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Optimal design and dynamic performance analysis of HMDV suspension based on bridge network
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Xiaofeng Yang1, 2, 3, Yan Yan1, Yujie Shen1, *, Xiaofu Liu4, Zhipeng Wang5, 6
Acta Mechanica Sinica | 2025, 41(12) : 524208
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Acta Mechanica Sinica | 2025, 41(12): 524208
• RESEARCH PAPER •
Optimal design and dynamic performance analysis of HMDV suspension based on bridge network
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Xiaofeng Yang1, 2, 3, Yan Yan1, Yujie Shen1, *, Xiaofu Liu4, Zhipeng Wang5, 6
Affiliations
  • 1School of Automotive and Traffic Engineering, Jiangsu University, Zhenjiang 212013, China
  • 2State Key Laboratory of Advanced Design and Manufacturing Technology for Vehicle, Changsha 410012, China
  • 3Jiangsu Province and Education Ministry Co-sponsored Synergistic Innovation Center of Modern Agricultural Equipment, Zhenjiang 212013, China
  • 4College of Engineering, China Agricultural University, Beijing 100083, China
  • 5State Key Laboratory of Intelligent Agricultural Power Equipment, Luoyang 471039, China
  • 6Luoyang Tractor Research Institute Co. Ltd., Luoyang 471039, China
Published: 2025-12-01 doi: 10.1007/s10409-024-24208-x
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In order to solve the vertical vibration negative effect problem caused by the increase of the unsprung mass in the hub motor driven vehicle (HMDV), a novel mechatronic suspension using the bridge electrical network is proposed. Firstly, the bridge electrical networks composed of two capacitors, two inductors, and one resistor are summarized and their impedance functions are analyzed forward through the structural method. Then a quarter HMDV model is constructed, and the optimal element parameters in the electrical networks are selected through the Pattern Search algorithm. The influence of element parameters perturbation of the optimal structure on the output response of HMDV suspension is further analyzed. Results show that the proposed bridge electrical network can be realized as a biquartic impedance. It can be equivalent to a mechanical impedance of the suspension through a linear motor. Compared with the conventional suspension, the root-mean-square values of the dynamic tire load and the suspension working space are reduced by 10.76% and 18.10%, respectively. The vibration at low and high frequencies of the unsprung mass is suppressed, effectively improving the grounding and handling stability of the vehicle.

Bridge network  /  Biquartic impedance  /  HMDV  /  Negative effects of vertical vibration
Xiaofeng Yang, Yan Yan, Yujie Shen, Xiaofu Liu, Zhipeng Wang. Optimal design and dynamic performance analysis of HMDV suspension based on bridge network[J]. Acta Mechanica Sinica, 2025 , 41 (12) : 524208 - . DOI: 10.1007/s10409-024-24208-x
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1.　Introduction
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Electric vehicles play a vital role in sustainable mobility. It is estimated that by 2030, battery-electric vehicles will become the mainstream of passenger vehicle transport [1]. The hub-driven technology offers several significant benefits to overcome the challenges in transportation electrification. As an ideal configuration for future electric vehicles, the hub motor driven vehicle (HMDV) has the advantages of high transmission efficiency and flexible operation. The motor is arranged directly in the wheel hub, and the output torque of the motor can be distributed independently so that the handling stability of the vehicle can be improved. Through the independent and accurate control of the motor, the control of the vehicle’s driving dynamics becomes faster and easier [2]. Despite the many advantages of using in-wheel motors, their application leads to an increase in unsprung mass, which negatively affects the ride comfort and handling stability. Under road excitation, the magnetic gap deformation of the hub motor will produce an unbalanced magnetic pull force, thus deteriorating the longitudinal and vertical dynamic performance of the vehicle [3]. Reference [4] established a dynamic model of a quarter-vehicle vibration system by using dynamic theory and state-space method. The simulation results showed that the vertical dynamic performance of the electric vehicle changes under the double excitation, and the electromagnetic force of the wheel motor led to a decrease in ride comfort. It also affects the grounding behavior of vehicles, causing the vertical vibration negative effect.
