The article considers nonstationary transverse vibrations of an elastic cantilever rod with a mass at the right end under controlled displacement of the left end along a vertical guide. The flexural vibrations of the rod are sought in the form of eigenmode vibration expansion. The following problem is posed: to find the law of controlled movement of the left end of the rod, governing its movement to the required distance in a given time, while simultaneously several lower eigenforms of the elastic vibrations are damped at the moment of stopping. The acceleration of the left end of the rod is considered to be an unknown function. The acceleration is supposed to be proportional to some finite function. This function, in turn, is written in the form of a series of sines or cosines with unknown coefficients at a given time of controlled motion. Examples of calculation with damping of elastic vibrations in the first three eigenforms are given