[Paper Review] Theoretical and experimental study of the normal modes in a coupled two-dimensional system

Marcos H. Giménez, Juan Carlos Castro-Palacio+3 more|arXiv (Cornell University)|May 4, 2016

Experimental and Theoretical Physics Studies9 references4 citations

TL;DR

This study presents a theoretical and experimental investigation of normal modes in a coupled two-dimensional oscillator system using smartphones as mass-spring units on an air table. By applying Hessian matrix formalism to predict normal frequencies and comparing them with acceleration sensor data from smartphones, the authors achieve excellent agreement within 1.7% discrepancy, demonstrating a practical, accessible method for teaching coupled oscillations in physics education.

ABSTRACT

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 remove 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, namely, within 1.7 % of discrepancy.

Motivation & Objective

• To bridge the gap in general physics education by providing a hands-on, accessible method for teaching two-dimensional coupled oscillations.
• To validate the Hessian matrix formalism in predicting normal frequencies for a real 2D coupled system.
• To demonstrate the utility of smartphone acceleration sensors as a fast, direct tool for collecting and analyzing 2D oscillation data.
• To provide a pedagogically effective experiment that connects classical mechanics with vibrational modes in solids and molecules.
• To offer a reproducible, low-cost experimental setup that enables students to explore normal modes and superposition of oscillations.

Proposed method

• Theoretical normal frequencies are derived using the Hessian matrix formalism applied to a two-particle system connected by springs in a two-dimensional geometry.
• An experimental setup uses two Samsung Galaxy S2 smartphones (each ~174.4 g) mounted on an air table to minimize friction, connected via springs with a force constant of 20.6 N/m.
• Acceleration data are collected using the built-in sensors of the smartphones and exported for analysis, capturing motion along both x- and y-axes.
• The data are fitted to harmonic functions of the form $ a(t) = A\sin(\omega t + \phi) $ to extract experimental angular frequencies.
• Nonlinear Levenberg-Marquardt fitting is applied to model arbitrary oscillations as superpositions of normal modes.
• Fourier analysis is used as an alternative method to identify dominant frequencies, though the primary validation relies on direct curve fitting to the superposition model.

Experimental results

Research questions

• RQ1 How accurately can the Hessian matrix formalism predict the normal frequencies of a two-dimensional coupled oscillator system?
• RQ2 To what extent do experimental frequencies from smartphone acceleration sensors match theoretical predictions in a 2D coupled system?
• RQ3 Can smartphone sensors serve as a reliable, low-cost tool for measuring and analyzing 2D coupled oscillations in a physics laboratory setting?
• RQ4 How well can the superposition of normal modes describe arbitrary, non-normal oscillations in such a system?
• RQ5 What is the quantitative discrepancy between theoretical and experimental normal mode frequencies in this 2D setup?

Key findings

• The theoretical normal frequencies derived from the Hessian matrix formalism show excellent agreement with experimental data, with discrepancies ranging from 0.6% to 1.7%.
• For the symmetric mode along the x-axis, the experimental frequency was $ 16.18 \pm 0.03 $ rad/s, compared to the theoretical value of 16.443 rad/s, a 1.6% discrepancy.
• For the antisymmetric mode along the x-axis, the experimental frequency was $ 22.14 \pm 0.02 $ rad/s, compared to the theoretical value of 22.508 rad/s, a 1.7% discrepancy.
• The y-axis symmetric mode showed a 1.5% discrepancy (experimental: $ 16.79 \pm 0.02 $ rad/s, theoretical: 17.038 rad/s), and the y-axis antisymmetric mode showed only 0.6% discrepancy (experimental: $ 19.92 \pm 0.04 $ rad/s, theoretical: 19.804 rad/s).
• The superposition model of four normal modes successfully described arbitrary oscillations, with $ R^2 $ values for the curve fits consistently around 0.99, indicating high-quality agreement.
• The study confirms that smartphone-based data acquisition enables fast, accurate, and pedagogically effective measurement of 2D coupled oscillations, suitable for integration into undergraduate physics labs.