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激光推进的基本原理是将激光能量转化为飞行器的动能,冲量耦合系数Cm就是衡量这一性能的重要参数,定义为在激光作用下产生的冲量∆J与入射激光能量E的比值,或者为产生的推力F与入射激光功率P的比值为:
$$ {C_m} = \frac{{\Delta J}}{E} = \frac{F}{P} $$ (1) 在烧蚀模式中引入比烧蚀能Q的概念,定义为烧蚀单位质量的工质所消耗的激光能量为:
$$ Q = \frac{E}{{\Delta m}} $$ (2) 式中:∆m为烧蚀质量。根据动量守恒原理,喷射过程中的工质和烧蚀产物满足下面关系:
$$ \Delta m{v_E} = m\Delta v $$ (3) 式中:vE为喷射速度;m∆v为工质在激光烧蚀喷射过程中产生的动量。
Cm和Q是实验中容易测量的参数,因此可以由如下关系得到喷射速度vE为:
$$ {v_E} = {C_m}Q $$ (4) 表述激光推进性能的另一个重要参数是比冲Isp,同化学火箭一样,比冲定义为单位质量工质产生的冲量,为:
$$ {I_{sp}} = \frac{{\Delta J}}{{\Delta mg}} $$ (5) 式中:g为重力加速度。由公式(1)~(5)可得到比冲同喷射速度的关系为:
$$ {I_{sp}} = \frac{{{v_E}}}{g} $$ (6) 无量纲参数烧蚀效率ηAB定义为激光脉冲能量转化为喷射动能的效率,为:
$$ {\eta _{AB}} = \frac{{{W_E}}}{E} = \frac{{\Delta mv_E^2}}{{2E}} $$ (7) 烧蚀效率ηAB、冲量耦合系数Cm和比冲Isp的关系为:
$$ 2{\eta _{AB}} = {C_m}{v_E} = {C_m}{I_{sp}}g $$ (8) 在推进的实际应用中,希望冲量耦合系数和比冲都具有高数值,但由公式(8)可知,两者乘积具有上限;而且在一定的烧蚀效率ηAB下,冲量耦合系数和比冲是一对相互制约的参数,高比冲势必会导致较低的冲量耦合系数,因此要根据不同的任务来选择适当的激光参数和工质。
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实验系统示意图和实验场景图如图2所示。整个实验系统由半导体激光器、真空舱、扭摆横梁、位移传感器、电磁阻尼器、信号接收器、位移台控制器和电磁阻尼控制器等部分组成。真空仓的真空度能够达到10−5 Pa量级。
图 2 实验系统。 (a) 示意图; (b) 实验场景图
Figure 2. Experimental system. (a) Schematic diagram; (b) Experimental scene graph
设扭摆系统的转动惯量为
$ J $ ,空气阻尼系数为$ c $ ,枢轴的刚度系数为$ k $ ,扭摆扭转角为$ \theta $ ,扭转角速度为$ \dot \theta $ ,角加速度为$ \ddot \theta $ ,外力作用力臂长为$ d $ ,$ t $ 时刻的外力大小为$ f(t) $ ;$ {T_0} $ 为外力作用时间,由动量矩定理,扭摆系统的运动方程表示为:$$ \left\{ \begin{gathered} J\ddot \theta + c\dot \theta + k\theta = f(t)d\;\;{\text{ }}0 < t < {T_0} \hfill \\ J\ddot \theta + c\dot \theta + k\theta = 0{\text{ }}\;\;t > {T_0} \hfill \\ \end{gathered} \right. $$ (11) 外力作用时间(
$ 0 < t < {T_0} $ )内,上述运动方程可改写为:$$ \ddot \theta + 2\xi {\omega _{{n}}}\dot \theta + \omega _{{n}}^2\theta = f(t)d/J\;\;{\text{ 0 < }}t < {T_0} $$ (10) $$ {\omega _{{n}}} = \sqrt {\frac{k}{J}}\;\;\;\; {\text{ }}\xi = \frac{c}{{2\sqrt {kJ} }} $$ 式中:
$ \xi $ 为阻尼比;${\omega _{{n}}}$ 为固有振动频率。利用拉普拉斯变换,令
$\varTheta (s){\text{ = }}L[\theta (t)]$ ,当初始扭转角和初始角速度都为零时,单位脉冲力$ f(t) = \delta (t) $ 作用下,有:$$ \varTheta (s) = (d/J)\frac{1}{{{s^2} + 2\xi {\omega _{{n}}}s + \omega _{{n}}^2}} = (d/J)\frac{1}{{{{(s + \xi {\omega _{{n}}})}^2} + \omega _{{d}}^2}} $$ 式中:
${\omega _{{d}}} = \sqrt {1 - {\xi ^2}} {\omega _{{n}}}$ 为振动频率。进行拉普拉斯反变换时,得到单位脉冲力的扭转角响应为:
$$ h(t) = {L^{ - 1}}[\varTheta (s)] = (d/J)\frac{1}{{{\omega _{{d}}}}}{{\rm e}^{ - \xi {\omega _{{n}}}t}}\sin {\omega _{{d}}}t $$ (12) 冲量为
$ I $ 的脉冲力$ f(\tau ) = I\delta (\tau ) $ 作用下,扭转角为:$$ \theta (t) = \frac{{Id}}{{J{\omega _{{d}}}}}{{\rm e}^{ - \xi {\omega _{{n}}}t}}\sin {\omega _{{d}}}t $$ (13) 由公式(13)可以看出,扭摆的扭转角是时间的函数,要根据时间
