Notations Glossary

Automatic Transmission Power Flow

Matrix Representation

MARTIN ÖUN

Master of Science Thesis

Stockholm, Sweden 2014

Automatic Transmission Power Flow Matrix

Representation

Martin Öun

Master of Science Thesis MMK 2014:26 MKN116

KTH Industrial Engineering and Management

Machine Design

SE-100 44 STOCKHOLM

Examensarbete MMK 2014:26 MKN116

Matrisrepresentation av effektflöde i

automatväxellådor

Martin Öun

Godkänt

2014-05-09

Examinator

Ulf Sellgren

Handledare

Stefan Björklund

Uppdragsgivare

AVL

Kontaktperson

Per Rosander

Sammanfattning

Projektet har behandlat epicykliska automatväxellådor och dess uppbyggnad och funktion. Idén

med projektet har varit att ta fram ett sätt för att på ett matematiskt sätt representera växellådans

struktur och dess möjliga effektflöden. Utöver detta har arbetet inneburit att alla teoretiskt

möjliga matrisrepresentationer för två enkla sammankopplade planetväxlar har tagits fram i

MATLAB som underlag för en framtida optimeringsmodell.

Resultatet av arbetet är en stor mängd uppställningar av dessa två planetväxlar och dessas

sammankopplingar. Resultatet från MATLAB har jämförts och verifierats genom manuell

beräkning av antalet variationer och dessas utseende. Resultatet från programmet kan anses som

komplett men för en utökad analys av epicykliska automatväxellådor med fler än två

planetväxlar och andra typer än den enklaste formen av planetväxel, rekommenderas en annan

typ av framställning av alla möjliga variationer. Den metoden för att generera sammankopplingar

som har använts i detta projekt är för komplex och tidskrävande.

Slutsatsen av projektet är att det finns möjlighet att generera och representera många epicykliska

automatväxellådor på matrisform. Ett optimeringsprogram baserat på denna typ av matris kan

förenkla utvecklingen av nya mer avancerade och mer effektiva epicykliska automatväxelådor

vilket leder till mer effektiva fordon.

Master of Science Thesis MMK 2014:26 MKN116

Automatic Transmission Power Flow Matrix

Representation

Martin Öun

Approved

2014-05-09

Examiner

Ulf Sellgren

Supervisor

Stefan Björklund

Commissioner

AVL

Contact person

Per Rosander

Abstract

The project has worked with the function and structure of epicyclical automatic transmissions.

The goal of the project has been to find a mathematical way of representing the transmissions

setup and possible power flows. Furthermore the project has included the generation of all

theoretically possible matrix representations of two simple planetary gear sets in MATLAB as

the base for a future optimization model.

The result of the project is a large quantity of matrix representations of the two planetary gear

sets and their connections and shafts. The result from the MATLAB program has been verified

by comparing the structure and the number of solutions to all manually derived setups. The result

from the program can be considered to be complete for two planetary gears but to extend the

analysis to more complex planetary gears and gearboxes with more than two sets, another

method is suggested. The generation process in this project has been rather complex and time

consuming.

The conclusions drawn from this project is that it is possible to represent many epicyclical

automatic transmissions in matrix form. An optimization program based on this type of matrix

would simplify the design of new, more complex and more efficient epicyclical transmissions

leading to more efficient vehicles.

Glossary

Notations

Roman Upper Case:

B Denes the input shaft in the matrix method in speed relationship

B1 Brake one in a gearbox

B2 Brake two in a gearbox

B3 Brake three in a gearbox

B4 Brake four in a gearbox

B5 Brake ve in a gearbox

C1 Carrier PGS 1

C2 Carrier PGS 2

D Denes the input shaft in the matrix method in torque relationship F1 Tangential force between S1 and R1

F2 Tangential force between S2 and R2

IP F Inverse possibility factor

K1 Clutch one in a gearbox

K2 Clutch two in a gearbox

M The M-matrix in the matrix method, describing connecting shafts and stationary ratios for speed relationship

N The N-matrix in the matrix method, describing connecting shafts and stationary ratios for torque realtionship

P Engine power P O Primary options P V Primary variations R1 Ring gear PGS 1 R2 Ring gear PGS 1 Rs Speed ratio Rt Torque ratio

S1 Sun gear PGS 1

S2 Sun gear PGS 2

SEO Shift element options

SV Secondary variations

T Engine torque

˜

T Torques in torque matrix method

TB1R Reaction torque on R1 by brake B1

TC1R2 Connection torque C1 to R2

Tin Input torque

Tout Output torque

TR1C2 Connection torque R1 to C2

V AR Variations of setups

Roman Lower Case:

˜

k Torque ratios in torque matrix method

r1 Inner radius of larger clutch plate

r2 Outer radius of the smaller clutch plate

rs1 /s1 Speed ratio of S1 to S1 rr1 /s1 Speed ratio of R1 to S1 rc1 /s1 Speed ratio of C1 to S1 rs2 /s1 Speed ratio of S2 to S1 rr2 /s1 Speed ratio of R2 to S1 rc2 /s1 Speed ratio of C2 to S1

zp Number of teeth, planet gear

zp1 Number of teeth, planet gear, PGS 1

zp2 Number of teeth, planet gear, PGS 2

zr Number of teeth, ring gear

zr1 Number of teeth, ring gear, PGS 1

zr2 Number of teeth, ring gear, PGS 2

zs Number of teeth, sun gear

zs1 Number of teeth, sun gear, PGS 1

zs2 Number of teeth, sun gear, PGS 2

Greek Symbols:

ω Engine rotational speed

˜

ω Rotational speeds of the planetary gear elements ωc Rotational speed of planetary carrier

