Research Article Coupling Mechanism and Decoupled ...downloads.hindawi.com/journals/mpe/2016/1932107.pdf · e half-car coupling mechanism between the front and rear quarter-car suspensions

Research ArticleCoupling Mechanism and Decoupled SuspensionControl Model of a Half Car

Hailong Zhang,1,2 Ning Zhang,1 Fuhong Min,2 Subhash Rakheja,3

Chunyi Su,3 and Enrong Wang2

1Magneto-Electronic Lab, School of Physics and Technology, Nanjing Normal University, Nanjing 210046, China2Vibration Control Lab, School of Electrical and Automation Engineering, Nanjing Normal University, Nanjing 210042, China3Center for Advanced Vehicle Engineering, Department of Mechanical Engineering, Concordia University, Montreal,QC, Canada H3G 1M8

Correspondence should be addressed to Enrong Wang; [email protected]

Received 21 November 2015; Revised 3 April 2016; Accepted 6 April 2016

Academic Editor: Reza Jazar

Copyright © 2016 Hailong Zhang et al.This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A structure decoupling control strategy of half-car suspension is proposed to fully decouple the system into independent frontand rear quarter-car suspensions in this paper. The coupling mechanism of half-car suspension is firstly revealed and formulatedwith coupled damping force (CDF) in a linear function. Moreover, a novel dual dampers-based controllable quarter-car suspensionstructure is proposed to realize the independent control of pitch and vertical motions of the half car, in which a newly addedcontrollable damper is suggested to be installed between the lower control arm and connection rod in conventional quarter-carsuspension structure.The suggested damper constantly regulates the half-car pitchmotion posture in a smooth and steady operationcondition meantime achieving the expected completely structure decoupled control of the half-car suspension, by compensatingthe evolved CDF.

1. Introduction

In the past decade, the active and semiactive control study ofintelligent vehicle suspension is a hot topic in internationalvehicle engineering [1–13], whereas the coordinate control onfull-car suspension has become a challenge bottleneck prob-lem and heavily restricts the application and further perfor-mance improvement of intelligent suspension, due to strongcoupling characteristic among the four quarter-car sub-suspensions in a full car. There have been a lot of valuablepapers reported in the research of decoupling [14–20]. Ref-erence [14] proposed an ideal output tracking controller toachieve the almost disturbance decoupled control in a halfcar, by combining both feedback linearization and feedfor-ward neural network control schemes, wherein the suspen-sion inner coupling property was neglected and more unde-termined parameters were required to synthesize controllerof the MIMO nonlinear system. Reference [15] proposed acontroller for decomposing the full-car system into two half-car models under a mild symmetry, and the roll/warp half

car was further decomposed into two quarter-cars under arigor assumed conditions and showed the effectiveness ofreducing dive and squat under acceleration and braking.Reference [16] focused on the decoupling and hierarchicalvibration control of a 6-DoF half-car suspension, in whichthere were more presumed conditions and only an approxi-mately decoupled control model was obtained. Reference [17]proposed a human-like intelligent controller by regarding thefull car as a quickly moving robot, in which eight poses wereclassified according to different coupling properties in thefull-car vertical, pitch, and roll motions. References [18, 19]proposed a new decoupled control model by decoupling thefour quarter-car suspensions in a full car into three quarter-car suspensions, which could greatly simplify the full-carcontroller synthesis complex and directly apply the pro-posed ripper control scheme for the quarter-car suspension,whereas the decoupled quarter-car suspensionmodel yieldeda large physical change. Reference [20] proposed a simplifiedmeasure and vehicle property index to permit a preliminary

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016, Article ID 1932107, 13 pageshttp://dx.doi.org/10.1155/2016/1932107

2 Mathematical Problems in Engineering

evaluation of different interconnected suspension configura-tions, through using qualitative scaling of the bounce-, roll-, pitch-, and warp-mode stiffness properties, which couldoffer considerable potential in realizing enhanced decouplingamong the different suspension modes. Above-mentionedstudy results provide ideas for decoupling the subsuspensionsystems in the full car and achieve an approximation decou-pling effect. Yet, until now, still few publications have deeplyinvestigated the coupling mechanism among the multiplequarter-car subsuspension systems in the half and full car,and above-mentioned studies have not achieved an idealizedfull-structure decoupled suspension control model, com-posed of the numbered independent quarter-car suspensionstandard dynamic models in the car. As a result, the pro-posed coordinate or decoupling control schemes seem morecomplicate and difficult in real application.The reason is thatthe studied half- and full-car suspensions are always aimed atthe conventional sprungmass damper-based quarter-car sus-pension structure, even though the linearization techniqueis utilized for modeling the suspension dynamic system byreasonably ignoring nonlinearities of the damper and carcomponents. However, the established original half- and full-carmodels are still a complicated asymmetricmultiple-input-multiple-output system, from which it is almost impossibleto achieve the full-structure decoupled suspension controlmodel, through utilizing the pure mathematical decouplingor advanced control methods.

