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1. 厦门大学 航空航天学院, 厦门 361005;
2. 山东科技大学 电气与自动化工程学院, 青岛 266510

Fusing method for dynamic derivatives and added mass of airships
LIN Xianwu1, WANG Shichao1, LI Zhibin2, LAN Weiyao1
1. School of Aerospace Engineering, Xiamen University, Xiamen 361005, China;
2. College of Electrical Engineering and Automation, Shandong University of Science and Technology, Qingdao 266510, China
Abstract: Dynamic stability derivatives and added mass represent an airship's aerodynamics in viscous flow and acyclic potential flow. To develop a method for fusing these two aerodynamic coefficients in airship modeling, the classification method for dynamic stability derivatives and added mass and the fusing method for same aerodynamic coefficient ingredient are studied. By introducing a unified theory for analyzing aerodynamics in incompressible flow, a conclusion is drawn that the viscous aerodynamics should be reserved and the corresponding results in acyclic potential flow should be abandoned in fusing the same ingredient of aerodynamics. By studying the relationship between aerodynamics and the present motion parameter of an airship, the classification method for aerodynamics and the aerodynamic coefficient is built. To keep the aerodynamic classification method same in the two flow fields, a new method for reconstructing the present motion parameters of airships is proposed so that the aerodynamic coefficients in the two flow fields can be both classified according to the new parameters, and each ingredient can be evaluated. Based on these studies, a new fusing method for dynamic stability derivatives and added mass is proposed and its difference with the present fusing method is discussed. A numerical example is presented to show the effect of different fusing methods on the dynamic characteristics of an airship's longitudinal perturbation motion and illustrate the necessity for adopting this new fusing method.
Keywords: airship    dynamic derivatives    added mass    fusing    vorticity dynamics    linearization    dynamic characteristics

1 不可压缩流中通用的气动力和力矩表达式

Lamb[1]所介绍无旋无环流场中对应的气动力和力矩方法是基于能量守恒的原理推导得到的；因为有黏流场中存在黏性损耗，因此这种方法难以被推广到有黏流中去。运动体在有黏流场中运动时所对应的气动力和力矩可以用Wu提出的涡量矩定理[24-26]来计算和分析，这种方法是基于动量守恒方法得到的。然而在流场黏性减少为零的情况下，涡量矩定理并不能收敛于Lamb的研究结果[27]。这种不兼容性的原因之一在于涡量矩定理的推导过程中将运动体和流场视为一体。在无旋无环流的情况下，滑移边界条件将导致速度等变量分布在流体和固体所占领的空间区域上不连续，从而导致涡量矩定理的推导过程和结果不成立。涡量矩定理不能兼容无旋无环流结果的另一个原因是目前许多涡动力学理论[28-31]中的气动力和力矩表达式并没有考虑力矩参考点可动的情况；然而在飞行力学中，运动体所受气动力矩的参考点一般是固联在运动体上的，因而是运动的。考虑到这些问题和不足，重新推导运动体的气动力和力矩表达式可得到能同时兼容涡量矩定理和无旋无环流结果的气动力FA和力矩MA的公式为

 $\left\{ {\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{F}}_{\rm{A}}} = - \frac{{{\rm{d}}{\mathit{\boldsymbol{Q}}_{f,\infty }}}}{{{\rm{d}}t}}}\\ {{\mathit{\boldsymbol{M}}_{\rm{A}}} = - \frac{{{\rm{d}}{\mathit{\boldsymbol{K}}_{f,\infty }}}}{{{\rm{d}}t}} - {\mathit{\boldsymbol{v}}_0} \times {\mathit{\boldsymbol{Q}}_{f,\infty }}} \end{array}} \right.$ （1）

