Simulink Yaw damping model

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The University of the Witwatersrand

School of Mechanical, Aeronautical and Industrial Engineering

Research Project: Simulink Yaw Damping Model

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RESEARCH PROJECT

Final year project report

Project title: Simulink Yaw Damping Model of Heavy Motor vehicle

Project supervisor: Dr. F. Kienhofer

Date: 18 October 2012

Student: Darryn Frerichs

Student number: 0600945H






ii

Declaration

University of the Witwatersrand, Johannesburg

School of Mechanical, Industrial and Aeronautical Engineering

Name: Darryn Frerichs Student no: 0600945H

Course no: MECN4006 Course Name: Research Project

Submission Date: 18 October 2012 Project Title: Simulink Yaw Damping Model of Heavy

Motor Vehicle

I hereby declare the following:

I am aware that plagiarism (the use of someone else’s work without their permission and/or withoutacknowledging the original source) is wrong;

I confirm that the work submitted herewith for assessment in the above course is my own unaided work

except where the I have explicitly indicated otherwise;

This task has not been submitted before, either individually or jointly, for any course requirement,

examination or degree at this or any other tertiary education institution;

I have followed the required conventions in referencing the thoughts and ideas of others;

I understand that the University of the Witwatersrand may take disciplinary action against me if it can be

shown that this task is not my own unaided work or that I have failed to acknowledge the sources of the

ideas or words in my writing in this task.

Signature: ___________________________ Date: _________________






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Abstract

It is known that a large portion of vehicle accidents involve heavy motor vehicles. Furthermore, it is

known that a large percentage of these accidents are caused by vehicles rolling over, where roll over

is initiated at the most rearward trailer. A computer model which is able to simulate the dynamic

response of an interlink truck system, where different parameters may be varied, would be useful in

the design and analysis of trailer and loading combinations. The model should be able to predict the

influence which different parameters, such as the location of a centre of gravity, have on the dynamic

behavior of the system.

A Simulink model was created which was able to predict the response of a system to a steering input

which simulated normal driving conditions on the road. The model allows parameters of the interlink

system to be varied and provides response curves in the form of yaw rate (rad/s), yaw (rad), and

lateral acceleration (m/ss). Initially a model describing a simpler truck and trailer system was

designed and compared to previously published results which were used to validate the model, the

same methodology was then employed to generate the more complicated model of the interlink

system.

It was found that yaw decreases rearward from the truck unit in the system due to yaw damping. The

most rearward trailer was exposed to the highest lateral acceleration, as predicted by the ‘rearward -

amplification’ phenomena. The tir e cornering stiffness’s of the system greatly affect the stability,

resulting in the yaw motion rapidly tending towards infinity when C2, C3, or C6, and C7 values are

were decreased beyond 300000N/rad.

The model was validated by using the same techniques used to generate simpler models which could

be validated using other publications. However, it is recommended that in future texts, the model be

compared to a real life data capture of the same system. Further investigation into the calculation of

tire cornering stiffness’s and the effects of a non-linear tire model is also recommended.






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Table of Contents

Declaration .............................................................................................................................................. ii

Abstract .................................................................................................................................................. iii

Table of Contents ................................................................................................................................... iv

List of Figures ....................................................................................................................................... vii

List of Tables .......................................................................................................................................... x

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

1.1 Motivation for Research.......................................................................................................... 1

2 Objectives ....................................................................................................................................... 3

3 Literature Review ........................................................................................................................... 4

3.1 Dynamic yaw response ........................................................................................................... 4

3.2 Two degree-of-freedom model ............................................................................................... 4

3.3 Understeer gradient ................................................................................................................. 5

3.4 Transfer functions ................................................................................................................... 5

3.5 Stability analysis ..................................................................................................................... 5

3.6 Tire cornering stiffness ........................................................................................................... 6

3.7 Stability ................................................................................................................................... 6

3.7.1 Root-locus plots .............................................................................................................. 6

3.7.2 Bode plots ....................................................................................................................... 6






v

3.7.3 Nyquist plots ................................................................................................................... 6

4 Analysis .......................................................................................................................................... 7

4.1 Single Vehicle Yaw Simulink Model ..................................................................................... 7

4.1.1 Assumptions .................................................................................................................... 7

4.1.2 Bicycle Model ................................................................................................................. 8

4.1.3 Equations of motion ........................................................................................................ 9

4.1.4 Simulink model ............................................................................................................... 9

4.2 Truck and trailer yaw simulink model .................................................................................. 10

4.2.1 Assumptions .................................................................................................................. 10

4.2.2 Bicycle model ............................................................................................................... 11

4.2.3 Equations of motion ...................................................................................................... 12

4.2.4 Simulink simulation model of truck and trailer system ................................................ 13

5 Experimentation ........................................................................................................................... 17

5.1 Assumptions .......................................................................................................................... 17

5.2 Bicycle model ....................................................................................................................... 18

5.3 Equations of motion .............................................................................................................. 20

5.4 Simulink model ..................................................................................................................... 24

5.5 Linear simulation of model ................................................................................................... 26

5.5.1 Parameters ..................................................................................................................... 26

5.5.2 Simulation results .......................................................................................................... 27






vi

5.6 Simulink model optimization ................................................................................................ 38

5.6.1 State space simulink model ........................................................................................... 38

5.6.2 Transfer function simulink model ................................................................................. 40

5.7 Stability ................................................................................................................................. 41

6 Discussion .................................................................................................................................... 47

7 Conclusion and Recommendations .............................................................................................. 53

8 Bibliography ................................................................................................................................. 55

9 References .................................................................................................................................... 56

Appendix A ........................................................................................................................................... 57

A.1 Derivation of equations of motion of one vehicle model ...................................................... 57

A.2 Derivation of equations of motion of truck and trailer model............................................... 58

Appendix B ........................................................................................................................................... 61






vii

List of Figures

Figure 1: Bicycle model [1] .................................................................................................................... 4

Figure 2: Single vehicle system .............................................................................................................. 7

Figure 3: 2-D.O.F. Bicycle model of single vehicle ............................................................................... 8

Figure 4: Yaw rate response of single vehicle system to chirp input .................................................... 10

Figure 5: Truck and trailer system ........................................................................................................ 10

Figure 6: Bicycle model of truck and trailer system ............................................................................. 11

Figure 7: Truck and trailer system vehicle lateral velocity response to sinusoidal input ..................... 14

Figure 8: Truck and trailer system vehicle yaw rate response to sinusoidal input ................................ 14

Figure 9: Vehicle response results published by [5] with sinusoidal input ........................................... 15

Figure 10: Truck and trailer system vehicle lateral velocity response to step input ............................. 15

Figure 11: Truck and trailer system vehicle yaw rate response to step input ....................................... 16

Figure 12: Vehicle response results published by [5] with step input .................................................. 16

Figure 13: Schematic of interlink system to be modelled [10] ............................................................. 17

Figure 14: Bicycle model of interlink system ....................................................................................... 19

Figure 15: Force diagram of interlink system showing coupling force representation ......................... 20

Figure 16: Simulink model of interlink system .................................................................................... 25

Figure 17: Yaw response of truck to sinusoidal lane change simulation input (rad versus time) ......... 27

Figure 18: Yaw response of first trailer to sinusoidal lane change simulation input (rad versus time) 28

http://c/Users/Darryn/Documents/Research%20Project/Research%20Report.docx%23_Toc338256898











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Figure 19: Yaw response of second trailer to sinusoidal lane change simulation input (rad versus time)

.............................................................................................................................................................. 28

Figure 20: Yaw rate response of truck to sinusoidal lane change simulation input (rad/s versus time) 28

Figure 21: Yaw rate response of first trailer to sinusoidal lane change simulation input (rad/s versus

time) ...................................................................................................................................................... 29

Figure 22: Yaw rate response of second trailer to sinusoidal lane change simulation input (rad/s versus

time) ...................................................................................................................................................... 29

Figure 23: Truck lateral acceleration response to sinusoidal lane change simulation input (m/s versus

time) ...................................................................................................................................................... 29

