Spectral recoverability of inverse loads for fractionally damped beams

Huaqing Ma; Jichen Zhang; Min Liu; Chen Gao; Zhiwei Yang; Huaqing Ma; Jichen Zhang; Min Liu; Chen Gao; Zhiwei Yang

doi:10.3934/era.2026386

Electronic Research Archive

2026, Volume 34, Issue 12: 9169-9189. doi: 10.3934/era.2026386

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Spectral recoverability of inverse loads for fractionally damped beams

1.
School of Qilu Transportation, Shandong University, Jinan 250002, China
2.
Shandong Inspur Science Research Institute Co., Ltd, Jinan 250101, China
3.
OriChip (ShanDong) Semiconductor, Qingdao 266104, China

Received: 22 July 2026 Revised: 14 August 2026 Accepted: 16 September 2026 Published: 27 October 2026

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.
- spectral recoverability,
- final-time inverse load reconstruction,
- Euler–Bernoulli beam,
- Tikhonov regularization,
- fractional damping,
- Caputo derivative
Citation: Huaqing Ma, Jichen Zhang, Min Liu, Chen Gao, Zhiwei Yang. Spectral recoverability of inverse loads for fractionally damped beams[J]. Electronic Research Archive, 2026, 34(12): 9169-9189. doi: 10.3934/era.2026386

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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.

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