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## How to Determine Gear Ratio

Last Updated: August 17, 2023 Fact Checked

This article was reviewed by Grace Imson, MA . Grace Imson is a math teacher with over 40 years of teaching experience. Grace is currently a math instructor at the City College of San Francisco and was previously in the Math Department at Saint Louis University. She has taught math at the elementary, middle, high school, and college levels. She has an MA in Education, specializing in Administration and Supervision from Saint Louis University. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 1,737,186 times.

In mechanical engineering, a gear ratio is a direct measure of the ratio of the rotational speeds of two or more interlocking gears. As a general rule, when dealing with two gears, if the drive gear (the one directly receiving rotational force from the engine, motor, etc.) is bigger than the driven gear, the latter will turn more quickly, and vice versa. We can express this basic concept with the formula Gear ratio = T2/T1 , where T1 is the number of teeth on the first gear and T2 is the number of teeth on the second.

## Finding the Gear Ratio of a Gear Train

- For now, let's look at a gear train with only two gears in it. To be able to find a gear ratio, these gears have to be interacting with each other — in other words, their teeth need to be meshed and one should be turning the other. For example purposes, let's say that you have one small drive gear (gear 1) turning a larger driven gear (gear 2).

- For example purposes, let's say that the smaller drive gear in our system has 20 teeth.

- Let's say that, in our example, the driven gear has 30 teeth.

- In our example, dividing the 30 teeth of the driven gear by the 20 teeth of the drive gear gets us 30/20 = 1.5 . We can also write this as 3/2 or 1.5 : 1 , etc.
- What this gear ratio means is that the smaller driver gear must turn one and a half times to get the larger driven gear to make one complete turn. This makes sense — since the driven gear is bigger, it will turn more slowly.

## More than Two Gears

- Let's say for example purposes that the two-gear train described above is now driven by a small seven-toothed gear. In this case, the 30-toothed gear remains the driven gear and the 20-toothed gear (which was the driver before) is now an idler gear.

- In our example, we would find the gear ratio by dividing the thirty teeth of the driven gear by the seven teeth of our new driver. 30/7 = about 4.3 (or 4.3 : 1, etc.) This means that the driver gear has to turn about 4.3 times to get the much larger driven gear to turn once.

- In our example, the intermediate gear ratios are 20/7 = 2.9 and 30/20 = 1.5 . Note that neither of these are equal to the gear ratio for the entire train, 4.3.
- However, note also that (20/7) × (30/20) = 4.3. In general, the intermediate gear ratios of a gear train will multiply together to equal the overall gear ratio.

## Making Ratio/Speed Calculations

- For example, let's say that in the example gear train above with a seven-toothed driver gear and a 30-toothed driven gear, the drive gear is rotating at 130 RPMs. With this information, we'll find the speed of the driven gear in the next few steps.

- Often, in these sorts of problems, you'll be solving for S2, though it's perfectly possible to solve for any of the variables. In our example, plugging in the information we have, we get this:
- 130 RPMs × 7 = S2 × 30

- In our example, we can solve like this:
- 910 = S2 × 30
- 910/30 = S2
- 30.33 RPMs = S2
- In other words, if the drive gear spins at 130 RPMs, the driven gear will spin at 30.33 RPMs. This makes sense — since the driven gear is much bigger, it will spin much slower.

## Community Q&A

## Video . By using this service, some information may be shared with YouTube.

- The power needed to drive the load is geared up or down from the motor by the gear ratio. The motor must be sized to provide the power needed by the load after the gear ratio is taken in to consideration. A geared up system (where load RPM is greater than motor RPM) will require a motor that delivers optimal power at lower rotational speeds. Thanks Helpful 10 Not Helpful 2
- To see the principles of gear ratio in action, take a ride on your bike! Notice that it is easiest to go up hills when you have a small gear in front and a big one in the back. While it's easier to turn the smaller gear with the leverage from your pedals, it takes many rotations to get your rear wheel to rotate compared to the gear settings you'd use for flat sections, making you go slower. Thanks Helpful 14 Not Helpful 5
- A geared down system (where load RPM is less than motor RPM) will require a motor that delivers optimal power at higher rotational speeds. Thanks Helpful 6 Not Helpful 7

## You Might Also Like

- ↑ https://www.sae.org/binaries/content/assets/cm/content/learn/education/motortoycar-samplelessonplan.pdf
- ↑ https://sciencing.com/calculate-gear-ratio-6495601.html
- ↑ https://www.omnicalculator.com/physics/gear-ratio
- ↑ https://sciencing.com/calculate-speed-ratio-7598425.html

## About This Article

To determine gear ratio of a gear train with 2 gears, start by identifying your gears. The gear attached to the motor shaft is considered the first gear, or the “drive gear”, and the other gear, whose teeth are meshed with the drive gear, is considered the second gear, or “driven gear.” Count the number of teeth on the drive gear and on the driven gear. Then, divide the number of teeth on the driven gear by the number of teeth on the drive gear to get the gear ratio. For example, if the drive gear has 20 teeth and the driven gear has 30 teeth, the gear ratio is 1.5. If you want to learn how to use the gear ratio to calculate the gears' speeds, keep reading the article! Did this summary help you? Yes No

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## What Is Gear Ratio? It’s Formula and Calculation on Gear Ratio

In this post, you will learn what is gear ratio in gears ? and how to calculate the gear ratio. Also, you can download the PDF file at the end of this article.

The gear ratio is the ratio of the number of turns the output shaft makes when the input shaft turns once. In other words, the Gear ratio is the ratio between the number of teeth on two gears that are meshed together, or two sprockets connected with a common roller chain, or the circumferences of two pulleys connected with a drive belt .

Don’t miss out: What are the Types of Gear Cutting Process? Their Advantages, Disadvantages [PDF]

## How Gears Transmit Power

The tooth and wheel of the gear are basic workings parts of all types of gears . The different types of gear are used to execute the transfer of energy in a different direction. For instance, when two gears of different sizes mesh and rotate, the pinion will turn faster and with less torque than the larger gear.

The teeth of the gear are principally carved on wheels, cylinders, or cones. Many devices that we use in our day-to-day life there working principles as gears.

Often gears that are meshed together will be of different sizes. In this case,

- The smaller gear is referred to as the pinion and
- The larger one is simply referred to as the gear.

Gear is different from a pulley. Gear is a round wheel that has teeth that mesh with other gear teeth, allowing the force to be fully transferred without slippage .

To overcome the problem of slippage as in belt drives, gear is used which produces a positive drive with uniform angular velocity. When two or more gears mesh together the arrangement is called a gear set or a gear train .

Read about: Gear Termi nology [This is one of the Easiest Guide on Gears ]

## Gear Ratio Calculation

For example, a pinion with 18 teeth is mounted on a motor shaft and is meshed with a larger gear that has 54 teeth.

During operation, the pinion makes three complete revolutions for every single revolution of the larger gear.

This relationship in which the gear turns at one-third of the pinion speed is a result of the number of teeth on the pinion and the larger gear. This relationship is called the gear teeth – pinion teeth ratio or the gear ratio.

This ratio can be expressed as the number of gear teeth divided by the number of pinion teeth. So in this example, since there are 54 teeth on the larger gear and 18 teeth on the pinion. There’s a ratio of 54 to 18 or 3 to 1 this means that pinion is turning at three times the speed of the gear.

Now often more than one gear set is used in a gearbox multiple gear sets may use in place of one large set because they take up less space.

However, the gear ratio can still be used to determine the output of a gearbox.

## Example of Gear Ratio

Let’s see how this illustration consists of two gear sets. This gear set has a pinion with 10 teeth and a gear with 30 teeth. The second gear set consists of an opinion with 10 teeth and a gear with 40 teeth.

In our example, the input shaft is turned by an external device such as a motor. And the output shaft is connected to a machine to drive, such as a pump or a fan it’s often called the output shaft.

The input shaft and output shaft are connected by the intermediate shaft.

Now by using the gear ratio formula we looked at earlier, we can determine the ratio across the gears. The first gear set is 30 over 10 or 3 to 1. And that the ratio across the second gear set is 40 over 10 or 4 to 1. This information can be used to determine the ratio across the entire series of gears.

That’s done by multiplying the ratio of the first gear set by the ratio of the second gear set.

So 3 / 1 times 4 / 1 results in a ratio of 12 / 1 this means that for every 12 revolutions of the input shaft the output shaft will complete one revolution. Or in other words, the motor shaft is turning 12 times faster than the pump shaft.

Well, so far we’ve looked at how a speed can be changed across the gear set and we’ve seen how this change can be described by you.

Gear ratios can be used to determine the speed of rotation of a gear set if the input or output speed of the gear set is known.

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## About Saif M

Saif M. is a Mechanical Engineer by profession. He completed his engineering studies in 2014 and is currently working in a large firm as Mechanical Engineer. He is also an author and editor at www.theengineerspost.com

## 8 thoughts on “What Is Gear Ratio? It’s Formula and Calculation on Gear Ratio”

Great learning

There is an error in your diagram showing Gear ratio = #teeth on pinion / # teeth on gear

Should be Gear ratio = #teeth on gear / # teeth on pinion.

