Tribology and dynamics of engine and powertrain fundamentals, applications and future trends ( TQL )

Dowson Introduction Part I Introduction to dynamics and tribology within the multi-physics environment 1 An introduction to multi-physics multi-scale approach 3 H.. Rahnejat , Loughbo

Tribology and dynamics of engine and powertrain

Materials, design and manufacturing for lightweight vehicles

(ISBN 978-1-84569-463-0)

Research into the manufacture of lightweight automobiles has led to the consideration of

a variety of materials, such as high-strength steels, aluminium alloys, magnesium alloys, plastics and composites This research is driven by a need to reduce fuel consumption to preserve dwindling hydrocarbon resources without compromising other attributes such

as safety, performance, recyclability and cost This important book will make it easier for engineers not only to learn about the materials being considered for lightweight automobiles, but also to compare their characteristics and properties It also covers issues such as crashworthiness and recycling.

Diesel engine system design

(ISBN 978-1-84569-715-0)

Diesel engine design is highly complex, involving many individuals and companies from original equipment manufacturers to suppliers A system design approach for setting up the right engine performance specifications is essential to streamline the processes This important book links everything a diesel engineer needs to know about engine performance and system design in order to master all the essential topics quickly; the focus is on how

to use advanced analysis methods to solve practical design problems Numerous case studies and examples illustrate advanced design approaches using engine cycle simulation tools The central theme is how to design a good engine system performance specification

at an early stage of the product development cycle.

Details of these and other Woodhead Publishing books can be obtained by:

∑ visiting our web site at www.woodheadpublishing.com

∑ contacting Customer Services (e-mail: sales@woodheadpublishing.com; fax: +44 (0) 1223 893694; tel.: +44 (0) 1223 891358 ext 130; address: Woodhead Publishing Limited, Abington Hall, Granta Park, Great Abington, Cambridge CB21 6AH, UK)

If you would like to receive information on forthcoming titles, please send your address details to: Francis Dodds (address, tel and fax as above; e-mail: francis dodds@woodheadpublishing.com) Please confirm which subject areas you are interested in.

Oxford Cambridge Philadelphia New Delhi

Published by Woodhead Publishing Limited, Abington Hall, Granta Park, Great Abington, Cambridge CB21 6AH, UK

www.woodheadpublishing.com

Woodhead Publishing, 525 South 4th Street #241, Philadelphia, PA 19147, USA

Woodhead Publishing India Private Limited, G-2, Vardaan House, 7/28 Ansari Road, Daryaganj, New Delhi – 110002, India

www.woodheadpublishingindia.com

First published 2010, Woodhead Publishing Limited

© Woodhead Publishing Limited, 2010

The authors have asserted their moral rights.

This book contains information obtained from authentic and highly regarded sources Reprinted material is quoted with permission, and sources are indicated Reasonable efforts have been made to publish reliable data and information, but the authors and the publisher cannot assume responsibility for the validity of all materials Neither the authors nor the publisher, nor anyone else associated with this publication, shall be liable for any loss, damage or liability directly or indirectly caused or alleged to be caused by this book Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming and recording, or by any information storage or retrieval system, without permission in writing from Woodhead Publishing Limited.

The consent of Woodhead Publishing Limited does not extend to copying for general distribution, for promotion, for creating new works, or for resale Specific permission must

be obtained in writing from Woodhead Publishing Limited for such copying.

Trademark notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe.

British Library Cataloguing in Publication Data

A catalogue record for this book is available from the British Library.

ISBN 978-1-84569-361-9 (print)

ISBN 978-1-84569-993-2 (online)

The publisher’s policy is to use permanent paper from mills that operate a

sustainable forestry policy, and which has been manufactured from pulp

which is processed using acid-free and elemental chlorine-free practices

Furthermore, the publisher ensures that the text paper and cover board used

have met acceptable environmental accreditation standards

Typeset by Replika Press Pvt Ltd, India

Printed by TJI Digital, Padstow, Cornwall, UK

Contributor contact details xix

D Dowson

Introduction

Part I Introduction to dynamics and tribology within

the multi-physics environment

1 An introduction to multi-physics multi-scale approach 3

H Rahnejat , Loughborough University, UK

1.3 Lagrange’s equation and reduced configuration space 9

1.9 Appendix: multi-physics analysis for investigation of

manual transmission gear rattle – drive/creep rattle 29Section I.I Fundamentals of tribology and dynamics

