Benchmarking

INTRODUCTION

This report has the following aim;

“To upgrade the current Land Rover Defenders driveline and transmission system”

The data from Project Stage 1 (concerning the upgrade of the Defenders engine) is carried forward to this report (Project Stage 2).

The following report firstly discusses the strengths and weaknesses of the current Defenders driveline and transmission compared to its competitors as well as advanced technologies being critically reviewed to see if they could be used for the upgrade.

The type and main parameters of the gearbox will be selected, leading to simulation analysis of the vehicles performance in several areas including; acceleration, fuel consumption, max ascendable slope etc. The simulation results will be able to show the advantages of the new Defender to the current model.

Design Sketches of the gearbox will be displayed along with justification of the designs capacity to work through bending fatigue damage analysis being carried out.

1. BENCHMARKING

The current Defenders transmission and drive is a dual-range six-speed transmission with locking centre differential.

Below shows the current Defenders data relating to driveline and transmission performance, compared to competitor vehicles;

Vehicle

Land Rover Defender

Jeep Wrangler

Mercedes-Benz G Class G550

Nissan Patrol

Toyota Land Cruiser

Engine Capacity L (type)

2.4 (D)

2.8 (D)

5.5 (Petrol)

3.0 (D)

Max Power (kw)

130

382

118

123

Max Torque (Nm)

360

400

391

380

410

Fuel Consumption (Combined mpg)

28.3

32.8

12.0

26.2

25.02

Engine Emissions g/km

266

227

322

288

214

Max Speed mph

112

131

109

0 – 62 mph time seconds

14.7

11.7

6.1

15.2

12.7

Gearbox Type

6-speed manual

6-speed auto

7-speed

automatic

6-speed manual

5-speed automatic

Gear Ratios;

1st

5.443

4.46

4.38

4.556

3.52

2nd

2.839

2.61

2.86

2.625

2.04

3rd

1.721

1.72

1.92

1.591

1.4

4th

1.223

1.25

1.37

5th

0.836

0.72

6th (if applicable)

0.742

0.84

0.82

(no data)

–

7th (if applicable

–

0.73

–

Final Drive

3.54

4.1

4.38

4.111

3.91

Reverse ratio

4.935

4.06

3.42

4.245

Differential

Locking

Limited slip

Electrically

Locking

Limited slip

Transfer box gear ratios

High Speed Ratio

1.211

0.87

1.1

–

Low Speed Ratio

3.269

2.16

2.48

–

From ‘1.0 it can be seen that the Defenders top speed is very low compared to its competitors with only a top speed of 82mph compared to its closest rival, the Jeep Wranglers top speed of 112mph. Other factors affect the top speed such as vehicle weight and aerodynamics, however the new engine designed in stage 1 has 50% more power available, thus this top speed can be raised and refined through modifying the current gearboxes ratios. A trade-off between achieving good torque and top speed will have to be carefully balanced.

The time it takes the current Defender to accelerate to 62 mph is currently the second slowest. Potential customers may be discouraged by this slow acceleration, therefore this characteristic will have to be improved.

The current Defender also has a 6-speed manual gearbox compared to its closest rival’s 6-speed automatic gearbox for the Jeep Wrangler. Manual gearboxes have the following advantages over automatic types;

– Up to 15% more fuel economic according to the US Department of Energy (2009).

– Do not require cooling

– Lower in weight than automatic transmissions

– Lower power consumption

– Manual transmission gives skilled drivers more control over the vehicle.

Automatic gearboxes also have their own advantages such as; shifting gear ratios faster, reducing the work load of the driver and making it generally difficult to select the wrong gear thus not damaging the gearbox. However the advantages for manual gearboxes still outweigh those of automatic gearboxes.

When comparing the Defenders gear ratios to the Jeep Wranglers, the following can be observed;

– The Defenders first gear ratio is greater than the Jeeps to provide more torque at lower speeds.

– The Defenders top gear is lower than the Jeeps to provide as economical drive when cruising as possible.

– Both the vehicles have wide-ratio gears. This means that there is a big difference between the gear ratios of the gears which is favoured for better acceleration at lower vehicle speeds such as when off-roading or stop-starting in traffic.

The Toyota Land Cruiser has the narrowest gear ratios due to it only having a 5-speed gearbox. This could have contributed to its top speed being among the highest, however the off road capability may have been compromised.

All but one of the vehicles is equipped with dual range (transfer box) capability which allows the driver to select two extra gear ratios which have the following purpose;

– A high speed ratio gives the driver more fuel economy for cruising or highway driving.

– A low speed ratio to help the driver when towing or driving off-road.

The dual range gear ratios for the Jeep Wrangler and Defender are currently the same and are the widest apart when compared to the Mercedes or Nissan Patrols transfer box.

The Defender also has a centre locking differential which can provide more traction then limited slip type differentials which are used by the Jeep Wrangler and Toyota Land Cruiser. Pietschmann (2010) explains how locking differentials are useful when passing through a tough off-road section to prevent from getting stuck.

To summarise the following areas need to be at least considered when selecting the new gear ratios for the new transmission;

– Need to maximise the torque at the first few gear ratios as this is where the vehicle is likely to be needed for its off-road performance purpose.

– The top speed of the vehicle needs to also be increased, which should be achievable as the new engine (designed in Project Stage 1) has 50% more power than the current Defender.

