Foundations of Physical Science Workshop: Gears

Gears CPO Science

Key Questions How do gears work? What is the Law of Gearing? How can Gear Ratios be used to design machines?

Overview Build gear machines Deduce the rule for calculating the number of turns for each gear in a pair of gears Apply ratios to design machines with gears Design a gear machine to solve a specific problem

Simple Machines Include: rope and pulley wheel and axle systems gears ramps levers screws Simple machines transform input forces into output forces. The concept of mechanical advantage is the measure of how much the forces are increased or possibly decreased. When we use simple machines, we apply an input force to accomplish some task, and the machine converts it into an output force that makes the task easier, or provides us with a more convenient option to accomplish the task. For instance, we can climb a ladder, or we can go up stairs( a kind of ramp ) to reach to top of a tower. Either way, we wind up the same height off the ground. However, the stairs allow us an easier option than the ladder to reach to the top.

Gears as Simple Machines Simple machines can change the direction and/or magnitude of an Input Force Mechanical systems and machines require an input force to achieve an output force. Gears are used often in the transmission of rotating motion. Two adjacent gears will turn in opposite directions, a fact must be considered when designing machines that use gears.

What are the variables? Teeth – Each sized gear has a particular # of teeth. How many does each size have? Turns – Each gear turns an exact amount of times, which we can count. How many times does each one turn? Go ahead and put two gears on the stand, it really doesn’t matter the sizes, as long as they are not the SAME size for this part of the Investigation. The most obvious thing to notice is the size difference in the gears. Do the gears have the same amount of teeth? If not, how many does each size have? (Some people may not notice the # of teeth is printed on the front of each gear, and they may go ahead and count them physically.) There are 3 sizes – small/12 teeth, medium/24 teeth, and large/36 teeth. As you spin one gear, as long as they are touching, the other one will spin as well. Do they spin the same # of times? Count the # of times the bottom one spins when the top one goes around once and vice versa. They do not spin the same # of times, and their directions of rotation are opposite too. We are going to perform an Investigation now to see if we can discover a rule for predicting the # of turns for each gear in a pair of gears. Questions about Size – Yes, size does vary. We can simply use the # of teeth as a value of size since all gears of the same size have the same # of teeth. It is actually the # of teeth that matter more than the size of the gear. It may be useful to point out that you could actually vary the # of teeth on the same sized gears, thereby removing the size as a useful predictor of behavior. For example, the large gear could be made to have two or three more or less teeth without a hugely noticeable difference in appearance. However, when it came to calculating the # of turns it would make produce on a smaller gear, # of teeth would matter and size would really not be a factor.

Gear Investigation #1 Top Gear – Input Gear Build 3 Gear machines Bottom Gear – Output Gear Follow Investigation 5.2 In our first gear investigation we are going to build three machines, each having two gears of different sizes as shown on the slide. When performing experiments it is vital to control what is being observed or measured. For this reason we will label the top gear the Input Gear and the bottom one the Output gear. This designation however, is completely arbitrary. We could, for the sake of argument apply force to either gear. Whichever one we choose though, is deemed the input gear, the gear to which input is provided in the form of force. We will see later that the decision of which gear to use for input will depend upon what we need to achieve with our gear machine.

Looking at the Results Can you derive a mathematical formula which relates the # of turns of the Input Gear to the # of Turns of the Output Gear? What is the relationship? Use the # of Turns Use the # of Teeth

Mathematical Relationship (Input gear turns x Input gear teeth) = (Output gear turns x Output gear teeth) Make the equation easier to use by substituting Let Ni = # of turns of the Input gear Let Ti = # of teeth of the Input gear Let No = # of turns of the Output gear Let To = # of turns of the Output gear Ni x Ti = No x To

The Law of Gearing Ni x Ti = No x To Can we write it another way? By taking the original equation and dividing both sides by T2 and then dividing both sides by N1 we can obtain the second equation known as the Law of Gearing. This law states that; The ratio of the number of teeth for a pair of gears is the inverse of the number of turns for a pair of gears. The term Gear Ratio refers to the ratio of output turns to input turns, which we can see from the mathematical relationship is also equal to the ratio of the number of teeth on the input gear to the number of teeth on the output gear.

