RANDOM VIBRATIONS OF AIRCRAFT ENGINE BLADES

A. D. Gershgorin and M. L. Kempner UDC 531:539.3

Experiments have shown [i, 2] that dynamic stresses in the blades of aircraft engines often contain not only a periodic component, but also a random component with a very broad- band spectrum and considerable energy. The vibrations of these structures should therefore be analyzed by the methods of statistical dynamics [3-5].

The present article examines a simple blade model with a long root, having a special insertion piece which is pressed by centrifugal force against the bottom flange and functions as a dry-friction damper (Fig. i). We used this model to study the optimum damping of bending vibrations of the blade in the plane of rotation. The problem was solved by the method of statistical linearization.

As the blade design scheme we took a system with two degrees of freedom -- a weightless, elastic, rigidly fixed rod with two concentrated masses (Fig. 2) [6]. The dynamic load ap- plied to the blade was nominally replaced by a concentrated force F(t) in the section i in the form of a steady centrifugal random process with a spectral density S(~). A damper with

�9 �9 ~ �9

an inelastlc reslstance R = R(y) nonlinearly dependent on the rate of displacement of the mass m2 was installed in section 2. Such a simplified scheme of blade, damper, and aero- dynamic forces allows us to discover several qualitative features of the random vibrations of the blade.

We will determine the dispersion of the bending stresses in sections 2 and 3 D[o2] and D[oa] and the rate of displacement of section 2 D[;2], on which depends the coefficient of statistical linearization (CSL). The dispersions are calculated from the formula

DIxl = ~ Ix" ( IQ)l 'S(~)d~, (1) m ~

where x * ( i ~ ) i s the f r e q u e n c y c h a r a c t e r i s t i c o f the p r o c e s s x.

I n the case in q u e s t i o n , the f r e q u e n c y c h a r a c t e r i s t i c s ( i n d e t e r m i n i n g the s t r e s s e s , the r e s t o r i n g e f f e c t o f the c e n t r i f u g a l f o r c e s i s n o t t a k e n i n t o a c c o u n t ) a r e e q u a l to

l s * ~ + luPt ~ = P,; ~ = 'w, " Y; =

whe re

Cjk (j, k = 1, in sections 2 and 3.

We write the equations of the blade vibrations in the plane of rotation in the form

n~t + ciiYl + ci~, = F; (2)

After statistical linearization R(;~) = B(D[;2])';2 and Fourier transformation of Eqs. (2) we obtain

y; = T , Y; ,

2) are stiffness coefficients; W2 and W3 are the bending moments of resistance

where

Moscow Institute of Railroad Engineering. Translated from Problemy Prochnosti, No. I, pp. 65-67, January, 1984. Original article submitted October 21, 1982.

0039-2316/84/1601- 0077508.50 @ 1984 P1enumPublishing Corporation 77

///,///I i//

F i g . 1

~'('t) m

),

Fig. 2

Fig. i. Diagram of blades with long roots and a damper.

Fig. 2. Design diagram of the blade as a system with two degrees of freedom.

AI = .% (tin)' + ~l~ + c,,; A, = -- c~,.

We will perform the calculations for the special case of a blade of constant stiffness E1 with a damper located in the middle of its length. Then

@9 8O 9,56 c. = 7-T ; c,, = c,, = -- -f~ ; c,, = ~ ;

8= 8-D-.

Let us examine two variants of distribution of the masses: ml = m', m2 = 0 one degree of freedom); ml -- ma = m" (system with two degrees of freedom) and the simplest model of random perturbation: S(m) = S = const (unlimited white noise).

We will use the following notation for dimensional quantities:

X ~ X~,

where x is a single value of the quantity x; Xo is a dimensionless coefficient.

Calculations with Eq. (i) for a system with one degree of freedom give

DId,I= ~S . (3)

D,o2]---. B(A + --~); D[o,] = B(A ~_ ~1, (4) where p = (Z/5W) 2, A = ~S8/m', B = (8~S~) 2 R2 = i/6m'

9 "

We take a damper characteristic in the form of a nondecreasing function (Fig. 3):

R----Q I .Vo I" sign.Vo, O~<n~<l , (5)

embracing a broad range of dampers with different nonconvex dependences R(;o) -- from linear (n = i) to dry friction (n = 0). Here, Q is the force of resistance at unit velocity.

