Mechanism and Machine Theory Vo[. 17, No, 4, 235-241, 1982 0094-114X/82/040235--07503.00/0 Printed in Great Britain. Pergamon Press Ltd

ANALYSIS OF LATERAL VIBRATIONS OF ROTATING CANTILEVER BLADES ALLOWING FOR SHEAR

DEFLECTION AND ROTARY INERTIA BY REISSNER AND POTENTIAL ENERGY METHODS "

K. B. SUBRAHMANYAM and S. V. KULKARNI

Department of Mechanical Engineering, Regional Engineering College, Kurukshetra, India

and

J. S. RAO Department of Mechanical Engineering, Indian Institute of Technology, New Delhi, India

(Received 17 November 1980; in revised form 8 July 1981)

Abstract--The lateral vibrations of a uniform rotating blade have been analysed applying the Reissner and the potential energy methods. Shear deflection and rotary inertia are taken into account. A convergence study of the two methods is made and the effects of shear deflection, rotary inertia, rotation and stagger angle on the blade vibration characteristics are discussed. Comparison of the results indicates a quicker convergence and better mode shapes by the Reissner method than the classical potential energy method.

INTRODUCTION

A turbomachine blade can be treated as a cantilever beam mounted on the periphery of a disc at a stagger angle. The cross section of the beam may be of asym- metric aerofoil shape and it may have pretwist and taper. Sometimes, a certain number of such blades are packeted or they may be laced at one or two locations. Accurate prediction of the natural frequencies and mode shapes of such blades when rotating becomes a complex problem. When the blades are stubby and short or when higher modes are required to be determined, the shear deflection and rotary inertia effects are to be considered, which makes the analysis further complicated. Other factors which are important are the root and disc flexi- bility, damping and a host of aerodynamic phenomena which enter into the blade problem when the tur- bomachine is under operation. The designer is forced to estimate the natural frequencies and mode shapes and also predict the stress and the displacement distributions in a vibrating blade as accurately as possible to avoid any possible resonances and consequent fatigue failures.

Several methods of solution of the blade vibration problem are well developed so far. Solution of the equa- tions of motion is possible under simplified conditions. The works of Rao and Carnegie[l], Rao[2] using Galer- kin process, Rao[3] using collocation method, Carnegie and Dawson[4] using transformation techniques, Car- negie, Dawson and Thomas[5], Carnegie and Thomas[6] using finite difference method are some such examples. In the continuum model approach the potential and the complementary energy principles have been used exten- sively. Rayleigh-Ritz principle was applied by Lo and Renbarger[7], Houboit and Brookes[8], Carnegie[9], Subrahmanyam, Kulkarni and Rao[10] et al. In the dis- crete model approach, the application of Holzer Myk- iestad, Polynomial frequency equation, station function, matrix and finite element methods are well known.

Targoff[ll], Rosard[12], Rao and Carnegie[13] used Holzer-Myklestad type of procedures; Rao and Banerjee[14] used polynomial frequency equation method; Mendelson and Gendler[15] applied the station function approach; Thomson[16] and Plunkett[17] used matrix methods while Dokainish and Rawtani[18], Gupta and Rao[19] and Putter and Manor[20] applied finite element technique in solving the blade vibration prob- lems.

Each of the above mentioned methods has its inherent advantages and disadvantages. For example, direct solu- tion of the differential equations of motion is possible under very much simplified conditions. The solution from the finite difference or transformation methods requires some sort of extrapolations or interpolations to avoid the matrix sizes becoming large, or else the round off errors in computation make the results unreliable. The potential and complementary energy principles do not give a simultaneously good distribution of stresses and displacements. The discrete model approaches yield lower bound solutions because of the discretisation process of the distributed mass and elasticity while the pure collocation method, though simple, is not definitely known to give accurate solutions and wherever it has been applied, lower bound results were obtained.