 $\left\{ {\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{Q}}_{f,\infty }} = \int_{{R_{f,\infty }}} \rho \mathit{\boldsymbol{v}}{\rm{d}}R = \frac{\rho }{{N - 1}}\int_{{R_{f,\infty }}} \mathit{\boldsymbol{r}} \times \mathit{\boldsymbol{\omega }}{\rm{d}}R + }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{\rho }{{N - 1}}\int_{{S_{\rm{i}}}} \mathit{\boldsymbol{r}} \times (\mathit{\boldsymbol{v}} \times \mathit{\boldsymbol{n}}){\rm{d}}S}\\ {{\mathit{\boldsymbol{K}}_{f,\infty }} = \int_{{R_{f,\infty }}} \rho \mathit{\boldsymbol{r}} \times \mathit{\boldsymbol{v}}{\rm{d}}R = - \frac{\rho }{2}\int_{{R_{f,\infty }}} {{\mathit{\boldsymbol{r}}^2}} \mathit{\boldsymbol{\omega }}{\rm{d}}R + }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \frac{\rho }{2}\int_{{S_{\rm{i}}}} {{\mathit{\boldsymbol{r}}^2}} (\mathit{\boldsymbol{n}} \times \mathit{\boldsymbol{v}}){\rm{d}}S} \end{array}} \right.$ （2）

 $\left\{ {\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{Q}}_{f,\infty }} = \int_{{R_{f,\infty }}} \rho \mathit{\boldsymbol{v}}{\rm{d}}R = \frac{\rho }{{N - 1}}\int_{{S_{\rm{i}}}} \mathit{\boldsymbol{r}} \times (\mathit{\boldsymbol{v}} \times \mathit{\boldsymbol{n}}){\rm{d}}S}\\ {{\mathit{\boldsymbol{K}}_{f,\infty }} = \int_{{R_{f,\infty }}} \rho \mathit{\boldsymbol{r}} \times \mathit{\boldsymbol{v}}{\rm{d}}R = \frac{\rho }{2}\int_{{S_{\rm{i}}}} {{\mathit{\boldsymbol{r}}^2}} (\mathit{\boldsymbol{n}} \times \mathit{\boldsymbol{v}}){\rm{d}}S} \end{array}} \right.$ （3）

 $\left\{ {\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{Q}}_{f,\infty }} = \frac{\rho }{{N - 1}}\int_{{R^\prime }_{f,\infty }} \mathit{\boldsymbol{r}} \times \mathit{\boldsymbol{\omega }}{\rm{d}}R + \frac{\rho }{{N - 1}}\int_{{S_{\rm{b}}}} \mathit{\boldsymbol{r}} \times }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (v \times n)dS}\\ {{\mathit{\boldsymbol{K}}_{f,\infty }} = - \frac{\rho }{2}\int_{{R^\prime }_{f,\infty }} {{\mathit{\boldsymbol{r}}^2}} \mathit{\boldsymbol{\omega }}{\rm{d}}R + \frac{\rho }{2}\int_{{S_{\rm{b}}}} {{\mathit{\boldsymbol{r}}^2}} (\mathit{\boldsymbol{n}} \times \mathit{\boldsymbol{v}}){\rm{d}}S} \end{array}} \right.$ （4）

2 气动系数计算与融合

2.1 无旋无环流中的气动系数计算

 $\left\{ {\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{Q}}_{f,\infty }} = {\mathit{\boldsymbol{\lambda }}_{Fv}} \cdot {\mathit{\boldsymbol{v}}_{\rm{D}}} + {\mathit{\boldsymbol{\lambda }}_{F\omega }} \cdot {\mathit{\boldsymbol{\omega }}_{\rm{b}}}}\\ {{\mathit{\boldsymbol{K}}_{f,\infty }} = {\mathit{\boldsymbol{\lambda }}_{Mv}} \cdot {\mathit{\boldsymbol{v}}_{\rm{D}}} + {\mathit{\boldsymbol{\lambda }}_{{M_\omega }}} \cdot {\mathit{\boldsymbol{\omega }}_{\rm{b}}}} \end{array}} \right.$ （5）

 $\left\{ {\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{F}}_{{\rm{A}},p}} = - \frac{{{\rm{d}}({\mathit{\boldsymbol{\lambda }}_{Fv}} \cdot {\mathit{\boldsymbol{v}}_{\rm{D}}} + {\mathit{\boldsymbol{\lambda }}_{F\omega }} \cdot {\mathit{\boldsymbol{\omega }}_{\rm{b}}})}}{{{\rm{d}}t}}}\\ {{\mathit{\boldsymbol{M}}_{{\rm{A}},p}} = - \frac{{{\rm{d}}({\mathit{\boldsymbol{\lambda }}_{Mv}} \cdot {\mathit{\boldsymbol{v}}_{\rm{D}}} + {\mathit{\boldsymbol{\lambda }}_{M\omega }} \cdot {\mathit{\boldsymbol{\omega }}_{\rm{b}}})}}{{{\rm{d}}t}} - }\\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\mathit{\boldsymbol{v}}_0} \times {\mathit{\boldsymbol{\lambda }}_{Fv}} \cdot {\mathit{\boldsymbol{v}}_{\rm{D}}} - {\mathit{\boldsymbol{v}}_0} \times {\mathit{\boldsymbol{\lambda }}_{F\omega }} \cdot {\mathit{\boldsymbol{\omega }}_{\rm{b}}}} \end{array}} \right.$ （6）

