1. Introduction

Commercial vehicles are an essential part of the modern transport network and are responsible for the bulk of freight transport around the world. The demand for better ride comfort and safety in these vehicles is increasing, as such commercial vehicle ride comfort optimization has been an active area of research. Ride comfort in commercial vehicles provides the following advantages: (1) increased driver’s comfort and driving safety, ensuring good ride ability, and reducing the incidence of traffic accidents; (2) assurance goods arrive in better condition, increasing the utility value of commercial vehicles; and (3) improved service life of commercial vehicle parts, as vehicles are subjected to reduced impact and vibration forces constantly [1–4].

Typically, software used for vehicle ride comfort simulation includes Adams and Matlab. The former adopts the structure-oriented modeling method, which is more accurate, and the simulation is reliable. However, the modeling procedures need to set characteristic parameters and curves for a large number of parts, resulting in a complex and time-consuming modeling solution. The latter establishes a ride comfort model from the perspective of mechanics and mathematics with lower accuracy but simpler modeling, a shorter simulation time, and convenient parameter setting and adjustment, with a powerful ability to perform algorithm-based optimization.

Many studies have been devoted to vibration control and ride comfort optimization of vehicles. Ding et al. [5] established an eleven degree-of-freedom (DOF) ride comfort model for a three-axle truck that assumes the body is slightly vibrating near the equilibrium position. Li et al. [6] studied the synthetic mechanical model of plane rigid-frame heavy-duty vehicles, considering only the microvibrations caused by road excitation. A dynamic model is developed that aims to reduce the road excitation transmitted to the vehicle body and improve the ride comfort by using optimization algorithms. The spring stiffness and damping coefficient are regarded as a fixed value [7–9]. The ride model is used to study the relationship between ride comfort and the in-wheel motor. There is a linear relationship between damping and damping force [10, 11]. Zhu et al. [12] established a ten-DOF ride model for analyzing vehicle ride comfort and driving safety, and the damping force is considered a linear parameter in this analysis.

Most of the research on simulation and optimization for vehicle ride comfort make the following assumptions on the actual situation: (1) the microvibration caused by road excitation is only considered and (2) the damping force is a linear function of its velocity and is regarded a constant value (equivalent damping). The research on ride comfort mainly focuses on equivalent damping when the model consists of passive suspension, with work based on nonlinear damping poorly established. However, in terms of the analysis for the ride comfort of commercial vehicles, the above assumptions are different from the actual situation. Commercial vehicle suspension damping is mostly a nonlinear damping curve that varies with extension or compression velocity. In addition, the driving conditions of commercial vehicles is far from ideal, with additional factors contributing to this, which goes against the assumption that only microvibration should be considered. Furthermore, this work is mainly focused on ride comfort and disregards the suspension working space and dynamic tyre load, which affect handling stability. The reduction of suspension stiffness has a negative effect on the suspension working space and dynamic tyre load.

In this paper, the ride comfort models of a commercial vehicle based on nonlinear damping and equivalent damping are developed. The responses of theoretical models are verified by simulation performed in Adams/Car on a multibody dynamic model of the target vehicle. To improve ride comfort, more performance criteria are considered and optimized. To achieve this, a ride comfort optimization method based on nonlinear damping of suspension and intelligent algorithms is proposed. The objective functions are optimized by algorithms, and the optimal design variables that consider nonlinear damping are obtained. Lastly, the theoretical model and optimization are validated by experiment. In short, the method is more practical and accurate for vehicle ride comfort optimization compared to those based on equivalent damping.

2. Ride Comfort Model

In this section, the equations are serially developed for the vibration system, road excitation model, nonlinear damping, and equivalent damping model. The effectiveness of the ride comfort model using nonlinear damping is demonstrated and compared with equivalent damping and multibody dynamic models.