Figure 24: Lateral accelerations of truck and trailer units due to sinusoidal input (m/s2)..................... 30

Figure 25: System yaw response to chirp input .................................................................................... 31

Figure 26: System yaw rate response to sinusoidal input with first and second trailers same length

(rad/s) .................................................................................................................................................... 32

Figure 27: System yaw response to sinusoidal input with first and second trailers same length (rad) . 33

Figure 28: System lateral acceleration response to sinusoidal input with first and second trailers same

length (m/s2) .......................................................................................................................................... 33

Figure 29: System yaw rate response to sinusoidal input with first and second trailers swapped (rad/s)

.............................................................................................................................................................. 34

Figure 30: System yaw response to sinusoidal input with first and second trailers swapped (rad) ...... 35

Figure 31: : System lateral acceleration response to sinusoidal input with first and second trailers

swapped (m/s2) ...................................................................................................................................... 35

Figure 32: System yaw rate response to sinusoidal input with 25% original tire stiffnesses (rad/s) .... 36

Figure 33: System yaw response to sinusoidal input with 25% original tire stiffnesses (rad) .............. 37






ix

Figure 34: System yaw rate response to sinusoidal input with longitudinal velocity of 34m/s (rad/s) . 37

Figure 35: System yaw response to sinusoidal input with longitudinal velocity of 34m/s (rad) .......... 38

Figure 36: State space representation of interlink model ...................................................................... 39

Figure 37: State space model truck lateral acceleration response to sinusoidal input .......................... 39

Figure 38: State space model truck yaw response to sinusoidal input .................................................. 40

Figure 39: State space model second trailer yaw response to sinusoidal input..................................... 40

Figure 40: Transfer function model truck lateral acceleration response to sinusoidal input ................ 41

Figure 41: Root-locus plot of unstable system due to smaller C2 and C3 values .................................. 43

Figure 42: Yaw response of unstable system due to smaller C2 and C3 values .................................... 43

Figure 43: Root-locus plot of unstable system due to smaller C6 and C7 values .................................. 44

Figure 44: Root locus plot of the unstable system due to further forward centre of gravity on first

trailer ..................................................................................................................................................... 45

Figure 45: Root-locus plot of stable system due to swapping of first and second trailer parameters ... 46

Figure 46: Simulink model of truck and trailer ..................................................................................... 60








1

1 Introduction

1.1

Motivation for Research

The dynamic instability in articulated vehicles is the cause of about 30% of accidents involving heavy

motor vehicles (HMVs) in New Zealand [1]. Most of the accidents are a result of vehicles rolling

over which is known to occur from the most rearward trailer rolling first, which in turn rolls the entire

vehicle onto its side [1]. The rolling over of the vehicle is a result of the rearward amplification of the

lateral acceleration of the trailers as a result of the turning frequency generated by the driver in a lane

change or evasive maneuver.

Motor vehicle accidents, in general, cause a large number of fatalities world-wide. Furthermore,

instability in an interlink HMV system (truck towing two trailers) may result in up to 56 tones

traveling out of control on the worlds highways with the potential to kill hundreds of people. In

addition to the fatalities caused, the loads carried by many of the vehicles are hazardous and may

contaminate the environment as a direct result of an accident. Finally, the traffic jams caused by

HMV accidents affect the economy of the country.

The poor state of the South African railway service has resulted in a large increase in the use of trucks

as the form of transport from the harbors to inland cities such as Johannesburg. The road from

Johannesburg to Durban is well maintained but the topology is not conducive to safe traveling for

HMVs as the winding and sharp descents on passes such as Van Reenen’s pass promote large steering

inputs of a high, regular frequency which can be exceptionally dangerous in an unstable HMV system.

The demand increase for trucks in South Africa has resulted in many new, inexperienced operators

taking to the road to take advantage of the market trend, however, the lack of experience and

knowledge of these operators could have catastrophic implications on the number of road accidents in

the country. The payload of a system is the mass of the load that the truck transports which the

supplier pays for, therefore, a larger payload on a system produces a larger turnover for the operator

per load transported. Unfortunately, a lack of knowledge of system characteristics results in loads

being placed dangerously on trailers and, although the policing of axle weights and loads is quite good

in the form of weigh bridges on all major traffic routes, the systems can be bypassed, and operators

are doing so, endangering the lives of all road users.

A simulation model where vehicle parameters can easily be changed to determine their influence on a

HMV system would be advantageous when designing a system or determining the best configuration






2

for a system in operation. Operators could use the model to understand the effects of the loadings on

the trailers and optimize their systems to attain the safest and most economical configuration for

themselves individually.






3

2 Objectives

Develop a simplified force model of an interlink South African truck and trailer system with

valid assumptions for force and impulse analysis:

o Constant friction on the tires

o No delay of input force

o The road surface is flat and level

o No aerodynamic forces influence the system

From the model, generate a model using Simulink to generate response curves relative to the

input frequency (steering):o Yaw rate and angle versus time

o Vehicle lateral acceleration versus time

Determine the yaw damping ratio of the system

Evaluate the influence of different vehicle parameters on the yaw motion of the system:

o Position and magnitude of centre of gravity of load

o Tire stiffness of vehicle

o Length of trailers

o Longitudinal velocity of vehicle

Make comparisons to other yaw damping models






4

3 Literature Review

3.1

Dynamic yaw response

A property that dominates the performance of the dynamic yaw response of a multi-articulated vehicle

is that known as rearward amplification. Other studies have found that during a transient turning

maneuver, the rear unit of the HMV may experience lateral accelerations far greater than that

experienced by the towing unit. The rearward amplification described is believed to be the property

which leads HMVs to roll from the rear trailer first. [3]

Rearward amplification is a frequency-sensitive phenomenon and seems to be more prominent when

the steering input has a high frequency. Multi articulated HMVs are multi-degree-of-freedom systems

with several lightly damped dynamic modes of oscillation and system excitement, due to evasive

maneuvers, in close proximity to these natural frequencies will cause an uncontrolled resonant

response. [4]

3.2 Two degree-of-freedom model

The dynamics of a vehicle may be represented in the form of a single track 2-degree-of-freedom

model. The model is known as a Bicycle Model. The bicycle model simplifies the complicated

system of a vehicle into accurate and measurable quantities for accurate modeling of a vehicles lateral

velocity and yaw dynamics. The coordinates of the model are fixed at the vehicles centre of gravity

[3]. The model below is an example of a Bicycle Model of a vehicle [3].

Figure 1: Bicycle model [1]






5

3.3 Understeer gradient

The understeer gradient is used to relate a vehicles weight distribution to its tires’ force generating

abilities. The understeer gradient is determined by modeling the bicycle model during a high speed

steady state turn and performing a sum of the forces and moments acting on the model. A vehicle

with a positive understeer gradient is known as an understeer vehicle and requires an increase in

steering input to negotiate a steady state turn as the speed of the vehicle increases. A vehicle with a

negative understeer gradient is known as an oversteer vehicle and the converse applies to that of an

understeer vehicle. [3]

3.4 Transfer functions

A multiply articulated vehicle is dynamically decoupled at the tow hitch if; for trains with more than

one full trailer results are consistent in that the two modes associated with a given full trailer tend to

be lightly damped and the addition of more trailers does not affect the dynamic behavior of units

ahead of the added trailers and the modes of motion associated with each full trailer become less and

less damped moving rearward. [4] The decoupling phenomenon allows each trailer unit to be

analyzed individually. The overall transfer function can be determined by multiplying the localized

transfer functions between centers of gravity and tow points along the length of the system in thisdecoupled system. [4]

3.5 Stability analysis

The state space form of the system can be used to calculate the stability parameters of the system.

The natural frequency of the undamped system (ωo), the natural frequency (ωn) and the damping ratio

(ξ) can be computed with use of the eigenvalues (λ ) of matrix A due to the time-independence of the

system matrices- forward velocity and tire cornering stiffness assumed constant. The parameters

necessary for a stability analysis can be calculated using Equations 3.1, 3.2, and 3.3 below [5]:

√ (3.1)

(3.2)

√ (3.3)






6

3.6 Tire cornering stiffness

The tire reactant force from the tire can be approximated to be a linear force with reference to the

stiffness coefficient of the tire, and the slip angle [4 – 9]. The tire cornering stiffness can be

approximated by Equation 3.4 below [8, 9]:

(3.4)

Where is the vertical load on the tire in Newtons [8, 9].

