3-D stress analysis generator rotor styudy.pdf

8/10/2019 3-D stress analysis generator rotor styudy.pdf

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3D Finite Element Stress Analysis of Two-PoleTurbogenerator Rotor

Alexey I. Borovkov

Eugeny V. PereyaslavetsDenis V. Shevchenko

Igor A. Artamonov

Computational Mechanics Laboratory,

St.Petersburg State Polytechnical University, Russia

Abstract

In the current paper the rotor (inductor) of two-pole turbogenerator is analyzed. This is the most loadedjoint by mechanical and thermal loading. One of the urgent problems during design and manufacturing

processes is the problem of rotor vibrations and balancing. Vibration of the two-pole rotor can be caused by

several reasons. Special attention should be paid to the vibration with double rotational frequency that can

be observed in general in rotors with significant l/D relation (relation of the rotor body length to its

diameter). This vibration is caused by different rotor body stiffness in two main axes directions: axis ofpoles (axis of "big tooth") and neutral normal to the axis of poles. In order to compensate rotor body

stiffness anisotropy in the current job the system of regular cross-slots in the zone of "big tooth" (Laffoon's

slots or slits) is used. An integral characteristic is considered to evaluate rotor body stiffness dissymmetry:

magnitude proportional to the difference of maximum static sags (located in the middle of rotor body) inthe directions of principle moments of inertia. One of the most topical problems after application of such

cross-slots is stress concentration at the bottom of slots. In the present paper the results of 3D multi-variant

finite element (FE) structural analysis of stress concentration zones at the bottom of cross-slots undergravity and centrifugal loading are presented. Together with the results of finite element analysis analytical

estimations after Neuber are quoted for stress concentration coefficients in U-like notches with arbitrary

depth. Various statements of the contact problems for rotor under gravity and centrifugal loading were

considered in order to analyze the effect of contact interaction consideration between various rotor parts onstressed state.

Introduction

Double rotational frequency vibration cannot be eliminated by any balancing, and the only method of itsreduction is maximum equalization of rotor stiffness in two directions. To equalize two-pole rotor stiffness

in main axes directions two methods are known and widely used: regular cross-slots (Laffoons slots) on

the rotor body poles (big teeth) or longitudinal slots on poles filled with ferromagnetic [1-7]. Bothmethods have their own advantages and weak points. In the current paper turbogenerator rotor with

stiffness equalization system in the form of Laffoons slots is considered (see Figure 1).


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Figure 1. Fragment of the turbogenerator rotor body with cross-slots on big teeth

System of rotor body cross-section anisotropy elimination by cross-slots suggested by US engineer morethen 60 years ago and having been used till present days is extremely easy. On the rotor body poles a

number of cross-slots is created by milling cutter of radius R (Figure 1). Bottom of the slot is made round

to lower stress concentration. Fillet radius ris commonly equal to half slot width b, i.e. slot bottom profileis semi-circle. In some cases milling cutter can be specially cloistered to get fillet radius more then b/2.

Rarely due to lower efficiency cross-slots are created as circle segment with rectilinear bottom.

Slots depth t and distance between them (spacing)sare defined first of all from the condition of

equalization system efficiency (extent of rotor body cross-section anisotropy decrease), secondly by safety

factor in the cross-section with stress concentrator and thirdly by limitations imposed on conditions of

electromagnetic parameters invariance. Slots width is usually equal to 1020 mm, but their influence is

obviously spread on much wider zone.

One of the most urgent problems at Laffoons slots usage for stiffness equalization is stress concentration

on the slots bottom. In the current paper the results of 3D multi-variant FE structural analysis of stress

concentration zones at the bottom of cross-slots under gravity and centrifugal loading are presented.

Together with the results of finite element analysis analytical estimations after Neuber are quoted for stress

concentration coefficients in U-like notches with arbitrary depth [8]. Comparison of stress concentrationcoefficients obtained with use of FE software ANSYS [9] and Neubers formulae will help to make

conclusions about analytical estimations applicability for this important class if industrial problems.

Finite element modeling and stress analysis of turbo-generatorrotor with stiffness equalization system in the form of Laffoonsslots

Statement of problem

During FE analysis two-pole turbogenerator rotor construction will be considered. 3D solid model and rotor

body cross-section are shown in Figures 2 and 3.


