Cantilever resonators serve as a building block in many sensors such as atomic force microscopes, sensors of biological substances, chemicals and more. In these sensors identification of a frequency shift indicates the effect of the measurand. However, frequency shifts may also be caused by changes in the amplitude of vibrations in resonating cantilevers. Therefore, accurate modeling of mechanical nonlinearities in the free vibration response of cantilevers is of great importance. Within the framework of the linear theory of elasticity, the transverse free vibration response of a vibrating cantilever is modeled using the Euler-Bernoulli beam theory. This analytic approach neglects nonlinear effects including the coupling of transverse and axial displacements. The predictions of the linear theory are adequate for low-amplitude free vibrations, but they cannot predict the response when the amplitude of vibrations is large.
The nonlinear free vibration response of cantilever beams has been extensively investigated, and surprisingly there are conflicting conclusions in literature. Some studies suggested that the frequency of free vibrations increases with increasing amplitude of vibrations (i.e., the response is stiffening) while other studies showed that the frequency of free vibrations decreases with increasing amplitude (i.e., the response is softening).
In this seminar, two models for analyzing the free vibration response are presented. Both methods are significantly more computationally efficient relative to transient dynamic simulations of the complete 3D structure.
