journal of sound and vibration volume 46 issue 4 1976 [doi 10.1016%2f0022-460x%2876%2990676-3]...

7/21/2019 Journal of Sound and Vibration Volume 46 Issue 4 1976 [Doi 10.1016%2F0022-460x%2876%2990676-3] B.M.E. de Silva; B. Negus; J. Worster -- Mathema

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502 H. M. F. I L SILVA, H. NtGUS /~\I J . WORSTLR

of control theory to the structural optimization area IS of recent originnd in the workreported here we have attempted to broaden its range of applicability to more complexproblems by determining the actual blade profiles on the assumption of elliptical cross-sectional shapes. This assumption was mad e purely for ease of numerical computation and

does not restrict the validity of the subsequent arguments. More general aerofoil shape s [8]consisting of algebraic curves are possible but with an increased degree of computationalcomplexity. The effects of taper and twist could also be considered. The elliptical forms a rerepresented by a set of discrete param eters and the optimization reduces to a non-linearprogramm ing type problem. The state differential equations are incorporated by using anintegral penalty function [9] and the resulting unconstrained problems are solved sequentiallyby using a combination of optimization techniques [ 10. I I]. The frequencies have the statusof a control param eter.

A different approa ch is to use a finite difference discretization of the state equations[ 12-141.These difference equations give a matrix eigenvalue problem for determining the frequencies

of vibration. The optimization again is handled by using a penalty function metho d [15].Sections 2 and 3 describe, respectively, the behaviour calculations and optimal control

problem statement. Sections 4 and 5 discuss the computational algorithms used. The paperis concluded by sections 6-8, which give a description of results and implementation of theoptimization routines on ICL 1904A /l906A digital comp uter.

2. BEHAVIOUR CALCULATIONS

The performan ce index to be minimized is the centrifugal loading on the blade whichmeasu res the interaction of the blade with the disc (Figure I):

J=pdV-P(z + R),

go (1)

whe re p is the specific weight, go the acceleration due to gravity, the radius of the disc andJ the spin of the blade-disc assembly. The forced transverse vibrations of a thick blade [2,3]

incorporating a Timoshenko correction term, k, are described by

(2)

whe re w(z,t) and Q(z,t) are the deflection and shear force, respectively,E and G are theelastic moduli, and A and Zare the areas of cross-section and second mom ent of area, respec-tively. These equations are derived on the assumption of flexural bending, with neglect of

_.____-c------__-__ ._ >

Figure 1. Typical cross section of turbine blade.


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OPTIMAL DESIGN OF TURBINE BLADE SHAPES 503

rotatory inertia terms or torsion or the stiffening effect of the centrifugal force field. Theseassum ptions are generally valid for short steam turbine blades with low spin. For wide chordgas turbine blading and for longer L.P. blading in steam turbines, the problem of torsion is amore significant factor. The variations in A must satisfy

ALGA


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504

The cross-sectionalelliptical, of the form

B. M. E. DE SILVA R. NEGUS AND J. WORSTER

3 STATEMENT OF PROBLEM

shape at each station along the blade axis (z-axis) is assumed to be

A X + B y + 2Hx~ + 2Gx + 2/;, + C = 0,

where A, B, C, F, G and Hare functions of z. A particular representation of this three dimen-sional surface is given by

,xz + by2 + dz2 + 2hxy -t 2gx + 2j-y + 2ez + c = 0, (11)

where

a > 0 , ab - h2 > 0 . (12)

The coefficients (a, b, . . ., c) in equation (11) are independent of (x,y,z) and are the designvariables. Rearranging equation (11) as a quadratic in x gives

ax2+2(hy+g)x+by2+dz2+2f_+2ez+c=0,

so that the zeros of equation (11) at a given pointz along the blade axis are given by

xi, x2 = - [ (hy + g) + d (h - ab)y2 + 2(hg - af ) y + g2 - a(dz2 + 2ez + c)] /a . (13)

Therefo re the thickness is defined by

h y, z ) = Ix 2 - x 1 I = 2/a) d(h2 - ab)y2 + 2(hg - af ) y + g2 - a(dz2 + 2ez + c)(14)

and the area by [16]

A(z) = 1 h(y,z)dy = (X/W) [dz + 2ez + c +(bg2 + af - 2gfh) / (ab - h ) ] . (15)

Similarly, after some simplification, the second m oment of area becom es[ 161

I z) = 1 2h(y, z) dy = (2/a) j y2d(h2 - ab)y2 + 2(hg - af )y + g2 - a(dz2 + 2ez + c) dy

