CHARACTERISTICS OF TRACKED VEHICLES ON€¦ · Tracked vehicle steering has been a subject of...
Transcript of CHARACTERISTICS OF TRACKED VEHICLES ON€¦ · Tracked vehicle steering has been a subject of...
HANDLING CHARACTERISTICS OF TRACKED VEHICLES ON
NON-DEFORMABLE SURFACES
by
Chi-Feng Chiang, B.Eng.
A thesis submitted to
the Faculty of Graduate Studies and Research
in partial fùlfillment of
the requirements for the degree of
Master of Engineering
Ottawa-Carleton lnstitute
for Mechanical and Aerospace Engineering
Department of Mechanicd and Aerospace Engineering
Carleton University
Ottawa, Ontario
June 1999
O Copyright Chi-Feng Chiang, 1999
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Abstract
Tracked vehicle steering has been a subject of interest to the off-road vehicle industry
since it was invented. Over the years a large nurnber of models for tracked vehicie steering
have been proposed. In most of the rnodels developed so far. Coulomb's law of fiction is
used in the prediction of the interacting forces between the tracked vehicle and the ground
dunng steering. This means that the full fictional force will be developed as soon as a smdl
relative motion takes place between the track and the ground. However, experimental
evidence has shown that the shear stress developed on the track-ground interface is
dependent upon the shear displacement. and the maximum shear stress will be developed
onIy afier a certain shear displacement has taken place.
A detailed study of the mechanics of tracked vehicle steering on non-defonnable
surfaces. taking into account the effects of shear displacement on the development of shear
stress has been carried out. Expenmental studies on the shear stress-shear dispiacement
relationship for a track link with rubber pad. used in the M l 13 armoured personnel carrier,
on asphah have been conducted. Based on the results of both anaiytical and experimentai
studies, a general theory for the handling of tracked vehicles on non-deformable surfaces has
been developed.
The basic features of the general theory are substantiated by available field measured
data. It is show that predictions of sprocket torquss and moments of turning resistance as a
function of turning radius, based on the general theory, bear a close resemblance to the
available experimental data.
With experimental substantiation of its basic features. the general theory has been
employed to evaluate the effects of major design parameters on the maneuverability of
tracked vehicles on non-defonnable surfaces. It is found that among the design parameters
exarnined. track contact length and tread of the vehicle have significant influence on tracked
vehicle steering while track width and location of center of gravity have less significant
etTec ts.
it is believed that this research makes a contribution to a better understanding of the
handling of tracked vehicles on non-deformable surfaces and that it provides a basis for the
further study of tracked vehicle maneuverability on deforrnable terrain.
ii i
Ackaowleàgements
The author wishes to extend his deepest gratitude to his thesis supervisor, Dr. J.Y.
Wong. for his invaluable guidance and suggestions throughout the study, as well as the
preparation of the thesis.
The assistance provided by Mr. Y.C. Wu in setting up the apparatus for investigating
track link-ground shearing charactenstics is also greatly appreciated.
Finally, the author wishes to thank his colleague. Dr. Peijun Xu. for his assistance in
conducting the experiments.
Table of Contents
Page
Acceptance Sheet
A bs trac t
Ackno wledgements
Table of Contents
List of TabIes
List of Figures
List of Symbols
iv
v
. S .
vi11
xv
xxvi
1. Introduction 1
2. Review of the State-of-Art on the Study of the Handling of Tracked Vehicles 3
2.1 . Steeds' Mode1 3
2.2. Crosheck's Mode1 9
2.3. Kitano's Mode1 10
2.4. Ehlert's Mode1 11
3. A General Theory for the Mechanics of Steering of Tracked Vehicles
on Non-deformable Surfaces 16
3.1. Skid-Steering on Non-deformable Surfaces Including the Effect of
Trac k W idth 16
3.1.1. Shear Displacement of the Outer Track 16
3.1.2. Shear Displacement of the Imer Track 21
3.1.3. Kinetics of Tracked Vehicles during a Steady-State Tuniing Maneuver 23
3.2. Normal Pressure Distribution Under the Tracks 28
4. Simulation of Handling Behaviour of Tracked Vehicles using the Proposed
General Theory
4.1. Basic Design Puameters of the Vehicle Simulated
4.2. Simulation Results
4.3. Simulation of Turning Resistance Moment
4.4. Summary
5. Evaluation of the Effects of Design and Operating Factors on the Handting
of Tracked Vehicles using the General Theory
5.1. Contact Length. L
5.2. Tread of Vehicle. B
5.3. Longitudinal Offset of Vehicle Center of Gravity. c,
5.4. Track Width. b
5.5. Height of Vehicle Center of Gravity, h
5.6. Summary
6. Enperimental Study of the Shearing Characteristics of a Representative
Track Link on Asphalt
6.1. Apparatus for Measuring the Shearing Characteristics
6.2. Analysis of Shear Stress-Shear Displacement Data
6.3. The Determination of Coeff~cient of Friction
7. Simulation of Steering Behaviour of an M l 13 Armoured Personnel Carrier
on Non-deformable Cround Based on Measured Shear Data
7.1. Basic Design Parameters of an M 1 13 Armoured Personnel Carrier
7.2. Simulation Results
7.3. Summary
8. Discussion and Conclusions
8.1. Conclusions
8.2. Future Work
Bibliography
Appendix
-4. The Effects of Design and Operating Factors on the Handling of
Tncked Vehicles under Continuous Trapezoidal Load Distribution
over the Entire Track
B. The Effects of Design and Operating Factors on the Handling of
Tracked Vehicles for Trapezoidal Load Distribution on the Track
Pitch under each Roadwheel
C. The Effects of Design and Operating Factors on the Handling of
Tracked Vehicles for Concentrated Load under each Roadwheel
vii
List of Tables
Table Description
Cornparison of steering models for tracked vehicles
Basic design parameters of the Jaguar tracked vehicle used in the simulation
Sprocket torques of the outer track at various turning radii with a speed
of 7.5 km/h
Sprocket torques of the inner track at various tuming radii with a speed
of 7.5 km/h
Sprocket torques of the outer track at various turning radii with a speed
of 14.3 km/h
Sprocket torques of the inner track ai various turning radii with a speed
of 14.2 kmh
Sprocket torques of the outer track at various tuming radii with a speed
of 2 1.3 km/h
Sprocket torques of the inner track at various turning radii with a speed
of21.3 km/h
Sprocket torques of the outer track at various turning radii with a speed
of 29 km/h
Sprocket torques of the inner track at various turning radii with a speed
of 29 km/h
Basic parameters of the M 1 1 3 tracked vehicIe used in the predictions
Sprocket torques at various turning radii with a vehicle speed of 7.5 km/h for an Ml 13 armoured personnel carrier
Sprocket torques at various tuming radii with a vehicle speed of 14.2 kmlh for an M 1 13 armoured personnel carrier
Sprocket torques at various tuming radii with a vehicle speed of 21.3 km, for an M 1 1 3 armoured personnel carrier
Sprocket torques at various tuming radii with a vehicle speed of 29 km/h
viii
Page
15
30
36
36
37
37
for an M 1 13 armoured personnel carrier 1 04
Sprocket torques at various turning radii with different track-ground contact
lengths at a vehicle speed of 7.5 kmlh under continuous trapezoidal load
distribution over the entire track
Sprocket torques at various turning radii with different track-ground contact
lengths at a vehicle speed of 14.2 km/h under continuous trapezoidal load
distribution over the entire track
Sprocket torques at various tuming radii with different traçk-ground contact
lengths at a vehicle speed of 2 1.3 kmlh under continuous trapezoidal load
distribution over the entire track
Sprocket torques at various turning radii with di fferent track-ground contact
lengths at a vehicle speed of 29 km/h under continuous trapezoidal load
distribution over the entire track
Sprocket torques at various turning radii with difTerent treads at a vehicle
speed of 7.5 kmh under continuous trapezoidal load distribution over the
entire track
Sprocket torques at various turning radii with different treads at a vehicte
speed of 14.2 krn/h under continuous trapezoidal load distribution over the
entire track 123
Sprocket torques at various turning radii with different treads at a vehicle
speed of 2 1.3 km/h under continuous trapezoidal load distribution over the
entire track 124
Sprocket torques at various tuming radii with different treads at a vehicle
speed of 29 km/h under continuous trapezoidal load distribution over the
entire track
Sprocket torques at various tuming radii with different longitudinal CG
offsets at a vehicle speed of 7.5 km/h under continuous trapezoidai load
distribution over the entire track
Sprocket torques at various turning radii with different longitudinal CG
offsets at a vehicle speed of 14.2 kmih under continuous trapezoidal load
distribution over the entire track
Sprocket torques at various turning radii with different longitudinal CG
offsets at a vehicle speed of 2 1.3 km& under continuous trapezoidal load
distribution over the entire track
Sprocket torques at various tuming radii with different longitudinal CG
offsets at a vehicle speed of 29 km/h under continuous trapezoidal load
distribution over the entire track
Sprocket torques at various turning radii with different track widths at a
vehicle speed of 7.5 kmlh under continuous trapezoidal Ioad distribution
over the entire track
Sprocket torques at various tuming radii with different track widths at a
vehicle speed of 14.2 km/h under continuous trapezoidal load distribution
over the entire track
Sprocket torques at various tuming radii with different track widths at a
vehicle speed of 2 1.3 km/h undrr continuous trapezoidal load distribution
over the entire track
Sprocket torques at various turning radii with different track widths at a
vehicle speed of 29 km/h under continuous trapezoidal load distribution
over the entire track
Sprocket torques at various turning ndii with different CG heights at a
vehicle speed of 7.5 km/h under continuous trapezoidal load distribution
over the entire track
Sprocket torques at various turning radii with different CG heights at a
vehicle speed of 14.2 km/h under continuous trapezoidal load distribution
over the entire track
Sprocket torques at various turning radii with different CG heights at a
vehicle speed of 2 1.3 km/h under continuous trapezoidal load distribution
over the entire track
Sprocket torques at various turning radii with different CG heights at a
vehicle speed of 29 km/h under continuous trapezoidal load distribution
over the entire track
S procket torques at various turning radii with different track-ground
contact lengths at a vehicle speed of 7.5 km/h for trapezoidal load
distri bution on the track pitch under each roadwheel
B-2
B-3
B-4
B-5
B-6
B-7
B-8
B-9
B-IO
B-1 1
B-12
Sprocket torques at various tuming radii with diflkrent track-ground
contact lengths at a vehicle speed of 14.2 kmh for trapezoidal load
distribution on the track pitch under each roadwheel
Sprocket torques at various turning radii with different track-ground
contact lengths at a vehicle speed of 2 1.3 km/h for trapezoidal load
distribution on the track pitch under each roadwheel
Sprocket torques at various tuming radii with different track-ground
contact lengths at a vehicle speed of 29 km/h Tor trapezoidal load
distribution on the track pitch under each roadwheel
Sprocket torques at various turning radii with different treads at a
vehicle speed of 7.5 km/h for trapezoidal load distribution on the
track pitch under each roadwheel
Sprocket torques at various tuming radii with different treads at a
vehicle speed of 14.2 km/h for trapezoidal load distribution on the
trac k pi tc h under each roadwheel
Sprocket torques at various tuming radii with different treads at a
vehicle speed of 2 1 -3 km/h for trapezoidal load distribution on the
track pitch under each roadwheel
Sprocket torques at various tuming radii with different treads at a
vehicle speed of 29 km/h for trapezoidal load distribution on the
trac k pitc h under each roadwhecl
Sprocket torques at various turning radii with different longitudinal CG
offsets at a vehicle speed of 7.5 krri/h for trapezoidal load distribution
on the track pitch under each roadwheel
Sprocket torques at various tuming radii with different longitudinal CG
offsets at a vehicle speed of 14.2 km/h for trapezoidal load distribution
on the track pitch under each roadwheel
Sprocket torques at various tuming radii with different longitudinal CG
offsets at a vehicle speed of 2 1.3 krnh for trapezoidal load distribution
on the track pitch under each roadwhee1
Sprocket torques at various turning radii with different longitudinal CG
offsets at a vehicle speed of 29 km/h for trapezoidal load distribution
on the track pitch under each roadwheel
Sprocket torques at various tuming radii with different track widths at a
vehicle speed of 7.5 kmih for trapezoidal load distnbution on the track
pitch under each roadwheel
Sprocket torques at various turning radii with different track widths at a
vehicle speed of 14.2 km/h for trapezoidal load distribution on the track
pitch under each roadwheel
Sprocket torques at various tuming radii with different track widths at a
vehicle speed of 2 1.3 h / h for trapezoidal load distribution on the track
pitch under each roadwheel
Sprocket torques at various turning radii with diflerent track widths at a
vehicle speed of 29 kmh for trapezoidal load distribution on the track
pitch under each roadwheel
Sprocket torques at various turning radii with dii'ferent CG heights at a
vehicle speed of 7.5 km/h for trapezoidal load distribution on the track
pitch under each roadwheel
Sprocket torques at various turning radii with different CG heights at a vehicle speed of 14.2 km/h for trapezoidal load distribution on the track
pitch under each roadwheel
Sprocket torques at various turning radii uith different CG heights at a
vehicle speed of 2 1.3 h h for trapezoidal load distribution on the track
pitch under each roadwheel
Sprocket torques at various turning radii with different CG heights at a
vehicle speed of 29 kmh for trapezoidal load distribution on the track pitch under each roadwheel
Sprocket torques at various tuming radii with different track-ground
contact lengths at a vehicle speed of 7.5 kmfh for concentrated load
under each roadwheel
Sprocket torques at various tuming radii with different track-ground
contact lengths at a vehicle speed of 14.2 kmh for concentrated load
under each roadwheel
S procket torques at various turning radi i with di fferent track-ground
xii
contact lengths at a vehicle speed of 2 1.3 km/h for concentrated load
under each roadwheel
Sprocket torques at various turning radii with different track-ground
contact lengths at a vehicle speed of 29 km/h for concentrated load
under each roadwheel
Sprocket torques at various turning radii with different treads at a vehicle
speed of 7.5 kmh for concentrated load under each roadwheel
Sprocket torques at various turning radii with different treads at a vehicle
speed of 14.2 km/h for concentrated load under each roadwheel
Sprocket torques at various turning radii with different treads at a vehicle
speed of 2 1.3 kndh for concentrated load under each roadwheel
Sprocket torques at varioüs tming radii with different treads at a vehicle
speed of 29 km/h for concentrated load under each roadwheel
Sprocket torques at various turning radii with different tongitudinai CG offsets
at a vehicle speed of 7.5 km/h for concentrated load under each roadwheel
Sprocket torques at various turning radii with different longitudinal CG offsets
at a vehicle speed of 14.2 km/h for concentrated load under each roadwheel
Sprocket torques at various tuming radii uith different longitudinai CG offsets
at a vehicle speed of 2 1.3 km/h for concentrated load under each roadwheel
Sprocket torques at V ~ ~ O U S turning radii with different longitudinal CG offsets
at a vehicle speed of 29 km/h for concentrated load under each roadwheel
Sprocket torques at various turning radii with different track widths at a vehicle speed of 7.5 krnh for concentrated load under each roadwheel
Sprocket torques at various tuming radii with different track widths at a
vehicle sjxed of 14.2 km/h for concentrated load under each roadwheel
Sprocket torques at various turning radii with different track widths at a
vehicle speed of 2 1.3 km/h for concentrated load under each roadwheel
Sprocket torques at various turning radii with different track widths at a
vehicle speed of 29 km/h for concentrated load under each roadwheel
Sprocket torques at various tuming radii with different CG heights at a
vehicle speed of 7.5 km.h for concentrated load under each roadwheel
C- 18 Sprocket torques at various tuming radii with different CG heights at a vehicle speed o f 14.2 km/h for concentrated load under each roadwheel
C- 19 Sprocket torques at various turning radii with different CG heights at a vehicle speed of 2 1.3 km/h tôr concentrated load under each roadwheel
C-20 Sprocket torques at various turning radii with different CG heights at a ~~ehic le speed o f 29 km/h for concentrated load under each roadwheel
xiv
List of Figures
Figure Description
Kinetics of a tracked vehicle during a steady-state turn
Track force ellipse from Micklethwait
Kinetics of tracked vehicles during a steady-state turn.