In this paper, a full-structure decoupling suspensioncontrol scheme is originally proposed to decouple the half-carsuspension into independent front and rear standard quarter-car suspensions. A 4-DoF half-car suspension dynamicmodel is firstly established and reorganized in a new for-mulation involving the front and rear 2-DoF quarter-carsuspension models with coupling characteristic, by usinga smart mathematical equivalent transformation method.The half-car coupling mechanism between the front andrear quarter-car suspensions is then conveniently revealedand quantitatively formulated with a coupled damping force(CDF). The CDF is an evolved damping force in a linearfunction of the half-car body pitch angular acceleration andonly supplements the front and rear quarter-car suspensiondamper forces with the same magnitude but an inversephase. In order to achieve the expected half-car suspensionstructure decoupled control model, a novel dual dampers-based controllable quarter-car suspension structure is furtherproposed, by installing another extra controllable damperbetween the lower control arm and connecting rod in theconventional quarter-car suspension structure.The suggesteddamper serves function of regulating the half-car body pitchmotion posture in a smooth and steady operation conditionand meantime fully compensating the evolved CDF, byyielding the damper force in response to the online measuredpitch angular acceleration.

The paper is organized as follows. In Section 2, thecoupling mechanism of half-car suspension is revealed. InSection 3, dual dampers-based controllable quarter-car sus-pension structure is introduced for the first time. Further,structure decoupled control model of half-car suspension isanalyzed in Section 4; meanwhile the decoupling controller

xs ms

ks

xu

xi

kt

mu

Fd

Figure 1: Dynamic model of conventional 2-DoF quarter-car sus-pension.

is derived. Section 5 shows the evaluation of the proposedfull-structure decoupling control method, by comparing thetime and frequency domain responses of the coupling anddecoupled half-car suspensions under different conditions.Finally, Section 6 presents the summary and forecast.

2. Coupling Mechanism ofHalf-Car Suspension

2.1. Conventional 2-DoF Quarter-Car Suspension Model. Acommon 2-DoF dynamic model of the sprung mass damper-based quarter-car suspension is shown in Figure 1, which isused to study the vertical motion suspension performance ofa car by linearizing the associated suspension components.Herein, the damper may be either the conventional passivedamper or the active and semiactive controllable dampers,which is generally characterized with 𝐹d for avoiding thedamper nonlinearity,𝑚s and𝑚u are the sprung and unsprungmasses, 𝑘s and 𝑘t are the stiffness of suspension spring andtyre, and 𝑥i, 𝑥s, and 𝑥u are the road excitation and sprung andunsprung mass vertical displacements, respectively. The lin-earized quarter-car suspension model is formulated as [4–6]

𝑚s��s = −𝑘s (𝑥s − 𝑥u) − 𝐹d,

𝑚u��u = 𝑘s (𝑥s − 𝑥u) − 𝑘t (𝑥u − 𝑥i) + 𝐹d,(1)

where (1) is considered as the quarter-car suspension stan-dard model, which will play an important role in deriving thehalf-car suspension full-structure decoupled control model.

2.2. Conventional 4-DoF Half-Car Suspension Model. Fig-ure 2 shows a common 4-DoF half-car suspension dynamicmodel, involving two front and rear quarter-car suspensionsas shown in Figure 1, which is mainly used to study bothvertical and pitch motion suspension performances of a car.Herein,𝑀g,𝑀uf , and𝑀ur express the car body mass and thefront and rear quarter-car suspension unsprung masses, 𝐽𝜃

Mathematical Problems in Engineering 3

xsr

ksr

xur

ktr

xir

b xg

J𝜃

𝜃

aMg

Fdr ksf

ktf

MufMur

xsf

Fdf

xuf

xif

Figure 2: Dynamic model of conventional 4-DoF half-car suspen-sion.

