Efficiency and stability of heavy vehicle’s control air suspensions on improving road friendliness and comfort level

Abstract

To study the working performance in heavy vehicles using the control air suspension system (CASS), a vehicle’s three-dimensional dynamic model was established and controlled by using the SIMULINK model and fuzzy control built into the software of MATLAB. Two indexes of road friendliness and comfort level were used for evaluating CASS’s stability and efficiency under all heavy vehicle’s different working cases. The study showed that all simulation cases including the moving speeds, load conditions, and road excitations of the heavy vehicle greatly affect road friendliness and comfort level. Especially at conditions of low speed from 7.5 to 12.5 m/s, vehicle’s half load, and random roads of ISO-D and ISO-E. Thus, to improve these two indexes, heavy vehicles should move at high speeds from 20 to 25 m/s and use full load. Besides, with the CASS applied, the working performance of the heavy vehicle was overall superior to the vehicle’s passive air suspensions in improving road friendliness and comfort level. Therefore, the CASS should be applied to heavy vehicles to enhance their working performance.

Keywords

Heavy vehicle mathematical model control air suspensions fuzzy’s optimal control road friendliness comfort level

Introduction

With heavy vehicles, ensuring road friendliness and comfort level (RF-CL) of vehicles are two important parameters in the vehicle design process. To solve these problems, the suspension systems designed for heavy vehicles ought to be able to isolate the vibrations from the road surface and decrease the dynamics loads of the wheel axles generated by the vehicle moving on the highway under various operation cases. With the vehicle’s passive suspension system, due to its low isolation performance, the air suspensions and control suspensions were studied and improved for providing better ride quality.^1–4 The researchers have reported that using control suspensions for vehicles can improve their comfort level.

With the control suspensions of vehicles, some optimal control approaches, such as the proportional-integral-derivative control, fuzzy-skyhook control, fuzzy control, skyhook-neuro fuzzy, or fuzzy-H_inf control were studied and applied to optimize the damping coefficients under the vehicle’s various working conditions.^5–7 The result showed that the control suspension systems have greatly improved the comfort level of vehicles, especially with the smart fuzzy control. However, in this research, a quarter model of vehicles was only established, thus, the vehicle’s comfort level was only analyzed in the vertical direction.^7,8 In order to fully assess the control performance of the active suspensions in different vehicle models, a half-vehicle model in cars, heavy vehicles, and vibratory rollers with their suspension system controlled via fuzzy control or machine learning approach were then used to improve the vertical vibration and pitching vibration of vehicles.^9–11 The research indicated that the vehicle’s suspension systems controlled by the fuzzy control method were also stable and effective on different vehicle models.

With heavy vehicles with the suspension system using the leaf springs, the dynamic loads generated by tires acting on the highway of the vehicle moving was very large.¹² Thus, in order to decrease these negative impacts of vehicles on the highway and improve the heavy vehicle’s road friendliness, the air suspensions with the air springs were chosen instead of the linear leaf springs.^13–15 Studies showed that the performance of the heavy vehicle’s suspensions using air springs not only decreased the road damage but also enhanced the comfort level. Besides, studies have also shown that controlling the heavy vehicle’s air suspension system can further enhance the RF-CL. However, there has been no comprehensive investigation in evaluating both the road friendliness and comfort level of heavy vehicles using control air suspensions.

Therefore, to improve all objective functions of the heavy vehicle including road friendliness and comfort level in all vibration directions (vertical direction, pitching direction, and rolling direction), this study proposes a heavy vehicle’s three-dimensional model with 16-degrees of freedom using air suspensions for establishing the vibration model. Then, the damping coefficients in the heavy vehicle’s air suspensions are controlled through the fuzzy control approach. The SIMULINK and control models of the vehicle and air suspensions built in MATLAB are used to solve the vibration equations and compute the RF-CL. Herein, the RF (road friendliness) has been evaluated via the dynamic load coefficient whereas the CL (comfort level) has been evaluated via the root-mean-square value of the vertical seat acceleration, cab’s pitch acceleration, and cab’s roll acceleration. The two indexes of the RF-CL are evaluated under all various working cases of the heavy vehicle.

The novelty of this work include (1) a three-dimensional dynamic model of a heavy vehicle is established to fully evaluate the vibration characteristics of the heavy vehicle, (2) the heavy vehicle’s air suspension system is controlled to simultaneously improve both RF and CL, and (3) the efficiency and stability of heavy vehicle’s control air suspensions are evaluated under all different vehicle operating conditions.

