Dynamic Simulation of Off-road Tracked Vehicles

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A detailed multi-body model fordynamic simulation of off-road tracked vehicles

D. Rubinstein a,*, R. Hitron b

a The Land Systems Unit, Technion Research & Development Foundation, Haifa 32000, Israel b Faculty of Agricultural Engineering, Technion – Israel Institute of Technology, Haifa 32000, Israel

Available online 23 April 2004

Abstract

Currently available models for dynamic simulation of tracked vehicles usually include su-

per-elements to describe the tracks and the suspension systems. In these models, the dynamics

of the track, the interaction between each track link and the ground, and their effect on thevehicle dynamics cannot be considered properly. The rapid increase in computing speed en-

ables the utilization of more complex models, including numerous bodies and force elements.

A three-dimensional multi-body simulation model for simulating the dynamic behavior of

tracked off-road vehicles was developed using the LMS-DADS simulation program. The

model incorporates detailed description of the track, the suspension system, and the dynamic

interaction between its components. The bodies of the model are the chassis, the wheel-arms,

the wheels, and each track link. Three-dimensional contact force elements are used to describe

the interaction of the track links with the vehicle’s road wheels, sprocket, and idler. Additional

force elements are used to simulate the bump stops and the dampers. User-defined force el-

ements are used to describe the interaction between each track link and the ground. The

normal and tangential forces are calculated using classical soil mechanics equations, such as

Bekker and Janosi correlations. Sinkage and slip are calculated separately for each track link.

Alternative correlations, based on recent studies of the dynamic variations of these forces, can

also be used. The model was first applied to the M113 armored carrier. Simulation results

under various road conditions were compared with the results of a super-element-based

model. It was concluded that the influence of the track dynamics and the soil–link interaction

on the vehicle dynamics can be better predicted with the newly developed model.

2004 ISTVS. Published by Elsevier Ltd. All rights reserved.

* Corresponding author. Fax: +972-4-8224580.

E-mail address: [email protected] (D. Rubinstein).

0022-4898/$20.00 2004 ISTVS. Published by Elsevier Ltd. All rights reserved.

doi:10.1016/j.jterra.2004.02.004

www.elsevier.com/locate/jterra

Journal

of

TerramechanicsJournal of Terramechanics 41 (2004) 163–173

http://mail%20to:%[email protected]/

http://mail%20to:%[email protected]/



Keywords: Multi-body simulation; Tracked vehicle; Track link; Track-spud

1. Introduction

The suspension unit of a tracked vehicle consists of many components, which need

to be optimized for proper design. Optimization based on field tests may significantly

increase development cost and time. Modern design of a tracked vehicle, based on a

simulation program that represents the suspension units, enables analysis of the ve-

hicles performance prior to its final design and field-testing. During the last decade,

many such simulation programs have been developed and adapted for this purpose.

However, the physical models, especially for tracked vehicles, still need improvement

[1,2]. Improving the simulation model may reduce the number of field tests required.Vehicle simulation programs are divided into two categories: the first consists of

special-purpose codes and the second is composed of multi-body programs [3]. As

computers have become increasingly powerful, the multi-body programs have be-

come a popular means for simulating the dynamic behavior of vehicles [4].

Available 2-D special-purpose programs for off-road vehicles, such as VEHDYN

I and II [5], are time-efficient, but include over-simplified assumptions (such as

constant forward velocity), and are not designed for detailed analysis of the sus-

pension system. The multi-body codes are much more powerful programs. However,

most of them are oriented to road-vehicle simulation [4,6], and are not suitable for

tracked-vehicle design. The DADS and ADAMS programs are currently the mostwidely used multibody codes [6].

Burt [7] has reviewed some of the soil–wheel and soil–wheel–track interaction

models. The present survey includes some of the empirical and semi-empirical soil–

wheel models, as well as models based on constitutive relations. Theradial spring model

is widely used in simulation programs to represent the soil–wheel interaction [1,5,8,9].

The purpose of the study is to present a detailed multi-body model for dynamic

simulation of off-road tracked vehicles.

2. Mathematical modeling

2.1. Single track-link and soil interaction model

The ground level is described using the XYZ world-coordinate system. The ground

level may be depicted in any profile as the following function.

Z ¼ f ð X ; Y Þ: ð1Þ

The track link is spatially oriented, with coordinate system x0 y 0 z 0 attached to the link.

Where y 0 is in the longitudinal direction of the link, x0 is the lateral direction and z 0 is

perpendicular to x0 and y 0. Another coordinate system X 0Y 0Z 0 is parallel to x0 y 0 z 0 andits origin is intercepted by XYZ origin. The link may penetrate into the soil when the

sinkage is measured along the z 0-axis. A two-dimensional plot of a ground profile

164 D. Rubinstein, R. Hitron / Journal of Terramechanics 41 (2004) 163–173



with a sinking link is presented in Fig. 1. A view of the link and its coordinate

systems from above is shown in Fig. 2.

