## What are workshop technology and workshop math?

Workshop technology is the application of scientific principles and methodologies for the manufacturing of machine components, tools, dies, etc. Workshop technology also provides the techniques, which can be carried out for metal joining and cutting processes. In a nutshell, workshop technology deals with metal works, and in a few cases, processing with plastics is also carried out. A mechanical workshop has seven major departments, each of the departments has sophisticated equipment and tools to carry out the manufacturing process. The major departments of mechanical workshops are plumbing shops, machining shops, carpentry shops, fitting shops, sheet metal shops, casting and foundry shops, and welding shops.

Each of the shops houses its specialized equipment to perform the required task. Basic concepts of calculus and algebra, which are the branch of mathematics, are applied to achieve precise and accurate manufacturing outputs. The application of mathematics also allows us to play with machine cutting speeds and feed to achieve excellent dimensional accuracy and tolerances.

In this article, some of the familiar workshop machines and processes have been outlined along with their associated mathematical approaches.

## Machinery in mechanical workshop

As outlined above, a mechanical workshop is divided into seven sub-divisions, out of which three are major and of utmost importance. These are machine shops, welding shops, and foundry shops. The details of which are outlined below with a brief introduction on the basic machinery used.

### Machine shop

This shop deals with sophisticated machinery to perform subtractive manufacturing of machine components. The components are manufactured by removing materials in the form of chips. The materials used are generally ductile materials such as mild steel, aluminum, brass, copper, and so on. In this shop, the machines remove materials by subjecting the materials to their failure point. Some of the machines used in this shop are lathe machines, milling machines, grinding machines, shaper machines, drilling machines, and so on.

### Welding shop

This shop specializes in performing metal joining operations. It utilizes specialized tools and equipment to join similar and dissimilar metals. The equipment subjects the metals to be joined to their melting point, which forms a permanent joint after solidification. Some of the operations in this category are arc welding, spot welding, friction welding, soldering, and brazing.

### Foundry shop

This shop manufactures machine components by converting the raw material to their molten state. The material is then poured into a specially designed mold cavity, which bears the shape of the component to be formed. Upon pouring the molten material into the mold, the material solidifies and takes the shape of the cavity. Casting is a popular method in this category.

## Material removal rate (MMR) of lathe and milling machine

The MMR is the direct indicator of the volume of material removed, the material is generally removed in the form of chips.

- It is measured per unit of time (minutes).
- The efficiency of a cutting machine directly depends on the MMR.
- The unit of MMR is ${m}^{3}/min$.

### MMR of lathe machine

The MMR of the lathe machine is based on the turning operation, which is given by, $MMR=V\times f\times d$, where, $V$ represents the cutting speed, $f$ represents the feed rate, and $d$ represents the depth of cut.

The cutting speed can be calculated by,

$V=\frac{D\times \mathrm{\pi}\times \mathrm{n}}{1000}$

where,

- $D$ represents the diameter of the machined component,
- $n$ represents the speed of the chuck.

### MMR for milling machine

The milling machine cuts material by use of special milling cutters known as end mill cutters. These cutters have multiple side cutting edges which cut material by shear action.

The MMR of a milling machine is given by,

$Q=\frac{D\times d\times {V}_{f}}{1000}$,

where,

- $D$ represents the general depth of cut,
- $d$ represents the radial depth of cut, and
- ${V}_{f}$ represents feed of the tool.

## Taylor's tool life equation

All the machines make use of special cutting tools to perform the cutting operation, but over time and due to harsh parameters, the tools fail due to excessive damage due to excessive wear. The tools generally fail by,

- Due to excessive forces and shocks.
- Plastic deformation by high temperature and high-stress generation.
- Gradual wear of tool's rake and flank.

Tool life generally indicates the amount of satisfactory performance a tool provides before the tool shows the first sign of wear. There are different methodologies to access tool life, one such methodology is by computing Taylor's tool life equation.

According to this equation, the wear of the tool is primarily governed by the cutting speed ${V}_{C}$, feed of the tool $f$, and depth of the cut of the tool ${d}_{C}$.

The Taylor's tool life equation is given by,

$V{T}^{n}=C$,

where,

- $T$ denotes tool life, and
- $V$ denotes cutting velocity,
- $C$ is constant,
- $n$ is Taylor's tool life exponent.

The value of $C$ and $n$ depends upon the tool material and the working environment.

## Welding equations

Two of the most popular welding methods used in a welding shop are arc welding and spot welding. Arc welding makes use of the high-temperature arc formed by the ionization of the air in between the electrode and the work material and is used as a very common welding technique. Spot welding on the other hand is achieved by the heat generated by providing resistance to the flow of current. The current user has a magnitude of 10,000 A.

### The heat input during arc welding

The net heat input is given by,

${H}_{net}=\frac{F\times VI}{E}$,

where,

- ${H}_{net}$ represents the net heat input,
- $V$ is the arc voltage,
- $I$ is the current,
- $E$ is the welding speed.

### Relation between voltage and current parameters in arc welding

The general relation between power source voltage ${V}_{p}$, open-circuit voltage ${V}_{o}$, power source current ${I}_{p}$, and open circuit current ${I}_{o}$ is given by,

$\frac{{V}_{p}}{{V}_{o}}+\frac{{I}_{p}}{{I}_{o}}=1$

### Determination of diameter of nugget in spot welding

The diameter of the nugget formed is given by, ${d}_{n}=6\sqrt{t}$, where, ${d}_{n}$ denotes the diameter of nugget and $t$ plate thickness which is to be welded.

### Heat generated in spot welding

The heat generated in spot welding which melts the material is given by,

$H={I}^{2}RT$

Where

- H denotes the heat generated,
- $I$ is the current passed,
- $R$ denotes resistance, and
- $T$ denotes the time of application of heat.

## Context and Applications

This topic is widely taught in many undergraduate and postgraduate degree courses of:

- Bachelors of Technology (Mechanical Engineering)
- Master of Technology (Mechanical Engineering)
- Master of Technology ( Manufacturing Technology)
- Master of Technology (Welding Engineering)

## Practice Problems

Q 1. Which of the following processes uses the heat of resistance to join metals?

- Arc welding
- Gas welding
- Friction welding
- Spot welding

Answer: Option d

Explanation: Spot welding uses the heat of resistance to join metals.

Q 2. Which of the following parameters govern the wear rate of a cutting tool?

- Cutting speed
- Feed rate
- Depth of cut
- All of these

Answer: Option d

Explanation: Feed rate, depth of cut, and cutting speed all govern the wear rate of a cutting tool.

Q 3. Which of the following is the correct Taylor's tool life equation?

a. $VT=C$

b. ${V}^{n}T=C$

c. $VT=0$

d. $V{T}^{n}=C$

Answer: Option d

Explanation: The correct Taylor's tool life equation is $V{T}^{n}=C$.

Q 4. What is the unit of MMR?

a. $\frac{m}{min}$

b. $\frac{{m}^{3}}{min}$

c. $\frac{{m}^{2}}{min}$

d.. ${m}^{3}$

Answer: Option b

Explanation: The unit of MMR is $\frac{{m}^{3}}{min}$.

Q 5. What is the correct equation for the diameter of the nugget in spot welding?

a. ${d}_{n}=6\sqrt{t}$

b. ${d}_{n}=\sqrt{t}$

c. ${d}_{n}=6t$

d. ${d}_{n}=6\sqrt[3]{t}$

Answer: Option a

Explanation: The correct equation for the diameter of the nugget for spot welding is ${d}_{n}=6\sqrt{t}$.

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