Modeling and Experimental Validation of an Off-Road Truck’s (4 × 4) Lateral Dynamics Using a Multi-Body Simulation

Abstract

In the automotive sector, the use of multi-body software for modeling of existing vehicles has become essential due to its advantages in understanding vehicle dynamics in different situations and improving vehicle performances. This paper aims to model an off-road truck (4 × 4) by using ADAMS software 2020. Several steps must be achieved, including experimentally identifying some truck characteristics such as the mass, center of gravity coordinates, and tire vertical stiffness. The truck features leaf springs in both the front and rear suspensions, which must be validated before their integration into the full model due to their modeling complexity. This validation is performed by comparing the force–displacement characteristics obtained experimentally with simulation results from ADAMS, showing a good agreement. Then, the full truck is modeled in ADAMS software and validated through an experimental test using a repeated double-lane change scenario with two tests for the validation of the truck’s lateral dynamics. The comparison between the results shows a good correlation, validating the multi-body truck model.

1. Introduction

Investigations related to motor vehicles are extremely versatile, including structural analysis of chassis [1] and tires [2], vehicle dynamics [3], and influence on traffic safety [4], to name but a few.

In recent years, computer-aided modeling and engineering have become the standard approach to developing and investigating vehicles, where suitable simulations are conducted to analyze the vehicle’s structure and performance in various scenarios. Commercial multi-body software packages are used to model different types of vehicles with varying complexity, allowing for a better study of their statics, kinematics, and dynamics. The created virtual models make it possible to improve vehicle performance or add systems as needed while maintaining the best vehicle characteristics, such as handling stability and passenger comfort, without resorting to costly and time-consuming experimental tests [5,6].

Furthermore, by combining numerical and experimental methods, researchers and engineers can develop accurate and reliable models that enhance understanding, predict behavior, and improve system performance [7,8].

The modeling process includes several essential steps, which begin with the determination of the main characteristics of the vehicle, such as the mass, the coordinates of its center of gravity, the moment of inertia, etc. The next step involves an analysis of the vehicle’s structure, including the type of suspension and its components, such as the shock absorber spring and anti-roll bar, along with their corresponding characteristics, and the links (joints and attachments) used for the connections between the elements. Finally, validation of the developed model is necessary through experimental tests to verify its applicability and accuracy.

The study of vehicle lateral dynamics is crucial in understanding their behavior in different driving situations. The simplest model is a bicycle model [9], which employs mathematical equations to approximate the vehicle’s motion and identify key parameters essential for understanding its stability. However, this model simplifies the vehicle as a single track with two wheels and does not account for roll motion, suspension dynamics, and detailed tire behavior [10], all of which are crucial for accurate stability analysis. In contrast, multi-body models offer advantages by integrating these factors into comprehensive mathematical equations that describe the full vehicle dynamics.

Driving stability is a multifaceted concept that involves the interplay of several vehicle dynamics factors, including steering, suspension, and braking systems. The major components of vehicles that play a crucial role in ensuring the best vehicle performance, such as passenger comfort, driving stability, and maneuverability, are suspension systems as well as powertrain suspension systems (because the powertrain of a vehicle accounts for 10–15% of the total mass) [11].

A good suspension system absorbs shocks from road irregularities, preventing excessive body roll and maintaining vehicle posture. The system must balance comfort and performance, adapting to various road conditions to provide a stable and smooth ride without sacrificing maneuverability [12]. By carefully designing and adjusting the characteristics of individual elements of these systems, conflicting goals can be achieved, including ride comfort, maneuverability, and stability.

Suspension systems have several classifications depending on the criterion used. Hence, they can be classified according to their configuration as dependent or independent, according to their function as passive and active, and according to the type of spring used which can be a coil spring, a leaf spring [13,14,15,16], etc.

The leaf spring is one of the types used in suspension systems to support light, medium, and heavy loads. It has demonstrated its effectiveness in maintaining the best vehicle performance under various conditions. These types of springs are among the oldest elastic elements [17]. They consist of several elements that interact, making their structure complex compared to the other elastic elements [18].

