Design and Optimization of a New Vibration Damping System for the Driving Characteristics of the Selected Case of Replacing Pneumatic Wheels with Nonpneumatic Wheels

Ding, Renkai; Wang, Ping; Wang, Ruochen; Sun, Dong

doi:https://doi.org/10.1155/2023/6050633

Shock and Vibration

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Abstract Introduction Discussion Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2023 | Article ID 6050633 | https://doi.org/10.1155/2023/6050633

Design and Optimization of a New Vibration Damping System for the Driving Characteristics of the Selected Case of Replacing Pneumatic Wheels with Nonpneumatic Wheels

Renkai Ding,^1,2Ping Wang,¹Ruochen Wang,³and Dong Sun³

Academic Editor: Peter Múčka

Received09 Aug 2023

Revised01 Dec 2023

Accepted07 Dec 2023

Published22 Dec 2023

Abstract

To cope with the deterioration of vehicle dynamic performance induced by the increased mass and vertical stiffness of selected nonpneumatic wheels (NPWs), this study proposed a new vibration damping system (NVDS) based on the dynamic vibration absorber (DVA). First, a quarter-vehicle model containing the effective vertical stiffness of the NPW is established. Then, the effect of increased wheel mass and stiffness on vehicle dynamic performance is investigated from various aspects. To improve the handling performance of the vehicle with NPWs, a DVA-based NVDS is proposed. The sensitivities of handling performance and ride comfort to parameter changes are investigated to determine the selection range of parameters. The multiobjective genetic algorithm (GA) is employed to determine the optimal parameters of NVDS. The results obtained under different road excitations indicate that the proposed NVDS can significantly enhance the handling performance and ride comfort of the vehicle.

1. Introduction

The wheel is a critical component of the vehicle system, which is utilized to support the vehicle load, ensure good contact with the road, and attenuate the road impact [1]. Owing to its good adaptability to road roughness and ability to absorb shock, the pneumatic wheel is widely used in passenger and commercial vehicles [2]. Although the pneumatic wheel has been proven to have superior performance, its deficiencies are also obvious, such as the complex manufacturing process, expensive production costs, difficult recycling, and the loss of vehicle control in the event of bursting [3]. In other words, the wheel performance not only influences the dynamic performance but also relates to the vehicle’s safety.

To overcome the shortcomings of traditional pneumatic wheels, various improved technologies have been put forward by manufacturers and research institutions, such as the run-flat technology [4], sealant technology [5], and internal support technology [6]. However, the above technology can only maintain the basic driving performance of the vehicle after a leak or puncture, which cannot fundamentally prevent punctures. To ensure the normal operation of the vehicle, it is essential to prevent punctures [7]. The emergence of NPW broke through the bottleneck of the current technology. Due to the special structure design and material application, it can not only realize the basic functions of a pneumatic wheel but also have the characteristics of puncture resistance and damage prevention, which fundamentally avoid the risk of wheel puncture. Thus, research on NPWs has been actively conducted recently. In terms of structure, various types of NPWs are developed, including Tweel NPW [8], honeycomb wheel [9–11], spoke NPW [12, 13], and nonpneumatic mechanical elastic wheel (ME-wheels) [14–16]. In terms of stiffness characteristics, Zhang et al. [17] designed a flexible spoke NPW based on the bionic concept, established its radial stiffness coupling model through numerical and theoretical derivation, and validated the accuracy of the model through test. Gasmi et al. [18] simplified the tread of the NPW as a bending beam and the spokes as springs and constructed a quasi-static, two-dimensional analysis model. Additionally, various studies regarding wheel ground contact behaviour [19–21], rolling resistance [22, 23], and fatigue life [24, 25] are also conducted by scholars.

