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Michigan Stave
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THEORY CF_TRACTIVK OF A HHEEL IOVILG

ON THIN VISCOUS IfiUD FILI

by

BONG—S IHG CHLUG

A THESIS

Submitted to
THE COLLEGES OF k3RICULTURE AND EVG'}75’1”
of Michige n State University
in partial fulfillment of the requiremr:nts
for the degree of

Mil “T R OF SCIEN CE

Department of Agricultural Engineering

1965

ABSTRACT

THEORY OF TRACTION OF A WHEEL MOVING
ON A VISCOUS MUD FILM

by Bong-sing Chang

A supersaturated viscous oil overlaying a hard,
plane ground is often critical to locomotion in many
areas due to the lubricating effect of the thin layer
of viscous mud.

In the theoretical analysis of this problem, equé
ations for the total lift force and the friction force
of a wheel moving on a thin layer viscous mud were ob-
tained. In order to illustrate the equations, a simpli-
fied numerical example was described.

A Ring-disk Viscometer was developed to measure the
viscosity of the mud. Soil was packed in the viscometer
and allowed to dry. After water was added a rotating
isteel disk was pressed against the soil. Strain gages
were used for the torque and axial force transducers,
which indicated the shear stress and normal pressure.
The result of the measurement showed that the shear
stress increases linearly when the normal pressure in-

creases. A considerable influence on the shear stress

Bong-sing Chang

from the velocity was also found. This means that
for a given load, a vehicle can obtain a greater pull

when its slip speed is increased.

Approved M W

Major Professor

Approved CEZL1L¢TCQéA<SQZ

Department Chairman

ACKNOWLEDGEMENT

The author wishes to eXpress his sincere thanks
to Dr. Sverker Persson for his continual suggestions
and guidance during the investigation.

Sincere appreciation is extended to Dr. T. H. Wu
of the Department of Civil Engineering and Prof. H. F.
McColly of the Department of Agricultural Engineering
for serving on the Guidance Committee and for their
helpful suggestions.

The author feels deeply indebted to Dr. A. w.
Parrall, former chairman of Agricultural Engineering
Department and Dr. C. w. Hall, chairman of Agricultural
Engineering Department for granting the graduate re-
search assistantship that made this thesis possible.

Partial support has also been received from the-
U. 8. Army, Detroit Procurement District which is also
deeply appreciated.

The efforts and assistance of Mr. James Cawood and
Mr. Glenn Shiffer are sincerely appreciated.

Special credit must be given to Mr. and Mrs.
Nung-shing Chang, the author's parents and Mr. and Mrs.
Jain-Li Kao, the author's best friends for their
encouragement and financial support that helped to
complete this work.

The author wishes to eXpress his deepest sorrow

to Mrs. Chei-Quen Chen for her husband was dead in an
air crash in Taiwan in 1964. Mr. and Mrs. Chen en-
courage and support me to come to the United States

to fulfill my advanced study.

May, 1965

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Fig. l; Tapered bearing moving over the
lubricant film.- ___________________ _

C\

Fig. 2; The deformatio on of the when? in the
thin layer 0: Y‘acou; Mud ------ - 9
Fig. 3; Analysis and trarsf ornaticn cf the

i...)
I.._J

velocities.- —————————— _ _____________

.)

Fig. 4; The element of the.f1uid,~a_-______ .

.42

“e - mu .
rig. a; lne forces on the el c.ent-~-————--- l4
.. "v- .r m, f‘ ‘ r ‘ ,3 ’f‘ i ’ v
Fig. o; he assumed contact zone.— ————— --—- l?
V‘- "" r-1 ('1‘ v ”5' ‘\ ‘3. ’\ ‘ fl. -‘ w ‘I ‘h”
rig. /; ire forces n any point 0: the saiiace
1"

Of tiAe Alt—)8]..-——-—-—-—-———————-—-——_— l_'_3
*1: ~. " (Tl‘ 1". . .- ' - “ . T. ,
rrg. 5; ins DGfOTnathfl 01 a a'neel on the

, n. t ~ - Ir 1‘ I :w 1" VYH’L 7“ flirt
mud lllm l: a 4—5 t:cord Hate Sdobeh cu

Fig. 9; Simplified di atran of Eir. 8 for
calculation exampl — ————————— ——-—— 22

I‘ . 4’ f7“ ._.: 1 V . ~z- ‘ 4" a” . v“ ' r7
lig.lJ; lne coaalal-cgliucer bJ‘c Visccmeter 2;
.1 r— {'1 ~4‘ '7 1" "0'1. m u- v n f) '1‘
rig.ll; ihe cone—plate t y3 rascanetcr.—-—-- a;
‘71 ' M r -"1‘ ' T, . ., .,2 , v—w J- 7 PM"
b15.12; lne disa type Visceneeer.—- —————— ~~~w a;
'1 - ,\ rm 1 - - ,° ,, a 4., . ,, m , ”
fl”.1/' he conz—c" Inoiicar type Vls CMQLCYS )0
‘ m“ r r ’1 7
£15.14, ihe Cone Viscomete3.— ——————————— --—~ )1
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rig.lj; he Hing-diam recomcte;.-————-—~-—- 2a
m- N - r'i -- .-: ., : 1 ~ . a n
lb.lu; lLe dealgn principle oi the Vis-
cometers .— ———————— — ------- - —————————— 3;
r F“ - 4-n‘ .-= P .1 'a ' . ‘
Fig.17; ne tObdi View oi the slag dick
' . . 1 . 4' ,r‘ - a, r ,‘ ~ "v. \*vr‘w" ‘2':
Viscometer ano its recording sttems 1)
fl: ‘. 1 Q ‘ - ‘r‘ 4' a a rv‘ r- m w ’ 7" fl
rib.l8; soil sample r t.c Lint—clan visco-
» r \ .0 , L ,fi —-,—
metrr belore UCuUL.€.— ————————————— — pg
‘ V - N 'v- r 1. we ‘ r\ P ‘1 ' Y ‘/~
£15.10; Durlnb tCStlflb \hlng—urrh <cometrr) ,c
‘ ’3 .‘ 4 4— ,‘ _..¢.: , ‘ 1-
Fig.a0; soil sample after t»serng (ring—bis;
hi lscometer --———~ ————————————————————— 20

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PART

F—i

‘ f‘ ' f-r'W"F ." *' v-xrf‘ r‘m
lHnOfiuilQnL mynululu

I. Introduction

A supersaturated viscous soil overlaWing a hard,
plane ground is often critical to locomotion in many
areas. The purpose of this thesis is to evaluate the
pressure and friction force under a wheel moving across
a relatively thin viscous mud overlaying a rigid,
flat bottom. A similar case is the journal bearing
which is supported by a loadcarrying thin film of
lubrication oil laying between the journal and the
bearing cell. The thin layer of viscous mud could be
considered as the lubricating film between a moving
wheel and the rigid ground.

Many similar examples are available for this
problem-—such as the vehicles moving on the wet high-

way, airplanes landing and taking off on a wet air trip,

0)

or vehicles crossing over field composed of a over—

saturated viscous mud overlaying a hard bottom.

Ci-

In the theoretical reatment of this studying only
two-dimensional flow is considered. The actual three-
dimcnsional flow will be considered in a more advanced
stage of study of this problem. Therefore, it is only

a very fundamental study of the traction of vehicles in
the viscous mud area.