There are several types of leaf springs used in utility vehicles. Single-leaf or mono-leaf springs are commonly found in light vehicles such as passenger cars, vans, and light trucks. Multi-leaf springs, made of several graduated leaves, are used across various vehicles including trucks, trailers, and heavy-duty vehicles. In contrast to conventional leaf springs that maintain a uniform thickness throughout, tapered leaf springs are thicker in the center and gradually their thickness decreases towards the ends. They are commonly used in various vehicles, including trucks, trailers, and passenger cars [19].

In modeling software, a coil spring is simpler than a leaf spring, which consists of direct input of characteristics such as rigidity, preload, and their coordinates into the model. However, modeling a leaf spring takes into account all elements that it contains, such as the number of leaves with their coordinates, the number of elements for each blade, characteristics of shackles, clips, and bushings, the spring’s position relative to the axis, etc., which makes its modeling more complex. Therefore, to ensure that the leaf spring model has the same characteristics as the real spring installed in the vehicle, it should be validated before being used in the full vehicle model.

It is challenging to design a leaf spring with all the desired characteristics such as vertical stiffness, torsional stiffness, etc. Several researchers have developed different methods to model leaf springs and find their characteristics.

The Fancher method [20], also known as the empirical method, uses equations or functions capable of representing experimentally obtained force–displacement characteristics. The mathematical model applies these equations to describe both the spring force and the Coulomb damping force, employing linear approximations based on experimental results of force–displacement tests conducted on the studied leaf springs.

The beam theory studies the leaf spring as a beam to determine its characteristics, and several beam theories have been used for this purpose. Timoshenko’s theory [21] considers the leaf spring as a cantilever beam and assesses its properties. In [22], a fusion of Timoshenko’s and Euler’s theories are employed to enhance the visualization of the physical dynamic behavior of leaf springs. Another theory applies the second theorem of Castigliano [23], which consists of determining the beam’s deformation energy when subjected to external force. The displacement of the beam is obtained as the derivative of the deformation energy expressed in terms of force and the expression of beam stiffness is also derived. This theory is used in [24] to replace the multi-leaf spring with five leaves with a tapered leaf consisting of three leaves, ensuring that the new leaf spring maintains the same stiffness as the original leaf spring. The accuracy of this method is verified experimentally and by simulation in ADAMS.

The finite element method (FEM) is widely used to solve complex physical problems for which obtaining analytical solutions is difficult [25], if not impossible. This method is used to analyze the leaf spring’s characteristics. In [26], a comparative study of leaf springs examined steel and composite materials. The leaf spring was modeled using CATIA, and static analysis was performed using ANSYS Workbench to compare it with the developed analytical model for the steel material. Then, a static analysis of a leaf spring with the same design but using composite materials was carried out using the same software. It was observed that the composite material performed better than steel.

The discrete method involves dividing the leaf spring into rigid elements, which are then linked together using components such as torsion springs and dampers [27]. This method is used in various multi-body software, like ADAMS, due to their simplicity and low degrees of freedom compared to FEM. A widely used approach is the SAE X-link model [28]. A simpler model, the 03-link model, is utilized in [29] to study the leaf spring. Experimental tests, including both static and dynamic tests, have validated the developed model. The advantage of this method lies in its ability to easily construct a simplified model that accurately replicates the kinematic and compliance properties of the leaf spring. However, this method does consider the hysteresis effect from friction between the leaves. The study [30] was based on the 03-link model, which introduces torsional friction at the revolute joints to address this issue. The optimization of key parameters is accomplished through a combination of ADAMS and OPTIMUS software. The comparison analysis between the simulation and test results of the leaf springs shows a good correlation. Another model used in [31] is a five-link model where the leaf spring is discretized in five rigid elements and mathematical development is used to find leaf spring performances. The results of this method are compared with those obtained using the sophisticated finite element method with SIMPACK Software, and a good correlation between the methods is observed.

