of the crack under the weight influence when the shaft is slowly turned. Please notice inspite of the steps in the steering function f(t), the deflection of the slowly rotating shaft is softly and continuously changing with Ot. With a small crack (a/Ro0.5) the crack opens and closes indeed very abrupt when passing 901, respectively, 2701, where the shift from the tension zone to the compression zone and vice versa occurs. Having deeper cracks the transition will be less abrupt. So Mayes suggested a more steady steering function f(t) ¼ (1+cos Ot)/2 for deeper cracks as it comes closer to reality [2,9,11,13]. For our basic considerations the step function of Fig. 8 is quiet sufficient as we focus our interest on the early state (a/Ro0.5). And by the way, the influence of the steering function on the results is astonishingly small [13]. Now we are prepared for the formal linearization procedure. Due to the weight dominance we re-write in Eq. (10) M€u þ D_u þ Sðu; tÞu ¼ p0 þ pu for the stiffnesses SðtÞ ¼ S0 þ DSðtÞ
(12)
using Eq. (9) with the step function as steering function. The deflections are split up into the static and the dynamic part according to Eq. (11). Inserting both, Eqs. (11) and (12) into the equations of motion yield
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a linear but periodically time-variant equation for the dynamic vibrations Du(t), MD€u þ DD_u þ ½S0 þ DSðtÞ Du ¼ DSðtÞu0 þ pu
(13)
as the static equilibrium can be separated: " # s0 wstat mg ; ¼ s0 0 0 S0
u0
¼ p0 :
The periodic time-variant stiffness matrix DS(t) appears twice. On the left-hand side it decides (together with the damping) whether the system is stable or not. On the RHS this matrix is multiplied with the weight deflections u0 producing an additional excitation force—the crack excitation—that is periodic in time. If the stability of the homogenous solution is guaranteed—this of course must be checked—a further simplification for the calculation of the forced vibrations due to the RHS in Eq. (13) is possible. Then the homogenous part of the complete solution DuðtÞ ¼ Duh ðtÞ þ Dup ðtÞ is fading out and the term DSðtÞDuðtÞ on the left-hand side will be very small. Neglecting this term for the calculation of the forced vibrations a timeinvariant differential equation MD€u þ DD_u þ S0 Du ¼ DSðtÞu0 þ pu
(14)
has to be solved. Actually this is the equation of motion of a round, uncracked shaft. On the RHS we find the unbalance excitation and the crack excitation forces DS(t)u0. 4. Stability The stability analysis was performed with Floquet’s theory. The homogenous system of the equations of motion (RHS zero in Eq. (13)) was transformed into first-order differential equations (state space) and the transition matrix U(T) was determined. This matrix tells us, how the vibrations due to initial conditions have developed after one cycle T ¼ 2p/O. So in order to find the transition matrix, we have to integrate all possible initial conditions over one period T 9 9 8 8 Dw > Dw > > > > > > > > > > > > > > > > > = = < Dv > < Dv > ¼ UðTÞ , > > Dw_ > Dw_ > > > > > > > > > > > > > > > > > ; ; : : D_v t¼T D_v t¼0 xT ¼ UðTÞx0 .
(15)
Having the transition matrix we demand: of which kind is the proportionality between the initial vector x0 and the state xt after one cycle t ¼ T xT ¼ mx0 .
(16)
These two Eqs. (15) and (16) yield Floquet’s eigenvalue problem ½UðTÞ mIx0 ¼ 0,
(17)
that gives an answer, whether the system is stable or not. For a stable system the magnitude of all eigenvalues must be less than 1, jmk jo1.
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To be more illustrative: |mk| gives us the factor of increase or decrease of the initial conditions after one revolution (cycle) of the rotor. Fig. 9 presents a survey on the stability of the cracked rotor. The stability depends on the rotational speed and the crack depth. Here damping was assumed to be zero—the worst case. The (dimensionless) crack-depth parameter in this figure is the frequency ratio oz/o0 (see Fig. 4). The rotational frequency O was made dimensionless with the natural frequency of the uncracked rotor o0U The unstable regions, |mk|41, are found at the dimensionless rotational speeds Z ¼ O=o0 ¼ 25; 24; 23; 22; 21.
(18)
Rotational speeds beyond Z ¼ 2 are always stable. With deeper cracks the unstable zone near to Z ¼ 2 becomes very broad. The unstable zone near to Z ¼ 1 is familiar to us from the rotor with unequal bending stiffnesses (EIz6¼EIZ). The regions of instability at Z ¼ 2/3, 2/4y are very small. As soon as we have some damping, they disappear [13].
|μ k|
3
2
1
0
0.5 0.66
1.0
1.5
Rot. speed Ω/ω0
0.52 0.76 ω ζ/ ω 0 pth e d k
2.0 1.0 c Cra
Fig. 9. Stability of the cracked Laval rotor versus rotational speed and crack depth, damping D ¼ 0. Unstable zones jmk j41. 0.7 stable
Crack depth ω ζ /ω 0
st able
unstable
0.8 D = 0.01
0.9 D=0 1.0
2 4
2 3
1
2
Rotational speed Ω /ω0
Fig. 10. Borderlines of stability with a damping of 1% (D ¼ 0.01) and no damping (D ¼ 0).
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Compared with instabilities due to fluid film bearings the crack instability can be very brutal: |mk| ¼ 2 is to say: after 10 revolutions of the rotor the amplitude has grown with a factor 210 ¼ 1024. Fortunately some small damping in the system is already very helpful to guarantee stability. Compare the width of the unstable zones for D ¼ 0 and 0.01 in Fig. 10. In [16] the borderlines of the unstable zones for zero damping, D ¼ 0 were calculated numerically. An analytical approach is given by
2 1 DsB O 2 1 , (19) o o n 4 s0 o0 n where n ¼ 1,2,3,y. The influence of fluid film bearings is discussed in [16,19].
5. Forced vibrations 5.1. Vibration forced by the crack influence Assuming stability of the system the forced vibration response may be calculated with the time-invariant equations of motion (Eq. (14)) " #( ) " #( ) " #( ) s0 d m Dw€ Dw_ Dw þ þ s0 d D_v m D€v Dv ( ) ) " #( cosðOt þ bÞ wstat 1 þ cos 2Ot sin 2Ot 1 . (20) ¼ DsB f ðtÞ þ mO2 2 sinðOt þ bÞ 0 sin 2Ot 1 cos 2Ot This equation can be simplified by introducing a complex notation (Fig. 11). Dr is the vector containing both deflections of the shaft Dv and Dw pffiffiffiffiffiffiffi Dr ¼ Dw þ jDv with j ¼ 1. (21) pffiffiffiffiffiffiffi Formally, we multiply the second equation in Eq. (20) with the imaginary unit 1 and add it to the first equation. Remembering Euler’s formula this yields the new second-order differential equation m D€r þ d D_r þ s0 Dr ¼ 12wstat DsB f ðtÞ 1 þ e2jOt þ mO2 ejðOtþbÞ . (22) We first focus our interest on the pure crack response (no unbalance, e ¼ 0), that is proportional to the static deflection wstat and the stiffness loss due to the crack Dsz. We develop the step function of opening and closing of the crack in Fourier-series f ðtÞ ¼ 12 þ ð2=pÞ½cos 1Ot ð2=3pÞ cos 3Ot þ ð2=5pÞ cos 5Ot .