Previous work which relates to this problem is

isted below.

t...J

limited. The studies found re

a
2

\N

resistance in extremely loose soils on the besis of
fluid dynamics since such soils behave like viscous
fluids rather than granular masses. The viscosity
resistance caused by fluid is called friction drag

( Df ). Another type of drag, Dp, is produced by the

. ,2 n . , .
dynamic pressure}7\ /a acting on tne normal progested

area of a body perpendicular to the flow lines. Thus

a

density of the fluid

velocitv

(.1

. ' ,i ., a-«h.,.‘,.‘ - .4..
normal progectec area peifienaieular to the
flow lines

H

A2 : wetted area
C? : friction drag coefficient
CD : pressure drag coefficient

The total drag D is sum of the above equations

V .
D I: C 1 f -—— r
t d 2 8

Where; A

mc+
p

*d

4

the immersed body

,4
{1.
c *-
b4
i“

st snag ar

'1”.
honest

- . l '
paraoolic

) depends on a constant value C and Reynolds num-

of dynamic vis—

loose mud indicate

rectangular shape

shape

F‘
5“
/

I‘.)

. Review of Literature on the Lubrication Theory
Lubrication and viscosity, which are interconnected,

play a most vital role in our great and ctmplex civili—

I ' " 7" _' ' 5.. ‘ ’fi 1‘ «‘21 5‘: " ‘e " a“ VI "
ration. ViscOSity is daiihed as the pL/rrcai property

* '1

1 'I 7 ' ’ z '\ "V i 1 . \ ‘ f I I ‘~ I f *’ . V V '7‘ Y -‘ "

of a fluid vnich offers resistance to relative motion

n . " ' " T “e A ‘ V“ ‘ “‘ " . ' ‘ . ’ 1‘ I‘ 1 . x

0: its parts. it corresponds to an inteinal fluic
. . V‘ ‘ r ‘ ‘ v 1 r" 1“ ' i ‘: ’ V" , In I ' xx .1‘ P i -‘ ’ I‘ “, ’ a '. ~. (— --,

friction Riguuveq sy the molecules oi the iu rrcant

as th

(5)

' 91 .V + ‘v ,~ ‘, «I - ‘ +‘I
v lira pest one anotne . The greater this

(1

—‘

I . - " - ‘4\ r 4' ' ,w l‘ :I . c‘ ' " " 1
relative motion, the greater lo tne internal resis—
. , .L. A‘ r '1 . .1 - "1‘” ‘
tance that CH5 lubricant Oil-rs.
m A 0" -: V V " V ' I K - ’ 2
in: earliest tne‘ry of viscosit/ mas rue l‘L‘u
‘ 1 / iii.“ v f‘ ‘1 . . ’ '() w r- ‘r-7K‘. A I r y . , ‘N
i: lord b/ air isaac newton (loan-i/:,, Oh tr.n risccus,
'~ r 1 I j v . n \ ' a ‘4‘ I '. - .r‘, A - ' V r‘ a
or raurgar, 110w. newton eseerted that there is a

.1 _ Inn” r, ..+ ' .',,2 ,.m'. V, “4,,
sclic surfaces. in concentric cylirlers anion acre

:'~ ' f‘b -._.'
‘Al,) -L K.’ (./ ALI-.—
"‘ r'ma In: I: To rv" 1.-., x? J-C
U4KJu, I‘lAv‘l .V'.‘LA—L- Lu b 4

. i . - : -, ' 1 : ., p ,. . I;
cause Inlatlte rotatscnar motion cl one cylinder aitn

fihere;

on- - 4, 9 -- l min .., ..--. ‘_‘r

r- z (30': I l i01{\ii L: 0—1- V1 SC( :1 a. "I, O]: YW‘I;£1 L; "I {I 12L) y'. l‘ulvbf.)“I-J
as acsclute :lscosrtf/u.

(‘ O 3

[1 e ar‘a

T . 'v 1r ,7 4—. " 4 4- I, ..

l : relative veiccltf oi the two '3'r lzdrvs.

KC

surfaces or the

fi'

l. W 0

he

tween t:

(-3

nce b

Ci

dist

h

A

J- I

water

tlle

CI

6

lat th‘

1

.4.
b

ume

ON a

a
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y.
b;

a.

uorica’

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tlie

of

uncticn

‘0 ‘r‘
t
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tirNAOLL

CC

is a c

O W

velocity in the fl

h

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rexlace

v
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uation

l

l

e

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4.

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ea

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A.

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uatio:

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aco

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45 .gl-tl

11 beawfivv

I
t C

Throw
(—Jvtlvu

1'3

1

r t

ly was developed fo

S

4

V'- "3
‘g (

a

O

0"
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US

rcblec

sq
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"9‘
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I‘ll’l'

bea

8.

e

f".
1

Tf‘f‘
UL _

f
)

U

 

 

 

 

”II/r

 

u. A

31,4,
L4.

I‘ I
-x,

‘J

a t

w; o.”

In this case, it has been found ( Fuller, 1956 )

—‘

that the total liftirt :orce to the

1 . : '
Ocarina l S

 

 

 

 

 

‘Q L)
6/uUfi
P = .K b
2 P
h
2
where;
, -
A +
‘ l m l 1
K : +
p m' — (2+ m')(l+m'x[2) (1+m'xéfl)
1
.L
2~ m'
h
l
m' : - l
h
2
S
U : velocity of too bearing
The total friction force on the bearing is

 

 

U
P :flbfl k
hm ‘
K.
macro;
4 6
if : log;e (1+m') -
m' 2 + m'

b: width of the element

C?

3. Some Basic Assumptions.
Some basic as sumptionsm must be made in order to
limit the difficulties of this problem.
a), The wheel has a smooth surface.
b), T e wheel is in a cylindrical shape.(not
o), The viscous mud (a clay suspension) acts as a
h ewtonian fluid.

d), The thickness of the viscous soil film is
d

relatively small a: it is OVerlaziog a rigid,
flat ground.

e), The ground is assumed rigid, thgrefore, the
deflection of the ground by applying a weight
(w ieel) ca) be neglected.

p m ‘ v " p o [‘1‘ ruw‘nf‘
i), The assumed ceiormea cnape
Y‘

rubber tire with an air pre u 4 “9) in the
viscous film is shown by Fig. 2. “

Before the theoretical analyst" being made, the

“J
r)
r_J
t-)
'3
C).

as state

f'-
( )
:3
U)
{3
fr“
0)
O
U)
:3
O
C
r. _J

wing ;importart assumpV

a), This problem is analyzed as a two—dimensional
f .1. C W o

b), The pressure in the viscous soil film under
tha state of ELg. 2 is a variable only in
X-direotion. It means the pressure in y-
dlrection r»m"in3 a constant. The reason ie
that the thiorness of the film compared to
the other dimensiogs is small.

0
v
a
w
('3
<
(D
H

o ity in the film is a function of
both X and i directions.

d), The viscosity of the mud susceiisich as it is
’ ing passed through by t-e uhe el remains
ta t

e), The viscous mud is income re ssible.

(II) (I)

 

 

 

B:
an
U:
H:

D:
h x
h 8
h:

‘3-
5'
p

 

 

 

radius of wheel

angular velocity of the wheel

velocity of the vehicle

thickness of the viscous soil £11m before passage
of the wheel

deformation distance of the wheel bottom
thickness of the film under point B

thickness of the film under point 3'

thickness of the film under any point

Fig.2. The deformation of the wheel in the
thin layer of viscous mud.