3.7 Stability

Linear systems are deemed to be unstable if either the real part of any one pole is positive, or any one

repeated pole has zero real parts, otherwise it is stable. Furthermore, a stable linear system having all

poles with negative real parts is asymptotically stable. [11]

3.7.1 Root-locus plots

Root-locus plots are used to plot the system roots over the range of a variable to determine if the

system becomes unstable. Positive real parts of roots will result in terms that grow exponentially and

become unstable while complex roots make a system oscillate. [12]

3.7.2 Bode plots

Bode plots are a useful way to represent the gain and phase of a system as a function of frequency,

known as the frequency-domain behavior of a system. The frequency response is shown with two

plots; one for magnitude, and one for phase. The phasor representation of the transfer function can be

easily determined at any frequency. The magnitude of the output is the magnitude of the phasor

representation of the transfer function (at a given frequency) multiplied by the magnitude of the input.

The phase of the output is the phase of the transfer function added to the phase of the input. [13]

3.7.3 Nyquist plots

Nyquist plots display both amplitude and phase angle on a single plot, using frequency as a parameter

in the plot. It is a polar plot of the frequency response of a system. [14]






7

4 Analysis

Yaw damping models describing the yaw motion of vehicle systems are available for different vehicle

system configurations in other research papers. Simulink models were initially created for a 1-vehicle

system followed by a truck and trailer system with the same properties as described in [3] and [5]

respectively and comparisons were made between the new Simulink model results and the results

published by [3] and [5] in order to validate the modeling method. The techniques used to build the

validated models described in Sections 4.1 and 4.2 below were used to generate a model describing

the behavior of an interlink South African truck system, discussed in Section 5.

4.1 Single Vehicle Yaw Simulink Model

A Simulink model describing the yaw characteristics of a single vehicle system was developed with

results compared to those published by [3]. The system modeled is shown below in Figure 2. The

derivation of the equations of motion and the Simulink model are described in Sections 4.1.1 through

4.1.4 below.

Figure 2: Single vehicle system

4.1.1 Assumptions

The assumptions used to generate the bicycle model of the system; the equations of motion and the

Simulink model are outlined below:

y

z

x






8

Vehicle mass and tire forces are symmetric about the x-z plane. Therefore, the vehicle can be

modelled as a single tracked vehicle where two front and two rear wheels can be represented

together as a single front and single rear wheel.

Longitudinal velocity is constant.

Tires roll without slipping in the longitudinal direction (no acceleration or braking forces).

Front and rear tires produce lateral forces which are linearly proportional to their respective

cornering stiffness’s (linear tire model).

Small angle approximations are valid: cosθ≈1, sinθ≈0.

4.1.2 Bicycle Model

The planar dynamics modeling of the vehicle were represented in the form of a single track, two-

degree of freedom model known as a ‘bicycle model’, consistent with the assumptions outlined in

Section 4.1.1 above. The bicycle model is an accurate model of the vehicles lateral velocity and yaw

dynamics and is described by a body-fixed coordinate system centered at the centre of gravity of the

vehicle. Figure 3 below is a schematic of the two-degree of freedom bicycle model modeling the

dynamics of Figure 2 previously, while Table 1 overleaf describes the symbols used.

δf

y

Vcg,t

u

v

FR

VR,t

-αR x

VF,t

-αF

β

FF

r

Figure 3: 2-D.O.F. Bicycle model of single vehicle






9

Table 1: Single vehicle list of symbols

Symbol Description

Vcg,t

Total velocity vector at centre of gravity

u Longitudinal velocity

v Lateral velocity

r Yaw about z-axis

β Side slip angle

δf Front wheel steering angle

αF Front wheel slip angle

VF,t

Total velocity vector at front tire

FF Lateral force generated at front tire

αR Rear tire slip angle

VR,t

Total velocity vector at rear tire

FR Lateral force generated at rear tire

4.1.3 Equations of motion

The full derivation of the equations of motion of the one-vehicle system is shown in Section 1 of

Appendix A. Meaningful expressions of the equations are listed below in Equations X and Y.

(4.1) (4.2)

The standard state matrix representation of the equations is shown below:

[

]

4.1.4 Simulink model

Figure 4 below shows the yaw rate response of the simulated system to a similar chirp input,

simulating an avoiding maneuver, to that in [3], which shows a similar trend to the response obtained

by [3], validating the designed model.






10

Figure 4: Yaw rate response of single vehicle system to chirp input

4.2 Truck and trailer yaw simulink model

The methods used to create the single vehicle model described in Section 4.1, previously, were

expanded upon to generate a model of a truck and trailer system of similar configuration to that shown

below in Figure 5. The model created was verified by comparing the yaw response of the system to

that published by [5].

4.2.1 Assumptions



Figure 5: Truck and trailer system






11








The connection between the truck and trailer is solid (does not bend) and operates without

friction. Small angle approximations are valid: cosθ≈1, sinθ≈0.

4.2.2 Bicycle model

The assumptions outlined in Section 4.2.1 above were used to simplify the system described by Figure

5 into an accurate bicycle model describing the planar dynamic behavior of the coupled system.

Figure 6 below is a schematic of the bicycle model which was used to determine the equations of

motion of the system described in Section 4.2.3.

Figure 6: Bicycle model of truck and trailer system

Table 2 overleaf describes the symbols used and their description in the model and during the

derivation of the equations of motion in Section 4.2.3 below.






12

Table 2: Truck and trailer system list of symbols

Symbol Description

a Distance between the centre of gravity (C.O.G) of truck and the front steering wheel axle

b Distance between the C.O.G of the truck and the truck rear axle

c Distance between the C.O.G of the truck and the first tow hitch

d Distance between the C.O.G of the first trailer and the first tow hitch

e Distance between the C.O.G of the first trailer and the trailer axle

F1 Force generated by the front tires of the truck

F2 Force generated by the rear tires of the truck

F3 Force generated by the trailer tires

m1 Mass of the truckm2 Mass of the trailer

J1 Moment of inertia of the truck about its C.O.G

J2 Moment of inertia of the trailer about its centre of gravity

ψ Yaw angle of truck

ϒ Articulation angle

δ Steering angle

αi Tire side slip angle

Ci Tire cornering stiffness

4.2.3 Equations of motion

The full derivation of the equations of motion of the truck and trailer system is shown in Section 2 of

Appendix A. Meaningful expressions of the equations are listed below in Equations 4.3, 4.4 and 4.5.

(4.3)

(4.4)

() (4.5)

The linear tire model approximation was used to calculate the force components applied by each tire:

, (4.6)

where i=1, 2, 3, for each individual tire.






13

The matrix form of the system can be written in the form , where j is the input

(steering input), as shown below:

[

]

4.2.4 Simulink simulation model of truck and trailer system

A simulink model was generated to determine the vehicle behavior when subjected to a steering input

using the assumptions and equations of motion outlined in preceding sections. The input parameters

chosen for the simulation were taken from [5] in order to validate the modeling technique and

derivations performed. Table 3 below is taken from [5] and shows the parameters used in the

Simulink simulation.

Table 3: Truck and trailer simulink model vehicle parameters [5]

Parameter Unit Value

a m 2.062

b m 2.723

c m 2.539

d m 7.483

e m 3.76

C1 N/rad 381930C2 N/rad 733390

C3 N/rad 881440

m1 kg 8812

m2 kg 16484

J1 kg.m2 46100

J2 kg.m2 452010

A schematic of the Simulink model designed is contained in Section 2 of Appendix A in Figure 46.






