Under complex conditions, the vertical, lateral and longitudinal dynamics of vehicles have obvious coupling and interaction. This paper aims to provide a suitable driver cab and a vehicle model for the study of vehicle coupling dynamic performance. In modeling the cab and body kinetic equation, two shock absorbers are considered in the front axle suspension system. In addition, the vertical, roll and pitch motion of the diver cab, vehicle body, the vertical and roll behavior of three wheel axles, the pitch angles of the left and right balancing pole on rear suspension, and roll angle the of each tire are considered. Finally, based on the above coupled motion characteristics, a diver cab and a vehicle model for three-axles heavy-duty vehicle with 26 degrees of freedom (DOF) are proposed.
Citation: Yongming Li, Shou Ma, Kunting Yu, Xingli Guo. Vehicle kinematic and dynamic modeling for three-axles heavy duty vehicle[J]. Mathematical Modelling and Control, 2022, 2(4): 176-184. doi: 10.3934/mmc.2022018
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Abstract
Under complex conditions, the vertical, lateral and longitudinal dynamics of vehicles have obvious coupling and interaction. This paper aims to provide a suitable driver cab and a vehicle model for the study of vehicle coupling dynamic performance. In modeling the cab and body kinetic equation, two shock absorbers are considered in the front axle suspension system. In addition, the vertical, roll and pitch motion of the diver cab, vehicle body, the vertical and roll behavior of three wheel axles, the pitch angles of the left and right balancing pole on rear suspension, and roll angle the of each tire are considered. Finally, based on the above coupled motion characteristics, a diver cab and a vehicle model for three-axles heavy-duty vehicle with 26 degrees of freedom (DOF) are proposed.
1.
Introduction
During the vehicle movement, the performance of the vehicle is affected by various vehicle structures and functions, such as power steering system, suspension system, braking system, etc. Moreover, in complex and high-speed environment, the vehicle's vertical, roll and pitch displacements contain a strong coupling relationship. Therefore, considering the motion and coupling characteristics of vehicle structures is meaningful to investigate vehicle dynamics. On the basis of geometric structure parameters of vehicle system and the nonlinear characteristics of shock absorber and leaf spring, the authors in [1] establish a nonlinear dynamic model for heavy vehicle. The correctness of the dynamic model is verified by testing the vertical acceleration data of the driver's seat, front wheel, middle wheel and rear wheel. To investigate the longitudinal driving behaviors of vehicle dynamics in the platoons, by taking the acceleration capability of heavy-duty vehicle into account, numerous heavy-duty vehicle platoon models are proposed [2,3,4,5,6,7,8,9,10]. Furthermore, by considering the lateral and longitudinal displacement characteristics of the vehicle, [11] represents a 2 DOF model of the vehicle and two diver cab models with time delays. In order to further describe the vehicle dynamics characteristics, in accordance with the two driver cab models in [11]. [15] and [16] further investigate the nonlinear lateral dynamics of a 2 DOF vehicle model. Based on longitudinal vehicle dynamics and by analyzing the dynamic of engine, torque converter, tire and capacitor pack, the authors of [17] present a dynamic model for a heavy-duty vehicle.
