﻿ 航空声呐浮标的水下减振系统研究
 舰船科学技术  2022, Vol. 44 Issue (20): 124-128    DOI: 10.3404/j.issn.1672-7649.2022.20.025 PDF

Research on underwater vibration damping system of aerial sonobuoy
CHENG Hao, ZHAO Hai-chao
The 715 Research Institute of CSSC, Hangzhou 310032, China
Abstract: Aerial sonobuoy realizes the detection function by collecting the radiated noise of underwater target by hydrophone. In order to avoid the interference caused by wave vibration to hydrophone, a set of vibration damping system composed of bungee cord and damping disk is designed. The system model is established, and the factors affecting the damping effect are analyzed. The natural frequency of the system can be effectively adjusted by choosing the appropriate bungee cord. The vibration amplitude of the system can be significantly adjusted by changing the damping disk. The characteristics of bungee cord are studied by test. According to the research method and conclusion, it can guide the design of underwater vibration damping system of new sonobuoy.
Key words: sonobuoy     hydrophone     bungee cord     damping disk
0 引　言

1 力学原理

1.1 无阻尼条件下的自由振动

 $m\ddot{x}+kx=0 。$ (1)

${\omega }_{0}=\sqrt{k/m}$ ，则 ${f}_{1}={\omega }_{0}/2\text{π}$ 为单自由度无阻尼振动系统的固有频率。

1.2 有阻尼条件下的自由振动

 $m\ddot{x}+R\dot{x}+kx=0。$ (2)

${\delta }^{2} > {\omega }_{0}^{2}（{R}^{2} > 4mk）$ ，系统为大阻尼系统，不会发生自由振动。

${\delta }^{2} < {\omega }_{0}^{2}（{R}^{2} < 4mk）$ ，系统为小阻尼系统， ${\mu }_{1}, {\mu }_{2}= -\mathrm{\delta }\pm \mathrm{j}\sqrt{{\omega }_{0}^{2}{-\delta }^{2}}$ ，令 $\mathrm{\Omega }=\sqrt{{\omega }_{0}^{2}{-\delta }^{2}}$ ，则 $x\left(t\right)={c}_{1}{e}^{(-\delta +j\Omega )t}+ {c}_{2}{e}^{(-\delta -j\Omega )t}=\sqrt{{c}_{1}^{2}+{c}_{2}^{2}}{e}^{-\delta t}\rm{cos}(\Omega t+\varphi )$ 。式中： ${c}_{1}$ ${c}_{2}$ $\varphi$ 由初始条件决定，系统振幅 $\sqrt{{c}_{1}^{2}+{c}_{2}^{2}}{e}^{-\mathrm{\delta }\mathrm{t}}$ 随时间衰减。系统的固有频率 ${f}_{2}=\mathrm{\Omega }/2\text{π} =\sqrt{{\omega }_{0}^{2}{-\delta }^{2}}/2\text{π}$ ，比无s阻尼时的固有频率降低。在极小阻尼条件下，即 ${\omega }_{0}^{2}{\gg \delta }^{2}$ 时，近似有 ${f}_{2}={f}_{1}$

1.3 有阻尼条件下的受迫振动

 $m\ddot{x}+R\dot{x}+kx=k{{A}}_{0}\mathrm{cos}\mathrm{\omega }{t}。$ (3)

 $T\left(\omega \right)=\frac{A}{{A}_{0}}=\frac{k}{\sqrt{{\left(\omega R\right)}^{2}+{(k-m{\omega }^{2})}^{2}}}。$ (4)
2 模型建立 2.1 浮标漂浮状态

 图 1 浮标漂浮状态示意图 Fig. 1 Schematic diagram of buoy floating
2.2 减振系统模型

 图 2 减振系统示意图 Fig. 2 Schematic diagram of vibration damping system
 \begin{aligned}\begin{cases}{m}_{1}\ddot{{x}_{1}}+R\dot{{x}_{1}}+{k}_{1}{x}_{1}+{k}_{2}\left({x}_{1}-{x}_{2}\right)={k}_{1}{{A}}_{0}\mathrm{cos}\mathrm{\omega }{t}，\\ {m}_{2}\ddot{{x}_{2}}+{{k}_{2}x}_{2}-{k}_{2}{x}_{1}=0。\end{cases}\end{aligned} (5)

