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doi: 10.3934/jimo.2022110
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Design and optimization of a novel supersonic rocket with small caliber

School of Energy and Power Engineering, Nanjing University of Science and Technology, Nanjing, 210094, China

*Corresponding author: Xiaobing Zhang

Received  November 2021 Revised  May 2022 Early access July 2022

It's difficult to meet the design requirements of a rocket with sensitive parameters using the traditional methods. This paper develops a design and optimization concept for a lightweight solid-propellant rocket with a small caliber and a high Mach number. The structural design and numerical modeling depend on the interior ballistic methodology. The model is validated by comparing pressure-time data from simulation and experiment. A modified evolutionary algorithm with constraints is applied because of the interactive design parameters and conflicting objects. The launch performance is improved by single- and multi-objective optimization. Under unchanged interior ballistic performance conditions, the peak pressure is reduced by 46.4%, and the erosive peak ratio is reduced by 46.4%. Despite the constraints, the Mach number is improved by 4.9%, the total impulse is improved by 6.3%, the peak pressure is reduced by 92.5%, and the erosive effects are reduced by 38.1% using different optimal solutions. A Pareto front is obtained by a constrained NSGA-Ⅱ, which reveals non-linear and non-uniform relations among design objects. A tidying method is proposed for a clear Pareto front. It indicates that, despite the sensitive parameters, launch safety and higher velocity are possible. The results help designers choose the best design schemes.

Citation: Hao Yan, Xiaobing Zhang. Design and optimization of a novel supersonic rocket with small caliber. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022110
References:
[1]

V. A. ArkhipovV. E. ZarkoI. K. ZharovaA. S. ZhukovE. A. KozlovD. D. Aksenenko and A. V. Kurbatov, Solid propellant combustion in a high-velocity cross-flow of gases (review), Combust. Explos. Shock Waves, 52 (2016), 497-513. 

[2]

J. S. Billheimer, Optimization and design simulation in solid rocket design, 3rd Solid Propulsion Conference, (1968), 488. 

[3]

A. Bougamra and H. Lu, Interior ballistics two-phase reactive flow model applied to small caliber projectile-gun system, Propellants, Explosives, Pyrotechnics, 40 (2015), 720-728. 

[4]

R. Cao and X. Zhang, Multi-objective optimization of the aerodynamic shape of a long-range guided rocket, Struct. Multidisc. Optim., 57 (2018), 1779-1792.  doi: 10.1007/s00158-017-1845-7.

[5]

C. Cheng and X. Zhang, Interior ballistic charge design based on a modified particle swarm optimizer, Struct. Multidisc. Optim., 46 (2012), 303-310. 

[6]

R. Courant, Variational methods for the solution of problems of equilibrium and vibrations, Bull. Amer. Math. Soc., 49 (1943), 1-23.  doi: 10.1090/S0002-9904-1943-07818-4.

[7]

E. J. CramerJ. E. DennisJr.P. D. FrankR. M. Lewis and G. R. Shubin, Problem formulation for multidisciplinary optimization, SIAM J. Optim., 4 (1994), 754-776.  doi: 10.1137/0804044.

[8]

K. DebA. PratapS. Agarwal and T. Meyarivan, A fast and elitist multiobjective genetic algorithm: NSGA-Ⅱ, IEEE Trans. Evol. Computat., 6 (2002), 182-197. 

[9] S. Dong and Z. Zhang, Principle of Solid Rocket Motors, Beijing Institute of Technology Press, Beijing, 1996. 
[10]

M. EbrahimiM. R. Farmani and J. Roshanian, Multidisciplinary design of a small satellite launch vehicle using particle swarm optimization, Struct. Multidisc. Optim., 44 (2011), 773-784. 

[11]

J. Eichler, Performance optimization of an air-to-air missile design, Journal of Spacecraft and Rockets, 14 (1977), 376-377. 

[12]

M. Gen and R. Cheng, Genetic Algorithms and Engineering Optimization, John Wiley & Sons, New York, 1999. doi: 10.1002/9780470172261.

