Formulas for the Motion of a Sensing Bar Due to Nearby Masses

This paper uses tensors to facilitate Newtonian gravity calculations. It presents the basic mechanics ("gravitostatics") for short range gravity gradiometry of static mass distributions, where the distance of the masses is not much greater than the size and spacing of the pair of sensing bars. Complete formulas are given for computing the 2nd and 4th order terms of the torque on a sensor bar, due to gravity gradients. (The odd-order terms are zero due to bar symmetries.) This gives us an equation of motion for the deflection angle of the bar on its flexure pivot, in the bar's body frame. Given the bar's moments of inertia, we then evaluate the angle difference between a pair of ideal twin-diamond sensor bars at static equilibrium. For each sensor position, this measured average angle difference is a linear combination of the unknown source masses. The combining coefficients are a function of the gradients of the gravity field and the bars' moments of inertia. Therefore, when the sensor scans the source masses, they can be recovered by optimization methods.