## 2 Model of the Chevron Type Microactuator
### Verification of the Numerical Model  
The optimal design of the chevron type thermal actuator requires building a numerical model of the actuator, which allows one to calculate the values of criteria with the satisfactory level of accuracy. It is possible to analytically calculate electrical, thermal and structural quantities for the actuator with straight arms. This subsection is dedicated to the verification of the numerical model prepared. Simplifying the verification procedure comparison between the numerical and analytical models only for mechanical quantities is shown. Since the thickness perpendicular to the midplane of the actuator (z-axis) and boundary conditions are constant, and thickness is very small compared to other dimensions in the model, only two-dimensional models will be considered. Consider the fundamental unit of the V-type actuator, a pair of angled beams (Figure 4). The average actuator temperature rise \(\Delta T\), caused by the flow of electric current, produces the output displacement \(u_{y}\) and the force \(F_{y}\). An external force \(F_{i}\) may be applied to the central shaft in y direction, affecting the \(u_{y}\) displacement. Analytical modeling of the V-type actuators can be found in [2; 5]. The equation below shows relationships between those quantities. \[u_{y}=\frac{\alpha\,\Delta TE\,A\,\sin\varphi}{\left(\frac{EA}{L}\sin^{2} \varphi+\frac{12 EI_{i}}{L^{3}}\cos^{2}\varphi\right)}+\frac{-F_{i}}{\left( \frac{EA}{L}\sin^{2}\varphi+\frac{12 EI_{i}}{L^{3}}\cos^{2}\varphi\right)} \tag{14}\]  
If the force constraint \(F_{i}\) is equal to zero, Equation (14) expresses the relationship for the free-moving actuator displacement. Taking into account--average temperature rise \(\Delta T=450\) K, E \(=158,000\) MPa, length of the beam \(L=220\) um and the square cross-section area \(A=25\) um\({}^{2}\), the FEM numerical model in MSC Mentat/Marc software was prepared and compared with the analytical solution. Plane stress model consisting of four node quad elements was assumed. The density of the mesh was established after some preliminary calculations. The number of 6-8 elements per width of the beam ensures correct results when the actuator bends, and therefore, the model consists of about 6000 elements (Figure 5). Figure 6 shows the characteristics of the output displacement with respect to the pre-bending angle. The numerical simulation results are in close agreement with the analytical results. The highest discrepancy was obtained for the peak value of the displacement, where relative error was equal to 2.3%. Figure 4: Symmetric beam pair of the V-type actuator. Figure 5: FEM model of the 2-beam V-type actuator. Figure 6: Pre-bending angle–displacement characteristics of the V-type actuator with straight arms. Analytical Equation (14) may be useful for the straight arms and basic design only; moreover, it underrepresents the interaction between constraint force and displacement when the displacement is close to zero or when a large force is applied [2]. From the practical point of view, actuation force should be modeled as a contact force, as in the model in [38]. Figure 7 shows the relationship between the pre-bending angle and the contact force calculated at the rigid surface, for the same model of the actuator. The gap is understood as a distance from the tip of the central shaft to the rigid surface. A small value of the pre-bending angle actuator cannot generate a high enough value of the force for it to be useful; a higher value of the pre-bending angle actuation force strongly depends on the value of the gap. The peak value of the contact force is decreasing while gap is increasing. Another important factor, which has to be considered in optimal design of the V-type actuator, is the possibility of buckling. Buckling appears if the internal force in the arms is greater than the critical buckling force.