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Solid State Phenomena
ISSN: 1662-9779, Vol. 265, pp 598-605
© 2017 Trans Tech Publications, Switzerland
Submitted: 2017-06-14
Accepted: 2017-06-19
Online: 2017-09-25
Determining Undeformed Chip Thickness Models in Milling
and its Verification during Wood Processing
A.A. Fomin
Vladimir State University, Gorky street 87, Vladimir, 600000, Russian Federation
Keywords: profile milling, undeformed chip thickness, mathematical model, the cutting tool, the
treated surface, depth of cut, feed rate.
Abstract. This paper presents a mathematical model of the maximum thickness and the chip area
for the processes of cylindrical and profile milling of various materials, including wood. The
analytical dependences connecting the geometry of the shear layer with the elements of the milling
mode and cutter design parameters are determined. Also, a model of the volume of material
removed from the surface of the workpiece during the milling profile is presented. The comparative
calculations of the previously known models and the models developed by the author were done. It
was found that the models of the geometric parameters of cutting layer presented in the article are
adequate and can be used to calculate the energy performance of the wood milling process with
cylindrical and shaping cutters. These models are suitable for use in the calculations of the
processing parameters for a wide range of material: metals, wood, plastic, glass and others.
For the production of metal and wood products different methods of mechanical treating are
used, including widespread milling, which provides high performance and desired quality of the
treated surfaces [1-7]. A significant part of an operation is aimed at obtaining products, the crosssection of which is convex and concave curved contours, treated by profile milling.
Profile milling of curved surfaces with shaping cutters is in no competition with other
technological methods, due to high performance and low cost of treating. When milling with the
shaping cutter its profile is copied to the treated surface of the product, greatly simplifying the
equipment, reducing its value and hence cost of manufacturing process [8-11].
Despite this, there are currently no accurate models of thickness, area and volume of the shear
layer at profile milling, including wood, whereby calculations of energy performance and process
performance, precision of products, production costs and other parameters of treating efficiency are
determined largely by subjective factor [12-14]. Increased dynamic of technological system activity
has a negative impact on the quality indicators of treated surface and performance of the treatment
process [15-18]. The use of shaping cutter allows to improve the utilization of the material [19, 20],
to reduce the waste occurring during milling of products widely used in various industries,
including in woodworking [21, 22]. In some industries, for example in wood working the recycling
is an actual economic problem [23-29].
Specificity of profile milling with shaping cutter is that the treating process is performed by
cutting contour (profile), its elementary cutting parts are arranged at different distances from the
center of rotation of the cutter [30-32]. These features are the reason for the change of elements of
the workpiece shear layer compared to the known cylindrical and angular milling.
The main parameters of the shear layer during the milling of materials, including wood are the
maximum thickness of cutting amax , layer area Fc and the volume of material Vc removed with one
tooth (single cut). Based on strict mathematical relationships it amax , Fc ,Vc is possible to make
accurate calculations of cutting force, torque, power of profile and cylindrical milling of different
types of wood, as well as the expected geometric accuracy of treated surfaces of finished products
[33-36]. It is important to have parameter's models amax , Fc ,Vc in the elements of cutting mode
function, the current treating time and the geometric characteristics of the cutting tool. The wood
milling process in scientific and technical literature was not considered in this formulation before.
All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of Trans
Tech Publications, (#103309480, University of Auckland, Auckland, New Zealand-11/11/17,22:22:37)
Solid State Phenomena Vol. 265
The current value of the thickness of the shear layer ai corresponding to the current angle of the
contact Ɛi between tool and workpiece, during the motion of the wedge cutting along a circular arc
varies from zero to amax , while the maximum thickness of shear layer amax in a cylindrical milling
on B height of the cutter does not change and is constant, that is, amax = const for any cross-section
of the cylindrical cutter.
For milling with concave profile the maximum thickness of shear layer varies in B height of the
cutter, ie amax = var: least of its value corresponds to the points of the cutting profile of mill lying in
both ends planes and the largest - in a transverse plane of the cutter of symmetry, in which the
current cutter radius takes a minimum value. Thus, the maximum thickness of cutting layer for
profile milling is to be determined.