[13]

R. J. HartfieldR. M. Jenkins and J. E. Burkhalter, Optimizing a solid rocket motor boosted ramjet powered missile using a genetic algorithm, 41st AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, (2005), 1720-1736.  doi: 10.1016/j.amc.2005.07.003.

[14]

P. R. HempelC. P. Moeller and L. M. Stuntz, Missile design optimization - experience and developments, 5th Symposium on Multidisciplinary Analysis and Optimization, (1994), 4344. 

[15]

W. B. Herbst and B. Krogull, Design for air combat, Journal of Aircraft, 10 (1973), 247-253. 

[16]

J. JodeiM. Ebrahimi and J. Roshanian, Multidisciplinary design optimization of a small solid propellant launch vehicle using system sensitivity analysis, Struct. Multidisc. Optim., 38 (2009), 93-100. 

[17]

Y. Kamm and A. Gany, Solid rocket motor optimization, 44th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, (2008), 4695. 

[18]

B. LiQ. LiY. ZengY. Rong and R. Zhang, 3D trajectory optimization for energy-efficient uav communication: A control design perspective, IEEE Transactions on Wireless Communications, (2021). 

[19]

B. LiY. WangK. Zhang and G. Duan, Constrained feedback control for spacecraft reorientation with an optimal gain, IEEE Transactions on Aerospace and Electronic Systems, 57 (2021), 3916-3926. 

[20]

B. LiJ. ZhangL. DaiK. L. Teo and S. Wang, A hybrid offline optimization method for reconfiguration of multi-uav formations, IEEE Transactions on Aerospace and Electronic Systems, 57 (2021), 506-520. 

[21]

C. LiH. Fang and C. Gong, Development of an efficient global optimization method based on adaptive infilling for structure optimization, Struct. Multidisc. Optim., 62 (2020), 3383-3412. 

[22]

K. Li and X. Zhang, Multi-objective optimization of interior ballistic performance using NSGA-Ⅱ, Propellants, Explosives, Pyrotechnics, 36 (2011), 282-290. 

[23]

K. Li and X. Zhang, Using NSGA-Ⅱ and topsis methods for interior ballistic optimization based on one-dimensional two-phase flow model, Propellants, Explosives, Pyrotechnics, 37 (2012), 468-475. 

[24] Y. Li, Principle of Solid Rocket Motors, Beijing University of Aeronautics and Astronautics Press, Beijing, 1991. 
[25]

C. LiuR. LoxtonK. L. Teo and S. Wang, Optimal state-delay control in nonlinear dynamic systems, Automatica, 135 (2022), 109981.  doi: 10.1016/j.automatica.2021.109981.

[26]

Y. LiuS. ChenF. Wang and F. Xiong, Sequential optimization using multi-level cokriging and extended expected improvement criterion, Struct. Multidisc. Optim., 58 (2018), 1155-1173.  doi: 10.1007/s00158-018-1959-6.

[27]

Z. Lv, Y. Zheng and T. Fang, Experimental study on erosive combustion of solid propellant, Journal of China Ordnance, (1981), 74–82,84.

[28]

Z. MichalewiczD. DasguptaR. G. Le Riche and M. Schoenauer, Evolutionary algorithms for constrained engineering problems, Computers & Industrial Engineering, 30 (1996), 851-870. 

[29]

J. D. Schaffer, Multiple objective optimization with vector evaluated genetic algorithms, Proceedings of the 1st International Conference on Genetic Algorithms, (1985), 93-100. 

[30]

Y. Shi, Research on Long Range Air-to-Air Anti-Radiation Missile General Design and Optimization, Ph.D thesis, Northwestern Polytechnical University in Xi'an, Shaanxi, 2015.

[31]

N. Srinivas and K. Deb, Muiltiobjective optimization using nondominated sorting in genetic algorithms, Evolutionary Computation, 2 (1994), 221-248. 