Δv Im y Δr
Δw
Re z Fig 11. Complex coordinates Dr for the deflections Dw and Dv.
(23)
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Reminding Euler’s equation cos nOt ¼ 12 eþjnOt þ ejnOt yields the crack force Fc on the RHS of Eq. (22) as a complex Fourier-series k¼þ1 X
F c ¼ DsB wstat
bk ejkOt ,
(24)
k¼1
where bk are the participation coefficients of the harmonics. In Table 1 they are presented with an interpretation of their physical meaning. Now the response of the rotating shaft due to a single harmonic bk of the crack excitation can easily be determined. With the approach Dr ¼ D^rk ekjOt we receive from Eq. (22) m D€r þ d D_r þ s0 Dr ¼ DsB wstat bk ejkOt
(25)
the equation O2 k2 m þ jOkd þ s0 D^rk ¼ DsB wstat bk
(26)
and from that the response amplitude D^rk ¼ wstat DsB
bk . ðs0 O k mÞ þ jðkOdÞ
(27)
2 2
By superposition of all components k the complete response of the rotor to the crack excitation turns out to be DrðtÞ ffi wstat
X DsB k¼þ3 bk ejkOt : s0 k¼3 1 k2 Z2 þ 2jDkZ
(28)
Since the high-order components in this series presentation are very small or exactly zero we truncate after k ¼ 73. The contribution k ¼ 0, b0 ¼ 0.25 is an additional static deflection due to the crack, averaged over one revolution with the magnitude 0:25wstat DsB =s0 , giving the reference level for the vibrations.
Table 1 Crack excitation force and its harmonics k 4 3 2 1
Backward
0 +1 +2 +3 +4
Additional static deflection Forward
Excitation
bk
Four times per revolution Three times per revolution Twice per revolution Once per revolution
0 0.021 0 0.106
Once per revolution Twice per revolution Three times per revolution Four times per revolution
0.250 0.318 0.250 0.106 0
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The contribution k ¼ 71, b1 ¼ 0.106 and b+1 ¼ 0.318 describes a forward and a backward circular orbit passed once per revolution DsB bþ1 eþ1jOt , 2 þ1 s0 1 ðO=o0 Þ þ 2jðO=o0 ÞD DsB b1 D r ðtÞ ¼ wstat e1jOt . 2 1 s0 1 ðO=o0 Þ 2jðO=o0 ÞD
D r ðtÞ ¼ wstat
(29)
They superimpose to an elliptic orbit passed once per revolution in a forward sense (Fig. 12). The resonance belonging to this ellipse is at O ¼ o0 (for deeper cracks at O ¼ o0 , see Section 5.3). The contribution k ¼ 72, b2 ¼ 0 and b+2 ¼ 0.25 describes a circular orbit passed forward twice per revolution of the shaft, Dr2 ðtÞ ¼ 0 þ wstat
DsB bþ2 eþ2jOt . 2 s0 1 ð2O=o0 Þ þ 2jð2O=o0 ÞD
(30)
Here the resonance occurs at O ¼ o0/2. Finally the contribution k ¼ 73, b3 ¼ 0.021 and b+3 ¼ 0.106 results in a somewhat elliptical orbit passed in a forward sense three times per revolution of the rotor. The resonance is at O ¼ o0/3. The magnification functions for k ¼ 3, +1, +2, +3 are shown in Fig. 13. More instructive however is Fig. 14, where the circular and elliptic orbits of the once, twice and thrice per revolution vibrations are shown over the dimensionless rotational speed Z ¼ O/o0 for the case of 5% damping. The curve touching the left arm of the cross (o) at Z ¼ 1 in the elliptic 1O-orbit is presenting the phase angle resulting from Eq. (29). The transparency of this presentation is lost as soon as the three contributions are superimposed to the real physical orbit of the system (Fig. 15). At Z ¼ 13, e.g. we have three loops per one revolution of the shaft, as this is the resonance of the three times per revolution excitation. With Z ¼ 12 there are two loops within one revolution of the shaft a.s.o. The result of the forward–backward decomposition of the orbits—the result of the two-sided spectral order analysis—is shown in Fig. 16. Combined with the phase development (not shown here) this is a very helpful instrument for early crack detection. Ω
^r e-1j Ωt -1
Im y ^r e+1j Ωt +1
Fig. 12. Elliptic orbit superimposed from the forward (+1O) and the backward (1O) circular vibration.
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Δr^k
s0 ) wstat ( Δs ζ 3
2 2Ω
3Ω
1Ω Forward whirl
1 0.32 0.25 0.11 0
0.33 Δr^-1 wstat
0.5
Ω ω0
1.0
s0 ( Δs ) ζ
Backward whirl
1
-1Ωt Ω ω0
0.11 0
0.5
1.0
Fig. 13. Magnification functions of the forward 1O-, 2O- and 3O-circular vibrations and the backward 1O circular vibration; D ¼ 0 and 0.05.
ed l spe
Ω /ω 0
=η
tiona
Rota 1.0
3Ω
ard
forw
bit, - Or
0.50 0.33
bit,
Δvy
3
Or Ω-
.
fwd
Δvy
bit,
1
Or Ω-
.
fwd
Fig. 14. Elliptical forward orbits (1O and 3O) and the circular 2O (forward) orbit versus dimensionless rotational speed; D ¼ 0.05. They superimpose to Fig. 15.
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η≈0
0.25
+
0.33
0.40
+
+
0.50
+
+
+
+
+
+ Ω/ω0 = 1
0.65
0.90
0.80
1.10
1.25
Fig. 15. Orbits of the shaft center versus (dimensionless) rotational speed Z, D ¼ 0.05. „Amplitude“ ^rk
6
6
5
5
4
4
3
3
-1Ω
-3Ω
2
+1Ω
+2Ω
2 1
+3Ω -3
-2
-1 0 Frequ ency
1 ω krit
2
nal spe
0 -4
Ω /ω krit
Rotatio
−ω
1 0
ed
-2Ω
3 +ω
Fig. 16. Two sided spectral order analysis of the Laval rotor with a crack. Dimensionless amplitude versus dimensionless rotational speed and frequency.
5.2. Crack- and unbalance response In practise there is always an overlay of crack- and unbalance response. Thus superimposing both solutions of Eq. (22) we receive DrðtÞ ¼ Dr þ Drcrack ¼ ejb
X Z2 DsB k¼þ3 bk jOt ekjOt . e þ w stat 1 Z2 þ 2jDZ s0 k¼3 1 ðkZÞ2 þ 2jkZD
(31)
The +1O response now has two contributions: one from the crack and one from the unbalance, compare Eqs. (22) and (29). It depends on the angular position b between eccentricity and crack (Fig. 6), whether these two contributions intensify or compensate each other, as the exciting +1O forces are now mO2 ejb þ wstat DsB bþ1 ejOt .
