# Stability Analysis of a Fractional-Order High-Speed Supercavitating Vehicle Model with Delay

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Fractional Calculus

#### 2.2. Mathematical Models and Behavior of the HSSV

#### 2.2.1. Integer-Order Mathematical Model

_{c}and F

_{f}caused by the cavitator and the fins, the gravity force F

_{g,}and the noncontiuous planing force F

_{p}. The equation for each force is provided as follows. First, the lift force caused by the cavitator is calculated as:

_{n}is the cavitator radius, V is the vehicle forward speed and ${\alpha}_{c}$ is the attack angle of the cavitator.

_{p}is quite complicated. Therefore, to simplify the equations describing the planing force, the following intermediate constants are firstly defined as:

_{c}is the diameter of the cavity at the planing location which can be calculated as:

_{c}= 17/28L and L

_{e}= −11/28L are the moment arm of the cavitator and fin forces, respectively. For simplicity, the planing force is assumed to act at the same position as fin forces.

_{f}in the aft and the cavitator deflection angle δ

_{c}in the front part. This model ignores the advection delay and takes into account the non-linear planing force, which describes the non-linear interaction between the body and the liquid out of cavity and a simplified description of the cavity dynamics [23].

#### 2.2.2. The Non-linear Planing Force and Advection Delay

_{o}, where the vehicle aft end pierces the super-cavity and contacts with water as illustrated in Figure 1. The force can cause vibration, shocks, and has the tendency of pushing the vehicle aft end toward the center of the super-cavity. Depending on the strength of the tail-moment generated by the planing force, the vehicle may be suffered from the tail-slap condition, when repeating shocks occur due to the contacts between the vehicle tail and the cavity wall in opposite sides.

#### 2.2.3. The Fractional-Order Mathematical Model

_{p}is calculated using Equation (10) with the immersion depth and angle are smoothly approximated by Equations (28) and (29) [24] without the delay term $\tau $.