[32]

K. L. Teo, B. Li, C. Yu and V. Rehbock, Applied and Computational Optimal Control: A Control Parametrization Approach, Springer, New York, 2021. doi: 10.1007/978-3-030-69913-0.

[33]

C. Tola and M. Nikbay, Solid rocket motor propellant optimization with coupled internal ballistic-structural interaction approach, Journal of Spacecraft and Rockets, 55 (2018), 936-947. 

[34]

T. J. Walsh and R. L. Wartburg, Ballistic missile sizing and optimizing, 14th Joint Propulsion Conference, (1978), 1019. 

[35]

J. WangJ. ZhuJ. HouC. Wang and W. Zhang, Lightweight design of a bolt-flange sealing structure based on topology optimization, Struct. Multidisc. Optim., 62 (2020), 3413-3428.  doi: 10.1007/s00158-020-02683-4.

[36]

P. Wang and X. Zhang, Optimized Bézier-curve-based command generation and robust inverse optimal control for attitude tracking of spacecraft, Aerospace Science and Technology, 121 (2022), 107183. 

[37]

M. A. WillcoxM. Q. BrewsterK. C. TangD. S. Stewart and I. Kuznetsov, Solid rocket motor internal ballistics simulation using three-dimensional grain burnback, Journal of Propulsion and Power, 23 (2007), 575-584. 

[38] X. WuJ. Chen and D. Wang, Solid Propellant Motor Gas Dynamics, Beijing University of Aeronautics and Astronautics Press, Beijing, 2016. 
[39] M. Xu, External Ballistics of Rockets, Harbin Institute of Technology Press, Harbin, 2004. 
[40]

Z. XuF. JiangC. ZhongY. Gou and H. Teng, Multi-objective layout optimization for an orbital propellant depot, Struct. Multidisc. Optim., 61 (2020), 207-223.  doi: 10.1007/s00158-019-02354-z.

[41]

F. YangK. L. TeoR. LoxtonV. RehbockB. LiC. Yu and L. Jennings, Visual MISER: An efficient user-friendly visual program for solving optimal control problems, Journal of Industrial and Management Optimization, 12 (2016), 781-810.  doi: 10.3934/jimo.2016.12.781.

[42]

S. M. Zandavi and S. H. Pourtakdoust, Multidisciplinary design of a guided flying vehicle using simplex nondominated sorting genetic algorithm Ⅱ, Struct. Multidisc. Optim., 57 (2018), 705-720.  doi: 10.1007/s00158-017-1776-3.

[43]

G. ZhaD. SmithM. SchwabacherK. RasheedA. GelseyD. Knight and M. Haas, High-performance supersonic missile inlet design using automated optimization, Journal of Aircraft, 34 (1997), 697-705. 

[44] X. Zhang, Interior Ballistics of Guns, Beijing Institute of Technology Press, Beijing, 2014. 
[45]

Y. ZhangC. Gong and C. Li, Efficient time-variant reliability analysis through approximating the most probable point trajectory, Struct. Multidisc. Optim., 63 (2021), 289-309.  doi: 10.1007/s00158-020-02696-z.

[46]

H. ZhuH. LuoP. WangG. Cai and F. Hu, Uncertainty analysis and design optimization of solid rocket motors with finocyl grain, Struct. Multidisc. Optim., 62 (2020), 3521-3537. 

show all references

References:
[1]

V. A. ArkhipovV. E. ZarkoI. K. ZharovaA. S. ZhukovE. A. KozlovD. D. Aksenenko and A. V. Kurbatov, Solid propellant combustion in a high-velocity cross-flow of gases (review), Combust. Explos. Shock Waves, 52 (2016), 497-513. 

[2]

J. S. Billheimer, Optimization and design simulation in solid rocket design, 3rd Solid Propulsion Conference, (1968), 488. 

[3]

A. Bougamra and H. Lu, Interior ballistics two-phase reactive flow model applied to small caliber projectile-gun system, Propellants, Explosives, Pyrotechnics, 40 (2015), 720-728. 