= (2n /a) [ 5 (hg - af ) + (ab - h ) (g - a{dz2 + 2ez + c})] x

x [ (hg - af ) + (ab - h2) g - a{dz2 + 2ez + c})] . (16)

A sufficient condition for I(z) > 0 is

g2 - a(dz2 + 2ez + c) > 0 V z E [0 , I ] . (17)

In addition, from machining and aerodynamic considerations the blade dimensions must liewithin a specified dom ain of the xv-plane (Figure 2(b)):

L, 0 . (19)

This implies y, < y < J ~, wh ere y1 and y, are the zeros of the quadratic form19). Thereforethe second of conditions (18) and condition (19) are satisfied if

L2 < [(hg - af)T d(hg - af )2 + (ab - h2) g - a{dz2 + 2ez + c})] / (ab - h2) d U , .


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506 B. M. E. DE SILVA B. NEGUS AND J. WORSTER

The problem consists essentially in determining (a, h. c, tl. t,.J:g, II) to minimize expression (23)subject to the inequality constraints (12), (17), (20))(22) and (3). The non-linear program sare solved by using penalty function techniques which transform the problem into a sequenceof unconstrained problems. The algorithms used in this paper differ in their handling of the

system differential equations (8)-( IO). These formulations are discussed below.

4. E-TECHNIQUE

In this approa ch a pow er series representation is assume d for the shear force, Q, of the form

Q=p+qz2+rz3+~z4. (24)

Substituting equation (24) in equation (9) and simphfying gives

2f a , c , e , f , , P, 4, r, 3; ~4 = - s+-

3ZdA 1dZ

po2A-AZdz+Adz (6r+24sz)+

1

+~~~-~+~~~~~+~-~~~~~2q+6rz+12sz2)t

2 dAd2A 4 dAdA

The boundary conditions (IO) correspond to

2q + (kpo2/g,G)P = 0, p + q12 + r13 + s14 = 0,

(25)

(26)

The derivatives ofA and /appearing in equations (25) and (26) are calculated from expressions(15) and (16 ). It should be noted that the first of boundary conditions (IO), dQ/d z = 0 atz=O , is automatically satisfied by expression (24). Representation (24) was found to beconvergent with decreasing coefficients. The behaviour requirements may be incorporatedbyintroducingtheadditional parametersp,q,randsintotheoptimization byusingthesquaredintegral of the left-hand side of equation (25) with a similar handling of the boundaryconditions (26).


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So that in the original form ulation the problem was represented by the unconstrainedfunction [9],

c-bg2+af2-2gfh +4er+K

h2-ab 3 2

+

bg2+af2-2gfhl

h2-ab

f2(a, b,c,d,e,~g,h; p,q,r,s;o;z)dz+ 2q+ spr+(p+q12+r13+s14)2+

+

+4s l3 + go GA(l) d2 z3

( Xpw ~(~)~)]2)~dz[ g2-a(dz2:2ez+c)+

1+

adz2 + 2aez - g2 + (ab - h2) U2 - 2(hg - af) U, + ac+

1+bdz + 2bez - f2 + (ab - h2) U, - 2(hf - bg) U1 + ac

+

1+

adz2 + 2aez - g2 + (ab - h2) L - 2(hg - af) L , + ac+

1+

bdz2 + 2bez -f 2 + (ab - h2) LZ, 2(hf - bg) L 1 + ac I .(27)

The penalty transformation (27) incorporates the performance index (23) in conjunction with

the differential equations (25) and boundary conditions (26) together with the inequalityconstraints (3), (22), (12), (15, (17,) (20) and (21). Some of the integrals appearing in expression(27) are calculated explicitly by using the formulae

sdx/(a + bx + cx ) = (2/d-) tan- ((2cx + b)/w}, 4ac - b2 > 0,

= (-2/d-) tanh-{2(x + b)/z/b2 - 4ac}, b2 - 4ac > 0. (28)

The unconstrained function (27) was minimized for a decreasing sequence E --f 0, theminim ization being based on the conjugate direction and variable m etric methods [15, 171.For convergence the method required a feasible starting po int. In addition, when non-

feasible regions w ere encountered it was difficult to correct back into the feasible regionsbecause the NAG-Loughborough University System library minimization routine exercisedvery little effective control on the maximum distance moved in one complete cycle. Theinequalities (20) and (21) are valid only if

