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Dynamic simulation of non-linear models of hydroelectric machinery
Mechanical Systems and Signal Processing (1993) 7(I), 29-44
D Y N A M I C SIMULATION OF N O N - L I N E A R MODELS OF H Y D R O E L E C T R I C M A C H I N E R Y R. CARDINAL1 University of Campinas, Geprom-Laboratorio de Projeto Mecanico, P.O. Box 6051, 13081 Campinas, Brazil R. NORDMANN
University of Kaiserslautern, Arbeitsgruppe Maschinendynamik, P.O, Box 3049, 6750 Kaiserslautern, Germany AND
A. SPERBER
BASF AG. ZE/IR, 6700 Ludwigshafen, Germany (Received 15 June 1991, accepted 17 November 199 I) This work describes the development of a non-linear model of the hydro-units at the power plant in Ilha-Solteira (Brazil). A computer program is presented that permits the calculation of the bending vibrations of vertical rotor systems. The program takes into account magnetic interactions, seals, journal bearings, as well as unbalanced, constant, harmonic and stochastic excitation forces. The shaft is modelled with the finite-element method including the gyroscopic effect for the discs. Some simulation results are presented. At the same time a linear model is established to compare with the results of the non-linear model, e.g. for the unbalanced response and the characteristic frequencies. This work has been carried out as part of a research programme Diagnosis of Vertical Machines.
1. INTRODUCTION With increasing size and performance of hydro-units for power generation plants, knowledge of their dynamic behaviour becomes more and more important. Computer-aided vibration simulations are indispensable for design. With the implementation of a predictive maintenance p r o g r a m m e one can utilise vibration simulations if it is necessary to know or to predict the dynamic response for the diagnosis of possible failures. The shafts of such machinery, including generators and water turbines, are submitted to axial, torsional and flexural vibrations. The range of critical frequencies for axial and torsional vibrations, however, is far higher than the frequency range of the excitation. Hence, the main interest is focused on flexural vibrations [1]. The flexural vibrations of vertically arranged turbomachinery with journal bearings show non-linear behaviour. As the static forces on the bearing are very small and the displacements are comparably large, linearisation may lead to errors [2-4]. 1.1. PHYSICAL MODEL OF A HYDROGENERATOR A Francis type hydro-unit with a vertical arrangement, Fig. 1, is modeled. It has a nominal power of 160 M W at a rotational frequency of 1.43 Hz. The vertical distance between the generator and turbine is 6.33 m, the diameter of the generator is 13.2 m. The hollow shaft with inner and outer diameter of 400 and 1300 mm, respectively is modeled 29
30
R. CARDINALI E T A L . Genera tar
1~7 ~torbearing
e bearing Seals
Figure 1. Basic model of the hydro-unit at llha-Solteira, Brazil.
as a Bernoulli beam. The generator and turbine are rigid disks of relatively large diameters, thus provoking gyroscopic effects. The axial support acts as a torsional spring [5]. The hydro-unit has two guide bearings: (1) generator guide bearing: a tilting pad journal bearing centrally pivoted, 20-shoe with 2400 mm journal diameter and 300 mm wide; (2) turbine guide bearing: a tilting pad journal bearing centrally pivoted, 12-shoe with 1650 mm journal diameter and 340 mm wide. To consider the non-linear effects arising due to the vertical arrangement, the forces on the bearings are calculated step-by-step, using a finite difference technique to solve the corresponding Reynolds equations. The seal effects between turbine and housing are considered by mass, damping and spring coefficients. They are also obtained by the use of a finite-difference method. The local magnetic pull of the generator was replaced by fictitious stiffness coefficients [5, 6]. The excitation forces of the system are of mechanical, hydraulic, electrical and electromagnetic origin. Unbalanced masses of the generator and turbine as well as the stochastically fluctuating forces due to the turbulent flow inside the turbine, are the most important excitation sources [7]. Other forces such as misalignment, change of revolving hydraulic radial forces, magnetic imbalance, etc. can also be simulated by harmonic, constant or stochastic forces. 2. EQUATIONS OF MOTION For the purpose of calculating the dynamic behaviour of a rotor system, an idealised model is needed. It may consist of beams, seals, journal bearings, springs and other elements. Each element equation is formed separately. The global equation of motion of the system is built up by superposition of these equations. The motion of each mode i is represented by two displacement coordinates x;, y; and two distortion angles ~P.~i, ~P~.i (Fig. 2). The dynamic behaviour of rotor systems can be described by a system of linear differential equations [M] {~} + [D] {t~}+ [Kl{q} = {F(t)} + {F(t2, q)}
(1)
consisting of individual element matrices. [M], [D] and [K] are the superposed element matrices [8]. {q} is a vector that contains the unknown co-ordinates. {F(t)} is the vector of the time dependent external forces like unbalanced, constant or hydraulic forces.
HYDROELECTRIC MACHINERY
31
c 9yiyf~'~xi"~- sDxi Figure 2. Co-ordinates at node L
{F((t, q)} are the external forces depending on the displacements and velocities of the unknown co-ordinates.
2.1. BEAM ELEMENT The shaft is the most important component of rotor systems. In the model it is divided into several beam elements, which are described by finite elements. Bending stiffness E1 and mass distributions p are taken into account. Gyroscopic effects as well as shear deformations are not considered (Bernoulli beam elements). External forces are only acting on the nodes, i.e. the element ends.
2.2. RIGID DISC ELEMENT Certain elements in a rotor system can be described as rigid elements. This means that their stiffness may be neglected. In this case the element equation has the following form: [MI {q} + [DI{q} = {F},
(2)
where [M] is a symmetric matrix that contains the inertia m and the transverse mass moment of inertia OA, and [D] is the skew-symmetric gyroscopic matrix built with the polar mass moment of inertia Op. As will be discussed later, in the case of non-linear simulations the terms on the damping matrix [D] are treated as velocity dependent external forces on the right side of equation (1).
2.3. SPRING ELEMENT Supports between shaft and housing or housing and foundation as well as the fictitious stiffness coefficients for the simulation of the magnetic pull are simulated using simple spring elements [K] {q} = {F}.
(3)
Also in the non-linear simulation, terms with negative coefficients (magnetic pull) are placed on the right side of equation (1) and treated as displacement dependent forces.
32
R. CARDINALI
ET AL.
2.4. SEAl. Et.EMENT It is well k n o w n that tile fluid forces o,1 the seals (Fig. 3), which are described by e q u a t i o n (4), have a s t r o n g influence on tile d y n a m i c b e h a v i o u r o f r o t a t i n g machinery.
.i: [M M]{i;l+[_ D. dD]{:~}+[_KK]{"I}=IE,. 0
F~
(4)
The d y n a m i c coemcients are o b t a i n e d from experilrlen!.a] investigations or theoretical calculations. Here the theory o f Dietzen and N o r d m a n n [9] is used to calculate the seal
~,~
~ ~,~
~
~
~\"
Seol " \ ~
Rotor
~Stotor
Figure 3. Displacements and forces acting on tile seals.
