This paper describes a physical and mathematical model of a new device called nanomechanical transistor able to control a current through a small pilot voltage. The novelty of the device relies in its mechanical working principle where nanopillars vibrate between electrodes providing a mechanical shuttling mechanism for electric charge transportation. The dynamics of the investigated system involves electromechanical phenomena with the addition of quantum effects due to the charge tunnelling appearing in contactless charge transfer from pillars to electrodes. The theory here presented has a general character and is an attempt to build a general model for those multiphysics phenomena (electrical-mechanical with presence of quantum effects) frequently met in nanotechnology that do not yet fit into a systematic frame.