You calculate correctly afterwards but the equation presented should also be corrected.

Thank you so much for letting me know, I’ve changed the image now you can see it, and thanks for reading 🙂

thanks for your clear explanation (for us gear head wanna be’s).

you’re welcome

Best information is provided by your site. I always go for your post Engeers Post to clear my concepts in easy way. Keep it up

Thank you so much 🙂

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## Gear Ratio Calculator

What is a gear, what is gear ratio and how to calculate gear ratio, understanding gear ratio and mechanical advantage values, important note on idler gears, real-life simple machines with gears.

This gear ratio calculator determines the mechanical advantage a two-gear setup produces in a machine . The gear ratio gives us an idea of how much an output gear is sped up or slowed down or how much torque is lost or gained in a system. We equipped this calculator with the gear ratio equation and the gear reduction equation so you can quickly determine the gear ratio of your gears.

Keep on reading to learn more about gear ratio calculation and how it is essential in making simple machines (and even complicated ones).

Prefer watching rather than reading? Learn all you need in 90 seconds with this video we made for you :

A gear is a toothed wheel that can change the direction, torque, and speed of rotational movement applied to it. Gears come in different shapes and sizes (even if the most common are involute gears – see involute function calculator ), and these differences describe the translation or transfer of the rotational movement. The transfer of movement happens when two or more gears in a system mesh together while in motion. We call this system of gears a gear train .

In a gear train, turning one gear also turns the other gears. The gear that initially receives the turning force, either from a powered motor or just by hand (or foot in the case of a bike), is called the input gear . We can also call it the driving gear since it initiates the movement of all the other gears in the gear train. The final gear that the input gear influences is known as the output gear . In a two-gear system, we can call these gears the driving gear and the driven gear , respectively.

The resulting movement of the output gear could be in the same direction as the input gear, but it could be in a different direction or axes of rotation depending on the type of gear in the gear train. To help you visualize this, here is an illustration of the different types of gears and their input-to-output gear relationships:

You might also like our chain length calculator and speedometer gear calculator .

The gear ratio is the ratio of the circumference of the input gear to the circumference of the output gear in a gear train. The gear ratio helps us determine the number of teeth each gear needs to produce a desired output speed/angular velocity, or torque (see torque calculator ).

We calculate the gear ratio between two gears by dividing the circumference of the input gear by the circumference of the output gear . We can determine the circumference of a specific gear in the same way we calculate the circumference of a circle. In equation form, it looks like this:

gear ratio = (π × diameter of input gear)/(π × diameter of output gear)

Simplifying this equation, we can also obtain the gear ratio when just the gears' diameters or radii are considered:

gear ratio = (diameter of input gear)/(diameter of output gear)

gear ratio = (radius of input gear)/(radius of output gear)

Similarly, we can calculate the gear ratio by considering the number of teeth on the input and output gears. Doing so is similar to considering the circumferences of the gears. We can express the gear's circumference by multiplying the sum of a tooth's thickness and the spacing between teeth by the number of teeth the gear has:

gear ratio = (input gear teeth number × (gear thickness + teeth spacing)) / (output gear teeth number × (gear thickness + teeth spacing))

But, since the thickness and spacing of the gear train's teeth must be the same for the gears to engage smoothly, we can cancel out the gear thickness and teeth spacing multiplier in the above equation, leaving us with the equation below:

gear ratio = input gear teeth number / output gear teeth number

The gear ratio, just like any other ratio, can be expressed as:

A fraction or a quotient – where, if possible, we simplify the fraction by dividing both the numerator and the denominator by their greatest common factor.

A decimal number – expressing the gear ratio as a decimal number gives us a quick idea about how much the input gear has to be turned for the output gear to complete one full revolution.

An ordered pair of numbers separated by a colon, such as 2:5 or 1:14 . With this, we can see the fewest number of turns required for both the input and output gears to return to their original positions at the same time.

From a different perspective, if we take the reciprocal of the gear ratio in its fractional form and simplify it to a decimal number, we get the value for the mechanical advantage (or disadvantage) our gear train or gear system has.

Gear ratios are pretty easy to understand, and now that we know how to calculate a gear ratio, wouldn't it be better to know how it affects the gears themselves? To better explain gear ratios, let us consider a two-gear system where the input and output gears have ten and forty teeth, respectively:

Following our gear ratio equation, we can say that this gear train has a gear ratio of 10:40, 10/40, or simply 1/4 (or 0.25). This gear ratio means that the output gear would only rotate 1/4 of a full rotation after the input gear has completed a full turn . Continuing in this fashion and keeping a consistent input speed, we see that the rate of the output gear is also 1/4 of that of the input speed. In other words, the speed of the input gear is four times the speed of the output gear, as can be seen in the animated image below:

While this setup demonstrates a gear reduction in terms of speed, in return it provides us with an output that has more torque , when compared to the input. The reciprocal of its gear ratio is 4/1, so we can say that we get four times the mechanical advantage when it comes to torque.

A spur gear of any number of teeth between the input and output gears does not change the total gear ratio of the gear train. However, this gear (or gears) can change the direction of the output gear. We call this in-between gear an idler gear . As an example, here is a 1:2.5 gear reduction system with an additional idler gear:

Without the idler gear, here is the same gear train. Note that the direction of the output gear is reversed:

We see gears in our everyday lives, and to gain an even better understanding of gear ratios, here are some real-life examples of simple machines with gears in them:

Mechanical advantage in terms of speed

Hand drills, though they seem less popular nowadays, are a great example of a simple machine that demonstrates a mechanical advantage in terms of speed. Cranking its handle will spin the drill bit at high speed.

Mechanical advantage in terms of torque

Going uphill, riding a bike is easier if you are in a low-speed gear . Doing so results in better torque, providing more power when going uphill. This may mean we have to pedal more, but our ascend will be much easier. A bicycle sprocket-and-chain mechanism is much like a rack-and-pinion setup. The chain acts as a rack gear, directly transferring the motion to the rear bike sprocket (see the bike gear calculator ).

A gear is a circular machine part that transmits torque when it meshes with its counterpart. Gears are usually a vital part of any machine with moving parts, such as a wristwatch and an automobile.

## What are different types of gears?

There are different types of gear depending upon the angle of power transmission. For parallel transmission, these include spur, helical, herringbone, and planetary gears. Bevel and spiral bevel gears are used for perpendicular transmission.

## What is the gear ratio?

Gear ratio is defined as the ratio of the circumference of two gears that mesh together for power transmission. This parameter determines if the amount of power transmission will increase or decrease.

## How do you calculate gear ratio?

To calculate the gear ratio:

Find the 1st (driving) gear's number of teeth or diameter.

Find the 2nd (driven) gear's number of teeth or diameter.

Divide the number of the driving gear by that of the driven gear to find the gear ratio.

Alternatively, you can also find out the gear ratio by dividing the speed of the 1st gear by the 2nd gear.

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## How Gear Ratios Work

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Key Takeaways

- Gear ratios are fundamental in understanding how gears work, with the ratio indicating how many times a gear has to turn for another gear to turn once; for instance, in a worm gear, a threaded shaft engaging the teeth on a gear can create a high gear ratio in a compact space.
- Planetary gear systems are specialized gear trains that can produce different gear ratios depending on which gear is used as the input, output, or held stationary; they are rugged due to multiple gears engaging simultaneously and are commonly found in automatic transmissions.
- The versatility of planetary gearsets allows for different gear ratios by rearranging the input, output, and stationary gears, with automatic transmissions utilizing this feature by using clutches and brake bands to change the stationary parts and alter inputs and outputs.

You see gears in just about everything that has spinning parts. Car engines and transmissions contain lots of gears. Wind-up, grandfather and pendulum clocks contain plenty of gears, especially if they have bells or chimes. You probably have a power meter on the side of your house, and if it has a see-through cover you can see that it contains 10 or 15 gears. Gears are everywhere where there are engines or motors producing rotational motion.

In this edition of HowStuffWorks , you will learn about gear ratios and gear trains so you'll understand what all of these different gears are doing. You might also want to read How Gears Work to find out more about different kinds of gears and their uses, or you can learn more about gear ratios by visiting our gear ratio chart .

## Putting Gears to Work

Understanding the concept of gear ratio, gear trains, other uses for gears.

Gears are generally used for one of four different reasons:

- To reverse the direction of rotation
- To increase or decrease the speed of rotation
- To move rotational motion to a different axis
- To keep the rotation of two axes synchronized

You can see effects 1, 2 and 3 in the figure above. In this figure, you can see that the two gears are rotating in opposite directions, that the smaller gear is spinning twice as fast as the larger gear, and that the axis of rotation of the smaller gear is to the right of the axis of rotation of the larger gear.