I
(o)
i
(b)
5.0
f=. r i~'~=/
Figure 4. Seals between turbine and housing (European machine, llha-Solteira).
HYDROELECTRIC MACHINERY
33
coefficients by the finite difference method (FDM). In spite of the resulting large CPUtime, this method allows the analysis of complicated seals to be performed. The example shown in Fig. 4 is for the European machine in IIha-Solteira/BR. 2.5. JOURNAL BEARING ELEMENT
The dynamic behaviour of a vertical machine has a great dependence on the journal bearing. As a horizontal machine possesses a stationary position, iinearised damping and stiffness coefficients can be used. In a vertical machine there are no static forces which result in a stationary position. Moreover, the relative displacements and velocities between shaft and housing have to be known in order to calculate the journal bearing forces. As a consequence the journal bearing forces will also be placed on the right side of equation (1) as non-linear forces (F(q, q} } that depend on the relative displacements and velocities. At the journal bearing the shaft is guided by a pressure field originating normally from a laminar circumferential fluid flow. The order of non-linearity of the journal bearing forces depends on the relative displacements and velocities on the oil film of the journal. As an example, Fig. 5 shows the behaviour of a 360 ° cylindrical bearing.
'NN~ ~ ~
Stator Rotor
~
distribution
Figure 5. Pressuredistribution and forceson a 360° cylindrical bearing. 2.5.1. Calculation of journal bearingforces To calculate the journal bearing forces, the pressure distribution on the oil film has to be determined. Thus, one starts the well known Reynolds differential equation. Due to its complexity, the Reynolds equation is usually solved numerically. In this work the FDM is used based on the works of Klump [ 10] and Varga [ I 1] As can be seen in the example of Fig. 6, large hydro-units are usually supported by tilting-pad journal bearings with a certain number of shoes. The solution of the pressure distribution should be computed simultaneously with the equation of motion for the rotor. Moreover as will be seen later, the solution of the equation of motion is carried out with the method of successive approximations. Consequently, the Reynolds equation also needs to be solved many times and it means very large CPU times. In spite of these considerations, the FDM was selected because of its better accuracy. Furthermore, at the same time a comparison of the non-linear simulation with a linear one is carried out. In the linear simulation (TURBO computer code [12]), the dynamic coefficients of journal bearing are also calculated by FDM.
34
R. CARDINALI E T A L .
Ii
Foundotion
Figure 6. Turbine guide bearing on the hydro-unit of the Ilha-Solteira power plant, Brazil.
From this point, there are two possibilities for applying the numerical solution : the first one is called one-dimensional FDM, which means that the grid line is constructed only in the circumferential direction and for the axial direction the pressure distribution is considered to be parabolic. The second method is the two-dimensional FDM, where the discretisation for the grid line is carried out for both axial and circumferential directions. The parabolic approximation form in the axial direction agrees very well with reality [11] and has the advantage that it needs less computer time. In order to solve the Reynolds equation, boundary conditions from the oil film pressure are needed. In the axial direction ( at the bearing edges, atmospheric pressure is assumed. In the circumferential direction tp, the well known Reynolds boundary conditions are used (Fig. 7). p 1"
/ /
l J -1
0
1
.~
_ Reynolds
/
boundory
L cOnditiOns ,
77"
.:cp
2Tr
Figure 7. Boundary conditions for oil film pressure [13].
2.5.2. Calculation of dynamic coefficients of journal bearings In the diagnosis programme being developed by the Universities of Campinas and Kaiserslautern linear and non-linear dynamic analysis is planned. Therefore, both methods are compared in order to evaluate the results of the dynamic response of the rotor. For that purpose stiffness and damping coefficients of the journal bearing are defined. If the rotor has sufficiently small vibration amplitudes around the steady-state operating eccentricity, fluid film forces may be replaced by their gradients. The equation of the journal bearing elements then becomes
[d,.
(5)
HYDROELECTRIC MACHINERY
35
The detailed method utilised in this work for the calculation of the eight reduced synchronous dynamic coefficients is presented in Parsell [14]. The calculated coefficients for some examples of multi-lobe and tilting-pad journal bearings were compared with the Journal-Bearing Databook by Someya [15]. The comparison is done in the form of curves of dimensionless dynamic coefficients vs Sommerfeld number. The results agree very well.
3. SOLUTION OF THE EQUATIONS OF MOTION Because there are non-linearities in the equation of motion (1), it will be solved in the time domain. As the system has many degrees of freedom, the modal analysis is used as a method to support the solution of the equations of motion. To apply the modal analysis in a non-linear differential equation like (1), one needs to rewrite it as: [M] {4} + [K]{q} = {F(t)} + {F( 0, q)} - [ D ] {0},
(6)
where the right side of equation (6) represents the unknown excitation forces. Now the problem is treated as a linear system with many degrees of freedom and external forces. Equation (6) then may be written as [M] {~/} + [K] {q} = {F}. 3 .1 .
(7)
D E C O U P L I N G A N D R E D U C T I O N OF T H E D I F F E R E N T I A L E Q U A T I O N
The solution of equation (7) is split into homogeneous and particular solutions. First, the eigenvalue problem is solved. With the calculated eigenvalues and eigenvectors, the modal matrix [0] is found. With the transformations [0] r[M ][0 ] = diag [M,],
(8a)
[O ] r[K ][0 ] = diag [K,],
(8b)
{q}=[@]{u}
(8c)
{q}=[.]{a}
(8d)
[M,] {t7} + [K,] {u} = {Qi},
(9)
and
equation (7) becomes
this means a decoupled differential equation system with the generalised forces {Q,} = [ * l r { F } . Usually, when vibration simulations are performed to find out the dynamic behaviour of rotor systems, one is interested in a special frequency range. In the case of hydro-units, interest may be restricted to vibration frequencies of less than 40 Hz [7]. The number of equations may then be reduced drastically, considering only the corresponding normal modes in the decoupled system (8).
R. C A R D I N A L I E T AL.
36
3.2. STEPWISE A P P R O X I M A T I O N O F T H E P A R T I C U L A R S O L U T I O N
The polygonal connection method [l 6] is used in order to solve the particular differential equation (9) in the time domain. It is possible to replace any excitation force by a polygon (Fig. 8).
F3
.
F'i+ 1
f
[_b_.to_4...~tl l n:2__ I Figure 8. Polygonal connection method.
