The Simple Pendulum is not so Simple
SIAM Review
Matrix analysis and applied linear algebra
Matrix analysis and applied linear algebra
Finite Differences And Partial Differential Equations
Finite Differences And Partial Differential Equations
Matrices, Moments and Quadrature with Applications
Matrices, Moments and Quadrature with Applications
Journal of Computational and Applied Mathematics
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To what extent do the vibrations of a mechanical system reveal its composition? Despite innumerable applications and mathematical elegance, this question often slips through those cracks that separate courses in mechanics, differential equations, and linear algebra. We address this omission by detailing a classical finite dimensional example: the use of frequencies of vibration to recover positions and masses of beads vibrating on a string. First we derive the equations of motion, then compare the eigenvalues of the resulting linearized model against vibration data measured from our laboratory's monochord. More challenging is the recovery of masses and positions of the beads from spectral data, a problem for which a variety of elegant algorithms exist. After presenting one such method based on orthogonal polynomials in a manner suitable for advanced undergraduates, we confirm its efficacy through physical experiment. We encourage readers to conduct their own explorations using the numerous data sets we provide.