Terramechanics and OffRoad Vehicle Engineering, Second Edition: Terrain Behaviour, OffRoad Vehicle Performance and Design

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First edition 1989

Second edition 2010

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ix

Since the publication of the first edition of this book in 1989, notable progress has been made

in terramechanics, which is the study of the dynamics of an off-road vehicle in relation to its

environment – the terrain. Understanding of the mechanics of vehicle–terrain interaction has

been improved. New techniques have been introduced into the modelling of terrain behaviour.

A series of computer-aided methods for performance and design evaluation of off-road vehicles

from the traction perspective, incorporating recent advancements in terramechanics, have been

further developed. These methods have been gaining acceptance in industry in the development

of new products. Continual interest in improving vehicle mobility over a wider range of envi￾ronments and renewed enthusiasm for the exploration of the Moon, Mars and beyond shown by

an increasing number of nations have given new impetus to the further development of terrame￾chanics. To reflect these and other advancements in the field and to serve the changing needs of

the professional and higher educational communities, time is ripe for this second edition.

While new topics are introduced and data are updated in this edition, the objective and format

remain similar to those of the previous edition. The fundamentals of terramechanics underlying

the rational development and design of off-road vehicles are emphasized. As the performance

of off-road vehicles over unprepared terrain constitutes a basic issue in vehicle mobility, this

book focuses on the study of vehicle–terrain interaction from the traction perspective.

To better serve the higher educational community in the fields of automotive engineering,

off-road vehicle engineering, and agricultural and biological engineering, examples of the

applications of the principles of terramechanics to solving engineering problems are given.

Practical problems that may be assigned to senior undergraduate or postgraduate students as

part of their study programme are also included in this new edition.

The number of chapters has been expanded to 12 in this edition from nine in the previous edi￾tion. Chapter 1 provides an introduction to the subject of terramechanics, outlines its roles, and

presents outstanding examples of its practical applications. A brief review of the modelling of

terrain behaviour is presented in Chapter 2. The fundamentals of the theories of elasticity, plastic

equilibrium and critical state soil mechanics, as applied to the study of vehicle–terrain interac￾tion, are outlined. The applications of the finite element method (FEM) and the discrete (distinct)

element method (DEM) to the modelling of terrain are reviewed. While these theories or model￾ling techniques provide a foundation for an understanding of some aspects of the physical nature

Preface to the Second Edition

x Preface to the Second Edition

of vehicle–terrain interaction, there are limitations to their applications in practice, particularly in

modelling behaviour of natural terrain. Chapter 3 describes the techniques and instrumentation

currently used for measuring terrain behaviour in the field. The responses of various types of nat￾ural terrain to normal and repetitive loading observed in the field are discussed in Chapter 4. This

provides the terrain information needed for predicting the sinkage of the vehicle running gear

and the normal pressure distribution on the vehicle–terrain interface. Chapter 5 describes the

shear strengths of various types of natural terrain measured in the field and their characterization.

This provides the required terrain information for predicting the tractive capability of off-road

vehicles in the field. Criteria commonly used for evaluating the performance of various types of

off-road vehicle are reviewed in Chapter 6. Empirical and semi-empirical methods for predicting

tracked vehicle performance are discussed in Chapter 7. Chapter 8 outlines the analytical basis

for the computer-aided method NTVPM for performance and design evaluation of vehicles with

flexible tracks, such as military and cross-country transport vehicles. The experimental validation

of NTVPM is also described. Applications of NTVPM to parametric analyses of vehicle designs

are discussed in Chapter 9. Examples of its applications to the development of new products in

off-road vehicle industry are presented. Chapter 10 outlines the analytical basis for the computer￾aided method RTVPM for performance and design evaluation of vehicles with long-pitch link

tracks, such as industrial and agricultural tractors. The experimental validation of RTVPM and

its applications to parametric analyses are presented. Chapter 11 presents empirical and semi￾empirical methods for predicting wheel and wheeled vehicle performance. The analytical basis

for the computer-aided method NWVPM for predicting the performances of wheels and wheeled

vehicles is outlined in Chapter 12. As an example, the application of NWVPM to the evaluation

of the performance of lunar vehicle wheels is presented.