The fact that one gear is spinning twice as fast as the other is because of the ratio between the gears — the gear ratio . In this figure, the diameter of the gear on the left is twice that of the gear on the right. The gear ratio is therefore 2:1 (pronounced "two to one"). If you watch the figure, you can see the ratio: Every time the larger gear goes around once, the smaller gear goes around twice. If both gears had the same diameter, they would rotate at the same speed but in opposite directions.

Understanding the concept of the gear ratio is easy if you understand the concept of the circumference of a circle—the distance around the circle's perimeter.

Let's say that you have circle whose circumference is 4 inches and a circle whose circumference is 2 inches. If you roll each circle along a 4-inch line, the first circle will cover the distance in a single full rotation; since the second circle's circumference is half that of the first circle, it has to complete two full rotations to cover the same distance. This explains why the two gears, one with half the circumference of the other, have a gear ratio of 2:1. The smaller gear has to spin twice to cover the same distance covered when the larger gear spins once.

Since the equation for calculating the circumference of a circle is simply pi multiplied by the circle's diameter, you can also calculate gear ratios by comparing two circles' diameters. Two gears, one with a diameter of 6 inches and another with a diameter of 3 inches, will have a gear ratio of 2:1.

Most gears that you see in real life have teeth . The teeth have three advantages:

- They prevent slippage between the gears. Therefore, axles connected by gears are always synchronized exactly with one another.
- They make it possible to determine exact gear ratios. You just count the number of teeth on the two gears and divide the two numbers. So if one gear has 60 teeth and another has 20, the gear ratio when these two gears are connected together is 3:1.
- They make it so that slight imperfections in the actual diameter and circumference of two gears don't matter. The gear ratio is controlled by the number of teeth even if the diameters are a bit off.

To create large gear ratios, gears are often connected together in gear trains , as shown on the left.

The right-hand (purple) gear in the train is actually made in two parts: A small gear and a larger gear are connected, one on top of the other. Gear trains often consist of multiple gears in the train, as shown in the next figures.

In the case above, the purple gear turns at a rate twice that of the blue gear. The green gear turns at twice the rate of the purple gear. The red gear turns at twice the rate as the green gear. The gear train shown below has a higher gear ratio:

In this train, the smaller gears are one-fifth the size of the larger gears. That means that if you connect the purple gear to a motor spinning at 100 revolutions per minute (rpm), the green gear will turn at a rate of 500 rpm and the red gear will turn at a rate of 2,500 rpm. In the same way, you could attach a 2,500-rpm motor to the red gear to get 100 rpm on the purple gear. If you can see inside your power meter and it's of the older style with five mechanical dials, you will see that the five dials are connected to one another through a gear train like this, with the gears having a ratio of 10:1. Because the dials are directly connected to one another, they spin in opposite directions (you will see that the numbers are reversed on dials next to one another).

If you want to create a high gear ratio, nothing beats the worm gear . On a worm gear, a threaded shaft engages the teeth on a gear. Each time the shaft spins one revolution, the gear moves one tooth forward. If the gear has 40 teeth, you have a 40:1 gear ratio in a very small package. Worm gears allow windshield wipers to function. A mechanical odometer is another place that uses a lot of worm gears.

## Planetary Gears

There are many other ways to use gears. One specialized gear train is called a planetary gear train . Planetary gears solve the following problem. Let's say you want a gear ratio of 6:1 with the input turning in the same direction as the output. One way to create that ratio is with the following three-gear train.

In this train, the blue gear has six times the circumference of the yellow gear (giving a 6:1 ratio). The size of the red gear is not important because it is just there to reverse the direction of rotation so that the blue and yellow gears turn the same way. However, imagine that you want the axis of the output gear to be the same as that of the input gear. A common place where this same-axis capability is needed is in an electric screwdriver . In that case, you can use a planetary gear system, as shown here.

In this gear system, the yellow gear (the sun ) engages all three red gears (the planets ) simultaneously. All three are attached to a plate (the planet carrier ), and they engage the inside of the blue gear (the ring ) instead of the outside. Because there are three red gears instead of one, this gear train is extremely rugged. The output shaft is attached to the blue ring gear, and the planet carrier is held stationary — this gives the same 6:1 gear ratio. You can see a picture of a two-stage planetary gear system on the electric screwdriver page , and a three-stage planetary gear system of the sprinkler page . You also find planetary gear systems inside automatic transmissions.

Another interesting thing about planetary gearsets is that they can produce different gear ratios depending on which gear you use as the input, which gear you use as the output, and which one you hold still. For instance, if the input is the sun gear, and we hold the ring gear stationary and attach the output shaft to the planet carrier, we get a different gear ratio. In this case, the planet carrier and planets orbit the sun gear, so instead of the sun gear having to spin six times for the planet carrier to make it around once, it has to spin seven times. This is because the planet carrier circled the sun gear once in the same direction as it was spinning, subtracting one revolution from the sun gear. So in this case, we get a 7:1 reduction.

You could rearrange things again, and this time hold the sun gear stationary, take the output from the planet carrier and hook the input up to the ring gear. This would give you a 1.17:1 gear reduction. An automatic transmission uses planetary gearsets to create the different gear ratios, using clutches and brake bands to hold different parts of the gearset stationary and change the inputs and outputs.

Imagine the following situation: You have two red gears that you want to keep synchronized, but they are some distance apart. You can place a big gear between them if you want them to have the same direction of rotation, as is shown in the image.

Or you can use two equal-sized gears if you want them to have opposite directions of rotation.

However, in both of these cases the extra gears are likely to be heavy and you need to create axles for them. In these cases, the common solution is to use either a chain or a toothed belt , as shown.

The advantages of chains and belts are light weight, the ability to separate the two gears by some distance, and the ability to connect many gears together on the same chain or belt. For example, in a car engine, the same toothed belt might engage the crankshaft, two camshafts and the alternator. If you had to use gears in place of the belt, it would be a lot harder.

For more information on gears and their applications, check out the links that follow.

## Gear Ratio FAQ

How is gear ratio calculated, is it better to have a higher or lower gear ratio, what is a good gear ratio for towing, do bigger gears give more torque, what is the purpose of gear ratio, lots more information, related howstuffworks articles.

- Gear Ratio Chart
- How Gears Work
- How Pendulum Clocks Work
- How Bicycles Work
- How Oscillating Sprinklers Works
- How Differentials Works
- Inside an Electric Screwdriver
- Inside a Bathroom Scale

## More Great Links

- Gears: An Introduction
- Some notes on a clock design
- Gears: Epicyclic Train Example
- Automobile Differential Gears

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## How to Calculate Gear Ratio

## Rack-and-Pinion: Gear Ratio

Gears are practically everywhere. They're in cars, both in the transmission and in the windshield wipers. They're in bicycles, in such kitchen utensils as the egg beater and even in watches – or at least they used to be. A gear is basically a set of toothed wheels coupled together to increase or decrease the speed of rotation of a motor drive shaft.

The amount a gear system can alter rotational speed is a function of the relative sizes of the gear wheels, and it's known as the gear ratio. The gear ratio formula turns out to be fairly simple. You basically count the number of teeth on the driven wheel and divide that by the number of teeth on the driver wheel, which is the one attached to the motor. It's a straightforward calculation, even when the gear system consists of several intermediate wheels called idlers.

## It's Easier to Calculate Gear Ratio Than You Think

When you couple two gear wheels, their relative sizes determine how fast each will spin. If the driver wheel is smaller than the driven wheel, it will spin more often than the larger one. If the driver wheel is larger, the driven wheel will spin faster.

You could calculate the amount of speeding up and slowing down that a simple gear system produces by comparing the radii of the wheels, but there's an easier way. Because the teeth of both gear wheels interlock, they have to be the same size on both wheels, so you can simply compare the number of teeth on the two wheels. This is actually how you calculate gear ratio. You count the number of teeth on both the driver wheel and on the driven wheel and express these numbers as a ratio, or a fraction.

For example, if the driver wheel has 20 teeth, and the driven wheel has 40, calculate the gear ratio as 40/20, which simplifies to 2/1, or 2:1. (The tooth count on the driven wheel always goes on top of the fraction or first in the ratio). This tells you that, for every rotation of the driven wheel, the driver wheel makes two rotations. Similarly, a ratio of 1/2 tells you that the driven wheel rotates twice for every rotation of the driver wheel -- in other words, the driven wheel spins faster than the motor shaft.

## How to Apply the Gear Ratio Equation to Complex Systems

Many gear systems incorporate one or more idler wheels, which are often there to ensure that the driver wheel and driven wheel spin in the same direction or to change the plane of rotation. You can apply the gear ratio formula to each pair of wheels in the gear system in succession to arrive at a final gear ratio for the system, but you don't need to do that. If you do, you'll find that the product of all the gear ratios is the same as the ratio between the driver wheel and the driven wheel.

In other words, the driver wheel and the driven wheel are the only two that matter. No matter how many idlers there are in the system, the final gear ratio is the ratio between the driver wheel and the driven wheel. This is true for all types of gears, including spur gears, bevel gears and worm gears.

