一种测量液浮陀螺仪中浮子所受扭矩的方法与装置

摘要

本发明公开了一种测量封闭球腔内浮子所受扭矩的方法与装置。本发明的方法首先是计算封闭球腔内浮子的扭矩，通过计算可知封闭球腔内浮子的扭矩与转子的扭矩大小相同，然后再利用本发明的装置检测转子的扭矩，便可得到封闭球腔内浮子的扭矩。所设计的装置包括底座、待测部件支撑装置、传感器支撑装置、电机固定装置及联轴器。待测部件支撑装置、传感器支撑装置、电机固定装置均固定在底座上。测量时，通过联轴器依次连接待测部件、动态扭矩传感器、电机，并调整三者的相对位置，使其同轴度达到实验允许值后固定。利用数据采集卡采集记录动态扭矩传感器输出的电信号，该电信号与转子所受力矩成正比，得到转子所受力矩数据后便可得到封闭球腔中浮子的受力情况。

主权项

1.一种测量液浮陀螺仪中浮子所受扭矩的方法，其特征在于，通过有效步骤得到液浮陀螺仪中浮子的扭矩与转子的扭矩大小相同，检测液浮陀螺仪中转子的扭矩即可得到浮子的扭矩；所述的有效步骤为：步骤一：所述液浮陀螺仪中浮子的状态为悬浮于封闭球腔内流体中，与球腔内的流体接触，其所受扭矩来自于球腔内的流体的摩擦切应力；封闭球腔内流体的应力与应变的变化率成正比，即为牛顿流体，已知在直角坐标系（x，y，z）下牛顿流体的应力应变关系为：$\left\{\begin{array}{c}{\sigma }_x=-p+2\mu \frac{{\partial u}_x}{\partial x}-\frac{2}{3}\mu \mathrm{div}u\\ {\sigma }_y=-p+2\mu \frac{{\partial u}_y}{\partial y}-\frac{2}{3}\mu \mathrm{div}u\\ {\sigma }_z=-p+2\mu \frac{{\partial u}_z}{\partial z}-\frac{2}{3}\mu \mathrm{div}u\\ {\tau }_{\mathrm{xy}}={\tau }_{\mathrm{yx}}=\mu (\frac{{\partial u}_y}{\partial x}+\frac{{\partial u}_x}{\partial y})\\ {\tau }_{\mathrm{xz}}={\tau }_{\mathrm{zx}}=\mu (\frac{{\partial u}_z}{\partial x}+\frac{{\partial u}_x}{\partial z})\\ {\tau }_{\mathrm{zy}}={\tau }_{\mathrm{yz}}=\mu (\frac{{\partial u}_y}{\partial z}+\frac{{\partial u}_z}{\partial y})\end{array}\right.---\left(1\right)$式（1）中p为封闭球腔内某点的静压强，σ_x、σ_y、σ_z分别为封闭球腔内流体在x，y，z三个方向的正应力，τ_xy、τ_xz为封闭球腔内流体在x平面上分别沿y轴和z轴的切应力，τ_yx、τ_yz为封闭球腔内流体在y平面上分别沿x轴和z轴的切应力，τ_zx、τ_zy为封闭球腔内流体在z平面上分别沿x轴和y轴的切应力；μ为封闭球腔内流体的粘度系数，$u=\left[\begin{array}{c}{u}_x\\ u_y\\ u_z\end{array}\right],$$\mathrm{div}u=\frac{{\partial u}_x}{\partial x}+\frac{{\partial u}_y}{\partial y}+\frac{{\partial u}_z}{\partial z},$u_x、u_y、u_z分别为在直角坐标系（x，y，z）下封闭球腔内流体在x，y，z三个方向的速度，ρ为封闭球腔内流体的密度；封闭球腔内的流体符合连续介质假设和质量守恒原理，即在单位时间t内流出与流入流体微团即边长为dx、dy、dz的六面体内的质量变化量等于该流体微团体内质量随时间的变化率，得到在直角坐标系（x，y，z）下封闭球腔内流体运动的连续性方程为：$\frac{\partial \rho }{\partial t}+\mathrm{di\rho vu}=---\left(2\right)$由于液浮陀螺仪中的流体不可压，即密度ρ为常数，则有$\frac{\partial \rho }{\partial t}=\frac{\partial \rho }{{\partial u}_x}=\frac{\partial \rho }{{\partial u}_y}=\frac{\partial \rho }{{\partial u}_z}=0---\left(3\right)$将式（3）代入式（2）有：divu＝0；则直角坐标系（x，y，z）下液浮陀螺仪中流体运动的连续性方程为：$\mathrm{div}u=\frac{{\partial u}_x}{\partial x}+\frac{{\partial u}_y}{\partial