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THESIS FOR THE DEGREE OF

LICENTIATE OF ENGINEERING

TRANSIENT STEERING OF TRACKED VEHICLES

ON HARD GROUND

DAG THUVESEN

MACHINE AND VEHICLE DESIGN

CHALMERS UNIVERSITY OF TECHNOLOGY

S-412 96 GÖTEBORG, SWEDEN

REPORT NO. 1997-06-09

• M

A C H I N

E A N D VEH I C L E D E S

I G N •

C H A L M

E R S • G Ö T E B O R

G





TRANSIENT STEERING OF TRACKED VEHICLES

ON HARD GROUND

DAG THUVESEN

MACHINE AND VEHICLE DESIGN

CHALMERS UNIVERSITY OF TECHNOLOGY

S-412 96 GÖTEBORG, SWEDEN





TRANSIENT STEERING OF TRACKED VEHICLES ON HARD GROUND

i

ABSTRACTA track module has been developed for analysis of steering of tracked vehicles on hard ground.The track module is especially developed to interact with an MBS (Multi Body System)program such as ADAMS or DADS. The track module is based on sound constitutiverelationships.

The track is regarded as a basic machine component which can be attached to a vehiclemodel. This module is developed for running on flat hard ground, which gives rise to a planarmotion in the ground plane. The model does not consider the width of the track, however thiswill not in general affect the steering analysis noticeably, at least not for the most commontypes of tracked vehicles.

The track module receives its input from the MBS software in terms of velocities, forces andmoments. The velocities are defined by a longitudinal, a transverse and a rotational componentfor the mid-point of the track. Due to the restriction to flat ground, the motion for the trackcould be described by the concept of a specific spin pole, i.e. an instantaneous centre of rotation. The equilibrium for the model of the vehicle-track system is solved by the software.

The model of the track has a linearly varying discrete normal track load distribution, whichcould easily be modified to some other more complex and realistic distribution. The magnitudeof the total normal load in the ground contact and its position are then applied to define thegeometry of the normal load distribution.

By applying consistent constitutive equations, the motion, the normal load distribution andthe resulting action of the friction forces in the ground plane could be calculated. In this case,Coulomb friction allowing for anisotropic friction, is used. These forces are returned from thetrack module to the model of the chassis in the MBS software. In the present case of planarmotion there is only one correspondingly acting reaction force along a specific line. However,in order to fit this force to the handling of data within the software, it is split into threecomponents: one longitudinal force, one transverse force, and a vertical moment acting aboutthe mid-point of the track. The longitudinal track force must be in equilibrium with the torquein the drive shaft.

The load changes due to tension in the tracks, which reduce the loads under the leading andtrailing road wheels, are also considered. The steering input could either be utilized by aspecified velocity of each track or by prescribing the sprocket torques. A realistic model of apowertrain can be part of the total vehicle model and control the track motion.

The track module has been tested in the MBS software ADAMS but can be modified to suitany MBS software.

Since the dynamic behaviour of a tracked vehicle is complicated, the track module presentedis a versatile and time-saving tool for the analysis and the prediction of the steeringperformance. Such analyses will provide much data useful in the early stages of the designprocess of tracked vehicles.

Key words: tracked vehicle, steering, hard ground, solid ground, simulation, multi bodysystem, MBS, ADAMS, DADS, MECHANICA MOTION



DAG THUVESEN

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iii

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Main objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Two theoretical approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 The work performed by Gerbert and Olsson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 MBS - Multi Body System approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 A vehicle model in an MBS program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 A tyre module for a vehicle model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.4 A track module versus a tyre module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 The track module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.1 Module interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.1.1 Ground - track module interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.1.2 Drive shaft - track module interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.1.3 Chassis - track module interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4 Special numerical considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5 Implementation of the track module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

6 Verification of the proposed method of analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236.1 Methods of verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236.2 The vehicle manoeuvre by Gerbert and Olsson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236.3 Comparison with the work by Gerbert and Olsson . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

6.3.1 Motion response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246.4 Comparison with the work by Jakobsson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266.5 Observed qualitative agreement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.6 Integration performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

7 Summary and future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Appendix A: Summary of the paper: On track vehicles running through curves . . . . . . . . . A1

CONTENTS



DAG THUVESEN

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ACKNOWLEDGMENTSThis work was carried out at the department of Machine and Vehicle Design at ChalmersUniversity of Technology under the supervision of Professor Mart Mägi. It took a long time tofinish not primarily due to its extreme difficulty but to the many intricate obstacles appearingon the way.

First I would like to thank my advisor professor Mart Mägi for his excellent support,guidance and great patience.

I would also like to thank Försvarets Materielverk (FMV, Defence Materiel Administration)for their financial support and, especially, Jonas F. Persson at FMV for his deep interest in thisproject, his belief in the future and positive thinking.

I also want to express my deep gratitude to Ph.D. Anders Hedman. Without him, this reportwould have been an impossibility. The article On track vehicles running through curves byGöran Gerbert and Karl-Olof Olsson, partially summarized in Appendix A, has been modifiedby Ph.D. Anders Hedman.

Special thanks to Ph.D. Lars Lindkvist and Lic.Eng. Johan Hultén who helped me to solvesome of the numerical problems that I have been faced with during my thesis work.

Ph.D. Rikard Söderberg and Lic.Eng. Magnus Evertsson are also remembered for theirmoral support and useful advice.

Last but not least, I want to express my deepest appreciation to my lovely supporting wifeKatarina and our two fascinating children Erik and Anna for putting up with me all this time,especially during the last few months.

Dag ThuvesenKungsbacka, May 1997




v

NOTATION

Mid-point on track

Distance to the track load centre (non dimensional)

Virtual diameter of driving wheel

Longitudinal track force

Transverse track force, in a specific point q

Track normal load

Ground contact length of track

Turning moment of a track

Brake torque, Brake torque by stationary motion

Driving torque on the drive shaft

Centre of curvature for the entire vehicle

Spin pole

Distance to the spin pole perpendicular to the track (Appendix A)

Distance along the track assembly from the mid-point of the track to the point A

Distance to the spin pole perpendicular to the track

Brake factor

Load distribution factorNumber of road wheels

Number of a specific road wheels

Radius to the spin pole

Moment arm for the transverse force acting about the point C

Distance from a specific contact point to the point A

Velocity of a track pad relative to the ground

Absolute velocity of the mid-point, C , of the track assemblySliding velocity component along and across the track

Absolute velocity of the track pad at the reference point C

Relative velocity of a track pad relative to the track assembly

Absolute velocity of the centre of gravity of the vehicle

Local coordinate system

Spin pole distance along track from one end to the point A

Degree of skewness of the track load distribution

C

C G

Ddrive

F L

F T F Tq,

G

L

M

M b M bs,

M drive

O

S

e

e x

e y

k b

k q

n

q

r 1 r 2 r 3, ,

r q

uq L 2 ⁄

v1 v2 v3, ,

vassembly

v L vT ,

v pad

vre l

vvehicle

x y z, ,

yL 2 ⁄

∆G



DAG THUVESEN

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Lifting component on leading or trailing road wheel

Integrating time step

Angle between the front (or rear) track links and the ground

Coefficient of friction, longitudinal, transverseAngular velocity of the chassis, yaw velocity

Angular velocity of the steering brake drum

Angular velocity of the planet carrier

∆k q

∆t

α

µ L µT ,ω z

ωb

ωc




1

1 INTRODUCTION

1.1 MAIN OBJECTIVE

There exist a number of ways to analyse a steering manoeuvre of a tracked vehicle. Jakobsson

analysed the steering performance of tracked and half tracked vehicles for a stationary curvemotion (Jakobsson 1947). The work by Jakobsson was performed in the early 1940’s. Muchlater Gerbert and Olsson continued this work, carrying out a more general analysis of atransient steering manoeuvre (Gerbert and Olsson 1982). They solved their equations for lowvelocities but were unable to find any stable solutions for higher velocities.