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Manipulating the unbalance in away that b becomes 1801 we may completely suppress the +1O response due to the crack for the rotational speed O considered. Then we balance the rotor so that b ¼ 1801 and mO2 ¼ wstat DsB b1 . By balancing in a way that mo20 ¼ wstat DsB b1 is fulfilled, even the resonance at O ¼ o0 is suppressed. However it is impossible to exert an influence by balancing on the 1O response or the 2O and 3O contributions of the crack and on the additional static deflection due to b0. 5.3. Deeper cracks In Sections 5.1 and 5.2 we used Eq. (20) for the calculation of crack- and unbalance response. As the left-hand side in this Eq. (20) describes nothing else but the uncracked shaft, we took its natural frequency pffiffiffiffiffiffiffiffiffiffi o0 ¼ s0 =m as reference frequency. So all results presented in these sections are valid only for small cracks. The well-known fact that with somewhat deeper cracks the ‘‘natural frequency’’ seen in the measured resonance diagrams is a little bit lower than that of the uncracked shaft o0, can be introduced into our results in the following way. Considering only the static part, k ¼ 0, b0 ¼ 0.25 in Eq. (28) and its solution we find the additional static deflection Drstat;ad ¼ 0:25wstat Ds=s0 averaged over one revolution. The total static deflection—including both the static deflection of the uncracked shaft wstat ¼ mg/s0 and its additional averaged contribution of the crack—will be
mg 1 Dsx mg wstat;tot ¼ wstat þ Drstat;ad ¼ 1þ ¼ . s0 4 s0 s0 So s0 is thepsomewhat reduced stiffness of the rotating shaft including now the weakening due to the crack. ffiffiffiffiffiffiffiffiffiffi From o0 ¼ s0 =m we find the somewhat lowered natural frequency
1 Dsx o0 o0 1 . 8 s0 For deeper cracks s0 , respectively, the crack-depth-dependent natural frequency o0 has to substitute o0 in Eqs. (26)–(30) of Section 5.1, describing the resonance behaviour. 6. Final remarks Although the dynamics of a rotating shaft with a crack is theoretically transparent, in practise early crack detection is still not easy. Hardly a crack was detected in a turboset running for month with constant rotational speed having a crack depth of less than 25% [5,6]. At that depth we have already a slight change in the natural frequencies (resonances) of some 2% or 3%. When manoeuvring this—and of course the 2O resonance at half-critical speed—becomes visible. The fluid film damping impedes the early crack detection, e.g. the 3O resonance is usually completely suppressed. However permanent monitoring enables to trend analysis (today’s vibration minus vibration a fortnight before), that is very helpful for successful early crack detection. Having a suspicious rotor balancing test weights that open or close the crack may give valuable hints (see Section 5). Much more information on early crack detection will be found in the following contributions to this monograph.
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Acknowledgement Thanks to Dr. Ing. Alexander Bormann, who helped me so much when organizing this paper. References [1] R.A. Henry, B.E. Okah-Avae, Vibrations in cracked shafts, in: on Institution of Mechanical Engineers Conference, Vibrations in Rotating Machinery, Cambridge, UK, 1976, Paper C 162/76. [2] I.W. Mayes, W.G.R. Davies, The vibrational behaviour of a rotating shaft system containing a transverse crack, in: on Institution of Mechanical Engineers Conference, Vibration in Rotating Machinery, Cambridge, UK, 1976, Paper C 168/76. [3] R. Gasch, Dynamic behaviour of a simple rotor with a cross-sectional crack, in: on Institution of Mechanical Engineers Conference, Vibration in Rotating Machinery, Cambridge, UK, 1976, Paper C 178/76. [4] R. Gasch, Kleiner Beitrag zur Behandlung des dynamischen Verhaltens einer rotierenden Welle mit angerissenem Querschnitt. 4 (A Short Paper on the Dynamics of a Rotating Shaft with a Crack), ILR-Bericht 8, TU Berlin, 1975 (ISBN 3 7983 0551 X). [5] A.F.P. Sanderson, The vibration behaviour of a large steam turbine generator during crack propagation through the generator rotor, in: on Institution of Mechanical Engineers Conference, Vibrations in Rotating Machinery, Bath, UK, 1992, Paper C 432/102. [6] Allianz-Berichte, Schwingungsu¨berwachung von Turbosa¨tzen—ein Weg zur Erkenntnis von Wellenrissen (Vibration monitoring in turbosets—a way to early crack detection), Bericht 24 (1987). [9] W. Theis, La¨ngs- und Torsionsschwingungen bei quer angerissenen Wellen, 9 (Longitudinal and Torsional Vibrations of Rotating Shafts with a Transverse Crack), Reihe 11, Nr. 131, VDI-Verlag, Du¨sseldorf, 1990. [11] J. Mayes, W.G.R. Davies, Analysis of the response of a multi-rotor-bearing system containing a transverse crack in a rotor, Journal of Vibration, Acoustics, Stress and Reliability in Design 106 (1984) 139–145. [13] R. Gasch, M. Person, B. Weitz, Dynamic behaviour of the Laval rotor with a cracked hollow shaft—a comparison of crack models, in: on Institution of Mechanical Engineers Conference, Vibration in Rotating Machinery, Edinburgh, UK, 1988, C314/88. [14] B. Schmalhorst, Experimentelle und theoretische Untersuchung zum Schwingungsverhalten angerissener Rotoren (Experimental and Theoretical Approach to the Vibrational Behaviour of Cracked Rotors), Reihe 11, Nr. 117, VDI-Forschungs-Berichte, 1989. [16] G. Meng, R. Gasch, Stabilita¨t eines gleitgelagerten, einfachen Rotors mit RiX (Stability of a Cracked Rotor in Fluidfilm Bearings), in: SIRM-Tagung ‘‘Schwingungen in rotierenden Maschinen,’’ Wien, February, Tagungsbericht Vieweg Verlag, 1992. [19] F. HiX, Nichtlineare Dynamik und Zustandsbeobachtung gleitgelagerter elastischer Rotoren mit angerissenem Wellenquerschnitt (Nonlinear dynamics and monitoring of flexible rotors in fluidfilm bearings having a transverse crack), Dissertation, TU Berlin, 1996.
