Appendix a Euler Angles

Euler Angles

Three angles are required to describe the attitude of a (body-fixed) Cxyz coordinate system with repsect to a (space-fixed) CXY Z coordinate sys tem. The sequence of angular rotations is significant. Euler angles involve one axis twice (once in its original attitude, then in its carried attitude) in this rotation process, one axis once, and one axis not at all. Altogether six kinds of Euler angles can be distinguished in this fashion. Since axes are, however, cyclically interchangeable, it suffices to investigate two kinds only. Selecting the z-axis for two rotations, they are shown in the following table.

Rotation First Kind Second Kind Sequence (Figure A.I) (Figure A.2) l. t/J about Zo t/J about Zo 2. 1/ about Xl J.l about Yl 3. (j about Z2 (j about Z2

Euler frequencies are integrable, thus

t/J = t/Jo + J~ dt

1/ = I/o + Jv dt or J.l = J.lo + JjJ, dt.

(j = (jo + JiT dt. 312 Appendix A. Euler Angles

3;0= X

yo= y

Zl = Zo= Z

...... , ,

Z , " \. \ ----- \

Sin 9 u 1 a r ity a t v = 0 0

FIGURE A.!. Euler angles of the first kind.

Relationshi pS

r = [e.. ey, e,,) [ : ] = [e., e" e,,) [ ~ ] = [~, ey, e,,) [ ~: ]

= [e., e" -,,) [ ~ 1

[ ~ ] =[113-2 [ ~ ] [ ~ ] = [~2_' [ ~:J

[~:J = !n,-o [ : ] Relationshi ps 313

Singularity at J,J=O·

FIGURE A.2. Euler angles of the second kind . 314 Appendix A. Euler Angles

Sample Rotation

e ~

o e ---.X 2 c X2

Transition Matrices

First Kind Second Kind cos t/J sin t/J 0 cos t/J sin t/J 0 [Th_o = - sin t/J cos t/J 0 ['111-0 = - sin t/J cos t/J 0 0 0 1 0 0 1

0 0 [ c~p -s~np 1 111,-1 = [ ~ cos v 1 ['112-1 = 1 -SlllV 'i~" cos V SlllJl 0 cos Jl

cos(J' sin (J' 0 COS(J' Slll(J' 0 ['113+-2 = -Slll(J' COS(J' 0 ['113+-2 = -Slll(J' COS(J' 0 0 0 1 0 0 1 ~ ~ .,...r!!. o· ~ s::: ~ :3. (') Transition Matrix for the First Kind ~

cos t/Jcos (j - sin t/Jcos v sin (j sin t/Jcos (j + cos t/Jcos v sin (j s~nV SIn (j 1 [T],~.=[ - cos t/Jsin (j - sin t/Jcos v cos (j - sin t/Jsin (j + cos t/Jcos v cos (j SIn v cos (j sin t/Jsin v - cos t/Jsin v cos v

Transition Matrix for the Second Kind

cos t/Jcos I-' cos (j - sin t/Jsin (j sin t/Jcos I-' cos (j + cos t/Jsin (j ~sin I-!'cos (j 1 [T],~o=[ - cos t/Jcos I-' sin (j - sin t/Jcos (j - sin t/Jcos I-' sin (j + cos t/Jcos (j SIn I-' SIn (j cos t/Jsin I-' sin t/Jsin I-' cos I-'

""...... Ol 316 Appendix A. Euler Angles

The angular velocity vector w may be expressed in several different co ordinates, such as body-fixed or space-fixed coordinates:

w =[e ... e,] [ :: ] =[ex ey ez] [ : ] .

It may also be expressed in (nonorthogonal) carried coordinates, i.e.,

w = [e" e.. e,,] [ r] for Euler angles of the 6mt kind and

w =[e" ... e,,] [ ;] for Euler angles of the second kind. ~ t:;3 \I> ~ ::I ::l gJ. .-.00 o·-- M-.-. ::I 0 :::::: ::l \I> Kind Second Kind .... First ;:;.-- cos U cos U (t) [ ~'inpco ~ fJ> [,in V ,in " .. ~ 11- 1/ Wy = SIn cos U -SlllU SlllJ.l COSU -SlllU M- 0' .-.1-1 .... W z cos 1/ 0 ~][!l [ ~: 1 cos J.l 0 ~][;1 (j <'tI ~ 00 0- £Q!£ s~n0' (") Sln v sin v sm J.' sm J.' -. [~l[_C~" [~~1= U 0 cos 1/ U [~n0 COSJ.l u U Q 0 coso/! _ coso/! [-~tan" tan II tanJ.' tanJ.' WX s cos'Ij; sin 'Ij; -sin'lj; cos'Ij; 0 Wy '"0 _cosy; cosy; ~ 1[ 1 0 ~ ~][~n[ ; 1 0 Wz [ 1 ~ ,.. $Jt ::l sin v sin v sin Sl~ __ <'tI ::l M- OO

c...:> ...... :J 318 Appendix A. Euler Angles Transition and Rotation

While both transition and rotation involve a transformation, there is a fundamental difference in the concepts of transition and rotation, and the experienced engineer takes great care in clearly stating from the outset which of the two applies in a given transformation. Transition is the process of changing the attitude of a coordinate system (for one and the same vector in absolute space). Rotation is the process of changing the attitude of a vector (in one and the same coordinate system). The issue is so confusing because of the similar appearance of the trans formation matrices containing the direction cosines, since one matrix is the transpose of the other. In Figure A.3 a transition is shown from a coordinate system 0 to a coordinate system 1. One and the same vector r can be expressed in either system

r =[e" ~, e"J [ ~: 1= [e .. e,o e"J [ ~ l· In order to transit from coordinate sytem 0 to coordinate system 1, we write

An inspection of Figure A.3 shows that the transition matrix is

COS (} sin (} 0 1 [TJ = [ - sin (} cos (} 0 . o 0 1 In Figure A.4 the rotation of a vector r through an angle (} is shown. After rotation, the vector is

r, = [e, e, e,J [ ~:]

Prior to rotation the vector was

r, = Ie. e, e,J [ ~n

In order to express the (new) coordinates 1 of the rotated vector in terms of the (old) coordinates 0 of the vector prior to rotation, we write Transition and Rotation 319

FIGURE A.3. Transition.

r ~ Yl~------~

FIGURE A.4. Rotation. 320 Appendix A. Euler Angles

An inspection of Figure A.4 shows that the rotation matrix is

COS () - sin () 0 1 [R] = [ sin () cos () 0 . o 0 1

The reader is invited to compare the transformation matrices [T] and [R], and to confirm that [R] = [T]T. The transition matrix [T] is also known as the direction cosine matrix, and is often given the symbol [C]. Thus Appendix B

Euler Parameters

As Euler has shown, any attitude of one coordinate system with respect to another can be expressed by a rotation axis and a rotation angle about this axis. The rotation axis is identified by a unit vector A along (or parallel to) the rotation axis; the rotation angle is given the symbol (J. To fully specify rotation axis and rotation angle, four items of information are required: the three components of the vector A, plus the rotation angle (J. Due to Euler is the computationally convenient concept of using four parameters. The first three f.x, f.y, and f.z, are the components of the Euler vector E, while the fourth, f.8, is directly related to the rotation angle.

, . (J E = A SIll- 2 with f.x = E . ex = f.x = E . ex = f.l f.y = E . ey = f.y = E . e y = f.2 and (J €IJ = cos 2" = €4· The Euler parameters are not independent, but are related by €I + f.~ + €~ + f.~ = l. The unit rotation axis vector A and the rotation angle () in terms of Euler parameters are

€y A = [ex ey e z]

f.Z 322 Appendix B. Euler Parameters

() = 2 arccos f8 0 $ () $ 360°. The entries 'Fij of the transition matrix [Tlj] are the direction cosines

with I = X, Y, Z, and j = :r:, y, z. The following relations also apply

{ell = [Tlj]{ej}

{edT = {ej }T[Tlj]T [Tlj] = {ed· {ej}T {ej}T = {edT[Tlj]. Direction cosines and Euler parameters are related by

Txy = 2(f1f2 - f3( 4) Txz = 2(f3f 1 + f2(4)

Tyx = 2(flf2 + f3(4)

Tyy = f~ - f~ - f~ + d = 1 - 2f~ - 2f~ Tyz = 2(f2f 3 - f1(4)

Tzx = 2( f3f 1 - f2(4) Tzy = 2(f2f 3 + fl(4) Tzz = f~ - f~ - f~ + f~ = 1 - 2f~ - f~. Euler parameters in terms of direction cosines are given by Euler Parameters 323

e ~

FIGURE B.1. Attitude of xyz coordinate system prior to and after rotation. 324 Appendix B. Euler Parameters