## Using Gear Ratio to Calculate Speed

If you know the rotational speed of the driver wheel, which is usually measured in revolutions per minute (rpm), the gear ratio tells you the speed of the driven wheel. For example, consider a system with a gear ratio of 3:1, which means the driver wheel spins three times as fast as the driven wheel. If the speed of the driver wheel is 300 rpm, the speed of the driven wheel is 100 rpm.

In general, you can calculate rotational speed using the following gear ratio equation:

S 1 • T 1 = S 2 • T 2 , where

S 1 is the speed of the driver wheel and T 1 is the number of teeth on that wheel.

S 2 and T 2 are the speed and tooth count of the driven wheel.

If you're designing a gear system, you will find a gear ratio chart handy. You can find the rpm of the motor in the specifications and use the chart to design a gear system that will produce whatever rotational speed in the driven wheel you require.

## Related Articles

How to calculate speed ratio, sprocket ratio calculations, how to calculate rack & pinion, simple gear ratio explained, how to calculate gear pitch, how to calculate wheel circumference, how to calculate reduction ratio, how to find belt and pulley speeds, how to convert rpm to mph with a calculator, how much weight does a pulley take off, how does a pulley system work, examples of gears and pulleys, how to calculate the mechanical advantage of a wedge, how to convert gallons, quarts, pints and cups, how do mechanics use math, adding & subtracting fractions, how to use pulleys for speed reduction, how to convert a fraction to a ratio.

- Southern California Home Shop Machinists: Gear Ratio Calculations
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About the Author

Chris Deziel holds a Bachelor's degree in physics and a Master's degree in Humanities, He has taught science, math and English at the university level, both in his native Canada and in Japan. He began writing online in 2010, offering information in scientific, cultural and practical topics. His writing covers science, math and home improvement and design, as well as religion and the oriental healing arts.

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## Gear Ratio Calculator

What is a gear and how do gears work, how to calculate the gear ratio, how to express the gear ratio, applications of the gear ratio formula: using the gear ratio to calculate speed and torque, connecting more than one gear: the gear ratio formula for complex gear trains.

Kick in high gear with our gear ratio calculator: here, you will learn how gear multiplies speed and torque using only a couple of simple mathematical formulas. Keep reading our article to find out:

- What is a gear ?
- What is the gear ratio ?
- How to find the gear ratio of a gear train;
- How to use the gear ratio to calculate the speed of the output gear; and
- The formula that relates the gear ratio and the torque.

A gear is one of the fundamental elements of every machine. Gears are everywhere : your car is full of them (and if you don't have a car, your bike literally is gears).

But what is a gear? A gear is a toothed wheel , a rotating device covered with cogs .

Each cog is a tooth of the gear. When you mesh two gears, you obtain a mechanism composed of two cogwheels (the name changes, but don't worry: we will keep calling them gears ). If you connect to gears with a chain (as in a bike), then you get two sprockets .

There are, of course, many types of gears. They can differentiate:

- In the shape ;
- In the shape of the cogs (helical, spur);
- In the orientation of the cogs: they can be vertical or horizontal, inside or outside;
- In the direction and type of the transferred motion (rotation, linear; clockwise, anticlockwise);

The classic gear we all know is, most likely, an involute gear of the type called a spur .

When you pair (mesh) two gears, some engineering magic happens: the torque and speed are transmitted from one gear to the other. What we just build is a gear train , and here it is:

Let's take a better look at that gear train: the most immediate thing is that turning one gear causes the other to turn too . The gear which causes the rotation is called the driving gear , while the one "receiving" it is, of course, called the driven gear .

In a more complex gear train, we can identify many pairs of driving-driven gears and an input and one or more output gears . An exemplary gear train is represented in the pictures below:

The gear ratio is a quantity defined for each couple of gears : we calculate the gear ratio as the ratio between the circumference of the driving gear to the circumference of the driven gear :

Where d i d_i d i is the diameter of the i th i^{\text{th}} i th gear . As you can clearly see, we can simplify this equation in two ways. As the ratio of the diameters:

And as the ratio of the radii :

These versions of the gear ratio formula require you to know the size of the gears , or to measure them. What if we told you that you could also learn how to calculate the gear ratio by counting the number of teeth ?

Measure the thickness of the gear's teeth , and their spacing . We can then write the gear ratio formula:

This is pretty much equivalent to calculating the gear ratio with the diameter of the gears; however, there's a lucky catch! The teeth's parameters must be the same: this allows to finally simplify the gear ratio formula to:

If you want, substitute driven with output and driving with input to use the alternative nomenclature.

Now you know how to find the gear ratio: but how do we express it? For now, you've seen the decimal form ; however, we can express it in three different ways :

- As a decimal number : this way, we immediately know how many turns of one gear correspond to a turn of the other.
- As a fraction : this is particularly useful in subsequent calculations.
- As a ratio : two integer numbers separated by a colon. The form a : b \text{a}:\text{b} a : b tells us that one gear needs a \text{a} a turns and the other b \text{b} b turns to return to the original position .

The gear ratio gives us indications of the rotational speed of the elements of a gear train. However, we often need to know also the effective advantage we'd get by using that particular combination of gears.

In this section, we will introduce and teach you how to calculate the mechanical advantage of a gear train.

Calculating the mechanical advantage is straightforward: we simply have to take the inverse of the reciprocal of the gear ratio . This means that we don't really need to calculate it to find out how to apply it: we can simply use the gear ratio formula!

We use the gear ratio to calculate the speed of the output gear given the speed of the input one. How?

Notice that we are calculating the angular velocity : we can use the gear ratio to calculate the RPM , rotation per minute: the tangential speed is necessarily the same: if this was not true, the gears would slip.

Take this example: an input gear with n in = 14 n_{\text{in}}=14 n in = 14 meshing with an output gear with n in = 14 n_{\text{in}}=14 n in = 14 .

Let's say the input gear is turning at a speed of 30 RPM 30\ \text{RPM} 30 RPM . Can you use the gear ratio to calculate the RPM of the output gear too?

As you can see, the bigger gear spins slowlier than the input one.

We can't use the gear ratio to calculate the torque in the same way: in fact, this quantity decreases with an increase in speed. We need to use the gears' mechanical advantage :

Back to the example. Let's say that you are feeding the smaller gear with a torque of 20 N ⋅ m 20\ \text{N}\cdot\text{m} 20 N ⋅ m . What's the torque on the bigger gear?

Thanks to the gear ratio, the torque increased!

If your gear train is made of more than one gear, what will happen to the speed of the output gear?

Let's consider the following gear train:

We are considering the input to be the bigger gear on the left. We can define two gear ratios :

For the first pair, and:

For the second pair. Say that the first gear is rotating at 60 RPM 60\ \text{RPM} 60 RPM . Calculate the speed of the last gear:

Now calculate the speed for the last gear:

Now, just for "fun", try to pass directly from the first to the last gear:

Same result! This is because gear ratios are multiplicative :

Do you want an easier explanation: the tangential velocity of the gears is conserved! The circumference of the middle gear moves as fast as the one of the bigger one: for the last gear, there's no difference... apart from the direction of the rotation . What we did is to insert an idle gear , which maintains the speed but inverts the direction of the output gear!

## Gear ratio speed

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## SMLease Design

Gear train : gear ratio, torque and speed calculations.

A Gear train consists of two or more gears in series. Its application is to increase or decrease the speed or torque of the output shaft. We use Gear Ratio to calculate the speed and torque of the output gear or shaft when the input shaft or gear torque is known.

For example, a gearbox is a type of gear train. Its application is to change (increase or decrease) the car’s speed by changing the engine torque. Any change in torque depends on what gear you are driving your vehicle. This article covers the gear train, gear ratio, speed, and torque calculations.

To understand the gear ratio, we suggest you read this article on gear Terminology ( Various Terms Used in Gears) and Various Types of Gears .

## Law of Gearing

The Law of Gearing states that the angular velocity ratio between mating gears remains constant. To achieve the law of gearing or constant angular velocity, a normal at the point of contact between mating gear teeth always passes through the pitch point. Here the Pitch point is the point of contact between mating gear pitch circles.

We can conclude the following relation if the angular velocity of the mating gear is constant :

Where ω1 and ω2: Angular Velocity in radian/sec for driver and driven gear.

n1 and n2 = Gear Speed in RPM for driver and driven gear.

d1 and d2 = driver and driven gear diameter.

T1 and T2 = Number of Teeth on driver and driven gear.

The Gear ratio is the ratio of the number of teeth of the driven or output gear and the driver or input gear. It is used to calculate the speed and torque of the output shaft when input and output shafts are connected using a gear train.

The driver or Input gear is the gear where we apply the torque. Driven or output gear is a gear where we use the applied torque. And the gears used in between the driver and driven gears are known as idler gears .

## Gear Ratio and Speed

Power transmission through the gear train affects the rotational speed of the output shaft as well.

## Speed of Output Shaft = Speed of input Shaft / Gear Ratio

As shown above, if the number of gears on the output shaft is greater than the gears on the input shaft. This arrangement is also known as reduction gear Drive. In this arrangement, the output shaft will have a low speed compared to the input shaft.

