Research Article
Network synthesis and parameter optimization for vehicle suspension with inerter
Advances in Mechanical Engineering 2017, Vol. 9(1) 1–7 Ó The Author(s) 2017 DOI: 10.1177/1687814016684704 journals.sagepub.com/home/ade
Long Chen, Changning Liu, Wei Liu, Jiamei Nie, Yujie Shen and Guotao Chen
Abstract In order to design a comfortable-oriented vehicle suspension structure, the network synthesis method was utilized to transfer the problem into solving a timing robust control problem and determine the structure of ‘‘inerter–spring–damper’’ suspension. Bilinear Matrix Inequality was utilized to obtain the timing transfer function. Then, the transfer function of suspension system can be physically implemented by passive elements such as spring, damper, and inerter. By analyzing the sensitivity and quantum genetic algorithm, the optimized parameters of inerter–spring–damper suspension were determined. A quarter-car model was established. The performance of the inerter–spring–damper suspension was verified under random input. The simulation results manifested that the dynamic performance of the proposed suspension was enhanced in contrast with traditional suspension. The root mean square of vehicle body acceleration decreases by 18.9%. The inerter–spring–damper suspension can inhibit the vertical vibration within the frequency of 1–3 Hz effectively and enhance the performance of ride comfort significantly. Keywords Suspension structure, inerter, network synthesis, parameter optimization, ride comfort
Date received: 30 June 2016; accepted: 25 November 2016 Academic Editor: Francesco Massi
Suspension directly affects the riding comfort, handling stability and driving safety1 of the vehicle. Spring and damper are the main components of traditional suspension system. However, there is continued to be numerous unsolved problems such as the contradiction between the riding comfort and handling stability. Smith2 proposed the inerter element based on electromechanical similarity theory and applied it into the vehicle suspension systems.3 Mechanical network with inerter has more abundant network features in comparison with the mechanical network just consisting of spring and damper.4 In addition, the emergence of the inerter makes the mechanical, as well as electrical, network strictly corresponding to each other. Therefore, RLC (resistor (R), inductor (L), and capacitor (C)) electrical network synthesis theory can be used to research inerter–spring–damper (ISD) mechanical
network to improve the performance of mechanical vibration isolation system. In recent years, the inerter becomes a research boiling spot in vehicle suspension systems, steering systems, and building vibration isolation system.5–14 Integrated and analysis network theory is applied to the study of ISD suspension system constantly. Based on network synthesis method, Wang and Chan15 applied a novel mechatronic network strut with ball screw inerter and
School of Automobile and Traffic Engineering, Jiangsu University, Zhenjiang, China Corresponding author: Long Chen, School of Automobile and Traffic Engineering, Jiangsu University, Zhenjiang 212013, China. Email:
[email protected]
Creative Commons CC-BY: This article is distributed under the terms of the Creative Commons Attribution 3.0 License (http://www.creativecommons.org/licenses/by/3.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/ open-access-at-sage).
2 permanent magnet motor to vehicle suspensions. Then, he discusses the benefits of system performance. Simulation and experimental results shown the performance of this suspension is superior to traditional passive suspension. Scheibe and Smith16 obtained the global optimum of riding comfort and handling stability from quarter-car vehicle suspension model via a method of network analysis. Chen et al.17–21 utilized graph theory and several existing results of n-port impedance network and concerned with the minimal realization problem of a third-order real symmetric matrix as the admittance of three-port impedance network. Hu and Chen utilized inerter into passive vehicle suspensions and semiactive suspensions.22,23 They analyzed and optimized the inerter-based isolators based on a ‘‘uni-axial’’ single-degree-of-freedom isolation system.24 Ming et al.25 systematically explained the concept, background and characteristics of the inerter and ISD suspension by summarizing its structure types, dynamics, frequency responses, nonlinear characteristics and breakdown phenomena, network synthesis theory and application layout and integrated design. The ISD suspension structure which consists of the inerter, spring, and damper is various. So far, the study of ISD suspension is to assume the suspension structure first and then utilize various methods to optimize the suspension structure parameters. This may limit ISD network into a few fixed structures; we might neglect a slice of good structures. To contain more suspension structures, this article will solve the robust of positive-real controller based on bilinear matrix inequality (BMI) instead of the problem of suspension structure. Genetic algorithm is combined with linear matrix inequality (LMI) algorithm for BMI, and the root mean square (RMS) of vehicle body acceleration is used to indicate the performance for single objective comprehensive. The aim of this article is to find out an ISD suspension structure which is comfort-oriented. Parameter sensitivity analysis and quantum genetic algorithm are used to optimize and simulate it. The dynamic response and the improvement of the ISD suspension, compared with traditional passive suspension, are investigated under the condition of random incentive.