Further reading [7] Schenck-Allianz, Unterlagen zum Symposium ‘‘Methoden, Nutzen und Trends der schwingungstechnischen U¨berwachung von Turbosa¨tzen in Kraftwerken und Industrieanlagen,’’ 7 (Symposium: Methods, Trends and Advantages of Vibration Monitoring in Industrial Plants and Turbosets), Sonthofen Ma¨rz, 1993. [8] I. Imam, et al., Development of an on-line rotor crack detection and monitoring system, ASME, Journal of Vibration, Acoustics, Stress and Reliability in Design 111/241 (1989) 12. [10] B. Grabowski, R. Po¨ppel, Das Schwingungsverhalten eines Rotors mit QuerriX—experimentelle und theoretische Ergebnisse (Vibrational Behaviour of a Rotor with a Crack—Theoretical and Experimental Results), VDI-Schwingungstagung, vol. 4456, VDI-Berichte, Du¨sseldorf, 1982. [12] H.D. Nelson, C. Nataray, The dynamics of a rotor system with a cracked shaft, ASME-Journal of Vibrations, Acoustics, Stress and Reliability in Design 108 (1986) 189–196. [15] M. Liao, R. Gasch, Crack detection in rotating shafts—an experimental study, in: on Institute of Mechanical Engineers Conference, Vibrations in Rotating Machinery, Bath, UK, Paper 1992–6, C 432/106. [17] W. Rothkegel, RiXerkennung bei rotoren durch Schwingungsu¨berwachung (Crack detection in rotors with the aid of vibration monitoring), Dissertation, Universita¨t, Fakulta¨t fu¨r Maschinenwesen, Hannover, Forschungs-Berichte, Reihe 11, Nr. 180, VDI, VDI-Verlag, Du¨sseldorf, 1993. [18] D. So¨ffker, P.C. Mu¨ller, Betriebsu¨berwachung und Schadensdiagnose an rotierenden Maschinen-Bewa¨hrte Methoden versus neue modellbasierte Ansa¨tze (Monitoring and fault detection in rotating machinery- classical methods and model based approaches), in: R. Irretier, et al. (Eds.), SIRM Tagung III Schwingungen in rotierenden Maschinen, Kassel, Vieweg Verlag Braunschweig, 1995. [20] R. Gasch, R. Nordmann, H. Pfu¨tzner, Rotordynamik (Rotordynamics), second ed, Springer, Berlin, 2002, 700pp. (ISBN 3-540-41240-9).
Dynamic behaviour of the Laval rotor with a transverse crack Robert Gasch Institut f. Luft-und Raumfahrt, Technical University Berlin, D-10587 Berlin, Germany
Abstract This paper presents an introduction to the dynamic behaviour of a one disc rotor (Laval rotor) having a transverse crack in the elastic shaft. With the aid of a simple crack model the non-linear equations of motion are derived. Due to the weight dominance in the elastic deflection of the horizontal shaft the equations can be simplified to linear but time-variant equations. The stability analysis is carried out with Floquet’s method and the results are presented in an overview diagram. The forced vibrations due to crack and unbalance are discussed in much detail. The orbit decomposition into forward and backward whirls (two-sided spectral order analysis) turns out to be a helpful tool for the understanding the complicated dynamic phenomena. Some hints for crack detection are given. r 2007 Elsevier Ltd. All rights reserved. Keywords: Cracked rotor; Crack model; Floquet’s stability analysis; Crack response; Early crack detection
1. Introduction In the beginning 1970th transverse cracks occurred in the heavy horizontal rotors of some turbosets. They were recognized more or less by chance during the revision after a 2 years run. The early papers presented at the IMECHE Conference on Rotating Machinery in Cambridge, UK, in 1976 [1–3,4], revealed already many peculiarities of the dynamics of a cracked rotor. Since that time plenty of papers were published dealing with crack models, the coupled transversal, torsional and longitudinal vibrations of cracked rotors and—of course—early crack detection to avoid catastrophic failure. Here, however, for a basic understanding we confine ourselves considering only the lateral vibrations of the Laval rotor with a transverse crack.
2. A simple crack model The deflection line of a shaft with a crack in the tension zone (Fig. 1) is superimposed from two parts: the deflection line of the uncracked shaft and an additional deflection from the local flexibility of the crack. This additional part cannot be found from the bending theory. Because for the beam theory a crack is a weakening Tel.: +49 30 314 21 337; fax: +49 30 314 22866.
E-mail address: [email protected] 0888-3270/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ymssp.2007.11.023
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Without crack
plus
Crack contribution Fig. 1. Deflection line; contributions of the uncracked shaft and the local crack flexibility without crack contribution.
1.5
^ H 22 ^ H
33
1.0
2R a ^ H 22
0.5
^ H 33 0.5
a/R
Crack depth a /R
0
^ H 22
0
0.028 0.144 0.374 0.734 1.248 1.947
^ H 33
0
0.001 0.009 0.038 0.107 0.245 0.493
0.1
0.2
0.2
0.4
0.5
0.6
Fig. 2. Main- and cross-flexibility of the open crack from [9].
of the bending stiffness EI on a length zero, only a three-dimensional consideration (or a two-dimensional approximation) is able to yield this additional weakening of the stiffness [9,13,14]. In fact, due to the loss of symmetry a cracked round shaft produces a coupling of lateral vibration, axial and torsional vibration. However for the sake of simplicity we ignore these effects and focus our interest only on the lateral vibration. In [9] we find the crack flexibilities for a general cracked beam element with 6 degrees of freedom at each end. We simplify and pick up only the main flexibility H22 and the cross-flexibility H33 (Fig. 2). These flexibilities are presented in a dimensionless form. To get the physical flexibilities they have to be divided by ER3, where E is the Young’s modulus an R the radius of the shaft, H 22 ¼ H^ 22 =ER3
and
H 33 ¼ H^ 33 =ER3 .
Fig. 2 shows these flexibilities versus crack depth a/R. For small cracks (a/Ro0.5) the cross-flexibility H22 is much smaller than the main flexibility H33, so we go ahead and ignore the cross-flexibility completely. This has the advantage that the crack flexibility is represented by only one parameter. So the understanding of our analytical results will be eased. Thus the deflection at the position of the disc with an open crack nearby can be written as #( ) ( ) " fB wB h0 þ DhB ¼ , (1) fZ vZ h0 þ 0
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fζ l/2
l/2
wζ Fig 3. Crack model (hinge model) for the ‘‘breathing’’ crack.
where h0 is the flexibility of the round uncracked shaft and DhB ¼ H^ 22 ‘2 =ER3 16
(2)
is the (translatoric) additional flexibility of the crack. Hereby the span of the shaft is l. Turning the shaft with the crack into the pressure zone the crack will be closed due to the pre-stressing of the weight and Dhz will become zero; the shaft is round again. This behaviour is represented by the hinge model shown in Fig. 3. The hinge opens in the tension zone and closes in the pressure zone. Thus the crack mechanism is non-linear, respectively, bi-linear due to our assumptions. So for our further calculations we re-write Eq. (1) in the following form: ( ) " # !( f ) wB h0 DhB B ¼ þ f ðtÞ , (3) fZ vZ h0 0 where f(t) is the steering function being f(t) ¼ 0, when the crack is in the pressure zone and f(t) ¼ 1, when the crack is in the tension zone. For very deep cracks a modification of the steering function is advisable [2,11,13]. We now invert Eq. (3), we change from the flexibility presentation to the more practical stiffness formulation for our equations of motion ( ) " # " #!( ) fB wB s0 DsB ¼ f ðtÞ , fZ vZ s0 0 f rot ¼ S0;rot þ DS rot ðurot ; tÞ urot , (4) whereby 1=ðh0 þ DhB Þ ¼ s0 DsB .
