Vehicle Model for Tyre-Ground Contact Force Evaluation

Vehicle model for tyre-ground contact force evaluation

Lejia Jiao

Master Thesis in Vehicle Engineering

Department of Aeronautical and Vehicle Engineering KTH Royal Institute of Technology

TRITA-AVE 2013:40 ISSN 1651-7660

Postal address Visiting Address Telephone Telefax Internet KTH Teknikringen 8 +46 8 790 6000 +46 8 790 6500 www.kth.se Vehicle Dynamics Stockholm SE-100 44 Stockholm, Sweden

Acknowledgment

I owe gratitude to many people for supporting me during my thesis work. Especially, I would like to express my deepest appreciation to my supervisor, Associate professor Jenny Jerrelind, for her enthusiasm and infinite passion for this project. Without her patient guidance and persistent help, this thesis would not have been possible.

I am particularly indebted to my parents for inspiring me to this work.

I would like to thank Associate professor Lars Drugge, who introduced me to vehicle-road interaction and gave me enlightening instruction.

In addition, I would like to give my sincere thanks to Nicole Kringos and Parisa Khavassefat, for helping me to understand the pavement and sharing model and data with me; to Ines Lopez Arteaga, for giving me feedbacks from tyre expert’s point of view. The great interdisciplinary cooperation and teamwork helped me to have a good understanding of the whole vehicle-tyre-pavement system, and get rational tyre and pavement parts included in my models.

Last but not least, I would like to thank all my friends, for their understanding, encouragement and support.

Stockholm June 26, 2013

Lejia Jiao

ii Abstract

Economic development and growing integration process of world trade increases the demand for road transport. In 2008, the freight transportation by road in Sweden reached 42 million tonne-kilometers. Sweden has a tradition of long and heavy trucks combinations. Lots of larger vehicles, with a maximum length of 25.25 meters and weight of 60 tonnes, are used in national traffic. Heavier road transport and widely use of large vehicles contribute to the damages of pavement. According to a recent research by the VTI, total cost of road wear by freight transport in Sweden in 2005 was about 676 million SEK. If the weights of all vehicles were limited to 40 tonnes, according to the new EU rules, the cost of wear in 2005 would have been 140 million SEK less.

Lots of studies about road damage caused by vehicle have been done since the last decades. It has been found that the dynamic tyre force plays an important role in the damages of pavement. However, the influence of vehicle-pavement interaction on pavement damage has not been investigated to any large extent yet. The aim of this study is to provide suitable computational truck models, study the influence of vehicle-pavement interaction and parameters of vehicle on pavement damage.

To fulfil the aims, this study presents vehicle models, including quarter, half, full vehicle models and quarter vehicle model coupled with pavement, used to compute the dynamic tyre force. The different models are then compared. Two actual road profiles measured by laser, a smooth one and an uneven one, are used for evaluation. The models are analysed to find out the vehicle parameters that influence the road damage most and to learn about how detailed models are needed.

It’s found that difference does exist between more detailed models and less detailed ones, and it’s non-negligible. It will increase with the increase of road unevenness. The dynamic tyre force will not be affected much by coupling the pavement, unless the road surface is very uneven or wheel hop exists. On uneven roads, energy mainly dissipates in vehicle suspension. However, on even roads, vibration can be well damped in tyre before it reaches suspension, so most of energy dissipates in tyre. Different components influence the tyre force differently. The influence varies with different frequency range of input signal (road profile) as well. The effects of sprung parts are mainly in low frequency range, while the effects of unsprung parts are mainly in high frequency range. Parameters of vehicle body influence the dynamic tyre force most. The effect of cabin is much smaller compared to vehicle body and unsprung part. Changes in parameters of pavement will not influence the road load, but its resonant frequency. Therefore, the best way to reduce dynamic tyre load is to design a more lightweight vehicle body, softer and better damped suspension.

iii

iv Contents 1 Introduction ...... 1 1.1 Background ...... 1 1.2 Problem description ...... 1 1.3 Aim ...... 3 2 Methodology ...... 4 3 Vehicle models ...... 5 3.1 Introduction ...... 5 3.2 Model establishment ...... 6 3.2.1 Quarter vehicle model ...... 6 3.2.2 Quarter vehicle model coupled with pavement ...... 8 3.2.3 Half vehicle model ...... 10 3.2.4 Full vehicle model ...... 13 4 Model comparison ...... 16 4.1 Parameters used in simulation ...... 16 4.1.1 Vehicle parameters ...... 16 4.1.2 Pavement parameters ...... 17 4.2 Quarter, half and full vehicle ...... 18 4.3 Influence of coupled pavement ...... 24 4.4 Energy dissipation ...... 27 5 Parametric study ...... 29 5.1 Typical response and frequency distribution ...... 29 5.2 Effect of mass ...... 32 5.3 Effect of stiffness ...... 35 5.4 Effect of damping ...... 38 6 Conclusions ...... 41 7 Future work ...... 44 8 References ...... 45

vi 1 Introduction This chapter gives a brief review of history and background, a short introduction to the subject and the goals of this study. 1.1 Background With the growing and deepening of the integration process of world trade, the demand for freight transport, especially by road, continues to increase. According to the Swedish Road Administration, the freight transport by road is continuously increasing, and arrived around 45 billion tonne-kilometres in 2008, which has exceeded train and marine transport [1]. Sweden has a tradition of long and heavy trucks combinations. Lots of larger vehicles, with a maximum length of 25.25 metres and weight of 60 tonnes, are used in national traffic [2]. Heavier road transport and widely use of larger vehicles will contribute to the damages of pavement, such a fatigue cracking, permanent deformation etc. The maintenances of road call for huge amount of investment. According to research performed by the Swedish national Road and Transport Research Institute (VTI), in Sweden, total cost of road wear by freight transport in 2005 was about 676 million SEK. If all the freight transportation carried out with vehicles weighing more than 40 tonnes is redistributed to vehicles that weigh a maximum of 40 tonnes, according to the new EU rules, the cost of wear in 2005 would have been 140 million SEK less [2].

However, limiting the maximum weight of vehicles isn’t the only and best measurement to reduce the pavement wear and thereby reduce the associated cost. If the mechanisms, which lead to the road surface damage, and the factors that affect them, could be figured out, it would be possible for vehicle industry, especially heavy vehicle manufacturers, to find out a way to optimize and improve their trucks in order to minimize the damage. It would also be good news for the road administration and the construction sector, since they can enhance roads with explicit target to minimize the damage from vehicle factors. 1.2 Problem description To accurately describe how the vehicle dynamics will interact with and influence the pavement, a large amount of work has been carried out from both vehicle dynamic and pavement point of views. Sun and Deng’s work [3] proved that pavement loads are moving stochastic loads whose power spectral density (PSD) is in proportion to the PSD of pavement roughness. Then Sun and Greenberg [4], [5] presented the theory to solve the dynamic response of pavement structure under moving stochastic loads.