The purpose of the present work is to develop a general track module for MBS (Multi BodySystem) software. Such a module would be based on the same type of assumptions that wereused by Jakobsson and by Gerbert and Olsson but would simplify the related analysis of steady-state and transient steering manoeuvres considerably.

1.2 TWO THEORETICAL APPROACHES

As was pointed out by Mägi, there are two theoretical approaches to the analysis of steering of tracked vehicles (Mägi 1994).

The first approach is based on the work performed by Merritt in the 1930’s (Merritt 1939).His theory assumes a completely rigid ground and track. He also assumes that Coulombfriction occurs in the entire contact area between the ground and the track. The main purposeof this approach is to analyse the steering behaviour of a tracked vehicle. These theories wereused by Jakobsson and later also by Gerbert and Olsson (Jakobsson 1947, Gerbert and Olsson

1982). Jakobsson analysed vehicle tracks in a steady-state motion in a curve at low speed. He

regarded a track as a machine component based on physically sound theories. Gerbert andOlsson applied the theories of Jakobsson but in a more general way including transient steeringnot restricted to low speed. Andersson also developed a model for a tracked vehicle sliding onhard ground based on the same concepts as Jakobsson (Andersson 1993). He introduced thisconcept of analysis of tracked vehicles to Swedish industry.

The second approach is that of Bekker (Bekker 1956, 1960 and 1969), whose theories are themost widely used today. These theories are further developed in the work by Wong (Wong

1989, 1993). The main intention of this approach is to analyse the tractive performance andslip, when driving straight ahead. Steering is then dealt with as an add-on effect. These theoriesare based on constitutive relations of elastoplastically deformable ground in the longitudinaldirection of the track. Perpendicular to the track, the forces are given by other constitutive

relationships, in this case Coulomb friction. To use different constitutive equationsperpendicular and parallel the tracks is obviously not consistent. It consists of thesuperposition of solutions to a nonlinear and nonconservative problem.

Kitano has also based his work on Bekker. He has analysed not only common skid steeredtracked vehicles (Kitano et al. 1976, 1977, 1988) but also articulated tracked vehicles, seeFigure 1.1 (Watanabe and Kitano 1986).



DAG THUVESEN

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1.3 THE WORK PERFORMED BY GERBERT AND OLSSON

Gerbert and Olsson elaborated the equations of Jakobsson, who restricted the analysis to astationary motion, and presented a more general analysis of the motion of tracked vehiclesincluding transient steering. The concept of spin poles according to Mägi was introduced in theanalysis (Mägi 1974). The spin pole is the instantaneous centre of rotation. The position of thespin pole is defined by the linear and angular velocity components of any point of a movingbody, see Figure 1.2. Furthermore, Gerbert and Olsson computerized the equations and solvedthem numerically. A more detailed description of the work is presented in Appendix A.

All loads were taken up as concentrated forces on the road wheels, giving the support loadon each wheel, where G is the total track load and n is the number of road wheels. There

was a pure Coulombian friction force beneath each support wheel, where the force iscounteracting the sliding velocity. The coefficient of friction was constant.

The model of the vehicle was given by analytical equations. The equations were solvednumerically only to a limited extent. At that time the computer capacity was low. For the caseof transient steering there were only stable solutions for velocities up to five metres per second.

The present paper is based on the work done by Gerbert and Olsson.

Figure 1.1 An articulated vehicle, BV 206, designed by Hägglunds

Vehicle AB in Sweden (Wong 1993).

Figure 1.2 The spin pole concept for a track in motion.

ω z

Spin pole

v2 ω zr 2=

r 2

v1 ω zr 1=

v3 ω zr 3=

r 3

r 1

G n ⁄




3

2 MBS - MULTI BODY SYSTEM APPROACH

2.1 INTRODUCTION

MBS software is used to model and analyse real world mechanical systems. Its advantage is to

create new or modify existing systems and to optimize them for different parameters. Using aset of data that describes the machine to be modelled, an MBS program builds a discretemathematical model of the system, corresponding to an idealized physical system as shown inFigure 2.1. Positions, orientations, velocities and accelerations of the various parts of themachine can be calculated, as well as resulting forces that act within the system. Externallyacting forces must be prescribed. By using such a program the behaviour of a wide range of alternative designs can be analysed. Thereby, the need for building and testing prototypes canbe reduced significantly. Such a technique may be referred to as virtual prototyping.

A computer model of a mechanical system, see Figure 2.2, is created by describing the realphysical system in terms of MBS software elements, see Figure 2.3.• A part in the physical system is defined by a discrete rigid body with mass and inertia.• A second type of element is a massless connector, obeying a constitutive relationship

between bodies. In some literature this element is referred to as a force generator (Haug

1992). A force generator could in the physical system be a force or torque like a spring ordamper applied to some part of the system. It could also be an actuator or gravity. The MBSsoftware defines the constitutive relationship as a massless element that produces a linear ornonlinear force or torque.

• A connection between parts in the physical system like a hinge or some other joint isdescribed by a massless constraint element. The constraint element is very theoretical in itsbehaviour. It reduces the allowed degrees of freedom between the two bodies it is attachedto and does not exhibit any kind of elasticity. It either allows a specific translation or rotationor prevents it. A particular kind of constraint is called a driver or motion generator . Thedriver is a massless constraint that will create a prespecified motion in terms of displacement,velocity or acceleration in an already defined constraint.

Figure 2.1 A discrete theoretical multi body system.

Ground

Rigid bodyConstraint

Force generator

Connecting point

Centre of gravity



DAG THUVESEN

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2.2 A VEHICLE MODEL IN AN MBS PROGRAM

A central idea of MBS modelling is to deal with and decide what parts the system analysedwill contain. Each of these parts will at first have all six degrees of freedom. The next step is todecide what constraints that have to be applied between the parts to reduce the number of degrees of freedom. The user must decide which degrees of freedom that are of greatestimportance and then introduce the constraints that suppress the non-existing motions. There

Figure 2.2 A simple physical system.

Figure 2.3 A physical system modelled in an MBS program with

wireframe representation of the bodies.

Piston

Connecting rod

Flywheel

Crankshaft

Bear ing

Gas force

Cylinder

Crankshaft part

Piston part

Cylinder constraint

Connecting rod part

Bearing constraint

Revo lu te constraint

Flywheel part

Force generator




5

are theoretically no restrictions to allow every degree of freedom in a system but the modelmight be too complex to analyse. The complexity of the model is limited by the computercapacity available during analysis. The model of the physical system must at least account forthe dominating degrees of freedom.

The physical system of a ground vehicle modelled as one rigid body has six degrees of

freedom, see Figure 2.4. There are three translational degrees of freedom: forward, lateral andvertical; and three rotational ones: roll, pitch and yaw. The default local coordinate system isoriented in such a way that x is forward, y lateral and z vertical.

Within the system there may be many other separately moving bodies, for example rotationsof the crankshaft or translations of the pistons in the engine, but this is not of very great interestwhen the overall steering behaviour is analysed. A computer model of such a vehicle shouldprimarily show this general behaviour, that is, motion along the six mentioned degrees of freedom.

2.3 A TYRE MODULE FOR A VEHICLE MODEL

A very simple model of a four-wheel vehicle would consist of four wheels connected to achassis. The chassis constitute one part, and four other parts define the wheels. Each part isdefined by its mass, inertia and position. Basically, each part has six degrees of freedom.

