Appendix A Review of Vector Algebra and Derivatives of Vectors

Appendix A Review of Vector Algebra and Derivatives of Vectors A physical vector is a three-dimensional mathematical object having both magnitude and direction, and may be conveniently represented as a directed line segment in three-dimensional space. Figure A-1 shows a vector P and a reference frame characterized by a set of right-hand orthogonal unit vectors {i,j, k}} It is clear that the magnitude of P (its length) is independent of the choice of reference frame, but that it's direction depends on this choice and will generally be different for different frames. kn Fig. A-1 Two vectors are equal if and only if they have the same magnitude and direction. This allows us to write P=\P\ep (Al) where P is the magnitude (length) of P and ep is a unit vector in its direction. It is easy to verify that the vectors on both sides of Eqn. 301

302 Newton-Euler Dynamics (A.l) have the same magnitude and direction. We can now state that if E. = \E.\^P a-nd Q = \Q\eQ are two vectors, then they are equal if both P = \Q\ and ep = iq (Fig. A-2). Note that the "point of application" of a vector is not a part of its definition. Thus, P_ = Q regardless of their points of application. Fig. A-2 The sum of two vectors P and Q^ R = P_-\- Q^ m defined by the parallelogram rule, or equivalently by placing the vectors "tip-to-tail" (Fig. A-3). kii or Fig. A-3 Two vectors may be multiplied in two different ways. The scalar or dot product^ denoted P Q, is defined as the magnitude of P times the magnitude of the projection of Q_ along P, or vice-versa. Thus P-Q = P IQI cos ^ (A2) where 6 is the angle between P and Q (see Fig. A-4). It is obvious that R- Q = Q P.- If 2, j, and k are orthogonal unit vectors.

Review of Vector Algebra and Derivatives of Vectors 303 Q Fig. A-4 Fig. A-5 i-i=j-j = k-k = l i-j=j-k = k-i = 0 {A.3) The vector or cross product is defined as PxQ= ( P Q sin^)e {AA) where e is a unit vector perpendicular to both P and Q_ and directed in such a way that P, Q, and e form a right-hand triad in that order (Fig. A-5). It is clear that P x Q = Q x P. For the right-hand triad of orthogonal unit vectors {i,j/k}, ixi=jxj=kxk=0 ixj = jxi = k j X k = k X j = i k X i = ~i X k = j ia.5) For the purpose of calculation, it is convenient to resolve vectors into components in specific directions. The vector P resolved into components along {i,j,k} is P = PJ + Pyj + Pzk (A6)

304 Newton-Euler Dynamics Px, Py, and Pz are called the rectangular components of P. Using Eqns. (A.3), they are Px^P-i Py=P-j Pz=P-k Equation (A.6) represents P_ as the sum of three vectors (Fig. A-6). (A7) Fig. A-6 Now suppose vector Q is also resolved into components along {i, j, k}: Adding Q and P_ gives g = Qj + Qyj + Q,k {A.8) R = P + Q r.... {A.9) Rxi + Ryj + Rzk = (Px + Qx)i + {Py + Qy)j + {Pz + Qz)k which implies Rx = Px + Qx, Ry = Py + Qy, Rz = Pz + Q Z {A.IO) IS Using Eqns. (A-3), the dot product of two vectors in component form P g Pr^i I Py^y "r i'^z^z (All) In particular, P-P = Px+Py+ Pz = P (A12)

