Simple Gears and Transmission Simple Gears and Transmission page: of 4 How can transmissions be designed so that they provide the force, speed and direction required and how efficient will the design be? Contents Initial Problem Statement arrative 3-9 otes 0-3 Solutions 4-38 Appendices 39-4 MEI 0
Simple Gears and Transmission Initial Problem Statement Many devices with moving components are driven by a motor of some sort. In early technologies this motor could be as simple as a hand-crank, a water wheel or a windmill. Modern technologies use electric motors or a combustion engine. The motor produces a circular motion. The motor generally spins at a given speed and provides a given amount of turning force, or torque. In many situations, though, the motor speed or torque is not appropriate for the machine it is driving. In some cases the direction of rotation may even be the wrong way or the rotating shaft may be at a right angle to the desired direction. In order to produce an appropriate speed and torque from a motor, gears are often employed. A series of gears connecting a motor to the part of a machine that requires moving is called a transmission. Simple Gears and Transmission page: of 4 How can transmissions be designed so that they provide the force, speed and direction required and how efficient will the design be? MEI 0
arrative Introduction Discussion Why would a motor spinning at 0 revolutions per minute (rpm) be inappropriate for direct connection to the second hand of a clock? What factor would you have to slow it down by to work as a second hand? A common gear design is the spur gear. Two inter-meshed spur gears are shown in the diagram below. From this point forward in this activity the word gear will be used to mean spur gear. Simple Gears and Transmission page: 3 of 4 Figure MEI 0
Zooming in on the point of contact Hint Activity Figure What physical characteristics of these two gears are important? Think about how the gears meet. Hint Think about where the gears meet. Simple Gears and Transmission page: 4 of 4 MEI 0
. Characterising gears The following diagram shows the same two gears. Figure 3 There are three important numbers that characterise a gear Pitch radius, r: Pitch, p: The pitch radius is the radius of the circle that passes through the points where two gears mesh. This circle is called the pitch circle and the pitch circles of two connected gears meet at a single point. The pitch is the distance around the pitch circle between the same two points on two adjacent teeth. Simple Gears and Transmission page: 5 of 4 umber of teeth, n: This is self-explanatory. Figure 4 MEI 0
Discussion In the above diagram the pitch radius of the large gear is r while that of the small gear is r. How far apart should you place the axles on which the gears are mounted? Discussion 3 Will the pitch be the same or different for the large and small gear? Activity The pitch of two gears must be the same regardless of the gear size for them to mesh correctly. Write an expression relating the pitch radius, r, the pitch, p, and the number of teeth, n. Hint Think about the circumference of the pitch circle. ote that n needs to be an integer Simple Gears and Transmission page: 6 of 4 MEI 0
3. Connecting gears The circumference of the pitch circle of a gear of radius r is given by: c πr If the pitch of the gears used is p then the number of teeth is given by n r π p Discussion 4 For a gear set using a fixed pitch, p, can the gears have any radius? Can a gear have any number of teeth? Activity 3 The two gears below have pitch radii of r 30 mm and r 5 mm. The larger gear has 0 teeth and the smaller gear has 0 teeth. Verify that these gears have the same pitch. Simple Gears and Transmission page: 7 of 4 Hint Figure 5 Activity 4 The larger gear turns clockwise through one full turn. How many times will the small gear turn and in which direction? First relate the answer to the radii of the gears, then the number of teeth. How far will a point on the circumference of the pitch circle of the large gear travel? What does this mean for a point travelling on the circumference of the smaller gear? Multimedia The resource Simple Gears and Transmission Interactive is available to demonstrate the coupling of connected gears. See appendix. MEI 0