14

A sinusoidal steering input with a frequency of 0.25 Hz and amplitude 0.1 radians to simulate a lane

change was applied as the steering input, consistent with the values used by [5], furthermore, a

longitudinal velocity component (u) was selected to be 20m/s. The dynamic response of the system

was plotted using scopes in the Simulink environment. The vehicle lateral velocity (m/s) and vehicle

yaw rate (rad/s) dynamic responses over an 8s simulation period are shown below in Figure 7 and

Figure 8 respectively.

Figure 7: Truck and trailer system vehicle lateral velocity response to sinusoidal input

Figure 8: Truck and trailer system vehicle yaw rate response to sinusoidal input






15

The responses published by [5] for the same system parameters and disturbance are shown below in

Figure 9.

Figure 9: Vehicle response results published by [5] with sinusoidal input

The trends shown in Figure 7 and Figure 8 accurately match the trends published by [5] shown in

Figure 9, proving the validity of the Simulink model generated at shown in Section 2 of Appendix A,

Figure 46. To further prove the accuracy of the designed model, the input was changed to a step input

for the steering angle of 1 radian after 1s, consistent with [5]. Figure 10 and Figure 11 below show

the dynamic response of the experimental system, while Figure 12 shows the results obtained and

published by [5]. The dynamic responses of the two models are identical, further proving the validity

of the designed truck and single trailer model derived in Sections 4.2.1 through to Sections 4.2.4.

Figure 10: Truck and trailer system vehicle lateral velocity response to step input






16

Figure 11: Truck and trailer system vehicle yaw rate response to step input

Figure 12: Vehicle response results published by [5] with step input






17

5 Experimentation

The methodology applied in Section 4 previously was used to derive an accurate bicycle model,

equations of motion, and a Simulink model able to accurately plot the dynamic response of a South

African interlink truck system to different steering inputs. The interlink system modeled was

comprised of an International 9800 model truck with two trailers built by Hendred Fruehauf, capable

of having a maximum gross weight of 56000kg. The maximum weights supported by each axle

comply with the South African National Roads Agency regulations. Figure 13 below is a schematic

of the system to be modeled.

Figure 13: Schematic of interlink system to be modeled [10]

A complete annotated derivation of the bicycle model, equations of motion, and Simulink model of

the system in Figure 13 is contained in Sections 5.1 to 5.4. Further experimentation using the

generated Simulink model is contained in Section 5.5 and beyond.

5.1 Assumptions








18








The connection between the truck and trailer is solid (does not bend) and operates without

friction. The centre of gravity location in the trailers can be approximated to be in the middle of the load.

i.e. the trailers are loaded symmetrically.

Small angle approximations are valid: cosθ≈1, sinθ≈0.

5.2 Bicycle model

The assumptions outlined in Section 5.1 above were used to simplify the system described by Figure

13 into an accurate bicycle model describing the planar dynamic behavior of the coupled system.

Figure 14 overleaf is a force diagram of the bicycle model which was used to determine the equations

of motion of the system described in Section 5.3, while Table 4: Description of symbols used in

interlink model is a nomenclature of the symbols used.

Table 4: Description of symbols used in interlink model

Symbol Description

a Distance between centre of gravity (C.O.G) of truck and front tire

b Distance between C.O.G of truck and first hitch point

c Distance between C.O.G of truck and first axle of truck

d Distance between first and second truck axles

e Distance between C.O.G of first trailer and first hitch point

f Distance between C.O.G of first trailer and second hitch point/first axle on first trailer

g Distance between first and second axles on first trailer

h Distance between C.O.G of second trailer and second hitch point

i Distance between C.O.G of second trailer and first axle on second trailer

j Distance between first and second axle on second trailer

Fi Force generated by tires






19

m1 Mass of truck

m2 Mass of first trailer

m3 Mass of second trailer

J1 Moment of inertia of truck

J2 Moment of inertia of first trailer

J3 Moment of inertia of second trailer

ϒ1 Articulation angle between truck and first trailer

ϒ2 Articulation angle between first and second trailer

ψ1 Yaw rate of truck

Ψ2 Yaw rate of first trailer

Ψ3 Yaw rate of second trailer

u longitudinal velocity

Figure 14: Bicycle model of interlink system








21

A sum of forces and moments about the centre of gravity of the second trailer gives:

(5.5)

(5.6)

Solving for Y1 using Equation 5.2 and substituting into Equation 5.1:

(5.7)

Solving for Y2 using Equations 5.1 and 5.5 and substituting into Equation 5.3:

(5.8)

Equation 5.8 can be written in terms of the trucks C.O.G coordinate system using:

(5.9)

(5.10)

Substituting Equations 5.9 and 5.10 into 5.8:

( ) (5.11)

Further combinations of the above equations produce the final two equations of motion:

(5.12)

(5.13)

The reactant force generated in the tires was again approximated to be linear:






22

The slip angles for each tire are given by:

(5.14)

(5.15)

(5.16)

(5.17)

(5.18)

(5.19)

(5.20)

The articulation rates were calculated as follows:

(5.21)

(5.22)

The tire stiffness constant Ci was calculated using the following from [8, 9]:

(5.23)

where zi is the load supported by the tire.

As in Section 4.2.3, the equations can be represented in matrix form where:

M = bm1 I1 0 0 0 0

m1+m2+m3 -b(m2+m3) -f(m2+m3)-m3e m3h 0 0

fm1-em3 em3b m3e(f+e)+I2 m3eh 0 0

hm3 m3bh m3h(f+e) m3(h2)-I3 0 0

0 0 0 0 1 0

0 0 0 0 0 1






23

K = 1/u -C1(a+b)-C2(b-c)-C3(b-(c+d)) -C1a(a+b)-m1bU2-C2(b-c)(-c)-C3(b-(c+d))(-(c+d))

-C1-C2-C3-C4-C5-C6-C7 -(m1+m2+m3)u2-C1a+C2c+C3(c+d)+C4b+C5b+C6b+C7b

f(-C1-C2-C3+C4)+C5(f+g)+C6e+C7e -(fm1-em3)u+f(-C1a+C2c+C3(c+d)-C4b-C5(b)(f+g)-C6eb-C7eb-C6(h+i)-C7(h+i+j) -m3u+C6b(h+i)+C7b(h+i+j)

0 1

0 0

0 0 0 0

2C4f+C5(2f+g)+C6(e+f)+C7(e+f) C6(h+i)+C7(h+i+j) -C4-C5-C6-C7 -C6-C7

f(-C4(2f))-C5(f+g)(2f+g)-C6e(e+f)-C7e(e+f) -C6e(h+i)-C7e(h+i+j) fC4+(f+g)C5+eC6+eC7 eC6+eC7

C6(e+f)(h+i)+C7(e+f)(h+i+j) C6(h+i)(h+i)+C7(h+i+j)(h+i+j) -C6(h+i)-C7(h+i+j) -C6(h+i)-C7(h+i+j)

-1 0 0 0

1 -1 0 0

B1 = (a+b)C1

C2

fC1

0

0

0

| | |

|

Manipulation of the matrix equation allows the response of the system to be written in the state space

form; , where and .

Four meaningful expressions are obtainable from the different forms of the equations of motion

describing the response of the system to an input function and are shown below in Equations 5.24 to

5.27.

(5.24)

() (5.25)






24

(5.26)

(5.27)

5.4 Simulink model

The equations of motion were used to build a Simulink model of the interlink system. The model was

designed to replicate the behavior of the real life system with given parameters in response to a

steering input over a time period. The Simulink model created is based on the idealized bicycle

model described in Section 5.2 and the derived equations of motion in Section 5.3 and is shown

overleaf in Figure 16.






25

Figure 16: Simulink model of interlink system






26

5.5 Linear simulation of model

Using Simulink and the model shown in Figure 16 of Section 5.4 above, simulations were made to

investigate the dynamic behavior of the model.

5.5.1 Parameters

The parameters of the system needed to be determined for the simulation. Parameters were obtained

from [10] for an ultra lightweight interlink combination in South Africa, operating at a speed of

22.2m/s which corresponds to the speed limit on national highways for heavy motor vehicles of

80km/h. Table 5 below shows the system parameters originally used.

