Dynamic Performance Analysis of Semiactive Vehicle ISD Suspension Based on the Power-Driven-Damper Strategy

. In this paper, the vehicle ISD (inerter-spring-damper) suspension and power-driven-damper control strategy are combined to the suspension design, and the power-driven-damper semiactive ISD suspension is proposed. Te dynamic models of the passive suspension S1 and two semiactive ISD suspensions S2 and S3 are established. Based on the port-controlled Hamiltonian theory, the power-driven-damper semiactive control strategy is designed by analyzing the power transfer of suspension S3. Ten, the parameters of the two models are optimized by the particle swarm optimization algorithm, and the optimization results show that the suspension S3 has better performance. Te infuence of the semiactive damping coefcient, the spring stifness, and the inertance on the vibration suppression performance is investigated based on the suspension S3. Te efect of parameter perturbation on power-driven-damper semiactive vehicle ISD suspension illustrates that the designed semiactive vehicle ISD suspension has better ride comfort in a wider range frequency and good robust performance.


Introduction
Vehicle suspension is an essential component of the chassis system, and it can bufer the impact of road roughness [1,2] and play a decisive role in providing passengers with excellent ride comfort performance [3,4].Terefore, improving the performance of suspension has important engineering signifcance.As we all know, the traditional passive suspension consisting of springs and dampers has been widely used for its low price and practicality [5].However, with the rapid development of automotive technology, the traditional passive suspension is brutal to meet people's demand for better comfort.Te performance enhancement of passive suspension has reached a bottleneck, and this situation is not improved until Smith [6] invented the inerter element, a passive mechanical structure that can generate a force proportional to the relative acceleration of its two ends.With the introduction of inerter into the feld of vehicle vibration isolation, the inherent form of traditional suspension "spring-damping" is broken, and a new "inerter-spring-damper" suspension (called ISD suspension) appears.Te vehicle ISD suspension has been widely studied by scholars, and it is proven to have good vibration suppression performance [7,8].
Te research illustrates that compared with the vibration suppression performance of traditional suspension, the performance of vehicle ISD suspension of many layouts is much better [9,10].Under diferent road excitations, the dynamic characteristics of suspension are studied, and the results illustrate that the vibration suppression performance of vehicle ISD suspension is very superior [11].Two types of vehicle ISD suspension layouts are designed and analyzed in the frequency domain, and it is shown that the vibration in the ofset frequency bands of the body resonance can be suppressed by vehicle ISD suspension [12].Moreover, the performance of heavy vehicles with inerter is greatly improved; the research indicates that the vertical vibration of heavy vehicle can be efectively reduced by the vehicle ISD suspension, and it is more prominent in the low frequency band [13,14].
Because the parameters of the passive suspension is not adjustable and the suspension can only obtain the best vibration suppression performance under a specifc road input, the control strategy [15,16] is introduced into the suspension system, and the suspension which can actively adjust the parameters is studied.Active suspension is to control the actuator output for power so that the body attitude can reach an ideal state under diferent road conditions.Compared with passive suspension, the ride comfort and handling stability of active suspension have been effectively improved [17][18][19], but the high manufacturing cost restricts its further wide application.In contrast to active control, semiactive control [20] necessitates a lower amount of external energy for the generation of control force and also diminishes structural complexity.Tis control approach has found application in areas such as earthquake resistance and suspension systems [21,22].Te concept of semiactive suspension is proposed by Karnopp [23], which can adjust the spring stifness and damping coefcient according to the road input to improve ride comfort.Research indicates that compared with passive suspension, semiactive suspension has better performance [24].Under the idea of semiactive suspension, a series of excellent semiactive control strategies come up and are applied to suspension systems in order to further improve the vibration suppression performance of suspension, including classic two-state skyhook (SH) [25], acceleration-driven-damper (ADD) [26], power-drivendamper (PDD) [27], mixed skyhook-acceleration driven damper (SH-ADD) [28], and mixed skyhook and powerdriven-damper (SH-PDD) [29].Te research shows that the ADD control strategy can suppress the vibration of the body in the high frequency band efectively, but the ADD control strategy is a control strategy that switches according to the value of acceleration, and obvious shake exists when the body acceleration is zero, while the PDD control strategy can suppress the vibration of the body efectively without producing obvious shake.Te vehicle ISD suspension has better vibration suppression efect in the low frequency band, but it does not have good performance in the high frequency band.By combining the vehicle ISD suspension with the powerdriven-damper control strategy, vibration suppression can be achieved over a wider range frequency, thus improving ride comfort.Tis paper is devoted to the study of vibration suppression mechanism of power-driven-damper vehicle ISD suspension.Te efects of the damping coefcient, variable spring stifness, and variable inertance on vibration suppression are also studied to draw some conclusions.Te arrangement of this paper is as follows.
In Section 2, three suspension models are built, the explicit expression of the power-driven-damper control strategy is derived based on the port-controlled Hamiltonian theory, and the optimal parameters of suspension are obtained by particle swarm optimization (PSO) which has fast convergence speed and is easy to obtain the global optimal solution [30].Efect of the damping coefcient on vibration suppression performance is analyzed in Section 3. In Section 4, the infuence of variable spring stifness and variable inertance on the semiactive vehicle ISD suspension is also studied.Finally, some conclusions are drawn in Section 5.

