Assessing isolation capacity of heavy vehicle’s control air suspensions with three different control cases

Abstract

Three different control cases in control air suspensions using air springs controlled (ASC), (2) using the damper controlled (DC), and (3) using both ASC and DC (ASDC) are proposed to ameliorate both riding quality and friendliness-road surface of heavy vehicle based on the heavy vehicle’s dynamic model and fuzzy logic control with its control rules optimized by genetic algorithm. The riding quality and friendliness-road surface are assessed through the root-mean-square cabin accelerations (a_wzc and a_wϕc) and coefficient of the dynamics load (CDL_n) at wheel axles. Research shows that the vehicle’s riding quality and friendliness-road surface are strongly affected by the moving speed and road surface roughness. The a_wzc, a_wϕc, and CDL_n of the control air suspensions are strongly decreased compared to passive air suspensions, particularly with ASDC. Additionally, a_wzc and a_wϕc of DC are smaller than that of ASC whereas the maximum CDL_n with DC is higher than that of ASC. Thus, DC could be used for improving vehicle’s riding quality, ASC should be applied for ameliorating vehicle’s friendliness-road surface, and ASDC should be applied for ameliorating both riding quality and friendliness-road surface of heavy vehicles.

Keywords

Heavy vehicle dynamic model air suspensions coefficient of dynamics load optimal control comfort level

Introduction

To enhance the riding quality and friendliness-road surface of heavy vehicle, air suspensions designed by air springs have been applied to replace the traditional suspension systems of heavy vehicle.^1–4 The dynamic characteristics of the air springs have been also researched for assessing the isolation ability and stability in air suspensions. Result of the dynamic stiffness in the air spring under the excitations of 2.0 Hz to 6.0 Hz has been strongly increased, thus, the deformation of the air spring and the impact force of the wheels on the road were significantly reduced compared to traditional suspensions using steel-springs and steel-leafs.^5–7 As a result, the vehicle’s friendliness-road surface was significantly ameliorated. However, under excitations of the frequency from 8.0 to 50 Hz researched, the dynamics stiffness in air springs was strongly reduced, so the air spring’s deformation was higher than the traditional suspensions and the heavy vehicle’s riding quality was low. Thus, the damper has been added to the air suspension to decrease the suspension system’s deformation and improve heavy vehicle’s vibration.

To further enhance the riding quality of heavy vehicles equipped with air suspensions, the damping coefficient in air suspensions using the magnetorheological (MR) fluid was also researched and controlled based on the optimal fuzzy control and Proportional-Integral-Derivative-fuzzy control.^8–10 The studies indicated that through the evaluation index of vehicle’s accelerations, the control air suspension using the damper controlled (DC) remarkably ameliorated vehicle’s riding quality. Besides, in some studies on control air suspension systems, the dynamics stiffness in air springs has been also researched and controlled by the valve in the air pipe used to connect the air spring and reservoir. The studies indicated that based on the coefficient of dynamic load (CDL) and road damage, the control air suspension using the air spring controlled (ASC) by the air valve reduced the wheel’s dynamic force and value of CDL, thus, the vehicle’s friendliness-road surface using ASC was better than that of uncontrol air suspensions.^5,7,11,12 However, in the above studies, some existing issues of the heavy vehicle that have not yet been resolved including (1) the control performance between the ASC and DC on improving both riding quality and friendliness-road surface of the vehicle has been not researched yet. (2) Heavy vehicle’s control air suspension using both the air spring controlled and damper controlled (ASDC) could ameliorate the riding ride quality and friendliness-road surface better than that of the single control method of the DC or ASC, however, this issue has been also not researched yet. (3) The advanced-control approaches of the fuzzy and genetic algorithm combined, skyhook-fuzzy combined, or fuzzy-neuro control combined have been used for controlling car’s active/semi-active isolation systems.^10,13–17 Especially, with control rules in the fuzzy logic control optimized based on the genetic algorithm, fuzzy logic control can further enhance its control performance.^14,15 However, this advanced control method has been also not investigated for controlling heavy vehicle’s pneumatic suspension systems yet.

In this study, three different control cases of the control air suspension using the ASC, DC, and ASDC are proposed for improving heavy vehicle’s riding quality and friendliness-road surface based on the advanced control method of fuzzy logic control and heavy vehicle’s dynamic model. In order to enhance the performance in control air suspensions, the fuzzy control’s rules are also optimized by genetic algorithm. The vehicle’s performance is assessed througth both riding quality and friendliness-road surface defined by the root-mean-square cabin acceperation responses and CDL_n at wheel axles n. Improving both riding quality and friendliness-road surface is the goal in this study.

This study’s contributions is summarized as follows:

• The heavy vehicle’s dynamic model and the advanced method of fuzzy logic control combined with the genetic algorithm have been applied to study the ride quality and road surface friendliness of heavy vehicle.

