[American Institute of Aeronautics and Astronautics 48th AIAA/ASME/ASCE/AHS/ASC Structures,...

American Institute of Aeronautics and Astronautics

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Robust Design Optimization for Multiple Responses Using Response Surface Methodology and Taguchi Approach:

Solid Rocket Motor Application

Bayindir Kuran*, M. Sinan Hasanoglu† and Kenan Bozkaya‡

The Scientific & Technological Research Council of Turkey - Defence Industries Research & Development Institute TUBITAK-SAGE, P.K. 16 Mamak, 06261 Ankara, Turkey

Robust design methodology is concerned with the process and the product design while

minimizing the effects of uncertainties in design and external noises. Designers are often required to explore the design space to predict the behavior of a multi-disciplinary system and to select optimum design(s). Selection of optimum design(s) may become very cumbersome (if not impossible) if the system has conflictive multi-objectives. In robust design, optimum system design is sought with special emphasis on the insensitiveness of objective functions due to uncertainties. This study presents the robust design of a solid rocket motor having multiple ballistic performance responses. Aim is to minimize the mean and the variation of the chamber pressure of the motor. Three approaches are compared. In the first approach, the means system objective and constraint functions along with the variation of the chamber pressure are approximated by response surface models. Control factors for the optimum design are sought by using Desirability Function approach. In the second approach, Signal to Noise (S/N) Ratios for the objective and constraint functions are approximated by response surface functions and maximized. Representative optimal solutions for the first and the second approaches are compared. In the third approach, S/N Ratio for the chamber pressure is maximized subject to probabilistic constraints. Probability of failure of the solid rocket motor is also calculated at the selected optimum states along with the confidence limits through Monte Carlo simulations.

Nomenclature aa = Arming Acceleration aL = Acceleration at the Launch Exit amax = Maximum Acceleration of the Rocket bi = Main Effect Coefficient in Response Surface Model COVP = Coefficient of Variation of the Chamber Pressure di = Desirability for the ith Performance Function dP = Desirability for the Chamber Pressure D = Desirability Function g(x) = Limit State Function g*(x) = Approximated Performance Function gmax = Upper limit for the performance function gmin = Lower limit for the performance function It = Total Impulse k = Number of Design variables in Response Surface Model l = Number of Experiments in Outer Array

* Chief Research Engineer, Systems and Specialty Engineering Division, [email protected]. † Research Engineer, Systems and Specialty Engineering Division, [email protected]. ‡ Senior Research Engineer, Systems and Specialty Engineering Division, [email protected].

48th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference<br> 15th23 - 26 April 2007, Honolulu, Hawaii

AIAA 2007-1977

Copyright © 2007 by Kuran, B., Hasanoglu, S., Bozkaya, K. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.


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P = Motor Pressure ( )•P = Probability

p = Number of Coefficients in Response Surface Model Rg = Radius of Grain S/N = Signal to Noise Ratio s, t = Weighting Factors for Desirability Functions x = Generic Design Variable ∆ = Variance Contribution due to Noise Variables µ = Mean Value σ = Standard Deviation

1gσ = Standard Deviation of the ith Performance Function z = Generic Noise Variable

I. Introduction Robust design methodology is used to optimize a system’s performance and to improve process such that the

outputs of the system or the process are immune to noises. The main objective is to set the system parameters (i.e. control variables) so that the system operates well over a wide range even the system parameters and the process variables have large variations. Robust design methodology has been originally developed by Taguchi to improve the quality of products6-7. Taguchi’s strategy is to make the system insensitive to variations rather than controlling the variations which is often more costly. Taguchi’s approach groups robust design problems in three distinct categories. These are larger-the-better, nominal-the better and smaller-the better according to their objectives. The optimal settings for the system/process parameters are found by maximizing the signal to noise (S/N) ratios. Optimum parameter settings are determined by using main effect plots and analysis of variance (ANOVA) tables. Block diagram for a product/process is given in Fig. (1) in which input parameters are classified as design variables (control factors) and noise factors. Control factors are determined by the designer to achieve optimum objectives. Noise factors are the parameters that are either inherently uncontrollable or impractical to control.