Dynamic Model of Vehicle ISD Suspension
Based on the Power-Driven-Damper

Dynamic Model of Vehicle ISD Suspension.
Quarter suspension is a two-degree-of-freedom model obtained by simplifying the vehicle suspension system according to its structure and type, and it is the most commonly used basic model to study the control law and vibration suppression mechanism of vehicle suspension, as shown in Figure 1.In this paper, three simple suspension layouts are established, which are traditional passive suspension S1, power-drivendamper vehicle ISD suspension S2, and power-drivendamper vehicle ISD suspension S3, as shown in Figure 2 [31].
In Figures 1 and 2, m s is the sprung mass, m u is the unsprung mass, z s is the vertical displacements of the sprung mass, z u is the vertical displacements of the unsprung mass, z r is the random road input, K t is the equivalent spring of the tire, and M represents the structure of the suspension, including the three structures shown in Figure 2.For suspension S1, the damper and the spring are arranged in parallel.For suspension S2, the semiactive damper and the inerter are arranged in parallel.For suspension S3, the semiactive damper and the intermediate structure T(s) are also arranged in parallel, where T(s) is an intermediate structure comprising an inerter and a spring connected in series.K is the support spring, b is the inerter, c is the damper, c p is the semiactive damper, and k is the auxiliary spring.Te dynamic motion equations of the suspensions are as follows:

Port-Controlled Hamilton System and the Power-Driven-Damper Control Strategy.
In suspensions S1 and S2, the controllable damper is the only energy dissipating element, and the others are the energy storage element, which is a typical port-controlled Hamiltonian system [27], and can be expressed as follows: where u is the pavement input disturbance variable, v is the input variable of controllable damping, y is the output variable, x is the state variable, J i (x, v) is the interconnection matrix of the system, R i (x, v) is the damping dissipation matrix, H (x) is the Hamiltonian energy function of the system, and g (x, v) is the function related to the state variable.Te power balance equation of the port-controlled Hamiltonian system can be expressed as follows: ( Te power dissipation equation of the system can be expressed as follows: In order to maximize the passenger riding comfort, the power-driven-damper control strategy gives priority to reducing body acceleration.Te control strategy can be expressed as follows: where W d denotes the desired level of power, which equals zero, and d (v, x) denotes all the energy dissipations.Taking the suspension S3 as an example, the storage power can be expressed as follows: Te dissipated power can be expressed as follows: Te total suspension power can be expressed as follows:

Shock and Vibration
Te control strategy set in this paper is as follows: when the net input power in suspension is less than zero, c p is the maximum value.When the net input power in suspension is greater than or equal to zero, c p takes the minimum value; When the net input power of suspension is zero but the relative displacement of suspension is not zero, that is, when the relative speed is zero and the relative displacement of suspension is not zero, c p is taken as the average value of the maximum and minimum values; this process is regarded as the transition stage.In the last mode, the c p value ensures zero net input power.Te power-driven-damper control strategy avoids the switching conditions associated with body acceleration and, therefore, can suppress shake to a large extent.Te conditions to be met for the controllable damping coefcient c p are as follows [32]:

Optimization of Suspension Structure Parameters.
To obtain the best performance of suspension, it is necessary to optimize the parameters of suspension.In this paper, the particle swarm optimization (PSO) with fast convergence speed and easy to fnd the global optimal solution is selected.Te basic idea of particle swarm optimization is to design a massless particle to simulate birds, set it in a certain space range, and each particle searches for the optimal solution p best in the space separately, and shares the individual optimal solution with other particles in the population.Finally, the best solution among all the individual optimal solutions is regarded as the global optimal solution g best [33].Te updated formulas for particle velocity and position properties are as follows: In the formula, ω represents the inertia weight, the general value is [0.8, 1.2]; c 1 and c 2 represent learning factors or acceleration constants; r 1 and r 2 represent random numbers in [0, 1]; m represents the number of iterations; p best represents the individual extreme value; g best represents the group extreme value; V m represents the velocity of the particle when the number of iterations is m; V m+1 represents the velocity of the particle when the number of iterations is m + 1; X m represents the position of the particle when the number of iterations is m; and X m+1 represents the position of the particle when the number of iterations is m + 1. Te fowchart of the optimization process is shown in Figure 3.
In this paper, the main spring stifness K, auxiliary spring stifness k, inertance b, maximum adjustable damping coefcient c max , and minimum adjustable damping coefcient c min are taken as the individual to be sought, the parameters are optimized with comfort as the guidance and the RMS (root mean square) of body acceleration as the optimization objective.Since the vehicle vibration response basically conforms to the normal distribution, according to the normal distribution probability integral principle, if RMS of the dynamic tire load is less than one third of the static load G, equaling m s plus m u , the odds that the wheel jumps of the road profle is less than 0.15%.If the RMS of the suspension working space is under one third of its threshold value [f d ], the suspension working space can maintain a safe range within 99.7% functioning time [34].Generally, the value of [f d ] is 7-9 cm, and 8 cm is selected in this paper.Terefore, the RMS of the dynamic tire load and the RMS value of the suspension working space are selected as constraints, and they are less than 1217 N and 0.027 m, respectively.Te boundaries of optimized design variables are as follows: main spring is the primary support spring, responsible for providing support, and its value needs to consider practical applications.Terefore, the value range of K is between 10000 N•m −1 and 25000 N•m −1 .Auxiliary spring stifness does not need to be excessively large.Terefore, the value of k ranges from 0 N•m −1 to 10000 N•m −1 .Te selection of the damping coefcient and inertance is appropriate for practical applications.Te range of damping coefcient is from 0 N•s•m −1 to 10000 N•s•m −1 , and the range of inertance is from 0 kg to 1000 kg.
In this paper, the pavement input model is a random pavement input model and the expression is as follows: In the formula, u is the vehicle speed, z r (t) is the vertical input displacement, w (t) is the white noise with an average value of 0, and G q (n 0 ) is the road roughness coefcient.In this paper, the set speed is 100 km/h, grade C pavement is selected, and the road roughness coefcient is 256 × 10 −6 m 3 .

Shock and Vibration
Te parameters of the quarter-car model are shown in Table 1 [35], and the optimized results are shown in Table 2.
Based on the parameter optimization, the three suspensions are simulated, respectively, and the comparison of their values are shown in the Table 3.
In this paper, the suspension is comfort oriented and the most important evaluation index is body acceleration; the responses are shown in Figure 4.
From Table 3 and Figure 4, it can be seen that the vibration suppression performances of the semiactive vehicle ISD suspensions based on the power-driven-damper are signifcantly improved.From the frequency domain diagram, it can be seen that compared with suspension S1, the gains of body acceleration of suspension S2 and suspension S3 are smaller in the whole frequency domain; this result indicates that that the semiactive ISD suspension based on power-driven-damper efectively achieves vibration suppression in the wider range frequency.From the time domain results, it can be seen that compared with suspension S1, the RMS of the body acceleration of suspension S2 is reduced by 25.3% and the RMS of body acceleration of suspension S3 is reduced by 29.8%.Compared with suspension S2, the RMS values of body acceleration of S3 is better in the whole time domain simulation process.It can be seen that the performance of suspension S3 is obviously superior to suspension S2 in terms of the frequency analysis and time domain analysis.

Impact of the Power-Driven-Damper Coefficient on Vehicle ISD Suspension Performance
Te core of the power-driven-damper control strategy is to change the damping coefcient according to the road input so as to improve the vibration suppression performance of the suspension.In order to further study the mechanism of the power-driven-damper control strategy in vehicle ISD suspension based on suspension S3, the body acceleration, the suspension working space, and the dynamic tire load are deemed as response variables while the c max and c min are deemed as independent variables, and the infuence of damping coefcient on vibration suppression performance is analyzed.Te responses are shown in Figures 5-7, and the asterisk in the fgure is marked as the optimal value point of optimization.
Figure 5 shows the change of body acceleration with the variation of c max and c min .It can be seen that when the value of c max increases gradually in the range of 0-10000 N•s•m −1 , the RMS of body acceleration decreases rapidly at frst and then decreases slowly after a turning point is reached.With an increase of the value of c min , the RMS of body acceleration frst decreases and then increases, and when the value of c max is small, the rate of change is large.
Figure 6 shows the change of the suspension working space with the variation of c max and c min ; it can be seen that with an increase of value of c max , the RMS of suspension working space decreases rapidly at frst and increases slowly after an infection point is reached.With an increase of the value of c min , the RMS of suspension working space  Shock and Vibration decreases gradually, and the decreasing speed decreases with an increase of value of c min .
Figure 7 shows the change of the dynamic tire load with the variation of c max and c min ; it can be seen that with an increase of the value of the c max , the RMS of dynamic tire load decreases rapidly at frst and then decreases slowly after an infection point is reached.With an increase of the value of c min , the RMS of dynamic tire load decreases gradually, and the decreasing speed decreases with an increase of value of c min .

