We study damped harmonic oscillations in mechanical systems like the loaded spring and simple pendulum with the help of an oscillation measuring electronic counter. The experimental data are used in a software program that solves the differential equation for damped vibrations of any system and determines its position, velocity and acceleration as a function of time. The method developed in this study can be easily adopted to study the properties of damped vibrations in any system. (Contains 5 figures and 2 tables.)