# Conversion of tool angles from one system to another: Mechanics of Machining

**Purposes of conversion of tool angles from one system to another**

• To understand the actual tool geometry in any system of choice or convenience from the geometry of a tool expressed in any other systems

• To derive the benefits of the various systems of tool designation as and when required

• Communication of the same tool geometry between people following different tool designation systems.

## Methods of conversion of tool angles from one system to another

• Analytical (geometrical) method: simple but tedious

• Graphical method – Master line principle: simple, quick and popular

• Transformation matrix method: suitable for complex tool geometry

• Vector method: very easy and quick but needs concept of vectors

## Conversion of tool angles by Graphical method – Master Line principle

This convenient and popular method of conversion of tool angles from ASA to ORS and vice-versa is based on use of Master lines (ML) for the rake surface and the clearance surfaces.

## Conversion of rake angles

The concept and construction of ML for the tool rake surface is shown in Fig. Master line for rake surface (with all rake angles: positive).

#### Figure: Master line for rake surface (with all rake angles: positive)

In Fig. Master line for rake surface (with all rake angles: positive), the rake surface, when extended along πX plane, meets the tool’s bottom surface (which is parallel to πR) at point D’ i.e. D in the plan view. Similarly when the same tool rake surface is extended along πY, it meets the tool’s bottom surface at point B’ i.e., at B in plan view. Therefore, the straight line obtained by joining B and D is nothing but the line of intersection of the rake surface with the tool’s bottom surface which is also parallel to πR. Hence, if the rake surface is extended in any direction, its meeting point with the tool’s bottom plane must be situated on the line of intersection, i.e., BD. Thus the points C and A (in Fig. Master line for rake surface (with all rake angles: positive)) obtained by extending the rake surface along πo and πC respectively upto the tool’s bottom surface, will be situated on that line of intersection, BD.

This line of intersection, BD between the rake surface and a plane parallel to πR is called the “Master line of the rake surface”.

From the diagram in Fig. Master line for rake surface (with all rake angles: positive),

OD = TcotγX

OB = TcotγY

OC = Tcotγo

OA = Tcotλ

Where, T = thickness of the tool shank.

The diagram in Fig. Master line for rake surface (with all rake angles: positive) is redrawn in simpler form in Fig. Use of Master line for conversion of rake angles for conversion of tool angles.

#### Figure: Use of Master line for conversion of rake angles

## Conversion of tool rake angles from ASA to ORS

Proof of Equation 4.1:

With respect to Fig. 4.2,

Consider, ΔOBD = Δ OBC + Δ OCD

Or, ½ OB.OD = ½ OB.CE + ½ OD.CF

Or, ½ OB.OD = ½ OB.OCsinφ + ½ OD.OCcosφ

Dividing both sides by ½ OB.OD.OC,

Similarly Equation 4.2 can be proved considering;

Δ OAD = Δ OAB + Δ OBD

i.e., ½ OD.AG = ½ OB.OG + ½ OB.OD

where, AG = OAsinφ

and OG = OAcosφ

Now dividing both sides by ½ OA.OB.OD,

Conversion of rake angles from ORS to ASA system

from which equation 4.4 and 4.5 are obtained.

The conversion equations 4.4 and 4.5 can also be proved directly from the diagram in Fig. Use of Master line for conversion of rake angles

Hints

To prove equation 4.4, proceed by taking (from Fig. Use of Master line for conversion of rake angles)

Δ OAD = Δ OAC + Δ OCD,

[involving the concerned angles γo, λ and γx i.e., OC, OA and OD]

And to prove Equation 4.5, proceed by taking

Δ OAC = Δ OAB + Δ OBC

[involving the concerned angles γo, λ and γy i.e., OC, OA and OB]

Maximum rake angle (γmax or γm)

The magnitude of maximum rake angle (γm) and the direction of the maximum slope of the rake surface of any single point tool can be easily derived from its geometry specified in both ASA or ORS by using the diagram of Fig. Use of Master line for conversion of rake angles. The smallest intercept OM normal to the Master line (Fig. Use of Master line for conversion of rake angles) represents γmax or γm as

OM = cot γm

Single point cutting tools like HSS tools after their wearing out are often resharpened by grinding their rake surface and the two flank surfaces.

The rake face can be easily and correctly ground by using the values of γm and the orientation angle, φγ (visualized in Fig. Use of Master line for conversion of rake angles) of the Master line.

Determination of γm and φγ from tool geometry specified in ASA system.

In Fig. Use of Master line for conversion of rake angles,

γm and φγ from tool geometry specified in ORS

Similarly from the diagram in Fig. Use of Master line for conversion of rake angles, and taking Δ OAC, one can prove

**Conversion of clearance angles from ASA system to ORS and vice versa by Graphical method.**

Like rake angles, the conversion of clearance angles also make use of corresponding Master lines. The Master lines of the two flank surfaces are nothing but the dotted lines that appear in the plan view of the tool (Fig. Master lines (ML) of flank surfaces).