The equivalent model consists of finding the corresponding leaf spring model, which contains simple elements such as a coil spring and a shock absorber, while retaining the same characteristics as the reference model. This approach aims to facilitate the study of leaf springs without requiring much time for modeling and simulation, as is the case with discretization methods and finite element methods (FEMs). This method is used in [18] to replace the transversal leaf spring and the longitudinal leaf spring, which are placed in the front and rear suspension of the vehicle, respectively, with coil springs using ADAMS software, and it is validated by simulation in ADAMS and experimentally by comparing the new model’s results with the original model.

All the previously cited methods have been validated through experimental tests, and each has its advantages and disadvantages.

Generally, the experimental testing procedure used to validate these methods consists of placing a leaf spring on a device known as a “Static Stiffness Test stand” (SST) [32]. This device includes a hydraulic actuator that applies a vertical force to the center of the leaf spring, and the associated vertical displacement is recorded via the displacement sensor [33]. Then, the results of the chosen method are compared to those found through the experimental tests to validate it.

To conduct this experiment, it is necessary to first disassemble the leaf springs from the vehicle and mount them on a SST device. Then, the leaf spring characteristics are determined.

To avoid disassembling the vehicle’s leaf springs, which is time-consuming, and to eliminate the need for using an SST device to carry out the test, we proposed a method that involves conducting experimental tests on the leaf springs as they are installed in the vehicle. Therefore, the experimental test is not performed directly on the leaf springs, but on the entire suspension system, where a lifting device, scales, and displacement sensors are necessary to conduct this test. The advantage of this method is that it gives more accurate suspension system results under real operating conditions. However, this method requires prior knowledge of all the characteristics of the elements constituting the suspension system and may require experimental tests on certain elements if certain characteristics are not known. In addition, the use of scales allows for the indirect measurement of forces, which can increase the measurement error, unlike the use of the SST device which allows for the direct measurement of the force applied by the actuator.

This method can be used to validate the ADAMS model of the leaf spring. However, the model must include all suspension elements, such as the leaf springs, shock absorbers, and anti-roll bars, to reproduce the real system best. This can be configured in the ADAMS software under “Suspension Assembly” and by using the “Parallel Travel Wheel” (PWT) test.

After validating the suspension systems, the multi-body truck model can be created in ADAMS software. This multi-body model provides a good understanding of truck dynamics in different situations [34]. However, before this, verifying and validating the truck model experimentally is essential to ensure its dynamic accuracy.

The validation process begins with selecting appropriate scenarios that meet the engineering team’s objectives, allowing the analysis of the truck’s maximum parameters. Then, a comparison between the simulation and experimental results is conducted to verify the accuracy of the truck model. The validated model can be used to improve truck performance without costly experimental tests.

Some examples of studies aimed at validating the multi-body model of a vehicle, where different scenarios are used to compare the simulation results with experimental tests, are given in [5,35,36]. The study carried out in [5] aims to model the off-road vehicle Defender 110 in the ADAMS software and validate it using a bump test and a double lane change maneuver, where the results showed good agreement. Another study [35] involves modeling the heavy vehicle ISUZU FSR two-axle Single-Unit Truck in the IPG TruckMaker software. The truck’s lateral acceleration and velocity are the parameters used to verify the truck’s dynamic. The simulation results closely match the experimental data, confirming the validity of the multi-body model. The methodology for validating the model has been discussed in [36], and modeling of the 2017 International 4300 SBA 4 × 2 truck has been conducted using dSPACE ASM software. Three scenarios have been chosen to compare the simulation results with experiment data and thus validate the developed model, leading to the conclusion that the truck model is valid within a limited range.

This paper aims to model a full truck (4 × 4) and validate it experimentally, focusing on its lateral dynamics. The organization of this paper is as follows: Firstly, experimental determination of truck characteristics such as the mass, center of gravity coordinates, and tire vertical stiffness is achieved. These parameters are essential inputs for the full truck model in ADAMS software. Secondly, the truck’s front and rear suspension systems are modeled, including leaf springs in each suspension. Due to their modeling complexity, validation is achieved through a static experimental test. Finally, a full truck is modeled and validated through the dynamic experimental test, with a focus on lateral dynamics. The results show a good correlation between the simulated model and the real truck, concluding that the truck model is validated in terms of lateral dynamics.