ωc1 Rotational speed of planetary carrier PGS 1

ωc2 Rotational speed of planetary carrier PGS 2

ωin Rotational speed of the input element

ωp Rotational speed of planetary gear

ωp1 Rotational speed of planetary gear PGS 1

ωp2 Rotational speed of planetary gear PGS 2

ωr Rotational speed of ring gear

ωr1 Rotational speed of ring gear PGS 1

ωr2 Rotational speed of ring gear PGS 2

ωs Rotational speed of sun gear

ωs1 Rotational speed of sun gear PGS 1

ωs2 Rotational speed of sun gear PGS 2

Abbreviations

AT Automatic transmission

CAD Computer aided design

CVT Continuously variable transmission

ICE Internal combustion engine

GUI Graphical user interface

MT Manual transmission

PF Power ow

PGS Planetary gear set

SE Shift element

Contents

1 Introduction 7 1.1 Background . . . 7 1.2 Objectives . . . 8 1.3 Delimitations . . . 8 1.4 Methodology . . . 9 1.4.1 Literature Study . . . 9 1.4.2 Requirements specication . . . 9

1.4.3 Software and Planetary Gear Set Representation Development . . . 9

2 Frame of Reference 11 2.1 Introduction . . . 11

2.2 Transmissions . . . 11

2.2.1 Transmission Types . . . 11

2.2.2 Epicyclic Automatic Transmissions . . . 12

2.3 Planetary Gear Sets . . . 14

2.4 Speed- and Torque Relationship of Planetary Gear Sets . . . 15

2.4.1 Speed- and Torque Relationship with the Standard Planetary Gear Set Equations . . . 15

2.4.2 Speed Relationship with the Matrix Method . . . 16

2.4.3 Torque Relationship . . . 18

2.4.4 Problems with Several Sets of Planetary Gears . . . 20

2.5 Shift Elements . . . 20

2.6 Shift Sequences and Power Flows in Epicyclic Automatic Transmissions . . 22

2.7 Losses in an Epicyclic Automatic Transmission . . . 24

3 Requirement Specication 25 4 Software- and Matrix Representation Development 27 4.1 Utilizing the Standard PGS equations . . . 27

4.2 Developing the Matrix Representation . . . 28

4.3 Finding All Solutions . . . 32

4.4 Stationary Ratios . . . 35

4.5 Generating Shift Elements . . . 35

5 Results and Discussion 37 6 Conclusion 39 7 Future Work 41 7.1 Matrix Generation and Structure . . . 41

1 Introduction

1.1 Background

The automotive industry is a competitive market and with more expensive fuels and tougher emissions regulations, the race for eciency has become reality for automotive companies.

To increase driveline eciency, every component of the drive line needs to be improved to minimize energy losses. Eorts have been made to make internal combustion engines (ICEs) more ecient and now the turn has come to the transmission. The British term transmission often refers to the clutch, gearbox, propeller shaft (for rear wheel drive cars), dierential and drive shafts. On the other hand, the American term transmission, in-stead refers only to the gearbox and clutch package. The American meaning of the word transmission that will be used in this thesis project.

There are possible improvements in most transmissions but in conventional epicyclic au-tomatic transmissions (ATs), where the eciency is lower than in manual transmissions (MTs) [1], the room for improvement is larger.

Looking into the future it seems that conventional ATs will keep a large market share meaning that the eciency needs to be increased. Losses in ATs occur mainly due to open clutches but also oil pump losses and losses in the gears. This means that the conguration and function of an AT can be constructed to minimize losses by reducing the number of clutches, gear sets and oil pump pressure. This will signicantly impact the overall eciency of the vehicle. Below is a chart that describes the average distribution of losses currently in ATs.

Figure 1.1: Average losses in an AT [2]

The optimization of the power ow (PF) eliminates unnecessary losses in an AT but also has to take into consideration a number of factors including: space, torque carrying capacity, churning losses and the losses in the open clutches. The system as a whole is interconnected and not only does the conguration options have varying losses but

also manufacture-ability, complexity and manufacturing cost. All that in the automotive industry needs to be weighed in. An optimization model can not work with CAD or drawings of the planetary gearbox, which is why it is needed a way to represent these gears in a MATLAB program. There are many ways of calculating ratios and speeds in planetary gearboxes, but there is no obvious way to represent a set of two planetary gear sets (PGSs) in a general fashion. In the project a way to represent a PF and PGS structure in matrix form is developed, making it possible to use it in mathematic equations enabling a future optimization of the PF. Furthermore, all possible matrix representations of PFs from a two planetary gear transmission have been generated for future optimization.

1.2 Objectives

The purpose of the project was to produce a matrix representation that can be modied to give all the parameters needed for an optimization model for ATs. A set of all theoretically possible combinations of two PGSs were generated in this matrix form that has been selected. These combinations have been veried to be able to run a future optimization model with these setups. The provided data suce as a base for the optimization of a PF in an AT, given the number of gears wanted and the wanted ratios on these gears. The base for an optimization model is the structure and representation of the PGS setup in a matrix form. The parameters needed for the optimization of a set of two regular planetary gears have been generated and structured. The matrix representation covers a range of parameters - which shafts are input, output, stationary and interconnected between the two PGSs and where the shift elements (SEs) are located. The objective was also to create a structure that can be used in the development of a more complex model.

The structure of the matrix is crucial for further work with the optimization, as the number of variations increase with an increased number of gears and PGSs.

1.3 Delimitations

For this project some limitations have been made to simplify the work process. To analyze and develop functional AT PFs and gear designs, costly and time consuming research is necessary. The tool that was developed is therefore a helping tool and a stepping stone for future projects and optimizing models. The provided matrices are not comprehensive but include complete structural setup of two sets of planetary gears and how they interact with each other. To develop the model into a functional gearbox design tool, the option of more and compounded PGSs have to be included and an optimization tool needs to be developed. This is left for future work.

Furthermore there is no physical testing or simulation in any software other than the created MATLAB program. Validation has been done by comparing to manually derived numbers meaning that the project has been limited to a small amount of variations of the PFs to be able to do this. Only two "simple" PGSs are represented in matrices with their interconnections and SEs.

1.4 Methodology

The project consists of three parts. The rst part of the project is a frame of reference in a literature study. This is followed by a requirement specication deciding what the customer needs. The program and structure developed for representing the PGSs is done in the third part.