*

10
4. Characteristics of velocities;

There are two velocities present in this problem,
namely U andzd; U is the forward velocity of the wheel
center,tois the angular velocity of the wheel. There-
fore, the linear velocity of any point on the wheel
surface is V=Rwahich is relative to the wheel center,
where R is the radius of the wheel.

In order to give a clear idea of the characteristics
of the velocities, it is helpful to consider U and V
separately. Fig. 3, shows the transformation of the
velocities U and V. f

In Fig. 3;

(a) is a normal picture of a moving wheel in the
viscous film.

(b) shows U and V separately and relative to the
ground.

From here below the velocities are relative
to the wheel center.

(0) 1. shows a flow with a constant velocity -U
passing a stationary wheel.

(c) 2. shows the velocity -V =-noR of the wheel.
V is assumed parallel to the ground surface.

(d) 1. shows the velocity distribution in the fluid
caused by the velocity U. In front of the
wheel at point a, the distribution of the
velocity is a straight line as shown in
(d) la. Point b is at the wheel surface
where the speed is zero and the velocity
distribution is a parabola which is shown by
(d) lb. See also Eq. (8).

or

 

11

 

 

(a)

A o

 

 

(b). Relative to ground

 

 

Fig.3:

(0). Relative to wheel center

Analysis and transformation of the
velocities

12

 

 

 

 

 

 

H
N

 

 

 

 

 

 

 

 

I?
‘ -V
V_,
v T
2:: YE. v h ( scale double )
v I
\1 —x
-U 0

 

(e)

Fig.3: Analysis and transformation of
the velocities ( continued )

15

(d) 2. shows the velocity distribution of the
flow corresponding to —V.

(e) shows the combination of the velocity dis-
tribution of (d) lb and (d) 2 which repre—
sents the velocity distribution of this
problem.

1

lhe absolute values of U and V wil usually not

F4

be equal. They have a relationship which is stated as

*4
v

wzm - s) = U or v =wR = (

 

where s is the slip.

11 ”1

If s: 0%, no slip eXis s nd U:V:wn. lhe velocity

distribution under this condition is the same as Fig. 3,

(d) la, But, actually, the slip can not be avoided.
If 3:100», U20, which means that the vehicle is "stuck".
By Fig. 3e, the.boundary condition of the velocity

function, v:F (U,V,h,y,x , under the ssumpticns they

are
A -" .
n v:—U ior ‘20 . . .
( ) J }and X values Within the ,
reiifln Ln;' 1? fly; £3. (5)

(B) Vt-V for VI“

v is the velocity in the fluid in the X-direction.

l4
5. The velocity function v=F (U,V,h,y,x)
In the general case, a small element as shown by

Fig. 4 is used

 

» (a)
dx f ‘
5* *5“ H

*3

0—1—4

fr/Ifi/rr;x

 

 

 

 

dY.

 

In Fig. 4b; b is the width of the wheel. The force
equilibrium diagram for the element in the x—direction

'1

is given by rig. S.

._..__. . “cu-31f )dy

+dx
dx

 

do
dy~-— My; )dx

1

 

 

. g;
‘

’t .
(Ho shear force at the end of theelement.)

Fig.5. The forces on the element.

15
The pres sure on the left side area (boy) of the ele-
ment is p, and the right srde area is p + (dp/dx)dx.
The shear stress on the bottom of the element is ’t, on
the tOp is T+ (aft/ByMy. The inertia forces resulting
from the acceleration of the liquid at: small compared
to the viscous shear forces and may be neb lected.
Set 2 Fx :0
[p + (fiflx dey +dex= [T+ (-2—— 3;.) ody]bdx+pbdy

OI‘

dp __b”t
YE?"EBT

A
\N
V

By the Newtonian definition of vise cosity
”z- —dv
/uchr (4)
As the velocity under the basic assumptions is a
function of both X an nd y, the expression dv/dy in (A)
should be changed to'bv/byu The shear stress we are

mainly interested in is the stre s on the

C')

H)

-urface o

the wheel. For this particular case, the equation (A)

should be stated as

lung}. (an (Fig. 4a) (4)'
Differentiatav (4)
:33: ,/ W( ay' ) , (5)

Combine Eq. (3) and Eq. (5)

.195 =/“—‘2'

lo

3? solving HQ. (6) the velocity function i found as

U

(0

vr : 1 ( EL}.2 + ‘Cly + C2 (7)

V 2/1 ‘x

To evaluate the cons ants

 

1"

cl , the boundary

 

 

- ‘2 . ‘ ° Lf, ,. Fina-.51 _-:+.. ' 4:“- +'
ordinary lubrication case. lLb \eiocltd distribution
- (1-: V 7. 4. ,— - s . . . .s is i
in z.-. ,e oJ razors i3 pFOKEJ to be a paranoia.
, fr“, ,-.. i ,
L. in” general DFQSWUCO equatlcl
fr} F: 7" I"_') “ “3‘91 7‘, + ‘1‘ 'f. V“ * (u H r 3" ‘ x ' 1. V} I: I' :r‘ a r‘ Y1?»
-& A- .1 {J _ .4 K; -J I r:; a U r .C .L. A A b 0 AA '. .. . u 1“ $. A n .. ‘1” . p.— f L _,
’ ~ :‘ - '~ ~ vv' 1 ‘ r 4 ."\ ~ ‘ n '*’ : " r . 4‘ V’. ‘r~ ' ' ~' ‘-
nor . rurtnernore, because the corolntity haraoteristic

"‘ V” \ ‘ ' '4' . « w. ‘ ‘ ‘7'. v ‘vr. r‘ -. -,. $-
una treasure, there must be a mamlflhu pressure pOint
.r A. 4'w,-p, C3 ", ~ f- c) )H ‘1 .' ":11 § r" Y} f" 'l y" ‘t (a I “I 1 )N A .1‘ ‘w ‘ ‘. r A} Mfr. r- y. ‘9’ .- }~,
UTUVV-‘.\,-. :1 club :1 . l u u elk/lb .‘a iiJ. u: let/(”u ”little 9.“,

J ‘ .c' 4‘ 1 " ‘ ~

- A A .,. - . . ~ . 1 w \ ' r-
rate Charge Cl one pressure (op/or, roannes more.

lhe velocity at that point is rivon by In. (9)

\J”

 

1 s
v 4 V! V ‘\
(1'51“ » v. r: “HI: I; ‘ ,-‘..’.‘r,(‘ 0+ trg P- <1 1.* '. ,3 WA ‘4 (‘14
‘ 4 “<-‘ -* Dal Ab-A—‘~1ajn»t.as-) 1.1 AA! Al-.. AU J-..\. i\ +-iL/ 14‘
A
J

17
maximum pressure.
Tf mainly the middle zone of the film along the

‘.~

X-direction (Fig. 6) is considered, and a very short

time interval required for the wheel to pass any see-

the film, the sideflow may be neglected and the

O m " ‘I ’ ‘L ' o‘ ‘
r ’” Lyrit tlLiE, can be

f'ow in the x-direction C, in an
assumed constant. , I2
I

— -. — ——
. .
:

 

 

\

 

 

 

+ V

L!

7.7

2
. h
S . C r
\1/ \) a S .. l .d 7- ma
2 7) C m. n h a,
l _ (J l r . l S u W. h
/\ h /( O t L n o S
n1 3 .1.“ 4 l flT. o. r.\
.i t. C NC
V4 n I do 0 a
O 0 VS a; 10 q. l
. . a .l t n“ t r d
it i .l a n u .

