﻿ 水下滑翔机的机翼位置与螺旋运动关系分析
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 浙江大学学报(工学版)  2017, Vol. 51 Issue (9): 1760-1769  DOI:10.3785/j.issn.1008-973X.2017.09.010 0

### 引用本文 [复制中英文]

dx.doi.org/10.3785/j.issn.1008-973X.2017.09.010
[复制中文]
LIU You, SHEN Qing, MA Dong-li, YUAN Xiang-jiang. Relationship of wing location and helical motion for underwater glider[J]. Journal of Zhejiang University(Engineering Science), 2017, 51(9): 1760-1769.
dx.doi.org/10.3785/j.issn.1008-973X.2017.09.010
[复制英文]

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orcid.org/0000-0002-1862-1652.
Email: yuan_xj18@163.com

### 文章历史

1. 北京航空航天大学 航空科学与工程学院, 北京 100191;
2. 中国航天空气动力技术研究院, 北京 100074

Relationship of wing location and helical motion for underwater glider
LIU You1,2 , SHEN Qing2 , MA Dong-li1 , YUAN Xiang-jiang2
1. School of Aeronautic Science and Engineering, Beihang University, Beijing 100191, China;
2. China Academy of Aerospace Aerodynamics(CAAA), Beijing 100074, China
Abstract: A mathematical model was constructed to describe the helical motion of underwater glider at steady state. Using the numerical method, the helical motion features of underwater glider were achieved corresponding to five wing locations. Numerical results indicate that the helical motion pattern of underwater glider changes with the wing location. And there is a transitional zone ("watershed" zone) determining the turning direction of underwater glider at steady state. The gliders with wing locations ahead of the zone turn in the same direction of lateral component of lift produced by wing and work in positive helical pattern. However, the gliders with wing locations behind the zone turn in the reverse direction of lateral component of lift produced by wing and work in anti-helical pattern. The glider with wing location within the zone can turn in any of the two directions dependent on its CG (centre of gravity) location. Furthermore, the gliders with wing locations far away from the zone turn faster than those with wing locations near it. The in-lake experiments indicate that wing location can affect the turning direction of helical motion and the percentage error between numerical results and experiment data at steady stage is less than 15%.
Key words: underwater glider    helical motion    hydrodynamic model    hydrodynamic coefficients    turning direction

1 平衡模型

1) 运动假设:正/反螺旋运动在稳态时可以简单看作是由一个水平面内匀速圆周运动和一个竖直方向的匀速直线运动的叠加运动.假设滑翔机为刚体, 滑翔机浮心的速度可以代表滑翔机的速度, 匀速圆周运动的角速度可以代表滑翔机的角速度.若考虑惯性离心力, 螺旋运动在非惯性系下(见2.1节)可以看作是平衡的, 即说合力和合力矩都为零向量.

2) 流动假设:当攻角、侧滑角、角速度都比较小时(见2.2节), 流体是定常的, 且没有发生分离.

1.1 参考系

1) 柱轴系.如图 1所示, 原点是浮心(CB), X轴与浮心的水平速度方向一致, Y轴垂直于XZ平面向右(参照飞行员视角), Z轴与重力加速度的方向一致.

 图 1 柱轴系与螺旋运动轨迹的关系示意图 Fig. 1 Relationship between cylinder coordinate system and spiral trajectory

2) 体轴系.该坐标轴的原点在浮心, 坐标轴xyz的方向如图 2所示.l为机翼离机头的距离, d为浮心(CB)到机头的距离.本文分析了5个机翼位置不同的滑翔机, 这5个滑翔机的名称及机翼位置如表 1所示.

 图 2 体轴系及滑翔机外形示意图 Fig. 2 Diagram of body coordinate system and gliderprofile

 图 3 滑翔机螺旋运动姿态角示意图 Fig. 3 Illustration of underwater glider's attitude angles during helical motion

3) 风轴系.如图 4所示, α为攻角, β为侧滑角.4个坐标转换矩阵[9]定义如下：

 图 4 风轴系与体轴系关系图 Fig. 4 Relationship between wind frame and body frame
 $\left. \begin{array}{l} {\mathit{\boldsymbol{R}}_{{\rm{CB}}}} = {{\rm{e}}^{{{\mathit{\boldsymbol{\hat e}}}_3}\psi }} \cdot {{\rm{e}}^{{{\mathit{\boldsymbol{\hat e}}}_2}\theta }} \cdot {{\rm{e}}^{{{\mathit{\boldsymbol{\hat e}}}_1}\theta }},\\ {\mathit{\boldsymbol{R}}_{{\rm{BC}}}} = \mathit{\boldsymbol{R}}_{{\rm{CB}}}^{\rm{T}},\\ {\mathit{\boldsymbol{R}}_{{\rm{BW}}}} = {{\rm{e}}^{ - {{\mathit{\boldsymbol{\hat e}}}_2}\alpha }} \cdot {{\rm{e}}^{{{\mathit{\boldsymbol{\hat e}}}_3}\beta }},\\ {\mathit{\boldsymbol{R}}_{{\rm{WB}}}} = \mathit{\boldsymbol{R}}_{{\rm{BW}}}^{\rm{T}}. \end{array} \right\}$ (1)

