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Design optimization of gas turbine blades with geometry and natural frequency constraints
CKkw7949/89 s3.00 + 0.00 Pergamon Press plc
Compurrrs & Swu~tures Vol. 32. No. I. pp. 113-l 17. 1989 Printed in Great Britain.
DESIGN OPTIMIZATION OF GAS TURBINE BLADES WITH GEOMETRY AND NATURAL FREQUENCY CONSTRAINTS* TSU-CHIEN CHEU,? Bo PING WANGI and TING-YU CHEN$ tTextron Lycoming, Stratford, Connecticut, U.S.A. IDepartment of Mechanical Engineering, The University of Texas at Arlington, Arlington, Texas, U.S.A. @epartment of Mechanical Engineering, National Chung Hsing University, Taichung, Taiwan, Republic of China (Received 18 March 1988)
Abstract-In this paper an automated procedure is presented to obtain the minimum weight design of gas turbine blades with geometry and multiple natural frequency constraints. The objective is achieved using a combined finite element-sequential linear programming, FEM-SLP technique. Thickness of selected finite elements are used as design variables. Geometric constraints are imposed on the thickness variations such that the optimal design has smooth aerodynamic shape. On the basis of the natural frequencies and mode shapes obtained from finite element analysis, an assumed mode reanalysis technique is used to provide the approximate derivatives of weight and constraints with respect to design variables for sequential linear programming. The results from SLP provide the initial design for the next FEM-SLP process. An example is presented to illustrate the interactive system developed for the optimization procedure.
NOTATION
element area blade cross sectional chord length natural frequency global stiffness matrix element stiffness matrix element stiffness matrix in modal coordinate stiffness matrix of modified system in modal coordinate global mass matrix element mass matrix element mass matrix in modal coordinate mass matrix of modified system in modal coordinate modal coordinate vector element thickness element kinetic energy element kinetic energy in modal coordinate maximum thickness of each blade cross section element strain energy element strain energy in modal coordinate total structural weight element structural weight element thickness ratio ratio of T,, to C eigenvalue matrix ith mode eigenvalue element mass density eigenvector matrix ith mode eigenvector Subscripts and superscripts i
j r
mode number element number or design variable number constraint lower limit
*This paper was presented at the 33rd Gas Turbine and Aeroengine Congress in Amsterdam, The Netherlands, 6-9 June 1988.
ND NF 0
l4 *
number of design variables number of frequency constraints original design constraint upper limit modal coordinate system
INTRODUCTION
In order to avoid resonances of natural frequencies and external excitation forces, structures must be designed properly to obtain desired distribution of natural frequencies. In this paper, an automated procedure is presented for the optimal design of gas turbine blades [l-5]. The objective of the procedure is to achieve the minimum weight design of gas turbine blades subject to geometry and multiple natural frequency constraints. The blades are modeled by thin shell elements. Thickness of selected elements are used as variables to control the optimization process. Thickness of each element may vary independently from others or the thickness of several elements may vary accordingly with the same scale to maintain the smoothness of structural contour shape. The optimal design problem is treated as one of the shape optimization and is solved by a combined finite element method and sequential linear programming [5-91, FEM-SLP technique. The general purpose finite element program MSC/NASTRAN is used to perform the free vibration model analysis. Based on the calculated eigenvalues and eigenvectors an assumed mode reanalysis technique [5, 10, 111 is used to perform approximate eigenvalue sensitivity analysis. Using the approximate eigenvalue sensitivities the SLP iterative procedure is then applied to 113
Tsu-CmtzNCXBUet al.
114
redesign the blades. After the SLP procedure converges finite element analysis is performed for the updated design to calculate eigensolutions and to check the validity of the optimized blades. The FEM-SLP procedure will be repeated until all design constraints are satisfied. FORMULATIONS
i=l,...,NF
(1)
(9)
Since the bending modes are mostly affected by the thickness variations, the stiffness and mass matrices [S] for the jth design variable can be written as K. = a3K”
and geometric side constraints U~~Uj$U~
(8)
Taking derivatives of eqn (7) then premuitiplying the results by 4’ and applying eqns (7a) and (7b) yields the eigenvalue derivatives shown in the reference [ 121. $=d:(+$)t#+.