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Figure 2. 3D solid model of the two-pole turbogenerator rotor

Figure 3. Cross section of the two-pole turbogenerator rotor body

Gravity is acting on the rotor as loading. In the zones of bearings rigid hinges constraint condition is

applied. System of loads and boundary conditions is presented in Figure 4. By virtue of homogeneity of

materials and symmetry of geometry, loadings and boundary conditions only half of the construction is

considered. To eliminate rotor rigid body displacements along longitudinal axis Ozouter point of the rotor

body was fully constrained (marked with in Figure 4).


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Figure 4. Scheme of loadings and boundary conditions: a horizontal big tooth axis; bvertical big tooth axis

It should be noted that turbogenerator rotor body is made of medium-carbon steel alloy with yield stress

2.0 900 MPa.

At this point and further FE models are created with use of 3D 20-node solid brick structural elementsSOLID95.

Rotor FE model (with vertical big tooth axis) and its fragment are presented in Figure 5, whereNE =

71268total number of finite elements,NDF = 741255total number of degrees of freedom. FE model isthe case of horizontal big tooth axis is similar.

Figure 5. 3D FE model of the two-pole turbogenerator rotor (taking into account symmetryplane Oyz)


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Influence of rotor components consideration during FE analysis onstressed and deformed state:

In order to analyze influence of various rotor components on its deformed state the hierarchical sequence of

rotor models was created: starting from the simplest and ending with maximum complete that includes all

structural elements of the rotor. Four main models of this sequence are the following: 1) model A shaft

machining, rotor body; 2) model B

shaft machining, rotor body, windings, wedges; 3) model C

shaftmachining, rotor body, windings, wedges, retaining and centering rings; 4) model D shaft machining,rotor body, windings, wedges, retaining and centering rings, cross slots.

In this part of the analysis 3D contact interaction between rotor body and windings, rotor body and wedges,

wedges and windings are simulated using overall ideal conjunction (Figure 6).

Figure 6. Modeling of 3D contact interaction using overall ideal conjunction

As a result of structural analyses deflections caused by gravity in the direction of axes Oxand Oywereobtained for each of four models. Based on the results obtained it is possible to make a conclusion that

retaining and centering rings practically do not influence the deflections value. Thus, to simplify the FE

models it is suggested not to take into account the mentioned above parts of the rotor design (retaining and

centering rings).

Contact problem

Influence of contact interaction between various parts of the rotor body under gravity and centrifugal

loadings on rotor stressed and deformed state is analyzed with use of the following problems.

3D friction contact test problem with cyclic symmetry and centrifugalloading:

As a test problem simplified rotor construction is considered under action of centrifugal loading. Numerical

model is selected so that cyclical symmetry condition can be used. 3D FE model with cyclic symmetry and

contact surfaces are shown in Figures 7 and 8, where 41 friction coefficients. Characteristics of 3D

FE model are presented in Table 1.


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Table 1

Name Description

Solid element type SOLID953D 20-Node Structural

Solid

Contact elements typeCONTA174 &

TARGE170

3D 8-Node Surface-to-Surface Contact, 3-D Target

Segment

Number of solid elements NE 19000

Number of contact elements NCE 17500

Number of nodes NN 112000

Number of equations NEQ 337000

Number of contact pairs NCP 5

The result of calculation, distribution of radial displacements Ur, is presented in Figure 9. One can see thatvalue of the solution time for this problem is too big. With the purpose of solution time reducing it was

offered to simulate 3D friction contact interaction windings and rotor body, rotor body and wedges,

windings and wedges by means of combination of free surfaces (gaps) and ideal conjunction (Figure 10).

Figure 9. Radial displacements Ur (with friction contact)


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Figure 10. Modeling of 3D friction contact interaction using free surfaces (gaps) & idealconjunction

The performed researches proved (see Figure 11) that the suggested approach of using free surfaces (gaps)

and ideal conjunction enables to reduce solution time from 50 hours to 15 minutes. It is also possible toconclude that in the suggested approach error in calculation of rotor body surface deflections is less than 1

% in comparison with initial model:

U = (Urcontact Ur) / Urcontact 100 % 0.5 %.

Figure 11. Radial displacements Ur (without friction contact)


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3D friction contact problem. Gravitational and centrifugal loads:

In this part we carry out the comparative analysis of results obtained from the solution of the following

problems:

1) Rotor (Figure 12) under the action of gravitation and centrifugal loads taking into account 3D contact

interaction between rotor body and windings, rotor body and windings (Figure 13);

Figure 12. Solid model of the turbogenerator rotor

Figure 13. Modeling of 3D friction contact interaction ( 1 , 2 friction coefficients)

2) Modeling contact by means of free surfaces (gaps) and ideal conjunction (see Figure 10) for real rotor

under gravitational and centrifugal loads.

