Simulating longitudinal vibrations of coupled oscillator using the fourth-order Runge–Kutta method by programming spreadsheet

Abstract

This paper presents a simple way of simulating the motion of two oscillators coupled via springs. A linear model using Hooke’s law describes the motion of a coupled oscillator through a set of two coupled second-order linear differential equations. A spreadsheet is programmed to obtain the numerical solutions of coupled differential equations using the fourth-order Runge–Kutta method. The main thrust behind programming the spreadsheet is to solve the differential equations for a physical system under different initial condition and to understand the motion of coupled masses graphically. The programmed spreadsheet can also be modified to include damping in the system and visualize a damped dynamical system. The algorithm developed for programming presents the use of inbuilt functions of the spreadsheet and is user-friendly, especially for those who are not familiar with the spreadsheets. Simulations of the coupled oscillator would help students to visualize the effect of variation of different parameters, such as mass, initial velocity, spring constant, and damping factor, on the motion of coupled oscillators.

Normal coordinates in a system of coupled oscillators and influence of the masses of the springs

“…Several proposals for experiments for analysis of normal modes and normal coordinates in undergraduate laboratories have been published [3][4][5][6][7][8][9][10][11][12]. A pioneering article [3] studied experimentally and theoretically the arbitrary movements of a system of two gliders in coupled oscillation.…”

Physics Simulation with Excel Spreadsheet

“…An Excel spreadsheet was used to illustrate Lissajous charts, Weibull distributions for wind speed, and damped harmonic oscillators (Uddin, Ahsanuddin, & Khan, 2017). Excel spreadsheets can also be used in teaching basic statistics (Warner & Judge, 2001).In addition, the Spreadsheet is also used to simulate the longitudinal vibration of the coupled oscillator using the fourth-order Runge-Kutta method and to study vertical motion under the influence of a gravitational field (Singh, Kaur, & Khun, 2019;Grigore & Barna, 2015).In other literature, spreadsheets can be used to teach electromagnetism by simulation (Squire, Barnett, Grant, & Higginbotham, 2004).…”