It can be seen that remarkable progress has been made in the research on the structure and basic characteristics of NPWs. However, how to realize the effective and comprehensive application remains to be explored. The radial stiffness of NPWs is significantly higher than that of pneumatic wheels and shows obvious nonlinearity [26, 27]. Therefore, its application to existing vehicles will inevitably lead to the deterioration of a series of dynamic performance, such as the ride comfort and handling performance. To address this problem, Dhrangdhariya et al. [28] studied the influence of different nonlinear materials and spoke structures on wheel stiffness and proposed the best combination of them that can enhance the ride comfort. Zhou et al. [29] developed an asymmetric spoke structure and optimized its structural parameters, which effectively improved the vibration damping performance of spoke NPWs. It should be noted that the optimization of structural parameters and materials has a limited effect on the improvement of ride comfort due to the contradiction between support and damping properties. On the other hand, the suspension and wheels work together to attenuate the road impact during driving. Thus, suspension matching and control strategy design may be another way to enhance the dynamic performance of the vehicle with NPWs, which is not involved in current research. Although Wang et al. [30] improved the dynamic performance of vehicles with ME-wheels through an adaptive feedback control strategy, the specific effects of ME-wheels on dynamic performance were not clarified. The main purpose of this work is to explore effective ways to improve the vehicle’s dynamic performance based on a clear understanding of the impact of the selected NPW.

The main contributions include the analysis of the effect of NPW mass and vertical stiffness on vehicle dynamic performance from different aspects, the design of an NVDS consisting of a passive suspension and a DVA, and the optimization of system parameters. It is shown that the proposed NVDS can reduce vehicle body and wheel vibration on different roads. The remainders are organized as follows: Section 2 describes the modelling of a quarter-vehicle model, and Section 3 analyses the effects of NPW mass and vertical stiffness on vehicle dynamic performance. Section 4 presents the design and optimization of the NVDS based on DVA using multiobjective GA. Section 5 gives the results and discussion. The general conclusions are drawn in Section 6.

2. Vehicle System Modelling

2.1. Identification of Effective Vertical Stiffness of NPW

This study focuses on the improvement of the vertical dynamic performance of the vehicle which directly related to the wheel’s vertical stiffness. To obtain the vertical stiffness of the NPWs, static load-deflection tests were performed. A commercially available 155/60R13 NPW designed for the automobiles, as shown in Figure 1(a), was used in this study. The overall diameter of the wheel was 508 mm, and it adopted a staggered O-type rubber structure. The multifunctional experimental bench used for the static load-deflection experiment is shown in Figure 1(b). It was mainly constructed of an inertial frame fixed to the ground, a moving frame restricted to move in a vertical direction, a force transducer placed between the moving frame and the inertial frame, and an oscilloscope. The force transducer is used to measure the vertical force of the wheel during the loading, and its value can be directly read by the oscilloscope. In the experiment, the vertical deflection of the wheel increased from 0 mm to 17 mm with a step of 0.5 mm, and the load was recorded. The same experiment was performed for a reference pneumatic wheel with the same size (155/60R13). The air pressure in the pneumatic wheel is 240 kPa, a commonly used air pressure in passenger vehicles.

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The relationship between the load and vertical deflection for both wheels is given in Figure 1(c). It is clear that the vertical stiffness of the NPW is greater than that of the pneumatic wheel and exhibits a significant nonlinearity. The vertical stiffness of the pneumatic wheel is linear, which is 128000 N/m according to the experimental data. The effective stiffness of the NPW calculated in the vicinity of the operating static load (∼362.9 kg) is adopted for the subsequent analysis, which is 332900 N/m. The masses of pneumatic wheel and nonpneumatic wheel were also measured, which were 17.4 kg and 23.3 kg, respectively. Considering that the measurement process is very simple and common, it is not described in detail here.

2.2. Quarter-Vehicle Model

To study the effects of NPW on vehicle dynamic performance, a quarter-vehicle model that can reflect the vertical vibration characteristics of the vehicle is established, as shown in Figure 2. The dynamics equation can be derived based on Newton’s second law as follows:where m₁ is the unsprung mass (herein, m₁ = 36.6 kg for the vehicle with pneumatic wheels and m₁ = 45.4 kg for the vehicle with NPWs); m₂ is the sprung mass (herein, m₂ = 317.5 kg); c₁ is the suspension damping (herein, c₁ = 1500 N·s/m); k₂ is the suspension stiffness (herein, k₂ = 20000 N/m); k₁ is the wheel/wheel stiffness (herein, k₁ = 128000 N/m for pneumatic wheel and k₁ = 332900 N/m for NPW); z₀ is the road excitation; and z₁ and z₂ are the motions of the unsprung and sprung masses, respectively.

