Automatic Transmission Power Flow Matrix Representation770002/FULLTEXT… ·  · 2014-12-09Automatic Transmission Power Flow Matrix Representation MARTIN ÖUN Master of Science Thesis

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

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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.

Nyckelord: automatisk växellåda, planetväxel, matrisrepresentation

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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.

Key words: automatic transmission, planetary gear, matrix representation

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Glossary

Notations

Roman Upper Case:

B De�nes 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 De�nes the input shaft in the matrix method in torque relationship

F1 Tangential force between S1 and R1

F2 Tangential force between S2 and R2

IPF 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 shaftsand stationary ratios for speed relationship

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

P Engine power

PO Primary options

PV Primary variations

R1 Ring gear PGS 1

R2 Ring gear PGS 1

Rs Speed ratio

Rt Torque ratio

1

S1 Sun gear PGS 1

S2 Sun gear PGS 2

SEO Shift element options

SV Secondary variations

T Engine torque

T̃ 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:

k̃ 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

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Greek Symbols:

ω Engine rotational speed

ω̃ 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

TCU Transmission control unit

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Contents

1 Introduction 7

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3 Delimitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.4.1 Literature Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.4.2 Requirements speci�cation . . . . . . . . . . . . . . . . . . . . . . . . 91.4.3 Software and Planetary Gear Set Representation Development . . . 9

2 Frame of Reference 11

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Transmissions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.1 Transmission Types . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2.2 Epicyclic Automatic Transmissions . . . . . . . . . . . . . . . . . . . 12

2.3 Planetary Gear Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.4 Speed- and Torque Relationship of Planetary Gear Sets . . . . . . . . . . . 15

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

2.4.2 Speed Relationship with the Matrix Method . . . . . . . . . . . . . . 162.4.3 Torque Relationship . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.4.4 Problems with Several Sets of Planetary Gears . . . . . . . . . . . . 20

2.5 Shift Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.6 Shift Sequences and Power Flows in Epicyclic Automatic Transmissions . . 222.7 Losses in an Epicyclic Automatic Transmission . . . . . . . . . . . . . . . . 24

3 Requirement Speci�cation 25

4 Software- and Matrix Representation Development 27

4.1 Utilizing the Standard PGS equations . . . . . . . . . . . . . . . . . . . . . 274.2 Developing the Matrix Representation . . . . . . . . . . . . . . . . . . . . . 284.3 Finding All Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.4 Stationary Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.5 Generating Shift Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5 Results and Discussion 37

6 Conclusion 39

7 Future Work 41

7.1 Matrix Generation and Structure . . . . . . . . . . . . . . . . . . . . . . . . 417.2 Creating an Optimization Software . . . . . . . . . . . . . . . . . . . . . . . 41

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1 Introduction

1.1 Background

The automotive industry is a competitive market and with more expensive fuels andtougher emissions regulations, the race for e�ciency has become reality for automotivecompanies.

To increase driveline e�ciency, every component of the drive line needs to be improvedto minimize energy losses. E�orts have been made to make internal combustion engines(ICEs) more e�cient and now the turn has come to the transmission. The British termtransmission often refers to the clutch, gearbox, propeller shaft (for rear wheel drive cars),di�erential 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 wordtransmission 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 e�ciency 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 sharemeaning that the e�ciency needs to be increased. Losses in ATs occur mainly due toopen clutches but also oil pump losses and losses in the gears. This means that thecon�guration and function of an AT can be constructed to minimize losses by reducingthe number of clutches, gear sets and oil pump pressure. This will signi�cantly impact theoverall e�ciency of the vehicle. Below is a chart that describes the average distribution oflosses 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 butalso has to take into consideration a number of factors including: space, torque carryingcapacity, churning losses and the losses in the open clutches. The system as a wholeis interconnected and not only does the con�guration options have varying losses but