Table 5: Interlink simulation system parameters

Symbol Value Units

a 40.00 mm

b 3311.00 mm

c 3719.00 mm

d 1400.00 mm

e 1275.00 mm

f 5225.00 mmg 1400.00 mm

h 4645.00 mm

i 3205.00 mm

j 1400.00 mm

m1 9200.00 kg

m2 16906.00 kg

m3 29894.00 kg

J1 36776.60 Kg.m2

J2 53285.60 Kg.m2

J3 376276.03 Kg.m2

u 22.20 m/s

L1 6926.00 mm

L2 6150.00 mm

L3 12290.00 mm

C1 891384.14 N/rad

C2 592777.40 N/rad

C3 592777.40 N/rad

C4 621796.91 N/rad






27

C5 621796.91 N/rad

C6 632902.54 N/rad

C7 632902.54 N/radZ1 32843.88 N

Z2 19239.86 N

Z3 19239.86 N

Z4 20386.41 N

Z5 20386.41 N

Z6 20832.76 N

Z7 20832.76 N

5.5.2 Simulation results

Using the Simulink model shown in Figure 16 in Section 5.4 with the parameters as tabulated in Table

5 the dynamic response of the system was plotted for different steering inputs. In Figure 17 to Figure

23 below, the dynamic response over a 15s period for a sinusoidal steering input comparable to a

vehicle lane change is plotted. The sinusoidal steering input has a frequency of 0.25Hz (1.5708 rad)

and amplitude of 0.1 [5].

Figure 17: Yaw response of truck to sinusoidal lane change simulation input (rad versus time)








29

Figure 21: Yaw rate response of first trailer to sinusoidal lane change simulation input (rad/s versus time)

Figure 22: Yaw rate response of second trailer to sinusoidal lane change simulation input (rad/s versus time)

Figure 23: Truck lateral acceleration response to sinusoidal lane change simulation input (m/s versus time)








31

The steering input frequency can vary from 0.15Hz in normal driving up to an approximate maximum

of 4Hz in an evasive maneuver [3]. Figure 25 below shows the system yaw response to a chirp signal

simulating an evasive maneuver with the input frequencies ranging from an initial frequency of

0.15Hz to 4Hz after 1s.

Figure 25: System yaw response to chirp input








33

Figure 27: System yaw response to sinusoidal input with first and second trailers same length (rad)

Figure 28: System lateral acceleration response to sinusoidal input with first and second trailers same length (m/s2)






34

The effect of changing the length of the first trailer to the length of the second and the second to the

length of the first (swap) was explored and the yaw rate and yaw system response to the lane change

sinusoidal input is shown below in Figure 29 and Figure 30, while Figure 31 shows the lateralacceleration response. The gross mass of the trailers, axle support weights and tire stiffness’s were

adjusted accordingly.

Figure 29: System yaw rate response to sinusoidal input with first and second trailers swapped (rad/s)






35

Figure 30: System yaw response to sinusoidal input with first and second trailers swapped (rad)

Figure 31: : System lateral acceleration response to sinusoidal input with first and second trailers swapped (m/s2)






36

The effect of tire cornering stiffness’s on the original system was analyzed. The system with

parameters outline in Table 5 was modified to have 25% of the original tire cornering stiffness’s and

the yaw rate and yaw responses of the system are shown overleaf in Figure 32 and Figure 33.

Figure 32: System yaw rate response to sinusoidal input with 25% original tire stiffness’s (rad/s)






37

Figure 33: System yaw response to sinusoidal input with 25% original tire stiffness’s (rad)

An investigation was made into the effect of the vehicles longitudinal velocity on the system response

to the sinusoidal steering input. Figure 34 and Figure 35 below show the yaw rate and yaw response

of the system with original parameters as outlined in Table 5 and a longitudinal velocity (U) of 34m/s

which is equivalent to approximately 120km/h.

Figure 34: System yaw rate response to sinusoidal input with longitudinal velocity of 34m/s (rad/s)






38

Figure 35: System yaw response to sinusoidal input with longitudinal velocity of 34m/s (rad)

5.6

Simulink model optimization

The time-domain simulink model shown in Figure 16 produces the most accurate, real-time system

response to a steering input. The model can be simplified to attain an approximate system response to

an input by determining the overall transfer function of the system or by using the state space

representation of the system response. The complicated time-domain model was simplified using

these theories and are discussed below in Sections 5.6.1 and 5.6.2.

5.6.1 State space simulink model

The general form of the state space representations of a continuous time-invariant linear system is

written in the form:

(5.28) (5.29)

where

is called the state vector, y is called the output vector, and u is the input or control vector. A

is the state matrix, B is the input matrix, C is the output matrix, and D is the feed forward matrix.






39

As discussed in Section 4, a system with the parameters shown in Table 5 can be represented by

setting matrix A = M-1K, and B = M-1B1. The input matrix u is the inputted steering function referred

to as δ. The Matrix C is the output matrix used to determine the output of the system and is a 1xp

matrix, where p is the number of variables in the state vector (6 in the interlink model system). For

example, when determining the vehicle lateral velocity response, matrix C = [1 0 0 0 0 0], and to

determine the yaw response of the first trailer, C = [0 0 1 0 0 0]. Matrix D is the zero matrix in the

case where the system model does not have direct feed through. For the system with parameters

described in Table 5, the values of matrices M, K, A, B1, and B are contained in Appendix B.

Figure 36 overleaf shows the simulink model used to determine the interlink system response to the

lane changing steering input.

Figure 36: State space representation of interlink model

Figure 37 below shows the truck lateral acceleration response given by the state space model due to a

sinusoidal steering input function representing a lane change, while Figure 38 and Figure 39 show the

yaw response of the truck, and second trailers respectively.

Figure 37: State space model truck lateral acceleration response to sinusoidal input






40

Figure 38: State space model truck yaw response to sinusoidal input

Figure 39: State space model second trailer yaw response to sinusoidal input

5.6.2 Transfer function simulink model

Using the matrices A, B, C, and D discussed in Section 5.6.1, the overall transfer function of the

system was calculated using the variations of C and Equation 5.30 [11]:

(5.30)

The transfer function for the system with parameters given in Table 5 and Appendix B with C= [1 0 0

0 0 0] was calculated as:

Figure 40 below shows the truck longitudinal acceleration response to a sinusoidal lane change

simulating steering input using the transfer function G1(s) in simulink.








42

The frequencies and damping of the system were calculated using Equations 3.1 to 3.3, where the

absolute values of the eigenvalues are used, and are tabulated below in Table 6.

Table 6: Stability parameters of interlink system

Re (λ) Im (λ) ωo (rad/s) ωn (rad/s) ξ

5.7256 23.2396 23.93453 23.2396 0.239219

5.7256 23.2396 23.93453 23.2396 0.239219

2.3315 1.3318 2.685067 1.3318 0.868321

2.3315 1.3318 2.685067 1.3318 0.868321

0.1135 0 0.1135 0 1

0.0917 0 0.0917 0 1

The same results for the eigenvalues, damping and undamped frequency of the system can be

determined by using the “damp (A)” command in Matlab, which yield the results shown below.

Eigenvalue Damping Freq. (Wo) (rad/s)-5.73e+000 + 2.32e+001i 2.39e-001 2.39e+001-5.73e+000 - 2.32e+001i 2.39e-001 2.39e+001-2.33e+000 + 1.33e+000i 8.68e-001 2.69e+000-2.33e+000 - 1.33e+000i 8.68e-001 2.69e+000-1.14e-001 1.00e+000 1.14e-001

-9.17e-002 1.00e+000 9.17e-002

The system remains stable while the real parts of the eigenvalues are negative [see Section 3.7]. The

parameters of the system were varied and their effects on the stability of the system examined. Root-

locus plots were discussed in Section 3.7.1 and will form the basis of the stability analysis; system

parameters were changed to determine if they can make the system unstable by plotting root-locus

plots and examining the location of the roots (positive side of the real axis results in an unstable

system). Firstly the tire stiffness’s were evaluated; it was found that no change in C1 would make the

system unstable. A decrease in stiffness’s C2 and C3 resulted in the system becoming unstable

between 290000N/rad and 300000N/rad. The root-locus plot for the system with all other parameters

the same as Table 5 with C2 and C3 values of 290000N/rad is shown overleaf in Figure 41.