[4]

R. Cao and X. Zhang, Multi-objective optimization of the aerodynamic shape of a long-range guided rocket, Struct. Multidisc. Optim., 57 (2018), 1779-1792.  doi: 10.1007/s00158-017-1845-7.

[5]

C. Cheng and X. Zhang, Interior ballistic charge design based on a modified particle swarm optimizer, Struct. Multidisc. Optim., 46 (2012), 303-310. 

[6]

R. Courant, Variational methods for the solution of problems of equilibrium and vibrations, Bull. Amer. Math. Soc., 49 (1943), 1-23.  doi: 10.1090/S0002-9904-1943-07818-4.

[7]

E. J. CramerJ. E. DennisJr.P. D. FrankR. M. Lewis and G. R. Shubin, Problem formulation for multidisciplinary optimization, SIAM J. Optim., 4 (1994), 754-776.  doi: 10.1137/0804044.

[8]

K. DebA. PratapS. Agarwal and T. Meyarivan, A fast and elitist multiobjective genetic algorithm: NSGA-Ⅱ, IEEE Trans. Evol. Computat., 6 (2002), 182-197. 

[9] S. Dong and Z. Zhang, Principle of Solid Rocket Motors, Beijing Institute of Technology Press, Beijing, 1996. 
[10]

M. EbrahimiM. R. Farmani and J. Roshanian, Multidisciplinary design of a small satellite launch vehicle using particle swarm optimization, Struct. Multidisc. Optim., 44 (2011), 773-784. 

[11]

J. Eichler, Performance optimization of an air-to-air missile design, Journal of Spacecraft and Rockets, 14 (1977), 376-377. 

[12]

M. Gen and R. Cheng, Genetic Algorithms and Engineering Optimization, John Wiley & Sons, New York, 1999. doi: 10.1002/9780470172261.

[13]

R. J. HartfieldR. M. Jenkins and J. E. Burkhalter, Optimizing a solid rocket motor boosted ramjet powered missile using a genetic algorithm, 41st AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, (2005), 1720-1736.  doi: 10.1016/j.amc.2005.07.003.

[14]

P. R. HempelC. P. Moeller and L. M. Stuntz, Missile design optimization - experience and developments, 5th Symposium on Multidisciplinary Analysis and Optimization, (1994), 4344. 

[15]

W. B. Herbst and B. Krogull, Design for air combat, Journal of Aircraft, 10 (1973), 247-253. 

[16]

J. JodeiM. Ebrahimi and J. Roshanian, Multidisciplinary design optimization of a small solid propellant launch vehicle using system sensitivity analysis, Struct. Multidisc. Optim., 38 (2009), 93-100. 

[17]

Y. Kamm and A. Gany, Solid rocket motor optimization, 44th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, (2008), 4695. 

[18]

B. LiQ. LiY. ZengY. Rong and R. Zhang, 3D trajectory optimization for energy-efficient uav communication: A control design perspective, IEEE Transactions on Wireless Communications, (2021). 

[19]

B. LiY. WangK. Zhang and G. Duan, Constrained feedback control for spacecraft reorientation with an optimal gain, IEEE Transactions on Aerospace and Electronic Systems, 57 (2021), 3916-3926. 

[20]

B. LiJ. ZhangL. DaiK. L. Teo and S. Wang, A hybrid offline optimization method for reconfiguration of multi-uav formations, IEEE Transactions on Aerospace and Electronic Systems, 57 (2021), 506-520. 

[21]

C. LiH. Fang and C. Gong, Development of an efficient global optimization method based on adaptive infilling for structure optimization, Struct. Multidisc. Optim., 62 (2020), 3383-3412. 

[22]

K. Li and X. Zhang, Multi-objective optimization of interior ballistic performance using NSGA-Ⅱ, Propellants, Explosives, Pyrotechnics, 36 (2011), 282-290. 

[23]

K. Li and X. Zhang, Using NSGA-Ⅱ and topsis methods for interior ballistic optimization based on one-dimensional two-phase flow model, Propellants, Explosives, Pyrotechnics, 37 (2012), 468-475. 

