# Free and Forced Vibration Modes of the Human Fingertip
## 1 Introduction  
Touch sensation occurs with the deformation of the skin under external forces and subsequent stimulation of the mechanoreceptors within the tissue. In the human hand, the fingertips exhibit the highest tactile sensitivity as they are densely innervated with sensory neurons [1; 2]. Hence, the deformation mechanics of the fingertips has received considerable attention in scientific efforts to understand touch sensation [3; 4; 5] as well as in the design of interfaces that provide haptic feedback [6; 7; 8]. The perception of dynamic forces applied on the fingertip significantly differs from sensation under static conditions. Human skin is sensitive to vibrations up to 1000 Hz, with the peak sensitivity observed around 250 Hz [9]. Many researchers have shown that the mechanical properties of the skin play an important role in its frequency-dependent sensitivity. In the case of fingertips, their intricate geometry and multi-layered internal anatomy restrict the accuracy with which analytical methods can predict deformations under dynamic excitation. Hence, many studies have utilized computational simulations to analyze the dynamic deformation mechanics of the human finger. For instance, [PERSON] et al. [10] performed two-dimensional (2D) finite element (FE) analyses to simulate the mechanical response of the human fingertip periodically pressing onto flat surfaces. Their results endorsed the suitability of FE simulations in analyzing finger deformation mechanics since their predictions on contact force histories and fingertip-surface separation periods agreed well with the reported experimental observations. Similarly, [PERSON] et al. [11] examined the dynamic strains in a 2D finite element modelof the fingertip section exposed to harmonic line indentation. Their results matched well with known experimental findings: they showed that excitation frequencies ranging from 63 to 250 Hz produce large dynamic strains in the deep finger tissue that accommodates the high-frequency mechanoreceptors, while frequencies lower than 31.5 Hz cause high strain values within the shallow layers that harbor the low-frequency mechanoreceptors. Later, [PERSON] et al. [12] measured the propagation of cutaneous vibrations in the hand during interactions with different objects. That study demonstrated that the spatial-temporal pattern of vibrations varies systematically across touch interactions, indicating the possibility of decoding the type of interaction from the recorded vibration data. Recently, [PERSON] et al. [13] obtained natural frequencies and mode shapes for a simple FE model in the form of a layered rectangular prism, which represents a slab cut out of the skin. The results of that work show that changes in skin elasticity can notably influence the resonance patterns and corresponding frequencies. The deformation of continuum bodies under dynamic loads is directly influenced by the object's natural frequencies and mode shapes. However, these modal features have remained unexplored for the fingertip, despite the high interest in the dynamic analysis of human tissue. To address this gap, we developed DigiTip, a detailed 3D finite element model of the human fingertip based carefully on prior studies of fingertip anatomy and biomechanics. We used our model first to determine the fingertip's natural modes with associated frequencies up to about 250 Hz. Seven modes were identified for the developed model, with several occurring near the frequency of peak sensitivity. Then, we performed frequency-response analyses to predict the forced vibration responses when distributed harmonic forces were applied at the fingerprint centroid in three principal directions. These analyses show that certain free vibration modes are excited significantly more efficiently than the others for the analyzed load cases. The results offer insights into the probable dynamic behavior of the fingertip in haptic applications involving artificial vibratory feedback or light sliding contact with undulating surfaces.