CRREL

CRREL REPORT 95-7

Paul W. Richmond March 1995

Motion Resistance of

Wheeled Vehicles in Snow
Cover: Humpty Dympty falling off of the very high wall.

For conversion of SI units to non-SI units of measurement consult ASTM Standard

E380-93a, Standard Practice for Use of the International System of Units,

published by the American Society for Testing and Materials, 1916 Race St.,

Philadelphia, Pa. 19103.

This report is printed on paper that contains a minimum of 50% recycled

material.

Cover: CRREL instrumented research

vehicle in deep snow.

Abstract

Before vehicle mobility in snow can be reliably predicted, a complete understanding of motion resistance in snow is required. This report examines several

aspects of wheeled vehicle motion resistance using results obtained with the

CRREL instrumented vehicle. Resistances of leading and trailing tires are

examined. Limited data are presented for undercarriage drag, and third and

fourth wheel passes in the same rut are initially analyzed, as is how snow

deforms around a wheel. For the CRREL instrumented vehicle, a trailing tire has

a resistance coefficient of about 0.017 for snow depths less than about 22 cm.

For deeper snow, the disruption of the snowpack caused by a preceding wheel

causes snow to fall into the rut, resulting in higher trailing tire coefficients. For

larger vehicles, which in some cases have trailing tires carrying larger loads

than preceding tires, the trailing tire coefficients are on the order of 0.048 and

0.025 for second and third trailing wheels respectively. Since there are no

trailing tire data available for these larger vehicles, these values are based on

nonlinear regression analysis, which includes a prediction of the leading tire

resistance. The results and observations of this study are applied in a reanalysis

of the towed resistance data obtained during the U.S. Army’s Wheels vs. Tracks

study. An improved algorithm is presented for predicting wheeled vehicle motion

resistance caused by snow.
Prepared for

OFFICE OF THE CHIEF OF ENGINEERS

Approved for public release; distribution is unlimited.

Paul W. Richmond March 1995

CRREL Report 95-7

U.S. Army Corps

of Engineers

Cold Regions Research &

Engineering Laboratory

Motion Resistance of

Wheeled Vehicles in Snow
PREFACE

This report was prepared by Paul W. Richmond, Mechanical Engineer, of the

Applied Research Division, Research and Engineering Directorate, U.S. Army Cold

Regions Research and Engineering Laboratory. Funding for this research was

provided by DA Project 4A762784AT42, Design, Construction and Operations Technology for Cold Regions; Task CS; Work Unit M04, Vehicle Snow Mechanics (Shallow

Snow).

Technical review of this report was provided by Sally A. Shoop and George L.

Blaisdell, both of CRREL. The author expresses his appreciation to Byron Young,

who assisted in conducting the tests and in the operation and maintenance of the

CRREL Instrumented Research Vehicle.

The contents of this report are not to be used for advertising or promotional

purposes. The citation of brand names does not constitute an official endorsement

or approval of the use of such commercial products.

ii
iii

CONTENTS

Page

Preface ..................................................................................................................... ii

Nomenclature ......................................................................................................... v

Introduction ............................................................................................................ 1

Experimental procedure ....................................................................................... 1

Results and analysis .............................................................................................. 4

Leading tire resistance .................................................................................. 4

Trailing tire resistance ................................................................................... 7

Deep snow ...................................................................................................... 9

Undercarriage drag ....................................................................................... 11

Multiple passes .............................................................................................. 13

Shallow snow resistance model ........................................................................... 13

Summary ................................................................................................................. 17

Literature cited ....................................................................................................... 17

Appendix A: Snow data ....................................................................................... 19

Appendix B: Observations of snow deformation by a wheel ......................... 23

Appendix C: Wheeled vehicle motion resistance data .................................... 35

Abstract ................................................................................................................... 41

ILLUSTRATIONS

Figure

1. CRREL instrumented vehicle .................................................................... 3

2. Off-center towing hitch .............................................................................. 3

3. Leading tire motion resistance .................................................................. 7

4. Snow and tire characteristic dimensions ................................................. 7

5. Normal probability plot ............................................................................. 8

6. Ninety-five percent confidence intervals for the means ....................... 10

7. Snow in front of wheel ............................................................................... 10

8. CIV in deep snow........................................................................................ 11

9. Comparison of values measured by Blaisdell and those obtained

here for trailing tire resistance behind a driven wheel ....................... 12

10. Forces acting on CIV................................................................................... 12

11. Net force and velocity vs. time ................................................................. 14

12. Loose snow in rut in front of trailing tire ................................................ 15

13. Plot of resistance coefficients for multiple passes .................................. 16

14. Comparison of two sinkage prediction algorithms ............................... 16

15. Comparison of predicted and measured resistance values .................. 17
TABLES

Table

1. Vehicle sinkage and snow density data .................................................. 2

2. Tire deflection data ..................................................................................... 4

3. CIV motion resistance data, 1993 ............................................................. 5

4. Average rear resistance coefficient and standard deviations

by data set .................................................................................................. 8

5. Analysis of variance ................................................................................... 9

6. Analysis of undercarriage drag, tests 0125h and 0125i ......................... 13

7. Multiple axle analysis of resistance, tests 0126 and 0202 ...................... 15

iv
NOMENCLATURE

Abbreviations

ANOVA Analysis of variance

CIV CRREL Instrumented research Vehicle

HMMWV High-Mobility Multipurpose Wheeled Vehicle

HEMTT Heavy Expanded Mobility Tactical Truck

LAV Light Armored Vehicle

Symbols

a arc length of the area of a wheel in contact with snow

h snow depth

hr snow depth ratio, see eq 8

L2 1/2 the load on the second axle

L3 1/2 the load on the third axle

r wheel radius

r

2 correlation coefficient

Rf motion resistance at a front wheel

Rr motion resistance at a rear wheel

Rs motion resistance attributable to snow

Vf vertical load on a front wheel

Vr vertical load on a rear wheel

w tire width

z sinkage or rut depth of a wheel in snow

ρ0 average undisturbed density

ρf theoretical final density of the snow under a wheel

v
Motion Resistance of Wheeled Vehicles in Snow

PAUL W. RICHMOND

INTRODUCTION

Models of terrain–vehicle interaction require

algorithms that can predict or estimate the resistance to motion caused by the terrain. As a vehicle

moves through snow, it must overcome the resistance caused by snow deformation to maintain

forward motion. There is resistance at each wheel

or track. As a vehicle moves through undisturbed

snow, the front wheels move and compact this

snow. Succeeding or trailing wheels move over or

through the snow compacted by the front wheels,

presumably encountering a smaller resistance

(since the snow has been compacted and moved

out of the way). If the vehicle’s undercarriage

drags, this adds another source of resistance that

must also be considered.

The motion resistance of a vehicle moving

through snow is a very complex phenomenon,

studied since the early 1940’s. However, a robust,

simple model of motion resistance due to snow

has eluded researchers.

Currently, the U.S. Army uses in its mobility

model an empirical equation for motion resistance

in snow developed by Richmond et al. (1990). This

equation is applicable to a large range of vehicles,

both wheeled and tracked, and does not require

elaborate measurements of snow properties. However, it is currently the weakest link of the overall

Cold Regions Mobility Model (Richmond et al., in

prep). When developing this equation, we analyzed a large quantity of total vehicle resistance

data by dividing the whole vehicle resistance by

the number of wheels, thus making the assumption that the trailing tire resistance was equal to the

resistance of preceding wheels. This assumption

was based on attempts to correlate data in which

motion resistance was measured on one axle

(CRREL Instrumented Vehicle [CIV]), on two axles (High Mobility Multipurpose Wheeled Vehicle

[HMMWV]), and on four axles (Heavy Expanded

Mobility Tactical Truck and Light Armored Vehicle [HEMTT and LAV]) during towing tests.

This approach was not without basis, as the density of the deformed snow behind each succeeding

axle generally increased, although axle load did

not. Density measurements in the wheel ruts (Table

1) were always well below what is considered the

critical density of snow (0.5 to 0.55 Mg/m3

) (Yong

and Fugue 1977), the point at which no further

compaction by typically loaded vehicle traffic is

observed. Additionally, Blaisdell (1987) made some

limited indirect measurements of trailing tire resistance using the CIV and found that trailing tire

resistance in snow could be as high as 84% of the

leading tire resistance.

