If the analogy between the oscillation of the moon and the pendulum
were applicable I would see an increase in the libration period (of
the double-moon) when the distance between the half-moons were
changing. But this did not happen in my simulations. The period are
obout 2/3 of one revolution no matter how far away from each other
the half-moons are.
Here are some details:
To simplify I reduced the mass of the Moon very considerably and
started with an ideal circular orbit (r=3.85*10^8 m, v=1019.5 m/s).
The center of the fixed Earth was at x=0, y=0. The center of the
rigid double-moon was always initially at x=3.85*10^8, y=0, that is
at the "3 o'clock" location. Each half-moon had the initial vx=0 and
vy=1019.5 m/s. The initial orientation of the double-moon was along
the x axis. The smallest separations between the half-moons was
0.4*10^7 m, the largest was 10^8 m. The orbits of the center of the
double-moon turn out to be slightly elliptical, for the imposed
initial conditions.
Ludwik Kowalski