Using Euler parameters, the direction cosine matrix [TIj], which is also called the attitude matrix, becomes 1- 2(t"~ + t"~) 2(t"1t"2 - t"3t"4) 2(t"1t"3 + t"2t"4) 1 [TIi] = [ 2(t"1t"2 + t"3t"4) 1 - 2(t"§ + t"?) 2(t"2t"3 - t"1t"4) . 2(t"1t"3 - t"2t"4) 2(t"2t"3 + t"1t"4) 1- 2(t"~ + t"n

As an example, take the xyz system shown in Figure B.1. Prior to rota tion icyz and XY Z are parallel, as shown in Figure B.1a. After rotation the xyz system has an attitude as shown in Figure B.1b. The direction cosine matrix is Txx Txy [Tai] = [ Tyx Ty y TxzTyz 1= [00 1 0 -10 1. Tzx Tzy Tzz -1 0 0 The Euler parameters are -1- 0 1 t"l = -2v'1+0+0+0 = 2

-1- 0 1 t"2 = - 2v'1 + 0 + 0 + 0 = 2 1-0 1 t"3 = - 2v'1 + 0 + 0 + 0 = - 2 1 1 t"4 = 2v'1 + 0 + 0 + 0 = 2' The Euler vector is 1 2

1 € = [ex ey ez] 2

The unit rotation axis vector is

The rotation angle is () = 2 arccos 1/2 = 1200 • The Euler rotation theorem states that the attitude of a vector r fixed in the xyz coordinate system, can be expressed by

rl=ro+2(t"8€ xro+ € x(€ xro)) Euler Parameters 325 with

" = [ex ey ez] [ ~ 1 representing the vector r after rotation, and

representing the vector r prior to rotation, i.e., when the xyz system and the XY Z system were parallel (Figure B.2). Take, as an example, the vector

shown in Figure B.2. Before rotation of the xyz system, the vector has an attitude (with respect to the XY Z system) given by

r, = [ex ey ez] [ n

After rotation of the xyz system given by the Euler vector

1 2'

1 f. = [ex ey ez] 2

-2'1 and the Euler parameter (8 = 1/2, the vector's new attitude (with respect to the XY Z system) is given by the Euler rotation theorem. The cross products occuring therein are

ex ey ez 2

1 1 1 3 f. X ro = 2' 2' -2' = [ex ey ez] -2'

1 1 2 2 2' and 1 ex ey ez -2'

1 1 5 f. 1 x (f. X ro) = 2 2' -2' = [ex ey ez] -4'

3 1 7 2 -2' 2' -4' 326 Appendix B. Euler Parameters

FIGURE B.2. Attitude of r vector prior to and after rotation of xyz system. Euler Parameters 327 such that

2 -'21

1 3 3 +2 - -'2 + -'2 [ ~ 1 2 Ul 1 7 '2 -4" that is

r, = [ex ey ez] [ =~ 1 as can be seen in Figure B.2. The angular velocity w of the xyz system with respect to the XY Z system is given by

1= {ej}T{wj}.

It can be shown that

. . . Wy = TYxTyz + TzxTzz + TXxTxz Wz = TzyTzx + TxyTxx + TYyih. The Poisson kinematic equations are

Txy = Txzwx - Txxwz

Txz = Txxwy - Txywx

Tyx = Tyywz - Tyzwy

Tyy = Tyzwx - Tyxwz Tyz = Tyxwy - Tyywx

Tzx = Tzywz - Tzzwy Tzy = Tzzwx - Tzxwz 328 Appendix B. Euler Parameters

The angular velocity in terms of Euler parameters is

with

E =[e. e, e,] [ :: 1

; = [e. e, e,] [ 1:1

The Euler parameter representation has as an advantage the fact that the absolute value of any of the four Euler parameters does not exceed unity. A disadvantage is that one of the four is redundant. By forming a so-called Rodrigues vector this disadvantage is removed. The Rodrigues vector, in terms of unit rotation axis vector A and rotation angle () is defined as

() p = A tan- 2 which, in terms of Euler parameters is 1 1 p = -E = -E. (9 (4 The Rodrigues parameters, i.e., the components of the Rodrigues vector, are 1 1 1 Px = -(X = -(:c = P:c = -(1 = P1 (9 (9 (4 111 py = -(v = -(v = Py = -(2 = P2 (9 (8 (4 1 1 1 pz = -(z = -(z = pz = -(3 = P3· (9 (9 (4 Rodrigues parameters can assume any value, including infinity. Problem

B.l. An Oxyz coordinate system's attitude with respect to a (fixed) OXY Z coordinate system is obtained by rotating it through the Euler angles 1/J = 00 , v = 40 , (J" = 100 • ( a) Determine the direction cosines; (b) find the four Euler parameters; (c) find and sketch the Euler rotation axis vector A; and (d) compute the Euler rotation angle (). Appendix C Cardan Angles

Depending upon the sequence of rotations, six kinds of Cardan angles can be defined (Figure C.l).

Rotation First Kind Second Kind Third Kind Sequence (Figure C.2) (Figure C.3) (Figure CA) 1. { about Xo T} about Yo ( about Zo 2. T} about Y1 ( about Zl { about Xl 3. ( about Z2 { about X2 T} about Y2

Fourth Kind Fifth Kind Sixth Kind (Figure C.5) (Figure C.6) (Figure C.7) 1. { about Xo ( about Zo T} about Yo 2. ( about Zl T} about Y1 { about Xl 3. T} about Y2 { about X2 ( about Z2

Cardan frequencies can be integrated:

T}=T}0+ J~dt

(= (0 + J(dt. 330 Appendix C. Cardan Angles

z; 1

11 , . '-=-----" 1)

FIGURE C.l. Gyro axes Cxyz rotated through Cardan angles of the first kind. Cardan Angles 331

Y • Ys Y2 YI Yo r

.& = Z S Z I II 0 = Z

...... " , " " , \ ,

Singu larity at n· 90·

FIGURE C.2. Cardan angles of the fi rst kind.

Zo - X

Yl YI-y,-r

1I. - Z

.1 Singularity at <. 90· ~

FIGURE C.3. Cardan angles of the second kind. 332 Appendix C. Cardan Angles . % • Z, Zz - %1 Zt .. I

!I - !I. - !lz !II !It • 1 • .a •• r 1/ • .s I/Z " I Z ...... " "' " , -\ ..

Singulartly at C· 90·

FIGURE CA. Cardan angles of the third kind.

c % • %, %2 %1 .. .1:, .. I l! !I • !I, .. !l2 !II !lo .. 1 \ \ , • lI, 'z .. '1 '" . Z , \ , , z , \

...n

Singularity at ~. 90·

FIGURE C.S. Cardan angles of the fourth kind. Cardan Angles 333 . t z % :C, OK %2 %1 :Co - X

Y .. y, Y2 .. VI Yo J

I z - Z, 112 Z I :z 30 IZ Z

Singula r i ty at ~ = 90·

FIGURE C.6. Card an angles of the fifth kind.

,. \ :c :c, :Cz '::1 :CD r YI ::z yo = 1

Z I Zo .. Z

L----y

Singularity at (= 90·

FIGURE C.7. Cardan angles of the sixth kind. 334 Appendix C. Cardan Angles Relationships

r = [e.. e" e,,] [ ~: 1= [e., e y, e,,] [ ~: 1= [e., e,. e•• ] [ ~: 1

= [e.. ey, e,,] [ : 1

[ ~ 1= [11'-2 [ ~ 1 [ ~ 1= [11'-1 [ ~: 1

[~:l = [11,-0 [ ~:l

[~:l = [11.-0 [ ~ 1

Sample Rotation

c XI Relationshi ps 335

First Kind Second Kind Third Kind

COS TJ 0 - sin TJ 1 [Th_o coseo sine0 1 [ 010 [~ - sine cose sin TJ 0 cos TJ

COS TJ 0 - sin 7] cos( 1 [ sine 0 1 0 [ o 1 0 -s~n( cos ( 0 coseo sine 1 sin TJ 0 cos TJ o 1 [~ -sine cose

COS ( sin ( 0 1 o COS TJ 0 - sin TJ 1 [ - sine cos( 0 cose [ o 1 0 o 0 1 [~ -sine sin TJ 0 cos TJ Fourth Kind Fifth Kind Sixth Kind

COS ( sine 1 0 0 [ o cose sine 1 [ - sine cos ( o - sine cose o 0

cos( sine 0 o 1 - s~n TJ 1 [ -sine cos ( 0 cose si~e 1 o 0 1 cos TJ [~ - sine cose co~e si~e 1 - sine cose 336 Appendix C. Cardan Angles

First kind:

cos T}cos~ cos ~ sin T} + sin ~ sin T} cos ~ sin ~ sin ( - cos ~ sin T) cos ( 1 [Th_o = [ -:- cos T} sin ( cos ~ cos ( - sin ~ sin T) sin ( sin ~ cos ( + cos ~ sin T) sin ( sm T) - sin~ cos T) cos~ cos T)

