Stability Criteria of Slosh Motion with Periodicity in a Spinning Spacecraft

a = transverse component of the system angular momentum C1, C = centers of mass of the main body and the system F = thrust vector (0 0 F3) along the symmetry axis passing through C1 H = angular momentum vector of the system I1, I2, I3 = moments of inertia of the main body about the centroidal principal axes x , y, and z m1, m2 = masses of spacecraft main body and pendulum bob O = pivot point of the pendulum r = length of massless rod connecting the pendulum bob r, rO , rp = position vectors from O to m2, C1 to O, and C1 to m2 r1, r2 = position vectors from C to C1 and m2, respectively xyz = body-fixed frame having its origin at C1 β, βd = resonant and detuned frequencies e = nondimensional small parameter θ , ψ = generalized coordinates of the pendulum θs , θ = stationary point of θ and variation from that point μ = ratio of m2 to the total mass, m1 + m2 σ = parameter describing the nearness of the detuned frequency to the resonance frequency τ = scaled time = angle of rotation of the main body ω = angular velocity of the coordinate system xyz