43

Figure 41: Root-locus plot of unstable system due to smaller C2 and C3 values

Figure 42 below proves the instability of the system due to smaller C2 and C3 values by showing the

yaw angles of the system tending to infinite values after a short time period.

Figure 42: Yaw response of unstable system due to smaller C2 and C3 values

Simulation results showed that a change of C4 and C5 values did not have an effect on the stability of

the system. It was discovered that a decrease in tire stiffness constants C6 and C7 resulted in an

unstable system when value of the constants were between 250000N/rad and 300000N/rad. Figure 43






44

shows the root-locus plot of the unstable system with two roots marginally on the right hand side of

the plane.

Figure 43: Root-locus plot of unstable system due to smaller C6 and C7 values

The longitudinal velocity of the truck was increased to 34m/s which corresponds to just over120km/h. It was found that the system did not become unstable due to this increase; however, the

damping ratio has decreased, as well as the undamped frequency of the system as shown below:

Eigenvalue Damping Freq. (Wo) (rad/s)

-1.10e+001 + 1.38e+001i 6.23e-001 1.76e+001-1.10e+001 - 1.38e+001i 6.23e-001 1.76e+001

-1.37e+000 + 1.11e+000i 7.77e-001 1.77e+000-1.37e+000 - 1.11e+000i 7.77e-001 1.77e+000-9.35e-002 1.00e+000 9.35e-002-1.15e-001 1.00e+000 1.15e-001

An analysis into the effect of the masses of the components on the system showed that instability

occurs when m1 is decreased to almost 1000kg, which is an unrealistic mass of a truck unit. Neither

lowering the mass of the first trailer to 4156kg, which is the mass of the empty trailer, nor did

increasing it to 36000kg render the system unstable. Neither a decrease in m3 to 4415kg, which is the

empty load mass of the second trailer, nor an increase in m3 was found to affect the stability of the

system.






45

The effect of the location of the centre of gravity of each trailer on the stability of the system was

investigated. The locations of the centre of gravity were adjusted forward and rearward on both the

first and second trailer individually, while, the total masses, length, and axle load weights were

approximated to those outline in Table 5. Moving the centre of gravity 1m forward on the first trailer

(e=0.2750m, f=6.225m) resulted in an unstable system. The root-locus plot of the unstable system is

shown below in Figure 44. An adjustment of the centre of gravity of the first trailer in the rearward

direction did not result in any instability of the system.

Figure 44: Root locus plot of the unstable system due to further forward centre of gravity on first trailer

A reasonable forward adjustment of the centre of gravity location of the second trailer unit was not

found to affect the stability of the system. Moving the centre of gravity of the second trailer 1m

rearward (h=5.645m, i=2.205m) resulted in the system having positive real roots as shown overleaf:

Eigenvalue Damping Freq. (rad/s)1.46e+000 + 2.38e+001i -6.13e-002 2.39e+0011.46e+000 - 2.38e+001i -6.13e-002 2.39e+001-2.09e+000 + 1.68e+000i 7.79e-001 2.68e+000-2.09e+000 - 1.68e+000i 7.79e-001 2.68e+000-9.27e-002 1.00e+000 9.27e-002






46

As a final investigation, the effect of replacing the parameters of the first trailer with that of the

second, and vice-versa, on the stability of the system was analyzed. It was found that the system

remained stable. The root-locus plot of the system is shown below in Figure 45.

Figure 45: Root-locus plot of stable system due to swapping of first and second trailer parameters






47

6 Discussion

The model derived in Section 5.2 of the interlink system shown in Figure 13 was based heavily on the

models derived in Section 4 of smaller, simpler systems. The model of a one truck and trailer system

derived in Section 4.2 produced identical simulation results to those published by [5],and this was

used to validate the designed simulink model of this system. The bicycle and Simulink model of the

interlink system was designed using the same methodology as that used to produce the valid model in

Section 4.2, and it was because of this, that the designed interlink Simulink model was determined to

be valid. No other validating texts of similar interlink model simulations could be found to verify the

model and no real life simulation were investigated, therefore, the entire model validation relies on the

model in Section 4.2 being valid.

The assumptions outlined in Section 5.1 were used to greatly simplify the real-life interlink system to

allow the simple bicycle model in Section 5.2 to be derived in order to be able to design a Simulink

model to, as accurately as possible, replicate the dynamic behaviour of the system due to a steering

input. The assumption about the symmetry of the vehicle mass and tire forces can be approximated to

be quite valid as a uniform load on a trailer would produce a symmetric mass distribution and, ideally,

the resultant reactant forces supplied by the tires would be the same. In a swerving manoeuvre, the

trailer would tend to roll from left to right in the restrictions of the suspension system which would

move the apparent centre of gravity of the load and decrease the accuracy of the model. The variation

of the longitudinal velocity of the vehicle was approximated to be negligible, which would hold

relatively true in situations where the magnitudes of the lateral velocities of the units are small relative

to the longitudinal velocity. The tires were assumed to roll without slipping in the longitudinal

direction which is true as the vehicle experienced no acceleration or breaking in this direction. The

lateral resultant forces from the tires was assumed to be linear which has some degree of accuracy

during small steering angles, however, some advanced texts have shown greater accuracy ofsimulation results by using a far more complicated non-linear tire model. The fixed connection

assumed at the kingpin between towing units is valid; however, a frictionless joint assumption could

prove to be far less accurate and should be investigated in further experimentation. The values of the

tire stiffness constants were calculated using Equation 5.23 published by [8] and [9], which makes use

of the loading supported by the tire. Without further means of calculating this value, the published

equation was deemed to be quite an accurate approximation, however, it is understood that different

tires may produce different stiffness’s which could change non-linearly over a range of loading

weights and, as discussed later, it was found that the stability of the system is to a large extent






48

dependent on the tire stiffness’s, therefore, a better analysis of these values should be performed in

future texts.

The bicycle model in Section 5.2 was used to derive equations of motion of the system detailed in

Section 5.3. The equations of motion were used to design the Simulink model to be used to

investigate the yaw rate, yaw angle, and lateral acceleration responses of the system to a steering

input. Furthermore, an investigation into the effects of varying the system parameters was performed

in Sections 5.5 and 5.7. The most accurate simulation results would be produced by a model in the

time-domain, which is shown in Figure 16. In addition to the time-domain model, far simpler

Simulink models were created using the state-space equations (Equations 5.28 and 5.29), and the

overall transfer functions of the system. Figure 23 shows the lateral acceleration response of the truck

to a sinusoidal steering input simulating a vehicle lane change produced by the time-domain model,

while Figures 37 and 40 show the same parameter response using the state space and transfer function

models respectively. It can be seen from the figures that the state-space and transfer function models

produce almost identical curves to the time-domain model after 0.5s of simulation, with magnitudes

being marginally different. A large instability and inaccuracy was noted in the initial response of the

simpler models as the initial conditions were inputted as zero; however, the steady state response was

only marginally inaccurate.