Fundamentais of tracked vehicle steering considered in Hock's mode1
Kinematics of a tracked vehicle during a turning maneuver
Kinematics of a tracked vehicle during a steady-state tum
A shear curve of a simple exponential fonn
Kinetics of a tracked vehicle during a steady-state turn
Normal pressure distribution
Sprocket torques vs theoretical turning radius for a Jaguar at a vehicle speed
of 7.5 km/h with K = 0.075 m and p = 0.9 during a steady-state turn
Sprocket torques vs theoretical tuming radius for a Jaguar at a vehicle speed
of 14.2 kmh with K = 0.075 m and p = 0.9 during a steady-state tum
Sprocket torques vs theoretical tuming radius for a Jaguar at a vehicle speed
of 21 -3 km/h with K = 0.075 m and C( = 0.9 during a steady-state turn
Sprocket torques vs theoretical tuming radius for a Jaguar at a vehicle speed
of 29 km/h with K = 0.075 m and p = 0.9 during a steady-state turn
Lateral shear stress distribution of the outer track dong the longitudinal
centerline of the track-ground contact area predicted fiom the proposed
general theory at a vehicle speed of 14.2 km/h with different tuniing radii
Lateral shear stress distnbution of the inner track dong the longitudinal
centerline of the track-ground contact area predicted fiom the proposed
general theory at a vehicle speed of 14.2 km/h with different turning radii
Lateral shear stress distnbution of the outer track dong the longitudinal
centerl ine of the track-ground contact area predicted from S teeds' mode1 (K = O m) at a vehicle speed of 14.2 km/h with different tuming radii
xv
Page
5
6
8
13
18
19
23
25
28
32
33
34
35
42
42
Lateral shear stress distribution of the inner track along the longitudinal
centerline of the track-ground contact area predicted fiom Steeds' mode1
(K = O m) at a vehicle speed of 14.2 km/h with different turning radii
Coefficient of turning resistance vs tuming radius for a Jaguar at a vehicle
speed of 7.5 km/h with K = 0.075 m under continuous trapezoidal load
distribution over the entire track
Coefficient of tuming resistance vs turning radius for a Jaguar at a vehicle
speed of 14.2 km/h with K = 0.075 m under continuous trapezoidal load
distribution over the entire track
Coefficient of tuming resistance vs turning radius for a Jaguar at a vehicle
speed of 2 1.3 km/h with K = 0.075 rn under continuous trapezoidal load
distribution over the entire track
Coefficient of tuming resistance vs turning radius for a Jaguar at a vehicle
spsed of 29 kmh with K = 0.075 m under continuous trapezoidal load
distribution over the entire track
Sprocket torques vs tuming radius for a Jaguar at a vehicle speed of
7.5 km% with different track contact lengths during a steady-state twn
under continuous trapezoidal load distribution over the entire track
Sprocket torques vs turning radius for a Jaguar at a vehicle speed of
14.2 km/h with different track contact lengths during a steady-state turn under continuous trapezoidal load distribution over the entire track
Sprocket torques vs tuming radius for a Jaguar at a vehicle speed of
2 1.3 krnh with different track contact lengths during a steady-state turn
under continuous trapezoidal load distribution over the entire track
Sprocket torques vs tuming radius for a Jaguar at a vehicle speed of
29 km/h with different track contact lengths during a steady-state tuni
under continuous trapezoidal load distribution over the entire track
Lateral forces vs tuming radius for a Jaguar at a vehicle speed of
7.5 km/h with different track contact lengths during a steady-state
turn under continuous trapezoidal load distribution over the entire track
Lateral forces vs tuming radius for a Jaguar at a vehicle speed of
1 4.2 km/h with different track contact lengths during a steady-state
\
xvi
turn under continuous trapezoidal load distribution over the entire track
Lateral forces vs turning radius for a Jaguar at a vehicle speed of
2 1.3 kmh with different track contact lengths during a steady-state
turn under continuous trapezoidal load distribution over the entire track
Lateral forces vs tuming radius for a Jaguar at a vehicle speed of
29 km/h with different track contact lengths during a steady-state
turn under continuous trapezoidal load distribution over the entire track
Moments of turning resistance vs turning radius for a Jaguar at a vehicle
speed of 7.5 km/h with different track contact lengths during a steady-state
turn under continuous trapezoidal load distribution over the entire track
Moments of turning resistance vs turning radius for a Jaguar at a vehicle
speed of 14.2 kmh with different track contact lengths during a steady-state
turn under continuous trapezoidal load distribution over the entire track
Moments of turning resistance vs turning radius for a Jaguar at a vehicle
speed of 2 1.3 km/h with different track contact lengths during a steady-state
turn under continuous trapezoidal Ioad distribution over the entire track
Moments of turning resistance vs tuming radius for a Jaguar at a vehicle
speed of 29 krnfh with different track contact lengths dunng a steady-state
turn under continuous trapezoidal load distribution over the entire track
Ratio of tuming resistance moment vs contact length L for a Jaguar at a
vehicle speed of 7.5 kmh with different turning radii during a steady-state
turn under continuous trapezoidal load distribution over the entire track
Ratio of turning resistance moment vs contact length L for a Jaguar at a
vehicle speed of 14.2 km/h with different turning radii during a steady-state
turn under continuous trapezoidal load distribution over the entire track
Ratio of turning resistance moment vs contact length L for a Jaguar at a
vehicle speed of 2 1 -3 km/h with di fferent tuming radii during a steady-state
turn under continuous trapezoidal load distribution over the entire track
Ratio of tuming resistance moment vs contact length L for a Jaguar at a
vehicIe speed of 29 km/h with different turning radii during a steady-state
turn under continuous trapezoidal load distribution over the entire track
Sprocket torques vs tuming radius for a Jaguar at a vehicle speed of
xvii
7.5 krnh with different vehicle treads during a steady-state turn under
continuous trapezoidal load distribution over the entire track
Sprocket torques vs turning radius for a Jaguar at a vehicle speed of
14.2 km/h with different vehicle treads during a steady-state turn under
continuous trapezoidal load distribution over the entire track
Sprocket torque vs turning radius for a Jaguar at a vehicle speed of
2 1 -3 krnh with different vehicle treads during a steady-state turn under
continuous trapezoidal load distribution over the entire track
Sprocket torques vs tuming radius for a Jaguar at a uehicte speed of
29 km/h with different vehicle treads during a steady-state turn under
continuous trapezoidal load distribution over the entire track 57
Lateral forces vs turning radius for a Jaguar at a vehicle speed of 7.5 km/h with different vehicle treads during a steady-state turn under continuous
trapezoidal load distri bution over the entire track 58
Lateral forces vs tuming radius for a Jaguar at a vehicle speed of 14.2 km/h with different vehicle trcads during a steady-state turn under continuous
trapezoidal ioad distribution over the entire track 58
Lateral forces vs turning radius for a Jaguar at a vehicle speed of 21 -3 km/h
with different vehicle treads during a steady-state turn under continuous
trapezoidal load distribution over the entire track 59
Laterai forces vs turning radius for a Jaguar at a vehicle speed of 29 km/h with different vehicle treads during a steady-state turn under continuous
trapezoidal load distribution over the entire track
Moments of tuming resistance vs tuming radius for a Jaguar at a vehicle
speed of 7.5 km/h with different vehicle treads during a steady-state tum
under continuous trapezoidal load distribution over the entire track
Moments of turning resistance vs turning radius for a Jaguar at a vehicle
speed of 14.2 km, with different vehicle treads during a steady-state turn
under continuous trapezoidal load distribution over the entire track
Moments of turning resistance vs turning radius for Jaguar at a vehicle
speed of 2 1.3 kmh with di fferent vehicle treads during a steady-state turn
under continuous trapezoidal load distribution over the entire track
xviii
Moments of tuming resistance vs t m i n g radius for a Jaguar at a vehicle
speed of 29 lun/h with diflkrent vehicle treads during a steady-state tum
under continuous trapezoidal load distribution over the entire track
Ratio of magnitude of the sprocket toques vs vehicle tread B for a Jaguar at
a vehicle speed of 7.5 km/h with different tuming radii during a steady-state
turn under continuous trapezoidal load distribution over the entire track
Ratio of magnitude of the sprocket toques vs vehicle tread B for a Jaguar at
a vehicle speed of 14.2 km/h with different turning radii during a steady-state
tum under continuous trapezoidal load distribution over the entire track
Ratio of magnitude of the sprocket toques vs vehicle tread B for a Jaguar at
a vehicle speed of 21 -3 km/h with different turning radii during a steady-state
turn under continuous trapezoidal load distribution over the entire track
Ratio of magnitude of the sprocket toques vs vehicle tread B for a Jaguar at
a vehicle speed of 29 km/h with different tuming radii during a steady-state
turn under continuous trapezoidal load distribution over the entire track
Sprocket torques vs turning radius for a Jaguar at a vehicle speed of
7.5 km/h with different longitudinal CG offsets dwing a steady-state
turn under continuous trapezoidal load distribution over the entire traçk
Sprocket torques vs tuming radius for a Jaguar at a vehicle speed of
11.2 km/h with different longitudinal CG offsets during a steady-state
turn under continuous trapezoidal load distribution over the entire track
Sprocket torques vs turning radius for a Jaguar at a vehicle speed of
2 1.3 km/h with different longitudinal CG offsets during a steady-state
turn under continuous trapezoidal load distribution over the entire track
Sprocket torques vs turning radius for a Jaguar at a vehicle speed of
29 km/h with different longitudinal CG offsets during a steady-state
turn under continuous trapezoidal load distribution over the entire track
Lateral forces vs tumine radius for a Jaguar at a vehicle speed of 7.5 km/h with different longitudinal CG offsets during a steady-state turn under
continuous trapezoidal load distribution over the entire track
Lateral forces vs tuming radius for a Jaguar at a vehicle speed of 14.2 kmh with different longitudinal CG offsets during a steady-state turn under
xix
continuous trapezoidal load distribution over the entire track 67
Lateral forces vs turning radius for a Jaguar at a vehicle speed of 21 -3 km/h with different longitudinal CG offsets dunng a steady-state hm under continuous trapezoidal load distribution over the entire track 68
Lateral forces vs turning radius for a Jaguar at a vehicle speed of 29 km/h with different IongitudinaI CG offsets during a steady-state t m under
continuous trapezoidal load distribution over the entire track 68
Moments of turning resistance vs turning radius for a Jaguar at a vehicie speed
of 7.5 km/h with different longitudinai CG offsets during a steady-state turn under continuous trapezoidal load distribution over the entire track 69
Moments of turning resistance vs turning radius for a Jaguar at a vehicle speed
of 14.2 kmh with different longitudinal CG offsets during a steady-state h m
under continuous trapezoidal load distribution over the entire track 69
Moments of turning resistance vs tuming radius for a Jaguar at a vehicle speed
of 21.3 km/h with different longitudinal CG offsets during a steady-state turn under continuous trapezoidal load distribution over the entire track 70
Moments of turning resistance vs tuming radius for a Jaguar at a vehicle speed
of 39 km/h with different longitudinal CG offsets during a steady-state turn
under continuous trapezoidal load distribution over the entire track
Ratio of tuming resistance moment vs longitudinal CG offset c, for a Jaguar at
a vehicle speed of 7.5 km/h with different tuming radii during a steady-state
turn under continuous trapezoidal load distribution over the entire track 71
Ratio of tuming resistance moment vs longitudinal CG offset c, for a Jaguar at
a vehicle speed of 14.2 kmh with different tuming radii during a steady-state
turn under continuous trapezoidal load distribution over the entire track 71
Ratio of turning resistance moment vs longitudinal CG offset c, for a Jaguar at
a vehicle speed of 2 1.3 km/h with different tuming radii during a steady-state
turn under continuous trapezoidal load distribution over the entire track 72
Ratio of turning resistance moment vs longitudinal CG offset cy for a Jaguar at
a vehicle speed of29 k m h with different turning radii during a steady-state
turn under continuous trapezoidal load distribution over the entire track 72
Sprocket torques vs tuming radius For a Jaguar at a vehicle speed of
7.5 km/h with different track widths during a steady-state tum under
continuous trapezoidal load distribution over the entire track
Sprocket torques vs tuming radius for a Jaguar at a vehicle speed of
14.2 km/h with different track widths during a steady-state turn under
continuous trapezoidal load distribution over the entire track
Sprocket torques vs turning radius for a Jaguar at a vehicle speed of
2 1.3 km/h with different track widths during a steady-state turn under
continuous trapezoidal load distribution over the entire track
Sprocket torques vs turning radius for a Jaguar at a vehicle speed of
29 km/h with different track widths during a steady-state turn under
continuous trapezoidal load distribution over the entire track
Lateral forces vs turning radius for a Jaguar at a vehicle speed of 7.5 krnh with different track widths during a steady-state tum under continuous
trapezoidal load distribution over the entire track
Lateral forces vs tuming radius for a Jaguar at a vehicle speed of 14.2 kmh with different track widths during a steady-state turn under continuous
trapezoidal load distribution over the entire track 77
Lateral forces vs turning radius for a Jaguar at a vehicle speed of 21.3 kmh with different track widths during a steady-state turn under continuous
trapezoidal load distribution over the entire track
Lateral forces vs t m i n g radius for a Jaguar at a vehicle speed of 29 km/h with different track widths during a steady-state tum under continuous
trapezoidal load distribution over the entire track
Moments of turning resistance vs turning radius for a Jaguar at a vehicle
speed of 7.5 km/h with different track widths during a steady-state turn
under trapezoidal load distribution over the entire track
Moments of turning resistance vs tuming radius for a Jaguar at a vehicle
speed of 14.2 kmh with different track widths during a steady-state turn
under trapezoidal load distribution over the entire track
moments of turning resistance vs turning radius for a Jaguar at a vehicle
speed of 2 1.3 km/h with different track widths during a steady-state turn under trapezoidal load distribution over the entire track
Moments of tuming resistance vs turning radius for a Jaguar at a vehicle
speed of 29 kmh with different track widths during a steady-state turn under trapezoidal load distribution over the entire track
Ratio of tuming resistance moment vs track width b for a Jaguar at a
vehicle speed of 7.5 km/h with different turning radii during a steady-state
turn under continuous trapezoidal load distribution over the entire track
Ratio of turning resistance moment vs track width b for a Jaguar at a vehicle speed of 14.2 km/h with different turning radii during a steady-state
turn under continuous trapezoidal load distribution over the entire track
Ratio of turning resistance moment vs track width b for a Jaguar at a
vehicle speed of 2 1.3 km/h with different turning radii during a steady-state
turn under continuous trapezoidal load distribution over the entire traçk
Ratio of turning resistance moment vs track width b for a Jaguar at a vehicle speed of 29 km/h with different turning radii during a steady-state
tum under continuous trapezoidal load distribution over the entire track
Sprocket torques vs tuming radius for a Jaguar at a vehicie speed of
7.5 km/h with different CG heights during a steady-state tum under
continuous trapezoidal load distribution over the entire track
Sprocket torques vs tuming radius for a Jaguar at a vehicle speed of
14.2 km/h with different CG heights during a steady-state tum under
continuous trapezoidal load distribution over the entire tmck
Sprocket torques vs tuming radius for a Jaguar at a vehicle speed of
2 1.3 km/h with different CG heights during a steady-state turn under
continuous trapezoidal load distribution over the entire track
Sprocket torques vs tuming radius for a Jaguar at a vehicle speed of
29 Iun/h with different CG heights during a steady-state turn under
continuous trapezoidal load distribution over the entire track
Lateral forces vs turning radius for a Jaguar at a vehicle speed of
7.5 km/h with different CG heights during a steady-state turn under
continuous trapezoidal load distribution over the entire track
Lateral forces vs turning radius for a Jaguar at a vehicle speed of
14.2 km/h with different CG heights during a steady-state tum under
xxii
continuous trapezoidal load distribution over the entire track
Lateral forces vs tuming radius for a Jaguar at a vehicle speed of
2 1.3 km/h with different CG heights during a steady-state turn under
continuous trapezoidal load distribution over the entire track
Lateral forces vs turning radius for a Jaguar at a vehicle speed of
29 kmh with different CG heights during a steady-state turn under
continuous trapezoidal load distribution over the entire track
Moments of turning resistance vs turning radius for a Jaguar at a vehicle
speed of 7.5 km/h with di fferent CG heights during a steady-state tum
under continuous trapezoidal load distribution over the entire track
Moments of turning resistance vs tuming radius for a Jaguar at a vehicle
speed of 14.2 km/h with different CG heights during a steady-state turn
under continuous trapezoidal load distribution over the entire track
Moments of tuming resistance vs turning radius for a Jaguar at a vehicle
speed of 2 1.3 km/h with different CG heights during a steady-state tum
under continuous trapezoidal load distribution over the entire track
Moments of turning resistancc vs turning radius for a Jaguar at a vehicle
speed of 29 km/h with different CG heights during a steady-state tum
under continuous trapezoidal load distribution over the entire track
Ratio of turning resistance moment vs CG height h for a Jaguar at a vehicle
sperd of 7.5 km/h with different turning radii during a steady-state tum
under continuous trapezoidal load distribution over the entire track
Ratio of turning resistance moment vs CG height h for a Jaguar at a vehicle
speed of 14.2 km/h with different tuming radii dunng a steady-state tum
under continuous trapezoidal load distribution over the entire track
Ratio of turning resistance moment vs CG height h for a Jaguar at a vehicle
speed of 2 1.3 kmh with different tuming radii dunng a steady-state tum
under continuous trapezoidal load distribution over the entire track
Ratio of turning resistance moment vs CG height h for a Jaguar at a vehicle
speed of 29 km/h with different turning radii dunng a steady-state turn under
continuous trapezoidal load distribution over the entire track
Schematic of the test setup for the measurement of track shear force and shear
xxiii
displacement on the asphalt
Block diagram of the data acquisition system for track shear force-shear
displacernent tester
Shear force-shear displacement curve at normal load 890 lb with sliding
speed O. 100 m/s
Shear force-shear displacement curve at normal load 890 lb with sliding
speed 0.145 m/s
Shear force-shear displacement curve at normal load 1000 lb with sliding
speed 0.100 m/s
Shear force-shear displacement curve at normal load 1000 lb with sliding
speed 0.145 m/s
Shear force-shear displacement c u v e at nomai load 1000 Ib with sliding
speed 0.195 m/s
Shear force-shear displacement curve at normal load 1 ZOO lb with sliding
speed 0.1 00 m/s
Shear force-shear displacement c w e at normal load 1 ZOO lb with sliding
speed 0.145 m/s
Shear force-shear displacement curve at normal load 1200 lb with sliding
speed 0.195 m/s
Shear force-shear displacement curve at normal load 1400 lb with sliding
speed 0.100 m / s
Shear force-shear displacement curve at normal load 1400 lb with sliding
speed 0.1 45 d s
Shear force-shear displacement curve at normal load 1400 lb with sliding
speed 0.195 m/s
The relation of maximum shear force and normal load
Sprocket torques vs tuming radius for an M 1 13 at a vehicle speed of
7.5 km/h with K = 0.0183 m and p = 0.684 during a steady-state tum
Sprocket torques vs tuming radius for an M 1 13 at a vehicle speed of 14.2 krn/h with K = 0.01 83 m and p = 0.684 during a steady-state tum
Sproc ket torques vs tuming radius for an M 1 1 3 at a vehicle speed of
2 1 -3 h / h with K = 0.0 183 m and p = 0.684 during a steady-state turn
Sprocket torques vs tuming radius for an M 1 13 at a vehicle speed of
29 km/h with K = 0.01 83 m and p = 0.684 during a steady-state tum
Lateral force of the outer track vs turning radius for an M 1 13 at a vehicle speed
of 7.5 kmh with K = 0.0 183 rn and p = 0.684 dunng a steady-state tum
Lateral force of the imer track vs tuming radius for an M 1 13 at a vehicle speed
of 7.5 kmfh with K = 0.01 83 m and p = 0.684 dunng a steady-state turn
Laterai forces vs tming radius for an M 1 13 at a vehicle speed of 14.2 krnh with K = 0.01 83 m and p = 0.684 during a steady-state tum
Lateral forces vs turning radius for an M 1 13 at a vehicle speed of 2 1.3 km/h with K = 0.01 83 m and p = 0.684 during a steady-state turn Lateral forces vs tuming radius for an M 1 13 at a vehicle speed of 29 km/h
with K = 0.0 183 m and p = 0.684 during a steady-state turn
Moment of turning resistance of the outer track vs tuming radius for an
M 1 13 at a vehicle speed of 7.5 kmlh with K = 0.01 83 m and p = 0.684 dunng a steady-state turn
Moment of turning resistance of the inner tarck vs turning radius for an
M 1 13 at a vehicle speed of 7.5 kmh with K = 0.0 183 m and p = 0.684
during a steady-state turn
Moments of turning resistance vs tuming radius for an Ml 13 at a vehicle speed
of 1 - 4 2 km/h with K = 0.0 183 m and p = 0.684 during a steady-state turn
Moments of tuming resistance vs tuming radius for an Ml 13 at a vehicle speed
of 2 1.3 krn/h with K = 0.01 83 rn and p = 0.684 during a steady-state tum
Moments of turning resistance vs tuming radius for an M 1 13 at a vehicle speed
of 29 km/h with K = 0.01 83 m and p = 0.684 during a steady-state turn
List of Symbols
slip radius of the i ~ e r track
slip radius of the outer track
track width
tread of vehicle
laterai offset of vehicle CG to the geometrical center of the vehicie
longitudinal offset of vehicle CG to the geometrical center of the vehicle
cone index of the soi1
track numeric of the j" track
coefticient of pull-slip equation
coefficient of pull-slip equation
coefficient of pull-slip equation
coefficient defining the extemal motion resistance
coefficient defining the extemal motion resistance
coefficient of extemal motion resistance
inner track longitudinal force
outer track longitudinal force
lateral force acting on the inner track
laterai force acting on the outer track
longitudinal force acting on the inner track
longitudina! force acting on the outer track
height of vehicle CG
track slip dong the longitudinal direction
shear displacement
shear displacement of the inner track
xxvi
shear displacement of the outer track
X direction of shear displacement on the inner track in contact with the ground
X direction of shear displacement on the outer track in contact with the ground
Y direction of shear displacement on the i ~ e r track in contact with the ground
Y direction of shear displacement on the outer track in contact with the gound
track exponent
shear deformation parameter
trac k-ground contact length
track pitch
turning moment acting on the imer track
tuming moment acting on the outer track
moment of turning resistance acting on the inner track
moment of tuming resistance acting on the outer track
moment of turning resistance
normal pressure distribution
normal pressure distribution on the inner track
normal load distribution under the i" roadwheel and j* track
normal pressure distribution on the outer track
pitch radius of the sprocket
turning radius of the vehicle
lateral distance between the center of turn and vehicle CG
extemal motion resistance of j" track
turning radius at which both track forces are zero
external motion resistance
theoretical turning radius
extemal motion resistance on the imer track
external motion resistance on the outer track
xwii
offset of turning center
time
mean speed of vehicle center of gravity
theoretical velocity of the inner track
sliding velocity of the i ~ e r track
sliding velocity of the outer track
lateral sliding velocity of a point (x,. y-) with respect to moving h m e of reference
iateral sliding velocity of a point (x,. y,) with respect to moving frame of reference
longitudinal sliding velocity a point (x,. y,) with respect to moving h e of
re ference.
longitudinal s!iding velocity of a point (x,. y,) with respect to moving frame of
reference
X direction of sliding velocity of a point (x2. y?) with respect to fixed fkme of
re ference
X direction of sliding velocity of a point (x,. y,) with respect to fixed fhme of
re ference
Y direction of sliding velocity of a point (x?. yi) with respect w fixed frame of
re ference
Y direction of sliding velocity of a point (x,. y,) with respect to fixed fi-ame of
re ference
theoretical velocity of the outer track
relative velocity of point ot, in y, direction with respect to O,
sliding velocity of point O[, along the longitudinal direction
absolute velocity of point O, in y, direction
absolute velocity of point O, in y, direction
vehicle weight
normal load acting on the inner track
normal load under the ilh roadwheel and jh track
xxviii
CL'
normal load acting on the j" track
normal load acting on the outer track
angle defining the direction of centrifuga1 force
angle between the resulting sliding velocity and the x, axis
angle between the resulting sliding velocity and the xZ axis
direction angle of shear force acting on the inner track
direction angle of shear force acting on the outer track
goodness-O f- fi t
yaw angle
angular speed of the vehicle
coefficient of fiction
coefficient of fiction along lateral direction
coefficient of fiction along longitudinal direction
coefficient of turning resistance
maximum coefficient of turning resistance
shear stress
calculated sprocket torque
shear stress on the imer track
measured sprocket torque
maximum shear stress
shear stress on the outer track
rotating speed of the sprocket driving the imer track
rotating speed of the sprocket driving the outer track
xxix
Chapter 1
Introduction
Tracked vehicles are widely used in construction, agriculture and military operattions.
While many studies (Bekker. 1956. 1960: Wong. 1989. 1993) on the mobility of tracked
vehicles have been perfonned. research on the handling behavior of tracked vehicles, has not
yet received the sarne level of attention.
Handling characteristics of tracked vehicles are quite different fiom those of wheeled
vehicles. Among the various methods that can accomplish the steering of a tracked vehicle,
such as skid-steering. steering by articulation. and curved track steenng, skid-steering is the
most widely used. In skid-steering. the tractive force of the outer track is increased and that
of the inner track is reduced to create a turning moment to overcome the moment of turning
resistance due to the skidding of the tracks on the ground and the rotational inertia of the
vehic te in gaw (Wong, 1993). Since the moment of tuming resistance is usually considerable,
braking of the imer track is ofien required in making a tum. This results in a reduction in the
forward thrust. Furtherrnore, skidding of the track on defonnable terrain causes additional
sinkage. Over weak terrain. these factors combined often lead to irnmobilization.
Skid-steering has been a subject of research for a considerable period of tirne. Steeds
( 1 950) exarnined the mechanics of skid-steering on rigid surfaces. His pioneering work laid
the foundation for the subsequent studies of this topic. In recent years* Weiss (1971), Kitano
et al. (1 976. 1977, 1992), and Ehlert et al. (1992) have expended considerable effort on the
fùrther study of the mechanics of steering of tracked vehicles, and a number of models have
been developed. It should be noted however, that in most of the previous studies on s t e e ~ g ,
the shear stress developed between the track and the ground is assumed to obey Coulomb's
law of friction. This implies that the full shear stress will be developed as soon as a srnall
relative motion between the track and the ground takes place. Experimental evidence (Wong,
1989. 1993) has shown that, the shear stress developed on the traçked vehicle-ground
interface is dependent upon the shear displacement. This means that the shear stress will
reach its maximum value T~, only after a certain shear displacement has k e n developed.
This research focuses on ihe detailed examination of the mechanics of vehicle-terrain
interaction on non-deformable surfaces during tuming maneuvers, taking into account the
effects of shear displacement on the developrnent of shear stress. Experimental studies have
been carried out to measure the shearing characteristics of representative track Iinks of
armoured personnel carriers on asphalt in the laboratory. Based on the results of both
analytical and experimental studies. a computer mode1 for simutating the steering behaviour
of tracked vehicles under steady-state conditions over non-defonnable surfaces has been
deveIoped. It is believed that this study makes a contribution to a better understanding of the
handling behaviour of tracked vehicles on non-deformable surfaces, and provides a basis for
the further study of tracked vehicle maneuvembility on deformable terrain.
C hapter 2
Review of the State-of-Art on the Study of the Handling of Tracked Vehicles
As mentioned previously. this research focuses on the handling characteristics of
tracked vehicles on non-deformable surfaces. As the forces and moments generated on the
track-ground interface control the tuming behavior. the study of the interaction between the
track and the ground is of importance to the understanding of the handling behavior.
Previous studies on tracked vehicle steering will be reviewed.
2.1. Steeds' Mode1
Steeds (1950) was one of the pioneers who conducted a systematic study of skid-
steering of tracked vehicles. Most of the subsequent studies on this topic were based on his
approac h. A brief outline of his method of approach is given below.