The wheels will interact with the ground and the chassis. The interaction between a wheeland the ground is accomplished with a tyre. The tyre is defined as the contact between thewheel and the ground, see Figure 2.5. The tyre is represented by a force generator previouslymentioned, see section 2.1. The wheel is allowed to move along its six degrees of freedomrelative to the ground, see Figure 2.5 and Figure 2.6. Therefore, there will be no constraintsthat reduce the allowed degrees of freedom.

Sliding in either direction produces frictional reaction forces between the tyre and theground. The force in the plane of the wheel, along the vehicle, is called tractive force and theperpendicular force, lateral force. The vertical translation represents the elasticity in the tyre.The rotations generate reaction moments between the tyre and the ground. These moments

represent overturning moment, rolling resistance and aligning moment, in the x, y and zdirection respectively. These forces and moments could be described by a force generator in

Figure 2.4 Principal degrees of freedom for any kind of ground vehicle.

y

Pitch

z

Ya w Roll

x La tera l

Vertical

Forward



DAG THUVESEN

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the MBS program, generating forces and moments in all directions defined by the six degreesof freedom.

The wheel will also interact with the chassis. There are at least two motions that have to beallowed between the wheel and the chassis. The wheel must be able to rotate about itsrotational axis and it must also be able to translate vertically relative to the chassis. Dealing

with an ordinary road vehicle, the front wheels can also rotate about the vertical axis or theking pin to enable a steering motion. This means that either two or three degrees of freedommust be allowed, depending on whether it is the front or rear wheel. The other motions must,for this simple vehicle model, be inhibited. The interface between the wheel and the chassiswill have constraints that inhibit these motions.

If the wheel is a driven wheel it will also interact with a drive shaft. This interaction could bedescribed by a force generator where a driving torque is defined.

The approach of describing a complex system in modules is very efficient. In the case of avehicle on wheels there are different ways to define the modules, which depend on where thesystem boundary is defined. In various MBS programs, the definition of the tyre modulediffers. One definition of the tyre module would be that only the massless force generatordefines the tyre module. Another variant would be that both the wheel and the force generatordefine the tyre module. In this case the numerical data that define the module would have to bethe mass of the wheel, its inertia and some description of the stiffness and damping in the tyre

in all six directions. A third type is defined by both the tyre, the wheel, the constraints and theforce elements that connect the wheel to the chassis.

Figure 2.5 Existing translational degrees of freedom.

Figure 2.6 Existing rotational degrees of freedom.

Tyre contact

Ground

Chassis

z

y

x

Wheel

Tyre contact

Ground

Chassis

z

y

x

Wheel

Only for a steered wheel




7

2.4 A TRACK MODULE VERSUS A TYRE MODULE

In order to enable versatile simulation of the performance of road vehicles on wheels, manydifferent models of the force interaction between the tyre and the ground have been developed.These models differ much in complexity and quality but are based on the same general ideas asare mentioned in the previous section.

With the basic principles of a tyre module on a vehicle as a background, it is not difficult todevelop a track module according to similar principles. The principles of motion for thechassis are the same but the track shows a different pattern of motion on the ground than thetyre does. A four-wheel vehicle would theoretically make contact with the ground in fourspots, creating contact lines while moving on the ground, see Figure 2.7. A tracked vehicle, onthe other hand, must, during a steering motion, slide its tracks over the ground, creating widecontact areas under each track. This indicates that the constitutive equations for the track toground contact are more complex.

By simplifying a model of a physical system, information will inevitably be lost, comparedto the real situation. Obviously, this will occur with a model of a track, too, but according toearlier results (Jakobsson 1947), a quite simple track model will predict steering performanceon hard ground surprisingly well. However, motion resistance and longitudinal slip mayrequire different or more complex models.

The tyre module did allow six degrees of freedom in the contact with the ground. Theproposed track module must be free to move in the surface plane of the ground with threedegrees of freedom; however, it must be constrained to remain in that plain, see Figure 2.8. If described by a line contact, however, the track may tilt about the longitudinal axis of the

vehicle, the roll axis. The two prevented components of motion are the vertical translation andthe rotation about the y axis, pitch. In the sense of MBS, constrained motions generatecompatible constraint reaction forces.

Figure 2.7 The traces of ground contact of a wheeled vehicle

compared to a tracked vehicle.

Tyres Tracks

Contact areaContact lines



DAG THUVESEN

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The proposed module will include the interaction with the ground, the chassis and the driveshaft.

Studying the vehicle model with four connected tyre modules, the system seems to bestatically overdetermined, however, this is avoided due to the vertical elasticity. The vehiclemodel with only two track modules connected must not be statically overdetermined either.This is accomplished by allowing certain motions between the chassis and the track module.Here the vertical motion and the rotation about the y axis are allowed, see Figure 2.9. This willalso allow the chassis its six degrees of freedom in order to demonstrate a normal performance.

Figure 2.8 The four existing degrees of freedom for a track module

relative to the ground.

Figure 2.9 The two existing degrees of freedom between the chassis and

the track module.

Track module

Ground

x

y

z

Ground

Chassis

Track module




9

3 THE TRACK MODULE

The main purpose of the track module developed in this thesis is to enable versatile analysis of a transient steering manoeuvre of any kind of tracked vehicle. The module is to be used inconjunction with an MBS program, like ADAMS or DADS.

When studying the interaction between the ground and the vehicle, it is important todistinguish between different relative motions.• Vehicle motion relative to ground• Track assembly motion relative to ground• Track pad motion relative to track assembly• Track pad motion relative to ground

Velocities relative to ground are in the present context absolute velocities both in translationand rotation. They may be identified as:

• at centre of gravity

• at reference point C in the middle of the ground-track contact

• at reference point C in the middle of the ground-track contact

which are observed at the points specified. The relative velocity between the pads and the trackassembly is always oriented along the track. This velocity is denoted and its magnitudeand direction are defined basically by the vehicle propulsion and steering system.

Schematically these motions could be illustrated as in Figure 3.1. The track module is heredefined to be composed of parts and elements according to the dashed line.

The physical track is an endless chain. There are track pads on each link. The track runsaround a number of wheels: the sprocket, the idler and the road wheels, see Figure 3.2. Thewheels may deflect relative to the chassis if they are connected to some kind of suspension.The model of the physical track assembly has to take into account all the parts mentioned.

There are no restrictions to modelling all the parts; wheels, links etc., see Figure 3.3. This hasbeen done when analysing a complete flexible link system (Murray, M., Canfield, T. R. 1992),

Figure 3.1 Schematic illustration of the existing motions within a

tracked vehicle model.

vvehicle

vassembly

v pad

vdiff

Chassis

Track

Track pad

Ground

6 DOF

2 DOF (min)

4 DOF

1 DOF

DOF = Degree of freedom

Aggregate

Track module



DAG THUVESEN

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but this system would not be practical for analysing steering behaviour due to excessiveanalysis time for the computer.

The model must be simplified but still reflect the general performance of a complete realvehicle. At the first stage, presented here, the internal friction losses within the track assemblycould be disregarded.

The characteristics of a physical wheel, referring to the mass, the inertia, the axis of rotationetc., are easy to apply to the tyre module. The definition of a track module is not that simple.The track assembly includes a number of road wheels, links of the track, etc. The developedmodule will in some simplified way comprise the whole track assembly, including theconnections to the ground, the chassis and the drive shaft.

3.1 MODULE INTERFACES

The proposed and developed track module is based on the same general ideas as the earlierdescribed tyre module. Therefore, each of the interfaces that the module of the track has with

its surroundings is studied.The module of the track is connected to the rest of the vehicle to create a tracked vehicle

system, used in the analysis. The module of the track interacts with the following parts:ground, drive shaft and chassis.

Figure 3.2 The track assembly of a tracked vehicle (Terry et al. 1991).

