# Implicit Subspace Iteration to Improve the Stability Analysis in Grinding Processes
## 3 Results  
### Application of the Method to Cylindrical Grinding Processes  
First application of the method is carried out for cylindrical infeed grinding process by comparing the results with the data proposed in [PERSON] et al. [14]. In this analysis, a validation of the accuracy of the new method was done and a comparison of the computational effort of both semidiscretization and subspace iteration methods is presented. The process conditions for stability analysis in [14] are shown in Table 1.  
Figure 2a shows the stability map obtained with the subspace iteration method for the process conditions of Table 1, which represents the stability degree of each combination of workpiece speed and ground length. Values below the stability limit of 1 are stable combinations and values over the stability limit are unstable combinations. Almost the same stability map as the one obtained with the semi-discretization method is achieved. Figure 2b shows isolines with different degrees of stability. Dash lines correspond to subspace iteration calculations, while solid lines correspond to semidiscretization calculations. Good correlation between isoline values can be noticed. Necessary computational effort for achieving the stability map has been reduced significantly. The time required for obtaining the map with the semi-discretization method is 11,441 s, while the time with the subspace iteration method is 256 s (45 times faster), considering that the map corresponds to a \(11\times 11\) matrix of stability values. For this comparison, an Intel (r) Core TM i7-4610M CPU 3.00 GHz with RAM 16 GB has been used. One important aspect is the dependency of the number of discretized points on the calculation time. As mentioned above, the integration time is calculated based on the considered maximum natural \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{4}{c}{**Modal Analysis**} \\ \hline
**Mode** & \(f_{\pi}\)** **(Hz)** & \(\xi\)** **(\%)** & \(\phi_{\pi}\) \\ \hline
1 & 200 & 5 & 0.1 \\
2 & 300 & 5 & 0.09 \\
3 & 400 & 4 & 0.1 \\ \hline Grinding wheel diameter (mm) & & 600 & \\ Grinding wheel speed (m/s) & & 50 & \\ Workpiece diameter (mm) & & 25 & \\ Equivalent stiffness (N/um) & & 50 & \\ Specific energy (J/mm\({}^{3}\)) & & 40 & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Process conditions for stability analysis in cylindrical infeed grinding [14]. Figure 2: (a) Subspace iteration stability map; (b) comparison of stability isolines between semi-discretization and subspace iteration methods. frequency. Figure 3 shows the comparison of the calculation time of both semi-discretization and subspace iteration methods related to that frequency. As it can be noticed, the higher the frequency the higher the difference between both calculations. For a natural frequency of 100 Hz, the subspace iteration method is six times faster, being 138 times faster for a frequency of 700 Hz. Therefore, the subspace iteration method is even more suitable for systems with higher natural frequencies. The subspace iteration method is also valid to calculate the stability maps with continuous workpiece speed variations, whose axes are the frequency and amplitude of the sinusoidal variation. Figure 4 shows a comparison of the method with that of the semi-discretization method following the same procedure. Stability maps are calculated with process conditions of [14]. Isolines of instability reduction ratio are presented for both methods and good correlation can be noticed. Black boxes show the theoretical values of isolines while white ellipses show the experimental values. Figure 4: Comparison of instability reduction ratio maps with the application of continuous workpiece speed variation, obtained with both methods. Figure 3: Comparison of time for calculation of stability maps between semi-discretization and subspace iteration methods for cylindrical grinding.