Correction

This article refers to:

Vibro-acoustic modulation-based bolt looseness detection method using frequency sweep probe waves

Article title: Vibro-acoustic modulation-based bolt looseness detection method using frequency sweep probe waves

Authors: Jingjing Fan, Wenfei Chen, Dingyue Chen, Hu Chen, Lijia Luo and Shiyi Bao

Journal: Nondestructive Testing and Evaluation

DOI: https://doi.org/10.1080/10589759.2023.2244123

This paper was initially published with two incorrect equations, (17) and (18). It has now been corrected and republished with the updated equations as below.

Equation 17:

$S_0={\displaystyle \sum _i{V}_0}(1+m_{a}cos{\omega }_{0}t)cos({\omega }_{i}t+m_{f}sin{\omega }_{0}t)$

Equation 18:

$\begin{array}{rl}& S_r={\sum }_i{S}_0\mathrm{c}\mathrm{o}\mathrm{s}{\omega }_{i}t\\ & =\sum _i\left[V_0(1+m_a\mathrm{c}\mathrm{o}\mathrm{s}{\omega }_{0}t)\mathrm{c}\mathrm{o}\mathrm{s}({\omega }_{i}t+m_f\mathrm{s}\mathrm{i}\mathrm{n}{\omega }_{0}t)\mathrm{c}\mathrm{o}\mathrm{s}{\omega }_{i}t\right]+\cdots \\ & =\sum _i\left[V_0\mathrm{c}\mathrm{o}{\mathrm{s}}^2{\omega }_{i}t\mathrm{c}\mathrm{o}\mathrm{s}(m_f\mathrm{s}\mathrm{i}\mathrm{n}{\omega }_{0}t)-V_0\mathrm{c}\mathrm{o}\mathrm{s}{\omega }_{i}t\mathrm{s}\mathrm{i}\mathrm{n}{\omega }_{i}tsin(m_f\mathrm{s}\mathrm{i}\mathrm{n}{\omega }_{0}t)\right.\\ & +m_a{V}_0\mathrm{c}\mathrm{o}{\mathrm{s}}^2{\omega }_{i}t\mathrm{c}\mathrm{o}\mathrm{s}{\omega }_{0}tcos(m_f\mathrm{s}\mathrm{i}\mathrm{n}{\omega }_{0}t)\\ & \left.-m_a{V}_0\mathrm{c}\mathrm{o}\mathrm{s}{\omega }_{i}tsin{\omega }_{i}tcos{\omega }_{0}t\mathrm{s}\mathrm{i}\mathrm{n}(m_f\mathrm{s}\mathrm{i}\mathrm{n}{\omega }_{0}t)\right]+\cdots \\ & =\sum _i\left[\frac{V_0}{2}\mathrm{c}\mathrm{o}\mathrm{s}\left(m_f\mathrm{s}\mathrm{i}\mathrm{n}{\omega }_{0}t\right)+\frac{m_a{V}_0}{2}\mathrm{c}\mathrm{o}\mathrm{s}{\omega }_{0}t\mathrm{c}\mathrm{o}\mathrm{s}(m_f\mathrm{s}\mathrm{i}\mathrm{n}{\omega }_{0}t)\right.\\ & +\frac{V_0}{2}\mathrm{c}\mathrm{o}\mathrm{s}2{\omega }_{i}t\mathrm{c}\mathrm{o}\mathrm{s}(m_f\mathrm{s}\mathrm{i}\mathrm{n}{\omega }_{0}t)+\frac{m_a{V}_0}{2}\mathrm{c}\mathrm{o}\mathrm{s}2{\omega }_{i}t\mathrm{c}\mathrm{o}\mathrm{s}{\omega }_{0}t\mathrm{c}\mathrm{o}\mathrm{s}(m_f\mathrm{s}\mathrm{i}\mathrm{n}{\omega }_{0}t)\\ & \left.-\frac{V_0}{2}\mathrm{s}\mathrm{i}\mathrm{n}2{\omega }_{i}t\mathrm{s}\mathrm{i}\mathrm{n}(m_f\mathrm{s}\mathrm{i}\mathrm{n}{\omega }_{0}t)-\frac{m_a{V}_0}{2}\mathrm{s}\mathrm{i}\mathrm{n}2{\omega }_{i}t\mathrm{c}\mathrm{o}\mathrm{s}{\omega }_{0}t\mathrm{s}\mathrm{i}\mathrm{n}(m_f\mathrm{s}\mathrm{i}\mathrm{n}{\omega }_{0}t)\right]+\cdots \end{array}$