As a crucial component of vehicle vertical motion, the suspension is of great significance to ride comfort. Research into suspensions centers on structural design [5,6], control strategies [7,8], and actuator development [9,10]. Some scholars have researched the suppression of HMDV vertical vibration negative effect. Meng et al. [11] proposed a new electric wheel configuration with a two-stage suspension and optimized suspension parameters. Li et al. [12] developed a direct wheel drive air suspension system and proposed evaluation indexes based on vertical and longitudinal dynamic coupling mechanisms. The vertical and longitudinal characteristics under different damping coefficients and motor speed are studied, and the effectiveness of the model is verified by experiments. Liu et al. [13] presented a two-pump hydraulic control system for a two-speed power shift transmission, and simulation results show that the energy consumption of the system is significantly reduced, and the dynamic and economic performance of electric vehicles is improved. Yu et al. [14] established a hub-motor electric vehicle model with air springs. Then the quasi-infinite horizon nonlinear model predictive controller was designed to improve the longitudinal and vertical dynamic behavior.
However, the existing suspension structure is still limited to the “spring-damping” binary frame. The mass block is rarely used in the suspension because of its large volume and inconvenient installation. That leaves the suspension system without an effective inertial element. In 2002, Smith proposed the inerter [15], which is a two-end mass component. Research shows that introducing the inerter into the suspension system to form an “inerter-spring-damper” suspension structure can improve the vertical dynamic performance of the vehicle [16-24]. There are many forms of inerter such as ball-screw, rack and pinion, hydraulic, and mechatronic inerter. The mechatronic inerter is a new type of inerter device that is coupled by a mechanical inerter and motor. Wang and Chan [25] first proposed the device and optimized the electrical network structure to improve the vibration isolation performance of the suspension. According to the different mechanical transmission modes, it can be divided into single motor type, linear inerter-motor type, and rotary inerter-motor type. The first type is a special mechatronic inerter, which directly couples the motor with the mechanical vibration isolation system [26]. It simulates the impedance characteristics of the system using the output force of the motor. The linear inerter-motor type combines a linear motion type inerter with a linear motor, such as a hydraulic electric inerter [27]. The rotary inerter-motor type combines a rotary inerter with a rotary motor, such as a ball-screw mechatronic inerter in Ref. [25]. The two latter types of system output impedance consist of the impedance of the mechanical device and the simulated impedance of the motor.
The advantage of the mechatronic inerter is that the target mechanical impedance can be simulated using electrical elements in the external circuit. This allows the implementation of over-complicated purely mechanical networks to be solved. The theory of network analysis in the field of electricity can also be used to solve the vibration isolation research of mechanical systems. On the one hand, the output impedance of different structures is studied forward through network analysis. On the other hand, the physical realization of specific impedance is studied by network synthesis. Many scholars have studied the realization of the biquadratic impedance function. Bott and Duffin [28] proposed that any biquadratic positive real impedance can be realized passively by nine elements at most. Subsequently, the realization with six [29], five [30], and four [31] elements were proposed.
In order to further improve the vibration isolation effect, the higher-order impedance function has been studied. For the bicubic impedance function, a series-parallel network consisting of 13 elements is synthesized using Bott-Duffin synthesis. Reference [32] defined the concept of essential regular and implemented the impedance function using three reactive elements and four resistive elements. Hughes [33] proved that any impedance that can be realized by a series-parallel network containing at most three energy storage elements can also be realized by a series-parallel network containing at most three energy storage elements and at most four resistors.
However with the increase of the impedance function order, the number of components required also increased. To reduce the number of components, more exacting realization conditions are required. According to Pantell [34] simplification, non-series-parallel networks can achieve higher impedance functions with fewer components. References [35,36] studied biquadratic impedance and bicubic impedance using bridge networks and also gave the realization conditions of them. When using five elements to implement the impedance function, the biquadratic impedance can be realized by a bridge network containing two energy storage elements, and the bicubic impedance can be realized by a bridge network containing three energy storage elements. It is found that the increase in order seems to be related to the number of energy storage elements. Therefore, this paper selects four energy storage elements and one energy consuming element as the basic elements to analyze their impedance functions. There are two types of components in energy storage components: inductor and capacitor. For a bridge network with five components four of which are energy storage elements, the following combination modes are included: (1) 4 inductors and 1 resistor; (2) 4 capacitors and 1 resistor; (3) 3 inductors, 1 capacitor, and 1 resistor; (4) 3 capacitors, 1 inductor, and 1 resistor; (5) 2 inductors, 2 capacitors, and 1 resistor. These combination methods can form 23 different structures, and analyzing them all would be a complex and massive task. Therefore, for the convenience of research, the combination of 2 inductors, 2 capacitors, and 1 resistor is chosen for study.