In an effort to increase the accuracy of the

motion resistance model for undisturbed snow, a

field study was conducted using the CIV with load

cells installed on all four wheels. Prior to this work,

only one set of load cells was available and these

were installed on the front axle of the CIV (from

1981 to 1991). By use of the data acquired from

these and other tests, a further understanding of

the motion resistance of wheeled vehicles in snow

was obtained, allowing an improved prediction

algorithm to be presented.

EXPERIMENTAL PROCEDURE

The experimental procedure was designed

around the capabilities of the CIV (Fig. 1 [Blaisdell

1983, Shoop 1992]). The instrumentation of the

CIV was recently enhanced through the addition

of load cells on the rear axle and a new data

acquisition system based on a personal computer

(Shoop et al. 1991). These triaxial load cells measure longitudinal, vertical and lateral forces at the

tire/terrain interface. Longitudinal forces represent net traction or motion resistance, depending

on whether the tire is driven or free rolling. Individual wheel and vehicle speeds are measured

simultaneously with the wheel forces. By towing

the CIV, resistance forces could be measured at all

four wheels, and a tow hitch that could be placed

off-center (Fig. 2) allowed the CIV to be towed

through undisturbed snow.
Table 1. Vehicle sinkage and snow density data (after Richmond et

al. 1990).

Contact Snow Initial Rut

pressure* depth Sinkage density density

Vehicle (kPa) (cm) (cm) (Mg/m3) (Mg/m3)

HMMWV† 101.8 15.0 8.0 0.20 0.38

123.3 15.0 8.0 0.20 0.48

101.8 19.5 11.9 0.20 0.47

123.3 19.5 13.8 0.20 0.50

101.8 23.5 19.1 0.20 0.535

123.3 23.5 21.0 0.20 —

101.8 19.5 14.5 0.20 0.455

123.3 17.5 13.1 0.20 0.470

101.8 18.0 14.5 0.20 0.445

123.3 18.0 13.0 0.20 0.475

101.8 13.0 9.0 0.12 0.320

123.3 12.0 8.5 0.12 0.410

101.8 19.5 14.0 0.12 —

123.3 18.0 14.0 0.12 —

123.3 10.0 9.0 0.19 0.575

HEMTT** 233 20.0 12.4 0.25 0.490

230 21.0 16.9 0.25 0.440

225 20.0 16.2 0.25 0.520

218 20.0 15.6 0.25 0.460

233 16.5 12.4 0.25 0.490

230 18.0 13.2 0.25 0.490

225 20.5 15.4 0.25 0.510

218 20.0 15.6 0.25 0.510

233 20.0 14.3 0.245 —

218 17.0 12.6 0.245 0.460

233 13.0 10.0 0.12 —

HEMTT†† 177 17.0 12.6 0.245 0.510

174 18.0 15.1 0.245 0.525

224 17.5 14.6 0.245 0.430

212 16.0 13.1 0.245 0.455

212 16.0 12.2 0.245 0.455

212 4.0 3.0 0.095 —

177 12.0 10.5 0.12 0.300

212 11.0 9.0 0.12 0.315

*Contact pressure of the wheel behind which the rut density measurement was made.

†With an inflation pressure of 137.9 kPa (front) and 151.7 kPa (rear).

**With an inflation pressure of 241.3 kPa (front two axles), 275.8 kPa (rear two axles).

††With an inflation pressure of 137.9 kPa (front two axles) and 206.8 kPa (rear two axles).

Tests were conducted at the Keweenaw Research Center of the Michigan Technological University, in northern Michigan, during January and

February 1993. The testing areas had been graded

free of vegetation prior to any snowfall, and were

nearly level. A layer of packed snow was beneath

the undisturbed snow in which the tests were

conducted. The CIV was equipped with Michelin

LT 235XCH4 tires for these tests.

Initial tests were done with the tires inflated at

103 kPa (15 lb/in.2), while most of the tests were

run at an inflation pressure of 179 kPa (26 lb/in.2).

Tire data are in Table 2.

A static calibration procedure (Shoop 1992) was

used. Prior to collecting data in the snow, we

conducted at least four tests on hard surfaces.

These hard surfaces were sometimes covered with

packed snow or ice; however, an attempt was

made to use areas that were as smooth as possible.

These hard surface data were averaged for each

load cell and used to remove from the subsequent

snow resistance data the hard surface rolling resistance and any zero offset (remaining from the

static calibration) in the load cells.

In these tests the vehicle was towed with all four

hubs locked out. During initial calibration tests,

2
Figure 2. Off-center

towing hitch.

Figure 1. CRREL

instrumented

vehicle.

the effects of the disk brake pads on the hard

surface rolling resistance values were examined.

As long as the brakes were not applied during the

hard surface tests, or during any subsequent tests,

the calibration and hard surface values remained

valid. It did not matter if the pads were installed or

not, although the hard surface resistance is higher

with the pads in place. By closing the valves in the

CIV brake system, inadvertent braking can be

avoided. Of course, with no effective brakes on the

vehicle, care must be taken with the tow vehicle

and tow hitch.

Most tests in undisturbed snow were conducted

by towing the vehicle at about 5 km/hr using the

off-set tow hitch shown in Figure 2. A few tests

were done without towing the CIV, using the front

wheels to drive the vehicle in deep snow. Additionally, some tests were conducted with the CIV

3
Table 2. Tire deflection data.

Maximum Maximum

Nominal Deformed undeformed section

load radius radius width

Wheel location (N) (cm) (cm) (cm)

Inflation pressure 103 kPa

left front 6800 30.5 36.5 27.2

right front 30.0 36.5 27.2

left rear 5510 31.5 37.0 27.2

right rear 32.5 37.0 27.2

Inflation pressure 179 kPa

left front 6800 33.0 36.7 25.6

right front 32.4 37.5 26.3

left rear 5510 33.5 37.5 25.5

right rear 34.2 38.0 25.5

following in the ruts made by the tow vehicle.

After each test run, the undisturbed snow depths

along both the left and right wheel ruts were

measured at 1-m intervals. These data were later

averaged to obtain an undisturbed snow depth

value for each side of the vehicle. At the conclusion

of a series of tests, several snow pits were used to

characterize the snow and to determine an average density for each test.

RESULTS AND ANALYSIS

Table 3 contains a summary of the data collected. The data are separated into sets based on

the day of the test and the tire inflation pressure.

Initial snow depths ranged between 10 and 37 cm

and average initial snow densities ranged from

0.11 to 0.25 Mg/m3

. All of the snow data collected

are given in Appendix A.

Leading tire resistance

The leading tire data from Table 3 are plotted in

Figure 3 using the resistance parameter (ρ0aw) in

kilograms per meter, developed by Richmond et

al. (1990). The variables in this parameter are illustrated in Figure 4; ρ0 is the average initial undisturbed density of the snow, a is the arc length of the

tire in contact with the snow, and w is the maximum tire width in meters. The arc length a is

determined from the sinkage z and the tire radius,

where the sinkage is determined from

 

z h = 









 1 0 – ρ

ρf

(1)

where h is the snow depth, and ρf is the theoretical

final density, which is determined from the following with pmax the maximum contact pressure:

ρf = 0.50 Mg/m3 for pmax ≤ 210 kPa

ρf = 0.55 Mg/m3 for pmax > 210 kPa

ρf = 0.60 Mg/m3 for pmax > 350 kPa

ρf = 0.65 Mg/m3 for pmax > 700 kPa.

A value of 0.50 Mg/m3 is used for the CIV with the

inflation pressure and tires used in these tests.

Three equations are plotted on Figure 3. These

are

Rs = 15.62 (ρ0aw)

1.2676 (2)

which is a fit of the data collected during the CIV

tests reported herein

Rs = 11.25 (ρ0aw)

1.58 (3)

which was obtained for the CIV alone in 1988 and

1989 (Richmond et al. 1990) and

Rs = 68.083 (ρ0aw)

0.9135 (4)

which is the currently used shallow snow resistance equation (Richmond et al. 1990). Rs is the

resistance per wheel (in Newtons) for a vehicle

traveling through primarily undisturbed snow.