expresses the car body pitch motion inertia, 𝑎 and 𝑏 are thefront and rear axle distances to the car body center, 𝑥g and𝜃 are the car body vertical motion displacement and pitchmotion angle, 𝐹df and 𝐹dr denote the front and rear quarter-car suspension damper forces, 𝑘sf , 𝑘sr, 𝑘tf , and 𝑘tr are stiffnessof the front and rear quarter suspension springs and tyres, and𝑥sf , 𝑥sr, 𝑥uf , 𝑥ur, 𝑥if , and 𝑥ir denote the sprung and unsprungmass vertical displacements and the road excitations of thefront and rear quarter-car suspensions, respectively. Thelinearized half-car suspension model is formulated as [14]

𝑀g��g + 𝑘sf (𝑥sf − 𝑥uf) + 𝑘sr (𝑥sr − 𝑥ur) = −𝐹df − 𝐹dr, (2)

𝐽𝜃�� − 𝑎𝑘sf (𝑥sf − 𝑥uf) + 𝑏𝑘sr (𝑥sr − 𝑥ur) = 𝑎𝐹df − 𝑏𝐹dr, (3)

𝑀uf ��uf − 𝑘sf (𝑥sf − 𝑥uf) + 𝑘tf (𝑥uf − 𝑥if) = 𝐹df , (4)

𝑀ur��ur − 𝑘sr (𝑥sr − 𝑥ur) + 𝑘tr (𝑥ur − 𝑥ir) = 𝐹dr. (5)

The dynamic properties of half-car body vertical andpitch motions are formulated in (2)–(5), as well as unsprungmass vertical motions in front and rear quarter-car suspen-sions, respectively.

By subtracting (2) multiplied by 𝑏 and (3) and subtracting(2) multiplied by 𝑎 and (3), respectively, we obtain

𝑏𝑀g��g − 𝐽𝜃�� + 𝑘sf (𝑎 + 𝑏) (𝑥sf − 𝑥uf) = − (𝑎 + 𝑏) 𝐹df ,

𝑎𝑀g��g + 𝐽𝜃�� + 𝑘sr (𝑎 + 𝑏) (𝑥sr − 𝑥ur) = − (𝑎 + 𝑏) 𝐹dr.(6)

Since the magnitude of pitch angle 𝜃 is usually a smallvalue, the vertical sprung mass displacements in front andrear quarter-car suspensions can be thus approximatelyexpressed as

𝑥g = 𝑥sf + 𝑎𝜃,

𝑥g = 𝑥sr − 𝑏𝜃.(7)

Substituting (7) into (6) leads to

𝑏

𝑎 + 𝑏𝑀g��sf = −𝑘sf (𝑥sf − 𝑥uf) − 𝐹df

− (𝑎𝑏

𝑎 + 𝑏𝑀g −𝐽𝜃

𝑎 + 𝑏) ��,

𝑎

𝑎 + 𝑏𝑀g��sr = −𝑘sr (𝑥sr − 𝑥ur) − 𝐹dr

+ (𝑎𝑏


𝑎 + 𝑏) ��,

(8)

where we denote 𝑀sf = (𝑏/(𝑎 + 𝑏))𝑀g and 𝑀sr = (𝑎/(𝑎 +𝑏))𝑀g, which just represent the sprung masses of front andrear quarter-car suspensions in the half-car, respectively.

We further set

𝐹sf = 𝐹df + 𝐹c (��) , (9)

𝐹sr = 𝐹dr − 𝐹c (��) , (10)

𝐹c (��) = 𝐾c��, (11)

𝐾c =𝑎𝑏


𝑎 + 𝑏. (12)

By replacing (9)–(12)with (8) and combiningwith (4) and(5), we obtain a new half-car suspension dynamic model asfollows:

𝑀sf ��sf = −𝑘sf (𝑥sf − 𝑥uf) − 𝐹sf ,

𝑀uf ��uf = 𝑘sf (𝑥sf − 𝑥uf) − 𝑘tf (𝑥uf − 𝑥if) + 𝐹df ,(13)

𝑀sr��sr = −𝑘sr (𝑥sr − 𝑥ur) − 𝐹sr,

𝑀ur��ur = 𝑘sr (𝑥sr − 𝑥ur) − 𝑘tr (𝑥ur − 𝑥ir) + 𝐹dr.(14)

