Vital functions, such as blood pressure, are regulated within a framework of neurovisceral integration in which various factors are involved under normal conditions maintaining a delicate balance. Imbalance of any of these factors can lead to various pathologies. Blood pressure control is the result of the balanced action of central and peripheral factors that increase or decrease. Special attention for blood pressure control was put on the neurovisceral interaction between Angiotensin II and the enzymes that regulate its activity as well as on nitric oxide and dopamine. Several studies have shown that such interaction is asymmetrically organized. These studies suggest that the neuronal activity related to the production of nitric oxide in plasma is also lateralized and, consequently, changes in plasma nitric oxide influence neuronal function. This observation provides a new aspect revealing the complexity of the blood pressure regulation and, undoubtedly, makes such study more motivating as it may affect the approach for treatment.
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
3.
Model building
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 $ F^e $ is the resultant external force of the rigid body, $ m $ is the rigid body mass, and $ a_c $ is the acceleration of center of mass.
Lemma 3.2. . The fixed point $ O $ (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 $ M_O $ is the is the external torque at point $ O $, $ L_O $is the angular momentum at point $ O $.
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 $ \{B\} $ is fixed to the vehicle's body and is called the body-fixed reference frame. The second coordinate frame $ \{C\} $ is fixed to the cab and is called the cab-fixed reference frame. The third coordinate frame $ \{E\} $ is fixed to the earth and is called the earth-fixed reference frame. The last coordinate frame $ \{T_i\} $ is fixed to the $ i-th $, $ (i = 1, 2, 3, 4, 5, 6) $ tire and is called the tire-fixed reference frame. In this paper, we assume that the body axes $ X_B, Y_B, Z_B, $ the tire axes $ X_{T_i}, Y_{T_i}, $ and the cab axes $ X_C, Y_C, Z_C, $ of heavy vehicle coincide with the principal axes of inertia, which are usually defined as:
$ \bullet $ $ X_B/X_{T_i}/X_C $ -longitudinal axis (directed from aft of the body/tire/cab to front).
$ \bullet $ $ Y_B/Y_{T_i}/Y_C $ -transverse axis (directed to right side of body/tire/cab).
$ \bullet $ $ Z_B/ Z_C $ -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 $ T_{i} $ 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
3.3. 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 $ \{C\} $ and Figs. 3-4, the spring force between the cab and body can be given as
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
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
where $ sign(\cdot) $ denotes the symbolic function, $ c(\cdot) = cos(\cdot) $ and $ s(\cdot) = sin(\cdot) $.
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
3.5. 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
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:
3.6. 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
where $ \eta_c = [x_c, y_c, z_c, \phi_c, \varphi_c, \delta_1]^T, $
$ J_c(\eta_c) = \left[J1(ηc)03×303×3J2(ηc)
\right], $
$ J_2(\eta_c) = \left[ 1s(ϕc)t(φc)c(ϕc)t(φc)0s(ϕc)−s(ϕc)0s(ϕc)/c(φc)c(ϕc)/c(φc)
\right], $
$ G_c(\eta_c, \upsilon_c) = \left[J3(ηc)03×2
\right], $
$ J_3(\eta_c) = \left[ c(δ1)c(φc)mt−s(δ1)c(ϕc)+s(ϕc)s(φc)c(δ1)mts(δ1)c(φc)mt+c(δ1)c(ϕc)+s(ϕc)s(φc)s(δ1)mt−s(φc)mss(ϕc)c(φc)ms
\right], $
$ F_c(\upsilon_c) = [F_{c1}(\upsilon_c), F_{c2}(\upsilon_c), F_{c3}(\upsilon_c), F_{c4}(\upsilon_c), F_{c5}(\upsilon_c), F_{c6}(\upsilon_c)]^T, $
$ J_1(\eta_c) = \left[c(δ1)c(φc)−s(δ1)c(ϕc)+s(ϕc)s(φc)c(δ1)s(δ1)c(φc)c(δ1)c(ϕc)+s(ϕc)s(φc)s(δ1)−s(φc)s(ϕc)c(φc)
\right. $
$ \upsilon_c = [u_c, v_c, w_c, p_c, q_c, r_c]^T $, $ F^*_{Xi} = \sum^6_{i = 1}F_{Xi} $, $ F^*_{Yi} = \sum^6_{i = 1}F_{Yi} $,
$ u_c = \left[05×1((FX2−FX1)l9+(FX4−FX3)l10+(FX5−FX6)l11Izzc+(FY1+FY2)l12−(FY3+FY4)l13−(FY5+FY6)l14))
\right], $ $ t(\cdot) $ represents the tangent function. The expansion equation of the matrix $ F_c(\upsilon_c) $ is
According to kinetic equations (18)-(42) and employing the Euler rotation theorem, the dynamic and kinetic equation of body are designed as
where $ \eta_b = [x_b, y_b, z_, b\phi_b, \varphi_b, \delta_1]^T, $ $ J_b(\eta_b) = \left[J1(ηb)03×303×3J2(ηb)
\right], $ $ G_b(\eta_b, \upsilon_b) = \left[
J3(ηb)03×2
\right], $ $ \upsilon_b = [u_b, v_b, w_b, p_b, q_b, r_b]^T $, $ F^*_{Yi} = \sum^6_{i = 1}F_{Xi} $, $ F_b(\upsilon_b) = [F_{b1}(\upsilon_b), F_{b2}(\upsilon_b), F_{b3}(\upsilon_b), F_{b4}(\upsilon_b), F_{b5}(\upsilon_b), F_{b6}(\upsilon_b)]^T, $ $ J_2(\eta_b) = \left[
1s(ϕb)t(φb)c(ϕb)t(φb)0s(ϕb)−s(ϕb)0s(ϕb)/c(φb)c(ϕb)/c(φb)
\right], $ $ F^*_{Xi} = \sum^6_{i = 1}F_{Xi} $, $ J_3(\eta_b) = \left[
c(δ1)c(φb)mt−s(δ1)c(ϕb)+s(ϕb)s(φb)c(δ1)mts(δ1)c(φb)mt+c(δ1)c(ϕb)+s(ϕb)s(φb)s(δ1)mt−s(φb)mbs(ϕb)c(φb)mb
\right], $ $ u_b = \left[
05×1((FX2−FX1)l25+(FX4−FX3)l26+(FX5−FX6)l27Izzb+(FY1+FY2)l28−(FY3+FY4)l29−(FY5+FY6)l30)
\right], $ $ J_1(\eta_b) = \left[
c(δ1)c(φb)−s(δ1)c(ϕcb)+s(ϕb)s(φb)c(δ1)s(δ1)c(φb)c(δ1)c(ϕb)+s(ϕb)s(φb)s(δ1)−s(φb)s(ϕb)c(φb)
\right. $
The expansion equation of the matrix $ {F_b(\upsilon_b)} $ is
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.