Figure 3.3 Detail of the track assembly (Terry et al. 1991).

Track

Suspension

Road wheel

Sprocket Idle r

Toproller




11

3.1.1 GROUND - TRACK MODULE INTERACTION

This interaction is the key issue of the track module. The module will in some way bedescribed by the motion of a line contact, which allows four degrees of freedom. Relatedmotions are the longitudinal and transverse translations in the ground plane as well as the rolland yaw rotations. The prevented translation is in the vertical direction and the prevented

rotation about the lateral direction. This is achieved by constraints that create a resultingnormal load and a moment which defines the virtual location of the normal load somewherealong the track.

Since the ground is only two-dimensional, the forces in the track-ground interface could beseparated into vertical and horizontal forces. Therefore, these two will hereafter be discussedseparately.

VERTICAL FORCES

The real pressure distribution on soft ground is very complex, see Figure 3.4. The actualdistribution is influenced not only by ground conditions but also by the design of thesuspension system and the track tension. The distribution on hard ground has, however, anaccentuated characteristics. On hard ground like asphalt and concrete, the contact pressure isconcentrated in small contacts areas under each road wheel. There are numerous ways toanalytically describe this distribution, see Figure 3.5. Fortunately, it has been found by severalauthors (Jakobsson 1947, Kitano 1976) that the details of the vertical force distribution alongthe track are not significant at the analysis of steering of tracked vehicles on hard ground.Therefore, a simplified model is chosen here, where the normal load is distributed asconcentrated discrete forces under each road wheel.

The distribution across the track is disregarded in the present analysis. This is appropriate tomost common tracked vehicles, except for snow-mobiles and similar applications.

Figure 3.4 Measured pressure distribution at a depth of 23 cm below

the soil surface under various tracked vehicles (Wong 1993).

Nominal

Nominal

Nominal

Nominal

Churchill v

Comet

Panther A

Sherman v



DAG THUVESEN

12

If the track is modelled as just one rigid body, interacting with rigid ground, the load

distribution along the track must be prescribed since it cannot be found by analysis. For anarrow track, the only significant vertical force is the resulting force and its location along thetrack, which also produces a moment about the pitch axis, see Figure 3.6. By definition, it isacting within the track width, which is negligible. The position and the magnitude of theresulting normal action must at all times be solved by the module.

In the most simple case, the load on a track is evenly distributed over all the road wheelsgiving the support load on each wheel, where G is the total track load and n is thenumber of road wheels on one track, see Figure 3.7. This implies that the centre of gravity ispositioned in the longitudinal centre of the track. However, the position of the resulting actionis determined both by the position of the centre of gravity and by the vehicle dynamics arisingfrom acceleration in any direction. Both the magnitude of the resulting action and its position

Figure 3.5 Different pressure distributions (Holmdahl 1989).

Figure 3.6 The position of the resulting force acting on the module.

Sinusoidal variation

Constant pressure

Forces on each road wheel

G

⇔ ⇔

G n ⁄




13

must at all times be correct. To enable the change in position of the resulting action, theinitially mentioned even distribution has to be reshaped. To modify this constant support load,the shape of the distribution could vary linearly, see Figure 3.8.

This variation is described in more detail in Appendix A. The individual support load will,according to this variation, be , where

(3.1)

In equation (3.1) the location parameter is defined according to Figure 3.8, and q is thesubscript for a specific road wheel, numbered one (1) through n. This linear default shapecould easily be modified to some other, possibly more realistic, distribution.

The tracks travel over the ground. Due to this motion the tracks are exposed to forces bothalong and across the tracks. A longitudinal force , either positive or negative, must alwaysbe present to produce either propelling or braking action. Due to this action, the leading andthe trailing road wheels on either side will be partially unloaded, see Figure 3.9. The liftingcomponent is assumed to be

(3.2)

where is the angle between the ground and the front (or rear) track link that is not in contactwith the ground, see Figure 3.9.

Due to this partial unloading of a support wheel, the shape must be modified so that themagnitude of the resulting action and its position remain unchanged. This is accomplished by

defining a steeper shape of the linear profile. The road wheel could be totally unloaded andthereby lift off the ground.

Figure 3.7 Constant support load on each road wheel.

Figure 3.8 Linearly varying support load.

Gn----G

q n= q 1=

L 2 ⁄

k qG

k q1n--- 1

3 C G 1–( )n 1+

------------------------ n 2q– 1+( )– =

C G

Gn----G

L 2 ⁄

∆G

C G L 2 ⁄

k qG

q n= q 1=

F L

∆k q F L αsin=

α



DAG THUVESEN

14

HORIZONTAL FORCES

Assuming the ground and the track to be rigid, and the surface dry, the friction forces couldthen be described by Coulomb friction. This module will allow anisotropic Coulomb frictionaccording to the model of Micklethwait (Micklethwait 1944). By defining the coefficient of friction in both the longitudinal and the transverse direction, and , respectively, thefriction forces could easily be dealt with analytically, see Figure 3.10. When relative slidingexists, then friction is developed component by component in the longitudinal and transversedirection. The fraction of longitudinal and transverse sliding is evaluated, and

, respectively. The same fractions of maximum friction forces will be obtained

(3.3)

The module treats motion with the concept of spin poles (instantaneous centre of rotation).The track is considered to rotate about an axis, normal to the plane of its motion. This axis is

Figure 3.9 Change in support load due to wheel lift.

Figure 3.10 An anisotropic friction model (Micklethwait 1944).

Gn----G

L 2 ⁄

αF L

∆k q F L αsin= ∆k q

k qG

C G L 2 ⁄

q n= q 1=

µ L µT

v L vto t ⁄ vT vto t ⁄

F L µ LGv L

vto t

--------=

F T µT GvT

vto t

--------=

Direction of motion

Direction of resulting

frictional force

µT

µ L

Friction force ellipse




15

called the instantaneous axis of rotation. The intersection of this axis with the ground plane isthen the instantaneous centre of rotation or spin pole. There are no restrictions on the locationof the spin pole. Knowing the normal loads and the coefficients of friction, the resulting actionof traction is uniquely defined.

The position of the track module is defined by a single point C located in the ground plane atthe longitudinal centre of each track assembly, see Figure 3.11. This point is the origin of thelocal coordinate system, which defines the orientation of the track and is fixed to the trackassembly. On the track assembly there are track pads that move relative to the assembly withthe velocity of .

The sliding motion of a track pad relative to the ground is obtained in terms of absolute andrelative motion, as was described earlier. Translational velocities are defined positive along thepositive axis in the local coordinate system at C . The relative velocity, , is defined positivefor the assembly motion relative to the pad motion, which is the normal case at driving in theforward direction. The motion of the reference point C of the assembly is absolute, as well asthat of the pad. At the reference point C is then obtained:

(3.4)

This corresponds to the velocity of a track pad relative to the ground positioned in the point C .With the absolute velocities , and , the spin pole location could be calculated,see Figure 1.2.

At a point A on the track, which could be anywhere along the track, there is no lateralvelocity, but only a longitudinal velocity, see Figure 3.11. The distance along the track fromthe centre C to the point A is

(3.5)

Figure 3.11 Geometrical definitions of the track module.

S

L

y L

2---

Oe x

e y

v pa d

y

x

ω z

ω z

C

A

uq

L

2---

vre l

vre l

v pad x, vassembly x, vre l–=

v pa d x, v pad y, ω z

e x

v pad y,

ω z

--------------–=






17

derived forces and moment (equations (3.9), (3.10) and (3.11)) could then be rearranged givingthe longitudinal friction force

(3.12)

The transverse force perpendicular to the track will then be

(3.13)

The moment which is created by the transverse components of friction forces, will be redefinedfrom point A to C , yielding

(3.14)

Here is the transverse force in a specific point q and the moment arm is

(3.15)

This gives the total moment

(3.16)

acting about the point C .