A large amount of work has been performed by researchers in order to reveal how the vehicle parameters affect the pavement load, and then affect the pavement performance [6– 1 12]. The importance of dynamic loads’ frequency and velocity was identified. Markov et al [8] found that the characteristics most important for dynamic loading include vehicle suspension type and characteristics, speed, height of pavement faults and joint spacing. Other factors (such as tyre pressure) contribute to a smaller extent. It was also found that under certain conditions dynamic loads are 40 % higher than static loads. Hudson et al [9] studied the impact of truck characteristics on pavements with truck load equivalency factors, and it was found that the frequency and speed of dynamic loads affects the pavement performance. Hardy and Cebon [10] studied the validated dynamic road response model and found out that the base strain and soil strain of flexible pavement are sensitive to vehicle speed, but not sensitive to the frequency of applied dynamic loads except for some resonance points. Collop and Cebon [13] used a simple road damage analysis based on the ‘fourth power law’. The result showed that road-friendly suspension (which is air-suspended in this study) does not have signiﬁcant e ect on thick pavement damage. However, it does reduce thin pavement damage. Cebon [14] studied the dynamic axle effects on road damage with a six-degrees-of-freedom, two dimensional vehicle model, which is similar to a walking beam model. Four road-damage-related wheel load criteria were developed, namely aggregate force criterion, fatigue weighted stress criterion, tensile strain fatigue criterion and permanent deformation criterion. He also proved that the dynamic component of wheel forces may reduce significantly the service lives of road surfaces which are prone to fatigue failure. Sun and Kennedy [12] investigated the effects of vehicle parameters, speed, and surface roughness on the PSD of stochastic pavement loads with quarter-vehicle model. They found that all these factors will influence the PSD loads. Their influence on the PSD loads were then given out based on frequencies. It was also found that passenger vehicles produce more high-frequency PSD loads than heavy vehicles do, and the frequency distribution of stochastic loads are quite different for these two kinds of vehicles. Sun [15] analysed the relation between suspension properties and tyre loads based on a walking beam suspension model. He used the probability that the peak value of the tyre load exceeds a certain given value to evaluate the road damage, which was based on the fourth power law. It was found that tyres with high air pressure and suspension systems with small damping will lead to large tyre loads and thus greater pavement damage. Elseifi et al [16] and Khavassefat et al [17] established finite element (FE) pavement model to analysis its behaviour under moving stochastic loads.

Although the vehicle-pavement interaction has been studied for several decades, the principle of interaction between vehicle and pavement and its influence on road wear haven’t been fully revealed yet. The study is still in a primary stage. It is noticed that, most of the studies use an existing moving load profile, or a stochastic one. A few recent studies used dynamic tyre loads from vehicle models, in which walking beam model or quarter vehicle model were used. Quarter vehicle model is a simple yet powerful model for most of vehicle dynamic 2 analysis, which concentrate their attention only on the most important characteristics of dynamic tyre forces. It provides details about vehicle suspension, but ignores the influence of yaw and pitch motion. Walking beam model represents the minority of suspensions which generate large dynamic tyre forces due to unsprung mass pitching motion as well as low frequency sprung mass motion. However none of them contain the detailed suspension nonlinearities and complexities of sprung mass motion that are typical of heavy vehicles [18]. To the best knowledge of author, the study of the vehicle related road damage using a more complex model than the quarter vehicle model has not been found in the literature. None includes a coupled vehicle-pavement model to study their interaction as well. 1.3 Aim The aim of this study is to solve the two problems mentioned in previous section: excluding the influence of yaw and pitch motion and ignoring the interaction between vehicle, tyre and pavement. It will provide more detailed vehicle models for moving load, which includes pitch and roll motion, and a vehicle model coupled with pavement mass to include the movement and force feedback from the pavement. It aims at building a more detailed yet simple model and more suitable model for further research regarding vehicle, tyre and pavement as a whole system.

There are three main aims in this study:

1. Build computational truck models, including quarter vehicle, half vehicle and full vehicle models, as a part of vehicle-tyre-pavement system to estimate road damage;

2. Build a vehicle model coupled with pavement to evaluate the characteristics of vehicle-tyre-pavement motion as a whole system;

3. Preliminary parameter analysis with the built models to find effects of different parameters and possible ways to reduce road damage caused by heavy vehicles and the huge associated cost.

3 2 Methodology This chapter explains the methods used in this study to reach the aims.

The work is divided into two major parts:

Building and validating the computational model of vehicle is one of the major parts of this study. In the first part, vehicle models suitable for vertical vehicle dynamics are studied. Differential equations for the systems are formulated. Computational models based on the equations of motion are constructed in Simulink. They are then compared to each other to evaluate advantages and disadvantages. In the second part, a parametric study is done with the selected model. Main parameters of the vehicle and the pavement, including mass, stiffness and damping, are variated to reveal the influence. Then regular patterns are summed up according to the results.

4 3 Vehicle models This chapter introduces the suitable vehicle models and their differential equations. 3.1 Introduction Dealing with vehicle dynamic problems, there are several models to choose from: from the simplest quarter vehicle model to the more complicated three-dimensional vehicle model. Each of them has its own scope of application and degree of precision.

In order to choose the suitable models, properties of concern should be reviewed from view of pavement engineering first. There are several types of pavements, including flexible, composite and rigid, used in modern road. Depending on type of pavement, different materials are used. No matter what type the pavement is, the most important types of road damage due to heavy vehicles are fatigue cracking and permanent deformation (or rutting) [19]. Examples are shown in Figures 1-2.

Figure 1 – Fatigue cracking [20]

Figure 2 - Permanent deformation-rutting [20]

5 Both kinds of failure mechanism are affected by several factors, such as construction method, material properties, environment and traffic load. In this study, only the vehicle load factor is investigated. Road vehicles interact with the pavement via the tyres that are in direct contact with the pavement. Tyre force, especially vertical force, and its distribution affect road wear to a large extent. While fatigue cracking is related to non-uniform contact traction distribution [21], rutting has a closer link with the vertical forces. Densification (compaction) and shear plastic deformation induced by vertical tyre force are two major mechanisms within the pavement materials contributing to permanent deformation [22]. So the vehicle model used to study pavement failure problems should at least reflect its vertical dynamics. Other properties, like horizontal motion and vehicle or wheel slip, are not that important.

The quarter vehicle model is the simplest one among models suitable for studying vertical dynamics of vehicle. It provides vertical dynamics only. The half vehicle model adds pitch characteristics compared to the quarter vehicle model, and the full vehicle (or four wheels) model adds the roll motion compared to the half vehicle model. The calculation amount will increase with the complexity of model. Even the full vehicle model is still a kind of very simplified model of a vehicle. With the help of a MBS-program like ADAMS, one can model the vehicle in more detail. However, as the complexity increases, so do the computation time and the complexity to analyse the results. In this study, the focus is on the three more simple models: the quarter vehicle, the half vehicle and the full vehicle, since those models are believed to provide sufficient results. 3.2 Model establishment In this section, the three vehicle models: the quarter vehicle, the half vehicle and the full vehicle models are presented. First, the equations of motion are derived under the assumption that springs and dampers are linear. Then the differential equations are implemented in Simulink models in order to simulate the models dynamic behaviour. Dampers and springs in the Simulink model can easily be replaced by nonlinear components to reveal vehicle’s nonlinear properties. 3.2.1 Quarter vehicle model The quarter vehicle model is often used in simple vehicle dynamics calculation when one is only interested in the vertical motion of the vehicle. It is the simplest vehicle model used to study vertical motion.