The first investigation performed was on the original interlink system with parameters as outlined in

Table 5. Figures 17, 18, and 19 show the yaw response of the system, over a 15s interval, to a

sinusoidal input with a frequency of 1.5Hz and amplitude of 0.1 which was deemed to simulate a lane

change by [5]. From the figures it can be concluded that the yaw angle amplitudes experienced by the

truck and the first trailer were similar with the trailer being marginally less. The yaw amplitudes

experienced by the second trailer were noticeably much smaller than the other two vehicles due to its

longer length and existence of yaw damping in the system. Interestingly, it is noted that the truck and

trailer were predicted to yaw symmetrically ‘left’ and ‘right’, while the second trailer has a tendency

to have a larger yaw magnitude in one direction than the other. The yaw rate responses shown in

Figures 20 to 22 are consistent with the yaw response curves with yaw acceleration decreasing

rearward along the system. With regards to the symmetry of the yaw experienced by the second

trailer, Figure 22 shows that the trailer experiences symmetrical yaw accelerations from 4 to -4x10-4

rad/s and it is therefore predicted that the phase difference between the yaw angles of the units results

in the second trailer not experiencing symmetrical yaw motion. As discussed in Section 3.1, the

lateral accelerations experienced by each unit are known to increase rearward from the towing unit,






49

known as the ‘rearward-amplification’ phenomena. The ‘rearward- amplification’ phenomena was

observed in the lateral acceleration responses produced by the interlink Simulink model, as shown in

Figure 24. The figure shows the lateral accelerations experienced by the truck (top curve) and the

first and second trailer units (second and third curves) in m/s2. From Figure 24, it was observed that

the truck experiences a maximum of approximately 5m/s acceleration in the lateral direction, while

the first trailer experiences marginally over 10m/s2 lateral acceleration and the third trailer

experiences a large maximum of 15m/s2 lateral acceleration from a simple lane change steering input

simulation. The rearward-amplification of the lateral acceleration responses shown in the figure is

consistent with known theory and results and can further be used to validate the accuracy of the

created Simulink model. It is known that the rolling over of heavy motor vehicles occurs initially

with the most rearward trailer rolling over first, this can be explained by the results shown in Figure

24 as the trailer with the largest mass is exposed to the largest lateral accelerations (more than twice

that of the towing unit), therefore, the momentum of the second trailer could easily cause the truck to

roll over in an evasive manoeuvre. As a final analysis, it is noted that the lateral acceleration response

curves are out of phase with one another and the second trailer experiences its largest acceleration

before the preceding units, even though the yaw response curves in Figures 17 to 19 only show a

marginal phase difference; again this is known to be due to the ‘rearward-amplification’ phenomena.

The steering input can vary from 0.15Hz in a normal lane change situation to approximately 4Hz in an

evasive manoeuvre [3]. A chirp signal in Simulink was used to vary a sinusoidal function linearly

over time from 0.15Hz to 4Hz to simulate the system response to an evasive manoeuvre. Figure 25

shows the yaw response of the system to this type of input over a 15s time period. From the figure, it

was noted that the yaw angle experienced by each unit again decreases rearward from the towing unit

and that the response decreases over time due to the damping of the system.

After the investigations of the dynamic responses of the initial system were completed, the effects of

varying different parameters of the system were explored. Firstly, the length of the second trailer was

changed to equal that of the first trailer, with masses and tire stiffness’s adjusted accordingly. Figure

26 shows the yaw rate response of the adjusted system to the same sinusoidal input as in Figures 20,

21, and 22 while Figure 27 shows the yaw response and 28 shows the lateral acceleration response.

Comparing Figures 26 and 27 to 20, 21, 22 and 17, 18, 19 it was noted that the truck and first trailer

experienced marginally lower yaw angles and yaw accelerations than in the original system, but the

second trailer experienced much larger accelerations and angles due to its decrease in length, mass

and tire stiffness’s. Furthermore, the lateral accelerations experienced by the truck and first trailer






50

were found not to have changed, while the lateral accelerations experienced by the second trailer were

found to have decreased, which would increase the stability of the system in a dynamic manoeuvre, as

shown by Figures 24 and 28. The consistency in the lateral accelerations experienced by the first two

units while the parameters of the third unit were varied show that a parameter change in rearward

units does not affect the lateral velocities experienced by leading units, there is therefore no feed

forward phenomenon and the lateral velocity systems can be said to be uncoupled.

The effects of increasing the length of first trailer to equal that of the second trailer and the second

trailer to equal that of the first (swapped) and adjusting masses and tire cornering stiffness’s

accordingly were also investigated. Figures 29, 30 and 31 show the dynamic responses of this system

to the sinusoidal lane change input. The most notable change in the system response can be found in

the lateral acceleration response curve in Figure 30, where it was noted that both the first and second

trailer experienced a large decrease in lateral acceleration (first trailer almost half) when compared

with the response shown in Figure 24. In terms of t a yaw angle analysis, it was noted that the

swapping of the trailers resulted in the second trailer experiencing almost twice the yaw angle than it

did in the original system with much larger yaw accelerations.

The tire cornering stiffness’s provide the reaction responses due to steering angles of the tires. As

discussed above, the stiffness’s were approximated using Equation 5.23. An investigation into the

effect that the tire cornering stiffness’s have on the dynamic response of the system was performed on

the original system with the parameters as outline in Table 5, but tire stiffness’s equal to 25% of the

originally estimated value. Figures 32 and 33 show the yaw rate and yaw response of the system to

the sinusoidal lane changing simulation. The yaw angles were found to decrease when compared to

the results in Figures 17, 18, and 19, while the yaw rates were found to increase due to the reduction

in tire cornering stiffness’s, resulting in the frequency of the yaw cycle to be increased. Furthermore,

instability can be seen during the initial part of the simulation until a steady state response was

attained. The tire cornering stiffness’s were predicted to have a large influence on the stability of the

system and it is noted that further experimentation should be performed to accurately calculate the

exact tire cornering stiffness’s of the model to validate the use of Equation 5.23.

Finally, an investigation into response of the system with a longitudinal velocity (u) of 34m/s

(approximately 120km/h) as opposed to 22.2m/s (approximately 80km/h) was performed. The yaw

rate response of the system with an increase in longitudinal velocity, subjected to the sinusoidal

steering input, is shown in Figure 34. Comparing the response to that shown in Figures 20, 21, and 22

shows a large increase in the yaw rate experienced by all units, while comparison of Figure 35 to






51

Figures 17, 18, and 19 show the corresponding increases in yaw angle experienced by the units with

the truck yaw angle increasing initially from approximately a maximum of 0.6rad to almost 1rad. The

first and second trailers have similar increases. Although it does not appear that the system has

become unstable due to the velocity increase, there is definitely a noticeable difference in the dynamic

behaviour of the system.

The system in comprised of many different parameters such as lengths and masses. The different

dynamic responses of the system that were observed during the experimentation when parameters

were changed led to the understanding that the changing of certain system parameters may render the

system to become unstable. A stability analysis was performed on the system with variable

configurations as outline in Section 5.7. Stability was summarised in Section 3.7 and root-locus plots

were used to determine the effect of the varying of different system parameters on the systems

stability. An observation of the root-locus plots while one parameter of the system was varied and

others were kept constant was performed to test whether the real part of the roots of the system

(marked with an ‘x’ on the figures) could become positive, or fall into the right hand side of the plane.

Firstly, the effect of the tire stiffness’s on the stability of the system was investigated because of the

prediction made previously that the system may become unstable due to a reduction in tire stiffness’s.

It was found that no amount of adjustment of stiffness C1 would make the system unstable. A

simultaneous decrease in stiffness’s C2 and C3, however, resulted in the root-locus plot shown in

Figure 41 which clearly shows a root in the right hand side of the plane and therefore an unstable

system. It was found that the system became unstable when C2 and C3 were decreased beyond

290000 and 300000N/rad. The time-domain Simulink model of the interlink system was used to test

the theory of the system becoming unstable with these parameters. It can be seen in Figure 42 that the

yaw response of the system due to the sinusoidal steering input rapidly tends to infinity after a short

period of time, proving that the system is thus unstable. Furthermore, the root-locus plot was

generated using the overall transfer function of the system discussed in Section 5.6.2 while the system

response plot was generated using the time-domain Simulink model, the correspondence between the

two completely separate methods again proves that very similar results can be attained from either

methodology. Simulation results showed that different values of stiffness’s C4 and C5 did not affect

the stability of the system, while Figure 43 shows that the system becomes unstable when C 6 and C7

were simultaneously reduced to between 250000 and 300000N/rad.






52

The increasing of the longitudinal velocity from 22.2m/s to 34m/s did not make the system unstable,

but the overall damping and damped natural frequency of the system were found to have decreased,

which would increase the susceptibility of the system instability if excited at a dangerous frequency.