Mathematical methods

Heavy vehicle’s mathematical model

To study the heavy vehicle’s RF-CL using control air suspensions, a heavy vehicle with three axles is applied for building the mathematical model as well as calculating the vibration equations of the vehicle. Based on the design parameters of the heavy vehicle listed in Table 1,¹⁶ a three-dimensional model with 16-degrees of freedom of the heavy vehicle is then plotted in Figure 1. Where (z_s, z_c, z_b, z_t, z_a1, z_a2, z_a3) and (m_s, m_c, m_b, m_t, m_a1, m_a2, m_a3) are vertical displacements and mass of the seat, cab, tractor driver, trailer and axles; (ϕ_c, ϕ_b, ϕ_t) and (θ_c, θ_b, θ_t, θ_a1, θ_a2, θ_a3) are angular displacements of the cab, the tractor driver, the trailer and the axles in pitch and roll directions; q_i are excitations at tires; l_n and (b₁, b₂, b₃) are distances of the heavy vehicle, (i = 1–7, n = 1–8).

Table 1.

Dynamic parameters of a heavy vehicle.

Parameters	Values
m_s/(kg)	85
m_c/(kg)	500
m_b/(kg)	3600
m_t/(kg)	12 500
m_a1/(kg)	270.1
m_a2/(kg)	520.4
m_a3/(kg)	340.0
l₁/(m)	0.2
l₂/(m)	0.8
l₃/(m)	1.3
l₄/(m)	4.1
l₅/(m)	6.9
l₆/(m)	4.0
l₇/(m)	1.2
l₈/(m)	4.8
b₁/(m)	0.3
b₂/(m)	0.4
b₃/(m)	1.0
A_01,02/(m²)	0.0281
A_03,04/(m²)	0.0906
A_05,06/(m²)	0.0906
V_01,02/(m³)	0.0078
V_03,04/(m³)	0.0322
V_05,06/(m³)	0.0322
p_01,02/(MPa)	0.134
p_03,04/(MPa)	2.860
p_05,06/(MPa)	1.035
α ₁	0.0590
α ₂	0.0186
α ₃	0.1270
α ₄	0.0307
λ	1.33

Figure 1.

Mathematical model of a heavy vehicle, (a) side view, and (b) front view.

To establish the equations of motion of the car and simplify the physical model of the heavy vehicle, (1) it is assumed that the vehicle vibration excitation is mainly from the vertical road surface roughness, (2) assuming the floors of the cab, tractor driver, and trailer are absolutely rigid, thus, their deformation is very small, and (3) the vibration excitations in the lateral and horizontal directions are very small and they are neglected.

Based on Newton’s second law and heavy vehicle’s dynamic model, the general vibration equations of the vehicle have been expressed by:

\begin{array}{l} m_{s} {\ddot{z}}_{s} = F_{s} \\ \begin{array}{l} m_{c} {\ddot{z}}_{c} = - F_{s} + F_{c 1} + F_{c 2} + F_{c 3} + F_{c 4} \\ I_{c y} {\ddot{ϕ}}_{c} = {l_{1} F}_{s} - l_{3} (F_{c 1} + F_{c 2}) + l_{2} (F_{c 3} + F_{c 4}) \\ \begin{array}{l} I_{c x} {\ddot{θ}}_{c} = - {b_{1} F}_{s} + b_{2} (F_{c 2} + F_{c 4} - F_{c 1} - F_{c 3}) \\ m_{b} {\ddot{z}}_{b} = \sum_{i = 1}^{4} (F_{i} - F_{c i}) - F_{0} \\ \begin{array}{l} I_{b y} {\ddot{ϕ}}_{b} = l_{3} (F_{3} + F_{4}) - l_{2} (F_{1} + F_{2}) - {l_{4} F}_{0} + \dots \\ + l_{7} (F_{c 1} + F_{c 2}) + l_{0} (F_{c 3} + F_{c 4}) \\ \begin{array}{l} \begin{array}{l} I_{b x} {\ddot{θ}}_{b} = b_{3} (F_{2} - F_{1} + F_{4} - F_{3}) - \dots \\ + b_{2} (F_{c 1} - F_{c 2} + F_{c 3} - F_{c 4}) \end{array} \\ \begin{array}{l} m_{t} {\ddot{z}}_{t} = F_{0} {+ F_{5} + F}_{6} \\ \begin{array}{l} I_{t y} {\ddot{ϕ}}_{t} = l_{6} (F_{5} {+ F_{6}) - l_{0} F}_{5} \\ I_{t x} {\ddot{θ}}_{t} = b_{3} (F_{6} - F_{5}) \end{array} \end{array} \\ \begin{array}{l} \begin{array}{l} m_{a 1} {\ddot{z}}_{a 1} = F_{t 1} {+ F_{t 2} - F}_{1} - F_{2} \\ I_{a 1 x} {\ddot{θ}}_{a 1} = b_{3} (F_{1} - F_{2}) + b_{3} (F_{t 2} - F_{t 1)} \end{array} \\ m_{a 2} {\ddot{z}}_{a 2} = F_{t 3} {+ F_{t 4} - F}_{3} - F_{4} \\ \begin{array}{l} I_{a 2 x} {\ddot{θ}}_{a 2} = b_{3} (F_{3} - F_{4}) + b_{3} (F_{t 4} - F_{t 30}) \\ \begin{array}{l} m_{a 3} {\ddot{z}}_{a 3} = F_{t 5} {+ F_{t 6} - F}_{5} - F_{6} \\ I_{a 3 x} {\ddot{θ}}_{a 3} = b_{3} (F_{5} - F_{6}) + b_{3} (F_{t 6} - F_{t 5}) \end{array} \end{array} \end{array} \end{array} \end{array} \end{array} \end{array} \end{array}