1.4.1 Literature Study

The project base is a literature study creating a frame of reference that describes the basic functions of an AT, the PGSs and the SEs that handle them. Calculations that need to be performed nding ratios are presented and methods for representing PGSs in matrices are studied.

The literature study leads into the development of a MATLAB program as a help for future designers of ATs with the matrix representation of an AT with two PGSs.

1.4.2 Requirements specication

The customer specied that the important part of the project was to nd a way to cre-ate and sort dierent PFs and to be able to evalucre-ate these after a set of criteria. The requirement specication will discuss the importance of the customer input and what will be expected of the nished product.

1.4.3 Software and Planetary Gear Set Representation Development

The program and the PGS representation structure was developed parallel to each other and the program for generating all PFs for two PGSs, the base for future optimization models, will use the structure that is used in the PGS representation matrices.

Verifying the program is done by comparing to manually calculated values and derived matrices.

2 Frame of Reference

2.1 Introduction

To understand the PF of an epicyclic AT, the basics in transmissions and vehicle compo-nents must be understood. The following chapters will give insight in transmissions, ATs and their components.

2.2 Transmissions

The function of the transmission is to convert engine power to tractive power. The trans-mission also allows the vehicle to start from a standstill through a clutch or coupling device while the engine is running. [3]

The basic idea is to get the power from the ICE to the wheels to be able to move the vehicle forward. The torque and power output of an ICE is dependent on the number of revolutions (RPM) and the torque of the engine multiplied with the rotational speed will give the power available, as seen in equation (2.1) below:

P = T ω (2.1)

The ICE has a range of RPM in which it is possible for it to operate. The range where the ICE can deliver sucient power and at the same time achieve eciency is very small. Therefore, to overcome the forces that act on the vehicle when driving (tractive-, gradient-and aerodynamic forces) the transmission needs to keep the engine RPM gradient-and output torque to match these and at the same time stay within the preferred range of RPM:s. This is done by changing the gear ratio between the ICE and the wheels and this is where the transmission comes in. [4]

2.2.1 Transmission Types

There are a number of transmission types on the market today, among which the traditional MT and AT still are very popular. The dierent transmission types can be categorized in several dierent ways but the easiest is to divide them into MTs or dierent types of automatic transmission with the most diversity on the automatic side. The MT is the simplest with a user operated clutch and mechanical gear shifting while the automatic transmission types need some sort of control unit. [5]

The fully automatic- and semi-automatic, automatic transmissions have a wider variety and can be seen in the gure below:

Figure 2.1: Automatic transmission types

The classic AT as well as the parallel shaft gearboxes are stepped transmissions in contrast to the continuously variable transmission (CVT), with a continuously variable gear ratio, giving an innite number of possible ratios in the gearbox. This means that the ICE can be run at an optimal RPM, increasing eciency. The same goes for the stepped gearboxes - the more steps, the closer to an optimal RPM the engine can run resulting in better eciency. [6]

2.2.2 Epicyclic Automatic Transmissions

ATs automatically vary the gear ratio between the engine and the wheels by using a set of PGSs. The PGSs are activated by a set of clutches and brakes making use of the versatility of a PGS. An oil pump is driven directly or by a chain pumping oil into the hydraulic system, creating a pressure that can engage or disengage the clutches and brakes. A transmission control unit (TCU) controls the valves that handle the gear changing, cooling, lubrication and the oil ow to the torque converter. The dierent components of an AT can be seen in gure 2.2.

Figure 2.2: Basic components of an AT

The torque converter is located between the engine and the gearbox. From standing still, the torque converter is used to smoothly transfer the power from the engine to the gearbox, enabling starting the vehicle from a standstill. The torque converter can be compared to the clutch in a car with a MT. In contrast to the manual transmission the torque converter transfers torque by pumping oil rather than putting two frictional plates together. A semi-stationary fan like piece called the stator is the center of the torque converter, enclosed by two rotating parts - the pump and the turbine. The pump is connected to the engine shaft and the turbine is connected to the gearbox. The stator sits on a one-way clutch, enabling it to freewheel in only one direction. When the vehicle is standing still, the stator acts to guide the oil ow to increase the torque on the turbine, see gure 2.3. The torque converter acts as a torque multiplier. As the vehicle picks up speed and the speed dierence between the turbine and the pump decreases, the stator starts freewheeling with the pump transferring torque more directly. As the gearbox changes gears the process starts over again. [7]

Torque converters with a lockup clutch locks the engine shaft directly to the gearbox when the vehicle has come up to speed, minimizing the drag losses in the torque converter.

2.3 Planetary Gear Sets

To be able to change the ratio in an AT, the gear ratio is changed using PGSs. A PGS consists of a set of gears with a sun gear in the middle and several planet gears that connect the sun gear to a ring gear. There can be one or more planetary gears but most common is three-ve planets in a standard PGS with no prole displacement. See gure 2.4 of a PGS with three planets:

Figure 2.4: Illustration of a PGS with three planets

More planets mean that more torque can be transferred by the planetary gear, but it also means that the inertia and losses in the PGS will increase. A PGS can have dierent stationary ratios by varying the number of teeth on the dierent gears. Due to the geom-etry, there will be a correlation between the dierent elements in the PGS. This means that for a standard PGS the ratio between the number of teeth is given by the following equation:

zr= zs+ (2zp) (2.2)

Where zs, zs and zs represent the number of teeth on the sun- ring- and planet gears.

Depending on which shaft is the input shaft, output shaft and stationary shaft, the PGS will have dierent gear ratios, increasing or decreasing the rotational speed, see table 2.1.

Table 2.1. PGS setups Stationary

element Inputshaft Outputshaft Rotationalspeed Torque Direction of ro-tation

sun ring carrier increases decreases same as input

sun carrier ring decreases increases same as input

ring sun carrier decreases increases same as input

ring carrier sun increases decreases same as input

carrier sun ring decreases increases opposite input

carrier ring sun increases decreases opposite input

With a known stationary ratio, the output speed can be varied by changing the stationary element, which element is connected to the input shaft and which element is connected to the output shaft.

