Automobile transmission system-1

Transmission system-1

                 Transmission is a speed reducing mechanism, equipped with several gears. It may be called a sequence of gears and shafts, through which the engine power is transmitted to the car wheels. The system consists of various devices that cause forward and backward movement of cars to suit different field condition. The complete path of power from the engine to the wheels is called power train. 

1.      Function of power transmission system

                Function of power transmission system: (i) to transmit power from the engine to the rear wheels of the car, (ii) to make reduced speed available, to rear wheels of the car, (ii) to alter the ratio of wheel speed and engine speed in order to suit the field conditions and (iv) to transmit power through right angle drive, because the crankshaft and rear axle are normally at right angles to each other. The power transmission system consists of:

(a) Clutch                   

 (b) Transmission gears        

(c) Differential

(d) Final drive              

(e) Rear axle                      

 (f) Rear wheels.

 Combination of all these components is responsible for transmission of power.

Necessity of Gear Box In an Automobile  

·         The  gear  box  is  necessary  in  the  transmission  system  to maintain engine  speed at the  most  economical value  under  all  conditions  of vehicle movement. 

·         An ideal gear box would provide an infinite range of gear ratios, so that the engine speed should be kept at or near that the maximum power is developed whatever the speed of the vehicle.

Function of a Gear Box

·         Torque ratio between the engine and wheels to be varied for rapid acceleration and for climbing gradients. 

·         It provides means of reversal of vehicle motion.

·         Transmission  can  be  disconnected  from  engine  by  neutral position of gear box

Types of Gear used in Gearbox

·         Spur Gears

·         Helical Gears (open or crossed)

·         Herringbone Gears

·         Bevel Gears

Gear efficiency

·         Spur gear =98 to 99%

·         Helical Efficiency of 96-98% for parallel and 50-90% for crossed

·         Herringbone 95% efficient

Fundamental Law of Gearing

·         The angular velocity ratio between 2 meshing gears remains constant throughout the mesh

·         Angular velocity ratio (m_V)

·         Torque ratio (m_T) is mechanical advantage (m_A).

Involute Tooth Shape

·         Shape of the gear tooth is the involute curve.

·         Shape you get by unwrapping a string from around a circle

·         Allows the fundamental law of gearing to be followed even if center distance is not maintained

Contact Geometry

•       Pressure angle (f): angle between force and motion.

Fundamental Law of Gearing

·         The common normal of the tooth profiles, at all contact points within the mesh, must always pass through a fixed point on the line of centers, called the pitch point

Change in Center Distance

·         With the involute tooth form, the fundamental law of gearing is followed, even if the center distance changes

·         Pressure angle increases

Backlash

·         Backlash – the clearance between mating teeth measured at the pitch circle

·         Whenever torque changes sign, teeth will move from one side of contact to another

·         Can cause an error in position

·         Backlash increases with increase in center distance

·         Can have anti-backlash gears (two gears, back to back)

Design of Spur gear

Gear Tooth Nomenclature

·         Circular Pitch, p_c=pd/N

·         Diametral Pitch (in 1/inch), p_d=N/d=p/p_c

·         Module (in mm), m=d/N

 Spur gear

Consider a spur gear with teeth=30 & pitch diameter =6

·         N = Number of Teeth DP x PD =30

·         OD = Outside Diameter (N+2)/DP =

·         PD = Pitch Diameter N/DP

·         DP = Diametral Pitch N/PD

·         RD = Root DiameterOD-2(HW)

·         CD= Centre Distance CD=PDgear1/2+PDgear2/2

·         A = Addendum1/DP

·         D = Dedendum HW-A

·         WD = Whole Depth2.157/DP

·         R=Tooth Radius¾(CP)

·         CT = Cordial Thickness PD sin (90/N)

·         CP= Circular Pitch3.1416/D

 Compound gear Train

Planetary Gear set with Ring Gear Output

·         Two inputs (sun and arm) and one output (ring) all on concentric shafts.