Results (Uy displacements) of the solution of these two problems (with and without 3D friction contact

interaction) are presented in Figure 14. Difference between two solutions is:

U = (Uycontact Uy) / Uycontact 100 % 3.0 %.


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Figure 14. Uy displacements: a with 3D friction contact interaction; b without 3Dfriction contact interaction, using free surfaces and ideal conjunction

Graphs of Uy and Ur displacements under gravitational and centrifugal loadings are shown in Figures 15and 16.

Figure 15. Uy displacements along lines AB, BC


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Figure 16. Uy displacements along line A1B1

From the obtained results it is possible to conclude that the suggested way of contact interaction modeling

with the help of free surfaces (gaps) and ideal conjunction with a good degree of accuracy describes real

behavior of object. As the advantages of this approach one can concern simplicity of realization and, what

is most important, decrease of calculation time of problems in dozens of times.

Algorithm of rotor body bending stiffness dissymmetrycompensation:

Results of rotor structural analysis under action of gravity without Laffoons slots (i.e. before stiffness

equalization procedure) are presented in Figure 17. Deformed state of the rotor without windings is shownfor better visualization of the results. The analysis was performed for 2 directions of big tooth axis

horizontal and vertical.


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concentrator, and thirdly by restrictions, caused by the necessity to keep electromagnetic parameters of

generator.

The width of saw cuts bis usually equal to 020 mm; the depth of saw cuts hshould be at least severaltimes less than diameter of the rotor body. The choice of optimum number of cross slots is characterized by

the relation of saw cuts depth hto their steps: n = h/s. For a choice of optimum number of cross slots this

relation should be less or approximately equal to 1.

Step 3.On the further steps of compensation for rotor body stiffness dissymmetry finer adjustment of cross

slots parameters is required.

During the analysis of rotor design described in the current paper the following parameters of Laffoonsslots were taken as initial (fillet radius ron the bottom of the cuts was not considered):

n= 10-16 number of slots;

s= 150-300 mm spacing (distance between slots);

t= 90-120 mm slot depth;

b= 10-20 mm slot width;

R = 250-350 mm milling cut radius.

Deformed states of the rotor under gravity loading for two positions of big tooth axis are shown in Figure18. The analysis was performed for the following parameters of cross slots: n= 12;s= 220 mm; t= 110mm; b= 10 mm;R = 300 mm.

Figure 18. Plot of Ux and Uy displacements in the rotor body with Laffoons slots forhorizontal (a) and vertical (b) big tooth axis

Rotor stiffness dissymmetry coefficients U for the rotor body without (0

U ) and with (L

U ) cross slots

are the following:

%.%)Uy,Uxinf(

UyUxmaxmax

maxmax

U 881000

=, %.%

)Uy,Uxinf(

UyUxmaxmax

maxmax

LU 02100

=.

Deformed central axes of the rotor body with and without Laffoons slots are presented in Figure 19.


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Figure 19. Deformed central axes of the rotor body at two positions of big tooth axis:

without Laffoons slots; b with Laffoons slots

The results presented enable to conclude that initial stiffness dissymmetry was equal to %8.80U ; after

application of Laffoons slots to equalize ending stiffness the dissymmetry was lowered to the value

%.L

U 02 , where points to the overcompensation of the rotor.

Further procedure of the compensation was aimed to find optimal values of cross slots parameters and lead

to the following results:

Slots optimal parameters: n= 12,s= 220 mm, t= 96 mm, b= 10 mm;R = 300 mm

Stiffness anisotropy coefficient at these parameters is equal to %.optL

U 10 .

Thus, algorithm of stiffness dissymmetry compensation is very simple, technological and efficient (low

cost). Moreover, Laffoons slots can be made on the assembled rotor. These advantages of the dissymmetry

compensation system with cross slots give it priority over others. Nevertheless, cross slots on the rotorbody poles cause a number of problems for design and exploitation, connected in general with deterioration

of electromagnetic parameters and operating characteristics [1-7].