With this method it is possible to obtain a stepwise exact solution of the differential equation, where the start condition for each interval is the solution from the interval before. If the start conditions for the first interval ( t = 0 ) are given, an arbitrary number of intervals can be solved. The equation of motion for the ith interval is then
mii+ d5 + ku = F, + (Fi+,- F~)t = F~+AF~ \ li+ I -- ti /
(10)
A ti '
which is easily solved. For the solution of the next interval both displacements and velocities must be known. With the knowledge of the external forces the solution for the (i+ 1)th interval becomes Ui+I = A F i +
BVi+l + C u i + D(li
(ll)
5i+ i = A 'Fi + B'Fi+ t + C'ui + D 'fii where:
A= B=
1
kco At 1
kco At
{2~'- ( 2 ( + co A t ) R + ( 1 - 2 ( 2 - o 9
{co A t + ( R -
C= R + (S, a '=
1
kAt
1)25-(1-2(2)S}
D= S/co
{R-l+((+coAt)}S
1
B'=--{I-R-(S} kAt C '= -coS,
D'= R - ~S
and R = e -¢°'a' cos (co Atx/l _(2) e - ¢ w At
S= ~
At()S}
(co A t v / ] - ~ )
(12)
HYDROELECTRIC
37
MACHINERY
with
foZ=k/m,
~" = d c 0 / 2 k .
Recall that usually a turbogenerator is supported by journal bearings, and here the journal bearing influence is input as external forces on the right side of the differential equation. Then the decoupled equation has the following form:
,nii= Fi+ (F'+' - Filt. \ ti+l
-
-
(13)
ti /
The solution for the (i+ 1)th interval is the same that was described by the equations (l l), but in the case of vanishing eigenvalues, the constants described by equation (l 2) have the following form : At 2
A=--; 3m A'-
At
• 2tn'
At 2
B= B'-
6m
;
At
• 2m'
C=I;
D=At;
C ' = 0;
D ' = 1.
(14)
3.3. M O D A L A N A L Y S I S F O R N O N - L I N E A R S Y S T E M S As described, all non-linear effects are substituted by external forces at the right side of the differential equation of motion. Moreover, the damping matrix D also may be written on the right side of the equation in order to solve the conservative system and reduce the number of equations. To calculate the dynamic behaviour at the time t + A t, the displacements and velocities at time t, as well as the forces at beginning and end of that interval have to be known. The displacements, velocities and the forces at the beginning of the interval are known from the previous interval. The problem now is to determine the forces at end of this interval. One can calculate in advance the time dependent forces such as unbalanced, constant, periodic and stochastic forces. The forces that depend on displacements a n d / o r velocities like journal bearings, negative stiffness, damping and gyroscopic forces can only be determined with the knowledge of the dynamic response at the end of each interval. An iterative procedure (Fig. 9) is performed where the starting values of the forces F(0, q) at the end of the interval are those of the beginning:
V*((l, q),,+a,= F(0, q),,.
(15)
With this start condition, one can calculate the new q and 0 and with that response one can update the value of the forces F( O, q). If there are no differences between the old value and the new one or else if a wished accuracy is attained, one can go to the next step, otherwise a new iteration needs to be performed. 3.4.
STOCHASTIC
FORCES
To the authors' knowledge, there is almost no literature dealing with the stochastic, mostly hydraulic forces exerted on turbine runners. Schwirzer [7] observed forces of statistically fluctuating amplitude on a pump-turbine. Its excitation spectrum extended from very low frequencies to about 100 Hz. A power spectral density function (s. co) 2 S(c0)-- (s 2 co-I)2 COc •
,
O~
,
09c <
CO
(16)
38
R. CARDINALI E T AL.
[F*(¢l,q)ti+z~
t = F(q,
I F*(q, q) =
F(¢l,q)
q)hI
Computation
q ti+Z~t and
r
~i,ti+z~ t
_Computation
F*(q,q)
ti +at
~next
step~
Figure 9. Iterative procedure for the calculation of forces F(q, q).
with cut-off-frequency coc in the range of the rotational frequency was established, s is an intensity factor of the fluctuations. Both of the parameters are expected to be highly dependent on operating conditions,the geometry of the water turbine and the inflow conditions like turbulence degree, mass flow and velocity. Hence the above mentioned model may only give a rough qualitative idea of the hydraulic excitation. For the simulation, it is assumed that stochastic forces may be represented by dynamic filter approximations [ 17] {(~,} = [F]{Q,} + {b}~,
(17)
where ~, is a Gaussian, centered, stationary white noise with spectral density S~¢(co)= 1. From a given power spectral density function SQQ(co), the size of the system and the filter matrix [F] are determined by optimisation. A simple example is the low pass filter
Q, = o~cQ, + 0.o~c~,
(18)
with spectral density function 0. 2
SQQ-- 1 + ((o/o9c) 2
(19)
HYDROELECTRIC MACHINERY
39
choosing a broad band excitation with intensity a = 50 000 and a cut off frequency roc = 40 Hz. In the general case, a stepwise simulation
{Q[(k+l)At]}=e[rla'{Q(kAt)}+[F]-I[etFlA'-I]{b}~(k),
k = 0 , 1,2 . . . .
(20)
is possible [which is derived from the corresponding convolution integral solution of (17)]. For a constant time step A t, only one calculation of the transition matrix e lvl'~' is necessary. [I] is the unity matrix, ~(k) are equally or Gaussian distributed random numbers with I~(k) l~<0-5. 4. NUMERICAL RESULTS The journal bearing housings shown in Fig. 6 were modeled by the FE-method with support of the ANSYS code in order to evaluate its dynamic behaviour. The radial stiffness calculated for both housings are comparably high and both show no radial natural modes in the frequency range of interest. Due to the described dynamic characteristics, the housings were not included in the global model. The initial conditions used in the numerical examples were always zero but the first four periods of integration are not shown in the results and were not utilised in the frequency analysis. The Fig. l0 shows a simulated model of the machine. Table I shows the calculated coefficients of the seals and the dynamic journal bearing coefficients utilised in the linear simulation described in section 4.2. All the other input data are the same as that in the work of Sperber and Weber [5].
I MG,IpG,IAG,
"11
MT ,IpT,IAT I
ks ,ms ,ds Figure 10. Model of Ilha-Solteira power plant, Brazil.
4.1. NON-LINEAR SIMULATIONS Two different simulations are presented here: (a) Unbalanced forces at nodes I (generator, eccentricity = 1.5 mm) and 7 (turbine, eccentricity = 1.0 ram) and stochastic forces in both directions at node 7 as external forces; (b) only the stochastic forces from example (a). The purpose of simulation (a) is to show the dynamic behaviour of a normal running machine in the presence of stochastic and unbalanced forces. The external force parameters were estimated as well as some other parameters such as fictitious stiffness coefficients of the magnetic pull, pressure at the seals, and so on. To refine the model some measurements must be done, but it is not the scope of the present work.