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also manufacture-ability, complexity and manufacturing cost. All that in the automotiveindustry needs to be weighed in. An optimization model can not work with CAD ordrawings of the planetary gearbox, which is why it is needed a way to represent thesegears in a MATLAB program. There are many ways of calculating ratios and speeds inplanetary gearboxes, but there is no obvious way to represent a set of two planetary gearsets (PGSs) in a general fashion. In the project a way to represent a PF and PGS structurein matrix form is developed, making it possible to use it in mathematic equations enablinga future optimization of the PF. Furthermore, all possible matrix representations of PFsfrom 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 modi�ed togive all the parameters needed for an optimization model for ATs. A set of all theoreticallypossible combinations of two PGSs were generated in this matrix form that has beenselected. These combinations have been veri�ed to be able to run a future optimizationmodel with these setups. The provided data su�ce as a base for the optimization of a PFin 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 setupin a matrix form. The parameters needed for the optimization of a set of two regularplanetary gears have been generated and structured. The matrix representation coversa range of parameters - which shafts are input, output, stationary and interconnectedbetween the two PGSs and where the shift elements (SEs) are located. The objectivewas also to create a structure that can be used in the development of a more complexmodel.

The structure of the matrix is crucial for further work with the optimization, as the numberof 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 analyzeand develop functional AT PFs and gear designs, costly and time consuming research isnecessary. The tool that was developed is therefore a helping tool and a stepping stonefor future projects and optimizing models. The provided matrices are not comprehensivebut include complete structural setup of two sets of planetary gears and how they interactwith each other. To develop the model into a functional gearbox design tool, the optionof more and compounded PGSs have to be included and an optimization tool needs to bedeveloped. This is left for future work.

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

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1.4 Methodology

The project consists of three parts. The �rst part of the project is a frame of referencein a literature study. This is followed by a requirement speci�cation deciding what thecustomer needs. The program and structure developed for representing the PGSs is donein the third part.

1.4.1 Literature Study

The project base is a literature study creating a frame of reference that describes the basicfunctions of an AT, the PGSs and the SEs that handle them. Calculations that need tobe performed �nding ratios are presented and methods for representing PGSs in matricesare studied.

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

1.4.2 Requirements speci�cation

The customer speci�ed that the important part of the project was to �nd a way to cre-ate and sort di�erent PFs and to be able to evaluate these after a set of criteria. Therequirement speci�cation will discuss the importance of the customer input and what willbe 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 otherand the program for generating all PFs for two PGSs, the base for future optimizationmodels, will use the structure that is used in the PGS representation matrices.

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

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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, ATsand 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 devicewhile the engine is running. [3]

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

P = Tω (2.1)

The ICE has a range of RPM in which it is possible for it to operate. The range where theICE can deliver su�cient power and at the same time achieve e�ciency 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 and output torqueto match these and at the same time stay within the preferred range of RPM:s. This isdone by changing the gear ratio between the ICE and the wheels and this is where thetransmission comes in. [4]

2.2.1 Transmission Types

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

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

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Figure 2.1: Automatic transmission types

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

2.2.2 Epicyclic Automatic Transmissions

ATs automatically vary the gear ratio between the engine and the wheels by using a setof PGSs. The PGSs are activated by a set of clutches and brakes making use of theversatility of a PGS. An oil pump is driven directly or by a chain pumping oil into thehydraulic 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 di�erent components ofan AT can be seen in �gure 2.2.

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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 tothe clutch in a car with a MT. In contrast to the manual transmission the torque convertertransfers 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 tothe 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 standingstill, the stator acts to guide the oil �ow to increase the torque on the turbine, see �gure2.3. The torque converter acts as a torque multiplier. As the vehicle picks up speedand the speed di�erence between the turbine and the pump decreases, the stator startsfreewheeling with the pump transferring torque more directly. As the gearbox changesgears the process starts over again. [7]

Figure 2.3: Torque converter �ow [6]

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Torque converters with a lockup clutch locks the engine shaft directly to the gearbox whenthe 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 PGSconsists of a set of gears with a sun gear in the middle and several planet gears thatconnect the sun gear to a ring gear. There can be one or more planetary gears but mostcommon is three-�ve planets in a standard PGS with no pro�le displacement. See �gure2.4 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 alsomeans that the inertia and losses in the PGS will increase. A PGS can have di�erentstationary ratios by varying the number of teeth on the di�erent gears. Due to the geom-etry, there will be a correlation between the di�erent elements in the PGS. This meansthat for a standard PGS the ratio between the number of teeth is given by the followingequation:

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 PGSwill have di�erent gear ratios, increasing or decreasing the rotational speed, see table2.1.

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Table 2.1. PGS setups

Stationaryelement

Inputshaft

Outputshaft

Rotationalspeed

TorqueDirection 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 stationaryelement, which element is connected to the input shaft and which element is connected tothe 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 thestationary ratio. The exact ratio can be calculated using a set of equations.