Some of the material included in this new edition has been presented at professional devel￾opment programmes and seminars in many countries. These included staff training pro￾grammes on the applications of terramechanics to the evaluation of planetary rover mobility,

presented at the European Space Research and Technology Centre (ESTEC) of the European

Space Agency (ESA) and at the Glenn Research Center, National Aeronautics and Space

Administration (NASA), USA.

This new edition includes some of the results of recent research on off-road vehicle mobility car￾ried out by the author together with his associates at Carleton University and at Vehicle Systems

Development Corporation (VSDC), Ottawa, Canada. The author wishes to express his apprecia￾tion to his former research staff, postdoctoral fellows and graduate students at Carleton, and to

his associates at VSDC for their contributions, particularly Jon Preston-Thomas, the late Michael

Garber, Yuli Gao, Mike Galway and Wei Huang. Appreciation is due also to many organizations,

in private and public sectors, for their generous support for our research over the years.

Jo Yung Wong

Ottawa, Canada

xi

In the past few decades, the continual demand for greater mobility over a wider range of ter￾rains and in all seasons by agricultural, construction and cross-country transport industries

and by the military has stimulated a great deal of interest in the study of vehicle mobility over

unprepared terrain. A large volume of research papers on this subject has been published in

journals and conference proceedings of learned societies. A variety of methods for predict￾ing and evaluating off-road vehicle performance, ranging from entirely empirical to highly

theoretical, has been proposed or developed. However, methods that will enable the design

engineer or the procurement manager to conduct a comprehensive and yet realistic evaluation

of competing vehicle designs appear to be lacking. This prompted the author of this book to

embark, more than a decade ago, on a series of research programmes aimed at filling this gap.

The objective is to establish mathematical models for vehicle–terrain systems that will enable

the engineering practitioner to evaluate, on a rational basis, a wide range of options and to

select an appropriate vehicle configuration for a given mission and environment. To be useful

to the design engineer or the procurement manager, the models should take into account all

major vehicle design and operational parameters as well as pertinent terrain characteristics.

After more than a decade of intense effort, a series of computer-aided methods (computer

simulation models) for predicting and evaluating the performance of tracked and wheeled

vehicles, which meet the basic objective outlined above, have emerged. These methods have

since been used to assist off-road vehicle manufacturers in developing new products and

governmental agencies in evaluating vehicle candidates with most encouraging results. The

encouragement that these developments have effected has convinced the author to put these

pages together, with the hope that this book may enhance the interest of the professionals

engaged in the field of off-road vehicle mobility.

This book summarizes some of the research and development work on the computer-aided

methods for evaluating off-road vehicle performance carried out by the author and his asso￾ciates at the Transport Technology Research Laboratory, Carleton University and Vehicle

Systems Development Corporation, Ottawa, Canada. Chapter 1 provides an introduction to the

subject of terramechanics, and outlines its roles and basic issues. Chapter 2 describes the tech￾niques and instrumentation for measuring terrain behaviour. An understanding of the mechani￾cal properties of the terrain is of importance in the prediction and evaluation of off-road vehicle

Preface to the First Edition

xii Preface to the First Edition

performance, as the behaviour of the terrain quite often imposes severe limitations to vehicle

mobility. Chapter 3 describes the responses of various types of natural terrain to normal and

repetitive loading. This provides information for predicting the sinkage of the vehicle running

gear and the normal pressure distribution on the vehicle–terrain interface. Chapter 4 describes

the shear strength of various types of natural terrain. This provides information for predicting

the tractive capability of off-road vehicles. Chapter 5 reviews some of the methods previously

developed for predicting the performance of tracked vehicles. Chapter 6 outlines the analytical

framework for the development of computer-aided methods for evaluating tracked vehicle per￾formance, while Chapter 7 illustrates some of the applications of the computer-aided methods

to the parametric analysis of tracked vehicle design and performance. Chapter 8 reviews some

of the methods previously developed for predicting the performance of tyres, while Chapter

9 outlines the recently developed computer-aided methods for evaluating the performance of

tyres and wheeled vehicles and illustrates their applications.