– The compromises between selecting gear ratios for maximising torque or speed needs to be balanced to achieve the best all-round off-road and highway performance as much as possible.

2. ADVANCED TECHNOLOGY REVIEW

In this section, the different types of transmission and driveline technology are reviewed

Dual Clutch Direct Shift Gearbox (DSG)

A Direct Shift Gearbox also known as dual- clutch transmission is an automated transmission which allows the driver to change gears faster than ever possible with any other types of geared transmission. Motor vehicle companies such Audi, Volkswagen, Mitsubishi, Porsche and Nissan are current using this technology. A DSG consists of two clutches which essentially is two separate gearboxes with a pair of clutches between them. According to Gold (2007) one of the clutches engages the odd numbered gears and the other even numbered gears, this allows for instant gear changes as the next gear up is already engaged via the twin-clutch transmission’s computerised controller which calculates the next likely gear change based on driver speed and behaviour.

The main advantages of such a gearbox are that it allows for quick gear changes (8ms) and provides the same characteristics as a conventional manual gearbox where traditional automated gearboxes normally lack throttle response when changing gear, where DSG’s allows for gear shifts without interrupting the power permitting for a faster and smooth ride.

Disadvantages include the complexity, and thus high cost, of manufacture. This type of gearbox according to myturbodiesel.com (2010) is also prone to bad reliability when highly stressed for long periods or transmitting large torques. This disadvantage is desired as the Defender when used in towing or off-road situations is expected to transmit high torque. The added cost also will not be suitable for our Defender as its price range needs to be competitive with existing models. The uses of DSGs are normally confined to luxury vehicles which require smoother gear changes and operation. Since the Defender doesn’t need this requirement a DSG gearbox will not be used.
Continuously variable transmissions (CVTs)

Traditionally automatic and manual transmissions have set numbers of gear ratios (normally up to 6), however CVTs can constantly change the relationship between engine speed to vehicle speed providing optimal power for acceleration from the gearbox to the cars speed. Vehicles such as the 2008 Mitsubishi Lancer, 2009 Nissan Maxima and the 2010 Honda Insight Hybrid to name a few use CVT transmissions.

Although there are many types of CVT most cars use a pair of variable diameter pulleys, one connected to the engine and the other connected to the wheels. The halves of each pulley are transferable as the pulley halves come nearer together the belt is forced to ride higher on the pulley, in effect making the pulley’s diameter larger. To change the transmission ratio, the diameter of the pulley is varied. As the car accelerates, the pulleys adjust their diameter to reduce the engine speed as the car speed increases. This is the same thing as what a conventional automatic or manual transmission carries out, but whilst a conventional transmission adjusts the ratio in phases by changing gears, the CVT incessantly varies the ratio so allows a selection of infinite number of ratios.

Continuously Variable Transmission (CVT) is 35% more efficient than the Manual Transmission (MT) as illustrated by ‘1.3. With same car and engine, the CVT takes only 75% of the time to accelerate to 100km/h.

The main advantages of such a transmission (CVT) is that engine runs at optimum revs per minute providing smooth and rapid torque which in turn provides excellent fuel efficiency which means less greenhouse emissions, also because of its simple design is easy to repair and build. Disadvantages include it being a purely mechanical operation so has inherent limitations and steel belts, hydrostatic and ratchet CVT advancements are expensive.

Nissan Extroid toroidal CVTThere is another version of CVTs which uses discs and power rollers instead of belts and pulleys called toroidal transmission. Instead of pulleys there are discs and instead of belts there are many rollers which transmit power from one disc to the other

Mercedes 7G-Tronic Transmission

7G-Tronic was the world’s first seven-speed automatic transmission. The amplified number of potential ratios brings many improvements like a faster shifting response, improved acceleration, greater driving pleasure, quieter running and better fuel consumption. With 7 forward gears and the ratios being spread over a wider range the transmission can react more rapidly and flexibly.

Dual Range Gearbox Capability

‘1.6 shows the dual range gearbox that many off road vehicles have. It is essentially a separate 2 speed gearbox which the driver can select between two gear ratios. One gear ratio is very high (low range) giving good torque characteristics for applications such as towing, off-roading. The other gives a very low gear ratio for optimising fuel economy and engine speed for cruising.

The new transmission will include dual range capability for the advantages described previously. The disadvantages include having to carry around more vehicle weight, however for vehicles such as the defender, the advantages of providing the customer with this feature can be useful to customers.

Double Stage Sliding Mesh Gearbox

In the sliding-mesh gearbox, the particular gear ratio is selected by sliding the elected gearwheel axially along the splined main output shaft until it meshes entirely with the corresponding lay shaft gear. The sliding main shaft gearwheels and their corresponding lay-shaft gearwheel clusters have to be spur straight tooth form so that when they are meshed there is no side thrust. The engine shaft (clutch shaft) holds the main drive gear A, which spins at the speed of the clutch shaft. The main drive gear is in constant-mesh with counter shaft (lay shaft) drive gear B. Since all the gears on the lay shaft are rigidly fixed, they also rotate along with the clutch shaft. The main shaft is held in line with the clutch shaft.

All the gears on main shaft can be slid back and forth on the main shaft spines using shifting forks dependant on the gear change being made. The transmission shafts and gears are made from low-alloy nickel-chromium-molybdenum steels.