The half-car front and rear quarter-car suspensiondynamic models with coupling characteristic are formulatedin (14) and (15), respectively, because they are quite similar tothe standard dynamic model of quarter-car suspension givenby (1), in which the only difference embodies the evolvedcoupling term 𝐹c(��) shown in (10) and (11). It is easily foundthat 𝐹c(��) quantitatively expresses the coupling characteristicbetween the front and rear quarter-car suspensions in half-car; for further discovering the half-car suspension couplingmechanism, we name 𝐹c(��) and 𝐾c as the coupled damp-ing force (CDF) and the coupling coefficient, respectively.Moreover, (12) shows that 𝐹c(��) is an evolved suspensiondamping force being a linear function of the half-car bodypitch angular acceleration �� and only supplements the yieldedfront and rear quarter-car suspension damper forces with thesame magnitude but an inverse phase. Equation (13) furthershows that 𝐾c is a constant ratio of 𝐹c(��) to ��, subject to fourparameters (𝑀g, 𝐽𝜃, 𝑎, and 𝑏) of a car design.

2.3. Coupling Mechanism of Half-Car Suspension. Considerthe following:

(i) From (9) and (10), it is found that the half-car frontand rear total suspension damping forces (𝐹sf , 𝐹sr)


are superposition of the yielded damper forces (𝐹df ,𝐹dr) and the evolved CDF (𝐹c(��)) with the samemagnitude but an inverse phase, which means that if𝐹sf increases, then 𝐹sr decreases and vice versa.

(ii) Equation (11) shows that if �� = 0, then 𝐹c(��) =0, which illustrates that when the car moves on anideally smooth road, the half-car body pitch motionwill never occur and its front and rear quarter-carsuspensions are mutually independent. In this case,the half-car suspension model, given by (13) and (14),can be easily simplified as the independent front andrear quarter-car suspension standard models. It isjust the reason that the quarter-car suspension hasbeen widely employed for the vehicle vertical motionsuspension study by international scholars.

(iii) Equation (11) further shows that if �� = 0 and 𝐾c = 0,then 𝐹c(��) = 0, it is said that when the car moves on arough road, the half-car body pitch motion and cou-pling of front and rear quarter-car suspensions willinevitably occur, and thismeans that the half-car bodypitchmotion is the essential cause leading to couplingcharacteristic between the front and rear quarter-carsuspensions. Moreover, the coupled serious degreemainly depends on the magnitude of coupling coef-ficient𝐾c, in which an idealized case is that if𝐾c = 0,the half-car suspension coupling will still never occur,even if the car moves on a quite rough road, whereas(12) shows that letting 𝐾c = 0 is rarely impossible,and one can onlymake𝐾c a small value, by reasonablymodifying the associated car design parameters.

(iv) In a car normal moving case such as �� = 0, 𝐾c =0, and 𝐹c(��) = 0, (13) and (14) further show that𝐹c(��)may be fully compensated, by artificially addinga control force (𝐹uf , 𝐹ur) in both front and rearunsprung mass motion equations and making 𝐹uf =𝐹c(��) and 𝐹ur = −𝐹c(��), such that 𝐹sf = 𝐹df + 𝐹uf and𝐹sr = 𝐹dr + 𝐹ur, which means that (13) and (14) canbe equivalently transferred to the independent frontand rear quarter-car suspension standardmodels.Theproposed decoupling control scheme of half-car sus-pension exhibits a clear physicalmeaning of indirectlyeliminating the suspension coupling characteristic,by effectively regulating the half-car pitch motionposture in a smooth and steady operation condition.

3. Dual Dampers-Based ControllableQuarter-Car Suspension Structure

In order to physically realize the above-proposed half-carsuspension decoupling control scheme, that is, adding controlforces (𝐹uf , 𝐹ur) in the unsprung mass motion equations ofhalf-car front and rear quarter-car suspensions and making𝐹uf = 𝐹c(��) and 𝐹ur = −𝐹c(��), respectively, a novel dualdampers-based controllable quarter-car suspension structureis initially proposed as shown in Figure 3. Herein, the original

Electriccontrolunit(ECU)

Moving postureof the car body

Angularaccelerationsensor

Drivermodel

Centerof mass

Controlcurrent

Controlcurrent

Figure 3: Dual dampers-based controllable quarter-car suspensionstructure.

sprung mass suspension damper is reserved for control-ling the car vertical motion suspension performance, whileanother unsprung mass controllable damper is suggested tobe installed between the low control arm and control rodalmost in parallel with the wheel [21], so as to control thehalf-car body pitch motion posture in smooth and steadyoperation condition. Moreover, a decoupling control unit,including the angular acceleration sensor and controller,is acquired to generate the front and rear unsprung massdamper forces (𝐹uf , 𝐹ur) in response to the online measuredhalf-car body pitch angle acceleration (��). The proposed dualdampers-based controllable quarter-car suspension structuremerits easy engineering implementation, due to maintainingthe original sprungmass-based quarter-car suspension struc-ture [22].

