Advanced Numerical Simulation of Pendulum Dynamics: A Comprehensive Analysis of Environmental Influences and Non-Linear Behavior

Abstract

This study explores the intricate dynamics of pendulum motion by numerically simulating the influence of environmental factors, including temperature, pressure, and humidity. Employing a high-precision Euler method with a time step of 0.0001 seconds, a 1-meter-long pendulum was modeled under varying conditions: temperatures ranging from 0°C to 40°C, pressures between 950 hPa and 1050 hPa, and humidity levels from 20% to 80%. The simulation incorporated key factors such as thermal expansion, air resistance, and non-linear oscillatory behavior for initial displacements up to 30°. Results reveal that a 40°C rise in temperature induces a 0.0007-second change in period and a 0.001 m/s² variation in calculated gravitational acceleration, predominantly due to rod expansion. Conversely, the effects of pressure and humidity were found to be negligible. Non-linear analysis at a 30° initial displacement indicated a 0.5% increase in period compared to 5°, underscoring the impact of the initial angle on pendulum dynamics. The model demonstrated remarkable accuracy for small-angle oscillations, aligning within 0.01% of theoretical predictions and 0.1% of experimental data. These findings offer valuable insights into pendulum behavior under diverse environmental conditions, providing a robust foundation for advancing the design and calibration of precision instruments and timekeeping mechanisms.