Output shaft speed will be high, compared to the input shaft speed, when the number of gears on the output shaft is less than the gears on the input shaft.

## Gear Ratio and Torque

According to the law of gears, in a Gear Train, the Ratio of output torque to input torque is also constant and equal to the Gear ratio. Therefore if the input torque is known, we can calculate the output torque by multiplying the input torque with the gear ratio.

## Gear Train Types and their Calculation

A gear train consists of a series of gears to transfer power from one shaft to another. For example, power from the engine is transferred to the wheels through the gearbox. Here are the four different types of the gear train.

- Simple Gear Train
- Compound Gear Train
- Reverted Gear Train
- Planetary Gear Train

## 1) Gear Ratio Calculations For Simple Gear Train

A simple gear train is a gear train with to or multiple gears between input and output shaft.

## 1.1) Two Gear Train

Two Gear Train is a type of Simple gear train with two connected gears. For Example, As shown below in a two-gear train. Gear-1 is the driver, and Gear-2 is the driven gear. When the the driver gear rotates in a clockwise direction, the driven gear starts rotating in an anti-clockwise direction.

Question: Calculate the Speed and torque of the output shaft for a simple gear train if the number of teeth on the driver and driven gear are 40 and 20. And Driver gear is rotating with 100 rpm and 10 N-m torque.

Number of teeth on driver Gear (T1) = 40

Number of teeth on driven Gear (T2) = 20

Speed of Driver Gear (n1) = 100 rpm

The torque acting on the driver gear = 10 N-m

Gear Ratio Calculation

GR = T2 / T1 = 20/40 = 0. 5

Output Gear Speed Calculation

Speed of Output Shaft/Gear = n1/GR = 100/0. 5 = 20 0 rpm

Output Shaft/Gear Torque Calculation

Torque generated by Driven gear = GR × Torque Generated by driver

= 0. 5 × 10 = 5 N-m

## 1.2) Multi Gear Train

Multi-gear trains consist of more than two gears to transfer motion from one shaft to another. The resultant gear ratio can be calculated by multiplying individual gear ratios.

Question: Calculate the gear ratio for multi-gear trains if the number of teeth on the driver, idler, and driven gear is 40, 20, and 10 .

Given Number of teeth

T1 = 40, T2 = 20, T2 = 10

Gear Ratio (GR) Calculation for Multi gear Train

Step-1: Calculate Gear-Ratio between Gear-1 and Gear-2 (Driver and Idler).

GR(1-2) = 20/40 = 0. 5

Step-2: Calculate GR in between Gear-2 and Gear-3 (Idler and Driven Gear).

GR(2-3) = 10/20 = 0. 5

Step-3: By Multiplying gear-ratio between 1 to 2 and 2 to 3. We will get resultant GR between Driver and Driven Gear.

Resultant Multi Gear Train GR = 0. 5 × 0.5 = 0.25

From the above, calculated gear ratio we can calculate the speed and torque at output gear.

## 2) Gear Ratio Calculations For Compound Gear Train

Compound Gears consist of more than one gear on a single axis. Therefore gears on the same shaft rotate at the same speed and torque.

Question: Calculate the gear ratio for the compound gear train if the number of teeth on the driver and driven gear are 40 and 10 with one compound gear. Compound gear has total of two gears. The 1st compound gear connected to the driver has 30 teeth, and the 2nd compound gear connected to the output has 20 teeth.

Given number of teeth

T1 = 40, T2 = 30, T3 = 20, T4 = 10

Gear Ratio Calculation For Compound Gear

In the above example, Gear-2 and Gear-3 are on the same shaft.

Step-1: Calculate Gear Ratio between Gear-1 and Gear-2

GR(1-2) = 30/40 = 0.75

Step-2: Calculate GR between Gear-3 and Gear-4.

GR(3-4) = 10/20 = 0.5

Step-3: Multiply GR(1-2) and GR(3-4)

Resultant Compound Gear GR = 0.375

## 3) Reverted Gear Train.

Reverted gear trains are a type of compound gear train in which input and output shafts are on the same axis. In the above example, gear-1 and gear-3 are on the same axis.

They are used to achieve a high gear ratio within a limited space. The reverted gear train gear ratio is calculated similarly to the compound gear train.

To sum up, the gear ratio is used to calculate the resulting gear speed and torque. The gear ratio of a gear train depends on the number of teeth on the driver, idler, and driven gear. We suggest you first read this article on Reduction Gears.

Got Questions?

We will be happy to help.

If you think we missed Something? You can add to this article by sending a message in the comment box. We will do our best to add it to this post.

After understanding this article, we suggest you take this exam to validate your knowledge and understanding.

## 6 Responses

Good write-up – thanks. Question related to my Renault (Dacia) Duster 1.5 dCi. My first gear ratio = 4.45 and my sixth gear ratio is 0.62. The final drive is 4.86. The outside diameter (OD) of my current 215/65/16 tyres is 685.9mm. I’d like to mount 225/75/16 tyres at OD of 743.9mm, but change the first and sixth gears for the final drive to me more and less than currently. At 2,000 rpm in sixth gear, my current speedometer reads 86 km/hr. I’d like the final speed to be between 110 and 120 km/hr. A “taller(?)” first gear would allow slower crawling speed with more torque over off-road rocky inclines. Any suggestions on the number of teeth (or gear ratios) for first and sixth gears? I assume that I should spread the ratios for the other gears, but my current spread makes no sense to me (4.45, 2.59, 1.63, 1.11, 0.81, 0.62). Cheers Brian Y

Do you have to consider negative signs here as well because we reverse directions?

I wih this was connected to an app as I use simular formula to thi speed numbers and torque. If you can help this with software or an app count m in!!!!

Please let us know what help do you need

Thank you for make it easier to calculate the final ratio. Today I was calculating the 5 gears train ration in a lathe machine and almost went crazy. Your “multiplication” of individual rations between gears made it easier. Thanks again. Considering A the driver gear and F the driven gear, and B/C and D/E in the same shaft, I was doing (B/A) * B * C / D * C / (F/E), I think. Your way is easier, B/A=J, D/C=K, F/E=L, then J*K*L is the answer. The only problem is remember that every division must be the intermediate Driven/Driver

In example 1.1 (Two gear train), If the driver gear meshes with two driven gears simultaneously, would that affect the torque of the driver gear?. I mean if it takes 10Nm input from driver for driven gear #2 to rotate at 200rpm and we connect an additional driven gear #3 of 20 teeth to the driver, how much torque is generated at gear #2 and #3?

Thanks in advance DS

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## Definition and purpose of gears

Gears are mechanical components with teeth that mesh with other gears to transmit rotational motion, force, or torque. They are used in various machines and mechanisms to:

- Change the speed of rotation: Gears can increase or decrease the rotational speed of an output shaft compared to an input shaft.
- Change the direction of rotation: Gears can reverse the direction of rotation.
- Transfer motion between non-parallel shafts: Some types of gears can transmit motion between shafts that are not parallel.
- Increase or decrease torque: Gears can amplify or reduce the torque, providing a mechanical advantage or disadvantage.

## Different types of gears

Helical gears have teeth that are cut at an angle to the gear's axis of rotation. The angled teeth engage gradually, resulting in smoother and quieter operation compared to spur gears. Like spur gears, helical gears transmit motion between parallel shafts but with improved efficiency and reduced noise.

Bevel gears

Bevel gears have conical shapes and are used to transmit motion between intersecting shafts, typically at a 90-degree angle. The teeth can be straight, spiral, or hypoid, with each type offering different performance characteristics. Bevel gears can change the speed, torque, and direction of rotation between non-parallel shafts.

Worm gears are a special type of gear used to transmit motion between non-parallel and non-intersecting shafts. A worm gear consists of a screw-like gear (the worm) that meshes with a larger gear (the worm wheel). Worm gear assemblies provide a high reduction ratio and can achieve a large mechanical advantage, but they usually operate with lower efficiency due to friction.

## Basic gear terminology

- Gear teeth: The protrusions on a gear's circumference that interlock with the teeth of another gear, allowing them to transmit motion. The shape and size of gear teeth can vary depending on the type of gear and the application.
- Pitch: The pitch of a gear refers to the distance between corresponding points on adjacent teeth, typically measured along the pitch circle (an imaginary circle that passes through the center of the gear teeth). The pitch determines how well gears mesh together and directly affects the gear ratio.
- Diameter: The diameter of a gear is the distance across the gear, usually measured at the pitch circle. Larger gears typically have more teeth and provide more torque, while smaller gears have fewer teeth and deliver higher rotational speed.
- Gear ratio: The gear ratio is the relationship between the number of teeth on two meshing gears. It is calculated by dividing the number of teeth on the output gear (the driven gear) by the number of teeth on the input gear (the driver gear). The gear ratio determines the mechanical advantage, speed, and torque of a gear system. A high gear ratio provides greater torque at lower speeds, while a low gear ratio provides higher speed at lower torque.

## Gear Ratios and Mechanical Advantage

Understanding gear ratios, definition and purpose.

