Rotor support systems interaction with parametric excitation is considered for both unequal principal shaft stiffness (generators) and offset disc rotors (ventilators). Instability regions and types of instability are computed in the first case, and parametric resonances in the second case. Computed and experimental results are compared for laboratory machine models. A field case study of parametric vibrations in industrial ventilators is reported. Computed parametric resonances are confirmed in field measurements, and some industrial failures are explained. Also the dynamic influence and gyroscopic effect of supporting structures are shown and computed

Krodkiemski, T.

Marynowski, K.

Parszewski, Z.

English

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PARAMETRICNSTABILITIESOFROTOR-SUPPORTSYSTEMS
WITHAPPLICATIONTOINDUSTRIALVENTILATORS
Zdzislaw Parszewski, Janusz Krodkiewski,
and Krzysztof Marynowski
Politechnika _odzka
_odz, Poland 90245
SUMMARY
Rotor-support systems interaction with parametric excitation is considered
for both unequal principal shaft stiffnesses (generators) and off-set disc
rotors (ventilators). Instability regions and types of instability are com-
puted in the first case, and parametric resonances in the second case. Com-
puted and experimental results are compared for laboratory machinemodels.
A field case study of parametric vibrations in industrial ventilators is re-
ported. Computedparametric resonances are confirmed in field measurements,
and someindustrial failures are explained. Also the dynamic influence and
gyroscopic effect of supporting structures are shownand computed.
INTRODUCTION
With growing rotor speeds or machine capacities the influence of support-
ing structures on stability and critical speeds is of increasing importance,
specially when somerotor parameters are variable. This is often the case in
machines used in mining and power generating industries, for example, large
generators, industrial ventilators, and turbopumps. The unequal rotor princi-
pal stiffnesses in the first instance and unequal principal momentsof inertia
of the ventilator disc in the second instance, with the flexible anisotropic
support interaction, cause the parametric effect.
A 120-MWgenerator of Dolmel-Wroc_awproduction is shownin figure I.
Its rotor is of 9.157 m length, with a span of 6.858 m between bearings. The
wiring slots distribution gives rise to small unequality of the rotor principal
stiffnesses kx,ky. Twotypes of mine ventilators are shownin figure 2:
large ventilator No. I of the type SIaM(fig. 2(a)) of nominal output of
320 m/min, and universal ventilator No. II of type FKD-30 (fig. 2(b)). The
first has an overhang shaft of full length 0.908 m. The disc is between bear-
ings in the second ventilator with a shaft of 0.773 m length. Production errors
cause the unequality of principal momentsof inertia of the disc.
GENERALTHEORY
The rotor-support system can be considered as two connected and dynamically
interacting subsystems. One (rotor) is a discrete parametric system (fig. 3,
bold lines). The second is nonparametric, contains the whole supporting struc-
ture, and is usually very complicated. A receptance (impedance) matrix is hence
used for description of its dynamic behavior. Its elements can then be mea-
sured if not calculated. All (usually infinity) but the connecting coordinates
383
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of the supporting structure can be eliminated (ref. 6) from the equations of
motion of the systems, giving their general matrix form
(t)
Here all the matrices are of the order equal to the sum of the number of de-
grees of freedom of the parametric subsystem (rotor) and the number of connect-
ing coordinates. In the cases considered, for instance, the last number is
four (fig. 2); _r and Yu are, respectively, real and imaginary parts of the
dynamic impedances of the supporting structure along the connecting coordinates;
G, P, and Q are force vectors. For an offset disc mounted flexibly (fig. 4)
or for a rotor of different principal stiffnesses (the two cases are here con-
nected only formally), the equations are
For a rotor of different principal stiffnesses
M(t)=M ; C(t)=C; K(t)=K+_.{Ecos_t*" Fsin _t) (2a)
For an offset disc and circular shaft
M_t)=M+ d(t); C(t)=C* Z*c(t) ; K(t)=K+ d(t) (2b)
where
ooII°o°°ii9-B'Bff° o o o o ZeB.,- 0 0M= o -_ s_ o Z- o oo s. -B.ol -,=Bo o
o o o- -mj o o o
I O
"& B COS2¢4t ", %Si n ¢91lt
bl 0 0
%cosk)t - S_i n (4t
-
O O O
o _ssio_,._os_,_os_,-%_in_,
O O O
o 6B cos2_t- E_in_t -S_sin _t - S_os_t '
o -s#io,.,,-S_o_,.,t o
Ii 0 0 0
26B_sin2et+2B.cr)s2_t o 2_,BQcos2et-2D_sin2_t _!
0 0 0 0
/
2A B_sin2_t _,2E_sin_t o -2,B_sin2_t-2 E_cos2et _/
_= o _
o0 0
(5)
(3)
X - (4)
[ -,.2s_f
-D}
p__] ,2s_s
- Dr t
