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Optimal design and experimental research of vehicle suspension based on a hydraulic electric inerter
Mechatronics 61 (2019) 12–19
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Optimal design and experimental research of vehicle suspension based on a hydraulic electric inerter Yujie Shen∗, Yanling Liu, Long Chen, Xiaofeng Yang Automotive Engineering Research Institute, Jiangsu University, Zhenjiang 212013, China
a r t i c l e
i n f o
Keywords: Vehicle Suspension Inerter Hydraulic electric Optimal design Bench test
a b s t r a c t As the mechanical dual of a capacitor via the force-current analogy, an inerter has been successfully applied in various fields such as automotive engineering, civil engineering and aerospace engineering. The introduction of an inerter not only allows the use of network synthesis analogy to design mechanical layouts but also opens the door to adopt electrical element impedances to simulate the corresponding mechanical elements. This paper combines both of these ideas and presents a new form of a mechatronic inerter, namely the hydraulic electric inerter (HEI), which consists of a hydraulic piston inerter and a linear motor. On the basis of the HEI device, a bicubic impedance function is considered in the optimal design of a vehicle suspension system employing both mechanical elements and electrical elements. In addition, a methodology for reducing the order of the bicubic impedance function is proposed, and the network is finally realized by utilizing inerter, spring and damper elements. Then, by comparison with a passive suspension, the advantages of the new vehicle ISD (inerter-spring-damper) suspension, called the vehicle HE-ISD (hydraulic electric-inerter spring damper) suspension, are demonstrated by numerical simulations. Finally, a HEI device is designed, and bench tests of the vehicle HE-ISD suspension are carried out. Experimental results indicate that, the vehicle HE-ISD suspension is superior to the passive suspension system, the RMS (root-mean-square) value of the suspension working space is improved by 19.97%, and the RMS value of the dynamic tire load is improved by 10.21%.
1. Introduction Since the first introduction of the inerter in 2002 [1], inerter-based vibration suppression has been a popular topic in mechanical fields. The feature of the inerter is that the force across its two terminals is proportional to the relative acceleration. Until now, the inerter element has been successfully deployed in various areas, including vehicle suspensions [2–5], civil engineering [6–8], railway suspensions [9–11], landing gear shimmy suppression [12,13], etc. The merit of the inerter is that it allows the use of electrical analogies, such as network synthesis, to design complex layouts of vibration isolation systems based on the inerter-spring-damper network. For the optimal design of vehicle suspension systems, low order admittance functions have been investigated [14–16] by using purely mechanical networks. In [17], different orders of positive real controllers were designed to improve the performance of the vehicle suspensions, and the results showed that the vibration suppression performance will be further improved when the admittance function order is increased. The same conclusion was also obtained in [18] that compared with the use of biquadratic impedances in railway suspension systems, a bicubic function can reduce the wheel load index by 36%. It is apparent that higher order impedance functions can provide a wider range of dynamic properties. However, the biquadratic ∗
and bicubic impedances can be realized by using at most nine elements and thirteen elements, respectively, in the Bott-Duffin procedure [19]. The bicubic impedances mentioned in [16] were all special third-order positive-real functions and were realized by only five elements using the methods in [1]. Although the minimal realizations problem of bicubic impedance has been explored in [20–22], the high complexities of the mechanical network may restrict the application of the inerter in vehicle suspensions. Another advantage of the inerter is that, it opens the door to utilize the electrical elements to simulate the corresponding mechanical network to obtain the target impedance output. Generally, there are many realizations of inerters, such as the ball-screw inerter [23], the rack and pinion inerter [24], the hydraulic inerter [25], and the fluid inerter [26–28]. In [29], a mechatronic inerter consisting of a PMEM (Permanent Magnet Electric Machinery) and a ball-screw inerter was proposed, and a bilinear and a biquadratic function was used for designing a vehicle suspension system on the basis of several simple mechanical networks. Recently, linear motors have been widely used in energy harvesting [30–32] and active control systems [33–35] due to their superior fast response, good controllability and easy energy-regeneration. From the above analysis, this paper will concentrate on the problem of designing a vehicle suspension system based on a new hydraulic-electric
Corresponding author. E-mail address: [email protected] (Y. Shen).
https://doi.org/10.1016/j.mechatronics.2019.05.002 Received 2 January 2019; Received in revised form 10 April 2019; Accepted 19 May 2019 Available online 29 May 2019 0957-4158/© 2019 Elsevier Ltd. All rights reserved.
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inerter (HEI), which includes a hydraulic piston inerter and a linear motor. Based on the features of the HEI device in the layout of the suspension system, a methodology for reducing the impedance function order is proposed based on the foster cycle procedure [36], in which a bicubic impedance function is considered. The paper is arranged as follows: in Section 2, a new hydraulicelectric inerter is introduced in detail, including its structural components and working principle. Then, the dynamic model of a vehicle suspension system using the inerter-spring-damper network adopting a bicubic impedance function is built. The methodology of reducing the impedance order is also presented and the network is finally realized via passive elements in Section 3. In Section 4, the dynamic model of the vehicle suspension incorporating both mechanical elements and electrical elements on the basis of the HEI device, namely, the vehicle HE-ISD suspension, is built. Numerical simulations are conducted in Section 5 to illustrate the validity of the newly proposed HE-ISD suspension. Finally, the HEI device is manufactured, and the vehicle HE-ISD suspension bench is built, the experiments are completed by comparing to the results with the passive suspension system. Some conclusions are presented in Section 7.
the linear motor and the velocity of the mover is shown as 𝑉𝑔 = 𝑘𝑒 𝑣𝑎
(2)
where Vg is the voltage generated by the linear motor, ke is the voltage coefficient, and va is the velocity of the mover. The force of the linear motor Ft is expressed as 𝐹𝑡 = 𝑘𝑡 𝐼𝑎
(3)
where kt is the force coefficient and Ia is the current in the external circuit of the HEI device. Similar to the mechatronic inerter in [29], the force and the velocity between the HEI terminals are expressed as [37]: 𝐹 (𝑠) = 𝑣(𝑠)
(
𝑆1 𝑆2
)2 𝑚𝑠 +
𝐾𝑚 𝑅𝑒 + 𝐿𝑒 𝑠 + 𝑍𝑒 (𝑠)
(4)
where F(s) and v(s) are the Laplace forms of the force and velocity, respectively, of the two terminals of the HEI device, Km is the motor coefficient, for the HEI device, Km = (S1 /S2 )2 ke kt , Re is the coil resistance, Le is the inductance of the linear motor. Ze (s) is the impedance of the external electric circuit. It is noted that, the impedance of the HEI device consists of two parts. One is the mechanical inerter impedance, and the other is the electrical impedance determined by the external circuit of the HEI device.