(1)

where l₀ = l₇ − l₂ − l₃, l₇ = l₁ + l₂; F_s, F_c1–c4, F₀, F_1–6, F_t1–t6 are the force responses in suspension systems of the seat, cab, articulation connection, vehicle body, and tires. These force responses are calculated based on their suspension models in Figure 2 as follows:

Figure 2.

Mathematical model of heavy vehicle’s suspension systems, (a) seat’s suspension, (b) cab’s suspension, (c) articulation connection, and (d) tires.

The vertical force generated in the seat’s suspension (F_s) designed by damping c_s and stiffness k_s is

\begin{array}{l} F_{s} = k_{s} (z_{c s} - z_{s}) + c_{s} ({\dot{z}}_{c s} - {\dot{z}}_{s}) \\ \begin{array}{l} = k_{s} (z_{c} - z_{s} - l_{1} ϕ_{c} + b_{1} θ_{c}) + \dots \\ {+ c}_{s} ({\dot{z}}_{c} - {\dot{z}}_{s} - l_{1} {\dot{ϕ}}_{c} + b_{1} {\dot{θ}}_{c}) \end{array} \end{array}

(2)

The force responses of cab’s suspensions (F_cj) using damping c_sj and stiffness k_sj are

\begin{array}{l} \begin{array}{l} F_{c j} = k_{c j} (z_{b c j} - z_{c j}) + c_{c j} ({\dot{z}}_{b c j} - {\dot{z}}_{c j}) \\ = k_{c j} [z_{b} - z_{c} - l_{x} ϕ_{b} + {(- 1)}^{β} l_{β} ϕ_{c} + \dots \end{array} \\ \begin{array}{l} {+ (- 1)}^{v} b_{2} (θ_{b} - θ_{c})] + c_{c j} [{\dot{z}}_{b} - {\dot{z}}_{c} - \dots \\ - l_{x} {\dot{ϕ}}_{b} {+ (- 1)}^{β} l_{β} {\dot{ϕ}}_{c} {+ (- 1)}^{j} b_{1} ({\dot{θ}}_{b} - {\dot{θ}}_{c})] \end{array} \end{array}

(3)

where

{\begin{cases} j = 1, 2 \\ j = 3, 4 \end{cases}

then

{\begin{cases} β = 2 \\ β = 3 \end{cases}

and

{\begin{cases} l_{x} = l_{2} + l_{3} + l_{4} \\ l_{x} = l_{4} \end{cases}

The force response of the articulation connection (F₀) using damping c₀ and stiffness k₀ is

\begin{array}{l} \begin{array}{l} F_{c j} = k_{0} (z_{0 b} - z_{0 t}) + c_{0} ({\dot{z}}_{0 b} - {\dot{z}}_{0 t}) \\ = k_{0} (z_{b} - z_{t} - l_{4} ϕ_{b} - l_{5} ϕ_{t}) + \dots \end{array} \\ + c_{0} ({\dot{z}}_{b} - {\dot{z}}_{t} - l_{4} {\dot{ϕ}}_{b} - l_{5} {\dot{ϕ}}_{t}) \end{array}

(4)

The force responses of the control air suspensions (F_i) using damping c_i, control damping c_ctrl-i, and stiffness of air spring k_ai are expressed in the next section.