Some of the material included in this book has been presented at seminars and professional

development programmes in Canada, China, Italy, Germany, Singapore, Spain, Sweden, the

United Kingdom and the United States. Some of these seminars were jointly offered with the

late Dr M.G. Bekker during the period from 1976 to 1985.

The computer-aided methods presented in this book represent recent advances in the method￾ology for predicting and evaluating off-road vehicle performance. This does not mean, how￾ever, that further development of the methods described is not required. If and when better

mathematical models for vehicle–terrain interaction and for characterizing terrain behaviour

are available, they could readily be fitted into the framework presented here to make an even

more comprehensive and precise picture.

Many organizations have supported the research upon which this book is based. In par￾ticular, the author wishes to record the support provided by the Canadian Department of

National Defence, National Research Council of Canada, Natural Sciences and Engineering

Research Council of Canada, and Vehicle Systems Development Corporation. In writing

this book, the author has drawn much on the experience acquired from working with many

industrial and research organizations, including Hagglunds Vehicle AB of Sweden, US Naval

Civil Engineering Laboratory, Institute for Earthmoving Machinery and Off-Road Vehicles

(CEMOTER) of the Italian National Research Council, and Vehicle Mobility Section,

Defence Research Establishment Suffield and other branches of the Canadian Department of

National Defence. This acknowledgement does not imply, however, that the views expressed

in this book necessarily represent those of these organizations.

The author acknowledges with gratitude the inspiration derived from collaboration and dis￾cussions with many colleagues in industry, research organizations and universities. He is

indebted to Dr A.R. Reece, formerly with the University of Newcastle upon Tyne and now

Managing Director, Soil Machine Dynamics Ltd, England, and the late Dr M.G. Bekker for

Preface to the First Edition xiii

their valued encouragement and stimulation. The author also wishes to express his appre￾ciation to the staff members and graduate students at the Transport Technology Research

Laboratory, Carleton University and to his associates at Vehicle Systems Development

Corporation for their contributions to the research work presented in this book. He is espe￾cially indebted to Mr J. Preston-Thomas of Vehicle Systems Development Corporation for

his contributions to the development of the computer-aided methods for evaluating off-road

vehicle performance and for reviewing the manuscript.

J.Y. Wong

Ottawa, Canada

June 1989

xiv

Quantity US customary unit SI equivalent

Acceleration ft/s2 0.3048 m/s2

Area ft2 0.0929 m2

in2 645.2 mm2

Energy ft·lb 1.356 J

Force lb 4.448 N

Length ft 0.3048 m

in 25.4 mm

mile 1.609 km

Mass slug 14.59 kg

ton 907.2 kg

Moment of a force lb·ft 1.356 N·m

Power hp 745.7 W

Pressure or stress lb/ft2 47.88 Pa (N/m2)

lb/in2 (psi) 6.895 kPa (kN/m2)

Speed ft/s 0.3048 m/s

mph 1.609 km/h

Volume ft3 0.02832 m3

in3 16.39 cm3

gal (liquids) 3.785 litre

Conversion Factors

xv

A area

Al rigid area of a track link as a proportion of its nominal contact area

Af vehicle frontal area

Au parameter characterizing terrain response to repetitive loading

a half width of loading area; distance defining the longitudinal location of

the centre of gravity; acceleration

B wheelbase

b smaller dimension of a rectangular plate or the radius of a circular plate;

width

bb belly width

bti tyre width

btr track width

C, CI cone index

c cohesion

D diameter

Dh hydraulic diameter

Dr relative density

d diameter

E modulus of elasticity

e void ratio; base for the natural logarithm

F function; thrust, tractive effort

Fd drawbar pull

Fv tractive effort developed on the vertical shear surfaces on both sides of

a track

fo yield strength of an ice layer in tension; coefficient of track internal

resistance

Nomenclature

xvi  Nomenclature

ft radial deflection of the roadwheels of a track system

G sand penetration resistance gradient

Ge effective sand penetration resistance gradient

Gey revised effective sand penetration resistance gradient

H horizontal component of a tension force

h thickness; tyre section height

hl lug height

i slip

is skid

j shear displacement

j0 shear displacement where shear stress peaks

K shear deformation parameter

K1, K2 parameters characterizing the shear stress–shear displacement relationship