 $\left\{ {\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{\lambda }}_{Fv}} = \left[ {\begin{array}{*{20}{c}} {{\lambda _{11}}}&0&0\\ 0&{{\lambda _{22}}}&0\\ 0&0&{{\lambda _{33}}} \end{array}} \right],{\mathit{\boldsymbol{\lambda }}_{F\omega }} = \left[ {\begin{array}{*{20}{c}} 0&0&0\\ 0&0&{{\lambda _{26}}}\\ 0&{{\lambda _{35}}}&0 \end{array}} \right]}\\ {{\mathit{\boldsymbol{\lambda }}_{Mv}} = \left[ {\begin{array}{*{20}{c}} 0&0&0\\ 0&0&{{\lambda _{26}}}\\ 0&{{\lambda _{35}}}&0 \end{array}} \right],{\mathit{\boldsymbol{\lambda }}_{{M_\omega }}} = \left[ {\begin{array}{*{20}{c}} {{\lambda _{44}}}&0&0\\ 0&{{\lambda _{55}}}&0\\ 0&0&{{\lambda _{66}}} \end{array}} \right]} \end{array}} \right.$ （7）

 ${\mathit{\boldsymbol{F}}_{{\rm{A}},p}} = - \left[ \begin{array}{l} {\lambda _{11}}\dot u + {\lambda _{35}}{q^2} - {\lambda _{26}}{r^2} - {\lambda _{22}}vr + {\lambda _{33}}wq\\ {\lambda _{26}}\dot r + {\lambda _{22}}\dot v + {\lambda _{11}}ur - {\lambda _{33}}w{\rm{p }} - {\lambda _{35}}pq\\ {\lambda _{35}}\dot q + {\lambda _{33}}\dot w - {\lambda _{11}}uq + {\lambda _{22}}vp + {\lambda _{26}}pr \end{array} \right]$ （8）
 ${\mathit{\boldsymbol{M}}_{{\rm{A}},p}} = - \left[ \begin{array}{l} {\lambda _{44}}\dot p - ({\lambda _{22}} + {\lambda _{33}})vw + ({\lambda _{35}} + {\lambda _{26}})vq - ({\lambda _{26}} + {\lambda _{35}})wr{\rm{ }} - ({\lambda _{55}} - {\lambda _{66}})qr\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\lambda _{55}}\dot q + {\lambda _{35}}\dot w + ({\lambda _{11}} - {\lambda _{33}})uw - {\lambda _{26}}vp - {\lambda _{35}}uq + ({\lambda _{44}} - {\lambda _{66}})pr\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\lambda _{66}}\dot r + {\lambda _{26}}\dot v + ({\lambda _{22}} - {\lambda _{11}})vu + {\lambda _{26}}ur + {\lambda _{35}}wp + ({\lambda _{55}} - {\lambda _{44}})qp \end{array} \right]$ （9）

2.2 有黏流中的气动系数计算

 $\left\{ {\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{F}}_v} = \bar QS({C_x}{\mathit{\boldsymbol{e}}_1} + {C_y}{\mathit{\boldsymbol{e}}_2} + {C_z}{\mathit{\boldsymbol{e}}_3})}\\ {{\mathit{\boldsymbol{M}}_v} = \bar QSL({m_x}{\mathit{\boldsymbol{e}}_1} + {m_y}{\mathit{\boldsymbol{e}}_2} + {m_z}{\mathit{\boldsymbol{e}}_3})} \end{array}} \right.$ （10）

 $\left\{ {\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{F}}_{qr,v}} = \bar QS({C_{yr}}\mathit{\boldsymbol{j}} + {C_{zq}}\mathit{\boldsymbol{k}})}\\ {{\mathit{\boldsymbol{M}}_{qr,v}} = \bar QU({m_{yq}}\mathit{\boldsymbol{j}} + {m_{zr}}\mathit{\boldsymbol{k}})} \end{array}} \right.$ （11）