From the foregoing discussion on the various classical methods, it is apparent that the simultaneous deter- mination of the natural frequencies, mode shapes and the stress and the displacement distributions is not pos- sible using any single method. These difficulties can be eliminated by the application of Reissner method where it is possible to determine accurately the stresses and the displacements [21-23] as also the natural frequencies and mode shapes [10, 24, 25] simultaneously. So far, the static and the dynamic cases of uniform, pretwisted and asymmetric blades have been studied by the Reissner method, but its application to a rotating blade allowing

235 MMT Vol, 17, No. 4---A

236

for the effects of stagger angle are not studied. The objectives of the present paper are, thus, to apply the Reissner method to a uniform blade mounted on the periphery of a rotating disc at a stagger angle, allowing for the effects of shear deflection and rotary inertia. The blade will be assumed to have coincident mass and elastic axes so that the torsional coupling can be eliminated, and further that the blade is untwisted, so as to have pure flexible vibrations. It is also proposed to develop the solution of the problem by the potential energy approach using the shape functions that are identical to those used in the Reissner method, so that a direct comparison will be possible. A convergence study is proposed to be made for the natural frequencies obtained by these two methods, and the present natural frequencies and mode shapes will be compared with those available in literature.

K. B. SUBRAHMANYAM and S. V. KULKARNI

where

= y cos 0+x sin 0 t5}

u = ~ ~ [(x 3 2 + (y ' f ] dZ. (6) .tO

For the present case of pure flexural vibrations in the YZ-plane, we have the y-deflection uncoupled with x, and thus, the total kinetic energy, T, is given by

T= T,+ T2= ~ foL [pas:2+ Oh~ z

- pa W2(R + Z) f : (y,)2 dZ + pa W2Y 2 cos 2 0 ] dZ

(7)

ENERGY FUNCTIONALS

A uniform blade of length L, area A, Young's modulus E, rigidity modulus G and mass density p is mounted on the periphery of a disc of radius R, rotating at a constant angular velocity W. xx-, yy-axes are principal centroidal axes of inertia and I~ is least moment of inertia, r l~ -~ are another set of orthogonal axes through the centroid of the cross section at root and are placed such that "0~7-axis lies in the plane of disc rotation and makes an angle 0 with xx-axis, thus 0 is stagger angle.

Any point on the blade axis is measured with the co-ordinate Z, along the longitudinal axis ZZ with the origin at root section. The blade is assumed to perform pure flexible vibrations in the YZ-plane. Denoting the dynamic deflection and bending slope by y and 4' and the bending moment and the shearing force by M and V, with dash over a quantity representing differentiation w.r.t. Z, and a dot over a parameter representing a time derivative, we can write the Reissner functional IR and Potential energy functional 7r as follows[10]

V 2 M 2

(1)

~r -- ,~ ( L [ELAn') 2 + KGA(y'- 402] dZ. (2) Jo

In writing the above equation, we have taken the shear deflection into account, thus K is the shear coefficient.

The kinetic energy for vibrations in YZ-plane, T1, allowing for rotary inertia effects[10] is given by

fO L T I = ½ [pAy: 2 + pI~ck 21 dZ. (3)

The additional kinetic energy due to centripetal effects, T2, as given by Carnegie[9] for the case of coupled vibrations, neglecting higher order effects, is

T2 = rio L [2pAW2( R + Z)u + pAWZ• z

+ 2pAW(~u - niJ) dZ (4)

Using eqns (1, 2, 7), we can formulate the dynamic Reissner functional and the Lagrangian as follows

LR = T - IR (8)

L,, = T - ~r. (9)

The time averaged values of the above func- tionals can be obtained by Ritz averaging process as

2w/p L~ = LR dt

fO 1rIp LTr = L~ dt.

the following Introducing parameters

and

(1o)

(11)

non-dimensional

Z R z = - - ; 1~ = £ L

~2= Lx/AL2; dz dZ L

2 a = hW 2 where m = pA (12)

= mL4/Eh~ ; ~2= GK/E, (13)

We can rewrite eqns (10, 11) as follows after performing the necessary calculus and noting the transformation of the integral terms under an in- tegral sign as given by Kilmister and Carnegie[26]

fo I [P2AEI,~, 2 a2EI,~ LR

- 2L~ (y')~{/~(l - z) + ½(I - z2)} + M~'

~2L2 2 E ~ ] - V(y'-~)+ ~ V2+ ~dZ (14)

Lateral vibrations of a uniform rotating blade

~rEI~, f ~ [p2Xy2 + p2A~2L2~2_ ot2L2(y,)2

x {/](1 - z) + ½(1 - z 2) + ot2y 2 cos 2 0

- L ' ( * ' ) - ~ ( y ' - * ) 2] dz. (15)

In the above equations, X is the nondimensional frequency parameter, a 2 is the rotational parameter, /~2 is the non-dimensional shear parameter and ~ is the radius of gyration per unit length.

