# The Application of Piecewise Regularization Reconstruction to the Calibration of Strain Beams

## TL;DR

This paper introduces a new method for calibrating strain sensors that improves accuracy across different frequencies.

## Contribution

A piecewise Tikhonov regularization method is proposed to enhance dynamic load reconstruction in strain sensor calibration.

## Key findings

- The PTR method achieves lower MREs (6.20% at 70 Hz and 5.86% at 80 Hz) compared to traditional methods.
- The proposed method also shows significantly lower PREs (3.54% at 70 Hz and 3.73% at 80 Hz).
- The method is effective for load reconstruction across various frequencies.

## Abstract

Standard beams are mainly used for the calibration of strain sensors using their load reconstruction models. However, as an ill-posed inverse problem, the solution to these models often fails to converge, especially when dealing with dynamic loads of different frequencies. To overcome this problem, a piecewise Tikhonov regularization method (PTR) is proposed to reconstruct dynamic loads. The transfer function matrix is built both using the denoised excitations and the corresponding responses. After singular value decomposition (SVD), the singular values are divided into submatrices of different sizes by utilizing a piecewise function. The regularization parameters are solved by optimizing the piecewise submatrices. The experimental result shows that the MREs of the PTR method are 6.20% at 70 Hz and 5.86% at 80 Hz. The traditional Tikhonov regularization method based on GCV exhibits MREs of 28.44% and 29.61% at frequencies of 70 Hz and 80 Hz, respectively, whereas the L-curve-based approach demonstrates MREs of 29.98% and 18.42% at the same frequencies. Furthermore, the PREs of the PTR method are 3.54% at 70 Hz and 3.73% at 80 Hz. The traditional Tikhonov regularization method based on GCV exhibits PREs of 27.01% and 26.88% at frequencies of 70 Hz and 80 Hz, respectively, whereas the L-curve-based approach demonstrates PREs of 29.50% and 15.56% at the same frequencies. All in all, the method proposed in this paper can be extensively applied to load reconstruction across different frequencies.