(1) [theta] = [alpha] + L, sin [theta] = sin[alpha] + L cos[alpha],
cos[theta] = cos[alpha] - L sin [alpha],
[Omega] = [mu] + N, r = R + Q,
where L, N, Q are small terms, involving a factor e^(nti), to express
the periodic nature of the nutation; and then if a, c denote the mean
value of x, z, at the point of contact
(2) x = a + L[rho] cos[alpha], z = c - L[rho] sin[alpha],
(3) x sin [theta] + z cos [theta] = a sin [alpha] + c cos [alpha]
+ L(a cos [alpha] - c sin [alpha]),
(4) x cos [theta] - z sin [theta] = a cos [alpha] - c sin [alpha]
- L(a sin [alpha] + c cos [alpha] - [rho]).
Substituting these values in (C*) with dq/dt = -d^2[theta]/dt^2 = n^2L,
and ignoring products of the small terms, such as L^2, LN, ...
/A \ /CR + K CQ\
(C**) ( -- + a^2 + c^2) Ln^2 - ([mu] + N)( ------ + -- )(sin[alpha] + L cos[alpha])
\M / \ M M /
/A \
+ ([mu]^2 + 2[mu]N)( -- + c^2 - 2L[rho]c sin[alpha]) (sin [alpha] cos[alpha] + L cos[alpha])
\M /
+ ([mu]^2 + 2[mu]N) [ac - L[rho](a sin [alpha] - c sin[alpha])] (sin^2 [alpha] + L sin 2[alpha])
- ([mu] + N)(R + Q)(a + L[rho]cos[alpha])[a sin [alpha] + c cos[alpha]
+ L(a cos[alpha] - c sin [alpha])] - g(a cos[alpha] - c sin[alpha])
+ gL(a sin[alpha] + c cos[alpha] - [rho]) = 0,
which is equivalent to
CR + K /A \
(5) -[mu] ------ sin[alpha] + [mu]^2 ( -- + c^2) sin [alpha] cos [alpha]
M \M /
+ [mu]^2 ac sin^2[alpha] - [mu]Ra(a sin[alpha] + c cos[alpha])
- g(a cos[alpha] - c sin[alpha]) = 0,
the condition of steady motion; and
(6) DL + EQ + FN = 0,
where
/A \ CK + K
(7) D = ( -- + a^2 + c^2) n^2 - [mu] ------ cos[alpha]
\M / M
- 2[mu]^2[rho]c sin^2[alpha] cos[alpha]
/A \
+ [mu]^2 ( -- + c^2) cos [alpha] - [mu]^2[rho](a sin [alpha]
\M /
- c cos[alpha]) sin^2[alpha] + [mu]^2ac sin 2[alpha]
- [mu]R[rho] cos [alpha](a sin [alpha] + c cos[alpha])
- [mu]Ra(a cos [alpha] - c sin[alpha])
+ g(a sin[alpha] + c cos[alpha] - [rho]),
C
(8) E = -[mu] -- sin [alpha] - [mu]a(a sin [alpha] + c cos [alpha]),
M
CR + K /A \
(9) F = - ------ sin [alpha] + 2[mu] ( -- + c^2) sin [alpha] cos [alpha]
M \M /
+ 2[mu]ac sin^2 [alpha] - Ra(a sin [alpha] + c cos [alpha]).