Gear ratio is the relationship between the number of teeth on two meshing gears. It determines how the speed and torque are transferred between the gears, allowing for various mechanical advantages. Gear ratios are essential in designing gear systems for specific applications, as they can be used to control the output speed and torque, based on the input speed and torque.

## Calculating gear ratios

To calculate the gear ratio between two meshing gears, divide the number of teeth on the output gear (the driven gear) by the number of teeth on the input gear (the driver gear). The formula is:

Gear Ratio = (Number of Teeth on Driven Gear) / (Number of Teeth on Driver Gear)

For example, if the driven gear has 40 teeth and the driver gear has 10 teeth, the gear ratio is 40/10 = 4. This means that the driven gear will rotate one-fourth as fast as the driver gear, but with four times the torque.

## Mechanical advantage

Definition and importance.

Mechanical advantage is the ratio of output force (or torque) to input force (or torque) in a mechanical system. In gear systems, the mechanical advantage is equal to the gear ratio. A higher mechanical advantage means that a system can amplify the input force or torque, making it easier to perform tasks that require significant force or torque. Mechanical advantage is crucial in the design of machines and mechanisms, as it allows engineers to optimize the system for a specific purpose or application.

## Examples of mechanical advantage in gear systems

A bicycle's gears

In a bicycle, the gear system enables the rider to adjust the mechanical advantage by changing the gear ratio. A lower gear ratio (smaller front chainring and larger rear sprocket) provides a higher mechanical advantage, making it easier to pedal uphill or start from a stop. A higher gear ratio (larger front chainring and smaller rear sprocket) provides a lower mechanical advantage, allowing the rider to achieve higher speeds on flat surfaces or downhill.

A car's transmission

In a car, the transmission uses different gear ratios to provide the engine with a range of mechanical advantages. Lower gear ratios provide higher mechanical advantage, which helps the car accelerate or climb steep hills. Higher gear ratios provide lower mechanical advantage, allowing the car to maintain high speeds with lower engine RPMs, improving fuel efficiency.

## Speed, torque, and power relationships

Speed and torque trade-offs.

In a gear system, there is a trade-off between speed and torque. When the gear ratio is increased (by using a larger driven gear or smaller driver gear), the output speed decreases while the output torque increases. Conversely, when the gear ratio is decreased (by using a smaller driven gear or larger driver gear), the output speed increases while the output torque decreases. This trade-off allows engineers to design gear systems that meet specific speed and torque requirements for various applications.

## Conservation of power in gear systems

The conservation of power principle states that the power input to a gear system is equal to the power output, assuming no energy losses due to friction or other factors. Power is the product of torque and rotational speed. In gear systems, the input power (Pin) equals the output power (Pout):

Input Torque (Tin) × Input Speed (ωin) = Output Torque (Tout) × Output Speed (ωout)

This relationship shows that while the speed and torque can be manipulated using gear ratios, the overall power remains constant (assuming no energy losses). This principle is crucial in understanding how gear systems can be designed to meet specific power requirements while manipulating speed and torque.

For example, if a gear system is designed to increase torque, the output speed will decrease to maintain the same power level. Conversely, if the gear system is designed to increase speed, the output torque will decrease to keep the power constant.

Understanding the conservation of power principle, along with the concepts of gear ratios and mechanical advantage, will help you analyze and solve various gear-related problems in mechanical aptitude tests and real-life applications.

## Gear Trains

Types of gear trains.

Gear trains are combinations of two or more gears used to transmit motion, force, or torque from one shaft to another. They are commonly used in various machines and mechanisms to achieve specific speed, torque, and direction requirements. There are three main types of gear trains:

## Simple gear trains

A simple gear train consists of two or more gears mounted on parallel shafts that mesh directly with each other. The input gear (driver) transmits motion to the output gear (driven) through intermediate gears, if present. Simple gear trains are used to change the speed, torque, and direction of rotation between the input and output shafts.

## Compound gear trains

A compound gear train consists of two or more gears mounted on the same shaft, with each pair of gears meshing with another pair on a different shaft. Compound gear trains allow for larger gear ratios than simple gear trains, making them suitable for applications requiring significant speed reductions or torque increases. They can also be more compact and efficient compared to simple gear trains with the same gear ratio.

Epicyclic (planetary) gear trains

Epicyclic or planetary gear trains consist of a central gear (sun gear) that meshes with one or more gears (planet gears) rotating around it. The planet gears are mounted on a carrier, which itself can rotate, and they also mesh with an outer gear (ring gear) that encircles the entire assembly. Planetary gear trains are used in applications requiring high gear ratios, compact size, or multiple output shafts with different speeds and torques.

## Calculating gear ratios in gear trains

To calculate the overall gear ratio of a simple gear train, divide the number of teeth on the output gear (driven gear) by the number of teeth on the input gear (driver gear). If there are intermediate gears in the gear train, multiply the gear ratios of each gear pair:

Overall Gear Ratio = (Teeth on Driven Gear 1 / Teeth on Driver Gear 1) × (Teeth on Driven Gear 2 / Teeth on Driver Gear 2) × ...

To calculate the overall gear ratio of a compound gear train, find the gear ratio of each pair of meshing gears and then multiply the gear ratios together:

Note that gears mounted on the same shaft have the same rotational speed, so their gear ratio does not affect the overall gear ratio of the compound gear train.

## Epicyclic gear trains

Calculating gear ratios in epicyclic gear trains can be more complex due to the multiple moving components. One common method is to use the tabular method, which involves fixing one component (sun gear, planet carrier, or ring gear) and treating the other two components as input and output. The gear ratio can then be calculated based on the number of teeth on each component and the fixed component's relationship with the others:

Overall Gear Ratio = (Teeth on Fixed Component ± Teeth on Input Component) / Teeth on Output Component

The choice between addition and subtraction depends on the relative direction of rotation of the input and output components when the fixed component is held stationary. If the input and output components rotate in the same direction, use subtraction; if they rotate in opposite directions, use addition. It's important to note that the overall gear ratio can change depending on which component is fixed, so it's necessary to specify the fixed component when calculating the gear ratio in an epicyclic gear train.

For example, consider a planetary gear train with a sun gear having 20 teeth, planet gears having 30 teeth each, and a ring gear having 80 teeth. To calculate the gear ratio when the planet carrier is fixed (input is the sun gear, and output is the ring gear):

Overall Gear Ratio = (Teeth on Fixed Component + Teeth on Input Component) / Teeth on Output Component = (30 + 20) / 80 = 50 / 80 = 5 / 8

In this example, the overall gear ratio is 5:8 when the planet carrier is fixed.

By understanding the types of gear trains and how to calculate gear ratios for each type, you can analyze and solve various gear-related problems in mechanical aptitude tests and real-life applications. By mastering these concepts, you'll be better equipped to tackle gear questions and design efficient gear systems for specific requirements.

## Solving Mechanical Aptitude Test Gear Questions

Identifying the type of gear question.

Gear questions in mechanical aptitude tests can be categorized into four main types:

Basic gear ratios: These questions involve calculating gear ratios for simple gear pairs or gear trains. They test your understanding of gear ratios and how they affect speed and torque.

Gear trains: These questions require you to analyze and calculate gear ratios for more complex gear systems, such as compound and epicyclic gear trains. Familiarize yourself with different types of gear trains and their properties to tackle these questions effectively.

Direction of rotation: Direction of rotation questions test your ability to determine the direction in which gears will rotate based on the arrangement and direction of the driving gear. Understanding the basics of how gears interact with each other is essential for answering these questions.

Speed and torque relationships: These questions involve analyzing the relationships between speed, torque, and power in gear systems. They require an understanding of mechanical advantage and the conservation of power principle.

## Strategies for solving gear questions

2) Breaking down complex problems into simpler parts For more complex gear problems, break the problem down into smaller parts. Analyze each part separately and then combine the results to solve the overall problem.

3) Using gear ratios and mechanical advantage concepts Leverage your understanding of gear ratios, mechanical advantage, and the conservation of power principle to tackle gear questions effectively. Remember the formulas and relationships between speed, torque, and power when solving problems.

By understanding the different types of gear questions, employing problem-solving strategies, and practicing various problems, you can improve your ability to solve gear questions in mechanical aptitude tests. The key is to familiarize yourself with gear concepts and principles, visualize gear systems, and break down complex problems into simpler parts. With practice and determination, you can excel at gear-related questions and boost your mechanical aptitude test performance.

## Free Sample Mechanical Aptitude Test Gear Question Quiz

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## Gear wheels

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## How to Calculate Gear Ratio: Explained Using Lego Gears

When first diving into the gears’ world, you may come across several terms: gear ratio and contact ratio. You do not have to know how to calculate gear ratio to choose the right one, as you can find it in the product description, but understanding how it works will make your choice more… calculated. Use this article as your personal flashlight and you won’t get lost.

To make it simple not only in word but also visually, we tried to explain the whole stuff using Lego. So, let’s start, or as scuba divers say, let’s get wet!