2.4 Speed- and Torque Relationship of Planetary Gear

Sets

Both the speed and torque relationship of a PGS depend on the PGS setup and the stationary ratio. The exact ratio can be calculated using a set of equations.

2.4.1 Speed- and Torque Relationship with the Standard Planetary

Gear Set Equations

The equations describing a PGS, in a static analysis, can be seen below [3]:

In the simplest of calculations the observer can be imagined sitting on the planet car-rier.

The ring gear speed compared to the carrier can then be described by:

ωr= ωp(zp/zr) (2.3)

With ωr representing rotational speed of the ring gear and ωp the rotational speed of

the planets. The speed of the carrier is ωc which gives the absolute speed of the ring

gear:

ωr= ωp(zp/zr) + ωc (2.4)

or:

ωr− ωc= ωp(zp/zr) → ωp· zp= (ωr− ωc)zr and ωs= −ωp· (zp/zs) + ωc (2.5)

The absolute rotational speed of the sun gear can then be calculated:

ωs− ωc= −ωp(zp/zs) → ωp· zp= (ωc− ωs)zs (2.6)

Which gives:

This gives a general relationship for a PGS:

ωr· zr+ ωs· zs= ωc(zs+ zr) (2.8)

Given a stationary element, an input speed and an output shaft the system is fully dened and the speed and torque of all the shafts can be calculated. The torque is given by the inverted relationship of the speed.

An example is when the ring gear is stationary, the sun gear is input and the planetary carrier is the output. The speed of nr = 0 which simplies the equation:

ωs· zs= ωc(zs+ zr) (2.9)

Which gives:

Rs= ωc/ωs= zs/(zs+ zr) (2.10)

Where the input is on the sun gear and the output on the planet carrier. The speed ratio, Rs, is given by the speed on the input shaft over the speed on the output shaft. The

torque ratio, Rt, is given by the following equation, assuming that no losses are present

in the system:

Rt= 1/Rs= ωs/ωc = (zs+ zr)/zs (2.11)

Speed and torque constraints for each element in the PGSs are parameters used in the design process of the PGS. It also aects the losses in the gears.

2.4.2 Speed Relationship with the Matrix Method

The set of two PGSs have more variations than the simple PGS calculator could handle. It expected a stationary element on each of the PGSs. In reality there were far more complex combinations of two PGSs meaning that the mathematical model for the system needed to be expanded. In [6] the matrix method was presented with an example. The following equations were found to describe a system with two PGSs with dierent connections and stationary elements:

ωs1zs1+ ωr1zr1− ωc1(zr1+ zs1) = 0 (2.12)

and:

ωs2zs2+ ωr2zr2− ωc2(zr2+ zs2) = 0 (2.13)

If the ring gear of PGS 1, R1, (closest to the engine) is connected to the carrier of PGS 2,

C2, (closest to the drive shaft) and the ring gear of PGS 2, R2 is connected to the carrier

of PGS 1, C2 the following equations will be true:

ωr1− ωc2 = 0 (2.14)

and:

ωr2− ωc1 = 0 (2.15)

Furthermore, if R1 is stationary:

ωr1= 0 (2.16)

The input shaft is set to be the sun gear of PGS 1, S1:

ωin= ωs1 (2.17)

Combining these six equations, 2.12 to 2.17, into a matrix system, the following equation was derived:

M ˜ω = Bωin (2.18)

The ˜ωrepresents the rotational speeds of the elements of the two PGSs shown in :

˜ ω =         ωs1 ωr1 ωc1 ωs2 ωr2 ωc2         (2.19)

The vector B denes that the last row of the matrix M species which element the input shaft is connected to and ωin gives the input speed.

B =         0 0 0 0 0 1         (2.20)

The matrix M is the interesting component in this equation describing the setup of the two PGSs.In the rst two rows the number of teeth of the dierent elements in the two PGSs are described, for PGS 1 on row one, columns one to three and PGS 2 on row two, columns four to six. The third and fourth row describe any connections of the two PGSs. M =         zs1 zr1 −(zs1+ zr1) 0 0 0 0 0 0 zs2 zr2 −(zs2+ zr2) 0 −1 0 0 0 1 0 0 1 0 −1 0 0 1 0 0 0 0 1 0 0 0 0 0         (2.21)

In this example the M matrix shows that the ring gear of PGS 1 is connected to the planet carrier of PGS 2 on the third row of the matrix. On the fourth row the carrier of PGS 1 is connected to the ring gear of PGS 2. The stationary element is the ring gear of PGS 1 shown on the fth row and the input shaft is on the sun gear of PGS 1 shown on the last row.

Equation 2.18 can be modied to nd the speeds of all the elements in the PGSs knowing the stationary ratio, in the example:

˜

ω = M−1Bωs1= ˜r ωin (2.22)

where ˜r is specic for this example:

˜ r =         rs1/s1 rr1/s1 rc1/s1 rs2/s1 rr2/s1 rc2/s1         (2.23)

In order to nd the torque relationship another set of equations are needed. The torques need to be found on all elements in both of the PGSs.

2.4.3 Torque Relationship

The torque relationship with the matrix method can be described with the following equations [6]:

N ˜T = DTin (2.24)

In this equation, N gives the gear ratios and the PGS setup. ˜T represents the torques of the dierent elements and D denes the input torque position in N and Tinis the input

torque, the same setup as above is used for the PGSs .

N =         zs1 0 0 0 0 0 zr1 0 −1 0 1 0 −(zs1+ zr1) 0 1 0 0 0 0 zs2 0 0 0 0 0 zr2 0 −1 0 1 0 −(zs2+ zr2) 1 0 0 0         (2.25)

Which means that ˜T will be specic for this example:

˜ T =         F1 F2 TR1C2 TC1R2 TB1R Tout         (2.26)

Where F1 is the tangential force between S1 and R1, F2the tangential force between S2

and R2. The terms Tr1c2 and Tc1r2 are the connection torques. Tin is the input torque

and TB1R is the reaction torque on the rst ring gear by the brake B1.