Four-speed-and-reverse double-stage sliding-mesh gearbox.

The key problem with this sort of gear engagement according to the-crankshaft.info (2009) is that, while attempting a gear change, the speeds of the input and output shafts are matched first, otherwise the sliding teeth of the meshing gearwheels does not line up and therefore smash into one other. Also there is a lot of noise associated with this type of gearbox.

Single Stage Constant Mesh Gearbox

Constant mesh refers to a gearbox in which all gears are always in mesh but only one of the pairs of meshed gears are locked to the shaft on which it is mounted on, while the other gears are allowed to rotate freely which reduces the difficulty in shifting gears.

Constant mesh type gearboxes, according to cartech.com (2009), allow the pinion gears to be in constant mesh with their corresponding layshaft wheels. The gears in constant mesh gearboxes use helical gears which are stronger and display quieter operation performance than spur gears. To select a Four-speed-and-reverse single-stage constant-mesh gearboxgear is locked to the main shaft via use of a dog clutch, and the output speed now relates to the locked gear, with the other gears free to rotate while in constant mesh. To then select a different gear, it needs to be locked to the shaft instead of the previously chosen gear.

Synchromesh Unit

A synchromesh transmission according to tpub.com (2010) is a type of constant mesh gearbox. Unlike other constant mesh gearboxes, a synchromesh transmission the speeds mating gears are matched so that they can engage without clashes or grinding occurring.

To select a gear, in this type of gearbox, a (dog) clutch with friction teeth slides onto the teeth of a coned section which is connected to the wanted gear. As the clutches fully engage, the output of the main shaft now depends on the gear that is selected. This type of gearbox will be used for the new proposed Defender.

differential-lockingLocking Differential

Nice (2009) states that locking differentials are very useful for ‘serious’ off-road vehicles. The differential can be ‘locked’ either manually by the driver or electronically by the vehicle itself using an ECU.

When locked the differential makes both wheels rotate at the same speed. Thus more traction can be experienced which is important for a vehicle such as the Defender when off-roading.

When ‘unlocked’ the differential acts like a common open differential and allows both wheels to rotate at different speeds for cornering capability.

Universal Joints

These joints are used in the driveshaft which allows it to bend and still transmit rotary motion through it.

This type of joint consists of two hinges which can move relative to each other depending on the movement of the vehicles suspension. For example if one wheel mounts a kerb, the hinges of the universal joints will move but still be able to transmit rotational power through it.

The advantages of universal joints;

– Simple to make

– Can be very strong

However the disadvantages of universal joints are;

– Difficult to rotate at extreme angles

– Need regular maintenance

– Require more complicated bearings when used in driven axles.

Constant Velocity Joints

Constant Velocity (CV) joints overcome the disadvantages of universal joints.

CV joints work by allowing the angle of the input and output joint to vary by allowing ball bearings (shown by ‘2.4) to slide in groves of the joint while transmitting the rotational motion through the joint. Due to using ball bearings, CV joints can bend at extreme angles with less friction occurring than universal joints. For those reasons CV joints are very well suited for off-road vehicles such as the Defender.
Chosen Technologies

The following transmission and driveline technologies have been chosen

– Double stage constant synchromesh gearbox

– Dual Range transfer case

– Locking Differential

– Constant Velocity Joints

3. MODIFICATION TO EXISTING TRANSMISSION

Driveline Configuration

The driveline includes transmission, propeller shafts, final drive (differential) and axle drives. The Land Rover Defender has longitudinal AWD with 3 differentials as shown in the diagram below:

The first differential from transmission (gearbox) is the transfer case (dual range) which provides the option of selecting off road (high torque) or normal (torque) driving conditions. Torque is then split equally to the front and rear differential. The final drive ratio at the front and rear differentials are the same. Hence all 4 wheels will have equal torque.

The two primary types of gearboxes are manual and automatic. Manual gearboxes are further categorized to sliding-mesh, constant-mesh and synchromesh.

Sliding-mesh gearboxes are noisier because only spur gears are used. Although they have higher mechanical efficiency, it is difficult to obtain a quick smooth change without great skill and judgment.

The primary advantage of another type of gearbox – constant-mesh gearbox is the use of stronger helical gears which leads to smoother and quieter operation, especially during gear change.

A better gearbox would be the constant-load synchromesh. Basically the layout of this type of gearbox is the same as constant mesh, with the exception that a cone clutch is fitted between the dog and gear members. The frictional cone will bring the output shaft a matching speed before the dog teeth actually matches. This allows an even smoother gear transmission. Baulk ring synchromesh is just an additional system designed to overcome the disadvantage of older synchromeshers. Baulk ring eliminates more noise and crashing of gears due to quick gear change.

The proposed gearbox type for the upgrade would be baulk ring synchromesh.

Transmission Ratios

The current type of Transmission for Land Rover Defender is GFT MT 82, 6-speed manual, 2-speed transfer case and full-time 4WD. Transfer case provides 2 additional ratios for off-road and highway driving options. A layshaft and two pairs of constant-mesh helical gears, attached to the end of the main gearbox, are driven via short coupling shaft from the gearbox mainshaft.