2. HEI device In contrast to the mechatronic inerter device introduced in [29], the hydraulic electric inerter consists of a hydraulic piston inerter and a linear motor. A schematic of the HEI device is presented in Fig. 1. In Fig. 1, 1 and 7 present the two terminals of the HEI device, 2 and 6 present the connecting tubes of the main cylinder and the auxiliary cylinder, 3 presents the main cylinder, 4 presents the piston of the main cylinder, 5 presents the piston rod of the main cylinder, 8 presents the moving rod of the linear motor, 9 presents the moving magnetic pole, 10 presents the winding of the linear motor, 11 presents the mover yoke, 12 presents the stator of the linear motor, 13 presents the piston rod of the auxiliary cylinder, 14 presents the piston of the auxiliary cylinder, and 15 presents the auxiliary cylinder. Note that the moving rod of the linear motor is integrated of the piston rod of the auxiliary cylinder so that the relative linear motion between the two terminals of the HEI device leads to the linear motion between the auxiliary cylinder and the moving rod of the linear motor. The different section areas of the main hydraulic cylinder and the auxiliary cylinder result in the motion transformation of the device, the inertance of the hydraulic piston inerter is expressed as [37]: ( )2 𝑆1 𝑏=𝑚 (1) 𝑆2
3. Optimal design of the vehicle ISD suspension In the optimal design procedure of the vehicle suspension system, a generalized quarter-car model, presented in Fig. 2, is considered. where ms is the sprung mass, mu is the unsprung mass, K is the supporting spring of the suspension, Kt is the tire stiffness, and zs , zu and zr are the vertical displacements of the sprung mass, unsprung mass and the random road input, respectively. The Laplace forms of the dynamic equations of the quarter-car model are expressed as 𝑚𝑠 𝑠2 𝑍𝑠 + [𝐾 + 𝑠𝑇 (𝑠)](𝑍𝑠 − 𝑍𝑢 ) = 0
(5)
𝑚𝑢 𝑠2 𝑍𝑢 − [𝐾 + 𝑠𝑇 (𝑠)](𝑍𝑠 − 𝑍𝑢 ) + 𝐾𝑡 (𝑍𝑢 − 𝑍𝑟 ) = 0
(6)
where Zs , Zu and Zs are the Laplace forms of zs , zu and zr . T(s) is the undefined suspension structure and is expressed by a bi-cubic impedance function as 𝑇 (𝑠) =
𝐴𝑠3 + 𝐵 𝑠2 + 𝐶𝑠 + 𝐷 𝐸 𝑠3 + 𝐹 𝑠2 + 𝐺𝑠 + 𝐻
(7)
where b is the mechanical inertance of the HEI device, m is the mass of the piston rod and the moving rod of the linear motor, and S1 and S2 are the sectional areas of the main cylinder and the auxiliary cylinder, respectively. For the linear motor, when there is a motion between the stator and the mover, the relationship between the voltage generated by
where A, B, C, D, E, F, G and H ≥ 0 and not all E, F, G and H = 0. The positive-real condition of the T(s) is introduced in [38] and can be shown as (1) (𝐶 + 𝐺)(𝐵 + 𝐹 ) ≥ (𝐷 + 𝐻)(𝐴 + 𝐸) (2) one of the following holds: √ (a) 𝑎3 = 0, a2 ≥ 0, a0 ≥ 0, −𝑎1 ≤ 2 𝑎0 𝑎2 ; (b) a3 > 0, a0 ≥ 0, and (b1) or (b2) holds:l
Fig. 1. Schematic of the HEI device.
Fig. 2. Quarter-car model. 13
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Table 1 Model parameters. Parameters
Values
Vehicle body mass ms (kg) Unsprung mass mu (kg) Suspension spring stiffness K (N m−1 ) Tire spring stiffness Kt (N m−1 )
320 45 22,000 190,000
Fig. 3. Schematic of the reducing order structure.
√ (b1) a1 ≥ 0 and −𝑎2 ≤ 3𝑎1 𝑎3 ; 2 (b2) 𝑎2 > 3𝑎1 𝑎3 and 2𝑎32 − 9𝑎1 𝑎2 𝑎3 + 27𝑎0 𝑎23 ≥ 2(𝑎22 − 3𝑎1 𝑎3 )3∕2 where a0 = DH, a1 = CG-DF-BH, a2 = BF-CE-AG and a3 = AE. The positive-real condition of the T(s) is very complex, and the bicubic impedance function is difficult to realize by using only resistors, capacitors and inductors (corresponding mechanical elements are dampers, inerters and springs, respectively). From the analysis in [29], it is noted that the mechatronic inerter or the HEI device is always in parallel with external circuits. According to the foster cycle procedure [36], the inerter, a dual of the capacitor, as a reactive element, can be extracted from the original impedance function to reduce the order of T(s). However, the damper, a dual of the resistor, as a resistive element, can also be extracted in parallel or in a series structure from the original function, and the order will not be reduced. Following the above two steps, the new structure is depicted in Fig. 3. where bm is the mechanical inertance, c0 and cm are the mechanical dampers and c0 = 0 or cm = +∞. Then, T(s) is changed to 𝑇 (𝑠) =
1
Fig. 4. Bode diagram of the optimal structure.
(8)
The optimal parameters are bm = 93.57, c0 = 0, cm = 1840, A0 = 494.4, B0 = 2.913 × 108 , C0 = 1.451 × 106 , D0 = 474.4, E0 = 2.697 × 104 , and F0 = 6.14 × 105 . The T(s) and T1 (s) are shown as:
(9)
𝑇 (𝑠) =
1.84 × 103 𝑠3 + 1.046 × 105 𝑠2 + 1.446 × 107 𝑠 + 6.015 × 104 𝑠3 + 76.53𝑠2 + 8.975 × 103 𝑠 + 2.548 × 104
(13)
where A0 , B0 , C0 , D0 , E0 and F0 ≥ 0 and not all D0 , E0 and F0 = 0. T1 (s) is positive-real if and only if [40] √ √ 𝐵0 𝐸0 − ( 𝐴0 𝐹0 − 𝐶0 𝐷0 ) ≥ 0 (10)
𝑇1 (𝑠) =
494.4𝑠2 + 2.913 × 108 𝑠 + 1.451 × 106 474.4𝑠2 + 2.697 × 104 𝑠 + 6.14 × 105
(14)
1 𝑏𝑚 𝑠+𝑐0 +𝑇1 (𝑠)
+
1 𝑐𝑚
where T1 (s) is a biquadratic transfer function and is expressed as 𝑇1 (𝑠) =
𝐴0 𝑠2 + 𝐵0 𝑠 + 𝐶0 𝐷0 𝑠2 + 𝐸0 𝑠 + 𝐹0
The Bode diagrams of the T(s) and T1 (s) are presented in Fig. 4. From Fig. 4, it is seen that the structures T(s) and T1 (s) function as inerters in the low frequency range, which is missing in the traditional passive vehicle suspension consisting of spring and damper elements. According to Jiang and Smith [40], T1 (s) can be realized by the fiveelement networks in Fig. 5.
The parameters of the suspension system are optimized by considering the vehicle suspension performance indexes, namely, the vehicle body acceleration, suspension working space and dynamic tire load. During the optimization procedure, the traditional passive suspension is set as a reference to optimize the parameters in the vehicle ISD suspension. Here, the three objectives of the ISD suspension are changed to the single objective shown as [39]: 𝑓 =
𝐽 𝐽1 𝐽 + 2 + 3 𝐽1pas 𝐽2pas 𝐽3pas
(11)
where J1 , J2 and J3 are the root-mean-square (RMS) values of the vehicle body acceleration, suspension working space and dynamic tire load, respectively, of the vehicle ISD suspension under the random road input. J1pas , J2pas and J3pas are the RMS of the vehicle body acceleration, suspension working space and dynamic tire load, respectively, of the traditional passive suspension. The variables need to satisfy the condition of Eq. (10) and their ranges are also set as: { 𝑏𝑚 > 0, 𝑐 0 ≥ 0, 𝑐 𝑚 > 0 (12) 𝐴0 , 𝐵0 , 𝐶0 , 𝐷0 , 𝐸0 , 𝐹0 ≥ 0
Fig. 5. Five-element networks [40].
where L is the inductor, R1 , R2 and R3 are the resistors, C is the capacitor, k is the spring, c1 , c2 and c3 are the dampers, and b is the inerter. 4. Vehicle HE-ISD suspension
The parameters of the quarter-car model are given in Table 1, and the compared traditional passive suspension damping comes from a mature passenger car and is also used in [39], where c = 1000 (N s m−1 ).