 $\left\{ {\begin{array}{*{20}{l}} {{C_{yr}} = C_y^rr,{C_{zq}} = C_z^qq}\\ {{m_{yq}} = m_y^qq,{m_{zr}} = m_z^rr} \end{array}} \right.$ （12）

 $\left\{ {\begin{array}{*{20}{l}} {{\mathit{\boldsymbol{F}}_{{a_y}{a_z},v}} = \bar QS({C_{y{a_y}}}\mathit{\boldsymbol{j}} + {C_{z{a_z}}}\mathit{\boldsymbol{k}})}\\ {{\mathit{\boldsymbol{M}}_{{a_y}{a_z},v}} = \bar QU({m_{y{a_z}}}\mathit{\boldsymbol{j}} + {m_{z{a_y}}}\mathit{\boldsymbol{k}})} \end{array}} \right.$ （13）

 $\left\{ \begin{array}{l} {C_{y{a_y}}} = C_y^{{a_y}}{a_y} = C_y^{{a_y}}(\dot v + ru - pw)\\ {C_{z{a_z}}} = C_z^{{a_z}}{a_z} = C_z^{{a_z}}(\dot w + pv - qu)\\ {m_{y{a_z}}} = m_y^{{a_z}}{a_z} = m_y^{{a_z}}(\dot w + pv - qu)\\ {m_{z{a_y}}} = m_z^{{a_y}}{a_y} = m_z^{{a_y}}(\dot v + ru - pw) \end{array} \right.$ （14）

2.3 动导数与附加质量的融合

 $\left\{ {\begin{array}{*{20}{l}} {{v_{\rm{D}}} = \sqrt {{u^2} + {v^2} + {w^2}} }\\ {\alpha = - {\rm{arctan}}\frac{v}{u}}\\ {\beta = {\rm{arcsin}}\frac{w}{{{v_{\rm{D}}}}}} \end{array}} \right.$ （15）

 $\left\{ {\begin{array}{*{20}{l}} {\dot \alpha = \frac{{v\dot u - \dot vu}}{{{u^2} + {v^2}}} \approx - \dot v/{v_{\rm{D}}}}\\ {\dot \beta = \frac{{\dot w{v_{\rm{D}}} - w{{\dot v}_{\rm{D}}}}}{{{v_{\rm{D}}}\sqrt {{w^2} + v_{\rm{D}}^2} }} \approx \dot w/{v_{\rm{D}}}} \end{array}} \right.$ （16）

 $\left\{ {\begin{array}{*{20}{l}} {\dot \alpha = {{\dot \alpha }_1} + {{\dot \alpha }_2}}\\ {\dot \beta = {{\dot \beta }_1} + {{\dot \beta }_2}} \end{array}} \right.$ （17）

 $\begin{array}{l} {\mathit{\boldsymbol{F}}_{{\rm{A}},p}} = - {\mathit{\boldsymbol{\lambda }}_{Fv}} \cdot \frac{{{{\rm{d}}_r}{\mathit{\boldsymbol{v}}_{\rm{D}}}}}{{{\rm{d}}t}} - {\mathit{\boldsymbol{\lambda }}_{F\omega }} \cdot \frac{{{{\rm{d}}_r}{\mathit{\boldsymbol{\omega }}_{\rm{b}}}}}{{{\rm{d}}t}} - {\mathit{\boldsymbol{\omega }}_{\rm{b}}} \times {\mathit{\boldsymbol{Q}}_{f,\infty }} = \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{l}} {{\kern 1pt} {\kern 1pt} - {\mathit{\boldsymbol{\lambda }}_{Fv}} \cdot \frac{{{\rm{d}}{\mathit{\boldsymbol{v}}_{\rm{D}}}}}{{{\rm{d}}t}} - {\mathit{\boldsymbol{\lambda }}_{F\omega }} \cdot \frac{{{\rm{d}}{\mathit{\boldsymbol{\omega }}_{\rm{b}}}}}{{{\rm{d}}t}} + {\mathit{\boldsymbol{\lambda }}_{Fv}} \cdot ({\mathit{\boldsymbol{\omega }}_{\rm{b}}} \times {\mathit{\boldsymbol{v}}_{\rm{D}}}) - }\\ {{\mathit{\boldsymbol{\omega }}_{\rm{b}}} \times ({\mathit{\boldsymbol{\lambda }}_{Fv}} \cdot {\mathit{\boldsymbol{v}}_{\rm{D}}}) - {\mathit{\boldsymbol{\omega }}_{\rm{b}}} \times ({\mathit{\boldsymbol{\lambda }}_{F\omega }} \cdot {\mathit{\boldsymbol{\omega }}_{\rm{b}}})} \end{array} \end{array}$ （18）

