Estimation of solar zenith passage on the Earth's surface by a simple harmonic motion

The kinematics of the Solar zenith passage on the Earth was shown as a simple harmonic motion. This analytical model used the uniform circular motion to describe the Earth's rotation relative to its axis and relative to the Sun, also this model considered the sunlight beams parallel and constant the obliquity of the ecliptic. We made a contrast between some data taken from of the software Stellarium 0.12.1 and the model, we obtained an uncertainty $\Delta t=1d$; this result allow to project the model as an astronomical context conducive to develop the kinematics of simple harmonic motion.