Figure 1. Block diagram for a product/process

Another approach in robust design is to model the response in terms of the control and noise factors. The main

objective is to select the control factors such that mean of the response is moved to target (or maximized, or minimized) and the variation of the response is minimized. Chen et al implemented Central Composite Design (CCD) for the experiment design and constructed second order response surface for the use of optimization8.

In many engineering problems designers are faced with optimization of multiple responses which may be conflicting in nature. One way of dealing with the multiple responses is to use the weighted sum with some drawbacks, which have been discussed by Das, I. And Dennis, J. E9. One may find various formulations for multi-objective robust design problems along with their solution approaches such as physical programming and genetic algorithms10-16. A detailed and well documented overview on the robust design methodology is given by Park et al17.

Optimization of multiple responses can be performed by building a composite response from individual responses. This composite response function is called as the desirability function3, 5. Desirability function expression is given by Eqn. (1) in which di and ri are the individual desirability function and importance factor for the ith response, respectively.


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⎟⎟⎠

⎞⎜⎜⎝

⎛

=

∑⎟⎟⎠

⎞⎜⎜⎝

⎛=

=∏n

i

rin

i

riidD 1

1

1

. (1)

The desirability function for response to be maximized is

s

ii gg

ggd ⎟⎟⎠

⎞⎜⎜⎝

⎛−−

=minmax

min , (2)

and the desirability function for response to be minimized is

s

ii gg

ggd ⎟⎟⎠

⎞⎜⎜⎝

⎛−−

=minmax

max . (3)

II. Robust Design for Solid Propellant Rocket Motor In this paper, robust design optimization of ballistic performance for a solid rocket motor is discussed. The

motor is designed against four limit state functions while keeping the maximum chamber pressure minimum. Ballistic performance of a solid propellant rocket motor is dependent on the propellant properties (propellant density, burning rate constant and enthalpy of combustion), grain geometry (burning area) and nozzle geometry (throat area and exit area). Limit state functions used to determine ballistic performance are given as follows1-2: i. Total impulse, It, of the motor shall be grater than 7300 Ns. 73001 −= tIg , (4) ii. Arming acceleration for fuze: aa>27 g, for a duration of at least 0.65 s. ( ) 65.0272 −>∆= gatg a (5) iii. Acceleration of the rocket at launcher exit shall be grater than 35g. gag L 353 −= (6) iv. Maximum acceleration of rocket shall be less than 105 g. max4 105 ag −= (7) Ballistic performance calculations are carried out by a code by which 1-dimensional steady conservation equations of mass, momentum and energy are solved in order to find the pressure, density and velocity distributions along the rocket motor case. Flow assumed to be in-viscid; therefore viscous fluxes were not modeled. The integral form of conservation equations with four different types of boundary conditions completely describes the flow inside the rocket motor. These boundary conditions are inert walls, regressing solid propellant grain surfaces, nozzle inlet and finally cell interface surfaces. Details of ballistic calculations can be found in study performed by Bozkaya et. al.1. Limit states for ballistic performance of the motor are approximated by response surface models3.

∑∑∑∑<==

+++=ji

jiij

k

iii

k

iii xxdxcxbaxg

1

2

1

* )( (8)


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Response surface methodology is a set of techniques in which approximate relations between the input variables

and the responses of a system are found. Response surface models are constructed by running a set of experiments (in the laboratory or by means of computer simulations) at prescribed combinations of the input variables. Number of experiments is governed by the order of the surfaces constructed and the type of the experiment design (such as the Box-Behnken design or the Central Composite design). In this study, quadratic response surface models are formed by employing the Central Composite design to generate solutions4. 143 runs ( 122 ++ kk ) are performed to establish response surfaces in terms of design variables. Ballistic performance functions are governed by 21 variables (propellant properties, environmental conditions such as ambient temperature, propellant grain geometry, nozzle geometric parameters such as nozzle throat radius and nozzle exit radius). Effective parameters are determined by means of screening runs and significant parameters are reduced to 9. List of the design variables (control factors) and noise variables which are effective on the ballistic performance of a solid propellant motor with a star-shaped grain are given in Table 1. Geometric variables for the star-shaped grain are illustrated in Fig. (2).

