Jiunn-Jyh Junz Wang

Convolution Analysis of Milling Force Pulsation

George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0405 This paper presents the establishment of a closed form expression for the dynamic forces as explicit functions of cutting parameters and tool/workpiece geometry in milling processes. Based on the existing local cutting force model, the generation of total cutting forces is formulated as the angular domain convolution of three cutting process component functions, namely the elementary cutting function, the chip width density function, and the tooth sequence function. The elemental cutting force function is related to the chip formation process in an elemental cutting area and it is characterized by the chip thickness variation, and radial cutting configuration. The chip width density function defines the chip width per unit cutter rotation along a cutter flute within the range of axial depth of cut_ The tooth sequence function represents the spacing between flutes as well as their cutting sequence as the cutter rotates. The analysis of cutting forces is extended into the Fourier domain by taking the frequency multiplication of the transforms of the three component functions. Fourier series coefficients of the cutting forces are shown to be explicit algebraic functions of various tool parameters and cutting conditions. Numerical simulation results are presented in the frequency domain to illustrate the effects of various process parameters. A series of end milling experiments are performed and their results discussed to validate the analytical model.

Mechanistic Modeling of Process Damping in Peripheral Milling

This paper extends analytical modeling of the milling process to include process damping effects. Two cutting mechanisms (shearing and plowing mechanisms) and two process damping effects (directional and magnitude effects) are included. The directional effect is related to vibration energy dissipation due to directional variation of cutter/workpiece relative motion. The magnitude effect is associated with change in force magnitude due to variation of rake angle and clearance angle. Process damping is summarized as containing these separate components: direction-shearing, direction-plowing, magnitude-shearing, and magnitude-plowing. The total force model including the process damping effect is obtained through convolution integration of the local forces. The analytical nature of this model makes it possible to determine two unknown dynamic cutting factors from measured vibration signal during milling. The effects of cutting conditions (cutting speed, feed, axial and radial depths of cut) on process damping are systematically examined. It is shown that total process damping increases with increasing feed, axial and radial depths of cut, but decreases with increasing cutting velocity. Predictions based on the analytical model are verified by experiment. Results show that plowing mechanism contributes more to the total damping effect than the shearing mechanism, and magnitude-plowing effect has by far the greatest influence on total damping.

Milling force convolution modeling for identification of cutter axis offset

Abstract This paper discusses the application of a convolution integral force model to the identification of the geometry of cutter axis offset in milling operations. This analysis builds upon the basis of linear decomposition of elemental local cutting forces into a nominal component and an offset-induced component. The convolution of each elemental local cutting force component with the chip width density in the context of cutter angular position provides an integral expression for the total cutting forces. By virtue of the convolution integration property, the total cutting forces in the frequency domain can be derived as closed-form functions of the cutting pressure constants, various cutting conditions, as well as the cutter offset geometry. Subsequently, the magnitude and phase angle of cutter axis offset are shown to be algebraic and explicit functions of the Fourier series coefficients of cutting forces at the spindle frequency. Following the theoretical analysis, experimental study is discussed to illustrate the implementation procedure for offset identification, and frequency domain data are presented to verify the analytical results. Potential industrial applications of this work include the real-time monitoring of dynamic cutter runout and the in-process compensation for the loss of tolerance or finish using automatic controls based on the feedback information of offset magnitude and phase angle.

Identification of shearing and ploughing cutting constants from average forces in ball-end milling

This paper presents an analytical model for the direct identification of global shearing and ploughing cutting constants from measured average cutting forces in ball-end milling. This model is based on the linear decomposition of elemental local cutting forces into a shearing component and a ploughing component. Then, a convolution integral approach is used to obtain the average cutting forces leading to a concise and explicit expression for the global shearing and ploughing cutting constants in terms of axial depth of cut, cutter radius and average milling forces. The model is verified by comparisons with an existing force model of variable cutting coefficients. Cutting constants are identified through milling experiments and the prediction of cutting forces from identified cutting constants coincides with the experimental measurements. A model for identifying the lumped shearing constants is obtained as a subset of the presented dual mechanism model. Experimental results indicate that a model with dual-mechanism cutting constants predicts the ball-end milling forces with better accuracy than the lumped force model.

An analytical force model with shearing and ploughing mechanisms for end milling

An analytical force model with both shearing and ploughing mechanisms is established for the end milling processes. The elemental forces are defined as the linear combination of shearing and ploughing forces in six cutting constants. The analytical model for the total milling forces in the angular and frequency domain are derived by convolution approach and Fourier transform respectively and are expressed as the superposition of the shearing force component and ploughing force component. This dual-mechanism model is analyzed and discussed in the frequency domain and compared with the lumped shear model. An expression is derived for identifying the cutting constants of the dual-mechanism model from the average milling forces. Explicit inclusion of ploughing force in the model is shown to result in better predictive accuracy and yields a linear force model with constant cutting coefficients. Experiments verify the accuracy and the frequency analysis of the dual-mechanism model and show that cutting constants for the dual-mechanism model are fairly independent of chip thickness.