Activity 5 Recall the problem of driving the second hand of a clock using a motor that turns at 0 rpm. Using the same pitch as calculated above and noting a minimum number of teeth of 0, how many teeth would you select for the gear on the motor and the gear on the second hand? What radius would these gears have? Draw a scale diagram of your result. Discussion 5 Do you think that there will be: (a) a small gear on the motor and a large gear on the second hand (b) equal sized gears on both motor and second hand (c) a large gear on the motor and a small gear on the second hand Hint Call the gear connected to the motor gear and the gear connected to the second hand gear. Gear will make 0 revolutions in one minute. How many full revolutions do you want gear to undergo in this time? Simple Gears and Transmission page: 8 of 4 MEI 0
4. Connecting three gears Recall, the ratio of the turns gear will make,, relative to the number gear will make,, is related to the ratio of the number of teeth through n n If the number of turns is measured per unit time (i.e. you are measuring the rpm), then this expression gives you the ratio of the speeds of the two gears. Activity 6 Consider the following set of gears. The largest gear turns clockwise through one full turn. How many times will the smallest gear turn and in which direction? Figure 6 Discussion 6 What do you notice about the result? What is the relationship between the direction of the final gear and the number of gears? Simple Gears and Transmission page: 9 of 4 MEI 0
Activity 7 Consider the following set of gears. The largest gear turns clockwise through one full turn. How many times will gear 3 turn and in which direction? How does this compare with the previous result? Figure 7 Discussion 7 Can you find a general relationship between the ratio of the turns of the first and last gears when there are many connected gears? Simple Gears and Transmission page: 0 of 4 MEI 0
5. Using two gears on the same axle Activity 8 Look at the following gears. Gear has 0 teeth and turns clockwise. Gear has 0 teeth and is connected to gear 3, which has 0 teeth, by an axle. Gear 4 has 0 teeth. How many times will gear 4 turn for each turn of gear? Figure 8 Discussion 8 When determining gear direction is it the number of gears that matters or the number of gear axles? Hint Gear and gear 3 are connected by an axle so will turn at the same speed. Simple Gears and Transmission page: of 4 MEI 0
6. Torque transmission You have previously seen that connecting two gears with a different number of teeth leads to an increase or decrease in the speed with which one rotates relative to the other. The ratio of the turns gear will make,, relative to the number gear will make,, is related to the ratio of the number of teeth through n n It was seen that a large gear will always turn more slowly than a small gear so that: Case, small motor wheel, large application wheel: Figure 9 Case, large motor wheel, small application wheel: Simple Gears and Transmission page: of 4 Figure 0 MEI 0
However, gears change more than just the speed one turns with respect to another. Because they change the distance from the axle about which the force is applied they change the torque that is applied. See "Torque" on page 0 Figure The torque, C, is given by the force F acting a perpendicular distance d from the point of rotation is given by: C Fd Activity 9 Write an expression for the torque supplied by the axle of the large gear and the torque supplied by the axle of the small gear. Find the ratio of the torques and relate this to the number of teeth on each gear. ext relate the ratio of the speed of rotation to the ratio of the torques. Discussion 9 What does the result mean? Simple Gears and Transmission page: 3 of 4 MEI 0
7. Transmission efficiency Whenever torque is transferred from one shaft to another using gears there is a loss of available torque due to friction between the gears and in the bearings that hold the axles. For the spur gears used in this example each gear pair will transmit 90% of the torque applied on the driven side. This is the gear efficiency. Activity 0 Look at the diagram below. Gear supplies a torque of 0 m. If the gears are 90% efficient what torque will be available on the shaft of gear? Figure Activity Look at the diagram below. Gear supplies a torque of 5 m. If the gears are 90% efficient what torque will be available on the shaft of gear 3? Simple Gears and Transmission page: 4 of 4 Figure 3 Discussion 0 Remember that when three gears were connected the speed ratio depended only on the first and last gear. Do you think torque will behave in the same way? How do the losses work for this system? MEI 0