Based on the HEI device, the detailed structure of the vehicle suspension system, called the vehicle HE-ISD suspension, is shown in Fig. 6. 14
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Table 2 Parameters of the electrical network. Parameters
Values
Coil resistance of the linear motor Re (Ω) Coil inductance of the linear motor Le (mH) Force coefficient ke (N/A) Voltage coefficient kt (Vs/m) Resistance of R1 (Ω) Resistance of R2 (Ω) Resistance of R3 (Ω) Inductance of L (H) Capacitance of C (F)
3.8 26 100 81 124,361.5 11.9998 98,134.3 0.2111 0.0037
The impedance of the external circuit from Fig. 5 is: Fig. 6. Vehicle HE-ISD suspension.
𝑍𝑒 (𝑠) =
where zb is the vertical displacement of the inerter. The dynamic equations of the HE-ISD suspension are as follows: ⎧𝑚 𝑧̈ + 𝐾(𝑧 − 𝑧 ) + 𝑏 (𝑧̈ − 𝑧̈ ) + 𝐹 = 0 𝑠 𝑢 𝑚 𝑠 𝑏 𝑡 ⎪ 𝑠 𝑠 ⎨𝑚𝑢 𝑧̈ 𝑢 + 𝐾𝑡 (𝑧𝑢 − 𝑧𝑟 ) − 𝐾(𝑧𝑠 − 𝑧𝑢 ) − 𝑏𝑚 (𝑧̈ 𝑠 − 𝑧̈ 𝑏 ) − 𝐹𝑡 = 0 ⎪𝑏𝑚 (𝑧̈ 𝑠 − 𝑧̈ 𝑏 ) + 𝐹𝑡 = 𝑐𝑚 (𝑧̇ 𝑏 − 𝑧̇ 𝑢 ) ⎩
1 + 𝑅1 𝐿𝑠 +
1 1 𝑅3
1 + 1 1 +𝑅2 𝐶𝑠
(17)
The parameters of the electrical networks are calculated by using the method proposed in [40] and the detailed coefficients of the linear motor are shown in Table 2.
(15)
5. Simulation
where Ft is the damping force generated by the linear motor. The Laplace form of Ft is: ( )2 ( ) [ ] 𝑘𝑡 𝑘𝑒 𝑆1 𝐹𝑡 (𝑠) = 𝑧̇ 𝑠 (𝑠) − 𝑧̇ 𝑏 (𝑠) (16) 𝑆2 𝑅𝑒 + 𝑠𝐿𝑒 + 𝑍𝑒 (𝑠)
In this section, numerical simulations are carried out to verify the effectiveness of the vehicle HE-ISD suspension. Figs. 7(a)–9(a) show the gain of vehicle body acceleration, suspension working space and
Fig. 7. Responses of vehicle body acceleration.
Fig. 8. Responses of suspension working space. 15
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Fig. 9. Responses of dynamic tire load. Table 3 Comparisons of peak values in frequency gains. Index 2
Peak gain values of vehicle body acceleration /(m/s /m) Peak gain values of suspension working space Peak gain values of dynamic tire load (kN/m)
Passive suspension
HE-ISD suspension
Improvement
213 615 2.91 2.79 69.7 558.7
122 598 1.85 1.99 40.4 432.5
42.7% 2.8% 36.4% 28.7% 42.0% 22.6%
dynamic tire load, respectively, in the frequency domain, and Figs. 7(b)– 9(b) show the responses of vehicle body acceleration, suspension working space and dynamic tire load, respectively, in the time domain compared to a traditional passive suspension. From Table 3, for the vehicle body acceleration, the peak of the HEISD suspension in the low frequency are obviously lower than that of the passive suspension, which decreases from 213 to 122 (m/s2 /m) (42.7%), but for the high frequency peak, the improvement is relatively small, from 615 to 598 (m/s2 /m) (2.8%). The improvement of the suspension working space is more apparent compared to the vehicle body acceleration. The low frequency peak decreases from 2.91 to 1.85 (36.4%), and the high frequency peak decreases from 2.79 to 1.99 (28.7%). For the dynamic tire load analysis, the low frequency peak decreases from 69.7 (kN/m) to 40.4 (kN/m) (42.0%), and the high frequency peak decreases from 558.7 (kN/m) to 432.5 (kN/m) (22.6%). For the responses in the time domain, the RMS values of the vehicle body acceleration of the passive suspension and the vehicle HE-ISD suspension are 2.2087 m/s2 and 2.1942 m/s2 , respectively. The improvements of the vehicle body acceleration are not obvious. The RMS values of the suspension working space of the passive suspension and the vehicle HE-ISD suspension are 0.0223 m and 0.0177 m, respectively, and the RMS values of the dynamic tire load of the passive suspension and the vehicle HE-ISD suspension are 1505.2 N and 1336.6 N, respectively. It can be seen that, there are 20.63% and 11.20% improvements of the vehicle HE-ISD suspension in the time responses, for the suspension working space and dynamic tire load, respectively. In general, the vibration isolation performance of the vehicle HE-ISD suspension is obviously improved by comparison with the passive suspension system. The suspension working space and the dynamic tire load are improved more apparently compared to the vehicle body acceleration.
Fig. 10. Prototype of the HEI device.
University in China. The prototype of the HEI device is depicted in Fig. 10. In Fig. 10, the piston rod of the auxiliary cylinder is connected with the mover of the linear motor so that they can keep the same moving condition. S1/S2 = 4.08, and the mass of the piston rod and the moving rod of the linear motor is m = 5.62 kg. The bench test of the vehicle HEISD suspension is then carried out on the INSTRON 8800 hydraulic servo exciting test bench, and the overall layout of the bench test is depicted in Fig. 11. In the bench test, assuming the vehicle is driving at speed of u on a Grade C road, the random road input is expressed as [41]: √ [ ] 𝑧̇ 𝑟 (𝑡) = −0.111 𝑢𝑧𝑟 (𝑡) + 40 𝐺𝑞 (𝑛0 )𝑢𝑤(𝑡) (18)
6. Experimental research
where zr (t) is the vertical displacement of the random road input, Gq (n0 ) is the road roughness, and w(t) is the white noise. Table 4 shows the
To further validate the vibration isolation performance of the vehicle HE-ISD suspension system, a HEI device was manufactured at Jiangsu 16
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Table 4 Performance indexes under different velocities. Suspension
Passive suspension
HE-ISD suspension
Velocity
Vehicle body acceleration 2
Suspension working space
Dynamic tire load
(m/s)
RMS (m/s )
Improvement
RMS (mm)
Improvement
RMS (kN)
Improvement
10 20 30 10 20 30
0.6595 1.0041 1.4856 0.6576 0.9976 1.4775
– – – 0.29% 0.64% 0.55%
4.4809 7.9547 11.5074 3.6733 6.3656 9.2123
– – – 18.02% 19.97% 19.95%
0.5421 0.9159 1.3551 0.4915 0.8223 1.2321
– – – 9.33% 10.21% 9.08%
Fig. 13. Suspension working space. Fig. 11. Bench test.