The problem of the blade shape optimi~tion can be stated as minimization of the blade weight, W, subject to the frequency constraints
rl5.fY4.f:
1(I_51.5,_ 1:.
I
j=l,...,ND
(2)
with
1
I
M,=a,M,“.
(11)
Therefore, eqn (9) can be reduced to (3)
where NF is the number of frequency constraints, ND is the number of design variables,fl is the natural frequency of the ith mode, rj and xj are the thickness and the thickness magnification factor of element j, the superscript o denotes the state of original design and the suverscrivts f and u denote the lower and upper limits of the associated items. The weight of the blade is w=c
wjcL/
t4)
i where Wj=pjAjt;.
w
j==l,...,ND.
The equation of free vibration modeled by finite elements is
@I = W#+
7-;i=fAi4fM;t#+
(14)
Using the first order Taylor series expansion W and & of the current design can be approximated as
(5)
p and Aj are density and area of element j. Therefore, W is a linear combination the weight sensitivities with respect to constants
aw_ atL/- j
where Vi/ and TV are strain and kinetic energies associated with the ith mode and the jth design variable
of 4, aj are c6j
for the structure
(7)
where K and M are global stiffness and mass matrices, d, and 4, are eigenvalue and eigenvector of the ith natural mode of the structure. The eigenvectors are normalized such that
09 (7b) Since A,= (2@J2, the natural frequency constraints specified in eqn (1) can be replaced by
With eqns (6), (12), (15) and (16) the objective function Wand frequency constraints can be linearized to form a linear programming problem. It has been shown that assumed mode reanalysis formulation (10, 1I] can use results obtained from finite element analysis and LP problem to calculate approximate frequency sensitivities for another LP operation. Therefore, a sequence of LP, SLP, can be performed after the finite element analysis of an initial design. When the SLP converges finite element analysis will be applied to confirm the results and provide data for the follow up SLP procedure. The expensive free vibration analysis for the full structure model thus can be avoided for each LP procedure. Using an assumed mode reanalysis method the eignevalue problem solved for each LP step becomes Rq1= l&3qi
(17)
Design optimization of gas turbine bfadcs where
where @ is a truncated orthonormal modal matrix and A0 is the diagonal eigenvalue matrix of the original design and I is an unitary matrix. From the solution of the reduced ei~nproblem~ the eigenvalue derivatives of the updated system can be approximated as
(22) where
v; = iq’Ki*qi
(231
T; = $lq;M~qi.
(241
Equation (22) is used to calculate z%%,/&~ in the SLP problem until the iterations converge. The updated design then becomes the initial design of the next FEM-SLP procedure. SOFTWARE IMP~ME~A~~N
MSC~NAST~N is used for &rite etement analysis. A program, BLADEOPT IS], written in FORTRAN has been developed for the SLP procedure. The system flowchart is shown in Fig. 1. One
I15
input file contains data for both NASTRAN and BLADEOPTl. NASTRAN provides the structural weight of each finite element and modal analysis results for BLADEOPTf to formulate and solve the SLP problem. Users can run the whole system interactively to adjust the input data for BLADEOPTl and check the results at the end of NASTRAN or BLADEOPTl run. Therefore, constraint limits can be adjusted such that a feasible solution can be easily obtained for the SLP problem. As the designs improve from one FEM-SLP to another, constraints can be moved toward desired limits to obtain an optimal design.
A gas turbine blade is used as an example to illustrate the system developed, Young’s modulus, Poisson’s ratio and weight density of the blade are 1.5 x lO’lb./in.‘, 0.3 and 0.16924 lb./in3 respectively. The blade is modeled by 192 quadratic shell elements, NASTRAN CQUADg elements, with 633 nodes. The tinite element model is shown in Fig. 2. All the degrees of freedom along the hub line are Sxed as boundary conditions. Initially, the total weight of the blade is 0.0634 lb., and the first three natural frequencies of the blade are 1524 Hz, 2863 Hz, and 4439 Hz. The objective is to find a minimum weight design such that
A = 2800 Hz,
with upper and lower limits imposed on the design variables.
Fig. 2. Blade modeled by she11finite demerits.
116
TSU-CHIEN CHEU et al.