‘Y
4-
U
1!“

s1

n

Q

"J
(Ann

r
. )
V .

qua
cnara(
uY‘l
l"
e
r.s
\
l
’t’

 

I e C O a. L
7\/ a wl my). . P l WI
1.7.“ ML a u ,4 u by u u
T ”J no a a . . 3
b ) U. 4b WJ uh 3 ”a. C
S .l l C a. ,J. C Q
J K S C a . l r o in r
.d .d C O u h F t C.
m... 1:. AU “.4. ,C (\
/l\ P1P. Pk . l— «I up .yu
_ 7nllz v. V b a T O D.
) ax n z o
2 1 or. .L a G d C C p . C S Q
l l . 7) r n n . i it C
(\ n a a .l n n... it 8
(IL h n S C C C R S
0 d C vs T M n l .l e
h n b I ~ g e 5 a C O F
b a h e o. P t
_ x) e t C G l S
\l v» .0. . .1. l r a
l : +U Kr.” m . r“ 1. Di nk
ii T +u he +U pl no w"
(\ O S _ L C d W t
2 h n .l x. .b n, P Q 0
. b (x O H F a Va
S U/ \I t r t S b
q V 2) C p C t
E 6 l e O T f 7 n
- J. n, a ( J S t u . l 0
mm 2 C Ii 00 fiL o
no no "no . no .nu Hm so ..a .nu .
r O X C. it 3 G G .l e \/
C. F _ d d E C .. l r h F t S.
7).. H .Ts U; .. l C u
1 «l a a (u
U u G T

. 3 . .i _ .
Q .m a G T. .w r: .l S

.77

//7’

A

/77f’77

 

‘.l

(3

caus

- \
“4 f)
.A_' \l

(

x
L

ir

iown

V

6&4
UL

'\ n - ~ 5 "
to 114.118

l

4

1"
R.
,l

‘
,.J\

”a
“A ,

.,
ODlQ

. v1
Fl

and

t

(l

i

\Il
Hit
l
(
K
,0
\Il
av ‘
.IL
or"
x).
A‘ \n
_'A
1...

 

\|./

.wl.
"11‘
1:;

4U
I.
1.4.

.
1k

.0

 

/\

é

Therefore

As we

umed

dp

 

dx

)hiJr

: 3(UTV) 0
fl hid

 

 

that the film thickness

U - V
p a (19)
h.
is relative-
region

ly small compared to the radius of the tire,

(II) will tare most lift force and friction resistance.

Therefore the maximum pressure point must exist in this

Fig. 8
ions for a

force and

in this figure, 8 1. £2

81/10)"; S

friction

the

‘

USSU

that tan Eizg.

l

resistance of the

9 E

39

 

“(he 8 l O

are small and we

important elements and assumpt-
for calculating the total lifting
Where,

assume

 

r»
-s-—£l3—--— \

 

7/f7/1

 

fi/////l////7////6/

///’7//

—-—X—>"

I/7////

 

 

 

X1 ____.,.)‘

 

 

 

d;

1
mad

 

 

The deformation of
film in a X—v

U

 

l/l/l/i’,’//

a wheel on the
coordinate system.

7x

From this figure, we get

2 ' ' '2 :29}:

 

 

Using Eq. (17) he _[hj +83 ( x _ X3 )] j
' OX
pi = 6/X(U+VZ[ . ‘ ,7 A‘
[hj 1' £3 ( X — k3 )]
( ) 2hi — hm
:: 3/“ U+V [ 2 ' l]
83:1 111

Eq. (16) therefore can be written 33

III , Zhi‘no + c. ) ax
P=2{3/X(UW)‘[(€ “2 l

 

 

 

 

1:1 5:11
ho — hi U — V
n+7 ’ ,7 x
vu£i[ [55(U‘\) h? + h. ]d
a c 3, (1]) becomes
a) - h
F: 3 (UtV)E ( —‘v C ) (K
1:1 # 1 £1~;hl l
d -“ .
h - h.
I _
T/ajfioww) o l Help.
2
lli Iii

The boundary conditions for Eqs. (19) and (21) are
a), For x:xo : H:hl+ElL1; (pl):0
b)’ For X:Xl : hlth2+£2L23 (pl)x=xlg:(pII)x=il

ror X20 : hOZhZ-EZXQ;

(p-).._ = )_
ll x—Xg (P111 x—xz

flf

 

(21)

(22)

A
1‘;

\fl

V

A
(\J
[.5

v

. '7)
c.

[‘3

H I ‘ ' - ,_ ' °
”nose boundary conditions for pi maxes it poss-

ible to solve C4 and h
.L

o’ hj’ two gjcan then be solved.

T"

sq. (22) gives the remainingéfi. The results will be

functions of L,, Ilptand velocities U and V.

8. A calculation example

in actural case, the pressure ianegion I and III
are small compared to Region II. If we consider that
the pressure in Regions I and III are negligible, and
tion etmponent of the friction resis-
tance it will simplify a great deal of calculation for
finding the lifting force in Regi n II. This lifting
force will represent approximately the total lifting
force caused by the relative motion of the wheel.
Ev this assumption, from Fig. 9, the boundary conditions

of the pressure p in Reg on ll ar

x=x2 ( p ) ":r2: 0 (3b)
Xzfll ( p ) x-xlIO

I
ha ho h hl

' 7f77T If], r
O —x—w-{

 

rfi/IVI’7

 

 

 

 

I"""—Xz--‘1 x1 ————-)-
Fig. 9. Simplified diagram of Fig. 8..
for calculation example.

23
From the figure, we get
+Ed ( X " X6.)
C 5.

sq. (21) therefore becomes

( \ 2': h2 + 52( x - x2 )]-hO
p::p:3 U+V / + C a
A [1 a2[.h2+52 ( X ‘ X2 )] 2 I;

(25) into Eq.

h : ht?

 

Substituting'Eq. (26), we get

2( h2+82D)—h

 

 

 

 

0 q
a + LII : 0
< we, D r-
Zh — h
2 O .L .—
8 2 C11 “ 0
2“2
Solving Eqs (27) and (2b), we obtain
2hp (h2+£2D)
hflr '
V 2h2-+EZD
2
CTI: _
‘ 52 (m2 +521) )
Substituting Fqs. (T;) ard (BC) into Eq. (21), we 5
2h - ——

 

 

 

 

The total lift

X
j._ l w
I— b.(X2 pox

c. . . u .2 ... n

7) 2 .fl 8 7) a

(K xh a me /\ C.
\ll‘lll) 1/ ”HE.
\n/ 11 L 0. .. WA.

a i 0 no
D _ \J O 1i ”it“ .1- i.
a); 2 .

n/L .2 (

h
111 r n.
.A.
a
i
00
\

 

’4

ft

\

1 x
\

,x

l..—

 

1 'mI/_ v.7. 7%.,
D Vail“ WHY-A ”K. O . 1W“
2 2 + u 2 P E A
DC h 5 S e D 5 6
+ Q d 5 ,/ .3 c a.
n, l .r .2 2 J. H
h e C 2 2 -. h F {k (x
I. Ix U. “w o
5 c h D 2 c 6
O r. ' 1 FC my.“ C _
1 o 3 e .. .. i . t . O
“1.1 l o n. We 1C / / / C 7;.
( fil. . O l L Q h F
a; s A. «(in . U 0' 6f. 1 a m e ”A

U.
V7
('V
N;
f‘ V
\/

 

 