Complex Gear Machines Make a machine that uses at least 2 pairs of gears- Not this one--- make your own Record the position and # of teeth on each gear in the Data Table Record and count how all the gears rotate Use what you have learned and the table to work out the gear ratios In this part of the Investigation, we’ll use what we have discovered to help us understand a more complex system. In this case, we are adding more gears in addition to stacking two gears on top of one another. Gears that are stacked with this system are linked, so they will rotate at the same rate. This can offer some interesting possibilities concerning gear ratios. In the last part of the investigation we saw gears that had ratios of 1:3, 2:3, and 1:2. While these ratios can be very useful, there are situations that require larger gear ratios. This can be done by having really large and really small gears, but this may not suit spatial limitations that may be encountered. Another way to acquire larger gear ratios is by building these complex machines, and we are going to find out how to go about calculating these ratios.

Gear Assembly for Complex Machine Here is an example of how to set up the gear machine. People should make a different one from the one in this picture. The long thumbscrews are needed for the stacked gears ( like on Axle #2), and a spacer is needed under the gear(s) that need to be away from the pole in order to make contact with another gear ( like on Axle #3) which would need a long thumbscrew as well.

Looking at the Pairs of Gears Use Table 2 to figure out how to calculate the final gear ratio By keeping track of the number of teeth on the gears we can figure out the Gear Ratio of each pair of interfacing gears. To do this, first record on Table 2 the number of teeth of each gear on each axle. Next, circle the gear pairs that interface, or in other words, have their teeth intermeshed directly. These two numbers then are used to make a fraction, from the example Table 2, the two fractions would be 12/24 and 12/24. (It may be helpful to mention that the ratio-fraction made by different gears will not necessarily always be 12/24 which may seem obvious, but may be helpful to those people new to working with gears.) These two fractions can both be reduced to ½. Once reduced, to calculate the final gear ratio. These two fractions are multiplied together. From the example, ½ x ½ = ¼. 1 to 4 is our final gear ratio

Final Gear Ratio This ratio (fraction) can be reduced Each pair of gears has a gear ratio This ratio (fraction) can be reduced There can be more than two pairs of interfacing gears in total Find the Gear Ratios of all pairs Each pair of gears has a gear ratio that can be calculated using the ratio of teeth input of the input gear to the number of teeth of the output gear. We need to figure out what all of these ratios are for the machine.

Final Gear Ratio ½ x ½ = ¼ = 1:4 Reduce each Gear Ratio fraction from Table 2 12:24 = 12/24 = 1/2 Multiply all Gear Ratio fractions ½ x ½ = ¼ = 1:4 This is the Final Gear Ratio Once reduced,the gear ratios of each pair of intermeshed gears are used to calculate the final gear ratio. In our example from Table 2 we have two sets of gears, but there are many different combinations that could be made with more than just two; three or even four. For this particular example, two fractions are multiplied together. From the example, ½ x ½ = ¼. 1 to 4 is our final gear ratio. This ratio means that for every 4 revolutions of the top gear on axle 1, bottom gear on axle 3 will go around 1 time.