On the other side of research, the above mentioned vehicle models are mostly used to evaluate vehicle lateral and longitudinal dynamics characteristics, the influence of vehicle lateral and yaw dynamics characteristics are not considered enough. In practice, the tires not only provide horizontal and vertical forces to the vehicle, but also give vertical forces to the suspension system, especially in complex driving situations such as lane changes, cornering, or obstacle avoidance. In these cases, the vehicle's vertical, roll, and pitch dynamics are clearly coupled with lateral and yaw motion. Due to large inertia, high center of gravity and high roll center, heavy vehicles have poor stability when entering a turn or lane change, and the three-way coupling effect is large. Therefore, it is necessary to establish a three dimensional coupled vehicle model and study the influence of steering process on vehicle dynamics. More recently, more and more works focus on the coupling property of the vehicle. To reflect the steering influence on the overall response of the vehicle, [18] designs a novel 4 DOF hydraulic power steering (HPS) system. Simultaneously, [18] develops a 24 DOF model by taking the HPS system, the steering hand wheel angle, rack displacement, and hand wheel angles into account. According to the nonlinear characteristics of suspension damping and tire stiffness, [19] establishes a nonlinear three-way coupled lumped parameter model, and an improved nonlinear delay preview driver model was proposed based on [11], which was connected with the TCLP model to form a driver-vehicle closed-loop system. [20] establishes a complete vehicle model of a heavy truck, which not only investigates the nonlinear characteristics of suspension damping and tire stiffness, but also contains a modified preview driver model with nonlinear time delays to calculate the right front wheel steering angle for driving the vehicle along the desired route. In this paper, the kinematics and dynamics equations of cab and body are established by analyzing the three-way coupling effect of cab and body, as well as the dynamic characteristics of tire and suspension. Firstly, the dynamic relation of the tyre with deflection angle is introduced. Secondly, the coupling dynamics equation of cab was established by analyzing the three-way coupling effect of cab. Then, considering the dynamic characteristics of the vehicle suspension, the three-way coupling dynamic equation of the vehicle body is established. Finally, the kinematic and dynamic equations of cab and body are established based on the dynamic characteristics of tire and suspension and the Euler rotation theorem.
2.
Nomenclature
Table 1.
The symbols of the heavy-duty vehicle.
Definition
Symbol
forward traction (lateral traction) of the tire
steering angle of the wheel
steering angular speed of the front axle tires
transverse (longitudinal) component of tire along the coordinate system
suspension force, damping coefficient and spring constant of the spring between cab and body
and
vertical displacement of the cab (body)
pitching angle of the cab (body)
roll displacements of the cab (body)
longitudinal distance between origin of coordinates and cab rear (front) spring
the distance between the origin of coordinates and
the angle between the origin of coordinate and the sprung mass bar center of suspension
transverse distance betweencab front spring and rear spring
resultant force of the cab inthe direction of axes , and
and
resultant moment of the cab in the direction of axes , and
and
total mass of the vehicle, cab and body
and
velocity vectors of the cab in the coordinate system and
and , and
roll angle rate, pitch angle rate and yaw angle rate of the coordinate system and
and , and
vertical and transverse distance from the origin of and to the center of gravity of cab
and , and
moment of inertia of a vehicle about axle and
and and
moment of inertia of the cab about the axis and ( and )
and and
integral of the product of the and ( and ) deviation of an area element in a vehicle
compression displacement of the spring between cab and body
cab center of gravity (body center of gravity) to cab front and rear spring transverse distance
and and
transverse (longitudinal) distance of cab center of gravity to set 1, set 2 and set 3 tires
and and
distance between the front axle and the rear axle in a suspension system
resultant force of the cab in the direction of axes and
and
suspension force, damping coefficient and springconstant of the spring between the front axle or rear axle and the body of the suspension system
and
longitudinal transverse distance from the center of gravity of suspension system to the front axis of the suspension system
pitching angle of the left and right balance bars of the suspension system
and
vertical displacement of the axle
angle of inclination of the wheel shaft
lateral distance between the left and right springs of the suspension system
and
damping forces of front suspension left and right springs
and
resultant moment of the cab in the direction of axes and
and
transverse (longitudinal) distance of cab center of gravity to set 1, set 2 and set 3 bearing spring
and and
transverse (longitudinal) distance of body center of gravity to set 1, set 2 and set 3 tires
and and
transverse distance (longitudinal distance) between the body center of gravity and the cab front (rear) spring
longitudinal distance from the center of gravity of the suspension system to the center of gravity of the rear axle of the suspension
In this paper, the kinematic characteristics of a heavy-duty vehicle are considered to construct a 26 DOF vehicle body and a cab model. As shown in Figs. 1-4, the considered heavy vehicle has one front axle and two rear axles, which is called a three-axial vehicle. The degrees of freedom are vertical, roll and pitch displacements of the diver cab, vehicle body, the vertical and roll motion of three wheel axles, the pitch angles of the left and right balancing pole on rear suspension, and roll angle the of each tire. To further study the coupling property with each part, the vertical, roll and pitch motion of cab and body is modeled independently. Before introducing the related coordinate systems, the Euler's laws of motion is firstly given.
