## 3 Measurement of Connecting Rod Journal
### Data Processing  
According to Equation (1) and the relation curve, it is found that the contour data collected by the connecting rod journal probe are non-equal interval samples. A method was proposed to multiply the amplitude of the original signal by the interval time between two signals, then take the Fourier transform [33]. In fact, the samples collected will change with time and should be processed first. The method based on the assumption of equal interval samples and Fourier transform in the frequency domain is no longer suitable. It is necessary to operate according to the actual distribution of the sampling angles of each sample and then deal with it by discrete Gauss weighting function.  
Figure 4: Relationship between included \(\theta\) and rotation angle \(\varepsilon\).  
Figure 5: Relationship between angular velocity \(\Omega\) and rotation angle \(\varepsilon\).  
In ISO 16610 standard, the weight function of the Gaussian low-pass filter for workpiece surface texture analysis can be given by Equation (2):  
\[s(x)=\frac{1}{\alpha\lambda_{\rm c}}\exp\left[-\pi\left(\frac{x}{\alpha\lambda_{ \rm c}}\right)^{2}\right] \tag{2}\]  
where \(x\) is the distance from the center of symmetry of the weight function, \(\lambda_{\rm c}\) is the cutoff wavelength of Gaussian low-pass filter and \(\alpha\) is a constant.  
In the filtering process of the circular profile with a cross-section, the cutoff wavelength \(\lambda_{\rm c}\) is usually replaced by the cutoff wave number \(N_{\rm C}\). In this scheme, the measured journal is compared with the calibration part, and the sampling data are the variations in diameter of all journals relative to the calibration part. The weight function of the Gaussian low-pass filter does not depend on the diameter of the journals, so Equation (2) can be expressed in cutoff wave number \(N_{\rm C}\). The weight function can be discretized in the case of equal interval sampling and is expressed as:  
\[s(p\Delta\theta)=\frac{N_{\rm c}}{\alpha}\exp\left[-\pi\left(\frac{p\Delta \theta N_{\rm c}}{\alpha}\right)^{2}\right] \tag{3}\]  
where \(\Delta\theta\) is the sampling interval angle, and \(p\) represents the \(p\)-th sampling angle. In fact, the sampling interval angle \(\Delta\theta\) of connecting rod journal varies non-uniformly, which is related to the distance \(L\) of the measuring point and the length \(R\) of the crank arm. Compared with the equal interval sampling, the weight function of the Gauss filter for non-equally samples is determined by the specific sampling angle in Equation (1). The corresponding weight function is different, while the actual sampling angle is different. Therefore, the weight function for each sampling angle can be obtained, which is expressed as follows.  
\[\begin{split} s(p-q)&=\frac{N_{\rm c}}{\alpha}\exp \left[-\pi\left(\frac{(\theta_{p}-\theta_{q})N_{\rm c}}{\alpha}\right)^{2} \right]\\ &(p,q=1,2,\ldots,M)\end{split} \tag{4}\]  
\(\theta_{p}\), \(\theta_{q}\) represent different sampling angles, and \(M\) represents the number of samples taken by a single probe around the connecting rod journal. By normalizing, Equations (4) and (5) can be obtained.  
\[\widetilde{s}(p-q)=\frac{s(p-q)}{\sum_{p=1}^{M}s(p-q)},p,q=1,2,\ldots,M \tag{5}\]  
The angle information is used to calculate the coefficients of the filter and the amplitude information is multiplied by the angle information. The discrete cyclic convolution sum is directly used for Gaussian filtering through the spatial domain, and the non-equidistant Gauss filtered data are:  
\[y(p)=\sum_{q=1}^{M}\widetilde{s}(p-q)\rho(q),p,q=1,2,\ldots,M \tag{6}\]  
where \(\rho(q)\) is the original contour data, and \(y(p)\) is the data filtered by unequal interval weight function.