Combining the old and new CIV data yields the

following equation

Rs = 21.313 (ρ0aw)

1.1918 (5)

which has an r

2

 value of 0.58 compared to 0.76 for

4
Table 3. CIV motion resistance data, 1993.

Snow Wheel Predicted Arc Resistance (N) Vertical load (N) Coefficient

Set Test Velocity depth Density sinkage Radius Width sinkage* length aρ wo

no. no. (km/hr) (cm) (kg/m3) (cm) (cm) (cm) (cm) (cm) (kg/m) Front Rear Front Rear Front Rear

1 0119g-r† 5.0 13.4 160 9.90 36.5 27.2 9.1 26.4 11.5 346.5 77.4 6931.2 5423.7 0.0500 0.0143

0119g-l 5.0 15.5 160 12.0 36.5 27.2 10.5 28.5 12.4 416.4 46.7 6646.1 5453.5 0.0626 0.0086

0119h-r 6.6 15 170 12.5 36.5 27.2 9.9 27.5 12.7 316.7 160.1 6886.7 5449.9 0.0460 0.0294

0119h-l 6.6 18.6 170 15.1 36.5 27.2 12.3 30.8 14.3 557.4 165.5 6800.4 5400.6 0.0820 0.0306

2 0119o-r 5.1 16.3 160 13.3 37.5 26.3 11.1 29.6 12.5 498.6 90.7 7032.2 5262.2 0.0709 0.0172

0119o-l 5.1 16.6 160 13.6 37.5 25.6 11.3 29.9 12.2 394.1 56.5 6787.5 5582.5 0.0581 0.0101

0119p-r 8.7 18.3 170 15.3 37.5 26.3 12.1 31.0 13.8 460.8 97.9 7119.3 5133.7 0.0647 0.0191

0119p-l 8.7 19.1 170 16.3 37.5 25.3 12.6 31.7 13.6 469.7 125.9 6775.9 5716.8 0.0693 0.0220

3 0122g-r 5.3 21.1 220 17.1 37.5 26.3 11.8 30.6 17.7 474.2 37.8 7071.3 5574.9 0.0671 0.0068

0122g-l 5.3 20.8 220 16.8 37.5 25.6 11.6 30.4 17.1 621.0 42.7 6461.5 5443.7 0.0961 0.0078

0122h-r 9.0 19.8 220 15.8 37.5 26.3 11.1 29.6 17.1 532.0 52.5 7146.0 5493.5 0.0744 0.0096

0122h-l 9.0 20.5 220 16.8 37.5 25.6 11.5 30.1 17.0 747.3 48.9 6453.9 5454.8 0.1158 0.0090

0122i-r 2.1 20.1 220 16.1 37.5 26.3 11.3 29.8 17.3 470.2 47.2 6716.8 5504.6 0.0700 0.0086

0122i-l 2.1 20.5 220 16.5 37.5 25.6 11.5 30.1 17.0 635.6 10.2 6853.8 5554.0 0.0927 0.0018

0122j-r 5.5 18 220 14.5 37.5 26.3 10.1 28.2 16.3 482.2 8.5 6876.0 5361.9 0.0701 0.0016

0122j-l 5.5 19 220 15.5 37.5 25.6 10.6 29.0 16.3 693.5 0.0 6662.5 5695.9 0.1041 0.0000

4 0122o-r 5.0 21.2 220 16.2 36.5 27.2 11.9 30.3 18.1 414.1 109.9 7265.2 5845.8 0.0570 0.0188

0122o-l 5.0 37.5 220 32.5 36.5 27.2 21.0 41.3 24.7 1176.5 139.2 6755.0 5007.3 0.1742 0.0278

0122p-r 6.8 22.2 220 17.2 36.5 27.2 12.4 31.1 18.6 525.8 93.0 7296.4 5700.8 0.0721 0.0163

0122p-l 6.8 20.5 220 15.5 36.5 27.2 11.5 29.8 17.8 451.5 64.9 6498.8 5236.4 0.0695 0.0124

0122q-r 5.5 17.9 220 12.9 36.5 27.2 10.0 27.7 16.6 340.7 83.6 7071.3 5648.3 0.0482 0.0148

0122q-l 5.5 17.8 220 12.8 36.5 27.2 10.0 27.6 16.5 374.5 57.4 6730.1 5315.2 0.0557 0.0108

0122r-r 4.8 20.3 220 15.3 36.5 27.2 11.4 29.6 17.7 473.3 55.2 7058.0 5627.4 0.0671 0.0098

0122r-l 4.8 20.7 220 15.7 36.5 27.2 11.6 29.9 17.9 480.9 92.1 6716.8 5371.6 0.0716 0.0171

0122s-r 5.3 22.2 220 17.2 36.5 27.2 12.4 31.1 18.6 520.9 13.3 6794.6 5344.5 0.0767 0.0025

0122s-l 5.3 21.6 220 16.6 36.5 27.2 12.1 30.6 18.3 456.8 37.8 7087.8 5574.0 0.0645 0.0068

0122t-r 8.2 21.8 220 16.8 36.5 27.2 12.2 30.8 18.4 379.0 83.6 7101.1 5284.5 0.0534 0.0158

0122t-l 8.2 22.9 220 17.9 36.5 27.2 12.8 31.6 18.9 505.8 82.3 6830.2 5578.0 0.0740 0.0148

5 0125f-l 6.0 35.7 160 30.7 37.5 25.6 24.3 45.4 18.6 1164.1 459.1 6994.8 5365.9 0.1664 0.0856

0125g-l 4.7 36 160 31 37.5 25.6 24.5 45.6 18.7 1314.0 492.0 7202.5 5479.3 0.1824 0.0898

0125h-l 7.6 32.4 160 27.4 37.5 25.6 22.0 43.0 17.6 –454.6** 368.3 6753.7 5747.1 -0.0673 0.0641

0125h-r 7.6 31.6 160 26.6 37.5 26.3 21.5 42.4 17.8 –634.0 347.8 6804.4 6065.6 -0.0932 0.0573

0125i-l 7.1 33.6 160 28.6 37.5 25.6 22.8 43.9 18.0 –492.7 368.3 6565.1 5796.9 -0.0751 0.0635

0125i-r 7.1 33.6 160 28.6 37.5 26.3 22.8 43.9 18.5 –771.0 456.4 6963.7 5954.4 -0.1107 0.0766

6 0126e-l 4.7 12.2 110 0†† 37.5 25.6 — — — 54.3 46.7 6897.8 5298.3 0.0079 0.0088

0126e-r 4.7 11.2 110 0 37.5 26.3 — — — 93.0 3.1 6811.1 5417.0 0.0136 0.0006

0126f-l 8.4 14.2 110 0 37.5 25.6 — — — 74.7 60.5 7004.6 5510.9 0.0107 0.0110

0126f-r 8.4 12.8 110 0 37.5 26.3 — — — 104.1 36.9 6728.8 5270.2 0.0155 0.0070

5
Table 3 (cont‘d). CIV motion resistance data, 1993.

Snow Wheel Predicted Arc Resistance (N) Vertical load (N) Coefficient

Set Test Velocity depth Density sinkage Radius Width sinkage* length aρ wo

no. no. (km/hr) (cm) (kg/m3) (cm) (cm) (cm) (cm) (cm) (kg/m) Front Rear Front Rear Front Rear

7 0126g-l 5.0 13.6 110 9.6 37.5 25.6 10.6 28.9 8.1 167.7 85.0 6968.4 5217.7 0.0241 0.0163

0126g-r 5.0 12.7 110 8.7 37.5 26.3 9.9 27.9 8.1 182.8 79.6 6907.1 5495.7 0.0265 0.0145

0126h-l 9.5 16.7 110 13.7 37.5 25.6 13.0 32.2 9.1 229.5 151.2 6865.1 5139.5 0.0334 0.0294

0126h-r 9.5 16.1 110 12.1 37.5 26.3 12.6 31.6 9.1 275.8 170.4 7068.3 5538.5 0.0390 0.0308