2. Experimental Measurement of Truck Characteristics

2.1. Determination of Truck Mass and Center of Gravity Coordinates

The purpose of this experiment was to measure the vehicle mass of the truck and the position of its center of gravity in three directions. This experiment is divided into two parts.

2.1.1. Determination of Vehicle Mass and the Coordinates of the Center of Gravity in the Horizontal Plane (x_G, y_G)

This part of the experiment aimed to measure the mass of the truck M_t and identify the position of its center of gravity in the horizontal plane (x_G, y_G). For this, two measurement tests were conducted. The truck was placed on four scales in each test, one under each wheel, as shown in Figure 1. The mass readings obtained from each scale were recorded.

These values are essential for knowing the mass distribution in the front/rear axles and left/right side of the truck denoted as m_front/m_rear and m_left/m_right, respectively, which allow the determination of the precise coordinates of the center of gravity in the horizontal plane. Additionally, the wheelbase (L) and width (l) of the truck were measured.

Table 1. Mass recorded by each scale.

Scales Position		Front Left	Front Right	Rear Left	Rear Right	Vehicle Mass	Mass Average
Mass value (kg)	T1	2120	2160	1360	1530	7170	7205
	T2	2170	2210	1340	1520	7240

presents the vehicle mass values from each scale for the two tests.

By applying Newton’s second law and using the data given in Table 1, the coordinates of the center of gravity in the horizontal plane were calculated by Equations (1) and (2), where x_G coordinate determines the distance from the front axle, and y_G coordinate the distance from the left truck wheels:

$$x_G={\displaystyle \frac{m_{\mathrm{rear}}·L}{M_t}}$$

(1)

$$y_G={\displaystyle \frac{m_{\mathrm{right}}·l}{M_t}}$$

(2)

2.1.2. Determination of Vertical Coordinate of the Center of Gravity (z_G)

This experiment aimed to determine the position of the center of gravity in the vertical direction z_G or h_G. This consisted of lifting the truck from its front as represented in Figure 2. The lifting point was horizontally located at a distance x_l from the front axle along the x-axis and vertically at a distance h_l from the point of contact of the wheels with the ground along the z-axis.

Scales were placed under the rear wheels to measure the mass distribution at different angles β. The results of these measurements for each angle β are detailed in

Table 2. Mass measurements at various angles of inclination.

β		6°10′	7°20′	9°	11°30′	14°10′	16°30′	18°30′
Mass Value (kg)	Left	1970	1980	1980	1990	2010	2010	2020
	Right	2130	2130	2140	2150	2160	2160	2170

.

The angles of β = 16°30′ and β = 18°30′ were used for calculating the vertical coordinates of the center of gravity. The problem was the accuracy of the scales, which had a measurement error of +5 kg. This error considerably influenced the results. For example, when 5 kg was added to the mass measured at an angle of 16°30′ the resulting change in height was 20 mm. Conversely, at an angle of 9°, the same addition of mass resulted in a change in height of more than 40 mm, i.e., larger angle values gave more accurate results. The coordinate of the center of gravity in the vertical direction, as provided by Equation (3), was calculated by applying Newton’s second law.

$$z_G=\left(L-x_G\right)\mathrm{ctg}\left(\beta \right)-\left(1-{\displaystyle \frac{m_{\mathrm{rear}}}{M_t}}\right)\left(L+x_l\right)\mathrm{ctg}\left(\beta \right)+\left(1-{\displaystyle \frac{m_{\mathrm{rear}}}{M_t}}\right)h_l+r_s+{\mathsf{\Delta }z}_{\beta }$$

(3)

Here, r_s—static radius of the wheels; Δz_β—the difference between the sum of the rear leaf spring’s vertical displacement and the rear tire’s vertical displacement when the truck is lifted at an angle β and those when in a static state β = 0. This vertical displacement value can be obtained from the force–displacement curves of the rear leaf spring and the tire, which are explained in the following sections. The characteristics of the truck are presented in

Table 3. Truck characteristics.