Figure 3 shows the quarter vehicle model, in which dynamics are simplified to vertical motion of sprung mass and unsprung mass. Sprung mass is the mass of the vehicle part which is supported above the vehicle suspension. In complex vehicles, like heavy truck in this study, it can be subdivided into cabin mass and vehicle body mass. Unsprung mass is a mass representing a part of the suspension, the wheels, the wheel axle and other components 6 connected to them. Sprung mass is coupled to unsprung mass via a spring and a damper, which represent the vehicle suspension. Likewise unsprung mass is coupled to the pavement via a spring and a damper, representing the tyre. [23] gives the typical quarter vehicle model, with and without damper, and methods to decouple and analyse. The quarter vehicle model can often provide acceptable predictions of vertical motion.

Figure 3 – 2-DOF Quarter vehicle model [12] In this study, the object is to model a heavy truck, which is a little different. Considering comfort of driver, the cabin of modern truck usually isn’t rigidly connected to chassis, but via cabin suspension. The mass of the cabin generally is close to the unsprung mass. The motion of cabin will influence the whole vertical dynamics of vehicle to some extent, and should be taken into consideration. It can easily be solved by connecting a mass-spring-damper system serially to the sprung mass (which represents the vehicle body mass now in the new truck model), as shown in Figure 4. Figure 5 shows the Simulink model of the 3-DOF quarter vehicle.

Figure 4 – 3-DOF quarter vehicle model representing a truck

7 The motion of the quarter vehicle model of a truck that includes the cabin dynamics can be described by the following equations of motion:

Where is mass, is spring stiffness, c is damping coefficient, is vertical displacement (positive direction is upward and measured from loaded position), is road unevenness as input, F is external force acting on each mass. Subscript c indicates cabin, subscript t indicates tyre/unsprung part, subscript s or no subscript indicates sprung part.

1 1 w Fzc 1/mc s s Road input zc' zc Fzc Gain8 -K-

Gain7 k_c

Gain6

1 1 s s du/dt z' z Derivative Saturation -C- 1 1 k_t s s Vehicle zt' zt Gain3 Force applied on road

c_t >= Scope1 Gain4 Switch

Gain simout1 In1 Out1 Fzt 1/mt To Workspace -K- c Fz 1/m Fzt Gain5 Gain9 Fz Gain2

Figure 5 – Simulink model of the 3-DOF quarter vehicle model representing a truck 3.2.2 Quarter vehicle model coupled with pavement By including a coupled pavement part, the influence of movement and force from pavement vibration (although small) can be included, and pavement movement can be roughly estimated. The integration will give a better understanding of pavement-vehicle interaction and evaluate the strategy of separating vehicle and pavement model. The pavement could be represented by a spring and mass combination as a basic assumption, as in Figure 6.

Figure 6 – 4-DOF quarter vehicle model representing a truck, coupled with simplified pavement The upper part is exactly the same as in the previous model. The pavement mass, , is connected to subgrade via spring , and tyre suspension directly. The road profile is still the input, and acts between tyre suspension and pavement mass. The road-tyre irregularity can be denoted as: [24]. The governing equations of motion of the system are:

where subscript p indicates pavement.

9 1 1 Fzc 1/mc s s zc' zc Fzc Gain8 c_c

Gain7 k_c dissipation

Gain6 To Workspace1

1 1 s s w du/dt z' z Derivative Saturation Road input 1 1 k_t -C- s s zt' zt Gain3 Vehicle

c_t >= Force applied on road (Dynamic) Scope1 Gain4 Switch

Gain

In1 Out1 Fzt 1/mt -K- simout Fzt c Fz 1/m Gain5 Gain9 To Workspace Fz Gain2

Gain10

k_p

1 1 Fzp 1/mp s s Fzp Gain1 zp' zp

Figure 7 - Simulink model of the 4-DOF quarter vehicle model representing a truck, coupled with simplified pavement

3.2.3 Half vehicle model A quarter vehicle model can provide vertical motion behaviour for a vehicle. However, it doesn’t take lateral and longitudinal dynamics as well as pitch and roll motion into account, as well as pitch, which may also be important. Figure 8 a) shows the 4-DOF half vehicle model without cabin dynamics. Figure 8 b) shows the 5-DOF half vehicle model with cabin dynamics. The structure of the front part and rear part of the half vehicles models are similar to the quarter vehicle model. Pitch motion is included in both models. Figures 9-10 show the Simulink models of the 4-DOF half vehicle model and 5-DOF half vehicle model respectively.

Figure 8 - Half vehicle model a) without cabin dynamics (4-DOF), b) with cabin dynamics (5-DOF) The equations of motion for the model without cabin dynamics are:

Where is pitch angle, is the moment of inertia around the y-axis, a and b are length from centre of gravity (COG) to front and rear axle respectively, subscript 1 indicates front part, 2 indicates rear part, all displacements are measured from loaded position.

The equations of motion for the model with cabin dynamics are:

11 For both models, the pitch angles are assumed to be small, thereby small angle approximations have been used in the equations of motion.

By assuming that the front and rear axle will be exposed to the same road profile but with a time delay, the road profile for the rear axle, , can be expressed as a function of , the road profile of the front axle, as follows:

Where is wheel base and is vehicle speed, which is assumed as a constant.

theta z zt1 theta' z' z zt1 theta

zt1' theta' zt1'

zt2 w1 z' zt2' w1' Fz1 Fz Fz Fzt1 Fzt1 Constant z Constant1 zt1

z z zt2 z' z' theta Scope theta zt1 theta' zt2'

zt1' w2

zt2 theta' w2' Fz2 w1 Fzt2 Fzt2 zt2' w1'

theta w2 Constant2 zt2 w2'

Road input

-K- HalfCar Scope1 Gain6 To Workspace

Figure 9 – Simulink model of half vehicle model without cabin dynamics (4-DOF)

12 theta z theta' zt1 z' zt1 z

zt1' theta

zt2 theta' zt1' zt2' w1 Fz z' w1' zc Fz1 Fz zc' Fzt1 Fzt1 Fz z Constant1 zt1

Fz z z Fzc zt2 z' z' theta zt1 Scope theta

zt1' theta' zt2' zt2 w2 theta zt2' theta' w2' zc zc Fz2 theta' w1 zc' Fzt2 Fzt2 w1' w2 z theta Constant2 zt2 w2'

z' Road input zc'

Fzc

-K- HalfCar Scope1 Gain6 To Workspace

Figure 10 - Simulink model of half vehicle model with cabin dynamics (5-DOF)

3.2.4 Full vehicle model The half vehicle model can be easily extended to a full vehicle model shown in Figure 11. Here the cabin dynamics is neglected. Similar to the half vehicle models, the pitch angle and the roll angle are assumed to be small, thereby small angle approximations have been used in the equations of motion. The Simulink model for the full vehicle model is shown in Figure 12.

Figure 11 - Full vehicle model without cabin dynamics (7 -dof) ThisThis modelmodel makesmakes itit possiblepossible to simulate the influencinflueinfluenncecee ofof aa fullfull 33 -D road profile, which is the reality. Equations of motion are:

Where , areare moment moment of inertia around x and y axle in body body -fixed coordinates, is pitch angle , is roll angle, is half of wheel track , all displacements are measured from loaded position .