TABLE I K~.x(N/m) Generator bearing 2.89E+09 Turbine bearing 1.95E+09 Seal A (Fig. 4) 4.53E+07 Seal B (Fig. 4) -5-39E+07 Global seal A + B - 8 - 6 5 E + 0 6
¢3
Kxy (N/m)
Ky.,. (N/m)
K~.~.(N/m)
D~.,. [kg/s]
D~y [kg/s]
Dyx [kg/s]
Dy,, [kg/s]
6.56E+06 2.61E+06 4.00E+06 1.12E+06 5.12E+06
-7.20E+06 -2.61E+06 -4.00E+06 -I.12E+06 -5.12E+06
2.92E+ 09 1-95E+09 4.53E+07 -5.39E+07 -8-65E+06
3-73E+07 3.81E+07 1.61E+06 4.79E+05 2.09E+06
-2.22E+04 -2.85E+04 2.17E+04 9.54E+03 3-07E+04
5.50E + 04 2.83E+ 04 -2.17E+04 -9.54E+03 -3.07E+04
3-64E+07 3-82E+07 !.61E+06 4.79E+05 2-09E+06
Z r-
HYDROELECTRIC
41
MACHINERY
Node 01
Node 02
J
X(~m)
Node 03
I,
Node 04
\_1¸o t ~op,~m~ Node 05
-4 ~
Node 06
4x(~.m)
~
xc,:m,
Node 07
zo
o
X
,A
m)
Figure I 1. Unbalanced and stochastic response of the hydro-unit model at llha-Solteira, Brazil.
42
R. C A R D I N A L I
ET AL.
Figure l I shows the orbit at each node. A frequency analysis of the responses of each node shows the predominance of the rotational frequency alone. The objective of simulation (b) is to identify some characteristic frequencies in the range from 0 to 40 Hz. Figure 12 shows a frequency spectrum of the response signal of node 4. The same characteristic frequencies as found at node 4 can also be encountered from the frequency analysis of the response signal of the other nodes, however with different emphasis. 4.2. L I N E A R S I M U L A T I O N S As mentioned before, the same model was simulated with support of the T U R B O computer code [12]. It permits the calculation of rotor journal bearing systems in the horizontal position, where the linear dynamic coefficients of journal bearings can be used. The other beam, seal, rigid disc and spring elements are the same as utilised in the nonlinear simulation. To calculate the dynamic coefficients of the journal bearings, the Sommerfeld number must be known. For vertical machines this number equals zero because there are no static forces on the bearing. In order to linearise the journal bearing behaviour a static force was imposed on each of the journal bearings. The static forces were chosen considering the results of the non-linear simulation such as characteristic frequencies and journal forces
of (b). Figure 13(a) shows the eigenvalues and natural modes of the linear model and Fig. 13(b) the unbalanced response. Figure 14 presents a comparison between the linear and non-linear responses of the same unbalanced forces. The parameters of the unbalanced forces utilized here were the same as example (a) above.
5. CONCLUSIONS The dynamic responses computed by non-linear simulation appears to be more accurate and realistic. However, from the point of view of CPU-time, the exclusive use of this method for the programme Diagnosis of Vertical Machines is not advised. Other methods should also be taken into account which need less CPU-time and furnish satisfactory results. So, simple linear models as described in [5] or the linear model described in this paper which uses information from the non-linear simulation should be considered as first steps. Comparing the results of the frequency analysis of the non-linear simulation (Fig. 12) with the eigenvalues found in the linear simulation [Fig. 13(a)] or the comparison of the
F1 = 2 - 2 7 -
1
(1
F2=
3.91
F3 ~-4
11.06 11.39
E I
J 10
20
30
40
Frequency (Hz) Figure 12. Frequency spectrum of the response signal at node 4 (Hanning window).
HYDROELECTRIC MACHINERY
018H,
047Hz]--~11.23 Hz ] 1 ~ - ~ ~
1006 Hz j 1
43
7
(
7
(b)
(a)
Figure 13. (a) Eigenvalues and natural modes of the hydrogenerator. (b) Unbalanced response of the hydrogenerater, maximum amplitude (lim).
Node amplitude (~m) 1 2 :5 4 ~J17 14 5 6 ~l 12 15 7
28
32
Figure 14. Comparisonbetween linear and non-linear responses of unbalanced forces, r-l, Non-linear simulation; l , linear simulation. unbalanced response (Fig. 14), the conclusion seems obvious that the linear model possesses a satisfactory accuracy. After having obtained linear results, a number of possible failure conditions may be selected. For these special cases, the non-linear simulation may be performed in order to obtain improved responses of the dynamic behaviour of the vertical machine.
REFERENCES I. F. SIMON 1982 Voith Forschung und Konstruktion, 38, paper 4. Zur Berechnung des dynamischen Verhaltens von Wellensystemen bei Wasserkraftanlagen. 2. W. DIEWALD, A. KLIMMEK and R. NORDMANN 1989 VDI Bericht Nr. 789. Die Berechnung des Biegeschwingungsverhaltens rotierender Maschinen. 3. H. J. W. MERKER 1981 Dissertation, Darmstadt. Ober den nichtlinearen EinfluB yon Gleitlagern auf die Schwingungen von Rotoren. 4. H. SPRINGER 1979 Forsch. Ing. Wes. 44 Nr. 4. Nichtlineare Schwingungen schwerer Rotoren mit vertikaler Welle und Kippsegmentradiallagern.
44
R. CARDINALI ET AL.
5. A. SPERBER and H. 1. WEBER 1990 Proceedings o/the IFToMM, Lyon. Modelling and estimation of hydroelectric machinery. 6. R. GASCH 1976 Journal o/Sound and Vibration 47, 53-73. Vibration of large turbo-rotors in fluid-film bearing on an elastic foundation. 7. T. SC14WlRZER 1977 Water Power 29, 39-44. Dynamic stressing of hydroelectric units by stochastic hydraulic forces on the turbine runner. 8. R. NORDMANN 1974 Dissertation, Darmstadt. Ein N/iherungsverfahren zur Berechnung der Eigenwerte und Eigenformen von Turborotoren mit Gleitlagern, Spalterregung, /iul3erer und innerer D/impfung. 9. F. J. DIETZEN and R. NORDMANN 1986 Proceedings of the 4th Workshop on Rotordynamic hlstability Problems in High Pelformance Turbomachinery, Texas A & M. Calculating rotordynamic coefficients of seals by finite difference techniques. 10. R. KLUMP 1978 Dissertation, TH Karlsruhe. Ein Beitrag zur Theorie yon Kip.psegmentlagern. 11. Z. E. VARGA 1971 Dissertation, ETH Ziirich. Wellenbewegung, Reibung und Oldurchsatz beim segmentierten Radialgleitlager yon beliebiger Spaltform unter konstanter und zeitlich ver/inderlicher Belastung. 12. W. D1EWALD 1989 Dissertation, Kaiserslautern. Das Biegeschwingungsverhalten yon Kreiselpumpen unter BeriJcksichtigung der Koppelwirkungen mit dem Fluid. 13. O. R. LANG and W. S'rE~NHILPER 1978 Gleitlager. Berlin: Springer Verlag. 14. J. K. PARSEL, P. E. ALt.AXRE and L. E. BARRET 1982 ASLE Transactions 26, 222-227. Frequency effects in tilting-pad journal bearing dynamic coefficients. 15. T. SOMEVA 1989 Journal-Bearing Databook. Berlin: Springer-Verlag. 16. E. KR~MER 1984 Maschhmndynamik. Berlin: Springer Verlag. 17. W. WEDIG 1987 Zeitschrift fiir angewandte Mathematik und Mechanik 67, 34-42. Stochastiche Schwingungen-Simulation, Sch/itzung und Stabilit/it.
