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Fault Modeling of Blade Tip Defect of a Quadrotor
DOI: 10.12677/JAST.2022.102002, PDF, HTML, XML, 下载: 59  浏览: 388  科研立项经费支持

Abstract: It is easy for the blade tip of quadrotor to have defect or fracture in real application. This would directly affect the generated thrust of quadrotor such that the flight quality or performance of quadrotor may be degraded. Therefore, the effect of blade tip defect, which is actually one kind of actuator faults, is analyzed from mechanism in this paper, and then the real generated thrust is calculated accordingly. Finally, the model of blade tip defect fault of quadrotor is obtained, which provides a foundation for designing the flight control scheme for quadrotor.

1. 引言

2. 基本知识

2.1. 桨叶动量定理

1) 穿过圆盘的空气流管与外部气流不存在相互作用；

2) 螺旋桨包含了无限多个桨叶元素；

3) 圆盘厚度无限薄；

4) 通过圆盘的气流的垂直速度是连续的；

5) 空气不可压缩。

${T}_{MT}=A\left({p}_{1}-{p}_{2}\right)={\stackrel{˙}{m}}_{A}\left({v}_{-\infty }-{v}_{+\infty }\right)={\rho }_{A}A{v}_{1}\left({v}_{-\infty }-{v}_{+\infty }\right)$ (1)

Figure 1. The diagram of blade model in momentum theory

2.2. 叶素理论

$\begin{array}{l}\text{d}{L}_{BET}=\frac{1}{2}{\rho }_{A}{v}_{H}^{2}{C}_{L}c\text{d}r\\ \text{d}{D}_{BET}=\frac{1}{2}{\rho }_{A}{v}_{H}^{2}{C}_{D}c\text{d}r\end{array}$ (2)

${C}_{L}=a\alpha =a\left({\theta }_{I}-{\varphi }_{I}\right)$

${\theta }_{I}={\theta }_{0}-{\theta }_{tw}\frac{r}{R}$

Figure 2. The diagram of force analysis on the blade section

${\varphi }_{I}=\frac{{v}_{V}}{{v}_{H}}$

$\text{d}{L}_{BET}=\frac{1}{2}{\rho }_{A}{v}_{H}^{2}a\left({\theta }_{0}-{\theta }_{tw}\frac{r}{R}-\frac{{v}_{V}}{{v}_{H}}\right)c\text{d}r$ (3)

$\text{d}{T}_{BET}=\text{d}{L}_{BET}\mathrm{cos}{\varphi }_{I}-\text{d}{D}_{BET}\mathrm{sin}{\varphi }_{I}\approx \text{d}{L}_{BET}$ (4)

$\text{d}{T}_{BET}$$\left[0,R\right]$ 区间求积分即可得到单片桨叶所能产生的推力 ${T}_{BET}$，具体为：

${T}_{BET}={\int }_{0}^{R}\frac{\text{d}{T}_{BET}}{\text{d}r}\text{d}r={\int }_{0}^{R}\frac{1}{2}{\rho }_{A}{v}_{H}^{2}a\left({\theta }_{0}-{\theta }_{tw}\frac{r}{R}-\frac{{v}_{V}}{{v}_{H}}\right)c\text{d}r$ (5)

3. 桨尖缺损故障建模

${v}_{H}=\omega r$

$\begin{array}{l}{p}_{-\infty }+\frac{1}{2}{\rho }_{A}{v}_{-\infty }^{2}={p}_{1}+\frac{1}{2}{\rho }_{A}{v}_{V}^{2}\\ {p}_{2}+\frac{1}{2}{\rho }_{A}{v}_{2}^{2}={p}_{+\infty }+\frac{1}{2}{\rho }_{A}{v}_{+\infty }^{2}\end{array}$

${v}_{1}=\frac{{v}_{+\infty }+{v}_{-\infty }}{2}$

${v}_{I}={v}_{1}-{v}_{-\infty }=\frac{{v}_{+\infty }-{v}_{-\infty }}{2}$

$T=2{\rho }_{A}A{v}_{1}{v}_{I}=2{\rho }_{A}A{v}_{V}^{2}$

$T=\frac{mg}{4}$

${v}_{I}={v}_{V}=\frac{1}{2}\sqrt{\frac{mg}{2{\rho }_{A}A}}$

$\lambda =\frac{{v}_{I}}{\omega R}$

${v}_{V}=\lambda \omega R$ (6)

$\begin{array}{c}{T}_{BET}\approx \frac{{\rho }_{A}ac}{2}{\int }_{0}^{kR}\left({\theta }_{0}{v}_{H}^{2}-{\theta }_{tw}\frac{r}{R}{v}_{H}^{2}-{v}_{V}{v}_{H}\right)\text{d}r\\ =\frac{{\rho }_{A}ac{\omega }^{2}{R}^{3}}{2}\left(\frac{{\theta }_{0}{k}^{3}}{3}-\frac{{\theta }_{tw}{k}^{4}}{4}-\frac{\lambda {k}^{2}}{2}\right)\end{array}$

$\begin{array}{c}{T}_{f}=2{T}_{BET}\approx {\rho }_{A}ac{\omega }^{2}{R}^{3}\left(\frac{{\theta }_{0}{k}^{3}}{3}-\frac{{\theta }_{tw}{k}^{4}}{4}-\frac{\lambda {k}^{2}}{2}\right)\\ ={\rho }_{A}ac{\omega }^{2}{R}^{3}{\rho }_{f}\left(\frac{{\theta }_{0}}{3}-\frac{{\theta }_{tw}}{4}-\frac{\lambda }{2}\right)\end{array}$

${\rho }_{f}=\frac{\left(\frac{{\theta }_{0}{k}^{3}}{3}-\frac{{\theta }_{tw}{k}^{4}}{4}-\frac{\lambda {k}^{2}}{2}\right)}{\left(\frac{{\theta }_{0}}{3}-\frac{{\theta }_{tw}}{4}-\frac{\lambda }{2}\right)}$

${T}_{f}={\rho }_{f}{T}_{n}$

4. 算例

${\theta }_{0}=0.67\text{\hspace{0.17em}}\text{rad},\text{\hspace{0.17em}}{\theta }_{tw}=0.29\text{\hspace{0.17em}}\text{rad},\text{\hspace{0.17em}}\lambda =0.05$

Figure 3. The curve of ${\rho }_{f}$ with respect to k

5. 结论

NOTES

*通讯作者。

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