Table 1. Design Variables and Noise Variables

# Variable Name

Variable Type

Distribution Type

1 Enthalpy of Propellant Noise Normal 2 Burn Rate Constant Design Normal 3 Density of Propellant Noise Normal 4 Throat Radius Design Normal 5 Grain Length Design Normal 6 Radius of Grain, Rg Design Normal 7 Grain Geometry Parameter, X1 Design Normal 8 Grain Geometry Parameter, X2 Design Normal 9 Number of Arms Design -

Figure 2. Geometric variables for star grain

Figure 3. Noise levels ( σµσµ 3,3 +− )

In this study, three approaches are proposed which are explained briefly as follows: Approach 1: 1.1. Perform experiments by employing Central Composite design with 7 control factors. L12 orthogonal array is

selected as the outer array which is the smallest two-level orthogonal array, (see Fig. (3)) so that 12 experiments are


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performed to develop outer array by considering variations in the control factors along with the noise variables (in total 8 variables). Cross product matrix is given in Fig.(4).

1.2. Determine the mean and the coefficient of variation for the Chamber Pressure, P, pCOV 1.3. Determine the mean values for ig performance functions: tI , ( )gat a 27>∆ , La and maxa . 1.4. Develop response surface models for means and ( )2log PCOV . 1.5. Maximize mean values of tI , ( )gat a 27>∆ and La .

Minimize mean value of maxa

Minimize Pµ

Minimize ( )2log PCOV Subject to UL xxx rvr

≤≤ 1.6. Optimization problem is formulated by defining a desirability function

( )⎟⎟⎠

⎞⎜⎜⎝

⎛∑=

++

×××××=

4

1

1

44

33

22

11

irirCOVprp

rCOVpCOVP

rpP

rrrr ddddddD (9) Approach 2: 2.1. Perform experiments by employing Central Composite design with 7 control factors. L12 orthogonal array is

selected as the outer array (the smallest two-level orthogonal array) so that 12 experiments are performed to develop outer array by considering variations in the control factors along with the noise variables (in total 8 variables).

2.2. Determine S/N ratio for Chamber Pressure, P (smaller the better)

( ) ⎥⎦

⎤⎢⎣

⎡−= ∑

=

l

jjP P

lNS

1

21log10/ (10)

2.3. Determine S/N ratios for performance functions

( )⎥⎥⎦

⎤

⎢⎢⎣

⎡−= ∑

=

l

j tjIlNS

121

11log10/ , ( )⎥⎥⎦

⎤

⎢⎢⎣

⎡

∆−= ∑

=

l

j jtlNS

122

11log10/ , ( )⎥⎥⎦

⎤

⎢⎢⎣

⎡−= ∑

=

l

j LjalNS

123

11log10/ ,

( ) ⎥⎦

⎤⎢⎣

⎡−= ∑

=

l

jja

lNS

1

2max4

1log10/ (11)

2.4. Develop response surface models for (S/N)P and (S/N)i, i=1-4. 2.5. Maximize (S/N)P Maximize (S/N)i


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Subject to UL xxx rvr

≤≤ 2.6. Optimization problem is formulated by defining a desirability function

Approach 3: 3.1. Same as 2.1. 3.2. Same as 2.2. 3.3. Same as 1.3. 3.4. Develop response surface models. 3.5. Maximize (S/N)P Subject to 11 )0( RgP >>

22 )0( RgP >>

33 )0( RgP >>

44 )0( RgP >>

UL xxx rvr≤≤

where, the specified reliability levels are ( )11 15.4 ∆+Φ=R , ( )22 15.4 ∆+Φ=R ,

( )33 15.4 ∆+Φ=R , ( )44 15.4 ∆+Φ=R . i∆ is the contribution of the variances for the ith response due to variances of noises variables.