Example

Given:

Sun gear N₂=40 teeth

Planet gear N₃=20 teeth

Ring gear N₄=80 teeth

W _arm =200 rpm clockwise

W _sun =100 rpm clockwise

Required:

Ring gear velocity w _ring

N₂=40, N₃=20, N₄=80

W _arm= -200 rpm (clockwise)  

W _sun= -100 rpm (clockwise)

Sign convention:

Clockwise is negative (-)

Anti-clockwise is positive (+)

w₄ = - 250 rpm

Constant mesh Gear box

Constant Mesh Gear Box

The construction or main components of constant mesh gearbox are:

1.      Shafts – Same as sliding mesh 3 shafts are there-

(i) Main shaft- Also known as the output shaft, the splined shaft over which the dog clutches along with gears is mounted.

(ii) Lay shaft- An intermediate shaft over which the gears which are in constant mesh with main shaft gears are mounted.

(iii) Clutch shaft- Same as sliding mesh clutch shaft carries engine output to the gearbox and transmits it through the constantly meshed lay shaft gear.

2. Gears –2 types of gears are used that are-

(i) Helical gears- having angular cut teethes over cylindrical cross-section metal body.

(ii) Bevel gears- having angular cut gear teethes same as helical gears but with conical cross-section.

3. Dog clutches- These are the special shifting devices responsible for transmitting appropriate gear ratio to the final output, the pair of gears with suitable gear ratio comes in contact with the sliding dog clutches which in turn transmit the gear ratio of the pair of meshed gears to the final output shaft.

4. Gear lever- It is the lever used for shifting or sliding the dog clutches over main

Working of Constant Mesh Gearbox

Since the gear of the main shaft are in constant mesh with the appropriate gear of the lay shaft ,so the selection of 1, 2, 3 , 4 and reverse gear is obtained with the sliding and meshing of the dog clutches with the appropriate pair of gears and process is as follows-

First gear

It is the gear which provides maximum torque and minimum speed to the final output shaft which helps the vehicle to start moving from its initial state, when the driver select the 1 gear by pushing or pulling the gear lever the dog clutch with corresponding pair of meshed gear i.e. smallest gear of lay shaft and largest gear of main shaft, slides right or left over the splined main shaft and make contact with the meshed pair and finally the 1 gear is obtained.

Second gear

It is the gear provides higher speed and lower torque than the first gear and is obtained by right or left sliding of the corresponding dog clutch towards the appropriate pair of meshed gears i.e. second smallest lay shaft gear and second largest main shaft gear, in order to make contact with the pair.

Third gear

It is the second highest speed gear having very low torque and is obtained by the right or left sliding of the corresponding dog clutch towards the appropriate pair of meshed gear i.e. second biggest lay shaft gear and second smallest main shaft gear.

Fourth gear

It is the highest speed gear of 4-speed manual transmission in which maximum speed of the clutch shaft is transmitted to the final output by right or left sliding of the dog clutch to make contact with the pair having largest gear of the lay shaft and smallest gear of the main shaft and very low torque and highest speed is obtained.

Reverse gear

It is the gear which reverses the direction of rotation of the output shaft in order to move vehicle in reverse direction, it is obtained with the special gear known as idler gear which mounted between the lay shaft and the main shaft when the reverse gear is selected the dog clutch makes contact with the idler gear and reverse gear is obtained.

Synchromesh Gearbox:

Principle:

In a gearbox, there is always a difficulty in engaging the stationary gear with the gears already rotating at a high speed. The principle states that “Before engaging the gears they are brought in frictional contact with each other and after equalizing the speed the engagement is done.”

 Construction:

The synchronizer is placed between two gears. So, we can use one unit for two gears.  G1 and G2 are the ring-shaped members which are having the internal tooth that fits onto the external teeth. F1 and F2 are the sliding members of the main shaft. H1, H2, N1, N2, P1, P2, R1, R2 are the friction surface.

