Ride comfort performance of heavy truck with three control cases of semi-active isolation systems

2.1. Heavy truck dynamic model with three cases of the control vibrations

A three-axle heavy truck with the dependent suspension for the steering axle (1st axle) and the trailing arm type suspensions for two rear vehicle axles (2nd axle and 3rd axle) is selected for the vehicle dynamic analysis. A nonlinear dynamic model of vehicle with 8-DOF is established to analyze the performance of semi-active isolation systems, as modeled in Fig. 1, where ($z_s$, $z_c$, $z_b$ and $z_j$) and ($m_s$, $m_c$, $m_b$, and $m_j$) are the vertical displacements and masses of the driver’s seat, the cab, the vehicle body, and the axles; ${\varphi }_{c,b}$ are the angular displacements of the cab and the vehicle body; $c_{wt}$ and $k_{wt}$ are damping and stiffness coefficients of wheels; $c_i$, $k_i$, and $u_i$ are damping, stiffness coefficients, and the control force of semi-active isolation systems, respectively; $F_i$ is the dynamic reaction forces of the driver’s seat suspension ($i=s$), cab’s suspensions ($i=$ $c_1$, $c_2$), and the vehicle’s suspensions ($i=$ 1-3); $q_{wj}$ is the excitation of the road on the tire; $l_v$ is the distances of the vehicle, ($j=$ 1-3 and $v=$ 1-7).

Fig. 1A heavy truck with semi-active isolation systems

a) The vehicle dynamic model

b) Semi-active isolation systems

The dynamic reaction forces $F_i$, and $F_{wj}$ of the suspension systems of the seat, cab, vehicle, and wheels are respectively written by:

where $n=$ 1-2, $l_p=$$l_2+l_3+l_4$; when $j=$ 1 then $m=$ 1, $l_p=l_5$; when $j=$ 2 then $m=$ 1, $l_p=l_4$; when $j=$ 3 then $m=$ 2, $l_p=l_6+l_7$.

Based on the vehicle dynamic model in Fig. 1, and by applying Newton’s second law of motion, the motion equations of the vehicle can be represented in the matrix form as follows:

where [M], [C] and [K] are ($a$×$a$) mass, damping and stiffness matrices, respectively, {$\mathbf{Z}$} is the ($a$×1) displacement vector, {$F\left(t\right)$} is the ($a$×1) exciting force vector, and $a$ is the number of DOF., ($a=$ 8).

2.2. Applying the fuzzy logic controller for semi-active isolation systems

In order to evaluate the ride comfort performance with three types of semi-active isolation systems of the heavy truck, three types of the control suspensions including the seat controlled ($u_s$), the cab controlled ($u_{c1,c2}$), and the vehicle controlled ($u_{\mathrm{1,2},3}$) are respectively given by:

Therefore, the different fuzzy logic controllers should be designed. However, design process of these controllers is same, thus a specific fuzzy control is designed. The design of a FLC is as follows: The deflection of the isolation systems “$E_i$” and its derivation “$E{C}_i$” are the two input variables while the variable damping forces “$u_i$” is the output value of the control isolation systems. The linguistic variables of input and output variables are defined by the positive big (PB), positive small (PS), zero (ZO), negative small (NS), and negative big (NB); and the membership functions for their variables are represented by a fuzzy set. The shape of membership functions is the Triangular function and their values are between 0 and 1, both input and output values are belonging to [–1, 1].

In this FLC, the if-then rules base are applied to describe according to expertise experiences and the designer’s knowledge, there are at most 25 possible rules, the fuzzy rules are given in Table 1, and written as follows $R_{i\alpha }$: If $E_{i\alpha }=A_{i\alpha }$ and $E{C}_{i\alpha }=B_{i\alpha }$ then $u_{i\alpha }=C_{i\alpha }$, ($\alpha =$ 1-25).

The fuzzy inference system is selected by the minimum function and the centroid method of Mamdani [14]. In this study, we used the FIS of Mamdani for control system model.