Lemma 3.1.Observed from an inertial reference frame, the force applied to a rigid body is equal to the product of the mass of the rigid body and the acceleration of the center of mass, i.e.
where is the resultant external force of the rigid body, is the rigid body mass, and is the acceleration of center of mass.
Lemma 3.2.. The fixed point (for example, the origin) of an inertial reference frame is set as the reference point. The net external moment applied to the rigid body is equal to the time rate of change of the angular momentum, i.e.
where is the is the external torque at point , is the angular momentum at point .
3.1. Establishing the related coordinate systems
To analyze the motion of heavy-duty vehicle, the corresponding coordinate frames are elaborated to describe the movement of the vehicle and indicated in Fig. 1. The moving coordinate frame is fixed to the vehicle's body and is called the body-fixed reference frame. The second coordinate frame is fixed to the cab and is called the cab-fixed reference frame. The third coordinate frame is fixed to the earth and is called the earth-fixed reference frame. The last coordinate frame is fixed to the , tire and is called the tire-fixed reference frame. In this paper, we assume that the body axes the tire axes and the cab axes of heavy vehicle coincide with the principal axes of inertia, which are usually defined as:
-longitudinal axis (directed from aft of the body/tire/cab to front).
-transverse axis (directed to right side of body/tire/cab).
-normal axis (directed from top to bottom).
3.2. Analysis of tire motion characteristics
To extract the kinetic model for the considered three-axial heavy-duty vehicle, the coordinate frame is designed for each tire, the corresponding schematic diagram is shown in Fig. 2. By taking the yaw angle into account, the forces produced by the engine are transformed into the forward traction and longitudinal traction on the suspension of heavy-duty vehicle. Based on the coordinate frame and Fig. 2, the forward and lateral traction of each tire can be expressed as
(3.1)
3.3. Analysis of cab motion characteristics
In this subsection, the vertical, roll and pitch motion of the cab are considered to further accurately reflect the performance of spring suspension force between the cab and the body in the actual scenario. In accordance with the coordinate frame and Figs. 3-4, the spring force between the cab and body can be given as
(3.2)
(3.3)
(3.4)
(3.5)
Furthermore, by viewing the diver cab as a rigid body and employing the Lemmas 3.1 and 3.2, the kinematical equation of diver cab model is
(3.6)
(3.7)
(3.8)
(3.9)
(3.10)
(3.11)
When the vehicle is moving, the spring will produce spring force whether it is in a state of compression or tension. However, the direction of the force is opposite, so this paper considers the sign function and the direction of the spring displacement to determine the direction of the spring force. In order to obtain the dynamic force equation of the vehicle cab and body, we assume that the cab and body mass are evenly distributed, that is, the transverse distance between the cab and body center of gravity from the left and right tires is the same. By considering the definition of resultant force and resultant moment, the kinetic formula of longitudinal, transverse and vertical forces acting on the cab, as well as the yaw, pitch and roll moments is described as
(3.12)
(3.13)
(3.14)
(3.15)
(3.16)
(3.17)
where denotes the symbolic function, and .