Synchromesh gearbox

Synchronising unit

1. Main shaft Gears:

A spline shaft is used as the output shaft over which the synchronizers and gears are mounted. According to the Fig. B, C, D, E are the gears that can freely rotate on the main shaft in mesh with corresponding gears in the lay shaft.  As long as shaft A is rotating all the gears in the main shaft and lay shaft rotates continuously.

2. Lay Shaft Gears:

It is the intermediate shaft over which gears with suitable size are mounted and is used to transmit the rotational motion from clutch shaft to the final output shaft. According to the Fig. U1, U2, U3, U4 are the fixed gears on the countershaft (lay shaft).

3. Clutch Shaft:

It is the shaft used as an input shaft in the gearbox as it carries the engine output to the gearbox.

4. Cone Synchromesh:

The side of the gear to be engaged has two features. One is hollow-cone, and the other is cone surrounded by the ring of dog teeth. The gear is made the cone and teeth that the synchromesh mechanism contacts.

Synchronizer unit

5. Synchronizers:

They are the special shifting devices used in the synchromesh gearbox which has conical grooves cut over its surface that provide frictional contact with the gears which is to mesh in order to equalize the speed of the main shaft, lay shaft and clutch shaft which in turn provides smoother shifting of gears.

6. Gear lever:

It is the shifting lever operated by the driver and is used to select the appropriate gear i.e. 1, 2, 3, 4, 5 or reverse gear.

Working of synchromesh gearbox

First gear

When the driver push or pull the gear lever in order to select the first gear which gives the maximum torque and minimum speed and is used to move the vehicle from its initial state, the synchromesh device attached with the pair of meshed gears having biggest gear of the main shaft and smallest gear of the lay shaft equalises the speed of the shafts by making frictional contact with the pair and finally the first gear is obtained.

Second gear

This is the gear having lower torque and higher speed than first gear and is obtained when the pair of gears having second largest gear of the main shaft and second smallest gear of the lay shaft is meshed by the corresponding synchromesh device.

Third gear

This gear having higher speed and lower torque than second gear is obtained when the corresponding synchromesh device attached to the pair of gear having intermediate size gear of main shaft and intermediate size of gear of lay shaft makes contact.

Fourth gear

It is the second highest speed gear which is obtained when the corresponding synchromesh device attached to the pair of meshed gears having second smallest gear of  main shaft and second largest gear of the lay shaft makes contact.

Fifth gear

It is the highest speed and lowest torque gear which transmit the maximum speed of the clutch shaft to the main or output shaft and is obtained when the corresponding synchromesh device attached to the pair of meshed gear having smallest gear of main shaft and largest gear of lay shaft makes contact.

Note – In some vehicle like ktm duke 390cc over drive is attached which directly obtained the output from the clutch shaft and transmit to the final drive when the vehicle is on a long run with high speed or when the vehicle is going down the hill.

Reverse gear

It is the gear that reverses the direction of the output shaft which in turn reverse the direction of the vehicle with the help of the idler gear which is usually fit in the middle of the lay shaft and main shaft and is obtained when the idler gear makes contact with the gears on the main shaft and lay shaft.

Note – The reverse gear does not have any synchronizer mechanism, so the gearbox shaft rotation is completely stop before engaging the reverse gear.

Advantages:

·         Smooth and Noise free shifting of gears which is most suitable for cars.

·         No loss of torque transmission from the engine to the driving wheels during gear shifts.

·         Double clutching is not required.

·         Less vibration.

·         Quick shifting of gears without the risk of damaging the gears.

Disadvantages:

·         It is extortionate due to its high manufacturing cost and the number of moving parts.

·         When teeth make contact with the gear, the teeth will fail to engage as they are spinning at different speeds which cause a loud grinding sound as they clatter together.

·         Improper handling of gear may easily prone to damage.

·         Cannot handle higher loads.

Application

It has a wide application as almost 50% of the vehicle on the road used synchromesh gearbox, some of them are-

·         In Maruti Suzuki swift it comes with 5-speed 1-reverse manual transmission configuration.

·         It is used in bikes like ktm duke 390cc.

