Prediction of ground pressure distribution under tracked vehicles—I. An analytical method for predicting ground pressure distribution

Journal ofTerramechanics, Vol. 18, No. 1, pp. I to 23, 1 9 8 1 0022-4898/81/010001-23$02.00/0 Printed in Great Britain. Pergamon Press Ltd. ~ 1981 International Society for Terrain Vehicle Systems.

P R E D I C T I O N OF G R O U N D PRESSURE DISTRIBUTION U N D E R T R A C K E D VEHICLESDI. A N A N A L Y T I C A L METHOD FOR P R E D I C T I N G G R O U N D PRESSURE DISTRIBUTION M. GARBER* and J. Y.

WONG'~

Smmnary--An analytical method for predicting the pressure distribution beneath a tracked vehicle under static conditions is presented. In the analysis, the track-suspension system which consists of the track, the suspension and the track tensioning device, is considered in its entirety. All major design parameters of the vehicle, as well as terrain characteristics, are taken into consideration. It is shown that the analytical method proposed can provide a means whereby the effects of vehicle design parameters and terrain conditions on ground pressure distribution can he assessed quantitatively.

a, c, d, e

b F

f

G,J H, H~p k k, kt L m

N n

p R1, R2, R3 r

rj rt S

To T~, Tp~, Te., T., To, t

t~ uo

V w (x~, zJ

NOTATION geometrical parameters of the track system, m width of the track, m auxiliary function, m-("+~)~2 auxiliary function, m" auxiliary expressions, non-dimensional horizontal component of the track tension, kN/m terrain stiffness, kN/m"÷~ combined stiffness of the suspension springs of one track, kN/m stiffness of the track tensioning device spring, kN/m distance between the centres of the front and rear road wheels, m number of supporting rollers number of the road wheels of one track terrain parameter, non-dimensional terrain pressure, kN/mI terrain reaction, kN radius of the road wheel, m radius of the sprocket, m radius of the tensioning wheel, m sinkage of the road wheel, m initial track tension per unit width of the track, kN/m track tension per unit width of the track, kN/m auxiliary function, non-dimensional track pitch, m auxiliary expressions, non-dimensional vertical component of the track tension, kN/m weight of the vehicle, kN coordinates of the inside junction point, m

*Senior Research Associate, Transport Technology Research Laboratory, Carleton University, Ottawa, Canada. "['Professor, Department of Mechanical and Aeronautical Engineering and Director, Transport Technology Research Laboratory, Carleton University, Ottawa, Canada. 1

2

M. GARBER and J. Y. WONG

coordinates of the minimum sinkage of the track, m coordinate of the intersection between the road wheel and horizontal surface, m x(z) or z(x) equation of track deflection, m angles between the track and horizontal surface on the side of the sprocket and a, 13 tensioning wheel, respectively, radian ~a, ~f~ differences between angles n and ~, t3 and [~0,radian Aa additional compression in suspension springs, m AC additional compression in tensioning device spring, m AL~, increase of the length of the track section A J, m AL,g reduction of the length of the track section AB-IJ, m 21 distance between the centres of two road wheels, m length of the track between a pair of supports (upper track section), m A~ 2~ distance between supports (upper track section), m weight of the track per unit track length, k N / m P v non-dimensional parameter FI auxiliary expression, non-dimensional angular coordinate of the inside junction point, radian tO, angular coordinate of the intersection between the road wheel and horizontal surface, radian. In the text, parameters with subscript "0" refer to the vehicle resting on a hard surface. Parameters with overbar correspond to those in non-dimensional form. (Xm, Zm)

Xs

INTRODUCTION THE SOFT GROUND performance of a tracked vehicle is greatly influenced by the pressure it exerts on the terrain, as sinkage and motion resistance of the vehicle are closely related to the ground pressure. Traditionally, the nominal ground pressure (NGP), which is defined as the vehicle weight divided by the product of track width and length on the ground, has been used as a design parameter of relevance to mobility. While the N G P may serve a useful purpose in comparing the soft ground performance of tracked vehicles with long pitch tracks and small diameter, closely spaced road wheels (such as agricultural and construction tractors), it can be misleading in assessing the mobility of vehicles with short pitch tracks and large diameter road wheels (such as high-speed military tracked vehicles). For these vehicles the actual ground pressure distribution may be far from uniform and depends on a large number of vehicle design parameters as well as terrain properties. Consequently, for these vehicles the N G P is not necessarily a parameter of relevance to soft ground mobility, as it is incapable of distinguishing between different track system designs giving drastically different performance. The inadequacy of the N G P as a general specification and design parameter of relevance to mobility has been pointed out by Bekker, Rowland and others [1], [2]. Recently Rowland suggested replacing the N G P with the mean maximum pressure (MMP) as a parameter for comparing the soft ground performance of tracked vehicles [2]. The M M P is defined as the mean value of the maxima occurring under all the road wheel stations. Based on experimental evidence, Rowland further proposed a set of empirical formulae for predicting the M M P of various types of track systems, including the link and belt tracks. In these formulae some of the design parameters, such as vehicle weight, track width, track pitch, diameter and number of road wheels, have been included. However, in Rowland's approach, terrain characteristics are not taken into consideration in the calculation of the MMP. Thus, the value of M M P calculated from Rowland's formulae is independent of terrain conditions. It will be