A. Kinematics
Consider a tracked vehicle in a tuming maneuver at low speeds with centrifuga1 force
negiected. as shown in Fig.2- 1. If the hull is turning about the center O with an angular speed,
6 . then the mean speed. V' of the vehicle a$ center of gravity (CG) is given by
V = R&
where R is turning radius.
Since the tracks are attached to the vehicle hull. the tracks are also rotating with the
3
sarne angular speed, 6 . If a track develops a fonvard thmst. then it will slip backward with
respect to the ground. The magnitude of the sliding velocity of the outer track, Vjo, can be
expressed by :
vI0 =a,$ (2-2)
where a, is called the slip radius of the outer track. which is the distance fiom the centerline
of outer track to the instantaneous center O' of that part of the outer track in contact with the
ground. 0' must lie on the line passing through the vehicle Nming center and perpendïcular
to the longitudinal centerline of vehicle hull. The instantaneous center O' is in fact the center
about which the part of the outer track in contact with the ground rotates. Consequently, the
direction and magnitude of the sliding velocity on the ground of a point on the outer track
c m be determined from the instantaneous center 0'. For instance, the direction of the sliding
velocity is perpendicular to the line connecting the point in question and 0' and its sense is
determined by the direction of the angular velocity. 6 . as s h o w in Fig.2-1. The magnitude
of the sliding velocity is equal to the product of 6 and the distance between the point in
question and 0'. Note that when the track skids forward relative to the ground. the sliding
veiocity. V,,. is negative (Le.. skid). and the instantaneous center 0 will lie between the
centerline of the track and the vehicle turning center, as s h o w in Fig.2-1. In this case, the
inner track develops a braking force.
From the above analysis, it can be seen that the speed of point Q on the outer track in
contact with the gound relative to the coincident point on the vehicle hull is (R + ~ / 2 ) 4 +
V,,. where B is the tread of the vehicle (i.e. the distance between the longitudinal centerlines
of the left- and right-hand tracks). This is equal to the theoretical speed of the outer track ro,,
where r is the pitch radius of the sprocket. and o, is the rotating speed of sprocket driving the
outer track. This is expressed by
Similady, for the i ~ e r track. the corresponding equation is
r q = ( R - B / ~ - ~ , W (2-4)
where a, is the slip radius of the imer track, assurning that the imer track skids, Le. sliding
fo ward .
Fig.2-1 Kinernatics of a tracked vehicle during a steady-state him.
B. Kinetics:
In analyzing the forces and moments generated by track-ground interaction, there are
three assumptions in Steeds' model:
1. The ground is non-deformable and the interacting forces between the tmck and the
ground obey Coulomb's law of friction: that is. the fictional force is proportional to the
normal load and acts in the opposite direction of the relative motion of the track with
respect to the ground as shown in Fi@-2. If the friction coefficient in the track
longitudinal direction, h. is different from that of in the lateral direction, pK, that is the
fiction is anisotropic. then the direction of the resultant shear force is modified as
suggested by Micklethwait (1944) and shown in Fig.2-2.
Direction of resu l tan t shear force
according to Y Micklethwait
Direction of resultant shear force according to
law
P ~ P ~ Y (p is normal load per unit track length)
Fig. 2-2 Track force ellipse from Micklethwait.
2. Normal pressure distribution on the track-yound contact area is considered to be
uni form.
3. The width of the track is neglected.
Based on Coulomb's law of isotropie friction, the forces and moments developed by
the track can be formulated as follows (see Fig.2-3). It should be noted that when the effect
of centrifuga1 force is taken into account. the tuming center O is at a distance, %, fiom the
center of gravity in the longitudinal direction. as shown in Fig.2-3, in order to satisfy the
condition of dynamic equilibrium in the lateral direction.
where Wo .Wi, F,,,. FKi. F,,. F?,. Mer, MTi. Ôo. and 6i are the normal loads, lateral forces,
longitudinal forces. moments of turning resistance. and direction angles of shear forces
acting on the outer and inner tracks. respectively (subscript O indicates outer track, and i
indicates inner track). p is the coefficient of friction. L is the track-ground contact length,
and so is the offset of turning center frorn the transverse centerline of the track-ground
contact area (the center of gravity is assumed to be located at the center of the track), as
shown in Fig.2-3.
As a result. under a steady-state turn the following equilibriurn equations can be
O btained.
W' WV' J R L ~ ~ F,, + F,, = - Co$ =
gR B ~ '
where R is the extemal motion resistance. and j3 is the angle defining the direction of the
centrifuga1 force with respect to the x-axis, as shown in Fig.2-3.
In Steeds' model. a set of monographs was provided, and the iteration method was
suggested to obtain the three unknowns &. ai and s,. Although, the model can be extended to
anisotropic case if coefficients of m'ction dong the longitudinal and lateral directions are
known. isouopic Coulomb fnction was used in the study. It should also be noted that there
was no measured data to support his analysis.
Y
Fig.2-3 Kinetics of tracked vehicles during a steady-state tum.
2.2. Crosheck's Model
Based on the study of skid-steering by Steeds, Crosheck (1975) provided a model
which is claimed to be applicable to deformable surfaces. in his model, the following
assumptions were made:
1 . The fiction between the track and the ground is anisotropic as suggested by Micklethwait.
The coefficient of friction. pi,. along the longitudinal direction is expressed by
where El. Ez and E; are coefficients for defining the longitudinal coefficient of friction,
derived from the pull-slip test of the track. and are 0.95. -0.1, and -1.0 for the particular
track on the specific terrain cited in the reference (Crosheck. 1975), respectively, and i, is
track slip along the longitudinal direction. In the lateral direction, full skid was assumed.
Therefore. the lateral coefficient of friction. p,. is equal to the value of H, at iy = 1. Cnj is
the track nurneric of the j" track and is expressed by
ClbL C", = -
*, where C, is the cone index of the soil. b is the track width, L is the track contact length,
Wj is the normal load acting on the j" track.
2. Triangular normal pressure distribution along track-terrain interface is considered.
3. The motion resistance is expressed by
where and E4 and Es are 0.45 and 0.045. for the particular track on the specific terrain
cited in the reference (Crosheck. 1 975). respectively.
4. The w-idth of the track is neglected.
Based on these assumptions and following the sarne approach to kinetic analysis as that
of Steeds. the forces and moments in a steady-state nirn can be detemined. In this model,
bulldozing effect due to skidding of the track on the terrain m d actual normal pressure
distribution have not k e n taken into consideration. In addition. no cornparison with
esperimentai resul ts was provided.
2.3. Kitano's Model
Similar to Crosheck's analysis. Kitano (1 976. 1977, 1992) provided a model for
predicting steering behavior of a tracked vehicle on level ground. and was considered to be
useful by Ehlert et al. (1992). In this model. the following assumptions were made:
1. The friction between the track and the ground is Coulomb friction, and it is anisotropic as
suggested by Micklethwait. The coefficients of friction dong longitudinal and lateral
directions. p? and p,. respectively. were derived from pull-slip relation for the track as
follows:
where El = 0.44. and Ez = 20.0 for a representative hard ground.
2. Normal load is concentrated under each road wheel.
3. Transverse distribution of normal load per unit track width is assumed to be of
trapezoidal form, which can be expressed by
1 -5 w, At middie of the track. p,, (O) = -
b
and at the edges.
where pij and Wu are the normal load F r unit width and normal load under the i'
roadwheel and j track. respectively.
4. Extemal motion resistance of the jth track. R,. is assumed to be proportional to normal
Ioad.
R I =f;W, (2- 1 9)
where f, and W, are the coefficient of extemal motion resistance and normal load on the
jIh tnck. respectively.
Kitano's model is usefùl in predicting the steeiing behavior of tracked vehicles on hard
ground and is substantiated by Ehlert et al. ( 1992) from the field measurement.
2.4. Ehlert's Model
In order to establish an adequate simulation model for tracked vehicle handling, three
different analytical models. including Kitano's model, were investigated by Ehiert et al.
( 1992). After certain modifications and improvernents. the results were verified by field
measurements. primarily on hard surfaces (Schmid. 1984). Although Kitano's mode1 offers
the possibility of predicting the tuming radius for a given set of inputs, it is time-consurning
and is not suitable for on-line test stand simulation. As a result, a model based on the
assumptions similar to those originally proposed by Hock (1970) was developed by Ehlert et
al. ( 1 992).
1. Center of gravity is located at the center of the vehicle.
2. Ground pressure on the track-ground contact area is uniformly distributed.
3. Outer track longitudinal force. F,. imer track longitudinal force, Fi, and moment of
turning resistance. M,. are predicted without considering the effects of centrifuga1 force,
and they are expressed by (as s h o w in Fig.2-4.)
W - L F, =-•
4B P w
W - L M W =-.
4 Ci,
where p, is the coefficient of turning resistance and is a iünction of theoretical tuMng
radius. Rh. which is the tuming radius if track slip is equal to zero. The parameters p,
and Rrh. are defined as
where p,,, is the maximum coefficient of tuming resistance. B is the tread of the vehicle.
RI. is the t m i n g radius at which both track forces are approaching zero, k is called track
exponent and is a function of Rb. and Vo and Vi are theoretical speeds of outer and imer
tracks when track slip is equal to zero. respectively. It should be pointed out that k is an
ernpirical parameter and is derived from experimental data.
4. Extemal motion resistance is a function of the theoretical tuming radius, Rh.
5. The radius enlargement factor, which is the ratio of actual turning radius, R, and
theoretical turning radius. Rih (as shown in Fig.2-4(c)). is taken to be 1.8.
The basic features of the mode1 have k e n substantiated by experimental data obtained
on non-deformable surfaces and is quite accurate in the prediction of sprocket torques for on-
line simulations.
12
Fig.2-4 Fundarnentals of tracked vehicle steering considered in Hock' s mode1 (Ehlert er al., 1 992).
The basic features of existing models for skid-steering of tracked vehicles are
summarized in Table 2- 1.
From the brief review given above. it can be said that most of the previous studies
focus on tracked vehicle steering on @id surfaces. and are based on simplifjing assumptions,
such as the normal pressure on the track k i n g either uniformly distributed or concentrated
under the roadwheels, and the negligible effects of track width on the moment of tuming
resistance. Furthemore. in most of the models developed, it is assurned that during steering
the shear forces on the track-ground interface obey Coulomb's law of friction. This impiies
that full shear force will be developed as soon as a small relative motion between the track
and the ground takes place. As noted previously, experimental evidence indicates that the
shear force developed on the track-ground interface is related to the shear displacement.
In this research. the et'fects of shear displacement on the development of shear force
will be taken into account. The effect on the handling behaviour of major vehicle design
parameters. such as track width. and the location of the center of gravity of tracked vehicles
on non-deformable surfaces will be examined in detail.
Table 2- 1 Cornparison of steering models for trackcd vchiclcs
Factors iiot iiiçluded Mode1
Steeds
[1950]
lrosheck
[ 19753
Kitano
[ 1975, 1977, 19921
Ehlert
il9921
Kiiieiiiatics and kiiictics of skid-steering on deforniable
terrain.
Track width effect.
Features coiisidered
Kitieiiiatics of stcady-statc tiirtiiiig oii level grauiid. Basic mcchanics of skid-steeriiig,
Coulomb friction representation of track-groiind
shearing.
O Uiiiform normal pressure distribution under thc tracks. Centrifugal force and motion resistance.
O Kinematics of steady-state turiiiiig on soft terrain.
O Anisotropic Coiiloiiib friction rcpreseiitatioii of track- groiiiid shearing.
a Location of C.G. of the vehicle, Triaiigular normal pressure distribution under the tracks. Centrifugal force and motion resistance.
O Kiiietics and kineiiiatics of skid-stcering on level ground. O Normal pressure conceiitrated under eacli roadwheel, an(
lateral pressure distribution of trapezoidal form.
O Anisotropic Coulomb friction representation of track- ground shearing.
O Centrifugal force and motion resistance.
0 Track tension.
Uni form ground pressure on track-ground contact area. O Longitudinal tractive forces and turning moments
dependent on the tuming resistance cwficient ~ r , which was a function of theoretical turning radius.
O Extemal motion resistance also a function of theoretical
turning radius.
O Centrifuga1 force.
Kinetics of skid-steering oii
deforniable surfaces. 1 'l'rack width effect.
Rciiiarks
No cxperiiiiental validatioii of analytical flndings.
Iteration method of solution.
No experiineiital validation of
Kinematics and kinetics of skid-steering on defonnable
terrain.
D Kinematics and kinetics of skid-steering on deformable
terrain, D Track width effect
a Experimental results on rigid surfaces provided to verif) simulation data.
Chapter 3
A General Theory for the Mechanics of Steeriag of
Tracked Vehicles on Non-deformable Surfaces
In most of the models described previously. shear force between the track and the
ground is assumed to obey Coulomb's law of friction and the coefficient of friction is either
isotropic or anisotropic. As mentioned previously. experimental evidence (Wong, 1989.
1993) indicates that the shear stress developed on the track-ground interface is dependent
upon the shear dis placement. In addition, the effects of track width on steenng have not k e n
considered in most of the previous studies. In this chapter. a general theory is developed
based on a detailed analysis of skid-steering on non-deformable surfaces. taking into account
the effects of the shear stress-shear displacement relationship and of track width.
3.1. Skid-Steering on Non-deformable Surfaces Including the Effect of Track Width
3.1.1. Shear Displacement of the Outer Track
Consider that a tracked vehicle with track width, b, is in a steady-state tum about O. as
shown in Fig.3-1 and Fig.3-2.
Let O, be the origin of a M e of reference (x,. y,) fixed to and moving with the vehicle
hull and located on the longitudinal centerline of the outer track and at a distance, s,,, fiom
the center of gravity (CG) of the vehicle. as shown in Fig.3-1. The offset distance, so, will be
deterrnined later from the dynamic equilibrium of the vehicle in the lateral direction during a
turn. As the vehicle hull is rotating about turning center O with an angular speed, 6 , the
absolute velocity of O:. V,,,,. in the y, direction can be expressed by
where R' is the lateral distance between the center of turn O and center of gravity of the
vehicle. and is equal to R-Cosp (or JR' - s j ). as shown in Fig.3-1, R is the tuming radius of
the vehicle. c, is the lateral distance between the vehicle CG and loagitudinal centerline of
vehicle hull (or the lateral offset of the center of gravity with respect to the geometrical
center o f the vehicle), and tread. B. is the distance between the centerlines of the outer and
inner tracks.
A point O,, on the outer track in contact with the ground coincident with 0, has a
relative velocity. V,,,,,, with respect to O,. which is expressed by
Vil'Ol = f o o (3-2)
where r is the pitch radius of the sprocket and o, is the angular speed of the sprocket of the
outer track.
As a result. the sliding velocity. V,,,. of point O,, on the ground dong the longitudinal
direction of the outer track is expressed by
Consider an arbitrary point defined by (x,. y,) on the outer track in contact with the
ground. Since the track is rotating with the vehicle at angular speed, 4 . during a steady tum,
the relative velocity components of the point (x,. y,) with respect to ot, in the longitudinal
and lateral directions of the track are given by x ,& and y,& respectively.
Hull
Fig.3-l Kinematics of a tracked vehicle during a tuming maneuver.
Fig.3-2 Kinematics of a tracked vehicle during a steady-state tum.
Based on the above analysis. the sliding velocity of a point defined by (x,, y,) on the
outer track in contact with ground. with respect to the fixed M e of reference (X. Y) c m be
expressed by (Fig.3- 1 )
X component of sliding velocity:
V,,, = -V ,,,, Sin4 + ro,Sin+ - x ,&~in$ - y,&os4
= -[(R'+B/~ +c, + x , ) 4 - ro,]~in$ - y,@os4
Y cornponent of sliding velocity:
V,,, = V ,,,,, C o 3 - ro,Cos4 + x ,(Cos+ - y ,+Sin+
= [(RI+ BI2 + c, + x, )Q - ru, ]COS+ - y ,&sin+
The angle, 4. as shown in Fig.3-1 is the angular displacement of the vehicle and can be
19
determined by the integration of the yaw velocity. 4 . with respect to the time. t, it takes for
the point (x,. y,) to mvel from the initial point of contact at the front of the track (at y, = U2
+ C- - so). that is
and
where c,. is longitudinal distance between CG and lateral centerline of the vehicle hull, or the
longitudinal offset of the center of p v i t y with respect to the geornehic center of the vehicle
(see Fig.3- 1 ).
-4s a result. the shear displacement. jso. at a point (x,, y,) on the outer track in contact
with the ground along the X direction with respect to fixed frame of reference (X, Y) can
therefore be determined by
and shear displacement, jY,. along the Y direction
B [ ( ~ , ' 2 + c , -so
= ( R f + l + c K + x , s i n faO ]-($+cy - S . ) +
The resultant shear displacement. j,. of a point at (x,. y ,) on the outer track in contact
with the ground is given by
3.1.2. Shear Displacement of the Inner Traok
Similarly let o2 be the origin for a frame of reference (x,, yJ fixed to and moving with
the vehicle hull and be located on the longitudinal centerline of the imer track. The absolute
velocity of O,. VeY2. in the y, direction can be expressed by
Following a similar approach described in 3.1.1, the sliding velocity of a point (x,, y 3
on the inner track in contact with the ground with respect to fixed h e of reference (X, Y)
can be expressed by (see Fig.3- 1 )
X component of sliding velocity:
V,,i = -V,2,2Sin+ + ro,Sin+ - x,&sin+ - y , & ~ o s +
= -[(R'-B/Z + c , + x2)& - r o , ] ~ i n + - y ,&os4
Y component of sliding velocity:
= [(RI-8/2 + c, + x 2 ) h - rw , ]~os+ - y , & ~ i n 4
where ai is the i ~ e r irack sprocket angular speed.
However. the contact time, t. for the point defined by (x,, y2) will be equal to (W2 + c, - s, - yL)/raI. and. dt = - dy J rmi. Therefore. the shear displacement, jxi, on the imer track of the
vehicle along the X direction with respect to fixed frame of reference (X. Y) can therefore be
deterrnined by
and shear displacement, jvi. along the Y direction
The resultant shear displacement. ji, of a point at (x', y,) on the imer tmck in contact
with the ground is given by
3.1.3. Kinetics of Tracked Vehicles during a Steady-State Turning Maneuver
As noted above. the shear stress developed on the track-ground interface is dependent
upon the shear displacement. On a non-defomable surface, in most cases the shear stress, T.
initial I y increases rapidly with an increase in the shear displacement, j, and then approaches a
constant value with a M e r increase in shear displacement, as show in Fig.3-3. This type
of shear stress-shear displacement relationship rnay be described by the following
exponential equation:
T=T,,(i-e-' K ) = p - p - ( ~ - e-JJl() (3- 1 7)
where K is the shear deformation parameter, and may be considered as a measure of the
magnitude of the shear displacement required to develop the maximum shear stress, p is the
normal pressure. and p is the coefficient of fiction between the track and the ground. K may
be determined by the intersection of the line tangent to the curve at the origin and the
horizontal line representing the maximum shear stress. as shown in Fig.3-3.
Accordiag to Coulomb's L a w
Actual Sbear Curve
1 ' I I I ' I 3 , -, I
Shear Dispîacement
Fig.3-3 A shear curve of a simple exponential form.
Therefore. the shear force developed on an element dA of the track in contact with the
ground c m be expressed as:
on the outer track d ~ , = s , d ~ = pop(l - e-jO " )- d~ (3-1 8)
on the inner track dF, = =,dA = p,p(l - e - ~ . ' " ) - d ~ (3- 19)
where t,, t,. p, and p, are the shear stress and normal pressure on elements of the outer and
i ~ e r tracks. respective1 y.
It should be noted that the assumption used in Steeds' mode1 that the shear force
between the track and the ground obeys Coulomb's law of friction is a special case of
equation (3-17). Coulomb's law assumes that full frictional force is reached as soon as a
small relative motion between the track and the ground takes place. This is equivaient to the
value of K in equation (3-1 7) equal to zero. This indicates that equation (3-17) is a more
general representation of the charactenstics of the shear force between the track and the
ground. C
As s h o w in Fig.3-4. the longitudinal forces. F,, and Fyi, acting on the outer and imer
tracks relative to the moving Frame of reference. can be expressed respectivAy by
where 6, and S2 are the angles between the resultant sliding velocities of the points on the
outer and inner tracks and the lateral directions of the tracks (Le., x, and x2 axes),
respectively. It should be noted that following Coulomb's law of fiction, the shear force
acting on the track will be in the opposite direction of the resultant sliding velocity.
Fig.3-4 Kinetics of a tracked vehicle during a steady-state tum.
The lateral forces. Fxo and FKi, acting on the outer and i ~ e r tracks, are given
respectively by
The tuming moments, ML, and ML,. acting on the outer and imer tracks respectively,
due to the longitudinal shear forces acting on the outer and inner tracks respectively with
respect to the longitudinal centerline o f vehicle hul1 can be expressed by
The moments of turning resistance. M T ~ and MTi, of the tracks due to the moments of
the lateral forces acting on the outer and inner tracks respectively, with respect to the line
passing through center of tum O and perpendicular to vehicle longitudinal axis c m be
expressed by
In order to determine 6, and 6?. the longitudinal sliding velocities, V,, and V,,,, of the
track elements on the outer and inner tracks with respect to the moving fiames of reference
(xI. y l ) and (x:. y,). respectively. are first calculated as follows (see Fig.3-2)
The iateral sliding velocities. V,, and V,,,. of the track elements on the outer and imer
tracks c m be expressed by
Therefore, the 6, and 62 c m be defined, respectively, by the following equations
26
coso, = vp - y Z i
- J = =z)+-ro,~ +(y,+y
From the above analysis. one can derive the equilibrium equations for the tracked
vehicle during a steady-state tum as follows (Fig.3-4)
CF, = O w'
FK0 + F, + - CosP = O (3-36) !ZR
whsre Rt, and Rt, are external motion resistance on the outer and imer tracks, respectively.