Fig. 14. Dynamic tire load. Fig. 12. Vehicle body acceleration.
and 0.55% improvements are obtained at speeds of 10 m/s, 20 m/s and 30 m/s, respectively. For the suspension working space, the improvements are very obvious: 18.02%, 19.97% and 19.95% can be seen at different speeds. In addition, the RMS values of the dynamic tire load are also decreased by 9.33%, 10.21% and 9.08% for the speeds of 10 m/s, 20 m/s and 30 m/s, respectively. For the frequency responses analyses of the PSDs, the PSDs of the vehicle body acceleration are very close to that of the passive suspension system, which indicates that the improvement is relatively small. For the PSDs of the suspension working space, the PSDs of the HE-ISD suspension are all lower than that of the passive suspension over the entire frequency range. The PSDs of the dynamic tire load of the HE-ISD suspension also obviously decrease. The
performance indexes of the vehicle HE-ISD suspension and the passive suspension under different velocities. Figs. 12–14 show the vehicle body acceleration, suspension working space and dynamic tire load in the time domain. Figs. 15–17 show the power spectral density (PSD) of the vehicle body acceleration, suspension working space and dynamic tire load in the frequency domain at a speed of 20 m/s. From Table 4 and Figs. 12–17, it is noted that in the bench test, the performance indexes of the vehicle HE-ISD suspension are all decreased compared to the passive suspension system. In terms of the vehicle body acceleration, the improvement is relatively small. Only 0.29%, 0.64% 17
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trends of the three performance indexes are the same as those in the simulations. The improvements of the suspension working space and the dynamic tire load are more obvious compared to the vehicle body acceleration. Although there is little difference between the simulations and experiments, it is within an acceptable range. The vibration suppression performance is verified by the experimental results. 7. Conclusion This paper investigates the optimal design problem of a vehicle suspension system considering both mechanical and electrical elements. A new hydraulic electric inerter was proposed in this paper, and the structure and the model were introduced in detail. On the basis of the HEI device, a bicubic impedance function was considered when designing a vehicle suspension structure. According to the structural feature of the HEI device, a methodology of reducing the bicubic impedance function order was proposed, and the impedance function was finally realized by seven passive elements. The dynamic model of the vehicle HE-ISD suspension system was built, and numerical simulations showed that, the vibration suppression performance of the HE-ISD suspension was superior to that of the passive suspension. Finally, the HEI device was manufactured, and bench tests of the vehicle HE-ISD suspension system were carried out. The advantages of the vehicle HE-ISD suspension were further validated by the experimental results.
Fig. 15. PSD of vehicle body acceleration.
Conflict of interest The authors declare that there is no conflict of interest regarding the publication of this manuscript. Acknowledgments This work is supported by the China Postdoctoral Science Foundation (Grant no. 2019M651723), the National Natural Science Foundation of China (Grant no. 51705209), the Natural Science Foundation of Jiangsu Province (Grant BK20160533), the Key Laboratory Project of New Technology Application of Jiangsu Transport Vehicle (Grant no. BM20082061510) and the New Energy Vehicle Discipline Project of Universities in Jiangsu Province. References Fig. 16. PSD of suspension working space. [1] Smith MC. Synthesis of mechanical networks: the inerter. IEEE Trans Autom Control 2002;47(10):1648–62. [2] Smith MC, Wang FC. Performance benefits in passive vehicle suspensions employing inerters. Veh Syst Dyn 2004;42(4):235–57. [3] Hu YL, Chen MZQ, Sun Y. Comfort-oriented vehicle suspension design with skyhook inerter configuration. J Sound Vib 2017;405:34–47. [4] Shen YJ, Chen L, Liu YL, Zhang XL, Yang XF. Improvement of the lateral stability of vehicle suspension incorporating inerter. Sci China Technol Sci 2018;61(8):1244–52. [5] Zhang XJ, Ahmadian M, Guo KH. On the benefits of semi-active suspensions with inerters. Shock Vib 2012;19(3):257–72. [6] Wang FC, Hong MF, Chen CW. Building suspension with inerters. Proc Inst Mech Eng Part C 2010;224:1605–16. [7] Lazar I, Neild SA, Wagg DJ. Using an inerter-based device for structural vibration suppression. Earthq Eng Struct Dyn 2014;43:1129–47. [8] Zhang SY, Jiang JZ, Neild S. Optimal configurations for a linear vibration suppression device in a multi-storey building. Struct Control Health Monit 2017;24(3):1887. [9] Wang FC, Liao MK, Liao BH. The performance improvements of train suspension systems with mechanical networks. Veh Syst Dyn 2009;47(7):805–30. [10] Jiang JZ, Matamoros-Sanchez AZ, Goodall RM, Smith MC. Passive suspensions incorporating inerters for railway vehicles. Veh Syst Dyn 2012;50(sup1):S263–76. [11] Chen HJ, Su WJ, Wang FC. Modeling and analyses of a connected multi-car train system employing the inerter. Adv Mech Eng 2017;9(8):1–13. [12] Li Y, Jiang JZ, Neild S. Inerter-based configurations for main-landing-gear shimmy suppression. J Aircr 2016;54(2):684–93. [13] Dong X, Liu Y, Chen MZQ. Application of inerter to aircraft landing gear suspension. In: Control conference IEEE; 2015. p. 2066–71. [14] Hu YL, Wang K, Chen MZQ. Semi-active suspensions with low-order mechanical admittances incorporating inerters. In: Control and decision conference, IEEE; 2015. p. 79–84.
Fig. 17. PSD of dynamic tire load. 18
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[15] Chen MZQ, Hu YL, Wang FC. Passive mechanical control with a special class of positive real controllers: application to passive vehicle suspensions. JDyn Syst Meas Control 2015;137:1–11. [16] Hu YL, Chen MZQ. Low-complexity passive vehicle suspension design based on element-number-restricted networks and low-order admittance networks. J Dyn Syst Meas Control 2018;140(10):1–7. [17] Papageorgiou C, Smith MC. Positive real synthesis using matrix inequalities for mechanical networks: application to vehicle suspension. IEEE Trans Control Syst Technol 2006;14(3):423–34. [18] Wang FC, Liao MK, Liao BH, Su WJ, Chan HA. The performance improvements of train suspension systems with mechanical networks employing inerters. Veh Syst Dyn 2009;47(7):805–30. [19] Bott R, Duffin RJ. Impedance synthesis without use of transformers. J Appl Phys 1949;20(8):816. [20] Tirtoprodjo S. Series-parallel RLC synthesis of bicubic impedance. Electron Lett 1973;9(16):370–2. [21] Tirtoprodjo S. Bicubic functions with real multiple poles and zeros. IEEE Trans Circt Theory 2003;18(4):470–1. [22] Zhang SY, Jiang JZ, Wang HL, Neild S. Synthesis of essential-regular bicubic impedances. Int J Circt Theory Appl 2017;45(2):1–15. [23] Papageorgiou C, Smith MC. Laboratory experimental testing of inerters. In: in: Proceedings of the 44th IEEE conference on decision and control, and the European control conference; 2005. p. 3351–6. [24] Papageorgiou C, Houghton NE, Smith MC. Experimental testing and analysis of inerter devices. J Dyn Syst Meas Control 2009;131:1–11. [25] Wang FC, Hong MF, Lin TC. Designing and testing a hydraulic inerter. Proc Inst Mech Eng Part C 2011;225:66–72. [26] Swift SJ, Smith MC, Glover AR. Design and modelling of a fluid inerter. Int J Control 2013;86(11):2035–51. [27] Liu X, Jiang JZ, Titurus B, Harrison A. Model identification methodology for fluid-based inerters. Mech Syst Signal Process 2018;106:479–94. [28] Shen YJ, Chen L, Liu YL. Optimized modeling and experiment test of a fluid inerter. J Vibroeng 2016;18(5):2789–800. [29] Wang FC, Chan HA. Vehicle suspensions with a mechatronic network strut. Veh Syst Dyn 2011;49(5):811–30. [30] Shi DH, Chen L, Wang RC, Jiang HB, Shen YJ. Design and experiment study of a semi-active energy-regenerative suspension system. Smart Mater Struct 2014;24:015001. [31] Zhang Y, Chen H, Guo K, Zhang X, Li SE. Electro-hydraulic damper for energy harvesting suspension: modeling, prototyping and experimental validation. Appl Energy 2017;199:1–12. [32] Zhu S, Shen WA, Xu YL. Linear electromagnetic devices for vibration damping and energy harvesting: modeling and testing. Eng Struct 2012;34:198–212. [33] Gysen BLJ, Paulides JJH, Janssen JLG, Lomonova EA. Active electromagnetic suspension system for improved vehicle dynamics. IEEE Trans Veh Technol 2010;59(3):1156–63. [34] Lee S, Kim WJ. Active suspension control with direct-drive tubular linear brushless permanent-magnet motor. IEEE Trans Control Syst Technol 2010;18(4):859–70. [35] Koch G, Kloiber T. Driving state adaptive control of an active vehicle suspension system. IEEE Trans Control Syst Technol 2014;22(1):44–57. [36] Brune O. Synthesis of a finite two-terminal network whose driving-point impedance is a prescribed function of frequency. J Math Phys 1931;10(14):30–6. [37] Yang XF, Shen YJ, Yang J. A hydraulic-electric impedance control device of vehicle suspension system. 2015, CN201520074528.3. [38] Chen MZQ, Smith MC. A note on tests for positive-real functions. IEEE Trans Autom Control 2009;54(2):390–3. [39] Shen YJ, Chen L, Yang XF, Shi DH, Yang J. Improved design of dynamic vibration absorber by using the inerter and its application in vehicle suspension. J Sound Vib 2016;361:148–58.