Figure 4. Inner and outer arrays

III. Numerical Results Response Surface Fittings: Range for the design variables in terms of the baseline design are given in Table 2. Quadratic response surface models for ballistic performance functions are established by employing 143 runs. Statistical results for the response surface fitting are shown in Table 3.


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Table 2. Range for the Design Variables

Variable Variable Name

Distribution Type

Range

X1 Grain Length Normal 000 12.1118.0 XXX ≤≤

X2 Burn Rate Constant

Normal 000 21.12285.0 XXX ≤≤

X3 Radius of Grain

Normal 000 31.1338.0 XXX ≤≤

X4 Number of Arms

- 5-6-7

X5 Throat Radius Normal 000 51.1559.0 XXX ≤≤

X6 Grain Geometry Parameter, 1

Normal 000 61.1669.0 XXX ≤≤

X7 Grain Geometry Parameter, 2

Normal 000 72.1778.0 XXX ≤≤

Table 3. Statistics for Response Surface Fitting

Response R2adj(%) Mean Square Error Mean of Response

Total Impulse 100 738 6577 Max. Acceleration 95.4 31.61 94.2 Arming Acc. Duration 96.5 0.003312 1.14 Launch Acceleration 98.9 5.12 51.5 Maximum Pressure 92.4 79.2 110.6 Log(Cov-Pressure2) 91.0 0.001455 -2.08 S/N-Total Impulse 100 0.0031 75.9 S/N-Max. Acceleration 96.6 0.1961 -392 S/N-Launch Acceleration 99.9 0.007 33.4 S/N-Arm. Acc. Duration 97.1 0.1836 0.735 S/N-Maximum Pressure 96.7 0.2081 -40.6

Figure 5. Correlation for total impulse


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Figure 6. Correlation for max. acc.

Figure 7. Correlation for arm. acc. duration

Figure 8. Correlation for launch acc.

Figure 9. Correlation for max. pressure

S/N-TOTAL IMPULSE

71

72

73

74

75

76

77

78

79

80

70 72 74 76 78 80

PREDICTED

AC

TUA

L

Figure 10. Correlation for S/N-total impulse

S/N-MAXIMUM ACCELERATION

-44

-42

-40

-38

-36

-34

-32

-30-45 -43 -41 -39 -37 -35 -33

PREDICTED

AC

TUA

L

Figure 11. Correlation for S/N-max. acc.

S/N-ARMING ACCELERATION DURATION

-4

-3

-2

-1

0

1

2

3

4

5

-4 -3 -2 -1 0 1 2 3 4 5

PREDICTED

AC

TUA

L

Figure 12. Correlation for S/N-arming

acceleration duration

S/N-LAUNCH ACCELERATION

25

27

29

31

33

35

37

39

41

43

25 27 29 31 33 35 37 39 41 43

PREDICTED

AC

TUA

L

Figure 13. Correlation for S/N-launch acc.


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S/N-MAXIMUM PRESSURE

-49

-47

-45

-43

-41

-39

-37

-35

-33-49 -47 -45 -43 -41 -39 -37 -35 -33

PREDICTED

AC

TUA

L

Figure 14. Correlation for S/N-max. pressure

Multi-Response Optimization: In the first and the second approaches the desirability function technique is employed. In this technique, lower

and upper limits for the performance functions have to be chosen. Lower and upper limits are given Table 4. For performance functions to be maximized, upper (target) limit is found by adding the mean value with ∆+15.4 σ of the corresponding response. For performance functions to be minimized, lower (target) limit is found by subtracting the ∆+15.4 σ value from the mean value of the corresponding response. ∆ is the contribution to variance of response due to noise variables’ variances. ∆ values for individual responses are found in the study carried out by Bozkaya et. al2. ∆ values for the total impulse, maximum acceleration, arming acceleration duration, launch acceleration and maximum chamber pressure are 0.175, 0.034, 0.001, 0.031, 0.0002, respectively. Optimum solutions on the limit lines of the corresponding objective functions are selected for further analyses (given in Table 5). Optimal solutions (composite desirability function is unity) for the first and the second approaches are illustrated in Figs (15) through (23).