The dynamic model of the proposed dual dampers-basedcontrollable quarter-car suspension structure, as shown inFigure 3, can be conveniently established in (15), by employ-ing the same modeling way and model parameter variablesfor the conventional quarter-car suspension given by (1), inwhich the only difference is adding a term of the unsprungmass damper force (𝐹u) in the unsprung mass motionequation, by comparing (15) with (1). It is easily found that theconventional sprung mass-based and novel dual dampers-based quarter-car suspensions are the same, when 𝐹u = 0:

𝑚s��s = −𝑘s (𝑥s − 𝑥u) − 𝐹d,

𝑚u��u = 𝑘s (𝑥s − 𝑥u) − 𝑘t (𝑥u − 𝑥i) + 𝐹d + 𝐹u.(15)

4. Structure Decoupled ControlModel of Half-Car Suspension

Figure 5 shows the controllable half-car suspension dynamicmodel, based upon the proposed dual dampers-based con-trollable quarter-car suspension structure shown in Figure 4,and its model is established in (16), by employing the same


xs ms

ks

xu

xi

kt

mu

Fd

Fu

Figure 4: Dynamic model of the dual dampers-based controllablequarter-car suspension.

modeling way andmodel parameter variables for the conven-tional half-car suspension given by (2)–(5), in which the onlydifference is adding the termof unsprungmass damper forces(𝐹uf , 𝐹ur) in its unsprung mass motion equations

𝑀g��g = −𝑘sf (𝑥sf − 𝑥uf) − 𝑘sr (𝑥sr − 𝑥ur) − 𝐹df

− 𝐹dr,

𝐽𝜃�� = 𝑎𝑘sf (𝑥sf − 𝑥uf) − 𝑏𝑘sr (𝑥sr − 𝑥ur) + 𝑎𝐹df

− 𝑏𝐹dr,

𝑀uf ��uf = 𝑘sf (𝑥sf − 𝑥uf) − 𝑘tf (𝑥uf − 𝑥if) + 𝐹df + 𝐹uf ,

𝑀ur��ur = 𝑘sr (𝑥sr − 𝑥ur) − 𝑘tr (𝑥ur − 𝑥ir) + 𝐹dr + 𝐹ur.

(16)

Furthermore, the similarly decoupled half-car suspensionmodel in (17) is derived from (16), by employing the samemathematical transformation method in Section 2.2. Thismodel is composed of the dual dampers-based controllablefull-car front and rear quarter-car suspensionmodels coupledwith 𝐹c(��). By comparing (17) with (13) and (14), it is foundthat the only difference is adding term of unsprung massdamper forces (𝐹uf , 𝐹ur) in their unsprung mass motionequations

𝑀sf ��sf = −𝑘sf (𝑥sf − 𝑥uf) − 𝐹df − 𝐹c (��) ,

𝑀uf ��uf = 𝑘sf (𝑥sf − 𝑥uf) − 𝑘tf (𝑥uf − 𝑥if) + 𝐹df + 𝐹uf ,

𝑀sr��sr = −𝑘sr (𝑥sr − 𝑥ur) − 𝐹dr + 𝐹c (��) ,

𝑀ur��ur = 𝑘sr (𝑥sr − 𝑥ur) − 𝑘tr (𝑥ur − 𝑥ir) + 𝐹dr + 𝐹ur.

(17)

To achieve the half-car decoupled suspension controlmodel, it is required to realize 𝐹uf = 𝐹c(��) and 𝐹ur = −𝐹c(��)by synthesizing a decoupling controller. Then the modelgiven by (17) can be rewritten as the decoupled form in (18),

xsr

ksr

xur

ktr

xir

b xg

J𝜃

𝜃

aMg

Fdr

Fur Fuf

ksf

ktf

MufMur

xsf

Fdf

xuf

xif

Figure 5: Dynamic model of dual dampers-based controllable half-car suspension.

which is composed of independent front and rear quarter-carsuspension standard models:

𝑀sf ��sf = −𝑘sf (𝑥sf − 𝑥uf) − 𝐹sf ,

𝑀uf ��uf = 𝑘sf (𝑥sf − 𝑥uf) − 𝑘tf (𝑥uf − 𝑥if) + 𝐹sf ,

𝑀sr��sr = −𝑘sr (𝑥sr − 𝑥ur) − 𝐹sr,

𝑀ur��ur = 𝑘sr (𝑥sr − 𝑥ur) − 𝑘tr (𝑥ur − 𝑥ir) + 𝐹sr.

