This question for the case of a homogeneous field—the gravity field—was discussed in detail by Prof. M.I. Akimov, who found all surfaces that allow the movement of a heavy point along helical lines with a vertical axis in the presence of friction, and investigated the conditions for these surfaces under which the movement in question would be stable.
In the previous work “On Inertial Screens,” I considered a vibrating screen as a system with three degrees of freedom, which corresponds to the assumption of the symmetry of all the masses of the screen and the forces applied to them relative to a certain vertical plane of symmetry intersecting with the screen sieve along the line of its greatest slope. The general case of small oscillations of an inertial screen, considered as a rigid body with all 6 degrees of freedom, is analyzed, and the conditions under which the problem can be reduced to three degrees of freedom are clarified (the case discussed in the previous article - “On inertial screens”). Euler's equations that determine the motion of a rigid body are simplified due to the smallness of its vibrations, and in a first approximation the problem is reduced to solving the System of 6 aggregate linear equations with constant coefficients or even to individual linear equations. If necessary, the result can be improved by successive approximation, also taking into account nonlinear terms."