(5)
So Dsz is the loss of stiffness due to the opening of the crack and s0 ¼ 1/h0. Having a real rotor Dsz can easily be determined by an experiment, measuring the natural frequencies with open and closed crack (Fig. 4). pffiffiffiffiffiffiffiffiffiffi Taking the natural frequency of the shaft with closed crack o0 ¼ s0 =m as a reference frequency to which we relate the natural frequency oz with an open crack (Fig. 4), we get the parameter 2 oB DsB ¼1 . o0 s0 The square root oz/o0 will be used later as dimensionless crack-depth parameter. Until now the stiffness matrix in Eq. (4) is defined in rotating, rotor fixed coordinates. We now shift to an inertial frame. From Fig. 5 we find the well-known transformation matrix for the deflections ( ) " #( ) wB cos Ot sin Ot w ¼ , vZ sin Ot cos Ot v urot ¼ TðtÞ uinertial
(6)
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Crack (S0 - ΔSζ) / m
ωζ =
ω0 =
S0 / m
Fig. 4. Experimental determination of the stiffness loss DsB due to the crack.
η vη v y
w wζ
Ωt ζ
z
Fig. 5. Rotating and non-rotating coordinates.
and analogously for the forces f rot ¼ TðtÞ f inertial , f inertial ¼ T1 ðtÞ f rot ðtÞ.
(7)
Inserting these transformations of forces and deflections into Eq. (4), yields the stiffness matrix, in the inertial frame SðtÞ ¼ S0 þ DSðtÞ ¼ T1 ðS0;rot þ DSrot ÞT. Written explicitly " S0 þ DSðtÞ ¼
and worked out further " s0 S0 þ DSðtÞ ¼
"
#
s0
# s0
cos2 Ot
sin Ot cos Ot
sin Ot cos Ot
sin2 Ot
1 þ cos 2Ot 1 f ðtÞ DsB 2 sin 2Ot
sin 2Ot
f ðtÞDsB
s0
(8)
1 cos 2Ot
#
.
(9)
This stiffness matrix is twofold time dependent: once due to the transformation from rotor-fixed to inertial coordinates and further due to the steering function f(t) being 1 or 0 depending from the crack position in the compression or the tension zone of the shaft. Since f(t) depends on the shaft deflection u as well, being more
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carefully we should write S ¼ Sðu; tÞ ¼ S0 þ DSðu; tÞ.
3. Equations of motion and its linearization for a horizontal shaft—the breathing crack We now consider the equations of motion for our simple Laval rotor with the disc mass m, the damping coefficient d and the stiffness matrix S(u, t) just derived (Eq. (9)). On the right-hand side (RHS) there are the vectors of weight and unbalance forces p0, respectively, pu ( ) " #( ) " #( ) " #( ) ( ) cos ðb þ OtÞ s11 s12 m d mg w€ w_ w 2 , þ þ ¼ þ mO sin ðb þ OtÞ s21 s22 m d 0 v€ v_ v M€u þ D_u þ Sðu; tÞu ¼ p0 þ pu .
(10)
The weight vector on the RHS is of great importance when linearizing the equations of motion. For with heavy crack endangered horizontal rotors the static deflection u0 is large compared to the additional deflections caused by the dynamics of the rotating shaft. This is confirmed by climps in Figs. 6 and 7, derived from wstat ¼ mg=s0 and o20 ¼ s0 =m0 , where o0 is the natural frequency (critical speed) of the uncracked system (Fig. 4).
η
v
v
y
ε
w
w
β z
Ωt
Ωt
ζ
Fig. 6. Laval rotor with a cracked shaft and the equations of motion (Eq. (21)).
1000
2000
3000 n0 [min-1 ]
wstat [mm]
100
200
300
ω0 [rad / s]
0.5
1.0
1.5 Fig 7. Static deflection and natural frequency o0.
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MG MG
MG
MG
MG
MG
Ωt 0
90 open
180 closed
360 ψ = Ωt
270 closed
open
wz stat wz
f (t) 1 open 0
open
closed 90
180
270
ψ = Ωt 360
Fig. 8. Static deflection and opening and closing of the ‘‘crack’’ versus angle of rotation; steering function f(t).
Having a (first) critical speed of 600–800 rpm like with heavy generators or low-pressure turbines we get static deflections of 1 mm or more (Fig. 7). Whilst dynamic deflections of 0.1 mm are already very serious. So it is obvious to split the total deflection mg=s0 uðtÞ ¼ u0 þ DuðtÞ ¼ þ DuðtÞ, (11) 0 knowing that Du(t) is small, Du(t)5u0. In other words the weight deflection is dominating and will govern the opening and closing of the crack, because the vibration amplitudes Du(t) are small compared to the static deflection. Fig. 8 shows the ‘‘breathing’’ of the crack under the weight influence when the shaft is slowly turned. Please notice inspite of the steps in the steering function f(t), the deflection of the slowly rotating shaft is softly and continuously changing with Ot. With a small crack (a/Ro0.5) the crack opens and closes indeed very abrupt when passing 901, respectively, 2701, where the shift from the tension zone to the compression zone and vice versa occurs. Having deeper cracks the transition will be less abrupt. So Mayes suggested a more steady steering function f(t) ¼ (1+cos Ot)/2 for deeper cracks as it comes closer to reality [2,9,11,13]. For our basic considerations the step function of Fig. 8 is quiet sufficient as we focus our interest on the early state (a/Ro0.5). And by the way, the influence of the steering function on the results is astonishingly small [13]. Now we are prepared for the formal linearization procedure. Due to the weight dominance we re-write in Eq. (10) M€u þ D_u þ Sðu; tÞu ¼ p0 þ pu for the stiffnesses SðtÞ ¼ S0 þ DSðtÞ
(12)
using Eq. (9) with the step function as steering function. The deflections are split up into the static and the dynamic part according to Eq. (11). Inserting both, Eqs. (11) and (12) into the equations of motion yield
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a linear but periodically time-variant equation for the dynamic vibrations Du(t), MD€u þ DD_u þ ½S0 þ DSðtÞ Du ¼ DSðtÞu0 þ pu
(13)
as the static equilibrium can be separated: " # s0 wstat mg ; ¼ s0 0 0 S0
u0
¼ p0 :
The periodic time-variant stiffness matrix DS(t) appears twice. On the left-hand side it decides (together with the damping) whether the system is stable or not. On the RHS this matrix is multiplied with the weight deflections u0 producing an additional excitation force—the crack excitation—that is periodic in time. If the stability of the homogenous solution is guaranteed—this of course must be checked—a further simplification for the calculation of the forced vibrations due to the RHS in Eq. (13) is possible. Then the homogenous part of the complete solution DuðtÞ ¼ Duh ðtÞ þ Dup ðtÞ is fading out and the term DSðtÞDuðtÞ on the left-hand side will be very small. Neglecting this term for the calculation of the forced vibrations a timeinvariant differential equation MD€u þ DD_u þ S0 Du ¼ DSðtÞu0 þ pu
(14)
has to be solved. Actually this is the equation of motion of a round, uncracked shaft. On the RHS we find the unbalance excitation and the crack excitation forces DS(t)u0. 4. Stability The stability analysis was performed with Floquet’s theory. The homogenous system of the equations of motion (RHS zero in Eq. (13)) was transformed into first-order differential equations (state space) and the transition matrix U(T) was determined. This matrix tells us, how the vibrations due to initial conditions have developed after one cycle T ¼ 2p/O. So in order to find the transition matrix, we have to integrate all possible initial conditions over one period T 9 9 8 8 Dw > Dw > > > > > > > > > > > > > > > > > = = < Dv > < Dv > ¼ UðTÞ , > > Dw_ > Dw_ > > > > > > > > > > > > > > > > > ; ; : : D_v t¼T D_v t¼0 xT ¼ UðTÞx0 .