3.4. Analysis of suspension
For the considered heavy duty vehicle, two hydraulic dampers are fixed to the left and right front suspensions, and the balanced suspension does not have any shock absorbers. Thus, to represent the force situation of leaf spring in suspension system, the damping force of the two hydraulic dampers is considered for the front axle. The kinetic equation is given by
(3.18)
(3.19)
(3.20)
(3.21)
(3.22)
(3.23)
3.5. Analysis of body motion characteristics
This is analogous to the diver cab part, taking the body as a rigid body and according to the Lemmas 3.1 and 3.2, the force equation of body model can be expressed as
(3.24)
(3.25)
(3.26)
(3.27)
(3.28)
(3.29)
Recalling the problem of spring force direction and the assumption of uniform distribution of body mass, the longitudinal, transverse and vertical forces acting on the body are:
(3.30)
(3.31)
(3.32)
(3.33)
(3.34)
(3.35)
3.6. Diver cab and body nonlinear model
In this final subsection, a common type of diver cab and body model for heavy-duty vehicle is proposed.
In accordance with 3.1-3.17 and invoking the Euler rotation theorem, the dynamic and kinetic equation of driver cab are designed as
(3.36)
where
, , ,
represents the tangent function. The expansion equation of the matrix is
(3.37)
(3.38)
(3.39)
(3.40)
(3.41)
(3.42)
According to kinetic equations (18)-(42) and employing the Euler rotation theorem, the dynamic and kinetic equation of body are designed as
(3.43)
where , , ,
The expansion equation of the matrix is
(3.44)
(3.45)
(3.46)
(3.47)
(3.48)
(3.49)
4.
Conclusions
In complex working conditions, there is a coupling relationship of the vertical, lateral and longitudinal dynamics of vehicles. By considering the kinetic character of the vertical, roll and pitch motion of the diver cab, vehicle body, the vertical and roll behavior of three wheel axles, the pitch angles of the left and right balancing pole on rear suspension, and roll angle the of each tire. In this paper, a common model of three-axles heavy-duty vehicle with 26 DOF have been proposed to extrude the kinetic characterization diver cab and vehicle body.
Acknowledgment
This work was supported by the National Natural Science Foundation of China under Grants U22A2043 and 62173172.
Conflict of interest
The author declares that there is no conflicts of interest in this paper.
References
[1]
Y. Lu, S. Yang, S. Li and L. Chen, Numerical and experimental investigation on stochastic dynamic load of a heavy duty vehicle, Appl. Math. Model., 34 (2010), 2698–2710. https://doi.org/10.1016/j.apm.2009.12.006 doi: 10.1016/j.apm.2009.12.006
[2]
Q. Deng, A general simulation framework for modeling and analysis of heavy-duty vehicle platooning, IEEE T. Intell. Transp., 17 (2016), 3252–3262. 10.1109/TITS.2016.2548502
[3]
S. Singh, S. Santhakumar, Modeling traffic parameters accounting for platoon characteristics on multilane highways, Transp. Dev. Econ., 69 (2022), 84–96. https://doi.org/10.1007/s40890-022-00166-3 doi: 10.1007/s40890-022-00166-3
[4]
M. Sala, F. Soriguera, Macroscopic modeling of connected autonomous vehicle platoons under mixed traffic conditions, Transportation Research Procedia, 47 (2020), 163–170. https://doi.org/10.1016/j.trb.2021.03.010 doi: 10.1016/j.trb.2021.03.010
[5]
Z. Ju, H. Zhang, Y. Tan, Distributed stochastic model predictive control for heterogeneous vehicle platoons subject to modeling uncertainties, IEEE Intel. Transp. Sy., 14 (2022), 25–40.
