# Determination of Clamping Force Using Bolt Vibration Responses during the Tightening Process
## 2 Nondestructive Method to Estimate Clamping Force  
This section describes the algorithm used to determine the bolt clamping force. The proposed algorithm is based on unsupervised learning using k-means clustering after extracting the cepstrum as a feature of the vibration responses. The cepstrum analysis is used to summarize the spectral characteristics [29]. The cepstrum analysis used to detect the echoes in seismic signals is defined as \"the power spectrum of the logarithm of the power spectrum\" [30]. It is widely used in speech analysis, communications, seismology, and geophysics [31]. Use of an inverse Fourier transform on the log spectrum instead of a standard Fourier transform was also proposed [32]. The \"complex cepstrum\" was defined as the inverse Fourier transform of the complex logarithm of the complex spectrum. The complex cepstrum contains phase information retrieved from the time domain. This mitigates the effects of harmonics and sidebands in the measured audio signals [33]. When using a rotating element such as a nutrunner to fasten bolts, sidebands associated with the fastening speed are generated at the control frequency of the servomotor and harmonics appear in the entire frequency band. These components are not related to the clamping force of the bolts. Elimination of the effects using cepstrum analysis benefits the interpretation of the signal. The calculation of cepstrum is available in various forms. When using LFCC or mel frequency cepstral coefficients (MFCC), a discrete cosine transform is widely applied instead of the Fourier transform or inverse Fourier transform [28]. The mel frequency was designed for frequency domain analysis when recognizing human speech signals. In this study, the real cepstrum was used as:  
\[\hat{x}_{r}(\tau)=DCT\Big{[}\log\Big{(}|X(f)|\Big{)}\Big{]}, \tag{1}\]  
where \(DCT\) is the discrete cosine transform and \(X(f)\) is the Fourier transform of the time signal. The discrete cosine transform used in this is study defined as:  
\[DCT(x(k))=\sqrt{\frac{2}{N}}\sum_{n=1}^{N}x(n)\frac{1}{\sqrt{1+\delta}}\cos \Big{(}\frac{\pi}{2N}(2n-1)(k-1)\Big{)} \tag{2}\]  
where \(k=1,2,\cdots,N\), \(x\) is the measured signal, \(N\) is length of signal, and \(\delta\) is Kronecker delta. The cepstrum was named by reversing the first syllable of spectrum. The quefrency was obtained from the Fourier transform of logarithmic frequency. An additional benefit of interpretation of the signal using the cepstrum includes the logarithmic computation. The signal is measured when the wave propagation possesses the combined characteristics of the source and the propagation path. In this study, the vibration excitation source is a fastening nutrunner. The bolt is included in the transmission path. The vibration due to transmissions reflect the change of the fastening force in the bolt. These properties are multiplicative in the frequency domain. When configured with the logarithmic transformation of the spectrum, the source and the propagation path become additives and are inherited in the calculation of the cepstrum. [PERSON] [28] applied triangular band pass frequency response for the filter bank. A triangular filter bank is used to simulate the subjective spectrum [34]. Triangular filters are converted into sinc\({}^{2}\) functions in the quefrency domain that are efficient for real-time calculations and anti-aliasing benefits. One hundred triangular filters are set at equal intervals in the frequency band between 4 kHz and 20 kHz to simulate a band pass filtering effect. The frequency band includes a change in natural frequency that occurs when the bolt is tightened. The linearly spaced triangular filters were utilized to characterize the cepstrum as feature coefficients. The feature was represented by the \(T\times 100\) dimensional vector, where \(T\) is the number of time windows. The data set constructed from 472 bolt fastening experiments was trained in machine learning algorithms.