The force responses of the tires using damping c_ti and stiffness k_ti are

\begin{array}{l} \begin{array}{l} F_{t i} = k_{t i} (q_{i} - z_{a i}) + c_{t i} ({\dot{q}}_{i} - {\dot{z}}_{a i}) \\ = k_{t i} [q_{i} - z_{a δ} + {(- 1)}^{i + 1} b_{3} θ_{a δ}] + \dots \end{array} \\ + c_{t i} [{\dot{q}}_{i} - {\dot{z}}_{a δ} + {(- 1)}^{i + 1} b_{3} {\dot{θ}}_{a δ}] \end{array}

(5)

where i = 1, 2 then δ = 1, i = 3, 4 then δ = 2, and i = 5, 6 then δ = 3.

Model of control air suspensions

With the control air suspension system (CASS) of the heavy vehicle designed by the damper with the damping value c and air spring with the stiffness value k_a, the active damping value c_ctrl of the CASS will be controlled by the fuzzy control approach to enhance the CASS’s working performance. Its model is shown in Figure 3.

Figure 3.

Mathematical model of a control air suspension system, (a) mathematical model, and (b) model of an air spring.

From the structure of the CASS in Figure 3(a), the force equation of the CASS could be written by:

F = (c_{c t r l} + c) ({\dot{z}}_{a} - {\dot{z}}_{b}) + k_{a} (z_{a} - z_{b})

(6)

where

z = z_{a} - z_{b}

is the relative displacement and

\dot{z} = {\dot{z}}_{a} - {\dot{z}}_{b}

is the relative velocity of the ACSS.

With the stiffness in air spring, k_a has been calculated through two methods, which are the experiment data method and the laws of the thermodynamics method.¹⁷ In this study, the k_a is calculated by the laws of the thermodynamics method based on the air spring’s area variation and volume variation as well as the air spring’s design parameters.

Based on the change of these parameters, the air pressure in the air spring is then also changed. Thus, the k_a can be calculated via a derivative of F_a and z in Figure 3(b) as follows:

k_{a} = \frac{d F_{a}}{d z} = \frac{d (p_{e} A_{e})}{d z} = p_{e} \frac{d A_{e}}{d z} + A_{e} \frac{d p_{e}}{d z}

(7)

where (V_e, A_e) are defined as the volume effective and area effective of the air spring and calculated as:

{\begin{cases} V_{e} = V_{0} - α_{1} z \\ A_{e} = A_{0} + α_{2} z \end{cases}

(8)

Based on the laws of thermodynamics,^13,17 it was assumed that the air spring’s expansion stroke or compression stroke was very rapid. Thus, it was defined by the adiabatic process and the pressure state of the air in the air spring can be defined by:

(p_{a} + p_{0}) V_{0}^{λ} = (p_{e} + p_{0}) V_{e}^{λ}

(9)

In Equations (8) and (9), (A₀, V₀) are the design area effective and design volume effective. (α₁, α₂) are the change of the volume effective and area effective. (p₀, p_e) are the air pressures at initial and final states in the air spring and p_a is the atmospheric pressure. γ is the specific heat ratio.

By substituting Equations (8) and (9) into Equation (7) and calculating Equation (7) for the stiffness of air springs, we have

\begin{array}{l} k_{a i} = [{(p_{a} + p_{0 i}) (\frac{V_{0 i}}{V_{0 i} - {α_{l} z}_{i}})}^{λ} - p_{a}] α_{l + 1} + \dots \\ + λ {(p_{a} + p_{0 i}) \frac{A_{0 i} + {α_{l + 1} z}_{i}}{V_{0 i} - {α_{l} z}_{i}} (\frac{V_{0 i}}{V_{0 i} - {α_{l} z}_{i}})}^{λ} α_{l} \end{array}

(10)

where i = 1, 2 then l = 1 and i = 3, 4, 5, 6 then l = 3.

The k_ai in Equation (10) is known as a nonlinear spring. This stiffness is then applied to simulate the CASS.

Fuzzy control and its application

Zadeh proposed the fuzzy control approach in 1965 to control the dynamic systems in various fields. This control method was also suggested and applied to control the vehicle’s isolations for ameliorating the comfort level. Its structure and control principle include a fuzzification used to transform the input values into the input linguistic variables. Then, these input linguistic variables were used to calculate the output linguistic variables based on the fuzzy rules in the fuzzy inference system. Finally, the defuzzification was used to transform the output linguistic variables to the output physical values to control the system.^5,18,19 The air suspension system of the heavy vehicle in Figure 1 is designed with six air suspensions. This means that six air suspensions need to be controlled. However, their control is similar. Therefore, a control model of fuzzy control has been designed and applied to control all the CASS in the heavy vehicle.

