Abstract
Problem about motion of a reservoir with liquid with a free surface is considered based on the compensation of a force response of the liquid on reservoir walls. Such an approach is selected since usual methods of control of mechanical system motion are mostly intended for linear systems of relatively small dimension. However, models of dynamics of the combined motion of reservoirs with liquid are described with relatively high-dimensional nonlinear systems ordinary differential equations. For obtaining the mathematical model of combined motion of a reservoir with liquid with a free surface we use the Hamilton–Ostrogradskiy variational principle, for which it is possible to determine analytically all internal forces of interaction of system component parts. Namely using this algorithm, we determine the main vector of forces of the liquid pressure on reservoir walls (force response of liquid). The algorithm of the motion control of the reservoir with liquid is based on the inclusion of the compensation of the liquid force response to controlling actions, this reduces the motion of the system reservoir–liquid, where the effect of forces from oscillating liquid on the reservoir motion is eliminated. This algorithm was tested for problems of impulse and vibration disturbance of the translational motion of the system in the horizontal plain. We consider the disturbance of the system motion by a force rectangular impulse applied to the reservoir wall, the duration of the impulse is lesser than a quarter of the period of a liquid free oscillations according to the first normal mode. Amplitudes of the impulse were selected with the purpose of analysis of the behavior of the controlled system in different ranges of manifestation of nonlinearities. We state the problem to verify the accuracy of this algorithm for three ranges of manifestation of nonlinear properties in the system, namely, for the linear range (amplitudes of waves on a free surface h do not exceed 0,1 of the radius of a free surface ( <0,1R); for the weakly nonlinear range ( <0,2R) and for the strongly nonlinear range with maximum amplitudes of waves about =0,32R. Numerical modeling enables the determination of errors of developed algorithm, which does not exceed 0,5 %, although they insignificantly increase with the increase of amplitudes of oscillations on a free surface of liquid. At the same time perturbations on a free surface of liquid for the controlled motion are always greater than for the uncontrolled motion.
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Schedule a callMore From: Bulletin Taras Shevchenko National University of Kyiv. Mathematics Mechanics
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