Advances in Mechanical Engineering
Figure 1. Quarter-car model.
Dynamic model of ISD quarter-car suspension Elastic element is indispensable to the vehicle suspension system to maintain the vehicle mass. Figure 1 represents a quarter-car suspension model. The suspension structure is divided into known and unknown parts: the known part is the spring ks and the unknown part is Y (s), which is the complex domain of mechanical impedance for spring, damper, and inerter. In Figure 1, mu is the unsprung mass, ms is the sprung mass, zr is the road random input, zu is the unsprung mass displacement, zs is the sprung mass displacement, F is the force generated of the unknown suspension structure portion which is impacted by sprung mass and unsprung mass, and kt is the tire stiffness. The dynamical equation is as follows ms€zs + ks (zs zu ) + F = 0 mu€zu + kt (zr zu ) ks (zs zu ) F = 0
The suspension system is viewed as a mechanical network, which is corresponded with electrical network synthesis. The positive-real impedance suspension of the transfer function is solved based on the road input and the output of vehicle performance we desired. Then, it is implemented with inerter, spring, and damper.
ð1Þ
The state variable is x = ½_zs zs z_ u zu T and the input variable is w = ½zr FT . The corresponding state equation is as follows x_ = Ax + B1 zr + B2 F Among them 2
Network synthesis for the ISD suspension system
3 ks ks 0 7 ms ms 7 7 0 0 0 7, ks ks + kt 7 7 0 mu mu 5 0 0 1 0 2 13 2 3 0 ms 6 0 7 607 6 7 6 7 B1 = 6 kt 7, B2 = 6 1 7 4 m 5 4m 5
60 6 61 A=6 6 60 4
u
u
0
0
ð2Þ
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The design of H2 robust positive-real controller for ISD suspension networks The sprung mass and unsprung mass relative velocity measurement are selected as output y = z_ s z_ u Select sprung mass velocity as controlled output z = z_ s . Figure 1 demonstrates the generalized suspension system that can be expressed as follows 8 < x_ = Ax + B1 zr + B2 F y = z_ s z_ u = C2 x ð3Þ : z = z_ s = C1 x To sweep into all kinds of ISD passive suspension system, we transform the problem of solving ISD suspension structure into robust control problem.26 Accordingly, Y (s) can be seen as a robust controller K(s), and the controller can be written in state-space form27 Ak Bk = Ck (sI Ak )1 Bk + Dk K(s) = ð4Þ Ck Dk x_ k = Ak xk + Bk uk ð5Þ yk = Ck xk + Dk uk In which xk is the state vector controller; yk is the controller output; uk is the controller input; and Ak , Bk , Ck , Dk is the positive-real controller parameter matrix to be determined. Combine the generalized suspension system, as well as robust controller, to form an augmented closed-loop system 8 9 2 3 x < x_ = Acl Bcl 4 5 = ð6Þ xk x_ Ccl Dcl : k; z z0 A + B2 Dk C2 B2 Ck B1 , Bcl = , In which Acl = B k C2 Ak 0n 3 1 Ccl = ½C1 0n 3 1 , and Dcl =0, n is the order of the controller. For the closed-loop system we mentioned above, Tzr !_zs represents the transfer function from input zr to output z_ s . The goal of H2 positive-real control is to design a linear time-invariant controller K(s), making the closed-loop system to meet the following criteria: (1) closed-loop system is asymptotically stable, (2) kTzr !_zs k2 \g 2 , and (3) K(s) is positive-real. In which g 2 is the target to be solved; according to LMI theory, there exists a positive definite matrix. The closed-loop system is asymptotically stable and kTzr !_zs k2 \g 2 is as follows: the existence of a symmetric positive definite matrix P1 = PT1 .0, such that "
ATcl P1 + P1 Acl P1 Ccl
BTcl P1 CclT .0, Z
P1 Bcl I
# \0
Trace(Z)\g 22 , P1 .0
ð7Þ
The necessary and sufficient condition for controller K(s) positive-real is: the existence of symmetric positive definite matrix P2 = PT2 .0, such that
ATk P2 + P2 Ak BTk P2 CkT
P2 Bk CkT 0, P2 .0 DTk Dk
ð8Þ
Equations (7) and (8) are bilinear matrix inequalities.