Through Laplace transform of equations (1a) and (1b), we can obtain the following equation:

Then, the transfer function between z₂ and z₁, z₁ and z₀, and z₂ and z₀ can be derived as follows:where , , and .

2.3. Road Excitation Model

As shown in Figure 2, the road excitation is the only disturbance source in the quarter-vehicle model. To analyse the dynamic performance of vehicles under different road excitations, the random road with statistical characteristics and the bump road with discrete characteristics are used as the input. Furthermore, a sinusoidal road excitation was also adopted to investigate the vibration response of the vehicle with NPWs under specific frequency excitation, which is a relatively simple model and is not elaborated here.

2.3.1. Random Road Excitation

Since the random road can be regarded as a white noise signal with a finite bandwidth, the road excitation signal can be generated by a white noise passed through a filter [31]. The model of random road excitation in the time domain constructed by the filtered white noise method can be expressed as follows:where f₀ is the lower cut-off frequency, generally taken as 0.011; is the vehicle speed; n₀ is the reference spatial frequency and n₀ = 0.1; G_q(n₀) is the spatial spectral density corresponding to the reference spatial frequency, also known as the road unevenness coefficient, and the corresponding relationship between it and the road classes is defined as ISO 8608 [32]; and is the white noise signal with zero mean value.

In this paper, the class B and C roads are taken as system input for dynamics simulation analysis, and the geometric mean of G_q(n₀) is adopted to generate the random road excitation. Specifically, G_q(n₀) = 64 × 10⁻⁶ m³ is selected for class B road, as well as G_q(n₀) = 256 × 10⁻⁶ m³ is selected for class C road. Assuming that the vehicle travels on class B and C roads at a speed of 60 km/h, the road excitations generated using equation (6) are shown in Figure 3(a). It is obvious that the amplitude of class C road is greater than that of class B road, which will lead to more severe vibration of the vehicle.

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(b)

2.3.2. Bump Road Excitation

To study the dynamic response characteristics of the vehicle on a discrete road, a model of bump road excitation is established as follows [33] (Jiang et al. 2021):where h and l are the height and length of the bump, respectively; t₁ is the starting time when the vehicle passes the bump; here, h = 0.05 m, l = 5 m, and t₁ = 0.5 s. Considering that the speed of vehicles passing through bumpy roads is usually not high, the speed is assumed to be 36 km/h here. Figure 3(b) depicts the generated bump road excitation.

3. Effects of NPWs on the Dynamic Performance of the Vehicle

The analysis shows that the selected NPW has greater mass and vertical stiffness compared to the reference pneumatic wheel, which will inevitably cause the variation in the vehicle vibration response and may even deteriorate the vehicle’s dynamic performance. Therefore, clarifying the effect of NPW on vehicle vertical vibration is essential to improving the dynamic performance and promoting the application of NPWs. In the following, the effects of NPW mass and stiffness on vehicle vibration response are comprehensively analysed from the perspective of frequency domain, time domain, and power flow.

3.1. Analysis of Frequency-Domain Response

For the quarter-vehicle model, the vehicle body acceleration (VBA), suspension rattle space (SWS), and relative dynamic wheel load (DWL) are usually used to evaluate its ride comfort and handling performance. These three indicators can well reflect the vertical dynamic performance of the vehicle at different frequencies. To acquire the effects of the NPW on the dynamic performance of the vehicle in the frequency domain, the amplitude-frequency characteristics were analysed. For the linear suspension system model, the frequency-domain transfer function is usually used to analyse the amplitude-frequency response characteristics [34, 35]. The relevant formulas for calculating the frequency-domain transfer function of a linear suspension system are provided below.

The amplitude-frequency characteristics of the VBA to the road excitation speed can be expressed as follows:

Substituting equation (4) into equation (8), we can obtain

The amplitude-frequency characteristic of the SRS to the road excitation speed can be expressed as follows:

Substituting equations (4) and (5) into equation (10), we can obtain

The amplitude-frequency characteristic of relative DWL to the road excitation speed can be expressed as follows:where is the DWL and is the static wheel load.