Hint First perform the calculation without losses. Activity Write an expression for the total efficiency of a gear set of g gears with an efficiency of f per gear pair. Activity 3 A gear set has 0 gears with each pair having 90% efficiency. The number of teeth on both the first and last gear is 30. A torque of 0 m is applied to the shaft of the first gear, what torque will be available on the shaft of the last gear? Discussion Do you think this is an efficient transmission? Simple Gears and Transmission page: 5 of 4 MEI 0
8. Gear boxes The previous activities have shown that gears of different size change not only the speed of rotation but also the torque applied. For two connected gears with n and n teeth the following have been found Ratio of rotation speed: n n Ratio of torque supplied: C C n n otice how torque increases as speed decreases. This characteristic is exploited in the design of the gearbox for a car. A high torque produces a high acceleration but high torques can only be produced at lower rotational speeds. This is why st gear of a car is used for pulling away as it produces the most torque. However, the limit of the engine s speed is reached while the car is still moving at a reasonably low speed. To avoid this, the driver changes into a higher gear. Higher gears do not provide as much torque but high accelerations are not usually required once the car is in motion. In this way the gearing is selected to produce the appropriate torque and rotation speed for the speed of the car. The following table gives a list of typical gear box transmission ratios for the different gears of a car as would be quoted in a sales brochure Gear Transmission ratio st 3.77 nd.05 3 rd.3 4 th 0.95 5 th 0.76 The transmission ratio gives the ratio of engine speed into the gear box to transmission speed out of the gear box. The above shows that for this car in st gear, the engine turns through 3.77 revolutions for every turn of the transmission shaft out of the gear box. Activity 4 The tyre on the car has an outer diameter of 0.6 m. If the engine revs to 6,000 rpm at what speed would the car be travelling in 5 th gear? Give your answer in ms - and mph and discuss the result. Simple Gears and Transmission page: 6 of 4 MEI 0
9. Gear boxes - part The last activity showed that the transmission ratios published for cars cannot be the ratio of the engine speed to the wheel speed. In fact they are not because the transmission shaft does not drive the wheels directly. Instead it feeds another gearbox called the differential gearbox before connecting to the wheels. See ote "Differential gear box" on page Multimedia The resource Simple Gears and Transmission Animation illustrates the workings of a differential gear box. For the car considered here the differential gear box will reduce the speed by a factor 4.4. This means that the gear box characteristics are Gear Transmission ratio To wheel ratio st 3.77 6.588 nd.05 9.0 3 rd.3 5.808 4 th 0.95 4.8 5 th 0.77 3.388 Activity 5 The tyre on the car has an outer diameter of 0.6 m. If the engine revs to 6,000 rpm what is the maximum speed the car could travel in each gear? Give your answer in ms - and mph both to d.p. Gear Speed (ms - ) Speed (mph) st nd 3 rd 4 th 5 th Simple Gears and Transmission page: 7 of 4 Discussion Discussion 3 Relate the results to how a car is driven. How would you make a reverse gear? MEI 0
0. Do all gears turn the same way? You have previously concluded that if there is an even number of gears the output rotation is reversed and if there is an odd number the output rotation is not reversed. Discussion 4 Look at the following capstan winch. If the crank were connected directly to the winch body with the rope wrapped around it which way would it turn? Figure 4 Simple Gears and Transmission page: 8 of 4 MEI 0
Internal gears are used to connect the crank to the winch body Seen from above the gears look like this: Figure 5 Simple Gears and Transmission page: 9 of 4 Figure 6 Discussion 5 If the inner gear, which is connected to the crank, turns clockwise which way will the outer gear turn? MEI 0
