91Ó°ÊÓ

The earth-scanning satellite is in a circular orbit of period \(\tau .\) The angular velocity of the satellite about its \(y-\) or pitch-axis is \(\omega=2 \pi / \tau,\) and the angular rates about the \(x\) - and \(z\) -axes are zero. Thus, the \(x\) -axis of the satellite always points to the center of the earth. The satellite has a reaction-wheel attitude-control system consisting of the three wheels shown, each of which may be variably torqued by its individual motor. The angular rate \(\Omega_{z}\) of the \(z\) -wheel relative to the satellite is \(\Omega_{0}\) at time \(t=0,\) and the \(x\) - and \(y\) -wheels are at rest relative to the satellite at \(t=0 .\) Determine the axial torques \(M_{x}, M_{y},\) and \(M_{z}\) which must be exerted by the motors on the shafts of their respective wheels in order that the angular velocity \(\omega\) of the satellite will remain constant. The moment of inertia of each reaction wheel about its axis is \(I\). The \(x\) and \(z\) reaction-wheel speeds are harmonic functions of the time with a period equal to that of the orbit. Plot the variations of the torques and the relative wheel speeds \(\Omega_{x}, \Omega_{y},\) and \(\Omega_{z}\) as functions of the time during one orbit period. (Hint: The torque to accelerate the \(x\) -wheel equals the reaction of the gyroscopic moment on the \(z\) -wheel, and vice versa.)

Step by step solution

01

Comprehend the System's Dynamics

The satellite orbits the Earth with a constant angular velocity \(\omega = \frac{2\pi}{\tau}\) about the \(y\)-axis. At \(t=0\), the \(x\)- and \(y\)-wheels are stationary, while the \(z\)-wheel has an initial angular velocity \(\Omega_0\). We need to maintain a constant \(\omega\) for the satellite by applying the correct torques \(M_x, M_y,\) and \(M_z\).

02

Relate Angular Momentum Principles

Employ the conservation of angular momentum. Since no external torque acts on the satellite, changes in angular momentum due to reaction wheels need to balance, ensuring the satellite's angular velocity remains constant. Calculate torques using inertia \(I\) and changes in angular velocity.

03

Express Inertial Torques for Wheels

For the wheels, the torque required on each wheel axis is related to the change in angular momentum: \(M_i = I \frac{d\Omega_i}{dt}\), where \(i\) corresponds to \(x, y, z\).

04

Specifically Derive Each Torque

Change in angular momentum around axis must match gyroscopic coupling. For wheel torques: * \(M_x(t) = I \frac{d\Omega_x}{dt}\) * \(M_y(t) = I \frac{d\Omega_y}{dt}\) * \(M_z(t) = I \frac{d\Omega_z}{dt}\). Functions are harmonic with \(\tau\). When one axis speeds up, gyroscopic effects act on another axis.

05

Implement Harmonic Relations in Speeds

Given harmonic functions have period equals orbital \(\tau\), use \(\Omega_x = A_x \sin(\frac{2\pi}{\tau}t)\) and \(\Omega_z = A_z \cos(\frac{2\pi}{\tau}t)\) with initial speeds \(\Omega_0\). Calculate, ensuring \(\omega_y\) constant.

06

Calculate Gyroscopic Reaction Balance

Gyroscopic interactions link \(x\)- and \(z\)-axis wheels: \(M_x (from \omega_z) = I \frac{d\Omega_z}{dt}\).Ensure balancing with harmonic shifts leading to required \(\omega_y\) changes.

07

Construct Label-axis Graphs for Torques and Speeds

Plot graphs for torques \(M_x, M_y, M_z\) and relative wheel speeds \(\Omega_x, \Omega_y, \Omega_z\) over time \(t\) within the full period \(\tau\): wheel speed harmonic changes dictate torque variation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Angular Momentum

Angular momentum is a crucial concept in satellite dynamics. It represents the rotational motion of an object and is defined as the product of the object's rotational inertia and its angular velocity. In mathematical terms, for a single axis, it is given by \( L = I\omega \), where \( L \) is the angular momentum, \( I \) is the moment of inertia, and \( \omega \) is the angular velocity.
For a satellite, maintaining a stable attitude involves ensuring that the angular momentum remains constant or changes in a controlled manner. This means the satellite's position with respect to its rotation axes doesn’t drift unpredictably.

• In the absence of external torques, the total angular momentum of the satellite system must remain constant. This is a result of the law of conservation of angular momentum.
• Any change in the angular momentum of the entire system (satellite plus reaction wheels) should ideally be zero.

Understanding this concept is critical, as it allows engineers to design satellite systems that can control their orientation effectively by using reaction wheels or other mechanisms.

Torque Calculation

Torque is the force needed to rotate an object around an axis. For satellites using reaction wheels, torques are calculated to manage changes in the angular velocities of these wheels.
To determine the required torque for stabilizing a satellite, one can use the relationship between torque \( M \), moment of inertia \( I \), and angular acceleration \( \alpha \): \( M = I \alpha \).

• This equation shows that the torque is directly proportional to the moment of inertia and the rate of change of angular velocity.
• In the context of reaction wheels, torque \( M_i \) for each wheel along axis \( i \) (where \( i \) could be \( x, y, \) or \( z \)) is given by the formula \( M_i = I \frac{d\Omega_i}{dt} \).

Calculating and applying the correct torques ensures the satellite maintains its desired orientation without external forces. This concept is pivotal for the functionality of autonomous satellite systems.

Gyroscopic Effect

The gyroscopic effect is a phenomenon that results from the conservation of angular momentum in rotating bodies, such as reaction wheels on a satellite.
When a rotating wheel (gyroscope) is subject to an external torque, the resulting motion is perpendicular to the direction of the applied torque. This causes interesting behavior in systems like satellites.

• When the speed of one axis changes, there are reactive forces exerted on perpendicular axes due to this effect.
• This is particularly important in satellites because when a reaction wheel changes speed, it induces a torque about the other axes, which must be balanced to maintain stability.

The gyroscopic effect can be used advantageously to control the orientation of a spacecraft by managing the reaction wheels' spin rates. Intuitive understanding of this effect underpins effective satellite attitude control systems.

Reaction Wheel System

A reaction wheel system is an essential component in satellite control dynamics. It consists of a set of wheels that can spin to produce torques that help maintain or change the satellite's orientation.
Each wheel in the system has its own motor to control the speed of rotation, and hence, the angular momentum.

• By changing the speed of the wheels, a satellite can change its orientation without using fuel or other propulsion mechanisms, which is very efficient.
• In a typical setup, there are three reaction wheels oriented along the satellite's principal axes (x, y, z), which enables control over each axis independently.

The moment of inertia of each wheel allows for precise control of the satellite's orientation by creating expected torques. Reaction wheel systems provide a means of attitude control that utilizes the fundamental principles of angular momentum and gyroscopic effects.