• Three different control cases in control air suspensions using ASC, DC, and ASDC are proposed to enhance the ride quality and road surface friend-liness of heavy vehicles.

• The performance of the heavy vehicle using the control air suspensions has been simulated and evaluated under all different working conditions in both the time and frequency domains.

Mathematical approach

Mathematical model in heavy vehicles

From the actual structure in heavy vehicles with suspension systems equipped with air suspensions, a five-axle heavy vehicle using the control air suspensions has been used. A mathematical model of the vehicle using 11 degrees of freedom has been built for evaluating the isolation efficiency of heavy vehicle’s air suspensions with three different control cases, as plotted in Figure 1. where M_c, M_td, M_t, and M_ai are the weight of the cabin, tractor-driver, trailer, and air suspension axles. Z_c, Z_td, Z_t, and Z_ai are the vertical vibration of the cabin, tractor-driver, trailer, and air suspension axles (front air suspension, middle air suspension, and rear air suspen-sion). ϕ_c, ϕ_td, ϕ_t, and ϕ_a2−3 are the angular vibration of the cabin, tractor-driver, trailer, and two air suspension axles (middle air suspension, and rear air suspension). F₀ is the articulation connection’s force of the trailer and tractor-driver. F_cj are the vertical forces in cabin’s front/rear isolations. F_ai are forces of the front air suspension, middle air suspension, and rear air suspension. F_tn are the vertical dynamic responses at the wheels. {C_c1, K_c1} and {C_c2, K_c2} are damping/stiffness parameters in cabin’s front and rear suspensions. {C_tn}, {K_tn}, and q_n are the damping parameter, stiffness parameter, and vibration excitation of the first-, second-, third-, fourth-, and fifth-wheel. l₀ and l_n are heavy vehicle’s longitudinal distances (n = 1, 2, 3, 4, 5; i = 1, 2, 3; j = 1, 2).

Figure 1.

(a) Mathematical model of heavy vehicle and (b) suspension models of the cabin and wheels.

By using Newton’s second law to calculate the vibration equations of the heavy vehicle via its mathematical model in Figure 1, the vibrational equations using air suspensions are presented by:

M_{c} {\ddot{Z}}_{c} = F_{c 1} + F_{c 2}

(1)

I_{c} {\ddot{φ}}_{c} = F_{c 1} l_{c 1} - F_{c 2} l_{c 2}

(2)

M_{t d} {\ddot{Z}}_{t d} = F_{a 1} + F_{a 2} - F_{c 1} - F_{c 2} - F_{0}

(3)

\begin{array}{l} I_{t d} {\ddot{φ}}_{t d} = F_{a 1} l_{3} + F_{0} l_{1} - F_{a 2} l_{4} . . . \\ - F_{c 1} (l_{0} + l_{c 1} + l_{c 2}) - F_{c 2} l_{0} \end{array}

(4)

M_{t} {\ddot{Z}}_{t} = F_{a 3} + F_{0}

(5)

I_{t} {\ddot{φ}}_{t} = F_{0} l_{2} - F_{a 3} l_{8}

(6)

M_{a 1} {\ddot{Z}}_{a 1} = F_{t 1} - F_{a 1}

(7)

M_{a 2} {\ddot{Z}}_{a 2} = F_{t 2} + F_{t 3} - F_{a 2}

(8)

I_{a 2} {\ddot{φ}}_{a 2} = F_{t 2} l_{5} - F_{t 3} l_{6}

(9)

M_{a 3} {\ddot{Z}}_{a 3} = F_{t 4} + F_{t 5} - F_{a 3}

(10)

I_{a 3} {\ddot{φ}}_{a 3} = F_{t 4} l_{9} - F_{t 5} l_{10}

(11)

The F_cj, F₀, and F_tn are calculated by:

The F_cj of the cabin,

\begin{array}{l} F_{c 1} = K_{c 1} [Z_{c} + φ_{c} l_{c 1} - Z_{t d} - φ_{t d} (l_{0} + l_{c 1} + l_{c 2})] + . . . \\ + C_{c 1} [{\dot{Z}}_{c} + {\dot{φ}}_{c} l_{c 1} - {\dot{Z}}_{t d} - {\dot{φ}}_{t d} (l_{0} + l_{c 1} + l_{c 2})] \end{array}

(12)

\begin{array}{l} F_{c 2} = K_{c 2} (Z_{c} - φ_{c} l_{c 2} - Z_{t d} - φ_{t d} l_{0}) + . . . \\ C_{c 2} ({\dot{Z}}_{c} - {\dot{φ}}_{c} l_{c 2} - {\dot{Z}}_{t d} - {\dot{φ}}_{t d} l_{0}) \end{array}

(13)