D Arnell , University of Central Lancashire, UK

Contents

2.2 The nature of engineering surfaces 41

P Prokopovich and H Rahnejat, Loughborough University, UK

and M Teodorescu, Cranfield University, UK

3.5 Estimation of interfacial tension between a liquid and a

3.7 Intermolecular interactions and near-surface effects 92

4 Fundamentals of impact dynamics of semi-infinite

M Teodorescu , Cranfield University, UK; V Votsios, Atos, Spain;

P M Johns-Rahnejat , (formerly) Imperial College London, UK

and H Rahnejat, Loughborough University, UK

R Gohar , Imperial College London, UK and M M A Safa,

Kingston University, UK

5.2 Reynolds equation 137

5.6 Externally pressurised (EP) gas journal bearings 154

5.9 Review of some unusual and recent applications of fluid

6.12 Application of elastohydrodynamic lubrication (EHL)

7 Measurement of contact pressure under

R Gohar , Imperial College London, UK and M M A Safa,

Kingston University, UK

7.4 Alternative methods of measuring contact pressure 241

Part II Engine and powertrain technologies and applications

Section II.I Overview

8 Tribological considerations in internal combustion

D R Adams , Ford Dagenham Development Centre, UK

8.2 Issues of cost, competition, and reliability in internal

8.3 Drivers for tribological design and innovation 2548.4 A systems view of the piston/ring/cylinder bore interface 2558.5 The development process in internal combustion (IC)

8.6 The piston in internal combustion (IC) engines 2648.7 Piston rings in internal combustion (IC) engines 270

8.9 Design validation of internal combustion (IC) engines 279

9 Predictive methods for tribological performance in

I McLuckie , AIES Ltd, UK

9.3 Application of integrated knowledge-based systems (IKBS)

and elastohydrodynamics (EHD) to a race engine crank pin 2899.4 Application of integrated knowledge-based systems (IKBS)

and right-hand drive (RHD) to piston and liner 3029.5 Application of integrated knowledge-based systems (IKBS)

and right-hand drive (RHD) to turbocharger bearings 3139.6 Engine friction: building a better understanding 331

Section II.II Tribology of piston systems

10 Fundamentals of lubrication and friction of piston ring

V D’Agostino and A Senatore, University of Salerno, Italy

10.2 Piston ring: history and basics 344

11 Measurement techniques for piston-ring tribology 387

I Sherrington , University of Central Lancashire, UK

11.7 Observation and measurement of lubricant movement and

12 An ultrasonic approach for the measurement of oil

R S Dwyer-Joyce , University of Sheffield, UK

13 Surface texturing for in-cylinder friction reduction 458

I Etsion , Technion, Israel

14 Optimised textured surfaces with application in

R Rahmani , Loughborough University, UK and A Shirvani and

H Shirvani , Anglia Ruskin University, UK

14.3 Application of surface texturing in tribology 473

14.5 The mechanisms behind tribological improvements

14.7 Surface texturing technology and internal combustion

14.12 Application of the optimum results in the piston

15 Transient thermo-elastohydrodynamics of rough

P.C Mishra, H Rahnejat and P King, Loughborough University, UK

15.3 Compression ring cylinder liner conformability 524

Section II.III Valve train systems

M Kushwahu , Ford Motor Company, UK

16.3 Lubrication analysis of the cam and tappet conjunction 548

17.2 Aspects of valve train geometry and construction 569

17.4 Valve train kinematics: cam-to-flat follower contact 573

Section II.IV Engine bearings

18 Fundamentals of hydrodynamic journal bearings: an

S Balakrishnan , Mercedes Benz High Performance Engines, UK;