The dotted line are the lines of intersection of the flank surfaces concerned with the tool’s bottom surface which is parallel to the Reference plane πR.

Thus according to the definition those two lines represent the Master lines of the flank surfaces.

Fig. Master line of principal flank shows the geometrical features of the Master line of the principal flank of a single point cutting tool.

From Fig. Master line of principal flank,

OD = Ttanαx

OB = Ttanαy

OC = Ttanαo

#### Figure: Master lines (ML) of flank surfaces

#### Figure: Master line of principal flank

The diagram in Fig. Master line of principal flank is redrawn in simpler form in Fig. Use of Master line for conversion of clearance angles for conversion of clearance angles.

The inclination angle, λ basically represents slope of the rake surface along the principal cutting edge and hence is considered as a rake angle. But λ appears in the analysis of clearance angles also because the principal cutting edge belong to both the rake surface and the principal flank.

#### Figure: Use of Master line for conversion of clearance angles

**Conversion of clearance angles from ASA to ORS**

Angles, αo and λ in ORS = f(αx and αy in ASA system)

Following the same way used for converting the rake angles taking suitable triangles (in Fig. Use of Master line for conversion of rake angles), the following expressions can be arrived at using Fig. Use of Master line for conversion of clearance angles:

**Conversion of clearance angles from ORS to ASA system**

Proceeding in the same way using Fig. Use of Master line for conversion of clearance angles, the following expressions are derived

The relations (4.14) and (4.15) are also possible to be attained from inversions of Equation 4.13 as indicated in case of rake angles.

The magnitude and direction of minimum clearance of a single point tool may be evaluated from the line segment OM taken normal to the Master line (Fig. Use of Master line for conversion of clearance angles) as OM = tanαm

The values of αm and the orientation angle, φα (Fig. Use of Master line for conversion of clearance angles) of the principal flank are useful for conveniently grinding the principal flank surface to sharpen the principal cutting edge.

Proceeding in the same way and using Fig. Use of Master line for conversion of clearance angles, the following expressions could be developed to evaluate the values of αm and φα

Similarly the clearance angles and the grinding angles of the auxiliary flank surface can also be derived and evaluated.

o Interrelationship amongst the cutting angles used in ASA and ORS

The relations are very simple as follows:

φ (in ORS) = 90o – φs (in ASA) (4.20)

and φ1(in ORS) = φe (in ASA) (4.21)

## Conversion of tool angles from ORS to NRS

The geometry of any single point tool is designated in ORS and NRS respectively as,

λ, γo, αo, αo’, φ1, φ, r (mm) – ORS

λ, γn, αn, αn’, φ1, φ, r (mm) – NRS

The two methods are almost same, the only difference lies in the fact that γo, αo and αo’ of ORS are replaced by γn, αn and αn’ in NRS.

The corresponding rake and clearance angles of ORS and NRS are related as ,

tanγn = tanγocosλ (4.22)

cotαn = cotαocosλ (4.23)

and cotαn’= cotαo’cosλ’ (4.24)

The equation 4.22 can be easily proved with the help of Fig. Relation between normal rake (γn) and orthogonal rake (γo).

Figure: **Relation between normal rake (γn) and orthogonal rake (γo)**

The planes πo and πn are normal to Yo and Yn (principal cutting edge) respectively and their included angle is λ when πo and πn are extended below OA (i.e. πR) they intersect the rake surface along OB and OC respectively.

Therefore,

∠AOB = γo

∠AOC = γn

where, ∠BAC = λ

The equation (4.23) relating αn and αo can be easily established with the help of Fig. Relation between normal clearance, αn and orthogonal clearance, αo.

#### Figure: Relation between normal clearance, αn and orthogonal clearance, αo

From Fig. Relation between normal clearance, αn and orthogonal clearance, αo,

## Tool geometry under some critical conditions

• Configuration of Master lines (in graphical method of tool angle conversion) for different tool geometrical conditions.

The locations of the points ‘A’, ‘B’, ‘C’, ‘D’ and ‘M’ along the ML will be as shown in Fig. Use of Master line for conversion of rake angles when all the corresponding tool angles have some positive values. When any rake angle will be negative, the location of the corresponding point will be on the other side of the tool.

Some typical configurations of the Master line for rake surface and the corresponding geometrical significance are indicated in Fig. Tool geometry and Master line (rake face) in some typical conditions.

#### Figure: Tool geometry and Master line (rake face) in some typical conditions

Tool angles’ relations in some critical conditions

From the equations correlating the cutting tool angles, the following critical observations are made:

• When φ = 90o; γx = γo for πX = πo

• When λ = 0 ; γn = γo

αn = αo

• When λ=0 and φ = 90o; γn=γo=γx pure orthogonal cutting

(πN=πo=πX)