The F₀ at the articulation connection,

\begin{array}{l} F_{0} = K_{0} (Z_{t} + φ_{t} l_{2} - Z_{t d} + φ_{t d} l_{1}) + . . . \\ + C_{0} ({\dot{Z}}_{t} + {\dot{φ}}_{t} l_{2} - {\dot{Z}}_{t d} + {\dot{φ}}_{t d} l_{1}) \end{array}

(14)

where K₀ is the stiffness parameter and C₀ is the damping parameter of the articulation connection.The F_tn at the wheels,

F_{t 1} = K_{t 1} (Z_{a 1} - q_{1}) + C_{t 1} ({\dot{Z}}_{a 1} - {\dot{q}}_{1})

(15)

F_{t 2} = K_{t 2} (Z_{a 2} + φ_{a 2} l_{5} - q_{2}) + C_{t 2} ({\dot{Z}}_{a 2} + {\dot{φ}}_{a 2} l_{5} - {\dot{q}}_{2})

(16)

F_{t 3} = K_{t 3} (Z_{a 2} - φ_{a 2} l_{6} - q_{3}) + C_{t 3} ({\dot{Z}}_{a 2} - {\dot{φ}}_{a 2} l_{6} - {\dot{q}}_{3})

(17)

F_{t 4} = K_{t 4} (Z_{a 3} + φ_{a 3} l_{9} - q_{4}) + C_{t 4} ({\dot{Z}}_{a 3} + {\dot{φ}}_{a 3} l_{9} - {\dot{q}}_{4})

(18)

F_{t 5} = K_{t 5} (Z_{a 3} - φ_{a 3} l_{10} - q_{5}) + C_{t 5} ({\dot{Z}}_{a 3} - {\dot{φ}}_{a 3} l_{10} - {\dot{q}}_{5})

(19)

The F_ai of the front, middle, and rear air suspension is calculated in the next section.

Models of control air suspension

The structure of each front air suspension, middle air suspension, and rear air suspension of the heavy vehicle is designed by an air spring and a damper. The air spring connects into the reservoir through a valve and pipe (the air valve is controlled). The damper is used by the MR fluid to control its damping coefficient, as plotted in Figure 2(a). To evaluate vehicle’s performance using the control air suspensions, from the model of passive air suspension (PS) using damper uncontrolled and air spring uncontrolled,^5,6,18 three different control cases of the air suspension system including (1) control air suspension using the ASC controlled by the air valve and the damper uncontrolled, (2) control air suspension using the DC via the MR fluid and the air spring uncontrolled, and (3) control air suspension using both the air spring controlled and damper controlled (ASDC) are then researched. The mathematical models of the control air suspension using ASC, DC, and ASDC have been described in Figure 2(b)–2(d), respectively. Where V_r and p_r are the reservoir’s volume and reservoir’s pressure. V_e, p_b, and A_b are the volume, pressure, and air spring’s effective-area. A_p, M_b, and L_p are the cross-section-area, air mass, and length in pipe. K_v is viscous stiffness parameter, K_e is static stiffness parameter, and M is air mass in air spring. C_β and C_p are passive damping coefficients. C_β-act and C_act are the active damping coefficients in heavy vehicle’s air suspension. δ is air’s movement in pipe. Z_b is heavy vehicle’s body vibration and Z_a is wheel axle’s vibration.

Figure 2.

(a) Air suspension structure, (b) with air springs controlled, (c) with damper controlled and (d) with ASDC.

The F_ai of the air suspensions is calculated by:

F_{a 1 - 3} = F_{a} = F_{a i r - s p r i n g} + F_{d a m p e r}

(20)

where F_air-spring is the air spring’s force response, F_damper is the damper’s force response.

Air spring’s dynamic characteristics have been researched in the existing studies.^1,2,5–7 Therefore, to simplify the calculation process the force response F_air-spring, dynamic parameters in air spring including the K_v, K_e, and M are calculated as follows: ^5,6

K_{v} = K_{e} (V_{r 0} / V_{b 0})

(21)

K_{e} = γ p_{0} Γ Λ (A_{p} / V_{r 0})

(22)

M = ρ L_{p} Γ Λ^{2}

(23)

where Γ = A_e²/A_p and Λ = V_r0/(V_b0 + V_r0). V_r0 is the reservoir’s design volume. V_b0 is air spring’s design volume. γ is the air’s polytropic rate. p₀ is air’s pressure.

Based on the calculation result of air spring’s viscous force (F_v) in Ref. 6 and damping parameter C_β-act controlled by the valve, the F_v is expressed by:

\begin{array}{l} F_{v} = K_{v} (Z_{b} - Z_{a} - δ) \\ = {\begin{cases} C_{β} Ψ + M \ddot{δ} with PS \\ (C_{β} + C_{β - a c t}) Ψ + M \ddot{δ} with ASC \end{cases} \end{array}

(24)

where

Ψ = {(Γ Λ / A_{e})}^{1 + β} {| \dot{δ} |}^{β} s i g n (\dot{δ})

C_{β - a c t} = ρ A_{p} C_{v - a c t} / 2

, and C_v-act is the loss parameters of the valve.