(15)
Having the transition matrix we demand: of which kind is the proportionality between the initial vector x0 and the state xt after one cycle t ¼ T xT ¼ mx0 .
(16)
These two Eqs. (15) and (16) yield Floquet’s eigenvalue problem ½UðTÞ mIx0 ¼ 0,
(17)
that gives an answer, whether the system is stable or not. For a stable system the magnitude of all eigenvalues must be less than 1, jmk jo1.
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To be more illustrative: |mk| gives us the factor of increase or decrease of the initial conditions after one revolution (cycle) of the rotor. Fig. 9 presents a survey on the stability of the cracked rotor. The stability depends on the rotational speed and the crack depth. Here damping was assumed to be zero—the worst case. The (dimensionless) crack-depth parameter in this figure is the frequency ratio oz/o0 (see Fig. 4). The rotational frequency O was made dimensionless with the natural frequency of the uncracked rotor o0U The unstable regions, |mk|41, are found at the dimensionless rotational speeds Z ¼ O=o0 ¼ 25; 24; 23; 22; 21.
(18)
Rotational speeds beyond Z ¼ 2 are always stable. With deeper cracks the unstable zone near to Z ¼ 2 becomes very broad. The unstable zone near to Z ¼ 1 is familiar to us from the rotor with unequal bending stiffnesses (EIz6¼EIZ). The regions of instability at Z ¼ 2/3, 2/4y are very small. As soon as we have some damping, they disappear [13].
|μ k|
3
2
1
0
0.5 0.66
1.0
1.5
Rot. speed Ω/ω0
0.52 0.76 ω ζ/ ω 0 pth e d k
2.0 1.0 c Cra
Fig. 9. Stability of the cracked Laval rotor versus rotational speed and crack depth, damping D ¼ 0. Unstable zones jmk j41. 0.7 stable
Crack depth ω ζ /ω 0
st able
unstable
0.8 D = 0.01
0.9 D=0 1.0
2 4
2 3
1
2
Rotational speed Ω /ω0
Fig. 10. Borderlines of stability with a damping of 1% (D ¼ 0.01) and no damping (D ¼ 0).
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Compared with instabilities due to fluid film bearings the crack instability can be very brutal: |mk| ¼ 2 is to say: after 10 revolutions of the rotor the amplitude has grown with a factor 210 ¼ 1024. Fortunately some small damping in the system is already very helpful to guarantee stability. Compare the width of the unstable zones for D ¼ 0 and 0.01 in Fig. 10. In [16] the borderlines of the unstable zones for zero damping, D ¼ 0 were calculated numerically. An analytical approach is given by
2 1 DsB O 2 1 , (19) o o n 4 s0 o0 n where n ¼ 1,2,3,y. The influence of fluid film bearings is discussed in [16,19].
5. Forced vibrations 5.1. Vibration forced by the crack influence Assuming stability of the system the forced vibration response may be calculated with the time-invariant equations of motion (Eq. (14)) " #( ) " #( ) " #( ) s0 d m Dw€ Dw_ Dw þ þ s0 d D_v m D€v Dv ( ) ) " #( cosðOt þ bÞ wstat 1 þ cos 2Ot sin 2Ot 1 . (20) ¼ DsB f ðtÞ þ mO2 2 sinðOt þ bÞ 0 sin 2Ot 1 cos 2Ot This equation can be simplified by introducing a complex notation (Fig. 11). Dr is the vector containing both deflections of the shaft Dv and Dw pffiffiffiffiffiffiffi Dr ¼ Dw þ jDv with j ¼ 1. (21) pffiffiffiffiffiffiffi Formally, we multiply the second equation in Eq. (20) with the imaginary unit 1 and add it to the first equation. Remembering Euler’s formula this yields the new second-order differential equation m D€r þ d D_r þ s0 Dr ¼ 12wstat DsB f ðtÞ 1 þ e2jOt þ mO2 ejðOtþbÞ . (22) We first focus our interest on the pure crack response (no unbalance, e ¼ 0), that is proportional to the static deflection wstat and the stiffness loss due to the crack Dsz. We develop the step function of opening and closing of the crack in Fourier-series f ðtÞ ¼ 12 þ ð2=pÞ½cos 1Ot ð2=3pÞ cos 3Ot þ ð2=5pÞ cos 5Ot .
Δv Im y Δr
Δw
Re z Fig 11. Complex coordinates Dr for the deflections Dw and Dv.
(23)
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Reminding Euler’s equation cos nOt ¼ 12 eþjnOt þ ejnOt yields the crack force Fc on the RHS of Eq. (22) as a complex Fourier-series k¼þ1 X
F c ¼ DsB wstat
bk ejkOt ,
(24)
k¼1
where bk are the participation coefficients of the harmonics. In Table 1 they are presented with an interpretation of their physical meaning. Now the response of the rotating shaft due to a single harmonic bk of the crack excitation can easily be determined. With the approach Dr ¼ D^rk ekjOt we receive from Eq. (22) m D€r þ d D_r þ s0 Dr ¼ DsB wstat bk ejkOt
(25)
the equation O2 k2 m þ jOkd þ s0 D^rk ¼ DsB wstat bk
(26)
and from that the response amplitude D^rk ¼ wstat DsB
bk . ðs0 O k mÞ þ jðkOdÞ
(27)
2 2
By superposition of all components k the complete response of the rotor to the crack excitation turns out to be DrðtÞ ffi wstat
X DsB k¼þ3 bk ejkOt : s0 k¼3 1 k2 Z2 þ 2jDkZ
(28)
Since the high-order components in this series presentation are very small or exactly zero we truncate after k ¼ 73. The contribution k ¼ 0, b0 ¼ 0.25 is an additional static deflection due to the crack, averaged over one revolution with the magnitude 0:25wstat DsB =s0 , giving the reference level for the vibrations.