[24] Y. Li, Principle of Solid Rocket Motors, Beijing University of Aeronautics and Astronautics Press, Beijing, 1991. 
[25]

C. LiuR. LoxtonK. L. Teo and S. Wang, Optimal state-delay control in nonlinear dynamic systems, Automatica, 135 (2022), 109981.  doi: 10.1016/j.automatica.2021.109981.

[26]

Y. LiuS. ChenF. Wang and F. Xiong, Sequential optimization using multi-level cokriging and extended expected improvement criterion, Struct. Multidisc. Optim., 58 (2018), 1155-1173.  doi: 10.1007/s00158-018-1959-6.

[27]

Z. Lv, Y. Zheng and T. Fang, Experimental study on erosive combustion of solid propellant, Journal of China Ordnance, (1981), 74–82,84.

[28]

Z. MichalewiczD. DasguptaR. G. Le Riche and M. Schoenauer, Evolutionary algorithms for constrained engineering problems, Computers & Industrial Engineering, 30 (1996), 851-870. 

[29]

J. D. Schaffer, Multiple objective optimization with vector evaluated genetic algorithms, Proceedings of the 1st International Conference on Genetic Algorithms, (1985), 93-100. 

[30]

Y. Shi, Research on Long Range Air-to-Air Anti-Radiation Missile General Design and Optimization, Ph.D thesis, Northwestern Polytechnical University in Xi'an, Shaanxi, 2015.

[31]

N. Srinivas and K. Deb, Muiltiobjective optimization using nondominated sorting in genetic algorithms, Evolutionary Computation, 2 (1994), 221-248. 

[32]

K. L. Teo, B. Li, C. Yu and V. Rehbock, Applied and Computational Optimal Control: A Control Parametrization Approach, Springer, New York, 2021. doi: 10.1007/978-3-030-69913-0.

[33]

C. Tola and M. Nikbay, Solid rocket motor propellant optimization with coupled internal ballistic-structural interaction approach, Journal of Spacecraft and Rockets, 55 (2018), 936-947. 

[34]

T. J. Walsh and R. L. Wartburg, Ballistic missile sizing and optimizing, 14th Joint Propulsion Conference, (1978), 1019. 

[35]

J. WangJ. ZhuJ. HouC. Wang and W. Zhang, Lightweight design of a bolt-flange sealing structure based on topology optimization, Struct. Multidisc. Optim., 62 (2020), 3413-3428.  doi: 10.1007/s00158-020-02683-4.

[36]

P. Wang and X. Zhang, Optimized Bézier-curve-based command generation and robust inverse optimal control for attitude tracking of spacecraft, Aerospace Science and Technology, 121 (2022), 107183. 

[37]

M. A. WillcoxM. Q. BrewsterK. C. TangD. S. Stewart and I. Kuznetsov, Solid rocket motor internal ballistics simulation using three-dimensional grain burnback, Journal of Propulsion and Power, 23 (2007), 575-584. 

[38] X. WuJ. Chen and D. Wang, Solid Propellant Motor Gas Dynamics, Beijing University of Aeronautics and Astronautics Press, Beijing, 2016. 
[39] M. Xu, External Ballistics of Rockets, Harbin Institute of Technology Press, Harbin, 2004. 
[40]

Z. XuF. JiangC. ZhongY. Gou and H. Teng, Multi-objective layout optimization for an orbital propellant depot, Struct. Multidisc. Optim., 61 (2020), 207-223.  doi: 10.1007/s00158-019-02354-z.

[41]

F. YangK. L. TeoR. LoxtonV. RehbockB. LiC. Yu and L. Jennings, Visual MISER: An efficient user-friendly visual program for solving optimal control problems, Journal of Industrial and Management Optimization, 12 (2016), 781-810.  doi: 10.3934/jimo.2016.12.781.

