# MATHEMATICAL PENDULUM MODEL WITH MOBILE SUSPENSION POINT

## Authors

• Viktor P. Legeza Igor Sikorsky Kyiv Polytechnic Institute

## Keywords:

mathematical pendulum, moving suspension point, small own oscillations, frequency, Appel’s formalism, linearization

## Abstract

Background. The new dynamic problem, which is posed and solved in this article, is a theoretical generalization of the well-known classical problem of free oscillations of a mathematical pendulum. In the proposed setting, it is new and relevant, and can be successfully used in such fields of technology as vibration protection, vibration isolation and seismic protection of high-rise flexible structures, long power lines, long-span bridges and other large-sized supporting objects.

Objective. The aim of the work is to derive the equations of own oscillations of a new mathematical pendulum-absorber, to find a formula for determining the frequency of its small own oscillations and to establish those control parameters that allow you to tune the single-mass pendulum absorber to the frequency of the fundamental tone of the carrier object.

Methods. To achieve this goal, we used the methods of analytical mechanics, namely, the Appel’s formalism, as well as the linearization of the obtained differential equations.

Results. A mathematical model is constructed in the work that describes the own oscillations of a new-design mathematical pendulum with a movable (spring-loaded) suspension point with length L. The model is a system of differential equations obtained using the Appel’s formalism. Based on them, after linearization of nonlinear equations, a formula is established for the frequency of small own oscillations of a pendulum with a mobile suspension point.

Conclusions. It is shown that the frequency of own oscillations of the new mathematical pendulum coincides with the frequency of own oscillations of the classical mathematical pendulum with an equivalent suspension length, which is equal to $L_{eq}=L+\frac{mg}{k}$. In the case where the suspension point is fixed (k ® ¥), the frequency formula turns into a well-known formula for the frequency of small own oscillations of a classical mathematical pendulum $\omega =\sqrt{\frac{g}{L}}$. If the stiffness coefficient of elastic elements tends to zero (k ® 0), then the frequency w of the damper also tends to zero. An important structural feature of the proposed pendulum is noted, consisting in the fact that due to the appropriate choice of the three control parameters of the pendulum (k, L, m), its frequency in magnitude can be made any in the range from zero to $\sqrt{\frac{g}{L}}$.

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