Thus, the air spring’s has been calculated by:

F_{a i r - s p r i n g} = {\begin{cases} F_{e} + C_{β} Ψ + M with PS \\ F_{e} + (C_{β} + C_{β - a c t}) Ψ + M \ddot{δ} with DC \end{cases}

(25)

where F_e = K_eZ_ba and Z_ba = Z_b − Z_a.

Besides, F_damper of the damper has been calculated by:

\begin{array}{l} F_{d a m p e r} = {\begin{cases} C_{p} {\dot{Z}}_{b a} \\ (C_{p} + C_{a c t}) {\dot{Z}}_{b a} \end{cases} & \begin{array}{l} with PS \\ with DC \end{array} \end{array}

(26)

By combining Equations (25) and (26), the force response F_a of the air suspension in equation (20) with three different control cases using the ASC in Figure 2(b), using the DC in Figure 2(c), and using the ASDC in Figure 2(d) has been rewritten by:

F_{a} = {\begin{cases} F_{e} + C_{β} Ψ + M \ddot{δ} + C_{p} {\dot{Z}}_{b a} with PS \\ F_{e} + (C_{β} + C_{β - a c t}) Ψ + M \ddot{δ} + C_{p} {\dot{Z}}_{b a} with ASC \\ F_{e} + C_{β} Ψ + M \ddot{δ} + (C_{p} + C_{a c t}) {\dot{Z}}_{b a} with DC \\ F_{e} + (C_{β} + C_{β - a c t}) Ψ + M \ddot{δ} + . . . \\ . . . + (C_{p} + C_{a c t}) {\dot{Z}}_{b a} with ASDC \end{cases}

(27)

Vibration of road surface

The excitation of the road greatly affects the vehicle’s vibration and it is used to evaluate the isolation ability in air suspension systems. The road surface could be represented with a realization of the random process described by the power spectral density.^7,9,17,19 The power spectral density in the random road has been expressed as follows:

S (n) = S (n_{0}) {(n / n_{0})}^{- 2}

(28)

where S(n) is the road’s power spectral density (m³). S(n₀) is roughness coefficient. n₀ = 1/2π is reference frequency (1/m). n is the spatial frequency (1/m).

Assuming the roughness of the road surface is built based on the random process of Gaussian. Thus, it is generated via the function of Fourier transformation by:

q (t) = \sum_{i = 1}^{N} \sqrt{2 S (n_{i}) Δ n} \cos (2 π n_{i} t + ϕ_{i})

(29)

where Δn = 2π/l, l is the length of the road segment, N is the number of intervals, and φ_i is the random phase uniformly distributed from 0 to 2π.

To build the road surface roughness for simulating the vibration of the heavy vehicle, based on the random function of q(t) and the different classes of the road from B-class to E-class carried out based on the International Organization for Standardization (ISO) 8068.¹⁹ The simulating results of the typically random surfaces of the road from ISO B-class to E-class are shown in Figure 3.

Figure 3.

Random excitations of heavy vehicle.

Evaluation index of the isolation performance

In the evaluation of the isolation ability in vehicle’s suspension systems, three indexes of the riding quality, friendliness-road surface, and suspension’s working space have been applied as the evaluating indexes, especially the riding quality for automotives and friendliness of the wheel and road for heavy vehicles.^7,9,14,17,20 With the heavy vehicle used in this study, the riding quality of heavy vehicle has been assessed through root mean square values of vertical and pitching acceleration responses of the cabin (a_wzc and a_wϕc). These two values are calculated based on ISO 2631-1 as follows²⁰:

a_{w z c} = \sqrt{\frac{1}{T} \int_{0}^{T} {\ddot{Z}}_{c}^{2} (t) d t}

(30)

a_{w φ c} = \sqrt{\frac{1}{T} \int_{0}^{T} {\ddot{φ}}_{c}^{2} (t) d t}

(31)

where

{\ddot{Z}}_{c}

is the cabin’s vertical acceleration.

{\ddot{φ}}_{c}

is the cabin’s pitching acceleration. T is the simulation time.

The friendliness-road surface of heavy vehicle is evaluated via the interaction between the dynamics force of the road and wheel defined by the CDL as^7,11:

{CDL}_{n} = \frac{F_{w z, t n}}{F_{s t a t i c, n}} = \frac{\sqrt{T^{- 1} \int_{0}^{T} F_{t n}^{2} (t) d t}}{F_{s t a t i c, n}}

(32)

where F_wz,tn is the root mean square force of the wheel n. F_static,n is the static load at the wheel n.

The research in Ref. 7 showed that with 0 < CDL < 0.05, this means that the heavy vehicle is traveling on a very smooth road, with 0.05 ≤ CDL ≤ 0.4, the heavy vehicle is traveling on a normal road, and with CDL > 0.4, this means that the heavy vehicle is traveling on a very poor road and the contact of road and wheel was broken.