Second kind:

COS T}cos ( sin ( - sin 1/ cos ( 1 [1"13_0 = [ - cos ~ cos T) sin ( + sin ~ sin T) cos~ cos ( cos ~ sin T) sin ( + sin ~ cos T) sin ~ cos T} sin ( + cos ~ sin T) - sin~ cos ( - sin~ sin 1/ sin ( + cos ~ cos T)

Third kind:

COS T} cos ( - sin ~ sin T) sin ( cos T) sin ( + sin ~ sin T) cos ( - cos ~ sin T} 1 [1"13-0 = [ -cos~sin( cos~ cos ( sin~ sin T) cos ( + sin ~ cos T) sin ( sin T) sin ( - sin ~ cos T) cos ~ cos~ cos T}

Fourth kind:

COS T} sin ( cos ~ cos T) sin ( + sin ~ sin 1] sin ~ cos 1] sin ( - cos ~ sin 1/ 1 [1"13-0 = [ - sin ( cos~ cos ( sin~ cos ( sin T) cos ( cos ~ sin 1] sin ( - sin ~ cos 1] sin ~ sin 1] sin ( - cos ~ cos 1]

Fifth kind:

COS 1] cos ( 1] cos sin ( - sin~ 1 [1"13-0 = [ - cos ~ sin ( + sin ~ sin 1] cos ( cos ~ cos ( + sin ~ sin 1] sin ( sin~ cos 1] sin ~ sin ( + cos ~ sin 1] cos ( - sin ~ cos ( + cos ~ sin 1] sin ( cos~ cos 1]

Sixth kind:

COS 1] cos ( + sin ~ sin 1] sin ( cos~ sin( - sin T} cos ( + sin ~ cos 1/ sin ( 1 [1"13-0 = [ - cos 1J sin ( cos~ cos ( sin 1] sin ( + sin ~ cos 1] cos ( cos~ sin T) -sin~ cos~ cos 1] Relationships 337

For very small angles, i.e., when each Cardan angle becomes so small that one can set cos{ = cOSTJ = cos, = 1, sine = {, sinTJ = TJ, sin' =" and {TJ = T]( = '{ = 0, the transition matrices of all six kinds become the same, l.e.,

[T]3-0 = [-~ 1 -~ l· TJ -{ 1 The vector w can also be expressed in terms of Cardan frequencies (which are not orthogonal): First kind: w ; [e.. ey• e,,1 [ n Second kind: w; [e., ~. e•• 1[! 1 Third kind:

w ; [e.. ~, e.. 1 D1 Fourth kind: w ; [e.. ~, e,,1 [ n Fifth kind: w; [e.. e,. e,,1 [ n Sixth kind: w ; [e., ~. e.. 1 [ H First kind:

[ C08~C08( sin, = - co~ TJsin, cos, [ ~: 1 sm TJ 0 nUl Second kind: sin, = cos { cos, [ ~: 1 [~ -sine cos, cos'i~e { 1P , 1 338 Appendix C. Cardan Angles

Third kind: 0 -~e~nq][ e1 [~~ 1 sme TJ [::] sm TJ 0 cose cos TJ ( Fourth kind: [ oosqoos( 0 = -sin( 1 -'~nql P1 [ :: 1 sin TJ cos ( 0 cos TJ ( Fifth kind: 0 -sine 1 = cose sine cos TJ [ :: 1 [~ -sine cose cos TJ Ul Sixth kind:

[ C~( cose sin ( -s~n( cose cos ( [ :: 1 -sine ~][H

Yaw, Pitch, and Roll Engineers describe the attitude of vehicles by angles of yaw, pitch, and roll. For ships, airplanes, railway cars, and automobiles yaw is the angle ( measured about the space-fixed axis Zo (Figure C.6) from vehicles mass center to the zenith, pitch is the angle TJ measured about the carried axis Yl from vehicle mass center to the horizon, and roll is the angle e measured about the body-fixed axis X2 from the vehicle mass center to the front (bow) of the vehicle. The system just described consists of Cardan angles of the fifth kind. For spacecraft, the normal to the orbital plane can be looked upon as fixed in space. Engineers therefore prefer the sequence pitch-yaw-roll for space vehicles. Pitch of a spacecraft is the angle ( about the space-fixed normal Zo (Figure CA) to the orbital plane, yaw is the angle e about the carried axis Xl from the spacecraft mass center and lying within the orbital plane, and roll is the angle about the body-fixed axis Y2 from the spacecraft mass center to the spacecraft bow. The system thus consists of Cardan angles of the third kind. In sports, the terms twist (yaw), somersault (pitch), and tilt (roll) are in common use, but for an athlete they typically are (non-integrable) angular velocity components of the w:z:, w,,' W z type. Problems 339 Problems

C.l. A gyro satellite with body-fixed Cxyz coordinate axes has rotated through Cardan angles of the first kind

TJ = 90° (= 90°. A small instrument is located inside the satellite at

(a) Express the position r of the instrument in space-fixed CXY Z coordinates. (b) If, say, for analysis purposes, it turns out that Cardan angles of the second kind are more suitable, determine their values. C.2. A sailing boat has a single vertical mast. (a) At a roll angle of 10°, a pitch angle of 8°, and a yaw angle of 5°, what is the angle between the mast and the local zenith? (b) At a roll angle of 90°, a pitch angle of 90°, and a yaw angle of 5°, what is the angle between mast and the local zenith?

C.3. A gyro with a body-fixed Cxyz coordinate system has an attitude given by the Euler angles 1/J = 0°, II = 15°, (J' = 30°, and at that very moment has Euler frequencies ~ = 2 rad/s, v = 0, ir = 2 rad/s. (a) What is the attitude expressed in Cardan angles of the first kind? (b) What are the Card an frequencies? 340 Appendix C. Cardan Angles

CA. Cardan angles of the first kind and Euler angles are related by

tan 6 =cos tP tan v sin 1/1 = sin tP sin v tan((1 - 0') = tan tP cos v. Sketch (a) the appropriate spherical triangles and derive equations that relate Cardan angles of the second kind to Euler angles, and (b) Cardan angles of the third kind to Euler angles. The subscript 1 has been used to indicate that the Cardan angles are of the first kind. Problems 341

C.5. The figure shows the Hermes (1976) satellite with roll, yaw, and pitch axes indicated. Sketch the (geosynchronous) orbit into the dia gram. Which kind of Cardan angle system is used, if x is chosen as roll axis, Zl as yaw, and Yo as pitch axis?

UItA T ILlYA flON .. f!NSIONING /oI\K"A.NISM

SI4, itA CO'" A. ... f!NN ... TT .. C liLT ANT&HN"

SA'I ...... OfVlCl~

tAl'" S.. SOtS

UTeNOIBLt SOLAR .... U T SOLA. C!US Appendix D Engineering Data trl (Jq= Systems of Units S· Cl> Cl> System Basic Units .... S· Length Mass Force Time (Jq t=' "..i:I' SI = physical MKS* meter, m kilogram 1 , kg second, s Force: 1 newton = 1 N = 1 kg m/s2 i:I'

Engineering MKS meter, m kilopond 1,2, kp second, s Mass: 1 hyl = 1 kp s2/m

Physical FPS foot, ft pound3 , Ibm second, s Force: 1 poundal = 1 Ibm ft/s2

Engineering FPS" foot, ft pound 3 , Ib second, s Mass: 1 slug = 1 Ib ft/s2

*In use throughout world. **In use in U.S. Standard gravitational attraction constant 9 = 9.980665 m/s 2 ~ 32.2 ft/s2 lREMEMBER: Weight force W = 1 kp when mass M = 1 kg (on Earth surface). 21 kp = 1 kgf = 1 kilogram-force 3REMEMBER: Weight force W = lIb when mass M = 1 Ibm (on Earth surface).

Notes: 1 = 0.3048 m/ft; 1 = 9.80665 N/kp; 1 = 0.453 592 37 kp/Ib; 1 ~ 4.448 N/Ib; 1 ~ 3.28 ft/m; 1 ~ 0.102 kp/N; 1 ~ 2.205 Ib/kp; 1 ~ 0.2248 Ib/N; 1 ~ 0.4536 kg/Ibm.

<:,..:> ~ ~ 344 Appendix D. Engineering Data

Larger and Smaller Units in the Metric System Factor Prefix Symbol Larger units 1018 Exa- E 1015 Peta- P or PA 1012 Tera- T 109 Giga- G 106 Mega- M 103 Kilo- k or'K 102 Hecto- h or H 101 Deka- da or D or dk

Smaller units 10-1 deci- d 10-2 centi- c 10-3 milli- m 10-6 mIcro- J.l 10-9 nano- n 10-12 PICo- p 10-15 femto- f 10-18 atto- a

Example: rSun-Earth = 150 Gm = 150 000 000 km = 150(109)m.

