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

The determination of the spring constants of atomic force microscope (AFM) cantilevers is of fundamental importance to users of the AFM. In this paper, a fast and nondestructive method for the evaluation of the spring constant which relies solely on the determination of the unloaded resonant frequency of the cantilever, a knowledge of its density or mass, and its dimensions is proposed. This is in contrast to the method of Cleveland et al. [Rev. Sci. Instrum. 64, 403 (1993)], which requires the attachment of masses to the cantilever in the determination of the spring constant. A number of factors which can influence the resonant frequency are examined, in particular (i) gold coating, which can result in a dramatic variation in the resonant frequency, for which a theoretical account is presented and (ii) air damping which, it is found, leads to a shift of -4% in the resonant frequency down on its value in a vacuum. Furthermore, the point of load on the cantilever is found to be extremely important, since a small variation in the load point can lead to a dramatic variation in the spring constant. Theoretical results that account for this variation, which, it is believed will be of great practical value to the users of the AFM, are given. © 1995 American Institute of Physics.

https://www.ampc.ms.unimelb.edu.au/afm/ref_files/Sader_RSI_1995a.pdf

Method for the calibration of atomic force microscope cantilevers

Preface Introduction 1. Fluid field equations and modal analysis in rigid containers 2. Linear forced sloshing 3. Viscous damping and sloshing suppression devices 4. Weakly nonlinear lateral sloshing 5. Equivalent mechanical models 6. Parametric sloshing (Faraday's waves) 7. Dynamics of liquid sloshing impact 8. Linear interaction of liquid sloshing with elastic containers 9. Nonlinear interaction under external and parametric excitations 10. Interactions with support structures and tuned sloshing absorbers 11. Dynamics of rotating fluids 12. Microgravity sloshing dynamics Bibliography Index.

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Liquid Sloshing Dynamics

The field of cantilever-based sensing emerged in the mid-1990s and is today a well-known technology for label-free sensing which holds promise as a technique for cheap, portable, sensitive and highly parallel analysis systems. The research in sensor realization as well as sensor applications has increased significantly over the past 10 years. In this review we will present the basic modes of operation in cantilever-like micromechanical sensors and discuss optical and electrical means for signal transduction. The fundamental processes for realizing miniaturized cantilevers are described with focus on silicon- and polymer-based technologies. Examples of recent sensor applications are given covering such diverse fields as drug discovery, food diagnostics, material characterizations and explosives detection.

http://iopscience.iop.org/article/10.1088/0034-4885/74/3/036101/pdf

Cantilever-like micromechanical sensors

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

: The presence of the liquid free surface has a significant effect upon the dynamic plate characteristics (frequencies) only when the surface is less than about one-half span length from the plate. The effect of the liquid free surface on the resonant frequency of a vertical surface peircing plate is highly dependent upon the relationship between the depth of immersion of the plate and the mode of vibration. The overall damping of cantilever plate vibration is increased significantly in water as compared with air. The results of this study would indicate that for those problem areas in the field of Naval dynamics wherein elastic plate vibrations are involved, and for which predictions of resonant frequencies and damping factors are derived, knowledge is either inadequate or unavailable. For cantilever plates, the present results should enable one to predict resonant frequencies in practical applications with some degree of confidence; but damping factors have been obtained only to within order-of-magnitude and to show trends. Clearly, similar information is very much needed for other plate geometries (i.e., skewed) and support conditions.

Elastic Vibration Characteristics of Cantilever Plates in Water

A mechanical model for representation of linear fuel sloshing with damping is derived by comparison and extension of available theoretical results in both translation and pitching without damping. Available experimental data indicates tha t the mechanical model representation is satisfactory for predicting total force response. A simple interpretation of the formulation also makes it possible to apply the mechanical model to the case of axial thrust acting in addition to the vertical gravitational field.

Representation of Fuel Sloshing in Cylindrical Tanks by an Equivalent Mechanical Model

Linear shell theory for nonlinear transverse coupled vibrations of partially filled circular cylindrical elastic tank

A theory for nonlinear transverse vibrations of a partially filled elastic tank.

Rigid top mass effect on longitudinal response of model vehicle propellant tank, determining natural frequencies and forced asymmetric response

Wen-Hwa Chu

Papers

Elastic Vibration Characteristics of Cantilever Plates in Water

Representation of Fuel Sloshing in Cylindrical Tanks by an Equivalent Mechanical Model

A theory for nonlinear transverse vibrations of a partially filled elastic tank.

Influence of a rigid top mass on the response of a pressurized cylinder containing liquid.

Some studies of liquid rotation and vortexing in rocket propellant tanks