 

1 I \If. \r: .. _ Au Hr} HI I
! .U 2 an a)... do. c. J -L b .l f.
V A}; E I) . a WU «I. I. i); o
.O my \./ H,“ (\ We w . .1; S X
a; w» 0 / 9 .. /\ 3 O , u ; i x .0
\/ CC _. C .8. u. i Z . U/ f _ l . i C
v. .+ NJ 8 «u .H“ w“ .1 fl Aw. um” PM o a.)
.+. 0c. n/L m/_ /l\ pk T1. .rU Pl. 0 74 hp. : 1; H” O u
U DC h a C 1.. a o C a X m .l , o d
{\ x \i. m.“ . O . ,7: I) H v . r1 i C
l 6 2 U. Q C u .. s Q A l c J 3 l :2 3.
h 3 S I O C i _
.0 /u\ S : .. : u rm __ : z c A” :
.3 : M 2 l u d an
: S .0: . h _ an .i... h

(N
.J
“\
u
I
T:
". :1
r3 "
:4 k)
'7
i
7‘
‘—\
U
Q
D
L;
(3
V

a
‘7
‘lue
e
a
11C

 

e n pl 1|; .4 .U. at e
.l Y4.
h“ I vi 1n "3. F . ‘0 r31.
.(Iml.N\ \ .. n x.t..IIFHI .L- >.t._. I) - .4 x. I I .I. II II a b}... | I.1I. -‘Vs‘|..
. . . ii... ”a . .. d... .. .>|.. t 1K». 11.4 “711.1... . 7w .1 h

 

n‘ f" \ O
u k.

A r\
Li \/

fjlxr
L.\.

25

= 0.000279 in2

Therefore
h2= 0.0167 in.
This value seems reasonable when compared to the result

of the experiment which is stated on page 49.

(P\

F 3
"1‘

(’1

LJ.

L,

‘ J

: ‘vy 09" 42' w‘1(7.'
" ‘l L2- “Li

’q
r7-
l

27
9. Review of Literature on the Viscometers

Several methods have been developed to measure the
absolute viscosity. The method we are interested in is the
rotational viscometer which includes coaxial-cylinder type,
cone-plate type, disk type and conicylinder type.

The coaxial-cylinder type is shown schematically in
Fig.10. A cylinder of radius Rb is suspended in the sample
fluid with the submerged height h in a container of radius
Rc' The inner cylinder, or core, rotates with a constant

angular velocity (i) in the outer cylinder.

 

 

 

 

 

\\\\ \‘ \ ‘\V \‘

 

 

r77///7”/ /f’/’/

\

 

Fig.10. The coaxial-cylinder type
viscometer

Several designs of this type are listed as following;
( wazer 1963 )

(a). MacMichal Viscometer

It has a motor-driven cup in which a cylinder of
slightly smaller diameter is suspended. The motor drives
:the cylinder and the viscous drag of the liquid is balanced

by the torque of the suspended wire. The test

b

reading has to be made as soon as the ovlinder reaches
qu; li brium position.
(b), Stormer Viscometer
This viscometer is similar to the MaeMichal's but
cylinder is much smaller than the cup and driven by a

wire an d a wei gh . The cup containing

4. ‘ n. ’
one sanple is

raised ur til the cylinder is sunny- ~1. A reading is

3
O
H)
i
,(5

taken of the time required for 100 revolutions
cylinder as indicated by the revolutionc20unter above

the cylinder. '

”D

(c), Brooxfield YieComet r

r3

hi s viscometer can only obtain the relative value.
The viscosity is measured as the toroue red~ired to
rotate a oyli nder or disk at constant sieed in a large
beaker of the test liquid. A scale indicates the
ransmitted to the spindle as a measure of vis—

torque

cosity.

’2

some more viscosity measurement instrumen
this type are the Rotovisso, the iolarad, the Fann i—G
and the Hercules High-Slear visccieters. (Laser 1055)

The Qpne— —ar djfilate type ‘ shown in Fifi. 11.

k-
U:

 

CFhe appeal i ng feature of the plate-and-cone principle is
t + p M m -, -o ,. ,_ D J

hat ioi small angles (CK ca.j ) tne rate or shear
Eicross the c nical gap may be eonsLdered Constant and

lihereforeLQflx_gives directly tlie true rate of shear.

I'\' f“

 

30K
27E

/A£

 

 

 

 

 

 

 

 

 

 

 

 

 

 

M: *v .,. ' .. 1 ., i (v * , '
lne nneoboniometer and the Ferrant i-ohirlx Vis-
VV‘ - .a n n )‘v '3 r [’7
ometer are this type. (wager iflsj)
L 3 05
V/
l
/ / l ‘ I d I
/ / //77f7 / //// ////////,ff“
"3' (a. 1 h. r‘ _ : ., #-
lléo ll. l.o.-.?-,,,l.te viscomete r
Vain <2 7, 4» , .. ., . ' ., .. i. 2.- :‘.:. a 4-
ii'Ufi Ci?“ UL. pe ViCC‘Ofl‘thr lb thunk UJ' 43-5. 4.5. In
.(‘J‘ ‘ V "w -—- ' ‘-' 1 1. ,q +eo
consists o a Jory thin Glbh iotat ti g betwee wa
‘ "‘ q‘V'r‘ ’i ' .1 .1. r" ""7 1*: ‘v'
equally; Spoiwefl paralltl platfjs. 4.118 C. L83.
. ' ‘ . *fl 1A“ .\ “A W l‘ - ‘.. V . “‘ r- x ("2‘
18 separate-:1 .LIUiLl these: plat: 3 by the Clstarice 1 . Tue
v‘ ‘ :r—p -: ' ’ I“ - ' ‘ . - 1 :1” V‘ 'v\
DTUOhlleld ornohro—Leetric, ano tne nior f Vl oxmeters
I" D ‘r a V, \ "I'r '; "C” ‘
aria oi, tzr.s ‘txymé (n( «o*‘ l wig).
(1L) (1.!
/ ' ' ‘lllé'éfic/cc.
fz‘LL‘LL“‘* fr
,4 mg
I LL
‘ S
A I
,A
1 {1'1—
r1
I/ff/7,777.7r/7f/7’T/17/i7/UA
"‘g. ld. wish viscometer.
7;“ : ~v- V '7 4" r s: ‘,"\ ‘ ”1 \ A I vvj -: I ~ ~,
.1. ._)3. .1.) S} IOMIS UIL t) p:*LLCApJ_e (.1... a Clierj-CdI‘L Lair“ 128-1
‘ ‘ v« ’+‘ t 1 : “ ' ‘ . " #
vi<eo e er. Several LOmmelClaill ailable instru—

 

I

ments have used Inodifl eat ,ns of this desirn. T

V

‘r ant

€.

COOUV

.
"‘. (3
d- V

:1

4.." .
tio

. O

“ulus

7 .q
.. 1-10

Ilindrica

U

C'.

VV l// (f/ I V/V///.////VMYV r/ /!

 

t)w

 

 

_
_
_
_
_
_
.
_
_
_

_
_
L
_
_
_
—
.
.
(la
_
.