Principal Moments of Inertia

1. Thin rod: A = (1/12)mP; B = (l/12)m12 ; C = 0 Engineering Data 345

3. Cylinder: A = (1/12)m(3a 2 + [2); B = (1/12)m(3a 2 + 12); C = tma2 Qb'Z y .... L 11 z x

4. Thin plate: A = (l/12)mb2; B = (1/12)ma 2; C = (1/12)m(a 2 + b2 )

5. Rectangular parallelepiped: If A> B > C then a < b < c A = (1/12)m(b 2 + c2 ); B = (1/12)m(c 2 + a2 ); C = (1/12)m(a 2 + b2 )

6. Sphere: surface = 411'a 2 ; volume = ~1I'a3 A = (2.5)ma 2; B = (2.5)ma 2 ; C = (2.5)ma 2

7. Spatial dumbbell: If A> B > C then a < b < c A = (1/12)m(b 2 + c2); B = (1/12)m(c2 + a2 ); C = (1/12)m(a 2 + b2 ) m = total mass

~ml'. Y' ~ Q

b 346 Appendix D. Engineering Data

8. Spatial cross boom ("inertia skeleton"): If A > B > C then a < b < c A = (1/12)m{b3 + c3 )/{a + b + c); B = (1/12)m{c3 + a3 )/{a + b + c); C = (1/12)m{a3 + b3 )/{a + b + c); m = total mass z I

9. Ellipsoid: V = {4/3)'Jrabc A = (1.5)m{b 2 + c2 ); B = (1.5)m{a 2 + c2 ); C = (1/5)m{a 2 + 62) r '~/; ~ 10. Thin-walled cylindrical shell: A = B = (1/12)m{6a 2 + /2); C= ma 2

11. Thin-walled spherical shell: V = 41f'a 2 h A = B = C = {2/3)ma 2

12. Simple dumbbell: A = B = tmc2; C = 0

y c x

'1'/2 Engineering Data 347

Material Properties

Density Young's Speed* Tensile Modulus of Sound Strength p E C Uu/t Material kg/m3 MPa mls MPa Air (O°C) 1.293 0.14 331 Aluminum (2014-T6) 2690 55000 5240 240 Beryllium copper (C17200) 8290 120000 4410 1 500 Brass (naval) 8450 100000 420 Brick 1 800 17300 3600 240

Bronze(phosphor ASTM B 159) 8930 100000 700 Concrete 2400 18 300 3200 175** Copper 9810 84900 3580 210-450 Dacron (polyester) 1 380 13800 1 120 Glass 2590 60000 5560 90(900**)

Gold 19300 59 100 2030 140 Ice 900 13** Iron (cast, gray, No. 60) 7210 130000 5 170 420 Kapton (polyimide film) 1 400 3000 4.8 172 Kevlar 29 (aramid fibre) 1 440 12000 2760

Lead 11 370 13200 1 250 14 Magnesium (AZ80A-T5) 1 800 43000 1 310 380 Mercury 13570 100 Nylon (728) 1 140 5520 985 Oil 90 830

Rubber 920 2 54 15 Soil, wet 1 760 Soil, dry 1 280 Steel (Spring alloy SAE 4068) 7830 200000 5850 1 720 Titanium 3080 110000 6420 800

Water, fresh 1 000 2000 1400 Water, salt 1 030 2000 1 560 Wood, oak 800 12000 3380 490** Wood, pine 480 14000 4 180 590** I-v E * c = (1+v)(1-2v)p **Compressive strength. 348 Appendix D. Engineering Data Series

1 2 3 4 (1 + x)2 = 1 - 2x + 3x - 4x + 5x - ...

1 234 (1 _ x)2 = 1 + 2x + 3x + 4x + 5x + ...

. 13 1 5 1 7 smx x - ,x + -x - ,x + ... = 3. 5! 7. 1 2 1 4 1 6 COS X = 1 - 2! x + 4! x - 6! x + ...

1 3 2 5 17 7 tan x = x +"3 x + 15 x + 315 x + ...

z 12 1 3 1 4 e =1+x+,x2. +3'x. +4'x . + ...

12 1 3 1 4 1 5 In(1 + x) = x - '2x +"3 x - 4x + 5x - ... n(n - 1) n(n - 1)(n - 2) (1 ± xt = 1 ± nx + 2! x2 ± 3! x3 + ...

Expansions of pi Pc = J1 + 2{xl Pc) cos 0: + (x 2 I pn

P x 1 x 2 - = 1 + - cos 0: + - 2(1 - cos2 0:) + ... Pc Pc 2 Pc

p2 X x 2 2 = 1 + 2- cos 0: + 2 Pc Pc Pc

p3 X 3 x 2 2 3=1+3-coso:+-22(1+cos 0:)+ ... Pc Pc Pc Appendix E

Nomenclature

Symbols

A area, principal inertia moment about x-axis Al principal inertia moment about xl-axis B principal inertia moment about y-axis Bl principal inertia moment about Yl-axis C centrifugal force C principal inertia moment about z-axis Cl principal inertia moment about zl-axis C' reduced principal inertia moment about z-axis C mass center D dissipation energy E total energy F force G universal gravitational constant [G] gyric matrix G gravity center H angular momentum [I] inertia tensor K Kepler force, internal angular momentum K complete elliptic integral of the first kind [K] stiffness matrix Kx, Ky, Kz interdependent constants, components of K M mass, moment M moment o ongm P function Px, Py,Pz interdependent constants Q function, effective dissipation ratio Qx, Qy,Qz interdependent constants R radius, nominal radius of master mass sphere [R] rotation matrix S satellite T kinetic energy [T] transition matrix U potential energy, potential, work done by external force or torque 350 Appendix E. Nomenclature

v elastic energy W weight force WII microweight force X,Y,Z coordinates

a acceleration a semi-minor axis of ellipsoid, distance, radius, reduced inertia moment ratio C' / C b semi-minor axis of ellipsoid, distance, radius c semi-axis of ellipsoid, damping coefficient e eccentric distance unit vector in x-direction gravitational attraction constant on surface of master mass microgravitational attraction coefficient JI/m = gyration radius k modulus of Jacobi elliptic function, spring stiffness, Smelt parameter length m mass r position vector r radial distance ro initial distance t time u Jacobi parameter u,v,z floating coordinates v velocity x ratio w~/wz X,y,z coordinates

direction angle with respect to x-axis, arcsink of Jacobi elliptic function, parameter elastic deformation coefficient f3 direction angle with respect to y-axis, internal energy dissipation coefficient direction angle with respect to z-axis deflection Kronecker delta Euler vector Euler parameter damping ratio spin change allotment factor Nomenclature 351

() polar angle, rotation angle J.L gravitational parameter, Euler angle .\ rotation axis unit vector v nutation angle ~, 1], ( coordinates, Cardan angles p Rodrigues vector p density, radial distance, Rodrigues parameter u spin angle T Kepler period ifJ angle between two vectors 1jJ precession angle n angular velocity of coordinate system, orbital angular velocity on circular orbits w angular velocity

Subscripts

C center, mass center G gravity center M Moon S satellite f final o original, initial r radial () transverse J.L mIcro P about p

Superscripts

E excess p of platform R of rotor Appendix F

Answers to Selected Problems

Chapter 1

12.5 0 0 1 1.1. (a) [I]c = [ 00 12.5 0 m2kg o 25

(b) [I]o = [~o Jo ~ 1m 2kg with Ao = 12.5 + 50P o 0 25

(d) !l = [e, e, e,] [ +1

(e) H = leu ev e z ] [ 0.~25l Ws2 75000

(f) Ii: = [e, e, e,] [ T1 w.