Therefore, to evaluate the control ability in air suspensions of heavy vehicles using the ASC, DC, and ASDC in ameliorating the riding quality and friendliness-road surface, the smaller values of the a_wzc, a_wϕc, and CDL_n have been used as the evaluation indexes.

Control of heavy vehicle air suspension

Applying fuzzy control

Fuzzy control is a controller that does not depend on the working conditions of the system, accordingly, it is applied in various control fields, especially automobile suspension control.^7,13–15 However, its control efficiency depends on control rules. In order to enhance the fuzzy control’s efficiency, in this study, its control rules are optimized by a genetic algorithm, and it is then applied to control vehicle’s air suspensions. In air suspensions of heavy vehicle, all the front air suspension, middle air suspension, and rear air suspension need to be controlled. However, the design and application of fuzzy logic control on the heavy vehicle’s control air suspensions are similar. Thus, only one fuzzy control has been designed and applied to control C_v-act of the ASC, C_act of the DC, or both C_v-act and C_act of ASDC in equation (27) as follows:

Based on two input signals of the displacement and speed in vehicle’s air suspension defined by e = Z_b − Z_a and ec = d(Z_b − Z_a)/dt the control rules in fuzzy-inference-system (FIS) optimized by genetic algorithm are then computed and given the optimal damping parameter of C_v-act with the ASC, C_act wiht the DC, or both C_v-act and C_act with the ASDC to control the heavy vehicle’s air suspension (see Figure 4). In the control process of the fuzzy control, two input signals {e, ec} in the fuzzification-interface have been transformed into linguistic variables, the fuzzy-inference-system then computes control values through the optimal inference rules, finally, the defuzzification-interface transforms the linguistic variables to the physical quantities of output signal “C_v-act with the ASC, C_act wiht the DC, or both C_v-act and C_act”.¹³

Figure 4.

Control model of the heavy vehicle’s air suspension using optimal control algorithm.

The fuzzy control is designed by three steps including (1) Establishing the linguistic variables and in/output values: Linguistic variables of input/output in fuzzy control have been designed by “Negative big (Nb), Negative small (Ns), Zero (Zo), Positive small (Ps), and Positive big (Pb)” and “Small (S), Medium small (Ms), Medium (M), Medium big (Mb), and Big (B)”. Besides, input values of e and ec are established by −0.1 ≤ e ≤ 0.1 and −1.0 ≤ ec ≤ 1.0 while the output values of C_v-act and C_act are established by 0.02 × 10⁴ ≤ C_v-act ≤ 1.00 × 10⁴ and 2.0 × 10⁴ ≤ C_act ≤ 4.0 × 10⁴, as provided in Table 1. (2) Establishing the membership function: The Triangular function is applied to built membership functions in input/output variables. (3) Establishing the initial control rules in FIS: Initial rules have been established by “if e is (Nb, Ns, Zo, Ps, or Pb) and ec is (Nb, Ns, Zo, Ps, or Pb) then C_v-act and C_act are (S, Ms, M, Mb, or B)”. Then, the centroid methods in Mamdani¹³ are used to compute the control quantities in FIS.

Table 1.

Input/output variables with the air springs controlled and damper controlled.

Input linguistics	Nb	Ns	Zo	Ps	Pb
e (m)	−0.1	−0.05	0.00	0.05	0.10
ec (m/s)	−1.0	−0.5	0.0	0.5	1.0
Output linguistics	S	Ms	M	Mb	B
C_v-act (kNs/m)	0.2	0.4	0.6	0.8	1.0
C_act (kNs/m)	20	25	30	35	40

In order to enhance the efficiency of the fuzzy control, FIS’s initial rules are then optimized by using the genetic algorithm in Optimizing control rules.

Optimizing control rules

Genetic algorithm is applied for searching random variables of n = [n₁, n₂, …, n_x]^T to obtain the minimum A(n) or maximum A(n) as follows^16,17:

A (n) = {[a_{1} (n), a_{2} (n), . . ., a_{x} (n)]}^{T}

(33)

Subject to p_x(n) ≤ 0, x = 1, 2, … , X and q_y(n) = 0, y = 1, 2, … , Y.

In this study, the genetic algorithm is used for optimizing the control rules in fuzzy control to obtain minimum values of a_wzc, a_wϕc, and CDL_n. The optimization process is performed via three steps as follows: (1) Establishing the initial population and encoding: To simplify the process of finding control rules in the FIS, the input variables of e and ec defined by “Nb, Ns, Zo, Ps, and Pb” and output linguistic variables of C_v-act or C_act defined by “S, Ms, M, Mb, and B” have been encoded by [0, 1, 2, 3, 4] = [a_i]_1×5 and [5, 6, 7, 8, 9] = [b_i]_1×5. In addition, the size in initial populations is set up by 100, in which individuals in 100 of initial populations have been randomly created and each gene in an individual has been also randomly selected by 0 ≤ a_i ≤ 4 and 5 ≤ b_i ≤ 9.

