is Mechanics of Machining and today is lecture - 4. The topic today is interrelations among

the tool angles expressed in different systems and what are the instructional objectives

or content of lecture today.

State the purposes of conversion of tool angles from one system to another. Why do we need

conversion of tool angles from ASA to ORS or ORS to NSA and so on? Name the different

methods of the conversion of tool angles. There are different methods what are those?

Convert tool angles by graphical method. Rake angle from ASA to ORS and vice versa, clearance

angles from ASA to ORS and vice versa. Next convert tool angles from orthogonal rake system

to normal rake system and last ascertain tool geometry in some specific conditions. Now

what are the purposes of conversion of tool angles?

To understand geometry if expressed in a different system. Suppose you are conversant with a

particular system of tool designation, say orthogonal rake system but you have got a

book written by a foreign author who dealt the cutting tool geometry in ASA system. You

may not be able to understand, then what you have to do? You have to learn both ASA system

and orthogonal system and also convert the tool angles given in the book in ASA system

in to ORS system for your understanding, your visualization and study. Now next, derive

benefits of different system. Now there are different tool designative system ASA system,

orthogonal system, normal rake system, maximum rake system, work reference system, each system

has got certain advantage.

Now we have to derive the advantages. To derive for example, say ASA system is very convenient

for tool manufacturing tool inspection and so on. Then orthogonal system, orthogonal

system is very simple. It is very useful say general study, research, analysis and so on

but orthogonal system does not provide the exact tool geometry if lambda inclination

angle is not zero. Thirdly if you want to resharpen the cutting tool then ORS system

is not that very convenient whereas normal rake system that gives the true picture of

the cutting tool and it enables very easy resharpening of the tool.

So if you want to derive the benefits we should convert tool angle from one system to another

to derive the benefits. Now communicate geometry to others who follow different tool designation

system. Suppose you do research on a machining process with a cutting tool in ORS system.

Now you go to a conference and say USA and there people are more conversant in ASA system.

How will you communicate the tool geometry that you have dealt with to them who are knowledgeable

about ASA system that means, you have to convert your tool angles from ORS to ASA, and then

tell them this is the geometry actually in ASA system for there understanding and this

is mutually true vice versa also. Now what are the different methods of conversion of

tool angles?

There are basically four methods. First is analytical method it is geometrical method

very simple but tedious. It takes lot of time and

monotonous. Graphical Method based are Masterline principle, it is very simple, quick

and very popular. So most of the people you know learn this method first. Transformation

matrix method is a little complex method but very suitable for complex tool

geometry. Say for example, say geometry of cutting, say drill geometry for horn tools

then for conversion of tool angles say hob cutters or gear shaping cutters for such complex

shaped cutters or cutting tools the transmission matrix method is appropriate

for conversion and vector method is Universal, very strong powerful method as

well as very simple and quick but it needs concept of vectors and matrix. So our concentration

will be on graphical method master line principle. So conversion of tool angles

by graphical method using master lines.

Now let us start with conversion of rake angles. You know the cutting tools have got three

important angles. One is rake angle, next is clearance angle and third is cutting angle

and beside that finally the tool nose radius. So let us start with conversion of rake angles

from ASA system to ORS and vice versa.

So first of all say rake angle, we shall deal with conversion of rake angles from ASA to

ORS and vice versa. So let us start with this. By graphical method, this is based on master

line principle, then what is master line? First of all you have to understand the master

line concept. See this is master line, let me show you the master line for rake surface.

First you draw a cutting tool. This is a top view of the cutting tool. It is drawn in orthogonal

rake sorry pi r reference plane, then you visualize the other planes. This is machine

longitudinal plane pi x, this is machine transfers plane pi y, this is cutting plane pi c and

this is orthogonal plane pi o. Now this is xm, this is ym, this is xo orthogonal, this

is yo and zo and zm perpendicular to the plane.

So again I repeat this is pi x plane, this is pi y plane - machine transverse plane,

this is orthogonal plane and this is cutting plane. Now you show the rake angles in different

planes. Say first of all, let us take the section of the tool in machine longitudinal

plane and show the cutting tool section. This is section of the cutting tool and in which

plane the diagram has been drawn, the section is taken on pi x plane. So this is pi x, machine

longitudinal plane and what is this axis this is xm, what is this is axis y this is zm.

So this is zm and this is xm and what is this angle? Angle between the reference plane and

the rake surface, so this is gamma x that is side rake.

Now what you have to, this is the bottom surface of the tool and these are rake surface you

extend the rake surface until it meets the bottom surface suppose it meets at point D

prime. Say at certain distance, this rake surface meets the bottom surface and if you

project it on this diagram on the view drawn in reference plane so this is point suppose

D. So D is situated on the bottom plane which is parallel to the reference plane or it is

horizontal plane. Now you take the cutting tool section along machine transfer plane

in this direction. So this is the cutting tool and which plane the diagram has been

drawn, pi y machine transverse plane by take a section by this machine transverse plane.

So this is obviously zm and this is ym axis, xm is perpendicular to the plane and what

is this angle? this is rake angle in which plane? pi y plane, so this is gamma y that

is back rake. If you extend this rake surface up to the bottom plane of the tool then meets

at a point say this is point B prime and on this diagram if you project it this is the

point this is the distance suppose this is o oD this is oB. So B and D both of situated

on the bottom surface and if I join B and D by a straight line then what is this BD.

BD is situated on the bottom plane which is also parallel to the reference plane. This

is reference plane, this surface is reference plane, this plate now what physically what

is the meaning of this.