 $\begin{array}{l} {\mathit{\boldsymbol{F}}_{qr,p}} = {\mathit{\boldsymbol{\lambda }}_{Fv}} \cdot [(q\mathit{\boldsymbol{j}} + r\mathit{\boldsymbol{k}}) \times {\mathit{\boldsymbol{v}}_{\rm{D}}}] - \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{l}} {(q\mathit{\boldsymbol{j}} + r\mathit{\boldsymbol{k}}) \times ({\mathit{\boldsymbol{\lambda }}_{Fv}} \cdot {\mathit{\boldsymbol{v}}_{\rm{D}}}) - }\\ {(q\mathit{\boldsymbol{j}} + r\mathit{\boldsymbol{k}}) \times [{\mathit{\boldsymbol{\lambda }}_{F\omega }} \cdot (q\mathit{\boldsymbol{j}} + r\mathit{\boldsymbol{k}})]} \end{array} \end{array}$ （19）

 $- {\mathit{\boldsymbol{\lambda }}_{Fv}} \cdot \frac{{{{\rm{d}}_r}{\mathit{\boldsymbol{v}}_{\rm{D}}}}}{{{\rm{d}}t}} = {\mathit{\boldsymbol{\lambda }}_{Fv}} \cdot ({\mathit{\boldsymbol{\omega }}_{\rm{b}}} \times {\mathit{\boldsymbol{v}}_{\rm{D}}})$ （20）

 ${\mathit{\boldsymbol{F}}_{qr1,p}} = {\mathit{\boldsymbol{F}}_{qr,p}} - {\mathit{\boldsymbol{F}}_{qr,p}} \cdot \mathit{\boldsymbol{i}}$ （21）

 $\begin{array}{l} {\mathit{\boldsymbol{F}}_{{a_y}{a_z},p}} = - {\mathit{\boldsymbol{\lambda }}_{Fv}} \cdot \frac{{{\rm{d}}(v\mathit{\boldsymbol{j}} + w\mathit{\boldsymbol{k}})}}{{{\rm{d}}t}} = \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} - {\mathit{\boldsymbol{\lambda }}_{Fv}} \cdot (\dot v\mathit{\boldsymbol{j}} + \dot v\mathit{\boldsymbol{k}}) - {\mathit{\boldsymbol{\lambda }}_{Fv}} \cdot [{\mathit{\boldsymbol{\omega }}_{\rm{b}}} \times (\dot v\mathit{\boldsymbol{j}} + \dot w\mathit{\boldsymbol{k}})] \end{array}$ （22）

 ${\mathit{\boldsymbol{F}}_{{a_y}{a_z}1,p}} = {\mathit{\boldsymbol{F}}_{{a_y}{a_z},p}} - {\mathit{\boldsymbol{F}}_{{a_y}{a_z},p}} \cdot \mathit{\boldsymbol{i}}$ （23）

 $\begin{array}{l} {\mathit{\boldsymbol{M}}_{{\rm{A}},p}} = - {\mathit{\boldsymbol{\lambda }}_{Mv}} \cdot \frac{{{\rm{d}}{\mathit{\boldsymbol{v}}_{\rm{D}}}}}{{{\rm{d}}t}} - {\mathit{\boldsymbol{\lambda }}_{M\omega }} \cdot \frac{{{\rm{d}}{\mathit{\boldsymbol{\omega }}_{\rm{b}}}}}{{{\rm{d}}t}} + \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{l}} {{\mathit{\boldsymbol{\lambda }}_{Mv}} \cdot ({\mathit{\boldsymbol{\omega }}_{\rm{b}}} \times {\mathit{\boldsymbol{v}}_{\rm{D}}}) - {\mathit{\boldsymbol{\omega }}_{\rm{b}}} \times ({\mathit{\boldsymbol{\lambda }}_{Mv}} \cdot {\mathit{\boldsymbol{v}}_{\rm{D}}}) - }\\ {{\mathit{\boldsymbol{\omega }}_{\rm{b}}} \times ({\mathit{\boldsymbol{\lambda }}_{M\omega }} \cdot {\mathit{\boldsymbol{\omega }}_{\rm{b}}}) - }\\ {{\mathit{\boldsymbol{v}}_0} \times {\mathit{\boldsymbol{\lambda }}_{Fv}} \cdot {\mathit{\boldsymbol{v}}_{\rm{D}}} - {\mathit{\boldsymbol{v}}_0} \times {\mathit{\boldsymbol{\lambda }}_{F\omega }} \cdot {\mathit{\boldsymbol{\omega }}_{\rm{b}}}} \end{array} \end{array}$ （24）