During design and creation of Laffoons slots several structural restrictions must be considered that can

dramatically hinder their application. First of all it is structural restrictions determined by windingcharacteristics of the rotor, i.e. its tooth zone size, body diameter, number of longitudinal slots with

winding. The specified parameters defined radial width of the pole (big tooth), i.e. free space for the

cross slot. Important technological restriction is the relation of mill radius and minimum size of mill

fastener construction.

In addition to mentioned the problems of quantitative evaluation of mechanical parameters of rotordissymmetry compensation system (i.e. the value of residual dissymmetry) and estimation of stress

concentration factor on the bottom of slots seem to be urgent at Laffoons slots design process.


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Finite element modeling and analysis of 3D stressed state andstress concentration in the Laffoons slots

The value of stress concentration factor on the bottom of cross slot is the defining parameter at evaluation

of workability of rotor body dissymmetry compensation by Laffoons slots. Until present days this valuewas estimated by theoretical coefficients obtained by Neuber [8]. This approach is questionable because

Neuber considered 2D problems, but stress distribution on the bottom of the cross slot is a priori not plane.Experimental determination of stress concentration (for example, with use of strain gages) is very difficult

task.

At present the most efficient way to analyze stress concentration in cross slots is the full-scale numericalsimulation with use of up-to-date FE software that enables to obtain maximum reliable results is a

reasonable time.

Multilevel Submodeling Method

To analyze local 3D stressed state on the bottom of Laffoons slot with use of ANSYS FE software [9] it is

necessary to apply multilevel submodeling method that is based on locality principle in composite

structures mechanics [10], developed in details in Computational Mechanics Laboratory of SPbSPU(Russia) and successfully evaluated by solution of various complex problems of structural mechanics.

Analysis of stresses in Laffoons slots will be carried out for one of central slots where maximum bending

stresses arise.

In multilevel submodeling method after obtaining solution for macro model (on the macro level i = 0) at

every next step i (i >1) 3D problems for i-level submodels should be solved. These problems can be

problems for elastic, elasto-plastic media, contact problems, and composite structures or, at last, elasto-plastic contact interaction of composite structures. It depends on the peculiarities of local stresses in the

zone where the correct FE solution is to be obtained. Deriving submodel of ilevel from (i-1)-level

submodel, it is necessary to apply kinematical boundary conditions on the submodel cut-boundary S(i-1, i).These boundary conditions are interpolated displacements obtained from the solution for (i-1)-level

submodel. Such boundary conditions bring about some edge-effect-like perturbation into stressed state of i-

level submodel, but this perturbation appears only close to the cut-boundary. If the cut-boundary is far

enough from stress concentration zone of interest there will be obligatory zone of results correlation for i-

level and (i-1)-level submodelsU

(i)

U

(i-1)

. In this way with use of multilevel submodeling method correctresults can be obtained even for the problems with singular stresses.

First level submodel:

3Dfirstandsecond levelsubmodelsof the rotor body zone containing Laffoons slot are created that

contain all geometrical features of the slot (see Figure 20).First level submodel(Figure 20a) is intended to

find out exactly local stressed state on the bottom of Laffoons slot. FEfirst level submodelhas the

following characteristics:NEs(1)= 63 082, NNs

(1)= 269 702, NDFs(1)= 809 106.


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Figure 20. Fragments of 3D solid models of the rotor body containing Laffoons slot first(a) and second (b) level submodels (taking into account symmetry plane Oyz)

As a basis for 3D stressed state analysis slot with the following initial parameters was chosen: t= 110 mm;

b= 10 mm; r= 5 mm. The solution was obtained for thefirst level submodel in Figure 21 equivalent von

Mises stress i field is presented originating in the rotor under gravity loading. It can be seen that

equivalent stresses i dont exceed 11MPa, while yield stress of the rotor body material is 0.2 = 900

MPa.

Figure 21. Distribution of equivalent von Mises stress i in the first level submodel


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Multilevel submodeling:

For the detailed analysis of local stressed and deformed state of the zone with Laffoons slot the 3Dsecond

level submodel(Figure 20b) was created. FEsecond level submodel (Figure 22) has the following

parameters:NEs(2) = 37 558,NNs

(2) = 163 790,NDFs(2) = 491 370.

