Dynamic Analysis of Nanomachining with a Multi- Cracked AFM Cantilever

The atomic force microscope (AFM) has become an essential tool for imaging the surface topography of conductors and insulators on a microand nanoscale level [1]. In addition, AFM has also been powerful in nanomachining [2]. Cracks may occur in the AFM cantilever during the nanomachining experiments or in the fabrication of cantilever. The cantilever with cracks will affect its performance in use. In this article, we analyze the dynamic displacement of a multi-cracked AFM-based nanomachining using the modified couple stress theory [3]. Unlike the classical continuum theory, the modified couple stress theory includes one additional material length scale parameter revealing the micro-scale effect on the response of structures to estimate the size-dependent behaviours. Here the effect of the length scale parameter on the dynamic response of nanomachining with a multi-cracked AFM cantilever is investigated. Assume an AFM cantilever probe, as shown in Fig. 1, has multiple cracks located at abscissa Cj from the fixed end for j =1,2,3,...p, where p is the number of cracks, and in the vicinity of the crack. The rotational spring is used to simulate the crack. The cantilever will be divided into p+1 segments by the spring. The spring is assumed to be massless. The probe has the Young’s modulus E, shear modulus G, moment of inertia I, density ρ, uniform cross-section A, and length L. When the machining is in progress, the cantilever tip contacts with the specimen and induces a vertical reaction force, Fy(t), and a horizontal reaction force, Fx(t), which are a function of time t. The cutting system of the AFM cantilever can be modeled as a flexural vibration problem. The governing equation of transverse vibration can be given by [4]