3.1.2 DRIVE SHAFT - TRACK MODULE INTERACTION

This interface does not have any degree of freedom, only the demand of equilibrium. The driveshaft interface has to allow for the equilibrium between the longitudinal track forces and thetorque from the drive shaft. This equilibrium must not include vertical forces like normal load.The local longitudinal forces on the track are in equilibrium with the torque at the drive shaft,giving

(3.17)

where refers to the total longitudinal force in the track, equation (3.12), and is thevirtual diameter of the driving wheels. The fictitious driving wheel diameter includes theheight of the track links. The internal frictional losses are disregarded in equation (3.17).

F L µ Lk qG=v pad x,

v pad x,( )2 2q 1–( )n 1–( )---------------- 1–

L

2---ω z v pad y,+

2

+

--------------------------------------------------------------------------------------------------------

q 1=

n

∑

F T

F T µT k qG

2q 1–( )n 1–( )

---------------- 1– L

2---ω z v pa d y,+

v pad x,( )2 2q 1–( )n 1–( )

---------------- 1– L

2---ω z v pad y,+

2+

--------------------------------------------------------------------------------------------------------q 1=

n

∑=

M F Tqr q=

F Tq r q

r q L 2q n– 1–( )

2 n 1–( )--------------------------------=

M µT k qG=

2q 1–( )n 1–( )

---------------- 1– L

2---ω z v pad y,+

v pad x,( )2 2q 1–( )n 1–( )

---------------- 1– L

2---ω z v pad y,+

2+

-------------------------------------------------------------------------------------------------------- L 2q n– 1–( )

2 n 1–( )--------------------------------

q 1=

n

∑

M drive F L Ddrive 2 ⁄ =

F L Ddrive



DAG THUVESEN

18

3.1.3 CHASSIS - TRACK MODULE INTERACTION

The last interface, the chassis track interface, must at least fulfil the total requirements of thesix degrees-of-freedom motion of the chassis. The degrees of freedom that are not lockedanywhere else must be in this interface. These degrees of freedom are the translations in the xand the y direction (forward and lateral) as well as the yaw and roll, the rotation about the x

and z axis. This means that the track must not move relative to the chassis in these fourdirections. In other words, the track assembly has to be fixed to the chassis in all but twodirections. These directions are the rotation about the y-axis (pitch) and the vertical translationin the z direction. This motion has to be reduced by means of some stiffness and damping toshow the real performance of a chassis in these two directions. This is accomplished by usingproper force generators and constraints.




19

4 SPECIAL NUMERICAL CONSIDERATIONS

The assumptions used in the previous chapter, rigid ground and track, imply a specialcondition for the existence of unique relationships between relative motion track-ground andrelated tractive action. Spin motion must be present to release frictional forces under controlled

slip conditions, i.e.,(4.1)

which will result in a unique definition of the location of the spin pole according to Figure 1.2.For any spin pole location, unique traction effort is defined according to equations (3.9), (3.10)and (3.11), and this defines the normal conditions prevailing when steering tracked vehicles.However, the special condition

(4.2)

may also occur, which requires other ways of determining the still existing tractive effort

between ground and track.Contrary to controlled slip conditions at normal steering conditions, two special slip

conditions may be identified:• zero slip• uncontrolled macro slip

According to the assumptions made, zero slip conditions occur when driving straight ahead.Then the maximum static friction is not utilized, which, in the case of isotropic friction, means

(4.3)

or modified to anisotropic friction

(4.4)

Uncontrolled macroslip means maximum utilization of friction with uncontrolled sliding inone direction only over the entire track assembly. This will most likely occur when brakingvery heavily and means

(4.5)

or modified to anisotropic friction

(4.6)

Uncontrolled macroslip is described by an infinitely distant spin pole location.

There are different ways of handling the problem of multiple definition conditions. One waywould be to alter the vehicle model by exchanging the force generators in the ground plane toconstraints when . The motion in the ground plane would, instead of generating forces,be constrained, and the calculated constraint forces would define the utilized friction forces,which must fulfil the condition of equation (4.3) or equation (4.4). Another alternative would

be to define a motion-dependent coefficient of friction, which conceptually allows somedistributed shear elasticity in the track-ground contact.

ω z 0≠

ω z 0≡

F L2 F T

2+ µG<

F L

µ L

------

2 F T

µT

------

2

+ G<

F L2 F T

2+ µG≡

F L

µ L

------

2 F T

µT

------

2

+ G≡

ω z 0≡



DAG THUVESEN

20

In the present module the second alternative is chosen. This implies that the equations fordefining the friction forces between the track and the ground are always the same; however,they are numerically reconstructed to produce the desired effects. The dependency of thefriction as a function of the sliding velocity relative to the ground is complicated. The modulecould be provided with different models of this sliding dependent friction; however, a simple

and versatile model has been chosen, see Figure 4.1.

Figure 4.1 Simple friction model, dependent on the relative sliding

velocity.

µ L µT ,( )

Sliding velocity

Utilized friction

vT v L,( )




21

5 IMPLEMENTATION OF THE TRACK MODULE

The implementation of a track module for MBS software is carried out in steps. The limits of atrack are established based on the concepts described. The module could be created in variousways depending on which MBS program it should fit to. The equations for the discrete normal

load distribution and the friction forces could either be written in FORTRAN or C code andcompiled with the main program, or defined directly by functions available in the MBSprogram.

The module must also contain some parts, force generators and constraints to resemble thedescribed track module. There are various ways of composing the track module. The module inFigure 5.1 is one suggestion. It is modelled in ADAMS, but could be created in any MBSprogram.

The ground contact is modelled by two constraints. An inplane joint is used to lock the

vertical translation and a perpendicular joint to lock the rotation about the transverse axis.There are two constraints between the track body and the chassis. There is one inline joint thatremoves two rotations. It allows rotation about the transverse axis, i.e., the pitch axis. A second

joint, a parallel joint, is used to remove two translations and here the vertical translation isallowed.

There are also four force generators within the module. In the ground plane a general force,which is a six-component force, defines the friction forces. Only three components of thegeneral force are defined as frictional components, the other three are being zero. Atranslational spring damper between the track body and the chassis resembles the verticalsuspension system. A rotational spring damper between the chassis and the track body allows

for the suspension about the pitch axis. The last force generator is a three-component torquebetween the track body and the chassis, which produces a driving torque on the track and

Figure 5.1 Schematic illustration of the track module modelled in

ADAMS.

Chassis

Track

Inline joint

Ground

Parallel axes joint

Perpendicular joint

Inplane joint

Body

Track module

Translational

Spring damper

Rotational

Spring damper

3 component torque

General force



DAG THUVESEN

22

reaction forces on the chassis. The other two are not applied.

The input to the track module must include:• Track contact length• Number of road wheels

• Coefficients of friction:

• Track assembly mass• Track assembly inertia• Stiffness and damping in the suspension system

A model of a vehicle is created in the program. The model is defined by parts described insection 2.1. If there are constraints or forces within the vehicle, for example between the twoparts of the chassis in an articulated steered vehicle, these constraints and forces must also becreated. The track module is attached to the vehicle to create a complete tracked vehicle model.

The result of the assembly is an analytical model of the tracked vehicle, which may bevisualized as in Figure 5.2. Arbitrary external loads and prescribed partial motion may beapplied and the resulting response is obtained. The motion may be animated and any internalforce may be plotted as a function of time.

Figure 5.2 A model of a tracked vehicle in ADAMS.