(2) Objective functions and fitness values: To obtain minimum values of a_wzc, a_wϕc, and CDL_n in equations (30)–(32), the fitness value of J₁ given to calculate the ride quality and the fitness value of J₂ given to calculate the friendliness-road surface are defined by:

J_{1} = 1 / (α_{1} a_{w z c} + α_{2} a_{w φ c})

(34)

J_{2} = 1 / \sum_{n = 1}^{5} β_{n} {CDL}_{n}

(35)

where α₁ and α₂ are the weight coefficients between a_wzc and a_wϕc, β_n is the weight coefficient of the wheel axle n. Additionally, a constant ratio of λ = J₁/J₂ is given to ensure optimal weighting between J₁ and J₂. In the optimization, the individuals obtained by the higher J₁ and J₂ values show that FIS’s rules obtained are also better. These individuals have been then updated. After the optimization process, the optimal control rules in fuzzy control could be reached. The optimization flowchart is given in Figure 5.

Figure 5.

Simulation and optimization flowchart of the heavy vehicle and genetic algorithm.

(3) Genetic operations: For enhancing the optimization efficiency and reducing the optimization time of genetic algorithm, a genetic operation with 5% of mutation probability and 95% of crossover probability in 1000 of the evolutionary generations has been established to perform the process of the crossover and mutation to create new individuals. Then, via the obtained results of J₁ and J₂, only optimal control rules that satisfy the condition will exist in the next generation. This process is repeated until the stopping condition of the genetic algorithm is satisfied.

Optimized results in fuzzy control’s rules

To optimize control rules, based on passive parameters of heavy vehicle and weight coefficients α, α₂, and β_n listed in Table 2,^8,12 random road surfaces of ISO C-class in Figure 3, and the moving speed 20 m/s of heavy vehicle, the optimization process is performed to search optimal rules to control C_v-act for ASC and C_act for DC. MATLAB software is used for the simulation. After the optimization process, the higher values of J₁,₂ and satisfying the condition of the constant ratio λ = J₁/J₂ have been saved and plotted in Figure 6(a) and 6(b).

Table 2.

Heavy vehicle’s dynamics parameters.

Parameter	Value	Parameters	Value
M_c (kg)	585	C_st1 (kNs²/m²)	4.127
M_td (kg)	4871	C_st2 (kNs²/m²)	32.58
M_t kg)	27542	C_st3 (kNs²/m²)	28.75
M_a1 (kg)	584	C_t1 (kNs/m)	1.105
M_a2 (kg)	1554	C_tn (kNs/m)	2.414
M_a3 (kg)	1314	K_c1 (kN/m)	100
C_c1 (kNs/m)	0.750	K_c2 (kN/m)	100
C_c2 (kNs/m)	0.750	K_v1 (kN/m)	190
C_p1 (kNs/m)	12.77	K_v2 (kN/m)	1500
C_p2 (kNs/m)	25.54	K_v3 (kN/m)	1324
C_p3 (kNs/m)	33.09	K_e1 (kN/m)	113
K_e2 (kN/m)	893	K_t1 (kN/m)	897
K_e3 (kN/m)	788	K_tn (kN/m)	179

Figure 6.

(a) Optimal rules for air springs controlled and (b) for damper controlled.

Figure 6(a) and 6(b) indicate that the maximum values of J₁,₂ for ASC have been reached after 861 of the generation and maximum values of J₁,₂ for DC have been reached after 835 of the generation. Therefore, optimal control rules of C_v-act for ASC and C_act for DC could be reached at the 861 and 835 generations. By recording the optimal control rules of the C_v-act for the ASC at the 861 generations and the C_act for the DC at the 835 generations. Their optimal rules have been provided in Tables 3 and 4. These optimal rules have been then applied for controlling heavy vehicle’s air suspensions as well as comparing the control performance between the ASC, DC, and ASDC.

Table 3.

Fuzzy control’s optimal rules with the air springs controlled.

C _v-act		e
C _v-act		Nb	Ns	Zo	Ps	Pb
ec	Nb	B	M	Mb	S	S
	Ns	Mb	Mb	S	S	S
	Zo	M	Ms	S	Ms	Ms
	Ps	S	S	Ms	Mb	B
	Pb	S	S	M	M	B

Table 4.

Fuzzy control’s optimal rules with the damper controlled.