0126i-l 5.1 17.6 110 15.1 37.5 25.6 13.7 33.2 9.3 213.1 74.7 6759.3 5281.4 0.0315 0.0141

0126i-r 5.1 15.7 110 12.2 37.5 26.3 12.2 31.2 9.0 418.6 78.7 7199.0 5427.6 0.0581 0.0145

8 0202f-l 3.4 18.5 250 0 37.5 25.6 — — — 654.8 167.3 6536.6 5509.5 0.1002 0.0304

0202f-r 3.4 13.2 250 0 37.5 26.3 — — — 311.8 0.0 7513.9 5294.2 0.0415 0.0000

9 0202g-l 4.2 16.1 230 13 37.5 25.6 8.7 26.1 15.3 876.7 115.7 6898.3 5362.3 0.1271 0.0216

0202g-r 4.2 12.8 230 5 37.5 26.3 6.9 23.1 14.0 521.3 138.3 7196.3 5609.2 0.0724 0.0247

0202h-l 5.8 18.6 230 8.5 37.5 25.6 10.0 28.1 16.5 831.8 327.4 7005.5 5130.1 0.1187 0.0638

0202h-r 5.8 10.8 230 9 37.5 26.3 5.8 21.2 12.8 435.0 240.2 7200.3 5839.2 0.0604 0.0411

*Predicted sinkage based on eq 1.

†Month and day of test are first four digits; last character (l,r) indicates whether data are from left or right side of vehicle.

**Traction measurements are shown as negative values.

††Measured wheel sinkage was zero because the CIV wheels traveled in the ruts of the tow vehicle.

6
Figure 4. Snow and tire characteristic dimensions.

h

r

w

Compacted Snow

Undisturbed

z Snow a

1000

2000

100

200

400

800

600

80

60

2 20 4 6 8 10

Density . Arc Length . Width (kg/m)

Resistance Per Wheel (N)

Eq 2

Eq 3

Eq 4

Figure 3. Leading tire motion resistance.

pendix B. Snow density (i.e., snow characteristics

other than depth) may have been too similar between tests to develop a clear effect.

Trailing tire resistance

The initial analysis consisted of examining scatter plots of the independent variables (snow depth,

density, tire width, sinkage and velocity) and several different combinations of these parameters vs.

several forms of the dependent variable: rear

wheel motion resistance (Rr). The following forms

of the trailing wheel motion resistance were considered—rear tire motion resistance (Rr), rear tire

motion resistance divided by vertical load (Rr/Vr)

(coefficient form), ratio of rear tire motion resistance to front tire motion resistance (Rr /Rf), and

eq 3. Further analysis of this data set (old and new

CIV data) using a stepwise regression procedure

showed that an equation including only snow

depth and the ratio of arc length a to 1/4 the

perimeter of the wheel provided a slightly better

correlation to the data. This equation

 

s a R h

r

= +











 66 148 51 915 490 26 4 . . –. π

(6)

has an r

2

 of 0.68. It is interesting that neither the

wheel width nor snow density is included in the

equation, even though wheel width varied from

0.156 to 0.274 m (albeit there are only a few data

points at the narrowest width). The lack of a wheel

width effect was also suggested by the analysis of

snow deformation by a wheel presented in Ap7
Table 4. Average rear resistance coefficient and

standard deviations by data set.

Average Average

Data Number of Avg. resistance Standard snow depth snow density

set observations coefficient deviation (cm) (Mg/m3)

1 4 0.0207 0.0095 15.6 0.165

2 4 0.0171 0.0044 17.6 0.160

3 8 0.0056 0.0036 20.0 0.220

4 12 0.0140 0.0061 22.2 0.220

5 6 0.0728 0.0120 33.8 0.160

6 4 0.0068 0.0039 12.6 0.110

7 6 0.0199 0.0072 15.4 0.110

8 2 0.0152 —— 15.8 0.250

9 4 0.0378 0.0168 14.6 0.230

Figure 5. Normal probability plot.

the ratio of rear tire motion resistance to front tire

motion resistance in coefficient form (RrVf /RfVr).

This analysis was done set by set and for all of the

data combined. No clear dependence was found

between any of the variables and any of the rear

tire motion resistance “parameters.”

It was next hypothesized that the rear tire motion resistance coefficient (Rr /Vr) may be a constant (at least for the CIV) under similar test conditions. To investigate this hypothesis, the average

and standard deviation for each data set were

calculated (Table 4); those sets that were collected

under clearly different conditions were not included in the analysis. Data set 5, which clearly has

a higher resistance coefficient, was neglected since

it was obtained in snow that was much deeper

than that in the other data sets. Data sets 6 and 8

were obtained by towing the CIV in the ruts that

were made by the tow vehicle, thus the trailing tire

data are actually the fourth pass over the snow

cover.

Inspection of the remaining data sets (1, 2, 3, 4,

7 and 9) showed that the average values of the

highest (set 9) and lowest (set 3) didn’t seem to

99.9

99.0

95.0

80.0

50.0

20.0

5.0

1.0

0.1

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

Trailing Tire Resistance Coefficient

Probability (%)

8
1, 2, 3, 4, 7 and 9. The data fell on nearly a straight

line, indicating that they are random and normally

distributed. Bartlett’s test for homogeneity of the

error variance showed that the variances were not

different at the 95% confidence level. A multiple

range test indicated that sets 3 and 9 should not be

included, and Table 5 contains the results of a

series of ANOVA tests to determine which of the

data could be combined.

Between each ANOVA test, a procedure following Grubbs (1969) was used to determine if the

extreme values could be considered outlying observations. This analysis also showed that sets 3

and 9 can be dropped from further analysis on the

basis of statistical arguments. The average rear tire

motion resistance coefficient Rr/Vr of the remaining data sets is 0.0169 with a 95% confidence

interval of 0.0138 to 0.0199. These values are shown

graphically on Figure 6. It should be noted that if

both sets 3 and 9 are included, the average Rr/Vr

is 0.0167 with a confidence interval of 0.0127 to

0.0207, which really isn’t much different from the

values obtained from the four data sets, although

it is serendipitous that the effects of these two tests

(3 and 9) average each other out.

From the above analysis, it appears that a trailing tire in dry snow that is less than 0.22 m deep

and that has a density less than about 0.250 Mg/m3

will have a resistance coefficient of about 0.017,

although values as low as 0.0 and as high as 0.64

were measured. No data were obtained for snow

densities greater than 0.250 Mg/m3. It should also

be noted that the rear wheels of the CIV carry less

weight than the front wheels, and the effect of a

trailing tire carrying a higher weight is not known.

Deep snow

Tests 0125f–0125i (data set 5) were done in

relatively deep snow compared to the other tests.

These deep snow tests resulted in limited data for

two different trailing tire configurations. Data were

obtained similar to those described above (all four

wheels of the CIV rolling freely in undisturbed

snow) and with the front wheels driving while the

rear wheels were rolling free. This second condition, freely rolling wheel trailing a driven tire, is

significantly different from that described earlier.

To tow the CIV through this deep (36 cm) snow,

a packed path was first made for the lead vehicle.

Additionally, the snow was deep enough such

that the undercarriage of the lead vehicle disturbed the snow in which the right side wheels of

the CIV traveled. Thus, in Table 3, for tests 0125f

and 0125g, data are only presented for the left side

Table 5. Analysis of variance.

Data Number of

set observations Mean

1 4 0.0207

2 4 0.0171

3 8 0.0056

4 12 0.0140

7 6 0.0199

9 4 0.0378

Total 38

For all data sets

SS df MS F

TOTAL 0.016013 38

CF 0.01059 1

TR 0.002986 5 0.000572

RES 0.002437 32 0.0000761 7.516

Since F0.95 (5,32) = 2.5, there is a difference between data sets.

For data sets 1,2,3,4 and 7

SS df MS F

TOTAL 0.009175 34

CF 0.0002698 1

TR 0.0075927 4 0.001898

RES 0.0013125 29 0.000045286 41.92

Since F0.95(4,29) = 2.7, there is a difference between data sets.