FREQUENCY EQUATIONS

Shape lunctions The following shape functions for y, ~b, M and V are

assumed in series form

y = ~ {A,z' + A,+~z '+t} (16)

$ = ~ {B,z' + B,+,z '+t} (17)

M = ~ {C,(I - z)' + C,+,(I - z) '÷~} (18)

V = ~ {D,(1 - z)' + D,+,(1 - z) '+'} (19)

which satisfy the boundary conditions

y = 4 ~ = 0 a t Z = 0

M = V = 0 a t Z = L . (20)

The arbitrary constants A,+I and B,+, are eliminated from the conditions

( y ' - ~) = $' = 0 at z = 0 (21)

and C,+, and D,+t are eliminated from the conditions

M' = V; V' = 0 (22)

throughout the length of blade.

Eigen value problems Substituting the shape functions developed above in

eqns (14) and (15), performing the necessary calculus and applying the Ritz process, wherein

3LR=0; OLR 3LR ^ 3L~ 0A---~- ~ = 0; ~ = U; ~ = 0 (23)

0L~r=0; OLzr ^ OAk ~ = O; k = 1., 2 . . . . . (24)

We get the familar eigenvalue problems which can be written in the following form

A + p 21] = 0. (25)

In the above equation, k and B are symmetric square

237

matrices and hence only the elements in the upper trian- gular portion are mentioned in Appendix 1 for each method treated, for a finite n-term solution.

METHOD OF SOLUTION

The eigen value problems defined by eqn (25) are solved using computer programs developed in fortran language. The program evaluates the magnitude and al- gebraic sign of the determinant A=IA+p2BI for an arbitrary chosen value p sufficiently lower than the fun- damental flexible mode of standstill blade. The value of p is increased in small steps till the determinant tends to zero. The true natural frequency lies at the value of p making A equal to zero, but to accelerate the frequency evaluation, we set the accuracy such that any two suc- cessive iterations separated by an interval equal to 0.0001 times the starting value and having opposite algebraic signs of the determinant will give the natural frequency of the mode. Next higher mode will be obtained in the same fashion as above but with a starting trial value being taken as 1.01 times the natural frequency of pre- vious mode. In present investigation, the starting trial frequency is taken as classical flexural frequency of the blade (in flexible direction) in cycles per second. The value obtained from the computer program will be in radians per second.

The mode shapes are obtained by solving any (n - 1) of the total n-equations in the matrix equations A+ p2B = 0 in terms of the nth one taken as unity, substitut- ing the values of the arbitrary parameters thus deter- mined for each natural frequency in the respective shape functions and normalizing them to represent the relative amplitude along the length of the blade.

NUMERICAL EXAMPLE AND RESULTS

The following numerical example relating to a typical turbomachine blade [2] is chosen to check the analysis

L = 91.948 mm A = 82.580 mm 2 p = 0.00783 kg/cm 3 E = 206.85 GPa hx = 577.729 mm 4 W = 540.350 rad/sec R = 263.652 mm G = 82.74 GPa K = 10(1+ v)/(12+ l lv)

(3.62 in.) (0.128 sq. in.) (0.283 Ib/in. 3) (30 x 106 Ib/in. 2) (0.001388 in?)

(10.38 in.) (12 x 106 lb/in. 2)

where v is Poisson's ratio.

The results obtained are discussed below. Table 1 shows the theoretical classical frequencies and

frequencies corrected for shear and rotary inertia effects[27] for the blade example considered. Tables 2 and 3 show the convergence pattern shown by the potential energy method and the Reissner method, res- pectively, for various number of terms in the assumed solution for a standstill blade. The convergence pattern observed in the case of a rotating blade is similar to the one shown for the standstill case and thus, the frequen- cies obtained with the 3- and 4-term solutions of the Reissner method and those obtained with a 5 term potential energy solution are presented in Table 4. Two

238 K.B. SUBRAHMANY~M

Table 1. Theoretical natural frequencies (rad/sec)

Mode Number 1 I1 I11

Uncorrected classical value 5654.07 35436JX) 99231.82 Corrected for Shear and rotary inertia effects[27/ 5608.84 33664.20 87323.28

cases of stagger angle setting are considered, that is 0 = 9 0 ° and 0=47.8696 ° . Figure 1 shows the mode shapes obtained from the potential energy approach and Figure 2 shows the mode shapes given by the Reissner method for a standstill blade. Figure 3 shows the mode shapes obtained from the Reissner method for both the stagger angle settings. From these results, the individual effects are discussed below.