D Y N A M I C SIMULATION OF N O N - L I N E A R MODELS OF H Y D R O E L E C T R I C M A C H I N E R Y R. CARDINAL1 University of Campinas, Geprom-Laboratorio de Projeto Mecanico, P.O. Box 6051, 13081 Campinas, Brazil R. NORDMANN
University of Kaiserslautern, Arbeitsgruppe Maschinendynamik, P.O, Box 3049, 6750 Kaiserslautern, Germany AND
A. SPERBER
BASF AG. ZE/IR, 6700 Ludwigshafen, Germany (Received 15 June 1991, accepted 17 November 199 I) This work describes the development of a non-linear model of the hydro-units at the power plant in Ilha-Solteira (Brazil). A computer program is presented that permits the calculation of the bending vibrations of vertical rotor systems. The program takes into account magnetic interactions, seals, journal bearings, as well as unbalanced, constant, harmonic and stochastic excitation forces. The shaft is modelled with the finite-element method including the gyroscopic effect for the discs. Some simulation results are presented. At the same time a linear model is established to compare with the results of the non-linear model, e.g. for the unbalanced response and the characteristic frequencies. This work has been carried out as part of a research programme Diagnosis of Vertical Machines.
1. INTRODUCTION With increasing size and performance of hydro-units for power generation plants, knowledge of their dynamic behaviour becomes more and more important. Computer-aided vibration simulations are indispensable for design. With the implementation of a predictive maintenance p r o g r a m m e one can utilise vibration simulations if it is necessary to know or to predict the dynamic response for the diagnosis of possible failures. The shafts of such machinery, including generators and water turbines, are submitted to axial, torsional and flexural vibrations. The range of critical frequencies for axial and torsional vibrations, however, is far higher than the frequency range of the excitation. Hence, the main interest is focused on flexural vibrations [1]. The flexural vibrations of vertically arranged turbomachinery with journal bearings show non-linear behaviour. As the static forces on the bearing are very small and the displacements are comparably large, linearisation may lead to errors [2-4]. 1.1. PHYSICAL MODEL OF A HYDROGENERATOR A Francis type hydro-unit with a vertical arrangement, Fig. 1, is modeled. It has a nominal power of 160 M W at a rotational frequency of 1.43 Hz. The vertical distance between the generator and turbine is 6.33 m, the diameter of the generator is 13.2 m. The hollow shaft with inner and outer diameter of 400 and 1300 mm, respectively is modeled 29
30
R. CARDINALI E T A L . Genera tar
1~7 ~torbearing
e bearing Seals
Figure 1. Basic model of the hydro-unit at llha-Solteira, Brazil.
as a Bernoulli beam. The generator and turbine are rigid disks of relatively large diameters, thus provoking gyroscopic effects. The axial support acts as a torsional spring [5]. The hydro-unit has two guide bearings: (1) generator guide bearing: a tilting pad journal bearing centrally pivoted, 20-shoe with 2400 mm journal diameter and 300 mm wide; (2) turbine guide bearing: a tilting pad journal bearing centrally pivoted, 12-shoe with 1650 mm journal diameter and 340 mm wide. To consider the non-linear effects arising due to the vertical arrangement, the forces on the bearings are calculated step-by-step, using a finite difference technique to solve the corresponding Reynolds equations. The seal effects between turbine and housing are considered by mass, damping and spring coefficients. They are also obtained by the use of a finite-difference method. The local magnetic pull of the generator was replaced by fictitious stiffness coefficients [5, 6]. The excitation forces of the system are of mechanical, hydraulic, electrical and electromagnetic origin. Unbalanced masses of the generator and turbine as well as the stochastically fluctuating forces due to the turbulent flow inside the turbine, are the most important excitation sources [7]. Other forces such as misalignment, change of revolving hydraulic radial forces, magnetic imbalance, etc. can also be simulated by harmonic, constant or stochastic forces. 2. EQUATIONS OF MOTION For the purpose of calculating the dynamic behaviour of a rotor system, an idealised model is needed. It may consist of beams, seals, journal bearings, springs and other elements. Each element equation is formed separately. The global equation of motion of the system is built up by superposition of these equations. The motion of each mode i is represented by two displacement coordinates x;, y; and two distortion angles ~P.~i, ~P~.i (Fig. 2). The dynamic behaviour of rotor systems can be described by a system of linear differential equations [M] {~} + [D] {t~}+ [Kl{q} = {F(t)} + {F(t2, q)}
(1)
consisting of individual element matrices. [M], [D] and [K] are the superposed element matrices [8]. {q} is a vector that contains the unknown co-ordinates. {F(t)} is the vector of the time dependent external forces like unbalanced, constant or hydraulic forces.
HYDROELECTRIC MACHINERY
31
c 9yiyf~'~xi"~- sDxi Figure 2. Co-ordinates at node L
{F((t, q)} are the external forces depending on the displacements and velocities of the unknown co-ordinates.
2.1. BEAM ELEMENT The shaft is the most important component of rotor systems. In the model it is divided into several beam elements, which are described by finite elements. Bending stiffness E1 and mass distributions p are taken into account. Gyroscopic effects as well as shear deformations are not considered (Bernoulli beam elements). External forces are only acting on the nodes, i.e. the element ends.
2.2. RIGID DISC ELEMENT Certain elements in a rotor system can be described as rigid elements. This means that their stiffness may be neglected. In this case the element equation has the following form: [MI {q} + [DI{q} = {F},
(2)
where [M] is a symmetric matrix that contains the inertia m and the transverse mass moment of inertia OA, and [D] is the skew-symmetric gyroscopic matrix built with the polar mass moment of inertia Op. As will be discussed later, in the case of non-linear simulations the terms on the damping matrix [D] are treated as velocity dependent external forces on the right side of equation (1).
2.3. SPRING ELEMENT Supports between shaft and housing or housing and foundation as well as the fictitious stiffness coefficients for the simulation of the magnetic pull are simulated using simple spring elements [K] {q} = {F}.
(3)
Also in the non-linear simulation, terms with negative coefficients (magnetic pull) are placed on the right side of equation (1) and treated as displacement dependent forces.
32
R. CARDINALI
ET AL.
2.4. SEAl. Et.EMENT It is well k n o w n that tile fluid forces o,1 the seals (Fig. 3), which are described by e q u a t i o n (4), have a s t r o n g influence on tile d y n a m i c b e h a v i o u r o f r o t a t i n g machinery.
.i: [M M]{i;l+[_ D. dD]{:~}+[_KK]{"I}=IE,. 0
F~
(4)
The d y n a m i c coemcients are o b t a i n e d from experilrlen!.a] investigations or theoretical calculations. Here the theory o f Dietzen and N o r d m a n n [9] is used to calculate the seal
~,~
~ ~,~
~
~
~\"
Seol " \ ~
Rotor
~Stotor
Figure 3. Displacements and forces acting on tile seals.