 $\left\{\begin{array}{c}{{x}_{1}=B}_{1}\mathrm{cos}\left(\omega t+{\varnothing }_{1}\right)={A}_{1}\mathrm{sin}\omega t+{A}_{2}\mathrm{cos}\omega t，\\ {{x}_{2}=B}_{2}\mathrm{cos}\left(\omega t+{\varnothing }_{2}\right)={A}_{3}\mathrm{sin}\omega t+{A}_{4}\mathrm{cos}\omega t。\end{array}\right.$ (6)

 \begin{aligned}\begin{cases}\left({k}_{1}+{k}_{2}-{m}_{1}{\omega }^{2}\right){A}_{1}-\omega {c}_{1}{A}_{2}-{k}_{2}{A}_{3}={k}_{1}{A}_{0}，\\ \left({k}_{1}+{k}_{2}-{m}_{1}{\omega }^{2}\right){A}_{2}+\omega {c}_{1}{A}_{1}-{k}_{2}{A}_{4}=0，\\ \left({k}_{2}-{m}_{2}{\omega }^{2}\right){A}_{3}-{k}_{2}{A}_{1}=0，\\ \left({k}_{2}-{m}_{2}{\omega }^{2}\right){A}_{4}-{k}_{2}{A}_{2}=0。\end{cases}\end{aligned} (7)

 \begin{aligned} &{T\left(\mathrm{\omega }\right)=\frac{{B}_{2}}{{A}_{0}}=\frac{{k}_{1}{k}_{2}}{\sqrt{{h}^{2}+{(e-{k}_{2}^{2})}^{2}}}=}\\ &{\frac{{k}_{1}{k}_{2}}{\sqrt{{({k}_{2}{R}_{1}\mathrm{\omega }-{m}_{2}{R}_{1}{\omega }^{3})}^{2}+{({k}_{1}{k}_{2}-{m}_{1}{{k}_{2}\omega }^{2}-{m}_{2}{{k}_{1}\omega }^{2}-{m}_{2}{{k}_{2}\omega }^{2}+{m}_{1}{{m}_{2}\omega }^{2})}^{2}}}。}\end{aligned} (8)
3 模型分析

 图 3 传递函数随阻力系数变化曲线 Fig. 3 Curve of transfer function with drag coefficient

 图 4 传递函数随弹力系数变化曲线 Fig. 4 Curve of transfer function with spring coefficient

 图 5 传递函数随水听器质量变化曲线 Fig. 5 Curve of transfer function with hydrophone mass
4 模型拓展

 ${m}_{e}=\frac{6}{5}\mathrm{\rho }{d}^{5/2}{A}^{1/2}，$ (9)
 ${R}_{e}=\frac{11}{15}\rho \omega {d}^{5/2}{A}^{1/2}。$ (10)

 $T\left(\mathrm{\omega }\right)=\frac{{k}_{1}}{\sqrt{{（\mathrm{\omega }{R}_{e}）}^{2}+{({k}_{1}-{m}_{e}{\omega }^{2})}^{2}}} 。$ (11)

 $T\left(\mathrm{\omega }\right)=\frac{1}{\sqrt{{0.3734\left(\mathrm{\omega }/{\omega }_{e}\right)}^{4}+{[1-{\left(\omega /{\omega }_{e}\right)}^{2}]}^{2}}}，$ (12)

 ${f}_{r}=\frac{\omega }{2\text{π} }=\frac{0.88{\varOmega }_{0}}{2\text{π} }=\frac{11}{25\text{π} }\sqrt{\frac{{k}_{1}}{1.2\rho {d}^{5/2}{A}_{0}^{1/2}}}。$ (13)

5 弹性绳的弹力系数分析

 $\tau =\frac{F}{\epsilon {L}_{0}}。$ (14)

 图 6 直径3.5 mm弹性绳的弹力系数 Fig. 6 Spring coefficient of 3.5 mm bungee cord

 图 7 直径2.5 mm弹性绳的弹力系数 Fig. 7 Spring coefficient of 2.5 mm bungee cord
6 结　语

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