Mathematical Modeling and Simulation of Spring Constant on the Behavior of Damping in a Mass-spring Damped Harmonic Oscillator

Abstract

Background and aims. Mass-Spring systems are second order linear differential equations that have variety of applications in science and engineering. They are the simplest model for mechanical vibration analysis. Damping of the oscillatory system is the effect of preventing or reducing its oscillations gradually with time. The damping ratio in physical systems is produced by the dissipation of stored energy in the oscillation. In this study, the effect of spring constant on the behaviour of damping in a mass-spring harmonic oscillator using simulation was investigated. Methods. The three cases of damping namely the underdamped, the overdamped and the critically damped cases were studied by varying the spring constant from 2 N/M to 10 N/M. (2, 4, 6, 8 and 10 N/M). Results. It was found that for the two damping cases; the underdamped case and the overdamped case, the effect of increasing the spring constant on the behavior of harmonic oscillator caused a decrease in the time of damping and an increase in the displacement, while in the case of critically damped harmonic oscillator, there was very little variation in the time of damping or it was almost steady while there was a negative decrease in the displacement. Conclusion. It was found that for the two cases of underdamped and overdamped harmonic oscillator, the effect of increase in the value of the spring constant causes a decrease in the time of damping, while the effect of increasing the spring constant on the displacement makes it increasing. In the case of critical damped harmonic oscillator, very little variation in the time of damping was noticed with increasing the value of the spring constant. The effect of increasing the spring constant cause a negative decrease in the value of displacement in the critically damped case.