Substituting equation (5) into equation (12), we can obtain

According to the above transfer functions of the amplitude-frequency characteristics, the curves of the dynamic responses in the frequency domain can be obtained by MATLAB software. The amplitude-frequency response curves of the VBA, SWS, and relative DWL are shown in Figure 4. It can be observed that the amplitude-frequency characteristics of the vehicle with pneumatic wheels and with NPWs are basically the same within the 2∼8 Hz. In case of the vehicle body resonance, the amplitudes of VBA, SWS and relative DWL are slightly lower for the vehicle with NPWs than for the vehicle with pneumatic wheels, which was caused by the increased wheel mass. However, the response amplitude of the vehicle with NPW was significantly larger in the case of wheel resonance due to the increased wheel stiffness. Furthermore, the vehicle with NPWs occurs for wheel resonance at a frequency of 13.5 Hz, which is higher than that of the vehicle with pneumatic wheels. Overall, NPW exacerbates high-frequency vibrations in the vehicle, leading to deterioration of ride comfort and handling performance.

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As shown in equation (6), the random road excitation is related to the vehicle speed and road roughness properties. The dynamic response of the vehicle varies for different road excitations. As a result, the effects of vehicle speed and road roughness variation on the frequency-domain response of the vehicle under random road excitation were investigated separately. Assuming that the vehicle runs on a class B road at speeds of 36 km/h, 72 km/h and 108 km/h, the calculated vertical vibration response is as shown in Figure 5. It can be seen that the amplitudes of the power-spectral-density (PSD) curves of three evaluation indicators increase with increasing vehicle speed. In other words, the increase in vehicle speed exacerbates the vertical vibration and deteriorates the ride comfort and handling performance. The overall trend of the PSD curves is similar to that of the corresponding amplitude-frequency characteristic curves.

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Similarly, assuming that the vehicle travels at 72 km/h on class A∼ C roads, the vibration response can be obtained as shown in Figure 6. It is clear that the amplitudes of the PSD curves for VBA, SRS, and relative DWL increase significantly with increasing road roughness in the full frequency band. According to Figures 5 and 6, it can be seen that the increase in road roughness has a greater effect on the vibration response of the vehicle than the increase in vehicle speed, which will drastically deteriorate the ride comfort and handling performance of the vehicle with NPWs.

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3.2. Analysis of Time-Domain Response

The vibration response of the vehicle was also calculated under different road conditions in the time domain to further study the effect of NPWs. The root mean square (RMS) values of the VBA, SRS, and DWL are adopted to evaluate the vehicle’s dynamic performance in the time domain. Assuming that the vehicle runs on the class B and C roads at different vehicle speeds, the statistical results of three evaluation indicators can be calculated.

As shown in Figure 7, the higher the vehicle speed, the greater the time-domain responses of the vehicle will be, and hence, the less ride comfort as well as handling performance will be. The RMS values of VBA and DWL of the vehicle with NPWs are significantly larger than those of the vehicle with pneumatic wheels, and the difference between them increases further as the vehicle speed increases. In other words, the ride comfort and handling performance of the vehicle with NPWs deteriorates faster as vehicle speed increases. On the other hand, the RMS values of SRS at different vehicle speeds are almost the same for both vehicles. This means that the SRS is not sensitive to the increase in NPW stiffness and mass. Similarly, it can be seen that an increase in road roughness has a greater impact on the vehicle’s dynamic performance. In addition, it can be seen that the deterioration of the DWL is greater than the VBA as the vehicle speed or road roughness increases, which will lead to a great deterioration of the handling performance. In general, the analysis in the time domain is consistent with that in the frequency domain.

3.3. Power Flow Analysis

To clarify the energy conversion and transfer relationship in the quarter vehicle-road coupled system and seek an effective method to improve the vehicle’s dynamic performance, a power flow analysis was conducted. As shown in Figure 2, the road excitation is the power source. The input power flow is transferred to the vehicle body through the wheel and suspension. To derive the calculation equation of the power flow, the equation (1b) can be rewritten as follows:

Multiplying both sides of equation (1a) by and both sides of equation (14) by , the following equations can be obtained:

Substituting equation (15a) into equation (15b), the energy conservation equation can be obtained as follows:where is the power flow input to the quarter-vehicle model by road excitation; and are the change rates of the kinetic energy for the vehicle body and wheel, respectively; is the energy dissipated by the suspension damper; and are the change rates of the potential energy for the suspension spring and NPW, respectively.

