One-to-One and Three-to-One Internal Resonances in MEMS Shallow Arches

The nonlinear modal coupling between the vibration modes of an arch shaped microstructure is an interesting phenomenon, which may have desirable features for numerous applications, such as vibration-based energy harvesters. This works presents an investigation into the potential nonlinear internal resonances of a Micro electro mechanical systems MEMS arch when excited by static (DC) and dynamic (AC) electric forces. The influences of initial rise and mid-plane stretching are considered in the governing equation. The cases of one-to-one and three-to-one internal resonances between the first and second modes and between the first and third modes are studied using the method of multiple scales and the direct attack of the partial differential equation of motion. It is interestingly shown that for distinct domain of actuation voltages, there exist three-to-one internal resonance between the first and third symmetric modes and one-to-one internal resonance case between the first symmetric and the second antisymmetric mode. These results can shed light on such interactions that are commonly found on micro and nano structures, such as carbon nano tubes.