GROUND PRESSURE DISTRIBUTIONU ~ E R TRACKED VEHICLES

3

shown later that in fact the pressure distribution under the track and hence the MMP are strongly influenced by terrain properties. A pioneering theoretical study of the pressure distribution under a track was made by Bekker more than two decades ago [l]. In his analysis the effects of vehicle weight, track-width, road wheel spacing and pressure-sinkage relationship of the terrain were taken into account. The track was considered as a flexible and inextensible belt and the pressure-sinkage relationship was assumed linear. The study was limited to the analysis of the shape of the track between two road wheels simplified as knife-edge supports. As the track was not considered an integral part of the track-suspension system consisting of the track, the suspension and the track tensioning device, certain design features such as the diameter of the road wheels and the characteristics of the suspension and the track tensioning device, were not included in the analysis. Nevertheless, Bekker obtained some interesting results and illustrated the complexity in predicting the pressure distribution under a track. Theories for predicting the sinkage of tracked vehicles in clay, based on the bearing capacity theory, have also been proposed by Evans, Sherratt and others [3]. The application of these theories is confined to a particular type of terrain and, as in the studies mentioned above, the effects of some significant parameters of the track-suspension system have not been taken into consideration. From the brief review given above, it appears that to provide a rational basis for evaluating the performance of tracked vehicles an improved method for predicting the ground pressure distribution is required. The track-suspension system must be considered as a whole. All major vehicle and terrain parameters that may influence the track-terrain interaction should be taken into account. Following this approach, an analytical framework for predicting the pressure distribution under the track is developed. In the analysis the design features of the tracked vehicle, such as vehicle weight, track dimensions, initial track tension, dimensions, spacing and number of road wheels and characteristics of the suspension and the track tensioning device are taken into consideration. The non-linear pressure-sinkage relationship of the terrain is incorporated into the analysis. The track is modelled as a flexible and inextensible belt. It is also assumed that the center of gravity of the vehicle is located at the midpoint of the contact area and that no driving or braking torque is applied to the sprocket. The analytical method thus developed, strictly speaking, is only capable of predicting the static pressure distribution under the track. It is believed, however, that the static pressure distribution can serve a useful purpose in differentiating the potential performance of vehicles of different designs, especially with respect to their sinkage and motion resistance. It is intended that the analytical framework described in this article will be extended later to examine the effects of tractive effort, drawbar pull and location of the center of gravity on the pressure distribution, and that experimental validation of the analysis will also be performed. MODELLING OF TRACK-TERRAININTERACTION When a tracked vehicle rests on a hard surface, the track lies flat on the ground. In contrast, when the vehicle rests on a deformable surface, the track between the road

4

M. GARBER and J. Y. WONG

wheels takes up more load. Consequently, the track deflects and has the form of a curve with troughs under the road wheels and crests in between. Hence the length of the track between the front and rear road wheels increases. Since the track is modelled as an inextensible belt, its overall length must remain constant. As a result, the sag in the top run o f the track decreases, the track tension increases, and additional compression of the springs in the suspension and/or the track tensioning device takes place. One of the basic tasks in the analysis of track-terrain interaction is to determine the shape of the deflected track, as it is closely related to the pressure distribution and sinkage. To achieve this, the changes in the geometry of the track-suspension system and the vertical equilibrium of the vehicle must be taken into account. This indicates that to examine the ground pressure distribution, the track-suspension system must be considered in its entirety. A system of equations describing the interaction of the track-suspension system with the terrain will be derived. The solution of this system of equations will define the actual shape of the track and consequently the ground pressure distribution. The actual shape of the deflected track will be a function of vehicle design parameters, such as vehicle weight, track dimensions, initial track tension, dimensions, spacing and number of road wheels, characteristics of suspension and track tensioning device, as well as terrain properties.

Shape of the deflected track The deflected track consists of portions that are in contact with the terrain only, and portions that are in contact with both the road wheels and the terrain (portions 1 and 2 in Fig. 1). The shape of portion 1 is defined from the interaction between the track and the terrain, while portion 2 necessarily has the form of the wheel. F

E

1 ~

1 2

1 2

1 ~ 2

1 2

FIG. 1. Schematic view of a track system. Two types of tracks, link and belt tracks, are currently in use. For link tracks, the track pitch may influence, to some extent, the shape of the track. To simplify the analysis, however, the track is modelled as a flexible and inextensible belt in the present study. With this idealization, the equilibrium equations for portion 1 are (see Fig. 2)

dV/dx = p, H = const

(1)

V/H = dz/dx,

(2)

and

GROUND PRESSURE DISTRIBUTION ~ E R

TRACKED VEHICLES

5

where H is the horizontal component of the track tension T per unit track width; Vis the vertical component of the track tension Tper unit track width; p is the vertical terrain reaction per unit area; z(x) is the deflection of the track. xm i Zm~ . . . . . . .

+dH

pdx

Z

V+dV

T*dT

FIG. 2. Forcesacting on a track element in contact with the terrain. The soil reaction is assumed to be vertical. This assumption is generally used for relatively flat contact surfaces. From the numerical results presented later, it can be shown that the average slope of the deflected track for the five vehicles considered over a wide range of terrain does not exceed 11°. It appears, therefore, that the assumption used may be considered reasonable. The pressure-sinkage relationship of terrain suggested by Bekker [1] is used in the present analysis

(3)

p = kz",

where k and n are terrain parameters. Differentiating equation (2) and using equations (1) and (3), one obtains the equation governing the shape of track portion 1 H d2z- : kz".

(4)

dx ~' Equation (4) is a second order differential equation. Its solution depends on three unknown parameters: the horizontal component of track tension H and two parameters of integration. The solution of equation (4) may be expressed in terms of elementary functions only for n = 0 or 1. The general solution for any positive n may be obtained using an intermediate function t(z) = d z / d x . Substituting this function into equation (4), one obtains /4

"'- tdt = z~dz.

