Size dependent vibrations of micro-end mill incorporating strain gradient elasticity theory

In this paper, a size-dependent formulation is presented for vibration analysis of micro-end mill tool. The formulation is developed based on the strain gradient elasticity theory in order to enhance the modeling capability of micro-size structures. Due to stubby geometry of micro-tool, the shear deformation and rotary inertia effects' are considered in the derivation of equations. Hence, based on the strain gradient Timoshenko beam theory, the extended Hamilton's principle is used to formulate a detailed dynamical model of the rotating micro-tool. The dynamical model includes a set of partial differential equations with gyroscopic coupling produced due to the spindle rotation. The governing equations of motion are reduced and solved by assumed mode model. To this end, an exact dynamic stiffness method is developed and employed to investigate the tool's free vibration characteristics such as structure mode shapes and natural frequencies. Also, the well-known Wittrick-Williams algorithm is utilized to guarantee that none of the natural frequencies are missed during the calculations. The mode shapes obtained from dynamic stiffness formulation can be utilized as base functions in the solution. Also, the proposed approach is applied to investigate the force vibration and chatter instability observed in micro-milling operations. (C) 2013 Elsevier Ltd. All rights reserved.