To control the CASS of the heavy vehicle, the relative displacement ( $e = z$ ) and the relative velocity ( $e c = \dot{z}$ ) of the CASS in Equation (6) are defined as the input signals whereas the c_ctrl in the same Equation (6) is defined as the output value of the fuzzy control. The control structure of the heavy vehicle and air suspensions is presented in Figure 4.

Figure 4.

Schematic fuzzy control for heavy vehicle.

To build the fuzzy control model, the input/output linguistic variables and the physical values to control the air suspensions are described and defined in Table 2. Besides, the membership function in input/output variables is used by the triangular functions with the degree of memberships (DOM) from 0 to 1, as shown in Figure 5.

Table 2.

The linguistic values.

Linguistic values	Descriptions	e	ec	Linguistic values	γ
pvb	Positive-very-big	0.8	1.2	o ₁	0
pb	Positive-big	0.6	0.9	o ₂	2
pm	Positive-medium	0.4	0.6	o ₃	4
ps	Positive-small	0.2	0.3	o ₄	6
ze	Zero	0	0	o ₅	8
ns	Negative-small	−0.2	−0.3	o ₆	10
nm	Negative-medium	−0.4	−0.6	o ₇	14
nb	Negative-big	−0.6	−0.9	o ₈	17
nvb	Negative-very-big	−0.8	−1.2	o ₉	20

Figure 5.

The membership function of input variable and output variable.

Additionally, to calculate the control values of the output signals, the control rules of “if-then” between the $e = z_{a} - z_{b}$ , $e c = {\dot{z}}_{a} - {\dot{z}}_{b}$ , and c_ctrl are also built based on the designer’s experience and provided in Table 3. These control rules are described as follows:

(1) If e = nvb and ec = nvb then γ = o₅,

(2) If e = vb and ec = nvb then γ = o₃,

…

(81) If e = pvb and ec = pvb then γ = o₅.

Table 3.

Rules for fuzzy control.

γ		e
γ		nvb	nb	nm	ns	z	ps	pm	pb	pvb
ec	nvb	o ₅	o ₉	o ₇	o ₆	o ₅	o ₄	o ₄	o ₃	o ₃
	nb	o ₃	o ₅	o ₈	o ₆	o ₅	o ₄	o ₃	o ₃	o ₃
	nm	o ₁	o ₂	o ₅	o ₇	o ₅	o ₃	o ₂	o ₂	o ₂
	ns	o ₁	o ₁	o ₁	o ₅	o ₅	o ₃	o ₂	o ₁	o ₂
	z	o ₁	o ₁	o ₁	o ₁	o ₅	o ₁	o ₁	o ₁	o ₁
	ps	o ₂	o ₂	o ₂	o ₃	o ₅	o ₅	o ₁	o ₁	o ₁
	pm	o ₂	o ₃	o ₃	o ₃	o ₅	o ₇	o ₅	o ₂	o ₁
	pb	o ₃	o ₃	o ₃	o ₄	o ₅	o ₆	o ₈	o ₅	o ₃
	pvb	o ₃	o ₃	o ₄	o ₄	o ₅	o ₆	o ₇	o ₉	o ₅

The centroid method and minimum function proposed by Mamdani were studied and designed in the fuzzy tool of MATLAB to calculate the output values.^5,18 Thus, in the fuzzy tool of MATLAB, Mamdani’s inference method is applied for calculating the control values of the CASS in the heavy vehicle.

Evaluation criteria and road excitation

Evaluation method of comfort level

Based on the international standard ISO 2631-1,20 the effect of vibrations on human’s comfort level was mainly assessed by the root mean square (rms) value of the acceleration. This index is then applied to evaluate the driver’s seat comfort level.^2–4,13,15 In this study, to evaluate the CASS’s performance in improving the heavy vehicle’s comfort level, the rms accelerations of the driver’s seat, cab’s pitch, and cab’s roll calculated based on ISO 2631-1²⁰ are also applied. These values are expressed by:

{\begin{cases} a_{w z s} = \sqrt{T^{- 1} \int_{0}^{T} {\ddot{z}}_{s}^{2} d t} \\ a_{w ϕ c} = \sqrt{T^{- 1} \int_{0}^{T} {\ddot{ϕ}}_{c}^{2} d t} \\ a_{w θ c} = \sqrt{T^{- 1} \int_{0}^{T} {\ddot{θ}}_{c}^{2} d t} \end{cases}

(11)

where

{\ddot{z}}_{s}

{\ddot{ϕ}}_{c}

, and

{\ddot{θ}}_{c}

are accelerations of the seat and cab in vertical, pitching, and rolling directions, T is the simulation time.