(18)

With the help of above-mentioned decoupling controlscheme, the half-car suspension can be ideally decoupled intotwo independent front and rear quarter-car suspensions byfully compensating the evolved 𝐹c(��). For this purpose, thekey work is to synthesize a decoupled controller as below:

𝐹uf = −𝐹ur = 𝐾c��, (19)

where the decoupled controller gain is just the proposedcoupling coefficient defined in Section 2.1.

5. Evaluation of the Proposed Full-StructureDecoupling Control Method

To verify the correctness of the proposed half-car suspensionfull-structure decoupling control method, a simulation plat-form for both coupling and decoupled half-car suspensiondynamic systems is built up. The responses of coupling anddecoupled half-car suspensions are thoroughly comparedunder harmonic and rounded pulse excitations inserted ona single tyre, and the employed half-car suspension modelparameters are listed in Table 1 [14]. It should be noticed thatthe passive dampers (𝐶sf , 𝐶sr) are employed as the sprungmass suspension dampers, and the added unsprung masscontrollable damper forces are generally characterized with𝐹uf and 𝐹ur in this study.

5.1. Coupling Characteristic of Conventional Half-Car Suspen-sion. Thehalf-car suspension coupling characteristic is firstlyanalyzed to evaluate correctness of the proposed couplingmechanism between the half-car front and rear quarter-car suspensions. According to (12) and parameters listed in


500

250

0

−250

−500

Dam

ping

forc

es o

fFdf

,Fc,F

sf(N

)

1.0 1.5 2.0 2.5 3.0

Time t (s)

FdfFcFsf

(a)

500

250

0

−250

−5001.0 1.5 2.0 2.5 3.0

Time t (s)

Dam

ping

forc

es o

fFdr

,Fc,F

sr(N

)

FdrFcFsr

(b)

Figure 6: The coupling characteristic of conventional half-car suspension system. (a) Damping forces in front quarter-car suspension. (b)Damping forces in rear quarter-car suspension.

Table 1: Model parameters of the half-car suspension system.

Symbol Quantity Value𝑀g Car body mass 575 kg𝐽𝜃

Car body pitch inertia 769 kg⋅m2

𝑀uf Front unsprung mass 60 kg𝑀ur Rear unsprung mass 60 kg𝑘tf Front tyre stiffness 190 kN/m𝑘tr Rear tyre stiffness 190 kN/m𝑘sf Front suspension stiffness 16.812 kN/m𝑘sr Rear suspension stiffness 16.812 kN/m𝐶sf Front damper coefficient 1.0 kN⋅s/m𝐶sr Rear damper coefficient 1.0 kN⋅s/m𝑎 Front axle distance to car body center 1.38m𝑏 Rear axle distance to car body center 1.36m

Table 1, the half-car suspension coupling coefficient, that is,the decoupled controller gain, is calculated as 𝐾c = 113.2.The conventional half-car suspension, as shown in Figure 2, isanalyzed under a harmonic excitation with amplitude 2.5 cmat 1.5Hz of the half-car sprung mass resonant frequency,which is inserted on the front tyre (𝑥if ). Figure 6 illustratesresponse comparisons of total suspension damping forces(𝐹sf ,𝐹sr), damper forces (𝐹df ,𝐹dr), and coupled damping force(𝐹c) in the half-car front and rear quarter-car suspensions.The results clearly show that the magnitude of 𝐹sf is muchlarger than that of𝐹sr, due to the harmonic excitation insertedonly on the front tyre.The slight vibration of rear quarter-carsuspension that occurred, as shown in Figure 6(b), is the sus-pension coupling effect transmitted from the front quarter-car suspension, although no excitation is inserted on the reartyre (𝑥ir = 0). Moreover, 𝐹sf and 𝐹sr are larger and lower than

their 𝐹df and 𝐹dr, because they are superposed by 𝐹c(��) on𝐹df and 𝐹dr with the same magnitude but an inverse phase,respectively. It is obvious that the total suspension dampingforces of half-car front and rear quarter-car suspensions arenot offered purely by the yielded suspension damper forces,due to the coupling characteristic defined by 𝐹c.