For data sets 1,2,4 and 7

SS df MS F

TOTAL 0.008817 26

CF 0.007399 1

TR 0.0002128 3 0.00007093

RES 0.001205 22 0.00005479 1.295

Since F0.95(3,22) = 3.05, there is no difference between data sets.

agree with the other data (Table 4). There was no

apparent reason for this on the basis of test conditions or procedure, although data set 9 may have

been collected in hard “crusty” snow. To legitimately combine these data sets, the data need to be

random and normally distributed and the variances need to be homogeneous. Once this is shown,

an analysis of variance (ANOVA) test or multiple

range test can be used to determine if the data sets

can be combined.

Figure 5 is a normal probability plot of data sets

9
Figure 7. Snow in front of wheel.







123479

Data Set Number

50 × 10–3

40

30

20

10

0

–10

Trailing Tire Resistance Coefficient, Rr/Vr Figure 6. Ninety-five percent confidence intervals for the means. Shaded area is the mean

and 95% confidence interval for combined sets 1, 2, 4 and 7.

10
Figure 8. CIV in deep snow.

of the CIV. The snow in these tests was about 36 cm

deep, which is about equal to the undeflected

radius of the tires. Figure 7 shows how the snow

piled up in front of a wheel, although it is not clear

whether this snow came from under the vehicle or

was pushed up from the area in front of the tire

(note the “bow wave” in front of the CIV in Fig. 8).

Because of the difficulty of towing the CIV in

snow this deep, it was disconnected from the tow

vehicle and driven in front-wheel drive. It became

immobilized when the snow piled up in front of

the bumper (Fig. 8), so that tire chains had to be

installed on the front wheels. This, and slightly

shallower snow in the succeeding area, allowed

some data to be collected. Tests 0125h and 0125i

yielded trailing tire resistance behind a driven tire.

The trailing tire resistance coefficients were 0.064,

0.057, 0.064 and 0.077 for the left and right sides of

tests 0125h and 0125i respectively. These values

and those of tests 0125f and 0125g are much higher

than those measured in the other tests; this is most

likely attributable to snow falling back into the rut

behind the front wheels in deep snow, or in the

case of tests 0125h and 0125i, attributable to the

disruption of the packed snow rut by a preceding

driven tire. (A preceding driven tire disrupts the

snow through the effects of wheel slip; compared

to the effects of a freely rolling wheel, the snow in

the bottom of the rut is made uneven and looser.)

Blaisdell (1987) measured trailing tire resistance

using the CIV with load cells mounted only on the

front axle. He used two different indirect methods

to accomplish this. In one procedure, he drove the

front wheels at a steady speed and assumed that

the measured force on the front load cells was

equal to the trailing tire resistance (the measured

forces for the CIV are analyzed below). A comparison of these measurements are made with those of

tests 0125h and 0125i in Figure 9, which shows all

the values falling in the same range.

To see if the rear coefficients were related to the

slip of a preceding driven wheel, the leading tire’s

Differential Interface Velocity (DIV) was compared

with the trailing tire resistance coefficients. There

did not appear to be any clear cause and effect.

Undercarriage drag

Figure 10 shows a force balance on the entire

vehicle (neglecting air resistance); once the vehicle

is moving at a steady speed, the sum of forces

should be zero. If the CIV is driven at a steady

speed, the load cells on the driven axles measure

the resistance to motion felt by the rest of the

vehicle (i.e., the net traction measured by these

load cells must equal the sum of all the other

resistances). Since some of the CIV undercarriage

was dragging in the deep snow tests, undercarriage drag can be estimated. Minimum ground

clearance for the CIV is 14.0 cm for the torsion bar

brackets and 19.1 cm for the front differential, and

11
0.12

0.10

0.08

0.06

0.04

0.02

0

Resistance Coefficient

Source :

Snow Depth (cm) :

Snow Density (Mg/m3) :

Blaisdell (1987) 0125h-l 0125h-r 0125i-l 0125i-r

7 16 32 32 34 34

0.257 0.100 0.160 0.160 0.160 0.160





+ Forces

– Forces

R(undercarriage)

R(terrain, internal)

R(gravity)

∆ Momentum

Net Traction Snow

Compacted Snow Layer

Figure 10. Forces acting on CIV (front wheel drive). Net traction + R(terrain, internal)

+ R(gravity)* + ∆momentum + R(undercarriage) = 0.

 *Horizontal component of gravitational force used when operating on a slope.

Figure 9. Comparison of values measured by Blaisdell (1987) and those obtained here for

trailing tire resistance behind a driven wheel. Blaisdell’s data are shown ± 1 standard

deviation.

vehicle sinkage was about 28 cm in the deep snow.

Table 6 is a summary of the forces on the CIV; hard

surface resistance was estimated using equations

developed by Shoop (1992) for the left and right

front tires of the CIV based on velocity. Since no

data are available on the hard surface resistance of

the rear CIV tires, it was assumed to be the same as

the front. From Figure 10, the sum of the measured

forces should equal the force ascribable to the

dragging undercarriage, if the momentum effect

can be assumed to be zero (i.e., the vehicle is not

accelerating or decelerating). Figure 11 gives plots

of the sum of the load cell output, reading by

reading, and the corresponding velocity. It does

not appear that momentum has an effect, since the

velocities are nearly steady, and the small variations in velocity don’t necessarily correspond to

changes in the sum of the forces on the vehicle.

The force exerted on the undercarriage is estimated to be 734 and 770 N for tests 0125h and 0125i

respectively. It can be seen that, without including

the effect of the slope, the vehicle would have been

immobilized. Using the information from tests

0215f and 0215g to obtain a value of resistance

12
Table 6. Analysis of undercarriage drag, tests

0125h and 0125i.

Test 0125h Test 0125i

Total average vertical force 25,371 N 25,280 N

Longitudinal forcesa:

Average left front 454.6 492.7

Average right front 634.0 771.0

Average left rear –368.3 –368.3

Average right rear –347.8 –456.4

Hard surface resistanceb –507.4 –505.6

Net force (available) w/o gravity –134.9c –66.6

Average force due to gravityd 869 837

Net force available with gravitye 734 770.4

aFrom Table 3.

bEstimated using an average coefficient of 0.02.

cA negative value infers that the vehicle would be immobilized

(no tractive reserve). dThe CIV tilt sensor measured an average angle (φ) of –1.96 and

–1.89°, indicating that the CIV was moving downhill. The

force due to gravity was calculated by multiplying the total

average vertical force by sin φ. eSince the vehicle was assumed to be traveling at a steady

speed, this value must be attributed to undercarriage drag.

This is most likely caused by snow falling back into

the rut, which is common for deep snow covers

(Fig. 12) and may have happened in this test.

Unfortunately, this effect was not identified early

enough during the testing to be fully investigated.

These data are shown graphically in Figure 13. The

data of test 0126 seem to agree well with the theory

that succeeding tires should have decreasing resistance, the average values for the third and fourth

“axles” being 0.0094 and 0.00685 respectively.

The implication of this analysis is that, for vehicles with more than two axles such as the HEMTT

and LAV, a rationale needs to be determined for

modeling these additional trailing tires. It does not

appear that individual wheel resistance on large

vehicles will be directly measurable in the near

future.

SHALLOW SNOW RESISTANCE MODEL

As mention earlier, Richmond et al. (1990) analyzed resistance data from a large number of tests

and vehicles, obtaining eq 4 from data for both

wheeled and tracked vehicles. Richmond (1990)

combined the data into groups of like vehicles and

developed a set of equations of the same form as eq

4, but no significant improvement in the correlation coefficients resulted. In these earlier analyses,

only the data from the CIV could be considered as

leading wheel data. The data obtained from the

other vehicles were “whole” vehicle resistances

(from snow) values. To try and reduce the data to

fit one equation, the whole vehicle resistance was

simply divided by the number of wheels for each

vehicle.

Appendix B contains a discussion of the deformation of snow by a wheel and concludes along

with eq 6 that wheel width may not be as an

important parameter as arc length and initial density. Additionally, the ratio of wheel radius to

sinkage should be considered to account for increased deformation (more bulldozing) in deeper

snow covers, possibly in the form (4a/rπ) noted

earlier. With these ideas in mind, and with the new

trailing tire information, an attempt was made to

improve the currently used algorithm (eq 4 and 1)

for predicting motion resistance for wheeled vehicles in snow.