It should be noted that the forces and moments are functions of the theoretical speeds,
ro, and roi, and the offset. so. With the other parameters known or given, such as tread, B,
shear deformation parameter. K. track-gound contact length. L, fonvard speed, V, tuming
radius. R. weight. W. track width. b. longitudinal offset. c,. and lateral offset, c,. of the
center of gravity with respect to geometrical center of the vehicle, CG height, h, coefficient
of motion resistance. f, and coefficient of friction. p. these three unknown parameters, ro,
ru, and so. c m be determined by solving the above three simukaneou equations. Thus al1 the
forces and moments during a given steady-state tum can be completely defined.
3.2. Normal Pressure Distribution under the Tracks
First consider the lateral load transfer of the imer and outer tracks due to the lateral
component of the centrifuga1 force s h o w in Fip.3-j(a). From force and moment equilibrium.
the normal load on the outer <rack. W,. and that on the imer track, Wi, c m be expressed by:
Y=-+ \z [iz2 -- CosP - " 1 B
Wl =-- 2 (BT2 -- COSP - c W ) B
( c )
Fig.3-5 Normal Pressure distribution.
Then consider the longitudinal load transfer due to the longitudinal component of the
centrifuga1 force and the longitudinal offset of vehicle CG as shown in Fig.3-5(b). Assuming
that the normal load distribution under the outer track has the form of a trapezoid, then the
normal pressure under the outer track. p,(y,). is given by
Similariy. normal pressure under the inner track. pi(y2). can be expressed as
If the normaf load is assumed to be supported only by the track link with pitch, L,,
under each roadwheel as shown in Fig.3-j(c). then p,(y,) and piO'z) can respectively be
espressed by
Chapter 4
Simulation of Handling Behaviour of Tracked Vehicles
using the Proposed General Theory
In order to demonstrate the utility of the general theory described above, the steering
behavior of a military tracked vehicle was simulated and the results were compared with
avaiIable msasured data (Ehlert er al.. 19%).
4.1. Basic Design Parameters of the Vehicle Simulated
The basic design parameters of a military tracked vehicle, known as a Jaguar, used in
the simulations are given in Table 4-1 (Foss. 1993-94).
rable 4-1 Basic design parameters of the Jaguar tracked vehicle used in the simulation. - - - - -
vehicle weight, kg
Height of CG, m Sprocket radius, m (Scaled from the drawing)
Track-ground contact length. m Longitudinal CG location from centerline of vehicle hult, rn
Track width. m
Track pitch, m
Tread of vehicle. m
; hearing characteristics
Shear deformation parameter. m Coefficient of friction between the track and the ground, Coefficient of external motion resistance,
The coefficient of friction, p. and shear deformation parameter, K, are dependent upon
properties of the track-ground interface. such as the matenal of tmck pad and the type of
ground. Since the values of p and K were not given in the reference (Ehlert et al.. 1992), the
values used in the simulations were derived fiorn fitting the simulated results to the
rneasured data using a optimization procedure. The goodness-of fit, c, achieved is 88.58%
and is defined by
where 5, and is the rneasured sprocket torque from the reference (Ehlert et al., 1992), N is
the nurnber of data points. and Tc is the calculated sprocket torque based on the model
developed.
4.2. Simulation Results
By solving the three simultaneous equations from (3-36) to (3-38), the circumferential
speed of sprocket on the outer track. ru,. and that on the iriner track, ro;, and the offset, %,
c m be obtained. The rneasured sprocket torque (Ehlen el al., 1992), the predicted data with
different types of normal load distribution under the tracks. as well as the predicted results
from Steeds' model (K = O m) are s h o w in Figs.4-1 to 4-4. Tables 4-2 to 4-4 show a
cornparison of the predictions obtained from Steeds' mode1 and those obtained fiom the
proposed mode1 with the measured data at different fonvard speeds.
: Measured (Speed Range: 5.8 - 9.2 kmh)
- - - - : Continuous trapezoidal load distribution over the entire track (Goodness-of-fit 90.55%)
---O. : Trapezoidal load distribution on the track pitch under
each road w heel (Goodness-of-fit 85.67%) ----. : Concentrated load under each roadwheel
(Goodness-of-fit 82.54%) --- -- : Steeds' mode1 under trapezoidal load distribution (K = O m)
Outer Track
Inner Track
- - - - , - , - - . - - - - - - - - - O - .
l I 1 1 1 1 1 1 I 1 1 I I I I I I I 1 I I ~ I I
1 10 100 1000
Theoretical Turning Radius (m)
Fig.4-1 Sprocket torques vs theoretical tuming radius for a Jaguar at
a vehicle speed of 7.5 kmh with K = 0.075 m and p = 0.9 during a steady-state tum.
: Measured (Speed Range: 1 1.9 -16.5 km&) - - - - : Continuous trapezoidal lord distribution over the
entire track (Goodness-of-fit 88.65%) ---o. : Trapezoidal load distribution on the track pitch under
each roadw heel (Goodness-of-fit 86.17%) ----- : Concentrated load under each roadwheel
(Goodness-of-fit 84.67%) - - - - - : Steeds' model under trapezoidal load distribution (K = O m)
-26000 1 I I I 1 1 I I I I I I I 1 1 1 1 I I I I I I I I
1 10 100 1 000
Theoretical Turning Radius (m)
Fig.4-2 Sprocket torques vs theoretical tuming radius for a Jaguar at a vehicle speed of 14.2 kmh with K = 0.075 m and p = 0.9 during during a steady-state turn.
: Measured (Speed Range: 18.6 - 24.0 kmR)
- - - - : Continuous trapezoidal load distribution over the
entire track (Goodness of Fit 88.3 1 %) ----. . . Trapezoidal load distribution on the track pitch under
each roadwheel (Goodness-of-fit 72.51%) - O * - . 0 . Concentrated load under each roadwheel
(Goodness-of-fit 73.93%) --*--a . Steeds' model uader trapezoidal load distribution (K = O m)
Outer Track
tnaer Track
1 10 f 00 1000
Theoretical Turning Radius (m)
Fig.4-3 Sprocket torques vs theoretical turning radius for a Jaguar at a vehicle speed of 2 1.3 km/h with K = 0.075 m and p = 0.9 uring a steady-state tum.
: Measured (Speed Range: 25.8 - 32.2 k m )
- - - - : Continuous trapezoidal load distribution over the
entire track (Goodness-of-fit 83.96%) -O--. . Trapezoidal load distribution on the t n c k pitch under
each roadwheel (Goodness-of-fit 83.71 %) -O--- . Concentrated load under each roadwbeel
(Coodness-of-fit 83.54%) - - * O - . Steeds' model under trapezoidal load distribution (K = O m)
1 10 100 11000
Theoretical Turning Radius (m)
Fig.4-4 Sprocket torques vs theoretical tuming radius for a Jaguar at a vehicle speed of 29 km/h with K = 0.075 m and p = 0.9 during a steady-state turn.
Table 4-2 Sprocket torques of the outer track at various turning radii with a speed of 7.5 kmh. -- -p
Theoretical
tuming
radius (m)
- -- - -
Predicted by the proposed general theory I I
(kN.m)
rable 4-3 Sprocket t o q u e s of the inner track at various turning radii with a speed of 7.5 kmh.
Theoretical
turning
radius (m)
Predicted by mode I
kN.m
23.898
18.947 23.89 1
12.895 23.88 1
6.71 1 23.870
3.816 23.866
Measured
(kN.m)
error
-9.58%
-26.1%
-85.2'/0
-256%
-525%
Predicted by Steeds mode1
Continuous
trapezoidal load over
the entire track
kN,m
20.8 14
17.875
13.345
7.682
4.860
kN.m 1 error
kN.m
20.6 12
18.149
14.189
8.561
5.462
error
4.56%
-5.66%
-3.59%
-15.5%
727.4%
Predicted by the proposed general theory
Trapezoidal load on
the tract pitch under
each roadwheel
Concentrated load
under each
roadw heel
error
-5.48%
-4.2 1%
+ 10.0%
+27.6%
+43.1%
kN.m 1 error 1 kN.m 1 enor 1 kN.m 1 e m r
- - - - - - -
Continuous
trapezoidal load over
the entire track
L
kN.m
19.156
17.437
13.926
8.522
5.468
error
- 12.2%
-7.97%
+7.96%
+27.0°/0
+43.3%
-
Trapezoidal load on
the mck pitch under
each roadwheel
Concenrrated load
under each
roadwheel
Table 4 4 Sprocket torques of the outer track at various turning radii with a speed of 14.2 kmh.
Predicted by the proposed generai theory 1 1 Theoretical
turning
radius (m)
Measured
(kN.m)
Predicted by Steeds'
mode1
Continuous
trapezoidal load over
the entire track
Table 4-5 Sprocket torques of the inner track a t various turning radii with a speed of 14.2 kmh.
- 3 1 .-- 735
18.947
12.895
6.711
3.8 16
I Predicted by the proposed general theory I
Trapezoidal load on the track pitch wider
each roadwheel
Theoretical
Concentrated load
under each roadwhee l
--
Concentrated load 1
kN.m
23.566
23.825
23.919
23.925
23.916
Continuous Trapezoidal load on Measured Predicted by Steeds'
trapezoidal load over the track pitch under (kN.m) mode1
the entire track eac h roadwheel
error
- 1 1 -0Yo
-25.7%
-85.5%
+256%
-527%
turning
radius (rn)
kN.m
19.709
17.168
11.796
7.347
4.673
under each roadwheel I
error
-7.14%
-9.39%
-0.77%
-9.48%
-22.5%
' k ~ . m error k ~ . m e m r
kN.m
20.1 69
17.956
13.949
8.348
5.324
error
- 16.6%
-3 1.7%
kN.m
19.060
17.375
13.728
8.323
5.339
error
-4.98%
-5.23%
+8.17%
+24.4%
+39.5%
emor
- 10.2%
-8.30%
+6.46% 1
+24.0%
+39.9??
Table 4-6 Sprocket torques of the outer track at various tuming radii with a speed of 2 1.3 kmh. -- -
Predicted by the proposed general theory
Theoretical
tuming
radius (m)
TabIe 4-7 Sprocket torques of the inner track at various tuming radii with a speed of 2 1.3 kmh.
turning
radius (m)
Measured
(kN.m)
1 O
2 O
50
1 O 0
Measured
(kNm)
- 10.658
15.789
12.500
6.7 1 1
3.816
Predicted by the proposed general theory I I
Predicted by Steeds'
mode1
kN.m
23.158
23.78 1
23.963
23.968
Continuous 'redicted by Steeds'
trapezoidal load over mode1
the entire track
error
-46.7%
-90.2%
~ 2 5 7 9 6
-54596
1
kN.m error kN-m error
1
k N m
15.802
1 1.895
6.765
4.294
Trapezoidal load on 1 Concentrated load (
Continuous
trapezoidal Ioad over
the entire track
error
-0.08%
4.84%
-O.80'!4
712.5%
the track pitch under 1 under each I
kN.m
17.254
13.5 14
each roadwheel roadwheel
Trapezoidal load on the mck pitch under
each roadwheel
error
+9-28%
+8.1 1%
kN.m
16.865
13.356
Table 4-8 Sprocket torques of the outer track at various tuming radii with a speed of 29 k d h .
Concentrated load
under each
roadwheel
7.974
5.031
error
+6.8 1 %
~ 6 . 8 5 %
Theoretical
tum ing
radius (m)
1 1 1 Predicted by the proposed general theory 1
i l 8.8% ) 7.962 1 +18.6%
+3 1.8%
. . -
Continuous
trapezoidal load over
the entire track r
5.055
Measured
(kN.m) Predicted by Steeds'
mode 1
Trapezoidal load on
the track pitch under
each roadwheel
Concentrated load
under each
roadwheel
20
50
100
12.895
6.71 1
3.816
kN.m
23.189
23.932
24.012
error
+79.8%
+257%
+529?6
kN.m
11.444
6.543
4.1 13
error
- 1 1.3%
-2.50%
+7.78%
kN.m
13.233
7.777
4.815
kN.m
13.197
7.804
4.836
error
12.62%
+15.9%
+26.2%
error
+2.34%
+16.3%
+26.7%
Table 4-9 Sprocket torque of the inner track at various turning radii with a speed of 29 km/h.
1 1 1 Predicted by the proposed general îheory 1 Theoretical
tuming
radius (rn)
It can be seen fiom Figs.4-1 to 4-4 and Tables 4-2 to 4-9 that the relationships between
sprocket torques and tuniing radius at various forward speeds predicted by the proposed
general theory bear a close resemblance to those measured. whereas the relationships C
predicted by Steeds' model are far from the measured data particularly at larger tuming radii.
This can be explained by the difference in the laterai shear stress distribution predicted by the
proposed general theory and by Steeds' model. as shown in Figs.4-5 to 4-8.
20
5 0
1 00
in Steeds' model. the maximum shear stress is developed whenever a small relative
motion between the track and the ground takes place. As a result. there is a large lateral shear
stress developed even at the front contact point of the track. where a track element just
cornes into contact with the ground. as shown in Figs.4-7 and 4-8. On the other hand, the
pro posed general theory takes into account the s hear stress-shear displacement relationship.
Consequently at the fiont contact point of the track. the shear displacement and hence the
shear stress is zero. as indicated in Figs.4-5 and 4-6. It should also be pointed out that the
lateral shear stress distribution predicted by Steeds' model varies little with the increase in
turning radius. except at very small turning radius. such as R = 10 m shown in Figs.4-7 and
4-8. This is the reason that the sprocket torques predicted by Steeds' model vary Iittle with
Measured (kN.m)
-9.367
4.2 1 I
-1.316
Predicted by Steeds' model
kN.m
-20.200
-2 1.165
Concentrated load under each roadw heel
Continuous
trapezoidal load over the entire track
error
7116%
4 0 3 %
Trapezoidal Ioad on the track pitch under
each roadwheel
kN.m
-8.409
-3.764
error
-10.2%
- 10.6%
kN.m
-10.379
-5.029
-;.196/o -21.261
error
+10.8%
+19.4%
-2.068 -1516OA -1.358
kN.m
-10.362
-5.060
+S7.I%
error
+10.6%
+20.2%
-2.090 +58.80/0
turning radius beyond a certain value. as shown in Figs.4-1 to 4-4. On the other hand, the
magnitude of the lateral shear stress predicted by the proposed general theory decreases with
the increase in t m i n g radius. As a result. the sprocket torques predicted by the proposed
gsneral theory also decrease with the increase in tuming radius.
I t c m . therefore. be concluded that the trac k-ground shearing characteristics.
specifically the shear stress-shear displacement relationship. must be taken into account in
predicting the handling behaviour of tracked vehicles. The normal pressure distribution is
s h o ~ n to have a certain influence on the relationships between sprocket torques and tuming
radius. It appears that among the three types of normal pressure distribution examined, the
continuous trapezoidal tàrm of normal pressure distribution provides the closest agreement
with the measured data.
4.3. Simulation of Turning Resistance Moment
For the computation of the tuming resistance moment. several analytical models were
developed in the past. The basic model is described in Section 2.4. In this model. the
interacting forces and moments between the tracks and the ground can be defined and
espressed in terms of coefficient of turning resistance. p,. as shown in equations (2-20) and
(2-2 1). It should be noted that p, in these equations is considered to be a fhction of track
exponent. k. which varies with turning radius. k is essentially an empirical parameter which
c m only be denved from experimental data (Ehlen et al.. 1992).
As described previously. using the general theory presented in Chapter 3. the moments
of turning resistance for the outer and inner tracks can be predicted analytically using
equations (3-26) and (3-27). respectively. Consequently. the equivalent coeficient of turning
resistance. p,, in equation (2-20c) can be derived quantitatively by equating the sum of the
moments of tuming resistance, Mr, and MTi- calculated fiom equations (3-26) and (3-27) to
.MN in equation (3-20c). The equivalent coeffkient of turning resistance, p, (for C( = 0.9),
using this method as a Cunction of turning radius is shown in Figs.4-9 to 4-12. The equivalent
coefficient of turning resistance. p,. derived from experimental data reported by Ehlert et al.
( 1 992) is also shown in the figures. It was calculated using equation (3-38), with measured
data on sprocket torques and external motion resistance and the calculated centrifuga] force
based on the forward speed and turning radius. It is shown that there is a reasonably close
agreement between the measured md predicted results obiained using the general theory
presented in Chapter 3.
4.4. Summary
It is shoun that there is a reasonably close agreement between the sprocket torques and
moments of turning resistance predicted using the general theory and the available measured
data. The variations of sprocket torques and equivalent coefficient of turning resistance with
turning radius at various forward speeds bear a close resemblance to those measured. On the
other hand. the predictions based on Steeds' mode1 are far from measured data. The mode1
proposed by Elhert et al includes a number of empirical coeflficients. such as track exgonent,
k. turning radius. Rk, where track forces approaching zero. etc. To obtain these values
requires extensively field tests. It can be concluded that the general theory developed offers a
significant improvement over al1 previous models.
rca r Outcr Track front
- - --- ; .
rcar lnncr Track + front
-1.90 -0.96 0.00 0.96 1.90 Location of Con tact Length (m)
Fig.4-5 Lateral shear stress distribution of the outer track along
the longitudinal centerline of track-ground contact arca
predicted from the proposed general theory at a vehicle
speed of 14.2 kmlh with different turning rridii.
-1.90 4.96 0.00 0.96 1.90 Location of Contact l m g t h (m)
Fig.4-6 Lateral shear stress distribution of the inner track along
the longitudinal centcrline of track-ground contact area
predictcd from the proposed general theory at a vehicle
speed of 14.2 kmlh with different tuming radii.
+ + +: Merisuretl data (Speecl range: 5.14 - 9.2 knilh) + + + : Mcasurcd datu (Spccd range: 11.9 -16.5 kmlh)
1 10 100 1000 1 10 100 1000 Theoretical Turning Radius (m) Theoretical Turning Radius (m)
Fig.4-9 CoefFicient of tuming resistance vs tuming radius Fig.4- 1 O Coefficient of tuming resistance vs tuming radius
for a Jaguar at a vehicle speed of 7.5 kmk with for a Jaguar at a vehicle speed of 14.2 kmlh with
K = 0.075 m under continuous trapezoidal load K = 0.075 m under continuous irapezoidal load
distribution over the entire track. distribution over the entire track.
Chapter 5
Evaluation of the Effects of Design and Operating Factors on
the Handling of Tracked Vehicles using the General Theory
In the previous chapter. a general theory for tracked vehicle steering on non-
deforrnable surfaces has k e n established. The predicted results of a particular vehicle during
a steady-state turning maneuver compare favourably with available field measurements.
With the basic features of the proposed theory substantiated, in this chapter, the effects on
handling of major vehicle design parameters, such as track-ground contact length, L, tread, B.
track width. b. longitudinal offset. c,, of vehicle CG with respect to the geometrical center of
the vehicle. and CG height. h. wiil be examined in detail.
5.1. Contact Lengtb, L
Using the tracked vehicle with design parameters given in Table 4-1 as a basis, the
effects of track-ground contact length. L, on steering behavior were exarnined using the
simulation mode1 developed. The sprocket torques under continuous trapezoidal normal load
distribution over the entire track contact length (as shown in Fig.3-5(b)) increase with the
increase of L as shown in Figs.5-1 to 5-4. This is because the moment of tuming resistance
generated during a tuming maneuver also increases with the increase of contact length, L, as
shown in Figs.5-9 to 5-12. The lateral force will be more or less the same with different
contact lengths as shown in Figs.5-5 to 5-8, since the centrifuga1 force during the tuming
maneuver does not change with L. The sprocket torques with different contact lengths and
vehicle speeds at various twning radii are tabulated in Tables A-1 to A-4 in Appendix A.
46
: Hcfcrencc (Contact Lcngth 1, = 3.8 m)
- - - :L=1.90m - - - : L = 2 . 8 5 m
30000L---, Outer Track
P---/#- lnner Track
10
Fig.5-3 Sprocket t
100 1000 Turning Radius (m)
orqi ues vs turning radius for a Jaguar at a
vehicle speed of 2 1.3 kmh with di fferent track contact
lengths during a steady-state turn undcr continuous
trapezoidal load distribution over the entire track.
: Hcfcrencc (Coiitact Lcngt h L = 3.8 ni)
- - - : L = 1.90 m --• : L = 2.85 m
30000
Outer Track -- --\. --.
Fig.54 Sprocket torques vs tuming radius for a Jaguar at a
vehicle speed of 29 kmlhr with different irack contact
lengths during a stead y-state turn under cont inuous
trapezoidal load distribution over the entire track.
- lnner Track
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 I
10 IO0 1000 Turning Radius (m)
: Rcfcrcncc (Contact Length L = 3.8 m)
- - - : L=1.90m - = : Id=2.85m
----O . L = 4.75 m ----- : L = 5.70 m
10 100 1000 Turaimg Radius (m)
Fig.5-7 Lateral forces vs tuming radius for a Jaguar ai a vehicle speed of 21.3 kmni with different track contact lengths during a steady-state tum under continuous trapezoidal load distribution over the entire track.
: Hekrcnce (Contact Length L = 3.8 m)
- - - : L = 1.90 m - . : L = 2.85 m
10 100 1000 Turning Radius (m)
Fig.5-8 Lateral forces vs tuming radius for a Jaguar at a vehicle speed of 29 kmlh with different track contact lengths during a steady ostate tum under continuous trapezoidal toad distribution over the entire track.
: Rcference (Con tact Lcngth L = 3.8 m)
- - - : L= 1.90m ----.: L=2.85m
- 0 - * . L = 4.75 ni ----- : L = 5.70 m
1 10 100 1 O00
Turning Radius (m)
Fig.5-9 Moments of tuming resistance vs turning radius for a Jaguar at a vehicle speed of 7.5 kmlh with different track contact Iengths during a steady-state turn under continuous trapezoidal load distribution over the entire
track,
O -
A
E 0 8 aoooo- e a Y V)
'1 * Da .O -120000- e e, O
8 3 -18000& E
E
: Heference (Contact Length L = 3.8 m)
- - - : L = I . W m ----m:L=2.8Sm
- - * . L = 4.75 m -O - - - : L = 5.70 m
Turning Radius (m)
Fig.5-IO Moments of tuming resistance vs turning radius for a Jaguar at a vehicle speed of 14.2 kmlh with different track contact lengths during a steady-state tum under continuous trapezoidal load distribution over the entire
track,
The effects of track-ground contact length, L. on the moment of turning resistance are
surnmarized in Figs.5- 13 to 5- 16. It shows the variation of the ratio of the moment of himing
resistance (Le., the surn of MT, and MTi at a particular contact length to that at the reference
length of 3.8 m) with the track contact length, L, at various turning radii. At the reference
contact length of 3.8 m, the ratio is equal to one.