C McMinn , Ford Motor Company, UK and C E Baker and

H Rahnejat , Loughborough University, UK

18.3 Simple analytical solutions for journal bearings 593

18.8 Tribological conditions 610

19 Practical tribological issues in big-end bearings 615

S Boedo , Rochester Institute of Technology, USA

19.7 Sample application of design charts: four-stroke

P.C Mishra and H Rahnejat, Loughborough University, UK

Section II.V Drivetrain systems

21 An introduction to noise and vibration issues in the

automotive drivetrain and the role of tribology 663

M Menday , Loughborough University, UK

21.1 Introduction to drivetrain noise, vibration, harshness

21.2 The application of multibody dynamics (MBD) analysis 664

21.3 Noise, vibration, harshness (NVH) characteristics 665

21.6 Noise and vibration paths from the driveline into the body 667

21.8 Examples of high-energy impacts in the drivetrain 668

21.11 Summary of drivetrain noise, vibration, harshness (NVH)

22 Friction lining characteristics and the clutch take-up

M Menday and H Rahnejat, Loughborough University, UK

22.6 Results and discussion of numerical findings of

22.7 Vehicle studies and design of experiments for

22.8 Overall conclusions of multi-body dynamics analysis of

23 Contact mechanics of tyre–road interactions and its

G Mavros , Loughborough University, UK

23.8 Relationship between the brush model and the magic

23.9 Transient tyre response: a first approach 722

23.11 Case study: the influence of transient tyre behaviour on

24 Tribology of differentials and traction control devices 735

S K Mohan , Magna Powertrain, USA

24.3 The basics of vehicle propulsion and dynamics 73824.4 The need for differentials and slip control devices 744

24.6 Advantages of electronically controllable ‘active’ slip

24.7 Tribological considerations in the design and development

24.8 Modelling and simulation of traction control devices 764

25.3 Typical numerical results from mechanical model and

26.3 Investigation strategy for rattling and clattering noises in

26.5 Peripheral instruments for measuring drag torque 811

26.9 Parameter studies for rattle and clatter noises in

27.3 Definition of rattle phenomenon in automotive

27.5 Types of rattle and their causes within automotive

27.6 Traditional rattle palliations in automotive powertrains 84827.7 Experimentation and evaluation method of rattle

27.8 Simulation of rattle phenomenon in automotive

28 Dual mass flywheel as a means of attenuating rattle 857

P Kelly , Ford Werke GmbH, Germany and B Pennec,

R Seebacher, B Tlatlik and M Mueller, LuK GmbH &

Co oHG, Germany

28.2 Dual mass flywheel (DMF) interactions at different

30.6 Results and discussion of driveline clonk experiment 92130.7 Some methods of palliation of driveline clonk 925

31 Tribo-elasto-multi-body dynamics of a single cylinder

M S M Perera, S Theodossiades and H Rahnejat,

Loughborough University, UK

31.5 Temperature effects in lubricated contacts 938

Part III Micro-systems and nano-conjunctions

M Teodorescu , Cranfield University, UK and H Rahnejat and

S Theodossiades , Loughborough University, UK

F W DelRio , National Institute of Standards and Technology, USA and C Carraro and R Maboudian, University of California at

Berkeley, USA

Jost Institute for Tribotechnology

University of Central Lancashire

School of EngineeringCranfield UniversityCranfield MK43 0AL UK

E-mail: m.s.teodorescu@cranfield.ac.uk

H Rahnejat*

Wolfson School of Mechanical and Manufacturing EngineeringLoughborough UniversityLoughborough

UKE-mail: H.Rahnejat@Lboro.ac.uk

Chapter 4

M TeodorescuDepartment of Automotive Engineering

School of EngineeringCranfield UniversityCranfield MK43 0ALUK

E-mail: m.s.teodorescu@cranfield.ac.uk (* = main contact)

West Lafayette, IN 47907USA

E-mail: sadeghi@ecn.purdue.edu

Chapter 8

Dr D R AdamsEngine EngineeringDagenham Development CentreDagenham RM9 6SA

UKE-mail: dadams69@ford.com

Chapter 9

Dr I McLuckieAIES Ltd

PO Box 7784Market Harborough LE16 7YHUK

Via Ponte don Melillo, 1

I 84084 Fisciano (SA) Italy

E-mail: a.senatore@unisa.it E-mail: dagostino@unisa.it

Chapter 11

I Sherrington

Jost Institute for Tribotechnology

University of Central Lancashire

Dynamics Research Group

Wolfson School of Mechanical and

Bishop Hall LaneChelmsford CM1 1SQUK

Email: Ayoub.Shirvani@anglia.ac.uk Hassan.Shirvani@anglia.ac.uk

UKE-mail: H.Rahnejat@Lboro.ac.uk

Chapter 16

M KushwahaLitens CanadaE-mail: manu.kushwaha@litens.com

Chapter 17

M TeodorescuDepartment of Automotive Engineering

School of EngineeringCranfield UniversityCranfield MK43 0AL UK

E-mail: m.s.teodorescu@cranfield.ac.uk

C E Baker and H Rahnejat*

Wolfson School of Mechanical and

Rochester Institute of Technology

76 Lomb Memorial Drive

UKE-mail: H.Rahnejat@Lboro.ac.uk

Chapter 21

M MendayLoughborough UniversityUK

Chapter 23

G MavrosAeronautical and Automotive Engineering

Loughborough UniversityEpinal Way

LoughboroughLeicestershire LE11 3TUUK

E-Mail: G.Mavros@lboro.ac.uk

E-mail: michael_menday@yahoo.co.uk

Chapter 28

P Kelly*

Ford Werke GmbHCologne

GermanyE-mail: pkelly7@ford.com

B Pennec, R Seebacher, B Tlatlik and M Mueller

LuK GmbH & Co oHGIndustriestrasse 377815

Germany E-mail: Roland.Seebacher@schaeffler.com

Chapter 29

S Theodossiades, O Tangasawi and H Rahnejat*

Wolfson School of Mechanical and Manufacturing EngineeringLoughborough University Loughborough

