Simple Gears and Transmission - Further Mathematicsfurthermaths.org.uk/docs/SimpleGearsandTransmissionStudentFinal.pdf · MEI ©2011 Supported by Student Simple Gears and Transmission

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ContentsInitial Problem Statement 2 Narrative 3-19 Notes 20-23 Appendices 24-26

Simple Gears and TransmissionHow can transmissions be designed so that they provide the force, speed and direction required and how efficient will the design be?


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How can transmissions be designed so that they provide the force, speed and direction required and how efficient will the design be?

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 TransmissionInitial Problem Statement


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NarrativeIntroduction

Discussion 1Why would a motor spinning at 10 revolutions per minute (rpm) be inappropriate for direct connection to the second hand of a clock?

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.

Figure 1


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Zooming in on the point of contact

Figure 2

Activity 1 What physical characteristics of these two gears are important?


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2. Characterising gearsThe following diagram shows the same two gears.

Figure 3

There are three important numbers that characterise a gear

Pitch radius, r: 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.

Pitch, p: The pitch is the distance around the pitch circle between the same two points on two adjacent teeth.

Figure 4

Number of teeth, n: This is self-explanatory.

Note that n needs to be an integer


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Discussion 2In the above diagram the pitch radius of the large gear is r1 while that of the small gear is r2. How far apart should you place the axles on which the gears are mounted?

Discussion 3Will the pitch be the same or different for the large and small gear?

Activity 2The 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.


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3. Connecting gearsThe circumference of the pitch circle of a gear of radius r is given by: c r= 2πIf the pitch of the gears used is p then the number of teeth is given by

n rp

=2π

Discussion 4For a gear set using a fixed pitch, p, can the gears have any radius?Can a gear have any number of teeth?

Activity 3The two gears below have pitch radii of r1 = 30 mm and r2 = 15 mm. The larger gear has 20 teeth and the smaller gear has 10 teeth. Verify that these gears have the same pitch.

Figure 5

Activity 4The 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.

MultimediaThe resource Simple Gears and Transmission Interactive is available to demonstrate the coupling of connected gears. See appendix 1.


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Activity 5Recall the problem of driving the second hand of a clock using a motor that turns at 10 rpm. Using the same pitch as calculated above and noting a minimum number of teeth of 10, 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 5Do 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


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4. Connecting three gearsRecall, the ratio of the turns gear 1 will make, N1, relative to the number gear 2 will make, N2, is related to the ratio of the number of teeth through

NN

nn

1

2

2

1

=

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 6Consider 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 6What do you notice about the result? What is the relationship between the direction of the final gear and the number of gears?


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Activity 7Consider 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 7Can you find a general relationship between the ratio of the turns of the first and last gears when there are many connected gears?


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5. Using two gears on the same axleActivity 8Look at the following gears. Gear 1 has 10 teeth and turns clockwise. Gear 2 has 20 teeth and is connected to gear 3, which has 10 teeth, by an axle. Gear 4 has 20 teeth. How many times will gear 4 turn for each turn of gear 1?

Figure 8

Discussion 8When determining gear direction is it the number of gears that matters or the number of gear axles?


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6. Torque transmissionYou 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 1 will make, N1, relative to the number gear 2 will make, N2, is related to the ratio of the number of teeth through N

Nnn

1

2

2

1

=

It was seen that a large gear will always turn more slowly than a small gear so that:

Case 1, small motor wheel, large application wheel:

Figure 9

Case 2, large motor wheel, small application wheel:

Figure 10


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See "Torque" on page 20

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.

Figure 11

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 9Write 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. Next relate the ratio of the speed of rotation to the ratio of the torques.

Discussion 9What does the result mean?


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7. Transmission efficiencyWhenever 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 10Look at the diagram below. Gear 1 supplies a torque of 10 Nm. If the gears are 90% efficient what torque will be available on the shaft of gear 2?

Figure 12

Activity 11Look at the diagram below. Gear 1 supplies a torque of 15 Nm. If the gears are 90% efficient what torque will be available on the shaft of gear 3?

Figure 13

Discussion 10Remember 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?


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Activity 12Write an expression for the total efficiency of a gear set of g gears with an efficiency of f per gear pair.

Activity 13A gear set has 10 gears with each pair having 90% efficiency. The number of teeth on both the first and last gear is 30. A torque of 10 Nm is applied to the shaft of the first gear, what torque will be available on the shaft of the last gear?

Discussion 11Do you think this is an efficient transmission?


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8. Gear boxesThe 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 n1 and n2 teeth the following have been found

Ratio of rotation speed: NN

nn

1

2

2

1

=

Ratio of torque supplied: CC

nn

1

2

1

2

=

Notice 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 1st 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

1st 3.77

2nd 2.05

3rd 1.32

4th 0.95

5th 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 1st gear, the engine turns through 3.77 revolutions for every turn of the transmission shaft out of the gear box.

Activity 14The tyre on the car has an outer diameter of 0.62 m. If the engine revs to 6,000 rpm at what speed would the car be travelling in 5th gear? Give your answer in ms-1 and mph and discuss the result.

See Note "Differential gear box" on page 21


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9. Gear boxes - part 2The 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.

MultimediaThe resource Simple Gears and Transmission Animation 1 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

1st 3.77 16.588

2nd 2.05 9.02

3rd 1.32 5.808

4th 0.95 4.18

5th 0.77 3.388

Activity 15The tyre on the car has an outer diameter of 0.62 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-1 and mph both to 1 d.p.

Gear Speed (ms-1) Speed (mph)

1st

2nd

3rd

4th

5th

Discussion 12 Discussion 13Relate the results to how a car is driven.

How would you make a reverse gear?


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10. 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 14Look 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 14


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Internal gears are used to connect the crank to the winch body

Figure 15

Seen from above the gears look like this:

Figure 16

Discussion 15If the inner gear, which is connected to the crank, turns clockwise which way will the outer gear turn?


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NotesTorqueA torque is the turning moment of a force about an axis of rotation.

Figure 17

The torque, C, of the force F acting a perpendicular distance d from the axis of rotation, O, is given by C = FdThe 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 1: F N d cm T Ncm= = ⇒ = × =100 3 100 3 300,Case 2: F N d cm C Ncm= = ⇒ = × =10 30 10 30 300,

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.

Figure 18

For a spinning object a high torque will lead to a high angular acceleration, if applied in the direction of 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.)

Rigid body


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NotesDifferential gear boxConsider a car driving in a curved path.

Figure 19

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.

Figure 20


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The inside can be shown schematically as below.

Figure 21

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.


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NotesConverting speedsTo convert a speed in ms-1 to mph the following steps are taken.

Converting ms-1 to metres per hour:

1 1 60 60 1 60 60 36001ms metres per minute metres per hour− = × = = × × =

Convert metres per hour to kilometres per hour (kph)

3600 3 6metres per hour kph= .

Convert kph to mph.

1 0 6214kph mph= . , so

3 6 3 6 0 6214 2 24. . . .kph mph 2 d.p.= × = ( )

So to convert ms-1 to mph multiply the speed in ms-1 by 2.24.


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Appendix 1using the interactive

Simple Gears and Transmission Interactive This resource is available to demonstrate the coupling of connected gears.

Figure 23

The display shows a 20 tooth gear connected to a 10 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.


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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 24

Now allow the motion to resume. Notice 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.

Figure 25


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Appendix 2mathematical coverageUse 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

Documents

Simple Gears and Transmission - Further Mathematicsfurthermaths.org.uk/docs/SimpleGearsandTransmissionStudentFinal.pdf · MEI ©2011 Supported by Student Simple Gears and Transmission