Truck Mass Mt (kg)	Wheelbase L (mm)	Width l (mm)	The Center of Mass Coordinates (xG, yG, zG) (mm)
7205	3600	2051.5	(1436.57, 1056.35, 1089)

.

2.2. Measurement of Tire’s Vertical Stiffness

This experiment aimed to determine the vertical stiffness of the tire, which plays an important role in the truck. This test procedure was carried out on one of the truck’s wheels. All four wheels of the truck had the same designation 13R22.5 149/146J, and the characteristics of the tire are presented in

Table 4. Tire characteristics.

Parameters	The Correspondent Value
Tire inflation pressure P (kPa)	6.7 × 102
Nominal section Width SN (mm)	13 × 25.4 = 330.2
Section height H (mm)	290
Aspect ratio AR (%)	87
Rim diameter DR (mm)	22.5 × 25.4 = 571.5

.

The experiment began with measuring the tire inflation pressure P, which significantly influenced tire stiffness compared to other parameters. The test consisted of lifting the front of the truck and slowly lowering it while recording the mass values indicated on the scale and the displacement values provided by the HBM-LVDT sensor (HBK, Darmstadt, Germany), as shown in Figure 3a.

LVDT is the abbreviation of Linear Variable Differential Transformer, which is a position/displacement sensor with a measuring range of 0 to 200 mm.

A force–displacement curve was plotted from the experiment results, as shown in Figure 3b. It can be seen that the curve is linear, which made it possible to determine the vertical stiffness of the tire.

To verify the value of the vertical stiffness of the tire obtained experimentally, we compared it with the value obtained analytically, provided by [37]. This analytical expression, given in Equation (4), is based on experimental results obtained from several tires of different dimensions.

$$K_t=9.81·\left[0.00028·P\sqrt{\left(-0.004·AR+1.03\right)S_N·\left({\displaystyle \frac{S_N·AR}{50}}+D_R\right)}+3.45\right]$$

(4)

Table 5. Vertical stiffness comparison.

	Measured Value	Analytical Value	Error %
Tire Stiffness (N/mm)	845.55	934.89	9.56

shows the error between the measured and analytical values, which is less than 10%. The authors believe such a difference between the measured and analytical value is acceptable. Therefore, the measured value can be used for the tire characteristics. This value was then inserted into the “Property file” of the tire, which holds the pneumatic characteristics of the wheels. This “Property file” was then used to simulate the suspension system and the full truck model.

3. Modeling and Experimental Validation of Truck Suspension Systems

3.1. Description of Vehicle Suspension System

The truck has a dependent suspension system type in both axles, with identical structures containing a rigid axle connecting the two wheels, a shock absorber, a leaf spring, and an anti-roll bar. The difference between them lies in the type of leaf spring, the location of the elements relative to the axles, and the presence of a drag link and steering arm for lying front suspension with the steering system.

The leaf spring in the front suspension is multi-leaf with 12 graduated leaves. In contrast, the rear suspension comprises a dual-rate helper spring consisting of a main leaf spring and an auxiliary leaf. The main leaf spring comprises three taper leaves, while the auxiliary leaf contains a single taper leaf supplementing the main spring during specific load conditions.

Before proceeding with truck suspension modeling, it was necessary to create a kinematic diagram illustrating the connections between the elements, which is presented in Figure 4.

3.2. Modeling of the Truck Suspension System

The procedure for modeling a system in ADAMS begins with generating a “template” where all the necessary details are provided. Then, a “subsystem” is generated based on the template already created. Finally, the simulations are carried out in the subsystem assembly.

To conduct a simulation on the leaf spring to compare the force–displacement characteristic with the experimental tests, an assembly was carried out between the leaf spring subsystem and another subsystem, including all the suspension elements such as the axle, anti-roll bar, shock absorber, etc. The force-velocity characteristics of the dampers, which are nonlinear, are provided in engineering drawings and are provided in the property file in ADAMS. The anti-roll bars were modeled as a beam by providing their characteristics, such as the diameter and their location. For the bushing parameters, a property file available in the ADAMS library was used.

