Appendix I: Derivation of instantaneous pole and angle of plate motion from analytic finite rotation functions (Plate Reconstruction Interpolation)
(Based on Smith, E. G. C., 1981, Calculation of poles of instantaneous rotation from poles of finite rotation, Geophysical Journal of the Royal Astronomical Society, 65, 223-227, and Pilger, R. H., Jr., 2003, Geokinematics: Prelude to Geodynamics, Springer-Verlag, Berlin.)The relation to be derived is:
wT = lT’ WT + sin lT WT’ + (cos lT –1 ) WT X WT’
in which:
wT = instantaneous angular velocity pseudovector for time T; magnitude is equal to the instantaneous rotation rate in radians,
lT = total finite rotation angle for time T,
WT = unit vector representation of the finite pole of rotation for time T,
X denotes the vector cross product, and the prime (‘) denotes the first time derivative.
Derivation follows the jump.
Derivation follows the jump.
Given a unit vector r (at time zero), rotate the vector to rT. rT is related to r as:
rT = cos l (WT ´ r) ´ WT + sin l WT ´ r + (r · WT) WT (eq. 1-1)
Differentiate rT:
rT' = - sin l l' (WT ´ r) ´ WT + cos l (WT' ´ r) ´ WT + cos l (WT ´ r) ´ WT' + cos l l' WT ´ r
+ sin l WT' ´ r + (r · WT') WT + (r · WT) WT' (eq. 1-2)
Define u:
u = WT' / ||WT'|| (eq. 1-3)
Define wT in terms of WT, with magnitudes a1, a2, and a3 to be determined.
wT = a1 WT + a2 u + a3 WT ´ u (eq. 1-4)
In addition to the form in eq. 1-2, rT' can be expressed in terms of wT and rT:
rT' = wT ´ rT (eq. 1-5)
Insert eq. 1-4 in eq. 1-5:
rT' = a1 cos l WT ´ [(WT ´ r) ´ WT] + a2 cos l u ´ [(WT ´ r) ´ WT] + a3 cos l (WT ´ u) ´ [ (WT ´ r) ´ WT] + a1 sin l WT ´ (WT ´ r) + a2 sin l u ´ (WT ´ r) + a3 sin l (WT ´ u) ´ (WT ´ r) + a1 (r · WT) WT ´ WT + a2 (r · WT) u ´ WT + a3 (r · WT) (WT ´ u) ´ WT = a1 cos l WT ´ [(WT ´ r) ´ WT] + a2 cos l u ´ [(WT ´ r) ´ WT] + a3 cos l (WT ´ u) ´ [ (WT ´ r) ´ WT] + a1 sin l WT ´ (WT ´ r) + a2 sin l u ´ (WT ´ r) + a3 sin l (WT ´ u) ´ (WT ´ r) - a2 (r · WT) WT ´ u + a3 (r · WT) u (eq. 1-6)
Let r = u:
rT' = a1 cos l WT ´ (WT ´ u) ´ WT + a2 cos l u ´ (WT ´ u) ´ WT + a3 cos l (WT ´ u) ´ [ (WT ´ u) ´ WT] + a1 sin l WT ´ (WT ´ u) + a2 sin l u ´ (WT ´ u) + a3 sin l (WT ´ u) ´ (WT ´ u) - a2 (u · WT) WT ´ u + a3 (u · WT) u = a1 cos l WT ´ u - a3 cos l WT - a1 sin l u + a2 sin l WT (eq. 1-7)
Simplify:
rT' = - sin l l' (WT ´ u) ´ WT + cos l (WT' ´ u) ´ WT + cos l (WT ´ u) ´ WT' + cos l l' WT ´ u + sin l WT' ´ u + (u · WT') WT + (u · WT) WT'
= - sin l l' u - cos l ||WT'|| WT + cos l l' WT ´ u + ||WT'|| WT (eq. 1-8)
Equate similar terms:
a1 cos l WT ´ u = cos l l' WT ´ u (eq. 1-9)
-a1sin l u = - sin l l'u (eq. 1-10)
a2sin l WT - a3 cos l WT = cos l ||WT'|| WT + ||WT'|| WT (eq. 1-11)
Let r = WT ´ u:
rT' = a1 cos l WT ´ [WT ´ (WT ´ u)] ´ WT + a2 cos l u ´ [WT ´ (WT ´ u)] ´ WT + a3 cos l (WT ´ u) ´ {[WT ´ (WT ´ u)] ´ WT} + a1 sin l WT ´ [WT ´ (WT ´ u)] + a2 sin l u ´ [WT ´ (WT ´ u)] + a3 sin l (WT ´ u) ´ [WT ´ (WT ´ u)] + a2 [(WT ´ u) · WT)] u ´ WT + a3 [(WT ´ u) · WT)] u
= - a1 cos l u + a2 cos l WT - a1 sin l WT ´ u + a3 sin l W T (eq. 1-12)
Simplify:
rT' = - sin l l' [WT ´ (WT ´ u)] ´ WT + cos l [WT' ´ (WT ´ u)] ´ WT + cos l [WT ´ (WT ´ u)] ´ WT' + cos l l' WT ´ (WT ´ u) + sin l WT' ´ (WT ´ u) + [(WT ´ u) · WT'] WT + [(WT ´ u) · WT] WT' (eq. 1-13)
rT' = - sin l l' WT ´ u - cos l l' u - sin l ||WT'|| WT (eq. 1-14)
Equate similar terms:
- a1 cos l u = - cos l l' u (eq. 1-15)
- a1 sin l WT ´ u = - sin l l'WT ´ u (eq. 1-16)
a2 cos l WT + a3 sin l W T = - sin l ||WT'|| WT (eq. 1-17)
Simplify and solve for a1, a2, and a3:
a1 cos l WT ´ u = cos l l' WT ´ u (eq. 1-18)
- a1sin l u = - sin l l'u - cos l u - cos l l' u (eq. 1-19)
- a1 sin l WT ´ u = - sin l WT ´ u (eq. 1-20)
a1 =l' (eq. 1-21)
a2 sin l WT - a3 cos l WT = cos l ||WT'|| WT + ||WT'|| WT (eq. 1-22)
a2 cos l WT + a3 sin l W T = - sin l ||WT'|| WT (eq. 1-23)
a2 sin l - a3 cos l = cos l ||WT'|| + ||WT'|| (eq. 1-24)
a2 cos l + a3 sin l = - sin l ||WT'|| (eq. 1-25)
a2 = (- sin l ||WT'|| - a3 sinl) / cos l (eq. 1-26)
a2 = - sin l ( ||WT'|| + a3) / cos l (eq. 1-27)
[-sin l ( ||WT'|| + a3) / cos l] sin l - a3 cos l = cos l ||WT'|| + ||WT'|| (eq. 1-28)
[-sin l ( ||WT'|| + a3) ] sin l - a3 cos2l = cos2 l ||WT'|| + cos l||WT'|| (eq. 1-29)- sin2 l ( ||WT'|| + a3) - a3 cos2l = cos2 l ||WT'|| + cos l||WT'|| (eq. 1-30)
- a3 sin2 l - a3 cos2 l = ||WT'|| cos2 l + ||WT'|| sin2 l + cos l||WT'|| (eq. 1-31)
- a3 = ||WT'|| + cos l||WT'|| (eq. 1-32)
a3 = - ||WT'|| (1 + cos l ) (eq. 1-33)
a2 = - sin q [ ||WT'|| + - ||WT'|| (1 + cos l ) ] / cosl (eq. 1-34)
a2 = - sin l ||WT'|| [1 - 1 - cos l ) ] / cosl (eq. 1-35)
a2 = sin l (eq. 1-36)
Substitute values of a1, a2, and a3 into equation 1-4:
wT = q' WT + sin q u + - ||WT'|| (1 + cos q ) WT ´ WT' / ||WT'|| (eq. 1-37)
Simplify:
wT = q' WT + sin q u - (1 + cos q ) WT ´ WT' (eq. 1-38),
which is the desired solution after conversion back to the same units as above:
wT = (lT)’ WT + sin (lT) WT’ + (cos lT –1) WT X WT’.
wT = (lT)’ WT + sin (lT) WT’ + (cos lT –1) WT X WT’.
Note that the signs of the last two terms are different from those in Smith (1981), because, for the relation between the instantaneous angular velocity pseudo vector and the instantaneous velocity vector, he used a non-standard form (rT' = rT´ wT = - (wT ´ rT), instead of the standard form (rT' = wT ´ rT, equation 5, above). [Note: there is a misprint in Pilger, 2003, p. 268, of these last two formulae.]
No comments:
Post a Comment