D.3 Tensor Property of the Rotational Time Derivative

In flight dynamics, only the tensors of rank zero, one, and two are needed. Special consideration, however, must be given to the axial vector of rank one, because it is derived by contraction from a skew-symmetric tensor of rank two. We will now proceed to prove that the operation of the rotational time derivative is covariant for these tensors, i. e., they remain tensors under all time dependent, Cartesian coordinate transformations.

Theorem DA: The rotational time derivative of vector v-, transforms like a

tensor of rank one under any Cartesian, time dependent coordinate transformations

іц* i-e.,

8Vi – 8vj

J7 = tijJ7

Proof: We introduce three Cartesian coordinate systems, x;,x,,x;, each of

which coordinates the tensor of rank one, іу, i>,, V/. The coordinate systems are related by the transformations

In addition, we need a tensor of rank one r, , whose coordinates are constant in the unbarred system, dr,/dr = 0. Now form the scalar product and take the ordinary time derivative in the three coordinate systems:

d d d _

T (riri) = — (Nri) = x Kv<r‘

dr dr dr

Expanding the first relationship, we obtain

‘For the remainder of this appendix the overbar is used to distinguish different Cartesian coordinate systems; i. e., serving a slightly different purpose than in Sec. D.2.

where we used the fact that the r, coordinates are constant. Let

d _ d _

rk – tkin and — (tkiri) – —tkiri

dr dr

and substitute into Eq. (D.25)

d _ d d.

ri — Vi = tsin—vs + vk—tkiri dr dr dr

Because r, is arbitrary, this relationship reduces to

d – d _ d _

~7~Vi = tsi~rVs -|- —tki)k

dr dr dr

Multiply with

$li — Гу//,

both sides of Eq. (D.28)

d _ d _ __d__

— Vs f tsltsl ~tk[Vk

dr dr dr

and contract

d. / d _ . d.

~7~^i — tsi I ~~Vs + tsj —~tkiVk

dr V dr dr

We recognize the term in parentheses as the rotational part of the time derivative. The second relationship of Eq. (D.24)

-7- Ып) = (vift) (D.32)

dr dr v ‘

yields a similar equation using the same procedure

d = (d = = d= = ,

— I); — tri I — D + trj-^tpjVpI (D.33)

Set Eq. (D.31) equal to Eq. (D.33)

/ d _ _ d _ _ = / d = = d = = ,

Ksl ( “t” ?si tki^k — trl I + t rj t pjV p 1 (D.34)

Multiply both sides with t, i, contract, and introduce from Eq. (D.23),

Г* = tJri (D.35)

and get

d _ _ d _ ^

— U; -|- tfl ~tkiVk = Г(г

d = = d = =

diVr+trjdttpjVp)

According to Eq. (D.21) both sides contain the rotational time derivative of the vector, expressed in the single – and double-barred coordinate systems, connected by their common transformation

Svr _ „5iv

J7 ~ ttt~sT

Because t* is any Cartesian coordinate transformation, the rotational time deriva­tive of a tensor of rank one is itself a tensor of rank one; and the proof is complete.

In the proof, the unbarred coordinate system, though arbitrary, has a key prop­erty, namely that in it the coordinates r, are constant. This coordinate system is associated with the reference frame of the rotational time derivative. To specify the rotational time derivative completely, such a reference frame must always be provided. Instead of the subscript notation, the symbolic notation is better suited to specify the rotational time derivative. Let the operator 5/5 f be replaced by D, the collection of all tensors of rank one by v, and the reference frame by A, then the rotational time derivative, valid in all Cartesian coordinate systems, is written as Da.

In flight dynamics, where coordinate transformations abound, identifying co­ordinate systems by overbars or other symbols is cumbersome. A matrix notation is preferred, with the coordinate system indicated by a capital letter, e. g., for two coordinate systems A and B, the transformation of В wrt A is [T]BA, and the transformation of tensor v of rank one is

[u]B = [T]BA[v]A (D.38)

In matrix notation, the rotational time operator of Eq. (D.21) takes on the form

Note that the coordinate system ]A is associated with the reference frame A, whereas ]B is entirely arbitrary. This equation is in the form as found in Eq. (4.36) of the main text. If in the current proof we associate the unbarred system with A, the one-barred with B, and the two-barred with C, then Eq. (D.37) becomes

[Dav]b = [T]BC[DAvf

Because ]B and ]c are any Cartesian coordinate systems, the rotational time deriva­tive of v is a tensor of rank one. Next, we prove that the rotational time derivative also preserves tensors of rank two.