µ L µT ,




23

6 VERIFICATION OF THE PROPOSED METHOD OF ANALYSIS

6.1 METHODS OF VERIFICATION

There are various ways of verifying the performance of the developed track module. One way

is to compare experimentally obtained and theoretically predicted data on vehicle response tosome characteristic steering control sequences. Physical experiments could be carried out witheither full-scale or downscaled tracked vehicles.

The verification could be done by full-scale experiments, where a model of a specifictracked vehicle is analysed in the MBS program, and the motion of the physical tracked vehicleis measured as the response to a specified steering manoeuvre. The position of the vehicle anda steering signal have to be recorded as functions of time. The measured steering signal wouldthen be the input to the model of the vehicle.

Another method of verification is to compare the performance of a computer model of atracked vehicle with experimental or analytical data from the literature.

To perform full-scale experiments at the Department of Machine and Vehicle Design wasnot possible due to lack of resources and time. FMV (Defence Materiel Administration)performed full-scale experiments; however, they lacked some important input data.

Complete experimental data is not available in the literature. Due to the problems of obtaining any reference data, the module has instead been compared to the results presented byGerbert and Olsson.

6.2 THE VEHICLE MANOEUVRE BY GERBERT AND OLSSON

Gerbert and Olsson produced numerical results to a limited extent covering two steering

manoeuvres using one specific vehicle with the travelling speed as a parameter.The first manoeuvre studied was a steady state motion in a curve at a constant velocity and

constant speed ratio between the two tracks, see equation (A.20) in Appendix A. This analysiswas performed to determine the steering brake torque and the engine torque required for aspecific stationary circular path motion. The velocity and the speed ratio for the tracks were theparameters used.

The second manoeuvre was the transition from straight ahead to circular path motion, whenthe steady-state steering control command was instantly issued, see Figure 6.1. The enginetorque was constant through the whole curve and set to the same torque as for the stationarymotion. The brake torque was applied as a multiple of the steady-state braking torque needed:

(6.1)

where is the brake torque in the stationary case and is a brake factor . Thebrake factor had to exceed a certain value to reduce the velocity of the brake drum.

This manoeuvre lasted until the speed ratio between the two tracks reached the speed ratio atthe stationary motion.

M b k b M bs

=

M bs k b k b 1>( )



DAG THUVESEN

24

6.3 COMPARISON WITH THE WORK BY GERBERT AND OLSSON

6.3.1 MOTION RESPONSE

The specific motion studied here to compare the prediction from the track module based

analysis to the work by Gerbert and Olsson was the application of full braking when travellingstraight ahead with constant speed. The initial velocity was four metres per second. Theexternal motion resistance was fully neglected and the vehicle data used is stated in Table 6.1.

To perform the comparison, two track modules have been connected to a model of a vehiclechassis. The vehicle data and the data to the track module are modified to a certain extent to fitas an input into the MBS model of the vehicle.

The powertrain which was used by Gerbert and Olsson was a Cletrac transmission, seeAppendix A, page A.6. The whole Cletrac transmission could have been modelled in ADAMSby creating parts, constraints etc. as described in section 2.1., however, it was modelled by aFortran subroutine.

The subroutine keeps track of the torques acting on the transmission from the left and rightside drive shafts, the torque from the engine and the brake torque. Two rotational velocities arecalculated, the steering brake velocity, , and the planet carrier velocity, . The rotationalacceleration, and in equation (A.28) and equation (A.29) in Appendix A, are defined astwo separate differential equations. These equations are integrated by the core software(ADAMS) in parallel with the other equations that define the mechanical system. At fullbraking, , the speed ratio between the two tracks is given by equation (A.21).

The complete vehicle model is given an initial velocity, , of four metres per second.A constant torque from the engine is applied, which is identical to the torque needed atstationary circular motion. A torque is also applied on the brake drum. The brake factor ,given in Appendix A, is the defining parameter for different runs, see Figure 6.2. This

manoeuvre is compared to the manoeuvre by Gerbert and Olsson and the pattern is found to besimilar, see Figure 6.2. The brake factor adjusts the braking torque applied to the brake drum,

Table 6.1 Vehicle data (Gerbert and Olsson 1982)

Vehicle weight = 5000 kg

Track contact length = 1.6 mDistance between tracks = 1.6 m

Centre of gravity location = 1

Height of centre of gravity = 0.6 m

Number of support wheels = 6

Radius of inertia (chassis) = L/3

Carrier shaft inertia = 70 kg

Brake shaft inertia = 0.7 kg

Speed ratio (planetary gears) = 0.2

Speed ratio (motor side) = 2.5

Speed ratio (track side) = 3.2

Driving wheel radius (fictitious) = 0.275 m

Coefficient of friction = 0.8

Length over width ratio = 1.0

Gravitational constant = 9.81

ωb ωc

ωc ωb

ωb 0=

m

L B

c

H

n

r

J c m2

J b m2

U p

U m

U t

r d

µ

υ

g

vvehicle

k b




25

and simulates variants of the basic steering manoeuvre. The brake drum will stop at differenttime instants depending on the brake factor.

The reported analysis started when the braking torque was applied at a stationary straight

Figure 6.1 Steering manoeuvres depending on brake torque (Gerbert

and Olsson 1982).

Figure 6.2 The response to a steering command at four metres per

second travelling speed but with different brake torques.

Indicates full braking

k b 1.3=

k b 1.4=

k b 1.5=




DAG THUVESEN

26

ahead motion. The analysis could alternatively have started earlier, applying a constant enginetorque when the vehicle is at rest. The vehicle is then accelerated straight ahead to somespecific speed, when the braking torque is applied instantly. What then follows, will beidentical to the reported steering response.

The performance of the vehicle is also studied by altering the initial velocity. A constant

brake torque is defined. The torque from the engine is the same as for the stationary motion. Atfour and five metres per second, the vehicle performs a stable manoeuvre but at six metres persecond the manoeuvre is unstable, see Figure 6.3.

Gerbert and Olsson were unable to find any stable solutions for velocities higher than fivemetres per second, which is supported and explained in the present study.

6.4 COMPARISON WITH THE WORK BY JAKOBSSON

Jakobsson studied stationary circular motion. He found from full-scale experiments that thevehicle could travel along a kind of spiral, when circular motion was expected. He explained

that this was the modification of the basic circular motion, when the vehicle travelled along aslope. The original results of Jakobsson are shown in Figure 6.4.

A qualitatively similar motion has also been studied here as shown in Figure 6.5. It wasfound that the deviation of the spiral orientation from the steepest slope depended on thelocation of the centre of gravity of the vehicle relative to the location of the tracks.

Figure 6.3 The trajectories of a vehicle at different velocities.

v x 5 m/s=

v x 6 m/s=

v x 4 m/s=





27

Figure 6.4 Trajectory of a vehicle in a slope (Jakobsson 1947).

Figure 6.5 Trajectory of a vehicle moving in a slope predicted by

ADAMS.

Direction of slope

Direction of slope



DAG THUVESEN

28

6.5 OBSERVED QUALITATIVE AGREEMENT

The reported comparisons demonstrate good qualitative agreement between results from thepresent study and the results available from other authors.

The result by Gerbert and Olsson also exhibit quantitative similarities with the results

obtained here, as is exemplified in Figure 6.6.Complete agreement is very unlike. Until MBS software was introduced, powerful tools for

integration of combined differential-algebraic equations were not generally available.

6.6 INTEGRATION PERFORMANCE

To test the convergence of the solutions from the integrator, the yaw velocity was especiallystudied. A specific motion was analysed, the one illustrated in Figure 6.2. The yaw velocity atthe time when the steering brake was locked was recorded. For the defined motion the time tofull braking was 1.1 seconds.

