﻿ 基于分叉理论的水下超空泡航行体运动特性研究
 舰船科学技术  2016, Vol. 38 Issue (12): 20-25,34 PDF

Motion characteristics of underwater supercavitating vehicles based on bifurcation theory
LV Yi-pin, XIONG Tian-hong, YI Wen-jun
National Key Laboratory of Transient Physics, Nanjing University of Science and Techology, Nanjing 210094, China
Abstract: To guarantee the steady motion of underwater supercavitating vehicles, the bifurcation analysis of underwater vehicles with variable cavitation numbers is conducted. Through numerical simulation and phase track diagram, the difference of motion characteristics with different cavitation numbers is revealed based on bifurcation theory. The conditions and ranges of parameters for the stable motion of vehicles are finally determined by a two-dimensional bifurcation diagram. The results indicate that the motion of supercavitating vehicles is found to have nonlinear dynamic characteristics, resulting in the phenomena with cavitation numbers varying, such as Hopf bifurcations, period-doubling bifurcations and chaos. The stable range of underwater supercavitating vehicles can be expanded by adjusting the gain of control law reasonably. Then the steady motion will be realized.
Key words: supercavitating vehicles     bifurcation theory     cavitation number     two-dimensional bifurcation diagram
0 引 言

1 水下超空泡航行体的动力学模型 1.1 空化数的描述

 $\sigma = \frac{{2\left( {{p_\infty }-{p_c}} \right)}}{{\rho {V^2}}}\text{。}$ (1)

1.2 水下超空泡航行体动力学建模

 图 1 超空泡航行体示意图 Fig. 1 The schematic diagram of supercavitating vehicles

 \begin{aligned} & \left( {\begin{array}{*{20}{l}} {\dot z}\\ {\dot w}\\ {\dot \theta }\\ {\dot q} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 0&1&{-V}&0\\ 0&{{a_{22}}}&0&{{a_{24}}}\\ 0&0&0&1\\ 0&{{a_{42}}}&0&{{a_{44}}} \end{array}} \right)\left( {\begin{array}{*{20}{l}} z\\ w\\ \theta \\ q \end{array}} \right) + \\ & \quad \left( {\begin{array}{*{20}{l}} 0&0\\ {{b_{21}}}&{{b_{22}}}\\ 0&0\\ {{b_{41}}}&{{b_{42}}} \end{array}} \right)\left( {\begin{array}{*{20}{l}} {{\delta _e}}\\ {{\delta _c}} \end{array}} \right) + \left( {\begin{array}{*{20}{l}} 0\\ {{c_2}}\\ 0\\ 0 \end{array}} \right) + \left( {\begin{array}{*{20}{l}} 0\\ {{d_2}}\\ 0\\ {{d_4}} \end{array}} \right){F_{\rm planing}}\text{。} \end{aligned} (2)

 \begin{align} & {{a}_{22}}=\frac{CVT}{m}(\frac{-1-n}{L})S+\frac{17}{36}nL\text{,} \\ & {{a}_{24}}=VTS(C\frac{-n}{m}+\frac{7}{9})-VT(C\frac{-n}{m}+\frac{17}{36})\frac{17}{36}{{L}^{2}}\text{,} \\ & {{a}_{42}}=\frac{CVT}{m}(\frac{17}{36}-\frac{11n}{36})\text{,} \\ & {{a}_{44}}=\frac{-11CVTL}{36m}\text{,} \\ & {{b}_{21}}=\frac{C{{V}^{2}}Tn}{36m}(\frac{-S}{L}+\frac{17L}{36})\text{,} \\ & {{b}_{22}}=\frac{-C{{V}^{2}}TS}{mL}\text{,} \\ & {{b}_{41}}=\frac{-11C{{V}^{2}}Tn}{36m}\text{,} \\ & {{b}_{42}}=\frac{17C{{V}^{2}}T}{36m} \\ \end{align}
 \begin{aligned} {c_2} = & g,{d_2} = \frac{T}{m}(\frac{{-17L}}{{36}} + \frac{S}{L}) \text{，}\\ {d_4} = & \frac{{11T}}{{36m}} \text{，}\\ S = & \frac{{11}}{{60}}{R^2} + \frac{{133}}{{405}}{L^2} \text{，}\\ T = &\frac{1}{{7S/9-289{L^2}/1296}} \text{，}\\ {C_x} = & {C_{x0}}(1 + \sigma ) \text{，}\\ C = & 0.5{C_x}\frac{{R{{_n^2}^{}}_{}}}{{{R^2}}} \text{，}\\ {R_c} = & {R_n}\sqrt {0.82\frac{{1 + \sigma }}{\sigma }} {K_2} \text{，}\\ {K_1} = &\frac{L}{{{R_n}(\frac{{1.92}}{\sigma }-3)}}-1 \text{，}\\ {K_2} = & \sqrt {1-(1-\frac{{4.5\sigma }}{{1 + \sigma }})} K_1^{40/17} \text{，}\\ {{\dot R}_c} = & {\frac{{\frac{{-20}}{{17}}(0.82\frac{{1 + \sigma }}{\sigma })}}{{{K_2}(\frac{{1.92}}{\sigma }-3)}}^{0.5}} \text{，}\\ V(1-& \frac{{4.5\sigma }}{{1 + \sigma }}){({K_1})^{23/17}}\text{。} \end{aligned}

