Effect of intramodal and intermodal nonlinearities on the flexural resonant frequencies of cantilevered beams

Sensing applications that utilize nanomechanical resonators require careful control of nonlinear effects in their eigenmodes to ensure robust measurement. While the effect of intra- and intermodal nonlinearities on the resonant frequencies of doubly clamped elastic beams have been widely studied using theory and experiment, commensurate studies on cantilevered beams are limited in comparison. Here, we present such a detailed study that includes an explicit and simple formula for the flexural resonant frequencies of slender cantilevered beams that accounts for intra- and intermodal nonlinearities. Using this general theory, numerical results for the modal nonlinear coefficients are tabulated for the first 20 flexural eigenmodes of cantilevered beams possessing uniform cross sections. The accuracy of this theory, and the effect of cantilever aspect ratio (length/width) on these nonlinear coefficients, is explored using high-accuracy laser Doppler vibrometry experiments. We anticipate that these results will find utility in single- and multimode applications, where the effect of finite oscillation amplitude on the cantilever resonant frequencies can significantly impact measurement design and interpretation.