1:4 or 4:1? The gear ratio depends on what gear you use as input and what gear you use as output. If the final gear ratio fraction is less than 1, like 1:2 = ½, the output gear turns less than the input gear. This is used for power, like when using a low gear to go uphill on a bike. If the final gear ratio fraction is more than one, like 4:1 = 4/1, the output gear will turn more than the input gear. This is used for speed, like using a high gear when going really fast downhill on a bike. We can see here that depending on what we want to have happen, we will need to use different gear ratios. If we have a giant 500 cubic inch engine to provide input to a gear system what could we do with it? Bulldozers and backhoes frequently have such large engines, and they utilize a very low gear ratio. You may have witnessed a bulldozers engine revving loudly as it plows along at about 1 mile an hour. In this situation, the gearing is used for power. The engine drive shaft turns a lot, but the lifting scoop only moves a little. The same engine may also be seen in a dragster. How can this be? In the dragster, the gear ratios used are quite high, and are used for speed. The same engine revs up in a similar way as the backhoe, but in this case the drive shaft spins and causes the wheels to spin many more times, propelling the dragster quickly down the drag strip. Interesting – Look at the gear machine you designed. Call the bottom gear the input gear and the top gear the output gear, the reverse of what we just did. Calculate the Final Gear Ratio this way. It will be the reciprocal of the first Final Gear Ratio. Instead of taking the whole gear assembly apart and re-attaching all the gears upside down, simply designate the bottom gear as the input gear. Used in this fashion, the gearing can be used for speed. Used in the original way the gearing can be used for power.

Designing Gear Machines What is the objective? Speed vs. Power What is the desired Final Gear Ratio? – factor it to lowest possible values Look at the gears you have to work with Use the available ratios to get the final gear ratio you need How about spin direction? When designing a machine that will employ gears, the most important factor is the objective. In other words, what needs to be accomplished, or what need to get done. We have seen from the previous activity that it is possible to reduce the # of rotations of the output gear from the # of rotations of the input gear, and when this is done the Final Gear Ratio turns out to be less that one. If we want to increase the # of rotations of the output gear compared to the # of rotations of the input gear the Final Gear Ratio will be more than one. When you know the desired Final Gear Ratio, simply factor it. If it can be factored into numbers that match the available pair gear ratios, you are in business. If not, the desired ratio can not be made. Spin direction is an interesting concept. The # of total gears that mesh together in the machine will affect the direction of spin of the final output gear. We’re not setting that design specification here, but notice that two meshing gears will always spin in opposite directions. How or why would that be used or needed in a design? Consider the transmission of a car. If there are the wrong amount of gears, you just may have wheels that spin in reverse instead of forward. It’s a good thing automotive transmission engineers are experts in Gearing. Interesting Aside - On the show Junkyard Wars on the Discovery Channel, teams were asked to use what they could salvage from a junkyard to construct a dragster. When they finished building the cars, they went to the drag strip to test them. One car had a manual transmission salvaged from a 1980 Monte Carlo. After their second of 3 races, the forward gears burned out. The solution? They disconnected the rear axle, flipped it around so the left tire became the right, and the right the left. They reconnected the axle and put the car in Reverse. They made it down the track for their 3rd try, but lost the race. By switching the axle they reversed the wheel’s spin!

Factoring Gear Ratios Example Gear Ratio Desired – 6:1 Factor 6/1  3 x 2 = 6 Check Available Ratios – 1:3, 1:2, 2:3, 3:1, 2:1, 3:2 Is it possible? Yes 3/1x 2/1= 6/1= 6:1 Attach Gear pairs that give desired Final Ratio It may require more than two pairs of gears

How Can You Make- 9:1 3/1 x 3/1 = 9/1 4:1 2/1 x 2/1 = 4/1 18:1 The highest possible ratio with the gears provided 3/1 x 3/1 = 9/1 2/1 x 2/1 = 4/1 3/1 x 3/1 x 2/1 = 18/1 You can see here that factoring the desired ratio will give the component ratios of pairs of gears that will result in the desired ratio. A ratio of 10:1 would be impossible with the gears we have here. 10 factors down to 2 x 5. We have no problem with getting a ratio of 2/1, we can use the 24 and the twelve toothed gears. Getting the 5/1 however is not possible, and that’s where it ends. We make final ratios that do not match with our available gear ratios. We can get a pretty good amount of Final Ratios using the gears in the set, but anything requiring 5:1 or 7:1 is not possible with the ones we have now. The Highest Possible with the extra set – 3 x 2 x 2 x 2 = 24 This is due to restrictions in the way the gears can physically be hooked up on the stand. Sometimes the mathematically figured ratios can’t actually be built since it would cause some gears to bind up.