I
(o)
i
(b)
5.0
f=. r i~'~=/
Figure 4. Seals between turbine and housing (European machine, llha-Solteira).
HYDROELECTRIC MACHINERY
33
coefficients by the finite difference method (FDM). In spite of the resulting large CPUtime, this method allows the analysis of complicated seals to be performed. The example shown in Fig. 4 is for the European machine in IIha-Solteira/BR. 2.5. JOURNAL BEARING ELEMENT
The dynamic behaviour of a vertical machine has a great dependence on the journal bearing. As a horizontal machine possesses a stationary position, iinearised damping and stiffness coefficients can be used. In a vertical machine there are no static forces which result in a stationary position. Moreover, the relative displacements and velocities between shaft and housing have to be known in order to calculate the journal bearing forces. As a consequence the journal bearing forces will also be placed on the right side of equation (1) as non-linear forces (F(q, q} } that depend on the relative displacements and velocities. At the journal bearing the shaft is guided by a pressure field originating normally from a laminar circumferential fluid flow. The order of non-linearity of the journal bearing forces depends on the relative displacements and velocities on the oil film of the journal. As an example, Fig. 5 shows the behaviour of a 360 ° cylindrical bearing.
'NN~ ~ ~
Stator Rotor
~
distribution
Figure 5. Pressuredistribution and forceson a 360° cylindrical bearing. 2.5.1. Calculation of journal bearingforces To calculate the journal bearing forces, the pressure distribution on the oil film has to be determined. Thus, one starts the well known Reynolds differential equation. Due to its complexity, the Reynolds equation is usually solved numerically. In this work the FDM is used based on the works of Klump [ 10] and Varga [ I 1] As can be seen in the example of Fig. 6, large hydro-units are usually supported by tilting-pad journal bearings with a certain number of shoes. The solution of the pressure distribution should be computed simultaneously with the equation of motion for the rotor. Moreover as will be seen later, the solution of the equation of motion is carried out with the method of successive approximations. Consequently, the Reynolds equation also needs to be solved many times and it means very large CPU times. In spite of these considerations, the FDM was selected because of its better accuracy. Furthermore, at the same time a comparison of the non-linear simulation with a linear one is carried out. In the linear simulation (TURBO computer code [12]), the dynamic coefficients of journal bearing are also calculated by FDM.
34
R. CARDINALI E T A L .
Ii
Foundotion
Figure 6. Turbine guide bearing on the hydro-unit of the Ilha-Solteira power plant, Brazil.
From this point, there are two possibilities for applying the numerical solution : the first one is called one-dimensional FDM, which means that the grid line is constructed only in the circumferential direction and for the axial direction the pressure distribution is considered to be parabolic. The second method is the two-dimensional FDM, where the discretisation for the grid line is carried out for both axial and circumferential directions. The parabolic approximation form in the axial direction agrees very well with reality [11] and has the advantage that it needs less computer time. In order to solve the Reynolds equation, boundary conditions from the oil film pressure are needed. In the axial direction ( at the bearing edges, atmospheric pressure is assumed. In the circumferential direction tp, the well known Reynolds boundary conditions are used (Fig. 7). p 1"
/ /
l J -1
0
1
.~
_ Reynolds
/
boundory
L cOnditiOns ,
77"
.:cp
2Tr
Figure 7. Boundary conditions for oil film pressure [13].
2.5.2. Calculation of dynamic coefficients of journal bearings In the diagnosis programme being developed by the Universities of Campinas and Kaiserslautern linear and non-linear dynamic analysis is planned. Therefore, both methods are compared in order to evaluate the results of the dynamic response of the rotor. For that purpose stiffness and damping coefficients of the journal bearing are defined. If the rotor has sufficiently small vibration amplitudes around the steady-state operating eccentricity, fluid film forces may be replaced by their gradients. The equation of the journal bearing elements then becomes
[d,.
(5)
HYDROELECTRIC MACHINERY
35
The detailed method utilised in this work for the calculation of the eight reduced synchronous dynamic coefficients is presented in Parsell [14]. The calculated coefficients for some examples of multi-lobe and tilting-pad journal bearings were compared with the Journal-Bearing Databook by Someya [15]. The comparison is done in the form of curves of dimensionless dynamic coefficients vs Sommerfeld number. The results agree very well.
3. SOLUTION OF THE EQUATIONS OF MOTION Because there are non-linearities in the equation of motion (1), it will be solved in the time domain. As the system has many degrees of freedom, the modal analysis is used as a method to support the solution of the equations of motion. To apply the modal analysis in a non-linear differential equation like (1), one needs to rewrite it as: [M] {4} + [K]{q} = {F(t)} + {F( 0, q)} - [ D ] {0},
(6)
where the right side of equation (6) represents the unknown excitation forces. Now the problem is treated as a linear system with many degrees of freedom and external forces. Equation (6) then may be written as [M] {~/} + [K] {q} = {F}. 3 .1 .
(7)
D E C O U P L I N G A N D R E D U C T I O N OF T H E D I F F E R E N T I A L E Q U A T I O N
The solution of equation (7) is split into homogeneous and particular solutions. First, the eigenvalue problem is solved. With the calculated eigenvalues and eigenvectors, the modal matrix [0] is found. With the transformations [0] r[M ][0 ] = diag [M,],
(8a)
[O ] r[K ][0 ] = diag [K,],
(8b)
{q}=[@]{u}
(8c)
{q}=[.]{a}
(8d)
[M,] {t7} + [K,] {u} = {Qi},
(9)
and
equation (7) becomes
this means a decoupled differential equation system with the generalised forces {Q,} = [ * l r { F } . Usually, when vibration simulations are performed to find out the dynamic behaviour of rotor systems, one is interested in a special frequency range. In the case of hydro-units, interest may be restricted to vibration frequencies of less than 40 Hz [7]. The number of equations may then be reduced drastically, considering only the corresponding normal modes in the decoupled system (8).
R. C A R D I N A L I E T AL.
36
3.2. STEPWISE A P P R O X I M A T I O N O F T H E P A R T I C U L A R S O L U T I O N
The polygonal connection method [l 6] is used in order to solve the particular differential equation (9) in the time domain. It is possible to replace any excitation force by a polygon (Fig. 8).
F3
.
F'i+ 1
f
[_b_.to_4...~tl l n:2__ I Figure 8. Polygonal connection method.
With this method it is possible to obtain a stepwise exact solution of the differential equation, where the start condition for each interval is the solution from the interval before. If the start conditions for the first interval ( t = 0 ) are given, an arbitrary number of intervals can be solved. The equation of motion for the ith interval is then
mii+ d5 + ku = F, + (Fi+,- F~)t = F~+AF~ \ li+ I -- ti /
(10)
A ti '
which is easily solved. For the solution of the next interval both displacements and velocities must be known. With the knowledge of the external forces the solution for the (i+ 1)th interval becomes Ui+I = A F i +
BVi+l + C u i + D(li
(ll)
5i+ i = A 'Fi + B'Fi+ t + C'ui + D 'fii where:
A= B=
1
kco At 1
kco At
{2~'- ( 2 ( + co A t ) R + ( 1 - 2 ( 2 - o 9
{co A t + ( R -
C= R + (S, a '=
1
kAt
1)25-(1-2(2)S}
D= S/co
{R-l+((+coAt)}S
1
B'=--{I-R-(S} kAt C '= -coS,
D'= R - ~S
and R = e -¢°'a' cos (co Atx/l _(2) e - ¢ w At
S= ~
At()S}
(co A t v / ] - ~ )
(12)
HYDROELECTRIC
37
MACHINERY
with
foZ=k/m,
~" = d c 0 / 2 k .
