Project 2 Charles Hunter AME 30381 02/21/2012
Introduction The goal of this project is to develop a preliminary design for a satellite which will be launched into Low Earth Orbit. The following are assumed for the calculations:
The volume of the COM array does not factor into the given volume of the satellite The solar panels will rotate to always be pointing toward the sun 4 m direction limit does not apply to solar arrays attached to the satellite body
The following information is given:
The volume of the satellite is 8 m3 The satellite body can be no more than 4 m in any direction The density of the satellite body is 34 kg/m3 At least 4 m2 of solar panel arrays must always be visible to the sun Satellite will be launched into LEO COM array must point toward Earth Solar panels are Gallium arsenide solar cells Solar panel cost is $10,000 per square meter Solar panel density is 7.7 kg/m2 with a thickness of 1.5 cm The mass of the COM array is 21.3 kg The diameter of the COM array is 1 m The focal length of the COM array is 0.5 m The inertia matrix of the COM array is: [
]
The information to be determined is:
The assembly of the satellite with solar panels and COM array The principal moments of inertia The cost of the total satellite The effect of perturbations in regards to attitude control
Outside information required:
Information on LEO perturbations affecting satellites
Procedure
Task 1: Satellite Design The satellite body must be restricted to an 8 m3 volumetric space. The solar arrays and solar panels do not factor into the 8 m3 restriction. The body of the satellite must also not fit within a 4 m diameter in any direction. Finally the satellite must have at least 4 m2 of solar panels always pointing towards the sun. To simplify the analysis of the design a cylindrical shape for the body of the satellite was chosen. Figure 1 is a sketch of the satellite design.
Figure 1: Satellite Design
The dimensions of the satellite were developed using the restriction on the satellite body volume. The following equation was used to dimension the cylindrical body of the satellite and the two rods on each side of the satellite. (
)
( )
Where V is the volume, rb is the radius of the satellite body, hb is the height of the satellite body, rr is the radius of the rod, and hr is the height of the rod. The volume is 8 m3 and the satellite body radius and the rod height are chosen to be 1.5 m and 0.5 m to remain within the 4 m limit. The rod radius is also chosen to be 0.1 m. Applying equation 1 the height of the satellite body and The COM array is attached to the bottom of the satellite to ensure constant alignment with the Earth. Two solar panels are attached to the sides of the satellite using the small cylindrical rods. The satellite is designed to be symmetric. This simplifies the calculations for inertia because the principal axis is located at the center of the satellite. To meet the 4 m2 solar panel power requirements the satellite is designed to have two 2 m x 1 m solar panels attached to the cylindrical rods. Each of the solar panels will rotate to remain in the sunlight and combined they will meet the 4 m2 area requirement. Task 2: Moments of Inertia Calculation The satellite is split into six separate bodies for the inertia calculations. The solar panels are bodies 1 and 2 and using the given density per area multiplied by the designed area the solar
panels each have a mass of 30.8 kg. The cylindrical rods are bodies 3 and 4 and using the given average density for the satellite have a mass of 0.1326 kg each. The main body of the satellite is body 5 and again using the given average density has a mass of 271.7348 kg. The COM array is body 6 and has a given mass of 21.3 kg. The principal moments of inertia for each body are calculated first, and then the parallel axis theorem is used to determine the principal moments of inertia for the satellite as a whole. An x, y, z Cartesian coordinate system is placed at the center of the satellite main body. The moments of inertia for the solar panels are determined by approximating them as rectangular slabs. The inertial equations for bodies 1 and 2 are as follows: ( ) ( ) (
)
( )
Where b and c are the length and width of the solar panels and M1 is the mass. The inertial moments in the x, y, and z direction are calculated to be 2.5667 kg-m2, 10.2667 kg-m2, and 12.8333 kg-m2, respectively. The moments of inertia for the cylindrical rods, which are used to attach the solar panels to the main body of the satellite, are calculated using the following equations: ( ) (
)
( )
Where the mass of the cylindrical rod is M3. The inertial moments in the x, y, and z direction are calculated to be 1.6575*10-4 kg-m2, 2.8454*10-3 kg-m2, and 2.8454*10-3 kg-m2, respectively. The main body of the satellite is approximated as a cylindrical rod. The moments of inertia for the main body are calculated as follows: (
)
( ) ( )
Where M5 is the mass of the satellite’s main body. The inertial moments in the x, y, and z direction are calculated to be 181.7657 kg-m2, 181.7657 kg-m2, and 305.7017 kg-m2, respectively. The moments of inertia for the COM array are given earlier as 5.74 kg-m2, 5.74 kg-m2, and 2.86 kg-m2 in the x, y, and z direction.