CONVERGENCE From Tables I-3(A), it can be seen that the Reissner

method indicates a quicker convergence. The fundamen- tal tlexural frequency is obtained with an error of about 3.6% for both standstill and rotating blades using only one term as compared to the exact value in the Reissner method and there is negligible error with a 2-term solu- tion. Corresponding errors in case of potential energy method are 128% with a l-term solution and about 1.05% with a 2-term solution. A 5-term Reissner solution gives the first three modes accurately while a 6-term solution shows a similar accuracy in the case of the potential energy method.

and S. V, KI I.K,',.RNI

Comparing the converged values of the naturat frequencies of the standstill blade with the corresponding classical values given in Table 1, we observe that there is a reduction in these frequency values and that the present standstill frequencies agree closely with those from Southerland and Goodman[27/.

Effects of shear deflection and rotary inertia Table 3(B) gives the rotating blade frequencies per-

taining to a 4-term Reissner solution of the present investigations and those by Galerkin process given by Rao[2] neglecting shear and rotary inertia effects. Also the squared values of the frequency ratio (theoretical frequency of rotating blade/standstill uncorrected clas- sical frequency) 2 are calculated and are compared with those of Rao[2] in Table 4. This table shows also a further comparison with the squared values of frequency ratios proposed in the form of frequency relations for the first three modes by Rao[2].

From these tables it can be seen that when the stagger angle is 90 °, the percentage reductions in the squared value of the frequency ratio calculated on the basis of the Reissner method results are 1.507, 10.508 and 22.774%, respectively, for the first three modes in com- parison with the Galerkin values, which ignores the shear and rotary inertia effects. When the stagger angle is 47.87°(cos 2 0 = 0.45) the corresponding reductions in the frequency ratio squared values are 1.4896, 10.4969 and 22.774%, respectively, for the first three modes. From these results, it is obvious that the shear and the rotary inertia effects when taken into account, lower the frequency values of the rotating blade, such reduction being more predominant af higher modes. Further, it may

Tabte 2. Convergence pattern: potential energy method standstill blade

Number of Natural frequency (rad/sec) terms in solution I Mode II Mode Ill Mode

1 12788.05 -- - - 2 5667.96 64773.1 - - 3 5612.30 34261.6 154548.2 4 5612.30 33843.7 91962.7 5 5612.30 33703.1 90244.0 6 5612.30 33703.1 89101.4

Table 3(A). Convergence pattern, Reissner method standstill blade

Number of Nature of frequency (rad]sec) terms in solution I Mode II Mode 1II Mode

1 5812.8 -- -- 2 5609.9 38368.2 -- 3 5612.3 33708.9 111778.2 4 5612.3 33671.8 89581.9 5 5612.3 33671.8 89019.9

Table 3(B). Rotating blade frequencies to = 540.35 rad/sec : ~ = 2.86743

Mode number

cos 2 0 Presented results by % increase Results by Reissner method over the Galerkin process allowing for shear corresponding Rao[2], neglecting and rotary inertia standstill shear and

frequency rotary inertia

% increase over the corres- ponding standstill classical frequency

1 0.0 5576.83 2.58 5800.04 II 0.0 33832.0 0.313 35565.0 III 0.0 89677.6 0.107 99365.6 I 0.45 5746.09 -- 5788.73 II 0.45 33832.0 -- 35563.3 III 0.45 89677.6 - - 99365.6

2.58 0.364 0.135

Lateral vibrations of a uniform rotating blade

Table 4. Comparison of rotating blade frequencies

[ Theoretical frequency of rotating blade ]2 P r= (Pt[Ps)2 = LClassi-~al ~ ~ u e n c - ~ - d ~ blade

w = 540.35 rad/sec; J~ = 2.86743

239

Present results allowing for shear and rotary inertia

Reissner Method (P2r) Potential energy (P~)

Mode cos 2 0 3-term 4-term 5-term number solution solution solution

Rao [21 Neglecting shear & rotary inertia (p2)