384
-_
"0 0 0 0 O"
0 0 0 0 0
0 0 P 0 0
0 0 0 0 0
(6)
(7)
--.
_2 s,t_J
D_
_* s_
- D_
where S_q,... Bn_ are the first and second moments of the rotor mass with re-
spect to the coordinate planes _q_ connected with the rotor; D_, Dq, D_ are
corresponding deviation moments; and
Finally
B" _.IBm,j* BI,;) (8)
(9)
where _ and 8 are the arguments of the solutions
x=eJ_t:_Caisin t+ bi cos t)
I--O
(to)
of the homogenous part of equation (I) at stability limits.
Instability
The conditions of existence of steady solutions (i0), giving the boundary
equations in the stability region, in the equivalent forms for odd or even
numerals i, are
T
!g. : ,...
i ee.
. ----D -- -- IQ• ,!
°'o I me.
ga' |e.e
=0 (tt)
Consecutive approximations are shown with broken lines in equation (ii). The
first approximation is
385
-- 0 (12)
The instabilities occur in the vicinity of circular frequencies
kj + Xk
n = i, 2, 3, (13)
@jk = n
The width of the instability region depends on _ and damping C. Here Xj
are the natural frequencies of the system, being the roots of equation (12) at
zero parametric excitation (_ = 0) and no damping (C = 0, Yu = 0).
Resonance Speeds
Forced vibrations analysis gives for the resonance speeds
and 6)
(2n- l)_cr = _i(_cr)_
and
=2n C0cr z (_cr)
_cr (refs. 3
(14)
Here is the axial natural circular frequency of the system.
LABORATORY MODEL TESTS AND COMPUTATIONS
Computations were done for a laboratory model (fig. 5) and compared with
the tests results.
Generator Model - Instabilities
The instability regions were computed for the model with the shaft of dif-
ferent principal stiffnesses kx, _ and rotor block between bearing (no disc 5
in fig. 5). The determinants (12) give equations of instability region bound-
aries in the form IW(_,@)I = 0. Their roots _i,8i define the stability
limits. An example of computed results is given in figure 6. This diagram
corresponds to the second-order combined instability region 0.5(8xi + @yl)
around point N (fig. 6). The solutions of the equation
lw = o
386
for zero parametric excitation (e = 0) and no damping (C = 0, Yu = 0) are
straight lines
intersecting at point N (fig. 6). For finite parametric excitation the lines
curve and do not intersect, separating in the direction 8 for frequencies
sums i/n(kj + kk) or in the direction _ for their differences I/n(%j - %k).
Hence there are no instabilities (also with no damping) corresponding to the
frequency differences. The curves in figure 6 correspond to constant values of
the determinant, and its zero values (denoted with the crosses) give the in-
stability region. They approach each other with increasing damping and, over
its limiting value, instability disappears. This was traced with changing
shaft length AL, and computedand experimental results are compared in fig-
ure 7. Corresponding oscillograms for the x and y directions are given in
figure 8. The instability region discussed here is denoted by 1 in the insta-
bility regions diagram (fig. 9). There are computed instability regions (with
the type of instability) shownin figure 9 and against them the observed experi-
mental instabilities shownwith the crosses.
Ventilator Model - ResonanceSpeeds
Analysis of forced vibrations was done for the model with circular shaft
and the disc 5 (fig. 5) slightly offset. Hence BD_and B_ are not equal
and Sn_,... , D_ as well as disbalance forces P and Q are not zero, but
e = 0. Graphical solution of equation (i) for the laboratory model is given in
figure i0. The continuous curved lines are the computednatural frequency
curves. Abscissas of their points of intersection with the straight lines
= _ (principal), k = 2_, k = 3_... give the parametric resonance speeds, with
one, two, three, ... vibration cycles per one revolution. Computedvalues of
i0 resonance speeds (taken from diagram, fig. I0) in s-I are
= 20, = 36, _ = 46 _ = 58, = 82,crl _cr2 cr3 ' cr4 _cr5
= 99, = 108 = 112 = 140 = 175cr6 _cr7 ' _cr8 ' mcr9 ' _crl0
Experimental resonance diagrams measured for the right support (fig. 5) are
given in figure ii as continuous broken and dotted lines for the directions
y, x, z, respectively. They comparewell with the computedvalues from fig-
ure i0. The vibrations recorded for speeds close to resonance, numberedin the
diagrams (figs. i0 and Ii) are shownin figure 12. It can be seen that close
to resonance speed of the type n_cr = _i(_cr) vibrations of frequency n_ domi-
nate. Comparisonof figures I0, ii, and 12 shows good consistency of theoreti-
cal and experimental results.
387
FIELD CASE STUDY
The above results were applied in field studies of vibrations of industrial
ventilators that developed repeated failures (shaft breakages). These were
ventilators No. II of the type FKI>-30 (fig. 2(b)) mentioned in the INTRODUCTION.
Analysis