Kr ratio of residual shear stress to maximum shear stress

K shear displacement where shear stress peaks

k stiffness; resultant pressure–sinkage parameter

kc, kφ pressure–sinkage parameters in the Bekker equation

kc



, kφ



, kc



, kφ

 pressure–sinkage parameters in the Reece equation

ke tyre carcass flexing resistance coefficient

k0 parameter characterizing terrain response to repetitive loading

kp1, kp2, kz1, kz2 pressure–sinkage parameters for snow cover

ku parameter characterizing terrain stiffness during the unloading–reloading

cycle

L perimeter; characteristic length for an ice layer

Lb belly contact length

Lt length of track in contact with terrain

l length

Mo limit bending moment per unit length of an ice layer

MI mobility index

m, mm parameters characterizing the relation between the strength of the muskeg

mat and that of the underlying peat

N number

Nc clay–tyre numeric

Ncs cohesive-frictional soil–tyre numeric

Nomenclature  xvii

Ns sand-tyre numeric

Nse, Nsey revised sand–tyre numerics

n exponent of sinkage

nav average exponent of sinkage

nr number of wheel stations in a track system

P load; power; spherical pressure

Pco collapse load for an ice layer

Pd drawbar power

Pe engine power

Pro transport productivity

Pus ultimate load due to local shear failure for an ice layer

Put ultimate load due to circumferential tension failure for an ice layer

p pressure

p reaction of sublayer

pb pressure on the belly–terrain interface

pc pressure due to carcass stiffness

pca pressure exerted on the carcass by the terrain

pc calculated pressure

pc0, pc1, pc2 collapse pressures for an ice layer

pcr critical inflation pressure

pg ground pressure

pgcr critical ground pressure

pi tyre inflation pressure

pm measured pressure

pp punching pressure

pu pressure at the beginning of unloading in a loading–unloading–reloading

cycle

pw pressure–sinkage parameter for a snow cover

q surcharge; pressure exerted on the muskeg mat by the underlying peat

R radius; deviatoric stress

Ra aerodynamic drag

Rbc belly drag

Rc resistance due to terrain compaction

Rf tyre carcass flexing resistance

Rg grade resistance

Rin internal resistance of gear running

Rob obstacle resistance

xviii  Nomenclature

Rt resistance due to vehicle running gear–terrain interaction

Rv motion resistance of vehicle running gear

RCI rating cone index

r radius

Sv shear force per unit track length developed on the vertical shear surfaces

on both sides of a track

s shear stress

sb shear stress on the belly–terrain interface

sc calculated shear stress

sm measured shear stress

smax maximum shear stress

sr residual shear stress

T tension

to thickness of muskeg mat

tt track pitch

V vertical component of a tension force; actual forward speed of a vehicle;

specific volume

Va absolute velocity

Vj slip velocity

Vr vehicle speed relative to wind

Vt theoretical speed

VCI vehicle cone index

W load, weight

Wp payload

wr weighting factor

z, zo sinkage

ze sinkage of a tyre in the elastic operating model

zm mean sinkage of a grouser

zr sinkage of a tyre in the rigid operating mode

zu sinkage at the beginning of unloading in a loading–unloading–reloading

cycle

zw pressure–sinkage parameter for a snow cover

 angle with the horizontal

b vehicle belly inclination angle; rake angle

 density

Nomenclature  xix

 inclination angle; interface friction angles; tyre deflection

t tyre deflection

ε goodness-of-fit; coefficient for tyre flexing resistance

 efficiency

d tractive (drawbar) efficiency

do tractive (drawbar) efficiency overall

m efficiency of motion

p propulsive efficiency

s slip efficiency

st structural efficiency

t transmission efficiency

tr transport efficiency

 angle

 ratio of total lug tip area to total tyre tread area

 concentration factor

 Poisson’s ratio; drawbar coefficient; coefficient of friction

�� normal stress

τ shear stress

r shear strength of muskeg mat

ø angle of shearing resistance

ϕ roadwheel contact angle

 angular speed



Man has a long history of involvement in off-road locomotion, perhaps since the invention of

the wheel about 3500 BC. Powered off-road vehicles have come into wide use in many parts

of the world in agriculture, construction, cross-country transportation and military opera￾tions since the turn of last century. In spite of rapid progress in technology, the development

of cross-country vehicles has, for a long period of time, been guided by empiricism and the