The integrator used by ADAMS can adjust the time step to obtain results within definedtolerances. To study the dependency of the time step, the integrator was forced to use aspecified time step. Depending on the time step used, the results obtained differed. Figure 6.6shows how the yaw velocity depends on the time step. The results are converging towards0.748 radians per second. The yaw velocity obtained by Gerbert and Olsson with the time stepof 0.1 second was 0.757 radians per second. The difference is about one per cent. Theintegrator in ADAMS did not produce the value 0.757 at any prescribed time step or errortolerance.

Figure 6.7 shows the consumed net integration time at different time steps for the specifiedmotion. However, the duration for the vehicle manoeuvre was ten seconds. If the defaultsettings are used for the integrator, the consumed computer time is 23.3 seconds. Most of that

time was used for the motion response, 23 seconds.




29

Figure 6.6 Convergence of the yaw velocity, , as a function of the

time step, , used for integration.

Figure 6.7 Consumed computer time as a function of the time step used

for integration.

10−4

10−3

10−2

10−1

0.72

0.725

0.73

0.735

0.74

0.745

0.75

0.755

0.76

z

ω

∆ t

s( )

rad s ⁄ ( )

The result by Gerbert and Olsson

ω z

∆t

10−4

10−3

10−2

10−1

0

50

100

150

200

250

300

350

∆t

C P U − t i m

e

s( )

s( )

Integrator default settings






31

REFERENCES

Andersson, B.: Mobility and Steering of Tracked Vehicles, Hägglunds Vehicle AB,Örnsköldsvik, Sweden, 1993.

Bekker, M. G.: Theory of Land Locomotion, The University of Michigan Press, Ann Arbor,

1956.Bekker, M. G.: Off-the-road Locomotion, Ann Arbor, The University of Michigan Press, 1960.

Bekker, M. G.: Introduction to Terrain-Vehicle systems, The University of Michigan Press,Ann Arbor, 1969.

Computer Aided Design Software Incorporated, DADS User’s manual, Revision 7.0,Coralville, Iowa, 1993.

Gerbert, G. and Olsson, K.-O.: On track vehicles running through curves, Joint report,Division of Machine Elements, Chalmers University of Technology, Göteborg, LundInstitute of Technology, 1982.

Haug, E. J.: Intermediate Dynamics, Prentice-Hall, Englewood Cliffs, New Jersey, 1992.

Holmdahl, L.: Anisotropic Dry Friction Models, Lic.Eng. Thesis, Machine and Vehicle Design,Chalmers University of Technology, Göteborg, 1989.

Jakobsson, B.: Styrning av bandfordon (Steering of Tracked Vehicles), Dissertation, ChalmersUniversity of Technology, Göteborg, 1947 (In Swedish).

Kitano, M. and Jyozaki, H.: A theoretical analysis of steerability of tracked vehicles, J. of

Terramechanics, Vol. 13, No. 4, pp. 241-258, 1976.

Kitano, M. and Kuma, M.: An analysis of horizontal plane motion of tracked vehicles, J. of

Terramechanics, Vol. 14, No. 4, pp. 211-225, 1977.

Kitano, M., Watanabe, K., Takaba, Y. and Togo, K.: Lane change maneuver of high speed

tracked vehicles, J. of Terramechanics, Vol. 25, No. 2, pp. 91-102, 1988.Mägi, M.: On Efficiencies of mechanical Coplanar Shaft Power Transmissions, Dissertation,Division of Machine Elements, Chalmers University of Technology, Göteborg, 1974.

Mägi, M.: On Constitutive Equations Used at Analysis of Steering of Tracked Vehicles,Proceedings of the 6th European ISTVS Conference, Vienna, Austria, 1994.

Mechanical Dynamics Incorporated, ADAMS/VIEW User’s Reference manual, ADAMS/

SOLVER Reference manual, Version 8.0, Ann Arbor, Michigan, 1994.

Merritt, H. E.: Some considerations influencing the design of high speed track-vehicles, The

Inst. of Automobile Engineers, pp. 398-430, January 1939.

Micklethwait, E. W.: Soil mechanics in relation to fighting vehicles, Military Coll. of Science,

Cobham Lane, Chertesy, 1944.Murray, M., Canfield, T. R.: Modelling a Flexible-link Power Transmission System,

Proceedings of the 6th International Power Transmission and Gearing Conference,Scottsdale, Arizona, 1992.

Terry, T. W., Jackson, S. R., Ryley, C. E. S., Jones, B. E. and Wormell, P. J. H.: Fighting

Vehicles, Brassey’s, UK, 1991.

Watanabe, K. and Kitano, M.: Study on steerability of articulated tracked vehicles-Theoreticaland experimental analysis, J. of Terramechanics, Vol. 23, No. 2, pp. 69-83, 1986.

Wong, J. Y.: Terramechanics and Off-Road Vehicles, Elsevier Science Publishers, Amsterdam,Netherlands, 1989.

Wong, J. Y.: Theory of Ground Vehicles (2nd Ed.), John Wiley & Sons, 1993.



DAG THUVESEN

32




A.1

APPENDIX A: SUMMARY OF THE PAPER

O N TRACK VEHICLES RUNNING THROUGH CURVES

This is a short summary of one chapter of the unpublished article On track vehicles running

through curves by G. Gerbert and K.-O. Olsson which was written in 1982. The original article

also includes a literature review and different driving transmissions. On the last page of thisappendix, the notation for equations is given.

TRACK BEHAVIOUR IN A CURVE

Figure A.1 shows a track with the contact length L between the track and the ground.

TRACK MOTION AND SLIP

Figure A.2 shows a track running through a curve of the radius R. The centre of the curve is

located at O. The velocity v represents the velocity of the track along the curve. The vectorfrom O and the longitudinal axis of the track are perpendicular in a point A. This point is

positioned at relative to the track and therefore moves along the track at the speed of

.

Figure A.1: Contact length of the track.

Figure A.2: Track motion along a curve.

L

yL 2 ⁄ y L 2 ⁄

ϕ v y L

2---+

y L

2---

L

RO A



DAG THUVESEN

A.2

The angular velocity of the track is then

(A.1)

In [1] it was shown that a track must slide against the ground while running in a curve. Theslip, either a positive or a negative nondimensional quantity, was defined in the following way:• circumferential velocity of the driven wheel w

• sliding velocity along the track ws

• velocity of the track relative to the ground w(1-s)

This gives the velocity of the track

(A.2)

At the point A the sliding velocity ws is directed only along the track. At other points on thetrack there is also a sliding velocity perpendicular to the track. This sliding velocity at a

distance from A is giving the total sliding velocity of

(A.3)

Figure A.3 shows the angular velocity of the track and the sliding velocity ws of the point A.The point S is referred to as the spinpole according to Mägi [2]. S is positioned on the line OA

at a distance e from the track.

(A.4)

In this point there is only a rotational speed relative to the ground.

Figure A.3: Location of the spin pole S.

ϕv y

L

2---+

R---------------=

v w 1 s–( )=

uL 2 ⁄ uL 2 ⁄ ϕ

vs ws( )2 u L

2---ϕ

2+=

ϕ

ews

ϕ------=

ϕ

A

vs

R

ws

e

u L

2---

O

S




A.3

TRACK LOAD

The contact area between the track and the ground is a rectangle. The normal load G isdistributed over this contact area. In reference [1] different distributions are discussed. Here thedistribution called 4-n is chosen which implies that:

all loads are taken up as concentrated forces beneath the n support wheels,

the normal load is evenly distributed on the support wheels giving the support load oneach wheel,

there is a pure Coulombian friction force beneath each support wheel. Here

is the coefficient of friction and is counterdirected to and counteracting the sliding veloc-

ity .

The conditions of the distribution are shown in Figure A.4. A discrete variable defines

the location to the q:th support wheel where

(A.5)

By using the distance the normal load G and the distance e to the spinpole S, the force

between the ground and the track can be calculated.