14 z z

z' zt11 z' zt12 theta z theta theta theta theta' theta' theta' z' theta' phi phi zt11' phi zt12' phi z z theta phi phi' phi' phi' phi' z' w11 w12 zt11 w11 zt11 zt11 w11' Fz11 w12' Fz12 zt11' w11' Fzt11 Fzt12 zt11' zt11' zt12 w12 zt11 zt12 zt12 zt12 zt12' w12' zt12' zt12' zt21 w21 zt21 zt21 zt21' w21' z z z' theta' phi' zt21' zt21' z' zt21 z' zt22 zt22 w22 zt22 zt22 theta theta zt22' w22' theta' theta' zt22' zt22' Fz Fz Road input phi zt21' phi zt22' z theta phi Constant1 phi' phi'

w21 w22

w21' Fz21 w22' Fz11

Fzt21 Fzt22 Fzt11 zt21 zt22 Fzt12 Fzt21 Fzt22

Road input1

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Gain To Workspace

Figure 12 – Simulink model of the full vehicle model

15 4 Model comparison This chapter compares the different models presented in the previous chapter, in order to choose a suitable model for further studies.

The higher model complexity doesn’t necessarily lead to matching improvement between measurements and simulated results. By comparing the models’ response when exposed to the same input, the use of a simpler model can be justified if the difference is small enough. The use of simpler model is often wanted since they reduce the amount of calculation and thereby reduces the calculation time. This chapter compares the outputs of different models to the same sinusoidal input and real road profile, and investigates their advantages and disadvantages. 4.1 Parameters used in simulation

4.1.1 Vehicle parameters Typical heavy truck parameters specified for the different models are shown in Tables 1-4. To make different models comparable, the parameters are estimated from the same truck. They are calculated for each model so that it will represent the front axle dynamics of a truck.

Table 1 – 2-dof quarter vehicle model parameters Parameters Value Parameters Value 3400 kg 350 kg 300000 N/m 1000000 N/m 2000 Ns/m 500 Ns/m 20000 Ns/m

Table 2 – 3-dof quarter vehicle model parameters Parameters Value Parameters Value 650 kg 2000 Ns/m 75000 N/m 20000 Ns/m 7500 Ns/m 350 kg 2750 kg 1000000 N/m 300000 N/m 500 Ns/m

16 Table 3 - Half vehicle model parameters Parameters Value Parameters Value 650 kg 400000 N/m 75000 N/m * 4000 Ns/m 7500 Ns/m * 40000 Ns/m 8800 kg 450 kg 300000 N/m 1800000 N/m * 2000 Ns/m 1000 Ns/m * 20000 Ns/m 50000 350 kg 2.54 m 1000000 N/m 1.16 m 500 Ns/m * com denotes compression ; ext denotes expansion.

Table 4 - Full vehicle model parameters Parameters Value Parameters Value 18900 kg , * 40000 Ns/m , 300000 N/m , 450 kg , * 2000 Ns/m , 1800000 N/m , * 20000 Ns/m , 1000 Ns/m , 350 kg 100000 , 1000000 N/m 20000 , 500 Ns/m 2.37 m , 400000 N/m 1.33 m , * 4000 Ns/m 1.1 m * com denotes compression ; ext denotes expansion.

4.1.2 Pavement parameters By adding a mass-spring-damper part to the system, the movement and force of pavement can be included as well. However, the equivalent damping is hard to estimate, and is not available during this study. Therefore pavement is simplified to a mass-spring system in this study, which can provide approximation of pavement movement but lacks damping properties.

The equivalent pavement mass and stiffness can be estimated by the Finite Element Method (FEM) [17]. The mass cannot be the mass of the model used in FE analysis, but it can be estimated by obtaining a ratio between the maximum displacement and average displacement in depth and using this coefficient as multiplier of vehicle mass. In this case:

Therefore the estimated pavement mass in the dynamic system would be:

To estimate the pavement stiffness, a selected pavement structure is analysed in order to obtain the displacement field caused by a uniform pressure on the surface. Figure 13 shows the vertical displacement of pavement and the corresponding radial distance. The load is applied with a 30 centimetres radius circular contact patch. Therefore the equivalent mean stiffness is about 160 MN/m.

Figure 13 – Pavement vertical displacement vs. radial distance

4.2 Quarter, half and full vehicle When the model is improved from quarter vehicle without cabin to half one without cabin, or from half vehicle without cabin to full one, its complexity increases, so does its computational amount. How about the improvement on estimation? Two actual road profiles, E4 Grimsmark longitudinal profile (E4) and 265 East longitudinal profile (265), are selected to test their responses. The first set of data E4 belongs to a highway 600 km north of Stockholm. It has high level of unevenness (International Roughness Index: IRI= 2.30 m/km). In Sweden the standard increment for longitudinal direction is 0.1 meter and thereafter the data is averaged for every 20 meters in order to obtain the IRI value. The second set of data 265 is the longitudinal roughness of a highway north of Stockholm. The highway is fairly new and also fairly even (IRI =0.99 m/km), which has lower level of unevenness compared to E4. It has however some short bridges which are visible on the

18 profile measurements with relatively higher roughness magnitude. The data for these measurements are from Laser 13 (a laser measuring device) which follows the right rut.

As mentioned in Chapter 1, force (or load) applied on the road from the tyre is commonly used as a measure of road damage. The force applied on the road from the tyre is chosen as output. The dynamic contribution of it can be derived by:

in model without pavement, or by:

in model with pavement. Where is stiffness of tyre, is damping coefficient of tyre, is vertical displacement of unsprung mass, is vertical displacement of pavement, and are both measured from loaded position, is position of road surface. According to the circumstances, may also include both static and dynamic contributions, which will be specified. The static part given in Equation 31, which equals to the gravity of the whole vehicle, should be added to the dynamic part.

where is mass of various parts of vehicle, is the acceleration of gravity.

There is only one set of data for each profile. If all wheels in all models use the same set of data as input, the output from different models will be exactly the same. To reveal the effect of roll and pitch motion in half or full vehicle models, the road profile data should be processed first for each wheel. The road profile data is used directly in quarter vehicle model. For half vehicle model, front wheel uses it directly, rear wheel uses data computed from it and vehicle speed according to Equation 17. The vehicle speed used in the simulation is selected as 20 m/s (~72 km/h). For full vehicle model, left wheels use the same data as half vehicle data. In order to make the input to right wheels different from left wheels, right wheels use the data shifted by time t. The forces applied on the road from each model are then compared. To make them comparable, the front wheel of the half vehicle model and the front left wheel of the full vehicle model are chosen and plotted.

The responses from the different models due to E4 road profile are shown in Figure 14, in which t=0.1 s. Subfigure b) is the partial enlarged drawings of Subfigure a). The root mean square (RMS) value of dynamic portion of force applied on the road from quarter, half and full vehicle model are 1.3617 kN, 2.0396 kN and 2.2537 kN respectively.

19 Figure 15 shows tyre force of the models due to the 265 road profile. The RMS value of the dynamic portion of force applied on the road from quarter, half and full vehicle model are 0.6516 kN, 0.9780 kN, and 0.9811 kN respectively.

The differences between models are caused by their differences in including the pitch and roll dynamic. Similarities can be observed in both amplitude and frequency of tyre force. It is seen that the differences greatly depend on the unevenness of road. The increased unevenness will generate larger dynamic tyre force, and bigger roll, pitch and bounce motion, which will increase output differences between the models. Profile 265 is evener compared to profile E4, so its RMS values are smaller than those of E4. The RMS difference between half vehicle model and quarter vehicle model of profile 265 is much smaller compared to E4 as well.