C _act		e
C _act		Nb	Ns	Zo	Ps	Pb
ec	Nb	B	B	Mb	S	S
	Ns	Mb	Mb	M	S	S
	Zo	M	Ms	M	S	Ms
	Ps	Ms	S	Ms	Ms	S
	Pb	S	S	S	Ms	M

Simulation and analysis results

Isolation ability of air suspensions using three various control cases

In order to assess the isolation efficiency of air suspensions using three different control cases of the ASC, DC, and ASDC, from design parameters of heavy vehicle provided in Table 2 and optimal control rules of C_v-act and C_act listed in Tables 3 and 4, the MATLAB/SIMULINK software is used for the simulation. The simulation results in the cabin’s accelerations and wheel axle’s dynamics force response when heavy vehicle is traveling at the road surface of ISO C-class at v = 20 m/s are plotted in Figure 7. From the cabin’s acceleration responses and dynamics forces at wheels simulated in Figure 7, the results of the a_wzc, a_wϕc, and CDL_n with the PS, ASC, DC, and ASDC are also computed and listed in Table 5. Figure 7 and Table 5 show that with heavy vehicle using the PS, both the cabin’s acceleration responses and dynamics force responses at the wheel axles are very high, thus, the riding quality and friendliness-road surface of the heavy vehicle are low. Moreover, the value of CDL₃ is higher than the values of CDL₁, CDL₂, CDL₄, and CDL₅.

Figure 7.

(a) Cabin’s vertical acceleration, (b) cabin’s pitch acceleration and (c) force response of 3rd wheel axle on ISO C-class.

Table 5.

Result of root mean square accelerations and CDL_n.

Three control cases	Root-mean-square accelerations		Value of CDL
Three control cases	a_wzc (m/s²)	a_wϕc (rad/s²)	CDL₁	CDL₂	CDL₃	CDL₄	CDL₅
PS	0.586	0.121	0.061	0.176	0.187	0.158	0.172
ASC	0.461	0.092	0.049	0.145	0.153	0.129	0.140
DC	0.407	0.088	0.052	0.157	0.163	0.139	0.148
ASDC	0.348	0.078	0.044	0.140	0.147	0.122	0.131
Reduction (%)
ASC versus PS	21.33	23.96	19.67	17.61	18.18	18.35	18.60
DC versus PS	30.54	27.27	14.75	10.79	12.83	12.02	13.95
ASDC versus PS	40.61	35.54	27.87	20.45	21.39	22.78	23.84
ASC versus DC	−13.26	−4.54	5.77	7.64	6.13	7.19	5.40

Thus, the influence of the third wheel axle on damaging road surface is high, so it needs to be evaluated in the design and control process of the heavy vehicle’s suspension systems. With heavy vehicle’s air suspensions applying the ASC, DC, and ASDC, both the cabin’s acceleration responses and dynamics force responses at wheel axles have been greatly decreased compared to the PS. This means that the riding quality and friendliness-road surface of the heavy vehicle is improved by using control air suspensions.

With using three different control cases of the ASC, DC, and ASDC, results in Table 5 show that both a_wzc and a_wϕc of the ASC, DC, and ASDC are strongly decreased by {21.33% and 23.96%}, {30.54% and 27.27%}, and {40.61% and 35.54%} compared to the PS. Besides, the maximum CDL₃ with the ASC, DC, and ASDC is also strongly decreased by 18.18%, 12.83%, and 21.39% compared to the PS, especially with ASDC. In addition, the a_wzc and a_wϕc of the DC are smaller than that of the ASC by 13.26% and 4.54% whereas the maximum CDL₃ with the DC is higher than that of the ASC by 6.13%. These results mean that the DC can improve the riding quality of heavy vehicle better than the ASC while ASC can improves the friendliness-road surface of heavy vehicle better than the DC. This is the reason why most damping coefficients in car’s isolations are mainly controlled to improve the riding comfort^9,13,14 while the heavy vehicle’s suspension systems are mainly equipped with air suspensions or control air suspensions to reduce the damaging road surface.^4,5,7

With using the ASDC, results in Table 5 indicate that all a_wzc, a_wϕc, and CDL_n are also lower than that of the ASC and DC. Therefore, in order to improve both riding quality and friendliness-road surface in heavy vehicle simultaneously, the control method of the ASDC should be applied.

Control performance of air suspensions under different speeds

With heavy vehicle traveling on the road, the moving speed of the vehicle can be changed and affect the control efficiency of the ASC, DC, and ASDC. To assess the stability of these three different control cases, the different speeds changed from 2.5 to 30 m/s of heavy vehicle are simulated. Results in a_wzc, a_wϕc, and CDL₃ have been shown in Figure 8(a)–8(c), respectively.

Figure 8.

(a) Cabin’s vertical acceleration, (b) cabin’s pitch acceleration and (c) CDL₃ at 3rd wheel axle under various speeds.