(g) M = [e, e, e,] [ T1 Nm

(h) 1.529 m

COS () sin () o~ 1 1.3. (a) [IiJ] = [ - sin () cos () o 0

1.8 [n = i2 rna' U[ n 1.9. c = V2a Answers 353

1.10. 2/a = va

1.12. a = 22.5° 1.13. Al = 2; A2 = 4; Aa = 4 0.1667 0 1.14. (a) [1] = [ 0 0.9167 o 1m 2kg o 0 0.9167 (b) a = f3 = 'Y = 54.74°

1.15. (a) m [~! ~~ ~198l m2kg 12 -9 -18 20 54.68 0 (b) ~ [ 0 50.49 o 0 (c) New "I" axis (associated with A = 54.68)

New "2" axis (associated with B = 50.49)

a = 82° f3 = 33.8° 'Y = 122.6° New "3" axis (associated with C = 6.826)

1.17. (b) [ex ey ez] [ 1:807~6l 3.950 (c) T = 12.68 mJ

0 cos t/; sin 1I sin t/; 1 1.18. [ 0 sin t/; sin 1I cos t/; 1 0 COSll

C?SO"smv 0 1 1.19. [ ;i~~ - sin 0" 0 _ sinu -~ 1 tan v tan v 354 Appendix F. Answers to Selected Problems

8 1.21. (a) [1]0 = ~ [ 0 0 5 0 1 Ws 3 005 40 0 (b) [Ih= ~ [ 0 37 w,' o 0 n 8(1 + t) (c) [I] = ~ [ ~ 8(1+t)001 0 Ws3 o 0 1.22. Mx = Awx + Awx - (B - C)WyWz My = Bwy + Bwy - (C - A)wzwx Mz = CW z + CWz - (A - B)wxwy

1.23. 2 . m c2 A = -m ( b2 +-t c ) A=-- 12 T 12 T

2 . m c2 B = -m (c-t+a2 ) B=-- 12 T 12 T

C = m (a 2 + b2) 12 1.26. kl = 0.1111 k2 = -0.6667 k3 = 0.6

0 1.28. (a) [I] = [40000 4000 ~ 1m 2kg o 0 0 (b) Ho = [e. e, e,j [ sr 1w,' (c) T = 80 J (d) v = 1 m/s

(e) [I] = [5~0 5~0 ~ 1m 2kg o 0 0

(I) He, = [e. e, e,j [ T1 w,'

(g) TTran3 = 30 J (h) TRotor = 10 J Answers 355

(i) Zero

(j) H = Ie. e, e,j [ T]Ws' (k) No loss of momentum

1.29. (a) H = [ex ey ez ] [ ~ ] Ws2 250 (b) Tl = 625 J

Chapter 2

[2 2.1. PG = Pc 4 1 +-2 4pc

1 [ 3 cos2 0: - 2 3 cos 0: cos f3 3 cos 0: cos 'Y ] 2.2. [Nr 1 =-2" 3coso:cosf3 3cos2 f3-2 3cosf3cos'Y 3 cos 0: cos 'Y 3 cos f3 cos 'Y 3 cos2 'Y - 2

16000 0 2.3. (a) [I] = [ 0 15300 ~ ] m2kg o 0 2500 0.976] (b) rG = [ex ey ez] [ 0.831 pm -1.808

1042 0 0] 2.6. (a) [I] = [ 0 6667 0 m2kg o 0 6375 -0.0073] (b) rG = [ex ey ez] [ 0.0033 mm 0.0028 356 Appendix F. Answers to Selected Problems

2500 0 0] 2.7. [I] [ 00 16 000 0 m2kg = 0 15300

-1.4] ra = [ex ey ezl [ 0 I'm 2.56

913 2.8. (a) Pc = lOa = 1.559 a (b) Pa = ~4V3a = 1.471 a (c) K = 4v'3"m = 0.462"m 15 a2 a2 sma 2.9. (a) Xc=r- a

(b) Xa = rJ .a sma (c) K= ~"m a r2

Table I a rc ra K

I'm 00 r r r2 1 90 0 = -7r 0.637r 1.253r 0.637"r; 2 r 5 1500 = -7r 0.191r 2.288r 0.191"~ 6 r

1800 = 7r 0 0 0

7 210 0 = -7r -0.136r -2.707r -0.136"~ 6 r

3 2.10. (a) Pc = 4"a (b) Pa = V2a Answers 357

(c) f{ = JLm 2a 2 a 2.11. (a) Pc = 2 (b) PG = a JLm (c) f{ = -2 a (d) Unstable, because PG > Pc = a/2 2.13. (a) C = mrn2 through mass center (b) C = mrn2 through mass center (c) PG = 1.409882 r

2.16. (a) f{ ~ -JLm ( 1---3 x2) pb 2pb 3 X2) (b) PG = Pc ( 1 + 4p~

2 2.17. (a) f{ ~ JLm (1- ! a ) p~ 2pb

(b) PG = Pc (1 + ~ ;~ )

218 h ~; [2+ Ja' + (rc -D' - a' + (rc+ D'

2.19 (a) T=2'~ (1+ 1~~:)

(b) H = m.jjir ( 1 + 4~ ~: )

(d) T = !JLm ~~) 2 r (1- 24 r2

(e) E = --JLm ( 1---1 12) 2r 24 r2

(f) 1'=211" ..1.3( 1---It2) ~JL 8 r~

(g) H2 = mJJLr 2 ( 1 + 2~ :~) 358 Appendix F. Answers to Selected Problems

(j) E = _Jlm + ~!:.) 2r (1 24 r2 (k) ~E = _~!:.Jlm 8 r2 r (1) 2.3JlW (m) ~r = 53.571 mm 2.20. U = -72.473 [1- 1.122 x 10-12] J

0.114] M = [ex e y ez] [ -0.649 10-9 Nm 0.804

2.21. (a) [1] = [~~a2 458~a2 ~ ] o 0 19 ma2 48 (b) XG = 89 mm (c) unstable because XG = +ve 2.22. M = 0.0306 Nm Jlm 1 2.23. (a) K= -2 P r (1 - 4r2 )

(b) rG = r (1 - ::2 )

. Jlm (d) TenslOn = -2 (3/)-- r 2 r 1 2.24. (a) rc = R+ 2

(b) rG ~ rcJI- 4~~ Jlm 12 (c) N = - - - - mw2rc r~ 4 (d) 1 = 143877 km Answers 359

(e) u = py [ (1' - ~) w2 - /I(/I~_ y)] x = /- y (f) 35 793 km from the surface of the Earth

(g) U max = 69832 MPa (exceeds U tilt)

GMm ( C-A c-a ) 2.26. (a) U=--p- 1+ 2Mp2(1-3sin2~)+2mp2(1-3sin28)

(b) K=_GMm(l+~C-A+~c-a) p2 2 Mp2 2 mp2 (c) K=_GMm(1+~C-A_3c-a) p2 2 Mp2 mp2

(d) K=_GMm(1_3 C - A _3 c - a ) p2 Mp2 mp2

(e) K=_GMm(l_~C-A+~c-a) M = ~Gm(A_ C) p2 4 Mp2 2 mp2 y 2 p3 3GM (f) K = _G M m(1 + ~ C- A _ ~ c- a) M = --(a-c) p2 2 Mp2 4 mp2 y 2 p3

2 2.27. (a) PG = Pc (1 _~ a2 ) 16 Pc

2 (b) PG = Pc ( 1 + --3 a ) 8 p~

1 2.28. (a) !]=!]o ( 1---/2) 16 r2

(e)Tl+Ul=-~m 2r 2.29. F = 145 mN

2.30. W~ = 53.312(10-6)N

Chapter 3

3.1. T = 56.08 min 360 Appendix F. Answers to Selected Problems

3.2. (= 3.982° 3.3. w = 2n

Chapter 4

4.3. (a) A * < 1/3 is insufficient (b) Other conditions: C* must not be intermediate inertia moment; B* > A* otherwise satellite would be turned by gravity gradient; C* > A * otherwise satellite would be turned by gravity gradient

4.4. Al=+V-¥+~ A'=-V-¥+~ A3=+V-~-~

P A·=-V-¥+V : -. are all always imaginary if p>O q>O p2 _ 4q > 0

4.5. b = 9 m; c = 11 m 4.6. See Appendix C

4.7. Fifth kind

COS 1/cos( sin esin 1/ cos ( + cos esin ( - sin esin 1/ cos ( + sin ~ 4.8. [Tla_o = [ - co~ 1/ sin ( sin esin 1/ sin ( + cos esin ( cos ~ sin 1/ sin ( + sin e sm 1/ - sin ecos 1/ cos ecos TJ 4.9. cos(xo, Y3) = sin esin TJ cos ( + cose sin ( 4.10. Second kind Wo C 4.11. n = 364.438; A = 1.003 Answers 361

Chapter 5

5.2. (a) H2 = 23125kg2m4 /s2 2TA = 68325 kg2m4 /s2 2TB = 45550kg2m4 /s2 2TC = 22775 kg2m4 /s2

(b) Wx = 5 = 7.6376cnuo ~ 7.6376cnuo Wy = 10 = -13.2288snuo ~ -13.2288snuo Wz = 150 = 150.333 dn Uo ~ 150.333

Uo = -0.857068627

to = 0.009874656s (c) t -to = 0.01- to = 0.000125344s u = 0.010879204 Wx = 7.637148 rad/s Wy = -0.143915974 rad/s W z = 150.333 rad/s (d) 0:0 = 84.399° 0: = 81.334° f30 = 82.442° f3 = 89.894° ')'0 = 9.462° ')' = 8.666° 5.3. (a) t = 0 Wx = 0.00 74 rad/s ~ +0