Development of a Model of the Cutting Layer Thickness and its Analysis
Prior to establishing a mathematical model of cutting layer thickness it's needed to know the
pattern of change of the current radius R =Ri along the height of shaping cutter. Most simply profile
treating can be performed at sharpening the blade of cutter on an arc of KLM circle (Fig. 1) by
radius Rp. To do this, we draw up design scheme, which is a shaping cutter and the workpiece that
are in the process of contact interaction. The distance from the center O of the arc KLM circle to the
axis I - I of rotation of the cutter 1:
Fig. 1. The contact area of shaping cutter and the treating workpiece (s): 1 – shaping cutter,
2 – workpiece, 3 – curve contour of the cutting blade; 4, 5, 6 – blade contours during changing of
radius profile Rp; Dr, Ds – working movements of the tool and the workpiece.
The arbitrary i - m cross section, detached at a distance from the beginning of the coordinate O
system YOZ, cutter radius is equal Ri and the equation (1) of this section is:
р cos
Equating the right hand sides of the expressions (1) and (2) and performing transformation, we
cos arcsin
Materials Engineering and Technologies for Production and Processing III
Current angle αi formed by the plane perpendicular to the tool axis, and the radius Rp of the
cutting blade
arcsin( ).
Its numerical values are in the range αi = 0 up to αmax = 0.5φ. At an angle α = arcsin
of coordinate zi = 0, and the plane perpendicular to the axis of the tool passes through the point L. In
accordance with the expression (3) cutter radius is equal to the minimum value Rmin at zi = 0. Now αi
= αmax = 0.5φ there are two transverse sections, one of which passes through the point K, the other through the point M.
These cross sections are arranged symmetrically respectively to the axis Y. Radius cutters in
these sections according to its maximum value Rmax, which can be determined if the minimum
radius of the cutter, the height and radius of the profile of the cutting blade:
р (1
− cos(arcsin
Expressions (4) and (5) allow to determine the radius of the cutter, in any of its cross-section at a
known the minimum radius Rmin and the radius of the profile of the cutting blade Rp. If Rmin not
known, and the maximum radius Rmax, the radius of the profile of the cutting blade Rp and height B,
then it is possible to determine Rmin, that is, to solve the inverse problem.
Profile cutter radius is given by operating drawing of the product. If to specify the dimensions of
the shaping cutter, it is possible to determine the radius of the profile milling cutter according to the
The central angle corresponding to the arc of a circle cutter profile, we will find from the
р cosα
The value of the central angle
= 2arccos
Profile of shaping cutter, outlined in an arc of KLM circle, is described by the equation:
) when − ≤
≤+ .
Thus, from the formulas (3) - (9) the structural dimensions of shaping cutter in any of its crosssection can be determined.
Known models of thickness amax and area Fc of shear layer are approximate and do not consider
the remaining 1-2-3-4-1 ridges on the treated surface of height ∆ (Fig. 2a) due to the kinematics of
the milling process. At a distance of half the feed to the milling tooth (0,5Sz) at the intersection
point of nearby cuts 1-4 and 4-3, in the plane 2-4-5, the height ∆ of remaining ridges is defined by
the formula:
Solid State Phenomena Vol. 265
where the current radius of the shaping cutter, is Ri (mm); cutting feed rate of the workpiece is vs
(m/min); z is the number of cutter teeth and the angular velocity of cutter – ω (radians/s).
Under normal milling conditions the height ∆ is negligible, but its number on the treated surface
of one meter length is measured in tens of thousands, which leads to errors in the calculation of
parameters amax and Fc, particularly at high cutting modes specific to the mechanical process of
wood treating.
Fig. 2. Kinematic wave, remaining on the treated surface after milling (a), the calculation scheme of
shear layer thickness (b) and 3D-model of undeformed chip thickness when profile milling (c).
The absolute value of the maximum thickness of the shear layer (Fig. 1b)
The length of the interval O2O3 to be determined, if we find the coordinates of points O2 and O3.
To the point O2 they will be x2=0, y2=0. Coordinates of the point O3 will be found by a simultaneous
solution of the equation of line 1 (Fig. 2b) with the equation of the arc of the circle O3O6 with center
in the point O1:
where t is the cutting depth; Ri is the current cutter radius; Sz is workpiece feed on the cutter tooth.
Solving together a system of equations (12), we obtain
Thus, the point coordinates O2 are (0, 0) and the point O3 has coordinates √2
Then the length of the segment is
2 √2
In view of (14) the expression (11) takes the form:
Materials Engineering and Technologies for Production and Processing III
Equation (15) includes the current shaping cutter radius Ri which depends on its geometrical
parameters on Eq. (3). To get a mathematical model of the shear layer thickness at shaping milling
of the workpiece a function of geometrical parameters of the shaping cutters Eq. (3) is substituted
into the expression (15).
