Modeling pneumatic bubble displacements with membrane theory

We have fabricated a small array of pneumatically actuated bubbles intended as a tactile stimulator that can be worn on the fingertip. While device operation is quite promising, we notice that the perceptibility of stimuli is a strong function of how much force a user applies. We surmise that membrane deflection is affected not only by valve-controlled pressure but by finger-applied pressure as well. With the eventual aim of developing an analytical model of membrane deflection as a function of multiple types of load, applied from both sides of the membrane and including varying contact area between membrane substrate and skin, we are developing expanded membrane equilibrium equations. In this paper, we present equilibrium equations suitable for obtaining the displacement of a thin, circular membrane of polydimethylsiloxane (PDMS) subjected to a uniform pressure on one side. A comparison of numerical solutions to an analytic solution available for linearized equations confirms operation of the numerical solver and singularity handling. Initial results from measurements made under an inverted microscope are also presented and compared to theory.