D =         1 0 0 0 0 0         (2.27)

Shifting equation 2.24 around the following equation can be derived: ˜

T = N−1DTin= ˜k Tin (2.28)

Introducing the ˜k vector:

˜ k =         kF 1/in kF 1/in kT R1C2/in kT C1R2/in kT B1R/in kT out/in         (2.29)

This gives an equation with torque ratios ˜k and the input torque Tinon one side and the

torques of the elements ˜T on the other side:         F1 F2 TR1C2 TC1R2 TB1R Tout         =         kF 1/in kF 1/in kT R1C2/in kT C1R2/in kT B1R/in kT out/in         Tin (2.30)

Equation 2.30 gives the torque on each element of the two PGSs.

2.4.4 Problems with Several Sets of Planetary Gears

In order to utilize several sets of planetary gears to achieve more gear ratios, some aspects need to be considered. The shafts that are rotating have no possible way to cross each other which means that some of the combinations of the gear sets are impossible. Furthermore, some of the combinations will lock up the whole gearbox by having two locked elements stopping the third one from rotating.

Combinations with too many stationary elements can be avoided even before the shafts are generated, but the impossible combinations with crossing shafts need to be removed after the shafts have been generated.

2.5 Shift Elements

Brakes and clutches are what disengages and engages the dierent elements in a PGS in an AT. The brakes and the clutches are therefore called SEs. The clutches and brakes are operated with hydraulic pressure generated by the oil pump and maneuvered by the TCU.

The brakes lock any shaft of the PGS to the housing of the gearbox, making it a stationary shaft. The clutches can connect dierent PGS elements within a PGS or between two separate PGSs. Clutches can also connect any element of a PGS to the output or input shaft of the driveline. The clutches are slightly more complicated than the brakes as the structure will be rotating with one of the shafts, meaning that the oil pressure, as the clutch is actuated with hydraulic pressure, will need to go through a rotary joint. The basics of the clutches and the brakes are the same. Clutches and brakes both consist of a number of friction plates layered with metal plates. The friction plates are splined to one of the shafts or the gearbox housing and the steel plates are splined to another shaft or the gearbox housing. The plates are submerged in oil to cool them and to get the correct friction coecient between the plates. As pressure is put on the plates, pushing the friction plates and the steel plates together, a frictional force is created transferring torque. The clutch package that can be seen in gure 2.5 is the cross section of the clutch and the plates are circular with the radius r1 to the inner edge of the larger plate and r2

to the outer edge of the smaller plate.

Figure 2.5: A typical clutch package

The clutches and brakes are maneuvered by the TCU. The TCU sends oil to the SE that needs to engage and releases pressure when a clutch needs to disengage. The torque transfers from one PF to the other. A schematic of the function of a gearbox can be seen in gure 2.6, [7]:

2.6 Shift Sequences and Power Flows in Epicyclic

Automatic Transmissions

The ratio steps of an AT can be linear or the steps can get smaller with the increasing speed of the vehicle. An example are heavy duty vehicles that normally have linear steps while passenger cars and trucks have smaller steps at higher speeds. [8]

In passenger cars the ratio steps usually get smaller as the speed increases and with it the rolling and wind resistance, see gure 2.7. The needed propulsion force goes up and a smaller range of the engine's RPM can be utilized. The range which has enough power to overcome the resistance gets smaller as the vehicle gets closer to its top speed.

Figure 2.7: Shift lines for a four speed gearbox [6]

The vehicle has specic ratios on each gear and to get the wanted ratio from an AT the correct clutches and brakes need to be activated. The PF that gives the wanted ratio is selected by the TCU and the correct ratio for the current speed and tractive load is chosen.

A ow chart of a gearbox can be seen in gure 2.8, showing a gearbox with six forward-and one reverse gear.

Figure 2.8: Gear box ow chart of a six speed AT [5]

This particular AT has ve PGSs, ve brakes and two clutches (seven SEs). When the rst gear is wanted the K1 clutch and the B5 brake are activated and in this specic gearbox

it gives a ratio of 6.154:1. The other gear and their engaged elements can be seen in table 2.2.

Table 2.2. Connected SEs of a six speed at, [5]

Table 2.2 shows which SEs are activated. For the forward gears, in this specic case, the clutch K1 is activated and the other SEs are varied. For the reverse gear, clutch K2 is

activated together with the B5 brake. The shifting is done by releasing one of the SEs and simultaneously activating the next SE.

If only one clutch is released and one clutch is engaged, the shift is called a single-transition shift and if two clutches need to be released and two clutches engaged, it is called a double-transition shift.

As mentioned earlier, the shifting is done with hydraulic pressure generated by an oil pump, meaning that the double-transition shifts require twice the capacity from the pump. This inevitably leads to less eciency in normal operation.

The PF also aects torques and speeds of the dierent elements, meaning that the PF has a direct correlation to the losses in an AT.

2.7 Losses in an Epicyclic Automatic Transmission

The main areas where losses in an AT can be found is the oil pump, open clutches, gear meshing, oil splashing, bearings, bushings and seals.

To be able to analyze these in a future PF optimization program the parts of the gearbox that contribute to the losses need to be represented in the base for the optimization. The oil pump losses depend heavily on the shift sequence since a more demanding shift pattern with double-transition shifts, require a pump with a larger capacity. This creates an overcapacity at times when the gearbox is not shifting. To be able to see these losses in the PF representation, a setup of the SEs need to be included.

The second largest losses are in the open clutches. The simplest way of reducing these losses is to reduce the number of open clutches.

In the meshing of the gears, there are churning losses and frictional losses. These losses depend on the speeds and torques of the dierent elements of the PGSs. To minimize these losses the number of planetary gears need to be minimized and the rotational speeds and torques need to be optimized.

3 Requirement Specication

The matrix representation of the AT needs to cover all the needed parameters for an optimization. The parameters that need to be represented are listed below:

• Which of the elements is the input shaft (comes from the engine-/torque converter shaft)?

• Which of the elements is the output shaft (to the drive-/propeller shaft)? • Which elements are stationary?