_

//Al/’/i////l.i/.A,4 1/4)

 

01" C0111-

. ’
g

rw
v.1;

, .
r) _h

\

L

1 (r
t":

Y
J

.i

l
J...A.,

"‘
b d‘

l i

‘O -

o .L.
d n ..
C u
.0 {o
T0 3.
3 .mt.
T r l
8
R4. U
m“ n .
u .. l
+u
S C
S a
0 w .
1i Jv
is

h 9
w h
C lo
Mm .
.l\/ H“.
S a
.1...
Vii
C
r. m1
a c
to
r e
C K
to C
C C
m S
O .i
C V
n a
4 v. G
it
a
1 l l
a p
C _
. i_ C
r R
.U 0
mi; C
onJ.
T; 8
VJ .n
C n l

{3
v

I
A

1 i;

pOflStaLu

'3’

or t. 17 ' " ”7'
i).L '0' Li'lb

1

3411611

nple

-
V v

C’.
Ull

. + p ,
instrument are

‘
'\
J

I

.
~<| .
hi...‘

ir ti

7"
.l.\,

equations

is

eter

C Crfil

fluid.

\
1
e

(1"

0
‘
.L.

A")

G ‘W’ t O

non—I

,3

6

V‘.
A-

study;

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I
2H: {11.5 {=41
:! n: In
—l_J L} ll

 

// f'l/‘///

 

 

AXIAL FC 33 A; TRAIJSDUJER

The CONE Viscometer

31

 

 

 

f_—__——_

 

 

 

 

 

 

 

 

 

 

 

(“:§;f>

 

 

 

--—e
L.

v-u——_-

T I i

 

 

AXIiL FORCE'TRAKSDUCER

Fig. 15: The Ring-disk Viscometer

32

 

.‘J

5 _ -

‘Y

Q. The Cone viscometer and the Ring—disk viscometer

Fl

‘
l
l

"“ ~ I. e 3/ A! l' .‘ ( 1. ‘ ‘ QA ,‘ - r ‘ h.~ I
ine tiscvmeters now ava_iaole are not designed for

($-
kr‘
p .1
(D

o
C
H

He
"J
O
U)
(D
O
F-fi
53

D
9:3
0
C.
P
y—
:S

O i
c +
>‘4
(U
,3.
(I)

., '-.v i‘ ‘ ‘ 4 '1
:oslty oi mud, especially

" ; 'L‘“ r :+ : ‘ I 1“ . -. . V ’\ 7’ J. 4 . ' “
unoer the condition which is assume‘ in this study.
I
r" 4‘ I p Q? ' w ’ I J. t l 1 " H ’\ /‘ r r‘ '
Two ypis at . econeuers were LCYOICPLG, Lanel; tor:
if. “a YY‘“+' ‘V‘ ’1 \ ID'; """ ‘4 (‘7' v'v': r‘ 3 "'1 “*er 1'.r"."' ""1 ' (”‘1 '~‘ ‘ - 1"
V J LJCOlslt: U6- (n; (A La... IXEJ-G_LUI‘L V ._ I." (10“.LO b'q .- 'NL;.‘I- (4AA (ire a.) -(j t‘UIL [)U'l

fig. 14 and rig. la.

 

 

I
Ir... a 1 “r, "‘ “ , - s a" 4 ":r- 1 fl 1 +.‘
The “esitr oi the VZEComobGFS rib. i6 ioilows one
‘ ‘ . 4 '
basic hootmpuitns.
——————q' V
r
[7‘ t
N" ”077
‘ 7‘. r", \ ‘. ~
gig lo; ine principle of the
.‘ ,... 4-
1'“ C‘tnzcu‘I‘S
" I ~ 1“, " . +1 . r..- .——,’ ,,t i
H ~ 8 :LC; r: 18.1 .1-‘ -1kl , \ - --() UF(1 I ‘a— \ (410‘: J. td’ , S 3 ‘1 {1.112;
p;—] "N: ‘,.'r (T1; ' ". I "r‘,' "4 / *~"
lilIH LIHth’lCSf-J. lilo) VlSCQTIEQtCI‘S were mounted CT] the

CI‘EiI‘Li‘.S Willflll are CECE??? l I"; Elf". .L
4" 1‘ :‘f‘ 'V‘( ‘f.fi vy'v‘rn fiar“ 't yarn (to trio 4“} 't "V 1')
Urdii‘. Del? 2, 0 Wk. - e U“) .flu O “hiauh- a bile oraliht,

L.)

and axial force and tne clearance. The torque and

(’3
ct
r‘5
{D
(I)
CL
C
0
(I,
‘ 1
in
W

axial fore ire also shown in Fl 14.

(7*:
O

34

A Brush Six Channel Oscillograph (Model BL 266)
was used‘with three Brush Amplifiers (Model BL 520, EL
320 and RD 5612), one each for torque, axial force
and clearance transducers. Fig. 17, 18, 19 and 20 show
the details of the Ring-disk Viscometer and its Opera-
tion.

In the viscometer, the core, or ring, corresponds
to the wheel which slips over the soil and exerts a
certain pressure equal to the contact pressure between
the tire and the rigid ground.' The soil used (a heavy
clay) was mixed with.water*to a moisture content correspond-
ing to the plastic limit. The soil then was applied in the
mantle in a uniform layer about 1/2 in. thick. After
a smooth surface was formed by pressing the core, or
ring, towards the mantle the soil was allowed to dry.
Two soil conditions were tested. One was with the
moisture content about the shrinkage limit of the
soil which was estimated about 10%. The other was
with an air dry soil which was estimated about 2%.

A sufficient amount of water was applied to the soil
surface in order to make it running with water. Then
the core, or ring, was started and the soil was lifted
up against it.

The Cone Viscometer failed to build up the pressure

35

81: channel
oscillograph

  
 

Amplifiers

Horizontal Axial increment
movement crank crank

Fig.17. Total View of the viscometer

 

Fig.18. Soil sample prepared
before test

 

Fig.19. Viscometer in testing position

Contact
area

 

Fig.20. Sample after test

' ‘ {’1 ‘fi r ‘h r . I
he deSlre‘. lhe reaSOn ~as tlzat tie 8 ll has deformed

too fast and was squeezed out of the contact rea before

C+
:7
(D
*d
‘1
«9
(.01
:1
H
5
z:
:s
c

built uo. The dry soil provided a hard

0)
(T:
0.:
e
‘pJ
0
DJ
1?
C.)
U}
“S
0
c #-
U}
,2
,—4
is
(D
(D
N
1)
C1
Q
Li
rt
(I J
'1
ct
c+
’o

EH1?1KK1T¥§<%3US€d

1

the soil sample to loosen from the mantle and rotate

The hing-disk Visccmetar gave more satl‘ sfactory

e23ults not only on the building up the pressure but

{D
F
33
ct
CD
l, J
:3
H
i“)

also on m vg the water film in the contact area..

A.)

3316 (fillb’ '

i l
C)
go

-dvahtage was that the relative velocity

.2

". 4‘ \ L '\ "7‘? ‘V P Y,’ '\ 1 . ‘V N.“ r" “.
is not constant, out the a.elage «eloc lit» should rive

L
e L)
r' -‘-. '
an scoopta*-e talus.
('11-. "'- ‘J‘ g ‘ '- P. -! v.) . N :,“., ' "" "' ‘
lhe test results on h_nC-dlbn scometsr snowed

4"“ . - 4 1’ ‘ ‘e I '.‘ +“ P "v ‘.. n‘. 3" r
all t lo.a1 plwxfsulnz a.mi oOIYNiC cuxr’es,llad £1 p1121o . cod

f‘.P - ‘ 1+ :- 9 I j+ +— » ." 4— - +-
t/rd-.. all-LA.) “1521'? .l.LI Gil f...Cl/1.LU UC‘ read {.1in J'rCU trail/iv,

esceclall" the torque value cor espon'
pressure peak. This was aitlltoe u to a oreahdown in
n order to obtain
a more constant torque reading, four lubrication grooves
were mad c on the rins as shown in F‘s. 21. ‘hls 5 v«
the watel a cotter Chane c to get into the contact area

I‘

,,: .1- . Al .q' l: , . 1": m r .,
and maintaln a constas tnl(hhcas Oi sud lllm oetwc

h; two surfaces.

ll. Procedures of Operating the Rinr—disk Viscometer.

