The rectilinear motion of a two-mass system in a resistive medium is considered. The motion of the system as a whole occurs by longitudinal periodic motion of one body (the internal mass) relative to the other body (the shell). The problem consists of finding the periodic law of motion of the internal mass that ensures velocity-periodic motion of the shell at a specified average velocity and minimum energy consumption. The initial problem reduces to a variational problem with isoperimetric conditions in which the required function is the velocity of the shell. It is established that, with optimal motion, the shell velocity is a piecewise-constant time function taking two values (a positive value and a negative value). The magnitudes of these velocities and the overall size of the intervals in which they are taken are uniquely defined, while the optimal motion itself is non-uniquely defined. The simplest optimal motion, for which the period is divided into two sections - one with a positive velocity and the other with a negative velocity of motion of the shell - is investigated in detail. It is shown that, among all the optimal motions, this simplest motion is characterized by the maximum amplitude of oscillations of the internal mass relative to the shell. © 2010 Elsevier Ltd. All rights reserved.
Yegorov, A. G., & Zakharova, O. S. (2010). The energy-optimal motion of a vibration-driven robot in a resistive medium. Journal of Applied Mathematics and Mechanics, 74(4), 443–451. https://doi.org/10.1016/j.jappmathmech.2010.09.010