D E T E R M I N A T I O N O F T H E

W H E N C A L C U L A T I N G T H E

T U R B I N E B L A D E S U N D E R

S E P A R A T E D F L O W S

V. A . A b l a m s k i i

P U L S E F R E Q U E N C I E S

F O R C E D V I B R A T I O N S

C O N D I T I O N S OF

OF

UDC 539.4.012:534 .i

When designing and making pract ica l use of axial turbodynamos it is essent ial to take a proper account of the t rans ient gasdynamic loads on which the operating rel iabi l i ty of the blades largely depends [1]. This is especial ly important when the various s tages of the turbedynamos a re working under conditions of separa ted flows [2-4]; p rocesses tending to inc rease the bending s t r e s s e s then occur on the working blades, and under resonance conditions these may exceed the nominal s t r e s s e s by a factor of ten or more .

The possibili ty of allowing for these charac te r i s t i c s of the s t r e s sed states of turbine blades involves the necess i ty of calculating thei r forced vibrat ions, e .g . , by the method descr ibed in [5]. However, owing to the absence of any recommendat ions as to the specif icat ion of the original frequencies of the gasdynamic pulses acting on the working blades, the rea l iza t ion of such vibrat ional calculations is at the present t ime beset by ser ious difficult ies.

Investigations show that the gasdynamic pulses of the separa ted flow Ppul [3] may act at frequencies equal to multiples of the frequencies of r emova l of the annular zones of rad ica l separa t ion fK, or may be determined by the periodici ty of the changes taking place in the velocity fields over the s tage during the rota t ion of the i - th quantity of vor t ica l zones in the c i rcumferen t ia l d i rec t ion (after their r emova l f rom the blades):

lpun = kdK; 0-)

u 2 - - u B ~pu~,, : , ,~ . i. (2)

c 0 Pp+\\ / fpul 0,6 \

0 ~l tl2 aJ o.4 6~ z

Fig. 1. Charac te r i s t i c s of the aeredynamic pulses on the working blades of the last s tage in a K-300-130 turbine [7] as functions of the re la t ive volumetr ic r a te of flow Gv2: 1) 72. = 1 2 . / l 2 [6]; 2) 0.002 Ppul N [3]; 3) 10 -4 fpulll for i = imax = z2 = 150 (z 2 -- number of working blades); 4) 10 -3 fpulI and 10 -4 fpuli for kf = 1.

Institute of Strength Prob lems , Academy of Sciences of the Ukrainian SSR, Poltava. Trans la ted f rom Problemy Prochnost i , No. 12, pp. 102-103, December , 1976. Original a r t i c le submitted November 27, 1973.

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Here kf = 1, 2, 3 . . . . ; d �9 = d2K + 12., d2K is the rad ica l d iameter of the blade crown; l . is the radial ex- tent of the region of radica l separa t ion [6]; u 2 is the c i rcumferent ia l velocity at the median diameter of the blade crown; UB - C2u is the velocity of the individual separa t ion zones in the c i rcumferent ia l direction, where c m is the c i rcumferen t ia l component of the absolute r a te of flow of the s team over the s tage, c 2. (The quantities with an as t e r i sk in the index a re pa ramete r s of the spearated flow.)

On the basis of the equations represen t ing the change in momentum, the cons ervation of mass , and the conservat ion of energy, we obtain

~,Ghp.T, f , - r . . B ~ / z ~ , ' ( 3 )

where the durat ion of the separa ted flow over the second cycle is

T , = F2* Apm" "}- Gkc*c2* �9 F~*APm*-{-G (kc'c~*--kcc2) ' (4)

F2, = ~(d2K + l~,) 12.;

G is the ra te of flow of the s team; hp , is the available energy of the s tage; v 2 is the specific volume of the s team over the s tage; Apm* is the mean p r e s s u r e drop in the vor t ical zone of separat ion; B 2 is the width of the cascade of working blades; ~. is the energy t r ans fe r coefficient f rom the flow of s team to the vort ical zone of separa t ion; kc a re the proport ionali ty coefficients between the momentum and the velocity of the s t eam.

Equations (1)-(4) sat isfy the principal physical boundary conditions fpul = fpulmax for l 2* ~ 0 and fpul = 0 for the development of the s eparat ion zone l 2" over the whole height of the blade l 2 (l 2" = l 2, if G = 0).

The resul ts of the calculations show that Eqs. (1) and (2) define the limits to the frequencies of the gas - dynamic pulses which vary in one direct ion with the flow of s team and a re not multiples of the numbers of revolutions (see Fig. 1). Hence for tow ra tes of s team flow these frequencies will coincide with the f requen- cies of the cha rac te r i s t i c (natural) vibrations of the blades, and in the absence of appropr ia te detuning may constitute a cause of dangerous resonance- induced increments in s t r e s s .

The proposed theore t ica l equations for determining the frequencies of the gasdynamic pulses fpul enable us to calculate the forced vibrations of the turbine blades; this is very important in solving problems as to vibrat ional adjustment , s t rength , and rel iabi l i ty when operating under conditions of separa ted flow in the final s tages of powerful modern s t eam turbines .

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4.

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6.

7.

L I T E R A T U R E C I T E D

G. S. Pisarenko and L. E. Ol 'shtein, "Problems of the aeroe las t ic i ty of turbodynamo blades," Probl. Proch. , No. 8 (1974). V. N. Ershov, Unstable Modes of Turbodynamos ~n Russian] , Mashinostroenie, Moscow (1966). V. A. Ablamskii and A. N. Girshberg , "Reasons for the increment in dynamic s t r e s s e s in turbine blades under light loads," Probl . Proch. , No. 4 (1974). E. I. Molehanov and S. M. Pervushin, "Study of the vibrations of the blades in the last s tages of large s t eam turbines ," Probl . Prochn. , No. 10 (1974). Yu. S. Vorob 'ev and N. G. Medvedev, "Calculation of the blades of turbodynamos in re la t ion to forced vibrations for various types of excitation," Probl . Prochn. , No. 11 (1972). #

V. A. Ablamskii , " P a r a m e t e r s of modes of separa ted flows in axial turbine s tages ," Energomashino- s t roenie , No. 9 (1973). A. V. Sheglyaev, Steam Turbines [in Russian], l~nergiya, Moscow (1967).

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