Table 1 Crack excitation force and its harmonics k 4 3 2 1
Backward
0 +1 +2 +3 +4
Additional static deflection Forward
Excitation
bk
Four times per revolution Three times per revolution Twice per revolution Once per revolution
0 0.021 0 0.106
Once per revolution Twice per revolution Three times per revolution Four times per revolution
0.250 0.318 0.250 0.106 0
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The contribution k ¼ 71, b1 ¼ 0.106 and b+1 ¼ 0.318 describes a forward and a backward circular orbit passed once per revolution DsB bþ1 eþ1jOt , 2 þ1 s0 1 ðO=o0 Þ þ 2jðO=o0 ÞD DsB b1 D r ðtÞ ¼ wstat e1jOt . 2 1 s0 1 ðO=o0 Þ 2jðO=o0 ÞD
D r ðtÞ ¼ wstat
(29)
They superimpose to an elliptic orbit passed once per revolution in a forward sense (Fig. 12). The resonance belonging to this ellipse is at O ¼ o0 (for deeper cracks at O ¼ o0 , see Section 5.3). The contribution k ¼ 72, b2 ¼ 0 and b+2 ¼ 0.25 describes a circular orbit passed forward twice per revolution of the shaft, Dr2 ðtÞ ¼ 0 þ wstat
DsB bþ2 eþ2jOt . 2 s0 1 ð2O=o0 Þ þ 2jð2O=o0 ÞD
(30)
Here the resonance occurs at O ¼ o0/2. Finally the contribution k ¼ 73, b3 ¼ 0.021 and b+3 ¼ 0.106 results in a somewhat elliptical orbit passed in a forward sense three times per revolution of the rotor. The resonance is at O ¼ o0/3. The magnification functions for k ¼ 3, +1, +2, +3 are shown in Fig. 13. More instructive however is Fig. 14, where the circular and elliptic orbits of the once, twice and thrice per revolution vibrations are shown over the dimensionless rotational speed Z ¼ O/o0 for the case of 5% damping. The curve touching the left arm of the cross (o) at Z ¼ 1 in the elliptic 1O-orbit is presenting the phase angle resulting from Eq. (29). The transparency of this presentation is lost as soon as the three contributions are superimposed to the real physical orbit of the system (Fig. 15). At Z ¼ 13, e.g. we have three loops per one revolution of the shaft, as this is the resonance of the three times per revolution excitation. With Z ¼ 12 there are two loops within one revolution of the shaft a.s.o. The result of the forward–backward decomposition of the orbits—the result of the two-sided spectral order analysis—is shown in Fig. 16. Combined with the phase development (not shown here) this is a very helpful instrument for early crack detection. Ω
^r e-1j Ωt -1
Im y ^r e+1j Ωt +1
Fig. 12. Elliptic orbit superimposed from the forward (+1O) and the backward (1O) circular vibration.
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Δr^k
s0 ) wstat ( Δs ζ 3
2 2Ω
3Ω
1Ω Forward whirl
1 0.32 0.25 0.11 0
0.33 Δr^-1 wstat
0.5
Ω ω0
1.0
s0 ( Δs ) ζ
Backward whirl
1
-1Ωt Ω ω0
0.11 0
0.5
1.0
Fig. 13. Magnification functions of the forward 1O-, 2O- and 3O-circular vibrations and the backward 1O circular vibration; D ¼ 0 and 0.05.
ed l spe
Ω /ω 0
=η
tiona
Rota 1.0
3Ω
ard
forw
bit, - Or
0.50 0.33
bit,
Δvy
3
Or Ω-
.
fwd
Δvy
bit,
1
Or Ω-
.
fwd
Fig. 14. Elliptical forward orbits (1O and 3O) and the circular 2O (forward) orbit versus dimensionless rotational speed; D ¼ 0.05. They superimpose to Fig. 15.
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η≈0
0.25
+
0.33
0.40
+
+
0.50
+
+
+
+
+
+ Ω/ω0 = 1
0.65
0.90
0.80
1.10
1.25
Fig. 15. Orbits of the shaft center versus (dimensionless) rotational speed Z, D ¼ 0.05. „Amplitude“ ^rk
6
6
5
5
4
4
3
3
-1Ω
-3Ω
2
+1Ω
+2Ω
2 1
+3Ω -3
-2
-1 0 Frequ ency
1 ω krit
2
nal spe
0 -4
Ω /ω krit
Rotatio
−ω
1 0
ed
-2Ω
3 +ω
Fig. 16. Two sided spectral order analysis of the Laval rotor with a crack. Dimensionless amplitude versus dimensionless rotational speed and frequency.
5.2. Crack- and unbalance response In practise there is always an overlay of crack- and unbalance response. Thus superimposing both solutions of Eq. (22) we receive DrðtÞ ¼ Dr þ Drcrack ¼ ejb
X Z2 DsB k¼þ3 bk jOt ekjOt . e þ w stat 1 Z2 þ 2jDZ s0 k¼3 1 ðkZÞ2 þ 2jkZD
(31)
The +1O response now has two contributions: one from the crack and one from the unbalance, compare Eqs. (22) and (29). It depends on the angular position b between eccentricity and crack (Fig. 6), whether these two contributions intensify or compensate each other, as the exciting +1O forces are now mO2 ejb þ wstat DsB bþ1 ejOt .
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Manipulating the unbalance in away that b becomes 1801 we may completely suppress the +1O response due to the crack for the rotational speed O considered. Then we balance the rotor so that b ¼ 1801 and mO2 ¼ wstat DsB b1 . By balancing in a way that mo20 ¼ wstat DsB b1 is fulfilled, even the resonance at O ¼ o0 is suppressed. However it is impossible to exert an influence by balancing on the 1O response or the 2O and 3O contributions of the crack and on the additional static deflection due to b0. 5.3. Deeper cracks In Sections 5.1 and 5.2 we used Eq. (20) for the calculation of crack- and unbalance response. As the left-hand side in this Eq. (20) describes nothing else but the uncracked shaft, we took its natural frequency pffiffiffiffiffiffiffiffiffiffi o0 ¼ s0 =m as reference frequency. So all results presented in these sections are valid only for small cracks. The well-known fact that with somewhat deeper cracks the ‘‘natural frequency’’ seen in the measured resonance diagrams is a little bit lower than that of the uncracked shaft o0, can be introduced into our results in the following way. Considering only the static part, k ¼ 0, b0 ¼ 0.25 in Eq. (28) and its solution we find the additional static deflection Drstat;ad ¼ 0:25wstat Ds=s0 averaged over one revolution. The total static deflection—including both the static deflection of the uncracked shaft wstat ¼ mg/s0 and its additional averaged contribution of the crack—will be
mg 1 Dsx mg wstat;tot ¼ wstat þ Drstat;ad ¼ 1þ ¼ . s0 4 s0 s0 So s0 is thepsomewhat reduced stiffness of the rotating shaft including now the weakening due to the crack. ffiffiffiffiffiffiffiffiffiffi From o0 ¼ s0 =m we find the somewhat lowered natural frequency
1 Dsx o0 o0 1 . 8 s0 For deeper cracks s0 , respectively, the crack-depth-dependent natural frequency o0 has to substitute o0 in Eqs. (26)–(30) of Section 5.1, describing the resonance behaviour. 6. Final remarks Although the dynamics of a rotating shaft with a crack is theoretically transparent, in practise early crack detection is still not easy. Hardly a crack was detected in a turboset running for month with constant rotational speed having a crack depth of less than 25% [5,6]. At that depth we have already a slight change in the natural frequencies (resonances) of some 2% or 3%. When manoeuvring this—and of course the 2O resonance at half-critical speed—becomes visible. The fluid film damping impedes the early crack detection, e.g. the 3O resonance is usually completely suppressed. However permanent monitoring enables to trend analysis (today’s vibration minus vibration a fortnight before), that is very helpful for successful early crack detection. Having a suspicious rotor balancing test weights that open or close the crack may give valuable hints (see Section 5). Much more information on early crack detection will be found in the following contributions to this monograph.