Review of Vector Algebra and Derivatives of Vectors 305 Similarly, using Eqns. (A.5), the cross product of P_ and Q is given by PxQ = [PyQ, - P,Qy) i + {PzQx - PxQz) ] + {PxQy - PyQx) k (A13) which is conveniently expressed in determinant form: PxQ = i Px Wx 3 Py Qy k Pz Qz (A. 14) Suppose P, Q, and R are three vectors. The following identities are then valid: P-QxR=^QRxP = R-PxQ (A.15) ipxq)xr = R-PQ-R-QP (A.W) P-QxRis called the triple scalar product and {Px Q) x Ris the triple vector product. The derivatives of vectors follow rules analogous to those for scalars. Let P(t) and Q{t) be vector functions oft and let u{t) be a scalar function oft. Then ~ dpjt) _ P{t + At) - P{t) _ APjt) dt At-^o At At->-o At dq{t),. Q{t + At)-Q{t),. AQ{t) ^=1 = lim = = = lim ^= dt At^o At At->-o At and the following rules apply d{p. + Q) _dp dq dt dt dt (A17) (^.18) d{p-q) ^^ dq ^ ^ dp dt dt dt d{pxq) dq dp ^, ~ = P X -~ + r ^ Q diup) dp du dt dt dt (A19) {A.20) (A21) dt at dt It must be kept in mind that the derivative of a vector is reference frame dependent.^ We recall that the value of a scalar function and of its derivative is independent of the choice of reference frame. This is not true, however,

306 Newton-Euler Dynamics for a vector. The derivatives of a vector w^ith respect to two reference frames, say {i, j, k} and {?', j', k'}, are equal only if these reference frames are fixed relative to one another, that is, if i', j' and k' are constant vectors in {i,j,k}. Also needed is the gradient vector. If f{x,y,z) is any function depending on the rectangular coordinates x, y and 2;, then the gradient of / is given in rectangular components as We conclude this Appendix with three examples. Example 1. A unit vector e rotates in a plane with rate 9 relative to orthogonal unit vectors {i,j}. We wish to show that e is perpendicular to e. From Fig. A-7, e = cos 9i + sin dj Fig. A-7 Differentiating, e = { sinoi + cos 6 j)6 We have so that e is perpendicular to e. Furthermore, the vector CQ { sinoi + cos9j) is a unit vector; thus we may write e = 9eg

Review of Vector Algebra and Derivatives of Vectors 307 This relation is used many times in this book. Example 2. Referring to Fig. A-8, we wish to find the unit vectors along AB and AC, and the unit vector perpendicular to the plane containing A, B, and C. We have AB = {A- 1)1 + (5 - l)j + (6-1)^ = 3l + 4j + 5k AC = {2-1)1 + (3-1)] + (7-1)^ = I + 2j + 6A; C(2, 3, 7) B(4, 5, 6) A(l, 1, 1) Fig. A-8 The unit vectors along AB and AC are then, respectively, AB 2,1 + Aj + 5^ \AB\ %/32 + 42 + 52 ^(3i + 4J + 5*) AC \AC\ ~ VI /I + -L. 4 + -I- 36 3fi ~ yit ^/IT \ ' ) A vector perpendicular to AB and AC is AS X AC i j k 3 4 5 12 6 = 14z - 13; + 2k Thus the unit vector perpendicular to AB and AC is ABxAC 1 (14i - 13i + 2k) AS X AC ^369

308 Newton-Euler Dynamics or 1 (14? - 13; + 2k) ^369 Example 3. We wish to find the equation of the plane containing points A(l,l,l), B(4,5,6), and C(2,3,7). Two vectors that lie in this plane are AB = (4-1)1 + (5 - l)j + (6-1)^ = 3z + 4j + 5k AC = 1 + 2] + 6k A vector normal to this plane is N = ABxAC = 14:i- 13] + 2k For any point P{x,y, z) in the plane, AP-N = 0 '{x - 1)1 + {y- 1)] + {z- l)k\ (l4«+ 13j + 2A:) = 0 14a; - 13y + 2^ - 3 = 0 which is the equation of the plane. Notes 1 In this Appendix, as in the rest of the book, vectors will underlined except that unit vectors will be denoted by ("). 2 Section 1.2 gives several examples in which the velocity and acceleration vectors of a point are different with respect to different frames. Problems A/1 Given an orthogonal triad of unit vectors {i,j,k}, express the position vectors r^ and rg with respect to the origin for the points A(3, 2,5) and B(l,4, 2). What is the position of S with respect to A?