In this paper, the impedance of the network structure is analyzed forward based on the bridge network, so as to avoid the complicated discussion of the realization conditions. The bridge network is coupled with a single motor to export force and improve the vibration isolation effect of HMDV suspension. The suppressing effect of different networks on the negative effect of vertical vibration of HMDV is studied. Aiming at improving vehicle grounding, the influence between the parameters of bridge network elements and suspension output response is analyzed. Then the suppression rule of the network structure on the negative effect of HMDV vertical vibration is studied. Firstly, the quarter HMDV model equipped with a switched reluctance motor (SRM) is built in Sect. 2. Then in Sect. 3, the impedance functions of the bridge network are solved by using the topological formulas. The element parameters in the network are optimized by using the Pattern Search algorithm. Moreover, in Sect. 4, the improvement effect of different structures of bridge network on HMDV vertical vibration negative effect is analyzed, and the influence between parameter changes and suspension output response is studied. Finally, some conclusions are drawn in Sect. 5.
2.　HMDV mechatronic suspension model
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2.1　SRM model
SRM has the characteristics of simple structure, large torque and small current at starting and low speed. It has high output and high efficiency in a wide speed and power range. It also has the advantages of good fault tolerance so it has a wide range of applications. In this paper, the external rotor 8/6 pole four-phase SRM is selected as the research object. The structure of the SRM is shown in Fig. 1 [37].
Radial electromagnetic force is the main source of noise and vibration of the SRM. Theoretically, the radial electromagnetic force is equal in magnitude and opposite in direction, so its resultant force is 0. However, because the air gap of the motor changes due to road excitation, the radial electromagnetic force is not really 0, and the motor eccentricity is defined as
where he is the absolute eccentricity of the motor, and gm is the air gap of the motor.
Therefore, the radial electromagnetic force Frm between the fixed rotor of the motor is
where Lj(θ, ij) is the inductance expressed in Fourier series [38-40] of the phase j, which corresponds to the rotor angular position and the winding current. Therefore, the unbalanced radial force of the phase j:
Where m and n are the pole number of stator in the opposite direction, gm(n) = (1-ε)gm, gm(m) = (1+ε)gm and |m-n| = 4.
For the HMDV system established in this paper, the vertical component of the unbalanced radial force is the main influencing factor. The total vertical electromagnetic force is the sum of the 4 phase electromagnetic forces, which is
2.2　Mechatronic suspension model
The 2-DOF vehicle suspension model considering the vertical vibration of sprung and unsprung mass is often used to study the vertical dynamic performance of vehicles. In this paper, a quarter vehicle dynamic model is selected as the research object, and the model is shown in Fig. 2. In the suspension part, the linear motor is used to equate the electrical impedance to the mechanical impedance. When the sprung mass moves relative to the stator, both ends of the motor also move and generate current. Then the current flows through the circuit at the external end of the motor to achieve the effect of equating the electrical network impedance to the mechanical impedance. This makes the original complex mechanical structure can be realized through the electrical network, which brings great convenience to the practical application. Through the design and analysis of the circuit structure, the influence of different structures on the vertical dynamic performance of the vehicle can be studied.