Chip load kinematics in milling with radial cutter runout

This paper presents the analytical modeling of chip load and chip volume distribution in milling processes in the presence of cutter runout. The understanding of chip load kinematics has a strong bearing on the prediction of milling forces, on the assessment of resulting surface finish and tool vibration, and on the identification of runout for multi-toothed machining process monitoring and control. In this study a chip thickness expression is analytically established in terms of the number of flutes, the cutter offset location and the ratio of offset magnitude to feed per tooth. The effects of runout geometry, feed rate, and depths of cut on the overall chip generating action is discussed through the illustration of cutting regions and chip load maps. Explicit solutions for the entry and exit angles are formulated in the context of milling parameters and configuration. Experimental measurement of the resulting chip volumes from machining with an offset cutter is compared to an analytical model formulated from the chip thickness expression. Additionally, an average chip thickness prediction, based on the chip volume model in combination with the entry/exit angle solutions, is compared to data reported in the literature for validity assessment.

Identification of cutter offset in end milling without a prior knowledge of cutting coefficients

This paper presents a method for the identification of cutter offset through milling force without requiring the specific cutting coefficients to be known as priori. The total milling force in the presence of cutter offset is first obtained on the basis of dual cutting mechanisms, where the local force is comprised of a constant plowing force and a linear shearing force proportional to the chip load under the cutter offset. The total milling force is synthesized through convolution and is shown to be the sum of three force components: the nominal chip shearing force component, the plowing force components and the offset related shearing force component. Fourier analysis of these force components reveals the effects of offset geometry and yields formulas for the identification of offset geometry. The identification process requires only two cutting tests and the evaluation of two algebraic expressions; the shearing constants are found from the average forces of cutting tests and the offset geometry is identified from the force component at the spindle frequency. Through numerical simulation and experimental results, the efficacy of the identification method is demonstrated; the effects of feed per tooth and cutting depths on the accuracy of the model are investigated and criteria for the appropriate selection of these parameters are suggested.

Critical Depth of Cut and Specific Cutting Energy of a Microscribing Process for Hard and Brittle Materials

This paper investigated the scribing process characteristics of the hard and brittle materials including single crystal silicon, STV glass, and sapphire substrate. Under various cutting angles, major process characteristics are examined including the groove geometry, specific cutting energy, and critical depth of cut at the onset of ductile-to-brittle cutting transition. As the cutting depth increases, groove geometry clearly reveals the ductile-to-brittle transition from the plastic deformation to a brittle fracture state. The material size effect in the ductile region as well as the transition in scribing behavior is well reflected by change in the specific cutting energy. Further, it is shown that the change of specific cutting energy as a function of the cutting depth can serve as a criterion for estimating the critical depth of cut. Such estimated critical depth of cut is confirmed by measurement from a 3D confocal microscope. The critical depths of cut for these hard materials are found to be between 0.1 μm and 0.5 μm depending on the materials and cutting angles.

Estimation of in-process cutting constants in ball-end milling

Abstract Two methods are presented for the estimation of tangential, radial and axial cutting coefficients for the shearing and ploughing mechanisms from a single set of cutting forces in ball-end milling. These estimation methods are based upon the invertibility of the analytical milling force model, which considers both the shearing and the ploughing mechanisms by incorporating their respective cutting constants in the local force model. The periodic milling forces are established as the convolution integral of the differential local cutting forces and their Fourier coefficients are derived and expressed in a matrix expression as a linear function of the unknown cutting constants in terms of cutting conditions and cutter geometry. This linear expression thus leads to a systematic formulation of the estimation methods allowing the six unknown cutting constants to be determined from the measured milling forces. The first method uses the first harmonic forces as the source signal while the second method extracts the six cutting constants from the average force as well as the first harmonics. Limitations of both estimation methods are discussed. The consistency and accuracy of the estimated cutting constants are confirmed by the experimental results.

The Effect of Harmonic Force Components on Regenerative Stability in End Milling

In a systematic manner, this paper investigates the effects of harmonic force components on the regenerative stability of an end milling process. By representing the milling force pulsation in a Fourier series expansion form, the dynamic force components and the average forces due to bi-directional dynamic feed rates are both included in the generalized system dynamics formulation. In the resulting expression for the stability criterion, the spectral features of the milling forces are integrated with the dynamics of the structure, showing the significance or insignificance of the dynamic components of the milling forces in affecting the stability of the milling process. Key system parameters discussed include the magnitude of the average and harmonic forces, the cutter helix angle and the spindle speed. It is shown that a low helix angle and a smaller number of cutting flutes increase the effect of dynamic forces on the system stability. The significance of the harmonic forces is exemplified by the special cutting conditions where the average force becomes zero and the stability limits would be infinite as predicted by models using the average force alone. Improvements in the accuracy of stability lobes resulting from the inclusion of the dynamic forces and the validity of the presented model in general will be illustrated by numerical simulation and verified by experiments as well as by comparison with published results.Copyright