Galerkin Frequency relation

I 0.0 1.036679 1.036679 1.036679 1.0523 1.0518 II 0.0 0.911837 0.911519 0.911519 1.0073 1.0073 III 0.0 1.271713 0.816706 0.829743 1.0027 1.0025 I 0.45 1.032815 1.032815 1.032815 1.0482 1.0478 II 0.45 0.911519 0.911519 0.911519 1.0072 1.0072 III 0.45 1.271713 0.816706 0.829743 1.0027 1.0025

also be observed that the percentage reduction in the natural frequency due to shear and rotary inertia cor- rection value for the stand still blade and the rotating blade with 0 = 90 o or 0 = 47.87 ° is almost same, if we compare the present results with the standard classical values in case of standstill blade with the results of Rao[2] using Galerkin process for the rotating blade. This implies that the percentage reduction due to shear and rotary inertias will be almost the same whether the blade is standstill, or rotating with any stagger angle setting.

Effects of rotation and stagger angle setting Comparing the results, given by the Reissner method

with a 4-term solution, of the rotating blade with the corresponding standstill values, we observe that the flexural modes are stiffened due to the rotational effect. The stiffening pattern can be understood by considering the percentage increase in the first two modal frequen- cies because of rotation, which are 2.58 and 0.313%, respectively for the first two modes as shown in Table 3(B). The corresponding values from the Galerkin process for the rotating blade and standstill classical frequencies are 2.58 and 0.364%, respectively.

From these results, it can be concluded that the first bending mode is having the largest percentage increase in

the frequency value due to rotation and higher modes show little variation in the frequency value due to rota- tion. Further, the present results show a similar trend of stiffening pattern as observed by Rao[2] using Galerkin process.

A comparison of the natural frequencies of the rotat- ing blade with different stagger angle settings indicates that there is a lowering of the fundamental frequency value when the stagger angle is changed from 90 to 47.870 . The second and the third modes show in- significant changes. Schilhansl[28], Rao[2], Rao and Carnegie[l], Carnegie, Stifling and Flemming[29] et al. have observed that the frequency decrease can be represented by a linear variation proportional to cos20 and that the effect of stagger angle setting becomes insignificant for modes other than the fundamental flexural mode. Present results are consistent with their observation.

Mode shapes Figure 1 and 2 show the mode shapes of a standstill

blade obtained from the potential energy method and the Reissner method, respectively. The nodal location shown by the modal curves of the Reissner method are close to the exact ones[30] while such agreement is relatively poor in case of the potential energy method. Considering

,o[ i0,t I .. . . j / z o2~ i .. J r ' - ' ,,. /

,, '..

' f -0"2 5612.3 rod/see. \ I ~

"~ Ill Mode -~'X"x j / x 06[ n Mode ~ ' , " " [ 89101"/.,3 rod/see. ~ - " / ~ x

J 33703"1 r a d i s h , x . I .0 / I I I I i i I ' ,

0 0.2 0.4 0-6 O.O 1.0 z = Z / t

Fig. I. Rexural mode shapes: potential energy method (Standstill blade).

I ~ Mode 3 . , i

~" I~ M o d e ~ ' \ /.'~ ~._0.6 I. 89019'9 rod/see. "\.... /'~f'~

| II ModJ-3" ',, _, .0 [ . . . . . o /se t ' ,

0 0.2 0"4 0"6 0.8 1 '0 z = Z/L

Fig. 2. Flexural mode shapes: Reissner method (Standstill blade).

240

1.0[ (5756.83 r o d / s e e /

/ I

o L-~' (~ ' ~

I BEN DING

K. B, SUBRAHMANYAM and S. V. KIilKARNI

w : SZ,03S rc~d/~ec co~ @: 0

1.0[ ( 33852.0 radlsec )

1 \1.o

:°?ol \ II BENDING

w : 540.35 rod/sec, col ~ :045

, o r (89677.~,radlsec) [

III BENDING

1.0

CtTS

Y o.5o

(9.25

o 05 1.0 I BENDING

1,0[ ( 33832.0 rad/sec ) /

I I BEND(NG

1.0 F ( 8967Z6 rctd~ec I I

/ o /,0

III BENDING

Fig. 3. Model curves of rotating uniform blade: Reissner Method.

the case of rotating blade, the mode shapes obtained from the potential energy method are similar to those from the Reissner method (Fig. 3) but the second mode shown slightly less relative magnitude at z = 0.5 and the third mode shape, being obtained with the 5-term solu- tion where the frequency has not converged, could not properly be compared. Thus only the modal curves given by the Reissner method for 0 = 90 ° and 0 = 47.87 ° are presented. It can be seen from these mode shapes that they resemble very closely the respective standstill modes (Fig. 2) and further that the stagger angle setting does not alter the mode shapes of the rotating blade.