Table 4. Lower and Upper Limits for Desirability Function Approach

Response Max/Min Lower Limit Upper Limit Total Impulse Maximize 7300 8300 Max. Acceleration Minimize 84 105 Arming Acc. Duration Maximize 0.65 0.806 Launch Acceleration Maximize 35 40.25 Maximum Pressure Minimize 94 115 Log(Cov-Pressure2) Minimize -2.2 -2.1 S/N-Total Impulse Maximize 77 77.23 S/N-Max. Acceleration Maximize -41 -40.84 S/N-Launch Acceleration Maximize 30 30.85 S/N-Arm. Acc. Duration Maximize -4 -3.77 S/N-Maximum Pressure Maximize -42 -41.22

Table 5. Selected Optimum Solutions for the First and the Second Approaches

RM BVA R N THRAD L1 L2 F-1 0,7071 -1,0000 0,6818 1,0000 1,0000 0,2525 1,0000 F-2 0,7071 -1,0000 0,5303 1,0000 1,0000 0,7828 1,0000 F-3 0,7071 -1,0000 0,5303 1,0000 1,0000 1,0000 0,8333 F-4 0,7071 -1,0000 0,7071 1,0000 1,0000 1,0000 1,0000

First

Approach

F-5 0,7071 -1,0000 0,7071 1,0000 1,0000 0,7576 1,0000 S-1 0,8081 -1,0000 -0,0326 -1,0000 0,7317 -0,7558 -0,3908 S-2 0,8838 -0,9848 1,0000 0,0000 0,7317 -0,7558 -0,3908 S-3 1,0000 -0,9848 0,1263 0,0000 0,0758 -0,7558 -0,3908

Second

Approach

S-4 1,0000 -0,9848 0,1263 1,0000 0,0758 -0,3030 0,1010


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8300 8400 8500 8600 8700 8800 8900 9000 9100

-2.23

-2.225

-2.22

-2.215

-2.21

-2.205

-2.2

-2.195

TOTAL IMPULSE (N-s)

CO

V-P

RE

SS

UR

E

MIN. COV-PRESSURE

MAX.TOTAL IMPULSE

Figure 15. Solutions for the first approach

(Cov-pressure vs. total impulse)

85 86 87 88 89 90 91 92 93

-2.23

-2.225

-2.22

-2.215

-2.21

-2.205

-2.2

-2.195

MAXIMUM ACCELERATION (g)

CO

V-P

RE

SS

UR

E

MIN.MAXIMUM ACC.

MIN. COV-PRESSURE

Figure 16. Solutions for the first approach (Cov-pressure vs. maximum acceleration)

1.46 1.48 1.5 1.52 1.54 1.56 1.58 1.6 1.62 1.64

-2.23

-2.225

-2.22

-2.215

-2.21

-2.205

-2.2

-2.195

ARMING ACCELERATION DURATION (s)

CO

V-P

RE

SS

UR

E

MIN. COV-PRESSURE

MAX. ARM. ACC. DUR.


(Cov-pressure vs. arming acceleration duration)

40 42 44 46 48 50 52

-2.23

-2.225

-2.22

-2.215

-2.21

-2.205

-2.2

-2.195

LAUNCH ACCELERATION (g)

CO

V-P

RE

SS

UR

E

MIN. COV-PRESSURE

MAX. LAUNCHACC.


(Cov-pressure vs. launch acceleration)

94 94.5 95 95.5 96 96.5 97 97.5 98

-2.23

-2.225

-2.22

-2.215

-2.21

-2.205

-2.2

-2.195

PRESSURE (bar)

CO

V-P

RE

SS

UR

E

MIN. PRESSUREMIN.

