^{1}and John E. Sader

^{1,a)}

### Abstract

Theoretical models for the frequency response of a cantilever beam immersed in a viscous fluid commonly assume that the fluid is unbounded. Experimental measurements show, however, that proximity to a surface can significantly affect the frequency response of a cantilever beam. In this article, we rigorously calculate the effect of a nearby surface on the frequency response of a cantilever beam immersed in a viscous fluid, and present a general theoretical model. Due to its practical relevance to applications of the atomic force microscope and microelectromechanical systems, detailed results are presented for cantilever beams with rectangular geometries executing flexural and torsional oscillations. It is found that dissipative loading in the fluid is primarily responsible for the observed variation in the frequency response, whereas inertial loading exerts a relatively weak influence.

This research was supported by the Particulate Fluids processing Center of the Australian Research Council and the Australian Research Council Grants Scheme. One of the authors (C.P.G.) gratefully acknowledges the support of an Australian Postgraduate Award.

I. INTRODUCTION

II. THEORY

A. Flexural frequency response

B. Torsional frequency response

III. RESULTS AND DISCUSSION

A. Flexural frequency response

B. Torsional frequency response

C. Complete frequency response

IV. CONCLUSIONS

### Key Topics

- Viscosity
- 29.0
- Structural beam vibrations
- 22.0
- Atomic force microscopy
- 15.0
- Atomic and molecular beams
- 12.0
- Hydrodynamics
- 11.0

## Figures

Rectangular cantilever beam a distance above a surface. The origin of the coordinate system is at the center of mass at the clamped end. Thickness of the beam is .

Rectangular cantilever beam a distance above a surface. The origin of the coordinate system is at the center of mass at the clamped end. Thickness of the beam is .

Normalized thermal spectra of flexural vibration for . (a) Fundamental mode in gas with , . Quality factors, obtained using Eq. (9), are for , respectively; (b) Fundamental mode in liquid with , . for all resonance peaks.

Normalized thermal spectra of flexural vibration for . (a) Fundamental mode in gas with , . Quality factors, obtained using Eq. (9), are for , respectively; (b) Fundamental mode in liquid with , . for all resonance peaks.

Peak and resonant frequencies of fundamental flexural resonance for immersion in gas. ; . (a) Peak frequency relative to frequency in vacuum . (b) Resonant frequency in the absence of dissipative effects in the fluid relative to frequency in vacuum . The gray area represents the region enclosed by the and the curves in (a).

Peak and resonant frequencies of fundamental flexural resonance for immersion in gas. ; . (a) Peak frequency relative to frequency in vacuum . (b) Resonant frequency in the absence of dissipative effects in the fluid relative to frequency in vacuum . The gray area represents the region enclosed by the and the curves in (a).

Peak and resonant frequencies of fundamental flexural resonance for immersion in liquid ; . (a) Peak frequency relative to frequency in vacuum . (b) Resonant frequency in the absence of dissipative effects in the fluid relative to frequency in vacuum . The gray area represents the region enclosed by the and the curves in (a).

Peak and resonant frequencies of fundamental flexural resonance for immersion in liquid ; . (a) Peak frequency relative to frequency in vacuum . (b) Resonant frequency in the absence of dissipative effects in the fluid relative to frequency in vacuum . The gray area represents the region enclosed by the and the curves in (a).

Quality factor of the fundamental mode of flexural vibration for . (a) Gas, and (b) liquid, .

Quality factor of the fundamental mode of flexural vibration for . (a) Gas, and (b) liquid, .

Normalized thermal spectra of torsional vibration for . (a) Fundamental mode in gas with , . Quality factors , obtained using Eq. (17), are for , respectively; (b) Fundamental mode in liquid with , . Note that in both instances, the thermal spectra lies atop the spectra.

Normalized thermal spectra of torsional vibration for . (a) Fundamental mode in gas with , . Quality factors , obtained using Eq. (17), are for , respectively; (b) Fundamental mode in liquid with , . Note that in both instances, the thermal spectra lies atop the spectra.

Peak and resonant frequencies of fundamental torsional resonance for immersion in gas. ; . (a) Peak frequency relative to frequency in vacuum . (b) Resonant frequency in the absence of dissipative effects in the fluid relative to frequency in vacuum . The gray area represents the region enclosed by the and the curves in (a).

Peak and resonant frequencies of fundamental torsional resonance for immersion in gas. ; . (a) Peak frequency relative to frequency in vacuum . (b) Resonant frequency in the absence of dissipative effects in the fluid relative to frequency in vacuum . The gray area represents the region enclosed by the and the curves in (a).

Peak and resonant frequencies of fundamental torsional resonance for immersion in liquid. ; . (a) Peak frequency relative to frequency in vacuum . (b) Resonant frequency in the absence of dissipative effects in the fluid relative to frequency in vacuum . The gray area represents the region enclosed by the and the curves in (a).

Peak and resonant frequencies of fundamental torsional resonance for immersion in liquid. ; . (a) Peak frequency relative to frequency in vacuum . (b) Resonant frequency in the absence of dissipative effects in the fluid relative to frequency in vacuum . The gray area represents the region enclosed by the and the curves in (a).

Quality factor of the fundamental mode of torsional vibration for . (a) Gas, and (b) liquid, .

Quality factor of the fundamental mode of torsional vibration for . (a) Gas, and (b) liquid, .

Ratio of peak noise levels of flexural and torsional vibration, for (solid line); (long dash, long space); (long dash, short space); (long dash, short dash); (short dash). (a) Gas: , and (b) liquid: , .

Ratio of peak noise levels of flexural and torsional vibration, for (solid line); (long dash, long space); (long dash, short space); (long dash, short dash); (short dash). (a) Gas: , and (b) liquid: , .

## Tables

Coefficients of , Eq. (A2a), for various values of .

Coefficients of , Eq. (A2a), for various values of .

Coefficients of , Eq. (A2b), for various values of .

Coefficients of , Eq. (A2b), for various values of .

Coefficients of , Eq. (A4a), for various values of .

Coefficients of , Eq. (A4a), for various values of .

Coefficients of , Eq. (A4b), for various values of .

Coefficients of , Eq. (A4b), for various values of .

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