otes Torque A torque is the turning moment of a force about an axis of rotation. Rigid body Figure 7 The torque, C, of the force F acting a perpendicular distance d from the axis of rotation, O, is given by C Fd The same torque can be achieved by several different sets of forces and distances. For example you can apply a large force at a short distance from the axis of rotation or a small force at a larger distance. Two torques will be the same if the product C Fd is the same: Case : F 00, d 3cm T 00 3 300 cm Case : F 0, d 30cm C 0 30 300 cm This is useful when, for example, a lid is stuck on a jar. If you cannot twist it off by hand (gripping the lid at its edge so the distance is the radius of the jar) you can use a device that gives you a longer lever - it lets you increase d, beyond the radius of the jar lid so that you can apply a higher torque and open the jar. Simple Gears and Transmission page: 0 of 4 Figure 8 For a spinning object a high torque will lead to a high angular acceleration, if applied in the direction or rotation, or a high angular deceleration, if applied against the direction of rotation. (A torque is a vector, like a linear force, and has both magnitude and direction. In the case of a torque the direction states whether it acts to turn clockwise or anticlockwise about the axis of rotation.) MEI 0
otes Differential gear box Consider a car driving in a curved path. Figure 9 The wheels on the outside will have to travel further than those on the inside and so must spin faster. This would be impossible if both the wheels were connected to a single axle so instead there are separate axles for each wheel. This causes a problem for driving the axles though; how can you make two axles spin at different speeds from a single drive shaft? This problem is solved by using a differential gear box to connect the drive shaft to the wheels. Simple Gears and Transmission page: of 4 Figure 0 MEI 0
The inside can be shown schematically as below. Figure The green gear is free to rotate and the red gears connect to the wheels. When the blue gear is driven in the direction shown by the engine the red and green gears lock which turns the red gears in the same direction as the blue gear. When a corner is turned the inside wheel slows down and the outside wheel speeds up. This causes the green gear to rotate. The blue driving gear continues to rotate at the given speed but its speed relative to the red drive gears in now modified by the speed of the green gear thus allowing the driving of both wheels at different speeds. Simple Gears and Transmission page: of 4 MEI 0
otes Converting speeds To convert a speed in ms - to mph the following steps are taken. Converting ms - to metres per hour: ms 60 60 metres per minute 60 60 3600 metres per hour Convert metres per hour to kilometres per hour (kph) 3600 metres per hour 36. kph Convert kph to mph. kph 0. 64mph, so 36. kph 36. 0. 64 4. mph d.p. ( ) So to convert ms - to mph multiply the speed in ms - by.4. Simple Gears and Transmission page: 3 of 4 MEI 0
Solutions Introduction Discussion solution As there are 60 seconds in one minute the second hand of a clock must make one full rotation of the clock face every minute. This means its rotation speed must be rpm, i.e. it must be slowed down by a factor of 0. Activity solution The teeth and the gaps must be of the same size and shape, otherwise the teeth would not mesh together correctly. The only parts of the gears that touch are the teeth as they mesh together. As the teeth must be the same size, a circle can be constructed a fixed distance in from the outside extent of the teeth for both gears. These circles will meet at a point between the centres of the two gears. Simple Gears and Transmission page: 4 of 4 MEI 0
. Characterising gears Discussion solution The gear axles must be placed a distance r + r apart. Discussion 3 solution The pitch must be the same otherwise the teeth on one gear would not fit into those on the other. Activity solution The circumference of the pitch circle is given by: c πr Around the circumference are n equally spaced teeth. The spacing between them is the pitch of the gear, p. If there are n teeth around the circumference c, then the pitch is given by r p π r or, making n the subject, n π n p Simple Gears and Transmission page: 5 of 4 MEI 0
3. Connecting gears Discussion 4 solution Clearly the gear must have a whole number of teeth. This means that the radius cannot take any value. Rearranging the expression for the number of teeth gives the radius as a function of the number of teeth and the pitch as r np π The pitch, p, is fixed for a given gear set and may take any value. The number of teeth, n, must be an integer so the radii of gears must follow p p p np r 3 π, π, π, π The first few entries are not practical as a one-tooth gear would not be very useful! There must be a minimum number of teeth for the gear to mesh efficiently. This number is usually about 0. Activity 3 solution Pitch is given by p r π. n For the large gear, r r 30 and n 0 so that πr π 30 60π p 3π n 0 0 For the small gear, r r 5 and n 0 so that The gears have the same pitch. πr π 5 30π p 3π n 0 0 Activity 4 solution In one full turn a point on the circumference