UK E-mail: s.theodossiades@lboro.ac.uk H.Rahnejat@Lboro.ac.uk

H Rahnejat* and S Theodossiades

Wolfson School of Mechanical and

National Institute of Standards and Technology

100 Bureau Drive, Stop 8520Gaithersburg, MD 20899USA

Email: frank.delrio@nist.govCarlo Carraro and Professor Roya Maboudian

Department of Chemical Engineering

University of California BerkeleyB78 Tan Hall

Berkeley, CA 94720USA

E-mail: carraro@yahoo.com maboudia@berkeley.edu

Future developments in engines and powertrain systems will necessarily

be sympathetic to nature There is a growing need to preserve the diminishing natural sources of energy; mainly fossil fuels The rate by which these resources are used far exceeds their natural recurrence For the foreseeable future the transport sector in all its variety will use these diminishing resources to some extent, even with the emergence of alternative sources of energy

The efficient use of these precious natural assets has never been as important

as it is today The quest to reduce the rate of depletion of these fuels would directly affect every facet of science and engineering One key discipline is tribology; the science of friction, wear and lubrication Its efficient and often empirical application to means of transport has been a subject interwoven with human civilisation from its very outset As our knowledge has grown, the seemingly empirical nature of rules governing the science of interacting surfaces has advanced towards a more fundamental understanding Tribology

is now one of the cornerstones in the quest for efficiency, conservation of resources and the protection of ecosystems These noble causes have driven

us from relatively large-scale observed phenomena such as friction down to the scale of minutiae to seek answers to many questions

Tribology cannot be viewed as an isolated discipline impervious to a host

of other phenomena that interact with it and determine its outcome It may, therefore, be regarded as a facet of physics of motion within a limited scale

At its upper boundary we encounter the behaviour of contiguous surfaces and beyond that the structure of solids themselves, and yet further on, the conglomerate of bodies that are subject to some form of constrained motions

At the lower bound of the tribological world we strive to understand the interaction of small matter; molecules of lubricant with each other and with the rough terrain on the surfaces themselves These interactions, minute or relatively large, are parts of the physics of motion at all physical scales and are currently described by a plethora of rules and laws Yet in the absence

of a unique law, these various facets act as our microscope pointing to one end of the scale to search for underlying causes and conversely as a telescope

Preface

to the other end to observe the effects In engineering this amounts to down-cascading to the root causes and up-cascading to a remedial solution This approach is best seen in problems associated with noise, vibration and harshness (NVH) refinement, a key concern in industry

Vibration and noise are parts of nature itself, as static equilibrium is only a mathematical convenience Vibration and noise appear to follow the principle of least action in all facets of dynamics Like water taking the path

of least resistance to conserve energy, all other matter from atoms to stars move just sufficiently to maintain the status quo We should learn from nature

in our inventions, including engines and powertrains, conserve energy and strive to maintain peace and tranquillity The affinity to nature translates to adherence to the principle of parsimony and a quest for the principle of least action (optimal conservation of energy)

In the absence of an adequate understanding of the laws and rules of nature, we should adopt our evolving knowledge to multi-physics, multi-scale problem-solving This is the approach used throughout this volume I would like to take this opportunity of thanking my contributing colleagues, who have enthusiastically strived to share their experiences with the readers

of this work

Homer Rahnejat

Since applications of several remarkable developments in science and engineering are eye-catching and attract much publicity, it is easy to overlook other equally impressive developments of mechanical systems because they are so familiar to us in everyday life The automobile is one such system Just

a few decades ago, the reliability of automobiles was far from satisfactory, fuel consumption was unacceptably high and frequent servicing was essential Engines now run smoothly and much more efficiently with pleasing ease and negligible wear Noise levels have been greatly reduced and the demands of frequent servicing much diminished

Advances in tribology have played a major role in this progress Modelling, analysis, experimentation, design, manufacture, materials and lubricants have all contributed to tremendous improvements in the performance, reliability and efficiency of modern engines How often do motorists experience problems with cylinder, piston ring, or valve train wear, or bearing failure?