$ t $ 以及对应的扭转角$ \theta (t) $ 计算出冲量$ I $ ,需已知式中的未知常量$ d $ 、${\omega _{{d}}}$ 、$ J $ 和$ \xi $ 。一般而言,由上式计算冲量的方法有两种,一种是利用可输出已知冲量的标定装置得到扭转角极值点与冲量的关系,另一种通过标定公式(13)中的四个未知参数得到扭转角与冲量关系。第一种方法对冲量测量的精度要求较低,认为阻尼接近零,对扭摆振动的影响非常小,式中的指数项近似为1。取冲量作用后扭摆的最大摆角数据,近似扭转角极值与冲量为线性关系为:
$$ {\theta _{\max }} \approx \frac{{Id}}{{J{\omega _{{d}}}}} $$ 实验采用的激光器为自行研制的半导体激光器,具有体积较小、功率密度高等特点,脉宽可实现50~1500 μs可调,可以通过改变输入激光器的电流大小来改变激光器的功率密度,功率密度最高可达106 W/cm2量级,实物如图3所示。采用扫描电子显微镜(SEM)来观察激光烧蚀工质的情况,采用微量天平来测量烧蚀工质的质量,质量测量时,为测量精准,采用多次烧蚀求平均值的方法。
Research on the influence of laser pulse width and power density on ablation performance
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摘要: 微纳卫星的飞速发展对微推力器的性能提出了更高的的要求。激光推进微推力器因其比冲高、推力控制精确、能耗低等特点,为微纳卫星提供了一种性能优异的微推力器选择方案。文中在透射式烧蚀模式下,研究了半导体激光器的激光功率密度和脉宽对激光烧蚀性能的影响。结果表明,在工质厚度为200 μm的工况下,随着激光功率密度的增加,单脉冲冲量和比冲都逐渐增大,而冲量耦合系数和烧蚀效率都存在一个最优值。随着激光脉宽的增加,单脉冲冲量逐渐增加,比冲呈现出先增大后减小的趋势,在250 μs时,比冲达到最大值,约为221.8 s;冲量耦合系数和烧蚀效率都随着脉宽的增大而减小;脉宽超过一定的临界值时,会对激光烧蚀工质的靶坑产生不良影响,使得激光能量和工质严重浪费。激光参数的优化对于激光推进微推力器的工程化应用提供了参考。Abstract: With the rapid development of micro-nano satellites, the performance of micro-thrusters is required to be higher. Because of its high specific impulse, precise thrust control and low energy consumption, the laser-propelled micro-thruster provides an excellent micro-thruster choice scheme for micro-nano satellite. In this paper, the influence of power density and pulse width of semiconductor laser on the performance of laser propulsion was studied in transmission ablation mode. The results show that when the thickness of the working medium layer is 200 μm, with the increase of the laser power density, the single pulse impulse and specific impulse gradually increase. And there is an optimal power density value to the impulse coupling coefficient and ablation efficiency. As the pulse width of the laser increases, the impulse of the single pulse gradually increases, and the specific impulse first increases and then decreases. At 250 μs, the specific impulse reaches its maximum value, which is about 221.8 s. The impulse coupling coefficient and ablation efficiency decrease with the increase of pulse width. When the pulse width exceeds a certain critical value, it will have a bad effect on the target pit of the laser ablation working medium, resulting in a serious waste of laser energy and working medium. The optimizing of laser parameters provides a reference for the engineering application of laser propelled micro-thruster.
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Key words:
- laser propulsion /
- specific impulse /
- impulse coupling coefficient /
- power density /
- pulse width
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