In Fig. 2, ms is the sprung mass, mus is the mass of the stator of the wheel hub motor, and mur is the mass of the rotor and tire. ks is the suspension spring stiffness, km is the equivalent stiffness of the motor, and Kt is the stiffness of the tire. zs is the vertical displacement of sprung mass, zus is the vertical displacement of stator and shaft, zur is the vertical displacement of rotor and tire, and zr is the road excitation. FEM is the electromagnetic force generated by the linear motor, V is the end voltage generated by the linear motor, ZSi (i = 1, 2, 3, 4, 5) is the electrical element in the bridge network circuit outside the motor, including 2 inductors, 2 capacitors, and 1 resistor, Fz is the vertical unbalance force caused by the wheel motor. The internal inductance and resistance of linear motor are ignored in modeling. According to Newton’s second law, the Laplace transform of the dynamic equation is shown as Eq. (5).
The calculation method of FEM is given by Eq. (6), where I is the current flowing through the external circuit of the motor, kt is the thrust coefficient of the linear motor, and ke is the electromotive force coefficient of the linear motor.
Then FEM is shown in Eq. (7):
where Zs, Zus, and Zur are the Laplace transform of the displacement, respectively, Ze(s) is the impedance of the external circuit, and s is the Laplace variable.
Conventional suspension consists of springs and dampers. Vehicle suspensions are tested and adjusted during the design process to ensure ride comfort and handling stability. To compare the dynamic performance of vehicles after the application of the bridge network, this paper takes a mature passenger car with conventional suspension as the comparison object, and the parameters of a quarter vehicle model are shown in Table 1.
3.　Bridge network and parameter optimization
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3.1　Impedance function of the bridge network
Chen et al. [35] used a five-element structure with two energy storage elements to establish biquadratic functions. Wang and Chen [36] gave 11 kinds of bridge networks with three energy storage elements to realize bicubic impedance. Therefore, this paper will establish a bridge network structure with four energy storage components, and the impedance function of the bridge network will be analyzed forwardly. From the component structure, the analysis avoids the complicated discussion of the conditions required for the synthesis of the positive real network. To facilitate the study, this paper selects 2 inductors, 2 capacitors, and 1 resistor to build a bridge network, and enumerates them to obtain the nine structures shown in Fig. 3.
The impedance functions of the above nine structures are calculated using the topological formula of the network function [41]. This section only shows the impedance of the three structures Z1, Z3, and Z7. To facilitate the calculation of linear motor thrust, the reciprocal expressions of the impedance function are shown in Eqs. (8)-(13). The impedance of other structures is in Appendix.
After analysis, it is found that the impedance functions of Z2, Z4, Z5, Z6, and Z7 are all of the strict biquartic types, that is, the highest order of numerator and denominator are 4. The impedance functions of other structures are non-strict biquartic functions. The highest order numerator in Z1 and Z9 is 3, and the highest order denominator is 4, which means A1 and A9 are equal to 0, the impedance functions of Z3 and Z8 are just the opposite, and F3 and F8 are equal to 0.
3.2　Dynamic performance optimization
In the suspension optimization process, assuming that the vehicle is driving at a speed of u = 20 m/s on a C-class road, the input displacement of road roughness can be expressed as follows:
where zr(t) is the vertical input displacement caused by the roughness of the model road surface, Gq(n0) is the road roughness coefficient, the value is 2.56×10-4 m3, and w(t) represents integral white noise.
3.2.1　Optimization algorithm
Pattern search algorithm is a general algorithm to solve the optimal value of the function. It does not need to use the derivative of the objective optimization function in the search process, so it can effectively solve the in-differentiable function and the complicated function optimization problem. The detailed steps of the pattern search method are shown in Fig. 4.
Each iteration of this algorithm alternates between axial and pattern movements. The purpose of axial movement is to detect the favorable direction of descent, while the purpose of pattern movement is to accelerate along the favorable direction. Axial movement is an exploratory search movement that starts from a point y and searches for probing movement with step size δk along the axis ej (j = 1,2,…, n) in turn. The probing movement along the axis is as follows:
(1) Positive axis detection
If f(y + δkej) < f (y), the detection is successful, take y = y + δkej; Otherwise, the detection will fail and the negative axis detection will be performed again.
(2) Negative axis detection
If f(y-δkej)<f(y), the detection is successful, take y = y-δkej; Otherwise, the detection fails and y remains unchanged.
The point obtained after each probe movement is used as the starting point for the next probe movement. After the n steps of detecting movement, the point at which the objective function value decreases can generally be obtained, thus an axial movement is completed. The starting point of each axial movement is called the reference point.