‘cut and try’ methodology. Systematic studies of the principles underlying the rational devel￾opment of off-road vehicles did not receive significant attention until the middle of the 20th

century. The publication of Dr M.G. Bekker’s classic treatises, Theory of Land Locomotion

in 1956 and Off-the-Road Locomotion and Introduction to Terrain–Vehicle Systems in the

1960s, stimulated a great deal of interest in the systematic development of the principles of

land locomotion mechanics (Bekker, 1956, 1960, 1969). His pioneering work and unique

contributions laid the foundation for a distinct branch of applied mechanics, which has now

become known as ‘Terramechanics’.

In a broad sense, terramechanics is the study of the overall performance of a machine in

relation to its operating environment – the terrain. It has two main branches: terrain–vehicle

mechanics and terrain–implement mechanics. Terrain–vehicle mechanics is concerned with

the tractive performance of a vehicle over unprepared terrain, ride quality over unprepared

surfaces, handling, obstacle negotiation, water-crossing and other related topics. Terrain–

implement mechanics, on the other hand, deals with the performance of terrain-working

machinery, such as soil cultivating and earthmoving equipment.

The aim of terramechanics is to provide guiding principles for the rational development,

design, and evaluation of off-road vehicles and terrain-working machinery. In recent years,

the growing concern over energy conservation and environmental preservation has further

stimulated the development of terramechanics. In addition to being a good engineering design

in the traditional sense, an off-road machine is now expected to attain a high level of energy

efficiency and not to cause undue damage to the operating environment, such as excessive soil

compaction in agriculture. Increasing activity in the exploration and exploitation of natural

resources in new frontiers, including remote areas and the seabed, and the growing demand

for greater mobility over a wider range of terrains and in all seasons have also given much

new impetus to the development of terramechanics.

Chapter 1

Introduction

  Chapter 1

Continuing interests of the USA, European Union and Russia, as well as programmes

initiated by China, Japan, India and other nations, in the exploration of the Moon, Mars and

beyond, have further stimulated advancements in terramechanics and its applications to the

development of extraterrestrial vehicles, including manned and unmanned rovers (Wong and

Asnani, 2008).

Terrain–vehicle mechanics is the prime subject of this book. It introduces the reader to the

basic principles of terramechanics, which include the modelling of terrain behaviour, mea￾surement and characterization of the mechanical properties of terrain pertinent to vehicle

mobility, and the mechanics of vehicle–terrain interaction. As the performance of off-road

vehicles over unprepared terrain constitutes a central issue in vehicle mobility, this book

focuses on the study of vehicle–terrain interaction from the traction perspective. It provides

the knowledge base for the prediction of off-road vehicle performance. Through examples,

this book also demonstrates the applications of terramechanics to parametric analyses of

terrain–vehicle systems and to the rational development and design of off-road vehicles from

the traction perspective. The handling and ride of off-road vehicles are discussed in a separate

book, Theory of Ground Vehicles (Wong, 2008).

1.1 Role of Terramechanics

The industries that manufacture and operate off-road equipment are multibillion dollar

businesses. By considering the number of tractors and soil-cultivating implements used

in agriculture, the number of earthmoving machines used in the construction industry,

the number of off-highway trucks used in the off-road transport industry, and the number

of combat and logistic vehicles used in the military, one can appreciate the scope for the

applications of terramechanics.

Terramechanics, coupled with a systems analysis approach, can play a significant role in the

development and evaluation of off-road equipment for a given mission and environment.

Systems analysis is a methodology that provides a quantitative and systematic assessment of

clearly defined issues and alternatives for decision makers. The knowledge of terramechan￾ics can be applied, directly or indirectly, to the development, evaluation or selection of the

following:

(a) vehicle concepts and configurations, defined in terms of form, size, weight and power;

(b) the running gear (or terrain-engaging elements) of a vehicle;

(c) the steering system of a vehicle;

(d) the suspension system of a vehicle;