2.1. Vibration Model

Generally, there are three types of vehicle ride comfort models, including the 1/4 vehicle model, half vehicle model, and full vehicle model. When a vehicle is symmetrical to its longitudinal axis, the main vibrations affecting ride comfort are vertical vibration along the z-axis and pitch vibration around the y-axis of the vehicle coordinate system. Therefore, a half commercial vehicle model that consists of five basic subsystems of the tires, suspension systems, frame, cab, and seat is selected in this paper, as shown in Figure 1. For this model, the vibration differential equation (1) is generated by using Newton’s second law of motion:where x_f and x_r are the displacement along the z-axis of the front wheel and the rear wheel, respectively; x_fu, x_ru, x_b, x_c, and x_p represent the displacement of the front axle unsprung mass, rear axle unsprung mass, sprung mass, mass of cab, and mass of the seat, respectively; θ_c, θ_b, and θ are the pitch angle displacements of the cab, sprung mass, and rear axle balance suspension around the y-axis, respectively. Moreover, the fixed parameters are presented in Table 1, and the nonlinear damping curves are shown in Figure 2(b).

$$\begin{array}{}\left\{\begin{array}{c}{m}_{\text{p}}{\ddot{x}}_{\text{p}}=-k_{\text{p}}\left(x_{\text{p}}-x_{\text{c}}-l_1{\theta }_{\text{c}}\right)-c_p\left({\dot{x}}_{\text{p}}-{\dot{x}}_{\text{c}}-l_1{\dot{\theta }}_{\text{c}}\right),\\ m_{\text{c}}{\ddot{x}}_{\text{c}}=k_{\text{p}}\left(x_{\text{p}}-x_{\text{c}}-l_1{\theta }_{\text{c}}\right)+c_{\text{p}}\left({\dot{x}}_{\text{p}}-{\dot{x}}_{\text{c}}-l_1{\dot{\theta }}_{\text{c}}\right)-k_{\text{fc}}\left(x_{\text{c}}-x_{\text{b}}-l_2{\theta }_{\text{c}}+\left(l_2+l_4\right){\theta }_{\text{b}}\right)-c_{\text{fc}}\left({\dot{x}}_{\text{c}}-{\dot{x}}_{\text{b}}-l_2{\dot{\theta }}_{\text{c}}+\left(l_2+l_4\right){\dot{\theta }}_{\text{b}}\right)-k_{\text{rc}}\left(x_{\text{c}}-x_{\text{b}}+l_3{\theta }_{\text{c}}+\left(l_4-l_3\right){\theta }_{\text{b}}\right)-c_{\text{rc}}\left({\dot{x}}_{\text{c}}-{\dot{x}}_{\text{b}}+l_3{\dot{\theta }}_{\text{c}}+\left(l_4-l_3\right){\dot{\theta }}_{\text{b}}\right),\\ J_{\text{c}}{\ddot{\theta }}_{\text{c}}=\left[k_{\text{p}}\left(x_{\text{p}}-x_{\text{c}}-l_1{\theta }_{\text{c}}\right)+c_{\text{p}}\left({\dot{x}}_{\text{p}}-{\dot{x}}_{\text{c}}-l_1{\dot{\theta }}_{\text{c}}\right)\right]l_1+\left[k_{\text{fc}}\left(x_{\text{c}}-x_{\text{b}}-l_2{\theta }_{\text{c}}+\left(l_2+l_4\right){\theta }_{\text{b}}\right)+c_{\text{fc}}\left({\dot{x}}_{\text{c}}-{\dot{x}}_{\text{b}}-l_2{\dot{\theta }}_{\text{c}}+\left(l_2+l_4\right){\dot{\theta }}_{\text{b}}\right)\right]l_2-\left[k_{\text{rc}}\left(x_{\text{c}}-x_{\text{b}}+l_3{\theta }_{\text{c}}+\left(l_4-l_3\right){\theta }_{\text{b}}\right)+c_{\text{rc}}\left({\dot{x}}_{\text{c}}-{\dot{x}}_{\text{b}}+l_3{\dot{\theta }}_{\text{c}}+\left(l_4-l_3\right){\dot{\theta }}_{\text{b}}\right)\right]l_3,\\ m_{\text{b}}{\ddot{x}}_{\text{b}}=k_{\text{fc}}\left(x_{\text{c}}-x_{\text{b}}-l_2{\theta }_{\text{c}}+\left(l_2+l_4\right){\theta }_{\text{b}}\right)+c_{\text{fc}}\left({\dot{x}}_{\text{c}}-{\dot{x}}_{\text{b}}-l_2{\dot{\theta }}_{\text{c}}+\left(l_2+l_4\right){\dot{\theta }}_{\text{b}}\right)-k_{\text{fs}}\left(x_{\text{b}}-x_{\text{fu}}-l_5{\theta }_{\text{b}}\right)-c_{\text{fs}}\left({\dot{x}}_{\text{b}}-{\dot{x}}_{\text{fu}}-l_5{\dot{\theta }}_{\text{b}}\right)+k_{\text{rc}}\left(x_{\text{c}}-x_{\text{b}}+l_3{\theta }_{\text{c}}+\left(l_4-l_3\right){\theta }_{\text{b}}\right)+c_{\text{rc}}\left({\dot{x}}_{\text{c}}-{\dot{x}}_{\text{b}}+l_3{\dot{\theta }}_{\text{c}}+\left(l_4-l_3\right){\dot{\theta }}_{\text{b}}\right)-k_{\text{rs}}\left(x_{\text{b}}-x_{\text{ru}}+l_6{\theta }_{\text{b}}\right)-c_{\text{rs}}\left({\dot{x}}_{\text{b}}-{\dot{x}}_{\text{ru}}+l_6{\dot{\theta }}_{\text{b}}\right),\\ J_{\text{b}}{\ddot{\theta }}_{\text{b}}=\left[k_{\text{fs}}\left(x_{\text{b}}-x_{\text{fu}}-l_5{\theta }_{\text{b}}\right)+c_{\text{fs}}\left({\dot{x}}_{\text{b}}-{\dot{x}}_{\text{fu}}-l_5{\dot{\theta }}_{\text{b}}\right)\right]l_5-\left[k_{\text{rs}}\left(x_{\text{b}}-x_{\text{ru}}+l_6{\theta }_{\text{b}}\right)+c_{\text{rs}}\left({\dot{x}}_{\text{b}}-{\dot{x}}_{\text{ru}}+l_6{\dot{\theta }}_{\text{b}}\right)\right]l_6-\left[k_{\text{fc}}\left(x_{\text{c}}-x_{\text{b}}-l_2{\theta }_{\text{c}}+\left(l_2+l_4\right){\theta }_{\text{b}}\right)+c_{\text{fc}}\left({\dot{x}}_{\text{c}}-{\dot{x}}_{\text{b}}-l_2{\dot{\theta }}_{\text{c}}+\left(l_2+l_4\right){\dot{\theta }}_{\text{b}}\right)\right]\left(l_2+l_4\right)-\left[k_{\text{rc}}\left(x_{\text{c}}-x_{\text{b}}+l_3{\theta }_{\text{c}}+\left(l_4-l_3\right){\theta }_{\text{b}}\right)+c_{\text{rc}}\left({\dot{x}}_{\text{c}}-{\dot{x}}_{\text{b}}+l_3{\dot{\theta }}_{\text{c}}+\left(l_4-l_3\right){\dot{\theta }}_{\text{b}}\right)\right]\left(l_4-l_3\right),\\ m_{\text{fu}}{\ddot{x}}_{\text{fu}}=k_{\text{fs}}\left(x_{\text{b}}-x_{\text{fu}}-l_5{\theta }_{\text{b}}\right)+c_{\text{fs}}\left({\dot{x}}_{\text{b}}-{\dot{x}}_{\text{fu}}-l_5{\dot{\theta }}_{\text{b}}\right)-k_{\text{ft}}\left(x_{\text{fu}}-x_{\text{f}}\right),\\ m_{\text{ru}}{\ddot{x}}_{\text{ru}}=k_{\text{rs}}\left(x_{\text{b}}-x_{\text{ru}}+l_6{\theta }_{\text{b}}\right)+c_{\text{rs}}\left({\dot{x}}_{\text{b}}-{\dot{x}}_{\text{ru}}+l_6{\dot{\theta }}_{\text{b}}\right)-k_{\text{rt}}\left(x_{\text{ru}}-x_{\text{r}}\right),\end{array}\right.\end{array}$$ ()

Table 1. Fixed parameters of the vibration model.