(5)

k

Track portion 1 necessarily has a point of minimum sinkage (xm, zm) (see Figs. 1 and 2). At this minimum point =0.

t=-dx

z = zm

6

M. GARBER and J. Y. WONG

Therefore, integrating equations (5) between the corresponding limits !

z

,d,=f z.d:. 0

(6)

z,n

one obtains t 2 --

2k (z.+l (n + 1)H-

--

.n+l) -m -'"

(7)

From equation (7), it follows t-

d z -dx

± V

I(

(z,,+~ - z ~ , - 2k n + 1)H

+') ] .

.J

(8)

Referring to Fig. 2, one can see that the minus sign corresponds to the left part of the track portion (x < x,,), while the plus sign corresponds to the right part x > x,,). Equation (8) may be rewritten as dx = qz V'[ (" 4-2_k_l)H]jF (z, z,,,) dz,

(9)

where F ( z , Zm) =

l/'~(Z

n+l

--

zn+l).

(10)

Integrating equation (9) between the corresponding limits x

2'

f dx = =7 W/[ ( n +2k1)H-] A f F(z, Zm) dz, Xw

(Jl)

Zm

one finally obtains z

x(z) = x,,,T w/I( n +2k I)H-]J 5 F(z, z,,,) dz {(--) for x <~ Xml (+) for x -~ xmj .

(12)

Zm

Thus, the shape of track portion 1 between two road wheels in the general case (n > 0) is described by two symmetrical parts, x(z) [equation (12)]. These two parts are smoothly connected at the minimum point (xm, Zm) (see Fig. 2). The function z

f F(z, Zm)dz Zra

GROUND PRESSURE DISTRIBUTION UNDER TRACKED VEHICLES

7

is an improper integral which converges for any n >__ 0. This integral can be expressed in elementary functions only for n = 0 or 1. For these two cases the solution can be presented as a function z(x) for n = O, z(x) = Zm + k

(x -- Xm)2 (parabolic function),

(13)

- xm)] (hyperbolic cosine function).

(14)

2/t

f o r n = 1, z ( x ) = Z m C O S h [ v ' k ( x

These solutions [equations (13) and (14)] may be obtained directly from equation (4), or they can be derived from the general solution [equation (12)]. Regardless of the particular value of n, the shape of the track depends on three unknown parameters: the horizontal component of track tension H and the coordinates of the minimum point, x m and z m. As stated in the introduction, the present analysis deals with the case where the center of gravity of the vehicle is located at the mid-point of the track contact area. As a result, for equally spaced road wheels the sinkages of all the wheels may be considered identical and the shape of the track will be a periodic function. An illustration of the shape of the track within the length of a period (2•) is presented in Fig. 3.

0 "-':'~/:17"~/ "

_

(O.Zm)'JI

'"

•

2

~

SOIL SURFACE

II

"¢ ~ ' ~ ' ~ ' ~ - ' ~ "

I

___. li ....
TRACK PORTION

1

~

2

,

:

i

ROAD WHEEL SINKAGE

r

:l

H

', R2

rYPICAL S E C T I O N ~ - - -

FIc. 3. Shape of the deflected track between two road wheels. The shape of track portion 2 is defined by the shape of the wheel. Therefore, the equations for defining portion 2 are written as follows (see Fig. 3) { x(q~) = =F (1 -- r sin q~) z(tp) = r c o s q ~ - - r

+s,

(--) for the left wheel } ( + ) for the right wheel

05)

8

M. GARBER and J. Y. WONG

where r is the radius of the road wheel; 2/is the distance between the centers of two adjacent road wheels; s is the sinkage of the road wheel; cp is the angular coordinate. As the track is modelled as a smooth and flexible belt, at the junction points portions 1 and 2 must have a common tangent. Thus,

{

(16)

x(~c) = x ( z c ) , z(~c) = z c

dx(~) [ __ dx(z) [ ~= ~ dz

(17) Z~

2c

where cpc and G are coordinates of the junction point (see Fig. 3). Referring to equations (12) and (15) and noting that x,, = 0 in the coordinate system used in Fig. 3, one may rewrite equations (16) and (17) in the following explicit form Ze

l -- r sin % = , /, [L( n_ +2k-I)H-] .... h f

F(z, 2m) dz,

(18)

Zm

rcosq~c-- r ÷ s

=G,

cot % = ~/[(n +2k-l)H]_l F(z, z,,,).

(19) (20)

Equations (12) and (15) together with equations (18-20) define the track in contact with the terrain between points A and J (see Fig. 1). As the track shape between A and J is a periodic function, parameters H, z m, z c and % are common to all track portions in this region. The track in contact with the terrain consists of A J and two segments outside A J, namely A C and H J (see Fig. 1). The contribution of these segments in supporting the vehicle weight is relatively small. Therefore, some simplification with respect to the shape of these two segments may be made. Sections B D and G I are assumed to be straight while A B and I J take the shape of the road wheels. It is further assumed that on the front and rear road wheels, the tension per unit track width T,.s at the inside junction point./'1 is equal to the tension per unit track width T0s at the outside junction point J2 (see Fig. 4). That is T0s = T;s = H / c o s ,,oc.

(21)

In the problem under consideration no torque is applied to the sprocket. Therefore, the tension in track segments B E and FI is assumed constant (see Fig. I). For a vehicle in contact with a deformable terrain, ?'BE =- Trl = To.~. For a vehicle resting on a hard surface, ThE = TFt = To, where TO is the initial tension per unit track width (or pre-tension).

GROUND PRESSURE DISTRIBUTION UNDER TRACKED VEHICLES

i¢111111

..... ~

I ~

xdr4

$

Fxo. 4. Shape of the deflected track under the front (or rear) road wheel.