Gear

Current Gear Ratio

5.443:1

2.839:1

1.721:1

1.223:1

1.000:1

0.742:1

Reverse

4.935:1

Final Drive

3.540:1

The Land Rover Defender is known for its off road high torque capabilities. In fact, when its Low Range transfer case engages first gear, it has enough torque to climb up any hill, without throttle input according to Maric (2008) .To enhance this capability, it has been decided to increase the existing gear ratios for even better off road performance. The transfer case, reverse and final drive ratios are retained, since all six gear ratios have already been modified to improve torque. The following table shows the upgraded ratios for all six gears.

Gear

Current Gear Ratio

Modified Gear Ratio

5.443:1

5.700:1

2.839:1

3.000:1

1.721:1

2.375:1

1.223:1

1.875:1

1.000:1

0.742:1

0.730:1

Reverse

4.935:1

Final Drive

3.540:1

One of the important parameters is the range of gear ratios, given by the formula:

Rg=RTRL=0.735.7=0.128

Where RT and RL are top gear ratio and lowest gear ratio respectively.

Since gear ratios, modules and number of gear teeth are all related, the process to obtain the appropriate values for each is iterative. We first assume 6th gear ratio of 0.730:1 from previous 0.742:1. We then assume Zc1 and Z11 of 16 each. Zc1 is the number of gear teeth for constantly meshed pinion (on input shaft) while Z11 is the number of gear teeth for pinion in 6th gear pair. Zc1 cannot be too small because it has to fit onto a bearing that’s on a shaft. A value of 16 seems sensible. Z11 is also set to be as small as possible for a more compact design. The smallest allowable size is 16 teeth. Lastly the 5th gear ratio of 1 is retained.

Next we calculate distance between shafts, A using the formula below:

A=KA3Te maxi1ηg=1034005.7(0.96)=129.8 mm

Where KA is assumed to be 10 and transmission efficiency, ηg = 0.96.

The formula yields a value of 129.8mm, which deviates from the suggested 60-80mm range. However, this range is only a guide and is meant for passenger cars. Since the Land Rover Defender is a tough off road 4WD, it is entirely feasible to have distance between shafts of 130mm because the powerful engine and large front bonnet can cope. Nonetheless, to be fair, we chose to have a distance between shafts of 110mm.

Next we assume pressure angle of 20o and helix angle of 30o.

Pressure Angle

Pressure angle is the profile angle of standard pitch circle. Standard values include 14.5o, 20o and 25o. A pressure angle of 20o is chosen because it has higher strength than 14o and more common than 25o.

Helical Angle

Helical angle is the angle between the axis of a helical gear and an imaginary line that is tangent to the gear tooth. Higher helical angles (according to Hombrechtikon, 2006) provide smooth-running and quieter gear sets at higher thrust loads (usually between 20-40o) while lower helical angles provide low thrust loads but results in fewer teeth in contact and more noise. An angle of 30o is chosen for the project because smooth running gears ensure longer life. Higher thrust loads can be overcome by properly locating the gears using shoulders and collars.

Face Width and Module

Other parameters for gears include face width and module. Face width is simply length of teeth in an axial plane. Module is the ratio of Pitch Circle Diameter to the number of teeth of a gear which basically means the size of a gear. Gears with smaller module have more teeth and produce more noise. Lower gears have larger module and fewer teeth. A gear pair must have equal module in order to mesh.

Now we have values of Zc1 and Z11, we have to assume values of Zc2 and Z12 to obtain 6th gear ratio of 0.730:1. Thus face width and modules can be calculated using the following formulae:

Face width, b=kcM

Kc is typically between 6 -8.5 for helical teeth.

First Gear Ratio, i1=Zc2Z11Zc1Z12

Module, M=2AcosβZn+Zn+1

This gives the following table:

Gear

Current Gear Ratio

Modified Gear Ratio

Pinion and Wheel, Z1, Z2

Number of Teeth, Z

Total Number of teeth, ZT

Module, M (mm)

Face Width, b (mm)

5.443:1

5.7

2.8

22.41

2.839:1

2.2

17.72

1.721:1

2.375

3.2

25.40

1.223:1

1.875

2.8

22.41

1.000:1

3.5

28.23

Z10

0.742:1

0.73

Z11

2.8

22.41

Z12

Detailed calculations for each gear ratio and respective modules can be found in Appendix 1.

Torque and Power Flow

Assuming a power transmission efficiency of 0.98, torque at the wheels can be calculated. Torque from engine has to undergo 3 gear reductions before reaching the wheels. The first is gearbox ratio, followed by dual range transfer case (High or Low) and finally final drive ratio. Using 2000 rpm which is max torque condition, we can calculate the torque that is transmitted to front or rear wheels using the following procedure:

First Gear, Low Transfer Case Ratio (@ 2000rpm; 83kW):

Gearbox output speed = 2000/5.700 = 350.88 RPM

Transfer case output speed = 350.88/3.269 = 107.33 RPM

Final Drive Speed = 107.33/3.54 = 30.32 RPM

Total wheel pair torque = 83000*0.98/ (2*π*41.82/60) = 25.62 kNm

The same method is used to calculate the torque for each wheel pair for each gear ratio at each transfer case ratios. Note that final drive ratios are the same for front and rear differentials.