Recall that usually a turbogenerator is supported by journal bearings, and here the journal bearing influence is input as external forces on the right side of the differential equation. Then the decoupled equation has the following form:
,nii= Fi+ (F'+' - Filt. \ ti+l
-
-
(13)
ti /
The solution for the (i+ 1)th interval is the same that was described by the equations (l l), but in the case of vanishing eigenvalues, the constants described by equation (l 2) have the following form : At 2
A=--; 3m A'-
At
• 2tn'
At 2
B= B'-
6m
;
At
• 2m'
C=I;
D=At;
C ' = 0;
D ' = 1.
(14)
3.3. M O D A L A N A L Y S I S F O R N O N - L I N E A R S Y S T E M S As described, all non-linear effects are substituted by external forces at the right side of the differential equation of motion. Moreover, the damping matrix D also may be written on the right side of the equation in order to solve the conservative system and reduce the number of equations. To calculate the dynamic behaviour at the time t + A t, the displacements and velocities at time t, as well as the forces at beginning and end of that interval have to be known. The displacements, velocities and the forces at the beginning of the interval are known from the previous interval. The problem now is to determine the forces at end of this interval. One can calculate in advance the time dependent forces such as unbalanced, constant, periodic and stochastic forces. The forces that depend on displacements a n d / o r velocities like journal bearings, negative stiffness, damping and gyroscopic forces can only be determined with the knowledge of the dynamic response at the end of each interval. An iterative procedure (Fig. 9) is performed where the starting values of the forces F(0, q) at the end of the interval are those of the beginning:
V*((l, q),,+a,= F(0, q),,.
(15)
With this start condition, one can calculate the new q and 0 and with that response one can update the value of the forces F( O, q). If there are no differences between the old value and the new one or else if a wished accuracy is attained, one can go to the next step, otherwise a new iteration needs to be performed. 3.4.
STOCHASTIC
FORCES
To the authors' knowledge, there is almost no literature dealing with the stochastic, mostly hydraulic forces exerted on turbine runners. Schwirzer [7] observed forces of statistically fluctuating amplitude on a pump-turbine. Its excitation spectrum extended from very low frequencies to about 100 Hz. A power spectral density function (s. co) 2 S(c0)-- (s 2 co-I)2 COc •
,
O~
,
09c <
CO
(16)
38
R. CARDINALI E T AL.
[F*(¢l,q)ti+z~
t = F(q,
I F*(q, q) =
F(¢l,q)
q)hI
Computation
q ti+Z~t and
r
~i,ti+z~ t
_Computation
F*(q,q)
ti +at
~next
step~
Figure 9. Iterative procedure for the calculation of forces F(q, q).
with cut-off-frequency coc in the range of the rotational frequency was established, s is an intensity factor of the fluctuations. Both of the parameters are expected to be highly dependent on operating conditions,the geometry of the water turbine and the inflow conditions like turbulence degree, mass flow and velocity. Hence the above mentioned model may only give a rough qualitative idea of the hydraulic excitation. For the simulation, it is assumed that stochastic forces may be represented by dynamic filter approximations [ 17] {(~,} = [F]{Q,} + {b}~,
(17)
where ~, is a Gaussian, centered, stationary white noise with spectral density S~¢(co)= 1. From a given power spectral density function SQQ(co), the size of the system and the filter matrix [F] are determined by optimisation. A simple example is the low pass filter
Q, = o~cQ, + 0.o~c~,
(18)
with spectral density function 0. 2
SQQ-- 1 + ((o/o9c) 2
(19)
HYDROELECTRIC MACHINERY
39
choosing a broad band excitation with intensity a = 50 000 and a cut off frequency roc = 40 Hz. In the general case, a stepwise simulation
{Q[(k+l)At]}=e[rla'{Q(kAt)}+[F]-I[etFlA'-I]{b}~(k),
k = 0 , 1,2 . . . .
(20)
is possible [which is derived from the corresponding convolution integral solution of (17)]. For a constant time step A t, only one calculation of the transition matrix e lvl'~' is necessary. [I] is the unity matrix, ~(k) are equally or Gaussian distributed random numbers with I~(k) l~<0-5. 4. NUMERICAL RESULTS The journal bearing housings shown in Fig. 6 were modeled by the FE-method with support of the ANSYS code in order to evaluate its dynamic behaviour. The radial stiffness calculated for both housings are comparably high and both show no radial natural modes in the frequency range of interest. Due to the described dynamic characteristics, the housings were not included in the global model. The initial conditions used in the numerical examples were always zero but the first four periods of integration are not shown in the results and were not utilised in the frequency analysis. The Fig. l0 shows a simulated model of the machine. Table I shows the calculated coefficients of the seals and the dynamic journal bearing coefficients utilised in the linear simulation described in section 4.2. All the other input data are the same as that in the work of Sperber and Weber [5].
I MG,IpG,IAG,
"11
MT ,IpT,IAT I
ks ,ms ,ds Figure 10. Model of Ilha-Solteira power plant, Brazil.
4.1. NON-LINEAR SIMULATIONS Two different simulations are presented here: (a) Unbalanced forces at nodes I (generator, eccentricity = 1.5 mm) and 7 (turbine, eccentricity = 1.0 ram) and stochastic forces in both directions at node 7 as external forces; (b) only the stochastic forces from example (a). The purpose of simulation (a) is to show the dynamic behaviour of a normal running machine in the presence of stochastic and unbalanced forces. The external force parameters were estimated as well as some other parameters such as fictitious stiffness coefficients of the magnetic pull, pressure at the seals, and so on. To refine the model some measurements must be done, but it is not the scope of the present work.
TABLE I K~.x(N/m) Generator bearing 2.89E+09 Turbine bearing 1.95E+09 Seal A (Fig. 4) 4.53E+07 Seal B (Fig. 4) -5-39E+07 Global seal A + B - 8 - 6 5 E + 0 6
¢3
Kxy (N/m)
Ky.,. (N/m)
K~.~.(N/m)
D~.,. [kg/s]
D~y [kg/s]
Dyx [kg/s]
Dy,, [kg/s]
6.56E+06 2.61E+06 4.00E+06 1.12E+06 5.12E+06
-7.20E+06 -2.61E+06 -4.00E+06 -I.12E+06 -5.12E+06
2.92E+ 09 1-95E+09 4.53E+07 -5.39E+07 -8-65E+06
3-73E+07 3.81E+07 1.61E+06 4.79E+05 2.09E+06
-2.22E+04 -2.85E+04 2.17E+04 9.54E+03 3-07E+04
5.50E + 04 2.83E+ 04 -2.17E+04 -9.54E+03 -3.07E+04
3-64E+07 3-82E+07 !.61E+06 4.79E+05 2-09E+06
Z r-
HYDROELECTRIC
41
MACHINERY
Node 01
Node 02
J
X(~m)
Node 03
I,
Node 04
\_1¸o t ~op,~m~ Node 05
-4 ~
Node 06
4x(~.m)
~
xc,:m,
Node 07
zo
o
X
,A
m)
Figure I 1. Unbalanced and stochastic response of the hydro-unit model at llha-Solteira, Brazil.