Analysis and design of linked engine components and a complete powertrain present fascinating challenges to the engineer, particularly in relation to tribology and multi-body dynamics The editor of this extensive volume is well placed to bring the reader up to date on these matters, having intimate experience of both fields The fundamentals of each topic are presented and followed by contributions by experienced engineers on analysis, experimental procedures and design

The reader is introduced to significant progress in both tribology and dynamics Effective performance and reliability of major engine components rely upon extensive analysis embracing current concepts of transient thermo-elastohydrodynamic lubrication, terminology that was unfamiliar a few decades ago A fascinating feature of the text is the mixture and interchange

of such engineering science vocabulary with practical terminology such as clonk, clatter, rattle, judder, shuffle and whoop!

The 33 chapters in this major text (each with many valuable references) are arranged in three parts, with the bulk of the work being presented in Part II:

Foreword

I Introduction to dynamics and tribology within the multi-physics

environment

II Engine and powertrain technologies and applications

III Micro-systems and nano-conjunctions

Basic aspects of dynamics, friction, wear, and lubrication, surface interactions and impact dynamics in Part I, contributed by authors from five UK universities, Spain and the United States, provide the foundations for subsequent topics Conventional approaches to bearing design, based upon the governing, classical Reynolds equation of fluid film lubrication, are presented and followed by detailed accounts of elastohydrodynamic lubrication

Intriguing discussion of the approach to engine and powertrain analysis, experimentation and design are presented in Part II Piston systems, valve trains and engine bearings receive special attention in Sections II.II, II.III and II.IV, while Section II.V is devoted to drivetrain systems The latter topic, which reflects long-standing research interests at Loughborough, the editor’s home university, provides a fascinating demonstration of the combined power of multi-body dynamics and tribology Problems with airborne and structural noise transmission, clutch take-up judder, tyre–road interactions, differential gears, traction control devices and flywheels are all addressed The authors of chapters in this balanced central core of the book are based

in industry in Germany, the UK and the USA and in universities in Greece, Israel, Italy and the UK

Special problems associated with the miniaturisation of mechanical drives have attracted much attention in recent years The two chapters in Part III introduce readers to developing surface interaction concepts in micro-systems and small-scale surface engineering

This fascinating and comprehensive book on current developments in the fields of tribology and multi-body dynamics related to vehicle problems will

be of interest to both the industrial and academic communities

Duncan DowsonSchool of Mechanical Engineering

The University of Leeds

Leeds LS2 9JT

andVisiting ProfessorLoughborough University

Fuel efficiency, environmental sustainability, reliability, quality and noise, vibration and harshness (NVH) refinement are key targets for the automotive industry into the future All these attributes require design and development, driven by fundamental and applied research in a multi-disciplinary environment Fuel efficiency is an important economic objective which also impinges on the longer-term environmental sustainability of the vehicle industry in terms

of conservation of energy resources Other aspects of the environmental impact of transport systems include emissions, which is a major growing concern This means that alternative methods of propulsion should be considered as well as better combustion strategies in internal combustion (IC) engines Whichever method is finally employed, friction, as the main source of parasitic losses, will always be a major problem and methods to mitigate its effects are of paramount importance

It is not surprising that with such a broad range of aforementioned ideals, downsizing of propulsion units has been favoured for some time now In fact, the future points to ever more compact high output power – to lightweight power train units However, one repercussion has been a plethora of NVH issues, which should be regarded as error states; or errant system dynamics The palliation methods are often quite costly and sometimes unnecessarily complex

The combination of parasitic losses and errant dynamic states are of concern in terms of fuel efficiency and represent undesired environmental impacts (emissions and noise pollution) Furthermore, these can be regarded

as being of poor quality and as potential sources of unreliability

The foregoing shows the importance of tribology and dynamics as key disciplines in design and analysis of future power trains These are the two aspects covered in this volume They interact with each other and other phenomena, also noted in this book, at all physical scales from the minutiae

of contacting surfaces to relatively large displacement of pistons and the drive train

The book covers many important practical engineering and technological issues that the industry faces in design and development today and into the

future Many of the authors are experienced technologists, specialists and component engineers from industry with many years of experience in the fields

of their expertise They are joined by established academics and promising researchers to make this a rather unique and comprehensive volume

Professor Richard Parry-JonesFormer Vice President of Ford Global Development and