[40] Jiang JZ, Smith MC. Regular positive-real functions and five-element network synthesis for electrical and mechanical networks. IEEE Trans Autom Control 2011;56(6):1275–90. [41] Sun XQ, Cai YF, Chen L, Liu YL, Wang SH. Vehicle height and posture control of the electronic air suspension system using the hybrid system approach. Veh Syst Dyn 2016;54(3):328–52. Yujie Shen is currently a lecturer in the Automotive Engineering Research Institute, Jiangsu University in China. His main research interests are the dynamic modeling and control of automotive engineering and the design and modeling of new mechatronic inerter elements.
Yanling Liu is currently a Ph.D. candidate in the School of Automotive and Traffic Engineering, Jiangsu University in China. Her main research interest is the dynamic modeling and control of automotive engineering.
Long Chen is currently a professor in the Automotive Engineering Research Institute, Jiangsu University in China. His main research interest is the dynamic modeling and control of automotive engineering.
Xiaofeng Yang is currently an associate professor in the School of Automotive and Traffic Engineering, Jiangsu University in China. His main research interest is the dynamic modeling and control of automotive engineering.
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Mechatronics journal homepage: www.elsevier.com/locate/mechatronics
Optimal design and experimental research of vehicle suspension based on a hydraulic electric inerter Yujie Shen∗, Yanling Liu, Long Chen, Xiaofeng Yang Automotive Engineering Research Institute, Jiangsu University, Zhenjiang 212013, China
a r t i c l e
i n f o
Keywords: Vehicle Suspension Inerter Hydraulic electric Optimal design Bench test
a b s t r a c t As the mechanical dual of a capacitor via the force-current analogy, an inerter has been successfully applied in various fields such as automotive engineering, civil engineering and aerospace engineering. The introduction of an inerter not only allows the use of network synthesis analogy to design mechanical layouts but also opens the door to adopt electrical element impedances to simulate the corresponding mechanical elements. This paper combines both of these ideas and presents a new form of a mechatronic inerter, namely the hydraulic electric inerter (HEI), which consists of a hydraulic piston inerter and a linear motor. On the basis of the HEI device, a bicubic impedance function is considered in the optimal design of a vehicle suspension system employing both mechanical elements and electrical elements. In addition, a methodology for reducing the order of the bicubic impedance function is proposed, and the network is finally realized by utilizing inerter, spring and damper elements. Then, by comparison with a passive suspension, the advantages of the new vehicle ISD (inerter-spring-damper) suspension, called the vehicle HE-ISD (hydraulic electric-inerter spring damper) suspension, are demonstrated by numerical simulations. Finally, a HEI device is designed, and bench tests of the vehicle HE-ISD suspension are carried out. Experimental results indicate that, the vehicle HE-ISD suspension is superior to the passive suspension system, the RMS (root-mean-square) value of the suspension working space is improved by 19.97%, and the RMS value of the dynamic tire load is improved by 10.21%.
1. Introduction Since the first introduction of the inerter in 2002 [1], inerter-based vibration suppression has been a popular topic in mechanical fields. The feature of the inerter is that the force across its two terminals is proportional to the relative acceleration. Until now, the inerter element has been successfully deployed in various areas, including vehicle suspensions [2–5], civil engineering [6–8], railway suspensions [9–11], landing gear shimmy suppression [12,13], etc. The merit of the inerter is that it allows the use of electrical analogies, such as network synthesis, to design complex layouts of vibration isolation systems based on the inerter-spring-damper network. For the optimal design of vehicle suspension systems, low order admittance functions have been investigated [14–16] by using purely mechanical networks. In [17], different orders of positive real controllers were designed to improve the performance of the vehicle suspensions, and the results showed that the vibration suppression performance will be further improved when the admittance function order is increased. The same conclusion was also obtained in [18] that compared with the use of biquadratic impedances in railway suspension systems, a bicubic function can reduce the wheel load index by 36%. It is apparent that higher order impedance functions can provide a wider range of dynamic properties. However, the biquadratic ∗
and bicubic impedances can be realized by using at most nine elements and thirteen elements, respectively, in the Bott-Duffin procedure [19]. The bicubic impedances mentioned in [16] were all special third-order positive-real functions and were realized by only five elements using the methods in [1]. Although the minimal realizations problem of bicubic impedance has been explored in [20–22], the high complexities of the mechanical network may restrict the application of the inerter in vehicle suspensions. Another advantage of the inerter is that, it opens the door to utilize the electrical elements to simulate the corresponding mechanical network to obtain the target impedance output. Generally, there are many realizations of inerters, such as the ball-screw inerter [23], the rack and pinion inerter [24], the hydraulic inerter [25], and the fluid inerter [26–28]. In [29], a mechatronic inerter consisting of a PMEM (Permanent Magnet Electric Machinery) and a ball-screw inerter was proposed, and a bilinear and a biquadratic function was used for designing a vehicle suspension system on the basis of several simple mechanical networks. Recently, linear motors have been widely used in energy harvesting [30–32] and active control systems [33–35] due to their superior fast response, good controllability and easy energy-regeneration. From the above analysis, this paper will concentrate on the problem of designing a vehicle suspension system based on a new hydraulic-electric
Corresponding author. E-mail address: [email protected] (Y. Shen).
https://doi.org/10.1016/j.mechatronics.2019.05.002 Received 2 January 2019; Received in revised form 10 April 2019; Accepted 19 May 2019 Available online 29 May 2019 0957-4158/© 2019 Elsevier Ltd. All rights reserved.
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inerter (HEI), which includes a hydraulic piston inerter and a linear motor. Based on the features of the HEI device in the layout of the suspension system, a methodology for reducing the impedance function order is proposed based on the foster cycle procedure [36], in which a bicubic impedance function is considered. The paper is arranged as follows: in Section 2, a new hydraulicelectric inerter is introduced in detail, including its structural components and working principle. Then, the dynamic model of a vehicle suspension system using the inerter-spring-damper network adopting a bicubic impedance function is built. The methodology of reducing the impedance order is also presented and the network is finally realized via passive elements in Section 3. In Section 4, the dynamic model of the vehicle suspension incorporating both mechanical elements and electrical elements on the basis of the HEI device, namely, the vehicle HE-ISD suspension, is built. Numerical simulations are conducted in Section 5 to illustrate the validity of the newly proposed HE-ISD suspension. Finally, the HEI device is manufactured, and the vehicle HE-ISD suspension bench is built, the experiments are completed by comparing to the results with the passive suspension system. Some conclusions are presented in Section 7.
the linear motor and the velocity of the mover is shown as 𝑉𝑔 = 𝑘𝑒 𝑣𝑎
(2)
where Vg is the voltage generated by the linear motor, ke is the voltage coefficient, and va is the velocity of the mover. The force of the linear motor Ft is expressed as 𝐹𝑡 = 𝑘𝑡 𝐼𝑎
(3)
where kt is the force coefficient and Ia is the current in the external circuit of the HEI device. Similar to the mechatronic inerter in [29], the force and the velocity between the HEI terminals are expressed as [37]: 𝐹 (𝑠) = 𝑣(𝑠)
(
𝑆1 𝑆2
)2 𝑚𝑠 +
𝐾𝑚 𝑅𝑒 + 𝐿𝑒 𝑠 + 𝑍𝑒 (𝑠)
(4)
where F(s) and v(s) are the Laplace forms of the force and velocity, respectively, of the two terminals of the HEI device, Km is the motor coefficient, for the HEI device, Km = (S1 /S2 )2 ke kt , Re is the coil resistance, Le is the inductance of the linear motor. Ze (s) is the impedance of the external electric circuit. It is noted that, the impedance of the HEI device consists of two parts. One is the mechanical inerter impedance, and the other is the electrical impedance determined by the external circuit of the HEI device.