Wy = 0.8rad/s

W z = 0.00192 rad/s ~ +0 (b) t = 20s Wx = 0.378 rad/s

Wy = 0

W z = 0.98rad/s

Wy = -0.799rad/s

W z = 0.06 rad/s (d) t = 40s Wx = 0.00074rad/s

Wy = -0.8 rad/s

W z = 0.00192rad/s 362 Appendix F. Answers to Selected Problems

5.6. W~ = 0.555 rad/s when w~ = 0 W~ = 0.141 rad/s when w~ = 0 Wy = 0.215 rad/s2 when Wy = 0 Wy = 0.803 rad/s when Wy = 0 Wz = 0.044 rad/s2 when W z = 0 W z = 0.659 rad/s when Wz = 0

5.7. (a) v; [e. e, e,] [ ! 1mls

(b) a = [e~ ej e z ] [ 17;7] m/s2 -125

5.8. a = 8.889 rad/s j b = 5 rad/sj C = 3.2 rad/s

5.9. a = 1.333 rad/sj b = 1 rad/sj C = 0.8 rad/s

5.10. (a) H = [e~ ey ez] [ !~~ 1Ws 2 1500 (b) H = 1641 Ws2 (c) T = 29800 J 5.11. (a) H = 25 Ws 2 (b) T = 127.50 J 5.12. (a) A = 18 WS3 j B = 32 WS3 j C = 50Ws3

(b) Hi = [e~ ey ez] [ !~~ 1Ws 2 1500 H = 1640.7Ws2 (c) Ti = 29800Ws (d) LlT = 2880Ws 5.14. (b) r = 2.582m 1= 3.162m (c) r = 3.651m 1= 3.162m (d) a = b = 105.2m C = 66.28m (e) a = b = 16.67m C = 8.33m Answers 363

Chapter 6

61.252 0 0] 6.1. (a) [I] = [ 0 61.252 0 m2kg o 0 122.5

0 2 (b) H = leu ev e z ] [ 1837] Ws 4901 (c) T = 125.57940 kJ (d) 40 rad/s about z-axis (e) 85.45 rad/s about H-axis 164800 0 6.2. (a) [I] = [ 0 164800 o 0 (b) (, = 0.0141078 rad/s (c) w = 0.0150564 rad/s

(d) H = leu ev ez] [ 249.g78 72] Ws2 143.81184 (e) T = 1.2654082 J 0.015114] (f) v = leu e v e z ] [ ~ m/s

(g) ~'= 0.003548 rad/s (h) (,' = 0.0117704 rad/s

(i) HI = leu ev e z ] [ ~3.85536] Ws 2 71.90592

6.3. (b) H = [ex ey ez] [~~~~ ] Ws 2 48000 (c) 12 rad/s (d) 5 rad/s (e) 24 rad/s

6.4. (al [I] ~ [1 ~ ~o 1m'kg 364 Appendix F. Answers to Selected Problems

(h) v = [ex ey ez ] [ ~5 ] m/s -30 sin lOt

6.5. ~ = 50.99 rad/s v = 78.690 rad/s iT = -4.99 rad/s

6.6. Force = {e}T [ 33~71 ] N -19440

20000 0 0] 6.7. (a) [1] = [ 0 20000 0 m2kg o 0 40000

(b) [Z: 1= [ {d W.'

6.8. (a) H = [eu e v e z ] [ ~ 125] Ws 2 4500 (b) 2.77778 rad/s (c) 1.7123 rad/s

6.9. (a) iT = -w~ = 8 rad/s (1' = 114.592 rev (b) ~ =6.403 rad/s tf; = 91.7 rev (c) v = 51.34° Answers 365

(d) prolate CIA = 1/3.

6.11. f= leu ev ez ] [ 153~.30] N -95.66 f = 1538.3 N 6.12. a = -8.1 m/s2 in u-direction

Chapter 7

24.0175 0.268 -0.2845] 7.1. [I] = [ 0.268 15.0226 0.0356 Ws3 -0.2845 0.0356 10.011

24.0256 -0.002 -0.2872] 7.2. [1] = [ -0.002 24.0226 0.1256 Ws3 -0.2872 0.1256 10.012 o 0 40.000905 o ] 3 7.3. (al [I] = [ 1 o Ws o 19.999095 (b) ~ = 0.286° "1 = 0.258° (c) for Al = 40: a = 42° {3 = 48° (J' = 90° for A2 = 40.000905: a = 132° {3 = 42° (J' = 89.615° for A3 = 19.999095: a = 89.742° {3 = 90.286° (J' = 0.385°

o ] 3 7.4. (al [11 = [1 40.~0405 o Ws 19.999595 (b) e= 0° "1 = 0.258° 366 Appendix F. Answers to Selected Problems

(c) for Al = 40: a = 900 f3 = 00 (1 = 90 0 for A2 = 40.000405: a = 179.7420 f3 = 90 0 (1 = 89.7420 for A3 = 19.999595: a = 89.7420 f3 = 90 0 (1 = 0.2580

0 7.5. (a) [J] = [804 84 o 0 84.006 (b) [J] = [ ~ ~4 0.~2 1Ws 3 0.12 120.006 (c) e= -0.190 (d) ~AA = 0.000007 ~B = 0 ~c = 0.000005 A C (e) Jvz = 0 Jzv = 0.12 Ws3

(f) Yvz = 0.17 Ws3

15714 0 0 1 7.6. (a) [I] = [ 0 15714 0 Ws3 o 0 720 15714.0006 0 0 1 (b) [I] = [ 0 15714 0.12 Ws3 o 0.12 720.0006 (c) e= 0.0004580 (d) ~A = 0.000000038 ~: = 0 ~~ = 0.000000833

(e) Jvz = 0 Jzv = 0.12 Ws3 (f) Yvz = 0.045819 Ws3 15690.14 0 7.7. (a) [J] = [ 0 15690.14 o 1Ws 3 o 0 740.28 Answers 367

15690.1406 (b) [1] = [ 0 15 69~.l406 0.~781 Ws3 o 0.078 740.28 (c) e= 0.0002990

(d) ~A = 3.824 (10-8) ~B = 3.824(10-8 ) ~C = 0 A A C (e) Jvz = 0.078 Ws3 Jzv = 0

(f) Yvz = 0.152 Ws3 15714.14 0 7.8. (a) [1] = [ 0 15714.14 o 1 Ws3 o 0 740.28 15714.142136 (b) [1] = [ 0 15714.141536001 0.2448 Ws3 o 0.2448 740.2806 (c) e=0.0009370

(d) ~A = 1.359 (10- 7) ~B = 0.977 (10-: 7 ) A A (e) Jvz = 0.1248 Ws3 Jzv = 0.12 (f) Yvz = 0.249 Ws3

15714.14 0 0 1 7.9. (a) [1] = [ 0 15714.14 0 Ws3 o 0 740.28 15714.142358 0 -0.0588 1 (b) [1] = [ 0 15714.141752 0.2448 Ws3 -0.0588 0.2448 740.286006 (c) e= 0.000937° 11 = 0.000225°

(d) ~ = 1.500 (10-7) ~: = 1.115(10- 7 )

(e) Jvz = 0.1248 Ws3 Jzv = 0.12 Ws3 Juz = -0.0468 Ws3 Jzu = -0.012 Ws3

(f) Yvz = 0.249 Ws3 Yuz = -2.092 Ws3

7.10. (.) w = Ie. e, e,j [ ~ 1raN, 368 Appendix F. Answers to Selected Problems

(b) H = [eu e v e z ] [ 4.~64] Ws2 23.818

(d) M= Ie. e. e,] [47f6] Nm

(e) U=O (f) U= 0.1 W

7.11. (a) F = 2mwvwz /sinwt w =W z (b) me + ce + ke = Fosinwt Fo = 2mwv w/ 2(r. w c (c) tan¢ = -- wIth r = - (= 2 'irm 1- r2 Wn yltm 1-r2 4m2/2 (d) Ivz = (1- r2) + (2(r)2 k WvWz 2(r 4m2/2 (e) luz = (1- r2)2 + (2(r)2 k WvWz

Chapter 8

8.1. (a) To = 123.370055 J (b) WI = 0.049087 rad/s (c) TI = 19.276539 J (d) flT = 104.093517 J (lost)

4000 0 0] 8.2. (a) [I] = [ 0 4000 0 m2kg o 0 800 (b) Wv = 5 rad/s (c) Vo = 64.359° ( d) To = 107600 J (e) T, = 61577 J (f) flt = 1.46 years 8.5. (a) 108 Nms (b) 46.477 J Answers 369

8.6. (a) Ho = 47.50 Ws2 (b) T = 27.807 J (c) U = -0.409 rad/s (d) ~ = 1.484 rad/s (e) WI = 0.95 rad/s (f) Tl = 22.563 J (g) To - Tl = 5.244 J (h) Ul = -0.534 rad/s (i) ~1 = 1.484 rad/s