Chairman of Premier Automotive Group

Part I

Introduction to dynamics and tribology within the multi-physics environment

An introduction to multi-physics multi-scale

approach

H RaHnejat, Loughborough University, UK

Abstract: all natural phenomena or those which are instigated by people

through devised mechanisms, machines or devices are subject to dynamics Dynamics, as physics of motion, is the broadest of sciences It happens that its many facets are explained by various laws of physics and theories, and methods of analysis therefore, one can view dynamics as a multi-physics science the underlying principles and methods describing the multi-physics nature of dynamics have their roots in laws and rules which belie the development of the various branches of physics these are often based on kinetic laws that seemingly govern interactions at various physical scales thus, dynamics is a multi-physics multi-scale science of interactions of all matter engines and powertrains are no exception, and perhaps very good examples of the interactive nature of multi-faceted dynamics this chapter introduces the fundamentals of dynamics, the science that has engaged human inquisition and has arguably contributed to the advancement of civilisation for the longest.

Key words: newtonian axioms, constraints, Lagrangian dynamics,

multi-body dynamics, elasticity.

Dynamics and tribology, described in this book, may be regarded as subsets

of physics of motion (in a multi-physics perspective) Dynamics is the study of motion of entities caused by the underlying forces Historically, in the discipline of dynamics and within engineering these entities have been considered to be an assembly of parts (a system), solid inertial elements

(a component) and rigid particles When the study of motion of a material point (a generic term used to describe these entities; a particle, a body: a

conglomerate of such particles or a system: an assembly or cluster of bodies)

is observation-based only (without regard to the underlying cause: force),

then the field of investigation is referred to as kinematics In the case of

a multi-body or a many-body system, kinematics refer to studies with no degrees of freedom; relative motions between their constituent material points (their motion is prespecified)

When a system undergoes no displacements with respect to a specified

frame of reference (a co-ordinate set; t, x, y, z), then it is regarded to be static

the forces applied to such a system are said to be in a state of equilibrium (no net force) If a multitude of such equilibria can be assumed, then these

various states of the system may be termed as quasi-static

Real systems are not rigid an example is the powertrain system, subject

of this book, where hollow driveshaft tubes undergo small amplitude elastic deformation under load, while they undertake much larger inertial motions

(see Chapter 30, explaining the clonk phenomenon) the same is true of all

material points, although in many cases, such as molecules, the deformation would be infinitesimal and thus almost insensible Therefore, flexible systems

are subject to elastodynamics Since the deformation amplitudes are different

in scale of measurement to the overall inertial displacements, the problem

is multi-scale If one disregards the larger scale and the study is confined

to small amplitude oscillations, then the problem at hand is regarded as

one of vibration In general all real material points, being compliant, can

assume many forms when vibrating these forms are known as modes For example, Chapter 30 shows various modes of flexible driveshaft tubes The same is true of very small material points such as electrons with their wavy motions with many spins at the other end of the scale, it is surmised that even heavenly bodies pulsate or quiver, spreading waves on the fabric of space, rather similar to the wave propagation on the surface of driveshaft tubes, explained in Chapter 30

When undertaking study of a problem in dynamics, the boundary of the system must be defined, because there is no generic system The interactions between the defined system and those material points extraneous to it are then ignored this is a fundamental rule of experimentation thus, for example,

in vehicle engineering, problems are defined as those of the powertrain system or vehicle–road interactions, and not a vehicle within the universe! With the system boundaries defined, interaction of key material points are considered these interactions are simply forces acting between them, causing motions in a multitude of physical scales therefore, the interaction scale(s) of interest should also be determined For example, powertrain dynamics problems may be in the scale of large displacements (inertial dynamics: shuffle, see Chapters 21, 23 and 30) or structural response (modal behaviour of driveshaft tubes or the transmission case, see Chapter 30) or noise propagation (acoustic response of thin-walled structures) these may

be regarded as wave motions from scales of metres to sub-millimetre and

on to nanometres respectively, but they are all part of dynamics of a defined system (as are the usual micro-scale deflections of load-bearing conjunctions

in tribology; see Chapters 4, 5 and 6) the environment outside the system boundary is considered to be rigid, to which a global frame of reference for measurement of multi-scale physics of motion is firmly attached (Rahnejat, 1998) In reality the extraneous environment is not rigid, nor is any place within the known or surmised universe the experiment carried out in a

laboratory within a defined system is positioned on Earth which moves around the Sun at 66 730 miles/h (average), while the Solar system is dragged by the Sagitarius a* at the centre of the Galaxy at 45 000 miles/h towards the constellation of Hercules However, one can consider the dominant forces

in the experiment to be because of material points of the defined system and

in some cases (bodies of significant size) due to the Earth’s gravitational pull only thus, in dynamics the motion of a material point is governed by