RPM

Power (kW)

Torque (Nm)

3500

135

368

2000

400

Engine Speed (RPM)

Gear

Gearbox Speed (RPM)

Transfer Case Speed (RPM)

Final Drive Speed (RPM)

Wheel Torque (kNm)

Low

High

Low

High

Low

High

2000

350.88

107.33

289.74

30.32

81.85

25.62

9.49

666.67

203.94

550.51

57.61

155.51

13.48

4.99

842.11

257.60

695.38

72.77

196.44

10.67

3.95

1066.67

326.30

880.81

92.17

248.82

8.43

3.12

2000.00

611.81

1651.53

172.83

466.53

4.49

1.66

2739.73

838.09

2262.37

236.37

639.09

3.28

1.22

Engine Speed (RPM)

Gear

Gearbox Speed (RPM)

Transfer Case Speed (RPM)

Final Drive Speed (RPM)

Wheel Torque (kN)

Low

High

Low

High

Low

High

3500

614.04

187.84

507.05

53.06

143.23

14.64

5.42

1166.67

356.89

963.39

100.82

272.14

7.70

2.85

1473.68

450.81

1216.92

127.35

343.76

6.10

2.26

1866.67

571.02

1541.43

161.31

435.43

4.82

1.78

3500.00

1070.66

2890.17

302.45

816.43

2.57

0.95

4794.52

1466.66

3959.14

414.31

1118.40

1.87

0.69

4. POWER FLOW DIAGRAMS

The following diagrams show a rough layout of the proposed gearbox and power flow for each gear ratio. The gear sizes are not to scale as the purpose is to illustrate the power flow from input to output.

First Gear

Second Gear

Third Gear

Fourth Gear

Fifth Gear

Sixth Gear

Reverse Gear

5. VEHICLE SIMULATIONS

By the using Microsoft Excel a simple model to carry out vehicle-power train simulations was developed. The following elements including the Engine map, transmission and driveline ratios, rolling tyre resistance, aerodynamic drag, max ascendable road gradient, as well as the transmission shift strategy (including shift delay) were simulated.
Required Data and Assumptions

Some essential data and assumptions from the new Defenders modified transmissions was required to carry out the analysis. They are shown in the table below;

Item:

Results:

Body Dimensions (L*W*H)

3894*1790*2021 mm

Weight (kg)

1797

Max laden weight (kg)

3500

Tyre Radius (mm)

320

Wheelbase Length (mm)

2360

Max speed (Km/h)

137

1st Gear

5.7

2nd Gear

3rd Gear

2.375

4th Gear

1.875

5th Gear

6th Gear

0.73

Final Drive Ratio

3.540

The table below shows the calculation and assumptions of data which was used to simulate the various elements of the transmission.

Items:

Theory

Results:

Frontal Area (m2)

A= H*W

3.61

Drag Coefficient

Sedan : 0.34-0.50 (Bowling 2004)

0.42

Rolling Resistance Coefficient

Average value for car on concrete and mud (Wikipedia 2010).

0.025

Density of Air (kg/m3)

20C

1.23

Overall Transmission efficiency

Assumed on usual conditions

0.98

Aerodynamic Drag and Tyre Rolling Resistance

When the Land Rover Defender is driven, it will inevitability suffer opposing forces which try to slow the vehicle down. One of those forces is the aerodynamic drag of vehicle, which is proportional to its speed. The other is the tyre rolling resistance, which was not dependent on vehicle speed.
Aerodynamic Drag

There have two formulas available to calculate the aerodynamic drag,

The first is; F=K*V, from Batchelor (1967)

Where:

K is the drag coefficient which is dependent on the shape of the moving object and also the medium the vehicle is travelling through.

V is the relative speed between the vehicles and the medium.

This formula from Batchelor (1967) is used when the relative speed between the vehicles and the medium is low.

The second formulae is; F = (also from Batchelor, 1967)

Where:

F – Aerodynamic Drag Force

– Density of the medium

– Relative speed between the vehicle and the medium.

A – Frontal area of vehicle

– Aerodynamic drag coefficient

This formula is used when the relative speed between the vehicle and the medium being travelled in is high. To find Aerodynamic Drag of the Land Rover Defender the second formulae shown above was chosen. ‘4.1 shows the aerodynamic drag force acting on the vehicle at different speeds.

Tyre Rolling Resistance

Another opposing acting force is the tyre rolling resistance which is not relative to the vehicle speed. The resistance is however a constant value for the vehicle providing its mass remains unchanged. The formula to calculate the tyre rolling resistance is:

F =

Where:

F – Tyre rolling resistance drag force

– Gravitational constant

m – Mass of the vehicle

– drag coefficient for tyre rolling resistance

By using the assumptions in ‘4.0, we can get the tyre rolling resistance for maximum vehicle laden weight;

F== 0.025*3500*9.8=856.5N

Now for the tyre rolling resistance at the vehicles gross weight;

F== 0.025*1797*9.8=440.265N

‘4.2 plots the tyre rolling resistance for the two conditions of maximum laden weight and vehicle gross weight.

Road Resistance Power

Based on the aerodynamic and rolling resistance drag force, by using the formula P=F*V, we can get the sum of the road resistance power, it provides the negative effects on the vehicle riding. We can see from the road resistance power below, the increase of road resistance power is attended by the speed of the vehicles.

Engine Performance

To get the engine power map, the engine performance data which from Project Stage 1 was required. An overall efficiency of 98% was assumed for the combination of the new gear transmission system and engine performance together.