Q

In order to obtain the value 05/1, the film

38

 

   
   

 

 

 

. as
‘_ A
as
P a
. -
‘” n
p
‘- 2
" a
||
'(lCCCI' ‘\\\\\‘\\

 

 

 

 

 

 

 

   
   

 

 

   
   

 

 

(ttttttt " tilt!!!)

 

\ttttttt(

 

 

"Ill!!l(

 

 

rounded

M

 

 

////

  

 

 

(b)

Fig.2l; Lubrication grooves on the Ring-disk-Visoometer

thic

movement

The size
movement
in

uated

In

axial force and film

(1). The

( ness g must be determined.

v
C
C}
{-3)

F .
9

The strain gaged

indicator did not give enoug h accuracy and

only to record hat the bed had been moved.
of the move was determined from the vertical

screw of the milling machine, which was grad-

0.0”l inch.

detail, the procedure of or ta: n

thi e‘cne s is follows:

0]

as

soil sample was prepared with the sur-

face as flat as possible.

). The soil surface was wetted to build up a
wet surface. The rinb mas run for a certain
period of time to make tne soil surface
smooth.

The ring was run at SOrpm and the ore Wear

point was determined by raising the mantle

up aLa‘" t the ring until the rec cords:

showed a pressure.

The mantle was raised 15 scale readings

( 0.01) in ) for l-2 secon" and then low-

ered to 0 again. This was repeated five

times and then the zero pelnt was determined
g.ain. ’This g;i”es iflh? valiua of tax: wear;

the axial forces and the corresponoihr

torques.

"Vr fl ‘ 1 J.
sane steps as j and 4 but Wish 25 scale
reau-rgs.

ha*~e the

a;(—-)

rrotaticn speed to
:eat steps ” '

3, 4, 5

a go the rotati.n speed to 450 rpm and
rep at steps 3, 4, 5 and 6 nee.

For each test some soil was worn off the solid
soil which had to be compensated for. Under the
influence of the normal load the device had an elastic
deflection which also had to be taken in account. ihe

Scale readines - Wear of soil + Film thickness

2 Elastic deflection + Initial film thickness (34)

he elastic Zeflectien w<s determined with no rotation

is

and is shown in Fig. 24 curve a. Hear is the di f rence

C”)

m
c
H
t 4
(t
(D
O
H)

between the zero points before and after a

12. Results, Discussions and Conclusions
Leo sets of data were eetained from two soil
conditions ( condition 1 is the sample with initially
estimated lO% moisture content which corresponds to the
shrinkage limit of the soil and condition 2 is air dry
soil ). Fig. 22 a, b and Fig. 2 a, b were plotted for
shear stress (Ti vs. normal pressure (0) to represent
the cendlt'ens l and 2 respectively. in these firures,
4 or 5 or 6 points ebtaine7 irom a same
increment of scale readings were ccnnected by a line.
This showed the variation properties o the data and

l

Fir. 22 and Fig. 23 showed that the shear stress

’ i‘ “i ' _ '1 3 4m- . 1 , i _‘- 4L a.
(77 increased linJariy w-un normal ressure ane that

#1

 
       
 

 

 

 

 

7-5 ‘
i hrs-1450RPM
V=188.1n/sec.
f/u
'/
z”\ I \ f/
a
go ' , w=i50L~iPM
VzoZ-in/sec..
5.0 " /a
I =5oapm
J V=21 in/sec.
l
of
all /A
2.5 ‘
A
..___. Detail Fig.22b
o 10 50

0"(3381)

Fig.22a Normal pressure (0) Vs. Shear stress (2)
( Soil Condition 1 )

1+2

 

 

 

 

 

w=4503PM
A w=15oam
3 ./° ¢
Z" I '
fihiiospm
I
/
v l/fA
/' ’ .
#/4.
/
./
2 1+ 6 8 10

0‘ ( psi )

43

zdzusoBPM
15 ..

V2188in/sec.

1f( psi)

10

U: 1 5 03PM
A V=6Zin/sec.

5.0

 

 

 

=503PM
V=211n/sec.

 

 

-<~——_.Detail Fig.23b

 

i0 '

20

 

50 do
0' ( psi ) '

Fig.23a Normal pressure (0) Vs. Shear stress (77
( 3011 Condition 2 )

an

 

 

 

A ”#5031314
'3
D.
N
5.0
A
\
\
\
\
\
\
23
/
I
w=15ORPM
/
I
7‘ x
205 1 p—‘ —D—-a I 5 .
a}: II)“ . , I 1
/ o w=50RPM
? //\ ,1
I 4—4 . 4/
r " . . .
. ,A/
0 10 20
0' ( Psi )

F18’23b 0'vs. T

 

\.

(D
*3
2: 4
,3
(D
.5
<1
i .1
r—J

the increase was larg ocitV increased. By

U

Comoaitii” the inwa fig

\1‘1

VV “K ‘ ‘ _‘_ _ _‘,
ufE", we leurd that, for the same

. '4- +2 '1 “:.. *' .5 r ' . J...
VClOCluf the siope of the lines niiiered between the unO

~~ D’+-: , 1‘ +‘ r p w- 1»
soil conditions. on; oi the reasons ior this probably

‘ u V ' , . \
normal pressure pFOddC?G a nigh-

the air or? seil condition.

"s n N 2 r \ r . ‘ wv fl ' 1, ~ «4'
ri.. a; or rte“ with respect to «elecitd, tnere must

, M- w- h 4" #fl. A‘ - ‘ - w r. ‘1' +- r‘
oe a posit .e efisct or one .eieCity os tie saear stic

r"\~

-- ,i. .. 1 ' . .-"‘: J.- .-i: r . ,. 1, a
1n Ctziarr.’ W'QTUS a VlSCC-US (-3l-,GCL.. Luis- LBWL-il’ls that f(l

U?

.V‘ 1" ’ V ‘A' ‘ W" :* . r—t "P~ ‘ a "r‘
a «itch load, a teniCie c n :bra-n a wreater pull Muufl

(D
(A ,

. ‘ ’V 1': \ 1 .r r ’. z 1‘
its CL~P SPCOG ( V - u ) Ls incree e .
T ‘ ' , .L _' ‘ t _ a ‘ A v 'fl 4 r‘ -. 'n . - fl "
in order b0 fine the constants :or this influe1ce
, .. 4- ' , . ' 3. < , ‘ -. .,- .. _ ' , i \ n um
a Lululple rfléTeSSlOJ hes been r.n on Eh; e.mputer cos
‘

”an . - , . - A \r. «~: .4- . m ... , 1 ~ *¥
Deco using progra; CON“. inis pivdfflw ha3 Case“ Oh ”“0

equation

2’: a + ”00+ Cl (3"

S

where a is a constant correso n

C.
h-
L:
m
c+
O
W
L4
:3
(D
(I?
F"
O
:3

b is the coefficient of friction

c is the absolute viscosity/u.

