X cos 1 + X 2 2 X sin 1 X sin

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(34) (35)

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1 + X 2 2 X sin Considering the triangle abc

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sin sin = f R d f sin R d = sin

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d = R

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Fig 341:

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Geometry of b and q

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f sin f cos = R sin sin

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f cos 1 + X 2 2 X sin X cos

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(36)

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Substituting for sin b from equation (34) d = R putting the value of X d = R R 1 + f 2 R 2 2 ( f R ) sin where d = chip thickness with f/R<<<1 we can write d = R R 1 2 ( f R ) sin (38) (37)

Precision Engineering

This equation can be simplified by using Taylor s series Let ( f/R ) sin q = a By using Taylor s series, the term can be expanded as

1 2a = (1 2a )2 = 1

2a 2 + + 2

This part of the expansion is very small as compared to the original value So the equation can be expressed as d = R R 1 2 ( f R ) sin d = R R (1 2 ( f R ) sin )

2 ( f R ) sin + + d = R R 1 2 d = R( f/R ) sin q d = f sin q (39) For typical conditions, q < 10 and sin q q d = fq The location of d is referenced by the distance Z measured from the tool centre to d on the plane of the cut as shown in the figure

Z = sin ( 90 ( + ) ) = cos ( + ) R

using the following identity: cos (a + b) = cos a sin b sin a cos b Substituting for sin a and cos b from equation (34) and (35) results in the following evolvement of Z: Z =

R sin (1 sin ) 1 + X 2 2X sin

R cos2 1 + X 2 2 X sin

R ( sin X sin2 X cos2 ) 1 + X 2 2X sin R ( sin X ) 1 + X 2 2X sin

(40)

Using the fact that X = f/R <<< 1 and q is small, Z becomes Z = R ( sin X ) = R ( sin f R )

Mechanics of Materials Cutting Substituting from equation 39 Z = R (d f f R) = d =

Rd f f

f (Z + f ) (41) R The equation contains a minor correction relative to Blake s result If the depth of the machining damage, Yc, at a critical depth, d = dc, were zero, that is, Yc = 0, then the measurement of Z = Zc would give a value of dc via equation 39 However, practically Yc is not zero, and also from productivity reasons, it is not desirable that it is zero Yc is proportionally dependent on the feed An increase in the feed causes an increase in the depth of damage and vice versa To continue with this fact, the geometry shown in Figure 342 is used As there is a nonzero Yc value, Z is shifted Zeff is now the measured value of the position for the onset of damage (ductile to brittle transaction) on the shoulder In the above Fig 342: The geometry used to derive values of @ figure, the chip has been moved from a shoulder distance h to allow an overlap of the damage from successive tool pass Zc is the value of Z corresponding to d = dc Zeff is the measured distance on the shoulder of the interrupted cut It can be seen that Zeff = Zc Z Figure 343 shows the enlarged key geometry From figure 343, Z = h Yc sin w h (42) Because both Yc and sin w are small relative to h, to evaluate h, distance p is added to Yc such that it forms a right angled triangle as shown in the figure, and the length of this new side can be expressed:

Fig 343: Triangle used to calculate ;

Precision Engineering n = R sin

so tan /2 = p/n p = R sin tan /2 Recognizing that and w are very small p =

R 2 2 h h cos = R R

(43)

is related to h using the data in figure 10 and by using the following equation: sin f = = so From the triangle p =

h2 2R

h sin w = Yc + p = Yc + where

h2 2R

sin w = Z/R

h 2Z c h + 2 RYc = 0 Using the quadratic formula,

h = Z c Z c 2 2 RYc The physically correct root corresponds to the minus sign so that h = Z c Z c 2 2 RYc We then have from equation 42 Zeff = Z c Z = Z c h = Z c 2 2 RYc