Geometry o f the track-suspension system on a deformable surface

The deflection of the track in contact with the terrain causes an increase in track tension as compared with the initial tension. The increase of the track tension causes an additional compression in the suspension springs as well as in the tensioning device spring (assuming an elastic tensioning device). For road wheels with independent suspensions, the additional compression in the springs of the front and rear road wheels may differ from that of the others. This difference in compression, however, will be very limited. Therefore, it is reasonable to assume that the compressions of all the suspension springs are identical. A similar assumption was made in the analysis of track-terrain interaction by Sherratt [3]. Thus, the additional compression of the suspension springs Aa may be calculated from (see Fig. 1) k, Aa = b [Tos (sin a + sin ~) -- To (sin % + sin ~o)],

(22)

where k s is the combined stiffness of the suspension springs of one track; b is the track width; ~t and [~ are the angles between the track and the horizontal surface on the side of the sprocket and the tensioning wheel, respectively, when the vehicle is in contact with a deformable terrain; % and I5o are the corresponding angles, when the vehicle is resting on a hard surface. As mentioned above, the deflection of the track will also affect the spring of the track tensioning device. For the tensioning arrangement shown in Fig. 1, the additional compression of the spring of the tensioning device Ac is given by k, &c = b [Tos (1 "4- cos 9) -- To (1 if- cos ~0)],

(23)

where k t is the stiffness of the track tensioning device spring. When the track is under load over a deformable terrain, it deflects and its length between points A and J (Fig. 1) increases as compared with that when the vehicle is resting on a hard surface. On the other hand, owing to the additional compression in the springs of the suspension and the track tensioning device, the length of track section AB--IJ (Fig. 1) decreases. However, since the track is modelled as an inextensible belt, the total track length has to remain constant.

10

M. GARBER and J. Y. WONG

The reduction in length A L , a of track section A B - I J Fig. 1) AL,a = AAE

may be expressed as (see

+ A F J -% A E F ,

(24)

where the symbol A denotes the reduction in length of the corresponding track section. Using the notations shown in Fig. 1, lengths A E , F J , e and e are AE

cos 9]/sin 9 -% r{5 -% r t ( n - - {5),

(25)

cos ~]/sin ~ -% r~ + r s (n -- ~),

(26)

= [a + (r - - rt)

F J = [d -% (r - - r~)

c = [a -% (r

--

rt)/cos

e = [d -% (r - - rs)/cos

9] cot 9,

(27)

~] cot ~.

(28)

The changes A A E , A F J , 2xc and Ae are defined as AAE

= AEla..~°

-- AE

[a.~ ,

AFJ = FJ]a .... -- FJla ,.,

~'~ = ~ Ioo.~o -

~1o.~,

A e = e leo. ao - - e ld. a = O.

(29) (30) (31)

(32)

Parameters a0, ~o, do and ~0 define the geometry of the track-suspension system when it is resting on a hard surface and are considered known. Parameters a, {5, d and 0c define the geometry of the track-suspension system when it is resting on a deformable terrain and are unknown. Equation (32) indicates that the sprocket is rigidly fixed to the vehicle frame. In addition, the following geometrical relationships should be noted d o = a o -% r t - - rs, d := a + r t

--

r~, a = ao - - A a .

(33)

The upper part of the track E F is supported by the sprocket, tensioning wheel and, normally, a number of supporting rollers (Fig. 1). The shape of the track between any two supports is a catenary with length Ai (see Fig. 5) as defined by Ai = 2bHeesin f3

h pLi

,

(34)

2b H E~.

where Hee is the horizontal component of the track tension T e F per unit track width; p is the weight of the track per unit track length; k i is the distance between the supports.

GROUND PRESSURE DISTRIBUTION UNDER TRACKED VEHICLES

11

The total length of the upper part E F is expressed as EF=

,,+l 2 b i l e F m+i P~'i E Al -Y~ s i n h - -

t-1

p

i=l

2briEr

,

(35)

where m is the number of supporting rollers. When the vehicle is in contact with a deformable terrain HEr is equal to T0s. When the vehicle is resting on a hard surface HEr is equal to To. The sprocket and the supporting rollers have fixed positions. Therefore, ~.~ = const, for i = 1, 2 . . . . m (see Fig. 5). The tensioning wheel, however, may move. Consequently, on a deformable surface the distance ~.,,+l (Fig. 5) is shortened by Ac from that on a hard surface. As a result, A E F is defined as A E F = 2b F m+l P~'l - - L TO iE=lsin h - - - - T 0 s p 2bTo

~sinh i=1

- P~.i -- T0s sin h p (km+t -- Ac)I. (36) 2bTo,

The distance ;~,.+1 in equation (36) is the initial distance between the tensioning wheel and the roller. All the ~,~(i = 1,2 . . . . m + l) may be considered known.

H E F ~ ::

3~1

.::

~2

~:

"'"

I

A

I

..-

Xm+l {

To

V E H I C L E IN CONTACT WITH A DEFORMABLE T E R R A I N

HEF = To

~

',To I

' HEF " ToS

II

~C

FIo. 5.

Schematic view of the upper track section.

In region A J (Fig. 1) the track consists of 2(N -- 1) identical sections, where N is the number of the road wheels of one track. A typical section is shown in Fig. 3. Thus, the increase in length ALa, of the track between .4 and J may be expressed as Z¢

(37) 7.m

As the total length of the track must remain constant, the reduction AL,a of track section AB--IJ must be equal to the increase ALi, of track section A J (Fig. 1). Thus,

12

M. GARBER and J. Y. WONG

equating AL, d and 5Lir [equations (24) and (37)] and referring to equation (9), one finally obtains the equation of track length conservation Ze

(n 4-2kl)HF2(z'zm)dz) 4-rtPc--I] gm

= AAE 4- AFJ 4- AEF.