The same trend is also observed for normal load distributed only on the track pitch (L,
= 15 cm) under each roadwheel (as shown in Fig.3-5(c)), as well as those for normal load
coricentratrd at a point (L, = 1 cm) under each roadwheel. These results are shown in Tables
B-1 to B-4 in Appendix B, and in Tables C-l to C-4 in Appendix C, respectively.
5.2. Tread of Vehicle, B
The s procket torques under con t inuous trapezoidal normal load distri bution over the
entire track increase with the decrease of B as shown in Figs.5-17 to 5-20. This is because
u-ith the decrease of B. the turning moment formed by the longitudinal shear forces on the
outer and inner tracks decreases. due to the decrease in the moment m. However, the lateral
force and moment of turning resistance generated during the tuniing maneuver remain
approximately the same. as shown in Figs.5-21 to 5-28. To balance the moment of tuming
resistance. the sprocket torques have to be increased with the decrease of B. The sprocket
torques with different treads and vehicle speeds at various turning radii are tabulated in
Tables A-5 to A-8 in Appendix A.
The effects of vehicle tread, B, on magnitude of the surn of the sprocket torques on the
outer and inner tracks are surnmarized in Figs.5-29 to 5-32. It shows the variation of the ratio
of the sum of sprocket torques at a particular vehicle tread. B, to that at the reference tread of
2.54 m. At the reference tread of 2.54 m. the ratio is equal to one.
0.95 1.90 2.85 3.80 4.75 5.70 0.95 1.90 2.85 3.80 4.75 5.70 Contact Length L (m) Contact Length L (m)
Fig.5-13 Ratio of turning resistance moment vs contact length L for Fig.5-14 Ratios of turning resistance moment vs contact length L for
a Jaguar at a vehicle speed of 7.5 kmlh with diffèrent a Jaguar at a vehicle speed of 14.2 kmlh with different
tuming radii under contiriuous trapezoidal load distribution tuming radii under continuous trapezoidal load distribut ion
over the entire track. over the entire track.
0.95 1.90 2.85 3.80 4.75 5.70 Contact Length L (m)
0.95 1.90 2.85 3.80 4.75 5.70 Contact LRngtb L (m)
Fig.5-15 Ratio of turning resistance moment vs contact length L for Fig.5-16 Ratio of turning resistance moment vs contact length L for
a Jaguar at a vehicle specd of 2 1.3 kinlh with diffcrent a Jaguar at a vehicle speed of 29 kmlh with different
turning radii under continuous trapezoidal load distribution turning radii under continuous trapezoidal load distribution
over the entire track. ovcr the cntire track.
: Rcfercncc (Trcad B = 2.54 ni) : Hcfcrcncc (Trcad B = 2.54 m)
lnner Track
10 100 1000 Turning Radius (m)
Fiy.5-19 Sprocket torques vs tuming radius for Jaguar at a vehicle speed of 21.3 km/h with different vehicle
treads during a steady-state tum under continuous trapezoidal load distribution over the entire track.
1 , Outer Track
I inner Track
10 100 1000 Turning Radius (m)
Fig.5-20 Sprocket torques vs tuming radius for a Jaguar at a vehicle speed of 29 kmh with different vehicle
treads during a steady-state turn under continuous trapezoidal load distribution over the entire track.
: Rcfcrence (Trelid B = 2.54 m)
- - - :B=1.90rn - : H = 2 . 1 7 m
- : B=3.04m - : B=3.80m
Outer Track
1 10 100 1000 Turning Radius (m)
Fig.5-21 Lateral forces vs turning radius for a Jaguar at a
vehicle speed of 7.5 km/h with different vehicle treads during a steady-state turn under continuous
trapezoidal load distribution over the entire track.
: Hcfcrcncc (Trcud B = 2.54 m)
- - - :B=1.90m ---:8=2.17m
- : B=3.04m : B=3.80 m
Outcr Track
d
1 10 100 1000 Turning Radius (m)
Fig.5-27 Lateral forces vs turning radius for a Jaguar at a
vehicle sped of 14.2 km/h with different vehicle treads during a steady-state turn under continuous
trapemidal load distribution over the entire trnck.
: Hcference (Tread B = 2.54 ni)
--- :B=1.90m : B = 2 . 1 7 m
- - : H=3.O4 m : B =3.80 m
Outer Track 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 I 1 1 1 1 1 1 1 1
1 10 100 1000 Turning Radius (m)
Fig.5-25 Moments of tuming resistance vs tuming radius
for a Jaguar at a vehicle speed of 7.5 kmh with different vehicle treads during a steady-state turn
under continuous tmpezoidal load distribution over the entire track.
: Hefcrence ('Trclid 13 = 2.54 m)
- - - : B = 1.90ni --:B=2.17m
: B = 3.04 m - : B = 3.80 m
Inter Track
1 10 100 1000 Turning Radius (m)
Fig.5-26 Moments of tuming resistance vs tuming radius
for a Jaguar at a vehicle speed of 14.2 kmlh with different vehicle treads during a steady-state tum under continuous trapezoidal load distribution over the eniire track.
: Refcrence (Trclid II = 2.54 m)
- - - : II = 1.90 m - 4 B = 2.17 m
10 100 1000 Turning Radius (m)
Fig.5-27 Moments of turning resistance vs tuming radius for a Jaguar at a vehicle speed of 21.3 km/h with different vehicle treads during a steady-state turn under continuous trapezoidal load distribution over the entire track.
: Hefcrcnce ( ï r c a d R = 2.54 m)
- - - :R=l.!Nhn - - - = : B = 2 . 1 7 m
- : R = 3 . 0 3 m : B = 3 . 8 0 m
10 100 1000 Turning Radius (m)
Fig.5-28 Moments of tuming resistance vs tuming radius
for a Jaguar at a vehicle spced of 29 km/h with different vehicle treads during a steady-state tum undcr cont inuous trapezoidal load distri but ion over the entire track.
1.90 2.38 2.85 3.33 3.80 Con tact Lengt h L (m)
1.90 2.38 2.85 3.33 3.80 Contact Length L (m)
Fig.5-29 Ratio o f magnitude o f the sprocket torques vs vehicle Fig.5-30 Ratio o f magnitude o f the sprocket torques vs vehicle
tread B for a Jaguar at a vehicle speed o f 7.5 km/h with tread B for Jaguar nt a vehicle speed of 14.2 kmh with
di fferent turning radi i under continuous trapczoidal load different turning rcidii under continuous trapezoidal load
distribution over the entire track. distribution over the cntire track.
1 .BO 2.38 2.85 3.33 3.80 Contact LRngth L (m)
1.90 2.38 2.85 3.33 3.80 Contact Length L (m)
Fig.5-31 Ratio of magnitude of the sprocket torqucs vs vehiclc Fig.5-32 Ratios of magnitude of the sprocket torques vs vehicle
tread B for a Jaguar ai a vehicle speed of 2 1.3 kiii/îi with tread B for a Jaguar at a vehicle speed o f 29 kmlh with
di fferent tuming radi i under cont inuous trapçzoidal load different tuming radii undçr continuous trapezoidal load
distribution over the entire track. distribution over thc entirc track.
The same trend is also observed for normal load distributed only on the track pitch (L, =
15 cm) under each roadwheel (as shown in Fig.3-5(c)). as well as those for normal load
concentrated at a point (L, = 1 cm) under each roadwheel. These results are shown in Tables
B-5 to B-8 in Appendix B, and in Tables C-5 to C-8 in Appendix C, respectively.
5.3. Longitudinal Offset of Vehicle Center of Graviîy, cr
Track forces are related to the shear displacement as mentioned previously. Ln general
the shear displacement increases from the front to the rear of the track-ground contact length.
Wirh the shifiing of the center of gravity of the vehicle to the rear or with the increase of cy (cy
is positive from the center of track contact length to the rear), the normal pressure under the
track is higher in the rear than in the front. Therefore. the vehicle will develop higher shear
force and as a result. track forces increase with the increase of c,. As a result, the sprocket
torques and moments of turning resistance will also increase with the increase in cy as shown
in Fig.5-33 to Fig.5-44. The sprocket torque at different c, values and vehicle speeds at
various tuming radii are tabulated in Table A-9 to Table A- 12 in Appendix A.
The same trend is also observed for normal load distributed only on the track pitch (L, =
t 5 cm) under each roadwheel (as shown in Fig.3-S(c)), as well as those for normal load
concentrated at a point (L, = 1 cm) under each roadwheel. These results are shown in Tables
B-9 to B- 1 2 in Appendix B. and in Tables C-9 to C- 12 in Appendix C, respectively.
The effects of the longitudinal offset of the center of gravity, c,, on the turning
resistance moment are summarized in Figs.5-45 to 5-48. It shows the variation of the ratio of
the turning resistance moment (i.e.. the tuming resistance moment at a particular value of cY to
that at c, = O m) with the value of c,. At c, = O m, the ratio is equal to one.
: Referencc (Longitudinal CG offset c, = O m)
- - - : C, = -0.20 in --- : c, = -0.40 m
----: C, = 0.20 m ----- : C, = 0.40 m
i 10 100 1000 Turning Radius (m)
Fig.5-33 Sprocket torques vs tuming radius for a Jaguar at a vehicle speed of 7.5 kmh with different longitudinal
CG offsets during a steady-state turn under continuous trapezoidal load distri but ion over the ent ire track.
: Rcfcrcnce (Longitudinal CG offsct c, = O m)
- 2 4 0 0 0 0 1
1 10 100 1000 Turning Radius (m)
Fig.5-34 Sprocket torques vs turning radius for a Jaguar at a vehicle speed of 14.2 kmlh with diffcrent longitudinal CG offsets during a steady-state turn under continuous
trapezoidal load distri but ion over the entire track.
: Reference (Longitudinal CG offset c,. = O m)
10 100 1000 Turning Radius (m)
Sprocket torques vs turning radius for a Jaguar ai a
vehicle speed of 2 1.3 km/h with different longitudinal
CG offsets durinp a steady-state tum under continuous
trapezoidal load distribution over the entirc track.
: Heferençe (Longitudinal CG offset c, = O m)
-- - : c,. = -0.20 m I . . c,. = -0.40 m
---- : C, = 0.20 n> ----- : C, = 0.40 m
1 lnner Track
10 100 tooo Turning Radius (m)
Sprocket torques vs turning radius for a Jaguar at a vehicle speed of 29 kmlh with different longitudinal
CG offsets during û steady-state tum under continuous
trapezoidal lood distribution over the entire track.
: Rcfcrcncc (Longitudinal CG offset c, = O 111) : Helerence (Longitudinal CC offset c, = O ni)
- - - : c, = -0.20 m - 0 . c,. = -0.40 m - - - : c,. = -0.20 m - 0 - 0 . c, = -0.40 m
----: C, = 0.20 m ----- : C, = 0.40 m ---- : C, = 0.20 m ----- : C, = 0.40 m
240001 Outcr Trirck
1 I O 100 1000 Turning Radius (m)
Outcr ~ r v c k
1 10 100 1000 Turning Radius (m)
Fig.5-37 Lateral forces vs tuming radius for a Jaguar at a vehicle Fig.5-38 Lateral forces vs turning radius for a Jaguar at a vehicle speed o f 7.5 kmlh with differcnt longitudinal CG offsets speed of 14.2 kmlh with diffcrcnt longitudinal CG offsets during a steady-state tum under continuous trapezoidal during a stcady-state turn undcr continuous trapezoidal Ioad distribution ovcr the entire track. load distribution over thc cntire track.
: Reference (Loiigituclinal CC offset cy = O ni) : Referençe (Longitudinal CG offset c,. = 0 m)
- 9 - : c, = -0.20 m . O . C , = -0.40 m - - - : c,. = -0.20 m -.-# , c,. = -0.40 m
---- : C, = 0.20 m ----- : C, = 0.40 m ---- : c,. = 0.20 m ----- : C, = 0.40 m
10 100 1000 Turning Radius (m)
10 100 100 Turning Radius (m)
Fig.5-39 Lateral forces vs tuming radius for a Jaguar at a vehicle Fig.5-40 Lateral forces vs turning radius for a Jaguar ai a vehicle
speed of 2 1.3 km/h with different longitudinal CG offsets speed of 29 kmlh with different longitudinal CG offsets
during a steady-state tum under continuous trapezoidal during a steady-state tum und& continuous trapezoidal
load distribution over the entire track. load distribution over the entire track.
: Heferencc (Longitudinal CG offset c, = O m) : Refercnïc (Longitudinal CC offset c,. = O ni)
-- --- : cy = -0.40 m - - - : c,. = -0.20 m - 7 - - : c, = -0.20 m . c,. = -0.40 in
---- : C, = 0.20 m ----- : C, = 0.40 m ---- : c,. = 0.20 m ----- : C, = 0.40 m
Fig.54 Moments of tuming resistance vs tuming radius for a
Jaguar at a vehicle speed of 7.5 kmlh with different
longitudinal CG offsets during a steady-state tum under
continuous trapezoidal load distribution over the cntire
track.
-100000 1 1 1 1 1 1 1 1 r r I I I I I I ~ I i r r i r i i l I
Fig.5-42 Moments of tuming resistancc vs tuming radius for a
Jaguar at a vehiclc speed of 14.2 km/h with different
longitudinal CG offsets during a steady-state turn under
continuous trapezoidal load distribution over the entire
track.
1 10 100 1000 Turning Radius (m)
-160000 Outer Track 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 I I I I I ~ I 1
1 10 100 1000 Turning Radius (m)
: Refercnce (Longitudinal CG offset ç, = O m)
-- - : c, = -0.20 m I . . c, = -0.40 m
---- : C, = 0.20 ni ----- : C, = 0.40 ni
lnner Track/ ./* 4
Track
10 100 1 O00 Turning Radius (m)
Fig.5-43 Moments of turning resistance vs turning radius for a
Jaguar at a vehicle speed of 21.3 kmlh with different longitudinal CG offsets during a steady-state turn under continuous trapczoidal load distributioii over the entire track.
: Kcfcrcnçe (Longitudinal CC offset c, = O m)
- 9 - : c, = -0.20 ni -.-O . C, = -0.40 m
-a-- : c,. = 0.20 ni ----- : C, = 0.40 m
Outer Track
10 100 1000 Turning Radius (m)
Fig.5-44 Moments of turning resistance vs turning radius for a
Jaguar at a vehicle speed of 29 kmlh with differeiit longitudinal CG offstes during a steady-state tum uiider continuous trapezoidal load distribution over the entire track.
-0.40 -0.20 0.00 0.20 0.40 Longitudinal CG Offset c, (m)
Fig.5-45 Ratio o f turning resistance moment vs longitudinal CG
offset c, for a Jaguar at a vehicle speed of 7.5 kmlh with
different turning radii under continuous trapezoidal load
distribution over the entire track.
0.5
-0.40 -0.20 0.00 0.20 0.10
Longitudinal CG Offset ce, (m)
Fig.546 Ratio o f tuming resistance moment vs longitudinal CG
offset ç, for a Jaguar at a vehicle speed o f 14.2 kmlh with
different tuming radii under continuous trapezoidal load
distribution over the entire track.
-0.40 -0.20 0.00 0.20 0.40 Longitudinal CG Offset c, (m)
4.40 4.20 0.00 0.20 0.40 Longitudinal CG Offset c, (m)
Fig.5-47 Ratio of turning resistance moment vs longitudinal CG Fig.5-48 Ratio o f turning resistance moment vs longitudinal CG
offset c, for a Jaguar at a vehicle speed o f 2 1.3 kmh with offset c, for a Jaguar at a vehicle speed of 29 kmlh with
different tuming radii under continuous trapewidal load di fferent turning radii under continuous trapezoidal load
distribution over the cntire track. distribution over the cntire track.
5.3. Track Width, b
The effects of track width, b. on the sprocket torque, lateral force and moment of
turning resistance during a steady-state tum under continuous trapezoidal normal load
distribution over the entire track are s h o w in Figs.5-49 to 5-60. It is noted that the traçk
width does not have much influence on the tracked vehicle steering on the nondefonnable
surfaces. The sprocket torques with different track widths and vehicle speeds at various
turning radii are tabulated in Tables A-1 3 to A-16 Appendix A.
The same trend is also observed for nonnal load distributed only on the track pitch (L,
= 15 cm) under each roadwheel (as shown in Fig.3-5(c)), as well as those for normal Ioad
concentrated at a point (Lp = 1 cm) under each roadwheel. These results are s h o w in Tables
B- 13 to B- 16 in Appendix B, and in Tables C- 13 to C- 16 in Appendix C, respectively.
The effects of track width on the turning resistance moment are sumrnarized in Figs.5-
6 1 to 5-64. It shows the variation of the ratio of the turning resistance moment (Le., the
turning resistance moment at a particular track width. b. to that at a track width, b9 of 0.45 m)
with track width. It appears that the track width has insignificant effect on the tuming
resi stance moment for the vehicle simulated.
5.5. Height of Vehicle Center of Gravity, h
The effects of height of vehicle center of gravity, h, on sprocket torques during a
turning maneuver under continuous trapezoidal nonnal load distribution over the entire track
are insignificant. as shown in Figs.5-65 to 5-68. The same observation can be made for both
IateraI forces and turning resistance moments on both outer and imer tracks of the vehicle.
except at a small turning radii, as s h o w in Figs.5-69 to 5-76.
The sprocket torques at different CG heights and forward speeds at various turning
73
radii are tabulated in Tables A- 17 to A-20 in Appendix A.
The same trend is also observed for normal load distributed only on the track pitch (L,
= 15 cm) under each roadwheel (as s h o w in Fig.3-5(c)), as well as those for normal load
concentrated at a point (L, = 1 cm) under each roadwheel. These results are shown in Tables
B- 17 to B-20 in Appendix B. and in Tables C- 17 to C-20 in Appendix C, respectively.
The effects of height of vehicle center of p v i t y on the turning resistance moments are
summarized in Figs.5-77 to 5-80. It appears that the ratio of the himing resistance moment
(i-e.. the tuming resistance moment at a particular CG height to that at a height of 1.3 m)
does not v u y significantly with the height of center of gravity.
5.6. Summary
Based on the parametric studies described above. it can be concluded that arnong the
vehicle design parameters, the vehicie contact length. L. and tread, B, have significant effects
on the steering of tracked vehicles. Increasing the track contact length, L, causes a
considerable increase in the turning resistance moment. On the other hand. reducing the tread.
B. would cause an increase in the sprocket torques and track longitudinal forces. It is found
that track width. b. longitudinal offset, c,, and CG height, h, of the vehicle have insignificant
effects on steering behaviour.
: Hcfcrcncc (Traïk witlth b = 0.45 m) : Hcfcrcncc (Track width b = 0.45 m)
1 10 100 1 O00 Turning Radius (m)
1 loner Track
1 10 100 1000 Turning Radius (m)
Fig.5-49 Sprocket torques vs tuming radius for a Jaguar at a Fig.5-50 Sprocket torques vs tuming radius for a Jaguar at a
vehicle speed of 7.5 km/h with different track widths vehicle speed of 14.2 kmlh with different track widths
during a steady-state turn under continuous trapezoidal during a steady-statc tum undcr continuous trapezoidal
load distribution over the entire track. load distribution ovcr the entire track.
: Hcfcrencc (Track width b = 0.45 m) : Rcference (Track width b = 0.45 m)
I I - * . b = 1.35 m
200001
Outer Track
h
2 l0OW- 8 - 3 P i: Y
O - Y 8 L P Fn
40000- lnner Track
10 100 1 O00 Turning Radius (m)
10 100 ~ 0 0 0 Turning Radius (m)
Fig.5-51 Sprocket torques vs turning radius for a Jaguar at a Fig.5-52 Sprocket torques vs tuming radius for a Jaguar at a
vehicle speed of 21.3 kmh with different track widths vehicle speed of 29 kmRi with different crack widihs during a steady-state tum under continuous trapezoidal during a steady-state tum under continuous trapezoidal load distribution over the entire track. load distribution over the entire track.
: Hcfcrcnce (Track width b = 0.45 m)
- - - :b=O.Olm : b = 0 . 9 0 m
- : b = 1.35 m
1 10 100 1000 Turning Radius (m)
Fig.5-53 Lateral forces vs tuming radius for a Jaguar at a vehicle
speed of 7.5 kmh with different track widths during a
steady-state tum under continuous trapezoidal load
: Hefercnce (Truck width h = 0.45 ni)
1 10 100 1000 Turning Radius (ni)
Fig.5-54 Lateral forces vs turning radius for a Jaguar at a vehicle
speed of 14.2 km/h with different track widths during a
steady-state tum under continuous trapezoidal load
distribution over the entire track. distribution over the entire track.
: Heferencc (Track width b = 0.45 m)
10 100 1000 Turning Radius (m)
Fig.5-55 Lateral forces vs tuming radius for a Jaguar at a vehicle speed of 2 1.3 k m h with different track widths during a steady-state tum under continuous trapezoidal load distribution over the entire track.
: Heferencc (Track width b = 0.45 m)
--- : h = 0.01 m --- : b = 0.90 m
- : b = 1.35 m
10 100 1000 Turning Radius (m)
Fig.5-56 Lateral forces vs tuming radius for a Jaguar at a vehicle speed of 29 kmni with different track widths during a steady-state turn under continuous trapezoidal load distribution over the entire track.
: Wcfcrcnce (Track witlth h = 0.45 m) : Wefcrence (Track width h = 0.45 m)
Fig.5-57 Moments of tuming resistance vs tuming radius for a Fig.5-58 Moments of tuming resistance vs tuming radius for a Jaguar at a vehicle speed of 7.5 km/h with different Jaguar at a vehicle speed of 14.2 krnh with different track widths during a stcady-state turn under continuous track widths during a steady-state turn under continuous trapemidal load distribution over the entire track. trapzoidal load distribution over the entire track.