 $\begin{array}{l} {\mathit{\boldsymbol{M}}_{qr,p}} = {\mathit{\boldsymbol{\lambda }}_{Mv}} \cdot [(q\mathit{\boldsymbol{j}} + r\mathit{\boldsymbol{k}}) \times {\mathit{\boldsymbol{v}}_{\rm{D}}}] - \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \begin{array}{*{20}{l}} {(q\mathit{\boldsymbol{j}} + r\mathit{\boldsymbol{k}}) \times ({\mathit{\boldsymbol{\lambda }}_{Mv}} \cdot {\mathit{\boldsymbol{v}}_{\rm{D}}}) - }\\ {(q\mathit{\boldsymbol{j}} + r\mathit{\boldsymbol{k}}) \times [{\mathit{\boldsymbol{\lambda }}_{M\omega }} \cdot (q\mathit{\boldsymbol{j}} + r\mathit{\boldsymbol{k}})] - }\\ {{\mathit{\boldsymbol{v}}_0} \times [{\mathit{\boldsymbol{\lambda }}_{F\omega }} \cdot (q\mathit{\boldsymbol{j}} + r\mathit{\boldsymbol{k}})]} \end{array} \end{array}$ （25）

 ${\mathit{\boldsymbol{M}}_{qr1,p}} = {\mathit{\boldsymbol{M}}_{qr,p}} - {\mathit{\boldsymbol{M}}_{qr,p}} \cdot \mathit{\boldsymbol{i}}$ （26）

 $\begin{array}{l} {\mathit{\boldsymbol{M}}_{{a_y}{a_z},p}} = - {\mathit{\boldsymbol{\lambda }}_{Mv}}\frac{{{\rm{d}}(v\mathit{\boldsymbol{j}} + w\mathit{\boldsymbol{k}})}}{{{\rm{d}}t}} = \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} - {\mathit{\boldsymbol{\lambda }}_{Mv}} \cdot (\dot v\mathit{\boldsymbol{j}} + \dot w\mathit{\boldsymbol{k}}) - {\mathit{\boldsymbol{\lambda }}_{Mv}} \cdot [{\mathit{\boldsymbol{\omega }}_{\rm{b}}} \times (v\mathit{\boldsymbol{j}} + w\mathit{\boldsymbol{k}})] \end{array}$ （27）

 ${\mathit{\boldsymbol{M}}_{{y^a}{z^1},p}} = {\mathit{\boldsymbol{M}}_{{a_y}{a_z},p}} - {\mathit{\boldsymbol{M}}_{{a_y}{a_z},p}} \cdot \mathit{\boldsymbol{i}}$ （28）

3 算例分析

 图 1 两种气动系数融合方法对比 Fig. 1 Comparison of two fusing methods of aerodynamic coefficients

 ${\mathit{\boldsymbol{M}}_{{\rm{ln}}}}{\mathit{\boldsymbol{\dot x}}_{{\rm{ln}}}} = {\mathit{\boldsymbol{a}}_{{\rm{ln}}}}{\mathit{\boldsymbol{\dot x}}_{{\rm{ln}}}} + {\mathit{\boldsymbol{b}}_{{\rm{ln}}}}{\mathit{\boldsymbol{u}}_{{\rm{ln}}}}$ （29）