An investigation into the effect that the mass of each unit has on the system was performed. The

moments of inertia and tire stiffness’s were adjusted accordingly. It was discovered that the mass of

the truck would have to be decreased beyond 1000kg to produce an unstable system, while the system

would remain stable under all other realistic loading situations where the masses could not be less

than that of an empty trailer or greater than that of any realistic value.

The positions of the centres of gravity on the trailers were altered and their effects on the stability of

the system recorded. The total mass of the load as well as the axle load weights and corresponding

tire stiffness’s were assumed to be unchanged from those outlined in Table 5 for simplicity. It was

discovered that moving the centre of gravity on the first trailer 1m forward resulted in an unstable

system as shown in the root-locus plot in Figure 44, while moving the centre of gravity 1m rearward

on the second trailer resulted in positive real roots and therefore an unstable system.

As a final investigation, the parameters of the trailers were swapped and the stability of the system

was analysed. Figure 45 shows the root-locus plot of the stable system produced which correspondsto the stability shown in Figures 29, 30, and 31 of the dynamic responses of the system with the

trailers swapped around.






53

7 Conclusion and Recommendations

The interlink truck system in Figure 13 was able to be accurately modeled in a simplified form of a

bicycle model as shown in Figures 14 and 15. The bicycle model was able to be created by making

assumptions with regards the behavior of the system and its surroundings. The model created was

used to be able to predict the dynamic response of the system with derived equations of motion with a

high degree of accuracy as discussed in Section 6.

The bicycle model of the interlink system was generated using the same methodology as that used to

generate the models of the one truck and truck and trailer systems as shown in Figures 3 and 6

respectively. The bicycle models shown in Figures 3 and 6 were used to accurately replicate the

dynamic responses of those systems using Simulink models and comparing the results to those

published by [3] and [5]. On the basis of the models generated in Sections 4.1 and 4.2 being accurate,

a Simulink model to predict the dynamic response of the interlink system was built using the bicycle

model in Figure 14 and validated methodology. The Simulink model created was able to predict the

yaw rate (rad/s), yaw angle (rad), and lateral acceleration (m/ss) responses of a system with the

parameters in Table 5 to different steering inputs simulating real driving conditions.

The yaw damping ratio of the system with parameters outlined in Table 5 operating at a longitudinal

velocity of 22.2m/s was able to be calculated. The yaw damping ratio between the truck and first

trailer was calculated to be 0.833, while the yaw damping ratio between the first and second trailer

was calculated to be 0.00102.

The time-domain Simulink model built in Section 5 was used to predict the dynamic behavior of the

system to steering inputs with different parameters to those outlined in Table 5. It was observed that

shortening the length and reducing the load mass of the second trailer increased the yaw response of

the second trailer to a sinusoidal input but decreased the lateral acceleration to which the unit was

exposed from 15m/ss to 12m/s

2. Replacing the first trailer with the properties of the second trailer and

vice versa, it was observed that the first trailer experienced marginally less yaw motion while the

second trailer yaw motion approximately doubled. The swapping of the trailers produced a reduction

in lateral accelerations to which the trailers were subjected to. A decrease in tire cornering stiffness’s

to 25% of the predicted value resulted in a large decrease in yaw experienced by the units, and the

system was able to be made unstable when the cornering stiffness’s C2, C3, C6, and C7 were decreased

beyond 300000N/rad as shown by the locus-plots in Figures 41 and 43 respectively. An increase in










56

9 References

1. Meuller, T.H., J.J., P.H., 1999. Heavy vehicle stability versus crash rates. 10-19.

2. Gerdes, J.C., 2002. Safety Performance and Robustness of Heavy Vehicle AVCS. California

PATH Program, w, 5-22.

3. Martini, R D, 2006. The Pennsylvania State University The Graduate School Department of

Mechanical and Nuclear Engineering- GPS/INS SENSING COORDINATION FOR VEHICLE

STATE IDENTIFICATION AND ROAD GRADE POSITIONING, 1, 22-30, 54-70.

4. Fancher, P.S., Segel, L., Winkler, C.B., Ervin, R.D. , 1984. Tracking and stability of multi-

unit truck combinations, UMTRI-84-25, 3-23, 50.

5. Maas, J. W. L. H. , 2006. Jackknife stability of a tractor semi-trailer combination. 9-11, 17-

25.

6. Dahlberg, E., Wideberg, J. P., 2004. INFLUENCE OF THE FIFTH-WHEEL LOCATION . 5-

11.

7. Ei-Gindy, M., N., X., 2001. Sensitivity of rearward amplification control of a truck/full trailer

to tire cornering stiffness variations. Journal of Automobile Engineering , Part D, 1-11.

8. Fancher, P.S., Generic Data for Representing Truck Tire Characteristics in Simulations of a

Braking and Braking-in-a-Turn Maneuvers. 1995, University of

Michigan Transport Research Institute (UMTRI).

9. Fancher, P.S., Bernard, J., Clover, C., Winkler, C., Representing Truck Tire Characteristics in

Simulations of Braking and Braking-in-a-Turn Maneuvers. Vehicle

System Dynamics, 1997. 27(Supp.): p. 207-220.

10. Henred Fruehauf, 2003. 6 and 12m Ultra lightweight Interlink Combination (All Steel).

11. Tewari, A, 2003. Modern Control Design with Matlab and Simulink . 1st ed. England: John

Wiley and Sons.

12. Claymore Engineering. 2004. Root Locus Analysis. [ONLINE] Available at:

http://claymore.engineer.gvsu.edu/~jackh/books/model/chapters/rootlocus.pdf.

[Accessed 10 September 12].

13. Cannon, R.H., Dynamics of Physical Systems, McGraw-Hill, 1967

14. Bucknell . 1818. Nyquist Plots. [ONLINE] Available at:

http://www.facstaff.bucknell.edu/mastascu/econtrolhtml/Freq/Freq6.html. [Accessed 05

September 12].

http://claymore.engineer.gvsu.edu/~jackh/books/model/chapters/rootlocus.pdf


http://www.facstaff.bucknell.edu/mastascu/econtrolhtml/Freq/Freq6.html









57

Appendix A

A.1 Derivation of equations of motion of one vehicle model

Σ forces in lateral direction:

Σ moments about z-axis:

Linear tire model approximation:

Slip angles (tire vector diagram):

ζ

y

x

Vf t

-αf

δf

Vf t

Vf

Vf x

ζ






58

Lateral velocity vector of front tire:

Longitudinal velocity vector:

Combining the above equations:

The rear tire has no steering angle, therefore:

Equations of motion with tire forces become:

Lateral acceleration in body-fixed coordinates:

Therefore, meaningful expressions of equations are given by:

A.2 Derivation of equations of motion of truck and trailer model

Performing a sum of forces and moments on the system produces three coupled equations of

motion:

( )

( )






59

The linear tire model approximation was used to approximate the resulting tire forces for each tire

model, using the method explained in Section A.1 of this appendix, previously:

The above equations can be combined to represent the behavior of the system in the matrix form, where:

,and j is the steering input.






60

Simulink model:

Figure 46: Simulink model of truck and trailer






Appendix B

M= 30461.2 36776.6 0 0 0 0

56000 -154955 -282645 138857.6 0 0

9955.15 126198.3 301032.1 177043.5 0 0

138857.6 459757.6 902574.6 268717.7 0 0

0 0 0 0 1 0

0 0 0 0 0 1

K= -75304.5 -969649 0 0 0 0

-206385 -636407 994221.3 487018.6 -112923 -56961.2

-84141.5 -336099 -4196529 -620949 404198.8 72625.57

-487019 948207.4 3165621 4191920 -487019 -487019

0 1 -1 0 0 0

0 0 1 -1 0 0

B= -111.273

173.3861

-89.3129

60.83398

0

0

A= -3.05137 19.87019 106.8542 48.27494 -11.8834 -5.6198

0.479761 -42.8239 -88.5049 -39.985 9.842783 4.654749

-0.24769 23.52745 44.9896 22.18026 -5.09334 -2.58103-0.22451 -12.4948 -43.1218 -15.4339 4.595575 1.796865

0 1 -1 0 0 0

0 0 1 -1 0 0

B1= 2987028

592777.4

4657482

0

0

0