-100000
1 10 100 1000 1 I O 100 1000
Turning Radius (m) Turning Radius (m)
Outer Track ' 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 7 1 1 1 1 1 1 1 I -t60000 1 1 1 1 1 1 1 1 1 1 I I I ~ I I I 1 1 1 1 1 1 1 1
: Refercncc (Track w idth 1) = 0.45 m) : Referencc (Truck width h = 0.45 m)
10 100 1000
Turning Radius (m)
460000 ; ' , 1 1 1 1 1 1 1 1 1
10 100 1000 Turning Radius (m)
Fig.5-59 Moments of tuming rcsistance vs tuming radius for a Fig.5-60 Moments of tuming rcsistance vs turning radius for a
Jaguar at a vehicle speed of 2 1.3 kmlh with di ffercnt Jaguar at a vehicle speed of 29 kmlh with different track track widths during a steady-state turn undcr continuous widths during a steady-state tum under continuous trapezoidal load distribution over the entire track. trapezoidal load distribution ovcr the entire track.
0.00 0.45 0.90 1.35 Track Widtb b (m)
Fig.5-63 Ratios o f tuming resistance moment vs track width b for a Jaguar at a vehicle specd of 21.3 km/h with
di fferent turning radii under cont inuous trapezoidal
load distribution over the entire track.
0.00 0.45 0.90 1.35 Track Width b (m)
Fig.5-64 Ratio of turning resistance moment vs track width b
for a Jaguar at a vehicle speed o f 29 kinh with
different turning radii under continuous trapezoidal
load distribution over the entire track.
: Hcfcrcnce (CG hcight h = 1.30 m)
--- : h = 0 . 8 0 m O - : h = 1 . 8 m
1 10 100 1000 Turning Radius (m)
: Hcfcrcnce (CG hcight h = 1.30 m)
--- : h=Q.$Om --- : h = 1 . 8 m
1 lnner Track
1 10 100 1000 Turning Radius (m)
Fig.5-65 Sprocket torques vs tuming radius for a Jaguar at a Fig.5-66 Sprocket torques vs tuming radius for a Jaguar at it
vehicle speed of 7.5 km/h with different CG heights vehicle speed of 14.2 km/h with different CG heights during a steady-state turn under continuous trapezoidal dwing a steady-state turn under continuous trapezoidal load distribution over the cntire track. load distribution over the entire track.
: Hcfcrcncc (CG hcight h = 1.30 m) : Hcferencc (CC hcight h = 1.30 m)
10 100 1000 Turning Radius (m)
I lnner Trrick
10 100 1000 Turning Radius (m)
Fig.567 Sprocket torques vs tuming radius for a Jaguar at a Fig.5-68 Sprocket torques vs tuming radius for a Jaguar at a
vehicle speed of 21.3 krnh with difl'erent CG heights vehicle speed of 29 kndh with different CG heights
during a steady-state turn under continuous trapezoidal during a steady-state turn under continuous trapezoidal
load distribution over the entire track. toad distribution over the entire trrick.
: Hefercnce (CG height h = 1 -30 m)
0 0 0 : h=0.80m --- : h = 1.8m
\ Outer Track
lnner Track \\
: Hefcrcncc (CG height h = 1.30 m)
--- : h=O.SOm ---: h = 1.8m
1 10 100 1000 Turning Radius (m)
1 10 100 1000 Turning Radius (m)
Fig.5-69 Lateral forces vs tuming radius for a Jaguar at a vehicle Fig.5-70 Lateral forces vs turning radius for a Jaguar at a vehicle speed of 7.5 km/h with different CG heights during a speed of 14.2 km/h with different CG heights dwing a steady-state tum under continuous trapezoidal load steady-state tum under continuous trapezoidal load
distribution over the entire track. distribution over the entire track,
: Refcrcnce (CG hcight h = 1 J O m)
10 100 1000 Turning Radius (m)
: Refcrencc (CG hcight h = 1.30 m)
--- : h = 0.80 m ---: h = 1.8 m
10 100 Io00 Turning Radius (ai)
Fig.5-71 Lateral forces vs tuming radius for a Jaguar at a vehicle Fig.5-72 Lateral forces vs tuming radius for a Jaguar at a vehicle
speed of 21.3 k m h with different CG heights during a speed of 29 kmh with different CG heights during a
steady-state tum under continuous trapezoidal load steady-state turn under continuous trapezoidal load distribution over the entire track. distribution over the cntire track.
: Heference (CC height h = 1.30 m)
O - - :h=0.80m ---:h=1.8m
1 10 100 1000 Turaing Radius (m)
: Refcrencc (CC height h = 1.30 m)
lnner
-180000 1 1 1 1 1 1
1 10 100 1000 Turniag Radius (m)
Fig.5-73 Moments of tuming resistance vs turning radius for a Fig.5-74 Moments of tuming resistance vs turning radius for a
Jaguar at a vehicle speed of 7.5 km/h with different CG Jaguar at a vehicle speed of 14.2 km/h with different
heights during a steady-staie turn under continuous CG hcights during a steady-state turn under continuous
trapezoiâal load distribution over the entire track. trapemidal load distribution over the entire track.
: Hcfercnïc (CG hcight Ii = 1.30 m)
--- :h=0.80m - - - :h=l .Um
Outer Track
10 100 1000 Turning Radius (m)
: Rcfcrencc (CG height h = 1 .JO m)
Fig.5-75 Moments of turning resistancc vs turning radius for a Fig.5-76 Moments of turning resistance vs turning radius for a Jaguar at a vehicle speed of 21.3 kinlh with different Jaguar at a vehicle speed of 29 km/h with different CG
CG heights during a steady-state turn under continuous heighis during a steady-state turn under çontinuous trapezoidal load distribution over the entire track. trapemidal load distribution over the entire track.
-200000 I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
10 100 1000 Turning Radius (m)
Chapter 6
Experimental Study of the Shearing Characteristics of a
Representative Track Link on AsphaIt
As mentioned previously. the shearing behaviour of a tracked vehicle is greatly
influenced by the shear force developed on the track-ground interface. A detailed
esperirnental study was canied out to measure the shearing characteristics between a
representative track link with rubber pad used on a curent amoured personnel carrier M 1 13
and asphalt in the laboratory.
6.1. Apparatus for Measuring tbe Sbearing Cbaracteristics
The apparatus used to measure the shearing characteristics. including the relation
between shear force and shear displacement. was a modification of an existing tire tester in
the Transport Technology and Research Laboratory. Carleton University. The apparatus
includes state-of-art data acquisition system. developed by Mr. Y.C. Wu for his Ph.D
research on the lateral dynamics of off-road tires.
The schematic of the test apparatus is illustrated in Fig.6-1. The major components of
the track link shear force-shear displacement tester include:
0 a horizontal motion carrier
0 a test bed on which an asphalt block was mounted
0 a double parallelograrn mechanism on which test track links were rnounted
0 a data acquisition system
1
Horizontal Motion Caniage
1- Double Parallelogram
/ Mec hanism
Dead Weight
x-y
1 a I - 1
1 - O 3 1 v Load Cell
Track Supporting Frame
Linear Bearing
Asphalt Track Links
Test Bed
Fig.6-l Schematic of the test setup for the rneasurement o f track shear
force and shear displacement on the asphalt.
two Lebow 6643 x-y load cells. and one displacement potentiometer.
Two test track links and their supporting h e are connected with a double
parallelogrm mechanism which is mounted on the horizontal motion carriage. The motion
carriage is driven by an electric motor through a chain drive mechanism. When the carriage
moves fonvard. shearing action between track links and the asphalt surface takes place. The
horizontal shear force was monitored by hvo Lebow 6643 x-y Ioad cells located at both lefi
and right hand sides of the carriage, as shown in Fig.6-1. A displacement potentiometer is
mounted to measure the horizontal travelling distance of the h e , which is the shear
displacement of the track links. These forces as well as horizontal displacement are al1
recorded by a data acquisition system (Fig.6-2).
Corn p u t e r
I - D i s p l a c e m e n t
P o t e n t i o m e t e r
Fig.6-2 Block diagram of the data acquisition system for the
shear force-shear displacement tester.
L e f t a n d R i g h t
x - y L o a d C e l l s
Tests were carried out under four normal loads on the track links of 890 , 1000 , 1200.
V
S t r a i n G a u g e
C o n d i t i o n e r A
w
A na Iogue/D i g i t C o n v e r t e r
D a t a A c q u i s i t i o n s o f t w a r e
and 1400 1 b. and at three different speeds. 0.195. 0.145. and 0.100 m/sec. Seven test nins
were conducted for each forward speed-normal load combination.
6.2. Analysis o f Shear Stress-Shear Displacement Data
The most commonly used semi-empirical equation to describe the shear stress-shear
displacement relationship on the track-ground interface is equation (3-1 5). To determine the
optimum value of shear deformation parameter. K. which minimizes the error in curve fitting.
the following equation is used (Wong. 1983)
where r, is measured stress. 5, is mê'timum shear stress under a particular load, and j is
measured shear displacement.
To solve for an appropriate value of K, the following procedure is used.
1 ). The maximum value of shear stress. rmx. in equation (6-1) was first identified fiom
the measured data, as shown in Fig.6-3.
2). K is calculated using equation (6- f ).
3). A goodness-of-fit is calculated using equation (3-43).
Foltowing the above procedure. the average value of the shear deformation parameter,
K. for the M l 13 track link with rubber pad on asphalt is found to be 0.0183 m and the
goodness-of-fit ranges from 86.72% to 93.52%. The measured data and fitted curves using
the calculated K value are shown in Figs.6-3 to 6-13.
0.0 O. 1 0.2 0.3 0.4
Shear Displacement (rn)
Fig.6-3 Shear force-shear displacement curve at normal load 890 Ib with sliding speed 0.100 mls.
Shear Displacemen t . .
Fig.6-4 Shear force-shear displacement curve at nomal load 890 Ib with sliding speed 0.145 mls.
0.0 0.1 0.2 0.3 0.4 0.6
Shear Displacement (m)
Fig.6-5 Shear force-shear displacement curve ai normal load 1 O00 Ib with sliding speed 0.100 mls.
0.0 0.1 0.2 0.3 0.4 0.6
Shear Displacement (m)
Fig.6-6 Shear force-shear displacement curve at nomial load 1 O00 Ib with sliding speed 0.145 m/s.
0.0 O. 1 0.2 0.3 0.4 0.6
Shear Displacement (m)
Fig.6-7 Shear force-shear displacement curve at normal load 1000 Ib with sliding speed 0.195 m/s.
0.0 0.1 0.2 0.3 0.4 0.5
Shear Displacement (m)
Fig.6-8 Shear force-shear displacement curve at normal load 1200 Ib with sliding speed O . I O m/s.
0.00 0.10 0.20 0.30 O.* 0.60
Shear Displacement (m)
Fig.6-11 Shear force-shear displacement curve at normal load 1400 Ib with sliding speed O. 100 m/s.
0.0 0.1 0.2 0.3 0.4 0.6
Shear Displacement (m)
Fig.6- 1 2 Shear force-shear displacement curve at nonnal load 1400 Ib with sliding speed 0.145 m/s.
0.0 0.1 0.2 0.3 0.4 0.5
Shear Displacement (m)
Fig.6- 13 Shear force-shear displacement curve at normal
load 1400 lb with sliding speed 0.195 m/s.
6.3. The Determination of the Coefficient of Friction
From the measured data shown in Figs.6-3 to 6-13, the maximum shear force
developed under various normal loads can be identified. Their relationship is illustrated in
O 400 M)O 1200 100
Normal Lord (N)
Fig.6-14 The relation of maximum shear force and normal load.
The coeficicnt of fiction. p. between the track Iink with rubber pad and the asphalt
surface. c m be found frorn the slope of the fitted line shown in Fig.6-11.
Based on the experimental data shown. it was found the average value of p was 0.684.
It should be mentioned that the shear tests were conducted at three different values of sliding
velocity. 0.10.0.145. and 0.195 m/s. The test results show that within this range, the sliding
velocity has insignificant effect on the coefficient of fiction, p, and the shear deformation
parameter. K. Within the range of speeds used. the value of p varies fiom 0.664 to 0.737, and
K varies in the range of 0.0 167 to 0.0225 m.
In the simulation of the steenng behaviour of Ml 13 on asphait described in the next
chapter. the average value of the shear defornation parameter K = 0.0183 m and p = 0.684
were used.
Chapter 7
Simulation of Steering Behaviour of an Ml13 Armoured
Personnel Carrier on Non-deformable Ground Based on
Measured Shear Data
7.1. Basic Design Parameters o f an M l 13 Armourd Personnel Carrier
Using the measured data on the shearing characteristics of an Ml 13 track link with
rubber pad on asphalt described in the previous chapter, the handling behaviow of an Ml 13
mored personnel carrier was predicted. The predictions were made using the general theory
described in Chapter 3. The basic vehicle and track parameten used in the predictions are
given in Table 7- 1 .
Table 7- 1 Basic parameters of the M 1 1 3 tracked vehicle used in the predictions
Sprung weight, kN 100.57 Unsprung weight, kN 10.08 ~ e i i h t of c.G.. rn Sprocket radius, m Trac k-ground contact length. m Longitudinal CG. location from centerline of vehicle hull, m Track width. m Track pitch Tread of vehicle, m
1 Shearing characteristic
Shear deformation parameter, m 0.0 183 Coeficient of friction between track-ground (asphalt) interface, 0.684 Coefficient of motion resistance, 0.03+0.000 1 5*V
*: The unit of forward speed V is in kilometers per hour.
7.2. Simulation Results
The predicted sprocket torques. lateral forces. and moments of tuning resistance for the
M l 13 dunng a steady-state tm on an asphalt surface are shown in Figs.7-1 to 7-14. A
cornparison of sprocket torques predicted by Steeds' model and by the proposed model at
different vehicle speeds and normal load distributions are shown in Tables 7-2 to 7-9.
Table 7-2 Sprocket torques at various turning radii with a vehicle speed of 7.5 km/h for an M 1 13
armoured personnel carrier.
Tuming 1 Predicted by Steeds' 1 Predicted by proposed general theory (kN.m)
1 1 the entire track lpitch under each madwheel 1
(kN.m)
(m) ndius- l
Outer lnner Outer
Continuous trapezoidal
load distribution over
Table 7-3 Sprocket torques at various turning radii with a vehicle speed of 14.2 kmih for an MI 13
armoured personnel carrier.
Inner
-3.678
-3 -3 94
-2.486
-1 567
Trapezoidal load distribution on the track
(ml 1 1 load distribution over 1 disaibution on the oack 1 under each roadwheel 1
Concentrated load under each roadwheel
Tuming
radius
1 1 the entire track 1 pitch under each roadwheel 1 1
Outer
4.664
4.338
3 -496
2.557
I ~ e r
-3 -938
-3 -607
-2.76 1
-1.821
Outer
4.304
4.123
3.427
2.538
Predicted by Steeds'
(kN.m)
lnner
-3.588
-3.393
-2.692
- 1 -802
Predicted by proposed general theory (kN.m)
Continuous trapezoidal [ Trapezoidal load 1 Concentrated load
1 O
20
50
1 O0
Outer
3.590
4.630
4.621
4.61 1
lnner
-3.748
-3.863
-3.861
-3.85 1
Outer
4.336
4.102
3.175
2.268
Inner
-3.504
-3 .340
-2.4 16
- 1 -508
Outer
4.679
4.386
3.52 1
2.579
Inner
-3.854
-3.63 1
-2.765
- 1.820
Outer
4.503
4.256
Inner
-3.700
-3.507
3.479
2.569
-2.723
-1.810
Table 7 4 Sprocket torques at various turning radii with a vehicle speed of 2 1.3 km/h for an M 1 13
armoured personnel carrier.
- m . - - - a - . - . a m . I I P L . - . n .- . I . .-
1 O
20
50
1 O0
I able 1-3 '4;proc)tet torques at vartous turning raaii witn a venicie speea or LY Kmn ror an M I i
Turning
radius
(ml
radius
im )
Predicted by Steeds'
model (kN-m)
Predicted by proposed general theory (kN.m)
Continuous trapezoidal load distribution over
I the entire track
1 theentiretrack lpitchwidereachroadwheell
armoured personnel carrier.
Trapezoidal load distribution on the track
Outer
4.5 18
4.259
3 -476
2.575
lnner
-2 -462
-2 -900
-2.257
-1 -377
pitch under each roadwheel
Outer
4.192
4.555
1.642
4.636
lnner
Concentrated load
under each roadwheel
Outer
4.380
4.24 1
3 -462
2.564
Turning
15
30
50
1 O0
inner
-3 .O25
-3.4 14
-2.680
- 1 -780
lnner
Predicted by Steeds'
model (kN-m) Predicted by proposed general theory (kN.m)
Outer
Concentrated Ioad under each roadwheel
Continuou traperoidal load distribution over
-2.88 1
-3.714
-3.855
Trapezoidal load distribution on the track
Outer
3.836
3 -670
3 .O87
2.353
Outer
3.732
4.474
4.621
4.656
3.769
3.738
3.043
1~
Outer
3.144
2.836
2.322
1.85 1
Inner
-2.346
-3.601
-3.797
-3.843
Inner
-1.707
-1 -941
-1.489
- 1 .O36
Imer
-2.346
-2.799
-2.264
-1 -541
-3.050
L
Outer
4.193
3.865
3.264
2.44 1
2.162
: Cantinuous trspe~oitlal load distribution aver the entirc truck
- - - : Canccntratcd load under each rolidwheel
: Trapczoidal load distribution on the trcick pitch unrlcr each roadwhecl
- - - - : Stceds' madcl
-1 Outer Trwck Outer Track
-
-6000 1 lnnet Track lnner Track
I I 1 1 1 1 1 1 I 1 1 1 1 1 1 1 I I 1 1 1 1 1 1 ) 1
1 10 100 1000 Theoretical Turning Radius (m)
1 10 100 1000 Theoreticml Turning Radius (m)
Fig.7- 1 Sprocket torques vs tuming radius for an M 1 13 at Fig.7-2 Sprocket toques vs tuming radius for an Ml 13 at
a vehicle speed of 7.5 kmlh with K = 0.01 83 m a vehicle speed of 14.2 km/h with K = 0.0183 m
and p = 0.684 during a stcady-state tum. and p = 0.684 during a steady-state tum.
: Cnntinuous trnpcroidal load distribution ovcr the cnqire track
- - - . : Concentratcd load under cach roadwhecl
- - : Trapzoidal loud distribution on the track pitch under eich rovdwhccl
- - - : Steeds' mode1
"1 Outer Track
Fig.7-3 Sprocket torques vs tuming radius for an MI 13 at a vchicle speed of 21.3 km/h with K = 0.01 83 m and )i = 0.684 during a steady-state turn.
-
-6000
"1 Outer Track
lnner Track
I 1 1 1 1 1 1 1 I 1 1 1 1 1 1 1 1 1 I 1 1 1 1 1 1 1 I
Inner Track
JO00 1 10 100 1000
Theoretical Turaiag Radiua (m)
1 10 100 1000
Theoretical Turning Radius (m)
Fig.7-4 Sprocket torques vs tuming radius for an M l 13 at a vehicle speed of 29 kmh with K = 0.01 83 rn and p = 0.684 duriny a steady-state tum.
: Continuous trapczoidal loarl distribution ovcr the cntirc track
- - - : Conccntrwtcd toatl under elrch ralrdwheel
: TrapmidaI load distribution on the track pitch under each road~hccl
1 10 100 1000
Theoretical Turning Radius (m)
Fig.7-7 Lateral forces vs tuming radius for an M 1 13 üt ü
vehicle speed of 14.2 km/h with K = 0.0183 m
and y = 0.684 during a steady-state tum.
1 qo 100 1006
Theoretical Turning Radius (m)
Fig.7-8 Lateral forces vs tuming radius for an M 1 13 at a vehicle speed of 21.3 kmlh with K = 0.0183 m and p = 0.684 during a steady-state tum.
: Continuous trapczoidal load distribution ovcr the cntire track
- - - : Concentratcd load undcr each roadwhccl
: Trupzoidal load distribulion on thc t r ~ c k pitch undcr euch roadwhcel
- - - - : Stceds' model
Fig.7-12 Moments of tuming resistance vs tuming radius Fig.7-13 Moments of tuming resistance vs tuming radius
for an MI 13 ai o vehicle speed of 14.2 k m h with for an Ml 13 at a vehicle speed of 21.3 k m h with
K = 0.01 83 m and p = 0.684 during a steudy-state K = 0.0183 m and )i = 0.684 during a steady-state
tum. turn.
-48m
1 10 100 1000 1 10 100 1OOO
Theoretical Turning Radius (m) Thearetical Turning Radius (m)
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 I 1 1 1 1 1 1 1 1 uooo 1 1 1 1 1 1 1 1 I 1 1 1 1 1 1 1 1 I 1 1 1 1 1 1 1 1 1
7.3. Summary
It cm be seen from Figs.7-l to 7 3 and Tables 7-2 to 7-5 that the relationships between
the sprocket torque and tuming radius at various vehicle speeds have similar trend to those
predicted for the Jaguar vehicle described in Chapter 4. The predictions using Steeds' mode1
again show that the sprocket torques and moments of twning resistance at higher ~ i n g
radii remain almost constant. This indicates again that the track-ground shearing
c harac terist ics. speci fical 1 y the shear stress-shear displacernent relation. must be taken into
consideration in predicting the handling of tracked vehicles. It also appears that normal
pressure distribution also has certain influence on the relationships between sprocket torque
and turning radius.
Chapter 8
Discussion and Conclusions
This research project focuses on a detailed study of the mechanics of steenng of
tracked vehicles on non-deformable surfaces. taking into account the effects of shear
displacement on the development of shear stress and of track width. Experimental studies
ha\.e been carried out to identi@ the shearing characteristics on asphalt of the track link with
the mbber pad used on the M l 13 annoured personnel carrier. A computer simulation model
for steering behavior of tracked vehicles under steady-state conditions over non-deforrnable
surfaces has been developed.