[6]
R. Merco, F. Ferrante, P. Pisu, A hybrid controller for DOS-resilient string-stable vehicle platoons, IEEE T. Intel. Transp., 22 (2021), 1697–1707. https://doi.org/10.1109/TITS.2020.29758151 doi: 10.1109/TITS.2020.29758151
[7]
L. Xu, W. Zhuang, G. Yin, C. Bian, H. Wu, Modeling and robust control of heterogeneous vehicle platoons on curved roads subject to disturbances and delays, IEEE T. Veh. Technol., 68 (2019), 11551–11564. https://doi.org/10.1109/TVT.2019.2941396 doi: 10.1109/TVT.2019.2941396
[8]
Y. Zheng, S. Li, K. Li, F. Borrelli, J. Hedrick, Distributed model predictive control for heterogeneous vehicle platoons under unidirectional topologies, IEEE T. Contr. Syst. T., 25(3) (2017), 899–910. https://doi.org/10.1109/TCST.2016.2594588 doi: 10.1109/TCST.2016.2594588
[9]
J. Zhan, Z. Ma, L. Zhang, Data-driven modeling and distributed predictive control of mixed vehicle platoons, IEEE Transactions on Intelligent Vehicles, (2022). https://doi.org/10.1109/TIV.2022.3168591 doi: 10.1109/TIV.2022.3168591
[10]
J. Guo, H. Guo, J. Liu, D. Cao, H. Chen, Distributed data-driven predictive control for hybrid connected vehicle platoons with guaranteed robustness and string stability, IEEE Internet of Things Journal, 9 (2022), 16308–16321.
[11]
T. Legouis, A. Laneville, P. Bourassa, G. Payre, Characterization of dynamic vehicle stability using two models of the human pilot behaviour, Vehicle Syst. Dyn., 15 (1986), 1–18. https://doi.org/10.1080/00423118608968837 doi: 10.1080/00423118608968837
[12]
M. Jeleva, B. Villeneuve N. Muralidhar, M. Himabindu, R. Ravikrishna, Modeling of a hybrid electric heavy duty vehicle to assess energy recovery using a thermoelectric generator, Energy, 148 (2018), 1046–1059. https://doi.org/10.1016/j.energy.2018.02.023 doi: 10.1016/j.energy.2018.02.023
[13]
E. Berton, N. Bouaanani, C. Lamarche, N. Roy, Finite element modeling of the impact of heavy vehicles on highway and pedestrian bridge decks, Procedia Engineering, 199 (2017), 2451–2456. https://doi.org/10.1016/j.proeng.2017.09.383 doi: 10.1016/j.proeng.2017.09.383
[14]
Y. Yang, H. Wang, G. Xia, Modeling and simulation analysis of wet multi-disk service braking system for heavy vehicles, IEEE Access, 8 (2020), 150059–150071. https://doi.org/10.1109/access.2020.3016663 doi: 10.1109/access.2020.3016663
[15]
Z. Liu, G. Payre, P. Bourassa, Nonlinear oscillations and chaotic motions in a road vehicle system with driver steering control, Nonlinear Dynamics, 9 (1996), 281–304. https://doi.org/10.1007/bf01833746 doi: 10.1007/bf01833746
[16]
Z. Liu, G. Payre, P. Bourassa, Stability and oscillations in a time-delayed vehicle system with driver control, Nonlinear Dynamics, 35 (2004), 159–173. https://doi.org/10.1023/b:nody.0000021080.06727.f8 doi: 10.1023/b:nody.0000021080.06727.f8
[17]
L. Serrao, C. Hubert, G. Rizzoni, Dynamic modeling of heavy-duty hybrid electric vehicles, ASME International Mechanical Engineering Congress and Exposition, 43106 (2007), 121–128. https://doi.org/10.1115/imece2007-41923 doi: 10.1115/imece2007-41923
[18]
H. Ahmadi Jeyed, A. Ghaffari, Modeling and performance evaluation of a heavy-duty vehicle based on the hydraulic power steering system, Simulation, 96 (2020), 297–311. https://doi.org/10.1177/0037549719866503 doi: 10.1177/0037549719866503
[19]
S. Li, J. Ren, Driver steering control and full vehicle dynamics study based on a nonlinear three-directional coupled heavy-duty vehicle model, Math. Probl. Eng., 22 (2014), 1–16. https://doi.org/10.1155/2014/352374 doi: 10.1155/2014/352374
[20]
S. Li, J. Ren, Investigation on three-directional dynamic interaction between a heavy-duty vehicle and a curved bridge, Adv. Struct. Eng., 21 (2018), 721–738. https://doi.org/10.1177/1369433217729516 doi: 10.1177/1369433217729516
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