Abstract
This paper presents a parametric investigation of the beat phenomenon in a two DOF mechanical oscillatory system coupled through elastic elements. The system consists of three masses connected by springs with stiffness coefficients c 1 , c 2 , and c 3 . Using Lagrange’s equations of the second kind, the mass and stiffness matrices were formulated, and the natural frequencies ω 1 and ω 2 were determined analytically. The beat period, defined as T b = 2 π / ω 2 - ω 1 , was studied as a function of stiffness parameters c 1 and c 2 . Numerical simulations in Python were conducted over a wide range of stiffness ratios, and contour maps were generated to illustrate how variations in stiffness coupling influence the dynamic interaction between modes. The results demonstrate that when c 1 and c 2 are close in magnitude, the modal frequencies converge, leading to a long beat period and pronounced energy exchange between oscillators. Conversely, greater stiffness asymmetry increases modal separation and shortens the beat cycle. The study provides valuable insights into tuning coupled oscillatory systems for vibration control and energy transfer optimization.