20 a) Force applied on the road 12 Quarter vehicle 10 Half vehicle Full vehicle 8

2 Fw Fw (kN) 0

-2

-4

-6

-8 0 2 4 6 8 10 12 Time (s)

b) Force applied on the road

12 Quarter vehicle Half vehicle 10 Full vehicle

2 Fw Fw (kN)

-2

-4

-6

4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 Time (s)

Figure 14 – The dynamic portion of the force applied on the road when the vehicle models are simulated with the E4 Grimsmark longitudinal road profile.

21 a) Force applied on the road 4 Quarter vehicle 3 Half vehicle Full vehicle

0 Fw Fw (kN) -1

-2

-3

-4 0 2 4 6 8 10 12 Time (s)

b) Force applied on the road 4 Quarter vehicle 3 Half vehicle Full vehicle

0 Fw Fw (kN)

-1

-2

-3

7.4 7.6 7.8 8 8.2 8.4 8.6 8.8 Time (s)

Figure 15 - The dynamic portion of the force applied on the road when the vehicle models are simulated with the 265 East longitudinal road profile.

22 Figure 16 shows the tyre force due to the E4 road profile. Inputs and outputs of the quarter vehicle model and the half vehicle model are exactly the same as in Figure 14. Here equals to 1 s instead of 0.1 s, which increases the unevenness level between left and right sides. The RMS values of the dynamic portion of force applied on the road are equal to 1.3617 kN, 2.0396 kN, and 6.5414 kN respectively. Larger difference can now be observed between models due to the increase in unevenness. For uneven road, the usage of simulation results from the quarter vehicle model or the half vehicle model may introduce large error when evaluating the dynamic tyre force. The quarter vehicle model can well represent the real dynamics only when the road is relatively even.

Force applied on the road 20 Quarter vehicle Half vehicle 15 Full vehicle

Fw Fw (kN) 0

-5

-10

-15 0 2 4 6 8 10 12 Time (s)

Figure 16 - The dynamic portion of the force applied on the road when the vehicle models are simulated with the E4 road profile. To the author’s knowledge, there isn’t any clear boundary for large tyre force with respect of road damage proposed in literatures. In [15], Sun used , the possibility of peak value of wheel load exceeds a certain value , to estimate the road damage from vehicle. is related to vehicle gravity:

where is mass of various parts of vehicle, is the acceleration of gravity, is the percentage of the static vehicle load for a given level of .

23 In this study, when . It’s in the same order of magnitude compared with the tyre force difference between models shown in Figures 14-16. The difference in amplitude isn’t negligible, especially when the road is relatively uneven. It can also be observed that the difference between the half vehicle model and the full vehicle model is smaller than the difference between the quarter vehicle model and the half vehicle model.

That is because the inertia and track width are much less than and wheelbase. There is no obvious change in frequency. 4.3 Influence of coupled pavement The quarter vehicle model is the most common model used when evaluating road damage. A quarter vehicle coupled with pavement is introduced in Section 3.2.2. How large difference is there in response between the models when evaluating tyre-road contact force? In this section, quarter vehicle models with and without pavement are compared to see the effect of coupled pavement.

Figures 17-18 show responses from model with and without coupled pavement to sinusoidal road profile and E4 road profile. They include both the static and dynamic contribution of the tyre contact force. The difference is small enough to be neglected. Figure 19 shows the amplitude of tyre and pavement motion in response to E4. To make the motion of tyre and pavement comparable, only dynamic contribution is shown in the figure. Pavement displacement is a high-frequency vibration. The amplitude of the displacement is so small, that it is enlarged by 50 times in the figure to be seen clearly. It is much smaller compared to the motion of the unsprung mass, even after being enlarged by 50 times. The pavement dynamics’ influence on the vehicle dynamics is rather small, so it is possible to use only the quarter vehicle model to compute the contact force. However, with the help of the coupled model, one can get a rough estimation for pavement dynamics.

24 Force applied on road -33 Without pavement -34 With pavement

-35

-36

-37 Fw Fw (kN) -38

-39

-40

-41 0 1 2 3 4 5 6 7 8 9 Time (s)

Figure 17 – The force applied on the road when the vehicle model is excited with 1 Hz sinusoidal input.

Force applied on road -28 Without pavement With pavement -30

-32

-34

Fw Fw (kN) -36

-38

-40

-42 0 1 2 3 4 5 6 7 8 9 10 Time (s)

Figure 18 - The force applied on the road when the vehicle models are excited with the E4 road profile.

25 Vetical motion 0.04 Unsprung mass(without pavement) Unsprung mass(with pavement) 0.02 Pavement*50

-0.02 Z (m) -0.04

-0.06

-0.08

-0.1 0 1 2 3 4 5 6 7 8 9 10 Time (s)

Figure 19 – The vertical movement of tyre and pavement when the vehicle models are excited with the E4 road profile. Wheel hop is a special condition which needs to be considered in all vehicle models. Large displacement of the road will induce large dynamic tyre forces. When the induced upward dynamics tyre force is greater than the downward vehicle gravity, the tyres will lose contact with road surface, and wheel hop will occur. For vehicle models without pavement dynamics, wheel hop can easily be handled by setting zero as the upper limit of tyre force (combination of both dynamic and static tyre force). For vehicle models including pavement dynamics, pavement should be decoupled from vehicle when wheel hop happens. The decoupling algorithm is implemented in the Simulink model. An extreme condition is designed to reveal it. It is assumed that the vehicle is running on a very uneven road, the height difference is about 1 meter. Vehicle speed is still 20 m/s. The vertical dynamics of the vehicle is so large that it’s possible for tyre to loose contact with the pavement. As shown in Figure 20, the model is decoupled when ( is the tyre force acting on the pavement, whose positive direction is upward), which means tyre has lost contact with the pavement, and is kept to be 0 until the tyre get in contact with road surface again. The pavement part will freely oscillate during the wheel hop. Compared to the model excluding pavement but with wheel hop taken into account, difference exists after wheel hops, which is introduced by the free oscillation of pavement mass during wheel hop. The model including pavement but without wheel hop taken into account will result in positive , which is impossible for a real vehicle.

26 Road profile 1.5

0.5 w w (m) 0

-0.5 0 1 2 3 4 5 6 7 8 9 10

Force applied on road 100

Fw Fw (kN) -100

-200 0 1 2 3 4 5 6 7 8 9 10 Time (s)

Figure 20 - The force applied on the road when the vehicle models are simulated with high-level unevenness, to activate wheel hop.

4.4 Energy dissipation The 4-DOF quarter vehicle model with simplified pavement is used in this section to study the energy dissipation in the vehicle. Figure 21 shows the power dissipation in each damper of the quarter vehicle model simulated with the road profile E4. The Root Mean Square (RMS) value of the power dissipation is 9.7 W in cabin suspension damper, 91.1 W in vehicle suspension damper, and 5.8 W due to the tyre damping. Figure 22 shows the power dissipation in each damper in the quarter vehicle model simulated with the road profile 265. The RMS value of power dissipation is 1.0 W in cabin suspension damper, 5.7 W in vehicle suspension damper, and 6.9 W due to the tyre damping. The dissipation in pavement is not included since its damping is not included in the model. With relatively uneven road profile, like E4 in Figure 21, loss in vehicle suspension is dominant, because most of the vehicle vertical dynamics are damped in main suspension. However, with relatively even road profile,

27 like 265 in Figure 22, loss in the tyre may be higher than in vehicle suspension, because the vertical dynamics are light enough to be well damped in tyre.