Table I. Distribution Section no. I
2 3 4 : 7 “9 10 II 12
Element
of maximum thickness and design variables
no.
Maximum initial
Thickness iinal
9 9 9 9 9 9 Q 9 9 9 IO 10
0.21308 0.20767 0.20058 0.19128 0.18038 0.16923 0.15586 0.13276 0.10443 0.08435 0.07501 0.07116
0.22800 0.22013 0.20585 0.18489 0.16415 0.15400 0.14105 0.11949 0.09959 0.08857 0.08101 0.06930
o : Initial Design + : final Dosign
Fiial 1.07002 1.06000
Y8
1.02627
z
0.96659 0.91002 0.91000 0.90498 0.9~05 0.95365 1.05003 1.08000 0.97386
0.01
0.06
0.02 0.00
l...,...,CP’,.~.,...I...,,..,.,,I...,.._) o
10
20 30 40
z
Fig. 3. Distribution of ~,,,/C Since critical speeds may occur at multiples of the blade rotating speed, according to Fig. 4, natural frequency constraints of this example are selected so as to increase the margin from critical speeds at 100% blade operational speed. In order to maintain smooth aerodynamic shape at each cross section chordwise of the blade, a single thickness magnification factor for each cross section is used as an independent design variable. Thickness variations in the longitudinal direction are less restrictive. However, these variations must be kept smooth and monotonic spanwise. One of the important parameters to define the blade aerodynamic shape is the distribution of the ratio
B=.in the longitudinal direction. This ratio is the maximum thickness, T,,, to the chord length, C, of each cross section chordwise. For an aerodynamically well-designed blade it is desired that the variations of p for a modified design do not exceed 10% of those of the initial design. Since chord iength of each cross section is fixed, the variations of T,,, or c( should be less than 10% of their initial values. Therefore, the geometric constraints of the design variables are o.9srjs
1.1
j = I,. . . , ND.
(24)
For the 192 finite element model shown in Fig. 2, there are 16 elements per cross section chordwise and
50
60
10
60
90
100
flow
in longitudinal direction,
12 sections spanwise. One design variable is used for each cross section, thus 16 elements in each cross section vary in the same scale. Initial values of the 12 design variables used for the mode are all one. The initial and final values of the maximum thickness at each cross section and final values of the design variables are shown in Table 1. The location of the maximum thickness is measured from the leading edge of each cross section and is presented as the element number in Table 1. Table 2 presents the iteration history of natural frequencies and blade weight. Data for design cycle number 0 are the analytical results for the original design. Data for cycle numbers 1, 2, and 3 are the results from the first, second and third FEM-SLP operations. Using IBM 3090 each FEM-SLP requires 157.8 CPU set of which 141.0 CPU set are for NASTRAN (FEM) and 16.8 CPU set are for BLADEOPTl (SLP). Total blade weight is reduced by 3.12%. All of the three frequency constraints are satisfied within 1% of design limits. The first IO modes are presented to illustrate the effect of the geometry and frequency constraints on the overall characteristics of the blade design. Figure 3 indicates that p of the optimal design is distributed spanwise as a smooth and monotonically decreasing function of air flow rate. Finally, a Camp bell diagram associated with the blade and its enviro~ent is presented in Fig. 4. Clearly, at engine full speed of 50,000 rev/min resonance of the natural frequencies and excitation forces are avoided with the new design.
Table 2. Iteration history of natural frequencies and weight Natural frequencies (Hz)
Design cycle no. 0 1 2 3
.f,
f2
Ji
f4
1524 1424 1410 1400
2863 2808 2800 2795
4439 4462 4449 4447
5023 4972 4965 4959
fJ z: 6412 6420
fa
57
fs
fp
fl0
6473 6457 6440 6458
8467 8306 8322 8393
9552 9503 9528 9627
10200 10218 10208 10231
11251 11230 11191
Weight Oh.)
