# Mathematical Modelling of Dynamics of the Vibroprotective System Equipped with Roller Absorber

Using the methods of mathematical modeling to develop numeral-analytical approach for determination of equations of ACHKH of the nonlinear vibroprotective system with a new isochronous roller absorber in the first approaching and to define the optimum parameters of its tuning. For the conclusion of equations ACHKH the averaging method of W. Ritz, adapted to the probed task, was used. For the solution of nonlinear algebraic equations of ACHKH of the system the special programmatic complex was developed in a non-obvious form.

# Mathematical Simulation of Oscillations of a String with Movable Bearing in a Vertical Plane

We consider the natural oscillations of the string whose left end is fixed and the right one has the ability to move in a vertical plane by a defined law. We should address this issue to construct an adequate mathematical model of the electric wire taking into account longitudinal displacement of one of its ends. The right-hand mobile support is a commuting pendant with electric wire in the form of insulator strings.

# Reasoning of the Use of Roller Dampers in an Extended Frequence Rangeт

The paper defines constraints in the form of inequalities for the minimal value of the coefficient of sliding friction ƒ  between the interacting elements of roller dampers, which set the boundaries, where pure rolling of roller or ball in the spherical cavities of their bodies can be implemented. We analytically justify the use of roller dampers in a wide frequency range both at 0<ω0≤1,6  rad/s, and at ω0>1,6 rad/s using as an example of roller and ball absorbers.

# Monitoring System of Sickness Rate of Acute Respiratory Diseases in Ukraine

Based on the multivariate regression analysis, we construct a mathematical model which allows analyzing and predicting the ARD incidence on the territory of Ukraine. The obtained results indicate a high accuracy of our calculations and the model adequacy to real-life conditions.

# The Conditions of Violation of Cylinder “Clean” Rolling along the Brachistochrone

We obtain the algebraic equation of the directive line of the fastest cylinder descent in the parametrical form. By utilizing the equation of cylinder motion with the binding reaction, we also determine the conditions of the cylinder rolling without slipping and tearing off along the brachistochrone. On the theoretical side, we get a significant theoretical result: a center of the cylinder masses circumscribes the cycloid when it moves along the brachistochrone.