Figure 22. FE second level submodel of the rotor body containing Laffoons slot (takinginto account symmetry plane Oyz)

Result of FE modeling field of bending stresses z, obtained with use of thesecond level submodelis

presented in Figure 23. Stress distribution along slot bottom is shown in Figure 24, where arc

coordinate along slot bottom. An important fact is noticed that when both gravity and centrifugal loading

are acting on the rotor stress values increase in many times, but character of stress distribution doesnt

change. At this point and further centrifugal loading in ANSYS was simulated by specifying angular

velocity z = 377s-1.

Figure 23. Stress zdistribution in the second level submodel


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Figure 24. Distribution of zand equivalent von Mises stress i along Laffoons slot

Comparative analysis of results obtained with use of macro model and submodels (Figure 25), shows that

on the boundaries of first- and second level submodels bending stress z coincide with stresses in macro

model, but every next submodel defines local stresses on the bottom of the slot more exactly. The presentedresults allows to conclude that thesecond level submodelused to analyze stresses in Laffoons slot is

chosen correctly.


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Figure 25. Plot of bending stresses zdistribution along line GHfor the macro model,

first and second level submodel

Estimation of stress concentration factors in the Laffoons slotsdepending on geometry of rotor body:

Computation of stress concentration factors in the Laffoons slots was carried out both with use of ANSYS

FE software [9] and theoretical estimations. At FE simulation bending stresses in the middle of rotor bodywithout Laffoons slots (in the cross-section in point O) were taken as nominal (see Figure 22). As it was

said above to determine theoretical stress concentration factorKt(analysis of stresses on the bottom of rotor

slot) estimations made by Neuber [8] were utilized.

It was found that factors that dramatically influence stress concentrator factor value are fillet radius ron thebottom of the slot, slot width band spacings. Dependence of stress concentration factorKtand relation of

slot depth tand spacingsis presented in Table 2 and Figure 26a. Dependences of stress concentrationfactorKtand fillet radius ron the bottom of the slot and slot width bare analogous. Other geometrical

features of the rotor body and slots (rotor body diameterD, mill radiusRand slot depth t) practicallydoesnt affect stress concentration factor value being varied in wide range.


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Table 2

t/s KtANSYS Kt

Neuber %K

K-KANSYSt

ANSYSt

Neubert 100=

0.37 8.48 8.84 4.2

0.39 8.27 8.7 5.2

0.46 7.79 8.34 7.1

0.50 7.59 8.12 7.0

0.55 7.38 7.87 6.6

0.61 7.03 7.59 8.0

0.73 6.69 7.09 6.0

Influence of secondary stress concentration on stresses inLaffoons slots

Analysis of stresses on the bottom of Laffoons slots has shown that large zone exists (Figure 24) that

adjoins slot bottom where bending stresses are constantly high. For any secondary concentrator scratches, marks, boreholes, etc. the mentioned above high stresses will be nominal and due to such surface

defects on the bottom of slot stresses can be many times as high as nominal bending stresses. To verify this

idea ANSYS was used to simulate and analyze one of ever-possible concentrators spherical depression(radius and depth of the depression are equal to 0.5 mm, see Figure 27). Analysis of stresses in the zone of

secondary concentrator (Figure 28) confirms that in this zone abrupt increase of bending stresses (in 1.5-

2 times, see Figure 29a), so that stress concentration factor value can reach 2025 and even more (Figure29b).

Figure 27. Submodel with secondary stress concentrator on the bottom of Laffoons slot


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Figure 28. Distribution ofz

stresses near Laffoons slot containing stress concentrator

spherical depression

Figure 29. Stress concentration on the bottom of Laffoons slot containing stress

concentrator spherical depression: a bending stresses zdistribution along slot

bottom; b stress concentration factor Ktdistribution along slot bottom


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It seems quite natural that cases of fatigue damage of rotors that are described in literature [11] in Laffoons

slots can be explained by appearance of secondary concentrators and stress increased concentration. It

should be noted that the considered case of spherical depression is not the most sharp concentrator and

also it is very difficult to control appearance of such surface defects. This fact is precisely the most serious

restriction on the Laffoons slots application to compensate rotor body stiffness dissymmetry for high-power turbogenerators. With turbogenerators power growth the rotor body length and distance between

supports are increasing what leads to the increase of maximum bending moment (in the middle of the rotor

body). If the nominal stresses in the cross-section are of order of 1520 MPa, the amplitude of stresses insmall concentrators can exceed limits and lead to the rotor damage [1].