0.06340 0.06118 0.06105 11184 0.06142
117
Design optimization af gas turbine blades a Initial
OIsign
finat Design -*-
100 2 Sp**d
7
to
20
30
Rotor Speed
40
50
- KRPM
Fig. 4. Campbeh diagram. CONCLUSION
The automated procedure presented in this paper is very effective to obtain the optimal design of gas turbine blades with geometry and natural frequency constraints. Since the use of geometry constraints and thickness magni~cation factors can keep the overall aerodynamic shape within an acceptable range, the number of design iterations between structural dynamics and aerodynamics can he greatly reduced. ltn the future aerodynamic performance optimization ought to be included in the automated procedure to further reduce the design effort. In order to reduce the effort of selecting frequency constraints to obtain feasible solutions from SLP procedure, adaptive constraint algorithms should be developed. With the adaptive constraint algorithms SLP may even be able to provide better designs for the next FEM-SLP procedure and further reduce the computation effort. At present shell elements are used to model thin blades. Further research is required for the optimal design of thick blades with other types of elements. RETRENCH 1. 0. M.
E. DeSiIva, Ii. Negus and 8. Worster. Penalty function type optimal control methods for the design of turbine blade profiles. NASA Report N76 11464 (1975). 2. B. M, E. DeSilva, G. N. C. Grant and B. L. Pierson, Conjugate gradient optimal control methods for the design ofturbine blades. NASA Report N76 1t463 (I975). 3. 3. M. E. DeSilva and G. N. C. Grant, Comparison of
some optimal control methods for the design of turbine blades. ASME Paper No. 77 Det. 43 (1987). J. P. Queau and P. Tompette, Optimal shape design of turbine and compressor blades. Proceedings of International Symposium on Optimum Strucrural Design, 19-22 Oct., Tucson, Arizona, pp. 5-9-S-14 (1981). B. P. Wang, T. C. Cheu and T. Y. Chen, Optimal design of compressor blades with multiple natural frequency constraints. Proceedings of A&WE Design Automation Conference, 27-30 Sept., Boston, Massachusetts (1987). P. Pederson, The integrated approach of FEM-SLP for solving problems of optimal design. In Optimization of DistributedParameter Structures (Edited by Hange and Cea), pp. 757-780. Sijthoff and Nordhoff (1981). 3. P. Wang, Synthesis of st~ctures with muItipie frequency constraints. AIAA Paper No. 09%CP. Proceedings of AIAA /ASMEIASCE/AHS 27th Structures, Structural Dynamicsand Materials Conference, Vol. 1, pp. 394-397 (1986). a. G. N. Vanderplaats, NumericalOptimizationTechniques for Engineering Design: With Applicatio,rs.McGrawHiif, New York (1984). 9. 0. C. Zienkiewicz and J. S. Campbell, Shape optimization and sequential Linear Programming. In Optimum Structural Design Theory and Applications(Edited by R, H. Gallagher and 0. C. Zienkiewicz). John Wiiey and Sons, New York (1973). 10, B. P. Wang, W. D. Pilkey and A. Palazzolq Reanalysis, modal synthesis and dynamic design. In State-Of-TheArt Survey on Finite Element Technology (Edited by Noor and Piikey), ASME (I9835 11. B. P. Wang, On Computing Eigensolution Sensitivity Data UsingFree vibrationSolutions.NASA Conference Publication 2457, National Aeronautics and Space Administration, pp. 233-246 (1987). 12. R. L. Fox and M. P. Kapoor, Rate of change of eigenvalues and eigenvectors. AK.&4 J. 6, 337-351 (1968).
Compurrrs & Swu~tures Vol. 32. No. I. pp. 113-l 17. 1989 Printed in Great Britain.
DESIGN OPTIMIZATION OF GAS TURBINE BLADES WITH GEOMETRY AND NATURAL FREQUENCY CONSTRAINTS* TSU-CHIEN CHEU,? Bo PING WANGI and TING-YU CHEN$ tTextron Lycoming, Stratford, Connecticut, U.S.A. IDepartment of Mechanical Engineering, The University of Texas at Arlington, Arlington, Texas, U.S.A. @epartment of Mechanical Engineering, National Chung Hsing University, Taichung, Taiwan, Republic of China (Received 18 March 1988)
Abstract-In this paper an automated procedure is presented to obtain the minimum weight design of gas turbine blades with geometry and multiple natural frequency constraints. The objective is achieved using a combined finite element-sequential linear programming, FEM-SLP technique. Thickness of selected finite elements are used as design variables. Geometric constraints are imposed on the thickness variations such that the optimal design has smooth aerodynamic shape. On the basis of the natural frequencies and mode shapes obtained from finite element analysis, an assumed mode reanalysis technique is used to provide the approximate derivatives of weight and constraints with respect to design variables for sequential linear programming. The results from SLP provide the initial design for the next FEM-SLP process. An example is presented to illustrate the interactive system developed for the optimization procedure.