• Which elements are connected to each other?

• How many SEs are needed and where are they located?

The matrix representation needs to be veried by showing that it covers all possible solu-tions of the two PGSs. This can be done by manually deriving all possible solusolu-tions. The matrix that represents a PF of an AT needs to be able, with a specied range station-ary ratio of the PGSs, to create a number of possible solutions that can be clustered into gear shifting sequences that can be analyzed. This means that the representation at some stage needs to consist of the M matrix in the matrix method representation, see equation 2.21

SEs are needed to be able to change the gears in the gearbox, but for every cluster of gear changing sequences dierent interconnecting shafts and stationary elements need to be switched into SEs. The matrix representation of the possible solutions therefore needs a SE representation that show where SEs are needed.

The user needs to specify the number of gears and their specic ratios to be able to create clusters of generated matrix representations. The number of gears will dene the number of PGSs and SEs that are needed and the specic ratios for each gear will dene the stationary ratios in the PGSs.

Starting o with the generation of the single PGS needs to include all possible options of a PGS, both including a single PGS and all the possible combinations of a PGS in a set of two PGSs. This means that all the variations of a single PGS needs to be included. The only limitation of a single PGS will be that the input shaft and the output shaft should not be stationary. The single PGSs need to be categorized and sorted to provide an overview and also to be counted.

4 Software- and Matrix

Representation Development

4.1 Utilizing the Standard PGS equations

The limitations in the project state that only the PGSs and their connections (shafts and SEs) are going to be represented.

Developing a representation of two PGSs and their connections and SEs need some un-derstanding of the concept of PGS calculations with MATLAB. The initial task of the project was to utilize the standard equations for calculating ratios and create a program that could create all possible ratios for all possible combinations of two PGSs.

In the beginning of the project a simple MATLAB GUI was created using equation (2.8). The initial program helped in the calculation of the ratio for each PGS and the combined ratio for the two sets together. The program was also prepared for calculating speeds on the output shaft.

The program input is dening which of the shafts that are stationary (one per PGS) and which is the input and the output shaft of each PGS, see gure 4.1. The output is the gear ratio of each PGS and the total ratio for two PGSs. The program can also calculate the output speed, given an input speed.

Figure 4.1: MATLAB program with GUI for calculating gear ratios for one and two stan-dard PGSs

Unfortunately the program only covered a small part of all the possible combinations that are possible and therefore a new method for generation of PGS representations was evaluated.

4.2 Developing the Matrix Representation

The matrix evaluation method [6] describes all possible variations of two PGSs with up to three interconnected shafts or stationary elements. However is it used to calculate the speeds of the dierent elements and not the total ratio of the PGSs. To manage this, it has to be known which element is the output shaft, but also the speed ratio between the input- and output shaft. Therefore the matrix representation was expanded to include the output shaft.

Furthermore the matrix method for two PGSs did not include the possibility of only utilizing one of the PGSs which, with the matrix method, gives an M matrix that is three by three, see the example in equation 4.1.

M =   zs zr −(zs+ zr) 0 1 0 1 0 0   (4.1)

To get the structure of the matrix correct the initial work was to organize and categorize the dierent setup possibilities - a preliminary matrix representation system was created. The system builds on a matrix representation of the PGSs.

Each PGS starts of as a three by three matrix with the rst column representing the sun gear, the second column the ring gear and the third column the carrier. The rows need to include information about input shaft, output shaft and all the stationary elements. The rst row consists of ones on the stationary elements and zeros on the non-stationary elements. The second row has a one on the element with the input shaft and the rest zeros. The same goes for the third row where the element with the output shaft, output from the gearbox, is represented with a one and zeros on the rest of the positions. For example a PGS with one stationary element, the ring gear (brown), and the input on the sun gear (red) and carrier of the color green would look like gure 4.2.

Figure 4.2: A single PGS on preliminary matrix form

In the example there is no output shaft on the PGS which is why the third row of the matrix is empty.

A PGS setup with two PGSs, can be built by combining two of these setups. These matrices can be visualized with block diagrams and are presented in gure 4.3 to 4.5. With zero stationary elements:

Figure 4.3: Zero stationary elements, mode one to ve

With one stationary element (The dots means that more variations exist but have not been written out):

The nal variations with two and three stationary elements:

Figure 4.5: Two and three stationary elements, mode one

If an element is connected, the input shaft it is shown by an arrow from the left and means that it is where the engine is connected. An arrow to the right means that it is the output shaft, where the power is transferred to the next step in the driveline, e.g. a propeller shaft. The number of stationary elements and the conguration of the input and output shafts decide which "mode" the PGS works in.

The "modes" for the PGSs decide how they can be combined. Mode one has no input or output shafts and can therefore only be combined with a PGS with both input and output shaft - which means mode four and ve. Mode two has only an input shaft which means it has to be combined with mode three that only has an output shaft. The number of stationary elements can vary within the modes which is why further categorizing is needed to create a structured way to generate shafts between the two PGSs. The following example shows two PGSs with one stationary setups, one from mode two and the other one from mode three. They go together because the input is on one of the PGSs and the the output is on the other one. There are in total one input and one output. This is important for the combinations to work. The dotted line between the two PGSs represents a possible shaft, but there is no representation in the matrix yet:

Figure 4.6: An example of a setup with one stationary per PGS and one of them from mode two and the other one from mode three

This translates to the matrix representation:

Matrix representation of gure 4.6 =   0 1 0 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0   (4.2)

To be able to generate shaft options into the matrix representation, two more rows need to be added. This gives the possibility to add two shafts to the setup. These two rows are added below the stationary row. This means that row one represents the stationary elements, row two and three can show connections between the PGSs, row four the input shaft and row ve the output shaft. The element that is connected in PGS 1 (closest to the engine) has a one on the connected element with a corresponding negative one on the connected element on PGS 2. So if the dotted line in the example in gure 4.6 represents a shaft the preliminary matrix representation would look like this:

Category 1.1 example =         0 1 0 0 1 0 0 0 1 0 0 −1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0         (4.3)

A number of categories to organize the generation of the possible solutions have been worked out. The number of categories and the sorting has been done starting with the simplest form of PGS combinations adding categories when it was needed. This means that the systematics of the sorting process are built on need rather than simplicity. The rst sorting factor is the number and placement of stationary elements. For example the main category one has two stationary elements, each on a separate PGS seen in gure 4.6.