Evaluation method of road friendly

The road friendliness of the heavy vehicles was defined as the impact of the wheel axle on the road damage when the vehicle moves on the highway. This road friendliness was assesed by the dynamic-load-coefficient (DLC) described by:^21–23

{\begin{cases} {D L C}_{i} = \frac{{(F_{t, r m s})}_{i}}{F_{s t i}} \\ {(F_{t, r m s})}_{i} = {(T^{- 1} \int_{0}^{T} {\ddot{F}}_{t i}^{2} d t)}^{0.5} \end{cases}

(12)

where F_t,rms is the rms dynamic load of the wheel axle. F_t is the wheel’s static load, i = 1–6.

The DLC of the heavy vehicle not only depends on the interaction of the wheel moving on the highway but also depends on the dynamic parameters of the suspensions, the vehicle’s load modes, the distribution of the stiffness between the vehicle axles etc. The existing studies showed that the DLC’s value has been limited of 0 to 0.4. It is often in the range of 0.05–0.3 in the normal operation case. DLC ≈ 0.0 in the case of the vehicle moving on a very good road surface and DLC >0.4 in the case of the vehicle moving on a very poor road surface.^23,24 Besides, DLC’s value also depended on different types of vehicle’s suspensions. DLC with walking beam suspensions is obtained by 0.04–0.25, DLC with leaf spring suspensions is obtained by 0.04–0.13, and DLC with air spring suspensions is obtained by 0.03–0.1. In addition, DLC with steel suspensions is always higher than that of the air suspensions for the same loads, vehicle speeds, and road surfaces.^22,24 Therefore, this DLC value is also applied to analyze the CASS’s performance in improving the heavy vehicle’s road friendliness.

Excitations of heavy vehicle

The surface roughness of the highway needs to concern in evaluating the vibration of the heavy vehicle because it not only impacts the interaction between the vehicle and highway but also impacts the fatigue life of the heavy vehicle. This random excitation was described with a periodic modulated random process based on the power-spectral-density (PSD) of the highway given in Equation (13):²⁵

S_{q} (n) = S_{q} (n_{0}) {(\frac{n}{n_{0}})}^{- 2}

(13)

where n is space frequency, n_o is reference space frequency, and S_q(n) is the PSD of the highway.

Assuming that the surface of the highway is a Gaussian’s random process with the zero-mean-stationary. Thus, the random road excitation can be calculated by an inverse Fourier transformation with a random phase ϕ_i random distributed between 0 and 2π as follows:

q = \sum_{j = 1}^{N} \sqrt{2 S_{q} (n) ∆ n} \cos (2 π n t + ϕ_{j})

(14)

According to ISO-8068,²⁵ the five different levels of the highway including from ISO-A to ISO-E have been simulated at 20 m/s. The results are presented in Figure 6. These random surfaces of the highway are then applied to evaluate CASS’s performance of the heavy vehicle.

Figure 6.

Profile of random road surfaces built via ISO 8068.

Analysis of numerical simulation results

The study mainly analyzes the performance and stability of CASS on the RF-CL of the heavy vehicle under various operation cases. From the heavy vehicle’s parameters listed in Table 1, the SIMULINK model of the vehicle established in MATLAB has been simulated to solve the vehicle’s vibration equations as well as compute the objective functions of the RF-CL.

Efficiency of control air suspension

To simulate the efficiency of the CASS, the fuzzy control tool and the SIMULINK model of the vehicle in MATLAB software are applied for controlling the heavy vehicle’s CASS on the highway of ISO-B at 20 m/s. From the control damping coefficients of the c_ctrl at the 1st, 2nd, and 3rd axles calculated and presented in Figure 7, both the results of the seat and cab’s acceleration responses are shown in Figures 8(a)–8(c).

Figure 7.

The control damping coefficients on three axles.

Figure 8.

The driver’s seat and cab acceleration, (a) seat’s heave, (b) cab’s pitch, and (c) cab’s roll.

