OSCILLATION OF A LINEAR SYSTEM UNDER THE ACTION OF A BEATING FORCE

Abstract

The article examines the dynamics of a linear oscillatory system under the influence of a beating force — an external periodic excitation with modulated amplitude. Linear oscillatory systems are widely used in various engineering and physical applications, where it is essential to consider the effects of external periodic forces, including those with temporal modulation. This study focuses on analyzing the impact of the parameters of the beating force, such as the base frequency, modulation frequency, and modulation depth, on the system's behavior. Particular attention is given to the study of resonance phenomena that occur when the system's natural frequency coincides with either the excitation frequency or the modulation frequency. It is shown that under such conditions, the system may exhibit complex oscillatory regimes, including beat phenomena and high-amplitude resonant states. These phenomena are analyzed both analytically and using numerical methods. The study investigates both transient and steady-state oscillation regimes of the system. Analytical expressions for amplitude-frequency characteristics are presented, allowing for the assessment of the influence of the beating force parameters on the amplitude and the phase of oscillations. Additionally, the analysis of beats arising from the interaction of two close frequencies is conducted, and the conditions for their occurrence are identified.

The results of the analysis demonstrate that the amplitude modulation of the external force significantly alters the system's dynamic behavior, which must be taken into account when designing such systems. Particular emphasis is placed on the practical applications of the research findings. Linear oscillatory systems subjected to modulated external forces frequently occur in mechanical, acoustic, and electrical systems, as well as in control and diagnostic tasks. The obtained results not only provide a deeper understanding of oscillatory processes in such systems but also offer recommendations for tuning their parameters to prevent undesirable resonant modes or optimize system performance under resonance conditions. The methods of analysis described in the study can be employed to predict the behavior of linear oscillatory systems in a wide range of applications.