Power dissipation in each damper 1800 Cabin suspension 1600 Vehicle suspension Tire 1400

1200

1000

800

Power (W) Power 600

400

200

-200 0 2 4 6 8 10 12 14 16 18 20 Time (s)

Figure 21 – Power dissipation in each damper in the quarter vehicle model simulated with Profile E4

Power dissipation in each damper 250 Cabin suspension Vehicle suspension 200 Tire

150

100 Power (W) Power

-50 0 2 4 6 8 10 12 14 16 18 20 Time (s)

Figure 22 – Power dissipation in each damper in the quarter vehicle model simulated with Profile 265

28 5 Parametric study In this chapter, the 4-DOF quarter vehicle model with simplified pavement is analysed to find the parameters that influence the road damage most. 5.1 Typical response and frequency distribution Road damage is the general term for deterioration of road conditions. It is caused by the combination of various factors including pavement material, construction, tyre force, temperature etc. Fatigue cracking and rutting are two major mechanisms causing road damage by heavy vehicles. Besides them, each kind of road damage has its unique mechanisms. There isn’t a widely-agreed unified standard to evaluate the combined effect of different kinds of road damage.

The ‘fourth power law’ is usually used in pavement design to aggregate the estimated traffic during the service life into the number of equivalent standard axle loads (ESALs)[25], [26]. It can be used to give out a rough estimation of road damage by static axle load. The number of ESALs N attributed to static load P is

where is generally taken to be 80 kN. It aggregates the traffic into a simple number of ESLAs, and uses ESLAs to indicate the road damage caused by the traffic loads. However, its validity is questionable. Current vehicle and pavement conditions, traffic volumes are figured out to be significantly different from the conditions of the AASHO road test, which is the basis and source of the ‘fourth power law’ [27], [28]. The results from most recent researches show that the damage exponent in Equation 33 may take a wide range of values [25].

There isn’t any widely recognized method to estimate the overall damage to road based on dynamic tyre force. According to Sun et al [5], Hardy et al [10] and Cebon [14], road damage is directly related to tyre force and its frequency, especially the tyre force. Cebon [26] gave out the common fatigue models developed from laboratory experiments:

where is the number of cycles to failure at strain level , is a constant that usually depends on the stiffness of the material, is a constant that depends on the material and the mode of distress. The strain can be calculated from dynamic tyre force by FEM model of pavement. As mentioned in previous chapter, Sun [15] took the times that is higher than a selected limit as an indicator of possible road damage. is quite important in analysis of road damage caused by vehicle, so how vehicle parameters and pavement 29 characteristics affects it will be studied in this chapter. The quarter car model with coupled pavement is chosen for this investigation.

Although there is inaccuracy because of the absence of roll, pitch motion in quarter vehicle model, the parameter study with quarter vehicle model can reveal the effects of major vehicle parameters investigated in this study.

Figures 23-25 show the model response to different road profiles. Tyre force is summation of the static force (the normal load is 36.75 kN) and the dynamic force. The amplitude of dynamic part is of concern and should be as small as possible to reduce the road damage. Pavement motion is high frequency resonance around its equilibrium position. The amplitude is quite small due to its high stiffness.

Road profile 0.01

0 w w (m) -0.01 0 1 2 3 4 5 6 7 8 9 10

Force applied on road -36

-37 Fw Fw (kN) -38 0 1 2 3 4 5 6 7 8 9 10

-6 x 10 Pavement motion 5

(m) 0 p Z -5 0 1 2 3 4 5 6 7 8 9 10 Time (s)

Figure 23 - The force applied on the road and the pavement motion when the vehicle models are simulated with the 0.5 Hz / 10 mm sinusoidal input.

30 Road profile 0.1

0 w (m) w

-0.1 0 5 10 15 20 25 30 35 40 45 50

Force applied on road 0

-20

-40 Fw Fw (kN)

-60 0 5 10 15 20 25 30 35 40 45 50

-4 x 10 Pavement motion 5

0 (m) p

Z -5

-10 0 5 10 15 20 25 30 35 40 45 50 Time (s)

Figure 24 - The force applied on the road and the pavement motion when the vehicle models are simulated with the E4 road profile.

Road profile 0.1

0 w w (m)

-0.1 0 5 10 15 20 25 30 35 40 45 50

Force applied on road -20

-30

-40 Fw Fw (kN)

-50 0 5 10 15 20 25 30 35 40 45 50

-4 x 10 Pavement motion 2

(m) 0 p Z

-2 0 5 10 15 20 25 30 35 40 45 50 Time (s)

Figure 25 - The force applied on the road and the pavement motion when the vehicle models are simulated with the 265 road profile.

31 Figure 26 shows the Bode diagram of the quarter vehicle model with the parameters in Chapter 4.1.1. Table 5 presents the systems’ natural frequencies and damping. There are four eigenvalues and their corresponding eigen-frequency. The eigen-frequency of vehicle body suspension and cabin suspension is too close that they locate at the same peak in lower frequency range.

Bode Diagram 160

140

120

100 Magnitude (dB) Magnitude 80

60 180

135

Phase (deg) Phase 45

0 -1 0 1 2 3 4 10 10 10 10 10 10 Frequency (Hz)

Figure 26 – Bode diagram of quarter vehicle model

Table 5 - Natural frequency and damping of the quarter vehicle model Part Eigenvalue Damping Freq. (Hz) Suspension -1.18e+000 ± 7.73e+000i 1.51e-001 1.2446 Cabin -7.15e+000 ± 1.05e+001i 5.64e-001 2.0213 Tyre -1.72e+001 ±5.80e+001i 2.85e-001 9.6289

5.2 Effect of mass By changing the parameters of the model, the effect of different parameters can be revealed. Mass, spring stiffness and damping coefficient are investigated separately in the following sections to study their effects. Tyre force is chosen as the indicator of their effects, because it affects pavement damage directly. Its dynamic part is influenced more by changes in parameters mentioned above and of more interests than the static part. In addition, in the investigation of effect of mass, tyre forces will be not comparable if static part is included. So only dynamic force is presented and discussed in following sections.

