|Title:||Split beam method for determining mode shapes of cantilever beams under multiple loading conditions||Authors:||Li, JJ
|Keywords:||Split beam method; Euler-Bernoulli beam equation; mode shape; inhomogeneous eigenvalue problem; atomic force microscope; DYNAMIC-ANALYSIS; BERNOULLI; ROBUSTNESS; ELEMENT; AFM||Issue Date:||2022||Publisher:||SAGE PUBLICATIONS LTD||Journal Volume:||28||Journal Issue:||21-22||Start page/Pages:||3207||Source:||JOURNAL OF VIBRATION AND CONTROL||Abstract:||
We have developed a general method, dubbed the split beam method, to solve Euler–Bernoulli equations for cantilever beams under multiple loading conditions. This kind of problem is, in general, a difficult inhomogeneous eigenvalue problem. The new idea is to split the original beam into two (or more) effective beams, each of which corresponds to one specific load and bears its own Young’s modulus. The mode shape of the original beam can be obtained by linearly superposing those of the effective beams. We apply the split beam method to simulating mechanical responses of an atomic force microscope probe in the “dynamical” operation mode, under which there are a stabilizing force at the positioner and a point-contact force at the tip. Compared with traditional analytical or numerical methods, the split beam method uses only a few number of basis functions from each effective beam, so a very fast convergence rate is observed in solving both the resonance frequencies and the mode shapes at the same time. Moreover, by examining the superposition coefficients, the split beam method provides a physical insight into the relative contribution of an individual load on the beam.
|Appears in Collections:||應用力學研究所|
Items in DSpace are protected by copyright, with all rights reserved, unless otherwise indicated.