Torque and Power at 3 operating conditions are as follows:

T = 368.5 Nm @ max power (3500rpm)

T = 400.0 Nm @ max torque (2000rpm)

T = 120.0 Nm @ idle (1000rpm)

The engine performance with the combined transmission efficiency is shown by ‘4.4.

Engine Power Map

By knowing the road resistance and engine power curves, the surplus power can be calculated by the relationship below;

Surplus Power = Engine Power – Road Resistance Power

From ‘4.5 (the new Defenders engine map), the intersecting point of engine power and total road resistance power (includes rolling and aerodynamic drag) indicates the maximum speed of the New Land Rover Defender which is 163km/h (101.3 mph). By comparing this with the current Defenders engine map (see ‘4.6), which shows a top speed of only 137 km/h (86 mph), the new Defenders engine and transmission increases the maximum top speed by 26 km/h (16.2 mph).

Transmission Power with Efficiency Losses

Below the formulas used to plot the transmission power at different gear ratios, with respect to the vehicles speed for the new modified transmission system are shown.

Tn = Tengine x Gear Ratio x Final Drive Ratio

Pn = ηn x Pengine

Vvehicle = Pn / Tn x Rtyre

Where:

Tn – Torque of the gear ratio.

Tengine – Torque from the engine produced.

Pn – Power output with efficiency losses.

Vvehicle – Velocity of the vehicle

Rtyre – Wheel radius.

ηn – efficiency loss of engine , assumed to be 2%.

The graph of the transmission power with against vehicle speed are plotted and shown by ‘4.7 for the new Defender. The same graph is also plotted for the current Defender model so a comparison between the sets of data can be made.

By comparison it can be concluded that in every gear the new Defender reaches its maximum power faster than the current Defender model by an improvement of approximately 4.58% in average. This means that the new Defender will accelerate quicker through each gear.

Traction Force for each Gear Ratio

The traction force enables the vehicle to grip the road and pull away from a standstill. The traction therefore needs to be highest at lower speeds to enable the vehicle to pull away by overcoming its own inertia. By using the following formula the traction force was calculated:

Traction Force =

Where

Pe – Power,

PL – Power Loss,

TF – Tractive Force

‘4.9 shows the tractive force against vehicle speed for each gear ratio related to the proposed new Defender where as ‘5.0 shows the same type of plotted data but for the current Defender model. It can be seen that the new proposed Defender has a larger tractive force of 25KN, especially in its first gear, compared to 21.5KN of tractive force for the current Defender. This is an increase of approximately 3.5KN which is advantageous to the potential market for the Defender where good traction for towing and off-road applications is expected.

Transmission shift strategy with assumption of shift delay

To determine the transmission shift strategy, the engine speed against the vehicle speed for all the gear ratios were plotted, which can be seen by ‘5.1 below.

From ‘5.1 the maximum speed that can be achieved at each gear ratio can be determined. For optimum transmission shift strategy the gears should be shifted at the maximum vehicle acceleration that can be achieved by each gear. The shift delay is also expected to be within the region of 0.5 to 2 seconds.
Displacement, Acceleration & Velocity histories

To calculate the displacement, acceleration and velocity histories, a MATLAB and Simulink model was used. The model with the data input can be seen in Appendix 3. The sequent graphs were thus plotted by the model.

Acceleration History

‘5.2 shows that the maximum acceleration achieved by the new proposed Defender is 3.7 m/s2.

Fuel Consumption

As previously mentioned (see pages 21 – 23) the power required varies with vehicle speed, and thus the fuel consumption rate can be related to the speed at which the vehicle is travelling. The equation for calculating the required power is as follows;

Power = (Drag force + Rolling Resistance force) × Vehicle speed

By using this formula and the previously calculated aerodynamic drag and rolling resistance forces (see pages 22 & 23), the following relationship between the required power and vehicle was derived.

equired

By knowing the required power from ‘5.5 for at any vehicle speed, the fuel consumption can be calculated by using the following formulae;

C =

Where;

C – Fuel consumption.

P – power of vehicle

Hu – (calorific diesel capacity) × (density of fuel) (42100000 × 0.876Kg/L),

V – vehicle speed

The following graph of Fuel consumption against vehicle speed was obtained;

From ‘5.6 it can be seen that at the maximum speed of 101.3 mph (163 km/h) the fuel consumption is 7.78 L/100km which is approximately 30.2 mpg. This value however is for extra urban fuel consumption specification.

To fully compare the new powertrains fuel consumption against the current Defenders, the fuel economy at the speed of 82mph (the current vehicles top speed) will be compared to the new vehicle at the same speed. It was found that the new vehicle has a fuel economy of 29.3 mpg which is lower than that for the current Defender which has an extra urban fuel economy of 30.1mpg (data from carpages.co.uk, 2009). The new Defender compared to the original transmission will have a higher top speed, greater tractive force and faster acceleration time from 0 – 60mph. However these improvements have come at the expense of a slightly greater fuel consumption rate.
Maximum Gradient (Fully Laden) at 30mph

It is assumed that the new Land Rover Defender has a fully laden mass of 3500kg.