[42]

S. M. Zandavi and S. H. Pourtakdoust, Multidisciplinary design of a guided flying vehicle using simplex nondominated sorting genetic algorithm Ⅱ, Struct. Multidisc. Optim., 57 (2018), 705-720.  doi: 10.1007/s00158-017-1776-3.

[43]

G. ZhaD. SmithM. SchwabacherK. RasheedA. GelseyD. Knight and M. Haas, High-performance supersonic missile inlet design using automated optimization, Journal of Aircraft, 34 (1997), 697-705. 

[44] X. Zhang, Interior Ballistics of Guns, Beijing Institute of Technology Press, Beijing, 2014. 
[45]

Y. ZhangC. Gong and C. Li, Efficient time-variant reliability analysis through approximating the most probable point trajectory, Struct. Multidisc. Optim., 63 (2021), 289-309.  doi: 10.1007/s00158-020-02696-z.

[46]

H. ZhuH. LuoP. WangG. Cai and F. Hu, Uncertainty analysis and design optimization of solid rocket motors with finocyl grain, Struct. Multidisc. Optim., 62 (2020), 3521-3537. 

Figure 1.  A design framework of the MDO problem
Figure 2.  An integrated solution flow for rocket design
Figure 3.  Structural sketch for small-caliber rocket
Figure 4.  The three combustion stages in solid-propellant rocket chamber
Figure 5.  Procedure of NSGA-Ⅱ with two modules
Figure 6.  Comparisons of P-t traces between numerical calculation and experiment
Figure 7.  Initial P-t and F-t curves of numerical simulation
Figure 8.  Convergence curves of single-object optimization when $ A $ = 5%
Figure 9.  Initial Pareto front of multi-objective optimization
Figure 10.  Pareto front after tidying up
Figure 11.  Comparison of special Pareto-optimal points
Figure 12.  Motor interior ballistic performance
Figure 13.  Three degree-of-freedom trajectory validation
Table 1.  Propellant ingredients in validation case
Ingredients Value
Ammonium perchlorate (AP) 83%
Hydroxy-terminated polybutadiene (HTPB) 11.9%
Oxamide 5%
Carbon black 0.1%
Ingredients Value
Ammonium perchlorate (AP) 83%
Hydroxy-terminated polybutadiene (HTPB) 11.9%
Oxamide 5%
Carbon black 0.1%
Table 2.  Charge & motor parameters in validation case
Parameters Value
Burning rate coefficient (mm/s/Pa$ ^n $) 0.0038
Burning rate pressure index 0.461
Nozzle throat diameter(mm) 26.416
Specific heat 1.217
Propellant web thickness(mm) 22.86
Propellant length(mm) 692
Parameters Value
Burning rate coefficient (mm/s/Pa$ ^n $) 0.0038
Burning rate pressure index 0.461
Nozzle throat diameter(mm) 26.416
Specific heat 1.217
Propellant web thickness(mm) 22.86
Propellant length(mm) 692
Table 3.  Initial charge parameters
Parameters Value Parameters Value
Burning rate coefficient (mm/s/Pa$ ^n $) 0.034 Pressure index 0.358
Characteristic velocity(m/s) 1341 Temperature factor 1.0
Propellant density(kg/m$ ^3 $) 1610 Flow rate factor 0.95
Propellant web thickness(mm) 27.3535 Specific heat 1.25
Parameters Value Parameters Value
Burning rate coefficient (mm/s/Pa$ ^n $) 0.034 Pressure index 0.358
