Study of milling stability with Hertz contact stiffness of ball bearings

Abstract

This present work examines the stability and nonlinear responses of a spindle milling system supported by ball bearings. A shaft finite element based on Timoshenko beam theory is employed to model the spindle, and modal reduction method is therefore adopted for saving the numerical calculating time. The issues of evaluating the effects of the ball bearing Hertz contact stiffness are consequently addressed. It is found that suitable constant bearing stiffness can be adopted to replace the nonlinear nonsmooth Hertz stiffness in prediction of the critical cutting depth of the milling system in certain bearing configuration conditions. For the constant bearing stiffness can be obtained by experiment, this replacement will undoubtedly simplify the spindle-bearing milling system. But with the increase in the bearing clearance, the spindle milling system will present obvious nonlinear behaviors, and the nonlinear Hertz contact bearing stiffness will take over. Isolated islands of chatter vibration, which are induced by the nonlinear nonsmooth bearing Hertz stiffness, can be found exist in milling processes in large bearing clearance conditions.

Unstable islands and bifurcation analysis in a spindle milling system supported by ball bearings

“…From the former researches by Gao et al [41], in some bearing interference fitting conditions, the non-linear and non-smooth bearing contacts can be assumed as linear springs in predicting the milling stability regions. Also, the chatter islands cannot be found.…”

“…The discretization of elastic systems using the finite element method often results in a large number Fig. 1 The finite element model of the spindle system with milling cutter [41] of nodal coordinates which are prone to numerical integration difficulties. Modal reductions have been introduced to alleviate this problem.…”

“…The treatment of higher frequency modes leads to a great reduction in the calculating time. The modal reduction method has been adopted successfully by Gao et al [41] to reduce the spindle bearing system from 28 DOF to 10 DOF. The spindle finite element system can then be reduced effectively to study the stability and vibration bifurcations with numerical simulation methods.…”

“…For the convenience of identifying the effects of the non-linear non-smooth ball bearing Hertz contacts on the milling stability, the front and rear positions of the spindle shaft are supported by the same angular contact ball bearings [41]. Without considering the axial forces and moment loadings [29,32] of the spindle system, the angular contact ball bearings can be assumed as deep-groove ball bearings, and the dynamic problem of the angular contact ball bearing is made as one of two DOFs in this article.…”

Applicability of the Hertz Contact Theory to Rolling Rubber Cylinders Model

“…The Figure 2 shows that two cylinders can be replaced by one equivalent cylinder and a rigid planar, the radius of two cylinders are R1 and R2 [5], if:…”