y}+\frac{{\partial u}_z}{\partial z}=0---\left(4\right)$步骤二：封闭球腔内运动的流体存在体积力和表面力，依据牛顿第二运动定律得到在直角坐标系（x，y，z）下封闭球腔内流体微团的加速度与质量、体积力、表面力的关系为：$\left\{\begin{array}{c}\frac{{\mathrm{Du}}_x}{\mathrm{Dt}}=f_x+\frac{1}{\rho }(\frac{{\partial \sigma }_x}{\partial x}+\frac{{\partial \tau }_{\mathrm{yx}}}{\partial y}+\frac{{\partial \tau }_{\mathrm{zx}}}{\partial z})\\ \frac{{\mathrm{Du}}_y}{\mathrm{Dt}}=f_y+\frac{1}{\rho }(\frac{{\partial \tau }_{\mathrm{xy}}}{\partial y}+\frac{{\partial \sigma }_y}{\partial x}+\frac{{\partial \tau }_{\mathrm{zy}}}{\partial z})\\ \frac{{\mathrm{Du}}_z}{\mathrm{Dt}}=f_z+\frac{1}{\rho }(\frac{{\partial \tau }_{\mathrm{xz}}}{\partial y}+\frac{{\partial \tau }_{\mathrm{yz}}}{\partial z}+\frac{{\partial \sigma }_z}{\partial z})\end{array}\right.---\left(5\right)$式（5）中f_x、f_y、f_z分别为在直角坐标系（x，y，z）下封闭球腔内流体微团在x，y，z三个方向的体积力；将式（1）代入式（5）得到封闭球腔内流体运动的纳维-斯托克斯方程：$\left\{\begin{array}{c}\rho \frac{{\mathrm{Du}}_x}{\mathrm{Dt}}={\mathrm{\rho f}}_x-\frac{\partial p}{\partial x}+2\frac{\partial }{\partial x}\left(\mu \frac{{\partial u}_x}{\partial x}\right)+\frac{\partial }{\partial y}\left[\mu (\frac{{\partial u}_x}{\partial y}+\frac{{\partial u}_y}{\partial x})\right]+\frac{\partial }{\partial z}\left[\mu (\frac{{\partial u}_x}{\partial z}+\frac{{\partial u}_z}{\partial x})\right]-\frac{2}{3}\frac{\partial }{\partial x}\left(\mu \mathrm{div}u\right)\\ \rho \frac{{\mathrm{Du}}_y}{\mathrm{Dt}}={\mathrm{\rho f}}_y-\frac{\partial p}{\partial y}+\frac{\partial }{\partial x}\left[\mu (\frac{{\partial u}_y}{\partial x}+\frac{{\partial u}_x}{\partial y})\right]+2\frac{\partial }{\partial y}\left(\mu \frac{{\partial u}_y}{\partial y}\right)+\frac{\partial }{\partial z}\left[\mu (\frac{{\partial u}_y}{\partial z}+\frac{{\partial u}_z}{\partial y})\right]-\frac{2}{3}\frac{\partial }{\partial y}\left(\mu \mathrm{div}u\right)\\ \rho \frac{{\mathrm{Du}}_z}{\mathrm{Dt}}={\mathrm{\rho f}}_z-\frac{\partial p}{\partial z}+\frac{\partial }{\partial x}\left[\mu (\frac{{\partial u}_z}{\partial x}+\frac{{\partial u}_x}{\partial z})\right]+\frac{\partial }{\partial y}\left[\mu (\frac{{\partial u}_y}{\partial z}+\frac{{\partial u}_z}{\partial y})\right]+2\frac{\partial }{\partial z}\left(\mu \frac{{\partial u}_z}{\partial z}\right)-\frac{2}{3}\frac{\partial }{\partial z}\left(\mu \mathrm{div}u\right)\end{array}\right.