2.4.1 Speed- and Torque Relationship with the Standard PlanetaryGear 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 ofthe planets. The speed of the carrier is ωc which gives the absolute speed of the ringgear:

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:

(ωr − ωc)zr = (ωc − ωs)zs (2.7)

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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 de�nedand the speed and torque of all the shafts can be calculated. The torque is given by theinverted relationship of the speed.

An example is when the ring gear is stationary, the sun gear is input and the planetarycarrier is the output. The speed of nr = 0 which simpli�es 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. Thetorque ratio, Rt , is given by the following equation, assuming that no losses are presentin 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 thedesign process of the PGS. It also a�ects 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 morecomplex combinations of two PGSs meaning that the mathematical model for the systemneeded 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 di�erentconnections and stationary elements:

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

and:

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

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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 carrierof 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 equationwas derived:

Mω̃ = Bωin (2.18)

The ω̃ represents the rotational speeds of the elements of the two PGSs shown in :

ω̃ =

ωs1

ωr1

ωc1

ωs2

ωr2

ωc2

(2.19)

The vector B de�nes that the last row of the matrix M speci�es which element the inputshaft is connected to and ωin gives the input speed.

B =

000001

(2.20)

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The matrix M is the interesting component in this equation describing the setup of thetwo PGSs.In the �rst two rows the number of teeth of the di�erent elements in the twoPGSs are described, for PGS 1 on row one, columns one to three and PGS 2 on rowtwo, columns four to six. The third and fourth row describe any connections of the twoPGSs.

M =

zs1 zr1 −(zs1 + zr1) 0 0 00 0 0 zs2 zr2 −(zs2 + zr2)0 −1 0 0 0 10 0 1 0 −1 00 1 0 0 0 01 0 0 0 0 0

(2.21)

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

Equation 2.18 can be modi�ed to �nd the speeds of all the elements in the PGSs knowingthe stationary ratio, in the example:

ω̃ = M−1Bωs1 = r̃ ωin (2.22)

where r̃ is speci�c for this example:

r̃ =

rs1/s1rr1/s1rc1/s1rs2/s1rr2/s1rc2/s1

(2.23)

In order to �nd the torque relationship another set of equations are needed. The torquesneed 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 followingequations [6]:

NT̃ = DTin (2.24)

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In this equation, N gives the gear ratios and the PGS setup. T̃ represents the torques ofthe di�erent elements and D de�nes the input torque position in N and Tin is the inputtorque, the same setup as above is used for the PGSs .

N =

zs1 0 0 0 0 0zr1 0 −1 0 1 0

−(zs1 + zr1) 0 1 0 0 00 zs2 0 0 0 00 zr2 0 −1 0 10 −(zs2 + zr2) 1 0 0 0

(2.25)

Which means that T̃ will be speci�c for this example:

T̃ =

F1

F2

TR1C2

TC1R2

TB1R

Tout

(2.26)

Where F1 is the tangential force between S1 and R1, F2 the tangential force between S2

and R2. The terms Tr1c2 and Tc1r2 are the connection torques. Tin is the input torqueand TB1R is the reaction torque on the �rst ring gear by the brake B1.

D =

100000

(2.27)

Shifting equation 2.24 around the following equation can be derived:

T̃ = N−1DTin = k̃ Tin (2.28)

Introducing the k̃ vector:

k̃ =

kF1/in

kF1/in

kTR1C2/in

kTC1R2/in

kTB1R/in

kTout/in

(2.29)

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This gives an equation with torque ratios k̃ and the input torque Tin on one side and thetorques of the elements T̃ on the other side:

F1

F2

TR1C2

TC1R2

TB1R

Tout

=

kF1/in

kF1/in

kTR1C2/in

kTC1R2/in

kTB1R/in

kTout/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 aspectsneed to be considered. The shafts that are rotating have no possible way to cross each otherwhich 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 elementsstopping the third one from rotating.

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

2.5 Shift Elements

Brakes and clutches are what disengages and engages the di�erent elements in a PGS inan AT. The brakes and the clutches are therefore called SEs. The clutches and brakesare operated with hydraulic pressure generated by the oil pump and maneuvered by theTCU.