 ${e^\mathit{\boldsymbol{Q}}} = \sum\limits_{i = 0}^n {\left( {\frac{1}{{n!}}{\mathit{\boldsymbol{Q}}^n}} \right)} .$ (2)
1.2 外力和外力矩

 $\begin{array}{l} {\mathit{\boldsymbol{F}}_{\rm{h}}} = \left[ {\begin{array}{*{20}{c}} {C_{\rm{X}}^\alpha }\\ {C_{\rm{Y}}^\beta + C_{\rm{Y}}^P + C_{\rm{Y}}^R}\\ {C_{\rm{Z}}^\alpha + C_{\rm{Z}}^Q} \end{array}} \right] \times 0.5\rho {v^2}S = \\ \;\;\;\;\;\;\;\left[ {\begin{array}{*{20}{c}} {X_\alpha ^0 + X_\alpha ^2{\alpha ^2}}\\ {Y_\beta ^1\beta + Y_P^1P + Y_R^1R}\\ {Z_\alpha ^1\alpha + Z_\alpha ^0 + Z_Q^1Q} \end{array}} \right] \times 0.5\rho {v^2}S, \end{array}$ (3)
 $\begin{array}{l} {\mathit{\boldsymbol{M}}_{\rm{h}}} = \left[ {\begin{array}{*{20}{c}} {C_{\rm{K}}^P}\\ {C_{\rm{M}}^\alpha + C_{\rm{M}}^Q}\\ {C_{\rm{N}}^\beta + C_{\rm{N}}^P + C_{\rm{N}}^R} \end{array}} \right] \times 0.5\rho {v^2}SL = \\ \;\;\;\;\;\;\;\;\;\left[ {\begin{array}{*{20}{c}} {K_P^1P}\\ {M_\alpha ^1\alpha + M_Q^1Q}\\ {N_\beta ^1\beta + N_P^1P + N_R^1R} \end{array}} \right] \times 0.5\rho {v^2}SL. \end{array}$ (4)

 $P = pL/v,Q = qL/v,R = rL/v.$ (5)

 $\left| \alpha \right|,\left| \beta \right| \le {10^ \circ },\left| P \right|,\left| Q \right|,\left| R \right| \le 0.06,v \le 1{\rm{m/s}}.$ (6)

 ${\mathit{\boldsymbol{F}}_{\rm{n}}} = {\left[ {\begin{array}{*{20}{c}} 0&0&B \end{array}} \right]^{\rm{T}}}.$ (7)

 ${\mathit{\boldsymbol{M}}_{\rm{G}}} = {{\mathit{\boldsymbol{\hat R}}}_{\rm{G}}}\left( {{\mathit{\boldsymbol{R}}_{{\rm{BC}}}}\mathit{\boldsymbol{G}}} \right).$ (8)

 ${\mathit{\boldsymbol{V}}_{\rm{C}}} = {\mathit{\boldsymbol{R}}_{{\rm{CB}}}}{\mathit{\boldsymbol{R}}_{{\rm{BW}}}}\mathit{\boldsymbol{v}}.$ (9)

 ${\mathit{\boldsymbol{\omega }}_{\rm{B}}} = {\mathit{\boldsymbol{R}}_{{\rm{BC}}}}\mathit{\boldsymbol{\omega }}.$ (10)

1.3 平衡方程组

 ${\mathit{\boldsymbol{F}}_{\rm{i}}} = - m{\omega ^2}{\left[ {\begin{array}{*{20}{c}} 0&R&0 \end{array}} \right]^{\rm{T}}},$ (11)
 ${\mathit{\boldsymbol{M}}_{\rm{i}}} = {{\mathit{\boldsymbol{\hat R}}}_{\rm{G}}}\left( {{\mathit{\boldsymbol{R}}_{{\rm{BC}}}}{\mathit{\boldsymbol{F}}_{\rm{i}}}} \right).$ (12)