 $\begin{array}{l} \begin{array}{*{20}{l}} {{\mathit{\boldsymbol{x}}_{{\rm{ln}}}} = [\Delta {v_{\rm{D}}},\Delta \alpha ,r,\Delta \vartheta ]}\\ {{\mathit{\boldsymbol{M}}_{{\rm{ln}}}} = } \end{array}\\ \left[ {\begin{array}{*{20}{c}} {m + {\lambda _{11}}}&0&{m{c_y}}&0\\ 0&{ - m{v_{\rm{D}}} - \bar QSC_y^{\dot \alpha }}&{ - m{c_x} - \bar QSC_y^{\dot r}}&0\\ {m{c_y}}&{m{c_x}{v_{\rm{D}}} - \bar QUm_z^{\dot \alpha }}&{ - \bar QUm_z^{\dot r} + {J_z}}&0\\ 0&0&0&{m{v_D}} \end{array}} \right] \end{array}$ （30）
 $\begin{array}{l} {\mathit{\boldsymbol{a}}_{{\rm{ln}}}} = \\ \left[ {\begin{array}{*{20}{c}} { - (\rho {v_{\rm{D}}}{C_{x0}} + 3\bar QC_x^{{v_{\rm{D}}}})S + 2{T^v}{\rm{cos}}{\mu _{\rm{e}}}}&0&{m{v_{\rm{D}}}{\alpha _{\rm{e}}}}&{ - (mg - \rho U){\rm{cos}}{\vartheta _{\rm{e}}}}\\ {2{T^v}{\rm{sin}}{\mu _{\rm{e}}}}&{\bar QSC_y^\alpha }&{\bar QSC_y^r + m{v_{\rm{D}}}}&{(mg - \rho U){\rm{sin}}{\vartheta _{\rm{e}}}}\\ {2{T^v}{d_x}{\rm{sin}}{\mu _{\rm{e}}} - 2{T^v}{d_y}{\rm{cos}}{\mu _{\rm{e}}}}&{\bar QUm_z^\alpha }&{\bar QUm_z^r + m{v_{\rm{D}}}{c_x} - m{v_{\rm{D}}}{c_y}{\alpha _{\rm{e}}}}&{\begin{array}{*{20}{l}} {(mg{c_y} - \rho U{b_y}){\rm{cos}}{\vartheta _{\rm{e}}} - }\\ {( - mg{c_x} + \rho U{b_x}){\rm{sin}}{\vartheta _{\rm{e}}}} \end{array}}\\ 0&0&{m{v_{\rm{D}}}}&0 \end{array}} \right] \end{array}$ （31）
 $\left\{ \begin{array}{l} {\mathit{\boldsymbol{u}}_{{\rm{ln}}}} = {[\Delta {\delta _z},\Delta p]^{\rm{T}}}\\ {\mathit{\boldsymbol{b}}_{{\rm{ln}}}} = \left[ {\begin{array}{*{20}{c}} 0&{2{T^p}{\rm{cos}}{\mu _{\rm{e}}}}\\ {\bar QSC_y^{{\delta _z}}}&{2{T^p}{\rm{sin}}{\mu _{\rm{e}}}}\\ {\bar QUm_z^{{\delta _z}}}&{2{T^p}{d_x}{\rm{sin}}{\mu _{\rm{e}}} - 2{T^p}{d_y}{\rm{cos}}{\mu _{\rm{e}}}}\\ 0&0 \end{array}} \right] \end{array} \right.$ （32）

 $\left\{ {\begin{array}{*{20}{l}} {C_y^r = C_y^{\bar r}l/{v_{\rm{D}}} + C_y^{{a_y}}{v_{\rm{D}}}}\\ {C_y^{\dot \alpha } = - C_y^{{a_y}}{v_{\rm{D}}},C_y^{\dot r} = C_y^{\dot {\bar r}}l/{v_{\rm{D}}}}\\ {m_z^r = m_z^{{\bar r}}l/{v_{\rm{D}}} + m_z^a{v_{\rm{D}}}}\\ {m_z^{\dot \alpha } = - m_z^{{a_y}}{v_{\rm{D}}},m_z^{\dot r} = m_z^{\dot {\bar r}}l/{v_{\rm{D}}}} \end{array}} \right.$ （33）

 $\left\{ {\begin{array}{*{20}{l}} {C_y^r = C_y^{\bar r}l/{v_{\rm{D}}},C_y^{\dot \alpha } = - C_y^{{a_y}}{v_{\rm{D}}},C_y^{\dot r} = C_y^{\dot {\bar r}}l/{v_{\rm{D}}}}\\ {m_z^r = m_z^{\bar r}l/{v_{\rm{D}}},m_z^{\dot \alpha } = - m_z^{{a_y}}{v_{\rm{D}}},m_z^{\dot r} = m_z^{\dot {\bar r}}l/{v_{\rm{D}}}} \end{array}} \right.$ （34）