of the pitch circle will travel a distance, d, that is equal to the circumference of the pitch circle of radius r (r 30 mm): d c πr 60π As the two gears are connected any motion of the large gear leads to an equal motion of the small gear. Therefore a point on the circumference of the small gear must also move the same distance. The circumference of the smaller gear is c (r 30 mm): c πr 30π The circumference is the distance travelled by a point on it during one revolution. Therefore the number of revolutions,, of the small gear to travel a distance d is given by d c πr r 30 c c πr r 5 Simple Gears and Transmission page: 6 of 4 In this example, the smaller gear turns through full revolutions. The ratio of the number of turns of the small gear to the number of turns of the large gear is equal to the ratio of the radius of the large gear to the radius of the small gear. MEI 0
Generally for two gears of radius r and r, the ratio of turns of gear to gear is given by: r r The way the gears are connected will mean that if the large gear turns clockwise then the small gear will turn anticlockwise. The radius of a gear set of given pitch p is given by np r π Substituting this into the above ratio gives np r np n π π r np π np n π So the ratio of the number of turns is also given by the inverse ratio of the number of teeth. 0 Activity 5 solution Consider the problem of the second hand of a clock driven by a motor running at 0 rpm. Call the gear connected to the motor gear and the gear connected to the second hand gear. Gear will make 0 revolutions in one minute. In this time gear two must make one full revolution. This means that The ratio of teeth must therefore be 0 n, or, 0n n. n From the above ratio it is seen that n 0 0n n n Simple Gears and Transmission page: 7 of 4 i.e. gear, the one connected to the second hand, must have 0 times as many teeth as gear, connected to the motor. The smallest number of teeth available is 0 so setting n 0 gives n as n 0n 0 0 00 The pitch, p, is given by p 3π. The radius of gear with 0 teeth is r np 0 3π π π 5 ( mm) The radius of gear with 00 teeth is r np 00 3π π π 50 ( mm) MEI 0
Drawn to scale these gears would appear as: gear gear Figure Simple Gears and Transmission page: 8 of 4 MEI 0
4. Connecting three gears Activity 6 solution First look at gear (30 teeth) driving gear (0 teeth). n 0 3 n 30 i.e. gear will turn one and a half turns for every turn of gear. ow look at gear (0 teeth) driving gear 3 (0 teeth). n3 0 3 n 0 3 i.e. gear 3 will turn twice for every turn of gear. Substituting the expression for previously found into this gives n3 0 3 3 n 30 3 Gear 3 will turn 3 times for every single turn of gear. Gear turns clockwise so gear will turn anticlockwise. This will turn gear 3 clockwise. Discussion 6 solution The ratio obtained is the same as that if the calculation were done only considering the first and last gear: 3 3 3 For two gears you found that the direction of the final output rotation was opposite to that of the input rotation. For three gears it was the same. In general if there is an even number of gears the output rotation is in the opposite direction to the input rotation and if there is an odd number the output rotation is in the same direction as the input rotation. Simple Gears and Transmission page: 9 of 4 MEI 0
Activity 7 solution First look at gear (30 teeth) driving gear (0 teeth). n 0 3 n 30 i.e. gear will turn three times for every turn of gear. ow consider gear (0 teeth) driving gear 3 (0 teeth). 3 n3 0 3 n 0 i.e. gear 3 will turn one half of a turn for every turn of gear. Substituting the expression for previously found into this gives 3 3 Gear 3 will turn one and a half times for every single turn of gear. otice again though that the ratio obtained is the same as that if the calculation were done only considering the first and last gear: 3 n3 0 3 3 n 30 Adjacent, connected gears turn in opposite directions. As gear turns clockwise, this means gear will turn anticlockwise and gear 3 will turn clockwise. Discussion 7 solution When there are many gears connected in this simple way the ratio of speeds is determined only by the ratio of teeth on the first and last gear. Simple Gears and Transmission page: 30 of 4 MEI 0