Results with PS, ASC, DC, and ASDC in Figure 8 indicate that the heavy vehicle’s moving speeds greatly affect the riding quality and friendliness-road surface. All values of the a_wzc, a_wϕc, and CDL₃ are low when the vehicle is moving at speeds of 2.5÷12.5 m/s. This means that the riding quality and friendliness-road surface in the vehicle are good. When the vehicle is moving at speeds of 12.5÷20 m/s, all values of a_wzc, a_wϕc, and CDL₃ are strongly increased, especially at 17.5 m/s. This means that both riding quality and friendliness-road surface in the vehicle are strongly decreased. Thus, the heavy vehicle should limit movement at this speed range to ameliorate the riding quality and friendliness-road surface. When the vehicle’s moving speed is increased from 20÷30 m/s, both values of a_wzc and a_wϕc are insignificantly changed while value of CDL₃ is strongly reduced from 20÷25 m/s and strongly increased from 25÷30 m/s. Therefore, in order to ameliorate the riding quality and friendliness-road surface, heavy vehicle’s speeds from 20÷25 m/s should be used when the vehicle is moving. This is also the reason why highways often regulate the heavy vehicles to travel at the speeds of 20÷25 m/s.^7,12

With using different methods of the ASC, DC, and ASDC in heavy vehicle’s air suspensions, all values of a_wzc, a_wϕc, and CDL₃ are strongly decreased in comparison with PS, especially, the a_wzc, a_wϕc, and CDL₃ with the ASDC are the smallest under the different moving speeds of the heavy vehicle. Besides, both a_wzc and a_wϕc with the DC are lower than that of the ASC (see Figure 8(a) and 8(b)) while CDL₃ with the DC is higher than that of the ASC (see Figure 8(c)) under the vehicle’s different moving speeds. This means that the control performance between the ASC, DC, and ASDC is relatively stable under various moving speeds.

Control performance of air suspensions under different road surfaces

With heavy vehicle traveling on the road, the road’s height of the road surface roughness can also affect the riding quality and friendliness-road surface of the vehicle as well as the control performance of the ASC, DC, and ASDC. To assess the performance and the stability of these three different control cases, four different road surface roughnesses of ISO B-class, C-class, D-class, and E-class have been simulated. Simulation results of the a_wzc, a_wϕc, and CDL₃ are shown in Figure 9(a)–9(c), respectively.

Figure 9.

(a) Cabin’s vertical acceleration, (b) cabin’s pitch acceleration and (c) CDL₃ at 3rd wheel axle under various road surfaces.

From Figure 9, we see that both the riding quality and friendliness-road surface of heavy vehicle with the PS, ASC, DC, and ASDC are greatly influenced by the road surface’s height. All values of a_wzc, a_wϕc, and CDL₃ are increased when the heavy vehicle is moving on the road from the B-class (good road) to E-class (very poor road). This means that the riding quality and friendliness-road surface of the vehicle are strongly reduced when the height of the road surface roughness is increased. Particularly, on the ISO E-class of the very poor road surface, the values of CDL₂ and CDL₃ are increased over 0.4. Thus, the contact between the second/third axles of the heavy vehicle and road surface has been broken and the road surface is strongly damaged.

With the heavy vehicle air suspensions controlled by the ASC, DC, and ASDC, the results in Figure 9 show that all values of the a_wzc, a_wϕc, and CDL₃ are strongly decreased in comparison with PS under all various road surfaces, especially with the ASDC. Besides, both a_wzc and a_wϕc with the DC are also lower than that of the ASC (see Figure 9(a) and 9(b)) while CDL₃ with the DC are also higher than that of the ASC (see Figure 9(c)) under all different road surfaces. This means that the control performance between the ASC, DC, and ASDC is also relatively stable under all various road surfaces of the vehicle moving.

Conclusions

The values of CDL₂ and CDL₃ at the second and third axle of the wheel are bigger than the values of CDL₁, CDL₄, and CDL₅ at the first, fourth, and fifth axle of the wheel. Therefore, the second wheel and third wheel bigger impact the road surface.

The road surface roughness and vehilce’s moving speed greatly affect two indexes of the riding quality and friendliness-road surface. To ameliorate these indexes, the moving speeds from 20÷25 m/s should be used in the moving process of heavy vehicle. Besides, the roughness of the road surface of ISO E-class should be also repaired to ensure safe movement of the heavy vehicle.

With ASC, DC, and ASDC applied, all a_wzc, a_wϕc, and CDL_n are strongly decreased in comparison with PS, especially with ASDC. Additionally, both a_wzc and a_wϕc of the DC are smaller than that of the ASC by 13.26% and 4.54% whereas the maximum CDL₃ with the DC are higher than that of the ASC by 6.13%. Therefore, in order to ameliorate the riding quality of the heavy vehicle, the control method of DC should be applied. To enhance the friendliness-road surface, the control method of ASC should be applied. In order to improve both riding quality and friendliness-road surface, the control method of the ASDC should be applied for the heavy vehicle’s air suspensions.

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 iD

Yang Liu

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