Network synthesis utilizing BMI for ISD suspension Take RMS value of the vehicle body acceleration optimization as objective and search ISD suspension structure of the vehicle to improve ride comfort. According to Yu,28 the transfer function Ty!z (jw) of H2 norm can be expressed as vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u +ð ‘ u Ty!z (j2pf ) = u Ty!z (j2pf )2 df t 2
ð9Þ
‘
If the input y is the road input z_ r (speed white noise), select the amount of vibration z response of body acceleration €zs , its variance is as follows E(€z2s ) = (2p)2 G0 vkTz_ r !€zs k22 = (2p)2 G0 vkTzr !_zs k2 ð10Þ The RMS value of the body acceleration J1 is as follows J1 = s(€z2s ) = 2p
pffiffiffiffiffiffiffiffi G0 vkTzr !€zs k2
ð11Þ
To minimize the body acceleration RMS J1, namely, to minimize the transfer function norm from zr to z_ s , let z_ s = z, and then C1 = ½ 1 0 0 0 . Select sprung mass m2 = 320 kg, unsprung mass m1 = 45 kg, tire stiffness kt = 192 kN/m, the suspension spring stiffness ks = 18 kN/m, vehicle speed = 30 m/s, and the irregularities coefficient G0 = 6.4 3 1027 m3/ cycle. In MATLAB, LMI toolbox and genetic algorithm are combined with path-following methods to solve BMI. Take ks = 18 kN/m, and the third-order controller is obtained by 512:8s3 + 4902:4s2 + 115:8s + 171:2 s3 + 4:356s2 + 41:387s + 1:3585 1 ’ 1 1 + 512:8 + (4892:6=s) 118:3s + 2:46 + (4:11=s)
K(s) =
ð12Þ
where g 22 = 1167:88. In the suspension machinery network, the controller K(s) is implemented in physical according to the speed rate impedance, as well as characteristics in series and in parallel of spring, damper, and inerter demonstrated in Figure 2. In which k1 = 4903.1 N/m, k2 = 4.11 N/m, c1 = 512.8 N s/m, c2 = 2.46 N s/m, and b = 118.3 kg.
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Advances in Mechanical Engineering Table 1. Center parameter values. Parameter
Center values
Spring stiffness, k1 (kN/m) Spring stiffness, k2 (kN/m) Damping coefficient, c1 (kN s/m) Damping coefficient, c2 (kN s/m) Inerter coefficient, b (kg)
22 22 1.5 4 600
Figure 2. Implementation structure of the third-order controller.
Figure 4. Sensitivity of suspension parameters.
model. Additional parameters take center elements’ parameter value. Utilizing the sensitivity coefficient as a measure of the sensitivity of the standard suspension parameters, the calculation formula is31 Figure 3. Quarter-car model of ISD suspension.