The model of the ventilator consists of two subsystems: the discretized
rotor (fig. 13) described in table i and the supporting structure containing
the ventilator body with bearings and foundation (fig. 14). The supporting
structure, for its dynamic interaction with the rotor, was dynamically defined
with its receptance matrix along the connecting coordinates Xl, Yl, x2, Y2
Cxlxl Cxlx2 Cxlyl Cxly2
Cx2xl Cx2x2 Cx2yl Cx2y2
Cylxl Cylx2 Cylyl Cyly2
Cy2xl Cy2x2 Cy2yl Cy2y2
= _C[ ie] (15)
The receptances Cik were measured on site. The measuring scheme is given in
figure 15. The notations in figure 15 are i - ventilator type II, 2 - electro-
dynamic exciter, 3 - displacement pickup, 4 - generator, 5 - vibration measuring
apparatus, 6 - two-ray cathode oscilloscope, 7 - frequency meter, 8 - phase
meter, 9 - ammeter.
The exciting force was measured indirectly (by current measurement). Re-
ceptance diagrams are given in figure 16. Because of symmetry all the cross
receptances for the perpendicular directions are zero. The other equal pairs
are denoted cii , cii. , , where i = I, 2. On the basis of the re-
ceptance matrix _15) the Cijx' Cijyimpedance (dynamic stiffness) matrix
-i
[Yik ] = [Cik]
was calculated and introducted in the computing program. Forced vibrations
were found as partial solutions of equation (I) for the case of equal rotor
stiffnesses by application of the small-parameter method. Their analysis gives
the formulas (14) for the resonance speeds. Graphical solution is given in
figure 17. The frequency lines are the solutions of the homogenous part of
equation (i). With the approximation used, there are computed two principal
resonance speeds (in s-l), 344 and 280; two subharmonic ones of second order
(of two cycles of vibration per revolution), 172 and 142; and two of third
order, 114 and 94.
388
Experimental Measurement on Site
An experimental resonance test was made on site (in the mine) in the ven-
tilator II in working condition. The measuring arrangement is shown in fig-
ure 18. Here 1 denotes the ventilator tested; 2 denotes the dc motor; 3 de-
notes the transformer displacement transducer; 4 denotes the vibration measur-
ing apparatus; 5 the cathode ray oscilloscope; and 6 the frequency meter.
The resonance diagram for the supports for the practically allowed speed
range from approximately 1200 to 2100 rpm is given in figure 19. The two ob-
served subharmonic resonances, 138 and 174 s-l, are in this speed range; both
of second order are very close to the corresponding computed values, 142 and
172 s-I .
CONCLUSIONS
The second-order subharmonic of frequency 142 s-I coincided with the nomi-
nal working speed of ventilator II, which was below 1500 rpm. Hence it was the
second-order parametric resonance that was responsible for the shaft breakages.
Supporting structure appreciably lowers critical speeds. The gyroscopic effect
was also considered and computed. It is specially visible in the ventilators
with the overhang rotor. The frequency diagram for that type of ventilator,
designated No. I in the INTRODUCTION, is given in figure 20. Important
gyroscopic-effect influence is visible. It was the principal resonance at
144 s-I that was responsible for the failures in those ventilators with nominal
working speed below 1500 rpm.
REFERENCES
i. Bolotin, W. W.: Dinamiczeskaja ustoicivost uprugich sistem. GIT-TL, Moskva,
1956.
2. Szempli6ska-Stupnicka, W.: Przyblizone metody analityczne badania
niestateczno§ci parametrycznej. Ossolineum, PAN, Warszawa, 1976.
3. Parszewski, Z.; Krodkiewski, J.: Stability and Critical Speeds of Non-Circular
Shafts on Flexible Anisotropic Supports. Archiwum Budowy Maszyn No. 2,
1975.
4. Krodkiewski, J.: Proc. V. World Congress TMM, Montreal, 1979.
5. Krodkiewski, J.: Habilitation thesis, Politechnika _6dzka, L6d_, 1980.
6. Krodkiewski, J.; Marynows_i, K.; Parszewski, Z.: Instabilities of Parametric
Rotor-Support Systems. Proc. Conf. Vibrations in Rotating Machinery,
Cambridge, 1980, I. Mech. Eng.
389
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Fig.4
392
_ _9 _ X - e _ 5 I3 __ ,o
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Fig.5 Scheme of model stand
I-DC motor,2-elastic coupling,
3-shaft, 4-rotor mass, 5-disc, 6-ball
bearing, 7-anls otroplc bearing,
8-flat springs, 9-inductive sensor,
10-piezo-electric sensors, 11-re-
corder, 12-recorder tape, 13-ele-
ctrodynamic exciter, 14-generator
of harmonic vibration, 15-cathode
oscilloscope •
Fig .6
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Fig. ii Resonance diagram of right
support vibration of model stand.
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Parametric instabilities of rotor-support systems with application to industrial ventilators

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