Theorem D.2: The rotational time derivative of tensor V, y transforms like a

tensor of rank two under any Cartesian, time dependent coordinate transformations

Proof: The proof is similar to that of Theorem D. 1. Again we introduce three

Cartesian coordinate systems, Xj, T, each of which coordinates the tensor of rank two, Vjj, Vij, Vjj. The coordinate systems are related by the transformations given in Eq. (D.23). We also need a tensor of rank two Rtj, whose coordinates are constant in the unbarred system, dRjj/dt — 0. Let’s form a scalar with the twice contracted product:

Ї (KA) = 5 = s (Mt.) Expanding the first relationship, we obtain d – d – – d-	(D.42)
R-ij ~7~ V[j — Rim ~7~ Vim “t – Vkn ~T~ Rkn dt dr dr Take the time derivative of	(D.43)
Rkn — ft/ tnj Rjj and get d – /_ d. d _ _	(D.44)
Rkn — Ifki ~ tnj T" t(j tfij J Rij	(D.45)
and then substitute into Eq. (D.43). Because Rjj is arbitrary, we get with Eq. (D.44) d __ __ d – _d_- d__-
і Vjj — tn tyfij Vim + t^i, tni Vkn ~~ . tin tnj Vkfi dr dr dr dr Multiplying by	(D.46)
Spi ^ trptri and Sqj — fijf, and contracting, yields d _ _ /d _ _ d _ – _ d_ _ >	(D.47)
Vpq tmqhp 1 “ 4“ f / ~ L, VlfH	) (D.48)

The expression in the parentheses is the rotational time operator of a tensor of rank two:

SV, m d _ _ d _ – _ d_ _

e TjL/m ”1“ Ui tki Vkm + Vln~rtnjtmj (D.49)

St dt d t dt

The first term is the ordinary time derivative, whereas the last two terms are the contributions of the rotating coordinate system to the rate of change of the tensor. The second relationship of Eq. (D.42),

yields a similar equation using the same procedure:

Set this equation equal to Eq. (D.48), multiply both sides by taqtbp, contract, and introduce the following:

tas — hqhq and — tbptfp

The result is

d – _d_- – d__

~~rVba "F hi T гкі Vka "F hj)n — tnj C / dr dr dr

/ d = =d== =d==

= hr his І ~j^ V rs "F t vn^ vr “F V tr~^t ruit sw

Both sides contain the rotational time derivative of the tensor, expressed in the single – and double-barred coordinate systems, connected by their common trans­formation:

Because the r*. is any Cartesian coordinate transformation, the rotational time derivative of a tensor of rank two is itself a tensor of rank two, and the proof is complete. Note that the unbarred coordinate system assumes the same special place as in Theorem D. l, namely that the coordinates Rjj are constant. It is associated with the reference frame of the rotational time derivative.

Following the matrix conventions of Eq. (D.39), the rotational time operator of a tensor of rank two, Eq. (D.49), assumes the form

which is the same as Eq. (4.37) of the main text. The tensor quality of the rotational time derivative of a tensor of rank two, Eq. (D.54), is stated in matrix notation,

[.DaV]b = [T}BC[DAVf[T}BC

where ]s and ]c are any two Cartesian coordinate systems.

In flight dynamics we deal not just with regular tensors of rank one and two, but also with a special skew symmetric tensor of rank two, which can be reduced to a tensor of rank one—an axial vector—under the restriction of right-handed Cartesian coordinate systems. Examples are angular velocity, angular momentum, and aerodynamic moments.

Theorem D.3: If the allowable coordinate systems are restricted to be right-

handed Cartesian, then the axial vector /,■ has the same rotational time derivative as the regular tensor of rank one.

Proof: Axial vectors are skew-symmetric tensors of rank two, which can be

contracted to tensors of rank one under the assumption of right-handed Cartesian coordinate systems. Let L,;be a skew-symmetric tensor, then, with the help of the

permutation tensor £, д the axial vector /; can be found:

І і = sijkLjk (D.57)

For the proof it is sufficient to show that, if Vjj of Eq. (D.46) is a skew symmetric tensor of rank two, say Eq. (D.46) can be reduced to Eq. (D.31) with axial vector /;. Rewrite Eq. (D.46) for Ьц

d _ _ d – – / d. d _ _

— tutmj—Lim -|- Lfcn І ^г”^иу ~^^ki^nj 1 (D.58)

and introduce the definition for the axial vector/, of the unbarred and single-barred coordinate systems,

Lij — Sijplp and L[j — Sjjplp (D.59)

to obtain

d d- _ – d _ d__

Sijp-lp — hitmj£lmq~^lq ”1“ &knAr ( / (D-60)

To further reduce the second term on the right-hand side, we use the transformation relationship

£ pqr — К I tpitqk trm ^ikm

and multiply both sides by tpstqt, while taking the time derivatives
. d. d. _

Now we insert the result into Eq. (D.60)

d d- __ d –

£(mq q t£ijs ~^trs*r

and use the inverse of Eq. (D.61)

— |-j Slmqtlitmjtqs

to combine the terms on the right-hand side of Eq. (D.63),

^— E-imq hi hn J I ^ q d – tqs

In view of

Slmqtlitmi — t£ijptq

For right-handed Cartesian coordinate systems |?| = +1. Furthermore, because i, j are free indices, there will be for each p = 1,2, 3 a combination of г, j such that є і//> ф 0. Therefore we have

dtp~tqp di 4+t4’dit, J’

which is exactly in the form of Eq. (D.31). This completes the proof.