We use the following formula [7] to find the CSL of function (5) from the criterion of the minimum of the dispersion of the error of the approximation, assuming normality of the process y2 :

.--I

= $ ,. (Do [$,]) , y

where ~ = ~(2/~)2n/aF(l + n/2); F is the gamma function.

Solving (3) and (6) simultaneously, we write:

(system with

(6)

78

Fig. 3.

o3

If/ ///

n,O _

v

r

Characteristic of inelastic resis- tance of the damper.

Then

where the expression in

(+ P = ~- " ~m'-'y

\y ! /

9Ji k

A =, "+' = "+' ~ ~ o-'~ /

parentheses is dimensionless.

In particular, for n << 1 (a damper close to an ideal dry-friction damper having the characteristic Ffr = Q) we obtain the following from [4]:

Dolosl m I+o V ~ + + ,

(7')

where

2 ~ . b = (8 .s .~, ) , . a~---~- ~o,

Similar calculations for the system with two degrees of freedom give

OotO,+ + +o + )" a ' (7")

(+sP w h e r e n o w ~+ = i / • m " .

Let us examine the problem of choosing an optimum friction force that will ensure a minimum for the larger of the dispersions of the stresses sin max (Do[a2], Do[~3]), limiting

" . (Ffro) ourselves to the system with one degre~ of fredom--since Eqs. (7') and (7") have similar structures. This optimum is achieved when Do [o2] = Do [o,] (Fig. 4). From this, with al- lowance for Eq. (7'), we obtain

then

Do W.]+pt = Do [Oslopt~ 9-O,~SoI+,. i.e., the square of the optimum friction force and the corresponding stress dispersion in sections 2 and 3 are proportional to the intensity of the incoming noise and the natural vibration frequency of the blade, as a system with one degree of freedom.

For example, Fig. 4 can be used to evaluate (for each level of perturbation):

a) the change in the stress dispersion compared to the minimum, resulting from the devi- ation of the friction force from the optimum value (due to instability of the damper, a change in the number of rotations of the turbine, etc.);

79

~ [ I t l \ ~ . ' /

~~ ~ \ ",, /.~ ,oFt,,,, , 8 I1 \ \ "'" 7 t ~ ~ I, X,, ' ~'A-~.-~" J "- ..... o / . \ ~ . J ' / - . . . . . . . s

0 2 ~ 0 8 2 FZ

F i g . 4 . D i s p e r s i o n o f t h e b e n d i n g s t r e s s e s i n s e c - t i o n s 2 (solid lines) and 3 (dashed lines) with n = 0.01, 8~o = i, and different values of S~: i) So 2 = 0.i; 2) S~ = I; 3) S~ = 2; 4) So 2 = 3; 5) S~ = 5; 6) S~ = 7; 7) S2o = I0. (The dot-dash line corresponds to the optimum damping.)

b) the permissible range of variation of Ff r for which D[o2] and D[o~] will not exceed their prescribed values. The solutions obtained here can be generalized to more complicated perturbation models differing from white noise.

LITERATURE CITED

i. I.A. Birger and B. F. Shorr (editors), Dynamics of Gas-Turbine Aircraft Engines [in Russian], Mashinostroenie, Moscow (1981).

2. I.V. Egorov, "Methods of spectral analysis in the study of coupled vibrations of tur- bine blading rings," Probl. Prochn., No. 12, 80-83 (1979).

3. V.A. Svetlitskii, Random Vibrations of Mechanical Systems [in Russian], Mashinostroenie, Moscow (1976).

4. M.Z. Kolovskii, Nonlinear Theory of Vibratonproofing Systems [in Russian], Nauka, Moscow (1966).

5. V.S. Pugachev, I. E. Kazakov, and L. G. Evlanov, Principles of a Statistical Theory of Automatic Systems [in Russian], Mashinostroenie, Moscow (1974).

6. M.L. Kempner, "Dynamic stiffnesses and simultaneous vibrations of axial-flow turbines in a centrifugal force field," in: Problems of Mechanics Applied to Transport and Con- struction [in Russian], Moscow (1971), pp. 48-58.

7. A.D. Gershgorin, "Statistical linearization of systems with exponential nonlinearities," Tr. Mosk. Inst. Inzh. Zheleznodorozh. Transp., No. 643, 87-92 (1979).

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