(38)

The integral in equation (38) may be evaluated analytically only in the case n = 0. Referring to equations (25-36), it can be seen that equation (38) provides a functional relation of six unknown parameters: 7, ~, zc, zm, Aa and T0s. In addition, two important geometrical relationships are established: equation (31) describing the relation of Aa, Ac and ~3,and equation (32) describing the relation of Aa and ~.

Vertical equilibrium of the vehicle The vertical reaction of the terrain on a typical track section (see Fig. 3) consists of reaction R~ over the section (0, xc) and reaction R2 over the section (xc, 1). From the vertical equilibrium of the track over the section (0, x¢), it follows that R~ is equal to the vertical component V of the track tension at the junction point x~ (Fig. 3)

b V Ix =xc = bH tan (Pc.

R 1 --

(39)

Using equations (3) and (15), one obtains the following expression for reaction R2

R 2 = b f p d x = b k r f f(cp, s)dcp, -~'c

(40)

0

where f(cp, s) = (1" cos q~ -- r + s) ° cos %

(41)

Reaction R3 of the terrain outside region AJ (Figs. 1 and 4) is relatively small as compared with that acting on section AJ of the track. As a first approximation reaction R3 may be calculated by (see Fig. 4) .v~

Ra = b

f

~0s

pdx = bkr

0

f

f(tp, s)dtp,

(42)

0

where x~ and tp,. are parameters as shown in Fig. 4. Angle ~p~ is defined as arc cos (1 -- s/r)

(for s/r ~ 1)

n/2

(for s/r ~ 1).

(43)

cps =

GROUND PRESSURE DISTRIBUTION UNDER TRACKED VEHICLES

13

The equation of the vertical equilibrium of the vehicle is written as W ---- 4 [(N -- 1) (Rx + R2) + Ra],

(44)

where W is the weight of the vehicle. Substituting equations (39-43) into equation (44), one obtains tp~ W/4bkr = ( N - - 1 ) [ H t a n t P c / k r

q~,

+ f f(~p, s) dtp] + f

0

f(tp, s) dtp.

(45)

0

The integrals in equation (45) can be evaluated analytically only for integervalues of n. In other cases, they must be evaluated numerically. SYSTEM OF EQUATIONS DEFINING THE TRACK-TERRAIN INTERACTION The case o f non-linear pressure-sinkage relationship (n > 0) From the analysis performed so far, it can be seen that the track-terrain interaction under static conditions is defined by a system of l0 equations:equations (18-20) defining the conditions at the junction point of the track and the road wheel; equation (21) defining the track tension outside the ground contact area; equations (22) and (23) defining the additional compression in the springs of the suspension and the track tensioning device; equations (31) and (32) defining the relations of the geometrical parameters of the track-suspension system; equation (38) defining the track length conservation and equation (45) defining the vertical equilibrium of the vehicle. The system contains 10 unknowns: z m, z c, H, s, q)c, To~, Aa, Ac, ~ and lB. Hence the system is consistent, that is, the number of equations is equal to the number of unknowns. It should be noted that it is a system of nonlinear transcendental equations. The direct solution of ten nonlinear equations is relatively complex. Fortunately, the original system may be reduced to a system of three equations with respect to three unknowns z,~, zc and s by substitution and by means of linearization of equations (31) and (32). The differences between angles ~t and g0 and [Band [Bo are usually small. Therefore, the following linearization is justified

= ~ o + & t , [3 = [ 3 o + ~ sin~t =sin~t o+8~teos~t o, sin~ = s i n ~ o + ~ c O S ~ o cos x -- cos xo -- 8~ sin ~to, cosfB = c o s [ 3 o-8[Bsin~B o

(46) ].

f

(47)

The final system will be presented in a non-dimensional form using the following non-dimensional parameters "X : x/r, ~ = z/r, "Xm = xm/r, "=,, = Zm/r, ~c : z,./r, ; : s/r, 7 = I/r, ~t = rt/r, 7s = rffr, ~i = )~ffr, A'~ = Aa/r,

(48) ~ c ~- Ac/r, 5o = aolr,

= k s r / w , "k, = ktr/IV, ~ = 9 8 2 W,

--m ,r 2kr n+l "! T O = Tob/W, Tos = To~b/W , W --- W(n + l)v2/4bkr "+I, v = ~ / L/ ( -n-+- _-=1)H1/ .

14

M. GARBER and J. Y. WONG E q u a t i o n s (19-21) o f the original system m a y be rewritten as ~c = arc cos (~c + 1 -- s)

(49)

v ---- F(~c, ~,~) tan q~o

(50)

Tos = 1/2W cos 9~,

(51)

where the n o n - d i m e n s i o n a l function F is defined as

T (3, ~ ) =

1 v,~+l_L:+l)

(52)

C o m b i n i n g equations (22), (23), (31) a n d (32) a n d using equations (46) a n d (47), one obtains 8*t =

~

(u12u23 -

ux3u~)/(unu22

-

u~u~l),

= (U13U21 - - U11U23)/(//11U22- H12/'/21),

(53) (54)

where u u ~ u2~ = cos *to cos ~o; u~2 = ~ o + (1 - - 7,) cos ~3o] k~/~o~ sin [3o + cosZ ~o - - G sin 13o + + ( sin2 ~o -- J cos [3o) k~/k,; u~s ---- G cos [3o - - J sin ~o

k~/k,;

(55)

u2x = [ao + 7t - - 7s + (1 - - 7~) cos ~o]'kJ~ro~ sin oco - - G sin ,t o + cos Z,to; U2a -~- G cos *to; G = (1 - -

To/To.3 (sin *to -- sin ~o); J : (1 -- To/To~) (1 + cos t3o).