<73
H .
(I)
(.4.
:34
(D
m

hear stress

2 is the film thickness

i
m
H
'1
Ci”
”3’
(D
<,‘
(D
F-
O
O
.J
T
0*}
'S
9.)
:22
(t)
ft
C t

The value/1was found between 0.000005 and
0.030175 lb-sec/inZ. This is in fairlv good asreenent
with the values given by.he5edus (l958).

However, since the standard errors were very lar5e,

the values obtained from the co.puter calculation should

:1;

c to fi

m
D
|- 1
:‘S
O
f”‘5
O;
(D

not be used for further calculation
the reason for the inaCCL racy, we checked the data of
soil conditions 2 and l in detail and plotted the film

nickness (S ) vs. normal oressurc (0') in Fig. 24 an

L3
ci-
:4
(D
(J
C')
H)
;J

Fig. 25 respectively. I ures, the sane method

of expressing the data es in F:5. 22 and Fig. 23 vas

used. The corresponding velocity is stated at the firs
point of each set. It shows how the normal pressure
decreases when the soil is worn off the solid surface.
Fi5. 24 was divided into two parts by the lQpS‘
normal pressure line. Above or below this pressure,
each part agreed with the lubrication the cry in so far
\that a 5r ater clearanc. was fo or2n3d for a hi5her speed.
The reason why this did not hold true when com Hlar n
across the l2psi line could not be found, therefore,
further invc fi5ati on is necess ar'f on this point.
Another discrepancy is that the film thickness increases

for increasing load if two sets of measurements are

4?

7OOJ

Ox
'0
Q)

U\
C)
‘9

 

Normal force (1b.)

400?

    

    

Xsoepm
6/ /

 

 

30m /" .
‘5}D1gpafim
*/'
200* )xSORPM 3, Elastic deflection
/x/* of the device
100 x50mm
/ "
,rxJ“
d 5 1b 15 ‘ 20 23. 36

Axial increment in 0.0011n.

Fig.24 Film thickness (8) vs. Normal pressure (0)
( Soil condition 2 )

Si is initial film thickness

 

 

  
    

 

 

#8
4508PM
00 a,Elastic deflection of
7 ‘ the device
A . a
n4
0
O.
f:6OO‘I;, ISORPM
.D O
r-i
0 u
0
3 Scale readings
:5004 - wear of the soil
3.
- lopsi ° 2.. $.81
400.
u 50 PM
300.; ___________________ _.....
200. b, Elastic de-
flection of the‘
device and the
deformation of
the soil.
100$
_o . {a £0 50 no - -5o 60

Axial increment in 0.0011n.

Fig.25 Film thickness (5) vs. Normal pressure (0)
( Soil condition 1 )

1 is the initial film thickness

49

compared but not within a set of measurements. This
indicates an inaccuracy in the determination of the‘
film thickness.

In Fig. 25, we used curve b as the reference line,
because we found that the soil deformed a large value
when the normal pressure was applied. Curve b is the
calibration curve of the elastic deflection of the
device and the deformation of the soil. In this figure,
the relationship of the sets are not so clear as that
in Fig. 24. They do not agree with the lubrication theory
as far as the velocity and the film thickness are con-
cerned. But within each set 0f,measurement, it agrees
with the lubrication theory when the normal pressure is
greater than 8 psi and disagrees With the theory when the
normal pressure is less than 8 psi. This again indi—
cates an inaccuracy in the determination of the film
thickness. '

Comparing the results from the calculation
example and the experiment, we obtained an interesting
result. In the example we assumed that0'= 40psi, V-U
=132 in/sec (slip speed) and obtained S: 0.0167 in.
From Fig. 24 8: 0.010 in. for the speed 188.6
in/sec andO’= 13psi. This indicates the calculated
value given in the early part of this thesis might

give a reasonable result.

In the further investigations,

b

IL}.

\_3"

the

test of the

y the Ring-disk Viscometer could be improved

More accurate determ inati
deflection of the device a

necessary.
More accurate
face is ne

cessary.

‘
1‘.

continuous water supply
arranged to 5ive a better

the mud film between ring
surface.

determinatio
clearance between rin5 and

on of the elastic
nd the scil is
n of the zero

soil sample sur-
system should be
chance to maintain

and soil sample

The temperature influence should be checked.
Temperature might become a problem when the
rotating speed is high and the normal
pressure is large.

The instrumentation snould be improved
re5ardin5

i, better de3i5n of the clearance transducer

i, Stabilizing the pen wh
is small

1, Jere point of the pen
tron l8 chan5ed.

LlIlCC .

when the attenua—

REFERENCES _
1. Aronson, M. H. and Nelson, R. C. (1964) ' Viscosity Mea-
lsurement and Control.“ Instruments Publishing Company,
Inc. Pittsburgh, Pa.
2. Baver L. D. (1963) ' 3011 Physics “ John Wiley & Sons,
Inc., New York.
3. Bekker, M. G. (1956) ' Theory of Land Locomotion ' Uni-
versity of Michigan Press. Ann Arbor.
4. ' Brush Six Channel Dynamic Recorder “, ' Oscillgraph “,
' Input Box “, ' Amplifier '. Brush ElectronicsCompany.
5. Czako, T. and Regedus E. (1958) ' Physical Soil and Snow
Values for the Determination of Vehicle Motion Resistance."
A Soil Value System for Land Locomotion Mechanics. Res.
Report No.5, Center Line, Michigan.
6. Dobie, W. B. and Peter, C. G. (19#8) ' Electric Resistance
Strain Gauges.” John Wiley & Sons, Inc., New York.
7. Eirich, F. R. (1956, 1958, 1960) ' Rheology ' Vol.1, II,
. III. Academic Press, New York.
8. Fuller, Duolly D. (1956) * Theory and Practice of Lubrica-
tion for Engineers.“ John Wiley & Sons, Inc., New York.
9. Harrison, W. L. (1957) ' A Study of Snow Values Related
to Vehicle Performance." Research Report No.23, Land Loco-
motion Research Branch, Center Line, Michigan.
10. Hegedus, E. (1958) ' Drag Coefficients in Locomotion over

Viscous Soils (Wheels)” Land Locomotion Research Branch,
51

11.

12.

13.

14.

15.

16.

17.

18.

52
Research Report No.25, Center Line,Michigan.
Hegedus, E. (1958) ' Wheels in Viscous Mud." A 8011
System for Land Locomotion Mechanics, Research Report
No.5, pp.61-67. Center Line, Michigan.
Kiel, D. F. and Ruble, W. L. "Use of the Core Routine
to Calculate Multiple Regressions ( Least Squares Fits
to Arbitrary Functions ) on the CDC 3600." Agricultural
Experiment Station, M.S.U. AES Program Description b.
Perry, Charles C. (1955) ' The Strain Gage Primer.“
McGraw Hill, New York.
' Proceedings of the First International Rheological
Congress.“ (1948) I
Rayleigh, (1918) " Notes on the Theory of Lubrication."
Phil. Mag. Vol.35, pp.1-12.
S. Oka, (1960) ' Rheology: Theory and Applications '_
( ed. F. R. Eirich ) Academic Press. Vol.3.
Streeter, V. L. (1951) ' Fluid Mechanics ' McGraw Hill,
New York.
Van Wazer, J. R., Lyons, J. W., Kim, K. Y. and Colwell,
R. E. (1963) ' Viscosity and Flow Measurement ' Inter-

science Publishers, New York.

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