32 In this section, mass of different part of the 4-DOF quarter vehicle model is changed to investigate its effect. Figures 27-29 a) show the tyre force in response to E4 road profile. The original mass is compared to the mass altered based on it. For instance, the original sprung mass, , is compared with 50 % of it, , and 200 % of it, . It can be seen from the simulation results that with greater mass, the amplitude of tyre force tends to be higher. The effects of cabin, vehicle body and unsprung mass are different. Vehicle body mass has the biggest effect, while unsprung mass has the smallest. Figures 27-29 b) show the Bode diagram of the vehicle with different masses. It’s clear that, for sprung mass (vehicle body and cabin), increasing mass will lead to increasing gain in low frequency range (0-2 Hz), and decreasing gain in medium frequency range (2-10 Hz). Sprung mass won’t influence high frequency behaviour. On the contrary, unsprung mass only affect high frequency part. Greater unsprung mass will lead to larger gain in high frequency range (>10 Hz). The road profile input is usually in the low frequency range (<2 Hz), where the effect of vehicle body mass is prior to cabin. So the difference in tyre force is most obvious in Figure 27 a), and nearly indiscernible in Figure 29 a).

a) b)

Force applied on road Bode Diagram 15 150 0.5m m 2m

100 Magnitude (dB) Magnitude

Fw Fw (kN) 180 0 0.5m m 135 2m

90 -5 Phase (deg) Phase 45

0 -10 -1 0 1 2 3 4 0 2 4 6 8 10 12 14 16 18 20 10 10 10 10 10 10 Time (s) Frequency (Hz)

Figure 27 - Effect of vehicle body mass a) The dynamic portion of the force applied on the road when the vehicle model is simulated with different body mass, b) Bode diagram of models with different body mass.

33 a) b)

Force applied on road Bode Diagram 10 160 0.5m c 8 m c 140 2m c 6 120

4 100 Magnitude (dB) Magnitude

2 80

0 60

Fw Fw (kN) 180 0.5m -2 c m 135 c 2m -4 c

90 -6 Phase (deg) Phase 45 -8

0 -10 -1 0 1 2 3 4 0 2 4 6 8 10 12 14 16 18 20 10 10 10 10 10 10 Time (s) Frequency (Hz)

Figure 28 – Effect of cabin mass a) The dynamic portion of the force applied on the road when the vehicle model is simulated with different cabin mass, b) Bode diagram of models with different cabin mass.

a) b)

Force applied on road Bode Diagram 8 160 0.5m t m 6 t 140 2m t 4 120

100 2 Magnitude (dB) Magnitude

80 0

Fw Fw (kN) -2 180 0.5m t m t -4 135 2m t

-6 90 Phase (deg) Phase

-8 45

0 -10 -1 0 1 2 3 4 0 2 4 6 8 10 12 14 16 18 20 10 10 10 10 10 10 Time (s) Frequency (Hz)

Figure 29 – Effect of unsprung mass a) The dynamic portion of the force applied on the road when the vehicle model is simulated with different unsprung mass, b) Bode diagram of models with different unsprung mass. Pavement mass will not influence tyre force at all, except on its natural frequency. As discussed in Section 4.1.2, the original pavement mass is 1.4 times of the total vehicle mass.

34 In Figure 30, where is the total vehicle mass, it is compared with pavement mass which is 0.6 times and 5 times of . Figure 30 b) shows the Bode diagram of models with different pavement mass. The gain is almost the same, and the natural frequency of pavement is high (>12 Hz). So for normal road-vehicle interaction studies, changes of pavement mass will not influence the tyre force. It is confirmed by the response to road profile in Figure 30 a).

a) b)

Force applied on road Bode Diagram 8 160 m =0.6m p vehicle 140 m =1.4m 6 p vehicle m =5m 120 p vehicle 4 100

80 2 Magnitude (dB) Magnitude 60

0 40

Fw Fw (kN) -2 360 m =0.6m p vehicle m =1.4m p vehicle -4 315 m =5m p vehicle

-6 270 Phase (deg) Phase

-8 225

180 -10 -1 0 1 2 3 4 0 2 4 6 8 10 12 14 16 18 20 10 10 10 10 10 10 Time (s) Frequency (Hz)

Figure 30 - Effect of pavement mass a) The dynamic portion of the force applied the on road when the vehicle model is simulated with different pavement mass, b) Bode diagram of models with different pavement mass. 5.3 Effect of stiffness Figure 31-33 show the responses and Bode diagrams of vehicles with different spring stiffness. Spring stiffness mainly influences the medium and high frequency behaviour. Effect of suspension spring is significant. Higher spring stiffness implies higher gain in 1-5 Hz frequency range, which means stiffer suspension spring will introduce higher damage. Cabin spring doesn’t have much effect. So change of cabin spring will not influence the road damage much. Tyre stiffness has similar effect as the suspension spring. Stiffer tyre gives higher road damage, but in high frequency range (>9 Hz).

35 a) b)

Force applied on road Bode Diagram 10 160 0.5k s 8 k s 140 2k s 6 120

4 100 Magnitude (dB) Magnitude

2 80

0 60 180 0.5k -2 s k 135 s 2k -4 s

90 -6 Phase (deg) Phase 45 -8

0 -10 -1 0 1 2 3 4 0 2 4 6 8 10 12 14 16 18 20 10 10 10 10 10 10 Time (s) Frequency (Hz)

Figure 31 – Effect of suspension spring stiffness a) The dynamic portion of the force applied on the road when the vehicle model is simulated with different suspension spring stiffness, b) Bode diagram of models with different suspension spring stiffness.

a) b)

Force applied on road Bode Diagram 8 160 0.5k c k 6 c 140 2k c 4 120

100 2 Magnitude (dB) Magnitude

80 0

Fw Fw (kN) -2 180 0.5k c k c -4 135 2k c

-6 90 Phase (deg) Phase

-8 45

0 -10 -1 0 1 2 3 4 0 2 4 6 8 10 12 14 16 18 20 10 10 10 10 10 10 Time (s) Frequency (Hz)

Figure 32 – Effect of cabin spring stiffness a) The dynamic portion of the force applied on the road when the vehicle model is simulated with different cabin spring stiffness, b) Bode diagram of models with different cabin spring stiffness.

36 a) b)

Force applied on road Bode Diagram 15 180 0.5k t k 160 t 2k 10 t 140

120

5 Magnitude (dB) Magnitude 100

0 60

Fw Fw (kN) 180 0.5k t k t -5 135 2k t

-10 (deg) Phase 45

0 -15 -1 0 1 2 3 4 5 0 2 4 6 8 10 12 14 16 18 20 10 10 10 10 10 10 10 Time (s) Frequency (Hz)

Figure 33 – Effect of tyre stiffness a) The dynamic portion of the force applied on the road when the vehicle model is simulated with different tyre stiffness, b) Bode diagram of models with different tyre stiffness. As show in Figure 34, pavement stiffness will only influence on its natural frequency, which is similar to the effect of pavement mass. It does not change the tyre force. That is because compared to the suspension or the tyre stiffness of the vehicle, the pavement is much stiffer and not comparable to them. The pavement can thereby be considered as rigid here.

37 a) b)

Force applied on road Bode Diagram 8 150 0.5k p k 6 p 2k p 100 4

2 Magnitude (dB) Magnitude 50

Fw Fw (kN) -2 360 0.5k p k p -4 315 2k p

-6 270 Phase (deg) Phase

-8 225

180 -10 -1 0 1 2 3 4 0 2 4 6 8 10 12 14 16 18 20 10 10 10 10 10 10 Time (s) Frequency (Hz)

Figure 34 - Effect of pavement stiffness a) The dynamic portion of the force applied on the road when the vehicle model is simulated with different pavement stiffness, b) Bode diagram of models with different pavement stiffness.