T h e expressions for ~5*t a n d 8[3 [equations (53) a n d (54)] in fact replace e q u a t i o n s (31) a n d (32) o f the original system. F u r t h e r m o r e , equations (22) a n d (23) m a y be rewritten as A~ = [~o~ (sin *t + sin [3) - - "~o (sin ,t o + sin ~o)]/k~,

(56)

±~ = [T'o~ (l + cos ~) -- ~o (1 + cos ~o)]/k,.

(57)

GROUND PRESSURE DISTRIBUTION UNDER TRACKED VEHICLES

15

The remaining 3 equations, that is, equations (18), (45) and (38) of the original system, may be rewritten as follows ~e

f

(Pt

(N-

(58)

ff~, [,.) d~ -- v (7-- sin 9c) = O,

2)[2 t a n % + f .7(9, 9, ~ ) d g ] + 0

Oa

f

~ = O,

(59)

0

~e

2(N--1)[

1 fv/(vz -q- ~ ~,-~m) d~) q-(pc - - ~ ] ~ I] : 0 ,

(60)

where ~and II are defined as 7((p, v, s') = (n q- 2) vz (cos q) -- 1 -q- s) n cos {p.

(61)

I] = (1 -- 7~) (cot ~ -- cot o% q- 8~) -I- (1 -- 7t) (cot [5 -- cot [3o -}- 6[3) -- ~o/sin [3o -+- (~0 -- A~)/sin [3 -(ao q- 7, -- 7,)/sin ~o q- (~o -- A~ -+- 7t -- 7,)/sin ~ -[~o "Z.. +' sinh 7'oP-~'_i 70, ,=,~ sinh P~--~os ~o, sinh ~~ ~m + , - - A~]/~.

(62)

It can be seen that using equations (46-57), all the parameters in equations (58-60) may be defined through three unknowns, ~m, ~', and ~. As a result, the original system of ten equations is reduced to a system of three equations [equations (58-60)] with three unknowns, ~'m, ~c and ~. The solution of the final system [equations (58-60)] requires numerical integration and may be obtained by a simple trial and error procedure or other numerical techniques. The analysis performed is valid for terrain with non-linear pressure-sinkage relationship, that is for any n ~ 0 [see equation (3)]. The numerical solution to a general problem with n > 0 will be discussed in a subsequent article. In this paper the application of the analytical method developed to the prediction of ground pressure distribution will be demonstrated through the specific case of n = 1. The case o f linear pressure-sinkage relationship (n = 1)

In this case, the shape of the deflected track between road wheels is described by a hyperbolic cosine function [equation (24)]. The system of equations defining the track

16

M. GARBER and J. Y. WONG

terrain interaction [equations (58-60)] is simplified to a great extent, since the integrals in equations (58) and (59) may be evaluated analytically. Equations (50), (58) and (49) take the following form --_~ == tg % / v sinh (x~),

(63)

-" ~ -?~1 = cosh (x,v)

(64)

-c

~ =-~ + 1 -- cos (pc,

(65)

.7c = 7 -- sin %.

(66)

where

Equations (51-57) remain unchanged, while equations (59) and (60) are transformed into

(N--1) [2 tan (p,. + ~2 ( 1 sin 2%

-- 2 sin (pc cos % + (pc) ] (67)

÷ v ~ ( % - - l - - s~i n 2' %=) 0 - ' 2

Xc

0 From equations (51-57) and (63-65) it can be seen that all the parameters in equations (67) and (68) may be defined through two unknown parameters % and v. As a result, for the case of n = 1 the final system reduces to a system of 2 equations [equations (67) and (68)] with respect to two unknowns tpc and v. The solution requires numerical integration in equation (68). A simple trial and error technique is quite effective for the solution. NUMERICAL RESULTS The analytical method described above has been applied to the calculation of the pressure distribution of five track systems (see Fig. 6). To apply the analytical method, 15 design parameters of the vehicle must be specified (see Table 1). All the numerical results presented below are obtained for a terrain with a linear pressure-sinkage relationship (n = 1). The values of terrain stiffness k used in the calculations (3508500 kN/m ~) cover a wide spectrum of terrains ranging from muskeg through snow to a very firm sand [4, 5]. Figure 7 shows the pressure distribution beneath the road wheels of the five track systems under consideration. The graphs in Fig. 7 represent the pressure distribution within the length of a period, which is equal to L / ( N - - 1). Beneath the front and rear

G R O U N D PRESSURE DISTRIBUTION U N D E R T R A C K E D VEHICLES

17

,,,

oooooP FIo. 6. (I) (II) (III) (IV) (V)

< h-

O"

Schematic views of the track systems under consideration. The tensioning wheel is located in the front. The tensioning wheel is located in the front and fixed. The tensioning wheel is located in the rear. The tensioning wheel is located in the rear and fixed. The tensioning wheel is located in the front.

$

O

A

/

IV

'~ i

z . ..~o-~As° f .

1 E

m

2'

UJ

z z

_o I-

O" 1'

g~

2 0

E I--

1 0

ul

E m E Q.

v

1

O' _ o

1 z

P

x 10 2 • k N / m 2

FIG. 7.

Pressure distribution beneath the track of the vehicles under consideration. I - V designate the types of track systems shown in Fig. 6.