After several attempts were made to extend the

CIV trailing tire data to the HMMWV data, without any clear success, the motion resistance data

base obtained during the Wheels vs. Tracks shallow snow tests was reexamined. Appendix C conattributable to the four wheels and the average

undercarriage resistance results in an undercarriage resistance that is 22% of the total vehicle

resistance or 1.14 times Rs. This is interesting in

that Richmond et al. (in prep) use a multiplier of

1.25 to estimate undercarriage drag for this condition.

Multiple passes

Data sets 6 and 8 were obtained by towing the

CIV such that the wheels traveled through the ruts

created by the tow vehicle. The forces measured

may be assumed to be estimates of third and

fourth trailing tires. It was further assumed that

tests 0126g–0126i could represent the first and

second wheels; together, these data represent a

complete set of resistance data for axles 1 through

4 (Table 7). Tests 0202f–0202h were analyzed similarly.

Comparing the average resistance coefficients

between the two data sets shows that the resistance data measured during the 0202 tests (snow

depth 15.8 cm with a density of 250 kg/m3

) are

much higher than the data from the 0126 tests

(snow depth 12.6 cm with a density of 110 kg/m3).

13
tains the data presented by Blaisdell et al. (1990),

Richmond et al. (1990), and Green and Blaisdell

(1991), combined with the data presented herein.

The data are presented in the form in which they

were used to develop the following algorithms.

The first step was to reevaluate the sinkage

relationship of Richmond et al. (1990). It was found

that, if the tracked vehicle data were ignored, a

linear trend exists between the highest contact

pressure of a preceding wheel and the theoretical

final density (ρf). The sinkage prediction algorithm for undisturbed snow remains as (eq 1)

 

z h = 









 1 0 – ρ

ρf

and the final density is now calculated using

 f 0.519 0.0023 max ρ = + p (7)

Velocity Test 0125h

Force due to Gravity

Sum of Load Cell Forces

–2

0

2

4

6

8

10

0246

Force (kN) or Velocity (km/hr)

Test 0125i

Velocity

Force due to Gravity

Sum of Load Cell Forces

–2

0

2

4

6

8

0 2 4 6 8 10

Force (kN) or Velocity (km/hr)

b. Test 0125i.

Figure 11. Net force and velocity vs. time.

a. Test 0125h.

Time (s)

Time (s)

14
where pmax is the maximum contact pressure of

the wheels in kilopascals. Figure 14 is a comparison of the two prediction methods. Much better

agreement is obtained using eq 1 and 7 as compared to eq 1 and the table of values for ρf.

Using the resistance data in Appendix C and a

nonlinear regression analysis combined with the

assumptions that 1) the resistance of the first wheel

through undisturbed snow is a function similar to

eq 4 and 2) that the resistance of trailing tires can be

Figure 12. Loose snow in rut in front of trailing tire.

Table 7. Multiple axle analysis of resistance, tests 0126 and 0202.

Snow Force (N) Coefficient (R/V)

Test depth Density

no. (cm) (Mg/m3) 1* 2 3 4 1 2 3 4

0126e-l 12.2 0.11 —— —— 54.3 46.7 —— —— 0.0079 0.0088

0126e-r 11.2 0.11 —— —— 93.0 3.1 —— —— 0.0136 0.0006

0126f-l 14.2 0.11 —— —— 74.7 60.5 —— —— 0.0107 0.0110

0126f-r 12.8 0.11 —— —— 104.1 36.9 —— —— 0.0155 0.0070

0126g-l 13.6 0.11 167.7 85.0 —— —— 0.0241 0.0163 —— ——

0126g-r 12.7 0.11 182.8 79.6 —— —— 0.0265 0.0145 —— ——

Average 12.7 0.11 175.3 82.3 81.5 36.8 0.0253 0.0154 0.0094 0.0069

0202f-l 18.5 0.25 —— —— 657.0 167.3 —— —— 0.1002 0.0304

0202f-r 13.2 0.25 —— —— 311.8 0.0 —— —— 0.0415 0.0000

0202g-l 16.1 0.23 876.7 111.2 —— —— 0.1271 0.0207 —— ——

0202g-r 12.8 0.23 521.3 138.3 —— —— 0.0724 0.0247 —— ——

0202h-l 18.6 0.23 831.8 327.4 —— —— 0.1187 0.0638 —— ——

0202h-r 10.8 0.23 435.0 240.2 —— —— 0.604 0.0411 —— ——

Average 15 0.23 666.2 204.3 484.4 83.7 0.0955 0.0376 0.0709 0.0152

*Axle number.

15
Figure 14. Comparison of two sinkage prediction algorithms.

Test 0202

Test 0126

012345

0

0.02

0.04

0.06

0.08

0.10

Number of Wheels through Snow

Resistance Coefficient

Figure 13. Plot of resistance coefficients for multiple passes.

represented as a coefficient multiplied by the vehicle wheel load, and 3) using the sinkage prediction algorithm of eq 1 and 7, resulted in the following equation for 1/2 the total vehicle resistance

(each axle is considered to be symmetrical).

 

Rs = 13 6041( ) o wa +

1 26 . . ρ

(8)

 

0 0475 0 0246 2 3 . . hr r L + h L

where

 

hr = 1 0. if h ≤ 0.22 m

 

hr

h = 0 22 . if h > 0.22 m

where

h = snow depth (m)

L2 = 1/2 the load (N) on the second axle

L3 = 1/2 the load (N) on the third axle.

Measured Sinkage (cm)

Calculated Sinkage (cm)

25

20

15

10

5

0

0 5 10 15 20 25

Eq 1 and 7

Eq 1 with Table

of Values for ρf

Over-predicted

Under-predicted

16
a. Using eq 1, 7 and 8. b. Using eq 4 and 1 with tabular values of ρf

.

Figure 15. Comparison of predicted and measured resistance values.

The term (ρ0wa) is in kilograms per meter. It is

interesting to note that adding a term for the fourth

axle adds little to the accuracy of the equation and

forces the coefficient for the third axle to be negative, a physical impossibility. The rationale for the

ratio of snow depth to the depth of 0.22 m is to

account for the effect of snow falling back into a rut

behind a leading wheel in deep snow. The trailing

tire coefficients obtained are higher than anticipated, and this is thought to be attributable to

higher loads on the trailing axles for most of the

vehicles in the data base, as compared to the CIV,

which has less weight on the trailing axle.

Other forms of the term (ρ0wa) were investigated using the ideas presented earlier; little or no

improvement was found in the regression analysis. Figure 15 shows a comparison of the predicted

vs. measured resistance. It can clearly be seen that

eq 8 produces excellent results and a marked improvement over earlier algorithms. Table C1 contains the actual predictions and residuals.

SUMMARY

Several aspects of the interaction between snow

and a wheeled vehicle were investigated. The

resistance of a trailing wheel can be considered a

constant in most cases; however, snow falling back

into the rut behind a leading wheel results in

higher values and seems to be related to initial

snow depth. The true effect of wheel width on

resistance of a leading wheel is still not known and

should be studied using a numerical technique

where other parameters can be held constant. Limited data were presented for undercarriage drag,

and since this will be the primary cause of immobilization on level surfaces, it deserves further

investigation.

A new and significantly improved algorithm

for predicting sinkage and motion resistance in

undisturbed snow was presented. This improved

algorithm should be implemented in current cold

regions mobility models.

LITERATURE CITED

Blaisdell, G.L. (1983) The CRREL instrumented

vehicle: Hardware and software. USA Cold Regions Research and Engineering Laboratory, Special Report 83-3.

Blaisdell, G.L. (1987) Trailing-tire motion resistance in shallow snow. Proceedings of the 9th Inter8 × 103

7

6

5

4

3

2

1

0

0 1 2 3 4 5 × 103

Predicted

Observed

Under-predicted

Over-predicted

Under-predicted

Over-predicted

8 x 103

6

4

2

0 2 4 6 8 10 x 103

Predicted

Observed

17
national ISTVS Conference, Barcelona, Spain, p. 296–

304.

Blaisdell, G.L., P.W. Richmond, S.A. Shoop, C.E.