5. Using two gears on the same axle Activity 8 solution First look at gear (0 teeth) driving gear (0 teeth). n 0 n 0 0 0 i.e. gear will turn through one half of a revolution for every full revolution of gear. As gear 3 is connected to gear by the axle, gear 3 will also turn through one half of a revolution. Gear will turn in the opposite direction to gear, i.e. anti-clockwise, as will gear 3 as they share an axle. ow look at gear 3 (0 teeth) driving gear 4 (0 teeth). 3 n4 0 4 n3 0 0 3 4 0 4 3 so that gear 4 turns half as much as gear 3. As gear 3 turns half as much as gear this means that for every full turn of gear, gear 4 will turn through one quarter of a revolution. Gear 4 will turn in the opposite direction to gear 3, i.e. clockwise. Discussion 8 solution In the previous cases, where each gear had its own, independent axle, you concluded that: Simple Gears and Transmission page: 3 of 4 In general if there is an even number of gears the output rotation is in the opposite direction to the input rotation and if there is an odd number the output rotation is in the same direction as the input rotation. However, in this case there are four gears and the rotation is not reversed. However, two of the gears share a common axle. In this case you can modify your conclusion to: In general if there is an even number of gear axles the output rotation is in the opposite direction to the input rotation and if there is an odd number the output rotation is in the same direction to the input rotation. MEI 0
6. Torque transmission Activity 9 solution For gear using T Fd : T Fr For gear using T Fd : T Fr The ratio of C :C is given by T T Fr r Fr r You have previously seen that r np π so that np T Fr r np n π π T Fr r np π np n π and the ratio of the torques is equal to the ratio of the number of teeth. The ratio of the speed of rotation is related to the number of teeth through n n Substituting this into the equation for the ratio of torques gives T T n T n T Discussion 9 solution This result shows that a large gear will make a smaller gear turn faster (n > n so > ) but the torque transmitted will be lower (n > n so C > C ). Simple Gears and Transmission page: 3 of 4 MEI 0
7. Transmission efficiency Activity 0 solution The torques are given in the ratio of the number of teeth for a fixed pitch gear set: C n C n In this example, C 0, n 0, n 0 and C is the unknown. Rearranging and substituting known values C n C n 0 0 C 0 0 0 C 0 C 0 m ( ) However, the gear efficiency is 90% so only 90% of this torque is transmitted: C 90% 0 09. 0 8 m ( ) Activity solution Method : Look first at gear and. In this example, C 5, n 30, n 0 and the torque at gear, C, is the unknown. Substituting known values in the torque ratio equation: C n C n 5 30 C 0 0 5 C 30 C 0 m ( ) Simple Gears and Transmission page: 33 of 4 However, the gear efficiency is 90% so only 90% of this torque is transmitted: C 90% 0 09. 0 9 ( m) MEI 0
ow look at gear and 3. In this example, C 9, n 0, n 0 and the torque at gear 3, C 3, is the unknown. Substituting known values in the torque ratio equation: C n C3 n3 9 0 C3 0 0 9 C3 0 C3 45. ( m) Again, the gear efficiency is 90% so only 90% of this torque is transmitted: C 3 90% 45. 09. 45. 405. ( m) Method : Ignoring torque loss, the above gives C 0 m. Recalculate C 3 based on this and again ignore torque losses: C n C3 n3 0 0 C3 0 0 0 C3 0 C3 5 ( m) ote, as with speed ratios, this is the same value that would be achieved if the calculation were performed with only the first and last gear: C n C3 n3 5 30 C3 0 5 0 C3 30 C 5 m 3 ( ) Discussion 0 solution You now need to take account of the efficiency! Each gear pair transmits 90% of the torque. So, transmitted fraction between gear and gear 0.9 transmitted fraction between gear and gear 3 0.9 transmitted fraction between gear and gear 3 through gear 0.9 0.9 0.8 Using this value for the above value of C 3 gives the actual torque as C 3 08. 5 405. m ( ) Simple Gears and Transmission page: 34 of 4 MEI 0
Activity solution For a gear set of g gears with an efficiency of f per gear pair the total transmitted torque is In the last example, f 0.9 and g, giving f total 09. 08. f total f g Activity 3 solution For a set of 0 gears the total torque without losses is given by ratio of teeth on the first and last gear. In this case the number of teeth is 30 on both the first and the last so that the torque on the last gear is the same as the torque on the first: C n C0 n0 0 30 C0 30 30 0 C0 30 C0 0 ( m) The transmission total is: g ftotal f 0 09. 035. d.p. so the actual torque at the last gear is ( ) C 0 035. 0 35. m ( ) Discussion solution The number of gears used gives an inefficient transfer of torque and a better method, such as a bicycle chain, should be considered. Simple Gears and Transmission page: 35 of 4 MEI 0