18

M. GARBER and J. Y. WONG

road wheels, the pressure distribution w i l l b e slightly different because of end effects. The graphs are for five values of terrain stiffness, k -----350, 650, 2000, 5000 and 8500 k N / m ~. The dotted lines shown in Fig. 7 represent the mean g r o u n d pressure ( M G P ) a n d are given for the purpose of comparison. In calculating the M G P , a more realistic contact area of 2b(L + 2r sin tps) than the n o m i n a l contact area 2bL is used. It is f o u n d that the relative difference between the m e a n g r o u n d pressure a n d the n o m i n a l g r o u n d pressure ( N G P - M G P ) / M G P , may be as high as 28.9 ° o (see track system III over a very soft terrain). F o r this reason, it is preferable to use the M G P instead of the N G P in assessing the u n i f o r m i t y of the pressure distribution. It can be seen from Fig. 7 that the m a x i m u m pressure occurs u n d e r the axes of the road wheels while the m i n i m u m pressure occurs at the mid-points between the centres o f adjacent road wheels. F o r a very firm terrain the m i n i m u m pressure is close to zero, which m e a n s the sinkage is very small. The graphs show that as the terrain becomes softer the pressure distribution becomes more uniform. F o r a very soft terrain the m a x i m u m pressure may be only 17% higher t h a n the mean g r o u n d pressure (see track system I over terrain with k = 350 kN/m~). In the present analytical framework, the pressure d i s t r i b u t i o n is a f u n c t i o n of 15 vehicle design parameters (see Table 1) a n d 2 terrain parameters (k a n d n). It is expected that some o f the design parameters may have more significant influence on the TABLE 1. DESIGNPARAMETERSOF THE TRACKSYSTEMSUNDERCONSIDERATION

Design parameters of the track system W weight of the vehicle, kN b width of the track, m L distance between the centres of the front and rear road wheels, m r radius of the road wheel, m N number of the road wheels ct0 angle between the track and horizontal surface on the side of the sprocket 8o angle between the track and horizontal surface on the side of the tensioning wheel a0 vertical distance between the centre of the tensioning wheel and the centre of the road wheel, m r, radius of the sprocket, m rt radius of the tensioning wheel, m k, combined stiffness of the suspension springs of the one track, kN/m kt stiffness of the track tensioning device spring, kN/m ,~ weight of the track per unit track length, kN/m m number of supporting rollers To initial track tension per unit width of the track, kN/m

Track System I

Track Track Track Track System II System Ill System IV System V

514 0.65

370 0.57

370 0.67

138 0.50

78 0.43

4.50 0.35 7

4.38 0.335 5

2.65 0.40 4

2.50 0.28 5

2.50 0.295 5

45°

22°

37°

33°

30°

35~

28.Y'

45°

28°

32°

0.525 0.28 0.30

0.335 0.335 0.285

0.34 0.24 0.22

0.22 0.17 0.11

0.175 0.175 0.175

1500

1075

1250

545

215

1000

co*

1250

oo*

500

1.2 4

0.9 5

1.1 2

0.55 3

0.3 0

44

25

35.5

12.8

2l

*oo indicates that the tensioning wheel is fixed.

GROUND PRESSURE DISTRIBUTION UNDF..R TRACKED VEHICLES

19

pressure distribution than the others. The relative importance of the various parameters will he discussed in Part II of this article. Figure 8 shows the variation of the mean maximum pressure (MMP) of the track systems with terrain stiffness k. As mentioned before, the maximum pressures under all tile road wheels of a vehicle in the present study are considered to be identical. The mean maximum pressure is, therefore, equal to the maximum pressure in this case. It can be seen from Fig. 8 that the value of MMP is strongly dependent on terrain stiffness. The ratio of the value of MMP over a very firm terrain to that over a very soft terrain may be as high as 3.3 (see track system II).

AN/re'

2501 ~ 2001 1501

V

1°°1 0

t

i

2

3 4

$

6

?

8

9 xl0akN/m 3

TERRAIN STIFFNESS k FIG. 8. Variation of the mean maximum pressure with terrain stiffness. I-V designate the types of track systems shown in Fig. 6.

The empirical formulae for calculating the value of MMP proposed by Rowland are based solely on vehicle design parameters [2]. For link and belt tracks on rigid road wheels, Rowland recommended the following formula MMP = where tp is track pitch.

1.26 W 2NB V~

.']-2' r"Z '

(69)

20

M. GARBER and J. Y. WONG

R o w l a n d ' s formula is based on five vehicle design parameters; vehicle weight, n u m b e r and radius o f the road wheels, track width and track pitch. Other design parameters (see Table i) that m a y be as important as those considered by Rowland are not taken into account. A shortcoming in Rowland's a p p r o a c h is that the effect o f terrain properties on the M M P is ignored. A comparison o f the values o f M M P by Rowland's formula and those obtained by the analytical method are given in Table 2.

TABLE2.

COMPARISON BETWEEN THE

M M P VALUES BY R O W L A N D ' S FORMULA AND BY THE ANALYTICAL APPROACH

MMP values obtained using the analytical approach (kN/m ~) Type of track system Track system I Track system lI Track system III Track system IV Track system V

MMP values by Rowland's formula (kN/m ~) 195.2 (pitch = 0.19 m) 249.8 (pitch = 0.16 m) 269.7 (pitch -- 0.13 m) 134.2 (pitch = 0.12 m) 89.7 (pitch ~- 0.I1 m)

very soft terrain --~ very firm terrain 89.3 ~ MMP ~ 238.1 78.8 _< MMP ~ 259.5 99.4 ~ MMP _~ 278.9 55.3 ~ MMP < 164.3 39.4 _~ MMP < 119.1