2. HEI device In contrast to the mechatronic inerter device introduced in [29], the hydraulic electric inerter consists of a hydraulic piston inerter and a linear motor. A schematic of the HEI device is presented in Fig. 1. In Fig. 1, 1 and 7 present the two terminals of the HEI device, 2 and 6 present the connecting tubes of the main cylinder and the auxiliary cylinder, 3 presents the main cylinder, 4 presents the piston of the main cylinder, 5 presents the piston rod of the main cylinder, 8 presents the moving rod of the linear motor, 9 presents the moving magnetic pole, 10 presents the winding of the linear motor, 11 presents the mover yoke, 12 presents the stator of the linear motor, 13 presents the piston rod of the auxiliary cylinder, 14 presents the piston of the auxiliary cylinder, and 15 presents the auxiliary cylinder. Note that the moving rod of the linear motor is integrated of the piston rod of the auxiliary cylinder so that the relative linear motion between the two terminals of the HEI device leads to the linear motion between the auxiliary cylinder and the moving rod of the linear motor. The different section areas of the main hydraulic cylinder and the auxiliary cylinder result in the motion transformation of the device, the inertance of the hydraulic piston inerter is expressed as [37]: ( )2 𝑆1 𝑏=𝑚 (1) 𝑆2
3. Optimal design of the vehicle ISD suspension In the optimal design procedure of the vehicle suspension system, a generalized quarter-car model, presented in Fig. 2, is considered. where ms is the sprung mass, mu is the unsprung mass, K is the supporting spring of the suspension, Kt is the tire stiffness, and zs , zu and zr are the vertical displacements of the sprung mass, unsprung mass and the random road input, respectively. The Laplace forms of the dynamic equations of the quarter-car model are expressed as 𝑚𝑠 𝑠2 𝑍𝑠 + [𝐾 + 𝑠𝑇 (𝑠)](𝑍𝑠 − 𝑍𝑢 ) = 0
(5)
𝑚𝑢 𝑠2 𝑍𝑢 − [𝐾 + 𝑠𝑇 (𝑠)](𝑍𝑠 − 𝑍𝑢 ) + 𝐾𝑡 (𝑍𝑢 − 𝑍𝑟 ) = 0
(6)
where Zs , Zu and Zs are the Laplace forms of zs , zu and zr . T(s) is the undefined suspension structure and is expressed by a bi-cubic impedance function as 𝑇 (𝑠) =
𝐴𝑠3 + 𝐵 𝑠2 + 𝐶𝑠 + 𝐷 𝐸 𝑠3 + 𝐹 𝑠2 + 𝐺𝑠 + 𝐻
(7)
where b is the mechanical inertance of the HEI device, m is the mass of the piston rod and the moving rod of the linear motor, and S1 and S2 are the sectional areas of the main cylinder and the auxiliary cylinder, respectively. For the linear motor, when there is a motion between the stator and the mover, the relationship between the voltage generated by
where A, B, C, D, E, F, G and H ≥ 0 and not all E, F, G and H = 0. The positive-real condition of the T(s) is introduced in [38] and can be shown as (1) (𝐶 + 𝐺)(𝐵 + 𝐹 ) ≥ (𝐷 + 𝐻)(𝐴 + 𝐸) (2) one of the following holds: √ (a) 𝑎3 = 0, a2 ≥ 0, a0 ≥ 0, −𝑎1 ≤ 2 𝑎0 𝑎2 ; (b) a3 > 0, a0 ≥ 0, and (b1) or (b2) holds:l
Fig. 1. Schematic of the HEI device.
Fig. 2. Quarter-car model. 13
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Table 1 Model parameters. Parameters
Values
Vehicle body mass ms (kg) Unsprung mass mu (kg) Suspension spring stiffness K (N m−1 ) Tire spring stiffness Kt (N m−1 )
320 45 22,000 190,000
Fig. 3. Schematic of the reducing order structure.
√ (b1) a1 ≥ 0 and −𝑎2 ≤ 3𝑎1 𝑎3 ; 2 (b2) 𝑎2 > 3𝑎1 𝑎3 and 2𝑎32 − 9𝑎1 𝑎2 𝑎3 + 27𝑎0 𝑎23 ≥ 2(𝑎22 − 3𝑎1 𝑎3 )3∕2 where a0 = DH, a1 = CG-DF-BH, a2 = BF-CE-AG and a3 = AE. The positive-real condition of the T(s) is very complex, and the bicubic impedance function is difficult to realize by using only resistors, capacitors and inductors (corresponding mechanical elements are dampers, inerters and springs, respectively). From the analysis in [29], it is noted that the mechatronic inerter or the HEI device is always in parallel with external circuits. According to the foster cycle procedure [36], the inerter, a dual of the capacitor, as a reactive element, can be extracted from the original impedance function to reduce the order of T(s). However, the damper, a dual of the resistor, as a resistive element, can also be extracted in parallel or in a series structure from the original function, and the order will not be reduced. Following the above two steps, the new structure is depicted in Fig. 3. where bm is the mechanical inertance, c0 and cm are the mechanical dampers and c0 = 0 or cm = +∞. Then, T(s) is changed to 𝑇 (𝑠) =
1
Fig. 4. Bode diagram of the optimal structure.
(8)
The optimal parameters are bm = 93.57, c0 = 0, cm = 1840, A0 = 494.4, B0 = 2.913 × 108 , C0 = 1.451 × 106 , D0 = 474.4, E0 = 2.697 × 104 , and F0 = 6.14 × 105 . The T(s) and T1 (s) are shown as:
(9)
𝑇 (𝑠) =
1.84 × 103 𝑠3 + 1.046 × 105 𝑠2 + 1.446 × 107 𝑠 + 6.015 × 104 𝑠3 + 76.53𝑠2 + 8.975 × 103 𝑠 + 2.548 × 104
(13)
where A0 , B0 , C0 , D0 , E0 and F0 ≥ 0 and not all D0 , E0 and F0 = 0. T1 (s) is positive-real if and only if [40] √ √ 𝐵0 𝐸0 − ( 𝐴0 𝐹0 − 𝐶0 𝐷0 ) ≥ 0 (10)
𝑇1 (𝑠) =
494.4𝑠2 + 2.913 × 108 𝑠 + 1.451 × 106 474.4𝑠2 + 2.697 × 104 𝑠 + 6.14 × 105
(14)
1 𝑏𝑚 𝑠+𝑐0 +𝑇1 (𝑠)
+
1 𝑐𝑚
where T1 (s) is a biquadratic transfer function and is expressed as 𝑇1 (𝑠) =
𝐴0 𝑠2 + 𝐵0 𝑠 + 𝐶0 𝐷0 𝑠2 + 𝐸0 𝑠 + 𝐹0
The Bode diagrams of the T(s) and T1 (s) are presented in Fig. 4. From Fig. 4, it is seen that the structures T(s) and T1 (s) function as inerters in the low frequency range, which is missing in the traditional passive vehicle suspension consisting of spring and damper elements. According to Jiang and Smith [40], T1 (s) can be realized by the fiveelement networks in Fig. 5.