The vehicle when ascending a gradient will have the following forces acting upon it;

– Aerodynamic Drag

– Tyre Rolling Resistance

– Gravity restricting the vehicles ability to ascend the slope

The forces due to tyre rolling resistance and gravity on the slope are also dependent on the degree of the gradient. The equation for the tractive force (F) required to overcome resistive forces to ascend the gradient is shown below;

Where:

F – Traction force required

– Density of the medium = 1.23kg/m3

– Relative speed between the vehicles and the medium. =13.9m/s

A – Frontal area of vehicle, A = 3.61 m2

– drag coefficient for aerodynamic = 0.42

– drag coefficient for rolling resistance = 0.025

m – mass of the fully laden Land Rover Defender ,here m = 3500kg

– Gravitational constant, g = 9.8N/kg

– Degree of the gradient.

By viewing the Traction force vs Vehicle speed graphs for the new and old Defender (Figs 4.9 & 5.0 on page 27), the tractive forces at 30mph can be determined. A comparison of the maximum gradient ascendable at 30mph will be carried out on the new and current vehicle so a clear concise comparison can be made.

The following was determined from Figs 4.9 & 5.0;

For New Defender – Tractive Force in 2nd gear at 30mph = 12.01KN

For Current Defender – Tractive Force in 2nd gear at 30mph = 7.42KN

From ‘5.7, it is seen that the new transmissions (red) line for the proposed new Defender can ascend a gradient of 18.8 degrees at a speed of 30mph when fully laden at a weight of 3500kg. Compared to the green line for the current defenders transmission, the current Defender can only ascend a gradient of approximately 11 degrees fully laden. A large improvement is therefore made through modifying the current transmissions system.

Compromises

It appears that as the performance of the vehicle regarding its acceleration, top speed, tractive force and maximum ascendable gradient can be improved, the fuel economy of the vehicle decreases. Thus not all characteristics can be improved at the same time.

The following question may be asked by the board of Directors of Land Rover;

“Is this compromise acceptable?”

Our design team would argue that the potential market that the Defender is aimed at will not regard fuel consumption as highly as other attributes of the vehicle such as the maximum torque of the vehicle and its off-road ability. While the fuel economy at the extra urban criteria has decreased by 0.8mpg, the improvements made in torque available, speed characteristics and ascendable slope that the vehicle can climb will be more appealing to potential customers.
Simulation Conclusion

The following improvements have been made over the current Defender;

– The top speed has increased from 82mph to 101.3mph

– The maximum power can be reached 4.58% faster through each gear ratio. This means that faster acceleration through the gears can be achieved.

– Modification to gear ratios and torque increase (see Project Stage 1) has led to a maximum traction force increase of 3.5KN to 25KN.

– The vehicles acceleration from 0 – 60mph will improve from 14.7s to 11.5s. This new time beats the performance of the Defenders closest competitor, the Jeep Wrangler. It has to be noted however that the results from simulations are estimates and may neglect some factors.

– The maximum ascendable gradient when the vehicle is fully laden has increased from 11 degrees to 18.8 degrees. This is highly advantageous when marketing the vehicle to customers.

The following aspects of the vehicle have been negatively impacted;

– The fuel economy of the proposed Defender has decreased by 0.8mpg. By referring to the benchmarking table, a decrease of 0.8mpg will still mean that the Defender is more fuel efficient than 3 out of 4 of the competitors researched.

6. GEARBOX DESIGN SKETCHES

2-D_view[1]

1[1]

2[1] 3[1]

7. GEAR FATIGUE ANALYSIS

Bending Stress Analysis

To determine the bending stress, the equation below was used:

Where:

Wt Transmitted load

Ko Overload correction factor = 1.5

Ks Size correction factor = 0.76

Kv Dynamic factor = 0.631

Km Load distribution correction factor = 1.0

J Geometry factor = 0.25

P Pitch diametral = 160

Pitch diametral, P = N/D where; N is the number of gear teeth

D is pitch diameter of the gear

Transmitted load, Wt = T/r where; T is the engine torque

r is the radius of pitch diametral

Gear

Pinion

Wheel

No. of teeth

Module

Pitch diameter (mm)

331

Transmitted load, Wt (Nm)

8602.2

2417

Bending stress, σ (Mpa)

622

175

Gears in final drive experience high cycle fatigue. Hence heat-treated alloy steel is a suitable material for selection. The selected steel has ultimate strength of 1500 MPa and yield strength of 800 MPa. Stress against number of cycles or S-N Curve can be plotted by applying Basquin’s Law:

log∆σ=-1alogNf+logSu

Where a=6; ultimate strength, Su=1500 MPa and Nf is Number of cycles.

Gear

Pinion

Wheel

No. of teeth

Module

Pitch diameter (mm)

331

Transmitted load, Wt (Nm)

7526.9

2115

Bending stress, σ (Mpa)

544

153

Gear

Pinion

Wheel

No. of teeth

Module

Pitch diameter (mm)

331

Transmitted load, Wt (Nm)

7914.0

2224

Bending stress, σ (Mpa)

572

161

Since the stress for both gears fall below the material’s endurance limit in all 3 conditions, the gears will have infinite life under calculated theoretical stress.
Gear life

To determine the lifetime or distance travel for the gear before it fails, the following formula can used:

Where, the number of cycles is assumed to be 108 and the final gear ratio is 3.54. Therefore, the gear revolution calculated to be 282485875 revolutions. And the distance travelled is equal to 162824km which means the gear will fail after 162824km without maintenance.