The traction force along the track is then

(A.6)

The transverse force perpendicular to the track is given by

Figure A.4: Load distribution and geometry.

G n ⁄

F q µG n ⁄ = µ

F q

vs

uq L 2 ⁄

uq L2--- q 1–( ) Ln 1–( )-------------------- yL2------–=

uq L 2 ⁄

G

n----

A e

S

L

F q µG

n

----=

L

n 1–------------

O R

uq

L

2---

y L

2---

2

1

q

n

ϕ

F L

F L µG

n----=

e

e2 uq

L

2---

2+

--------------------------------q 1=

n

∑

F T



DAG THUVESEN

A.4

(A.7)

The moment around the point A on the track is

(A.8)

VARYING GROUND PRESSURE

The load distribution, previously described, has a constant support load . This means that

the resulting load G is located in the middle of the track. To be able to model a less simpledistribution, the position of centre of gravity and all forces in a plane perpendicular to theground and along the vehicle have to be considered. A continuous redistribution of the supportload is desirable. The distribution is affected by ground conditions, the suspension system andthe track tension. A linear distribution would allow the resulting load G to move along thetrack.

Figure A.5 shows the linearly distributed support load . Vertical equilibrium requires

(A.9)

Turning the load line, , around the midpoint gives

Figure A.5: Linearly varying support wheel load.

F T µG

n----

uq

L

2---

e2 uq

L

2---

2+

--------------------------------q 1=

n

∑=

M µG

n----=

uq

L

2---

2

e2 uq

L

2---

2+

--------------------------------q 1=

n

∑

G n ⁄

k qG

G k qG

q 1=

n

∑= k qq 1=

n

∑⇒ 1=

C G L 2 ⁄

k qG

G

n----

G

∆G

k nG k qG

∆G

L

q 1n 2

G n ⁄




A.5

(A.10)

In this equation ∆ is the degree of skewness. There are support forces at q=1 to q=n. At the q:thsupport we have

(A.11)

Equilibrium yields

(A.12)

Eliminating and performing the summation gives

(A.13)

Thus according to equation (A.11)

(A.14)

This equation is valid as long as i.e. which results in

(A.15)

Once is obtained, the frictional forces , and the moment can be calculated.

k 11n--- ∆–=

k n1n--- ∆+=

k q1n--- ∆– 2∆q 1–

n 1–------------+=

C G L

2--- k q

q 1–n 1–------------ L

q 1=

n

∑=

k q

∆ C G 1–( )3 n 1–( )n n 1+( )--------------------=

k q1n--- 1

3 C G 1–( )n 1+

------------------------ n 2q– 1+( )– =

k 1 0> 1

n

⁄ ∆>

C G 1n 1+

3 n 1–( )--------------------+<

k q F L F T M



DAG THUVESEN

A.6

DRIVING TRANSMISSION

The most common principle for steering a tracked vehicle is to drive the tracks at differentvelocities relative to each other. In a transmission with a differential the velocities of the twodriving wheels will change but as long as the engine speed is constant the mean velocity willbe constant.

A transmission that works in this way is the Cletrac transmission, see Figure A.6. Referringto Figure A.6 there are six shafts labelled 1 through 6. There are also a planet carrier c, steering

brakes b and an engine m. The inner driving wheel is indexed i and the outer u. The indexes Athrough H refer to the gears and have tooth numbers through .

While the vehicle runs straight forward, the rotational velocities are

(A.16)

Making a left turn implies that the left steering brake will be applied. This will reduce therotational velocities , and . The speed relationship for the differential is

(A.17)

and for the planetary gear

(A.18)

The mean velocity discussed is given by equation (A.17) and is expressed

(A.19)

By combining the equations and assuming , the speed ratio is

Figure A.6: A differential steering transmission.

za zh

ω2 ω4 ω5 ω6 ωc= = = =

ω5 ω2 ω4

ω4 ωc–

ω2 ωc–

------------------ 1–=

ω2 ωc–

ω5 ωc–------------------

zC

z D-----

z E

zF

----- U p= =

ω4 ω2 2ωc=+

ωb ω5=




A.7

(A.20)

At full braking giving

(A.21)

In the stationary case, equilibrium yields

(A.22)

Analysing the rolling power relative to the carrier giving

(A.23)

and applying the speed relationship in the stationary case given

(A.24)

In the dynamic case, while braking, , the inertia loads present are

(A.25)

(A.26)

where is the reduced inertia of the tracks, the engine the brake drum and the transmission

applied to the carrier shaft, and is the inertia of the transmission, brake drum and tracks

reduced to the brake shaft.

Defining the torque above as positive implies the following signs for the torques.

(A.27)

This gives

(A.28)

(A.29)

where the speed ratios are defined as

U ω4

ω2------

ωu

ωi

------

1 U p 1ωb

ωc

------– +

1 U p 1ωb

ωc

------– –

-------------------------------------= = =

ωb 0=

U 1 U p+

1 U p–----------------=

M c M 2 M 4 M 5+ + + 0=

M 2 ω2 ωc–( ) M 4 ω4 ωc–( ) M 5 ω5 ωc–( )+ + 0=

M 2 M 4–( )U p M 5+ 0=

ωb 0≠

J cωc M c M 2 M 4 M 5+ + +=

J b ωb ωc–( ) M 2 M 4–( )U p M 5+=

J c

J b

M c

M m M c U m ⁄ =

M b M 5–=

M i M 2U t –=

M u M 4U t –=

M u M i+ U t M mU m M b– J cωc–( )=

M u M i–U t

U p------- M b J b ωb ωc–( )+( )=



DAG THUVESEN

A.8

(A.30)

The driving torques and are related to the longitudinal frictional track forcesand by the following

(A.31)

where is the fictitious radius of the driving wheel.

REFERENCES[1] Jakobsson, B.: Styrning av bandfordon. Dissertation, Chalmers University of Technology,

Göteborg 1947 (In Swedish).

[2] Mägi, M.: On efficiencies of mechanical coplanar shaft power transmissions. Dissertation,Division of Machine Elements, Chalmers University of Technology, Göteborg 1947.

U m z H

zG

------=

U t z A

z B-----=

M u M i F Lu

F Li

M u F Lur d =

M i F Lir d =

r d




A.9

NOTATION

Distance to track load centre (non-dimensional)

Traction force

Transverse track force

Frictional force beneath a support q

Track load

Moment of inertia of transmission reduced to brake shaft

Moment of inertia of transmission reduced to carrier shaft

Contact length of track

Turning moment of a track about reference point A

Brake torque

Engine torque

Driving torque on inner and outer track

Centre of curvature

Turning radius of reference point A

Spin pole

Speed ratio of outer track speed over inner track speed

Speed ratio at engine side

Basic speed ratio of planetary gear

Speed ratio at engine side

Spin pole distance transverse track

Brake factorLoad distribution factor

Number of road wheels (support wheels)

Integer

Virtual radius of driving wheel

Slip (non-dimensional)

Road wheel (support wheel) coordinate

Velocity of vehicle

Sliding velocity

Velocity of trackSpin pole distance along track (non-dimensional)

Degree of skewness of track load distribution

Coefficient of friction

Angular coordinate

Derivative with respect to time

Angular velocity

C GF LF T

F qG

J b J c L

M

M b M m M

i M

u,O

R

S

U ωu ωi ⁄ =

U mU pU t e

k bk qn

q

r d

s

u

v

vs

w

y

∆µϕϕ d ϕ dt ⁄ =

ω



DAG THUVESEN

INDICES

Along track

Transverse track

Brake

Planet carrier

Driver wheel

Inner track

Engine

Planetary gear

Support q

Slip, stationary

Track

Outer

L

T

b

c

d

i

m

p

q

s

t

u

Documents

transient steering.pdf