5.2. Frequency Domain Response Analysis of Decoupled Half-Car Suspension. The frequency domain response analysis isan important tool for evaluating vehicle suspension perfor-mance due to the larger dominant frequency band 0–20Hz ofroad vehicle [2]. For evaluating engineering feasibility of theproposed novel dual dampers-based controllable quarter-carsuspension structure, frequency responses of the decoupledhalf-car suspension are thoroughly compared with thoseof the conventional coupling half-car suspension based onsprungmass damper-based quarter-car structure. Herein, theevaluation indexes are selected as half-car body center ver-tical displacement acceleration transmissibility (𝑇g), sprungmass vertical displacement acceleration transmissibility (𝑇sf ,𝑇sr) and dynamic travel transmissibility (𝑇df , 𝑇dr) of the frontand rear quarter-car suspensions, and Root-Mean-Square(RMS) of the half-car body pitch angular acceleration (𝑎𝜃);the calculation method is given in [7].

Figure 7 illustrates the frequency domain response com-parisons of both coupling and decoupled half-car suspen-sions, under discrete harmonic excitations inserted on thefront tyre in frequency range 0–20Hz. The results show thatthe suggested unsprung mass dampers in the decoupled half-car suspension yield obvious regulation role of the smoothand steady pitch motion posture, due to its much lower RMSmagnitudes of 𝑎𝜃 than those of the conventional couplinghalf-car suspension as shown in Figure 7(b), and realizethe expected half-car full-structure decoupling suspension


0.16

0.12

0.08

0.04

0.00

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fTg

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Coupling suspensionDecoupled suspension

(a) 𝑇g

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RMS

of(r

ad·s−

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ty o

fTdr

0 5 10 15 20

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(f) 𝑇dr

Figure 7: Frequency domain response comparisons of coupling and decoupled half-car suspensions: (a) acceleration transmissibility ofhalf-car body center 𝑇g; (b) RMS of pitch angular acceleration 𝑎𝜃; (c) acceleration transmissibility of front quarter-car suspension 𝑇sf ; (d)acceleration transmissibility of rear quarter-car suspension 𝑇sr; (e) travel transmissibility of front quarter-car suspension 𝑇df ; and (f) traveltransmissibility of rear quarter-car suspension 𝑇dr.


control function, because the rear quarter-car suspensionof decoupled half-car does not work as shown in Figures7(d) and 7(f), while the rear quarter-car suspension ofcoupling half-car still occurs with slight vibration due tothe transmitted suspension coupling effect from its frontquarter-car suspension, although nonexcitation is insertedon the rear tyre (𝑥ir = 0). Moreover, for both couplingand decoupled half-car suspensions, 𝑇g does not change anymore as shown in Figure 7(a). Slight vibrations of only 𝑇sfand 𝑇df do occur as shown in Figures 7(c) and 7(e), whichmeans that the proposed dual dampers-based quarter-carsuspension structure does not affect the fundamental vehiclesuspension performance, while contributing to develop a newhalf-car suspension full-structure decoupled control model.

5.3. Evaluation of Half-Car Suspension Full-Structure Decou-pled Control Model. The time domain response comparisonsof the coupling and decoupled half-car suspensions arefurther conducted, under a rounded pulse excitation insertedon the front tyre and meantime zero excitation inserted onthe rear tyre, given by (20). The employed rounded pulseexcitation is used to emulate the usually encountered harshroad surfaces such as hills and hollows, which makes thecar body pitch serious. The evaluation indexes are selectedas the half-car body center vertical displacement and pitchangular accelerations (𝑎g, 𝑎𝜃), the sprung mass displacementaccelerations (𝑎sf , 𝑎sr), and dynamic travels (𝑥df , 𝑥dr) of frontand rear quarter-car suspensions, respectively [7]:

𝑥if = 0.25𝑎me2(𝜇𝜔0𝑡)

2 e−𝜇𝜔0𝑡,

𝑥ir = 0,(20)

where parameters of the rounded pulse excitation are chosenas amplitude 𝑎m = 2.0 cm, fundamental harmonic frequency𝜔0 = 10.4 rad/s, and pulse stiffness 𝜇 = 3.