Cab h2) U2 - (hg - af) > 0, (hg - af) - (ab - h2) L2 2 0,

(ab - h2) U1 - (hf - bg) > 0, (hf - bg) - (ab - h2) L 1 > 0. (2


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5 8 R. M. t.. DE SILVA, R. NEGUS AND J. WORSTER

The inequalities (29) were incorporated into expression (27) by using SUM T type penalties[18]. In addition the constraints 17), (20) and (21) were replaced by a simple constraint ofthe form

min (AZ + 2 Bz + c) 2 0. (30)LO.ll

To improve th e convergence characteristics, a different formulation was subsequently used[IO, I I], which remove d the necessity of starting from a feasible estimate and which also hadbuilt-in correction procedu res to ensure feasible points. For examp le, a design constraint ofthe form

fi(x> 3 O* irl, (31)

was handled by using a penalty term

--E 2 log (h(x)) (32)i

for feasible designs, while for non-feasible designs the corresponding term was

(l/e) 1 {min [OAx)l12.L

(33)

Equality constraints were represented by

(l/s) Xff(x). (34)

These formulations have the effect of preventing the synthesis from moving too far from the

feasible region. Th e algorithm has three different ch oices for the unconstrained minimizationcycles. Sixth order polynomial extrapolation techniques are also available for testing ultimateconvergence.

5. MATRIX EIGENVALUE FORMULATION

From expression (8),

(35)

where

ff = -@o/b, /3 = GIkE, B = l/A. (36)

The fourth order differential equation (35) is reduced to a set of difference equations byusing the central finite difference approximations

(d4 Q/dz4)i = (Pi+2 -4Qi+l+ 6Qi - 4Q,-1+ Qi-2)lH4T

(d3 Q/dz3), = Qi+2 - 2Qi+l+ 2Qt-,- Qi-J/2H3, Cd2 Q/@)i = Qi+l Qi + Qi_l)/H,

(dQ/dZ)i = (Qi+l - Q,_,)/2H , 0 = Zo < Z1 < . . . < Z, = I, H = l/n. (37)

Substituting expressions (37) in expression (35) and using the boundary conditions (10) gives,on simplification, the eigenvalue equation

MQ = wNQ, (38)


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OPTIMAL DESIGN OF TURBINE BLADE SHAPES

where

& = cr(ZB+ ZB)/H, & = c((3ZB+ 21B) /H2,

& = a (3Z B + I B )/ H 3 , q 5 4 aZB/H4 ,

8, = Z B + Z B , t = ( 2Z B - I - B )/ H , e2 = Z B I H 2

and the non-zero elements of the (n x n)-matrices M and N are given by

(MW) = 644 - 242 (j=2,...,n- l), M(l, 1) = 244,

M(n, n ) = -~c#J~ 4 a Z B l H 3 - a Z B l H 2 ,

M ( j ,j - 1 ) = -494 43 42 - &I ( j=2, . . . ,n- I) ,

M ( n , n - 1) = 3$4 + 2 aZB /H3 , M .Lj - 2) = 44 - t43 (j=3,...,n- l),

M ( n , n - 2) = -44, M(jLZ + 1) = -444 - 43 + 42 + 341 (j=2,. . . ,n- l),

M(l, 2) = 244, M(U + 2) =$4 + 393 (j=2,...,n-2),

N(j,j) = -282 + 80 - /I (j=2,...,n-l), ~(1~1) = e2,

N (n , n ) = -e2, w,j + 1 =e, +3e,, ~ j,j - 1 =8, - 38,.

509

The frequencies of vibration o = ~(a,b, c,d,e,f,g,h) are given by the eigenvalues of thematrix N-M. The eigenvalue calculations are based on reducing N-M to upper H essenbergform by using similarity transformations [19, 201. The eigenvalues are then calculated byusing the QR algorithms for real Hessenberg matrices. The condition (22) corresponds to the

smallest positive eigenvalue of expression (38). The constraints (12) were dealt with by usingslack variables

a = a& b = ( h 2/ a 2) + / I &

while the frequency constraint (22) was handled by using a penalty function

{min [0, (w - o,)]}~.

(39)

(40)

All other constraints are of the form

Az+2Bz+c>Ov z E [O,Z].

These are handled by using a penalty function

P = i dz{min [0, Az2 + 2Bz + cl}.0

This is evaluated by defining

(41.)

F(u ,u ) = [ (A Z + 2 Bz + c)2dz = (A 2/5)(u5 - us) +AB(u 4 - u 4 )+u

+ (2/3) (AC + 2B2) (v3 - u3) + 2Bc(u2 - u2) + c(u - u).

