Computation of drag force Consider a small atmospheric mass element m that hits the spacecrafts cross-sectional area A in a time interval t, m = Avrelt with the atmospheric density at the satellites location and vrel the speed of the satellite relative to

ENAE404 Spring 2007 Answers to Exam #1
March 13, 2007 75 minutes L. Healy 1. [20 points] For each of the following earth-orbiting satellites, identify whether the orbit is circular, elliptical, parabolic, or hyperbolic. Give a brief justication. (a) Posit

CHAPTER 12 DAMAGE TOLERANCE
In this chapter the following topic will be covered: Safe life design Damage tolerant (Fail-Safe) design Stress intensity factors Mode I fracture, mode II fracture and mixed mode fracture Fatigue crack growth: Paris equation
Ae

CHAPTER 11 TEMPERATURE EFFECT
Structural components are often subjected to temperature changes during flight operation. high-speed aircraft, especially along the wing leading edge reentry vehicles jet engine combustion chamber and exhaust nozzle spacecraf

CHAPTER 10 FORCE METHOD
In this Chapter we will introduce the Castiglianos 2nd theorem and apply it to various structural systems of slender bodies. The Castiglianos 2nd theorem is convenient in handling statically indeterminate, flat and curved structure

CHAPTER 9 FINITE ELEMENT MODELING
The finite element method is a very powerful tool for modeling and analysis of various structures (aircraft, spacecraft, launch vehicles, engines, automobiles, bridges, buildings, machine components) with complicated geom

Pressurized Cylindrical Shells under Compression
Internal pressure stiffens cylindrical shells. For short cylindrical shells under compression, buckling stress can be expressed as
c = E
or
R t p f , t R E
Nc t p = f , ER R E where
(1)
c : buckling stress

CHAPTER 6 SHEAR FLOWS IN THIN-WALLED SECTIONS
Once resultant shear forces are given over a cross-section of a thin-walled slender structure, shear flows over the cross-section can be determined by considering the force equilibrium in the axial direction a

CHAPTER 5 BENDING OF THIN-WALLED STRUCTURES
5.1
Bending in 3-D space Kinematics of displacement Strain-displacement relationship Stress-strain relationship Centroidal bending axes and principal axes Centrodal bending (CB) axes Centroidal bending and princ

Chapter 3
Wing Box Sizing
3.1 Shear center: Decoupling of bending and torsion 3.2 Sizing of wing box 3.3 Wing divergence 3.4 Aileron reversal 3.4 Flutter
1
3.1 Shear Center: Decoupling of Bending and Torsion
Analysis of wing bending and torsion can be dec

CHAPTER 2 INTERNAL LOADS IN AEROSPACE STRUCTURES
2.1 Force and Moment Distributions Slender body under axial force Slender body under torque Slender body under lateral loads 2.2 Inertia Loads Load factor Examples
1
2.1 Force and Moment Distributions
Aircr

Special cases have usable solutions There are two approximately solvable cases of interest to satellite orbit study. The perturbative (lunisolar or planetary perturbation) approximation for earth-orbiting satellites we have seen. Restricted three body pro

Application: main problem Consider the simplest geopotential perturbation problem: J2 only. In this case the potential is: R2 U (r, ) = J2 3 (3 sin2 1) . r 2r
Disturbing function=R
The satellite latitude is not a convenient quantity. Using spherical trigo

Perturbations Until now, weve dealt almost exclusively with two-body motion, a pure Keplerian orbit. It is now time to consider other forces that act on a satellite. Perturbations are small forces that are not included in the point-mass (= spherically sym

Torques and their sources Torques may be External: interaction with the outside world. May be natural or may be generated inside spacecraft, e.g., thrusters. Internal: acting totally inside the spacecraft; they must all cancel out. Nevertheless, they are

Attitude dynamics Ref.: Curtis Sec. 9.4 Attitude dynamics is the study of how spacecraft orientation changes in time with or without external torques. The principal result we will obtain are the dierential equations describing how the angular velocity cha

Examine equations for type of solution
Notice that the x and y directions are coupled together, and the z direction is not coupled to either. If you look at the z equation rst, youll see that it can be solved by a function whose second derivative is the

Assumptions of two-body motion
Only two bodies (satellite and planet), point masses (spherical distribution of mass), no other forces acting, satellite has much smaller mass than planet, so center of mass is essentially at the planets center.
L. Healy EN

Orbit propagation: the Kepler problem With an understanding of Keplers equation, we can solve the position of a satellite for a given time. This is called orbit propagation. In the two-body model, it is sometimes called the Kepler problem. With perturbati

Summary of true, eccentric, and mean anomalies
tan = 2
1+e E tan 1e 2
M = E e sin E a3 Note quadrant of , E , and M ; at 0 and , they are all the same, = E = M = 0 and = E = M = . Quadrant is never ambiguous: M =n If any of , E , M is between 0, , they a

Satellite centered coordinate systems The nal two coordinate systems well discuss, RSW and NTW, are satellite-centered, Type: usually Cartesian. Center: satellite position. Orientation: polar axis W = h perpendicular to orbital plane, RSW principal axis R

Earths rotation and velocity vectors When two frames are moving with respect to one another, time derivates in one do not transform to time derivatives in the other in the same way that postion vectors do, because the time rate of change of the coordinate

Earth-centered earth xed (ECEF) The rotation of the earth means that all of inertial space appears to be spinning around us at approximately one revolution per day. This means earth-relative positions and observations will have to be converted to inertial

Altitude and apogee x perigee specication Although well develop a more precise notion of altitude a little later for the purpose of handling ground stations, it is often useful to specify the (nominal) altitude of a satillite above a spherical earth, part

Circular orbit special case
For circular orbits e = 0, there is no perigee, so is undened. Argument of latitude is the angle from the ascending node n to the satellite position vector r , and is dened even for circular orbits. Argument of latitude is den

Computation of ellipse quantities Ellipse quantities may be computed from algebraic relations we have: b = a 1 e2 is the semiminor axis (put x = 0 into Cartesian conic section equation and solve for y ), p = a(1 e2) is the semilatus rectum, and is the dis

Conservation of ang. mom. = Keplers 2nd law
Consider a satellites motion over a short period of time t. The center of the earth and its positions at the two times form a triangle
Position vector time t later r r Postition vector at initial time l
The area

Syllabus This course is ENAE 404, Space Flight Dynamics. It is an introduction to orbit and attitude dynamics (astrodynamics) and a little attitude dynamics. Topics we will cover: Physics of orbiting satellites: Keplers and Newtons laws, conservation laws

ENAE404 Spring 2006 Answers to Exam #2
April 27, 2006 75 minutes L. Healy 1. [20 points] For each of the following sets of dierential equations with dynamical variables x, y , z , determine whether the equations divide into sets of equations that are deco