Let xk+1 be the point obtained by a single axial movement with point xk as the reference point. If f(xk+1) < f (xk), then start from point xk+1 and move in a pattern with a step size of αdk along the acceleration direction dk = xk+1-xk to obtain a new reference point. Then, start from the new reference point y and continue to move axially with the same step size of δk.
3.2.2　Optimization objectives and constraints
The negative vertical vibration effect of HMDV is caused by the significant increase of sprung mass, which leads to the deterioration of dynamic tire load. In order to improve its tire grounding, this paper selects the root-mean-square (RMS) value of the dynamic tire load as the optimization objective and establishes the objective function, which is given by Eq. (15). The optimization object is the element parameters in nine kinds of electrical network structures. Then the structure values under the optimal dynamic response can be obtained.
where DTLi represents the RMS value of dynamic tire load with bridge network suspension structure, DTLpas represents the RMS value of dynamic tire load of conventional suspension, J represents the ratio of them, and q represents the set of electrical network element parameters to be optimized.
To obtain electrical network parameters with better performance, this paper establishes certain constraint conditions to ensure that the optimized performance is better than that of the traditional structure. The DTLi of the bridge network suspension structure should be smaller than the DTLpas of the conventional suspension. The RMS values of body acceleration BAi and suspension working space SWSi of bridge network suspension are smaller than those of conventional suspension. At the same time, considering the rationality of the components, the optimization constraints are set as follows:
3.3　Optimization results and time domain response
After several optimizations, the optimal parameters of the suspension structure were obtained, as shown in Table 2. The parameters from Tables 1 and 2 were combined and substituted into the model to analyze the suspension’s output response, as shown in Table 3.
In Table 3, Z0 represents the conventional suspension structure, and the percentage in parentheses represents the comparison results between this indicator and the conventional suspension. The upward arrow indicates that the indicator has deteriorated, while the downward arrow indicates that the indicator has been optimized compared to conventional suspension. As shown in Table 3, after optimizing all structures, the RMS value of dynamic tire load is improved, and the RMS values of suspension working space and body acceleration are also reduced. It is worth noting that the impedance functions of Z1, Z3, Z8, and Z9 are not strictly biquartic, which also makes the performance improvement of these structures for HMDV less obvious than the others. The body acceleration of Z3 and Z8 even slightly deteriorated. Among several structures with strict biquartic impedance function, Z7 has the best improvement effect. Its time-domain response is shown in Fig. 5.
Compared with conventional suspension, its RMS value of dynamic tire load is reduced by 10.76%, and the RMS value of suspension working space is reduced by 18.10%. Moreover, the body acceleration is also improved by 1.45%. The application of this structure can improve the handling stability of the vehicle based on ensuring ride comfort. Then suppress the negative vertical vibration effect of HMDV.
To further verify the effectiveness of the established mechatronic suspension, the parameters of the Z7 structure in Table 2 and different road inputs were used to analyze its time domain response. According to Eq. (14), the road displacement input is not only related to road roughness but also to the vehicle’s driving speed. Therefore, by changing one of the values within a certain range, a series of different road input conditions can be obtained. In this article, the road roughness is set as a constant value, and the vehicle speed u is changed to 10 m/s, 20 m/s, and 30 m/s. The time domain response at 20 m/s is shown in Fig. 5, the time domain response at 10 m/s is shown in Fig. 6(a)-(c), and the time domain response at 30 m/s is shown in Fig. 6(d)-(f). The data in Tables 4 and 5 are the RMS values of the time domain response of performance indexes at different speeds and their improvement effects compared to conventional suspension.