Since the symbol of power flow only represents the direction, its absolute value is used for analysis. Furthermore, the average power is usually more meaningful for the power flow analysis. The time-averaged input power flow can be defined as follows:

Similarly, the time-averaged kinetic energy of vehicle body and wheel is defined as follows:

The time-averaged energy dissipated by the suspension damper is defined as follows:

The time-averaged potential energy of spring and wheel is defined as follows:

Based on the above analytical formula, the average power of both vehicles is calculated on class B and C roads at different vehicle speeds, as shown in Figure 8. It can be seen that the average power of each part for both vehicles increases with the vehicle speed. The average power of all parts of the vehicle with NPWs was greater than that of the vehicle with pneumatic wheels, and the difference between them increased with the vehicle speed. In addition, it can be seen that the greatest difference in wheel kinetic energy is between the two vehicles. The above differences in the average energy of the vehicle components are all caused by the wheels, which is the only difference between the two vehicles.

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Specifically, as shown in Figure 8(a), the average input power increases significantly with the vehicle speed. The increase of the input power led to an increase in the average power of all parts, including the wheel, spring, and damper. Similarly, it is clear that the increase in road roughness has a greater effect on the average input power compared to the increase in vehicle speed. As shown in Figure 8(c), on both class B and C roads, the average change rate in wheel kinetic energy is significantly greater for the vehicle with NPWs than for the vehicle with pneumatic wheels, i.e., the NPWs lead to further deterioration of vehicle handling performance. As shown in Figure 8(f), the difference in vehicle body kinetic energy is less than that in wheel kinetic energy. The main reason for this is that the suspension damper dissipates more of the vibration energy. As shown in Figure 8(d), the average dissipation energy of the damper in a vehicle with NPWs increases rapidly with increasing vehicle speed and road roughness. The more energy dissipated by the suspension damper, the greater the potential for vibration energy regeneration. This can provide a new idea for the research on vibration energy regeneration, which will not be further discussed here. Similarly, the difference in spring potential energy between both vehicles (shown in Figure 8(e)) is smaller than that in wheel potential energy (shown in Figure 8(b)), also due to the suspension damper. In summary, the increased mass and stiffness of the NPW result in more input energy and greater change rates of the kinetic energy of the vehicle body and wheel, which deteriorate the ride comfort and handling performance of the vehicle.

It can be observed from the above analysis that the suspension stiffness and damping have an important effect on the power flow of the quarter vehicle. Therefore, the effect of suspension stiffness and damping on the change rates of the kinetic energy of the vehicle body and wheel was further investigated, as shown in Figure 9. It can be seen that the average change rate of wheel kinetic energy decreases as the suspension damping increases. The greater the suspension damping, the slower the change rate of wheel kinetic energy increases with the vehicle speed. In other words, the tuning of suspension damping can help improve the handling performance of the vehicle. However, the increasing suspension damping may lead to an increase in the change rate of vehicle body kinetic energy (as shown in Figure 9(a)), which deteriorates the ride comfort of the vehicle. As shown in Figures 9(c) and 9(d), the average kinetic energy of the vehicle body and wheel tends to change in opposite directions as the suspension stiffness increases. The average kinetic energy of the vehicle body increases with suspension stiffness, while that of the wheel decreases. In addition, the suspension stiffness has very little effect on wheel kinetic energy. In summary, the suspension stiffness and damping have an important effect on the kinetic energy of the vehicle body and wheel. The reasonable selection of suspension stiffness and damping can help improve the comprehensive performance of the vehicle with NPWs.