Green and R.G. Alger (1990) Wheels and tracks in

snow: Validation study of the CRREL shallow

snow mobility model. USA Cold Regions Research and Engineering Laboratory, CRREL Report 90-9.

Green, C.E. and G.L. Blaisdell (1991) U.S. Army

wheeled versus tracked vehicle mobility performance test program. Report 2: Mobility in shallow

snow. U.S. Army Waterways Experiment Station,

Technical Report GL-91-7.

Grubbs, F.E. (1969) Procedures for detecting outlying observations in samples. Technometrics, 11(1):

1–21.

Richmond, P.W. (1990) Vehicle motion resistance

due to snow. Proceedings of the 1990 Army Science

Conference.

Richmond, P.W., G.L. Blaisdell and C.E. Green

(1990) Wheels and tracks in snow: Second validation study of the CRREL shallow snow mobility

model. USA Cold Regions Research and Engineering Laboratory, CRREL Report 90-13.

Richmond, P.W., S.A. Shoop and G.L. Blaisdell

(in prep.) Cold regions mobility models. USA Cold

Regions Research and Engineering Laboratory,

CRREL Report.

Shapiro, L.H., J.B. Johnson, M. Sturm and G.L.

Blaisdell (in prep.) Snow mechanics: Review of

the state of knowledge and applications. USA

Cold Regions Research and Engineering Laboratory, CRREL Report.

Shoop, S.A. (1992) Precision analysis and recommended test procedure for mobility measurements

made with an instrumented vehicle. USA Cold

Regions Research and Engineering Laboratory,

Special Report 92-7.

Shoop, S.A., E. Berliner and S. Decato (1991) An

experimental method for vehicle mobility research

on freezing/thawing soil. Proceedings of the 1st

Meeting on Winter Mobility. Santa Barbara, California: Engineering Foundation.

Yong, R.N. and M. Fugue (1977) Performance of

snow in confined compression. Journal of

Terramechanics, 14(2): 59–82.

18
APPENDIX A: SNOW DATA

Tests 0119 H, 0119 O

Tests 0119 G, 0119 P

Density (Mg/m3)

Density (Mg/m3)

0.14

0.18

0.20

0.20

0.13

0.19

0.17

Ave. Density:

Ave. Density: 0.16

17 cm

16 cm

Air Temp.:

Snow Temp.:

Grain Size:

–5°C

–6°C

0.5 - 1.0 mm2

Air Temp.:

Snow Temp.:

Grain Size:

–5°C

–6°C

1.0 - 2.0 mm2

Figure A1. Snow data.

a. 19 January 1993.

19
22 cm 23 cm

17 cm 18.5 cm

13.5 cm

17 cm

20 cm

Density

(Mg/m3)

Density

(Mg/m3)

Density

(Mg/m3)

Density

(Mg/m3)

0.21

0.15

0.13

0.22

0.28

0.21

0.22

0.27

Ave. Density:

0.22

0.24

0.33

0.24

0.15

0.13

0.24

0.22

0.38

Ave. Density:

Ave. Density: 0.21 Ave. Density: 0.23

Grain Size

(mm2)

2.0 - 3.0

1.0 - 2.0

Ice Layer

(1 cm) 

Ice Layer

Ice Layer

Ice Layer

(3 - 4 mm) 

Air Temp.:

Snow Temp.:

2°C

0°C

b. 22 January 1993.

Figure A1 (cont'd). Snow data.

20
Air Temp.: –8°C

Density (Mg/m3)

0.05

0.03

0.06

0.17

0.12

0.22

0.24

0.23

0.14

0.21

Ave. Density: 0.18

38 cm

37 cm

16 cm

Temp. (°C)

Density (Mg/m3) Temp. (°C)

–5°C

–5°C

–3°C

1.0 mm2 Grains

1 - 2 mm Flakes

1.0 mm2 Grains

0.22

0.23

Ave. Density: 0.14

16 cm –2°C

Ice Layer

(1 cm)

Ice Layer

(1 cm)

c. 25 January 1993.

Figure A1 (cont'd).

21
13 cm

16.5 cm

Density

(Mg/m3)

Density

(Mg/m3)

0.08

0.09

0.12

0.18

0.08

0.09

0.14

Ave. Density:

Ave. Density: 0.10

Air Temp.:

Snow Temp.:

–4°C

–4°C

12 cm

Density

(Mg/m3)

0.24

0.25

0.24

0.20

Ave. Density: 0.23

Ice Crust

(1 cm) 

Air Temp.:

Snow Temp.:

3°C

–2°C

Figure A1 (cont'd). Snow data.

d. 26 January 1993.

e. 2 February 1993.

22
APPENDIX B: OBSERVATIONS OF SNOW DEFORMATION BY A WHEEL

The purpose of this study was to examine the deformation of snow around a

rolling wheel and to identify mechanisms and parameters that could lead to an

improved motion resistance model for shallow snow.

Procedure

Deformation experiments were conducted in areas of undisturbed snow that

were marked by chalk dust. Holes in the snow, which were punched both perpendicular to and in the direction of vehicle travel, were coated with carpenter’s chalk

dust. This was done by giving a plastic bottle of chalk dust with a tapered nozzle a

quick squeeze into the hole. A 12.5-mm (1/2-in.) diameter dowel was used to punch

the holes, which were placed 76.2 mm (3 in.) apart (Fig. B1). After emplacing the

dust, a vehicle was driven into the marked area such that the axle passed through

the row of perpendicular holes, then the vehicle was backed out. Whenever possible,

the wheel was undriven. The snow was then carefully removed to reveal a cross

section of the deformed area and the lines formed by the chalk (Fig. B2). Density

measurements were made using a 100-cm3 sampler, which had a 6- by 3-cm cross

section. Air and snow temperatures were recorded and, in some cases, so was the

grain size.

Three different vehicles were used, although most of the experiments used the

CRREL instrumented vehicle. Tire data for each vehicle are in Table B1.

Results and observations

Photographs and documentary data are presented in Figures B3–B5. From these

figures it can be seen that, as the wheel rolls forward in the snow pack, deformation

occurs in all three directions. In the following discussion, forward is the direction in

which the vehicle is moving, perpendicular refers to the direction perpendicular to

Figure B1. Snow marked with chalk-dust-filled holes.

23
Table B1. Vehicle and wheel characteristics.

CRREL Instrumented Vehicle (CIV), 1977 Jeep Cherokee

Tires: Michelin LT 235XCH4, 179 kPa

Radius: 37.5 cm

Maximum deformed width: 26 cm

CRREL Support Truck, 1989 Chevrolet 3/4 ton pickup

Tires: LT 245/75R16 M&S

Radius: 37 cm

Maximum deformed width: not recorded

HEMTT, M997 Heavy Expanded Mobility Tactical Truck

Tires: Michelin 16.0 R20

Radius: not recorded

Maximum deformed width: 48 cm

Figure B2. Removing snow to show deformed chalk holes.

the direction that the vehicle is moving but parallel to the ground, and downward

or lower infers motion towards the ground. The deformation around a rolling wheel

is extremely complex and there seem to be four areas to consider. These are the area

under the wheel (the rut area), the area in front of the wheel, and the upper and lower

areas adjacent to the wheel sides. The deformation in front of the wheel and to the

sides of the wheel in the upper part of the snow cover are related to each other, to

the snow depth and the wheel sinkage.

Looking first at the snow under the wheel rut, one can see that the snow at the

centerline of the wheel is pushed forward and downward as the wheel rolls forward.

Snow that is off the centerline is pushed off at an increasing angle by the compacting

snow closer to the centerline. This is seen in the chalk exposed in the area of the wheel

rut. The chevron shapes are created because the snow near the center of the snow

pack is pushed out further (perpendicularly and forward) than the snow near the

snow or ground surfaces. As forward motion continues, the upper snow is pushed

24
Figure B3. Snow deformation and density distributions.

5 cm

5 cm

*Snow density (Mg/m3) typical.

0.15*

0.19

0.21

0.28 0.51 0.30

0.20

0.20

0.15 0.15

0.19

a. 25 January 1992. Snow depth: 19 cm; snow temperature: –10°C; grain size: 0.5–0.75 mm2; air

temperature: –8°C; wheel size: LT 235XCH4, 179 kPa; wheel sinkage: 11 cm.