Symbol	Description	Value	Unit
mp	Driver and seat mass	100	kg
mc	Cab mass	850	kg
mb	Sprung mass	14238	kg
mfu	Unsprung mass of front axle	607	kg
mru	Unsprung mass of rear axle	1054	kg
Jc	Rotational inertia of cab mass around y-axis	560	kg·m2
Jb	Rotational inertia of sprung mass around y-axis	115000	kg·m2
J	Rotational inertia of rear axle balance suspension mass around y-axis	615	kg·m2
kp	Seat stiffness	20000	N/m
kfc	Front suspended stiffness	25000	N/m
krc	Rear suspended stiffness	25000	N/m
kfs	Front suspension stiffness	410000	N/m
krs	Rear suspension stiffness	1476000	N/m
kft	Front wheel stiffness	1800000	N/m
krt	Rear wheel stiffness	3600000	N/m
cp	Equivalent damping coefficient of driver seat	800	N·s/m
cfs	Equivalent damping coefficient of front suspended	5000	N·s/m
crs	Equivalent damping coefficient of rear suspended	5000	N·s/m
cft	Equivalent damping coefficient of front suspension	12000	N·s/m
crt	Equivalent damping coefficient of rear suspension	15000	N·s/m
l1	Distance between the center and seat of the cab	0.2	m
l2	Distance between front suspended and the center of the cab	0.76	m
l3	Distance between rear suspended and the center of the cab	0.87	m
l4	Distance between the center of the cab and the center of target commercial vehicle	2.47	m
l5	Distance between the front suspension and the center of target commercial vehicle	2.1	m
l6	Distance between the rear suspension and the center of target commercial vehicle	3	m

2.2. Road Excitation Model

As the most important excitation source when driving, the acquisition of road excitation is highly significant for ride comfort analysis. The road level and vehicle speed are primary factors affecting vehicle vibration. Power spectral density (PSD) is usually used for statistical characteristics of road excitation [13, 14]. The PSD of road excitation satisfies

$$\begin{array}{}{G}_{\text{q}}\left(n\right)=G_{\text{q}}\left(n_0\right){\left(\frac{n}{n_0}\right)}^{-w},\end{array}$$ ()

where n is the spatial frequency; n₀ is the reference spatial frequency, n₀ = 0.1 m⁻¹; G_q(n₀) is the road surface PSD corresponding to the reference spatial frequency n₀, or the road excitation coefficient; and w is the frequency index that determines the frequency structure of the road surface PSD.

The time-domain model of road excitation based on filtered white noise is as follows:

$$\begin{array}{}\dot{q}\left(t\right)=-22\pi u{n}_{\text{q}}q\left(t\right)+\pi n_0\sqrt{G_{\text{q}}\left(n_0\right)u}\omega \left(t\right),\end{array}$$ ()

where u is the vehicle speed (m/s); n_q is the cutoff space frequency, n_q = 0.011 m⁻¹; q(t) is the displacement excitation of the road surface; n₀ represents the spatial reference frequency and is equal to 0.1 m⁻¹; G_q(n₀) is the road excitation coefficient, G_q(n₀) = 256 × 10⁻⁶ m³; and ω(t) is the Gauss white noise with zero mean.

Figure 3 shows a road profile when the vehicle travels on the C-level road at 60 km/h. The profile PSD and standard road PSD according to ISO 8608 are presented in Figure 4 and show that a simulation model of road excitation is feasible.

2.5. Verification of Ride Comfort Model

Equations (1), (3), and (5) have been solved using Matlab/Simulink and validated through a multibody dynamic model in Adams/Car. The multibody dynamic model is shown in Figure 5. The response of the driver seat, obtained from the Simulink models based on nonlinear damping, equivalent damping, and multibody dynamics model, is compared through time-domain curves and frequency-weighted RMS to verify the accuracy of the nonlinear damping ride model.

The time-domain curves are shown in Figure 6. Overall, the Matlab-based model has the same level of acceleration amplitude as the Adams-based model. The acceleration amplitude of the model based on nonlinear damping is greater than the one based on equivalent damping, leading to a higher frequency-weighted RMS, and the situation is consistent with the RMS in Figure 7. The difference in the curves is typically caused by the different methods of road modeling in Adams and Matlab and the bushings in the Adams model.

