## 4 Numerical Results and Discussion  
In this section, dynamic for design procedure presented in previous sections is used to design small-scale grating device system using MATLAB symbolic toolbox. The system is fully constructed using CAD software with suitable dimensions and bodies masses and inertias are listed in Table 2 and used as initial simulation parameters. The external applied forces due to ceramic piezoelectric actuators are F = 4700 N. By applying external forces, the corresponding frame displacement of the system is obtained by dominant movements of piezoelectric actuators. Figure 7 shows constraints violation due to the prismatic joint  
Figure 6: Fixed joint between flexible body2 and flexure body2. between lower plate and grating base. The violation does not exceed \(2\times 10^{-12}\) which indicates the computational efficiency of multibody system model. Table 3 illustrates the degree of freedom movement that the system are able to do. Figure 8 shows the global position vector of the point **P** associated with translation coordinate. The translation along the Z-axis about 3 \(\upmu\)m in case of applying four forces at flexible bodies 1, 3, 4 and 5. The translation along the X-axis reaches 0.5 \(\upmu\)m at the same forces while there is no movement along Y-axis which indicates the accuracy of the formulation of multibody constraints. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Components** & **Mass (kg)** & \(\mathbf{I}_{xx}(\mathbf{kg.m}^{2})\) & \(\mathbf{I}_{yy}(\mathbf{kg.m}^{2})\) & \(\mathbf{I}_{zz}(\mathbf{kg.m}^{2})\) \\ \hline Grating base & 86.34 & 4.92 & 3.64 & 2.8 \\ Disk\({}_{1}\) & 0.15 & 0.000035 & 0.000035 & 0.000018 \\ Disk\({}_{2}\) & 0.11 & 0.000024 & 0.000013 & 0.000013 \\ Grating mass & 78.26 & 1.8 & 1.3 & 0.985 \\ Flexure part & 0.068 & 0.0001 & 0.0001 & 0.000025 \\ Flexible part & 0.012 & 0.00001 & 0.00001 & 0.0000063 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Grating device parameters employed in the numerical simulation. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**DOF** & **X (\(\upmu\)m)** & **Z (\(\upmu\)m)** & \(\boldsymbol{\phi}\) (\(\upmu\)rad)** & \(\boldsymbol{\theta}\) (\(\upmu\)rad) & \(\boldsymbol{\psi}\) (\(\upmu\)rad) \\ \hline Displacement & \(\pm 1.5\) & \(\pm 3.00\) & \(\pm 2.5\) & \(\pm 1.5\) & \(\pm\) 2.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Displacement range of grating parameters. Figure 7: Constraints violation of the prismatic joint. Figure 9 shows the orientational coordinate of the grating mass with the same applying forces. The grating mass can rotate 2.5 \(\upmu\)rad about X and Z axis. In order to optimize the grating device design, a multibody model provides all reaction forces acting on system components with maximum load condition. Figure 10 shows reactions forces acting on grating base, the values of the forces in X-axis and Z-axis are altered and the only reaction force in Y-axis due to the system weight. Figure 8: Grating mass displacement. Figure 9: Orientation of grating mass around three axes. Another useful type of data from multibody constraints are reaction moments. Figure 11 shows reactions moments acting on grating base. Likewise, other system bodies' reaction forces can be computed. The flexure and flexible bodies should be considered in the design of the tiling system because its the most important factors for system stability. Flexible bodies are an elastic element for the movement of system degree of freedom. The vector of acceleration and Lagrange multipliers associated with flexible bodies can be computed from the multibody model. The acceleration vector is integrated forward in order to calculate elastic displacement of flexible bodies. Figures 12 and 13 shows displacement and orientation of local frame at the middle of the flexible body four due to generalized coordinates associated with flexible bodies. Figure 11: Reaction moments acting on grating base. Figure 10: Reaction forces acting on grating base.