Conclusion

In the current paper the analysis is carried out for the most frequently used in the past till the presentmoment system of two-pole turbogenerator rotor body stiffness dissymmetry compensation methods of

regular cross slots creation on the rotor body poles, so-called Laffoons slots. Stiffness dissymmetry is one

of the main factors defining vibration state of the rotor during exploitation. With turbogenerators power

growth and consequent increase of sizes necessity of stiffness dissymmetry is becoming urgent.

One of the most suitable ways of the minimization of stiffness dissymmetry influence on its vibration state

is elimination of rotor body cross-sections anisotropy at every anisotropic part.

In the present work the advantages and weak points of rotor body stiffness dissymmetry compensation by

Laffoons slots, potential restrictions of its applications are pointed out. With use of FE analysis of stressed

and deformed state of the two-pole turbogenerator rotor body quantitative estimations of compensation

system efficiency were obtained. Example procedure of rotor body stiffness dissymmetry compensation ispresented with determination of residual dissymmetry coefficient and optimal parameters of the Laffoons

slots.

One of he most topical problem that should be solve at rotor design development with Laffoons slotscompensation system is stress concentration on the bottom of slots. In order to analyze this problem FE

models were created in ANSYS that enabled to analyze stresses state in the Laffoons slots and obtain

dependency of stress concentration factor on geometrical characteristics of the rotor body.

Together with multi-variant FE modeling theoretical estimation of stress concentration factors with use of

Neubers formulae was carried out. Comparison of FE and theoretical results shows that analytical

estimations are applicable with high degree of accuracy for a wide range of turbogenerator rotorsgeometrical parameters variation.

Influence of secondary concentrators on the bottom of Laffoons slots on strength of rotor body is

considered and evaluated quantitatively. Such concentrators (small defects like scratches, marks, boreholes,

etc. appearing due to mechanical processing of the slots surface) are seriously restricting applicability of

Laffoons slots for rotor body stiffness dissymmetry compensation in high-power turbogenerators.

Influence of contact interaction consideration between various rotor elements is also analyzed in the current

work. The research performed showed that contact between rotor body, winding and wedges can be

simulated with high degree of accuracy with use of free surfaces (gaps) and ideal conjunction combinationwhat can be explained by specific loadings applied to rotor.

Finally it should be noted that all obtained results are of great practical importance and can be used at

design development of two-pole turbogenerators rotors.

References

1) Shtilerman I.Z. Two-pole turbogenerator rotor body stiffness dissymmetry and its compensation.

PhD thesis. SPbSPU, St. Petersburg, 2003 (in Russian).

2) Iogansen, V.I., Kadi-Ogly, I.A. and Shtilerman, I.Z., Twofold stiffness of rotor body of turbogen-

erator and methods of its compensation. Theses of reports of the scientific-practical conference

Electroenergo-2002, 9-11 September 2002, St. Petersburg, OJSC Electrosila, 2002 (in


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Russian).

3) Dovger, N.E., Sokolov, D.Y. and Shtilerman, J.Z.,Slots of Laffoon: efficiency and workingcapacity. Theses of reports of the scientific-practical conference Electroenergo-2002, 9-11

September 2002, St. Petersburg, OJSC Electrosila, 2002 (in Russian).

4) Titov, V. V., Hutoretsky, G. M., Zagorodnaya, G.A. et al.,Turbogenerators. Analysis and con-struction. Energy, Leningrad, 1967 (in Russian).

5) Glebov, I. A., Danilevich, Y. B.,Scientific design basis of turbogenerators. Nauka, Leningrad,1986 (in Russian).

6) Detinko, F. M., Zagorodnaya, G.A. and Fastovsky, V. M., Strength and vibrations of electrical

machines. Energy, Leningrad, 1969 (in Russian).

7) Komar, E.G. Turbogenerators design. Moscow, Leningrad, Gosenergoizdat. 1955 (in Russian).

8) Neuber, H., Kerbspannungslehre, Springer, Berlin, 1937.

9) ANSYS Users Manual, Release 6.0, 2001 SAS IP, Inc.

10) Borovkov A.I., Palmov V.A. Locality principle in mechanics of composite structures. Preprints 3rdInt. Workshop Nondestructive Testing and Computer Simulations in Science and Engineering

(NDTCS'99). St. Petersburg. Russia. 1999. H6-H7.

11) Stanislavskiy L.Ya., Gavrilov L.G., Osternik E.S. Vibrations safety of powerful turbogenerators.

Moscow, Energy. 1975 (in Russian).

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