Determination of the rational parameters of the low-frequency oscillation damper of large-sized engineering structures

Abstract

The article is aimed at determining the rational parameters of a low-frequency vibration damper with rolling elements, which have a simple design and high reliability. The parameters of the damper should be such that its frequency is close to the frequency of the main tone of the structure's vibrations. For this, the natural frequency of the vibration damper is determined under the assumption of small amplitude of the oscillations, which allows the equations of motion to be linearized. At large amplitudes, the nonlinear differential equation of motion is solved by numerical methods that allow finding only private solutions for specific conditions. There is a need to generalize private decisions. For this purpose, one of the simplest dampers was considered, which consisted of a uniform cylinder and a recess (the axes of the cylinder and recess are parallel). It is assumed that there is no energy dissipation and slippage. The cylinder makes free oscillations, which are described by a nonlinear differential equation of the second order. As a result of calculations, the natural frequency and period of natural oscillations of the linearized system were determined. Then, from the condition of conservation of energy, the natural frequency is determined, taking into account the nonlinearity of the system. As a result of numerical integration using the Newton-Cotes method, the dependence of the relative natural frequency and the error caused by the linearization of the differential equation of the cylinder's motion on the amplitude was obtained. Dependence analysis demonstrates a characteristic property of nonlinear systems ‒ non-isochronism. With linearization, an increase in the amplitude of oscillations leads to a quadratic increase in the relative error. As a result of the approximation of the numerical results in the range of values of the angle, which was calculated from the vertical passing through the center of curvature of the depression, from 10 to 900, the dependences of the determination of the relative natural frequency and the error were obtained. In order to determine the accuracy of the obtained dependencies, a comparison was made with approximations obtained by other methods. The comparison shows that the maximum relative error of 0.216039 of the obtained dependence is less than that obtained by other methods – 0.272803, while the calculation is simpler. As a result of calculations and analysis, the following was established. The influence of the amplitude of free oscillations on the frequency is characterized by the relative natural frequency, which shows how many times the natural oscillation frequency of the linearized system is greater than the oscillation frequency of the nonlinear system. The relative frequency depends only on the amplitude of the oscillations. In the case of nonlinear oscillations of a mathematical pendulum and a cylinder, their relative natural frequencies are described by the same function. This allowed the cylinder to use the approximation methods developed for the pendulum. The relative natural frequency of oscillations of the cylinder is determined as a result of numerical integration approximation, as well as by the method of harmonic balance. The maximum relative error of the first method (0.22%) is smaller than that of the second (0.27%), while the calculation is simpler. The error, which is due to the linearization of the differential equation of motion of the cylinder, increases quadratically with an increase in the amplitude of oscillations.