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According to the above analysis from different aspects, the following conclusions can be drawn:(1)The increased mass and stiffness of NPW will lead to the deterioration of the dynamic performance of the vehicle in the case of wheel resonance, especially the handling performance. Moreover, the increase of the vehicle speed and road roughness will further aggravate the deterioration.(2)The increased vehicle speed and road roughness led to the faster increase of the average input power of the vehicle with NPWs than the vehicle with pneumatic wheels, resulting in the greater average kinetic energy of the vehicle body and wheel, which is the fundamental reason for the deterioration of ride comfort and handling stability. Similarly, the increased average energy dissipated by the suspension damper reveals the greater potential for energy regeneration in the vehicle with NPWs.(3)The suspension stiffness and damping have a significant effect on the average kinetic energy of the vehicle body and wheel. The optimization of suspension stiffness and damping may be a feasible way to improve the dynamic performance of the vehicle with NPWs.

It should be emphasized that the conclusions drawn above only apply to the selected cases in this paper and cannot be generalized to all situations where nonpneumatic wheels are used instead of pneumatic wheels.

4. NVDS Design and Parameter Optimization

4.1. Proposal and Parameter Sensitivity Analysis

Based on the analysis, it is obvious that the first thing to address is the handling performance of vehicles with NPWs, followed by ride comfort. Currently, there are two main approaches used to attenuate vehicle vibration: one is the active approach that realizes the vibration attenuation through the controllable suspensions with advanced control strategies, and the other one is the passive approach that attenuates vibrations by means of properly designed DVA. It should be noted that the active approach requires not only actuator matching and control strategy design but also additional vehicle sensors, which greatly increases production costs. Therefore, a passive approach was adopted in this study, i.e., a DVA-based NVDS was designed. The NVDS consists of a passive suspension and a DVA placed on the unsprung mass, as shown in Figure 10, and its dynamics differential equation can be expressed as follows:where m₃, c₂, and k₃ are the mass, damping, and stiffness of the DVA to be determined, respectively.

The DVA is composed of an elastic element, a damping element, and a mass element. From the perspective of power flow, part of the energy flowing to the wheel will be absorbed by the DVA and transformed into the potential energy of the elastic element, the internal energy of the damping element, and the kinetic energy of the mass element. The three parameters of the DVA directly determine its vibration suppression effect. Therefore, including the suspension stiffness and damping, there are a total of five parameters to be determined.

To narrow down the range of parameter optimization to improve efficiency, the sensitivities of VBA, SRS, and DWL to system parameter changes were investigated. The initial mass of DVA is taken to be 0.1 times the unsprung mass, i.e., m₃ = 0.1m₁ = 4.54 kg. To attenuate the high-frequency vibration of the wheel, the inherent frequency of the DVA () should theoretically be equal to the offset frequency of the wheel (). Thus, the initial stiffness of DVA can be determined as k₃ = 35290 N/m. The damping ratio of the DVA () is generally taken as 0.1 to 0.3, and 0.2 is taken here, corresponding to an initial damping of c₃ = 160 Ns/m. Assuming that the vehicle is travelling at 60 km/h on a class C road and the variation range of the five parameters is 0.2 to 1.8 times the initial values, the RMS values of the three evaluation indicators were calculated. Figure 11 plots the normalized performance of the changes in suspension stiffness, suspension damping, DVA mass, DVA stiffness, and DVA damping, respectively.

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As shown in Figures 11(a) and 11(b), the VBA increases with the increasing suspension stiffness, while the DWL does the opposite. As the suspension damping increases, the VBA first decreases and then increases, and the optimal suspension damping for ride comfort is shown at 0.4c₁. Furthermore, it can be seen that the suspension damping has the greatest effect on the SRS, which should be constrained during the parameter optimization. As shown in Figures 11(c)–11(e), the trend of VBA with the change of DVA parameters is similar to that of DWL. However, the SRS is hardly affected by the change in the DVA parameters. Specifically, the VBA and DWL decrease with increasing DVA mass and first decrease and then increase with increasing DVA stiffness and damping. The optimal DVA stiffness for ride comfort is shown at 0.9k₃ while it is 1.3k₃ for the best handling performance concern. The optimal DVA damping for ride comfort and handling performance is shown at 0.7c₂ and 0.5c₂, respectively. The optimization range of DVA parameters can be finally reduced to m₃∼1.8m₃, 0.5k₃∼1.5k₃, and 0.2c₂∼c₂.