[ 1300 0 8.8. (a) [I] = ~ 1300 ~ ] W,' 0 2500 (b) W = 13 rad/s Wv = 5 rad/s

8.9. (a) w~ = 27.143 rad/s Wv = 32.991 rad/s W = 44.614 rad/s ~ = -27.143 rad/s "p = 65.982 rad/s Hv = 3464.1 Ws2 Hz = 6000Ws2 H = 6928.2 Ws2 T=147141J 370 Appendix F. Answers to Selected Problems

(b) W z = 34.641 rad/s w~ = 31.341 rad/s Wv =0 W = 34.641 rad/s iJ = -31.341 rad/s ;p = 65.982 rad/s Hv =0 H = Hz = 6928.2Ws2 T = 120000 J

8.11. Jt = 19.47°

8.12. Jt = -41.81°

8.13. (a) Wn = 100 rad/s (b) (=0.5 [ 1859.12 0 (c) [1] = ~ 1859.12 o 1 Ws3 0 118.24

(d) Wv = 3.227 rad/s ,,":z = 87.891 rad/s tP = 6.455 rad/s iJ = 82.301 rad/s (e) F = 878.736 N 6 = 0.099406 m

(f) Jvz = 0.116 Ws3 Jzv = 0 Juz = -0.296 Ws3 Jzu = 0 ,p = 68.59° (g) V = 99 J (h) ~1/ = 0.007645°

(i) 1/ = 0.047 rad/s (j) D = 13386 W (k) Trigid = 466380 J TdeJ = 466281 J

8.14. (a) Wn = 82.3 rad/s (b) (= 0.5 [ 1859.12 0 (c) [1] = ~ 1859.12 o 1 Ws3 0 118.24 Answers 371

(d) Wv = 3.227 rad/s W z = 87.891 rad/s ir = 82.301 rad/s ~ = 6.455 rad/s (e) F = 878.736 N (f) Jvz = Jzv = 0 Juz = -0.4152 Ws3 Jzu = 0 ¢ = 90° (g) V = 114 J (h) ~v = 0.013282° (i) v = 0.019629 rad/s (j) b = 9692 W (k) Trigid = 466380 J Tdej = 466266 J

8.15. (a) Wn = 60 rad/s (b) J = 0.5 1859.12 o (c) [I) = [ 0 1859.12 o 1Ws 3 o o 118.24 (d) Wv = 3.227 rad/s W z = 87.891 rad/s ir = 82.301 rad/s ~ = 6.455 rad/s (e) F = 878.736 N (f) Jvz = -0.1982 Ws3 Jzv = 0 Juz = -0.4301 Ws3 Jzu = 0 ¢ = 114.7° (g) V = 79 J (h) ~v = 0.0156° (i) v = 0.02 rad/s (j) b = 9950 W

8.17. (a) Wn = 100 rad/s (b) (= 0.01 1886 o (c) [I] = [ ~ 1886 o 372 Appendix F. Answers to Selected Problems

(d) Wv = 3.181 rad/s W z = 96.225 rad/s iT = 90.715 rad/s ¢ = 6.363 rad/s (e) F =70.112 N 6 = 0.007010 m (f) t/J = 0.009425 0 Jzv = 0.056 Ws3 Jvz = 0 Juz = 0 Jzu = -0.000009225 Ws3 (g) V = 70 J (h) ilv=0.007277° (i) v = 26.95 (10-9) rad/s = 0.133°/day (j) iJ = 0.014716 W (k) Trigid = 509544 J Tdej = 509474 J 8.18. Wn = 1.2 rad/s; Wz = 0.6 rad/s; ro = 0.5 for stability

r2 C> A + 2mt2~; 1800 '11750 + 80 1- ro

not satisfied, unstable!

Chapter 9

9.1. (a) 31.66 m (b) F=31142N

9.2. (a) 1= 21.287 m (b) t = 10.164 s 9.3. 549.3 m

9.4. (a) H:c = J{:c + I:cw:c + I:cywy + I:czwz Hy = Ky + I:cyw:c + Iywy + Iyzwz Hz = Kz + I:czw:c + Iyzwy + Izwz Answers 373

(b) Me = Hx +wyHz -wzHy My = 1!x + wyHz - wzH" My = 1(y + wzHx - wxHz Mz = Hz + wxHy - wyHy CWo 9.5. (a) W - --;:;----: - C+l!!!.x2 3 I CWo (b) W C· 2 [2 = +"3 m 9.7. (a) M = m(xw + 2xw - ew 2 )(x - b)

2 (b) e -!?- 2( C + me ) = constant if x = const. - Wo C+2m(e2 +b2)

Chapter 10

10.1. (a) Ho. = 10000 m kg/s Ho. = 17320 m kg/s (b) Woo = 10 rad/s Wo. = 8.66 rad/s (c) Iwol = 13.23 rad/s (d) w~. = 8.66 rad/s (h) HJ• = 10000 m kg/s Hfz = 27472 m kg/s (i) IHI = 29235 Ws2 (j) wJ. = 10 rad/s (k) wJ = 17 rad/s (1) wiz = 13.736 rad/s (m) wiz = 11.20 rad/s (0) 0.94 Nm

10.2. (a) wJv =

(b) Nothing has changed

10.3. Wzo = 0.4 rad/s w~o = 0.2 rad/s R = 1 rad/s Wx = 0.809 rad/s Wy = 0.412 rad/s W z = 0.4 rad/s 374 Appendix F. Answers to Selected Problems

Chapter 11

11.1. (a) H = [eu ev ez ] [ ~9.787l Ws2 50.161

P (b) H ~ [eu ev ez ] [ ~6.775l Ws2 1.461

(d) w = [eu ev ez ] [ ~.757l rad/s 0.487

(e) wR = [eu ev ez] [ ~.757l rad/s 48.70 (f) T = 1208.8 J (g) T P = 21.8 J (h) TR = 1187 J (i) u = -0.148 rad/s (j) uR = 48.065 rad/s (k) ~ =0.988 rad/s (1) ~R = 0.988 rad/s

11.2. (a) w~ = 0.148 rad/s (b) 0.635 rad/s (c) F = 87.358 N

11.3. (a) f = [eu ev ez] [ ~i!~; 1N 0.81

(b) f = [eu ev ez] [ 3~.1 1N -0.24 Answers 375

Chapter 12

12.1. (a)!).v = -10 = -0.017453 rad !).wv = -0.000377 rad

(b) !).wz = 0.002181 rad (c) !).w~ = 0 (d) !)'TR = -0.047 mJ (e) !)'TP = -3.518 mJ (f) !)'D = 3.565 mJ (g) v = 0.101 (10-6) rad/s (h) YJz = 0.019 Ws3 (i) M;; = 0.261 Nm M:; = 0.000005 Nm (j) f3P = 17701 Ws6 (k) T = 44.695 J

(1) !).wv = -0.000377 rad/s !).wz = 0.002181 rad/s !).w~ = 0 !)'TR = -0.047 mJ !)'TP = 0.027 mJ !)'D = 0.020 mJ v = -0.101 (10-6) rad/s

12.4. (a) Wv = 3.68 rad/s (b) stable, since A < G' < Ax and TJ < 0 (c) -18.52 W (d) 18.36 W (e) -9.86 W (f) 10.02 W (g) -0.16 W means the energy flow is out of the rotor, into the platform

12.5. (a) 0.828 W (b) 0.171572 Gw 2 (c) 32.8%

12.6. (a) 0.166 w (b) 46.6% 376 Appendix F. Answers to Selected Problems

12.7. (a) Cw 1 1 C2w2 (b) "2Cw2 + "2Cw2(1- COSV)2 + """4A sin2 v (c) w (d) 0 (e) ~Cw2

(f) (1 - cos v)2 + 2~ sin2 v (g) 26.8% (h) 8% (i) 1.8%

Appendix B

0.985 0.173 0.012 B.l. (a) -0.174 0.982 0.069 o -0.070 0.998 (b) f1 = 0.035 f2 = -0.003 f3 = 0.087 f4 = 0.0996

Appendix C

C.l. (a) X = 3 m Y=-l m Z=2m (b) 11 = -90° ( = 0° e= 180° C.2. (a) 1=12.78° (b) 1=90° C.3. (a) e= 15° 7J = 0° (= 30° Answers 377

~ = -0.517 rad/s

( = 3.932 rad/s

SIn V SIn (j C.4. (a) tan 7J2 = . 1/1 . 1/1 SIn cos V SIn (j - cos cos (j

sin (2 = sin 1/1 cos (j + cos 1/1 cos v sin (j

SIn V tan6 = . cos v cos (j - tan 1/1 SIn (j

I" tan (j + tan 1/1 cos v (b) tan.,3 = ------':-- cos V - tan 1/1 tan (j

sin 6 = sin v cos (j

tan 7J3 = - tan v sin (j C.5. Second kind Author Index

Annett, R. 293, 305 Hagedorn, P. 265, 287 Arnold, R.N. 1,27 Halfman, L. 65 Hamilton, W.R. 230 Hetenyi, M. 65 Bauer, H.F. 65 Hughes, P.C. 1, 27, 100, 155, 160 Beletskii, V. V . 85 Huygens, C. 8 Bianchi, G. 306 Blanton, J.N. 132 Buckens, F. 175 lorillo, A.J. 305 Ishlinskij, A.Y. 1,27

