# Maths

## Contents

## Maths:

### Measurments And Explanations - Automotive Mathematics:

(The numbers you NEED to be familiar with) This document aims to give an overview of this subject for the home mechanic so that you can understand the basics of some aspects of your vehicle. It is not provided to be the last word on these subjects. They are much too complex to cover in great depth here. Disclaimer: This information is provided free of charge and it is worth every cent! Use the information provided here at your own risk, it is provided as a guide only. Errors and omissions are excepted. Any corrections, please E-mail qutewa@tpg.com.au

### Tyres:

So, you want to put wider tyres on your car or you want to go for a Plus1, Plus2 or even Plus3 fitment…and wider of course. How do you do this while not affecting your speedo reading (so you can avoid getting stopped by those people in funny blue suits who dispute the accuracy of your speedometer)? First things first…what do the size markings on your tyres really mean and what effect do they have on your choice of tyres? Let’s take a typical example: Your original tyre size is 185/75-14. The 185 is the width of the tyre in millimetres, but not necessarily the width of the tread. It is the maximum width of the TYRE. The 75 is the Aspect Ratio or the ratio of height to width, Also known as "Profile" as in, "Low Profile Tyres". Low Profile Tyres are much wider than they are high, are usually expensive, and deliver better handling characteristics, but of course a less smooth ride. The 14 is your rim size in inches. Yes, there are metric diameter rims, but they are only used on some exotics like Mercedes and BMW and even then not on all models so we will ignore them in this document. So far we know that our tyre is 185mm wide, the sidewall is 75% of 185mm high and the tyre goes on a 14" rim. These 3 pieces of information should determine what replacement tyre you choose. Why? Because the combination of them determines the overall diameter of your wheel and therefore its rolling diameter. So, you want to stay with a 14" rim, but go to a 235 width tyre (with the appropriate increase in rim width). What size tyre do we need to keep the speedo reasonably accurate and what effect does aspect ratio (or sidewall height) have on your speedo reading? If you put a tyre with a much lower sidewall height on your car and you don’t compensate by putting larger diameter rims on your car, you decrease the rolling diameter of your tyre, so it has to do more revolutions per kilometre. Because your speedo gives you a reading based on the number of revolutions that your tyre does per kilometre, this will affect your Speedo’s accuracy, sometimes by quite a lot. This is how to work it out: A 185/75 tyre has a sidewall height of 185 x 0.75 or 138.75mm, so we want to keep this sidewall height in the new tyre as well. If you want to put 235 tyres on the car, what ratio of 235 is 138.75? To find out divide 138.75 by 235 and the answer is: 0.594. So we need a 60 series Aspect Ratio tyre of 235mm width to maintain our wheel’s rolling diameter. With an appropriate rim width, a 235/60-14 tyre is a very close match to a 185/75-14 tyre’s rolling diameter. If you want to go for a Plus1 fitment (replace the 14" rims with 15" rims) you work things out the same EXCEPT that you need to take 12.7mm off your sidewall height to allow for ½ the extra inch in rim diameter (the other ½" is taken up by the other side of the rim). So, to keep it easy, we will replace our 185/75-14 tyres with 235 tyres on 15" rims. What Series or Aspect Ratio do we need? The sidewall height on the 185 tyre is 138.75 mm, so we need to take 12.7 mm off that figure to allow for ½ the increased rim diameter. This means we need a tyre with a sidewall height of 126.05 mm. We then divide 126.05 by 235 and we get 0.536 which means we need a 55 Series or Aspect Ratio tyre. This will give you a very close rolling diameter to your original tyre. Want to go to a Plus2 fitment (replace the 14" rims with 16" rims)? Same principle, but you need to deduct 25.4 mm off your sidewall height (½ the increase in rim diameter expressed in millimetres). Using the same tyre widths as above, we would need to go from a 185/75-14 tyre to what Aspect Ratio in a 16" tyre? The sidewall height we are looking for is 138.75 mm – 25.4 mm, which equals 113.35 mm. To get a 113.35 mm sidewall in a 235 tyre, we need an Aspect Ratio of 48.2%. The closest size available to that is a 50 series tyre or 235/50-16. This tyre will have close to the same overall diameter (and rolling circumference) as your original 185/75-14 tyre. But, what if you want to fit 275 tyres on your 16" rims? Easy…divide your desired sidewall height by the tyre width (113.35/275) and you get 41.2%. Therefore, you will need a 275/40-16 tyre to give you close to the same rolling diameter as a 185/75-14 tyre. The other thing you need to be aware of with replacement tyres is the Load Rating. On your Tyre Placard in your Glove-box or on your "A" pillar, the original tyre sizes/pressures will be recorded. The other piece of information listed on there that you need is the Load Rating. If your vehicle had original tyres with a Load Rating of 600 Kg, your replacement tyres will need to have a Load Rating of 600 Kg or more. If they have a lesser Load Rating, your car will be illegal and you will void your Insurance Policy.### Compression Ratio:

What is your Compression Ratio? It is the volume of your cylinder with the piston at the top of its stroke (Top Dead Centre or TDC) compared to the cylinder’s volume with the piston at the bottom of its stroke (Bottom Dead Centre or BDC). Why is it important? The greater the compression of the fuel/air mixture in your cylinder, the more power your motor will produce. But, there are limits to the amount of compression the fuel/air mixture will allow. In the 1970s, it was not uncommon for engines to have compression ratios around 10:1. These days, compression ratios of around 9:1 are far more common. On the other hand diesels run as high as 22:1 compression, but diesels are a whole other subject and not covered in this document. The compression ratios of modern engines have gone down because of the lower Octane Rating of most of today’s fuels. Without getting too technical, Octane Rating is the resistance of the fuel to spontaneous combustion or, if you prefer, its stability. When the fuel/air mixture in your motor is compressed (when the piston rises on its compression stroke), it heats up. How much it is compressed, or squeezed, determines how hot it gets before the sparkplug ignites it. If you compress it too much (have too high a Compression Ratio) it will self ignite from the heat build-up before your piston reaches TDC and before your spark plug fires. You will then have your conrod pushing the piston up and the ignited fuel/air mixture trying to push it down at the same time. This is known as pinging or pinking (named after the sound that it makes). If this pinging occurs for a period of time, your pistons will either: melt, break, break the conrod or come out through the side of your engine block. None of these are a good look! How do you work out what your compression ratio is? Mostly, you do not have to. The engine manufacturer will tell you. But what if you want to change the head on your motor? For example, what happens if you put a 173 small combustion chamber head on a 202 (which normally has a large combustion chamber head)? For this example, we will assume that both heads are the High Compression head for the respective motors. The High Compression 173 has a compression ratio of 9.4:1. This means that the combustion chamber size is 3.07 cubic inches for each cylinder. To find this out, divide the motor size by the number of cylinders and then divide the result by the compression ratio for the engine: 173/6 = 28.83 cubic inches per cylinder. 28.83/9.4 = 3.07 cubic inches per combustion chamber. To then work out the compression ratio of this head on a 202, we divide 202 by 6 (33.67 cubic inches per cylinder) and divide this by our combustion chamber size (33.67/3.07) which will give us a compression ratio of 10.97:1. This is WAY too high to run on ULP. It is even be a bit high to run normal PULP. If you did this conversion, you would need to run one of the 98 Octane PULP fuels, such as Ultimate98 from BP. Even then, an almost 11:1 compression ratio is pushing the envelope with normal, street legal fuels. Raising the compression ratio will give you a bit better power, but it will also make your motor run a bit hotter and put more pressure on the bottom end of your motor. Make sure your motor is in good condition and that your cooling system is up to the task before you consider doing anything like this.#### Compression Ratio Calculation Amendment:

By Circlotron 23/07/16@22:29 The correct way is (cylinder + combustion chamber) divided by combustion chamber. That is to say, the largest combined volume divided by the smallest combined volume. End of submission by Circlotron.