## The Big Gears Family

First there was the Wheel. Then came a Gear. Gears come into play for one of the following reasons:

- to reverse the direction of rotation
- to keep the rotation synchronized
- to increase or decrease the speed of rotation
- to move rotational motion to a different axis

Humanity has known the gear systems (known as ‘trains’) since at least four ages BC in China, so the first invention may even be much older. During these 2,000 years, they developed and divided into numerous dynasties. Let’s mention some of the most powerful representatives. Three big train groups exist based on the orientation of the gear axes:

## I. Parallel Axes Trains: Where the teeth are parallel to the axis

When someone hears the word ‘gear’, in most cases, a simple spur gear comes to their mind. Spur type is applied in a wide range of industries and it is the easiest to find on the market in all the sizes.

A cylindrical gear with helicoid teeth (they are at an angle to the axis of rotation of the gear) can handle more load force than spur one, but they are less efficient. During the motion, there are sliding contacts between two helical gears, which produce axial thrust of gear shafts and increased heat.

## 3. Rack and Pinion

This is a linear gear (the rack) comprised with a spur gear (the pinion). Rotary driving of the pinion causes the linear motion of the rack. Driving the rack causes the pinion to rotate. This system is the simplest tool to convert rotary motion into linear motion and back. You can read more on linear drives and its advantages here .

## 4. Internal Gears

This is another one with cylindrical shape, but the teeth here are inside the ring. As a rule of thumb, these gears are found in planetary mechanisms. Thanks to this design, the vibrations and losses in planetary drives are minor. The rotational stability improves the reliability and repeatability of the movement. Planetary drives offer greater efficiency and accuracy than other systems.

## II. Intersecting Axes Trains: Where the gear axis cross at a point

In a bevel gear, the shaft axes intersect. Usually, the angle between the shafts is 90 degrees, but the train exists with other angles as well (except for 0 and 180 degrees). This angle flexibility lets us widely use the bevel system to change the force direction. For example, from horizontal to vertical. According to the shape and the number of teeth, there are miter, spiral, and straight bevel gears.

## III. Nonparallel Axes Systems:

1. worm and worm wheels.

Worm train consists of a normal spur gear (worm wheel) and a gear with one tooth (the worm), but this one tooth is like a screw thread. The motion of the wheel and the gear is a mixture of sliding and rolling actions. Sliding moves cause friction and heat that decrease the efficiency of the train up to 50%. At the same time, this type of meshing ensures very quiet performance, which makes worm suitable for usage in environments where noise should be minimized. Even if the efficiency is relatively low, worm trains provide very high reduction ratios.

The screw train is a pair of crossed helical gears that transmit motion and power between perpendicular but not-intersecting shafts. They do not have a standard rolling movement during their interaction but a screw motion (that’s why screw).

Every mentioned train type has its strengths or weaknesses, but the main parameter of meshing gears still is the gear ratio . Since each part in the train has a different number of teeth, each of the gears rotates at a different speed. Gear ratio shows this difference.

Knowing a ratio in a robotic project is crucial for:

- Finding the rotational speed of drive gear.
- Reaching speed and torque balance. If the gear ratio is 1:1, the amount of torque is the same, and the speed is the same. With the ratio 1:4, for example, you’ll get less torque but more speed. With the ratio 4:1, you would cut the speed but boost the torque.
- Optimal servo motor sizing. If the motor inertia is too high relative to the load inertia, the motor is bigger, meaning that it was more expensive than it’s necessary to produce it, and the motor uses more energy than needed for the application.
- Minimizing errors and increasing accuracy. An ideal gear ratio is the lowest inertia that can produce maximum acceleration without making the system unstable, overheated, or inaccurate.

## How to Calculate Gear Ratio?

Let’s start with the simplest gear train with only two meshing gears. The first gear attached to the motor shaft is called a "drive". The second attached to the load shaft gear is the "driven" one.

To calculate the gear ratio:

- Count the number of teeth on each part. In our example, the smaller drive has 21 teeth and the driven has 28 teeth. Besides, when we speak about spur type, the one with more teeth is called the “gear” and the one with the fewer teeth is the “pinion”.
- Divide the number of driven gear teeth by the number of drive gear teeth. In our example, it’s 28/21 or 4 : 3. This gear ratio shows that the smaller driver gear must turn 1,3 times to get the larger driven gear to make one complete turn.

Now, stay calm and brace yourself: A high numerical gear ratio is called a “low gear”. A low numerical gear ratio is a “high gear”. Low gears cause fast acceleration and are suitable for smaller engines. High cause better cruising and higher top speed so they are applicable for more powerful engines.

But what if the system incorporates more than two elements? The intermediate gears or so-called "idlers’’ are used to keep or to change the direction of rotation. You can apply the ratio formula to each pair of wheels, but you don't actually need to do that. No matter how many idlers there are in the train, the final gear ratio is the ratio between the driver and the driven wheel. For easy calculation, you can use this Lego Online ratio calculator .

## Contact Ratio

Another important parameter when choosing gears is the contact ratio. It represents the number of teeth meshing at the same time. If multiple teeth are in contact, the load is shared and the operational life of the mechanism is improved.

Knowing a contact ratio in a robotic project is crucial for:

- Torque: more teeth in contact = the mechanism transmits and withstands more torque.
- Stiffness: higher contact ratio = fewer stiffness variations. This means less deflection of the teeth and transmission errors.
- Accuracy: higher contact ratio = good accuracy, fewer mistakes, and a uniform manner of system performance.
- Noise level: higher contact ratio = increased stiffness = reduced noise.

How to Calculate Contact Ratio?

Formulas for calculating the contact ratio will depend on the gear type. But when considering all types in general, contact ratio consists of two components: Radial contact ratio, εα + Overlap contact ratio, εβ (applied only for helical or spiral types).

More on calculating gear contact ratio you can read here and here .

So, hopefully, you now know a bit more about gears, it’s differences and parameters, and can easily dive deeper into the Big Ocean called “Motion Control’’.

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# of teeth input gear = 8

# of teeth output gear = 40

What is the gear ratio (in lowest terms)?

What's the ratio?

If the small gear is input gear, what is happening to the torque?

Remains constant

Not enough information

# of teeth input gear = 60

# of teeth output gear = 36

The input gear in the image is the larger gear. Calculate the gear ratio (in lowest terms)

# of teeth input gear = 24

# of teeth output gear = 4

# of teeth input gear = 10

# of teeth output gear = 6

- Multiple Choice Edit Please save your changes before editing any questions. 5 minutes 1 pt When gears are equal in size, what happens to the speed and torque? Speed increases; torques decreases Speed decreases; torque increases Speed and torque are constant

# of teeth input gear = 6

# of teeth output gear = 48

What is the best explanation for a gear train with a 7:1 gear ratio?

The input gear turns 1 time for every 7 turns of the output gear.

The input gear turns 7 times for 1 turn of the output gear.

# of teeth input gear = 30

# of teeth output gear = 60

# of teeth input gear = 25

# of teeth output gear = 75

# of teeth input gear = 20

# of teeth output gear = 80

If the pink gear is the input gear and the blue gear is the output gear, what will happen with speed and torque?

increase speed, decrease torque

decrease speed, increase torque

speed & torque remain constant

Which of the following pics shows a simple gear train that will turn the gears at a constant speed/torque?

If this gear ratio is 5:3, we have increased:

If we have a gear ratio of 1:2, there is increased:

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## The two biggest problems with electric bikes aren’t even about e-bikes

The rise of electric bicycles is leading to a critical shift in urban transportation, bringing with it the potential for cleaner cities, reduced traffic congestion, and a boost to riders’ physical and mental health. However, there are still two significant barriers preventing many from adopting this green mode of transportation. The strange thing though is that neither of the two biggest problems with e-bikes are even about e-bikes themselves.

## The deadly risk of cycling on roads

Riding an e-bike is almost entirely a positive experience in and of itself – at least if you ignore outside factors. Riders get where they’re going faster, they save money, and they get healthy.

But if you ask those who are bike-curious why they haven’t made the switch from their car to a bike, the most common answer will be something to do with getting hit by a car.

And it’s not an irrational fear. While most cyclists are likely never to get hit, especially when employing safe riding tips and practices , there’s no getting around the fact that the rate of cycling injuries and deaths is increasing in the US.

There’s plenty of blame to go around for this problem. Part of it can be related to car bloat , where every year we see cars growing a little bit bigger and heavier, resulting in today’s massive SUVs and trucks. Part of it can be attributed to distracted drivers who are increasingly tied to their phones instead of looking out for other road users. But the biggest culprit of all – and the one thing that can help negate all of those other issues – is a distinct lack of widescale safe cycling infrastructure.

Every year, countless cyclists lose their lives in collisions with motor vehicles. The juxtaposition of lightweight bicycles and heavy, fast-moving vehicles, especially in areas without designated bike lanes, means that cyclists are always the losers regardless of who’s at fault in a crash.

Cities worldwide must recognize this urgent issue and invest in creating safer infrastructure for cyclists. This means constructing protected bike lanes, especially those separated from traffic by barriers or a safe distance.