Characteristic velocity(m/s) 1341 Temperature factor 1.0
Propellant density(kg/m$ ^3 $) 1610 Flow rate factor 0.95
Propellant web thickness(mm) 27.3535 Specific heat 1.25
Table 4.  Initial motor parameters
Parameters Value Parameters Value
Length-to-diameter proportion 18.75 Nozzle throat diameter(mm) 18.0
Nozzle expansion ratio 1.6 Nozzle exit diameter(mm) 28.8
Parameters Value Parameters Value
Length-to-diameter proportion 18.75 Nozzle throat diameter(mm) 18.0
Nozzle expansion ratio 1.6 Nozzle exit diameter(mm) 28.8
Table 5.  Parameters of initial performance with erosive effects
Parameters Value Parameters Value
Equilibrium pressure (MPa) 31.3275 Average thrust(kN) 12.8273
Peak pressure(Mpa) 105.3515 Total impulse(kN $ \cdot $ s) 10.2872
Max thrust(kN) 40.5700 Burning time(s) 0.801
Erosive peak ratio 3.3629 Max Mach number 2.5510
Parameters Value Parameters Value
Equilibrium pressure (MPa) 31.3275 Average thrust(kN) 12.8273
Peak pressure(Mpa) 105.3515 Total impulse(kN $ \cdot $ s) 10.2872
Max thrust(kN) 40.5700 Burning time(s) 0.801
Erosive peak ratio 3.3629 Max Mach number 2.5510
Table 6.  Parameters of initial performance theoretically
Parameters Value Parameters Value
Equilibrium pressure (MPa) 31.3275 Average thrust(kN) 11.2740
Peak pressure(Mpa) 31.3275 Total impulse(kN $ \cdot $ s) 10.2815
Max thrust(kN) 12.0287 Burning time(s) 0.911
Erosive peak ratio 1.0000 Max Mach number 2.5517
Parameters Value Parameters Value
Equilibrium pressure (MPa) 31.3275 Average thrust(kN) 11.2740
Peak pressure(Mpa) 31.3275 Total impulse(kN $ \cdot $ s) 10.2815
Max thrust(kN) 12.0287 Burning time(s) 0.911
Erosive peak ratio 1.0000 Max Mach number 2.5517
Table 7.  Optimal design parameters. The size of nozzle throat becomes smaller to enhance the interior ballistic performance, with reduced erosive effects
$ A $ $ r_p $ $ {P_{\rm{eq}}} $/MPa $ B $ $ e $/mm $ d_{\rm{t}} $/mm $ A_{\rm{t}} $/mm$ ^2 $
0.1% 1.8018 31.3586 + 0.1% 19.8536 19.7633 306.7656
1% 1.7993 31.6407 + 1% 19.8536 19.7043 304.9377
5% 1.7884 32.8938 + 5% 19.8536 19.4502 297.1239
10% 1.7748 34.4605 + 10% 19.8536 19.1497 288.0125
15% 1.7619 36.0264 + 15% 19.8536 18.8663 279.5513
20% 1.7492 37.5927 + 20% 19.8536 18.5984 271.6681
$ A $ $ r_p $ $ {P_{\rm{eq}}} $/MPa $ B $ $ e $/mm $ d_{\rm{t}} $/mm $ A_{\rm{t}} $/mm$ ^2 $
0.1% 1.8018 31.3586 + 0.1% 19.8536 19.7633 306.7656
1% 1.7993 31.6407 + 1% 19.8536 19.7043 304.9377
5% 1.7884 32.8938 + 5% 19.8536 19.4502 297.1239
10% 1.7748 34.4605 + 10% 19.8536 19.1497 288.0125
15% 1.7619 36.0264 + 15% 19.8536 18.8663 279.5513
20% 1.7492 37.5927 + 20% 19.8536 18.5984 271.6681
Table 8.  Optimal rocket parameters. The peak pressure and maximum propulsion force are increased but the change of average thrust ($ {F_{\rm{ave}}} $) and maximum Mach number ($ M_{\rm{max}} $) fluctuate because of different burn time. $ {F_{\rm{m}}} $ is the maximum thrust