5.4 Effect of damping Changes in damping will affect the tyre force as well, but in a different way compared with changes in mass or spring stiffness. With smaller suspension damping, the resonance peaks at the resonance frequencies of the sprung mass and the tyre tend to be higher, and the valley between them tend to be lower (Figure 35 b). With increasing suspension damping, the gain between sprung mass and tyre resonant frequency will increase as well. However, when the damping is increased to its saturation, it will stop increasing and keep to be a gradual slope. I.e. a better damped road will result in smaller tyre force as shown in Figure 35 a). In Figure 36, it can be seen that changes in the cabin damping don’t affect the tyre force much, and the difference is small for the real road response as well. Figure 37 shows that the tyre damping only effect the gain in the high frequency range (>40 Hz), and has no effect on the lower part. Increasing the tyre damping will increase the gain in the high frequency range, but for the low or medium frequency range, the gain will remain the same. So the tyre forces in Figure 37 a) are almost the same. Compared to mass and stiffness, damping does not affect the tyre force significantly.

38 a) b)

Force applied on road Bode Diagram 8 160 0.5c s c 6 s 140 2c s 4 120

100 2 Magnitude (dB) Magnitude

80 0

Fw Fw (kN) -2 180 0.5c s c s -4 135 2c s

-6 90 Phase (deg) Phase

-8 45

0 -10 -1 0 1 2 3 4 0 2 4 6 8 10 12 14 16 18 20 10 10 10 10 10 10 Time (s) Frequency (Hz)

Figure 35 – Effect of suspension damping a) The dynamic portion of the force applied on the road when the vehicle model is simulated with different suspension damping, b) Bode diagram of models with different suspension damping.

a) b)

Force applied on road Bode Diagram 8 160 0.5c c c 6 c 140 2c c 4 120

100 2 Magnitude (dB) Magnitude 0.5c c c 80 c 0 2c c

Fw Fw (kN) -2 180

-4 135

-6 90 Phase (deg) Phase

-8 45

0 -10 -1 0 1 2 3 4 0 2 4 6 8 10 12 14 16 18 20 10 10 10 10 10 10 Time (s) Frequency (Hz)

Figure 36 – Effect of cabin damping a) The dynamic portion of the force applied on the road when the vehicle model is simulated with different cabin damping, b) Bode diagram of models with different cabin damping.

39 a) b)

Force applied on road Bode Diagram 8 160 0.5c t c 6 t 140 2c t 4 120

100 2 Magnitude (dB) Magnitude 0.5c t c 80 t 0 2c t 60

Fw Fw (kN) -2 180

-4 135

-6 90 Phase (deg) Phase

-8 45

0 -10 -1 0 1 2 3 4 0 2 4 6 8 10 12 14 16 18 20 10 10 10 10 10 10 Time (s) Frequency (Hz)

Figure 37 - Effect of tyre damping a) The dynamic portion of the force applied on the road when the vehicle model is simulated with different tyre damping, b) Bode diagram of models with different tyre damping.

40 6 Conclusions This chapter concludes the work done in this study, discusses the results and their implication to road damage caused by heavy vehicle.

In this study, vehicle models used in vehicle-road interaction are investigated. The quarter vehicle model, the half vehicle model and the full vehicle model are selected. The equations of motion are derived for each of them, which are then implemented into Simulink models in order to simulate the vehicle motions for different road inputs. To study the interaction between pavement and vehicle, a simplified pavement model is added to the quarter vehicle model.

All models are then compared. From quarter vehicle to half vehicle and then to full vehicle, the complexity increases. In section 4.2, it is showed that how detailed the model is does affect its estimation results. The difference in response between the models is non-negligible, and it will increase with the increase of road unevenness. For simple road damage analysis, the quarter vehicle model is sufficient. However, for more complicated road damage analysis, especially with uneven road profile, more complex model should be used to get better results. Adding the pavement part won’t change the dynamic behaviour of vehicle part much. The difference between models coupled with and without pavement is negligible. However, with the help of the coupled pavement part, the movement of pavement mass can be estimated. The coupled pavement will also influence the tyre force, when the road surface is greatly uneven and wheel hop exists.

The performed parametric study reveals the relation between pavement loads and vehicle/pavement parameters, which can help to understand the dynamic pavement loads explicitly. A summary of the results are shown in Table 6 below.

Table 6 – Effects of different parameters on tyre force Parameters ↑ Gain Around Around Cabin mass ↗ ↓ －－－ Vehicle body mass ↑ ↑ ↘ －－ Unsprung mass －－ ↑ ↑ － Pavement mass －－－－－ Cabin spring － ↗ －－－ Suspension spring － ↑ ↑ ↘ － Tyre stiffness －－－ ↑ ↑ Pavement stiffness －－－－－ Cabin damping －－－－－ Suspension damping － ↓ ↑ ↓ － Tyre damping －－－－ ↑ ↑↓: Significantly increase or decrease; ↗↘: Slightly increase or decrease; －: No change. : Natural frequency of sprung mass; : Natural frequency of unsprung mass.

From the results in Table 6 it can be concluded:

1. The tyre force will change with change of vehicle parameters. Vehicle body and suspension part have the greatest effect, while cabin and its suspension have the least effect;

2. The effects of vehicle body and suspension are mainly in low frequency range (<2 Hz). Cabin and its suspension’s effects are mainly in medium frequency range (2-10 Hz). Unsprung mass and tyre’s effects are mainly in high frequency range (>10 Hz). Normal road profile is mainly in low frequency range, so the effect of changes in cabin and tyre is not obvious compared to vehicle body and suspension part;

3. Different components influence the tyre force differently. Increasing the mass or the spring stiffness will increase the gain from road input to tyre force. Increasing the damping will lower the peak on its corresponding natural frequency and level up the valley between them, until the system is well damped. After it’s well damped, the gain between peaks should be a gradual slope, and will not change according to increasing of damping anymore;

42 4. Changes of pavement parameters will not influence the dynamic road load. It will only change its resonant frequency, which is quite high and hard to be reached during normal road transport.

From vehicle point of view, the best way to reduce dynamic tyre force and road damage is to decrease the mass, especially vehicle body mass, and the suspension stiffness, especially main suspension. Having vehicle well damped helps to reduce road damage as well. More lightweight vehicle body design and restricting vehicle load can help to reduce vehicle mass. Using softer suspension can lower dynamic tyre load, but will degrade roll stiffness and hence reduce static roll-over performance at the same time. It also increases static suspension deflection, and therefore increases the sensitivity of ride height to static load, which is undesirable in a truck [29]. However these problems can be solved by using anti-roll bars, independent suspension, active or semi-active suspensions. Compared with mass and spring stiffness, it’s more complicated to optimize dynamic tyre force by changing damping, because changing damping will increase gain in some frequency and decrease it in other frequency. Active damper provides a good solution for the problem. By using an actuator instead of a passive damper, active damper can provide different damping to difference force which can be controlled and optimized by designer. Thus it can be adjusted to minimize the dynamic tyre force.

43 7 Future work This chapter gives some suggestion for future work.

In this study, different models suitable for analysis of vehicle road dynamic and road damage caused by it are developed and analysed. Although the quarter vehicle model is used because of its simplicity and acceptable accuracy when performing the parameter analysis, half- and full vehicle can still be used to handle more complex problems or problems needing higher accuracy, especially for uneven road. The pavement model used in this study contains only mass and spring. The damping part should also be added in further studies to improve the accuracy of model. However, it’s tricky to find out a value for pavement damping.

If available, the tyre model can replaced by more advanced tyre model, which can compute force distribution. And then pavement model can be replaced by an FE model to give out detailed damage estimation.

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