COV-PRESSURE

Figure 19. Solutions for the first approach (Cov-pressure vs. maximum pressure)


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77.5 78 78.5 79 79.5 80 80.5 81 81.5-41.5

-41

-40.5

-40

-39.5

-39

-38.5

-38

S/N-TOTAL IMPULSE

S/N

-PR

ES

SU

RE

MAX. S/N-PRESSURE

MAX.S/N-TOTAL IMP.

Figure 20. Solutions for the second approach

(S/N-pressure vs. S/N total impulse)

-41 -40.5 -40 -39.5 -39 -38.5 -38-41.5

-41

-40.5

-40

-39.5

-39

-38.5

-38

S/N-MAXIMUM ACCELERATION

S/N

-PR

ES

SU

RE

MAX. S/N-PRESSURE

MAX. S/N-MAX. ACC.

Figure 21. Solutions for the second approach

(S/N-pressure vs. S/N maximum acceleration)

1.5 2 2.5 3 3.5 4 4.5 5 5.5 6-41.5

-41

-40.5

-40

-39.5

-39

-38.5

-38

S/N-ARMING ACCELERATION DURATION

S/N

-PR

ES

SU

RE

MAX. S/N-PRESSURE

MAX. S/N-ARM. ACC. DUR.

Figure 22. Solutions for the second

approach (S/N-pressure vs. S/N arming acceleration duration)

30.5 31 31.5 32 32.5 33 33.5 34 34.5 35 35.5-41.5

-41

-40.5

-40

-39.5

-39

-38.5

-38

S/N-LAUNCH ACCELERATION

S/N

-PR

ES

SU

RE

MAX.S/N-PRESSURE

MAX. S/N-LAUNCH ACC.

Figure 23. Solutions for the second

approach (S/N-pressure vs. S/N launch acceleration)


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Figure 24. Optimum solution F-4

Figure 25. Optimum solution S-4

The third approach is robust & reliability based design optimization technique. In this study, the Performance Measure Approach (PMA) and the Sequential Optimization and Reliability Assessment (SORA) method is utilized. The SORA method is developed by Du and Chen18. They have demonstrated the effectiveness of the method on two numerical examples: the reliability based design for vehicle crashworthiness of side impact and the integrated reliability and robust design for the speed reducer of a small aircraft engine. In this method, optimization and reliability assessment are decoupled from each other. In other words, reliability assessment is conducted only after the optimization. The proposed method shifts the boundaries of violated deterministic constraints to the feasible


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direction based on the reliability information obtained in the previous iteration step. The flowchart of the SORA method is shown in Figure 26.

Figure 26. Flowchart of SORA Method

Optimum solution is reached after 5 iteration loop. RBDO yields an optimum solution for S/N Ratio for the

chamber pressure as -40.339. Optimum settings for the design variables are x*=(0.542, -1, 0.986, 0, 1, 1,-0.351). The second (arming acceleration duration) and the fourth (maximum acceleration) probabilistic constraints are found to be inactive.

Reliability Predictions: Reliability of the solid rocket motor at the optimum settings is calculated by establishing response surfaces

around each optimum setting and through Monte Carlo simulations. 10,000,000 MCS are performed. ¼ fractional CCD is utilized to construct the response surfaces with 81 runs. R2-adj values are presented in Table 6 and the probabilities of failures are given in Table 7. Upper and lower confidence limits (95 %) for probability of failure values are found by utilizing t-statistics and Mean Square Errors for individual responses. Determination of confidence intervals for reliabilities is presented in the study performed by Vittal and Hajela19.