CONCLUSIONS

The Reissner and the potential energy methods are successfully applied to the case of a uniform rotating blade executing lateral vibrations. It has been observed that the Reissner method indicates a quicker con- vergence and gives better mode shapes in comparison with the classical potential energy approach. It is expec- ted that the present investigation would be of consider- able use in developing the Reissner method to a more complicated blade vibration problem involving pretwist, taper and asymmetry, and in determining the forced response of the blading since the method shows excellent convergence for stresses [22] and displacements.

Acknowledgements--The authors gratefully acknowledge the financial assistance rendered by the University Grants Com- mission, Government of India, for this investigation.

REFERENCES I. J. S. Rao and W. Carnegie, Aero. J. Roy. Aero. Soc. 74, 161

(1970). 2. J. S. Rao, J. Aero. Soc. India 22, 257 (1970).

3. J. S. Rao, Proc. Ind. Soc. Theo. App. Mech. 17, 211 (1972). 4. W. Carnegie and B. Dawson, Aero. Q 20, 178 (1969). 5. W. Carnegie, B. Dawson and J. Thomas, Proc. Inst. Mech.

Engrs 180, 71 (1965). 6. W, Carnegie and J. Thomas, Aero. Q. 18, 309 (1%7). 7. H. Lo and J. Renbarger, Ist. Proc. U.S. Nat. Cong. AppL

Mech. 75 (1952). 8. J. C. Houbolt and G. Brookes, NACA Report 1346 (1958). 9. W. Carnegie, J. Mech. Engng Sci. 1 235 (1959).

10. K. B. Subrahmanysm, S. V. Kulkarni and J. S. Rao, J. Sound and Vib. 75, Preprint I (1981).

11. W. P. Targoff, J. Aero. Sci. 14, 579 (1947). 12. D. D. Rosard, J. App. Mech. Trans. ASME 20, 241 (1953). 13. J. S. Rao and W. Carnegie, Int. J. Mech. Engng Educ. 1, 37

(1973). 14. J. S. Rao and S. Banerjee, Mechanism and Machine Theory

12, 271 (1977). 15. A. Mendelson and S. Gendler, NACA TN 2185 (1949). 16. W. T. Thomson, J. App. Mech. Trans ASME 17, 337 (1950). 17. R. Plunket, J. Aero. Sci. 18, 278 (1951). 18. M. A. Dokainish and S. Rawtani, AIAA J. 1O, 1397 (1972). 19. R. S. Gupta and S. S. Rao, 3". Sound Vib. 56, 187 (1978). 20. S. Putter and H. Manor, J. Sound Vib. 56, 175 (1978). 21. K. B. Subrahmanyam, S. V. Kulkami and J. S. Rao, Proc.

23rd Cong, ISTAM, 113 (1978). 22. K. B. Subrahmanyam, S. V. Kulkarni and J. S. Rao, Proc.

24th Cong. ISTAM Paper No. MS-33 (1979--80). 23. K. B. Subrahmanyam, S. V. Kulkarni and J. S. Rao, Sub-

mitted to J. Aero. Soc. India (1980). 24. J. S. Rao, S. V. Kulkarni and K. B. Subrahmanyam, To

appear in J. Appl. Mech. ASME shortly. 25. K. B. Subrahmanyam, S, V. Kulkarni and J. S. Rao, Proc.

Int. Syrup. Nonlin. Cont. Mech. and Silver Jub. Cong. ISTAM. Kharaqpur, Dec. 17-20 (1980).

26. C. W. Kilmister and W. Carnegie, Bulletin Mech. Eng. Edu. 4, 49 (1965).

27. R. L. Southerland and L. E. Goodman, Rep. N6-ORI-71 TO-6 Project NR-064-183. 1 (1951).

28. M. J. Schilhansi, J. App. Mech. Trans ASME 25, 28 (1951). 29. W. Carnegie, C. Stifling and J. Flemming, Proc. Inst. Mech.

Eng. 180, 124 (1%5). 30. C. M. Harris and C. E. Crede, Shock and Vibration Hand-

book, Vol. 1, p. 7. McGraw-Hill, New York (1%1).