8.1. Conclusions
The handling characteristics of tracked vehicles predicted using the mode1 developed
in this project bear close resemblance to those measured in the field, as shown in Figs.4- t to
4-4 and in Tables 4-2 to 4-9. It is also shown that this model represents a significant
improvement on Steeds' model. This is because the model developed takes into account the
effect of shear displacement on the development of shear stress on the track-ground interface.
whereas Steeds' model assumes that full shear stress develops as soon as a small relative
motion between the track and the ground takes place. However, Steeds' model may be
considered a special case of the general theory presented in this thesis where the shear
defonnation parameter, K, is zero.
Using the model developed, the effects of design and operating parameters on the
stéering of tracked vehicles have been systematically examined. It is found that increasing
tnck-ground contact length. L. the sprocket torques required in a steady-state h n g
maneuver increase significantly. This is caused by the considerable increase of the NNng
resistance moment.
It is also found that the sprocket torques required to execute a steady-state hirn increase
u.ith the decrease of tread. B. This is primarily due to the decrease in the moment arms of the
longitudinal track forces. which tom the tuniing moments and initiate the turn of the
vehicle.
Longitudinal offset of vehicle center of gravity, c,. also affects the sprocket torques
required during a tuming maneuver. General speaking, sprocket torques will increase with
the shifting of CG rearward. As a result of shifiing CG rearward the normal pressure at the
rear is higher than that at the front. This causes an increase in the tuming resistance moment
and hence sprocket torques.
in general. the height of vehicle CG and track width do not have significant effects on
the steering characteristics of tracked vehicles over non-deformable surfaces.
Experimental studies have also been carried out to identiQ the shearing characteristics
between the track link used in the armoured personnel carrier M 1 1 3 and asphalt. Based on
experimental data the steering characteristics of Ml 13 over asphalt are predicted using the
model developed. and compared with those predicted using Steeds' model.
In conclusion. the general theory for tracked vehicle steering on non-deformable
surfaces presented in this thesis is a significant improvement on existing models. The
predictions based on this model have a very similar trend to measured field data. It is
believed that the general theory developed provides a basis for the M e r study of tracked
vehicle maneuverability on deformable terrain.
8.2. Future Work
Based on the experimental and analytical studies mentioned previously, the following
recommendations for future work can be made:
1 ). Further study on the factors affecting the shearing characteristics between the track and
the ground. such as sliding velocity. temperature. moisture content of the terrain, etc.
1). Initiation of systematic study of tracked vehicle steering on deformable terrain based on
the studies presented in the thesis. with the following factors included in the analysis:
O Vehicle sinkage on deformable terrain
a Shearing characteristics between the track and the deformable terrain
a Normal pressure distribution under the track on deformable terrain during a tuming
maneuver
O Bulldozing effect of the track in lateral skidding on turning resistance
Bibliography
Baladi. G.Y. and Rohani, B. 1978. A Mathematical Model of Terrain-Vehicle Interaction for
Predicting the Steering Performance of Track-Laying Vehicles, Proceedings o f the 6th
International Conference o f the International Society for Terrain-Vehicle Systems, pp285-332.
Baladi. G.Y. and Rohani. B.. 198 1. Analysis o f Steerability of Tracked Vehicles: Theoretical
Pred ic t ions Versus Field Measurements. Proceedings of the 7m Internat ional Con ference of the
lnternational Society for Terrain-Vehicle Systems, pp. 1 175- 1220.
Bekker. M.G., 1956. Theory o f Land Locomotion, Ann Arbor, MI: University o f Michigan
Press.
Bekker. M.G.. 1960.Off-the-Road Locomotion. Ann Arbor, MI: University of Michigan Press.
Bekker. M.G.. 1969. Introduction to Temin-Vehicle Systems, Ann Arbor, MI: University of
Michigan Press.
Crosheck. J.E.. 1973. Skid-Steering o f Crawlers, SAE Paper No.750552, pp. 1 - 15.
Ehiert. W. Hug. B. and Schmid, I.C., 1992, Field Measurements and Analflical Models as a
Basis of Test Stand Simulation of the Turning Resistance o f Tracked Vehicles, Journal of
Terramechanics.Vol.39( 1 ), pp.57-69.
Eiyo, F. and Kitano, M., 1984. Study on Controllability and Stability o f High Speed Tracked
Vehicles. Proceedings o f the 8" International Conference of the International Society for
Terrain-Vehicle Systems, pp.789-8 19.
Foss. C.F.. 1993-94, Jane's Amour and Artillery, 1 4 ~ Ed.. Jane's Information Group Inc..
Alexandria, USA.
Hayashi. J. 1975, Practical Analysis o f Tracked Vehicle Steering Depending on Longitudinal
Track Slippage. Proceedings of the 5'h lnternational Conference o f the International Society for
Terrain-Vehicle Svstems. DD-493-5 1 2.
I 1 . Kar. M.K., 1987, Production of Track Forces in Skid Steering of Military Tracked Vehicles,
Journal of Terramechanics. Vo1.23( 1 ), pp.75-86.
12. Karafiath, L.L.. 1981. Analytical Model for the Turning of Tracked Vehicles in Soft Soils.
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Systems. pp. 1385-141 1.
13. Kitano. M. and Jyozaki, H., 1976. A Theoretical Analysis of Steerability of Tracked Vehicles.
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! 5. Kitano. M. Watanabe. K. Takaba, Y. and Togo. K.. 1 988. Lane-Change Maneuver of High Speed
Tracked Vehicles. Journal Terramechanics. Vo1.25(2), pp.91-102.
16. K. Kitano. Watanabe. M. and Nagatomo, N.. 1990. Stability and Controllability of High-Speed
Tracked Vehicles-Linear Model and Vehicle Response. Proceedings of the 10" lntemational
Con ference of the International Society for Terrain-Vehicle Systems, pp.659-670.
17. Merritt. H.E.. 1939. Some Considerations Influencing the Design of High Speed Tracked
Vehicles, Proceedings of Institute of Automobile Engineers, Vol.33, pp.398-430.
18. Micklethwait. E.W.E., 1944, Soil Mechanics in Relation to Fighting Vehicles, Military Coli. of
Science. Chobham Lane. Chertsey,
19. Murakami. H. Watanabe, K. and Kitano, M.. 1992, A Mathematical Model for Spatial Motion
of Tracked Vehicles on Soft Ground, Journal of Terramechanics, Vo1.29( 1 ), pp.7 1-8 1.
20. Ogorkiewicz, R.M., 199 1, Technolow of Tanks. London: Jane's Information Group.
2 1 . Schmid. I.C., 1984, A tracked Vehicle Test Plant for the Simulation of Dynarnic Operation,
Proceedings of the 8" In ternational Con ference of the 1 ntemational Society for Temin-Veh icle
Systems. pp.835-853.
22. Shiller, Z and Serate, W. 1995, Trajectory Planning of Tracked Vehicles, Journal of Dynamic
Systems. Measurement, and Control. Vol. 1 1 7, pp.6 19-623.
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190 and pp.2 19-222.
24. Stoer. J. and Bulirsch, R.. 1980, Introduction to Numerical Analysis, Spronger-Verlag: New
York. Inc.
25. Stroud. A.H.. 1966. Gaussian Quadrature Formulas, Prentice-Hall, Inc.
26. Sugiyama. N. and Kondo. H., 1984, Basic Study on the Turning Resistance o f Track,
Proceedings of the 8& International Conference of the International Society for Terrain-Vehicle
Systems. pp.889-899.
27. Watanabe. K. Kitano, M. and Katahira, T.. 1993, Controllability and Stability of Tracked
Vehicles on lnclined Ground, Proceedings of the 1 2 ' ~ International Conference of the
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28. Watanabe. Murakami. H. Kitano, M. and Katahira. T., 1993, Experimental Characterization of
Dynamic Soil-Track Interaction of Dry Sand, Journal of Terramechanics, Vol-30(2), pp.1 I l -
131.
29. Weiss. K.R., 197 1. Skid-Steering, Automobile Engineer, pp.22-25.
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3 1 . Wong. J.Y. and Preston-Thomas. J.. 1983, On the Characterization of the Shear Stress-Shear
Displacement Relationship of Terrain. Journal of Terrarnechanics, Vol. 19(4), pp.225-234.
32. Wong. J .Y. and Preston-Thomas. J., 1988, Investigation into the Effects of Suspension
Characteristics and Design Parameters on the Performance of Tracked Vehicles using an
Advanced Computer Simulation Model, Proceedings of Institution of Mechanical Engineers.
V01.202(D3), pp.143-161.
33. Wong. J.Y .. 1989. Terramechanics and Off-Road Vehicles. Elsevier. Amsterdam.
34. Wong. J.Y.. 1993. Theory of Ground Vehicles, 2"' Edition, John Wiley & Sons, Inc.
35. Wong. J.Y.. 1994. Cornputer-Aided Methods h r Design Evaluation of Track Systems, SAE
Paper No.94 1675, pp. 1-9.
Appendix A.
The Effects of Design and Operating Factors oa the Handling of Tracîced
Vehicles under Continuous Trapezoidal Load Distribution over the entire track
A.1. Contact Lengtb, L
Table A-1 Sprocket torques at various turning radii with different track-ground contact lengths at a vehicle speed of 7.5 km/h under continuous trapezoidal load distribution over the entire track.
Table A-2 S procket torques at various turning radi i with di fferent track-ground contact lengths at a vehicle speed of 14.2 km/h under continuous trapezoidal load distribution over the entire track.
1
Tuming
Radius
(ml
5
1 O
20
50
1 O0
Turning
Rad i us
<ml
5
1 O
20
50
L = 1.90 m
N.m
L = 4.75 m.
N.m
Outer
7810
5460
3886
2407
1947
L = 3.8 m
(Reference)
N-m
L = 2.85 m
N.m
Outer lnner
1 O0
Outer
25438
24033
21689
15153
L = 1.90 m
N.m
L = 5.7 m.
N.m
Inner
-5191
-2959
-1195
83.70
544.6
Outer
21742
19740
15762
9509
6012
15598
12401
8665
4980
3363
Inner
-18997
-21082
-19125
-12594
Outer
8833
5483
3534
2291
Outer
28654
27295
23302
20740
14765
L = 4.75 m
N-m
lnner
-19110
-17273
-13286
-7021
-3522
-13022
-9920
-6181
-2491
-871.8
[mer
4010
-2470
-868.8
292.2
L = 2.85 m.
N.m
Inner
-20999
-24163
-23313
-18183
-12199
Outer
25777
24532
L = 5.7 m
N.m
1909
Outer
14837
11042
7431
4283
r
L = 3.8 m.
(Reference)
N.m
lnner
-23043
-22049
21866
15230
9983
Outer
28519
27670
664.5
lnner
-9919
-8137
-4804
-1707
Outer
20948
18883
15211
9144
1
lnner A
-25665
-25164 .
3049
Inner
-15451
-16047
-12641
-6578
-476.8
-23505
-18226
-12242
-19391
-12744
-7493
-7368 5798
25983
20712
14732
-3228 9936
Table A-3 Sprocket torques at various turning radii with different track-ground contact
lengths at a vehicle speed of 2 1.3 km/h under continuous trapezoidal load
Table A-4 Sprocket torques at various turning radii with different track-ground contact lengths at a vehicle speed of 29 kmh under continuous trapezoidal load
distribution over the entire track.
distribution over the entire track.
N.m 1 (Reference)
1
Outer [nner Outer Inner Outer Inner Outer Inner 1
-6132 15489 -10149 19678 -13972 22952 -16586
7358 4175 11765 -8684 17881 -14848 22854 -19772
-2346 8682 -5846 14625 -1 1833 20410 -17629
3404 1 -622.3 5396 -2628 9680 -6929 14790 -12048
Tuming
Radius
(m.)
1 O
20
50
100 ,
L = 1.90 m
N.m
L = 2.85 m.
N.m
Outer
7874
4720
2756
1085
Outer
1 1896
7918
43 15
lnner
-3096
-1638
-39.98
583.0
L = 4.75 m.
N.m
L = 3.8 m.
(Reference)
N.m
Inner
-7317
4936
-1617
Outer
22286
20887
14948
Outer
17482
14317
8581
5368
L = 5.7 m.
N.m
2932 -265.91
lnner
-17078
-18016
-12300
Inner
-12781
-11440
-5912
-2709
Outer
25615
25184
20688
14788 9818 , -7166 ,
Inner
-19660
-22224
-18045
-12140
A.2. Tread of Vehicle, B
Table A-5 Sproc ket torques at various turning radii with different treads at a vehicle speed of 7.5 km/h under continuous trapezoidal load distribution over the entire track.
Table A-6 Sprocket torques at various turning radii with different treads at a vehicle speed of 14.2 krnh under continuous trapezoidal load distribution over the entire track.
Outer
25877
23969
19946
12497
7845
inner
-23231
-21502
-17473
-10010
-5355
Outer
24047
22066
18010
11056
6938
Inner
-21412
-19601
-15536
-8569
4448
Outer
21742
19740
15762
9509
6012
Inner
-19110
-17273
-13286
-7021
-3522
Outer
19003
17063
13322
7968
5128
lnner
-16363
-14591
-10843
-5479
-2638
Outer
15692
13903
10554
6318
4201
lnner
-13029
-11419
-8071
-3828
-1711
Table A-7 Sprocket torques at various turning radii with different treads at a vehicle speed of 2 1.3 km/h under continuous trapezoidal load distribution over the entire track.
Turning
Radius (Reference)
N.m
Outer Inner
17482 -12781
14317 -11440
8581 -5912
Outer Outer Inner Outer 1 Inner Inner
1 able A-u sprocKet torques at various rurning raaii witn airrerent ueaas at a venlcie speea
of 29 km/h under continuous trapezoidal load distribution over the entire track.
Tum ing
Radius
(ml
B = 2.54 m
(Reference)
N.m 1
Outer Inner Outer Outer
18316 -12546 17225 15489
15786 -12745 13893 11765 r
Inner Outer lnner
11356 -5966
7805 -4627
5638 -2757
- - -
Inner Outer 1 Inner
A.3. Longitudinal Offset of Vehicle Center of Gravity, c,
Table A-9 Sprocket torques at various turning radii with different longitudinal CG offsets at
a vehicle speed of 7.5 km/h under continuous trapezoidai load distribution over the entire track.
Radius
(m)
(Re ference) 1 N.m 1 N.m I Inner Outer lnner
1
Inner Outer Outer Inner
Table A- 1 0 Sprocket torques at various turning radii with di fferent longitudinal CG offsets at
a vehicle speed of 14.2 km/h under continuous trapezoidal load distribution over
the entire track.
1 Radius 1 N.m 1 N.rn (Reference) 1 N.m 1 N.m l
Table A- 1 1 Sprocket torques at various turning radii with different longitudinal CG offsets at a vehicle speed of 2 1.3 km/h under continuous trapezoidal Ioad distribution over the entire track.
Turning
Radius
(m)
1 able A- 1 f Sprocket torques at vanous turning rad11 with dltterent Longitudinal Cti oïtsets at a vehicle speed of 29 km/h under continuous trapezoidal load distribution over the entire track.
Turning
Radius
(m.)
N.m I N.m ( (Reference) 1 N.m
c, = -0.4 m
Outer Inner & c, = -0.2 m
Outer
14156
10410
7781
4967
c ,=Om
lnner
-8606
-7298
4940
-2200
c,. = 0.2 m
Outer lnner Outer
15489
11765
8682
5396
Inner
-10149
-8684
-5846
-2628
Outer
15356
11511
8437
5248
15049
Inner
-10102
-8416
-5591
-2478
-9605
11371
8462
53 18
-8283
-5628
-2552
A.& Track Width, b
Table A- 1 3 Sprocket torques at various t u m i n g radii with di fferent track widtbs at a vehicie
speed of 7.5 km/h under continuous trapezoidal load distribution over the entire track.
1 Radius 1 N.m 1 (Rrference) I N.m I N.m
Table A- 13 Sprocket torques at various tuming radii with different track widths at a vehicle
5
1 O
20
50
speed of 14.2 km/h under continuous trapezoidal load distribution over the entire
Outer
21695
19660
15619
9327
trac k.
L
1 O0
Turning
Radius
(ml
5
10
20
50
1 O0
Inner
-19068
5868
Outer
-39 15 6937
b = 0.01 m
N.m
-4447 -3377
Outer
20883
18833
15099
897 1
5646
6012 -3522
lnner
-15432
-16007
-12530
-6405
-3076
b = 0.45 m
( Reference)
N.m
lnner
-17504
-13711
-7549
20353
16861
10795
6406
Outer
20948
18883
15211
9144
5798
-17877
-14385
-8307
Outer
-17194
-13143
-6839
lnner
-15451
-16047
-12641
-6578
-3228
b= 0.90 m
N.m
Outer Inner
22163 21742
19740
15762
9509
-17273
Outer
21185
19053
15551
966 1
6218
b = 1.35 m
N.m
lnner
21879 -19500 -19110
19975
lnner
-15565
-16192
-12978
-7096
-3649
Outer
21691
19441
16166
10464
6803
-19232
Inner
-15931
-16547
-13591
-7900
-4234
-13286 16190
-7021 ' 10037
Table A- 15 Sprocket torques at various tuming radii with different track widths at a vehicle
Tuming
Radius
(m.)
speed of 2 1.3 km/h under continuous trapezoidal load distribution over the entire
trac k.
N.m 1 (Reference) 1 N.m N.m
inner Outer lnner Outer lnner Outer Inner
- 1 2694 17482 -12781 17835 -13057 18409 -13539
- 1 1342 14317 -11440 14638 -11748 15218 -12309
-577 1 858 1 -59 12 9029 -636 1 9817 -7151
-2572 5368 -2709 5810 -3 153 6435 -3778
Table A- 16 Sprocket torques at various twning radii with different track widths at a vehicle speed of 29 km/h under continuous trapezoidal load distribution over the entire track.
Turning b=O.OI m
1 Radius ( N-m
b = 0.45 m b= 0.90 m b = 1.35 m
(Reference) N.m N.m
N.m
A S . Height of Vebicle Center of Cravity, b
Table A- 17 Sprocket torques at various twning radii uith different CG heights at a vehicle speed of 7.5 hdh under continuous trapezoidal load distribution over the entire track.
Radius
(ml
5
10
20
Table A- 18 Sprocket torques at various turning radii with different CG heights at a vehicle speed of 14.2 km/h under continuous trapezoidal load distribution
N.m
over the entire track.
Outer
2 1778
19747
Tuming
Radius
(m)
( Reference)
N.m
Inner
-19147
- 1 7281
lnner
Outer
2 1742
19740
N.m
( Reference)
lnner
-191 10
- 1 7273
- 13286
-702 1
-3522
15767
9506
6010
Outer
21681
19728
15767
9513
6014
Outer inner
20525 -14935
1 867 1 - 1 5822
Inner
- 19047
-17261
-13291
-7025
-3523
-13281 15762
-70 1 8 9509
-3520 1 60 12
Table A- 19 Sprocket torques at various turning radii with different CG heights at a
vehicle speed of 2 1.3 krn/h under continuous trapezoidal load distribution
over the entire track.
Table A-20 Sprocket torques at various turning radii with different CG heights at a
vehicle speed of 29 kmdh under continuous trapezoidal load distribution
Turning
Radius
(m)
1 O
20
50
1 O0
over the entire track.
h = 0 1 . 8 r n
N.m
Turning
Radius
(m)
15
30
50
1 O0
Outer
17565
1430 1
85 10
5328
h = 1.3 m
(Reference)
N.m
lnner
-12915
- 1 1425
-5840
2669
Outer
17482
1-4317
858 1
5368
h = 1.8 rn
N.m
h = 0.8 m
N. rn
lnner
-12781
- 1 1440
-59 12
-3709
Outer
17199
14268
8649
5407
Outer
15239
1 1648
8563 L
h = 1.3 rn
(Reference)
N.m
lnner
-12376
- 1 1385 A
-5980
2748
lnner
-9946
-8565
-5725
Outer
15489
1 1765
8682
Inner
-10149
-8684
-5846
-2628 532 1 -2553 5396
Appendix B.
The Effects of Design and Opernting Factors on the Handhg of Tracked
Vehicles for Trnpezoidal Load Distribution on the Track Pitch under each Roadwheel
B. 1. Contact Lengtb, L
Table B- 1 Sprocket torques at various turning radii with different traçk-ground contact lengths
at a vthicle speed of 7.5 kmh for trapezoidal load distribution on the track pitch
under eac h roadwheel
Outer Inner Outer lnner Outer
Turning L = 1.90 rn L = 2.85 m 1 Radius 1 N..
L = 3.8 m L = 4.75 m L = 5.7 rn
(Reference) 1 N.m 1 N m
I
L able B-2 Sprocket torques at various turning fadi1 w t n ditterent track-grouna contact lengfns
i Inner
-22819
-21231
Inner
-19282
-17495
at a vehicle speed of 14.2 km/h for trapezoidal load distribution on the track pitch
Outer
2533 1
23624
-19194 24609 -22162
Outer
28045
26139
-14349 21643
Inner
-25376
-23740
under each roadwheel
Turning
Radius
(m)
5
10
20
50 1
L=1.90m
N.m
L = 3.8 m
(Reference)
N.m
L = 5.7 m
N.m
Outer
9144
5865
3724
2355
Outer
21206
19414
16583
10730
6936
Outer
28877
26105
24752
21094
15947
L = 2.85 m
N.m
L = 4.75 m
N.m
lnner
-4418
-2926
-1080
223.5
616.7
lnner
-15864
-16774
-14083
-8178
-4370
lnner
-20776
-23174
-22254
-18555
-13386
Outer
1561 3
12311
8196
4699
3258
Outer
25467
23441
21727
16433
11294 1 O0
lnner
-10946
-9617
-5619
-2128
-686.9
lnner
-18914
-20697
-19240
-13891
-8731 1956
Table B-3 Sprocket torques at various turning radii with different track-ground contact lengths at a vehicle speed of 2 1.3 kmih for trapezoidal load distribution on the track pitch
under each roadwheel
Table B-4 Sprocket torques at various turning radii with different track-ground contact lengths
Tuming
Radius
(m)
at a vehicle speed of 29 kmih for trapezoidal load distribution on the track pitch
L = 1.90 m
N.m
under each roadwheel
Outer
1 IO0 1 21321 535.1 1 31371 -476.31 65871 -39361 113441 -87021 16127 1-134901
lnner
-3153
-1794
-116.3
L = 2.85 m
N.m
1 O
20
50
Turning
Radius
(m)
Outer
12752
9060
4819
7873
4836
2825
L = 1.90 m
N.m
L = 2.85 m
N.m
L = 3 . 8 m
(Reference)
N.m
lnner
-8325
-6192
-2141
15
30
50
1 O0
L = 4.75 rn
Outer
18226
16016
10337
L = 5.7 m
Inner
-13602
-13280
-7700
L = 5.7 m
N.m
l
L = 3 . 8 m
(Reference)
N.m
-19068
-21526
-18661
22221
21296
16501
L = 4.75 m
N.m
N.m
Outer
8290
5017
3639
N.m
i
Inner
-2517
-1716
-712.0
Outer
12091
8224
5876
-16933
-18548
-13888
Outer
22951
Outer
16390
Inner
-6725
-5116
-3024
25289
24368
21267
2544
Inner
-16016
Inner
-10960
11436
Outer
20092 --
Inner Outer
245.2
13641
IO428
-8703 16365 -13643 d
Inner
-14071
lnner Outer
-10676
-7649
-3762
19227
16517
3735
-13784 21287 -185601
-962.2 6513
B.2. Tread of Vehicle, B
I able 15-3 >proclcet torques ot vanous tuming radil wim ditterent treaas a[ a vemcie speea or 7.5 hh for trapezoidal load distri bution on the track pitch under each roadwheel.
