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# SPACE TECHNOLOGY I AERO 423

Texas A&M

GPA 3.66

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This 14 page Class Notes was uploaded by Carlotta Hermann on Wednesday October 21, 2015. The Class Notes belongs to AERO 423 at Texas A&M University taught by Daniele Mortari in Fall. Since its upload, it has received 22 views. For similar materials see /class/226159/aero-423-texas-a-m-university in Aerospace Engineering at Texas A&M University.

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Date Created: 10/21/15

Chapter 1 Introduction AERO423 Notation Reference frames are indicated by capital letters Example E frame with axes E1 E2 E3 Unit vectors are indicated with lower cases 61762763 Referred to means the reference frame in which the vector is being described or the reference frame whose vectors are being used to describe the vector 61 ocle1 Ot262 013e3 4 Example It means that the vector a is referred to the E frame Relative to means the reference frame in which the vector is being measured Subscripts denote the reference frame the vector is referred to Right superscripts indicate which body is being considered and which reference frame the vector is relative to Example TagO is the angular velocity of B relative to 0 expressed in the B frame Reference Frames Spacecraft reference or body frame B This is often chosen to be coincident with o1 o3 pitch Airplane the inertial principal axes However the 02 02 P reference presence of important mission pointing instruments may cause a different choice Orbit reference Orbit reference frame 0 Two orbit frames are used One is the rollpitchway 123 airplane reference and another is the more classic orbit reference with 1 along the radius and 3 normal Inertial Reference frame I Usually built on the Earth spin axis and For an Earth pointing satellite on the Earth orbit plane Two I frames the angles of B wrt 0 should be can be built They share the vernal zeros in the nominal equinox or vernal axis con guration Coordinate Transformations Consider a vector g with components a11a120l13 in the 1 frame and components aSIaS2aS3 in the S frame gt Cl 2 611111 611212 611313 and as ClSlS1 CZSZS2 615353 anArias39Eia31i hasz 2 ias3 3 i an23 Asiaa31 1 a32 2 a53 3 ananis39gZami famgz39g ia gfg 11 gz39h g5 aSl an 61 12 gz39iz 339i2 aSZ CISaS an 43 43 339i3 aS3 This is the Direction Cosine Matrix DCM The DCM transforms a vector from one 5 frame to another 1 frame It is denoted by C Is Orthogonal Transformations C15 represents an orthogonal transformation C e S0n if detC 1 Orthogonal transformations have nice properties a CISaS implies as CI SIaI C3161 CI S1 2 CS and also CST C5 1 T 1 The 1nverse of a DCM1s equal to 1ts transpose CIS CS CIS 2 The determinant of a DCM is always det C13 1 Orthogonal transformations keep invariant distancesangles bl C311 1 71sz IETCEI CBIrz TF2 b2 CHI2 bkll2 kabk rkTC1CBIrk rkTrk llrkIIZ 5 Some properties of square matrices transpose MAB gt MT BTAT inverse M 2 AB gt M 1 2 3 11 1 determinant detAB detA detB trace trA trAT and trAB trBA 6 Derivative in a rotating frame d 1 1tdt 1t grime2 dt dt dt dm 2 2 sin 0336 033dt 2 gawk 204 cogdo z 10A wall dt dt 3 2 d 2 033dt 1 0 dt dt a Hat f 05 6 ml l Dz 2 033 3 and 17 vl l vz 2 v3 3 mo 83 due to ml l 3 a 2 dt 031 caw Qg ym due to mz z 3 d zdt 0 gma wg due to 033 3 a 2 dt 033 1 gwo V V161 V262 V363 dv dvl A dv 2 A dV3 A 1 61 62 63V1 V v d d z d 3 dt dt dt dt dt 2dr 3dr 8 0 17 a v1 A A e eZ e dt dt 1 dr dr 3 dvl z dv3A d17 dr 1 dt 2 dt 3 dr d l A A v o o 1dr 1 32 23 d z A A v v 0 De 2dr 2 13 31 d A gt V V 06 De 3 3 21 12 d d d Transport Theorem Vector derivative in a rotating ref frame The derivative of a vector can be expressed in any rotatinq frame The rule is dt dt I B dV dV6BI XV Remember vector is an abstract mathematical concept characterized by modulus and direction Both are independent from the system of coordinates used Example Problem The Space Station SS of radius R is rotating at angular speed 0 as shown 03 I An astronaut is walking on the inner ring at speed v relative to the SS Find the inertial velocity of the astronaut Assume the SS is only rotating and not in orbit I inertial reference frame S reference frame xed to the SS P Reference frame rotating with astronaut such that P1 points to astronaut Let 9 represent the angle between P1 and SI Example first method gtA I Is the pos1tion of astronaut in any reference frame Referred to P frame FPA 2 R131 Referred to S frame FSA Rcos 63 sin 63 Performing the derivative sAr R9 sjn 6E1 cos 6E2 Rcos 63 sin 63 2 d 1 oadl 2 and 61 oadl 1 Example rst method S20 2tdt if RG sin 6E1 cos6 2Rcos6 1sin6 2 RG sin 6E1 cos6 2Roa sin 6E1 cos6 2 Sic Fair RGoa sin6 1 cos6 2 SIG E1 cos 12 sin 112 cos 1 sin 2 ltgt 2 sin 1 cos 1 sin 1 cos 2 if RG 0 sin 6cos bi sin 112 cos 6 sin bi cos 119 RG 0 sin 6 cos d cos 6 sin 1i cos 6 cos d sin 6 sin 112 R6 0 sin6 1 cos6 12 Example second method dC I gt S gt air 63m x derived from szr and 03C dl dl dl zd CrC CrC bC dt dt t dt dt dr db ax C 0 dt dt P frame 71 12132 033 gtlt R131 2 R9132 R 132 Rm W32 gtAI sin 9E COS 9 S frame VS 14

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