 $\left. \begin{array}{l} {\mathit{\boldsymbol{R}}_{{\rm{CB}}}}{\mathit{\boldsymbol{F}}_{\rm{h}}} + {\mathit{\boldsymbol{F}}_{\rm{n}}} + {\mathit{\boldsymbol{F}}_{\rm{i}}} = 0,\\ {\mathit{\boldsymbol{M}}_{\rm{h}}} + {\mathit{\boldsymbol{M}}_{\rm{G}}} + {\mathit{\boldsymbol{M}}_{\rm{i}}} = 0,\\ {v_Y} = 0. \end{array} \right\}$ (13)

x={v ω α β ψ θ}Ty ={xG yG zG B}T, 式(13) 可以简化为

 $\mathit{\boldsymbol{G}}\left( {\mathit{\boldsymbol{x}},\mathit{\boldsymbol{y}}} \right) = {\bf{0}}.$ (15)

y给定时, 式(15) 可以进一步简化为

 $\mathit{\boldsymbol{F}}\left( \mathit{\boldsymbol{x}} \right) = 0.$ (16)

2 数值结果 2.1 机翼位于过渡区域前面

1) 设B=4.9 N, zG=0.005 m, 则

 $\begin{array}{l} {\mathit{\boldsymbol{y}}_{ij}} = \left( {{x_{Gij}},{y_{Gij}},0.005,4.9} \right),\\ {x_{Gij}} = {z_{\rm{G}}} \times \tan \left[ {{{15}^ \circ } + \left( {i - 1} \right) \times {5^ \circ }} \right],\\ {y_{Gij}} = {z_{\rm{G}}} \times \tan \left( {j * {{4.5}^ \circ }} \right). \end{array}$

2) 对于每一个yij, 使用信赖区域反射算法(trust-region reflective algorithm)[17]求得对应的xij ={vij ωij αij βij ψij θij ij}的值.

3) 将所得的结果(xGij yGij vij), (xGij yGij ωij), (xGij yGij αij), (xGij yGij βij), (xGij yGij ψij), (xGij yGij θij), (xGij yGij ϕij)呈现在7个不同的坐标系中.

 图 5 滑翔机A和滑翔机B的运动特性与重心位置关系图 Fig. 5 Relationship between CG location and motion features of glider A and glider B

2.2 机翼位于过渡区域内

 图 6 滑翔机C的运动特性与重心位置关系图 Fig. 6 Relationship between CG location and motion features of glider C
 图 7 滑翔机C转向相速度与重心位置的关系 Fig. 7 Relationship between turning velocity of Glider C and CG location

2.3 机翼位于过渡区域后

 图 8 滑翔机D和滑翔机E的运动特性与重心位置关系图 Fig. 8 Relationship between CG location and motion features of glider D and glider E

3 湖试结果

 图 9 在湖试中的滑翔机 Fig. 9 Glider during in-lake experiment

 图 10 试验中滑翔机净浮力的时间变化历程 Fig. 10 Time variation history of glider net buoyancy during experiment

 图 11 试验中竖直下潜深度、航向角、俯仰角、滚转角、竖直下潜速度以及航向角速度的时间历程 Fig. 11 Time history of depth, yaw angle, pitch angle, roll angle, depth rate and yaw angle rate during experiment

4 结论

(1) 存在1个决定水下滑翔机转弯方向的过渡区域, 当机翼位于这个区域的前面时, 滑翔机按正螺旋方式转弯；当机翼位于这个区域的后面时, 滑翔机按照反螺旋方式转弯；当机翼位于这个区域内时, 滑翔机的转弯速率很小, 且转弯方向由重心位置决定.

(2) 过渡区域的概念代表了一种特殊的水动力外形布局, 设计师必须避免将滑翔机的机翼设计在这个特殊区域内.每个滑翔机都有其自身固有的过渡区域, 在滑翔机设计阶段, 可按照本文所提供的方法分析并确定这个区域.

(3) 无论滑翔机是按何种方式运行, 机翼距离过渡区域越远, 转弯速率越高.在一定的范围内, 减小xG的绝对值并增大yG的绝对值可以显著提升转弯速率的绝对值.

(4) 相比于正螺旋模式, 采用反螺旋转弯模式的滑翔机的机翼更加靠后, 其俯仰方向的稳定性能更优, 因而反螺旋模式在水下滑翔机领域是一个新的设计方向.

(5) 湖试结果表明：机翼位置可以影响螺旋运动的转弯方向, 且稳态试验数据与数值结果比较吻合, 误差≤15%.

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