Distance travelled = Gear revolution*Wheel perimeter

= 282485875*0.576

= 162824858m = 162824

8. CONCLUSION

The following transmission and driveline technologies were chosen

– Double stage constant synchromesh gearbox

– Dual Range transfer case

– Locking Differential

– Constant Velocity Joints

The gear ratios were altered to maximise the torque of the vehicle, while still improving the vehicles other speed characteristics. The achievement of these aims was easily met by using the characteristics of the new engine for the Defender which was designed in Project Stage 1.

Specification

New proposed Defender

Current Defender

Top Speed (mph)

101.3

Traction Force (KN)

25.0

21.5

0-60 mph Acceleration (s)

11.5

14.7

Maximum Ascendable Gradient (°)

18.8

Fuel Economy (mpg)

29.3

30.1

It can be seen that all but one of the vehicles specification is improved. The fuel economy of the vehicle decreased. It is argued by the design team however that the Defenders fuel economy is not the main reason potential customers purchase the vehicle. Even with the decrease in fuel economy, the defender is still more fuel efficient than most of its competitors shown by ‘1.0.
Appendix 1 – Detailed Gear Ratio Calculations
Gear Ratio Calculations

Let A=110 mm and β=30⁰ for all gears.

Sixth Gear Ratio, i6

Assume: i6=0.73, Z11=16 and Zc1=16

I6 = (Z2/Z1)*(Z11/Z12)

0.73 = (Zc2/16)*(16/Z12)

Zc2/Z12 = 0.73

Reasonable values of

Zc2=18

Z12= 52

The module, m11-12:

A= m11-12*(Z11+Z12)/ 2cos (β)

110 = m11-12*(16+52)/ 2cos (30⁰)

M11-12= 2.8

First Gear Ratio, i1

Assume i1=5.7

5.7 = (Zc2/Zc1)*(Z1/Z2)

= (38/16)*(Z1/Z2)

Z1/Z2 = 12/5

Reasonable values:

Z1 = 48

Z2 = 30

Hence, the module, m1-2:

A= m1-2*(Z1+Z2)/ 2cos (β)

110 = m1-2*(48+30)/ 2cos (30⁰)

m1-2 = 2.8

Second Gear Ratio, i2

Assume i2=3

3 = (Zc2/Zc1)*(Z3/Z4)

= (38/16)*(Z3/Z4)

Z3/Z4 = 24/19

Reasonable values:

Z3 = 48

Z4 = 38

Hence, the module, m1-2:

A= m1-2*(Z3+Z4)/ 2cos (β)

110 = m3-4*(48+30)/ 2cos (30⁰)

m3-4 = 2.2

Third Gear Ratio, i3

Assume i3=2.375

2.375 = (Zc2/Zc1)*(Z5/Z6)

= (38/16)*(Z5/Z6)

Z5/Z6 = 1

Reasonable values:

Z5 = 30

Z6 = 30

Hence, the module, m1-2:

A= m5-6*(Z5+Z6)/ 2cos (β)

110 = m5-6*(48+30)/ 2cos (30⁰)

M5-6 = 3.2

Forth Gear Ratio, i4

Assume i4=1.875

1.875 = (Zc2/Zc1)*(Z7/Z8)

= (38/16)*(Z7/Z8)

Z7/Z8 = 15/19

Reasonable values:

Z7 = 30

Z8 = 38

Hence, the module, m7-8:

A= m7-8*(Z7+Z8)/ 2cos (β)

110 = m7-8*(30+38)/ 2cos (30⁰)

m7-8 = 2.8

Fifth Gear Ratio, i5

Assume i5 = 1

1= (Zc2/Zc1)*(Z9/Z10)

= (38/16)*(Z9/Z10)

Z9/Z10 = 16/38

Reasonable values:

Z9 = 16

Z10 = 38

Hence, the module, m9-10:

A= m9-10*(Z9+Z10)/ 2cos (β)

110 = m9-10*(16+38)/ 2cos (30⁰)

M9-10 = 3.5

Reverse Gear Ratio, iR

Using iR=4.935:1 and assuming m=2.0

4.935 = (Zc2/Zc1)*(Z13/Z14)

= (38/16)*(Z13/Z14)

Z13/Z14= 2.0778

Assume Z13= 16

A= m13-14*(Z13+Z14)/ 2cos (β)

110 = 2*(16+ Z14)/ 2cos (30⁰)

Z14= 52

Note that the Pitch Center Distance (PCD) for these 2 gears should not meet as they require an idler gear in between to reverse direction. For the idler gear, using module of 2.0, we can assume a reasonable PCD and teeth number. Since the smallest allowable teeth number is 16, we assume it to be 20. Hence PCD for idler gear is 40 mm.

Appendix 2 – Nomenclature

Rg Range of Gear Ratio

RT Top Gear Ratio

RL Lowest Gear Ratio

A Distance between Shafts (mm)

Te max Max Torque (Nm)

I1 First Gear Ratio

In nth gear ratio

ηg Transmission of Efficiency

KA Coefficient of Experience

Zn Number of Teeth of nth Gear

Zc1 Number of teeth for pinion on input shaft

Zc2 Number of teeth for wheel on layshaft (constant-mesh pair)

b Face width (mm)

M Module (mm)

β Helical Angle (o)

Appendix 3 – MATLAB & Simulink Models

MATLAB data input file
Simulink Model

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