# Free and Forced Vibration Modes of the Human Fingertip
## 2 Discretized 3D Fingertip Model: DigiTip
\begin{table}
\begin{tabular}{l l l l l l} \hline \hline
**Variable** & **Description** & **Value (mm)** & **Variable** & **Description** & **Value (mm)** \\ \hline \(l_{f}\) & Fingertip length & 28.0 [14] & \(t_{sij}\) & Soft-tissue thickness at the & 2.9 [15] \\ \(w_{f}\) & Fingertip width & 18.0 [14] & \(l_{b}\) & Bone length at the distal & 22.5 ** [16] \\ \(h_{f}\) & Fingertip height & 15.0 [14] & \(l_{bd}\) & Bone length at the distal & 8.4 *** [17] \\ \(l_{n}\) & Total nail plate length & 13.0 [18] + 4.6 [19] & \(w_{bd}\) & Bone width at the distal & 8.6 *** [20] \\ \(w_{n}\) & Nail plate width & 12.3 [18] & \(h_{bd}\) & Bone height at the distal & 4.2 *** [20] \\ \(h_{n}\) & Nail plate height & 2.5 * [18] & \(w_{bp}\) & Bone width at the proximal & 13.0 *** [20] \\ \(t_{n}\) & Nail plate thickness & 0.4 [21] & \(h_{bp}\) & Bone height at the proximal & 8.0 *** [20] \\ \(t_{sc}\) & Thickness of stratum corneum & 0.4 [22] & \(w_{bn}\) & Bone width at the neck & 5.3 [23] \\ \(t_{ep}\) & Thickness of epidermis & 0.1 [22] & \(h_{bn}\) & Bone height at the neck & 3.8 [23] \\ \(t_{de}\) & Thickness of dermis & 1.5 [24] & & & \\ \hline \hline \end{tabular}  
* The nail height was calculated according to the reported nail curvature radius and nail width. ** The total length of the distal phalanx was determined by preserving the reported ratio between the fingertip’s soft-tissue thickness and the bone length. *** The bone length at the distal section was determined by preserving its reported proportion to the total length of the distal phalanx. *** The width and height at the distal and proximal sections of the bone were calculated by preserving their reported proportions with respect to the total length of the distal phalanx.  
\end{table} 
Table 1: The descriptions and values of the model dimensions shown in Figure 1. The selected values are given along with the references from which they were taken. Figure 1: The variables that govern the dimensions of DigiTip: (**a**) Exterior fingertip dimensions; (**b**) thicknesses of tissue layers; (**c**) dimensions of the distal phalanx. \begin{table}
\begin{tabular}{l l l l l l l} \hline \hline  & **Stratum** & **Epidemis** & **Dermis** & **Hypodermis** & **Nail Bed** & **Fingermail** \\  & **Corneum** & & & & & \\ \hline Elastic modulus (MPa) & 1.000 [25] & 0.140 [25] & 0.080 [25] & 0.034 [25] & 1.000 [25] & 170.0 [25] \\ Poisson’s ratio & 0.30 [25] & 0.3 [25] & 0.48 [25] & 0.48 [25] & 0.30 [25] & 0.30 [25] \\ Density (g/cm\({}^{3}\)) & 1.100 [26] & 1.106 [26] & 1.274 [26] & 0.920 [27] & 1.100 * & 1.300 [28] \\ \hline \hline \end{tabular}  
* The nail bed was assigned the density value of the stratum corneum since these two layers were modeled with the same material properties in the study by [PERSON] et al. [25].  
\end{table} 
Table 2: The values of elastic modulus, Poisson’s ratio, and density for six fingertip tissue layers. The selected values are given along with the references from which they were taken. formulated with linear elasticity [29]. For reference, this 3D FE analysis code was used together with experiments in a previous study by the authors [30]. The portions of the code related to dynamic analysis were also previously used to compute the free and forced vibrations of shells [31]. The bone was not explicitly modeled since it was assumed to be rigid due to its significantly higher elastic modulus compared to the other tissue layers [32]. We numerically imposed this condition by immobilizing the innermost nodes of the hypodermis along the bone's surface. To allow for tissue deformation, we applied a free boundary condition along the surface where the fingertip would connect with the rest of the finger, and we compared the resulting behavior with the effects of fixing this boundary. Therefore, if we denote the number of nodes within the entire model and on the fixed boundary as \(N_{n}\) and \(N_{nb}\), respectively, the system has \(N_{d}=3(N_{n}-N_{nb})\) total degrees of freedom.