Si = In Figure 2, the implementation of the third-network controller structure is two-stage in series: the first stage consists of the spring k1 and damper c1 connected in parallel as the same traditional passive suspension; the second stage consists of the spring k2, the inerter b, and damper c2 connected in parallel. Considering the thirdnetwork controller structure has involved two springs in series and it did not need independent spring ks to support the vehicle static weight. The spring ks is removed and the parameters are re-optimized. The new structure is applied to vehicle suspension system. The structure of quarter-car model of ISD suspension is manifested in Figure 3.
d€z dpi
ð13Þ
where Si is the ith parametric sensitivity of suspension parameter; pi is the change in the ith suspension parameter, center parameter values demonstrated in Table 1; and €z is the body acceleration RMS. Through simulation, the sensitivity of parameters is obtained as shown in Figure 4. As can be seen from Figure 4, different parameters have different effects on the body acceleration of the vehicle. The spring k1 has little effect on the body acceleration of the vehicle. So, only the value of spring k2, dampers c1, c2, and inerter b are taken into consideration.
Quantum genetic algorithm Parameter optimization of ISD suspension Sensitivity analysis The parameters of ISD suspension obtained from network synthesis are complex. In order to optimize the key parameters and reduce the workload of model calibration, sensitivity analysis29,30 is utilized. Just a single parameter is tested for the influence extent of the
In traditional automobile design, the determination of the suspension parameters is considered the riding comfort of the vehicle on the road separately or for the purpose of reducing dynamic tire load. The aim of this article is to compare the overall performance between ISD suspension and traditional suspension and explore the potential advantages of ISD suspension. Comprehensively consider body acceleration,
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Table 2. Limit values of the decision variables.
Table 3. Quantum genetic algorithm optimization results of the ISD suspension parameters.
Variable
Lower limit
Upper limit
Spring stiffness, k1 (kN/m) Damping coefficient, c1 (kN s/m) Damping coefficient, c2 (kN s/m) Inerter coefficient, b (kg)
15 1 4 50
25 3 6 1000
suspension working space, and dynamic tire load and utilize quantum genetic algorithm to determine the parameters of ISD suspension structure. Quantum genetic algorithm is a genetic algorithm32 on the base of the quantum computing theory. Through the time-domain simulation, the body acceleration RMS BA0, suspension working space RMS SWS0, and dynamic tire load RMS DTL0 of traditional passive suspension in the random input are obtained. The ISD suspension parameters are obtained in the same road input above. This mathematical model of quantum genetic algorithm problem can be expressed as follows
BA SWS DTL +0:253 +0:253 minY = 0:5 3 BA0 SWS0 DTL0 s:t LB xi UB
ð14Þ
In which xi is the decision variable, and UB and LB represent its upper and lower limits, considering the actual suspension parameters of most cars; the limit values of the decision variables are demonstrated in Table 2. In the progress of population initialization, the quantum code of each individual is initialized p1ffiffi , p1ffiffi , 2
2
and the chromosome is expressed by the probability of all possible states’ superposition. The adjustment of quantum revolving gate is given as follows
cos (ui ) sin (ui ) U (ui ) = sin (ui ) cos (ui )
ð15Þ
The update process is as follows
0 ai ai ) = U (u 0 i bi bi 0
0
ð16Þ
In which (ai , bi )T and (ai , bi )T represent the ith quantum rotation gate probability amplitude before and after the update; ui is the rotational angle; and its magnitude and sign utilize the default policy of quantum algorithm to adjust. The population size at set at 40; the maximum hereditary algebra is set as 200. The parameters of ISD suspension were determined as show in Table 3.
