# Free and Forced Vibration Modes of the Human Fingertip
## 4 Modal Analysis
### Low-Order Free Vibration Modes  
In this section, we present the mode shapes for the natural frequencies computed in Section 4.1. Figure 4 shows the first seven fingertip mode shapes and corresponding natural frequencies. The mesh contours demonstrate relative displacement magnitudes; the blue and yellow colors show the lowest and highest displacement regions, respectively. Readers can refer to Video S1 provided in the supplementary materials to see the evolution of the modal deformations over time. The first mode primarily exhibits torsional deformation around the distal phalanx. The second mode shape involves a rotational motion around the lateral axis. In the third mode, a lateral movement is observed around the distal section, while some torsional deformation appears at the proximal side. The fourth mode shape includes contraction around the fingerprint centroid and expansion at the sides or vice versa. In the fifth mode, an out-of-phase motion normal to the surface is observed in the lower-left and -right regions of the finger. The sixth mode shape contains out-of-phase vertical motions around the distal and proximal sections. In the seventh mode, distal-left and proximal-right sides contract toward the fingerprint centroid, while distal-right and proximal-left sections expand outward, or vice versa.  
As described before, the rear boundary of the modeled fingertip is free to move except the innermost nodes, which are fixed as they are in contact with the bone. We explored the  
Figure 4: DigiTip’s first seven mode shapes and corresponding natural frequencies. The contours indicate equivalent displacement magnitudes.  
Figure 3: A section view of the finite element model that resulted from the convergence analysis; it consists of 11,637 nodes and 10,080 elements.  
implications of this modeling decision by performing additional modal analyses for the other extreme condition, in which the rear boundary is entirely restrained. Figure 11 in Appendix A shows the results, which can be compared with Figure 4. This investigation shows that immobilizing the nodes at the proximal boundary of DigiTip causes the modal frequencies to rise, but the general shapes and the order of the modes do not change. This boundary condition has a limited influence on the modal dynamics since the bone's cross-section expands toward the joint and decreases the soft-tissue thickness. The observed insensitivity also supports the simplification of further anatomical details (e.g., tendons) around the interphalangeal joint. However, one should note that a real finger joint might exhibit additional damping mechanisms.  
The influence of the presented natural vibration modes depends on the application of interest. For example, the modes that involve normal motion at the skin surface (e.g., the fifth and sixth modes) probably play an essential role in scenarios that provide normal excitation, such as the ultrasonic cues provided during mid-air haptics. The modes that exhibit tangential skin motion are likely to have a major impact in situations where the finger is exposed to lateral vibrations of a contacted object. The first and second modes may be particularly important since the bulk in-phase motion of the skin can cause significant resonances. However, the modal attributes may vary due to prestressing effects, which depend on the magnitude of the pressing force.