The method chosen to conduct vehicle dynamic modeling has a significant impact on the evaluation and optimization of ride comfort. This paper summarizes the current modeling methods of ride comfort and their limitations. Then, models based on nonlinear damping and equivalent damping and the multibody dynamic model are developed and simulated in Matlab/Simulink and Adams/Car. The driver seat responses from these models are compared, showing that the accuracy of the ride comfort model based on nonlinear damping is higher than the one based on equivalent damping. To improve the reliability of ride comfort optimization and analysis, a ride comfort optimization method based on nonlinear damping and intelligent algorithms is proposed. The sum of the frequency-weighted RMS of the driver seat acceleration, the RMS of dynamic tyre load, and suspension working space is taken as the objective function in this article, using nonlinear damping coefficients and stiffness of suspension as design variables. By applying the particle swarm optimization (PSO), cuckoo search (CS), dividing rectangles (DIRECT), and genetic algorithm (GA), a set of optimal solutions are obtained. The method efficiency is verified through a comparison between frequency-weighted RMS before and after optimization. Results show that the frequency-weighted RMS of driver seat acceleration, RMS values of the suspension working space of the front and rear axles, and RMS values of the dynamic tyre load of front and rear wheels are decreased by an average of 27.4%, 21.6%, 25.0%, 19.3%, and 22.3%, respectively. The developed model is studied in a pilot commercial vehicle, and the results show that the optimization method proposed in this paper is more practical and features improvement over previous models.
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 [
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. [
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.
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.
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
Seven-DOF vibration model.
Fixed parameters of the vibration model.
Symbol | Description | Value | Unit |
---|---|---|---|
|
Driver and seat mass | 100 | kg |
|
Cab mass | 850 | kg |
|
Sprung mass | 14238 | kg |
|
Unsprung mass of front axle | 607 | kg |
|
Unsprung mass of rear axle | 1054 | kg |
|
Rotational inertia of cab mass around |
560 | kg·m2 |
|
Rotational inertia of sprung mass around |
115000 | kg·m2 |
|
Rotational inertia of rear axle balance suspension mass around |
615 | kg·m2 |
|
Seat stiffness | 20000 | N/m |
|
Front suspended stiffness | 25000 | N/m |
|
Rear suspended stiffness | 25000 | N/m |
|
Front suspension stiffness | 410000 | N/m |
|
Rear suspension stiffness | 1476000 | N/m |
|
Front wheel stiffness | 1800000 | N/m |
|
Rear wheel stiffness | 3600000 | N/m |
|
Equivalent damping coefficient of driver seat | 800 | N·s/m |
|
Equivalent damping coefficient of front suspended | 5000 | N·s/m |
|
Equivalent damping coefficient of rear suspended | 5000 | N·s/m |
|
Equivalent damping coefficient of front suspension | 12000 | N·s/m |
|
Equivalent damping coefficient of rear suspension | 15000 | N·s/m |
|
Distance between the center and seat of the cab | 0.2 | m |
|
Distance between front suspended and the center of the cab | 0.76 | m |
|
Distance between rear suspended and the center of the cab | 0.87 | m |
|
Distance between the center of the cab and the center of target commercial vehicle | 2.47 | m |
|
Distance between the front suspension and the center of target commercial vehicle | 2.1 | m |
|
Distance between the rear suspension and the center of target commercial vehicle | 3 | m |
Damping curves. (a) Damping ratio. (b) Damping coefficient. (c) Damping force.
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 [
The time-domain model of road excitation based on filtered white noise is as follows:
Figure
Road excitation time-domain profile.
Road excitation frequency-domain profile.
To capture the damping nonlinear characteristic, a set of damping ratio curves are provided by a commercial vehicle manufacturer and shown in Figure
The damping force lines presented in Figure
Usually, the equivalent damping coefficient is calculated through the energy dissipated during one vibration period, which is equal to the energy dissipated by the equivalent damping. When a system is forced to vibrate, the viscous damping force can be expressed as the following equation:
Energy consumption by equivalent damping in a vibration cycle is
When the damping of the system is nonlinear, it can be replaced by equivalent damping:
Equations (
Multibody dynamic model.
The time-domain curves are shown in Figure
Time-domain responses of the driver seat. (a) 40 km/h. (b) 80 km/h.
Frequency-weighted RMS.
As shown in Figure
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.
