#### 期刊菜单

The Investigation about the Flying Drag Force of Sphere Type Body
DOI: 10.12677/APP.2019.92008, PDF, HTML, XML, 下载: 826  浏览: 2,363  国家自然科学基金支持

Abstract: Due to the mechanical analysis of a flying bbadminton, some phenomena of the badminton sports were discussed and the theoretical explanation is presented. The experiments on the drag coefficient of the flying badminton were conducted. Based on the experiments, the badminton flight phenomenon was analyzed via the commercial software of Matlab. The results can be expanded into the other field and it can be added in the teaching process of fluid dynamic or the other class.

1. 引言

2. 控制方程

Figure 1. The analytical layout of the flying badminton

$\left\{\begin{array}{l}m\frac{{\text{d}}^{2}{s}_{x}}{\text{d}{t}^{2}}=f\cdot \mathrm{cos}\theta \\ m\frac{{\text{d}}^{2}{s}_{y}}{\text{d}{t}^{2}}=f\cdot \mathrm{sin}\theta +mg\end{array}$ (1)

$\left\{\begin{array}{l}m\frac{\text{d}{v}_{x}}{\text{d}t}=-\frac{1.225}{2}{C}_{d}\left({v}_{x}^{2}+{v}_{y}^{2}\right)\frac{{v}_{x}}{\sqrt{\left({v}_{x}^{2}+{v}_{y}^{2}\right)}}\\ m\frac{\text{d}{v}_{y}}{\text{d}t}=-\frac{1.225}{2}{C}_{d}\left({v}_{x}^{2}+{v}_{y}^{2}\right)\frac{{v}_{y}}{\sqrt{\left({v}_{x}^{2}+{v}_{y}^{2}\right)}}+mg\end{array}$ (2)

$\frac{1.225{C}_{d}}{2m}=c$ ，则方程可化为：

$\left\{\begin{array}{l}\frac{\text{d}{v}_{x}}{\text{d}t}=-c{v}_{x}\sqrt{\left({v}_{x}^{2}+{v}_{y}^{2}\right)}\\ \frac{\text{d}{v}_{y}}{\text{d}t}=-c{v}_{y}\sqrt{\left({v}_{x}^{2}+{v}_{y}^{2}\right)}+g\end{array}$ (3)

$t+\text{const}=\frac{1}{2\sqrt{gc}}\mathrm{ln}|\frac{\sqrt{g/c}+{v}_{y}}{\sqrt{g/c}-{v}_{y}}|$

$t=\frac{1}{2\sqrt{gc}}\mathrm{ln}|\frac{\sqrt{g/c}+{v}_{y}}{\sqrt{g/c}-{v}_{y}}|$ (4)

${e}^{2\sqrt{gc}\cdot t}=\frac{\sqrt{g/c}+{v}_{y}}{\sqrt{g/c}-{v}_{y}}$

${s}_{y}={\int }_{0}^{T}2\sqrt{g/c}\frac{{e}^{2\sqrt{gc}\cdot t}}{1+{e}^{2\sqrt{gc}\cdot t}}-\sqrt{g/c}\text{\hspace{0.17em}}\text{d}t=\sqrt{g/c}\cdot t-{c}^{-1}\mathrm{ln}\left(\frac{1+{e}^{2\sqrt{gc}\cdot t}}{{e}^{2\sqrt{gc}\cdot t}}\right)$ (5)

3. 实验结果

Table 1. Test data

Table 2. The velocity result of the test

4. 数值计算

Figure 2. The plot of velocity vs time

Figure 3. The plot of height vs displacement (left) and velocity vs time (right)

Figure 4. The plot of height vs time

Figure 5. The plot of displacement vs time

5. 结论

NOTES

*第一作者。

#通讯作者。

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