The brakes lock any shaft of the PGS to the housing of the gearbox, making it a stationaryshaft. The clutches can connect di�erent PGS elements within a PGS or between twoseparate PGSs. Clutches can also connect any element of a PGS to the output or inputshaft of the driveline. The clutches are slightly more complicated than the brakes as thestructure will be rotating with one of the shafts, meaning that the oil pressure, as theclutch 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 consistof a number of friction plates layered with metal plates. The friction plates are splinedto one of the shafts or the gearbox housing and the steel plates are splined to anothershaft or the gearbox housing. The plates are submerged in oil to cool them and to get thecorrect friction coe�cient between the plates. As pressure is put on the plates, pushingthe friction plates and the steel plates together, a frictional force is created transferringtorque. The clutch package that can be seen in �gure 2.5 is the cross section of the clutchand the plates are circular with the radius r1 to the inner edge of the larger plate and r2to the outer edge of the smaller plate.

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Figure 2.5: A typical clutch package

The clutches and brakes are maneuvered by the TCU. The TCU sends oil to the SEthat needs to engage and releases pressure when a clutch needs to disengage. The torquetransfers from one PF to the other. A schematic of the function of a gearbox can be seenin �gure 2.6, [7]:

Figure 2.6: The power losses and the oil movement in an AT

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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 increasingspeed of the vehicle. An example are heavy duty vehicles that normally have linear stepswhile 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 itthe rolling and wind resistance, see �gure 2.7. The needed propulsion force goes up and asmaller range of the engine's RPM can be utilized. The range which has enough power toovercome 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 speci�c ratios on each gear and to get the wanted ratio from an AT thecorrect clutches and brakes need to be activated. The PF that gives the wanted ratiois selected by the TCU and the correct ratio for the current speed and tractive load ischosen.

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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 �rstgear is wanted the K1 clutch and the B5 brake are activated and in this speci�c gearboxit gives a ratio of 6.154:1. The other gear and their engaged elements can be seen in table2.2.

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 speci�c case, theclutch K1 is activated and the other SEs are varied. For the reverse gear, clutch K2 is

23

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

If only one clutch is released and one clutch is engaged, the shift is called a single-transitionshift 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. Thisinevitably leads to less e�ciency in normal operation.

The PF also a�ects torques and speeds of the di�erent elements, meaning that the PF hasa 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, gearmeshing, oil splashing, bearings, bushings and seals.

To be able to analyze these in a future PF optimization program the parts of the gearboxthat 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 shiftpattern with double-transition shifts, require a pump with a larger capacity. This createsan overcapacity at times when the gearbox is not shifting. To be able to see these lossesin 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 theselosses is to reduce the number of open clutches.

In the meshing of the gears, there are churning losses and frictional losses. These lossesdepend on the speeds and torques of the di�erent elements of the PGSs. To minimizethese losses the number of planetary gears need to be minimized and the rotational speedsand torques need to be optimized.

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3 Requirement Speci�cation

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

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

• 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 veri�ed by showing that it covers all possible solu-tions of the two PGSs. This can be done by manually deriving all possible solutions.

The matrix that represents a PF of an AT needs to be able, with a speci�ed range station-ary ratio of the PGSs, to create a number of possible solutions that can be clustered intogear shifting sequences that can be analyzed. This means that the representation at somestage needs to consist of the M matrix in the matrix method representation, see equation2.21

SEs are needed to be able to change the gears in the gearbox, but for every cluster ofgear changing sequences di�erent interconnecting shafts and stationary elements need tobe switched into SEs. The matrix representation of the possible solutions therefore needsa SE representation that show where SEs are needed.

The user needs to specify the number of gears and their speci�c ratios to be able to createclusters of generated matrix representations. The number of gears will de�ne the numberof PGSs and SEs that are needed and the speci�c ratios for each gear will de�ne thestationary ratios in the PGSs.

Starting o� with the generation of the single PGS needs to include all possible options ofa PGS, both including a single PGS and all the possible combinations of a PGS in a set oftwo 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 shaftshould not be stationary. The single PGSs need to be categorized and sorted to providean overview and also to be counted.

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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 andSEs) 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 theproject was to utilize the standard equations for calculating ratios and create a programthat 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 combinedratio for the two sets together. The program was also prepared for calculating speeds onthe output shaft.

The program input is de�ning which of the shafts that are stationary (one per PGS) andwhich is the input and the output shaft of each PGS, see �gure 4.1. The output is thegear ratio of each PGS and the total ratio for two PGSs. The program can also calculatethe output speed, given an input speed.