In these figures:

u1; u2; u3 ¼ unit vectors of the X ; Y ; Z axes directions, respectivelyu0

1; u02; u0

3 ¼ unit vectors of the X 0; Y 0; Z 0 axes directions, respectively

‘ ¼ origin of the x0 y 0 z 0 coordinate system

g ¼ point at which the z 0-axis passes through ground-surface level

x‘; y ‘; z ‘ ¼ location of point ‘ in the XYZ coordinate system

x0‘; y 0‘; z 0‘ ¼ location of point ‘ in the X 0Y 0Z 0 coordinate system

x g ; y g ; z g ¼ location of point g in the XYZ coordinate system

x0 g ; y 0 g ; z 0 g ¼ location of point g in X 0Y 0Z 0 coordinate system

a ¼ link length

b ¼ track width

D ¼ sinkage of the track link in z 0-direction.

Fig. 2. View of the link and its coordinate systems from above.

Fig. 1. Two-dimensional description of a track-link sinkage.

D. Rubinstein, R. Hitron / Journal of Terramechanics 41 (2004) 163–173 165



An unique definition of X 0Y 0Z 0 orientation (the same as the track link) may be

obtained using the transformation matrix ½ A, where

½ A ¼a

11 a

12 a

13a21 a22 a23

a31 a32 a33

24 35 and aij ¼ ui u0 j i; j ¼ 1; 2; 3: ð2Þ

Theoretically, the values of matrix ½ A components may be calculated based on three

independent parameters. In our case, we used four dependent Euler parameters

(e0; e1; e2; e3) as follows:

a11 ¼ e20 þ e2

1 e22 e2

3;

a22 ¼ e20 e2

1 þ e22 e2

3;

a33 ¼ e2

0 e2

1 e2

2 þ e2

3;aij ¼ 2 eie j

e0ek

i; j; k ¼ 1; 2; 3; i 6¼ j; k 6¼ i; k 6¼ j:

ð3Þ

The dependency equation of the Euler parameters is

e20 þ e2

1 þ e22 þ e2

3 ¼ 1: ð4Þ

The unit vectors of the relative coordinate system X 0Y 0Z 0 is obtained based on

world coordinate system directions and Euler parameters as follows:

u0

i ¼X

3

j¼1

a ji

u j

i ¼

1; 2; 3: ð

5Þ

The world coordinates x‘; y ‘ and z ‘, as well as the Euler parameters, are gen-

eralized coordinates and their values are known in each time step of the simulation.

Therefore, point ‘ location is calculated as a function of the known parameters.

x0‘ ¼ x‘u1 þ y ‘u2 þ z ‘u3

u0

1;

y 0‘ ¼ x‘u1 þ y ‘u2 þ z ‘u3

u0

2;

z 0‘ ¼ x‘u1 þ y ‘u2 þ z ‘u3 u03:

ð6Þ

The X 0 and Y 0 coordinates of the point g are identical to the coordinate of the x0 y 0 z 0

origin and the Z 0 coordinate is z g . Thus, the world coordinates of g are:

x g ¼ x0‘u0

1 þ y 0‘u02 þ z 0 g u

03

u1;

y g ¼ x0‘u0

1 þ y 0‘u02 þ z 0 g u

03

u2;

z g ¼ x0‘u0

1 þ y 0‘u02 þ z 0 g u

03

u3:

ð7Þ

Substitution of Eq. (7) into Eq. (1) yields a non-linear algebraic equation where

the unknown is z 0 g . The equation is solved simply and efficiently using iterations. Wefound that in most of the cases no more than four iterations were required for the

solution. The sinkage distance along the z 0 coordinate is




D ¼ z 0 g z 0‘: ð8Þ

When the track link is in contact or sinking into the soil, the value of D is positive.

Otherwise, when the link is above the soil, the value is negative. The velocity v‘ of thepoint ‘ on the track link is given in the world coordinate system as follows:

v‘ ¼ _ x‘u1 þ _ y ‘u2 þ _ z ‘u3: ð9Þ

The sinkage velocity _ D is the projection of the link velocity on the negative

direction of the z 0-axis.

_ D ¼ v‘ u03: ð10Þ

The track-soil model proposed by Bekker [10] is based on the stress–displacement

relationship for a single application of load to the soil. The original formula has beenmodified by adding a viscous friction element. The normal direction of the pressure–

sinkage and sinkage–velocity relationship is

p ¼k cb þ k u

Dn þ C _ D for D > 0 and p > 0;

0 otherwise;

ð11Þ

where p is the pressure, b the track width, C the damping per unit area coefficient and

k c; k /; n are the empirically determined constants.