NOTATION
element area blade cross sectional chord length natural frequency global stiffness matrix element stiffness matrix element stiffness matrix in modal coordinate stiffness matrix of modified system in modal coordinate global mass matrix element mass matrix element mass matrix in modal coordinate mass matrix of modified system in modal coordinate modal coordinate vector element thickness element kinetic energy element kinetic energy in modal coordinate maximum thickness of each blade cross section element strain energy element strain energy in modal coordinate total structural weight element structural weight element thickness ratio ratio of T,, to C eigenvalue matrix ith mode eigenvalue element mass density eigenvector matrix ith mode eigenvector Subscripts and superscripts i
j r
mode number element number or design variable number constraint lower limit
*This paper was presented at the 33rd Gas Turbine and Aeroengine Congress in Amsterdam, The Netherlands, 6-9 June 1988.
ND NF 0
l4 *
number of design variables number of frequency constraints original design constraint upper limit modal coordinate system
INTRODUCTION
In order to avoid resonances of natural frequencies and external excitation forces, structures must be designed properly to obtain desired distribution of natural frequencies. In this paper, an automated procedure is presented for the optimal design of gas turbine blades [l-5]. The objective of the procedure is to achieve the minimum weight design of gas turbine blades subject to geometry and multiple natural frequency constraints. The blades are modeled by thin shell elements. Thickness of selected elements are used as variables to control the optimization process. Thickness of each element may vary independently from others or the thickness of several elements may vary accordingly with the same scale to maintain the smoothness of structural contour shape. The optimal design problem is treated as one of the shape optimization and is solved by a combined finite element method and sequential linear programming [5-91, FEM-SLP technique. The general purpose finite element program MSC/NASTRAN is used to perform the free vibration model analysis. Based on the calculated eigenvalues and eigenvectors an assumed mode reanalysis technique [5, 10, 111 is used to perform approximate eigenvalue sensitivity analysis. Using the approximate eigenvalue sensitivities the SLP iterative procedure is then applied to 113
Tsu-CmtzNCXBUet al.
114
redesign the blades. After the SLP procedure converges finite element analysis is performed for the updated design to calculate eigensolutions and to check the validity of the optimized blades. The FEM-SLP procedure will be repeated until all design constraints are satisfied. FORMULATIONS
i=l,...,NF
(1)
(9)
Since the bending modes are mostly affected by the thickness variations, the stiffness and mass matrices [S] for the jth design variable can be written as K. = a3K”
and geometric side constraints U~~Uj$U~
(8)
Taking derivatives of eqn (7) then premuitiplying the results by 4’ and applying eqns (7a) and (7b) yields the eigenvalue derivatives shown in the reference [ 121. $=d:(+$)t#+.
The problem of the blade shape optimi~tion can be stated as minimization of the blade weight, W, subject to the frequency constraints
rl5.fY4.f:
1(I_51.5,_ 1:.
I
j=l,...,ND
(2)
with
1
I
M,=a,M,“.
(11)
Therefore, eqn (9) can be reduced to (3)
where NF is the number of frequency constraints, ND is the number of design variables,fl is the natural frequency of the ith mode, rj and xj are the thickness and the thickness magnification factor of element j, the superscript o denotes the state of original design and the suverscrivts f and u denote the lower and upper limits of the associated items. The weight of the blade is w=c
wjcL/
t4)
i where Wj=pjAjt;.
w
j==l,...,ND.
