Spectral recoverability of inverse loads for fractionally damped beams

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Abstract

This paper investigates the spectral recoverability of a final-time inverse load problem for a Caputo fractionally damped Euler–Bernoulli beam. The unknown source is assumed to be separable in space and time, and the aim is to recover the spatial load profile from terminal displacement data. For a simply supported beam with constant coefficients, the terminal input-output map is diagonalized by the beam eigenfunctions, leading to a modal relation between the unknown load and the observation. The associated singular values are determined by the terminal responses of fractional-order modal equations and depend explicitly on the damping order. A truncated spectral reconstruction method with Tikhonov filtering is used for stable recovery from noisy data. Several effective indicators, including the minimum retained singular value, the effective condition number, and the maximum regularized amplification factor, are introduced to assess reconstruction stability. Numerical experiments show that the fractional damping order can reshape the effective terminal spectrum and influence the stability of the inverse problem. In suitable parameter regimes, fractional memory damping improves recoverable modal responses and reduces noise amplification compared with the classical integer-order damping model.

Keywords

spectral recoverability / final-time inverse load reconstruction / Euler–Bernoulli beam / Tikhonov regularization / fractional damping / Caputo derivative

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