Imagine you have a piece of wood. And you have to cut it into a cuboid of dimensions 2cm x 4cm x 8cm. You only have an axe to work with. Given 30 mins, how close to the measurements can you make your cuboid? Tolerance is how much difference can be allowed between the dimensions in the design of a part and the manufactured dimensions. And tolerance analysis is how you determine the tolerance of individual parts in a design and how they’ll fit together in the overall design.

**Importance of tolerance analysis. **

Take the case of that piece of wood. Imagine you have to make a simple model house from it. You design the house on a piece of paper, or using software. And then you proceed to cut out the parts of the house from wood. Odds are you won’t get all parts cut to the last millimeter, there will be some differences. But this is ok, to an extent. It becomes a problem when you can’t close the door, or open the windows, or in short, when the piece you produced is affecting the functionality of the design.

This is the importance of tolerance analysis. Manufacturing processes, particularly with mass production, are not perfect (and they don’t have to be). So you have to determine, at the design stage itself, how much imperfection is ok. You have to determine, at what point the produced part will fail.

As you can imagine, tolerance analysis becomes more complex as the number of parts increases. With one or two parts for a design, its easy enough to bring down the tolerances to a large extent. But with hundreds of parts, the tolerances of the individual parts kinda add up(an oversimplification of events, but yes), and the situation is more complex.

**Factors affecting tolerance analysis**

The specified requirements for a product or equipment translates through the design and into the tolerances. Designs are made according to the requirements set, and for manufacturing the tolerances are set so that the pieces produced work as specified.

An easy way to understand these tolerances is to compare the manufacturing processes of common road cars, luxury cars, and F1 cars. All three have different requirements and therefore different tolerances.

An average road car comes out of the factory in large numbers and is a general purpose vehicle. The goal is to create a car for the average customer, for everyday use. The usage conditions vary a lot among the users. And the users are not experts.

The use of a luxury car is more defined. It could be a supercar, intended for high speeds, or it could be a chauffeured experience, like a Rolls Royce or a Bentley. Either way, use cases are much more defined. You’re not buying a Rolls Royce to take it to a track and step on it. Either way, users are willing to pay good money for it.

In an F1 car, the requirements are even tighter. Each car is moulded for the individual driver, and the goal is to be the fastest on a track. And not just in a straight line.

So let’s see how tolerances apply for the different cars.

**Resources**

The cost of a vehicle is part of the requirement. It defines the resources available for design and manufacture.

Of the three, resources available for an average road car are the lowest. And these are produced in larger numbers. So the emphasis on tolerance is low. Not that designers will compromise on safety or anything, everything still works(that’s putting it very low), but compared to the other two, tolerances are mild. Even though cars may all look very similar, a closer inspection will show significant differences between individual cars. But that’s ok.

Now let’s have a look at luxury cars, or the higher end of road cars. These cars sell for much higher prices. And even in a traditional assembly line process(not made custom by hand), the differences between individual cars will be much lower. An oft-noted factor is the gap between the panels, which are either non-existent or consistent in a batch of luxury cars.

## And then there are tolerances on F1 cars.

One can observe the tightest tolerances on F1 cars. From the bodywork to the engines and transmission, every part has tight tolerances. On these engineering beauties, the designers(and/or the engineers) try to eke out the best performance, to get the maximum downforce, maximum speed and there’s plenty of resources available to do this. And even small variations can have large impacts on the performances. If you’re an F1 enthusiast, you must have noticed that sometimes when two cars touch just lightly, they bring the car back to the pit and change the front wing. Sometimes no damage will be visible to the spectators, but the smallest damage can affect the performance.

Take the case of an F1 engine. Their tolerances are so tight and account for the expansion when the engine heats up. They’re so tight that unless the engine is hot, the pistons won’t move. That’s why you can’t start an F1 car with the turn of a key. Engineers heat the engine to the right temperatures before starting it.

**So how do more resources translate to more tight tolerances?**

As you can imagine, to get parts of tight tolerances you need better equipment and better tolerances. If you want to cut a piece of wood into 2m square, and the tolerances are +-20 cms, you can probably use an axe. When the tolerances are +- 2mm, you’ll probably need something more expensive.

Same thing happens in manufacturing. When the tolerances are low, you need better equipment, which directly translates to higher manufacturing costs.

And its not enough to make it, you need good quality control. After manufacturing something, you have to make sure that it fits the specs. And for tighter tolerances, you need better tools for inspection, and the cost increases here.

The last part is the scrap rate. When the tolerances are high, even after using good tools, many of the final products won’t fit the specs. And you’ll have to discard them. This further drives up the cost of production.

But as technologies evolve, tighter tolerances are possible and may even bring the cost down.

**How is tolerance analysis performed?**

There are mainly two ways of performing tolerance analysis: Worst case analysis and statistical analysis.

**Worst case analysis**

Picture yourself designing a car for mass production. Each individual part is produced in large quantities, each part with its own tolerance limitations. And these pieces are randomly assigned to individual cars. Among these parts produced, some pieces would be at the extremes of the tolerance limits. Now imagine that all the pieces are at the extremes of the tolerance limits for one car by chance. The height of the windshield is specified as 50 cms+-10 cms, and the piece you have is 60cms, and for the door, the height is specified as 100cms+-5cms the piece you have is 95cms and like that for all parts.

Will you be able to produce a car using these parts? This is what the worst-case analysis considers. It ensures that even the pieces with dimensions at the extreme end of their tolerance is within the requirements.

#### An example

For example let’s say there are 3 parts for a door. For the sake of simplicity, let’s assume they only have one dimension(never happens in reality) specified as

30cm +- 1cm

20cm +- 2cm

35cm +- 3cm

Now if these tolerances were set by worst-case analysis, and you get a set of parts by the dimensions 31cm, 22cm, and 38cm, these parts will work together.

Why the worst case analysis is not always the best?

The problem with the worst-case analysis is that it doesn’t consider the probability of the worst-case scenario. In actual manufacturing situations, the worst-case scenario happens rarely. For example, if the tolerance is set to 30cms +- 5cms, the odds of a part coming out with 35cms is very low. And the odds of all parts for a specific product having extreme imperfections are infinitesimally low.

Using the worst-case tolerance also has monetary implications. When you’re going by the worst-case situation, you need to go for tighter tolerances. If all parts in the extreme cases need to work together, you need really tight tolerances, and as shown above, tighter tolerances translate to more expenses.

This is where statistical analysis comes in.

**Statistical analysis**

Statistical analysis is much more complicated than a worst case analysis. But the advantage is that it gives more options to designers. Designers get more flexibility for the same resources.

The statistical analysis takes advantage of the probabilities to use more loose tolerances. As we talked about earlier, the worst-case situation is a rarity. The variations mostly follow a normal distribution. That is, if the tolerance is specified as 35cms +-5cm, most of the products that come out of the production will have been 35cm. There will be fewer with 36cm and 34cm. And even lesser with 33cm and 37cm.

Such concepts can drastically reduce the cost of production without affecting quality.

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