The vibrational characteristics of a cantilever beam are well known to strongly depend on the fluid in which the beam is immersed. In this paper, we present a detailed theoretical analysis of the frequency response of a cantilever beam, that is immersed in a viscous fluid and excited by an arbitrary driving force. Due to its practical importance in application to the atomic force microscope (AFM), we consider in detail the special case of a cantilever beam that is excited by a thermal driving force. This will incorporate the presentation of explicit analytical formulae and numerical results, which will be of value to the users and designers of AFM cantilever beams.

Frequency response of cantilever beams immersed in viscous fluids with applications to the atomic force microscope

We have designed and tested a family of silicon nitride cantilevers ranging in length from 23 to 203 μm. For each, we measured the frequency spectrum of thermal motion in air and water. Spring constants derived from thermal motion data agreed fairly well with the added mass method; these and the resonant frequencies showed the expected increase with decreasing cantilever length. The effective cantilever density (calculated from the resonant frequencies) was 5.0 g/cm3, substantially affected by the mass of the reflective gold coating. In water, resonant frequencies were 2 to 5 times lower and damping was 9 to 24 times higher than in air. Thermal motion at the resonant frequency, a measure of noise in tapping mode atomic force microscopy, decreased about two orders of magnitude from the longest to the shortest cantilever. The advantages of the high resonant frequency and low noise of a short (30 μm) cantilever were demonstrated in tapping mode imaging of a protein sample in buffer. Low‐noise images were tak...

Short cantilevers for atomic force microscopy

Detailed measurements of the frequency responses of a series of rectangular atomic force microscope (AFM) cantilever beams, immersed in a range of fluids, have been performed to test the validity and accuracy of the recent theoretical model of Sader [J. Appl. Phys. 84, 64 (1998)]. This theoretical model gives the frequency response of a cantilever beam, that is immersed in a viscous fluid and excited by an arbitrary driving force. Very good agreement between experimental measurements and theoretical calculations is found for all fluids considered. Furthermore, a critical assessment of the well-known inviscid model is presented, which demonstrates that this model is not applicable to AFM cantilever beams in general.

https://researchbank.swinburne.edu.au/file/69c82204-bd13-4c5e-8a3e-a49cbffd64e3/1/PDF (Published version).pdf

Experimental validation of theoretical models for the frequency response of atomic force microscope cantilever beams immersed in fluids

Linear vibration analysis of cantilever plates partially submerged in fluid

This paper is concerned with the virtual mass effect on the natural frequencies and mode shapes of rectangular plates due to the presence of the water on one side of the plate. The approximate formula, which mainly depends on the so-called nondimensionalized added virtual mass incremental factor, can be used to estimate natural frequencies in water from natural frequencies in vacuo. However, the approximate formula is valid only when the wet mode shapes are almost the same as the one in vacuo. Moreover, the nondimensionalized added virtual mass incremental factor is in general a function of geometry, material properties of the plate and mostly boundary conditions of the plate and water domain. In this paper, the added virtual mass incremental factors for rectangular plates are obtained using the Rayleigh-Ritz method combined with the Green function method. Two cases of interfacing boundary conditions, which are free-surface and rigid-wall conditions, and two cases of plate boundary conditions, simply supported and clamped cases, are considered in this paper. It is found that the theoretical results match the experimental results. To investigate the validity of the approximate formula, the exact natural frequencies and mode shapes in water are calculated by means of the virtual added mass matrix. It is found that the approximate formula predicts lower natural frequencies in water with a very good accuracy.

M.A. Veluswami

Papers

Vibrations of stepped orthotropic square plates

Damped vibrations of orthotropic square plates having variable thickness (linear variation)