The equation of free vibration modeled by finite elements is
@I = W#+
7-;i=fAi4fM;t#+
(14)
Using the first order Taylor series expansion W and & of the current design can be approximated as
(5)
p and Aj are density and area of element j. Therefore, W is a linear combination the weight sensitivities with respect to constants
aw_ atL/- j
where Vi/ and TV are strain and kinetic energies associated with the ith mode and the jth design variable
of 4, aj are c6j
for the structure
(7)
where K and M are global stiffness and mass matrices, d, and 4, are eigenvalue and eigenvector of the ith natural mode of the structure. The eigenvectors are normalized such that
09 (7b) Since A,= (2@J2, the natural frequency constraints specified in eqn (1) can be replaced by
With eqns (6), (12), (15) and (16) the objective function Wand frequency constraints can be linearized to form a linear programming problem. It has been shown that assumed mode reanalysis formulation (10, 1I] can use results obtained from finite element analysis and LP problem to calculate approximate frequency sensitivities for another LP operation. Therefore, a sequence of LP, SLP, can be performed after the finite element analysis of an initial design. When the SLP converges finite element analysis will be applied to confirm the results and provide data for the follow up SLP procedure. The expensive free vibration analysis for the full structure model thus can be avoided for each LP procedure. Using an assumed mode reanalysis method the eignevalue problem solved for each LP step becomes Rq1= l&3qi
(17)
Design optimization of gas turbine bfadcs where
where @ is a truncated orthonormal modal matrix and A0 is the diagonal eigenvalue matrix of the original design and I is an unitary matrix. From the solution of the reduced ei~nproblem~ the eigenvalue derivatives of the updated system can be approximated as
(22) where
v; = iq’Ki*qi
(231
T; = $lq;M~qi.
(241
Equation (22) is used to calculate z%%,/&~ in the SLP problem until the iterations converge. The updated design then becomes the initial design of the next FEM-SLP procedure. SOFTWARE IMP~ME~A~~N
MSC~NAST~N is used for &rite etement analysis. A program, BLADEOPT IS], written in FORTRAN has been developed for the SLP procedure. The system flowchart is shown in Fig. 1. One
I15
input file contains data for both NASTRAN and BLADEOPTl. NASTRAN provides the structural weight of each finite element and modal analysis results for BLADEOPTf to formulate and solve the SLP problem. Users can run the whole system interactively to adjust the input data for BLADEOPTl and check the results at the end of NASTRAN or BLADEOPTl run. Therefore, constraint limits can be adjusted such that a feasible solution can be easily obtained for the SLP problem. As the designs improve from one FEM-SLP to another, constraints can be moved toward desired limits to obtain an optimal design.
A gas turbine blade is used as an example to illustrate the system developed, Young’s modulus, Poisson’s ratio and weight density of the blade are 1.5 x lO’lb./in.‘, 0.3 and 0.16924 lb./in3 respectively. The blade is modeled by 192 quadratic shell elements, NASTRAN CQUADg elements, with 633 nodes. The tinite element model is shown in Fig. 2. All the degrees of freedom along the hub line are Sxed as boundary conditions. Initially, the total weight of the blade is 0.0634 lb., and the first three natural frequencies of the blade are 1524 Hz, 2863 Hz, and 4439 Hz. The objective is to find a minimum weight design such that
A = 2800 Hz,
with upper and lower limits imposed on the design variables.
Fig. 2. Blade modeled by she11finite demerits.
116
TSU-CHIEN CHEU et al.
Table I. Distribution Section no. I
2 3 4 : 7 “9 10 II 12
Element
of maximum thickness and design variables
no.
Maximum initial
Thickness iinal
9 9 9 9 9 9 Q 9 9 9 IO 10
0.21308 0.20767 0.20058 0.19128 0.18038 0.16923 0.15586 0.13276 0.10443 0.08435 0.07501 0.07116
0.22800 0.22013 0.20585 0.18489 0.16415 0.15400 0.14105 0.11949 0.09959 0.08857 0.08101 0.06930
o : Initial Design + : final Dosign
Fiial 1.07002 1.06000
Y8
1.02627
z
0.96659 0.91002 0.91000 0.90498 0.9~05 0.95365 1.05003 1.08000 0.97386
0.01
0.06
0.02 0.00
l...,...,CP’,.~.,...I...,,..,.,,I...,.._) o
10
20 30 40
z
Fig. 3. Distribution of ~,,,/C Since critical speeds may occur at multiples of the blade rotating speed, according to Fig. 4, natural frequency constraints of this example are selected so as to increase the margin from critical speeds at 100% blade operational speed. In order to maintain smooth aerodynamic shape at each cross section chordwise of the blade, a single thickness magnification factor for each cross section is used as an independent design variable. Thickness variations in the longitudinal direction are less restrictive. However, these variations must be kept smooth and monotonic spanwise. One of the important parameters to define the blade aerodynamic shape is the distribution of the ratio
B=.in the longitudinal direction. This ratio is the maximum thickness, T,,, to the chord length, C, of each cross section chordwise. For an aerodynamically well-designed blade it is desired that the variations of p for a modified design do not exceed 10% of those of the initial design. Since chord iength of each cross section is fixed, the variations of T,,, or c( should be less than 10% of their initial values. Therefore, the geometric constraints of the design variables are o.9srjs
1.1
j = I,. . . , ND.