All Figures 8(a)–8(c) indicated that the seat’s heave, cab’s pitch, and cab’s roll accelerations with heavy vehicle’s CASS have been strongly reduced compared with heavy vehicle applying the passive air suspensions (PASS). This result is achieved because the damping coefficients of the CASS have been varied by the fuzzy control based on the road surface excitation to achieve optimal damping forces, thereby significantly increasing the suspension’s ability to isolate vibrations. The computation result provided in Table 4 shows that the a_wzs, a_wϕc, and a_wθc with the CASS are reduced by 24.5%, 25.0%, and 30.0% in comparison with the PASS. The result implies that the comfort level of the heavy vehicle has been significantly improved by the CASS in all vertical, pitch, and roll directions.

Table 4.

Computation results of rms accelerations.

Parameters	PASS	CASS	Reduction (%)
a_wzs/m s⁻²	0.34	0.26	24.5
a_wϕc/rad s⁻²	0.08	0.06	25.0
a_wθc/rad s⁻²	0.10	0.07	30.0

Besides, the DLC’s computation results at the 1st, 2nd, and 3rd axles are also provided in Table 5. We can see that DLC values at the 1st, 2nd, and 3rd axles with the CASS are lower than DLC values with the PASS by 12.5%, 15.1%, and 11.5%, respectively. This means that the interaction between the road surface and wheel axles is significantly reduced and the wheel’s road damage in the heavy vehicle using CASS has been improved. Besides, Table 5 also indicates that DLC value at the 2nd axle is higher than DLC value at the 1st axle and 3rd axle. This means that the impact of 2nd axle on the highway is the greatest. Based on the simulation result, the force response, F_t,rms, and F_t,max at the 2nd axle have been plotted in Figure 9 and listed in the same Table 5.

Table 5.

The computation results of the vehicle on ISO-B.

Parameters	PASS	CASS	Reduction (%)
DLC at 1st axle	0.024	0.021	12.5
DLC at 2nd axle	0.073	0.062	15.1
DLC at 3rd axle	0.052	0.046	11.5
F_t,rms at 2nd axle/kN	3.78	2.81	25.7
F_t,max at 2nd axle/kN	12.5	10.5	16.0

Figure 9.

Dynamic load on 2nd axle.

The results show that with the CASS applied, the force response of the 2nd axle is significantly reduced compared to the PASS. Especially, both F_t,rms and F_t,max on the 2nd axle with the CASS are decreased by 25.7% and 16% in comparison with the PASS. Consequently, the heavy vehicle’s CASS could improve the RF-CL.

Stability and efficiency of CASS under different speeds

CASS’s efficiency has been evaluated when the heavy vehicle operates on the highway of ISO-B at 20 m/s. However, in actual conditions of heavy vehicle operation, the speed, road surface, and vehicle load can be changed and affect the stability of the CASS. To clarify these issues, under the change of the speeds of v = {5, 10, …, 30} m/s and the vehicle’s load m_t = {half load, full load, and over load} when the heavy vehicle operates on the highway of ISO-B, the heavy vehicle is then simulated to evaluate CASS’s efficiency and stability. The results of both seat and cab’s rms accelerations have been presented in Figures 10(a)–10(c), respectively.

Figure 10.

Seat and cab’s rms accelerations, (a) seat’s heave, (b) cab’s pitch, and (c) cab’s roll.

Figure 10 provides that all rms values of the a_wzs, a_wϕc, and a_wθc with CASS have significantly decreased in comparison with PASS under all different load and speed conditions of the heavy vehicle. Thus, CASS is relatively stable and efficient compared to PASS under various speeds and loads. Additionally, the change in the vehicle’s loads significantly affects the comfort level. When the vehicle’s load has been increased by over load - CASS, all a_wzs, a_wϕc, and a_wθc are lower than that of the full load -CASS. Conversely, when the vehicle’s load is reduced by half load - CASS, all the value of a_wzs, a_wϕc, and a_wθc are increased in comparison with the full load - CASS. Besides, in the case of the heavy vehicle moving at the low speed below 20 m/s, a_wθc is small while both a_wzs and a_wϕc are high. Therefore, the comfort level of the heavy vehicle has been decreased. Conversely, when the heavy vehicle moves at high speeds of over 20 m/s, a_wzs and a_wϕc have been strongly decreased while a_wθc is insignificantly increased. The result implies that the comfort level of the heavy vehicle has been greatly improved. This is also the reason why road designers recommend that heavy vehicles running on the road should run at high speed of 20 to 25 m/s and use full load to improve the vehicle’s comfort level.^17,23,24

Because the 2nd axle impact on the highway is the greatest, therefore, the computed results of DLC, F_t,rms, and F_t,max at the 2nd axle under different speeds and loads of the heavy vehicle are presented in Figures 11(a)–11(c).

