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Geometric models of non-standard serrated end mills

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Abstract

Serrated end mills reduce process forces and improve chatter-free material removal rates. Improvements in cutting performance are governed mainly by the serration profile on these cutters. The geometric models of serration profiles are necessary to guide the design of improved cutters. Since these geometric models are usually not available a priori, this paper presents two methods to reconstruct geometric models from scanned measurements of eight different serrated cutter types available commercially. One representation is parametric-based, and another is NURBS-based. Reconstructed serration profiles are classified as the standard sinusoidal, circular, and trapezoidal profile types; and the non-standard semi-circular, circular-elliptical, semi-elliptical, inclined semi-circular; and the inclined circular types. For all eight profiles, the variation in local radius and irregular chip thickness distribution that are a characteristic of serrated cutters are captured by both approaches to approximate the geometry. Force models with the proposed geometric models as inputs are used to predict forces and those forces were experimentally validated. Validated forces confirm that the proposed geometric models are indeed correct. Comparing the cutting performance of all eight serrated cutters suggests that the circular and non-standard serrated end mills can preferentially reduce cutting forces as compared to the standard sinusoidal and/or trapezoidal profiles. However, in terms of the ratios of maximum to minimum peak resultant cutting force, we find that the standard sinusoidal profiled cutter outperforms the other four-fluted cutters, whereas the non-standard inclined circular profiled cutter retains its advantages over other three-fluted cutters.

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Abbreviations

a p :

Axial depth of cut

A :

Amplitude of the serration profile

c :

Total number of input cloud data points in the optimization problem

C(u):

NURBS curve having parameter u

C b :

Coordinates of the bth output NURBS point

d :

Total number of output NURBS points in the optimization problem

dFrta, i(z, t):

Differential cutting forces in the rta frame for the ith flute at the height z at time t

dFxyz, i(z, t):

Transformed differential forces in the xyz frame for the ith flute at the height z at time t

D :

Shank diameter of the serrated cutter

e :

Index for knot values for NURBS

E :

Objective function

f :

Feed rate per minute

fi, l(z, t):

Feed per revolution between ith and (i + l)th flute at height z at time t

f t :

Feed per tooth per revolution

Fxyz (t):

Total lumped cutting force vector in the x, y and z directions at time t

gi(z, t):

Screening function for the ith flute at the height z at time t

\( {h}_{\mathrm{g},i}^{st}\left(z,t\right) \) :

Elemental geometric static chip thickness for the ith flute at the height z at time t

\( {h}_i^{st}\left(z,t\right) \) :

Actual elemental physical static chip thickness for the ith flute at the height z at time t

i :

General notation for any flute of the N- fluted cutter

I a :

Coordinates of the ath cloud point data

j :

Index for control points, and weights for NURBS

k :

Order of the NURBS profile

K c ,K e :

Primary and edge cutting force coefficient vectors

l :

Dummy index

m + 1:

Number of control points in NURBS curve

ni(z):

The normal vector for the ith flute at the height z

N :

Number of flutes (teeth) of the serrated cutter

Ne, k (u):

NURBS basis function of the order k calculated at u in the eth knot span [ue, [ue + 1)

P j :

Coordinates of the jth control point

P s :

Array of horizontal coordinates of all control points along s direction

P n :

Array of vertical coordinates of all control points along normal to s direction

Q e :

eth segment of NURBS curve

Ri(z):

Local radius of the serrated cutter for the ith flute at the height z

\( {R}_i^s(s) \) :

Equivalent term to local radius \( \left({R}_i^s(s):= {R}_i\ (z)\right) \)

\( {\mathbf{\mathfrak{R}}}_i\left(z,t\right) \) :

Position vector of an element P located in the ith cutting edge at the height z at time t

s :

Equivalent serration height along the tangent of the cutting edge flute

t :

Time

Txr, i (z, t):

Force transformation matrix from rta to xyz coordinate for the ith flute at height z at time t

u :

NURBS parameter in the eth knot span

u e :

eth knot in the NURBS profile

U :

Knot vector for the NURBS profile

w j :

Weight corresponding to jth control point

W :

Weight vector for the NURBS profile

z :

Serration height along tool rotation axis (z-axis)

NURBS:

Non-uniform rational basis spline

rta :

Rotating radial-tangential-and-axial frame

xyz :

Non-rotating body-fixed coordinate system

δi(z):

Variation in the lag angle for the ith flute at the height z

ΔRi(z):

Variation in local radius for the ith flute at the height z

ηi(z):

Local helix angle for the ith flute at height z for variable pitch and helix cutter

\( {\overline{\eta}}_i \) :

Mean helix angle for the ith flute

θ D :

Slope angle at the endpoints of the NURBS

κi (z):

Axial immersion angle for the ith flute at height z

λ :

Wavelength of the serration profile

φ en :

Entry radial immersion angle

φ ex :

Exit radial immersion angle

φi(z, t):

Instantaneous radial immersion angle for the ith flute at the height z at time t

φp, i(z):

Local pitch angle between ithand (i + 1)th flute at height z for variable pitch and helix cutter

φη, i(z):

Local lag angle for the ith flute at height z for variable pitch and helix cutter

\( {\varphi}_{\overline{\eta},i}(z) \) :

Mean lag angle for the ith flute at the height z

χ :

Distance along the s-direction

ψ i :

Initial phase shift for the ith flute at the starting of the serration profile

Ω:

Clockwise spindle speed

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Acknowledgements

We acknowledge Forbes & Company Limited, India, for helping us with geometric measurements on their profilometer and for providing us their serrated cutters. We also acknowledge Kennametal India Limited for providing us their serrated cutters. We further acknowledge support from the Government of India’s Impacting Research Innovation and Technology (IMPRINT) initiative through project number IMPRINT 5509.

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Authors and Affiliations

  1. Machine Tool Dynamics Laboratory, Department of Mechanical Engineering, Indian Institute of Technology Kanpur, Uttar Pradesh, 208016, India

    Pritam Bari & Mohit Law

  2. Department of Mechanical Engineering, Indian Institute of Technology Kanpur, Uttar Pradesh, 208016, India

    Pankaj Wahi

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  1. Pritam Bari
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Correspondence to Mohit Law.

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Bari, P., Law, M. & Wahi, P. Geometric models of non-standard serrated end mills. Int J Adv Manuf Technol 111, 3319–3342 (2020). https://doi.org/10.1007/s00170-020-06093-0

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