Erhan Budak

Analytical Prediction of Stability Lobes in Milling

Abstract A new method for the analytical prediction of stability lobes in milling is presented. The stability model requires transfer functions of the structure at the cutter - workpiece contact zone, static cutting force coefficients, radial immersion and the number of teeth on the cutter. Time varying dynamic cutting force coefficients are approximated by their Fourier series components, and the chatter free axial depth of cuts and spindle speeds are calculated directly from the proposed set of linear analytic expressions without any digital iteration. Analytically predicted stability lobes are compared with the lobes generated by time domain and other numerical methods available in the literature.

Prediction of Milling Force Coefficients From Orthogonal Cutting Data

The mechanistic and unified mechanics of cutting approaches to the prediction of forces in milling operations are briefly described and compared. The mechanistic approach is shown to depend on milling force coefficients determined from milling tests for each cutter geometry. By contrast the unified mechanics of cutting approach relies on an experimentally determined orthogonal cutting data base (i.e., shear angle, friction coefficient and shear stress), incorporating the tool geometrical variables, and milling models based on a generic oblique cutting analysis. It is shown that the milling force coefficients for all force components and cutter geometrical designs can be predicted from an orthogonal cutting data base and the generic oblique cutting analysis for use in the predictive mechanistic milling models. This method eliminates the need for the experimental calibration of each milling cutter geometry for the mechanistic approach to force prediction and can be applied to more complex cutter designs. This method of milling force coefficient prediction has been experimentally verified when milling Ti 6 Al 4 V titanium alloy for a range of chatter, eccentricity and run-out free cutting conditions and cutter geometrical specifications.

Analytical prediction of chatter stability in milling-Part I: General formulation

A new analytical method of chatter stability prediction in milling is presented. A general formulation for the dynamic milling system is developed by modeling the cutter and workpiece as multi-degree-of-freedom structures. The dynamic interaction between the milling cutter and workpiece is modeled considering the varying dynamics in the axial direction. The dynamic milling forces are governed by a system of periodic differential equations with delay whose stability analysis leads to an analytical relation for chatter stability limit in milling. The model can be used to determine the chatter free axial and radial depth of cuts without resorting to time domain simulations.

Analytical Prediction of Chatter Stability in Milling—Part II: Application of the General Formulation to Common Milling Systems

The general formulation for the milling chatter prediction developed in Part I of the paper is applied to common milling systems. Three cases are considered: a workpiece with single-degree-of-freedom, a face milling cutter with two-degree-of-freedom, and peripheral milling ofa cantilevered thin web. The general milling stability formulation is further simplified for the less complicated models. For each case, an analytical expression which explicitly relate the chatter limit to the milling conditions and tool-workpiece dynamics are derived. The analytical predictions are compared with numerical and time domain solutions proposed by previous research. It is shown that the proposed method can accurately predict the chatter limits in milling and thus eliminates the time consuming numerical solutions.

Sources of nonlinearities, chatter generation and suppression in metal cutting

The mechanics of chip formation has been revisited in order to understand functional relationships between the process and the technological parameters. This has led to the necessity of considering the chip–formation process as highly nonlinear, with complex interrelations between its dynamics and thermodynamics. In this paper a critical review of the state of the art of modelling and the experimental investigations is outlined with a view to how the nonlinear dynamics perception can help to capture the major phenomena causing instabilities (chatter) in machining operations. The paper is concluded with a case study, where stability of a milling process is investigated in detail, using an analytical model which results in an explicit relation for the stability limit. The model is very practical for the generation of the stability lobe diagrams, which is time consuming when using numerical methods. The extension of the model to the stability analysis of variable pitch cutting tools is also given. The application and verification of the method are demonstrated by several examples.

Analytical stability prediction and design of variable pitch cutters

Art analytical prediction of stability lobes for milling cutters with variable pitch angles is presented. The method requires cutting constants, number of teeth, and transfer function of cutter mounted on the machine tool as inputs to a chatter stability expression. The stability is formulated by transforming time varying directional cutting constants into time invariant constants. Constant regenerative time delay in uniform cutters is transformed into nonuniform multiple regenerative time delay for variable pitch cutters. The chatter free axial depth of cut is solved from the eigenvalues of stability expression, whereas the spindle speed is identified from regenerative phase delays. The proposed technique has been verified with extensive cutting tests and time domain simulations. The practical use of the analytical solution is demonstrated by an optimal tooth spacing design application which increases the chatter free depth of cuts significantly.

Modeling and avoidance of static form errors in peripheral milling of plates

Abstract Peripheral milling of very flexible, cantilevered plates with slender end mills is modeled. The plate has varying structural properties in the axial and feed directions due to metal removal. The cutter is modeled as a cantilevered continuous elastic beam with flexible clamping in the collet. The plate structural properties are updated using the finite element technique as the cutter and plate interact at changing contact zones along the feed direction. The analytical cutting force and surface generation model considers partial separation of tool and plate structures due to static displacements. The experimentally verified model predicts the cutting forces and dimensional surface error on the plate. The model identifies the required feed variation along the plate in order to keep the static form errors within the specified tolerance of the part. The model is developed to improve dimensional accuracy in peripheral milling of very flexible aerospace components.

An Analytical Design Method for Milling Cutters With Nonconstant Pitch to Increase Stability, Part I: Theory

Chatter vibrations result in reduced productivity, poor surface finish and decreased cutting tool life. Milling cutters with nonconstant pitch angles can be very effective in improving stability against chatter. In this paper, an analytical stability model and a design method are presented for nonconstant pitch cutters. An explicit relation is obtained between the stability limit and the pitch variation which leads to a simple equation for determination of optimal pitch angles. A certain pitch variation is effective for limited frequency and speed ranges which are also predicted by the model. The improved stability, productivity and surface finish are demonstrated by several examples in the second part of the paper.

Analytical prediction of stability lobes in ball end milling

The paper presents an analytical method to predict stability lobes in ball end milling. Analytical expressions are based on the dynamics of ball end milling with regeneration in the uncut chip thickness, time varying directional factors and the interaction with the machine tool structure. The cutting force coefficients are derived from orthogonal cutting data base using oblique transformation method. The influence ofcutting coefficients on the stability is investigated. A computationally efficient, an equivalent average cutting force coefficient method is developed for ball end milling. The prediction of stability lobes for ball end milling is reduced to the solution of a simple quadratic equation. The analytical results agree well with the experiments and the computationally expensive and complex numerical time domain simulations.

Peripheral milling conditions for improved dimensional accuracy

Abstract Peripheral milling with flexible helical cutters is analyzed and modelled. The end mill is modelled as a cantilevered beam clamped to the collet with linear springs. Cutting forces and resulting tool deflection marks on the surface are analytically expressed. It is shown that by proper selection of cutting conditions, the material removal rate can be increased significantly without sacrificing the dimensional accuracy of the finished product. A method of identifying optimal feedrate and width of cut for given cutter dimensions and cutting constants is presented with experimental verification.