The PSO is an iterative algorithm formed by a family of particles in which each particle keeps track of its coordinates and shares them with the other particles. The particles fly in the
CS is a nature-inspired algorithm developed based on the evolution of cuckoo birds [ Each cuckoo has only one egg and selects the nest position randomly The best nests with high-quality eggs will be carried to the next generation The host nest number is constant during iteration, and the probability of the host finding eggs is
The particle
A simple version of Levy distribution can be mathematically defined by
The DIRECT algorithm is composed of potentially optimal hyper-rectangles and the dividing strategy for hypercubes [
We define the potentially optimal hyper-rectangle as follows: let
Furthermore,
Then,
The GA is an optimization method based on the principles of natural genetics and natural selection. The basic elements of GA are performed by three operations as reproduction, crossover, and mutation. Solutions are evaluated with respect to their fitness value that indicates how well the individual will solve the problem [
The vibration perception of the human body is related to the frequency, response of the driver seat, and vibration acceleration and has been weighted according to the ISO 2631:1997 standard. This standard evaluates human exposure to whole-body vibration and is usually based on one objective function. The frequency-weighted RMS value
The dynamic tyre load is defined as the RMS value of the change in tyre load relative to the static equilibrium position. The mechanism of tyre adhesion loss caused by the tyre load is described as the following: when a tyre is required to generate lateral or longitudinal force, the contact portion between the tyre and the ground must be deformed before the force is sufficiently generated. However, this deformation requires a roll distance, so there is a time delay before the tyre force is fully obtained. When the tyre load fluctuates with the suspension motion, the effective lateral or longitudinal force available is reduced because of the tyre dynamic mechanism. Therefore, if a stable normal load of the tyre can be maintained, a large tyre force can be obtained; if the dynamic load fluctuation of the tyre increases, the tyre grip ability weakens as the tyre jumps. The dynamic tyre load can be expressed as
The suspension working space is defined as the difference between the wheel and the body displacement. According to the Gaussian distribution of random roads, for linear systems, the response should have a Gaussian property and can be described by a normal distribution. Therefore, the suspension working space can be considered the probability that the relative displacement of the wheel and the body remains within the SWS, 2SWS, and 3SWS under static equilibrium position conditions 68.3%, 95.4%, and 99.7% of the time, respectively. According to the RMS value of the suspension working space, the suspension working space required by the vehicle under certain road input conditions can be determined:
In summary, there are the following objective functions:
Figure
The range of variables should not be too large or too small. When the variation ranges are too large, other performances of the vehicle may be drastically reduced after optimization, and the matching and installation of the entire vehicle will be affected. In contrast, the effect of optimization is not obvious. Therefore, variables are altered up and down by 10% as the optimization interval, and the bounds are defined in Table
Bounds of design variables.
Variables | Original | Lower | Upper |
---|---|---|---|
kfs (N/m) | 410000 | 369000 | 451000 |
krs (N/m) | 1476000 | 1328400 | 1623600 |
|
36.54 | 32.886 | 40.194 |
|
1.487 | 1.338 | 1.636 |
|
0.5996 | 0.540 | 0.660 |
|
49.66 | 44.694 | 54.626 |
|
1.487 | 1.338 | 1.636 |
|
0.5996 | 0.540 | 0.660 |
The static deflection of the suspension is the ratio of the sprung mass to the stiffness under full loads. Typically, the static deflection of the suspension is 50–110 mm. At the same time, having a static deflection of the rear suspension
The natural frequency of suspension usually ranges from 1.5 to 2.2. Thus,
The probability that the wheel jumps off the ground is 0.15%, and the RMS value of the dynamic tyre load must satisfy the following equation:
A ride potimization method based on nonlinear damping and intelligent lagorithms is presented, and the procedure is shown in Figure
Ride comfort optimization procedure based on nonlinear damping.
In this section, the results of ride optimization are discussed. First, the iterations of algorithms are shown in Figure
Objective function iteration.
Optimization variables.
Variables | kfs (N/m) | krs (N/m) |
|
|
|
|
|
|
---|---|---|---|---|---|---|---|---|
Original | 410000 | 1476000 | 36.54 | 1.487 | 0.5996 | 49.66 | 1.487 | 0.5996 |
CS | 369000 | 1330000 | 40.18 | 1.635 | 0.6599 | 54.63 | 1.636 | 0.6600 |
DIRECT | 371000 | 1345000 | 40.06 | 1.631 | 0.6578 | 52.97 | 1.608 | 0.6578 |
GA | 381000 | 1340000 | 40.19 | 1.636 | 0.6600 | 54.62 | 1.636 | 0.6600 |
PSO | 369000 | 1328000 | 40.19 | 1.636 | 0.6600 | 54.63 | 1.636 | 0.6600 |
Evaluation indexes of objective function before and after optimization.
