Machinery's Handbook, 31st Edition
1200 MACHINING ECONOMETRICS The Generalized Taylor Equation Using Equivalent Chip Thickness (ECT): Due to the compression of the aforementioned geometrical variables ( f , a , LA , r , etc.) into ECT , Equation (2) can now be rewritten: (3) Experimental data confirm that Equation (3) holds, approximately, within the range of the test data, but as soon as the equation is extended beyond the test results, the error can become very great because the V - ECT curves are represented as straight lines by Equation (3) and the real curves have a parabolic shape. The Colding Tool Life Relationship.— This relationship contains 5 constants, H , K , L , M , and N 0 , which attain different values depending on tool grade, work material, and the type of operation, such as longitudinal turning versus grooving, face milling versus end milling, etc. This tool life relationship is proven to describe, with reasonable accuracy, how tool life varies with ECT and cutting speed for any metal cutting and grinding operation. It is expressed mathematically as follows either as a generalized Taylor equation (4a), or, in logarithmic coordinates (4b): V T A ECT n m # # =
− (
) ( ) M H 4 − K
ln 4 +
M ECT
H
(4a) (4b)
2
( N
0 – L # ln ECT )
M
=
V T #
ECT
e
#
y K M x H z N L 4 x 0 = − − − − ^ h
where x = ln ECT z = ln T M = vertical distance between maximum point of cutting speed ( ECT H , V H ) for T = 1 minute and speed V G at point ( ECT G , V G ), as shown in Fig. 5. 2 M = horizontal distance between point ( ECT H , V G ) and point ( V G , ECT G ) H and K = logarithms of coordinates of maximum speed point ( ECT H , V H ) at tool life T = 1 minute, thus H = ln( ECT H ) and K = ln ( V H ) N 0 and L = variation of Taylor slope n with ECT : n = N 0 − L 3 ln ( ECT ) y = ln V
1000
H-CURVE
V H
G-CURVE
K = ln( V
H )
M
2M
V G
100
Constants N 0 and L define the change in the Taylor slope, n , with ECT
T = 1 T = 100 T = 300
H = ln( ECT H )
10
ECT G
ECT , mm ECT H 0.1
0.01
1
Fig. 5. Definitions of the Constants H , K , L , M , and N 0 for Tool Life Equation in the V-ECT Plane with Tool Life Constant
The constants L and N 0 are determined from the slopes n 1 and n 2 of two Taylor lines at ECT 1 and ECT 2 , and the constant M from three V - ECT values at any constant tool life. Constants H and K are then solved using the tool life equation with the above-calculated values of L , N 0 and M .
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