如何计算卫星的干扰扭矩

【原标题:How to Calculate Satellite Disturbance Torques。原网页: https://www.valispace.com/how-to-calculate-satellite-disturbance-torques/ 。原文章遵循CC-BY 3.0协议】

In the first two tutorials of this series we discussed components of the satellite’s power system. We will now explain the necessary steps for designing the Attitude Determination and Control System (ADCS). During the lifetime of an Earth-bound satellite, its attitude (the direction it is pointed towards) gets continually disturbed by different forces, called disturbance torques. These torques have to be mitigated by the ADCS, to make sure the satellite is always pointed in the right direction. In this tutorial, we will discuss how to calculate these disturbance torques.

在本系列的前两个教程中,我们讨论了卫星电源系统的组件。现在,我们将解释设计姿态确定和控制系统(Attitude Determination and Control System,ADCS)的必要步骤。在地球卫星的生命周期中,其姿态(指向的方向)会不断受到称为干扰转矩的不同力的干扰。 ADCS必须减轻这些扭矩,以确保卫星始终指向正确的方向。在本教程中,我们将讨论如何计算这些干扰转矩。

Different types of torques

We distinguish four different types of disturbance torques, namely gravity gradient, solar radiation pressure, magnetic field and aerodynamic torques. Because the ADCS must handle the maximum possible torques, we will try to find the worst-case torques for all different cases. Note that some of these forces can also cause orbit perturbations, but in this tutorial we will just focus on the attitude of the satellite. Below you can see the influence of the different torques on a typical satellite over a range of altitudes. It shows that generally, up to an altitude of around 500 km, the aerodynamic torque is the maximum torque, but going higher up the gravity gradient will take over.

我们区分了四种不同类型的扰动转矩,即重力梯度,太阳辐射压力,磁场和空气动力转矩。因为ADCS必须处理最大可能的扭矩,所以我们将尝试查找所有不同情况下最坏情况的扭矩。注意,这些力中的一些也会引起轨道扰动,但是在本教程中,我们将只关注卫星的姿态。在下面,您可以看到不同扭矩对典型卫星在一定高度范围内的影响。它表明,一般来说,在海拔约500 km处,空气动力扭矩是最大扭矩,但在重力梯度上升时,空气动力扭矩将占主导地位。

不同干扰转矩对典型卫星的影响。从PawełZagórski检索。对影响地球轨道卫星的干扰进行建模。 2012年5月。 重力梯度扭矩

The Earth’s gravitational field varies inversely with distance from the Earth, thus the gravitational field varies slightly across a satellites length. This difference in acceleration causes a torque on the satellite. As a first order approximation, this torque is constant along the satellite’s orbit for an Earth-orientated vehicle and can be approximated as follows:

地球的引力场的影响与距地球的距离成反比,因此引力场在卫星的整个长度上略有变化。这种加速度差异会在卫星上产生扭矩。作为一阶近似值,对于面向地球的飞行器,该扭矩沿卫星轨道是恒定的,可以近似如下:

[\ T_g = \frac{3 \mu}{2 R^3} |I_z – I_y| sin(2\theta) \; \; [Nm] \]

In here, μμ is Earth’s gravitational constant (398600km3/s2398600km3/s2) and R is the orbital radius (dependent on the satellite’s altitude). IzIz and IyIy are the moments of inertia referring to the z- and y-axis. The moments of inertia are highly variable depending on the exact satellite design, to find out how to calculate them, give these different sources a look. Note that IyIy is interchangeable with IzIz, and as we are trying to find the worst-case torque, the moment of inertia which gives the highest torque should be used. ΘΘ is the maximum angle the local vertical makes from the z-axis, as visualized below.

在这里,μ是地球的重力常数(398600km^3 / s^2),R是轨道半径(取决于卫星的高度)。 IzIy是相对于z轴和y轴的惯性矩。惯性矩的变化很大,具体取决于卫星实际上的设计,要了解如何计算惯性矩,就需要注意到这些不同的源。请注意,Iy可与Iz互换,并且当我们试图找到最坏情况的转矩时,应使用能提供最高转矩的惯性矩。 Θ是局部垂直线与z轴的最大夹角,如下图所示。

Solar radiation pressure torque

Radiation (electromagnetic waves) carries momentum according to Maxwell’s theory of electromagnetism. Upon hitting a surface, this momentum can be transferred to the surface. The amount of momentum transferred depends on the type of surface being illuminated. It will be lowest for transparent surfaces, higher for absorbent surfaces and highest for reflective surfaces. In general, it can be said that solar arrays are absorbers and a spacecraft’s body is a reflector. The total force exerted on a spacecraft by solar radiation is as follows:

根据麦克斯韦的电磁学原理,辐射(电磁波)具有动量。在撞击表面时,该动量可以转移到表面上。传递的动量大小取决于被照明表面的类型。透明表面最低,吸收表面最高,反射表面最高。一般而言,可以说太阳能电池板是吸收器,航天器的主体是反射器。太阳辐射施加在航天器上的总力如下:

[\ F = \frac{J_s}{c} A_s cos(I) (1 + q) \; \; [N] \]

Here, Js=1367W/m2Js=1367W/m2 is the solar constant at Earth, c=3⋅108m/sc=3⋅108m/s is the speed of light, AsAs is the total surface area and II is the angle of incidence of the solar radiation. Finally, qq is the reflectance factor. Obviously, this value is not constant over the spacecraft’s body, but with a first order approximation it can be assumed to be constant and equal to 0.6. Now, the torque exerted on the satellite will be:

J_s = 1367W / m^2是地球的太阳常数,c =3⋅108m/ s是光速,A_s是总表面积,I是入射角太阳辐射。最后,q是反射系数。显然,该值在飞船的整个身体上并不是恒定不变的,但是对于一阶近似值,可以假定为恒定且等于0.6。现在,施加在卫星上的扭矩将为: [\ T_{sp} = F ( c_{ps} – cg) \; \; [Nm] \]

Here, cgcg is the center of gravity and cpscps is the center of solar pressure. This center can be found in a similar way to finding the center of gravity, namely by finding the surfaces which are hit by radiation and their areas as follows:

在这里,c_g是重心,c_{ps}是太阳压力的中心。可以通过类似于找到重心的方式来找到该中心,即通过以下方式找到受辐射撞击的曲面及其面积: [\ c_{ps} = \frac{\Sigma A_n x_n}{\Sigma A_n} \; \; [m] \]

Magnetic field torque

The magnetic field of the Earth causes a cyclic torque across the spacecraft no matter its orientation due to interactions with the spacecraft’s magnetic dipole. It can be approximated using the following formula:

由于与航天器的磁偶极子相互作用,地球磁场无论航天器的方向如何,都会在航天器上产生循环转矩。可以使用以下公式进行近似:

[\ T_m = DB = \frac{\mu_E}{c R^3} \; \; [Nm] \]

In here, BB is the Earth’s magnetic field strength, which is approximated using the magnetic moment of Earth  μE=7.96⋅1015tesla⋅m3μE=7.96⋅1015tesla⋅m3, c is an approximation constant, which can be taken to be 1 for a polar orbit and as 2 for an equatorial orbit, and R is the distance to Earth’s dipole center in meters. DD is the satellite’s residual dipole, which is caused by electric currents and magnetic material within the vehicle. In the design of your satellite, it is important to minimize this value, for instance by including twist in the cables and avoiding large loops on the electronics boards! Due to different currents over time, this is hard to calculate. However, DD can be estimated as a function of mass as:

在这里,B是地球的磁场强度,它是使用地球的磁矩μE=7.96⋅1015tesla⋅m^3近似得出的,c是一个近似常数,对于一个极地轨道它可以取为1,对赤道轨道取为2,R为到地球偶极子中心的距离(以米为单位)。 D是卫星的残留偶极子,它是由车辆内的电流和磁性材料引起的。在设计卫星时,重要的是要最小化此值,例如,通过在电缆中加入绞线并避免电子板上的大环路!由于随时间变化的电流不同,因此很难计算。但是,可以将D作为质量的函数估算为: [\ D = c \cdot 10^{-3} \cdot m_{sc} \; \; [A m^2] \]

Here, mscmsc is the satellite mass and c is a constant in a range of 1 to 10 depending on the level of magneticity of the spacecraft (NASA source). Also, the residual dipole can be measured with a magnetic moment test after the satellite is integrated!

在这里,m_{sc}是卫星质量,c是1到10范围内的常数,具体取决于航天器(来源NASA)的磁性水平。同样,在集成卫星之后,可以通过磁矩测试来测量剩余的偶极子!

Aerodynamic torque

An object flying through air is subjected to aerodynamic drag. For some low-flying satellites, this means that it is necessary to calculate torques generated aerodynamically across the vehicle. For satellites above an altitude of 500km500km this can generally be neglected. This can be done using the following approximation:

空气中飞行的物体会受到空气动力学阻力。对于一些低空飞行的卫星,这意味着有必要计算整个卫星的在空气动力学上产生的扭矩。对于海拔500 km至500 km的卫星,通常可以忽略不计。可以使用以下近似值完成此操作:

[\ T_a = 0.5 [\rho C_d A V^2](c_{pa} – cg) \; \; [Nm] \]

In here, ρρ is the atmospheric density, CdCd is the drag coefficient (usually between 2 and 2.5), AA is the area of the front-facing satellite, VV is the spacecraft’s velocity and cgcg is the center of gravity. cpacpa is the center of aerodynamic pressure, which can be calculated in a similar way as cpscps is calculated.

在这里,ρ是大气密度,Cd是阻力系数(通常在2到2.5之间),A是前置卫星的面积,V是航天器的速度,而c_g是重心。 c_{pa}是空气动力压力的中心,其计算方法与c_{ps}相似。

If you followed the steps correctly, you have now calculated the different disturbance torques on your satellite and you are ready to start the sizing of your ADCS, congratulations!

如果正确地遵循了这些步骤,那么现在您已经计算出卫星上的不同干扰扭矩,并且可以开始调整ADCS的大小了,恭喜!

We hope you liked this mini-tutorial! If you want to learn how the disturbance torques and the ADCS are related to other subsystems in a satellite or how to design a complete satellite using Valispace and practical examples, also check our Satellite Tutorial.
Stay tuned for more and feel free to give us feedback at contact-us@valispace.com!

我们希望您喜欢这个迷你教程!如果您想了解干扰转矩和ADCS与卫星中其他子系统的关系,或者如何使用Valispace和实际示例设计完整的卫星,请查阅我们的卫星教程。 请继续关注更多信息,并随时通过contact-us@valispace.com向我们提供反馈!

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