But it also means more comprehensive road safety campaigns, focusing on educating drivers about sharing the road. And lastly, penalizing dangerous driving. Speed cameras are a simple and easy way to enforce the most common cause of accidents (and the largest risk that an accident leads to injury or death), speeding cars. Other methods should also be explored. There’s no reason to allow dangerous driving that threatens other road users to go unchecked.

## The lurking threat of bike theft

Even if we suddenly solved the issue of dangerous car drivers killing cyclists, there’d still be one other major hurdle to promoting widespread e-bike adoption: bike theft. It’s one of the main concerns potential e-bike riders face, with rampant theft of bicycles in urban areas now increasingly focusing on electric bikes for their higher value and ability to part out the expensive components.

E-bikes are often pricier than their traditional counterparts due to their motors and battery systems, making them especially attractive targets for thieves. The anxiety of leaving a pricey investment locked in public has deterred many potential riders.

One solution to this problem lies in strengthening security infrastructure. Cities can invest in more secure bike parking stations, equipped with surveillance systems and secure locking structures. Property owners can also contribute by providing safe indoor storage spaces for residents and employees. I recently spoke with LeGrand Crewse, the CEO of California-based e-bike company SUPER73, who told me about a local project where the company partnered with high schools to help build secure locking rooms on campus to ensure students wouldn’t have to worry about an expensive bike being stolen during the day.

Focusing on locking education, especially on which locks are higher quality and how to use multiple locks in unison, can help riders feel more confident about protecting their rides. I’ve used a Foldylock Forever from Seatlylock for the last few months and found it to be one of the best, most secure locks I’ve ever tried.

Education is key, but the best and most effective option relies upon cities helping to create safer locking locations. You don’t have to go full-Amersterdam, though that’d be a good place to draw inspiration. Check out the impressive bike parking garages I saw on my last visit to The Netherlands.

## A call to action

While e-bikes offer a promising solution to many urban transportation problems, their potential remains needlessly limited by theft concerns and safety issues. These are issues with solutions, and we should work to implement those solutions.

As cities look to a more sustainable future, it’s imperative to tackle these challenges head-on. Safe and secure bike parking and dedicated protected bike lanes are not luxuries; they are necessities in a world where we rely on smaller and more efficient alternatives to the cars and SUVs that have taken over our cities.

It’s time to fix these problems and reimagine our urban spaces to ensure that everyone, from e-bike riders to pedestrians, can move about safely and confidently.

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Micah Toll is a personal electric vehicle enthusiast, battery nerd, and author of the Amazon #1 bestselling books DIY Lithium Batteries , DIY Solar Power, The Ultimate DIY Ebike Guide and The Electric Bike Manifesto .

The e-bikes that make up Micah’s current daily drivers are the $999 Lectric XP 2.0 , the $1,095 Ride1Up Roadster V2 , the $1,199 Rad Power Bikes RadMission , and the $3,299 Priority Current . But it’s a pretty evolving list these days.

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1 Start with a two-gear train. To be able to determine a gear ratio, you must have at least two gears engaged with each other — this is called a "gear train." Usually, the first gear is a "drive gear" attached to the motor shaft and the second is a "driven gear" attached to the load shaft.

One can determine the gear ratio either by dividing the output speed of a mechanism by its input speed (i.e., i = Ws/We), or by dividing the number of teeth on the driving gear by the number of teeth on the driven gear (i.e., i = Ze/Zs). What is Gear Ratio Calculation?

The gear ratio is the ratio of the number of turns the output shaft makes when the input shaft turns once. In other words, the Gear ratio is the ratio between the number of teeth on two gears that are meshed together, or two sprockets connected with a common roller chain, or the circumferences of two pulleys connected with a drive belt.

Simply divide 60 teeth by 30 teeth to work out the number of revolutions of the driven gear. Work out the Velocity Ratio (Gear Ratio); Work out the Velocity Ratio (Gear Ratio); Work out the Velocity Ratio (Gear Ratio);

Space Math http://spacemath.gsfc.nasa.gov Answer Key 1 Problem 1 - When Gear A completes one revolution, how many revolutions does Gear B make? Answer: 20/60 = 1/3 Problem 2 - When Gear B makes one complete revolution, how many revolutions does Gear C make? Answer: 60/10 = 6

Students derive the formula for finding the gear ratio of a pair of gears: gear ratio = teeth in driven gear/teeth in driver gear. Rationale Students express the gear rotation relationships mathematically. They investigate the concept of ratio. They find the gear ratio for trains of simple gears. Students will engage in the following:

gear ratio = (π × diameter of input gear)/ (π × diameter of output gear) Simplifying this equation, we can also obtain the gear ratio when just the gears' diameters or radii are considered:

Key Takeaways Gear ratios are fundamental in understanding how gears work, with the ratio indicating how many times a gear has to turn for another gear to turn once; for instance, in a worm gear, a threaded shaft engaging the teeth on a gear can create a high gear ratio in a compact space.

Newton's laws state: (1) an object at rest will stay at rest until an unbalanced force acts upon it. Every object moves in a straight line unless acted upon by a force. (2) The acceleration of an object is directly proportional to the net force exerted and inversely proportional to the object's mass.

You basically count the number of teeth on the driven wheel and divide that by the number of teeth on the driver wheel, which is the one attached to the motor. It's a straightforward calculation, even when the gear system consists of several intermediate wheels called idlers. It's Easier to Calculate Gear Ratio Than You Think

https://engineers.academy/This video introduces gear ratios for simple gear systems, or simple gear trains. You will learn how to calculate gear ratios from ...

Find the worksheet here:https://www.pdf-archive.com/2016/09/26/gear-practice-problems-yt/

Calculating the mechanical advantage is straightforward: we simply have to take the inverse of the reciprocal of the gear ratio. This means that we don't really need to calculate it to find out how to apply it: we can simply use the gear ratio formula! We use the gear ratio to calculate the speed of the output gear given the speed of the input one.

gear ratios, focuses on how to incorporate these concepts into their ... that the ratio is [MATH PROBLEM] 3. Slides 10 - 13 will talk about torque and speed. Have the students try to define both of ... Have the students solve the equation assuming the power of the motor stays constant but you increase the torque. Then do the same thing with ...

Take Exam Law of Gearing The Law of Gearing states that the angular velocity ratio between mating gears remains constant. To achieve the law of gearing or constant angular velocity, a normal at the point of contact between mating gear teeth always passes through the pitch point.

What is the gear ratio? Why should you use a Gear Ratio Calculator? How can a Gear Ratio Calculator help you in your everyday life?Watch our video to see if ...

START PRACTICING Definition and purpose of gears Gears are mechanical components with teeth that mesh with other gears to transmit rotational motion, force, or torque. They are used in various machines and mechanisms to: Change the speed of rotation: Gears can increase or decrease the rotational speed of an output shaft compared to an input shaft.

Solved Problems - Gear Motion and Spur Gear Forces Problem 1 A pinion gear with 22 teeth and a module of 6 mm has a rotational speed of 1200 rpm and drives a gear at 660 rpm. Determine: The number of teeth on the gear, and; The theoretical centre distance Solution Data zp = 22 teeth = 6 mm Np = 1200 rpm Ng = 660 rpm zg = ? teeth = ?mm

Correct answer: n 1 = 348.2 RPM Step-by-step explanation: z1 = 117 z2 = 70 n2 = 582 RPM s1 = s2 n1⋅ z1 = n2⋅ z2 n1 = z1z2 ⋅ n2 = 11770 ⋅ 582 = 11770⋅ 582 = 11740740 = 3913580 RPM= 348.2 RPM Try another example Did you find an error or inaccuracy? Feel free to write us. Thank you! Tips for related online calculators

To calculate the gear ratio: Count the number of teeth on each part. In our example, the smaller drive has 21 teeth and the driven has 28 teeth. Besides, when we speak about spur type, the one with more teeth is called the "gear" and the one with the fewer teeth is the "pinion".

Computers 6th - 8thgrade Gear Ratio Practice Sharon Lennstrom 161 plays 21 questions Copy & Edit Show Answers See Preview Multiple Choice 5 minutes 1 pt # of teeth input gear = 8 # of teeth output gear = 40 What is the gear ratio (in lowest terms)? 5:1 40:8 1:5 8:40 Multiple Choice 2 minutes 1 pt What's the ratio? 1:1 84 : 60 60 : 84

Things to remember. A ratio is a comparison of two quantities. A proportion is an equality of two ratios. To write a ratio: Determine whether the ratio is part to part or part to whole. Calculate the parts and the whole if needed. Plug values into the ratio. Simplify the ratio if needed.

Three problems are mentioned which are facing quite frequently in Gear machining processes 1. Oil Mist and Smoke in Working Area (Increases the Oil Consumption) 2. Excessive Heating of Tools...

The two biggest problems with electric bikes aren't even about e-bikes. Micah Toll | Nov 21 2023 - 4:58 am PT. 19 Comments. The rise of electric bicycles is leading to a critical shift in urban ...