---\left(6\right)$在液浮陀螺仪中，流体不可压缩,设其粘度系数μ为常数，同时将式（4）代入式（6）得到液浮陀螺仪中流体运动的纳维-斯托克斯方程为：$\left\{\begin{array}{c}\rho (\frac{{\partial u}_x}{\partial t}+u_x\frac{{\partial u}_x}{\partial x}+u_y\frac{{\partial u}_x}{\partial y}+u_z\frac{{\partial u}_x}{\partial z})={\mathrm{\rho f}}_x-\frac{\partial p}{\partial x}+\mu (\frac{{\partial }^2{u}_x}{{\partial x}^2}+\frac{{\partial }^2{u}_x}{{\partial y}^2}+\frac{{\partial }^2{u}_x}{{\partial z}^2})\\ \rho (\frac{{\partial u}_y}{\partial t}+u_x\frac{{\partial u}_y}{\partial x}+u_y\frac{{\partial u}_y}{\partial y}+u_z\frac{{\partial u}_y}{\partial z})={\mathrm{\rho f}}_y-\frac{\partial p}{\partial y}+\mu (\frac{{\partial }^2{u}_y}{{\partial x}^2}+\frac{{\partial }^2{u}_y}{{\partial y}^2}+\frac{{\partial }^2{u}_y}{{\partial z}^2})\\ \rho (\frac{{\partial u}_z}{\partial t}+u_x\frac{{\partial u}_z}{\partial x}+u_y\frac{{\partial u}_z}{\partial y}+u_z\frac{{\partial u}_z}{\partial z})={\mathrm{\rho f}}_z-\frac{\partial p}{\partial z}+\mu (\frac{{\partial }^2{u}_z}{{\partial x}^2}+\frac{{\partial }^2{u}_z}{{\partial y}^2}+\frac{{\partial }^2{u}_z}{{\partial z}^2})\end{array}\right.---\left(7\right)$步骤三：液浮陀螺仪中的浮子位于封闭球腔内，为了便于计算，在液浮陀螺仪中建立柱坐标系（r，θ，z），即令x=rsinθ，y=rcosθ，将x=rsinθ，y=rcosθ代入式（7）得到柱坐标系（r，θ，z）下液浮陀螺仪中流体运动的纳维-斯托克斯方程为：$\left\{\begin{array}{c}\rho (\frac{{\partial u}_r}{\partial t}+u_r+\frac{{\partial u}_r}{\partial r}+\frac{u_{\theta }}{r}\frac{{\partial u}_r}{\partial \theta }+u_z\frac{{\partial u}_r}{\partial z}-\frac{u_{\theta }^2}{r})={\mathrm{\rho f}}_r-\frac{\partial p}{\partial r}+\mu (\frac{{\partial }^2{u}_r}{{\partial r}^2}+\frac{1}{r}\frac{{\partial u}_r}{\partial r}+\frac{1}{r^2}\frac{{\partial }^2{u}_r}{{\partial \theta }^2}+\frac{{\partial }^2{u}_r}{{\partial z}^2}-\frac{2}{r^2}\frac{{\partial u}_{\theta }}{\partial \theta }-\frac{u_r}{r^2})\\ \rho (\frac{{\partial u}_{\theta }}{\partial t}+u_r\frac{{\partial u}_{\theta }}{\partial r}+\frac{u_{\theta }}{r}\frac{{\partial u}_{\theta }}{\partial \theta }+u_z\frac{{\partial u}_{\theta }}{\partial z}+\frac{u_r{u}_{\theta }}{r})=\rho f_{\theta }-\frac{1}{r}\frac{\partial \rho }{\partial \theta }+\mu (\frac{{\partial }^2{u}_{\theta }}{{\partial r}^2}+\frac{1}{r}\frac{{\partial u}_{\theta }}{\partial r}+\frac{1}{r^2}\frac{{\partial }^2{u}_{\theta }}{{\partial \theta }^2}+\frac{{\partial }^2{u}_{\theta }}{{\partial z}^2}+\frac{2}{r^2}\frac{{\partial u}_r}{\partial \theta }-\frac{u_{\theta }}{r^2})\\ \rho (\frac{{\partial u}_z}{\partial t}+u_r\frac{{\partial u}_z}{\partial r}+\frac{u_{\theta }}{r}\frac{{\partial u}_z}{\partial \theta }+u_z\frac{{\partial u}_z}{\partial r})={\mathrm{\rho f}}_z-\frac{\partial p}{\partial z}+\mu (\frac{{\partial }^2{u}_z}{{\partial r}^2}+\frac{1}{r}\frac{{\partial u}_z}{\partial r}+\frac{1}{r^2}\frac{{\partial }^2{u}_z}{{\partial z}^2}+\frac{{\partial }^2{u}_z}{{\partial z}^2})\end{array}\right.---\left(8\right)$式（8）中：u_r为液浮陀螺仪中流体的径向速度，u_θ为液浮陀螺仪中流体的周向速度，u_z为液浮陀螺仪中流体的轴向速度；f_r为液浮陀螺仪中流体的径向体积力；f_θ为液浮陀螺仪中流体的周向体积力；f_z为液浮陀螺仪中流体的轴向体积力；r为液浮陀螺仪中封闭球腔的半径；在液浮陀螺仪中，当浮子转速稳定后，流将体在球腔中的流动设为定常流动，即u_z=0；由几何条件的轴对称性和流场中没有点源和点汇的特性得到液浮陀螺仪中u_r＝0，式（8）中液浮陀螺仪中某点的静压强p和周向速度u_θ均是半径r的函数，则有将u_z=0，u_r＝0，和代入式（8），得到柱坐标系（r，θ，z）下液浮陀螺仪中流体纳维斯托克斯方程的简化形式，即：$\left\{\begin{array}{c}{f}_r-\frac{1}{\rho }\frac{\mathrm{dp}}{\mathrm{dr}}+\frac{u_{\theta }^2}{r}=0\\ \frac{\mu }{\rho }(\frac{d^2{u}_{\theta }}{{\mathrm{dr}}^2}+\frac{1}{r}\frac{{\mathrm{du}}_{\theta }}{\mathrm{dr}}-\frac{u_{\theta }}{r^2})+f_{\theta }=0\end{array}\right.---\left(9\right)$步骤四：已知液浮陀螺仪中流体运动的边界条件为：在半径处的环形截面上，转子内表面处的径向速度u_r＝0，周向速度为即有：忽略液浮陀螺仪中流体的体积力，此时式（9）中可变形为：$\left\{\begin{array}{c}\frac{u_{\theta }^2}{r}=\frac{1}{\rho }\frac{\mathrm{dp}}{\mathrm{dr}}\\ \frac{d^2{u}_{\theta }}{{\mathrm{dr}}^2}+\frac{1}{r}\frac{{\mathrm{du}}_{\theta }}{\mathrm{dr}}-\frac{u_{\theta }}{r^2}=0\end{array}\right.---\left(11\right)$将边界条件式（10）代入微分方程得到：将u_r＝0和式（12）代入已知的柱坐标系下牛顿流体的应力与应变关系式得到液浮陀螺仪中流体的摩擦切应力为：式中τ_rθ指在半径为r的切面上指向旋转方向的切应力；则液浮陀螺仪中流体与浮子接触面上所受的扭矩为：液浮陀螺仪中流体与转子接触表面上所受的扭矩为：式中：r_外为转子半径；r_内为浮子半径；w_外为转子转速；w_内为浮子转速；通过式（13）和式（14）可以看出液浮陀螺仪中流体与浮子接触面上的扭矩与体与转子接触表面上的扭矩大小相等，方向相反，根据牛顿第三定律，得到液浮陀螺仪中浮子所受的扭矩为：M_浮子＝-M_内＝M_外。