Celestial and common mechanics

Gravitational evaluations

The basic concept is that, due to the combined effect of Earth's and Moon's gravitational forces, the weight of a mass is different when Moon is over the mass location and when it is conversely at the opposite side of the Earth. We can write this as follows:

$$F_{up}=G\frac{mM_E}{r_E^2} - G\frac{mM_M}{(d-r_E)^2} \tag{1}$$
$$F_{dw}=G\frac{mM_E}{r_E^2} + G\frac{mM_M}{(d+r_E)^2} \tag{2}$$

where the used symbols has this meaning:

$m$: mass that is the object of our engine manipulation
$F_{up}$: mass weight with the moon over the mass
$F_{dw}$: mass weight with the moon in opposition at the mass
$G$: the universal gravitational constant
$M_E$: mass of Earth
$M_M$: mass of Moon
$r_E$: Earth's radius
$d$: Earth - Moon central distance

Mechanical evaluations

Since we are interested in evaluating the energy that this engine could give us we go on writing the work done by the motor during the lifting and lowering phases that occurs respectively when $F_{up}$ and $F_{dw}$ is affecting the mass. Noting with $h$ the elevation of the manipulated mass we can write the works produced by the mass movements:

$$W_{lift}=-F_{up}h$$
$$W_{lower}=F_{dw}h$$

If we consider a full cycle of lifting and lowering, each one occourring at proper time (moving towards the moon), than we have the total cycle work:

$$\begin{align}
W &= W_{lift} + W_{lower} \\
   &= F_{dw}h-F_{up}h \\
   &= (F_{dw}-F_{up})h \tag{3}
\end{align}$$

Substituting $(1)$ and $(2)$ in the previous we have:

$$\begin{align}
W&=h((G\frac{mM_E}{r_E^2} + G\frac{mM_M}{(d+r_E)^2})-(G\frac{mM_E}{r_E^2} - G\frac{mM_M}{(d-r_E)^2})) \\
  &=h(G\frac{mM_M}{(d+r_E)^2}+ G\frac{mM_M}{(d-r_E)^2}) \\
  &=hGmM_M(\frac{1}{(d+r_E)^2}+\frac{1}{(d-r_E)^2}) \\
  &=hGmM_M\frac{(d-r_E)^2+(d+r_E)^2}{(d^2-r_E^2)^2} \\
  &=hGmM_M\frac{2(d^2+r_E^2)}{(d^2-r_E^2)^2} \\
  &=2hGmM_M\frac{(d^2+r_E^2)}{(d^2-r_E^2)^2} \tag{4}
\end{align}$$
 

Further if we note as $f$ the frequency at which we're going to lift and lower the mass (that could be at most the same of the Moon apparent cycle frequency) than we have also an evaluation of the power generated by the engine:

$$P=fW=2fhGmM_M\frac{(d^2+r_E^2)}{(d^2-r_E^2)^2} \tag{5}$$