Review of Vector Algebra and Derivatives of Vectors 309 A/2 A vector starts at the origin and extends outward through the point (4,-1,3) to a total length of 10 units. Give an expression for the vector. A/3 The sum of the two vectors P and Q is given by R = P + Q lfp = 30l + 40j and Q = 20l - 20j, find the vector R. What angle does R make with the direction of the unit vector z? A/4 Given: P = ZOi + 40J - 7k Determine: Q = Qi 5j (a) P.+ Q {h)p-q (c)z-q (d)(p + Q)-(P-g) A/5 Given a vector PQ defined by the line segment beginning at point P( 3,1,5) and terminating at Q(4,0, 2), express PQ in terms of the rectangular unit vectors. Find the commponent of PQ in the direction defined by a line passing through the points (3,1, 1) and (-1,2,7). A/6 For the right-handed rectangular triad {i,j,k}, show that AxB = i j k ai a2 03 h 62 ^3 where A aii + a2j + a^k and B_ = bii + 62J + 63A;. A/7 For the right-handed rectangular triad {i, j, k}, show that i x {i x j) = -3- A/8 Show that the time derivative of the vector e of Example 1 can be represented by the vector w^e, if ^ = cj is constant. A/9 A vector is given by the following expression: u{t) = 2t^l + (3t^ - l)j -{t- t^)k Find the time derivative of this vector as a function of time t. What is the value of this derivative at i = 2?

310 Newton-Euler Dynamics A/10 Verify Eqns. (A.18) - (A.21) where A = aii + a2j + a^k and B_ = bii + b2j + bzk, and where u, ai, bi{i = 1, 2, 3) are functions of t. A/11 Find the gradient of each of the following scalar functions of x, y, and z: (a) V = 6x + 9y-6z (b) V = -Ax'^y (c) V = Axy + Byz + Cxz where A, B, and C are constants. A/12 Find the gradient of the function V=-{A-B) at thepoint(l,-l,2) where A = xyi + yzj + zxk and B_ = yi + zj + xk. A/13 Find the vector projection of B_ onto Aii A = 2i + 3j k and S = 3J - ; + 2k. A/14 Given vectors A{ai, a2,as) and Bjbi^b^, 63), find the magnitude of vector 0, = B_ A. A/15 Do the following vectors ^4, B, and C_ form a right-handed triad? A = 2i-''j + k B_=-i-\-2j+ C = i+j-k 2k A/16 Choose x, y, and 2; such that i + j + 2k, i + zk, and 2i + xj + yk are mutually orthogonal. A/17 likl = 0J.xR, where w = tt + t'^j + 3A: and E = 3^2 + t^j + t^k, find du/dt. A/18 For any vector E, prove that d / dr\ d'^r Rx -=] =Rx - dt \r dt J ~ dt"^ A/19 Find the unit vector which is perpendicular to both 2>i 2j + k and i+j -2k. A/20 Prove that Ax {B_x Q) + B_x {Qx A) + C_x (Ax B_) =^ Ohv any three vectors A, B_, and C_.

Review of Vector Algebra and Derivatives of Vectors 311 A/21 lia = i + j-k,b = 2i+j + k, and C = ~i-2j+ 3k, find (a) (b) (c) AxiBxC) A-iRxC) B-{AxC) A/22 Show that iaxb)-{cxd) = A-C A-D B-C B-D A/23 Given A = 2tH + 5tj - 8k and B = Si + 6j, find {B_-A) and dt (R X A). dt A/24 Integrate A x S with respect to time where A = bti (4i^ + 6)j and B_ = (-6*^? + 12A;). A/25 Integrate the vector A = [l^ + 12t^)j + 8k from time t = 3 to time ^-10. A/26 Find the equation of the line which is parallel to the vector R = 2i + 3j + bk and passes through the point Q(3,4, 5). A/27 Find the equation of the plane which is normal to the vector N_ = lai 13j + 2k and contains the origin.