This is the line of intersection between the rake surface of the tool if extended and the

bottom surface of the tool. So this is the line of intersection between the rake surface

and the bottom surface if it is extended on this direction it will meet a point D and

B. Now you can extend it. So this line is called Master Line for rake surface for a

Master Line for rake surface, similarly Master Line for clearance surface will also appear.

Now what is this point? Say this is point C. It means if the rake surface is extended

along this orthogonal plane this will meet the bottom surface at point C and that point

C will be always situated on the line of intersection.

Physically it means that if you take the section draw the view of the cutting tool from here extend it. So this

is orthogonal plane and this is the cutting tool, this is the rake surface, this is rake

angle orthogonal rake and this axis is zo, this is xo and so on. Now if you extend this

rake surface then it will meet the bottom surface of the tool at point say C prime and

if it is extended it meets point C. Now what is the thickness of the tool? This is the

thickness of the tool say T, here this is the thickness of the tool T. Now look at this

tool from this side along this direction along xo and now but this view that we will get

of the tool will be in the cutting plane. So you draw the diagram in the cutting plane.

So this is yo and this is zo and what is the view?

Now if this is extended like this, this will also meet certain point and if it is extended

this will be point, this will be somewhere A prime and this will be extended here. This

will be A. So we get DCBA were this rake surface if extended meets the bottom surface along

machine longitudinal plane, machine transverse plane, orthogonal plane and cutting plane.

Now here you can see, that what this length OD? If we put it here OD equal to this OD

this intercept. This intercept is equal to this intercept how much is this? This is equal

T what is this angle same gamma x, so this OD. OD is equal to T multiplied tangent cotangent

of gamma x so T cotangent gamma x.

Similarly OB is how much? OB this is OB this equal to T, this is gamma y, so this is equal

to T cotangent of gamma y, T cot gamma y then what is OC? OC will be T what is this angle?

gamma O, so this will be T cotangent of gamma O and what is OA? This much this angle is

lambda for inclination angle of the cutting tool. So OA will be equal to T cotangent lambda.

So this will be very useful. Now you get the concept of master line. Now this concept will

be know utilized to convert the tool angles from to one system to another.

Now say conversion. First you start say convert rake angle from ASA to ORS. Now let us draw

the diagram again. This is the cutting tool top view drawn in reference plane pi and then

you visualize the different planes - machine longitudinal plane, machine transverse plane,

cutting along the cutting edge and orthogonal plane and mind that this is ninety degree

and this is xm, ym. This is xo, this is yo, and zo zm perpendicular to this point o.

Now here suppose this is the master line, suppose this is the master line for rake surface

and this point is D, this point is C, this point B and this point is A. What we observed

last time that OD is equal to T, the thickness of the tool multiplied by cotangent gamma

x. If T is one, say unity for T is equal to unity OD is equal to cotangent of gamma x.

OB is equal cotangent of gamma y. OC is equal to cotangent of gamma O and OA is equal to

cotangent of lambda we get it now.

We have to proof now we have to convert from ASA to ORS. Now what is the meaning of conversion

from ASA to ORS that means if the tool angle the value of the tool angles are given in

ASA system then what will be the value of the rake angles of the same tool that is the

question that means, the meaning is determine gamma O and lambda which is also a rake angle

from the given values of gamma x and gamma y that means, this is a function of gamma

O and lambda is an this is a ORS system and this is ASA system. So from ASA system to

ORS we have to convert now how this will be converted. Now we are discussing about conversion

of cutting tool angles. So first we shall show the conversion of rake angles.

You know the rake angles, there are side rake, back rake or orthogonal rake even lambda intersecting

angle is also a rake angle. So first conversion of rake angles from ASA to ORS that means,

let us write say tangent of gamma o is equal to tangent of gamma x sin phi plus tangent

of gamma y cosine phi. The other one is tangent of inclination angle lambda is equal to minus

tangent of gamma x cosine phi plus tangent of gamma y sin phi instead of cosine phi.

Say equation number one and equation number two.

Now actually what we are going to do now, what we are going to do that if the values

of gamma x, the gamma x and gamma y gamma x and gamma y are known to me then and phi

is also known that is the tool angles are given in ASA system we have to determine the

rake angles gamma o and lambda in ORS system that means to find out rake angle orthogonal

rake and lambda that is in ORS system as function of gamma x, gamma y and the cutting angle

phi. The principle cutting angle phi and we know that principle cutting angle phi is equal

to ninety degree minus phi s where phi s is approach angle and this approach angle is

given in ASA system.

So if phi s know from where you can determine phi and put the values of gamma x gamma y

and phi in to these equations number one and two we can determine the values of orthogonal

rake and inclination angle lambda that is from the given values of the rake angles in

a ASA system, we are determining the rake angles in ORS system. So now, how to proof

this equation? We have to proof these equations. See equation number one proof of equation

number one. How shall we do it? Let us take the help of this drawing. Draw this diagram

this is the principle cutting edge, auxiliary cutting edge. This is the tool and this is

machine longitudinal plane pi x, machine transverse plane pi y, cutting plane pi c and orthogonal

plane pi o and now you draw this master line. Suppose this is the master line for the rake

surface, now suppose this is point O, this is point A, this is point B, this is point

C and this is point D.

What are the intercepts I remind you that OD stands for cotangent of gamma x OB cotangent

of gamma y, OC cotangent of gamma O orthogonal and O is cotangent of lambda and this is phi

and this is phi s. Now, we have to proof with the help of this equation. The equation number

one now how we shall proceed? Consider triangle OBD this triangle is a right angle triangle

OBD. This area is equal to area of the triangle OBC plus the triangle OCD. Now all these are

triangles. So what is OBD area of the triangle OBD this is a right angle triangle so this