Vibrations in engineering and technology

Olshanskiy Vasyl Pavlovych – Doctor of Physical and Mathematical Sciences, Professor of the Department of Physics and Theoretical Mechanics, Petro Vasilenko Kharkiv National Technical University of Agriculture (44, Alchevskyi str., Kharkiv, Ukraine, 61002, e mail: OlshanskiyVP@gmail.com).

Burlaka Volodymyr Vasilovych – Candidate of Technical Sciences (Ph. D.), Associate Professor of the Department of Physics and Theoretical Mechanics, Petro Vasilenko Kharkiv National Technical University of Agriculture (44, Alchevskyi str., Kharkiv, Ukraine, 61002, e mail: Burlaka2V@ukr.net).

Slipchenko Maksym Volodymyrovych Candidate of Technical Sciences (Ph. D.), Associate Professor of the Department of Physics and Theoretical Mechanics, Petro Vasilenko Kharkiv National Technical University of Agriculture (44, Alchevskyi str., Kharkiv, Ukraine, 61002, e mail: Slipchenko_M@ukr.net).

A theoretical study of the movement of a beam with binary boundary conditions at a short-term pulsed load is carried out in the article. The non-stationary oscillations of the beam, caused by short-term force loading, are evenly distributed along its length. Beam movement is considered in two stages. It is assumed that the conditions for fixing the ends of the beam are associated with the direction (sign) of their rotation when the beam is bending. It is assumed that under the action of the load, the edges of the beam are rigidly clamped, that is, the angles of their rotation are equal to zero, and after unloading and changing the sign of the curvature of the beam they become pivotally supported in cylindrical hinges. According to the second boundary conditions, the free oscillation of the beam passes. Due to the use of the method of compensating loads, the analytical part of the isolation is expressed in rows of sines. Since the system has an asymmetric elastic characteristic with respect to the position of static equilibrium, the method of adding solutions is used to achieve this goal. As a result, the calculation is reduced to the numerical solution of the Volterra type integral equation on a computer. In addition, an approximation is proposed, which gives simpler calculation formulas than a piecewise linear approximation. The proposed approximation allows us to calculate the value of the boundary moments and obtain the formula for the beam deflections. Numerical analysis showed that for certain durations of force loading, the beam deflection amplitude with pinched edges in the direction of the external impulse is less than the deflection amplitude in the opposite direction during the movement of the unloaded beam with hinged edges. This dynamic effect is characteristic of systems with an asymmetric characteristic of elasticity, which occurs when a binary fixing of the edges of the beam or when it is supported by unilateral elastic supports. Formulas are derived by which it is possible to calculate the time when the effect of asymmetry will be most clearly expressed, and also in which case it will be absent. Numerical results confirming the adequacy of the proposed dependencies are presented.

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