Parameter
Values
Spring stiffness, k1 (kN/m) Spring stiffness, k2 (kN/m) Damping coefficient, c1 (kN s/m) Damping coefficient, c2 (kN s/m) Inerter coefficient, b (kg)
17 17 1.21 5.32 525
Table 4. RMS values of random response outputs. Performance index
Traditional suspension
ISD suspension
Amplitude reduction
Body acceleration, RMS (m s22) Suspension working space, RMS (m) Dynamic tire load, RMS (kN)
3.7437
3.0324
18.9%
0.0409
0.0403
1.47%
1.7690
1.7639
0.3%
Performance analysis of ISD suspension system To analyze the performance of the ISD suspension, random road is taken as input to the traditional suspension for the comparative target. The parameters of the traditional suspension is k = 19 kN/m and c = 1.4 k N s/m. The system response is analyzed. Taking integral white noise of time-domain expression as the road input model, the input equation is given by z_ r (t) = 2p
pffiffiffiffiffiffiffiffi G0 uw(t)
ð17Þ
In which w(t) is a mean of zero Gauss white noise; G0 is the road roughness coefficient; and u is the vehicle speed. When the vehicle has u = 30 m/s passing roughness factor of G0 = 5 3 1026 m3/cycle road. The system output power spectral density of random response is manifested in Figure 5. The response RMS value is demonstrated in Table 4. As can be seen from Figure 5, in the high-frequency portion above 3 Hz, the power spectral density between ISD suspension and traditional suspension performance substantially coincide. But within the frequency of 0–3 Hz, compared with the traditional suspension, ISD suspension of body acceleration power spectral density and suspension working space power spectral density decreased. This indicates that the body resonance is significantly suppressed. Dynamic tire load power density within the frequency of 2–4 Hz also declined. Only within the frequency of 0–1 Hz, there is a slight deterioration. This is due to the inerter making
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Advances in Mechanical Engineering value decreased by 0.3%. Obviously, ISD suspension significantly improves vehicle ride comfort.
Conclusion 1.
2.
3.
The network synthesis approach can solve an optimum ISD suspension structure from a number of unknown compositions, which is a general solution method. Sensitivity analysis can determine the fundamental parameters and reduce the workload of model calibration. The quantum genetic algorithm is applied to optimizing parameters of vehicle suspension, and the simulation consequences are profound satisfaction. ISD suspension effectively inhibits the lowfrequency vertical vibration of the vehicle to improve the ride comfort performance. At the same time, steady-state response time of the ISD suspension is short and the overshoot of response output is truncated, with superior dynamic performance.
Declaration of conflicting interests The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
Funding The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Nature Science Foundation of China (grant no. 51405202) and the Nature Science Foundation of Jiangsu Province (BK20130521).
References
Figure 5. Power spectral density of random response outputs: (a) body acceleration, (b) suspension working space, and (c) dynamic tire load.
the body resonance frequency point move forward. This indicates that the body resonance is significantly suppressed. Table 4 manifests that compared with traditional suspension, body acceleration RMS of ISD suspension fell by 18.9%, suspension working space RMS fell by 1.47%, and the dynamic tire load RMS
1. Shieh MY, Chiou JS and Liu MT. Design of immunealgorithm-based adaptive fuzzy controllers for active suspension systems. Adv Mech Eng 2015; 6: 916257. 2. Smith MC. Synthesis of mechanical networks: the inerter. IEEE Decis Contr P 2002; 47: 1648–1662. 3. Papageorgiou C, Houghton NE and Smith MC. Experimental testing and analysis of inerter devices. J Dyn Sys Meas Control 2009; 131: 011001. 4. Chen MZQ and Smith MC. Electrical and mechanical passive network synthesis. In: Blondel VD, Boyd SP and Kimura H (eds) Recent advances in learning and control. London: Springer, 2008, pp.35–50. 5. Shen Y, Chen L, Yang X, et al. Improved design of dynamic vibration absorber by using the inerter and its application in vehicle suspension. J Sound Vib 2016; 361: 148–158. 6. Chen MZQ, Hu Y and Du B. Suspension performance with one damper and one inerter. In: Proceedings of the 2012 24th Chinese control and decision conference
Chen et al.
7.
8.
9.
10.
11.
12.
13.
14. 15.
16.
17.
18.
19.