 图 2 浸没深度 Fig. 2 Immersion length

 ${F_{\rm planing}} =-{V^2}[1-{(\frac{{{R_c}-R}}{{h'R + {R_c}-R}})^2}](\frac{{1 + h'}}{{1 + 2h'}})\alpha \text{。}$ (3)

 h' = \left\{ \begin{aligned} & 0,\;\;\left| w \right| ＜ {w_{th}} = \frac{{\left( {{R_c}-R} \right)V}}{L}\text{，}\\ & \frac{{L\left| w \right|}}{{RV}}-\frac{{{R_C}-R}}{R},\;\;\;{\rm otherwise}\text{；} \end{aligned} \right. (4)
 \alpha = \left\{ \begin{aligned} & \frac{{w-{{\dot R}_c}}}{V},\;\;\;\frac{w}{V} ＞ 0\text{，}\\ & \frac{{w-{{\dot R}_c}}}{V},\;\;\;\;{\rm otherwise}\text{；} \end{aligned} \right. (5)

2 水下超空泡航行体运动状态分析 2.1 分叉理论

 ${\dot x} = {f}({x},{\mu })\text{。}$ (6)

 ${f}({x},{\mu }) = 0\text{，}$ (7)
 $det{{D}_x}{f}({x},{\mu }) = 0\text{。}$ (8)

 $x(t) = A(\mu )x(t) + F(x(t),\mu )\text{。}$ (9)

1）若 Aμ）的特征根均存在负实部，可判定系统稳定；

2）若 Aμ）的特征根中有 1 对共轭纯虚特征根，而其他特征根均有负实部，可利用中心流形定理对系统进行降维约化后分析其稳定性；

3）若 Aμ）的特征根有 1 对纯虚特征根，则说明系统在此处发生 hopf 分叉，开始从平衡态失稳并产生周期振荡。

2.2 超空泡航行体分叉分析

 图 3 系统随空化数 σ 变化的分岔图 Fig. 3 Bifurcation diagram of system for σ

2.3 不同空化数下航行体运动状态分析

 ${ { J{ s}}_1}\!\!=\!\!\left[\! {\begin{array}{*{20}{c}} {239.92}\!\! & \!\!{495.62}\!\! & \!\!{40011.91}\!\! & \!\!{-20005.95}\!\!\\ \!\!{-276.88}\!\! & \!\!{-326.42}\!\! & \!\!{-3195.02}\! & \!\!{15975.10}\!\!\\ 0 & 1 & 0 & 0\\ 1 & 0 & \!\!{-89.64}\!\! & 0 \end{array}}\! \right]\text{，}$

 $\det \left( {1\lambda-{{J}_S}} \right) = 0\text{，}$

 图 4 σ = 0.021 0 时系统响应示意图 Fig. 4 Time response for σ = 0.021 0

 ${ J{ s}_2}\!\!=\!\!\left[ \!{\begin{array}{*{20}{c}} \!\!\!{353.88}\!\!\! & \!\!\!{397.28}\!\!\! & \!\!\!{31303.26}\!\!\! & \!\!\!{-15651.63}\\ \!\!\!{-413.03}\!\! & \!\!{-256.10}\!\! & \!\!\!{-24996.18}\!\!\! & \!\!\!{12498.10}\!\!\\ 0 & 1 & 0 & 0\\ 1 & 0 & {-79.06} & 0 \end{array}} \!\!\right]\text{。}$

 图 5 σ = 0.027 0 时系统相轨图 Fig. 5 Phase track diagram for σ = 0.027 0

 图 6 σ = 0.027 0 时系统响应示意图 Fig. 6 Time response for σ = 0.027 0

 ${ J{ s}_3}\!\!=\!\!\left[ \!{\begin{array}{*{20}{c}} \!\!\!{902.74}\!\!\! & \!\!\!{338.27}\!\!\! & \!\!\!{26145.05}\!\!\! & \!\!\!{-13072.53}\!\!\!\!\\ \!\!\!{-1061.92}\!\!\! & \!\!\!{-214.40}\!\!\! & \!\!\!{-20877.27}\!\! & \!\!\!{10438.64}\!\!\!\\ 0 & 1 & 0 & 0\\ 1 & 0 & {-72.06} & 0 \end{array}} \right]\!\text{。}$

 图 7 σ = 0.032 5 时系统相轨图 Fig. 7 Phase track diagram for σ = 0.032 5

 图 8 σ = 0.032 5 时系统响应示意图 Fig. 8 Time response for σ = 0.032 5
3 超空泡航行体的稳定运动

 图 9 系统动力学行为分布图 Fig. 9 Regions of different dynamical behaviors in the space of the bifurcation parameters σ and k

4 结 语

1）运用分叉理论可以确定超空泡航行体在任一参数下的运动稳定性，为探讨求解超空泡的非线性问题，提供了分析途径；

2）随着空化数的变化，超空泡航行体的运动具有稳定、周期和混沌 3 种状态；相轨图揭示了极限环和混沌吸引子的出现，说明超空泡航行体的纵向运动具有非线性动力学特性；

3）二维分岔图能够确定航行体稳定运动条件和参数范围，根据航行体的运动速度，可以对应设置恰当的控制率，从而有效抑制航行体的振动与冲击，实现超空泡航行体的稳定航行。

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