### Carburettor Size:

How do you know what size carbie to put on your motor? The short answer is that it depends on the size of your motor, how high it revs, and how good it is as a pump. Most older motor were an inefficient pump. Their adiabatic efficiency was in the region of 60% – 75%. Most multi-valve modern motors have an adiabatic efficiency in the region of 80% - 90%. If you can improve the adiabatic efficiency of the motor, you will, potentially, get better power. There are several ways to increase the efficiency of your motor (in no particular order): * A more free flowing exhaust. This does not have to mean louder. * Head work…cleaning up your ports and around your valves inside the head (in this area size does not matter…bigger is not necessarily better). Just remember that a carbie that is too big is just as bad, or worse, than one that is not big enough! How do we work out what size carbie we need (as opposed to want)? Maximum Cubic Feet per Minute (CFM) is important for ultimate power. However, how often do you really sit on 5,500 RPM? You spend a LOT more time in the 2,000 to 3,000 RPM range, so light and mid throttle size is also important. What we are chasing at these revs is airspeed through the carbie. If the carbie is too big at this point, the speed of the air through the carbie will drop and it will not drag enough fuel through the jets. Therefore, your motor will hesitate or, at worst, stall through lack of fuel. What is the maximum CFM that you need for your motor to produce the maximum power? If your motor will rev to 5,500 RPM, you need to work out how many CFM of air it draws at those revs. To do this, multiply your engine size by the revs (308 x 5,500 = 1,694,000 cubic INCHES per minute of air), then multiply this by the best guess of the adiabatic efficiency of your motor. Assuming you have worked heads, a good exhaust and a free-flowing air filter, we’ll apply an arbitrary 80% Adiabatic Efficiency rating (1,694,000 x 0.80 = 1,355,200 cubic INCHES of air per minute). Convert that to cubic FEET per minute and we have 784 CFM (1 cubic foot = 1728 cubic inches). So, in a racing application, where we are revving the 308 to its redline all the time, we should put a 790 CFM carbie on it. For street driving, it is way more complicated than that, even if we do not take the emission laws into consideration. As I said above, you spend much more time in the mid-range revs than you do at redline (or you should when you are on the street anyway…) so midrange size is more important than ultimate size. That is why many 308s run 600 or 650 CFM carbies very happily. That size carbie will still allow the motor to rev to its redline (with a slight loss of top end power), but in the "usual" rev-range, it suits the motor well. My recommendations for carbies? For a 253 or 308, use a [Quadrajet_id_codes Quadrajet]. For a 202, use a Webber (from an XD – XF Falcon), a 390 or 465 4 barrel Holley or EFI VK Injection . For a 186, use a Webber, a 390 or 465 4 barrel Holley or VK Injection . Also refer to EFI VK Conversion . In any case, get the carbie setup by someone who REALLY does know what they are doing and preferably get it setup on a dyno. If your car is subject to Emission Laws (ADR27, ADR27A, etc), check with your registration people and/or the EPA in your state to make sure you can legally do what you want to do. Also, talk to your insurance company about any modifications you want to do to your car BEFORE you do them. You don’t want to crash into a Mercedes and find your insurance won’t cover you because you have changed your carbie!### Diff and Gearbox Ratios:

Before looking at what effect a Diff Ratio change has on your car, what does the diff do? In its basic form the diff allows the two driven wheels to rotate at different speeds. This is important when cornering and the sharper the turn, the more important it is. The diff also multiplies the torque of your motor. One problem with normal or "open" diffs is that, if they are given the option, they will direct the torque to the wheel with the least traction. A Limited Slip Diff (LSD) does exactly what its name implies; it limits the difference (or slip) between the speeds of the wheels. In basic terms, an LSD will operate as a normal diff until one wheel starts to lose traction, then it will limit the amount of torque going to the wheel that has broken traction by starting to feed it to the other wheel (the one with traction). There are various methods of achieving this limited slip. Cones, clutches and hydraulic pressure are the most common. No matter what method is used to activate the limited slip mechanism, LSD’s will try to keep the proportion of torque going to each wheel relatively even. What do diff ratios do? If you are doing approximately 120 KMH, your wheels will be turning at about 1,000 rpm. Assuming you are in 4th, and that 4th is a 1:1 ratio, with a 2.6:1 diff, your motor will be turning at approximately 2,600 rpm, with a 3.08:1 diff your motor will be turning at approximately 3,080 rpm and with a 4.44:1 diff your motor will be turning at approximately 4,440 rpm. A low diff ratio is higher numerically. The "low" part of the name refers to it being a low speed diff ratio. From the above, you can see that you should choose your diff ratio carefully. A stock 202 with a 4.44:1 diff will not be a happy motor above about 80 KMH, but it will get to this speed quite quickly. Put a 3.08:1 diff in the same car and it will accelerate more slowly (due to less multiplication of the available torque) but it will happily sit on 110 KMH. If you have a worked motor with a higher comfortable rev-range, a lower ratio diff may be a viable option for you. Don’t fall into the trap of putting a higher ratio (lower numerically) diff in your 173 HQ to break the land speed record! Wind resistance has a lot to do with top speed too. If you double your speed, you need 4 times as much power to propel your car because you are pushing twice as much air out of the way twice as fast. Now let’s look at what your gearbox does. Basically, your gearbox allows you to drive your car over a wide range of speeds. The lower gears (the ones with a ratio greater than 1) allow you to start the car from rest and accelerate up to what we consider cruising speeds. Imagine (do not try this!) starting your car from rest in your highest gear…IF your car will do it at all, it will torture your clutch. Then imagine (do not try this either!) starting from rest in your highest gear going up hill. That is why we have gearboxes. The spread of the ratios in the gearbox is important too and should be matched to the "power band" of your motor. For example, if your motor develops good power from 3,500 to 5,000 rpm, you don’t want your gearbox to drop the revs back to 3,000 rpm when you change up at 5,000 rpm…you will no longer be "on the cam" or in the power band of your motor and acceleration will suffer. A better option, in this example, will be if your gear spread allows the motor to drop to 3,500 rpm when you change up at 5,000 rpm. Most, but not all, 5 speed gearboxes have an overdriven 5th gear (it has a ratio of les than 1:1, commonly in the region of 0.8:1). This allows a reduction in revs at cruising speeds, while still allowing good acceleration in the lower gears because we can use a lower diff ratio while maintaining reasonable revs at cruising speeds. Your gearbox ratios act in concert with your diff ratio (and your tyre size) to give you overall gearing. In some racing applications (Formula1, for example), all ratios in the gearbox can be changed as well as the diff ratio…this gives the driver lots of options to alter his overall gearing to suit a particular track. On the road, we tend to choose a gearbox with a good spread of ratios and a diff that allows us to comfortably cruise at the speeds we normally travel at. How do gearbox and diff ratios work together? I remember reading that motorcycle manufacturers take more care than other companies over choosing their gearbox and final drive (diff) ratios. This is because most humans need each gear to "last" about 4 seconds. If your car will accelerate from the entry speed for a gear to that gear’s maximum speed in less than 4 seconds, you are VERY likely to over-rev the motor. This is just a point to remember when choosing your gearbox and diff…do not choose too low a diff ratio, especially if your gearbox has a low first gear. We looked at direct drive (1:1) ratios above, but what do the lower gears do to our overall gearing? It can be worked out quite easily. For this example, we’ll look at 2 Holden "All Aussie" gearboxes, the M20 and M21.