From Fig. 6, Tables 4 and 5, it can be seen that the Z7 structure has also achieved significant performance improvements at different vehicle speeds. When the vehicle travels at a speed of 10 m/s, using a suspension with the Z7 structure can reduce the RMS value of dynamic tire load by 10.66%, and the body acceleration can also be maintained at a level close to that of conventional suspension. Although the improvement effect of the Z7 structure on three indexes is slightly worse than 20 m/s at 10 m/s, it significantly reduces the RMS value of dynamic tire load under the same conditions, effectively improving tire grounding. When driving at a speed of 30 m/s, there may be a slight deterioration of less than 1% in vehicle body acceleration, but it is still within an acceptable range. The RMS values of suspension working space and dynamic tire load were reduced by 18.22% and 10.79%, respectively. Overall, the Z7 structure can also suppress the negative vertical vibration effect of HMDV at this speed, and improve tire grounding and handling stability. Therefore, the suspension with Z7 is selected as the research structure, and its dynamic characteristics are analyzed in the next section.
4.　Dynamic performance analysis
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4.1　Frequency domain analysis
The Z7 is selected as the research object, and the suspension structure parameters are set according to the optimization parameters in Table 2. The Z1, Z3, and conventional suspension were compared. The gain of body acceleration gain, suspension working space, and dynamic tire load of the four suspensions in the frequency domain are shown in Fig. 7.
As can be seen from Fig. 7, the gain of dynamic tire load with bridge network in low frequency band and high frequency band is smaller than that of conventional suspension. It indicates that the equivalent mechanical impedance of the bridge network structure can suppress vehicle vibration. It also points out that the suppression effect is better in the low frequency band and high frequency band, and the Z7 structure has the best effect. Although slightly higher than the conventional suspension between 2-7 Hz, the overall effect is still better than the conventional suspension. The advantage of the bridge network is more remarkable in the gain of suspension working space. As this paper focuses on the negative vertical vibration effect of HMDV, the improvement of dynamic tire load is quite obvious. In terms of body acceleration, low-frequency vibration can be suppressed. But in the middle and high frequency bands, it is almost the same as that of conventional suspension and even has a slight worsening trend.
On the one hand, it is because of the selection of optimization focus. On the other hand, it is the increase of energy conversion process. Conventional suspension converts kinetic energy into spring potential energy and the internal energy of the oil in the damper. The linear motor, as an energy conversion device, converts kinetic energy into electric energy first. Then consumes energy in the form of resistance heating. Under the high frequency excitation, the energy input speed is accelerated, and the energy dissipation component fails to totally absorb the input energy in time. Part of the energy is transferred to the sprung mass, increasing the gain about body acceleration. But in general, the use of bridge network structure to equivalent mechanical impedance still has the advantages of simple, no complex mechanical components, and can effectively inhibit vehicle vibration.
4.2　Discussion on frequency domain characteristics of three structures
In vehicle vibration analysis, the damping ratio is commonly used to describe the amplitude-frequency characteristics of vibration response to input, which is defined as Eq. (18). Vehicles with lower damping ratios can achieve high comfort while achieving better tire adhesion performance requires a higher damping ratio. Sports cars typically have a large damper and harder tire stiffness, while luxury cars typically have a small damper and softer springs. The force between the two ends of a linear motor is related to the relative velocity between the sprung mass and the unsprung mass, which is similar to the working mode of a damper. Compared with the concept of “damping ratio”, the damping coefficient in the damping ratio is replaced by the impedance of the electrical network of the outer end of the linear motor. And the ratio obtained can be considered an“equivalent damping ratio”, as shown in Eq. (19).
Let T(s) = kekt/Ze(s), then the T(s) of three structures can be expressed as Eq. (20).
Let s = jω, then T(s) can be written as Eq. (21). T(s) contains the impedance function of the external electrical network, which has both real and imaginary parts. In order to compare the equivalent damping ratio, its modulus is taken for calculation as Eq. (22).
The equivalent damping ratio is shown in Eq. (23).
Unlike the fixed value of the damping ratio, the equivalent damping ratio of the bridge network structure is a function of frequency. Let ω = 2πf then analyze the equivalent damping ratio of the suspension with Z1,Z3, and Z7, and the results are shown in Fig. 8.