The parameters of the suspension system are optimized by considering the vehicle suspension performance indexes, namely, the vehicle body acceleration, suspension working space and dynamic tire load. During the optimization procedure, the traditional passive suspension is set as a reference to optimize the parameters in the vehicle ISD suspension. Here, the three objectives of the ISD suspension are changed to the single objective shown as [39]: 𝑓 =
𝐽 𝐽1 𝐽 + 2 + 3 𝐽1pas 𝐽2pas 𝐽3pas
(11)
where J1 , J2 and J3 are the root-mean-square (RMS) values of the vehicle body acceleration, suspension working space and dynamic tire load, respectively, of the vehicle ISD suspension under the random road input. J1pas , J2pas and J3pas are the RMS of the vehicle body acceleration, suspension working space and dynamic tire load, respectively, of the traditional passive suspension. The variables need to satisfy the condition of Eq. (10) and their ranges are also set as: { 𝑏𝑚 > 0, 𝑐 0 ≥ 0, 𝑐 𝑚 > 0 (12) 𝐴0 , 𝐵0 , 𝐶0 , 𝐷0 , 𝐸0 , 𝐹0 ≥ 0
Fig. 5. Five-element networks [40].
where L is the inductor, R1 , R2 and R3 are the resistors, C is the capacitor, k is the spring, c1 , c2 and c3 are the dampers, and b is the inerter. 4. Vehicle HE-ISD suspension
The parameters of the quarter-car model are given in Table 1, and the compared traditional passive suspension damping comes from a mature passenger car and is also used in [39], where c = 1000 (N s m−1 ).
Based on the HEI device, the detailed structure of the vehicle suspension system, called the vehicle HE-ISD suspension, is shown in Fig. 6. 14
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Table 2 Parameters of the electrical network. Parameters
Values
Coil resistance of the linear motor Re (Ω) Coil inductance of the linear motor Le (mH) Force coefficient ke (N/A) Voltage coefficient kt (Vs/m) Resistance of R1 (Ω) Resistance of R2 (Ω) Resistance of R3 (Ω) Inductance of L (H) Capacitance of C (F)
3.8 26 100 81 124,361.5 11.9998 98,134.3 0.2111 0.0037
The impedance of the external circuit from Fig. 5 is: Fig. 6. Vehicle HE-ISD suspension.
𝑍𝑒 (𝑠) =
where zb is the vertical displacement of the inerter. The dynamic equations of the HE-ISD suspension are as follows: ⎧𝑚 𝑧̈ + 𝐾(𝑧 − 𝑧 ) + 𝑏 (𝑧̈ − 𝑧̈ ) + 𝐹 = 0 𝑠 𝑢 𝑚 𝑠 𝑏 𝑡 ⎪ 𝑠 𝑠 ⎨𝑚𝑢 𝑧̈ 𝑢 + 𝐾𝑡 (𝑧𝑢 − 𝑧𝑟 ) − 𝐾(𝑧𝑠 − 𝑧𝑢 ) − 𝑏𝑚 (𝑧̈ 𝑠 − 𝑧̈ 𝑏 ) − 𝐹𝑡 = 0 ⎪𝑏𝑚 (𝑧̈ 𝑠 − 𝑧̈ 𝑏 ) + 𝐹𝑡 = 𝑐𝑚 (𝑧̇ 𝑏 − 𝑧̇ 𝑢 ) ⎩
1 + 𝑅1 𝐿𝑠 +
1 1 𝑅3
1 + 1 1 +𝑅2 𝐶𝑠
(17)
The parameters of the electrical networks are calculated by using the method proposed in [40] and the detailed coefficients of the linear motor are shown in Table 2.
(15)
5. Simulation
where Ft is the damping force generated by the linear motor. The Laplace form of Ft is: ( )2 ( ) [ ] 𝑘𝑡 𝑘𝑒 𝑆1 𝐹𝑡 (𝑠) = 𝑧̇ 𝑠 (𝑠) − 𝑧̇ 𝑏 (𝑠) (16) 𝑆2 𝑅𝑒 + 𝑠𝐿𝑒 + 𝑍𝑒 (𝑠)
In this section, numerical simulations are carried out to verify the effectiveness of the vehicle HE-ISD suspension. Figs. 7(a)–9(a) show the gain of vehicle body acceleration, suspension working space and
Fig. 7. Responses of vehicle body acceleration.
Fig. 8. Responses of suspension working space. 15
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Fig. 9. Responses of dynamic tire load. Table 3 Comparisons of peak values in frequency gains. Index 2
Peak gain values of vehicle body acceleration /(m/s /m) Peak gain values of suspension working space Peak gain values of dynamic tire load (kN/m)
Passive suspension
HE-ISD suspension
Improvement
213 615 2.91 2.79 69.7 558.7
122 598 1.85 1.99 40.4 432.5
42.7% 2.8% 36.4% 28.7% 42.0% 22.6%
dynamic tire load, respectively, in the frequency domain, and Figs. 7(b)– 9(b) show the responses of vehicle body acceleration, suspension working space and dynamic tire load, respectively, in the time domain compared to a traditional passive suspension. From Table 3, for the vehicle body acceleration, the peak of the HEISD suspension in the low frequency are obviously lower than that of the passive suspension, which decreases from 213 to 122 (m/s2 /m) (42.7%), but for the high frequency peak, the improvement is relatively small, from 615 to 598 (m/s2 /m) (2.8%). The improvement of the suspension working space is more apparent compared to the vehicle body acceleration. The low frequency peak decreases from 2.91 to 1.85 (36.4%), and the high frequency peak decreases from 2.79 to 1.99 (28.7%). For the dynamic tire load analysis, the low frequency peak decreases from 69.7 (kN/m) to 40.4 (kN/m) (42.0%), and the high frequency peak decreases from 558.7 (kN/m) to 432.5 (kN/m) (22.6%). For the responses in the time domain, the RMS values of the vehicle body acceleration of the passive suspension and the vehicle HE-ISD suspension are 2.2087 m/s2 and 2.1942 m/s2 , respectively. The improvements of the vehicle body acceleration are not obvious. The RMS values of the suspension working space of the passive suspension and the vehicle HE-ISD suspension are 0.0223 m and 0.0177 m, respectively, and the RMS values of the dynamic tire load of the passive suspension and the vehicle HE-ISD suspension are 1505.2 N and 1336.6 N, respectively. It can be seen that, there are 20.63% and 11.20% improvements of the vehicle HE-ISD suspension in the time responses, for the suspension working space and dynamic tire load, respectively. In general, the vibration isolation performance of the vehicle HE-ISD suspension is obviously improved by comparison with the passive suspension system. The suspension working space and the dynamic tire load are improved more apparently compared to the vehicle body acceleration.
Fig. 10. Prototype of the HEI device.