4.2. Optimization of System Parameters

The above determination of system parameters is a typical multiobjective optimization problem with five optimization variables, and the optimization objectives include handling performance and ride comfort. To improve handling performance, the RMS value of DWL should be minimized, while to improve ride comfort, the RMS value of VBA should be minimized. The optimization objectives can be expressed as follows:where is the parameter vector to be optimized and and are the RMS values of the DWL and VBA.

Then, the multiobjective optimization problem is formulated as follows:subject to the following constraints:where and are the constraints of working space for the suspension and DVA and l = 0.06 m denotes the maximum travel.

In this study, a GA is used to determine the optimal parameters, which have been proven to be suitable for solving the multiobjective optimization problems [36]. The relevant settings of the GA are as follows:(1)The population size is set to 100, and the number of elites is set to 10;(2)The competitive selection method is used, and the competitive scale is set to 10;(3)The crossover method selects the arithmetic crossover method;(4)The single point genetic mutation is adopted, and the mutation probability is set to 0.02 to coordinate the solution speed and the diversity of the population;(5)The termination condition adopts the combination of termination algebra and evolution trends. The GA will be terminated when the change of the fitness function value is less than 0.1% for 10 consecutive generations or after 100 consecutive generations of evolution.

The quarter-vehicle model described by equation (21) was used as the iterative model, the class C road was selected as the model input, and the vehicle speed was set to 60 km/h. The execution of the GA yields the pareto-optimal front given in Figure 12. As shown in the figure, an improvement in handling performance will inevitably lead to a deterioration in ride comfort. Considering that the increase in wheel mass and stiffness mainly affects vehicle handling performance, followed by ride comfort, an optimal solution considering 80% handling performance and 20% ride comfort was chosen to enhance the comprehensive performance of the vehicle with NPWs. The optimized parameters are shown in Table 1.

5. Simulation Results and Discussion

To validate the correctness of the parameter optimization and the effectiveness of the proposed NVDS, the frequency-domain response analysis and dynamics simulation were performed. For the sake of the simplicity, the systems to be compared are named as follows:(i)Passive system: quarter-vehicle system modelled by equation (2)(ii)NVDS-based system: quarter-vehicle system with NVDS, as shown in equation (21).

5.1. Analysis of Frequency-Domain Response

For the NVDS-based system described in equation (21), the Laplace transform yields

Then, the transfer function between z₂ and z₁, z₃ and z₁, and z₁ and z₀ can be obtained by equation (25) as follows:where , , , , and .

The amplitude-frequency characteristic of the VBA to the road excitation speed can be expressed as follows:

Substituting equations (26) and (28) into equation (29) we get

The amplitude-frequency characteristic of the SRS to the road excitation speed can be expressed as follows:

Substituting equations (26) and (28) into equation (31) we get

The amplitude-frequency characteristic of the relative DWL to the road excitation speed can be expressed as follows:where is the DWL and is the static wheel load.

Substituting equation (28) into equation (33) we get

According to above derivation, the amplitude-frequency characteristics of the two systems are compared in Figure 13.

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Due to the introduction of DVA, the amplitude-frequency characteristic curves of VBA and DWL for the NVDS-based system show three resonance peaks, as shown in Figures 13(a) and 13(c). The amplitude-frequency characteristic curve of the SRS shown in Figure 13(b), on the other hand, only has one obvious resonance peak. Furthermore, it can be seen that the amplitude of the NVDS-based system is significantly less than that of the passive system in the case of body resonance and wheel resonance, although it is slightly larger within the 3∼9 Hz frequency range. In general, the ride comfort and handling performance of the NVDS-based system are effectively improved. Within the 1∼50 Hz, the amplitude-frequency characteristic curve of the SRS of the NVDS-based system is always below that of the passive system. Therefore, the proposed NVDS can reduce the probability of a strike between suspension springs and the limit block.

5.2. Dynamics Simulation under Random Road Excitation

Assuming that the vehicle runs on the class C road at the speed of 60 km/h, Figure 14 gives a comparison of VBA, SRS, and DWL for the two systems.