It is suggested by R o w l a n d that the M M P may be considered as a basic parameter of relevance to soft g r o u n d performance. It implies that the lower the value o f the M M P , the better the soft ground mobility of the vehicle will be. The results presented in Fig. 8 indicate, however, that while the concept o f the M M P may be useful in differentiating the potential mobility o f different vehicles over a given terrain, the value o f M M P alone is not sufficient in assessing the mobility o f a given vehicle over different types o f terrain. As can be seen from Fig. 8, for a given vehicle the value o f M M P o v e r a firm terrain is always higher than that over a soft terrain. There is no reason to expect, however, that in general the performance o f a vehicle over a firm terrain is necessarily lower than that over a soft terrain. Figure 9 shows the variation of the ratio of mean maximum pressure to mean ground pressure ( M M P / M G P ) with terrain stiffness k. The ratio o f M M P to M G P characterizes the non-uniformity of the pressure distribution under the track. As can be seen from Fig. 9, for a very firm terrain the ratio may be as high as 3.7 (see track system II over a very firm sandy terrain with k ~ 8500 kN/m3). For a given vehicle, as the terrain becomes softer the pressure distribution becomes more uniform. F o r a very soft terrain the ratio o f M M P to M G P may be as low as 1.17 (see track system 1 over a muskeg with k ---- 350 kN/m3). Figure 10 shows the variation of the maximum sinkage s with terrain stiffness k. The m a x i m u m sinkage occurs under the axes o f the road wheels and is proportional to the m a x i m u m pressure exerted by the track.

GROUND PRESSURE DISTRIBUTION UNDER TRACKED VEHICLES

21

MMP MGP

I!

III

3

2

1

0

J

2

3

4

$

6

?' 8

9 x lOakN/m 3

TERRAIN STIFFNESS k

FIG. 9. Variation of the ratio of mean maximum pressureto mean ground pressure with terrain stiffness. I-V designate the types of track systems shown in Fig. 6. CLOSING REMARKS (1) An analytical method for predicting the pressure distribution beneath a tracked vehicle under static conditions has been developed. In the analysis, the tracksuspension system which consists of the track, the suspension and the track tensioning device is considered in its entirety. All major design parameters of the vehicle as well as terrain characteristics have been taken into consideration. The results of the analysis show that the track-terrain interaction under static conditions is defined by a system of ten nonlinear equations. However, by means of linearization and substitution this system may be reduced to a system of three equations. For terrain with a linear pressure-sinkage relationship, it may be further reduced to a system of two equations. (2) The analytical procedure developed has been applied to the calculations of the pressure distribution of five track systems over a wide spectrum of terrains. The quantitative relationships between ground pressure distribution and vehicle design parameters and terrain stiffness are illustrated. It is shown that the mean maximum pressure is not only a function of vehicle design parameters, but also strongly dependent upon terrain characteristics. It is found that the value of the mean maximum pressure of a vehicle decreases with the decrease of terrain stiffness. Consequently, as the terrain becomes softer the pressure distribution becomes more uniform.

22

M. G A R B E R and J. Y. W O N G

S,cm 30"

25

20

15

!o III I

IV V 1

2

3

4

5

6

7

TERRAIN STIFFNESS

FIG. 10.

8

9 x103kN//m 3

k

Variation of the maximum sinkage with terrain stiffness. I-V designate the types of track systems shown in Fig. 6.

(3) The empirical formulae for calculating the mean maximum pressure proposed by Rowland have been examined. In Rowland's approach the effect of terrain properties on the mean maximum pressure has been ignored. Thus, the value of the mean maximum pressure determined by Rowland's formulae is independent of terrain conditions. It has been suggested by Rowland that the mean maximum pressure can effectively replace the nominal ground pressure as a general parameter of relevance to soft ground performance. The results of the analysis presented in the article indicate, however, that while the concept of the mean maximum pressure may be useful in differentiating the potential mobility of different vehicles over a given terrain, the value of the MMP alone is insufficient in assessing the mobility of a given vehicle over different terrains. This illustrates the limitation of the concept of the mean maximum pressure in evaluating vehicle mobility. (4) It is shown that the analytical method described in this paper can provide a means whereby the effects of vehicle design parameters and terrain conditions on ground pressure distribution may be assessed quantitatively. Thus, it can serve a useful purpose in differentiating the potential mobility of vehicles of different design, particularly with respect to vehicle sinkage and motion resistance.

GROUND PRESSURE DISTRIBUTION UNDER TRACKED VEHICLES

23

I t is intended t h a t the analytical f r a m e w o r k described in this p a p e r will be extended to include the effects o f tractive effort a n d d r a w b a r pull a n d t h a t e x p e r i m e n t a l validation o f the analysis will also be p e r f o r m e d . Acknowledgements--The work described in this paper was supported by the Natural Sciences and Engineering Research Council of Canada through Grant No. A5590 awarded to Professor J. Y. Wong. REFERENCES ll] M.G. BEKKEg, Theory of Land Locomotion. University of Michigan Press, Arm Arbor (1956). [2] D. ROWLAND, Tracked vehicle ground pressure and its effect on soft ground performance, Proc. of the 4th Int. Conf. of the Int. Soc. for Terrain-Vehicle Systems, Vol. 1. Stockholm, Sweden (1972). [3] I. EVANS,The sinkage of tracked vehicles on soft ground, J. Terramechanies I (2) (1964). [4] J.Y. WONG, M. GARNER,J. R. RAaFORTHand J. T. DOWELL,Characterization of the mechanical properties of muskeg with special reference to vehicle mobility, J. Terramechanics 16 (4) (1979). [5] J.Y. Woso, Data processing methodology in the characterization of the mechanical properties of terrain, J. Terramechanics 17 (1), 13 (1980).