$ A $ $ {P_{\rm{m}}} $/MPa $ {F_{\rm{m}}} $/kN $ {F_{\rm{ave}}} $/kN $ I $/kN$ \cdot $s $ t $/s $ M_{\rm{max}} $
0.1% 56.5028 26.2026 15.2652 9.0061 0.589 2.3231
1% 56.9306 26.2440 15.1632 9.0065 0.593 2.3221
5% 58.8283 26.4259 15.4245 9.0075 0.583 2.3236
10% 61.1599 26.6331 15.3745 9.0091 0.585 2.3225
15% 63.4747 26.8312 15.5617 9.0098 0.578 2.3232
20% 65.7580 27.0144 15.4561 9.0105 0.582 2.3219
$ A $ $ {P_{\rm{m}}} $/MPa $ {F_{\rm{m}}} $/kN $ {F_{\rm{ave}}} $/kN $ I $/kN$ \cdot $s $ t $/s $ M_{\rm{max}} $
0.1% 56.5028 26.2026 15.2652 9.0061 0.589 2.3231
1% 56.9306 26.2440 15.1632 9.0065 0.593 2.3221
5% 58.8283 26.4259 15.4245 9.0075 0.583 2.3236
10% 61.1599 26.6331 15.3745 9.0091 0.585 2.3225
15% 63.4747 26.8312 15.5617 9.0098 0.578 2.3232
20% 65.7580 27.0144 15.4561 9.0105 0.582 2.3219
Table 9.  Relevant optimal motor parameters. The ratio of nozzle throat area to sectional area of charging channel is decreased and the ratio of burning surface area of solid propellant to nozzle throat area is increased
$ A $ $ m $/kg $ \Delta $/(g/cm$ ^3 $) $ {\text{æ}}_0 $ $ J $ $ K_{\rm{N}} $
0.1% 4.2276 1.2890 388.4005 0.4504 862.2808
1% 4.2276 1.2890 388.4005 0.4478 867.4496
5% 4.2276 1.2890 388.4005 0.4363 890.2617
10% 4.2276 1.2890 388.4005 0.4229 918.4257
15% 4.2276 1.2890 388.4009 0.4105 946.2236
20% 4.2276 1.2890 388.4005 0.3989 973.6809
$ A $ $ m $/kg $ \Delta $/(g/cm$ ^3 $) $ {\text{æ}}_0 $ $ J $ $ K_{\rm{N}} $
0.1% 4.2276 1.2890 388.4005 0.4504 862.2808
1% 4.2276 1.2890 388.4005 0.4478 867.4496
5% 4.2276 1.2890 388.4005 0.4363 890.2617
10% 4.2276 1.2890 388.4005 0.4229 918.4257
15% 4.2276 1.2890 388.4009 0.4105 946.2236
20% 4.2276 1.2890 388.4005 0.3989 973.6809
Table 10.  Primary optimum solutions. The four special points, A, B, C, and D, represent the boundary points of Pareto front. The goal for minimum peak pressure, minimum erosive effects, and maximum Mach number can be attained by different points of the Pareto front, and their threshold is determined by these special points
$ {P_{\rm{m}}} $/MPa $ r_p $ $ I $/kN$ \cdot $s $ e $/mm $ d_{\rm{t}} $/mm $ d_{\rm{e}} $/mm $ t $/s $ t_{\rm{max}} $/s $ M_{\rm{max}} $
A 7.9447 2.2001 8.1926 19.854 40 50 1.161 1.161 2.0913
B 59.977 1.7818 9.1996 19.854 19.301 33.745 0.594 0.578 2.3668
C 59.993 4.3926 10.93 28.251 23.372 50 1.024 1.021 2.6762
D 18.792 2.082 7.7942 19.854 29.745 32.239 0.849 0.849 2.0185
$ {P_{\rm{m}}} $/MPa $ r_p $ $ I $/kN$ \cdot $s $ e $/mm $ d_{\rm{t}} $/mm $ d_{\rm{e}} $/mm $ t $/s $ t_{\rm{max}} $/s $ M_{\rm{max}} $
A 7.9447 2.2001 8.1926 19.854 40 50 1.161 1.161 2.0913
B 59.977 1.7818 9.1996 19.854 19.301 33.745 0.594 0.578 2.3668
C 59.993 4.3926 10.93 28.251 23.372 50 1.024 1.021 2.6762
D 18.792 2.082 7.7942 19.854 29.745 32.239 0.849 0.849 2.0185
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