According to Fig. 8, the equivalent damping ratio of the suspension with Z7 can stabilize and remain around 0.2 after 2 Hz. The suspension with Z1 reaches its maximum equivalent damping ratio around 0.9 Hz, but then rapidly decreases and tends towards 0.17. The equivalent damping ratio of the suspension with Z3 rapidly reaches 0.13 at 0.1 Hz and continues to increase thereafter. Although its equivalent damping ratio exceeds that of the Z7 at 12 Hz, it is lower than that of the Z7 before 12 Hz. The purpose of this paper is to reduce the dynamic tire load of HMDV to improve the negative effects of vertical vibration. Compared to the other two types of suspension, the Z7 can achieve a higher equivalent damping ratio, which is beneficial for improving tire adhesion and reducing dynamic tire load. Therefore, the suspension with Z7 can reduce the vibration peak of dynamic tire load at low and high frequencies, but it will bring a slight deterioration in comfort, which is consistent with the analysis conclusion of the damping ratio.
4.3　Influence of interaction between components on suspension performance
Because the bridge network structure is different from the series-parallel network structure, there are some influence relationships between the component parameters. Therefore, the Z7 structure is taken as the research object. The vertical output response of the suspension is analyzed in the time domain when the energy storage elements and the energy dissipation elements change parameters. Then the interaction law between the elements is studied. L1, L2, C1, and C2 are energy storage elements, and R is energy dissipation elements. Table 6 shows the parameter ranges of circuit components in the Z7 structure. Figures 9-12 show the analysis results.
Figure 9(a)-(c) shows the influence of capacitors C1 and C2 on suspension performance, and the other pictures show the influence of inductors L1 and L2 on suspension performance. It can be seen that the combination of capacitors has a low influence on suspension performance, and the RMS values of body acceleration, suspension working space, and dynamic tire load change little. The combination of inductors has a certain influence on the suspension performance. When the two inductors are set to smaller values, the RMS value of the dynamic tire load and the suspension working space can be reduced. The handling stability and grounding of HMDV can be improved. However the RMS value of the body acceleration will be increased, and the ride comfort will be worsened. Therefore, appropriate values should be selected based on actual needs. To study the interaction between different types of energy storage elements and analyze the influence of capacitor and inductor elements on suspension performance, the combination relationship of C1-L1 and C2-L1 are taken as examples. The analysis results are shown in Fig. 10.
Figure 10 shows the influence of C1 and L1 on suspension performance on (a)-(c), and the influence of C2 and L1 on suspension performance on (d)-(f). As can be seen from Fig. 9, the RMS value of body acceleration decreases with the increase of C1 or C2 and remains unchanged with the increase of L1. The RMS values of suspension working space and dynamic tire load increase with the increase of C1 or C2, and has no obvious relationship with the change of L1. The variation range of the RMS values of the three evaluation indexes did not change greatly. In general, the parameter variation of the energy storage component combination had little influence on the suspension performance.
The bridge network contains both energy storage elements and energy dissipation elements. Then the R-C1 and R-L1 combinations are taken as examples to analyze the interaction between different types of elements and study the impact of parameter changes on suspension performance. The results are shown in Fig. 11.
Figure 11(a)-(c) shows the influence relationship of R and C1 change on suspension, and the other pictures show the influence relationship of R and L1 change on suspension. As can be seen from Fig. 11, the RMS value of body acceleration decreases with the increase of R but remains basically unchanged with the increase of C1 or L1. The RMS value of suspension working space and dynamic tire load increase with the increase of R, and slightly increases with the increase of C1 or L1. To further study the influence of energy consuming components on suspension performance, the output response of suspension when resistor parameters change is analyzed, and the results are shown in Fig. 12.
As can be seen from Fig. 12, the RMS value of body acceleration decreases with the increase of resistor parameters. The RMS value of suspension working space increases with the increase of resistor, and the RMS value of dynamic tire load first decreases and then increases with the increase of resistor. In terms of energy, when the energy consuming element takes a larger value, the input energy will be dissipated and absorbed at a faster speed. Then the energy transferred to the sprung mass becomes less, the body acceleration decreases and better riding comfort can be obtained. However, while energy consuming components absorb energy, there is still some energy in the energy storage elements. Even if the relationship between energy storage components and suspension output response is not very obvious, the existence of this part of the energy will affect the unsprung mass. That will increase the dynamic tire load, and affect the grounding and handling stability of the vehicle. Therefore, to suppress the negative effect of HMDV vertical vibration and reduce the dynamic tire load, it is necessary to take the appropriate resistor value. Thus the grounding of the vehicle could be improved based on ensuring the ride comfort.