- -
B = 2.54 m
(Retèrence)
Inner
Tuming
Radius
(ml
Inner Outer Inner Outer Imer Outer 1 r
B = 3.04 m
N.m
B = 1.90 m
N.m
Table 8-6 Sprocket torques of various turning radii with different treads at a vehicle speed of
B = 3.80 m
N.m
11.2 kmh for trapezoidal load distribution on the mck pitch under each roadwheel.
Tuming l- B = 2.54 m
(Reference)
N-rn
Radius
(ml
Outer Inner Outer Inner
18985 -13799 16271 -11105
16909 -14269 13622 -10957 1
Outer lnner lnner
Table B-7 Sprocket torques of various turning radii with different treads at a vehide speed of
Table B-8 Sprocket torques o f various turning radii with different treads at a vehicle speed o f 29 km/h for trapezoidal load distribution on the track pitch under each roadwheel.
2 1.3 km/h for trapezoidal load distribution on the track pitch under each roadwheel.
Turning
Radius
(m)
10
20
50
1 O0
B = 3-04 m
N.m
B = 3.80 m
N.m
Outer
16189
13305
7726
4719
Outer
13683
10267
5499
3684 A
B = 1.90 m
N.m
lnner
-11660
-10540
-5070
-2063
B = 2.54 m
(Reference)
N-rn
Inner
-9107
-7451
-283 1
-1025
Outer
20678
19863
1.103 1
9082
B = 2 . 1 7 m
N.m
Outer
18226
16016
10337
6587
Inner
-15586
-17115
- 1 1306
-6436
Outer
19749
18215
12355
7937
Inner
-13602
-13280
-7700
-3936
Inner
-14919
-15482
-9726
-5290
B.3. Longitudinal Offset of Vehicle Center of Gravity, c,.
Table B-9 Sprocket torques of various tuming radii with different longitudinal CG offsets at a vehicle speed o f 7.5 km/h for trapezoidal load distribution on the track pitch
Table B- 10 Sprocket torques of various turning radii with different longitudinal CG offsets
at a vehicle speed of 14.2 km/h for trapezoidal load distribution on the track pitch under each roadwheel.
Radius
(m.)
3
10
20
50
1 O0
N.m
Outer
16340
15671
12925
7521
4553
Inner
-11241
-13036
-10425
-4967
-1986
N.m
Outer
19190
lnner
-13913
(Re ference)
N m
Outer
21206
19414
16583
10730
6936
N.m
Inner
-15864
-16774
-14083
-8178
-4370
Outer
22546
20644
17892
11895
7755
17804
14897
9166
57iO
ïnner
-17229
-17991
-15391
-9344
-5189
-15175
-12397
-6613
-3143
Table B- 1 1 Sprocket torques of various turning radii with different longitudinal CG offsets at a vehicle speed of 2 1.3 km/h for trapezoidal load distribution on the track pitch under each roadwheel.
1 Radius 1 N.m N.m 1 (Reference) 1 N.m 1 N.m 1
at a vehicle speed of 29 km/h for trapezoidal load distribution on the track pitch under each roadwhesl.
Turning
Radius
(m)
Outer
20831
18404
12545
8246
Outer
16047
14405
9076
5503
c,=Om c, = 0.20 rn + = 0.40 rn
( Re ference) N.m N.m
lnner
-16251
-15660
-991 1
-5597
f
lnner
-11446
-11663
-6438
-285 1
Outer
18226
16016
10337
6587
I
Outer 2
19801
17313
11538
7534
lnner
-13602
-13280
-7700
-3936
lnner
-15187
-14574
-8902
-4885
BA. Track Width, b
Table B- 1 3 Sptoçket torques o f various turning radii with different track widths at a vehicle speed o f 7.5 km/h for trapezoidal load distribution on the track pitch under each
Table 8-1 4 Sprocket torques of various tuming radii with different track widths at a vehicle speed of 14.2 km/h for trapezoidal load distribution on the track pitch under each
roadwheel.
roadwheel.
h = 0.45 m
( Reference)
N-m
I
Turning
Radius
(m.)
b=0.01 m
N.m
b = 0.90 m
N.m
5
10
20
50
100
b = 1.35 m
N.m
Outer
21818
20049
17069
1 1299
7366
Tuming
Radius
(m.)
5
10
20
50
100
b = 1.35 m.
N.m
Outer
21988
20296
17480
11860
781 1
Inner
-19389
-17648
-14614
-8815
-4877
Outer
21736
19620
17084
Inner
-19538
-17890
-15024
-9376
-532 1
Inner
-16101
-16938
-14571
b=O.OI m .
N.m
b = 0.45 m
(Reference)
N .m
Outer
21655
19838
16711
10792
6960
Outer
21142
19387
16517
10604
6826
Outer
21206
19414
16583
10730
6936
b = 0.90 m.
N.m
lnner
-19243
-17443
-14258
-8308
-4470
Outer
21698
19892
16803
10925
7067
Inner
-15840
-16753
-14018
-8053
-4260
lnner
-15864
-16774
-14083
-8178
-4370
Outer
21402
19492
16776
11094 11659
7715
Inner
-19282
- 1 7195
-14349
-8441
1577
Inner
-15941
-16834
-14271
-8543 -9108
-5 150 7247 -4681 1
Table B-15 Sprocket torques of variaus tuming radii with different track widths at a vehicle
speed of 2 1.3 km/h for trapezoidal load distribution on the track pitch under each
Table B- 1 6 Sprocket torques of various tuming radii with di fferent track widths at a vehicle speed of 29 km% for trapezoidal load distribution on the track pitch under each
roadwheel.
Turning
Radius
(m)
10
20
50
1 O0
b = 1.35 m
N.m
b = 0.01 m
N.m
b = 0.45 m
Outer
17149 -
14412
1 1 176
7129 L
Outer
16287
13541
10333
6439
b = 0.90 m
Inner
-11543
-11425
-8394
-4378
lnner
-10881 .
-1Q579
-7554
-3688
( Retèrence)
N.m
N.m
Outer
16390
13641
10428
6513
Outer
16687 .
13937
10713
673 5
Inner
-10960
-10676
-7649
-3762
Inner
-11189
-10963
-7932
-3984
B.1. Height of Vehicle Center of Gravity, h
Table B- 1 7 Sprocket torques of various tuming radii with different CG heights ai a
vehicle speed of 7.5 km/h for trapezoidal load distribution on the track
pitch under each roadwheel
Radius 1 1 (Reference)
N.m
h = 1.8m h = 1.3 m Turning
Table B-18 Sprocket torques of various tuming radii with different CG heights at a
h = 0.8 m
5
1 O
20 - -
50
1 O0
vehicle speed of 14.2 km/h for trapezoidal load distribution on the track
Outer
2 1702
19892
16798 - - - - -
1092 1
7065
pitch under each roadwheel
Inner
- 19287
- 1 7496
-14334
-843 8
-4576
Turning
Radius
(m)
5
10
20
50
1 O0
Outer
2 1698
19892
16803 . - - .
10925
7067
Outer
2 1 669
19887
16808 ~-
10928
7068
fnner
- 19282
-1 7495
- 14349 - - -
-844 I
-4577
lnner
- 19250
- 1 7490
- 14354
-8444
-4579 L
h = 0.8 m
N.m
Outer
2 1325
19571
16573
10712
6927
inner
-15991
-16941
- 14073
-8161
-436 1
h = 1.3 m
( Reference)
N.m
h = 1.8 m
N.m
Outer
S 1206
19414
16583
10730
6936
Outer
2088 1
19162
16578
10747
6944
Inner
-1 5864
- 16774
- 14083
-8 178
-4370
In ner
-15418
- 1 6506
-14077
-8 195
-4378
Table B- 19 Sprocket torques of various turning radii with different CG heights at a vehicle speed of 2 1.3 k m , for trapezoidal load distribution on the track pitch under each roadwheel
Turning
Radius
(m)
h = 0.8 m.
N-m
Table B-20 Sprocket torques of various turning radii with different CG heights at a
vehicle speed of 29 km/h for trapezoidai load distribution on the track pitch under each roadwheet
h = 1.3 rn
( Reference)
N.m
Outer
1846 1
16088
10266
6543
1 Radius I N.m
h = 1.8 m.
N.m
Outer
17727
15845
10399
6628
Inner
- 13922
-13358
-7628
-3 893
lnner
- 12923
- 13099
-7762
-3978
(m)
15
30
50
I
Outer
18226
16016
10337
6587
lnner
- 13602
- 13280
-7700
-3936 i
Outer
16414
13630
10325
h = 1.3 m
(Reference)
N.m
h = 1.8m
N.m
Inner
-1 1078
- 1 0669
-7544
Outer
16390
13641
1 0428
Outer
16000
13572
10505
I
lnner
- 1 0960
- i 0676
-7649
Inner
-103 14
- 10597
-7727
Appendix C.
Tbe Effects of Design and Operating Factors on the Handling of Tracked
Vebicles for Conceatrated Load under eacb Roadwheei
C.1. Contact Lengtb, L
Table C- 1 Sprocket torques at various turning radii with di fferent track-ground contact lengths at a vehicle speed of 7.5 kmh for concentrated load under each roadwheel.
Tuming
Radius
(ml
5
1 O
20
50
1 O0
L = 1.90 m L = 2.85 m L = 3.8 m L = 4.75 m L = 5.7 m
N.m N.m ( Reference) N.m N.rn
Outer Inner Outer Inner Outer lnner Outer Inner Outer lnner
-20276 23560 -22382
-19571 23928 -21608
-18261 23361 -20926
1 ame L-1 sprocicer torques ar vanous rurning raaii witn airrerenr uacu-grouna contacr iengms at a vehicle speed of 11.2 km/h for concentrated load under each roadwheel.
Tuming
Radius
(m.)
N.m 1 N.rn 1 (Reference) 1 N.m
Outer Inner Outer
6251
3930
2470
2018
Inner
-3398
-1311
105.8 r
-553.9
Outer
12139
8112
4734
3364
Inner
-9504
-5532
-2164
-793.3
N.m
Outer
18732
16250
10688
6960
Outer
22182
20993
16169
11215
Inner
-16189
-13769
-8139
-4394
Inner
-19601
-18532
-13629
-8652
Table C-3 Sprocket torques at various tuming radii with different track-ground contact lengths
at a vehicle speed of 2 1.3 km/h for concentrated load under each roadwheel.
Tuming
Radius
1 ame L-u 3procKer torques at vanous tuming raaii wtn airrerenr uaclt-grouna conracr ienguis
(ml
1 O
20
50
IO0 1
at a vehicle speed of 29 k r n h for concentrated load under each roadwheel.
L=1.90m
N.m
Radius 1 N.m 1 N.m 1 (Reference) 1 N-m 1 N-III
L = 2.85 rn
N.m
Outer
9131
5514
3 130
2277 1
L = 3 . 8 m
(Reference)
Inner
-4542
-2517
-437.4
388.6 1
15
30
50
100
Outer Inner
N.m
Outer
18523
16100
IO496
6752 1
Outer
10129
5900
4174
2827
lnner
-14012
-13405
-7864
4104
13696
9573
5052
Inner
-4349
-2635
-1262
-40.19
-9412
-6755
-2382
Outer
13665
9064
6423
4028
3290 1 -630.8 1
Inner
-8311
-6002
-3589
-1260
Outer
17525
14219 -
10877
6844
lnner
-12014
-1 1289
-8111
-4097
Outer
20585
I9344
16628
Il567
Outer
22905
22590
21119
16333
lnner
-14331
-16436
-13906
-8838
Inner
-1555%
-19610
-18405
-13613
C.2. Tread of Vehicle, B
Table C-5 Sprocket torques at various tuming radii with different treads at a vehicle speed of
7.5 krnh for concentrated load under each roadwheel.
i ame L-O 3procket torques at vanous turning raciii wxtn dilterent treaas at a venicie speed ot
14.2 k m , for concentrated load under each roadwheel.
Turning
Radius
5
I O
20
50
1 O0
Radius1 N.m 1 (Reference)
B = 3.80 m.
N.m
B = 2.54 m.
(Reference)
B = 1.90 m.
N-m
B = 3.04 m.
N.m
B = 2.17 m.
N.m
Outer
22930
22147
19771
13591
8871
Outer Inner Outer Inner Outer lnner Outer Inner Outer Inner
Inner
-20703
-19783
-17323
-1 II08
-6381
5
10
Outer
21494
20631
18165
12245
7967
Inner
-19280
-18268
-15717
-9762
-5478
22751
21994
Outer
19638
18735
16255
10782
7029
Inner
-17426
-16371
-13806
-8299
4540
-16940
20
50
1 O0
Outer
17370
16500
14124
9290
61 I l
20042
13792
8992
Inner
-15135
-14130
-11673
-6807
-3622
Outer
14402
13764
11652
7710
5177
21919
-17567
- 1 1246
-6427
lnner
-12084
-11378
-9198
-5227
-2688
- 1 5879
-9783
-5444
18354
12330
8009
-16686
-19404 -18077
16250
10688
6960
20626
20328
18732
-13769
-8139
-4394
-15448
-16189
13687
8873
5857
18216
16275
-1 1186
-6320
-13557
-13718
9369
5490
-6803
-2920
-3291 3951
15510
-1381
-10949
12846 -10240
Table C-7 Sprocket torques at various tuming radii with difièrent treads at a vehicle speed of
Turning
Radius
(m)
I 1 O
20
50
1 O0
2 1.3 km/h for concentrated load under each roadwheel.
Outer tnner Outer Inner
Table C-8 Sprocket torques at various tuming radii with different treads at a vehicle speed of
29 km/h for concentrated load under each roadwheel.
Radius
1 O0
(Reference)
Outer Inner Outer inner
15478 -10345 13040 -7866
11873 -8914 9458 -6445
C.3. Longitudinal Offset Vehicle Center of Cravity, c,
Table C-9 Sprocket torques at various turning radii with dinèrent longitudinal CG offsets at
a vehicle speed of 7.5 km/h for concentrated load under each roadwheel.
Table C- 1 0 Sprocket torques at various twning radi i with di fferent longitudinal CG offsets at
a vehicle speed of 14.2 iunh for concentrated load under each roadwheel.
1 Radius 1 N.m 1 (Reference) 1 N.m [ N-m I
5
10
20
50
1 O0
Outer
15953
15100
12449
7199
4449
Inner
- 1 1341
-12610
-9974
-4646
-1882
Outer
18412
17094
14482
9098
5780
lnner
-13637
-14585
-12002
-6547
-3214
Outer
20328
18732
16250
10688
6960
Inner
-15448
-16189
-13769
-8139
-4394
Outer
21850
20103
17642
11867
7767
Outer
23035
21204
Inner
-16891
-17526
-15156
-9318
-5202
fnner
-17964
-18599
18700 -16205 k
12743
8345
-10192
5779
Table C-11 Sprocket torques at various turning radii with different longitudinal CG offsets at
Table C- 1 2 Sprocket torques at vanous turning radii with different longitudinal CG offsets at
a vehicle speed o f 21 -3 km/h for concentrated load under each roadwheel.
a vehicle speed of 29 km/h for concentrated load under each roadwheel.
Turning
Radius
(ml
Turning
Radius
(ml
15
30
50
1 O0
c, = -0.40 rn
N.m
Outer
Outer
19170
15260
1 1674
c , = -0.40 m
N.m
Inner
-9516
-9810
-5259
-2202
c, = -0.20 m
N.m
1 O
20
50
1 O0
Outer
13248
1 1069
8712
5595
Outer
16484
14487
9163
5612
13901
12513
7891
4852
c, = -0.20 m
N.m
Inner
-7529
-8094
-5938
-2847
lnner
c,=Om
( Re ference)
N.m
Outer
15566
12843
9923
6247
hner
-15125
-13112
-9640
Inner
c,=Om
(Reference)
N.m
Outer
20138
17471
11706
7656
Inner
-9980
-9899
-7157
-3500
Outer Outer
17525
14219
10877
6844
Inner
-14012
-13405
-7864
4104
-15597
-14761
-9075
-5009
c, = 0.20 m
N.m
21391
18658
12718
8354
-12029
-11797
-653I
-2962
Inner
-12014
-1 1289
-81 t 1
-4097
Outer
-16834 I
-15932
-10085
5706
c,, = 0.40 m
N.m
18523
16100
10496
6752
-13714
-12329
-8905
tnner Outer
20525
16056
12415
lnner
CA. Track Width, b
Table C- 1 3 Sprocket torques at variuus tuming radii with different track widths at a
vehicle speed 7.5 of km/h for concentrated load under each roadwheel.
1 Radius N.m 1 (Reference) I N.m l N.m
Table C-14 Sprocket torques at various tuming radii with different track widths at a vehicie speed of 14.2 km/h for concentrated load under each roadwheel.
5
1 O
20
50
1 O0
Tuming b = 0.01 m
Radius N.m
Ou ter lnner
Outer
19602
18678
16158
10646
6920
lnner
-17396
-16316
-13710
-8163
-443 1
Outer Inner Outer
f9743
18900
16531
11161
733 1
lnner
-17513
-16533
-14082
-8679
-4842
Outer
19919
19161
16950
1 1722
7775
19638
Inner
-17661
-16790
-14501
-9239
-5286
-17426
18735 -16371
16255
10782
7029
-13806
-8299
-4540
Table C-15 Sprocket torques at various turning mdii with different track widths at a
vehicle speed of 2 1.3 km/h for concentrated load under each roadwheel.
Table C- I 6 Sprocket torques at various turning radii with different track widths at a
vehicle speed of 29 kmh for concentrated load under each roadwheel.
Tuming
Radius
(ml
10
20
50
1 O0
Tuming
Radius
(m)
1 O
20
50
1 O0
b = 0.45 m
( Reference)
N.m
b=0.01 m
N.m
b = 0.90 m
N.m
Outer
18523
16100
10496
6752
Outer
18455
16052
10397
6648
b = 0.01 m
N-m
b = 1.35 m
N.m
Inner
-14012
-13405
-7864
4104
Inner
-13974
-13361
7766
-3999
Outer
17438
14130
1 0787
677 1
Inner
-11955
-11203
-8022
-4024
Outer
18725
16243
10791
7064
b = 0.35 m
(Reference)
N.m
b = 1.35 m
N.m
Outer
17525
14219
10877
6844
b = 0.90 m
N.m
Outer
18163
14904
11571
744 1
lnner
-12014
-11289
-8111
-3097
Outer
17775
14482
Il142
7064
,
lnner
-12447
-11952
-8802
-4693
Inner
-14326
-13763
-8652
4908
Outer
Inner
-12185
-11543
-8376
-4317
-14128
-13535
-8159
-44 1 6
19052
16486
11285
7555
CS. Height of Vehicie Center of Cravity, h
Table C- 17 Sprocket torques at various tuming radii with different CG heights at a
vehicle speed of 7.5 kmh for concentrated load under each roadwheel.
1 Radius I N.m 1 (Reference) I N.m
Turning h = 0.8 m
(ml
5
10
20
50
1 O0
h = 1.3 rn h = 1.8m
Outer Inner
N.m
Outer Outer Inner
- 1 7426
-16371
- 13806
-8299
4540
19654
18735
16250
10779
7028
Inner
19599
1873 1
16259
10784
703 1
-17380
- 16367
-13810
-8301
-454 1
- 1 7147
-16371
- 13802
-8296
-4539
1
1 9638
18735
I6255
IO782
7029
Table C- 19 Sprocket torques at various tuming radii with different CG heights at a vehicle speed of 2 1 -3 km/h for concentrated load under each roadwheel.
1 Tuming 1 h =O.8 rn 1 h = l . 3 m 1 h = 1.8m
Table C-20 Sprocket torques at various tuming radii with different CG heights at a vehicle speed of 29 km/h for concentrated load under each roadwheel.
Radius
(ml
10
20
50
1 O0
Tuming
Radius
(ml
15
30
50
1 O0
N.m
Outer
18713
16161
10432
6716
( Reference)
N.m
lnner
-14327
-13471
-7800
-4067
h = 0.8 m
N.m
Outer
18523
16100
10396
6752
N.m
h = 1.3 rn I h = 1.8m
( Reference) N.m
N.m
Outer
17443
14188
10769
6753
lnner
-13012
- 13405
-786.)
-4104
Outer
17953
1 593 5
10552
6787
Outer
17525
14219
10877
6844
Inner
-121 1 i
-1 1262
-8002
-4005
lnner
-13172
- 13228
-792 1
-4 139
lnner
-12014
-1 1289
-81 1 1
-4097
Outer
1701 7
Inner I
-1 1005
14162
10955
6929
-1 1220 1
-8191 1
-4182