·         Most of the race cars like formula-1 uses synchromesh gearbox with suitable modification in shifting lever as they required sudden shifting of gears from high torque to high speed because they have to race onto the zig-zag track having sharp turns.

Design of helical gear

Profile of helical gear

Formulae to calculate helical gear dimensions

What is AMT?

AMT stands for Automated Manual Transmission. It is a type of semi-automatic transmission. ‘Clutch less Manual Transmission’ is another name for AMT. It is gaining rapid popularity among the vehicle manufacturers and the customers. This is because it offers the cost-effectiveness over the conventional automatic transmission. In addition, this technology is also very convenient to use.

Operation

The Easy-R transmission has the following gears: R, N, D, M+, and M-

·         R is the reverse gear. It is similar to R in both traditional manual and in full automatic transmissions.

·         N is the neutral gear. It is similar to N in both traditional manual and full automatic transmissions.

·         D is the drive gear. It is equivalent of D in a full automatic transmission. The gearbox in an AMT car is a manual gearbox, instead of one with a torque converter as in a traditional automatic transmission.

·         M- Downshifts a gear in sequential fashion, from M5 (M6 in 6-speed AMT cars) to M1.

·         M+ up-shifts a gear in sequential fashion, from M1 to M5 (M6 in 6-speed AMT cars).

An automated manual transmission (AMT) is basically a manual transmission (MT) with electronic controlled clutch and gear actuators. To convert a manual transmission into an automated manual transmission, the clutch pedal (1) and the gear shift lever (11) are replaced by electrohydraulic or electric actuators.

First generations of AMTs were based on the concept of “add-on“, which means that an existing, already designed MT was converted into an AMT by adding external electronic controlled actuator mechanisms. Later generations of AMTs had the actuators embedded into them from the early stages of the design phases.

A conversion from a MT to an AMT requires:

• replacement of the clutch actuation mechanism with an electrohydraulic / electrical actuator
• replacement of the gear actuation mechanism with an electrohydraulic / electric actuator
• integration of an electronic control module
• integration of: input shaft speed sensor, clutch position sensor, gear selection and engagement position sensors, shift lever position sensor, fluid pressure and temperature sensor (in case of an electrohydraulic actuation system)
• engine control software which allows torque control during gearshift

Driving modes

Nowadays, for a driver, is quite difficult to distinguish between an AMT, AT or DCT. If at a hardware level there are different in terms of layout and components, at the function (software) level, they all behave similarly.

In an AMT vehicle the driver has an accelerator pedal, a brake pedal, a program / gear selector lever and (optional) steering wheel paddle shifters. With the lever, the driver can select at least four modes:

·         Automatic (also called Drive) (A, D)

·         Manual (M or +/-)

·         Neutral (N)

·         Reverse (R)

 

               In the Automatic mode (also called Drive mode), both the decision to shift gears and the actual gear shifting is performed by the transmission control module, without any intervention or input from the driver. The main criteria for a gear change are calculated function of the vehicle speed and engine load (accelerator pedal position).

               In Manual mode, the driver can decide when to shift gears. By tipping “+” an upshift if requested and by tipping “-” a downshift is requested. In this mode, there are some protection functions active, which will shift the gears even if the driver didn’t requested it. For example, if the engine speed is too high, an upshift will be performed and, if the engine speed is too low (not enough engine torque), a downshift will be performed.

Most of the vehicles with AMT have a Snow mode. This mode is useful in driving conditions with low friction of the road. In this mode, for vehicle launch, the 2^nd gear is selected instead of the 1^st gear. This way, the traction force at the wheel is limited and wheel slip is avoided.

The main advantages of an automated manual transmission (AMT), compared with a manual transmission (MT), are:

·         more comfortable driving (the gear shifting is done automatically)

·         better fuel economy (the engine is kept in the most fuel efficient operating zone, through the gear ratio)

·         wear diagnostic (the electronic controlled actuators can measure clutch wear and inform the driver)

The disadvantages are the higher price of the transmission which translates in slightly higher price of the vehicle.