For category two there are three to four stationary elements, one of the PGSs having one stationary element and the other PGS has the rest of the stationary elements.

Sub categorizing has been about placing the input- and output shafts on the same or dierent PGSs. Category 1.1 has the input on one of the PGSs and the output on the other one, while 1.2 has the input and the output shaft on the same element.

The categories are presented in the table 4.1:

Table 4.1 All the created categories and their descriptions

4.3 Finding All Solutions

The program created in this project aims to nd all theoretical solutions to the combi-nation of two regular PGSs. This means two gear sets that have negative ratio and are independent.

The simplest example is the solution with a stationary element on each PGS and the input shaft on the PGS closest to the engine, PGS 1, and the output shaft on PGS 2, the PGS closest to the nal drive. This simple setup can be combined in a number of ways. The stationary elements have three dierent positions, each in their respective PGS. The input shaft has in turn two shafts left to chose from and same goes for the output.

The basic layout that has been the example earlier, has the ring gear on both PGSs stationary and the input on the sun gear of PGS 1 and the output on the sun gear of PGS 2, gure 4.7.

Figure 4.7: Category 1.1 example block diagram

Here it is clear that the stationary elements can move about and all in all creating three times three combinations just by changing which shafts are stationary. This gives the number of stationary element options (SEOs), which in the end also will determine the number of shaft combinations.

By switching the stationary elements around some of the options are found, see gure 4.8:

Figure 4.8: Category 1.1 example block diagrams

For a single shaft the input and output shafts need to be considered as well. There needs to be a connection between the two PGSs and the PGSs need to be able to rotate. This excludes a number of options, for example to have no connection between the two PGSs. Furthermore it is not possible to connect a stationary shaft to any of the moving shafts in this case (category 1.1). It is not possible to connect anything to a stationary shaft in this specic setup.

For category 1.1, this gives us four primary options with one shaft. The primary options (POs) is a multiplication factor that can be used to determine the total number of single shaft solutions for category 1.1.

The number of possible shaft solutions increase when the output and the input shafts are varied. The output shaft has two modes for each stationary setup which gives it a multiplication factor two, This is a primary variation (PV) and the input shaft also has two options giving it a factor of two as well, secondary variation (SV).

The nal possibility to nd more variations is to switch the positions of the input and the output, meaning that the input is on PGS 2 and the output on PGS 1. If this option is possible, which it is in this case, there will be twice as many solutions adding a factor of two. This option is called inverse possibility factor (IPF).

V ar = SEO ∗ P SO ∗ P V ∗ SV ∗ IP F (4.4)

The total number of variations in this case is 288 given nine SEOs, four POs, two PVs, two SVs and an IPF of two. In simpler terms, nine SEOs and 32 possible shaft setups per stationary setup.

This needs to be redone with two shafts in the setup. In this simple case the only possibility to add another shaft is to do so between the stationary elements, see gure 4.9:

Figure 4.9: Category 1.1 example block diagrams, two shafts

This means that the number of shaft options will be the same as for the single shafts: nine SEOs, four POs, two PVs, two SVs and an IPF of two giving 288 options.

For this example setup with the input on one of the PGSs and the output on the other, there are no possible shaft setups with 0 shafts nor possible solutions with the input- and output shaft on the same element.

For this to happen the input- and output shafts need to be on the same PGS. This is the case in sub category 1.2. Each PGS has a stationary element and one of the PGSs has both the input and the output shaft. The number of variations for a single shaft is 324 and for two shafts 216. The setup that is used as example is shown in gure 4.10:

Figure 4.10: Category 1.2 example block diagram

For the options with no shafts and the option of having the input- and output shaft on the same element, some of the factors dier. In the case of a no shaft setup (NSS), there is only one P O which is no shafts at all. This means that compared to the single shaft setup this NSS will have a factor nine less solutions, which means a total variation of 36 setups.

4.4 Stationary Ratios

To generate the possible stationary ratios the limitations on the gearbox design need to be known. If the stationary ratio is given within a certain range, all the possible variations of all the possible PFs can be calculated with the speed ratio equation:

˜

ω = ˜rωin (4.5)

By knowing the output speed, or in other words, the correlation between the input speed and the output speed, the total ratio can be calculated as long as both PGSs are used.

This means that the calculated ratio can be compared to the wanted ratio, making it possible to cluster solutions with the same stationary ratios.

4.5 Generating Shift Elements

PGS setups with the same number of teeth can then be clustered into groups with possible solutions. The groups that have a complete set of gear ratios, meaning that all the wanted ratio steps are accounted for, are saved for further evaluation.

The clusters can then be compared depending on possible SEs and creating a gear sequence where no double shifting occurs. Only gear changes with one SE releasing and one SE engaging are allowed. To save the number of SEs needed, one SE can be used for several gear shifts.

After this step the matrix representation of all possible solutions is completed. The number of possible solutions depend on the specied wanted ratios and the accuracy they are specied with.

5 Results and Discussion

The number of individual shaft and planetary gear setups generated are presented in the table below. The number of solutions were veried by comparing to the list with manual calculation seen in APPENDIX A, that was calculated using equation 4.4. The generated solutions were on the preliminary matrix form, see table 5.1.

Table 5.1: Number of shaft setups Category Number of solu-_tions

1.1 576 1.2 1080 2.1 504 2.2 168 3.1 1872 3.2 252 3.3 888 3.4 810 4.1 810 4.2 828 5.1 450 5.2 150 total 7788

The number of shaft setups generated by the MATLAB program are coherent with the manually derived solutions. This indicates that the program include all possible solu-tions.

The equation that was created to verify the MATLAB solutions is part of the result, see equation 4.4.

