On the Origin of the Nonclassical Softening Nonlinearity in MEMS/NEMS Cantilevers

Cantilevered configurations of thin beams are used in numerous conventional and emerging applications of mechanical, aerospace, and civil engineering, as well as material science. In many resonant scenarios, nonlinearities are inevitably pronounced under moderate to large intensity excitations. Other than dissipative nonlinearities due to internal and external damping mechanisms, and in the absence of material nonlinearities such as those of piezoelectricity, two types of nonlinearities are pronounced for large amplitude dynamics of Euler-Bernoulli cantilevers in the first bending mode: stiffness hardening due to nonlinear curvature and inertial softening. While these two counteracting nonlinearities are both of cubic order, it can be shown theoretically that the geometric hardening always dominates inertial softening for the response of the first bending mode, and therefore one does not expect softening in the resulting dynamics. Most of the existing experiments in conventional structures are aligned with this theoretical expectation. Recent efforts in NEMS cantilevers resulted in experimental data with a softening nonlinearity instead of the classical overall hardening for the first bending mode. The perplexing results made some authors suggest that the Euler-Bernoulli theory was not sufficient to capture the resulting dynamics, and that a new theory would be required. In the present work, we hypothesize that the observed softening effect in the fundamental resonance may be a result of the asymmetric boundary condition of the NEMS cantilever (due to its fabrication) at the clamped end such that the cantilever experiences two different boundary conditions in the two halves of the oscillation cycle. We show both theoretically and experimentally that such uneven boundary conditions indeed result in softening behavior in the first bending mode, while the evenly clamped case yields the expected hardening.