## 3 Mechanical Error Analysis
### Case Study  
A case study is given to demonstrate the derived analytical equations and the established evaluation indices. Table 1 lists the geometric parameters of a nine-link type double-toggle mold/die clamping mechanism, which are approximately one-third sizes of the clamping system of a real die casting machine. Figure 6 shows the variation trend of the linear position of the output link, \(L(x)\), with respect to the input variable \(x\) that varies from \(-3.439\) to \(51.566\) mm. The theoretical stroke of the output link is \(45.003\) mm. As seen, when \(x=51.566\) mm, the mechanism is in its toggle position as similar to that shown in Figure 3c. In order to perform the mechanical error analysis, the dimensional errors of all geometric parameters in Table 1 are given based on the ISO tolerance (IT) grades. For the geometric parameters in the lower sub-mechanism, the lower deviations of their specified bilateral tolerance ranges according to the IT grade are given, and for those in the upper sub-mechanism, the upper deviations of their specified bilateral tolerance ranges also according to the IT grade are given. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Term** & **Value** & **Term** & **Value** \\ \hline \(r_{1L}\) & 100.000 mm & \(r_{1U}\) & 100.000 mm \\ \(r_{2L}\) & 45.162 mm & \(r_{2U}\) & 45.162 mm \\ \(r_{3}\) & 36.573 mm & \(r_{6}\) & 36.573 mm \\ \(r_{4a}\) & 64.071 mm & \(r_{7a}\) & 64.071 mm \\ \(r_{4b}\) & 100.000 mm & \(r_{7b}\) & 100.000 mm \\ \(r_{4c}\) & 41.563 mm & \(r_{7c}\) & 41.563 mm \\ \(r_{5}\) & 110.000 mm & \(r_{8}\) & 110.000 mm \\ \(e_{L}\) & 10.000 mm & \(e_{U}\) & 10.000 mm \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Geometric parameters of a nine-link type double-toggle mold/die clamping mechanism. Figure 6: Variation trend of the linear position of the output link with respect to the input variable. For this case study, Figure 7 shows the results for the mechanical error analysis of the lower sub-mechanism, in which, a tolerance grade of IT10 is specified to each geometric parameter. In Figure 7a, output error curves induced by each negative dimensional error are presented, and their resultant worst-case and maximum expected deviations, \((\Delta L_{\text{wor}})_{L}\) and \((\Delta L_{\text{rss}})_{L}\), are presented in Figure 7b. Similarly, Figure 8 shows the results for the mechanical error analysis of the upper sub-mechanism, in which, a tolerance grade of IT10 is also specified to each geometric parameter. In Figure 8a, output error curves induced by each positive dimensional error are presented, and their resultant worst-case and maximum expected deviations, \((\Delta L_{\text{wor}})_{U}\) and \((\Delta L_{\text{rss}})_{U}\), are presented in Figure 8b. As shown in Figures 7a and 8a, when the mechanism is in its toggle position, the dimensional errors of links 4, 5, 7, and 8 will considerably influence the output errors, while those of other links have quite less effects on the output errors. In other words, links 4, 5, 7, and 8, which are the critical link members to form the double-toggle configuration, are also most sensitive to the mechanical error of the output link at that instant. As shown in Figures 7b and 8b, when a tolerance grade of IT10 is specified to each geometric parameter, the stochastic deviations of the output link at the toggle position are about 0.2 to 0.3 mm. Considering that the theoretical stroke of the output link is 45.003 mm, such stochastic deviations imply that the output link will have a degraded positioning accuracy ranging from IT11 to IT13. Figure 7: Results for the mechanical error analysis of the lower sub-mechanism: (**a**) output error curves induced by negative dimensional errors, and (**b**) the resultant worst-case and maximum expected deviations.