8. Gear boxes Activity 4 solution If the engine turns at 6,000 rpm then the transmission ratio means that the transmission shaft will turn at 6000 076. 7894. 7 rpm( d.p. ) ext calculate how far the car will travel when the wheel turns once. This is given by the circumference of the tyre as c πd 06. π. 947 (m) The distance travelled in minute is then the rate of rotation in rpm multiplied by the distance travelled per rotation: x 7894. 7 947. 5377. 3 ( m) ( d.p. ) The speed in ms - is the distance travelled in second which is the distance travelled in minute divided by 60 5377. 3 v 56. 3 ( ms ) ( d.p. ) See Converting 60 speeds on page 3 Converting this to mph by multiplying by the factor.4 gives This is clearly far too fast! Possible causes are v 56. 3. 4 574. ( mph) ( d.p. ) (a) A mistake in the algebra check your workings (b) Something stops the car reaching this speed (c) Something's missing It is not (a)! For (b) students may talk about drag due to the ground or the air. While this does limit speeds it would not slow a car from 574 mph down to about the 0 30 mph limit of a car. There must be something missing. Simple Gears and Transmission page: 36 of 4 MEI 0
9. Gear boxes - part Activity 5 solution The solution method is the same as for the previous activity except that a different gear ratio is used. The results are Gear Speed (ms - ) Speed (mph) st.7 6.3 nd.6 48.4 3 rd 33.5 75. 4 th 46.6 04.4 5 th 57.5 8.8 Discussion solution You can see that each gear has an upper limit of speed. A car is driven so that the engine is revving at a reasonably low level for fuel efficiency. However, if the engine speed is too low it is in danger of stalling. If it does not stall, the torque transmitted will be low so acceleration will be poor. Discussion 3 solution To make a reverse gear you have to add an extra gear to the system. If the engine and wheels both turn clockwise when the car is in forward motion then the gear box must have an odd number of gears connecting the engine to the wheels, see Activity 4. To make a reverse gear then, the gear box must introduce another gear between the engine and the wheels so that there is an even number. Simple Gears and Transmission page: 37 of 4 MEI 0
0. Do all gears turn the same way? Discussion 4 solution It would turn in the same direction as the crank. Discussion 5 solution If the inner gear turns clockwise then the one immediately next to it will turn anticlockwise. If you look carefully at the figure you will see that this gear then turns the outer gear in the same sense, anticlockwise. Internal gearing does not change the direction of rotation. The rule of direction only works in some cases! Multimedia The resource Simple Gears and Transmission Animation is available to demonstrate internal gearing. Simple Gears and Transmission page: 38 of 4 MEI 0
Appendix using the interactive Simple Gears and Transmission Interactive This resource is available to demonstrate the coupling of connected gears. Figure 3 The display shows a 0 tooth gear connected to a 0 tooth gear. The red, green and yellow dots on the gears provide a point of reference as the gears turn. When first started the gears rotate at a speed determined by the position of the red slider at the top of the screen. Moving the slider to the right increases speed while moving it to the left decreases the speed so that the motion can be observed more carefully. The motion can be paused at any point by clicking on the pause button at the bottom right of the screen. A second click of this button resumes motion. Simple Gears and Transmission page: 39 of 4 MEI 0
You can use the display to observe the relationship between the number of rotations made by the small gear in relation to the number made by the large gear. Pause the display in the following position Figure 4 ow allow the motion to resume. otice that when the yellow dot has returned to its original position (i.e. the small gear has made one complete revolution) the red dot is on the opposite side of the large gear; the yellow dot now touches the green dot. The large gear has therefore only made half a revolution. Simple Gears and Transmission page: 40 of 4 Figure 5 MEI 0
Appendix mathematical coverage Use trigonometry and coordinate geometry to solve engineering problems Solve problems involving angular motion, converting between units of revolution speed Use algebra to solve engineering problems Evaluate expressions Work with fractions Solve problems involving ratio and proportion Understand and work with percentages Use scale drawings Simplify and evaluate expressions involving the use of indices Change the subject of a formula Simple Gears and Transmission page: 4 of 4 MEI 0