Answered: Assume you have a 1U cubesat, which has…

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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Question

Assume you have a 1U cubesat, which has a total mass of 2kg. It has one reaction wheel
to control its orientation around its nadir pointing axis. The mass of the reaction wheel is
200g. The wheel itself has a diameter of 30mm and a thickness of 10mm. You may ignore
any translational motion for the time being.
(a) Write down the kinematic and dynamic equations of the system and compute all relevant
constant values.
(b) Where do you want to mount your wheel? Make a drawing and a qualitative argument.

Expert Solution

Step 1

(a)

The kinematic equation for the system can be written as:

I $\times$ w₀= M_rw

where I is the moment of inertia of the cubesat about its nadir pointing axis,

w is the angular velocity of the cubesat about its nadir pointing axis, and

M_rw is the torque applied by the reaction wheel.

The dynamic equation for the system can be written as:

I $\times$ w₀ + Bw = M_rw

where B is the damping coefficient, which represents the effect of any internal or external friction that may oppose the rotation of the cubesat.

The moment of inertia of the cubesat about its nadir pointing axis can be calculated as:

I = (1/12) $\times$ m $\times$ ((3 $\times$ r²)+ h²)

where m is the mass of the cubesat,

r is the radius of the cubesat, and

h is the height of the cubesat.

In this case, m = 2kg, r = 0.5U = 0.5 $\times$ 10cm = 5cm, and h = 1U = 10cm.

Therefore:

I = (1/12) $\times$ 2(3 $\times$ (0.05)² + (0.1)²) = 0.002916 kg.m²

The torque applied by the reaction wheel can be calculated as:

M_rw = I_wheel $\times$ $α$

where I_wheel is the moment of inertia of the reaction wheel and $α$ is the angular acceleration of the reaction wheel.

The moment of inertia of the reaction wheel can be calculated as:

I_wheel= (1/2) $\times$ m_wheel $\times$ (r_wheel)²

where m_wheel is the mass of the reaction wheel and r_wheel is the radius of the reaction wheel.

In this case, m_wheel = 200 g = 0.2kg and r_wheel = 0.015m.

Therefore:

I_wheel= (1/2) $\times$ 0.2 $\times$ (0.015)² = 0.0000225 kg.m²

The damping coefficient B can be estimated as:

B = k $\times$ w

where k is a damping constant that depends on the internal and external friction of the system.

The constant values are:

Moment of inertia of the cubesat about its nadir pointing axis: I = 0.002916 kg.m²
Moment of inertia of the reaction wheel: I_wheel = 0.0000225 kg.m²
Mass of the reaction wheel: m_wheel = 0.2kg
Radius of the reaction wheel: r_wheel = 0.015m
Kinetic damping coefficient: k (unknown)

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