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

Harmonic and superharmonic liquid surface motions observed for longitudinally excited rigid cylindrical tank

Liquid surface oscillations in longitudinally excited rigid cylindrical containers.

A combined spherical pendulum and linear pendulum system is developed to produce the same dynamic in-line and cross-axis reaction weight as liquid exhibiting rotary liquid slosh. An approximate solution of the dynamic equations for the spherical pendulum is used along with cross-axis weight measurements from slosh experiments to develop mass and damping parameters for the spherical pendulum. Similar measurements of in-line weight responses are used to develop the corresponding parameters for the linear pendulum solution. A verification of the spherical pendulum solution is accomplished by comparing the approximate solutions with those predicted by a time-step integration of the nonlinear governing equations. It is found that a constantparameter combined system model cannot be used to represent typical rotary slosh over the entire frequency range within which this type of liquid motion response occurs. Therefore, a model is developed with some parameters constant, and others are allowed to vary as necessary to match the force data.

Validated spherical pendulum model for rotary liquid slosh

1 Yao, J. C., "Stability of a Circular Cylinder Under Dynamic Radial Pressure," ARS Journal, Vol. 31, No. 12, Dec. 1961, pp.1705-1708. 2 Coale, C. W. and Nagano, M., "Axisymmet ic Modes of an Elastic Cylindrical Hemispherical Tank Partly Filled w'th a Liquid," AIAA Symposium on Structural Dynamics and Aeroelasticity, American Institute of Aeronautics and Astronautics, New York, 1965, pp. 169-188. 3 Daniels, V. R., "Dynamics Aspects of Metal Bellows," The Shock and Vibration Bulletin, Bull. 35, Pt. 3, Jan. 1966. 4 Lamb, H., Hydrodynamics, 6th ed., Dover, New York, pp. 471-473. 5 Lindholm, U. S. et al., "Breathing Vibrations of a Circular Cylindrical Shell with an Internal Liquid," Journal of Aerospace Sciences, Vol. 29, No. 9, Sept. 1962, pp. 1052-1059. 6 Reiner, M., "Second Order Effects," Research Frontiers in Fluid Dynamics, edited by R. J. Seeger and G. Temple, pp. 208210. 7 Berry, J. G. and Reissner, E., "The Effect of an Internal Compressible Fluid Column on the Breathing Vibrations of a Thick, Pressurized, Cylindrical Shell," Journal of the Aeronautical Sciences, Vol. 25, May 1958, pp. 288-294. 8 McLachlan, N. W., Theory and Application of Mathieu Functions, 1st ed., Clarendon Press, Oxford, England, 1947. 9 Handbook of Math Functions with Formulas, Graphs and Mathematical Tables, Applied Mathematics Series 55, National Bureau of Standards, U.S. Dept. of Commerce.

Parametric oscillations of a longitudinally excited cylindrical shell containing liquid.

A compound pendulum model is developed to predict rotary slosh of liquid in a scale model Centaur propellant tank at low fill level. A portion of the liquid behaves as a spherical pendulum that experiences rotary motion throughout a frequency range below, at, and above first mode resonance. The remainder of the fluid behaves as an ordinary linear pendulum. Phasing of the rotary motion is such that it subtracts from the effects of normal slosh below resonance and adds to the effects above resonance. The compound pendulum experiences analogous responses to that observed experimentally for the liquid, including bivalued states and jump phenomena, in a range just above resonance. Its responses are used to predict the effective weight response of the liquid by assigning pendulum mass, frequency, and damping parameters based on experimental measurements. An especially important result is that a significant magnitude cross-axis force exists for the rotary motion. This force is predicted by the compound pendulum model, but it is not predicted by the usual linear theory models.

Daniel D. Kana

Papers

Elastic Vibration Characteristics of Cantilever Plates in Water

Liquid surface oscillations in longitudinally excited rigid cylindrical containers.

Validated spherical pendulum model for rotary liquid slosh

Parametric oscillations of a longitudinally excited cylindrical shell containing liquid.

A model for nonlinear rotary slosh in propellant tanks