In the next step, a “Parallel Wheel Travel” analysis was conducted to determine the force–displacement curve of each leaf spring and to compare it with the results from experimental testing, where the force should be measured at the wheels, and the displacement should be taken at the same sensor position as in the experimental test. In this simulation, vertical displacement was imposed by defining the upper and lower limits of wheel center displacement, known in the software as “Bump travel” and “Rebound travel”, respectively, as shown in Figure 5. Given that leaf springs are subjected to compression forces during experimental tests, the “Rebound Travel” value was set to zero in the simulation, and the upper limit was set at 120 mm.

3.3. Experimental Testing of Truck Suspensions

To experimentally determine the force-displacement characteristic of the front leaf springs, which are multi-leaf types, two displacement sensors were placed on the master leaf, one on the right and one on the left, at a distance of l_sf = 150 mm from its center to measure the vertical displacement. Scales were also used to measure the mass and, consequently, the force applied to the springs, as shown in Figure 6. The test began by raising the front part of the truck until the scales showed a value of 0, indicating no contact between the wheels and the ground. We measured the initial curvature of the leaf spring as a reference value. Then, the truck gradually descended, and the mass values were displayed on the scales, while the corresponding displacements indicated by the sensor were recorded. The curvature of the leaf spring was determined as the difference between the values recorded at each moment and the reference value.

The sensor used was the WDS-500-P60-CR-P produced by the company “MICRO-EPSILON” (με) (Ortenburg, Germany). It is an industrial draw-wire sensor with a measuring range of 0 to 500 mm.

The same procedure was applied to the main rear leaf spring, which is a taper leaf spring type. However, instead of placing the displacement sensor directly onto the master leaf spring, it was positioned on the auxiliary leaf spring, at a distance of l_sr = 150 mm. During testing, the auxiliary leaf did not function, so it can be assumed that the recorded displacement corresponds to that of the main leaf spring. Additionally, only one displacement sensor was used on the left side of the suspension, as shown in Figure 7. We also took advantage of the front spring test to measure the vertical displacement of the rear spring and the mass values displayed when the front part of the truck was raised, using these data as additional measurements.

The force–displacement curves for both the front and rear suspensions, determined through experimental tests and simulations, are represented in Figure 8.

Figure 8 represents the force–displacement curves for the front suspension, both simulated and tested, which are closely aligned. The force–displacement curves for the rear suspension align well from 0 to 50 mm, but discrepancies increase beyond this range. This discrepancy can likely be attributed to the positioning of the displacement sensor on the auxiliary leaf spring, implying a small displacement imposed by the auxiliary leaf spring on the sensor. Despite this, the error between the values does not exceed 10%. Given these observations, we can assert that the models for both the front and rear leaf springs are validated.

The leaf spring model must have the same curvature as the installed leaf spring in the truck and, for this, it is necessary to know the initial curvature of each leaf spring before their assembling in the truck. To avoid disassembling this leaf spring, which is time-consuming, the leaf spring was modeled as installed in the truck by adding the static force (preload). This force needed to be included in the ADAMS model of each leaf spring and a static simulation needed to be conducted prior to dynamic simulation of the full truck. If the mentioned static force is neglected, the leaf spring model changes its curvature due to the truck’s weight, thus making the dynamic simulation invalid. By adding these forces, the curvatures of the leaf springs remained unchanged, as their effect compensated for the truck’s weight.

Considering the truck’s symmetry relative to its roll axis (x-axis), the preload for each leaf spring, whether front or rear, is the same on both the right and left sides, as detailed in Equations (5) and (6), respectively.

$$F_{fpl}={ F}_{fpr}={\displaystyle \frac{M_t · g · (L-x_G)}{2 · L}}$$

(5)

$$F_{rpl}={ F}_{rpr}={\displaystyle \frac{M_t · g · x_G}{2 · L}}$$

(6)

Here, F_fpl and F_fpr—preloads applied to the front left/right leaf springs, respectively; F_rpl and F_rpr—preloads applied to the rear left/right leaf springs, respectively; M_t—truck mass; g—gravity acceleration, which is equal to 9.81 m/s²; x_G—coordinate of the center of mass along the x-axis; and L—truck wheelbase.