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

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Unfortunately the program only covered a small part of all the possible combinationsthat are possible and therefore a new method for generation of PGS representations wasevaluated.

4.2 Developing the Matrix Representation

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

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

M =

zs zr −(zs + zr)0 1 01 0 0

(4.1)

To get the structure of the matrix correct the initial work was to organize and categorizethe di�erent 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 sungear, the second column the ring gear and the third column the carrier. The rows needto 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-stationaryelements. The second row has a one on the element with the input shaft and the restzeros. The same goes for the third row where the element with the output shaft, outputfrom the gearbox, is represented with a one and zeros on the rest of the positions. Forexample a PGS with one stationary element, the ring gear (brown), and the input on thesun 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 thematrix is empty.

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A PGS setup with two PGSs, can be built by combining two of these setups. Thesematrices 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 notbeen written out):

Figure 4.4: One stationary element, mode one to �ve

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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 meansthat it is where the engine is connected. An arrow to the right means that it is the outputshaft, where the power is transferred to the next step in the driveline, e.g. a propellershaft. The number of stationary elements and the con�guration of the input and outputshafts decide which "mode" the PGS works in.

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

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Figure 4.6: An example of a setup with one stationary per PGS and one of them frommode 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 01 0 0 0 0 00 0 0 1 0 0

(4.2)

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

Category 1.1 example =

0 1 0 0 1 00 0 1 0 0 −10 0 0 0 0 00 0 0 0 0 01 0 0 0 0 00 0 0 1 0 0

(4.3)

A number of categories to organize the generation of the possible solutions have beenworked out. The number of categories and the sorting has been done starting with thesimplest form of PGS combinations adding categories when it was needed. This meansthat 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 examplethe main category one has two stationary elements, each on a separate PGS seen in �gure4.6.

For category two there are three to four stationary elements, one of the PGSs having onestationary 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 ordi�erent PGSs. Category 1.1 has the input on one of the PGSs and the output on theother one, while 1.2 has the input and the output shaft on the same element.

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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 areindependent.

The simplest example is the solution with a stationary element on each PGS and the inputshaft on the PGS closest to the engine, PGS 1, and the output shaft on PGS 2, the PGSclosest to the �nal drive. This simple setup can be combined in a number of ways. Thestationary elements have three di�erent positions, each in their respective PGS. The inputshaft 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 PGSsstationary and the input on the sun gear of PGS 1 and the output on the sun gear of PGS2, �gure 4.7.

Figure 4.7: Category 1.1 example block diagram

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

By switching the stationary elements around some of the options are found, see �gure4.8:

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 needsto 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 twoPGSs. Furthermore it is not possible to connect a stationary shaft to any of the movingshafts in this case (category 1.1). It is not possible to connect anything to a stationaryshaft in this speci�c 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 singleshaft solutions for category 1.1.

The number of possible shaft solutions increase when the output and the input shaftsare varied. The output shaft has two modes for each stationary setup which gives it amultiplication factor two, This is a primary variation (PV) and the input shaft also hastwo 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 theoutput, meaning that the input is on PGS 2 and the output on PGS 1. If this option ispossible, which it is in this case, there will be twice as many solutions adding a factor oftwo. This option is called inverse possibility factor (IPF).

The total number of combinations with one shaft can then be calculated:

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V ar = SEO ∗ PSO ∗ PV ∗ SV ∗ IPF (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 perstationary setup.

This needs to be redone with two shafts in the setup. In this simple case the only possibilityto 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: nineSEOs, 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- andoutput shaft on the same element.

For this to happen the input- and output shafts need to be on the same PGS. This is thecase in sub category 1.2. Each PGS has a stationary element and one of the PGSs hasboth the input and the output shaft. The number of variations for a single shaft is 324and 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 onthe same element, some of the factors di�er. In the case of a no shaft setup (NSS), thereis only one PO which is no shafts at all. This means that compared to the single shaftsetup this NSS will have a factor nine less solutions, which means a total variation of 36setups.

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4.4 Stationary Ratios

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

ω̃ = r̃ωin (4.5)

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

This means that the calculated ratio can be compared to the wanted ratio, making itpossible 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 possiblesolutions. The groups that have a complete set of gear ratios, meaning that all the wantedratio steps are accounted for, are saved for further evaluation.