The M20 and M21 | |||||

1st Ratio |
2nd Ratio | 3rd Ratio | 4th Ratio | ||

M20 | 3.05:1 | 2.19:1 | 1.51:1 | 1.00:1 | |

M21 | 2.54:1 | 1.83:1 | 1.38:1 | 1.00:1 | |

If we were travelling at about 120 KMH, our wheels would be doing approximately 1,000 RPM. What will the gearbox and diff ratios mean to our motor? Assuming we have a 3.08:1 diff ratio, our tail-shaft will be turning at 3,080 rpm when the wheels are turning at 1,000 rpm. If we are in 4th gear, our motor will be turning at 3,080 RPM as well. What if we change back to 3rd at this speed? With an M20 gearbox, we will increase the motor’s revs to 4,650 rpm (1.51 x 3080); with an M21 gearbox we will increase the motor’s revs to 4,250 rpm (1.38 x 3,080) because of the different ratios in the two gearboxes. Lets change back to 2nd gear at the same speed (in theory…do not try this in reality!). The revs will rise to 6,745 rpm (2.19 x 3080) with the M20, or 5636 with the M21. If we then changed back to first at this speed, the motor would (VERY briefly) rev to 9,394 rpm (3.05 x 3080) with the M20 or 7,823 (2.54 x 3080) with the M21. Of course, in reality, our motor would have blown itself to bits long before approaching these revs… Our overall gearing with an M20 gearbox with a 3.08:1 and a 3.55:1 ratio diff can be summed up as follows: {| border="1" cellpadding="5" cellspacing="0" align="center" | **Gear M20 Ratio Overall gearing** |- ! style="background:#efefef;" | Gear/Diff ! style="background:#ffdead;" | with 3.08:1 diff ! style="background:#ffdead;" | with 3.55:1 diff |- | style="border-bottom:3px solid grey;" | 1st 3.05:1 | style="border-bottom:3px solid grey;" | 9.3940:1 | style="border-bottom:3px solid grey;" | 10.8275:1 |- | style="border-bottom:3px solid grey;" | 2nd 2.19:1 | style="border-bottom:3px solid grey;" | 6.7452:1 | style="border-bottom:3px solid grey;" | 7.7745:1 |- | style="border-bottom:3px solid grey;" | 3rd 1.51:1 | style="border-bottom:3px solid grey;" | 4.6508:1 | style="border-bottom:3px solid grey;" | 5.3605:1 |- | style="border-bottom:3px solid grey;" | 4th 1.00:1 | style="border-bottom:3px solid grey;" | 3.0800:1 | style="border-bottom:3px solid grey;" | 3.5500:1 |- |} What does this tell us? Something that we already know from experience: The 3.55:1 diff multiplies the torque of our motor more and therefore will accelerate more quickly. But, now we should be able to work out what diff ratio we need for our particular application and also what effect the gearbox we choose will have on the driveability of our car.