Note that Bekker [10] originally proposed a vertical sinkage. In our case, the

sinkage is perpendicular to the track link plane surface. When the link penetrates

into the soil, the forces of the two components of Eq. (11) are in the same di-rection. During the rebound, the forces are in opposite directions. Therefore, a

value of zero pressure may be obtained while geometrically the link is still sinking

into the soil. This is the point at which the link and the soil actually part from

one another.

The attainable locomotion of the whole vehicle over a terrain is based on the shear

forces that develop between track links and soil in the longitudinal directions of the

links. The shear stress–displacement relation is obtained as follows:

s ¼ cð þ p tan/Þ 1 e rj j

k ; ð12Þ

where s is the shear stress, D the shear displacement, c the cohesion, u the angle of

internal friction, k the empirically determined constant.

The value of the shear displacement is

r ¼

Z v‘ u

02 dt : ð13Þ

Integration of Eq. (13) is not activated unless a contact between the link and the soil

is detected. The integration starts from the beginning with each contact. Assuming

constant distributed shear stress yields the shear force F s as follows:

F s ¼ sign v‘ u02

sab; ð14Þ




where

signð x

Þ ¼

1 x > 0;

1 x < 0;

0 x ¼ 0:8<: ð

15Þ

For side-slide modeling, we assume a model for lateral direction that is similar to

the longitudinal direction. Thus:

F Ls ¼ sign v‘ u01

ab cð þ p tan/Þ 1

e

rLj j

k L

ð16Þ

and

rL ¼ Z v‘ u01 dt : ð17Þ

The model can be refined by dividing the link area into n sub-areas (n ¼ 4 is

reasonable) and applying the model described above to each sub-area. This modi-

fication has been checked using a single-link simulation model without any signifi-

cant effect. However, it has not been checked on the whole vehicle model.

2.2. Vehicle model

The M113 armored carrier vehicle was selected as a typical model of tracked

vehicle. The vehicle consists of a main body (chassis) and two track systems. Each

track system contains a track with 63 links, 5 road-wheels and road-arms, a sprocket,

and an idler. The road-arms of the M113 are relatively parallel. Each road-arm is

attached by a torsional rod to the chassis, and there is a translational damper be-

tween the road-arm and the chassis. While riding over an obstacle, the road-arm/

wheel may receive an impact. The impact moves the wheel toward the chassis. The

suspension components (torsional rod and damper) restrain the relative motion of

the road-arm and reduce the force exerted onto the chassis. When the force of the

impact is very high, the relative motion is halted by the bumper. A description of

these components is provided in Fig. 3.

The LMS-DADS multi-body simulation program is used for the vehicle model-ing. The soil–link interaction model (described in the previous section) was added to

the program as a user-force subroutine. The links are connected to each other using

revolute joints. In order to prevent excessive constraints, the joint between the last

link and the first link is defined by preventing the relative motion in y 0- and z 0-

directions. Additional planer joints between one link and the sprocket or idler is

required for keeping the track in the sprocket-idler plane. The friction in each axis

connected between links is considered by adding force elements (RSDA) to the

joints. The contact of the links with the road-wheel, sprocket, and idler is formulated

with a contact element as a non-linear spring and damper or Hertz theory. The

friction between the contact surfaces is also considered. The connections of the road-wheels with the road-arms and the road-arms with chassis are implemented with

revolute joints. Force elements with non-linear characteristics (TSDA and RSDA)




are used for the modeling of the torsional bars and the dampers. The implementation

of the bump-stops can be created based on the available components of the contact

elements. A user can add its own model when a more comprehensive one is required

(for example, for bump-stops with hydropneumatic components).

2.3. Power train

The engine torque is applied on the sprocket through the power train system.

Every engine type has a characteristic curve with a specified working zone. The curve

is a torque or power vs. rotating speed. Note that the torque value should be defined

by the riding conditions and not by the characteristic curve. The characteristic

specifies only the maximum available torque or power in a certain condition. The

rotating mass of inertia of the engine and the power train components is relatively

small (compared to those of the chassis and track links), and can be neglected

without any significant effect on the whole vehicles dynamic behavior. Therefore, the

use of a detailed and accurate model for the power train system is not important

unless the simulation goal is to explore the behavior of the power train components.The engine and the power train system are considered part of the main body. The

equivalent rotating mass of inertia can be added to that of the sprocket.

In the simulation model, the engine torque is applied directly on the sprocket.

Therefore, the torque of the characteristic curve should be multiplied by the trans-

mission ratio and the rotation speed divided by the transmission ratio. Each gear has

a different transmission ratio. Thus, instead of one characteristic curve, one can

obtain as many curves as the number of the gears. We propose a propulsion model as

described in the block diagram (Fig. 4).