It can be observed from the figure that under the random road excitation, the NVDS-based system generates a good improvement on both the VBA and DWL compared to the passive system. The SRS of the two systems is always within the maximum travel. The specific comparison of the RMS values of the three evaluation indicators is shown in Table 2. It can be seen that the DWL and VBA of the NVDS-based system decreased by 12.4% and 11.8%, respectively, compared to the passive system. This suggests that the proposed NVDS can effectively enhance the handling performance and ride comfort of the vehicle with NPWs on the random road.

5.3. Dynamics Simulation under Bump Road Excitation

Using the bump road excitation described in equation (7) as the system input and the vehicle speed set to 36 km/h, a comparison of the dynamic response of different systems are obtained, as shown in Figure 15.

It can be seen from Figure 15 that the proposed NVDS-based system outperforms the passive system with a smaller VBA, SRS, and DWL amplitude. To better evaluate the performance differences between the two systems, a quantitative analysis was performed using the peak-to-peak values, representing the difference between the maximum and minimum values, as shown in Table 3. According to Table 3, the application of the NVDS can reduce 9.1% of the VBA, 3.5% of the SRS and 7.7% of the DWL. Besides the peak-to-peak value, the convergence time is also an important indicator to evaluate the dynamic performance of the system under the bump road excitation. As shown in Figure 15, it is clear that the convergence time of the dynamic response of the NVDS-based system is shorter than that of the passive system. In other words, the NVDS-based system can recover the steady-state response at a faster rate after driving over a bump. The comparison of peak-to-peak values and convergence times indicates that the NVDS-based system provides better handling performance on bumpy roads as well as ride comfort compared to the passive system.

5.4. Dynamics Simulation under Sinusoidal Road Excitation

A sinusoidal signal with an amplitude of 0.01 m and a frequency of 13.5 Hz was selected as the input to analyse the dynamic response of the system in the case of wheel resonance. The dynamic response of the two systems is shown in Figure 16.

It can be seen that the proposed NVDS-based system outperforms the passive system in terms of both VBA and DWL under sinusoidal road excitation with wheel resonance frequency. The peak-to-peak values representing the difference between the maximum and minimum values were used to quantitatively evaluate the dynamic response of the vehicle, as shown in Table 4. It is clear that compared to the passive system, the NVDS-based system yields 25.3%, 32.2%, and 34.1% reductions in the VBA, SRS, and DWL, respectively. In other words, the application of NVDS significantly enhances the ride comfort and handling performance of the vehicle in cases of the wheel resonance, reducing the damage to the vehicle and passengers induced by wheel resonance.

6. Conclusion

In this article, a DVA-based NVDS is proposed for the vehicle with NPWs to improve the dynamic performance. First, the effective vertical stiffness of the selected NPW was identified through a static load-deflection experiment. Next, the effects of NPW mass and vertical stiffness on vehicle dynamic performance were investigated in terms of frequency domain response, time domain response, and power flow. Then, a DVA-based NVDS was designed, and its optimal parameters are determined by multiobjective GA. In the end, the handling performance and ride comfort of the vehicle with NPWs are improved. The major conclusions can be drawn as follows:(1)Compared with the reference pneumatic wheel, the NPW selected in this case presents greater mass and higher vertical stiffness, which will aggravate the high-frequency vibration of the vehicle and cause an increase in wheel resonance frequency.(2)Compared to vehicle speed, road roughness has a greater impact on vehicle dynamic performance, and its increase leads to a significant deterioration of vehicle dynamic performance, especially handling performance.(3)The proposed NVDS can effectively reduce the vibration response in the case of vehicle body and wheel resonance, enhancing the ride comfort and handling performance of vehicles with higher wheel mass and stiffness. Specifically, the RMS/peak-to-peak values of the DWL and VBA for the NVDS-based system are improved by 12.4% and 11.8% on the random road and by 34.1% and 25.3% on the sinusoidal road, respectively.

This study could provide a reference for the design of vehicle chassis and the performance matching of NPWs (with higher mass and stiffness) with vehicle vibration damping systems. In future work, the DVA prototype will be fabricated for experiments to explore the effects of NVDS in practical applications.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

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

Acknowledgments

The work described in this paper was fully supported by a grant from Open fund of State Key Laboratory of Automotive Simulation and Control (Project 20210204).

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Copyright © 2023 Renkai Ding et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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