Figure 11.

The results under the effect of the speed and load, (a) DLC at 2nd axle, (b) F_t,rms at 2nd axle, and (c) F_t,max at 2nd axle.

The results show that all DLC, F_t,rms, and F_t,max at the 2nd axle have been lightly changed when the change of the vehicle’s speed. Besides, the DLC, F_t,rms, and F_t,max with the CASS are lower than that of PASS under different speeds of the vehicle’s full load. However, these values are strongly influenced by the change in vehicle’s loads. In the case of the heavy vehicle’s half load - CASS used, all DLC, F_t,rms, and F_t,max at 2nd axle are higher than that of the full load - CASS. Conversely, in the case of the heavy vehicle’s over load - CASS used, all DLC, F_t,rms, and F_t,max at 2nd axle are lower than that of the full load - CASS under various speeds of the vehicle. This result implies that when the heavy vehicle travels on the highway with half load mode, the vehicle’s influence on the road damage is large. Accordingly, the vehicle’s half load mode should be limited to improve the road friendliness of the heavy vehicles.

Stability and efficiency of control air suspension under different road surfaces

During the heavy vehicle moving on the highway, the quality of the highway will be reduced, which greatly affects the heavy vehicle’s RF-CL. Therefore, to evaluate CAS’s stability and efficiency under different highways, five different highway types from ISO-A to ISO-E in Figure 6 have been applied for the simulation. The results of DLC, F_t,rms, and F_t,max at 1st, 2nd, and 3rd axles are plotted in Figures 12(a)–12(c).

Figure 12.

Computation results under effect of highways, (a) DLC at 2nd axle, (b) F_t,rms at 2nd axle, and (c) F_t,max at 2nd axle.

The result indicates the random surface of the highway greatly affects the heavy vehicle’s road friendliness. When the high roughness of the highway increases (the highway’s surface quality reduces), all DLC, F_t,rms, and F_t,max are also increased and vice versa. The result implies that when the surface quality of the highway is good, it contributes positively to increasing the vehicle’s friendliness to the highway. Conversely, when the surface quality of the highway is poor, it also contributes to increasing the road damage of vehicles. In order to ensure the vehicle’s friendliness to the highway, the surface of the highway in ISO-A and ISO-B should be maintained. Besides, Figure 12 also indicates that all DLC, F_t,rms, and F_t,max of 2nd axle are higher than that of 1st and 3rd axles under all different highways from ISO-A to ISO-E. Thus, the heavy vehicle’s 2nd axle greatly impacts the road damage.

However, with the CASS applied, all DLC, F_t,rms, and F_t,max at the 1st axle, 2nd axle, and 3rd axle have been strongly reduced compared to PASS under all different highways from ISO-A to ISO-E, especially at ISO-D and ISO-E. Therefore, the CASS is also stable and effective under all different excitations of the highways.

Conclusions

By applying the CASS for the heavy vehicle, all the a_wzs, a_wϕc, and a_wθc with the CASS are reduced by 24.5%, 25.0%, and 30.0% in comparison with the PASS. Besides, the DLC value at the 1st, 2nd, and 3rd axles with the CASS are lower than that of the PASS by 12.5%, 15.1%, and 11.5%, respectively. Therefore, the heavy vehicle’s RF-CL is significantly ameliorated compared to the heavy vehicle’s PASS.

Heavy vehicle’s moving speeds greatly affect the comfort level while these speeds insignificantly affect the road friendliness in heavy vehicles. Conversely, the vehicle’s load strongly affects both the heavy vehicle’s RF-CL, especially in the case of the heavy vehicle using the half load and moving at a low speed from 7.5 to 12.5 m/s. Thus, in order to ameliorate the heavy vehicle’s RF-CL, heavy vehicles should move at high speeds of 20 to 25 m/s in the case of the vehicle’s full load.

Besides, under the excitations of the highway from ISO-A to ISO-E, the heavy vehicle’s 2nd axle strongly influences the road damage. Thus, to reduce this road damage, the good surface of the highway in ISO-A and ISO-B should be maintained.

Under all simulation cases of the heavy vehicle, the investigation indicates that the heavy vehicle’s CASS is overall superior to the heavy vehicle’s PASS in improving the vehicle’s RF-CL. Thus, the control air suspensions should be applied to the heavy vehicle to enhance the working performance of the vehicle.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Beibei Su

Cuifang Wang

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