Canavin, J.R. 169, 198 CardanQ, G. 87,329 Jacobi, C.GJ. 106 Cherchas, D.B. 155, 160 Jacobson, D. 132 Chesser, H.C. 27, 178, 198 Jahnke, E. 108, 132 Cochran, J.E. 305 Jarmolow, K. 256, 262 Johnson, T.V. 86 Junkins, J.L. 115, 132 DeBra, D.B. 24, 27 deCoriolis, G. 153 Delp, R.H. 24, 27 Kane, T.R. 1,27,94, 100,273,287 Doetsch, K.H. 156, 160 Kaplan, M.H. 1,27, 287 Dong, W.N. 230,233 Kelvin, W.T. 264 Duffing, G. 112 Kepler, J. 40,46 Kim, Y.I. 28 Kohler, P. 85 Elias, J.H. 86 Kolbe, O. 65 Emde, F. 132 Euler, L. 1, 13, 14, 15,311,321 Landon, V.D. 207,233,293,305 Leech, C.M. 28 Faulkner, M.G. 306 Leipholz, H.H.E. 230, 233 Levinson, D.A. 27 Li, Y.S. 299, 306 Giacaglia, G.E.O. 85 Liapunov, A.M. 230 Goldeich, P. 85 Likins, P.W. 27, 132, 169, 198,269, G6rtler, H. 101 287, 293, 306 Gossain, D.M. 155, 160 Ling, F.W. 189,199,291,306 Graham, J.D. 155, 160 Liu, Y.Z. 1,27 Grammel, R. 256, 262 Lohmeier, P. 129, 132 Greenwood, D.T. 27 Losch, F. 132 380 Author Index

MacCullagh, J. 118, 129,224 Schlack, A.L. Jf. 230, 233 MacNaughton, J.D. 155, 160,253 Shackcloth, W.J. 305 Magnus, K. 1,22,27,98, 100, 127, Sherman, B.C. 155, 160 132, 184, 198, 287 Smelt, R. 24 Marsh, E.L. 100 Steiner, J. 8 Maunder, L. 1,27 Stewart, B. 293, 305 McCord, T.B. 86 Meirovitch, L. 230, 234 Misra, A.K. 86 Tabarrok, B. 1,28 Modi, V.J. 86 Thomson, W.T. 1,28,95, 101 Tisserand, F. 175, 177, 198 Tonkin, S.W. 305 Newton, I. 1 Torzhevsky, A.P. 101 Truesdell, C. A. 1, 28 Tyc, G. 96, 101 Okhotsimsky, D.E. 101

Vigneron, F. 287 Peale, S.J. 85 Vincenti, W.G. 65 Poinsot, L. 115 Poisson, S.D. 327 Wagner-Bartak, C.G. 156, 160 Wilson, W.G. 100 Rasmussen, H. 27 Wittenburg, J. 1,28,87, 101,287 Rimrott, F.P.J. 27,83,86, 123, 132, 178, 189, 198,234,253,299,301, 302,306 Xu, D.M. 86 Robe, T.R. 1,27 Roberson, R.E. 273, 287, 306 Rodrigues, O. 328 Yu, Y. 189, 198, 234, 306

Sarychev, V.A. 101 Zhang, W. 189, 198, 199,291,306 Schiehlen, W. 65, 287, 306 Ziatoustov, V.A. 99, 101 Subject Index

Angular momentum Damping ratio 183, 189 Angular momentum ellipsoid 117 Deformable 124 Angular momentum law I Deformable gyros 166 Angular velocity 18 Degeneracy 19 Apparent libration 83 Despin 240 Approach 156 Deviation moment 5 Attitude 40, 87 Direct precession 146 Attitude diagram 130, 150, 152 Dissipation energy 132 Attitude drift 205, 209 Dissipation ratio 302 Attitude matrix 324 Dissipative gyros 198 Attitude stability 83,95,98, 122, 205, Dissipative gyros tats 290 268 Dissipative platform 292, 299 Auxiliary frequency 140, 270 Dissipative rotor 297, 299 Auxiliary inertia moments 170 Dissipative solid bodies 198 Auxiliary inertia products 170 Diurnal libration 85 Axisymmetric gyros 139 Drift rate 205 Axisymmetric gyrostats 270 Dual spinners 265 Duffing equation 112

Bearing friction 273 Body cone 146 Eccentric booms 244 Boom extension 244 Effective dissipation ratio 302 Booms 59 Eigenvalue 9 Elastic deformation 182, 187, 225 Capture 156 Elastic deformation coefficient 182, Cardan angles 87, 329 187 Carrier 264 Elastic energy 132 Center of gravity 40 Elliptic functions 106 Center of mass 40 Elliptical orbit 98 Central force field 87, 281 End effector 156 Centrifugal moment 5 Energy dissipation rate 132, 186, 205, Changing inertia moments 154 297, 299, 302 Characteristic equation 10, 93 Epicycloidal motion 106 Circular orbit 91 Euler angles 15, 311 Collinearity theorems 131 Euler equations 13 Complete elliptic integral 108 Euler frequencies 15 Configuration change 249 Euler parameters 321 Constant configuration 170 Euler's law I Coordinate transformation 17 Excess angular velocity Coriolis force 153, 187 267 382 Subject Index

Floating coordinates 140, 169, 273 Locked rotation 41,55, 76, 95, 264 Frame rate of change 4, 5, 14 Long run (attitude stability) 124 Frequency ratio 183, 189

MacCuUagh ellipsoid 118 Grapple fixture 156 Magnus shape triangle 22, 97, 123 Gravitational attraction constant 40 Mass center 40, 95 Gravitational force 40, 52 Maximum inertia moment 123 Gravity center 40 Microgravitational attraction Gravity gradient 59, 62 coefficient 64 Gravity gradient stabilization 83 Microgravity 63 Guide rate of change 4 Microweight force 64 Gyric matrix 95 Minimum inertia moment 123 Gyro 1,265 Modified inertia product 173, 301 Gyroscope 1, 265 Moment of inertia 5 Gyroscopic stability 95, 98 Momentum wheel 264 Gyrostats 265 Moon 83

Huygens-Steiner parallel axis Natural frequency 183 theorem 8 Near-rigid solid 167 Node line 15, 16, 19 Nutation 15 Inertia ellipsoid 117 Inertia moment 5 Inertia product 5 Oblate 117 Inertia tensor 5 Ogives 210, 212 Inertial force field 153, 278 Orbital period 40, 55 Integrability 20 Interdependent constants 112 Internal angular momentum 249 Parallel axes theorem 8 Internal energy dissipation Parametric coefficients 100 coefficient 183, 192, 208, 211 Partial inertia products 173 Internal torque 273, 293, 297, 298, Pericycloidal motion 108 299, 300, 303 Period 40 Phase plane 113 Pitch 338 Jacobi elliptic functions 106 Platform 264 Poinsot ellipsoid 115 Poisson's kinematic equations 327 Kepler force 40, 52, 87 Polar inertia moment 5 Kepler period 40 Potential 54 Kinematic equations 327 Practical stability 123 Kinetic energy 11 Precession 15 Kinetic energy ellipsoid 115 Principal axes 10, 14, 123 Principal inertia moments 7, 8, 9, 344 Libration 76 Product of inertia 5 Line of nodes IS, 16, 19 Prolate 117 Subject Index 383

Quasi-locked rotation 98 Smelt parameters 24 Somersault 338 Space cone 146 Reduced inertia moment 267 Spacecraft 59 Relative rate 5, 13 Spin 15, 183, 189 Remote manipulator system 156 Spin change allotment factor 299 Retrograde precession 146 Stability 95, 98, 122, 124, 205, 268 Rigid 122 Static stability 95, 98 Rigid platform 297 Steiner parallel axes theorem 8 Rigid rotor 292 STEM tubing 83 Rigidification 131 Stiffness matrix 95 Rigidization 156 RMS 156 Rod satellite 41 TEE tubing 83 Rodrigues parameters 328 Tethered bodies 85 Rodrigues vector 328 Theta functions 135 Roll 338 Tilt 338 Rotation matrix 320 Tisserand-type frame 178 Rotation theorem 324 Torque 1,53,273,293,297,298,299, Rotor 264 300, 303 Torquefree gyros 105 Transition matrix 89, 318 Satellite 40 True libration 83 Secular attitude drift 205, 292, 297, Twist 338 299 Self-excited gyros 256 Shape petal 27 Wheel 264 Shape triangle 22 Short run 123, 268 Shuttle 156 Yaw 338 Singularity 19 Yo-yo masses 240