Mathematical equations (6) and (7) describe the change of the maximum thickness of the shear
layer in function of elements in cutting mode (t, vs, ω), dimensional parameters and the number of
cutter teeth (Rp, Rmin, zi and z). It is the most valuable and complex, as is applicable not only to the
cylindrical processes but also for profile milling with shaping cutters in a wide range of variable
independent factors and geometrical parameters of the cutting tool. Based on it, it is possible to
perform calculations of energy performance of the milling processes, without the need for the
implementation of time-taking experiments.
Maximum thickness of cutting layer for the cutting point of the tooth, the rotating in a transverse
plane of symmetry,
Based on the model (16) the graphs of dependences amax from independent factors of processing
were plotted: minimum radius of the profile cutters Rmin = 50 ... 80 mm, the feed rate of the
workpiece vs = 3 ... 50*103 mm/min, rotary velocity of the tool n = 4000 ... 6000 min-1, the number
of cutter teeth z = 2 ... 6, cutting depth t = 3 ... 25 mm. The calculations are performed in software
environment Advanced Grepher. Effect of depth of cut t on the thickness of the shear layer is
described by curves 1-3 of the second order (Figure 3.).
Fig. 3. Effect of depth of cut on the thickness of the shear layer during the wood processing (Curve
1 - at cutting modes Rmin = 50 mm, vs = 204 mm/min; n = 4880 min-1, z = 4, curve 2 - at modes Rmin=
50 mm, vs = 304 mm/min; n = 4880 min-1, z = 2, curve 3 - at modes Rmin= 65 mm, vs = 304 mm/min;
n = 4880 min-1, z = 2).
Development of the Model Area of the Shear Layer and its Analysis
The area of the shear layer in the longitudinal section of the workpiece is determined by
calculating of definite integrals of functions limiting the curved shape,
( (
− ) − 0,5 )
− ) + 0,5
( − ∆),
where the expression under the integral sign is a function describing a circle centered at the points
O1 and O2 , respectively.
According to the known data shear layer area in the longitudinal section of the workpiece is
Solid State Phenomena Vol. 265
Equation (17) is an exact expression of area of the shear layer when milling with both cylindrical
and shaping cutters, and the formula (18), as previously noted, is approximate. The area of the shear
layer in a longitudinal section in the workpiece at applicate zi = 0 (for the transverse plane of
symmetry of the shaping cutter):
For the mathematical model of shear layer area at shaping milling of workpiece it should be (3)
substituted with the expression (19) instead Rmin. Formulas (3) and (19) represent a rigorous
mathematical dependences of the shear layer area Fc from the elements of milling mode and the
dimensional parameters of cutting tool. Based on (3) and (19) a specific degree of influence factors
on the area Fc of the cutting process and the cutting tool can be set.
With increasing feed Sz per tooth the shear layer area of the workpiece increases in a linear
relationship, which is explained by the growth of the segment length O3O4 (Fig. 2b).
The minimum radius Rmin of the milling cutter, cutting feed speed vs , the number of teeth z and
the angular speed ω of the cutter have little effect (less than 0.5%) on the area of shear layer Fc , so
you can take their impact as insignificant.
The volume of sheared layer with a single tooth
с р arccos
where lklm is the length of the cutting profile of the cutter; B is is the height of the cutter.
The reliability of the developed models is confirmed using rigorous mathematical methods of
integral calculus, by the convergence of the results of the geometric parameters' calculation of shear
layer by known and proposed models.
The reliability of the models is also tested by measuring in PRO ENGINEER software
environment for different mode factors of the process and dimensional parameters of the cutting
tool. The discrepancy between the values calculated on the developed models and measured in the
PRO ENGINEER was not more than 0.5%. When zooming in plotting the shear layer in the
discrepancy between theory and practice it is reduced to 0.1%.
Thus, we can conclude on the basis of these data, that the developed theoretical models of
geometrical parameters of shear layer are adequate and can be used for the calculation of power and
energy parameters of wood milling processes.
As a result of mathematical modeling of the parameters of shear wood layer the scientific
information about the nature and limits of variation of the parameters required to predict and control
the output performance of wood profile milling in a wide range of variable kinematic factors of the
process: elements of cutting mode and geometry of shaping cutting tool was obtained.
The developed models are valid for cylindrical and profile milling of different types of wood, are
not restricted to cutting modes, take into account the parameters of the shaping cutting tool, the
presence of the kinematic wave on the treated surface, and are necessary for qualitative
performance of design procedures.
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