Shock and Vibration
It can be concluded that the trend of body acceleration is diferent, but the trends of RMS of the suspension working space and RMS of the dynamic tire load are similar.When the value of c max is large and the value of c min is small, the RMS of body acceleration is small, but when the value of c max and the value of c min are large, the RMS of the dynamic tire load and RMS of the suspension working space are small.Since it is a comfort-oriented suspension system, the value of c max should be as large as possible and the value of c min should be as small as possible in order to obtain a small RMS of body acceleration within the constraints.In this paper, when the RMS of body acceleration reaches the minimum value, the RMS of the dynamic tire load reaches the boundary value of the constraint condition, and the RMS of the working space is still within the constraint range.Te abovementioned studies show the efect of the damping coefcient variation on the vibration suppression performance and also verify the correctness of the parameter optimization.

Impact of the Parameters Perturbation on Vehicle Suspension Performance
In this section, the efect of the parameters perturbation on suspension performance will be analyzed in detail.According to the above research, the performance of the suspension S3 is the best, thus suspension S3 is selected as the research object.According to the parameter optimization results, the values of auxiliary springs are increased and decreased by 50%, while other parameters are unchanged, as shown in Table 4. Figure 8 shows the gains of body acceleration, suspension working space, and dynamic tire load with variable stifness while the damping coefcient and the inertance are constant; the comparison of peak data is shown in Table 5. Figure 8(a) shows the gains of body acceleration of variable stifness.It can be seen that the peak values become larger with the increase of the stifness both in the low frequency and high frequency band.Figure 8(b) shows the gains of the suspension working space of variable stifness.Te peak values in the low frequency band become larger with the increase of the stifness, but the peak values in the high frequency band become smaller with the increase of the stifness.Figure 8(c) shows the gains of the dynamic tire load of variable stifness.With the increase of stifness, the peak values in the low frequency band become larger, but with the increase of stifness, the peak values in the high frequency band become smaller.It can be concluded that the variable stifness results in diferent change trends of the three performance indices, but all of the peak value variations are less than 10% in Table 5, which indicates that the suspension of this structure has good robustness.
Figure 9 shows the changes of RMS of body acceleration, RMS of suspension working space, and RMS of dynamic tire load with the inertance varying from 0 kg to 1000 kg.
It can be seen that the RMS of body acceleration undergoes a rapid increase with the rise of the inertance, followed by a sharp decline.Subsequently, it undergoes a slight increase before gradually decreasing again.With the increase of the inertance, the RMS of suspension working space undergoes a brief period of slight decrease, followed by a rapid increase and then a swift decrease.Subsequently, it undergoes two stages of frst increasing and then decreasing, and the magnitude of the change gradually decreases.Eventually, it slowly rises.With the increase of the inertance, the RMS of the dynamic tire load undergoes three stages of rapid decrease followed by rapid increase, with the magnitude of change gradually diminishing as the inertance increases.Following this stage, the RMS of the dynamic tire Shock and Vibration load decreases slowly.Since it is a comfort-oriented suspension, the body acceleration should be as small as possible within the constraint conditions, and it can be seen that when the inertance is 1000 kg, the body acceleration is the minimum value that satisfes the constraint conditions.

Conclusion
In this paper, the performance of the semiactive vehicle ISD suspension based on the power-driven-damper strategy is investigated.Tree-type suspension dynamic models are established, and the expression of the power-drivendamper control strategy is obtained according to the portcontrolled Hamiltonian theory.Te particle swarm optimization is used to optimize the parameters of the suspension, and the optimal structure is suspension S3.Te suspension S3 is selected as the research object, and the infuence of the semiactive damping coefcient on suspension performance is analyzed.Results illustrate that the designed semiactive vehicle suspension has a better vibration isolation performance in the wider range frequency, and larger c max value and smaller c min value are more benefcial to obtain good ride comfort performance within the constraint range.Ten, the impact of the spring stifness perturbation and the inertance variation on suspension performance is also studied; the results illustrate that variable stifness results in diferent variation trends of the body acceleration, suspension working space, and dynamic tire load, but all of the peak value changes are less than 10%.A big inertance value is also suggested for the comfort-oriented suspension design, but the actual application cost of the device should also be considered.Te research results may provide certain engineering-guiding signifcance for the parameter design of the semiactive vehicle ISD suspension.

Figure 7 :Figure 8 :
Figure 7: Te change of dynamic tire load with the variation of c max and c min .

Table 3 :
RMS values comparison of suspension.

Table 5 :
Comparison of peak data.