all those within the same system this is the essence of Mach’s principle

and is fundamental to the subject of dynamics now with this philosophical

basis and within any system of any conglomerate of material points i, any

one such point has an acceleration due to its interactions with others as:

this is Newton’s second law of motion, where m is the mass of any material

point newton called this an axiom, because in his perspective it was a natural observation for which no proof was required at the time of his enunciation, same as Euclid’s geometrical axioms The second law is the foundation upon which all the field of dynamics resides Later a fundamental proof for this axiom is provided through energy consideration: Lagrange’s equation The notion of material points is not confined to those of a solid nature, but all matter in any physical state including fluids Thus, a system may be defined as a volume of fluid bounded by solid surfaces such as a river and its impervious banks The volume of fluid may be considered as a series of elemental volumes progressing through the system in the same manner (but not exactly) as the deformation wavefronts progress in the hollow driveshaft tubes in Chapter 30 The study is, therefore, one of continuous fluid flow due

to a pressure gradient and velocity profile (both as a result of forces) through

the assumed system the subject is called hydrodynamics (see Chapter 5)

this is also a multi-scale problem, like other forms of dynamics For a large expanse of fluid, the elemental volumes may be considered large, but finite within which a state of equilibrium may be assumed relative to the interactions between any pair of such elements themselves Large elemental volumes mean significant body forces (weight) and inertial forces On the other hand, in tribological conjunctions, the narrowness of the gap between the boundary solids means small elemental volumes and their assumed uniform motion through the system (the conjunction) thus, the problem simplifies to that of flow induced by changes in pressure gradient and any

relative motion of bounding surfaces, forming a wedge effect (Gohar and

Rahnejat, 2008; also see Chapter 5) Therefore, the complex flow dynamics

is reduced to a manageable problem in hydrodynamics, with a fundamental equation:

a pair of close solid boundaries (Chapter 5 describes the equation in detail) Therefore, as in the case of solids, a question of scale exists in the case of fluids

as well If the size of the system is reduced, the incremental computational

volume must also decrease accordingly, where the conditions within such a volume may be considered to be in equilibrium It is, therefore, clear that in the extreme cases (ultra-thin film tribology), with molecular interactions and surface energy effects, no bulk properties such as a computational elemental volume may be assumed (see Chapter 3)

therefore, one may surmise that physical interactions regardless of the state of matter are functions of size of the assumed system and that of a

material point considered, or the ratio e/ (Rahnejat, 2008) It turns out

that the nature of physical interactions (force) changes according to scale

(the same ratio) However, the size of the material point e is explained by

a host of physical attributes such as mass or charge and that of the system

 by density, viscosity, permittivity, elasticity, coefficient of friction, etc this means that forces other than gravity are related to kinematic quantities (displacement, velocity and acceleration) by physical properties of material points and the environment of the system the introduction to Chapter 3 describes the philosophical concern about the multiplicity of forces of nature this means the current knowledge is based on acceptance of a multi-physics character for interactions of material points at multi-scale within defined

systems, thus the increasingly used phrase: multi-scale multi-physics analysis

Finally, this brief introduction has shown that both dynamics and tribology are subsets of physics of motion

Kinematics, being the study of motion without regard to the underlying cause (force), is one of the oldest sciences Its roots can be traced back to the ancient studies of heavenly bodies, such as Homer’s Earth-centred universe

in the Iliad as an observation-based science, kinematics is concerned with

measurement of the state of motion of a material point (displacement, velocity and acceleration) with respect to a frame of reference as already discussed,

it is particularly convenient to firmly attach this frame of reference to a fixed (static) object therefore, it was particularly convenient to assume earth

to be the fixed central entity about which all the heavens would revolve (presumably in the adoration of humanity!) the heavenly bodies would then describe curvilinear paths whose slope at a given position would yield

their relative velocity with respect to the frame of measurement Much later, Galileo understood that deviation from a straight-line motion corresponded

to non-uniform velocity and the curvature was due to accelerated motion therefore, kinematics is the study of curves; their local slope and curvature

