# Numerical and Experimental Analysis of Torsion Springs Using NURBS Curves
## 3 Torsion-Spring Displacement Analysis  
### Torsion-Spring Displacement Analysis  
To verify the correlation between the interpolation point and control point, the coordinates of the interpolation point generated through the NURBS curve are applied to finite-element analysis to create the torsion-spring shape for analysis [11; 12; 13]. Figure 2 shows the finite-element model of the torsion spring that was used to derive the displacement of the control point as it shifted according to the external force [14; 15]. Table 1 shows the parameters of the model used for analysis. Commercial software HyperWorks Optistruct (Altair, United States) was used for finite-element analysis, and the shifted displacement data were acquired for generated moment \(M_{z}\). The displacement values shifted after the generation of the initial coordinates, and loads were analyzed for a total of 100 elements (\(N_{i}\)). Figure 1: Representation of nonuniform-rational-basis-spline (NURBS) curves coupled in 3D space. The displacement of the element determined through finite-element analysis (\(\mathbf{x_{disp}}\), \(\mathbf{Y_{disp}}\), \(\mathbf{z_{disp}}\)) was added to the coordinates of the interpolation point generated through the NURBS curve to express the deformed shape of the spring. This is expressed using Equations (9) to (11). \[P_{Ndisp}(x) =P_{N}(x)+x_{disp} \tag{9}\] \[P_{Ndisp}(y) =P_{N}(y)+y_{disp}\] (10) \[P_{Ndisp}(z) =P_{N}(z)+z_{disp} \tag{11}\]  
where \(P_{N}\) are the coordinates of the interpolation point generated through the initial NURBS curve, and \(P_{Ndisp}\) are the coordinates of the interpolation point of the NURBS curve after deformation. The coordinates of \(P_{Ndisp}(x,\ y,z)\), derived from Equations (9) to (11), were next applied to the inverse method to determine the displacement of the control point. This process is shown in Equation (12), which was derived with reference to Equations (4) and (5). Figure 3 shows the control point derived using Equation (12),  
\[P_{(x,y,z)}=\frac{\sum_{i=1}^{n}R_{i,p}(u)}{P_{Ndisp}(x,y,z)}, \tag{12}\]  
where \(P_{x}\), \(P_{y}\) and \(P_{z}\) are the coordinates of the control point, and \(R_{i,p}(u)\) is the NURBS curve-based function. In Figure 3, \((x,y,z)_{disp}\) represents the \(x\), \(y\), and \(z\) displacements of the point; \(P_{N}(x,y,z)\) represents the coordinates of the point before interpolation; and \(P_{Ndisp}(x,y,z)\) represents the coordinates of the point after interpolation. The inverse method can be applied to determine the displacement and load as \(K_{NURBS}\). \[a[D]=a[K]^{-1}[F], \tag{13}\]  
where \([D]\) is the displacement matrix of the control point derived from the existing stiffness matrix, \(a\) is the stiffness-correction constant, \([K]^{-1}\) is the inverse of the stiffness matrix, and \([F]\) is the external force matrix. Stiffness-correction constant a can be expressed as follows:  
\[a=[a_{i}]^{T}. \tag{14}\] \begin{table}
\begin{tabular}{l l l} \hline \hline
**Parameters** & **Unit** & **Dimension** \\ \hline Spring cross-section diameter (\(d\)) & mm & 3 \\ Spring radius (\(R\)) & mm & 20 \\ Spring length (\(I\)) & mm & 125.6 \\ Moment in the \(z\)-axis direction (\(M_{z}\)) & N m & 1000 \\ Elastic modulus & GPa & 206 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Design parameters. Figure 2: Finite-element model of torsion spring. Thus, the following is obtained by applying Equation (14) to Equation (13). \[[a_{i}]^{T}[D]=[a_{i}]^{T}[K]^{-1}[F] \tag{15}\]  
Equation (15) can then be expressed as in Equation (16). \[[D_{\texttt{E}v}]=[K_{NRBS}]^{-1}[F] \tag{16}\]  
where \([D_{\texttt{E}v}]\) is the control point displacement determined using the inverse method, and \([K_{NRBS}]^{-1}\) is the inverse of the stiffness matrix of the control point. Using Equation (16), the \(M_{z}\) load was applied to the NURBS curve-based spring structure that was composed of a total of nine control points, and analysis was conducted.