As shown in Figure 7, there is a similar tendency that ride comfort increases with speed among the three models. The results show that simulation results of the model based on the nonlinear damping are more consistent with the Adams model than the model based on equivalent damping. Therefore, the ride comfort model based on nonlinear damping has improved accuracy compared to equivalent damping models.

3. Multiobjective Optimization

In this section, the optimization of suspension systems based on nonlinear damping is presented. The efficiency and robustness of the optimization method affect the results. PSO, CS, DIRECT, and GA are chosen as the optimization methods to optimize performance with respect to comfort, dynamic tyre load, and suspension working space.

3.5. Nonlinear Ride Optimization Method

A ride potimization method based on nonlinear damping and intelligent lagorithms is presented, and the procedure is shown in Figure 8.

4. Optimization Results

In this section, the results of ride optimization are discussed. First, the iterations of algorithms are shown in Figure 9, and the optimal design variables are listed in Table 3. The responses, which represent the time-domain signals of objective functions based on original design variables and optimized ones, are shown in Figure 9. The indexes for objectives are calculated and presented in Table 4.

The iterations of ride comfort optimization are shown in Figure 9. Obviously, the CS, DIRECT, and PSO get the similar and ideal results besides GA, and the iteration of PSO is better than others, which is usually caused by the following reasons: (1) different algorithms have different characteristic for various problems and (2) the generation of individuals and population is random during the optimization processing. Consequently, it is necessary to compare the results among different algorithms that can ensure a relatively better result. Then, we listed the original variables and optimized ones in Table 3.

To verify the feasibility and effectiveness of the ride comfort optimization method, nonlinear damping based on intelligent algorithms is proposed in this paper. A comparative analysis of ride comfort based on the original and optimized design variables is performed. Design variables are substituted into the ride comfort model based on nonlinear damping. Then, the optimized responses of the time history of the acceleration of driver seat, dynamic tyre load, and suspension working space at 40 km/h and 80 km/h are obtained. Data are presented in Figure 10 in comparison with the original response. The evaluation indexes for objective functions are calculated from 30 to 100 km/h and shown in Table 4 according to equations (15), (19), and (21)

As shown in Table 4, the target commercial vehicle ride comfort is significantly improved after optimization. The proposed optimizer with intelligent algorithms and nonlinear damping successfully optimized the design variables with an average fall ratio of 27.4%, 21.6%, 25.0%, 19.3%, and 22.3% for f₁, f₂, f₃, f₄, and f₅. Additionally, the evaluation indexes for objective increase with increased vehicle speed; that is, ride comfort deteriorates totally as speed increases.

5. Experiment

To further verify the feasibility and efficiency of the ride optimization method in this paper, the ride comfort experiment of target commercial vehicle traveling on the C-level road with 30–100 km/h was conducted in accordance with GB/T 4970-2009 “Method of running test—Automotive ride comfort.” The experimental vehicle is shown in Figure 11(a). A section of the cement road is selected as the ride comfort experiment road, and its surface is shown in Figure 11(b). The sensor is used to measure the vertical vibration acceleration of the seat. The installation position is shown in Figure 11(c). The CPCI signal acquisition instrument is used for data acquisition, as shown in Figure 11(d).

The ride comfort experiment test requirements are as follows: (1) The road surface is even and straight, and there is an acceleration section initially. (2) The target commercial vehicle is fully loaded with standard driving equipment. The components of the vehicle are in good condition, and the tyre pressure complies with regulations. (3) When the test truck speed reaches the speed to be analyzed, the tester starts timing for 60 s. (4) The speed of vehicle ride comfort experiment is 30–100 km/h, and the speed is increased by 5 km/h each test. The target commercial vehicle drives evenly at the specified speed during the experiment.

Figure 12 shows that the frequency-weighted RMS of optimized seat vibration acceleration is significantly lower than the original situation. The results in Figure 12 are similar to those in Figure 7, demonstrating that the ride comfort model based on nonlinear damping constructed in this paper is feasible and the optimization method is effective. In addition, the fluctuations of the experimental result after 70 km/h may be affected by engine vibration because of increased rotating speed.

6. Conclusions

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.