To generate more than the preliminary matrix representation user input is needed, specied range of stationary ratios and wanted gear ratios, which unfortunately was not possible to do in the time frame of this project. The preliminary matrix representation was generated and the formulas for deriving the rest of the needed variables for an optimization model exist.

Figure 5.1: Matrix setup structure

The rst six by six is the preliminary representation of the structure, connected shafts and stationary shafts. The M matrix will be lled in with stationary ratios once the wanted ratios are specied. The SE-matrix denes the possible SE used when clustering the dierent shaft setups.

The speed ratio equation (2.18) rewritten gives the possibility to calculate preliminary stationary ratios enabling the clustering process. This means that the torque equation (2.28) is only needed in a future optimization program where torque on each individual element is a design parameter. The same goes for the choice of the torque converter setup and the chosen ratios on the dierent gears in the gearbox.

To create the perfect AT the number of PGSs and SEs need to be minimized and the oil pump needs to be optimized, meaning that there will be an optimized AT for each application with a given number of gears and optimal ratios on the gears. A future optimization tool should give the best option for a pre-dened number of gears and the ratios on these gears. The possibilities with this setup is that many variations in number of gears and ratios can be tested before a decision on the transmission design is made.

6 Conclusion

Using manual matrix generation methods are time consuming and unfortunately it is far from simple to generate them as well. In this project one way of creating and representing two PGSs and their shafts in an AT was investigated. The representation proved accurate and can be used for development of an optimization model. To generate solutions on the preliminary matrix representation form, another method is suggested.

The main diculties in this project have been to nd ways of structuring and to overview the vast quantity of data. The problem has not been the generation of data but how to store it and to grasp how much it actually is. This is done for only two PGSs which are actually possible to calculate by hand to check if the number of solutions are correct. Problems would immediately arise if another PGS were added. This means that more advanced model needs to be developed.

To conclude an eective way of storing the data in a matrix representation has been found and all the possible solutions of two simple PGSs has been found. The time consuming way that the generation process of the shaft setups was performed, required more time than expected and a more realistic time plan or a longer time span for this type of project would have been needed.

The obvious gains of a project like this are, among other things, that a known PF setup when starting the design process of an AT, giving many of the variables required in the design process simplies the process a lot. Combining the optimization model with a design tool could shorten the time for a new gearbox design considerably, giving new exciting possibilities for manufacturers to create super ecient ATs, saving resources and minimizing the impact on the environment.

7 Future Work

The project has been a study in matrix representation of epicyclic ATs meaning that the PGSs in the ATs have been translated to ones and zeros in matrices enabling further computer and MATLAB based analysis and optimization of the data.

7.1 Matrix Generation and Structure

The data sets generated in the project covers all solutions of a gearbox consisting of two standard PGSs. This means that only a small part of the possible solutions are covered. The data generated also needs sorting and analysis before it can be used in an optimization program. The ultimate goal of this project was to provide sucient data for a program deciding the optimal PF with a given set of input parameters. A lot of manual work was used to nd a structure for the matrix representations consuming a lot of the time in the project.

The rst part of the future work could be to create a mathematic equation based on boundary values generating the PGS setup and shaft options rather than using logic equations and manual matrix generation. This would mean that the manual work for setting up the base for the optimization program would be more general and enabling it to be expanded to include more types of PGSs and more than two PGSs in a row. Another aspect of the generation of optimization data that was not covered in this project is the fact that all solutions for a PGS was considered in the generated data. There is no deletion of illegal shaft structures included in this project. Given previous work, see [9], the shaft deletion process is a complex programming task. A MATLAB program to detect and delete illegal shaft structures within a generated set of data is needed to be able to create a useful optimization model.

7.2 Creating an Optimization Software

The matrix representation and the data generated in this project can be used to create an optimization model nding the optimal PF. This future work needs to consider several aspects within gearbox design that this project does not include.

Further research within gearbox losses and gearbox design will have to be performed and an equation or optimization algorithm will have to be developed to complete the task. Some of the aspects that need to be considered are presented below:

• All losses in the planetary gears and SEs need to be calculated for each PF. • Design parameters and structural eort need to be quantied and made comparable.

• SE options need to be quantied and evaluated.

• The number of wanted gears, gear ratios and stationary ratios will give the number of brakes, clutches and PGSs that are needed.

• Speeds and torques of the dierent components in the PGSs need to be weighed in. When all the parameters have been made comparable an optimization algorithm needs to be developed to be able to nd the best option for the given setup of the gearbox. The gear stepping and ratios of the gearbox is not determined by the optimization model but are input parameters for the model. Parameters that might be interesting could include:

• The structural eort, shift logic, gear ratios, speeds- and torques in the gearbox and the dierent losses.

• Relative speeds in the clutches, space- and weight of the gearbox and torque carrying capacity limitations.

• An evaluated and optimized PF can then be derived and presented.

With an optimization tool for the PF in the AT, it would be possible to simplify the design process for ATs and increase the complexity and functionality of future transmissions. Integrating an optimization tool with standard design tools open new doors and exciting possibilities.

Bibliography

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[5] M. K. K. Venu, Wet clutch modelling techniques, Master's thesis, Chalmers Univer-sity of Technology, 2013.

[6] S. Bai, J. Maguire, and H. Peng, Dynamic Analysis and Control System Design of Automatic Transmissions. 400 Commonwealth Drive Warrendale, PA 15096-0001 USA: SAE International, 2013.

[7] H. Naunheimer, J. Ryborz, B. Bertsche, and W. Novak, Automotive Transmissions, Fundamentals, Selection, Design and Application. Springer, second edition ed., 2011. [8] P. Rosander, Stepping in transmissions. AVL, Södertälje, 2012-05-25.

[9] D.-I. B. Mueller, D.-I. F. H. Ubben, D.-I. F. W. Gantner, and B. Dipl.-Ing. (FH) G. Rathke LuK GmbH & Co. KG, Ecient components for ecient transmissions, in 12th CTI Symposium and Exhibition, 2013.

APPENDIX A