Benefits of AMT

·         Relieves driver from clutch and shifting operations

·         Achieves up to 5% fuel savings vs. manual transmission

·         Optimal gear shifts reduce maintenance cost, clutch wear and downtime

·         Minimizes the performance gap between experienced and non-experienced drivers

·         Helps the driver to give full attention to the road

·         Enables comfortable, smooth and safe driving like an automatic transmission

·         High degree of modularity significantly reduces development cost and time

Features of AMT

·         Complete system includes AMT hardware and shift software

·         System modularity easily adapts to existing gearboxes

·         Minimizes space requirements at gearbox

·         Available for 5-speed up to 18-speed applications

·         Designed for constant and synchro-mesh gearboxes

·         Offers a wide range of driver functions

Application of AMT

·         Heavy, medium and light duty truck

·         Bus, coach

·         Off-Highway

Automated transmission system

Purpose of an Automatic Transmission

Just like that of a manual transmission, the automatic transmission's primary job is to allow the 

for an introduction to planetary gearsets.

Any planetary gearset has three main components:

1.      The sun gear

2.      The planet gears and the planet gears' carrier

3.      The ring gear

Each of these three components can be the input, the output or can be held stationary. Choosing which piece plays which role determines the gear ratio for the gearset. Let's take a look at a single planetary gearset.

Planetary Gearset Ratios

One of the planetary gearsets from our transmission has a ring gear with 72 teeth and a sun gear with 30 teeth. We can get lots of different gear ratios out of this gearset.

 

Also, locking any two of the three components together will lock up the whole device at a 1:1 gear reduction. Notice that the first gear ratio listed above is a reduction -- the output speed is slower than the input speed. The second is an overdrive -- the output speed is faster than the input speed. The last is a reduction again, but the output direction is reversed. There are several other ratios that can be gotten out of this planetary gear set, but these are the ones that are relevant to our automatic transmission. You can try these out in the animation below:

First Gear

In first gear, the smaller sun gear is driven clockwise by the turbine in the torque converter. The planet carrier tries to spin counterclockwise, but is held still by the one-way clutch (which only allows rotation in the clockwise direction) and the ring gear turns the output. The small gear has 30 teeth and the ring gear has 72, so the gear ratio is:

Ratio = -R/S = - 72/30 = -2.4:1

So the rotation is negative 2.4:1, which means that the output direction would be opposite the input direction. But the output direction is really the same as the input direction -- this is where the trick with the two sets of planets comes in. The first set of planets engages the second set, and the second set turns the ring gear; this combination reverses the direction. You can see that this would also cause the bigger sun gear to spin; but because that clutch is released, the bigger sun gear is free to spin in the opposite direction of the turbine (counterclockwise).

Second Gear

This transmission does something really neat in order to get the ratio needed for second gear. It acts like two planetary gearsets connected to each other with a common planet carrier.

The first stage of the planet carrier actually uses the larger sun gear as the ring gear. So the first stage consists of the sun (the smaller sun gear), the planet carrier, and the ring (the larger sun gear).

The input is the small sun gear; the ring gear (large sun gear) is held stationary by the band, and the output is the planet carrier. For this stage, with the sun as input, planet carrier as output, and the ring gear fixed, the formula is:

1 + R/S = 1 + 36/30 = 2.2:1

The planet carrier turns 2.2 times for each rotation of the small sun gear. At the second stage, the planet carrier acts as the input for the second planetary gear set, the larger sun gear (which is held stationary) acts as the sun, and the ring gear acts as the output, so the gear ratio is:

1 / (1 + S/R) = 1 / (1 + 36/72) = 0.67:1

To get the overall reduction for second gear, we multiply the first stage by the second, 2.2 x 0.67, to get a 1.47:1 reduction. This may sound wacky, but if you watch the video you'll get an idea of how it works.

Third Gear

Most automatic transmissions have a 1:1 ratio in third gear. You'll remember from the previous section that all we have to do to get a 1:1 output is lock together any two of the three parts of the planetary gear. With the arrangement in this gear set it is even easier — all we 

parts of the planetary gear. With the arrangement in this gear set it is even easier — all we have to do is engage the clutches that lock each of the sun gears to the turbine.