(CCDC), Taiyuan, China, 23–25 May 2012, pp.3534– 3539. New York: IEEE. Sharma A and Limebeer DJN. Dynamic stability of an aerodynamically efficient motorcycle. Vehicle Syst Dyn 2012; 50: 1319–1340. Wang F-C, Hong M-F and Chen C-W. Building suspensions with inerters. Proc IMechE, Part C: J Mechanical Engineering Science 2010; 224: 1605–1616. Matamoros-Sanchez AZ and Goodall RM. Novel mechatronic solutions incorporating inerters for railway vehicle vertical secondary suspensions. Vehicle Syst Dyn 2015; 53: 113–136. Xiaoliang Z, Long C, Jiamei N, et al. Analysis and experiment of frequency response characteristics of twostage series-connected ISD suspension. J Jiangsu Univ 2012; 33: 255–258. Hu Y and Chen MZQ. Performance evaluation for inerter-based dynamic vibration absorbers. Int J Mech Sci 2015; 99: 297–307. Chen MZQ, Hu Y, Huang L, et al. Influence of inerter on natural frequencies of vibration systems. J Sound Vib 2014; 333(7): 1874–1887. Hu Y, Chen MZQ, Xu S, et al. Semiactive inerter and its application in adaptive tuned vibration absorbers. IEEE T Contr Syst T 2016; 1–7. Wang F-C, Hong M-F and Chen C-W. Building suspensions with inerters. P I Mech Eng 2010; 224C8. Wang FC and Chan HA. Vehicle suspensions with a mechatronic network strut. Vehicle Syst Dyn 2011; 49: 811–830. Scheibe F and Smith MC. Analytical solutions for optimal ride comfort and tyre grip for passive vehicle suspensions. Vehicle Syst Dyn 2009; 47: 1229–1252. Chen MZQ, Wang K, Zou Y, et al. Realization of a special class of admittances with one damper and one inerter for mechanical control. IEEE T Automat Contr 2013; 58(7): 1841–1846. Chen MZQ, Wang K, Shu Z, et al. Realizations of a special class of admittances with strictly lower complexity than canonical forms. Circuits & Systems I Regular Papers IEEE Transactions on 2013; 60(60): 2465–2473. Chen M. Realization of Three-port Spring Networks with Inerter for Effective Mechanical Control. IEEE T Automat Contr 2015; 60: 1473–1478.
7 20. Wang K and Chen MZQ. Minimal realizations of three-port resistive networks. Circuits & Systems I Regular Papers IEEE Transactions on 2015; 62(4): 986–994. 21. Chen MZQ, Wang K, Li C, et al. Realization of Biquadratic Impedances as Five-Element Bridge Networks. IEEE Transactions on Circuits and Systems I: Regular Papers 2016; 1–13. 22. Hu Y, Chen MZQ and Shu Z. Passive vehicle suspensions employing inerters with multiple performance requirements. J Sound Vib 2014; 333(8): 2212–2225. 23. Chen MZQ, Hu Y, Li C, et al. Performance benefits of using inerter in semiactive suspensions. IEEE T Contr Syst T 2015; 23(4): 1571–1577. 24. Hu Y, Chen MZQ, Shu Z, et al. Analysis and optimisation for inerter-based isolators via fixed-point theory and algebraic solution. J Sound Vib 2015; 346(1): 17–36. 25. Mao M, Wang L, Chen Y-j, et al. Research progress in inerter and inerter-spring-damper suspension. Acta Armamentarii 2016; 37: 525–534. 26. Gaspar P and Kuti I. Robust control design of active suspensions using modal descriptions of vehicle finite element models. Int J Veh Auton Syst 2003; 1: 405–420. 27. Baihuai L. The network synthesis of passive mechanical system employing inerters: LMI approaches and applications to train suspension systems. Taipei, Taiwan: National Taiwan University, 2006. 28. Yu L. Robust control: linear matrix inequality approach. Beijing, China: Tsinghua University Press, 2002. 29. Tian G, Zhang H, Jia H, et al. Automotive style design assessment and sensitivity analysis using integrated analytic hierarchy process and technique for order preference by similarity to ideal solution. Adv Mech Eng 2016; 8(5): 1–10. 30. Zhou B, Geng Y and Huang XT. Global sensitivity analysis of hydraulic system parameters to a hydraulically interconnected suspension’s dynamic responses. J Vib Shock 2015; 11(2): 185–197. 31. Chonggang X, Yuanman H, Yu C, et al. Sensitivity analysis in ecological modeling. Chin J Appl Ecol 2004; 15: 1056–1062. 32. Deng G, Wei M, Su Q, et al. An effective co-evolutionary quantum genetic algorithm for the no-wait flow shop scheduling problem. Adv Mech Eng 2015; 7(12): 1–10.