42
R. C A R D I N A L I
ET AL.
Figure l I shows the orbit at each node. A frequency analysis of the responses of each node shows the predominance of the rotational frequency alone. The objective of simulation (b) is to identify some characteristic frequencies in the range from 0 to 40 Hz. Figure 12 shows a frequency spectrum of the response signal of node 4. The same characteristic frequencies as found at node 4 can also be encountered from the frequency analysis of the response signal of the other nodes, however with different emphasis. 4.2. L I N E A R S I M U L A T I O N S As mentioned before, the same model was simulated with support of the T U R B O computer code [12]. It permits the calculation of rotor journal bearing systems in the horizontal position, where the linear dynamic coefficients of journal bearings can be used. The other beam, seal, rigid disc and spring elements are the same as utilised in the nonlinear simulation. To calculate the dynamic coefficients of the journal bearings, the Sommerfeld number must be known. For vertical machines this number equals zero because there are no static forces on the bearing. In order to linearise the journal bearing behaviour a static force was imposed on each of the journal bearings. The static forces were chosen considering the results of the non-linear simulation such as characteristic frequencies and journal forces
of (b). Figure 13(a) shows the eigenvalues and natural modes of the linear model and Fig. 13(b) the unbalanced response. Figure 14 presents a comparison between the linear and non-linear responses of the same unbalanced forces. The parameters of the unbalanced forces utilized here were the same as example (a) above.
5. CONCLUSIONS The dynamic responses computed by non-linear simulation appears to be more accurate and realistic. However, from the point of view of CPU-time, the exclusive use of this method for the programme Diagnosis of Vertical Machines is not advised. Other methods should also be taken into account which need less CPU-time and furnish satisfactory results. So, simple linear models as described in [5] or the linear model described in this paper which uses information from the non-linear simulation should be considered as first steps. Comparing the results of the frequency analysis of the non-linear simulation (Fig. 12) with the eigenvalues found in the linear simulation [Fig. 13(a)] or the comparison of the
F1 = 2 - 2 7 -
1
(1
F2=
3.91
F3 ~-4
11.06 11.39
E I
J 10
20
30
40
Frequency (Hz) Figure 12. Frequency spectrum of the response signal at node 4 (Hanning window).
HYDROELECTRIC MACHINERY
018H,
047Hz]--~11.23 Hz ] 1 ~ - ~ ~
1006 Hz j 1
43
7
(
7
(b)
(a)
Figure 13. (a) Eigenvalues and natural modes of the hydrogenerator. (b) Unbalanced response of the hydrogenerater, maximum amplitude (lim).
Node amplitude (~m) 1 2 :5 4 ~J17 14 5 6 ~l 12 15 7
28
32
Figure 14. Comparisonbetween linear and non-linear responses of unbalanced forces, r-l, Non-linear simulation; l , linear simulation. unbalanced response (Fig. 14), the conclusion seems obvious that the linear model possesses a satisfactory accuracy. After having obtained linear results, a number of possible failure conditions may be selected. For these special cases, the non-linear simulation may be performed in order to obtain improved responses of the dynamic behaviour of the vertical machine.
REFERENCES I. F. SIMON 1982 Voith Forschung und Konstruktion, 38, paper 4. Zur Berechnung des dynamischen Verhaltens von Wellensystemen bei Wasserkraftanlagen. 2. W. DIEWALD, A. KLIMMEK and R. NORDMANN 1989 VDI Bericht Nr. 789. Die Berechnung des Biegeschwingungsverhaltens rotierender Maschinen. 3. H. J. W. MERKER 1981 Dissertation, Darmstadt. Ober den nichtlinearen EinfluB yon Gleitlagern auf die Schwingungen von Rotoren. 4. H. SPRINGER 1979 Forsch. Ing. Wes. 44 Nr. 4. Nichtlineare Schwingungen schwerer Rotoren mit vertikaler Welle und Kippsegmentradiallagern.
44
R. CARDINALI ET AL.
5. A. SPERBER and H. 1. WEBER 1990 Proceedings o/the IFToMM, Lyon. Modelling and estimation of hydroelectric machinery. 6. R. GASCH 1976 Journal o/Sound and Vibration 47, 53-73. Vibration of large turbo-rotors in fluid-film bearing on an elastic foundation. 7. T. SC14WlRZER 1977 Water Power 29, 39-44. Dynamic stressing of hydroelectric units by stochastic hydraulic forces on the turbine runner. 8. R. NORDMANN 1974 Dissertation, Darmstadt. Ein N/iherungsverfahren zur Berechnung der Eigenwerte und Eigenformen von Turborotoren mit Gleitlagern, Spalterregung, /iul3erer und innerer D/impfung. 9. F. J. DIETZEN and R. NORDMANN 1986 Proceedings of the 4th Workshop on Rotordynamic hlstability Problems in High Pelformance Turbomachinery, Texas A & M. Calculating rotordynamic coefficients of seals by finite difference techniques. 10. R. KLUMP 1978 Dissertation, TH Karlsruhe. Ein Beitrag zur Theorie yon Kip.psegmentlagern. 11. Z. E. VARGA 1971 Dissertation, ETH Ziirich. Wellenbewegung, Reibung und Oldurchsatz beim segmentierten Radialgleitlager yon beliebiger Spaltform unter konstanter und zeitlich ver/inderlicher Belastung. 12. W. D1EWALD 1989 Dissertation, Kaiserslautern. Das Biegeschwingungsverhalten yon Kreiselpumpen unter BeriJcksichtigung der Koppelwirkungen mit dem Fluid. 13. O. R. LANG and W. S'rE~NHILPER 1978 Gleitlager. Berlin: Springer Verlag. 14. J. K. PARSEL, P. E. ALt.AXRE and L. E. BARRET 1982 ASLE Transactions 26, 222-227. Frequency effects in tilting-pad journal bearing dynamic coefficients. 15. T. SOMEVA 1989 Journal-Bearing Databook. Berlin: Springer-Verlag. 16. E. KR~MER 1984 Maschhmndynamik. Berlin: Springer Verlag. 17. W. WEDIG 1987 Zeitschrift fiir angewandte Mathematik und Mechanik 67, 34-42. Stochastiche Schwingungen-Simulation, Sch/itzung und Stabilit/it.