Latera l v ibra t ions of a un i form rota t ing blade 241

APPENDIX

ELEMENTS OF MASS AND b'l'ltl,~f..g.S MATRICES

Reissner method A,.~ = T A ( T 4 + T I . cos 2 0 ) / L

Ai, i+. = T A ( T 2 . cos 2 0 - R 4 . $ 3 . T5) Ai, i .2, = - T7

A~+.,~+. --- L . TA{R2 . $2 . cos2/(Q + 3) - T6} Ai+.,i+2. = T8 Ai+.,i~3. = T7 A~+2.,i+3. = U I. R3. S3/(Q + I) + U2. R2. S2/(Q + 3) Ai+2,,j+3, = - U 1 . R 3 . S21L(Q + 1) Ai+3,,i*~. = R 3 . $3 . UI /L~(Q+ 1)

Bi, j+. = T B . T2 Bi, j = TB. Tl lL

B~+.,~+. = T B . R 2 . S2/ (Q + 3) + 52 . T I l L T A = a 2EL~/L ~

TB = hEI,~/L ~ T I = I (Q+ I ) - ( R I + S1)I(Q+ 2)+ R I . S I / ( Q + 3) T 2 = S 2 / ( _ Q + 2 ) - R 1 . S2/ (Q + 3) T4 = i.i[6R(Q - 2)!/(Q + 2 ) ! - (Q - 2)!/(Q + 1)!

- Q!I(Q + 3)!]

T5 = 2 R ( Q - 1)!/(Q + 2)! + (Q - 1)!/2(Q + 1)! - (Q + I)I/2(Q + 3)!

T6 = R 3 . S3{RQ!I(Q + 2)! + 0.5/(Q + 1 ) - 0.51(Q + 3)} T7 = i ! j ! / L ( Q + 1)! T 8 = - R 3 . S 3 . L . T 7 U 1 = L2~21EIxxk 2

U 2 = Le/ELx; Q = ( i + j ) R1 = i(i + i); R2 = 1(i + i) i R3 = l ( i + 1); R4 = i S1 = j/(j + 1); S2 = 1(] + 1) ; S3 = 1(] + 1); S4 = j

Potential energy method

Ai,~ = - c t 2 • T 4 - [ z • T9/~Z + a 2 • T I cos 2 O Ai,j÷. = - a 2 • L . R 4 . $ 3 . T5 + / ~ 2 U3/P2

+ a ~ . L . T 2 . c o s 20 Ai÷,,~+. = - a2L 2 • T 6 - L ~ . T 9 - ~2, T5/~2

+ ~2L2. R 2 . $2 COS 2 O/(Q + 3) B~,j= - h . T l

Bi, j+. = - h . L , T2 Bi+. , i÷. = - ~.. L ~ , R 2 . S2/ (Q + 3) - L 2 • ~ . T I

T9 = 2. R 4 . S 4 ( Q - 2)!/(Q + 1)! u3 = 2. L . R 4 . S I ( Q - 1)!/(Q + 2)!

AN~E DER SEIT~h~NGUNG~N ~OT~EEEND~ BALEEN UNTER ~SRUCEBICHTIGUNG DER SCHUBVERSCHIEB~NG

ROTATI~TI~GHE~T IIIT~JILB ANS£TZ N~CH ,P~SNER UND I~Y~ENTIELLE-ENE~GIE-m~ETHODE

K. B. Bub~ahman~am, S. V. Kulkarni und J. B. Rao

K~GCassunH - Die Beitensohwingtuagen elnez gleichmKSig rotle~enden Schaufel warden un~er Anwen-

dung des Ansatzes naoh ReissneE und des klasslschen Ansatzes dee potentlellen Eme~gie analyllert.

Die 8ohubve~sohieb~ng und Rot&tlonstr~helt wezden be~Gckslahtigt. Die Konvergenz der zwei Metho-

den wiEd un~e~muaht 0 und die Einflt~sse der 8ohubveEschiebungp der Rotations- und 8taffelungswln-

kel auf die Sohwingun~selgensohAften we~den besp~ochen. Elm Verglelch der Ergebnisse zeigt~ da~

de~ RelssneEansatz sohnelle~e Konve~genz bzin~t und besse~e 8ohwlngungsfo~men im Vergleich zum

klassis~hen Ansatz de~ po~entlellen Ene~ie e~glbt.