By the late sixteenth and early seventeenth centuries kinematics had finally attained the status only hitherto afforded to geometry as a fundamental science, because of the historical prominence of ancient and Middle age geometers such as Homer, Pythagoras, archimedes, Ptolemy, Khayam, tusi and Copernicus, among others Using astronomical observations and a cursory understanding of non-uniform motion, Galileo and Kepler put an end to the concept of the Earth-centred universe and obviously put a sizeable dent in the human vanity! Kinematics had its greatest moment in history with the

acceptance of the heliocentric system

Using Kepler’s observations and his laws of motion, Newton explained the elliptical orbit of planets around the Sun by a central force due to gravitational attraction the cause belying kinematics was found: force In the case of planetary motions, the force of gravity caused the non-uniform accelerated motion; curvature of the path the law of universal gravitation states:

where G is the universal gravitational constant, M the mass of a source

(such as the Sun) and m that of a target body (such as earth) thus, the radius of curvature r at any position along the path is r = √(GM/g) for a

two-body system (Sun and Earth) Since the Earth’s path is elliptical (only

slightly) on the ecliptic plane, then r is not a constant, which means that g

varies accordingly

the simple calculations here assume a two-body system, but the path of a body within a system (earth in the solar system) is subject to all material points within it (other planets); remember Mach’s principle More comprehensive treatment of this problem is given by Chandrasekhar (1995)

Newton then stated that in general equation (1.3) can be extended to his

second axiom; equation (1.1) If there is no net force; F = 0, a body at rest

remains stationary, while one in motion pursues a straight-line path; Newton’s first axiom a straight-line path is an extremal path (shortest path) due to

uniform motion of a material point relative to an observer One can surmise from equation (1.3) that attraction between two bodies necessitates equal

and opposite forces this is Newton’s third axiom; for every action there

is an equal and opposite reaction

the assertion of these axioms by newton, in addition to the law of universal gravitation, resulted in scientific disputes, some of which persisted beyond his lifetime One concerned a fundamental proof for the second axiom, to

render the same as a law of physics an axiom is an assertion which appeals

to all observers who would all agree on the cause of a phenomenon this definition does not put the onus of acceptance on a mathematical proof; such as the existence of the Sun Some have proposed intangible proofs for

certain axiomatic concepts such as Descartes’ for ‘life’: I think, therefore, I

am Mathematical discourse has increasingly been viewed as a requirement

for proof since the seventeenth century In this respect, Lagrange’s equation

is the proof of Newton’s second axiom as:

where K is the kinetic energy (considered to be independent of displacement

q with the right choice of co-ordinate system), U is the potential energy and

F a j the component of net applied force in the co-ordinate direction q j (a generalised proof is given in Section 1.3)

In general a completely unconstrained material point in space has six

degrees of freedom, therefore, the generalised co-ordinate set: q j Πx, y, z,

y , q, j the kinetic energy has, therefore, components: K j = 1mq j

2 2

ÍÍÍÍ

q j= – ∂ j

Thus, a material point falling freely under the influence of gravity towards

the centre of earth from any height x, with the frame of reference q aligned

with the direction of motion has a body force:

Using Lagrange’s equation and noting that there is no applied force (free

fall); F ax = 0, then: F ax = – ( = )mx mg x g , which is Newton’s second axiom

Therefore, Lagrange’s equation is essentially the determination of net force, causing an acceleration (same as equation (1.1)):

U q

where W Πm, I according to the degree of freedom (translational or rotational) thus, if a potential f can be specified, then acceleration of all material points

within such a field can be determined One can now revisit the same example

of the falling matter above, this time attaching the frame of reference to the

material point itself, falling within a field, where f = –(GM/x) In this case,

q = x, W = m, F ax = 0 as before and a q = x , then:

which yields the same results as previously two important observations should be made Firstly, the potential used is due to gravitation, thus equation (1.3) is proven from first principles If the field is due to Earth’s gravity then

M represents its mass x = r + H is the distance to the centre of earth, r its radius and h the height of the falling matter above the Earth’s surface Since usually H << r, g hardly changes near the surface of Earth This justifies the use of a constant value for g in engineering Secondly, the above alternative

analyses yield the same result, indicating the equivalence of the two systems; one in a gravitational field and the other falling uniformly with an equivalent

inertial acceleration this was noted by einstein as the equivalence principle,

the implication being that inertial acceleration produces gravitational action there are many examples, such as a material point in curvilinear motion or planetary motion or a vehicle cornering this means that motion on curves induces gravitational action This became clear with Einstein’s general relativity; after all physics of motion in all its forms could be reduced to study of curves and motion of material points upon them this appears to be true apart from various electromagnetic phenomena in the scale of minutiae

as described in Chapter 3 the problem is that general relativity is based

on a theory for gravity (macroscopic material points), thus, seemingly prevalent potentials at very small scale deviate from it the next section discusses Lagrange’s equation Readers should note that inertial and body forces which are dominant in the equation play an insignificant role in the scale of minutiae (see also Chapter 3)

space

Lagrange’s equation (1.4) is for unconstrained systems, where any material

point within the defined system enjoys six degrees of freedom as already