Table 6. Statistics (R2-adj.,%) of Response Surface Fits Around Optimum Solutions

Optimum Solution

Total Impulse

Max. Acceleration

Arm. Acc. Duration

Launch Acceleration

Maximum Pressure

F-1 99.999 99.997 99.920 100 100 F-2 99.999 99.997 99.969 100 100 F-3 99.999 99.996 99.968 100 99.999 F-4 99.999 99.998 99.965 100 99.999 F-5 99.999 99.997 99.968 100 100 S-1 86.250 85.233 83.465 99.994 99.975 S-2 99.999 99.998 99.967 100 100 S-3 99.999 99.999 99.965 100 99.997 S-4 99.999 99.997 99.966 100 99.866 T-1 99.999 99.998 99.968 100 99.404


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Table 7. Reliability Values for Optimum Solutions Optimum Solution

Pf

F-1 23E-7 F-2 49E-7 F-3 36E-7 F-4 0/10,000,000 F-5 (Pf)L=0.0057575, (Pf)M=0.0058167, (Pf)U=0.0058882

S-1 (Pf)L=0.1027024, (Pf)M=0.1049344, (Pf)U=0.1073066 S-2 (Pf)L=0.3858049, (Pf)M=0.4302860, (Pf)U=0.6810981 S-3 0/10,000,000 S-4 0/10,000,000 T-1 0/10,000,000

Mean values of the performance functions along with the standard deviation of the maximum chamber pressure

are given in Table 8.

Table 8. Performance Functions at the Optimum Settings First App.(F-4) Second App. (S-4) Third App.

Mean of Total Impulse (N-s) 8448 8001 7891 Mean of Maximum

Acceleration (g) 89.03 87.96 81.72

Mean of Arming Acceleration Duration (s)

1.423 1.348 1.388

Mean of Launch Acceleration (g)

50.64 57.99 53.57

Mean of Maximum Chamber Pressure (bar)

94.31 98.86 91.32

Standard Deviation of Chamber pressure

4.267 5.220 4.391

IV.Conclusions In this study, robust design optimization of a solid rocket motor is performed using the response surface methodology to establish the surrogate model. Three approaches are employed to obtain the optimized solutions. In the first approach, mean and the variation of the maximum chamber pressure is maximized simultaneously subject to 4 limit states. In the second approach, S/N ratios for the chamber pressure and the limit state functions are formed by means of Taguchi’s approach and response surface models are fitted to S/N ratios instead of mean values. In the first and the second approaches, the desirability function approach is utilized to obtain optimum solutions. In the third approach, robust & reliability based design optimization is performed such that each limit state reliability goal is set to 0.9999966 to achieve 4.5σ quality level. The following conclusions and suggestions for future work are noted:

• Accuracy of response surface models for the mean of the maximum pressure, the mean of the chamber pressure and the variation of the chamber pressure is inferior when compared to other responses. This can also be observed by examining the cross- correlation results. Authors’ suggestion is to use a modern design of experiments technique (so called the space filling approach such as Latin Hypercube Sampling or Uniform Sampling) or a non-linear regression approach (such as Artificial Neural Networks) which can better model the system behavior with large number design variables20.


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• Optimum solutions are obtained at the lower or the upper limits of some design variables. Hence, accuracy of the optimum solutions should be examined in detail. In addition, larger design space should be explored if possible.

• Although the accuracy of the response surface model for the maximum acceleration is questionable, reliability analysis gives satisfactory results. This is due to fact that the constraint associated with the maximum acceleration is found to be inactive.

• Coefficient of variation for the maximum chamber pressure is as 4.52 %, 5.28% and 4.81% for the first, the second and the third approaches, respectively. Upper limit for the maximum chamber pressure is assigned as 115 bars. Hence, means of the optimum solutions are 4.85σ, 3.09σ and 5.39 σ are away from the limit, for the first, the second and the third approaches, respectively. The optimum solution found by means of the third approach can be selected if only the structural reliability is taken as the primary concern.

• The reliability goal for the ballistic reliability of the rocket motor (considering 4 critical limit states) specified as 99.999 % are achieved through robust design methodology. Inclusion of structural reliability of the motor closure and the propellant grain are left as the future work.

References 1Bozkaya, K., Sumer, B., Kuran, B. and Ak, M.A “Reliability Analysis of Solid Rocket Motor Based on Response Surface

Method And Monte Carlo Simulation”, 41st AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, 10-13 July 2005, Tucson, Arizona.

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