## 2 Materials and Methods
### Analytical Modelling of the Proposed IAC Actuator

Table 3: Phase transformation characteristics. where \(A_{S},\;\;A_{F},\;M_{S},\;M_{F}\) are the starting austenite temperature, final austenite temperature, starting martensite temperature, and final martensite temperature. These parameters are listed in Table 3. \(\alpha_{1,2,3}\) represent the crystal variable speed coefficients of nitinol, \(\beta_{A}\), \(\beta_{M}\) are slope limit curves of austenite and martensite, and the other mathematical symbols are defined in Tables 2 and 3. The twinned crystals' components are separated during detwinned martensite. However, introducing the stresses in the transformation process of the \(M_{tw}\) to \(M_{dt}\) defines the deformation of the SMA spring in the martensitic condition. The kinetic model is used to find the total stroke that is moved by the pintle. It is defined by the displacement of the SMA spring from the axial pitch. This displacement includes both the change in the axial pitch of the coils (\(\Delta_{P}\)) and the initial pitch (\(P_{s}\)) of the SMA spring. Hence, this kinetic model determines the change in vertical pitch (\(\Delta_{P}\)) of each coil in a reasonable working strain of spring wire. The product of (\(\Delta_{P}\)) with the number of active coils (\(N_{sc}\)) results in the total displacement of the spring if it is added or deducted from the original vertical pitch of the SMA spring. The change in axial pitch of the coils (\(\Delta_{P}\)) is a function of working stress, working strain, SMA spring size and the SMA spring's generated force, as expressed by the first term of Equation (8):  
\[S=\left(P_{s}+\left(4\frac{F_{s}P_{s}\varepsilon_{w}}{\pi D_{s}^{2}\varepsilon _{w}}\right)\right)N_{sc} \tag{8}\]  
where \(S\) represents the total stroke moved by the actuator's pintle, \(\varepsilon_{w}\) stands for the working strain, and \(\sigma_{w}\) is the working stress. Other variables were defined in the previous paragraphs or in Appendix A at the end part of the article. The working strain (\(\varepsilon_{w}\)) used in this model is from the strain due to elasticity, strain induced by thermal, and induced strain by phase transformation. It is expressed as follows:  
\[ \begin{gathered}\varepsilon_{w}=\varepsilon_{i}\frac{\sigma_{w}}{ E(\theta_{A},\theta_{A})}-\frac{\sigma_{i}}{E(\theta_{Ai},\theta_{ali})}+ \xi(\theta_{dt}-\theta_{dti})+\mathcal{O}(\theta_{A})T_{w}-\mathcal{O}( \theta_{Ai})T_{e}\\ E(\theta_{A},\theta_{dt})=\varepsilon_{tw}+(E_{A}-E_{tw})\theta_{ A}+(E_{dt}-E_{tw})\theta_{dt}\end{gathered}  \tag{9}\]  
In the above, \(\xi,\mathcal{O},\varepsilon_{i}\) are the transformation modulus, temperature coefficient from the material's resistance, and initial strain of the SMA spring, respectively. \(E_{A},E_{tw},E_{dt}\), \(E(\theta_{A},\theta_{dt})\) represent the austenite young's modulus, twinned young's modulus, detwinned young's modulus, and young's modulus in austenite and detwinned martensite phase transformations, respectively. Equations (4)-(9) were used to develop the MATLAB algorithms that define the operation of the actuator in the analytical mean. The model-based system design approach (Simulink (R2023b)) was adopted and linked with those algorithms because of the multidomain models of the proposed actuator. Simulation outputs are presented in the results section. Figure 6: Maximum reference of fraction volume in phase transformation.