Now the parallel axis theorem may be used in order to determine the principal moments of inertia for the entire satellite. First the center of mass for the whole system must be computed. Due to symmetry in the satellite’s design and Cartesian coordinate placement the center of mass in the x and y-directions remain at the zero location. The center of mass in the z-direction is calculated using the following equation: ( ) Where z is the distance on the z-axis of each respective body’s center of mass. Using equation 9 the satellite’s center of mass in the z-direction is located at 0.0339 m. Now with the moments of inertia for each body and the location of the center of mass of the satellite, the parallel axis theorem is used. The following three equations use this theorem in order to translate the separate moments of inertia into the total moments of inertia of the satellite along the principal axis. (
)
(
)
(
)
Where the Δ symbol corresponds to the distance from the center of mass from a body to the center of mass of the satellite. The results from the parallel axis theorem are given in the Results section. Task 3: Cost Analysis The intention for the design of this satellite is to reduce the production cost. This is an 8 m3 solar powered satellite that is capable of communication with Earth. The solar panels cost $10,000 per square meter so it is imperative that the satellite uses no more than the required 4 m2 of panels. The solar panels alone add $40,000 to the cost of this satellite. The COMM array on the satellite only needs to be able to send and receive signals from a Low Earth Orbit. The satellite meets the minimum required volume for the body of the satellite and since the density of the material is supplied the satellite mass is the lowest possible. This is important when considering the cost of the rocket this satellite will have to use to travel into orbit. A lighter mass means a lighter rocket. The cylindrical design will also allow for easier attitude control because thrusters could be placed in any location 360o around the main body. This will save cost because the thrusters would be able to fire at will. The satellite can easily be stabilized if the COMM array is not facing the Earth directly. Task 4: Perturbation Analysis There are a few orbit perturbations which may affect how this satellite maintains a given attitude. They are the non-spherical Earth gravity effects (J-2 Effects), atmospheric drag, third bodies, and smaller effects such as solar radiation pressure and relativistic effects. The J-2 Effects are based upon the satellites orbit in regards to the semi-major axis, inclination, and eccentricity. The J-2 Effects will alter the acceleration of the satellite along its orbit. It is necessary to minimize this
effect especially if it is meant to pass over locations at the same time every day. Gravity from additional planetary bodies could alter the satellites orbit over the Earth as the distance between large bodies changes often as they rotate around the sun. In LEO the primary effects are due to the sun and moon. Atmospheric drag is a dominant perturbation in LEO. This drag constantly decelerates the satellite throughout its orbit. Drag lowers the semi-major axis and decreases the eccentricity of the orbit. This perturbation may be useful however when the satellite is decommissioned and will help the satellite fall back towards Earth. Solar radiation and relativistic effects can add a torque to the satellite but are usually negligible in LEO. See also Appendix Source 1
Results A satellite which meets the given volume and power requirements is assembled and shown in Figure 1. A cost analysis of the design as well as a perturbation analysis is provided above. The principal moments of inertia for the satellite were calculated using the parallel axis theorem and are: [
]
Discussion There are many different solutions for designing this satellite. The body did not have to cylindrical and the placement of the solar panels is completely dependent upon the designer as long as the requirements are met. The satellite could have been designed to have the solar panels on the satellite body instead of being attached using cylindrical rods. The results show that the largest moment of inertia occurs along the z-axis. It is important that the largest moment be on this axis because the satellite is only intended to rotate around the z-axis. Rotation about the x or y-axis would render this satellite useless because the COM array would not be facing the Earth and communication would be lost. The symmetry of the satellite as well as the short and wide main body aids this satellite to remain rotating along the z-axis. The cost of any satellite is never cheap so it is imperative that great care is taken when developing a new satellite because each one requires sizeable financial backing. Several perturbations will affect this satellite along its orbit, but with attitude control programming the satellite should be able to remain along its intended course.
Appendix 1) Perturbation Effects on Satellites http://www.rfcafe.com/references/articles/Satellite%20Comm%20Lectures/SatelliteComms-Orbital-Perturbation.pdf