Flexural Vibration Of An Atomic Force Microscope Cantilever

Atomic Force Microscope (AFM), highlighted to show the importance of extending and enhancing microscopic observation techniques as a function of the human visual senses, is a molecular and atomic microscopic tool to observe the non-conductive material which is developed on the basis of the scanning tunneling microscope. In contrast to other existing microscopic tools, AFM, because of its high resolution, simple sample preparation, easy operation and other characteristics, is greatly concerned. It has been a major role in the life sciences, materials science and other fields, which greatly promoted the development of nanotechnology and promoted human into the nanometer era.Functionally graded materials (FGMs) are the inhomogeneous composite materials whose composing and structural constitution element is continuously gradient from one side to another, leading to the continuous gradient of the material properties and functions. This continuous gradient of the material component volume fraction will result in the continuous gradient of structural performance, thus eliminating the mutation of the material properties, and avoiding the stress concentration. In this thesis, FGMs are employed in the AFM micro-cantilever for the analysis of the vibration performance.Both Euler beam theory and Timoshenko beam theory will be used in the analysis of the AFM micro-cantilever based on the modified couple stress theory. The modified couple stress theory is a non-classical continuum theory, which includes an additional material parameters to describe the size effect. By using the Hamilton’s principle, the governing equation of motion and the boundary conditions are derived. The equation is solved by using the analytical method or numerical method for the natural frequencies, sensitivity and mode shapes. The effects of the sample surface contact stiffness, length scale parameter, gradient index and location of the sensor tip on the flexural vibration characteristics of the AFM cantilever are discussed. The main contents of this paper can be divided into the following two parts:(1) Based on the Euler beam theory, the effects of the sample surface contact stiffness, length scale parameter and gradient index on the flexural vibration characteristics of the AFM cantilever that is made of either homogeneous material or functionally graded material are discussed.(2) Based on the Timoshenko beam theory, the effects of the sample surface contact stiffness, length scale parameter, gradient index and location of the sensor tip on the flexural vibration characteristics of the AFM cantilever that is made of either homogeneous material or functionally graded material are discussed.In the theoretical analysis and mathematical derivation, the Hamilton principle is used in deriving the governing equations and the corresponding boundary conditions. The analytical method and numerical method (Differential Quadrature Method) are used to discretize the governing equations. The main conclusions contain:the size effect is significant when the thickness of the micro-cantilever has a similar value to the material length scale parameter; an increase in gradient index n diminishes both the frequency and sensitivity; the tip location has no effect on the frequency for low values of contact stiffness, whereas the effect is great for high values of contact stiffness; the effect of the length-to-thickness ratio on the frequency becomes more pronounced when the contact stiffness becomes larger; the effect of the length scale parameter on the mode shape is pronounced near the free end of micro-cantilevers.