cos2X + 4sin2A

where for a circular, polar orbit, X=<•>„. Substituting Eq. (18-66) into Eq. (18-54) and using Table 18-1 with hx = hz =0, hy=-h, and Ix = Iy=I gives the result

4A:Dsin2X

-2UokD

The pitch equation has the form of a forced harmonic oscillator which may be solved by expanding the right-hand side of the equation or the forcing function in a Fourier series [Repass, et al., 1975],

-2uakD

l + 3sin 2uat - uakD ( 1 + f cos 2uat + fcos 4u0t+ • • • ) a0 "

where* A = w„f and an and bn are the Fourier coefficients given by

1 r2v

Taking the Laplace transform of the pitch equation and rearranging yields (see Appendix F)

lp(s) = L%(/))= | C[ -«.*D(1 + §cos2u0/ + |cos4«„i + ■••+)].

IT Jn

where the time constants and frequencies associated with the decay of pitch oscillations are given by the zeros of the characteristic equation f(s)=/52+ft0s + 3Uo2(/-/,) (18-71)

and the time constant, r, and oscillation frequency,/, are*

The steady-state solution, ip(t)>'s obtained using Eq. (F-33) and the principle of superposition, j -2 ao fl|COs(«„/-v,)

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