The reference velocity of the vehicle vref can be constant or a function of time,

location, and riding conditions. Subtracting actual velocity v from the referencevelocity yields the error signal. The required torque T r is obtained by applying the

cruise controller on the error signal. The controller can be proportional where the

Fig. 3. Description of a track system.




gain is the driver behavior. The required torque signal goes through the character-

istic limitation box. The signal is checked; if it is above the characteristic curve, its

valve is changed to the maximum accessible. The output of characteristic limitationis the actual torque T a that is applied on the sprocket. The current vehicle velocity v

and the sprocket rotational speed x are the output of the DADS plant. The rota-

tional speed is used for proper gear selection and definition of the current torque

limit. The velocity returns to the beginning of the control loop.

3. Case study

The M113 armored carrier vehicle was selected for examination of the simulation

results. The case study focuses on single track-link behavior, a comparison withWong [11], and a comparison of the current track model with the DADS super-

element model.

3.1. Single track link

The single link was checked by simulation of shearing and sinkage tests. In the

shearing simulation, the link was constrained with uniaxial quasistatic motion with

three levels of applying loads. The simulation results are shown in Fig. 5. In the

figure, the shear stress is normalized with the normal stress.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

shear displacement [m]

n o r m a l i z e d s h e a r s t r e s s

Vertical load - 1 kN



Fig. 5. Normalized shear stress vs. shear displacement.

Fig. 4. Block diagram of the propulsion model.




explained above, only qualitative agreement between the test results and the model

can be achieved.

3.3. Riding simulation

Riding simulation over a ditch (5 m long and 20 cm depth) was performed with

5 m/s forward velocity. The results of the new model were compared with those of the existing DADS model. The FFT transformations of the simulation results are

presented in Fig. 8. The results of the two models are fairly similar until 20 Hz.

Higher frequency results in significant differences between the models. This is

probably due to the dynamic behavior of the track link, which is effectively taken

into account in the new model.

0.00

0.01

0.10

1.00

10.00

0 20 40 60 80 100

frequency [Hz]

A c c e l e r a t i o n [ m / s

2 ]

NEW MODEL

DADS MODEL

Fig. 8. FFT Transformation of the mass center vertical acceleration – riding over a ditch with forward

velocity 5 m/s.

0

50

100

150

200

250300

350

400

450

0 1 2 3

Distance [m]

P r e s s u r e

[ k P a ]

4

Wong Measured Wong Predicted

New model

Fig. 7. Comparison between measured and predicted ground pressure.




4. Conclusion

Accurate model for track link and terrain interaction based on Bekker and Ja-

nosis approaches was developed. A detailed model of a tracked-vehicle suspensionwas built using the LMS-DADS simulation program. The link model was included

in the suspension model.

The single-link simulations show that the link model operates reasonably well.

The normal stress under the track was predicted using the model and compared with

test results [11]. A good qualitative matching between the predicted stress and the

measured stress was obtained.

Comparison of the new model with the DADS model in riding over an obstacle

was performed. The comparison results indicate a good match at lower frequency

and a significant difference at higher frequency.

References

[1] Rubinstein D, Galili N. REKEM – a design-oriented simulation program for off-road track vehicles. J

Terramech 1994;31:329–52.

[2] Rubinstein D. An upgraded off-road track model for multibody simulation. In: Proceeding of the

Eighth European Conference on ISTVS, Umeo, Sweden, 2000.

[3] Kortum W, Sharp RS. Multibody computer codes in vehicle system dynamics. Amsterdam: Swets &

Zeitlinger; 1993.

[4] Schiehlen W. Multibody system handbook. Berlin: Springer-Verlag; 1990.

[5] Creighton DC. Revised vehicle dynamics module: Users guide for computer program VEHDYN II,

Technical Report No. SL-86-9, VSAE-WES, Vicksburg, MS, 1986.

[6] Sharp RS. The application of multi-body computer codes to road vehicle dynamics modeling

problems. J Automobile Eng 1994:55–61.

[7] Burt EC. Soil-tire/track interaction-current and future research needs. J Terramech 1993;30:317–23.

[8] Shneor Y. Dynamic modeling of off-road tracked vehicle suspension system, MSc Thesis, Technion –

Israel Institute of Technology, 1988.

[9] McCullough MK, Haug EJ. Dynamics of high mobility track vehicles. J Mech, Trans Automat Des

1986;108:189–97.

[10] Bekker MG. Off-the-road locomotion. Ann Arbor: The University of Michigan Press; 1960.

[11] Wong JY. Terramechanics and off-road vehicles. New York: Wiley-Interscience; 1989.


Dynamic Simulation of Off-road Tracked Vehicles

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