The exact solution of the differential equation of the compelled cross vibrations of the core with any continuous parameters

Abstract

The compelled cross vibrations of a core with any continuous variable parameters, loaded with evenly distributed harmonious loading are considered. The exact solution of the corresponding differential equation of vibrations in partial derivatives is constructed for the first time. As a result, in an analytical look formulas for dynamic movements and internal efforts in any section of a core are received. A practically important case is considered when the external dynamic load acting on the core is harmonic. For the external friction, the hypothesis is accepted according to which the resistance force is proportional to the mass and speed of the core, and the internal friction is taken into account according to the Kelvin-Voigt hypothesis, where the internal resistance force is proportional to the first degree of the strain rate. As is known, the Kelvin-Voigt hypothesis in its pure form has several disadvantages. The main one is that it leads to a contradictory experimental data conclusion about the frequency-dependent internal friction in the material. This drawback can be eliminated if we accept the adjusted Kelvin- Voigt hypothesis, according to which the coefficient of internal friction is chosen inversely proportional to the frequency with which the structure oscillates. The dynamic parameters of the core are fully defined. The obtained formulas contain unknown constants in the form of initial values of real and imaginary components (initial parameters). Additionally, the formulas which are equivalent to them are proposed. They are recommended for practical use during the study of the oscillations, which are different, the amplitude functions of dynamic parameters are clearly distinguished in them.