Method for determining the rational parameters of dynamic dampers of low-frequency vibrations

The problems have been considered of natural nonlinear vibrations of an absolutely rigid semiball and a semicylinder on a horizontal plane, assuming that there is no energy dissipation, sliding and tipping on foundation. To adjust the damper to a frequency close to the fundamental tone of vibrations, it is necessary to assess the natural frequency of the damper, which is determined under the assumption on smallness of the vibrations amplitude. This paper represents the comparison of the natural frequency of linearized and nonlinear system. The relative error has been estimated of the natural frequency calculation, which is caused by linearization. It is shown that the ratio of the natural frequency of the linearized system to the natural frequency of the nonlinear system does not depend on the mass and radius. This conclusion made it possible to generalize the results of particular computational solutions and to obtain a formula which takes into account the amplitude influence on the natural vibrations frequency and helps to determine the natural frequency for the initial angles to ninety degrees.


Introduction
The dynamic dampers of vibrations, which are characterized by low-frequency natural vibrations (less than 10 Hz, and often less than 1 Hz), are widely used to reduce the loads in different mechanisms and engineering structures [1] when mining operations and underground space development. In this case, the vibration dampers with rolling bodies are used [2][3][4][5][6], which have high reliability. Their overview is represented in the works [2,6]. The damper parameters must be such that natural frequency is close to the frequency of fundamental tone of vibrations [2]. Therefore, it is important to know the natural frequency of the vibration damper, which, as a rule, is determined under the assumption on smallness of the vibrations amplitude, which makes it possible to linearize the motion equation. At high amplitudes, the nonlinear differential motion equation is solved by numerical methods, which, however, allow to find only particular solutions for specific conditions. There is a need to generalize the particular solutions.
The objective of work is to develop a generalization method of numerical results for determining the rational parameters of dynamic dampers of vibrations, which provide the required natural frequency of nonlinear vibrations.

Main part
Let us consider one of the simplest dampers [6], made in the form of a semicylinder or a semiball 1, vibrating on a plane 2 (Fig. 1). When solving a number of applied problems related to the vibrational impact on the loose medium [7,8], the vibrations of a semicylinder and a semiball can serve as a model representation of the solid particles motion.
When a semicylinder swings, we assume that there is no energy dissipation, sliding and tipping on foundation.
where  -angle; g -gravitational acceleration; 1 r -semicylinder radius (identical symbols used in the description of vibrations of a semicylinder and a semiball, will be recorded with 1 and 2 indices, respectively). A point above the letters means time differentiation. Initial conditions: . Since the dissipation is not considered, then 0  is the amplitude of natural vibrations.
The nonlinear equation (2) has no analytical solution. For its linearization, it is assumed that the angle  is low and take , sin Dividing (3) by a factor before the second derivative, we get -the natural angular frequency.
Then the frequency and period of natural vibrations From (5) we determine the radius at which the semicylinder has the natural frequency 0 In order to determine the natural frequency of nonlinear vibrations for each specific value r and 0  , the equation (2) can be solved numerically. However, this makes it difficult to generalize the results. The task may be simplified if to consider that, based on the theorem on the change in the kinetic energy, we have The semicylinder is turned through an angle  d during the time By integrating both sides of the equation (7), we determine the vibration period   When integrating, symmetry of a phase portrait with respect to the coordinate axes is taken into account, which made it possible to set the limit values 0 and 0  ; and for obtaining the period, the result should be quadrupled. Figure 2 represents the characteristic form of a phase portrait, which has been obtained by numerical integration of equation (2) using the Runge-Kutta-Fehlberg method of an order 4-5 with 1 .
60 and 90. Hereinafter, the calculations are performed through mathematical Maple package. Let us find the natural frequencies ratio This value will be called the relative natural frequency. Pay attention to the from the mass and radius of the semicylinder -the first provision of the method.
The relative error is caused by linearization of equation (2)         % 100 1 % 100 In Figure 3, the round markers represent the results of numerical integration of the expression (9) by Newton-Cotes method, and the triangular markers represent the errors. At  does not depend on the mass and radius of the semicylinder, taking into account (9) and (10) Having substituted (5) into (9), and solving relatively   0 1  r , we obtain the radius at which the semicylinder has the natural frequency   In formula (13), in contrast to (6), there is a factor   Let us analyse the vibrations of a damper [6], made in the form of a semiball. When modelling the semiball motion, we use the assumptions and numerical methods described above.
The equation of semiball vibrations [10-12] 0 sin cos sin where p I -the moment of inertia of a semiball relative to the instantaneous centre; mthe semiball mass; l -the distance of the mass centre C from the semiball base O . The moment of inertia p I and a distance l are calculated by the formulas: -the moment of semiball inertia relative to the horizontal axis, passing through the centre of inertia, and which is perpendicular to the drawing plane.
Having divided (15) by c I , we get Having done the same way as when determining the period of nonlinear vibrations of a semicylinder, we have: Pay attention to the fact that in this case as well the   In Figure 3, the results are represented of the numerical integration of expression (19) and the error   With account of (19) and (20), we obtain an expression for determining the natural frequency of nonlinear vibrations