(24)
For the 192 finite element model shown in Fig. 2, there are 16 elements per cross section chordwise and
50
60
10
60
90
100
flow
in longitudinal direction,
12 sections spanwise. One design variable is used for each cross section, thus 16 elements in each cross section vary in the same scale. Initial values of the 12 design variables used for the mode are all one. The initial and final values of the maximum thickness at each cross section and final values of the design variables are shown in Table 1. The location of the maximum thickness is measured from the leading edge of each cross section and is presented as the element number in Table 1. Table 2 presents the iteration history of natural frequencies and blade weight. Data for design cycle number 0 are the analytical results for the original design. Data for cycle numbers 1, 2, and 3 are the results from the first, second and third FEM-SLP operations. Using IBM 3090 each FEM-SLP requires 157.8 CPU set of which 141.0 CPU set are for NASTRAN (FEM) and 16.8 CPU set are for BLADEOPTl (SLP). Total blade weight is reduced by 3.12%. All of the three frequency constraints are satisfied within 1% of design limits. The first IO modes are presented to illustrate the effect of the geometry and frequency constraints on the overall characteristics of the blade design. Figure 3 indicates that p of the optimal design is distributed spanwise as a smooth and monotonically decreasing function of air flow rate. Finally, a Camp bell diagram associated with the blade and its enviro~ent is presented in Fig. 4. Clearly, at engine full speed of 50,000 rev/min resonance of the natural frequencies and excitation forces are avoided with the new design.
Table 2. Iteration history of natural frequencies and weight Natural frequencies (Hz)
Design cycle no. 0 1 2 3
.f,
f2
Ji
f4
1524 1424 1410 1400
2863 2808 2800 2795
4439 4462 4449 4447
5023 4972 4965 4959
fJ z: 6412 6420
fa
57
fs
fp
fl0
6473 6457 6440 6458
8467 8306 8322 8393
9552 9503 9528 9627
10200 10218 10208 10231
11251 11230 11191
Weight Oh.)
0.06340 0.06118 0.06105 11184 0.06142
117
Design optimization af gas turbine blades a Initial
OIsign
finat Design -*-
100 2 Sp**d
7
to
20
30
Rotor Speed
40
50
- KRPM
Fig. 4. Campbeh diagram. CONCLUSION
The automated procedure presented in this paper is very effective to obtain the optimal design of gas turbine blades with geometry and natural frequency constraints. Since the use of geometry constraints and thickness magni~cation factors can keep the overall aerodynamic shape within an acceptable range, the number of design iterations between structural dynamics and aerodynamics can he greatly reduced. ltn the future aerodynamic performance optimization ought to be included in the automated procedure to further reduce the design effort. In order to reduce the effort of selecting frequency constraints to obtain feasible solutions from SLP procedure, adaptive constraint algorithms should be developed. With the adaptive constraint algorithms SLP may even be able to provide better designs for the next FEM-SLP procedure and further reduce the computation effort. At present shell elements are used to model thin blades. Further research is required for the optimal design of thick blades with other types of elements. RETRENCH 1. 0. M.
E. DeSiIva, Ii. Negus and 8. Worster. Penalty function type optimal control methods for the design of turbine blade profiles. NASA Report N76 11464 (1975). 2. B. M, E. DeSilva, G. N. C. Grant and B. L. Pierson, Conjugate gradient optimal control methods for the design ofturbine blades. NASA Report N76 1t463 (I975). 3. 3. M. E. DeSilva and G. N. C. Grant, Comparison of
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