4. Modeling and Experimental Validation Dynamics Characteristics of the Truck

4.1. Truck Modeling

After modeling the front and rear suspensions and validating the leaf springs as described in the previous section, the next step involved complete truck modeling. This included the integration of the steering system, the front and rear wheels, the chassis, the powertrain, and the driveline. The ADAMS software provides template models in its library, which could be used in the truck model with some modifications according to the truck’s requirements. This involved adjusting the hardpoints coordinates and the element properties, such as the wheels, to ensure that the characteristics of the created model are identical to those of the real model.

The chassis model is represented as a rigid body with some details such as mass and moment of inertia. For the front and rear wheels, the Pacejka 2002 tire model was used, which is based on the Magic Formula, developed by Pacejka [38], a widely used model in tire dynamics. This model uses semi-empirical formulas to define the wheel–ground contact [39].

In the property file of the Pacejka 2002 tire model, we modified the characteristics of the tire, including the dimensions and vertical stiffness measured previously. The other parameters were kept unchanged.

In the ADAMS library, there are models of the powertrain, driveline, and steering system that are similar to those used in the truck. For the powertrain, its mass and transmission ratios were modified. For the driveline, the front and rear transmission ratios were adjusted.

Some modifications were made to ensure the same steering ratio as the real systems for the steering system. First, we measured the wheel rotation angles (steer angle) when the steering wheel was turned at specific angles and plotted the steering wheel angle versus the steer angle. Then, we assembled the front suspension presented in Figure 5a with the steering subsystems. The assembly is represented in Figure 9a. The measured maximum and minimum steering wheel angles were 650° and −650°, respectively, and these data were added to the model.

We adjusted the gear ratio within the steering model and conducted simulations. We compared the curve of the steering wheel rotation angle as a function of the wheel angles (steer angle) obtained by simulation with the one obtained by measurements. This iterative process was repeated until the acceptable alignment between simulated and measured curves was achieved, as illustrated in Figure 9b.

The steering ratio has a crucial role in truck validation, especially in scenarios such as open-loop steering events in ADAMS, including single-lane changes or during free-driving scenarios.

The truck model can be assembled using the ten subsystems of the vehicle mentioned previously (front suspension, front leaf spring, front tire, rear suspension, rear leaf spring, rear tire, steering system, the chassis, powertrain, and driveline) and is presented in Figure 10b.

Before performing the simulation of the truck, it was essential to analyze its characteristics, such as the mass, center of gravity coordinates, and moment of inertia. For the truck mass and the coordinates of its center mass, we used the values presented in Table 3. However, the moment of inertia for the full truck was difficult to find experimentally, so we created a 3D truck model, as shown in Figure 10a, to determine the approximate values of I_xx, I_yy, and I_zz.

From there, we proceeded to analyze the dynamics of the truck model and validate it through experimental tests.

4.2. Dynamic Validation of the Truck Model

For the experimental test, a scenario repeated double-lane change was chosen to verify the lateral dynamics of the truck model, where the truck follows a defined trajectory, as shown in Figure 11.

These experiments were carried out at the Military Academy in Belgrade on a dry asphalt road. Two tests were performed. Sensors were essential for signal acquisition. Tree types of sensors were used to extract dynamic information and these were the following:

• A Potentiometer sensor is placed on the truck’s steering wheel to record the steering wheel rotation angle applied by the driver as a function of time during the maneuver.
• SST 810 dynamic inclinometer (Vigor Technology, Shanghai, China) is placed in the truck’s center of gravity and is used to measure the angle and acceleration.
• Kistler Correvit S-350 sensors (Winterthur, Switzerland) are used for the direct, slip-free measurement of a vehicle’s longitudinal and transverse dynamics, placed on the left side of the truck to measure the truck’s longitudinal and lateral velocities.

For data acquisition, an HBM Quantum MX840B universal amplifier was used together with HBM Catman easy 5.4.2.11 software for measurement and result analyses (HBK, Darmstadt, Germany).

The entities measured by each sensor are presented in

Table 6. Description and position of sensor used for experimental test.