If r,s ( r < s) are the zeros of Az2 +2Bz + c = 0 , then

r = min{(-B f -)/A}, s = max{( -B f - ) /A} ;

if A = 0 , then t = -c/2B.

(42)

(43)


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510 B. M. F. F SILVA,B. NEGUS Ahi . ORSTER

The values of P are as follows :

(a) A=B=O:ifcO,P=O:(b) .~=O.B 0 Initial values Final values

w - Qg 0.5680 x lo4 0.5340 x lo4A-AL 0.63585 0.10819 x 1O-4Au-A -0.5585 x 10-l 0.5799Scaled boundary conditions - 105-lo9 10-3-100Scaled integral of expression (25) z 102 loo

7. DISCUSSION

Case 1 presented considerable computational difficulties. The metho d [ 10, 111 was generallysatisfactory in the sense that it automatically ensures that once a feasible point is found thenfuture iterations will re.ject non-feasible points. Thus, in many cases, it was possible to startthe synthesis with a more or less arbitrary design. How ever, on occasion, the starting pointgave an area less than the prescribed minimum, A,_. In this instance, as the optimizationprogres sed this constraint was more and more violated and the performanc e index, J, -lo2as the area decreased to zero. Ho weve r the metho d was able to satisfy the system differentialequation and boundary conditions (25) and (26). The efficiency w as dependent not only onthe initial design estimate but also on the choice of scaling factors for the various terms in themodified performance index. Unlike Case 2, the formulation incorporated additionaloptimization variables, p,q, r, s and o, to simulate the differential equation. This feature alsotended to reduce the robustness of the computations. Another disadvantage is that in generalthe metho d is unable to lock in to the lowest frequency of vibration as stipulated in expression(22). In this respect Case 2 formulation is superior as it always generates the comp leteeigenvalue spectrum 119, 201 from wh ich the lowest frequency is picked. In addition, theproblem works n a lower order design space. Initially, Case 2 was handled by using a variable


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13. M. I [ I: SILVA. 13. Nt AXI J. WORSrER

Figure 3. (a) Initial case-Case 2(a) (Table 2); (b) final shape-Case 2(b); (c) final shape-Case 2 (initialpoint not shown) (Table 2); (d) case I(a)-final; (e) case l(b)-final.

metric optimizer [17]. How ever, the metho d found considerable difficulty in moving into the

feasible regions when a non-feasible design was encountered, as the algorithm did not providean effective control on the maximum distance moved in one iteration. Unlike Case 1, Case 2depende d critically on the choice of a starting point. T he positive definiteness properties ofthe eigenvalue equation on occasion tended to depend on the finite difference discretizationof the boundary conditions (26): higher order finite difference app roximations tended toexhibit ill-conditioning. The optimization program s were characterized by severe non-linearities in both the performanc e index and constraints. As indicated in TableI, the problems


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have also regions of relative minim a. In spite of this high degree of non-linearity, the optimiza-tion tiethods worked well to reduce the area to its minim um A,. A n examination of thefirst of expressions (23) shows that this result is to be expected for J to be a minim um. Thefinal values for the optimization for both cases gaved = e = 0. From expressions (15) and

(16) this implies that the cross-sectional shapes are independent of z. The optimal bladeconfiguration is uniform in the z-direction, with zero twist.Due to the intrinsic non-linearity of the problem, it is to be expected that the modified

objective function of the form (27) would have several local minim a and the unconstrainedminimizer may appear to converge to designs which are non-optimal in some physical sense.The presence of relative minim a can only be established with any degree of confidence byscanning as representative a region of design space by starting from different points. Relativeminim a give different configurational representations for the final blade profiles and aselection between these would have to be made by using additional design efficiency criteria.The system dynamics involves only the area and its second moments and the synthesis can

give information primarily with regard to these variables.While Case 1 generally converged towards AL on occasion it gave higher values for A andJ

as recorded in Table 2. This is explained in terms of the multimodality of the performanceindex (23) and the numerical difficulties in handling the differential constraints (25) and (26).In this instance, the convergence was only apparent and the synthesis tended to move towardsa solution of max imum area. The initial design started with an area near to A, but with thedifferential constraints large. Therefore, to reduce these the method tended to go to the otherextreme AU, thus giving a deterioration in performance. The method has homed inminimum that did not represent physical convergence.

to a local

8. CONCLUSIONS

This paper describes the successful development of an automated structuralsynthesiscapability for turbine blade profiles characterized by a high degree of non-linearity in theproblem formulation. In spite of the very different formulations all the methods tend to givevalues for A and Z independent of z, corresponding to blades wh ich are uniform. Currentlythese optimal control representations are being further generalized to include the presence ofdam ping factors in the system dynam ics, resulting in complex differential equations. Theoptimization techniques which are presently undergoing program testing include second

variation methods [21] incorporating systems of Riccati-type inhomogeneous two pointmatrix boundary value problems in the presence of control parameters, and quasilinearizationm 1

1

2.