If both sun gears turn in the same direction, the planet gears lockup because they can only spin in opposite directions. This locks the ring gear to the planets and causes everything to spin as a unit, producing a 1:1 ratio.

Overdrive

By definition, an overdrive has a faster output speed than input speed. It's a speed increase — the opposite of a reduction. In this transmission, engaging the overdrive accomplishes two things at once. If you read How Torque Converters Work, you learned about lockup torque converters. In order to improve efficiency, some cars have a mechanism that locks up the torque converter so that the output of the engine goes straight to the transmission.

In this transmission, when overdrive is engaged, a shaft that is attached to the housing of the torque converter (which is bolted to the flywheel of the engine) is connected by clutch to the planet carrier. The small sun gear freewheels, and the larger sun gear is held by the overdrive band. Nothing is connected to the turbine; the only input comes from the converter housing. Let's go back to our chart again, this time with the planet carrier for input, the sun gear fixed and the ring gear for output.

Ratio = 1 / (1 + S/R) = 1 / (1 + 36/72) = 0.67:1

So the output spins once for every two-thirds of a rotation of the engine. If the engine is turning at 2000 rotations per minute (RPM), the output speed is 3000 RPM. This allows cars to drive at freeway speed while the engine speed stays nice and slow.

Reverse Gear

Reverse is very similar to first gear, except that instead of the small sun gear being driven by the torque converter turbine, the bigger sun gear is driven, and the small one freewheels in the opposite direction. The planet carrier is held by the reverse band to the housing. So, according to our equations from the last page, we have:

So the ratio in reverse is a little less than first gear in this transmission.

Hybrid Transmissions system

working

Steps 1 - The hybrid drive-train consists of two separate power sources to propel the vehicle at various modes of driving. Depending on load conditions, the operating system will switch controls from one power source to another which increases efficiency.

Step 2 - The hybrid transmission consists of gears, shafts and clutches that perform various operations such as gear selection much like a conventional transmission. This transmission also contains an additional electrical motor(s) contained inside the transmission housing, besides propelling the vehicle this electrical motor is used the start the conventional gasoline engine, no external starter motor is used in this application. A torque converter which is used in conventional automatic transmissions has been replaced by an electrically controlled coupler that can be engaged and disengaged. A manual hybrid transmission has yet to be developed.

Step 3 - Electric engine(s) are integrated and located inside the transmission case, which is turned on at optimal times in the vehicles operation. These electrical engines 

also provide vehicle braking and battery charging capabilities depending on system mode.

Step 4 - An electrical connector is used to transmit sensor feedback and solenoid control functions supplied and used by the main PCM.

Step 5 - Gears inside the hybrid transmission are much like conventional transmission gears and provide much of the same operations.

Step 6 - A gear selector which controls the transmission mode of operation which is used to maneuver the vehicle in various conditions.

Hybrid electric vehicles

A hybrid electric vehicle is distinguee from a standard ICE driven by four different parts:

 a) A device to store a large amount of electrical energy,

b) An electrical machine to convert electrical power into mechanical    

   Torque on the wheels,

c) A modified ICE adapted to hybrid electric use,

d) A transmission system between the two different propulsion techniques.          

    The possible subsystems of a hybrid vehicle configuration.

The architecture of a hybrid vehicle is defined as the connection between the components of the energy flow routes and control ports. Hybrid electric vehicles were classified into two basic types: series and parallel. But presently HEVs are classified into four kinds: series hybrid, parallel hybrid, series-parallel hybrid and complex. The primary power source (steady power source) is made up of fuel tank and ICE and battery-electric motor is taken as secondary source (dynamic power source).