 参数 数值 m/kg 453 U/m3 370 l/m 7.18 cx/m 0 cy/m -2.35 bx/m 0 vD/(m·s-1) 10 μe 0 by/m 0 Jz/(N·m2) 145 000 Tv/(N·s·m-1) 15 dx/m 0 dy/m -3.70 ρ/(kg·m-3) 1.225 ϑe/(°) 0 αe/(°) 0

 $\left\{ \begin{array}{l} \begin{array}{*{20}{l}} {{\lambda _{11}} = {K_{11}}\rho U = 35.34{\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{kg/}}{{\rm{m}}^{\rm{3}}}}\\ {C_y^{\dot \alpha } = - {v_{\rm{D}}}C_y^{{a_y}} = - {v_{\rm{D}}}\frac{{ - {K_{22}}\rho U}}{{\rho {U^{2/3}}v_{\rm{D}}^2/2}} = 1.338{\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{s}}} \end{array}\\ \begin{array}{*{20}{l}} {C_y^{\dot r} = \frac{{ - {K_{26}}\rho {U^{4/3}}}}{{\rho {U^{2/3}}v_{\rm{D}}^2/2}} = 0.098{\kern 1pt} {\kern 1pt} {\kern 1pt} 95{\kern 1pt} {\kern 1pt} {\kern 1pt} {{\rm{s}}^{\rm{2}}}}\\ {m_z^{\dot \alpha } = - {v_{\rm{D}}}m_z^{{a_y}} = - {v_{\rm{D}}}\frac{{ - {K_{62}}\rho {U^{4/3}}}}{{\rho Uv_{\rm{D}}^2/2}} = - 0.137{\kern 1pt} {\kern 1pt} {\kern 1pt} 8{\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{s}}} \end{array}\\ \begin{array}{*{20}{l}} {m_z^{\dot r} = \frac{{ - {K_{66}}\rho {U^{5/3}}}}{{\rho Uv_{\rm{D}}^2/2}} = - 0.485{\kern 1pt} {\kern 1pt} {\kern 1pt} 5{\kern 1pt} {\kern 1pt} {\kern 1pt} {{\rm{s}}^2}}\\ {{C_{x0}} = 0.055{\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{m/s}},C_x^{{v_{\rm{D}}}} = - 0.000{\kern 1pt} {\kern 1pt} {\kern 1pt} 4{\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{m/s}}} \end{array}\\ \begin{array}{*{20}{l}} {C_y^\alpha = 0.82,m_z^a = 0.52}\\ {C_y^r = C_y^{\bar r}\frac{l}{{{v_{\rm{D}}}}} = 0.581{\kern 1pt} {\kern 1pt} {\kern 1pt} 5{\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{s}}}\\ {m_z^r = m_z^{\bar r}\frac{l}{{{v_{\rm{D}}}}} = - 0.7538{\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{s}}} \end{array} \end{array} \right.$ （35）

 $\left\{ {\begin{array}{*{20}{l}} {C_y^r = C_y^{\bar r}\frac{l}{{{v_{\rm{D}}}}} - C_y^{\dot \alpha } = - 0.756{\kern 1pt} {\kern 1pt} {\kern 1pt} 5{\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{s}}}\\ {m_z^r = m_z^{\bar r}\frac{l}{{{v_{\rm{D}}}}} - m_z^{\dot \alpha } = - 0.616{\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{s}}} \end{array}} \right.$ （36）

 图 2 纵向扰动系统的根轨迹 Fig. 2 Root locus of longitudinal perturbation system

 图 3 复特征根随飞行速度的变化 Fig. 3 Variation of complex characteristic roots with flight velocity
 图 4 实特征根随飞行速度的变化 Fig. 4 Variation of real characteristic roots with flight velocity

4 结论

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http://dx.doi.org/10.7527/S1000-6893.2020.23648

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#### 文章信息

LIN Xianwu, WANG Shichao, LI Zhibin, LAN Weiyao

Fusing method for dynamic derivatives and added mass of airships

Acta Aeronautica et Astronautica Sinica, 2020, 41(8): 123648.
http://dx.doi.org/10.7527/S1000-6893.2020.23648