University in China. The prototype of the HEI device is depicted in Fig. 10. In Fig. 10, the piston rod of the auxiliary cylinder is connected with the mover of the linear motor so that they can keep the same moving condition. S1/S2 = 4.08, and the mass of the piston rod and the moving rod of the linear motor is m = 5.62 kg. The bench test of the vehicle HEISD suspension is then carried out on the INSTRON 8800 hydraulic servo exciting test bench, and the overall layout of the bench test is depicted in Fig. 11. In the bench test, assuming the vehicle is driving at speed of u on a Grade C road, the random road input is expressed as [41]: √ [ ] 𝑧̇ 𝑟 (𝑡) = −0.111 𝑢𝑧𝑟 (𝑡) + 40 𝐺𝑞 (𝑛0 )𝑢𝑤(𝑡) (18)
6. Experimental research
where zr (t) is the vertical displacement of the random road input, Gq (n0 ) is the road roughness, and w(t) is the white noise. Table 4 shows the
To further validate the vibration isolation performance of the vehicle HE-ISD suspension system, a HEI device was manufactured at Jiangsu 16
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Table 4 Performance indexes under different velocities. Suspension
Passive suspension
HE-ISD suspension
Velocity
Vehicle body acceleration 2
Suspension working space
Dynamic tire load
(m/s)
RMS (m/s )
Improvement
RMS (mm)
Improvement
RMS (kN)
Improvement
10 20 30 10 20 30
0.6595 1.0041 1.4856 0.6576 0.9976 1.4775
– – – 0.29% 0.64% 0.55%
4.4809 7.9547 11.5074 3.6733 6.3656 9.2123
– – – 18.02% 19.97% 19.95%
0.5421 0.9159 1.3551 0.4915 0.8223 1.2321
– – – 9.33% 10.21% 9.08%
Fig. 13. Suspension working space. Fig. 11. Bench test.
Fig. 14. Dynamic tire load. Fig. 12. Vehicle body acceleration.
and 0.55% improvements are obtained at speeds of 10 m/s, 20 m/s and 30 m/s, respectively. For the suspension working space, the improvements are very obvious: 18.02%, 19.97% and 19.95% can be seen at different speeds. In addition, the RMS values of the dynamic tire load are also decreased by 9.33%, 10.21% and 9.08% for the speeds of 10 m/s, 20 m/s and 30 m/s, respectively. For the frequency responses analyses of the PSDs, the PSDs of the vehicle body acceleration are very close to that of the passive suspension system, which indicates that the improvement is relatively small. For the PSDs of the suspension working space, the PSDs of the HE-ISD suspension are all lower than that of the passive suspension over the entire frequency range. The PSDs of the dynamic tire load of the HE-ISD suspension also obviously decrease. The
performance indexes of the vehicle HE-ISD suspension and the passive suspension under different velocities. Figs. 12–14 show the vehicle body acceleration, suspension working space and dynamic tire load in the time domain. Figs. 15–17 show the power spectral density (PSD) of the vehicle body acceleration, suspension working space and dynamic tire load in the frequency domain at a speed of 20 m/s. From Table 4 and Figs. 12–17, it is noted that in the bench test, the performance indexes of the vehicle HE-ISD suspension are all decreased compared to the passive suspension system. In terms of the vehicle body acceleration, the improvement is relatively small. Only 0.29%, 0.64% 17
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trends of the three performance indexes are the same as those in the simulations. The improvements of the suspension working space and the dynamic tire load are more obvious compared to the vehicle body acceleration. Although there is little difference between the simulations and experiments, it is within an acceptable range. The vibration suppression performance is verified by the experimental results. 7. Conclusion This paper investigates the optimal design problem of a vehicle suspension system considering both mechanical and electrical elements. A new hydraulic electric inerter was proposed in this paper, and the structure and the model were introduced in detail. On the basis of the HEI device, a bicubic impedance function was considered when designing a vehicle suspension structure. According to the structural feature of the HEI device, a methodology of reducing the bicubic impedance function order was proposed, and the impedance function was finally realized by seven passive elements. The dynamic model of the vehicle HE-ISD suspension system was built, and numerical simulations showed that, the vibration suppression performance of the HE-ISD suspension was superior to that of the passive suspension. Finally, the HEI device was manufactured, and bench tests of the vehicle HE-ISD suspension system were carried out. The advantages of the vehicle HE-ISD suspension were further validated by the experimental results.
Fig. 15. PSD of vehicle body acceleration.
Conflict of interest The authors declare that there is no conflict of interest regarding the publication of this manuscript. Acknowledgments This work is supported by the China Postdoctoral Science Foundation (Grant no. 2019M651723), the National Natural Science Foundation of China (Grant no. 51705209), the Natural Science Foundation of Jiangsu Province (Grant BK20160533), the Key Laboratory Project of New Technology Application of Jiangsu Transport Vehicle (Grant no. BM20082061510) and the New Energy Vehicle Discipline Project of Universities in Jiangsu Province. References Fig. 16. PSD of suspension working space. [1] Smith MC. Synthesis of mechanical networks: the inerter. IEEE Trans Autom Control 2002;47(10):1648–62. [2] Smith MC, Wang FC. Performance benefits in passive vehicle suspensions employing inerters. Veh Syst Dyn 2004;42(4):235–57. [3] Hu YL, Chen MZQ, Sun Y. Comfort-oriented vehicle suspension design with skyhook inerter configuration. J Sound Vib 2017;405:34–47. [4] Shen YJ, Chen L, Liu YL, Zhang XL, Yang XF. Improvement of the lateral stability of vehicle suspension incorporating inerter. Sci China Technol Sci 2018;61(8):1244–52. [5] Zhang XJ, Ahmadian M, Guo KH. On the benefits of semi-active suspensions with inerters. Shock Vib 2012;19(3):257–72. [6] Wang FC, Hong MF, Chen CW. Building suspension with inerters. Proc Inst Mech Eng Part C 2010;224:1605–16. [7] Lazar I, Neild SA, Wagg DJ. Using an inerter-based device for structural vibration suppression. Earthq Eng Struct Dyn 2014;43:1129–47. [8] Zhang SY, Jiang JZ, Neild S. Optimal configurations for a linear vibration suppression device in a multi-storey building. Struct Control Health Monit 2017;24(3):1887. [9] Wang FC, Liao MK, Liao BH. The performance improvements of train suspension systems with mechanical networks. Veh Syst Dyn 2009;47(7):805–30. [10] Jiang JZ, Matamoros-Sanchez AZ, Goodall RM, Smith MC. Passive suspensions incorporating inerters for railway vehicles. Veh Syst Dyn 2012;50(sup1):S263–76. [11] Chen HJ, Su WJ, Wang FC. Modeling and analyses of a connected multi-car train system employing the inerter. Adv Mech Eng 2017;9(8):1–13. [12] Li Y, Jiang JZ, Neild S. Inerter-based configurations for main-landing-gear shimmy suppression. J Aircr 2016;54(2):684–93. [13] Dong X, Liu Y, Chen MZQ. Application of inerter to aircraft landing gear suspension. In: Control conference IEEE; 2015. p. 2066–71. [14] Hu YL, Wang K, Chen MZQ. Semi-active suspensions with low-order mechanical admittances incorporating inerters. In: Control and decision conference, IEEE; 2015. p. 79–84.
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[40] Jiang JZ, Smith MC. Regular positive-real functions and five-element network synthesis for electrical and mechanical networks. IEEE Trans Autom Control 2011;56(6):1275–90. [41] Sun XQ, Cai YF, Chen L, Liu YL, Wang SH. Vehicle height and posture control of the electronic air suspension system using the hybrid system approach. Veh Syst Dyn 2016;54(3):328–52. Yujie Shen is currently a lecturer in the Automotive Engineering Research Institute, Jiangsu University in China. His main research interests are the dynamic modeling and control of automotive engineering and the design and modeling of new mechatronic inerter elements.
Yanling Liu is currently a Ph.D. candidate in the School of Automotive and Traffic Engineering, Jiangsu University in China. Her main research interest is the dynamic modeling and control of automotive engineering.
Long Chen is currently a professor in the Automotive Engineering Research Institute, Jiangsu University in China. His main research interest is the dynamic modeling and control of automotive engineering.
Xiaofeng Yang is currently an associate professor in the School of Automotive and Traffic Engineering, Jiangsu University in China. His main research interest is the dynamic modeling and control of automotive engineering.
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