N On the stability of nonlinear motions of a mechanical model of a body with two fluids


Аuthors

Win K. K.1*, Temnov A. N.1, 2**

1. Bauman Moscow State Technical University, MSTU, 5, bldg. 1, 2-nd Baumanskaya str., Moscow, 105005, Russia
2. Peter the Great St. Petersburg Polytechnic University, 29, Polytechnicheskaya str., St. Petersburg, 195251, Russia

*e-mail: win.c.latt@gmail.com
**e-mail: antt45@mail.ru

Abstract

The article considers a mechanical model with a spherical pendulum, corresponding to nonlinear oscillations of the interface of two liquids that completely fill a cylindrical vessel. It is shown that when using direction cosines that determine the position of a spherical pendulum as generalized coordinates of oscillations, the equations of motion of an equivalent body with a pendulum will strictly coincide with the equations of considerable motions of a solid body with two liquids up to the second order of smallness inclusive. When taking into account a higher order of smallness, to compare the movements of a mechanical analogue and a solid body with two liquids, one has to use amplitude-frequency characteristics.  
A mechanical analogue of a rigid body with a cavity partially or completely filled with one liquid in the case of small oscillations was proposed by G. N. Mikishev, B. I. Rabinovich in the form of an equivalent rigid body and a set of mathematical pendulums [1]. A slightly different model was considered by H. Bauer, who replaced the moving part of the liquid with a set of oscillators in the form of springs with masses [2]. When studying nonlinear oscillations of a body with a liquid, the equations of motion are significantly complicated and a number of authors have attempted to study such oscillations using a visual nonlinear mechanical model. However, due to the complexity of the phenomenon and the cumbersomeness of equations such as [3-4], it is necessary to consider special cases of motion and accept additional hypotheses. Thus, in work [5-7] for a flat case, the model of a mathematical pendulum is used even for large deflection angles. In work [8], the spatial motion of a pendulum replacing a liquid in a sphere is studied under the assumption that the free surface of the liquid remains flat during oscillations. The same hypothesis for the general form of the cavity, but for flat motion, is used in the work [9-10], where a model in the form of a pendulum with a variable length determined by the surface of the "metacenters" is proposed. In the work [2], for a cylindrical tank, a semi-empirical model is proposed in the form of a spring with nonlinear stiffness and mass moving along a parabolic surface, in which two parameters are subject to determination from the experiment.

Keywords:

spherical bearing, pendulum, direction cosines, skeletal line, amplitude-frequency characteristics

References

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