_{0}= 2.8

## 3. Results and Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

## Appendix B

Parameter | Value | Unit |
---|---|---|

C_{x}_{0} | 0.82 | Unitless |

g | 9.81 | m/s^{2} |

L | 1.18 | m |

m | 2 | Unitless |

n | 0.5 | Unitless |

R_{n} | 0.0191 | m |

R | 0.0508 | m |

V | 75 | m/s |

$\sigma $ | 0.03 | Unitless |

## References

- Vanek, B.; Bokor, J.; Balas, G.J.; Arndt, R.E. Longitudinal motion control of a high-speed supercavitation vehicle. J. Vib. Control
**2007**, 13, 159–184. [Google Scholar] [CrossRef] - Munther, A.H.; Nguyen, V.; Balachandran, B.; Abed, E.H. Stability analysis and control of supercavitating vehicles with advection delay. J. Comput. Nonlinear Dynam.
**2013**, 8, 021003. [Google Scholar] [CrossRef] - Kirschner, I.; Rosenthal, B.J.; Uhlman, J. Simplified dynamical systems analysis of supercavitating high-speed bodies. In Proceedings of the Fifth International Symposium on Cavitation (CAV2003), Osaka, Japan, 1 November 2003. [Google Scholar]
- Dzielski, J.; Kurdila, A. A benchmark control problem for supercavitating vehicles and an initial investigation of solutions. J. Vib. Control
**2003**, 9, 791–804. [Google Scholar] [CrossRef] - Han, Y.; Geng, R. Active disturbance rejection control of underwater high-speed vehicle. In Proceedings of the IEEE International Conference on Automation and Logistics, Zhengzhou, China, 15–17 August 2012. [Google Scholar]
- Ahn, S. An Integrated Approach to the Design of Supercavitating Underwater Vehicles. Ph.D. Thesis, Georgia Institute of Technology, Atlanta, GA, USA, 2007. [Google Scholar]
- Li, D.; Zhang, Y.; Luo, K.; Dang, J. Robust control of underwater supercavitating vehicles based on pole assignment. In Proceedings of the International Conference on Intelligent Human-Machine Systems and Cybernetics, Hangzhou, China, 26–27August 2009. [Google Scholar]
- Choi, J.; Ruzzene, M.; Bauchau, O.A. Dynamic analysis of flexible supercavitating vehicles using modal based elements. Simulation
**2004**, 80, 619–633. [Google Scholar] [CrossRef] - Nguyen, V.; Balachandran, B. Supercavitating vehicles with noncylindrical, nonsymmetric cavities: Dynamics and instabilities. J. Comput. Nonlinear Dynam.
**2011**, 6, 041001. [Google Scholar] [CrossRef] - Solteiro Pires, E.J.; Tenreiro Machado, J.A.; Moura Oliveira, P.B. Fractional order dynamics in a GA planner. Signal Process
**2003**, 83, 2377–2386. [Google Scholar] [CrossRef] - Sheng, H.; Chen, Y.Q.; Qiu, T.S. Fractional Processes and Fractional-Order Signal Processing; Springer: New York, NY, USA, 2012. [Google Scholar]
- Ionescu, C.M. The Human Respiratory System an Analysis of the Interplay between Anatomy, Structure, Breathing and Fractal Dynamics; Springer: Berlin/Heidelberg, Germany, 2013. [Google Scholar]
- Yuste, S.B.; Acedo, L.; Lindenberg, K. Reaction front in an A + B → C reaction-subdiffusion process. Phys. Rev. E
**2004**, 69, 036126. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Lin, W. Global existence theory and chaos control of fractional differential equations. J. Math. Anal. Appl.
**2007**, 332, 709–726. [Google Scholar] [CrossRef] [Green Version] - Lo, A.W. Long-term memory in stock market prices. Econometrica
**1991**, 59, 1279–1313. [Google Scholar] [CrossRef] - Gabano, J.-D.; Poinot, T. Fractional modeling applied to nondestructive thermal characterization. In Proceedings of the 18th IFAC World Congress, Milano, Italy, 28 August–2 September 2011; pp. 13972–13977. [Google Scholar]
- Hartley, T.T.; Lorenzo, C.F.; Qammar, H.K. Chaos in a fractional order chua system. IEEE Trans. Circuits Syst. I
**1995**, 42, 485–490. [Google Scholar] [CrossRef] - Chen, W.C. Nonlinear dynamics and chaos in a fractional-order financial system. Chaos Solitons Fractals
**2008**, 36, 1305–1314. [Google Scholar] [CrossRef] - Fathalla, A.R. Numerical modeling of fractional-order biological systems. Abstr. Appl. Anal.
**2013**, 2013, 816803. [Google Scholar] - Oustaloup, A.; Levron, F.; Mathieu, B.; Nanot, F.M. Frequency band complex noninteger differentiator: Characterization and synthesis. IEEE Trans. Circuit Syst. I Fundam. Theory Appl.
**2000**, 47, 25–39. [Google Scholar] [CrossRef] - Logvinovich, G. Hydrodynamics of Free-Boundary Flows; Russian (NASA-TT-F-658), Translator; US Department of Commerce: Washington, DC, USA, 1972.
- Liberzon, D. Switching in Systems and Control; Birkhauser: Cambridge, MA, USA, 2003; pp. 3–9. [Google Scholar]
- Lin, G.; Balachandran, B.; Abed, E.H. Nonlinear dynamics and bifurcations of a supercavitating vehicle. IEEE J. Ocean. Eng.
**2007**, 32, 753–761. [Google Scholar] [CrossRef] - Lin, G.; Balachandran, B.; Abed, E.H. Supercavitating body dynamics, bifurcations and control. In Proceedings of the American Control Conference, Portland, OR, USA, 8–10 June 2005. [Google Scholar] [CrossRef]
- Matignon, D. Generalized fractional differential and difference equations: Stability properties and modelling issues. In Proceedings of the Mathematical Theory of Networks and Systems Symposium, Padova, Italy, 6–10 July 1998. [Google Scholar]
- Diethelm, K.; Freed, A.D. The Frac PECE subroutine for the numerical solution of differential equations of fractional-order. In Forschung und Wissenschaftliches Rechnen; Heinzel, S., Plesser, T., Eds.; Gessellschaft fur Wissenschaftliche Datenverarbeitung: Gottingen, Germany, 1999; pp. 57–71. [Google Scholar]

**Figure 3.**Closed-loop system response for delay dependent system with trivial initial condition and ${\delta}_{c}=-0.3q-30\theta +15z,{\delta}_{f}=0$. (

**a**,

**d**) delay-free model and corresponding control signal; (

**b**,

**e**) delayed model and corresponding control signal; (

**c**,

**f**) FOM and corresponding control signal.

**Figure 4.**Planing forces in three models being respective to the time responses in Figure 3: (

**a**) delay-free model; (

**b**) delayed model; (

**c**) FOM.

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**MDPI and ACS Style**

Doan, P.T.; Bui, P.D.H.; Vu, M.T.; Thanh, H.L.N.N.; Hossain, S.
Stability Analysis of a Fractional-Order High-Speed Supercavitating Vehicle Model with Delay. *Machines* **2021**, *9*, 129.
https://doi.org/10.3390/machines9070129

**AMA Style**

Doan PT, Bui PDH, Vu MT, Thanh HLNN, Hossain S.
Stability Analysis of a Fractional-Order High-Speed Supercavitating Vehicle Model with Delay. *Machines*. 2021; 9(7):129.
https://doi.org/10.3390/machines9070129

**Chicago/Turabian Style**

Doan, Phuc Thinh, Phuc Duc Hong Bui, Mai The Vu, Ha Le Nhu Ngoc Thanh, and Shakhawat Hossain.
2021. "Stability Analysis of a Fractional-Order High-Speed Supercavitating Vehicle Model with Delay" *Machines* 9, no. 7: 129.
https://doi.org/10.3390/machines9070129