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

After this step the matrix representation of all possible solutions is completed. The numberof possible solutions depend on the speci�ed wanted ratios and the accuracy they arespeci�ed with.

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5 Results and Discussion

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

Table 5.1: Number of shaft setups

CategoryNumber 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 themanually 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, seeequation 4.4.

To generate more than the preliminary matrix representation user input is needed, speci�edrange of stationary ratios and wanted gear ratios, which unfortunately was not possible todo in the time frame of this project. The preliminary matrix representation was generatedand the formulas for deriving the rest of the needed variables for an optimization modelexist.

The �nal version of the total matrix representation can be seen in �gure 5.1:

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Figure 5.1: Matrix setup structure

The �rst six by six is the preliminary representation of the structure, connected shaftsand stationary shafts. The M matrix will be �lled in with stationary ratios once thewanted ratios are speci�ed. The SE-matrix de�nes the possible SE used when clusteringthe di�erent shaft setups.

The speed ratio equation (2.18) rewritten gives the possibility to calculate preliminarystationary 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 individualelement is a design parameter. The same goes for the choice of the torque converter setupand the chosen ratios on the di�erent gears in the gearbox.

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

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6 Conclusion

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

The main di�culties in this project have been to �nd ways of structuring and to overviewthe vast quantity of data. The problem has not been the generation of data but howto store it and to grasp how much it actually is. This is done for only two PGSs whichare 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 moreadvanced model needs to be developed.

To conclude an e�ective way of storing the data in a matrix representation has been foundand all the possible solutions of two simple PGSs has been found. The time consumingway that the generation process of the shaft setups was performed, required more timethan expected and a more realistic time plan or a longer time span for this type of projectwould have been needed.

The obvious gains of a project like this are, among other things, that a known PF setupwhen starting the design process of an AT, giving many of the variables required in thedesign process simpli�es the process a lot. Combining the optimization model with adesign tool could shorten the time for a new gearbox design considerably, giving newexciting possibilities for manufacturers to create super e�cient ATs, saving resources andminimizing the impact on the environment.

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7 Future Work

The project has been a study in matrix representation of epicyclic ATs meaning that thePGSs in the ATs have been translated to ones and zeros in matrices enabling furthercomputer 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 twostandard 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 optimizationprogram. The ultimate goal of this project was to provide su�cient data for a programdeciding the optimal PF with a given set of input parameters. A lot of manual work wasused to �nd a structure for the matrix representations consuming a lot of the time in theproject.

The �rst part of the future work could be to create a mathematic equation based onboundary values generating the PGS setup and shaft options rather than using logicequations and manual matrix generation. This would mean that the manual work forsetting up the base for the optimization program would be more general and enabling itto 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 projectis the fact that all solutions for a PGS was considered in the generated data. There is nodeletion 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 detectand delete illegal shaft structures within a generated set of data is needed to be able tocreate a useful optimization model.

7.2 Creating an Optimization Software

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

Further research within gearbox losses and gearbox design will have to be performed andan 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 e�ort need to be quanti�ed and made comparable.

41

• SE options need to be quanti�ed and evaluated.

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

• Speeds and torques of the di�erent components in the PGSs need to be weighed in.

When all the parameters have been made comparable an optimization algorithm needsto 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 modelbut are input parameters for the model. Parameters that might be interesting couldinclude:

• The structural e�ort, shift logic, gear ratios, speeds- and torques in the gearbox andthe di�erent losses.

• Relative speeds in the clutches, space- and weight of the gearbox and torque carryingcapacity 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 designprocess for ATs and increase the complexity and functionality of future transmissions.Integrating an optimization tool with standard design tools open new doors and excitingpossibilities.

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[2] K. Martin and B. Warner, �Transmission e�ciency developments,� in SAE Interna-

tional, 2012.

[3] H. o. T. D. Lars Bergkvist, �Introduction to transmissions.� AVL, Södertälje, 2012-05-25.

[4] T. K. Garrett, K. Newton, and W. Steeds, The Motor Vehicle. Society of AutomotiveEngineers Inc, 13th ed., December 2000.

[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, �E�cient components for e�cient transmissions,� in12th CTI Symposium and Exhibition, 2013.

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APPENDIX A

The complete manual count of possible shaft and PGS setup variations

45