Speed (km/h) | 30 | 40 | 50 | 60 | 70 | 80 | 90 | 100 | |
---|---|---|---|---|---|---|---|---|---|
|
Original | 0.4107 | 0.5190 | 0.4430 | 0.6077 | 0.6988 | 0.6914 | 0.8738 | 0.7832 |
Optimized | 0.3067 | 0.3531 | 0.3527 | 0.4318 | 0.4742 | 0.5284 | 0.5786 | 0.6012 | |
Fall ratio (%) | 25.3 | 32.0 | 20.4 | 28.9 | 32.1 | 23.6 | 33.8 | 23.3 | |
Average (%) | 27.4 | ||||||||
|
|||||||||
|
Original | 0.2442 | 0.2903 | 0.3118 | 0.3740 | 0.4085 | 0.4042 | 0.4731 | 0.4755 |
Optimized | 0.1922 | 0.2263 | 0.2557 | 0.2908 | 0.3122 | 0.3275 | 0.3584 | 0.3701 | |
Fall ratio (%) | 21.3 | 22.0 | 18.0 | 22.2 | 23.6 | 19.0 | 24.2 | 22.2 | |
Average (%) | 21.6 | ||||||||
|
|||||||||
|
Original | 0.1413 | 0.1672 | 0.2050 | 0.2405 | 0.2577 | 0.2674 | 0.3045 | 0.3021 |
Optimized | 0.1075 | 0.1314 | 0.1567 | 0.1756 | 0.1875 | 0.2014 | 0.2196 | 0.2282 | |
Fall ratio (%) | 23.9 | 21.4 | 23.6 | 27.0 | 27.2 | 24.7 | 27.9 | 24.5 | |
Average (%) | 25.0 | ||||||||
|
|||||||||
|
Original | 0.0142 | 0.0166 | 0.0171 | 0.0215 | 0.0233 | 0.0218 | 0.0271 | 0.0261 |
Optimized | 0.0113 | 0.0132 | 0.0148 | 0.0172 | 0.0183 | 0.0186 | 0.0207 | 0.0209 | |
Fall ratio (%) | 20.4 | 20.5 | 13.5 | 20.0 | 21.5 | 14.7 | 23.6 | 19.9 | |
Average (%) | 19.3 | ||||||||
|
|||||||||
|
Original | 0.0086 | 0.0101 | 0.0125 | 0.0150 | 0.0159 | 0.0161 | 0.0188 | 0.0183 |
Optimized | 0.0068 | 0.0082 | 0.0099 | 0.0114 | 0.0119 | 0.0127 | 0.0139 | 0.0144 | |
Fall ratio (%) | 20.9 | 18.8 | 20.8 | 24.0 | 25.2 | 21.1 | 26.1 | 21.3 | |
Average (%) | 22.3 |
The iterations of ride comfort optimization are shown in Figure
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
Comparison between original and optimized objective functions. (a, b) Acceleration of the driver seat at 40 and 80 km/h, respectively. (c, d) Dynamic tyre load of the front wheel at 40 and 80 km/h, respectively. (e, f) Dynamic tyre load of the rear wheel at 40 and 80 km/h, respectively. (g, h) Suspension working space of the front axle at 40 and 80 km/h, respectively. (i, j) Suspension working space of the rear axle at 40 and 80 km/h, respectively.
As shown in Table
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
Target commercial vehicle ride comfort experiment.
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
Frequency-weighted RMS of experiment.
In this paper, we summarized the situation and shortcoming for the ride comfort modeling method, currently. Furthermore, the vibration model of commercial vehicle, road excitation model, nonlinear damping model, and equivalent damping model are successively developed. The accuracy of a ride comfort model using nonlinear damping is demonstrated and compared with other models. Then, the ride comfort model based on the nonlinear damping is optimized using four intelligent algorithms. Lastly, the effectiveness of the optimization method is tested according to the performances before and after optimization. The main conclusions are as follows: The frequency-weighted RMS values of seat vibration acceleration based on models with nonlinear damping and equivalent damping are obtained and compared with the response of a multibody dynamic model. The result indicates that the ride comfort model based on nonlinear damping has higher accuracy than a model based on equivalent damping. The comparison of driver seat vibration acceleration and relevant responses before and after optimization is performed, and the evaluation indexes at different speeds are obtained. The frequency-weighted RMS of driver seat acceleration, RMS of suspension working space of front and rear axle, and RMS of dynamic tyre load of front and rear wheel are decreased by an average of 27.4%, 21.6%, 25.0%, 19.3%, and 22.3%, respectively, through the optimization. Thus, the ride comfort of the commercial vehicle is improved. In addition, the optimization method developed in this paper has theoretical and practical significance for similar problems.
The relatedly Simulink models developed in the paper are illustrated in Figures (
Nonlinear damping model.
Road excitation model.
Ride comfort model.
The data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This research was financially supported by the project of the National Natural Science Foundation of China (Grant no.51565008), the Innovation-Driven Development Special Fund Project of Guangxi (Grant nos. Guike AA18242033 and AA18242036), the Scientific Research and Technology Development in Liuzhou (Grant no. 2016B020203), the Basic Ability Promotion Project for Young and Middle-Aged Teachers in Guangxi Province (Grant nos. 2017KY0869 and 2018KY0205), the Innovation Project of GUET Graduate Education (Grant no. 2019YCXS001), and the GUET Excellent Graduate Thesis Program (Grant no. 17YJPYSS02).