Series HEV

             A series hybrid drive train uses two power sources which feeds a single power plant (electric motor) that propels the vehicle.  A series hybrid electric drive train where: fuel tank is an unidirectional energy source and the ICE coupled to an electric generator is a unidirectional energy converter. The electric generator is connected to an electric power bus through an electronic converter (rectifier). Electrochemical battery pack is the bidirectional energy source and is connected to the power bus by means of a power electronics converter (DC/DC converter). Also the electric power bus is connected to the controller of the electric traction motor. The traction motor can be controlled either as a motor (when propels the vehicle) or as generator (to vehicle braking). A battery charger can charge batteries with the energy provided by an electrical network.

Parallel HEV

           In the parallel configuration the power of the ICE and the electric motor are added into mechanical coupling, and operate the drive train by the mechanical transmission. There are different combination of the engine and electric motor power:

In parallel hybrid electric vehicle various control strategies can be used. In the most common strategies, ICE is used in on mode and operates at almost constant power output at its area of maximum efficiency. If the power requested from drive train is higher than the output power of ICE, the electric motor is turned on, ICE and electric motor supply power to the drive train. If the power requested from drive train is less than the output power of the ICE, the remaining power is used for charging the batteries. In this configuration, regenerative braking power on a down slope driving is used to charge the batteries. Examples of the parallel hybrid electric vehicles. Insight model introduced by Honda, Ford Escape Hybrid SUV and Lexus Hybrid SUV.

Wheel torque can be calculated function of engine torque if the parameters and status of the transmission are known. In this tutorial, we are going to calculate the wheel torque and force for a given:

§  engine torque

§  gear ratio (of the engaged gear)

§  final drive ratio (at the differential)

§  (free static) wheel radius

Also, we are going to assume that there is no slip in the clutch or torque converter, the engine being mechanically linked to the wheels.

This method can be applied to any powertrain architecture (front-wheel drive or rear-wheel drive) but, for an easier understanding of the components, we are going to use a rear-wheel drive (RWD) powertrain.

Image: Vehicle rear-wheel drive (RWD) powertrain diagram

As depicted in the image above, the engine is the source of torque. The gearbox is connected to the engine through the clutch (on a manual transmissions) or torque converter (on an automatic transmissions). We consider that there is absolutely no slip in the clutch (fully closed) or in the torque converter (lock-up clutch closed). In this case the engine torque T_e [Nm] is equal with the clutch/torque converter torque T_c [Nm].

Tc = Te

Further, the engine torque is transmitted through the gearbox, where is multiplied with the gear ratio of the engaged gear i_x [-] and outputs the gearbox torque T_g [Nm].

Tg = ix⋅Te

The propeller shaft is transmitting the torque to the rear axle, where is multiplied with the final drive gear ratio i₀ [-]. This gives the torque at the differential T_d [Nm].

Td=i0⋅T

If the vehicle is driven on a straight line, the torque at the differential is equally split between

The left wheel T_lw [Nm] and the right wheel T_rw [Nm].

Tlw=Trw=Td/2

The sum of the left and right wheel torque gives the torque at the differential.             

Tlw +Trw=Td

Replacing (2) in (3) in (4) gives the mathematical expression of the wheel torque function of the engine torque, for a given gearbox ratio i_x and a final drive ratio i₀.

Tw=ix⋅i0⋅Te/2

The formula of the wheel torque applies to a vehicle which is driven on a straight line, where the left wheel torque is equal with the right wheel torque.

Tlw=Trw=Tw

From mechanics (static), we know that the torque is the product between a force and its lever arm length. In our case, the wheel torque is applied in the wheel hub (centre) and the lever arm is the wheel radius r_w [m]. For this example we assume that both wheels have the same radius r_w.

Tlw=Flw⋅rw

The same principle applies to the right wheel torque.

Trw=Frw⋅rw

Assuming that both left and right wheel torque and radius are equal, we can write a generic expression of the wheel force F_w [N], function of wheel torque T_w [Nm] and wheel radius r_w [m].

Tw=Fw⋅rw

 We can extract the formula of the wheel force function of the wheel torque and wheel radius.

Fw=Twrw

Replacing will give the mathematical expression of the wheel force function of engine torque, gearbox gear ratio, final drive ratio and wheel radius.

Fw=ix⋅i0⋅Te2⋅rw