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Acknowledgement Thanks to Dr. Ing. Alexander Bormann, who helped me so much when organizing this paper. References [1] R.A. Henry, B.E. Okah-Avae, Vibrations in cracked shafts, in: on Institution of Mechanical Engineers Conference, Vibrations in Rotating Machinery, Cambridge, UK, 1976, Paper C 162/76. [2] I.W. Mayes, W.G.R. Davies, The vibrational behaviour of a rotating shaft system containing a transverse crack, in: on Institution of Mechanical Engineers Conference, Vibration in Rotating Machinery, Cambridge, UK, 1976, Paper C 168/76. [3] R. Gasch, Dynamic behaviour of a simple rotor with a cross-sectional crack, in: on Institution of Mechanical Engineers Conference, Vibration in Rotating Machinery, Cambridge, UK, 1976, Paper C 178/76. [4] R. Gasch, Kleiner Beitrag zur Behandlung des dynamischen Verhaltens einer rotierenden Welle mit angerissenem Querschnitt. 4 (A Short Paper on the Dynamics of a Rotating Shaft with a Crack), ILR-Bericht 8, TU Berlin, 1975 (ISBN 3 7983 0551 X). [5] A.F.P. Sanderson, The vibration behaviour of a large steam turbine generator during crack propagation through the generator rotor, in: on Institution of Mechanical Engineers Conference, Vibrations in Rotating Machinery, Bath, UK, 1992, Paper C 432/102. [6] Allianz-Berichte, Schwingungsu¨berwachung von Turbosa¨tzen—ein Weg zur Erkenntnis von Wellenrissen (Vibration monitoring in turbosets—a way to early crack detection), Bericht 24 (1987). [9] W. Theis, La¨ngs- und Torsionsschwingungen bei quer angerissenen Wellen, 9 (Longitudinal and Torsional Vibrations of Rotating Shafts with a Transverse Crack), Reihe 11, Nr. 131, VDI-Verlag, Du¨sseldorf, 1990. [11] J. Mayes, W.G.R. Davies, Analysis of the response of a multi-rotor-bearing system containing a transverse crack in a rotor, Journal of Vibration, Acoustics, Stress and Reliability in Design 106 (1984) 139–145. [13] R. Gasch, M. Person, B. Weitz, Dynamic behaviour of the Laval rotor with a cracked hollow shaft—a comparison of crack models, in: on Institution of Mechanical Engineers Conference, Vibration in Rotating Machinery, Edinburgh, UK, 1988, C314/88. [14] B. Schmalhorst, Experimentelle und theoretische Untersuchung zum Schwingungsverhalten angerissener Rotoren (Experimental and Theoretical Approach to the Vibrational Behaviour of Cracked Rotors), Reihe 11, Nr. 117, VDI-Forschungs-Berichte, 1989. [16] G. Meng, R. Gasch, Stabilita¨t eines gleitgelagerten, einfachen Rotors mit RiX (Stability of a Cracked Rotor in Fluidfilm Bearings), in: SIRM-Tagung ‘‘Schwingungen in rotierenden Maschinen,’’ Wien, February, Tagungsbericht Vieweg Verlag, 1992. [19] F. HiX, Nichtlineare Dynamik und Zustandsbeobachtung gleitgelagerter elastischer Rotoren mit angerissenem Wellenquerschnitt (Nonlinear dynamics and monitoring of flexible rotors in fluidfilm bearings having a transverse crack), Dissertation, TU Berlin, 1996.
Further reading [7] Schenck-Allianz, Unterlagen zum Symposium ‘‘Methoden, Nutzen und Trends der schwingungstechnischen U¨berwachung von Turbosa¨tzen in Kraftwerken und Industrieanlagen,’’ 7 (Symposium: Methods, Trends and Advantages of Vibration Monitoring in Industrial Plants and Turbosets), Sonthofen Ma¨rz, 1993. [8] I. Imam, et al., Development of an on-line rotor crack detection and monitoring system, ASME, Journal of Vibration, Acoustics, Stress and Reliability in Design 111/241 (1989) 12. [10] B. Grabowski, R. Po¨ppel, Das Schwingungsverhalten eines Rotors mit QuerriX—experimentelle und theoretische Ergebnisse (Vibrational Behaviour of a Rotor with a Crack—Theoretical and Experimental Results), VDI-Schwingungstagung, vol. 4456, VDI-Berichte, Du¨sseldorf, 1982. [12] H.D. Nelson, C. Nataray, The dynamics of a rotor system with a cracked shaft, ASME-Journal of Vibrations, Acoustics, Stress and Reliability in Design 108 (1986) 189–196. [15] M. Liao, R. Gasch, Crack detection in rotating shafts—an experimental study, in: on Institute of Mechanical Engineers Conference, Vibrations in Rotating Machinery, Bath, UK, Paper 1992–6, C 432/106. [17] W. Rothkegel, RiXerkennung bei rotoren durch Schwingungsu¨berwachung (Crack detection in rotors with the aid of vibration monitoring), Dissertation, Universita¨t, Fakulta¨t fu¨r Maschinenwesen, Hannover, Forschungs-Berichte, Reihe 11, Nr. 180, VDI, VDI-Verlag, Du¨sseldorf, 1993. [18] D. So¨ffker, P.C. Mu¨ller, Betriebsu¨berwachung und Schadensdiagnose an rotierenden Maschinen-Bewa¨hrte Methoden versus neue modellbasierte Ansa¨tze (Monitoring and fault detection in rotating machinery- classical methods and model based approaches), in: R. Irretier, et al. (Eds.), SIRM Tagung III Schwingungen in rotierenden Maschinen, Kassel, Vieweg Verlag Braunschweig, 1995. [20] R. Gasch, R. Nordmann, H. Pfu¨tzner, Rotordynamik (Rotordynamics), second ed, Springer, Berlin, 2002, 700pp. (ISBN 3-540-41240-9).