3.

4.5.

6.

7.

REFERENCES

B. M. E. DE SILVA and G N C GRANT 1975 International Journal of Numerical Methods inEngineering 9, 509-533. Optimal frequency-weight computations for a disc.G. CAPRIZ 1961 English Electric Com pany, Nelson Research Laboratory Report No. NsV 19 4.Vibrations of turbine blades, Part I: Single thick b lades.G. CAPRIZ 1961 English Electric Com pany, Nelson Research Laboratory Report No. NsV 200.Vibrations of turbine blades, Part VI: Forced tangential vibrations in blades with continuousshrouding.P S BRASS 1965 English Electric Com pany Report No. W/M(6B) 1022.Disc and blade vibrations.C. C. Fu 1974 International Journal of Numerical Methods in Engineering 8,569-588. Computeranalysis of rotating axial turbo machine bladein coupled bending-bending torsion vibration.W. CARNEGIE 1959 Journal of Mecha nical Engineering Science 1 235-240. Vibrations of rotatingcantilever blading: Theoretical approaches to the frequency problem based on energy methods.W . CARNEGIE 1964 Journal of Mecha nical Engineering Science 6, 105-109. Vibration s of pre-twisted cantilever blading allowing for rotary inertia and shear deflection.


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5 4 B. M. E. DE SILVA B. NEGUS AND J. WORSTER

8. J. D UNHAM 1974 American Society of Mecha nical Engineers Paper 74-GT-I 19.A parametricmethod of turbine blade profile design.

9. A. P. JONES and G. P. MCCORM ICK 1970SIAM Jo urnalqf Control 8,218-225. A generalizationof the method of Balakrishnan : inequality constraints and initial conditions.

10. F. A. LOOTSM A 1972 inNumerical methods for non-linear optimization (editor F. A. Lootsma),

3 13-34 7. A survey of methods for solving constrained minimization problems via unconstrainedminimization. New York : Academic Press.

11. F. A. LO~T SMA 1972Philips National Research Laborarory Report 4761.The Algol 60 procedureminifun for solving non-linear optimization problems.

12. B. L. PIERSON 1969Astronautica Acta 14, 157-1 69. A discrete variable approxim ation to optimalflight paths.

13. J. B. ROSEN 1966 SIAM Journal of Control 4, 223-244. Iterative solution of non-linear optimalcontrol problems.

14. C. STOREY and H. H. ROSENBROCK 1964 inCom puting m ethods in Optimization Problems(editors A. V. Balakrishnan and L. W. Neustadt), 23-64. O n the computation of the optimaltemperature profile in a tubular vessel. New York: Academic Press.

15. M. J. D. POW ELL 1964Computer Journal7,155-162. An efficient method for finding the minimumof a function of several variables without calculating derivatives.

16. R. C. WEAS-~and S. M. SELBY 1970Hand book of tablesfor mathematics. Cleveland: ChemicalRubber Company . See pp. 561-5 62.

17. R. FLETCH ER and M. J. D. PO WELL 1963Computer Journal 6, 163-1 68. A rapidly convergentdescent m ethod for minimization.

18. R. L. Fox 1971 Optimization M ethodsfor Engineering Design.Addison-Wesley. See pp. 124-1 63.19. R. S. MARTIN and J. H. WILKINSO N 1968Numerical Mathema tics 12, 349-368. Similarity reduc-

tion of a general matrix to Hessenberg form.20. R. S. MA RTIN. G. PETERS and J. H. WILKINSO N 1970Numerical Mathema tics 14, 219-231. The

QR-algorithm for real Hessenberg matrices.21. S. K. MITTER 1966Automatica 3. 135-144. Successive approximation methods for the solution

of optimal control problems.22. A. E. BRYSON JK. and Y. C. Ho 1969 Applied Optimal Control. Blaisdell Publishing Co. See

pp. 234-236.

journal of sound and vibration volume 46 issue 4 1976 [doi 10.1016%2f0022-460x%2876%2990676-3]...

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