Sensor Name	Measured Entity	Position on Truck
Potentiometer	Steering wheel angle	Steering wheel
SST 810 dynamic inclinometer	Roll angle, lateral acceleration roll acceleration, and yaw acceleration	Truck center of gravity
Kistler S-350 non-contact optical sensor	Truck velocity, lateral velocity	The left side of the truck

and their positions on the truck are shown in Figure 12.

Before the test started, it was necessary to calibrate the sensors. The truck ha to be driven in a straight line to reach the desired speed, and the driver had to maintain this speed during the test. The data from the steering sensor and the average truck’s velocity were an input for the simulation. Two event files were created in the ADAMS software to control the steering wheel of the truck model, and to impose a constant truck velocity. These files included the curves of the steering wheel angle as a function of time, obtained from the experiments, as shown in Figure 13, and the imposed truck velocities, which were 47 km/h and 45 km/h for test 1 and test 2, respectively.

The results obtained from the experimental test and the simulation for test 1 and test 2 are presented in Figure 14 and Figure 15, respectively, where the truck velocity (v), lateral velocity (v_y), yaw acceleration (α_z), roll acceleration (α_x), roll angle (θ_x), and lateral acceleration (a_y), are plotted. From these figures, it can be observed that the curves obtained from the experiment and the simulation are in phase. This enables better validation than the one presented in [5]. In the case of a double lane change maneuver [5], the simulation results demonstrate a phase shift in time compared to the experimental ones, which is due to the inability to control the steering wheel rotation during the simulation in order to reproduce the same maneuver performed by the driver during the experiment. On the other hand, validation of the model presented here was carried out by comparing five parameters, which provided a higher degree of model validity compared to [35], where only two parameters were used for the validation process.

From the comparison of the Root Mean Square (RMS) of each parameter between the simulation and the experiment, as detailed in

Table 7. RMS value comparison for test 1.

Parameter	ay (g)	vy (km/h)	θx (deg)	αz (deg/s2)	αx (deg/s2)
Experiment	0.18	0.22	1.15	9.21	3.87
Simulation	0.17	0.21	1.08	8.91	4.23
Deviation (%)	7.74	5.40	6.51	3.33	6.86

and

Table 8. RMS value comparison for test 2.

Parameter	ay (g)	vy (km/h)	θx (deg)	αz (deg/s2)	αx (deg/s2)
Experiment	0.21	0.26	1.26	10.43	4.90
Simulation	0.20	0.24	1.15	10.55	5.18
Deviation (%)	7.59	8.65	8.37	1.09	5.44

, the deviation is within the acceptable range, and the maximum deviation does not exceed 10%. The deviation between the simulated model and experimental test due to several parameters such as the truck model is not ideal due to their structural complexity, the initial conditions during the experiment, road conditions, and the sensor errors. Based on the lateral dynamics results, the multi-body model can be considered validated. This validation allows us to proceed with the complete validation in future works focusing on the vertical dynamic and further suspension optimization to gain better off-road characteristics of the truck.

5. Conclusions

This paper addressed the modeling and validation of an off-road truck (4 × 4) using commercial software for multi-body dynamics, ADAMS 2020. The modeling process involved the determination of certain truck characteristics important for later simulations: mass, center of gravity coordinates, and tire vertical stiffness.

In the next step, validation of the leaf spring model developed for the multi-body software ADAMS was performed experimentally by considering the entire suspension system to extract the characteristic force–displacement dependence of the leaf spring. The experimental results were compared to those obtained in the simulation by using the “Parallel Wheel Travel” test in ADAMS. A good agreement between the results was observed, thus validating the leaf spring models.

Further validation was achieved by comparing the lateral dynamics of the model with the real truck dynamics determined experimentally in a repeated double-lane change scenario conducted in two tests. The simulated values deviated from the experimental values by 10%, which the authors consider acceptable.

Based on these results, it is concluded that the truck model developed in ADAMS delivers acceptable results in simulations. This study provides a solid basis for the next step, which would be the optimization of the suspension system in order to achieve better characteristics of the truck in off-road conditions.