Appendix B - Mass Moments of Inertia of Selected Homogeneous Solids 1 Thin rod ^ Thin rectangular plate z^ ^ [ Ixx = Y^"^(^^ + C^) T 1 2 Rectangular prism z-^ y ^. 4x = ^"^(&^ + C^) 313

314 Newton-Euler Dynamics y Thin disk i T 1 2 Izz = lyy = -mr^ Circular cylinder y 9^ Ixy = -ztna Circular cone. - ^ y r 3 2 Sphere / X. I XX = j^y = /^i = ma 5

Index 315 Index acceleration, 4 aircraft equations of motion, 137-139,155-158 angular acceleration, 74,85 angular momentum, 166,195 angular velocity, 18,71,83 assumptions of dynamics, 2 azimath, 150 balancing, static and dynamic, 222,223 basic kinematic equation, 20,80, 86 basis vectors, 3 center of mass, 110,111 centripetal acceleration, 82 components cylindrical, 48,139 intrinsic, 39,41 radial-transverse, 16,17,136 rectangular, 8,136,139 spherical, 51,52,139 tangential-normal, 10,12,136 composition relations for angular velocity and acceleration 84,85 conservative force, 272 continuum of mass, 115 coordinate transformations, 52 coordinates, cylindrical, 46 polar, 16 rectangular, 7,37 spherical, 50,150 Coriolis acceleration, 82 Coriolis effect, 149 dead weight, 227 degrees of freedom, 124-126 dynamic loads, 216,221 energy equation, 267, 282 eigenvalue problem, 67,173 elevation, 150 Euler's angles, 232-234 Euler's equations, 197 Euler's theorem, 64 equivalent force systems, 204 flexure, 40 forces, external and internal, 110,193 frame of reference, 3 inertial, 2,140,141,144-148 Prenet formulas, 41 Frenet triad, 40 gradient vector, 306 gravitation, 122-124, 272,273 gravitational force per unit mass (g), 123, 148 gyrocompass, 255 gyroscopic motion, 239 equations of, 242 gyroscope, 254,255 holonomic constraint, 125,126 inertia matrix, 168 instantaneous axis of zero velocity 231 kinematics, 1 kinetic energy, 267,274 for rigid body, 281,285,286 for systems of rigid bodies, 287

316 Newton-Euler Dynamics kinetics, 1 rotational mass symmetry, 175 latitude 145 line of nodes, 239 linear momentum, 109,166 longitude, 145 mass, 2, 107 mass properties engineering, 186 matrix, identity, 54 inverse, 54 matrix algebra, 53 mechanical energy, 274,282 moments of forces, 194 moments of inertia, 167 Newton's laws of motion, 107, 108,111,135,194,197 nutation, 239 osculating plane, 42 parallel axis theorem, 179 particle, 107 potential energy, 271-272 power, 265 principal axes of inertia, 172 principal moments of inertia, 172 precession, 239 problems, basic dynamic, 1 products of inertia, 167 speed, 10 spin, 239 state variable form, 138 steady precession, 243,244 about axis normal to spin axis, 249 with zero moment, 246 time, 2 top, motion of, 239 torsion, 41 unit vector transformations, 11 universal law of gravitation, 122 vectors, 301 derivatives of, 305 scalar product, 302 triple scalar product, 305 triple vector product, 305 vector product, 303 velocity, 4 weather patterns, 153-155 weight, 124 work, 261 done by force couple, 265 done by internal forces in a rigid body, 277 work integral, forms of, 262-264 radius of curvature, 12 radius of gyration, 181 relative velocity and acceleration, 21,22,81,82,86 rigid body, 2,113-115,118-122 rigid body motion, 197 rotation matrix, 66