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The process of suppressing transverse vibrations of an elastic rod spinning in a horizontal plane and fixed at one of its ends is studied. It is supposed that the rod spins around the vertical axis at a constant angular velocity and performs transverse vibrations in the vertical plane, the vibrations are assumed small in amplitude. Transverse vibrations of the spinning rod are performed under external mechanical action. Lateral vibrations are described by the displacement function and considered in a rotating plane by using the classical beam model. The necessary conditions of optimality are derived and applied for suppressing elastic vibrations on a finite time-interval. The problem of optimal suppression of lateral vibrations caused by initial disturbances is formulated as a variational problem with constraints that take into account the suppressing effects on the rod. The limiting restrictions are presented in the form of inequalities. With the introduction of an additional variable, these restrictions are reduced to standard integral equality, while taking into account the energy constraints imposed on the control actions. The proposed iterative algorithm for solving the formulated problem is a numerical-analytical algorithm and consists in minimizing the quadratic quality criterion. This criterion characterizes the vibration suppression process and allows the implementation of improving variations. As a result of the operations carried out, the dependence of the vibration suppression process on the determining parameters, such as the angular velocity of rotation, the isoperimetric energy constant, and the length of the considered process of vibration suppression in time, has been clarified. An example that illustrates the implementation of the proposed algorithm and shows the effectiveness of this method for suppressing lateral vibrations is given.

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