2.3. Excitation vibration sources

In order to analyze the ride comfort performance of heavy truck with different control cases of semi-active isolation systems, two types of excitation road including the step and random road surfaces are considered as follows:

The excitation of step road: The set of typical step road disturbances be of the form:

The excitation of a random road: A random road surface represented by a randomly modulated periodic is carried out in this study. The general form of the displacement power spectral density (PSD) of the road surface is determined by the experimental formula [15]:

where $n$ is space frequency; $n_0$ is reference space frequency, $n_0=$ 0.1 m^-1; $S_q\left(n\right)$ is the PSD of the road surface under the reference space frequency $n_0$; $\omega$ is the frequency index which determines the frequency configuration of the PSD of the road surface ($\omega =$ 2 rad.s^-1). The random road is assumed to be a zero-mean stationary Gaussian random process, and it can be generated by the Fourier transformation as follows:

where $N$ is the number of intervals, $\mathrm{\Delta }n=2\pi /L$ and $L$ is the length of the road segment, ${\varphi }_i$ is a random phase uniformly distributed from 0 to 2$\pi$, and $\mathrm{\Delta }\omega =\mathrm{\Delta }n{v}_0$ is the fundamental temporal frequency, $v_0$ is the vehicle speed.

Time histories of the excitation vibrations at speed $v_0=$ 20 m.s^-1 are shown in Fig. 2.

Fig. 2Time history of the excitation road sources

a) Step road surface

b) Random road surface of ISO level C [16]

3.1. The performance on a step road surface

Three control cases of the semi-active isolation systems including the seat controlled, the cab controlled, and the vehicle controlled are respectively simulated under the condition of the vehicle traveling on a step road surface. The simulation results of the acceleration responses of the vertical driver’s seat and cab’s pitch angle are plotted in Fig. 3(a) and 3(b).

With the seat controlled, the simulation results are shown that the acceleration response of the vertical driver’s seat is greatly reduced in comparison with uncontrolled. However, the acceleration response of the cab’s pitch angle is insignificantly improved because the only ability of the seat controlled is to control the vibration in the vertical direction of the seat.

With the vehicle controlled, both the acceleration responses of the driver’s seat and cab’s pitch angle are significantly improved in comparison with uncontrolled. Besides, the acceleration responses of the driver’s seat and cab’s pitch angle with the cab controlled are strongly reduced comparable with both the vehicle controlled and uncontrolled. Therefore, in order to enhance the vertical ride comfort of the driver’s seat, the seat controlled should be used, whereas, in order to improve the vertical ride comfort of the driver’s seat and controlling the cab’s shake, the cab controlled or the vehicle controlled should be used.

Fig. 3The acceleration responses under the excitation of a step road surface

a) The vertical driver’s seat

b) The cab’s pitch angle

3.2. The performance on a random road surface

In the condition of the vehicle traveling on a road roughness surface, the simulation results of the acceleration responses of the vertical driver’s seat and cab’s pitch angle with three control cases are shown in Fig. 4(a) and 4(b). The comparison results are also similar in the condition of the vehicle traveling on a step road surface. The vertical driver’s seat is clearly improved by the seat controlled whereas both the vertical driver’s seat vibration and the cab’s shake are greatly reduced with the cab controlled.

Fig. 4The acceleration responses under the excitation of a random road surface

a) The vertical driver’s seat

b) The cab’s pitch angle

According to the international standard ISO 2631-1, the effect of vibration on ride comfort is evaluated based on the weighted root-mean-square (RMS) acceleration responses ($a_{wz}$) [16]. The weighted RMS values of the vertical driver’s seat ($a_{ws}$) and the cab’s pitch angle ($a_{w\varphi c}$) are listed in Table 2. Comparing with uncontrolled, the $a_{ws}$ and $a_{w\varphi c}$ are decreased by 31.4 % and 7.1 % with the seat controlled, decreased by 24.5 % and 30.3 % with the cab controlled, and decreased by 19.0 % and 21.2 % with the vehicle controlled. Therefore, both $a_{ws}$ and $a_{w\varphi c}$ values of the cab controlled are greatly reduced. The simulation and analysis results show that all the heavy vehicles should be controlled by the cab’s suspension systems or the vehicle’s suspension system to improve the vehicle ride comfort. This is the reason that most heavy truck’s cab suspension systems are researched and controlled [1, 10]. Besides, the road friendliness of the heavy trucks is being concerned about using the vehicle’s air suspension systems or the vehicle’s semi-active suspension systems [3-6].