25
5 cm

5 cm

*Snow density (Mg/m3) typical.

0.06*

0.14

0.18

0.29

0.24

0.32 0.19 0.175

0.18

0.11

0.06 0.06

0.07

0.18

0.26

b. 27 January 1992. Snow depth: 21.6 cm; snow temperature: –3°C; grain size: 1.0 mm2;

air temperature: –7°C; wheel size: LT 235XCH4, 179 kPa; wheel sinkage: 16 cm.

Figure B3 (cont'd). Snow deformation and density distributions.

26
Figure B3 (cont'd).

c. 29 January 1992. Snow depth: 11.5 cm; snow temperature: –0°C; air temperature: –2°C; wheel size: LT 235XCH4,

179 kPa; wheel sinkage: 8 cm.

d. 3 February 1992. Snow depth: 31.5 cm; snow temperature: –1°C; air temperature: 1°C; wheel size: LT 235XCH4,

179 kPa; snow was removed from the surface so that the wheel could be driven into the deep snow. The axle did not

pass the chalk dust holes (4–5 cm short).

33 cm

0.26

0.23

0.27

0.25

0.30

0.27

Initial Snow

Density (Mg/m3)

0.11

0.17

0.18

11 cm

Initial Snow

Density (Mg/m3)

27
0.20

0.305

0.225

0.25

0.30

46 cm

Initial Snow

Density (Mg/m3)

Figure B3 (cont'd). Snow deformation and density distributions.

e. 4 February 1992. Snow depth: 46 cm; vehicle: HEMTT; wheel sinkage: 24 cm.

28
0.05

0.05

0.15

16 cm

Initial Snow

Density (Mg/m3)

f. 7 February 1992. Snow depth: 16 cm; snow temperature: –1°C; air temperature: –11°C; wheel size: LT 235XCH4,

179 kPa; wheel sinkage: 10 cm.

Figure B3 (cont'd).

29
25 cm

0.06

0.06

0.17

Initial Snow

Density (Mg/m3)

g. 8 February 1992. Snow depth: 25 cm; snow temperature: –13°C; air temperature: –11°C; wheel size: LT 245/

75 R16, M&S; wheel sinkage: about 14 cm.

Figure B3 (cont'd). Snow deformation and density distributions.

30
Figure B4. Deformation in direction of vehicle travel, same snow properties as in Figure B3f.

downward (see Fig. B3a, c and B4).

For the snow located to the side of the wheel, it appears that the forward

displacement component is smaller than the perpendicular displacement, since the

chalk at further distances from the centerline has little forward displacement

compared to chalk near the wheel (Fig. B3d). This may occur as the full load on the

wheel is transferred to the snow, causing additional plastic deformation in the

direction perpendicular to the wheel. Maximum deformation seems to occur at or

slightly below the bottom of the rut. Additionally, there appears to be little

displacement of the snow near the surface compared to snow deeper in the snow

cover. This suggests that the strength of the snow lower in the snowpack is more

important in resistance calculations.

As wheel sinkage approaches the radius of the wheel, the forward part of the

wheel acts more as a bulldozer than a compactor (as in the case of relatively shallow

snow) and deformation (both forward and perpendicular) is higher in the snow pack

(compare Fig. B3b and c). This is also observed in Figures B4 and B5 for snow

deformation in front of the wheel. In shallow snow the deformation area does not

extend beyond the leading edge of the tire (Fig. B5), while for deeper snow it does

(Fig. B4). This effect can be illustrated by examining the velocity vector components

of a rigid, non-slipping wheel. Figure B6 shows the forward, vertical and resultant

vectors. One can see how the forward component is reduced as one moves along the

tire surface towards the supporting surface.

Most of the experiments were done in relatively low-density snow. When the

snow had a higher density, greater deformation was observed (compare Fig. B3b

and c). This agrees with observations of soil compaction and suggests that there may

be a controlling density (maximum packing density without significant crystal

deformation). Recent analysis by Shapiro et al. (in prep.) suggests that intergranular

bonding (between snow crystals) is more important than previously suspected, and

will greatly control this type of displacement. There is currently no proven method

of measuring bond strength and density remains the primary field index of snow

31
14.5 cm

0.14

0.12

Initial Snow

Density (Mg/m3)

Figure B5. Deformation (horizontal and lateral) and initial snow density, 8 February 1992. Snow depth: 14.5 cm;

snow temperature: –13°C; air temperature: –11°C; wheel size: LT 245/75 R16, M&S; wheel sinkage: 10.5 cm.

32
Figure B6. Forces on a rigid rolling wheel.

a. Forward component unit vectors.

b. Vertical component unit vectors.

c. Resultant forces.

strength.

Two tests were done in fairly deep snow with significantly different wheels.

Figures B3c and d show the results for the CIV wheel (tire width of 26 cm) and the

HEMTT (tire width of 48 cm). For both tests the maximum perpendicular displacement is about 18–20 cm. This indicates that the effect of tire width on resistance may

not be linear.

Summary

The results of some simple tests to examine the deformation of snow around a

rolling wheel were presented. Although they were primarily qualitative, some

interesting phenomena were observed. Snow deformation caused by a tire is threedimensional and, depending on the depth and density, will primarily occur at or

below the wheel rut in shallow snow and in the upper area if the snow is deep.

Increasing wheel width does not seem to increase deformation in the direction

perpendicular to vehicle motion, at least in the one comparison available from these

tests. The ratio of wheel radius to sinkage should be considered to account for

increased deformation in deeper snow covers.

33
APPENDIX C: WHEELED VEHICLE MOTION RESISTANCE DATA

Table C1 contains the data for the CIV reported here plus the wheeled vehicle data

presented in Blaisdell et al. (1990), Richmond et al. (1990) and Green and Blaisdell

(1991). Some corrections were made to the data presented in the earlier reports. The

resistance values are presented as the measured value minus the hard surface

resistance divided by two, and thus represent the 1/2 of the resistance attributable

only to the snow. For each vehicle, normal loads are 1/2 the axle loads; some of the

CIV resistance data are for the first axle only and thus it is considered a one-axle

vehicle. The sinkage is calculated using eq 1 and 7. The shaded data were not

included in the regression analysis for the reasons stated in the Table C1 notes.

34
47

UNCLASSIFIED UNCLASSIFIED UNCLASSIFIED UL

March 1995

Motion Resistance of Wheeled Vehicles in Snow

Paul W. Richmond

U.S. Army Cold Regions Research and Engineering Laboratory

72 Lyme Road

Hanover, N.H. 03755-1290

Office of the Chief of Engineers

Washington, DC 02314-1000

CRREL Report 95-7

Approved for public release; distribution is unlimited.

Available from NTIS, Springfield, Virginia 22161

Before vehicle mobility in snow can be reliably predicted, a complete understanding of motion resistance in snow

is required. This report examines several aspects of wheeled vehicle motion resistance using results obtained with

the CRREL instrumented vehicle. Resistances of leading and trailing tires are examined. Limited data are presented

for undercarriage drag, and third and fourth wheel passes in the same rut are initially analyzed, as is how snow

deforms around a wheel. For the CRREL instrumented vehicle, a trailing tire has a resistance coefficient of about 0.017

for snow depths less than about 22 cm. For deeper snow, the disruption of the snowpack caused by a preceding wheel

causes snow to fall into the rut, resulting in higher trailing tire coefficients. For larger vehicles, which in some cases

have trailing tires carrying larger loads than preceding tires, the trailing tire coefficients are on the order of 0.048 and

0.025 for second and third trailing wheels respectively. Since there are no trailing tire data available for these larger

vehicles, these values are based on nonlinear regression analysis, which includes a prediction of the leading tire

resistance. The results and observations of this study are applied in a reanalysis of the towed resistance data obtained

during the U.S. Army’s Wheels vs. Tracks study. An improved algorithm is presented for predicting wheeled vehicle

motion resistance caused by snow.

PR: 4A762784AT42

TA: CS

WU: M04

Instrumented vehicles Snow Vehicles

Mobility predictions Snow strength Trafficability

Motion resistance Vehicle traction Off road mobility

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