5.　Conclusion
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In this paper, the deterioration of dynamic tire load caused by the vertical vibration negative effect of HMDV is studied. The high order impedance function is used to suppress the negative effect of vertical vibration, and a bridge network structure is established. The biquartic impedance of an electrical network is equivalent to that of the mechanical network by a linear motor. The bridge network structure with 2 inductors, 2 capacitors, and 1 resistor is constructed from the perspective of forward structural analysis. Then the impedance function of different structures is analyzed by topological formulas. To improve the tire grounding, the optimization algorithm is used to find the best parameters corresponding to the bridge network structure. The simulation analysis shows that the strict biquartic impedance is better than the non-strict biquartic impedance. Compared with the conventional suspension, the RMS values of the dynamic tire load and suspension working space can be reduced by 10.76% and 18.10%, respectively. The body acceleration is also maintained to the same degree as that of the conventional suspension. The suspension with a bridge network can also suppress the vibration at low and high frequencies of HMDV. The Z7 structure was selected as the research object to analyze the influence of parameter perturbations of different components on suspension performance. It was found that the parameter changes of energy storage components had a general effect on the improvement of suspension performance. The parameter variation of energy consuming components has a great influence on the performance of the suspension. Selecting the appropriate parameters for the bridge network structure can improve the grounding of the vehicle and keep the ride comfort to a good degree.
The mechatronic suspension using the bridge electrical network proposed in this paper only uses four energy storage elements and one energy dissipation element to achieve the biquartic impedance function. Compared to the implementation of series-parallel networks with high order impedance functions, the bridge network can be achieved with only 5 components. Compared to the bicubic impedance function containing five components, this paper achieves a higher biquartic impedance by changing the type and quantity of components. In addition, this mechatronic suspension uses a linear motor and external circuit to simulate the impedance function. This reduces the space occupied by mechanical components, providing great convenience for the layout and maintenance of practical applications. The proposed analysis method provides a new idea for the suppression of the negative effects of vertical vibration of HMDV. It also provides a research direction for the realization and analysis of higher-order impedance functions. The types and connection modes of electrical elements proposed in this paper are only part of the electrical network. Future research will include different types and quantities of components, and based on simulation analysis, experiments are conducted to verify the simulation results.
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doi: 10.1007/s10409-024-24208-x
  • Receive Date：2024-07-13
  • Online Date：2026-03-24
  • Published：2025-12-01
Article Data
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History
  • Received：2024-07-13
  • Accepted：2024-10-23
Affiliations
    1School of Automotive and Traffic Engineering, Jiangsu University, Zhenjiang 212013, China
    2State Key Laboratory of Advanced Design and Manufacturing Technology for Vehicle, Changsha 410012, China
    3Jiangsu Province and Education Ministry Co-sponsored Synergistic Innovation Center of Modern Agricultural Equipment, Zhenjiang 212013, China
    4College of Engineering, China Agricultural University, Beijing 100083, China
    5State Key Laboratory of Intelligent Agricultural Power Equipment, Luoyang 471039, China
    6Luoyang Tractor Research Institute Co. Ltd., Luoyang 471039, China

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* E-mail address: (Yujie Shen)
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表12种不同金属材料的力学参数
科
Family		 属数
Number of
genus		 种数
Number of
species		 占总种数比例
Percentage of
total species (%)		 属
Genus		 种数
Number of
species		 占总种数比例
Percentage of total
species (%)
鹅膏菌科Amanitaceae 2 11 5.26 鹅膏菌属 Amanita 10 4.78
小菇科 Mycenaceae 2 12 5.74 丝盖伞属 Inocybe 5 2.39
多孔菌科 Polyporaceae 8 14 6.70 蜡蘑属 Laccaria 5 2.39
红菇科 Russulaceae 3 23 11.00 小皮伞属 Marasmius 6 2.87
小菇属 Mycena 11 5.26
光柄菇属 Pluteus 5 2.39
红菇属 Russula 17 8.13
栓菌属 Trametes 5 2.39
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