In this work, the normal modes of a two-dimensional oscillating system have been studied from a theoretical and experimental point of view. The normal frequencies predicted by the Hessian matrix for a coupled two-dimensional particle system are compared to those obtained for a real system consisting of two oscillating smartphones coupled one to the other by springs. Experiments are performed on an air table in order to largely reduce the friction forces. The oscillation data are captured by the acceleration sensor of the smartphones and exported to file for further analysis. The experimental frequencies compare reasonably well with the theoretical predictions, specifically, within 1.7% of discrepancy.