Figure 8 shows the time domain response comparisons ofboth coupling and decoupled half-car suspensions, under therounded pulse excitation inserted on front tyre. It is foundthat 𝑎g remains the same, while magnitudes of 𝑎𝜃 of thedecoupled half-car suspension are obviously lower than thoseof the coupling half-car suspension, as shown in Figures 8(a)and 8(b). The results maintain identical tendency as shownin Figures 7(a) and 7(b), due to the achieved ideal regulationrole of half-car body pitch motion posture in smooth andsteady operation condition. Furthermore, effectiveness ofthe proposed half-car suspension decoupled control modelcan be well verified by comparing the responses of frontand rear quarter-car suspensions, as shown in Figures 8(c)–8(f), in which the magnitudes of 𝑎sf and 𝑥df show onlyslight variations for both coupling and decoupled frontquarter-car suspensions, while the magnitudes of 𝑎sr and 𝑥drshow quite obvious difference with larger and zero valuesfor the coupling and decoupled front quarter-car suspen-sions, respectively, because the coupling half-car suspensioninevitably transmits vibration from the front quarter-carsuspension to the rear quarter-car suspension, while thedecoupled half-car suspension has effectively eliminated thecoupling characteristic between the half-car front and rearquarter-car suspensions.

5.4. Exploratory Research on the Semiactive Decoupling. InSections 5.1–5.3, all the results are based on the assumptionthat the unsprung mass damper provides the active force forfully compensating the CDF. If the unsprung mass damperis a semiactive damper, there may be some problems in thedynamic equations of the suspension, for the reason that thesemiactive damper cannot provide the active force and canonly dissipate vibration energy. In this section, we intend todiscuss the results of the decouplingwithmagnetorheologicaldamper (MRD), which is known as a classic semiactivedamper.

The MRD is certainly able to provide variable dampingcoefficient rather than wanted active force. Despite that, thedesired forces of the front and rear unsprung mass damperare just the opposite at the same time from (19). If one oftwo unsprung mass dampers serves the desired force withspecific driven current, while the other one is working at zerodriven current at the same time, we think that an approximatedecoupling control is achieved by partly compensating theCDF. Next, the experimental study is carried out to validatethe idea. We build up the dedicated semiphysical experimentplatform, using hardware-in-the-loop (HIL) simulation tech-nology [23, 24]. Nonphysical model is built in real-time sim-ulation platform with virtual components, including𝑚s,𝑚u,𝑘s, 𝑘t, and 𝑥i. For the restrictions in experimental conditions,the front unsprung mass damper uses the real MRD, whilethe rear unsprungmass damper uses the modified Bouc-Wenmodel identified from the real MRD, as shown in Figure 9.The modified Bouc-Wen model structure and parametersare elaborated in [25]. 𝐹uf is the actual damping force, and𝐹ur is the calculated damping force of the Bouc-Wen model.The schematic of the experimental set-up is illustrated inFigure 10, where the instruments are numbered from 1 to 11.The platform is composed of four sections, namely, real-timesimulation platform (1, 2), MR damper (8), vibration system(3–7), and an interface circuit (10).

In accordance with the analysis in Section 5.3, the smoothpulse is used as road excitation, which is only insertedon the front tyre, given by (20). The system responses areselected as 𝑎g and 𝑎𝜃 𝑎sf and 𝑥df 𝑎sr and 𝑥dr. The results ofsemidecoupled suspension with MRD are shown by compar-ing with coupling and decoupled suspensions in Figure 11.Obviously shown in Figures 11(a) and 11(b), the amplitudeof both 𝑎g and 𝑎𝜃 of the semiactive decoupled suspensionreduces significantly, which is quite different from those of thefull-decoupled suspension. It is also found that stabilizationtime delays by about 0.3 s due to regulation of the drivencurrent for MRD. As shown in Figures 11(c) and 11(e), thevibration amplitude of the front suspension response of thesemidecoupled suspension reduces, while the stabilizationtime also delays by about 0.3 s. We should note that the start-ing nonlinear oscillations are suppressed effectively in thesemidecoupled suspension. By observing the rear suspensionsystem with no excitation, as shown in Figures 11(d) and11(f), it is found that the suspension vibration is significantlyweakened compared with the coupling suspension by partlycompensating the CDF though not excellent as the full-decoupled suspension. In conclusion, the semidecoupled


1.8

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amic

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)

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amic

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el o

fxdr

(cm

)

0.0 0.5 1.0 1.5 2.0

Time t (s)


(f) 𝑥dr

Figure 8: Time domain response comparisons of coupling and decoupled half-car suspensions: (a) acceleration of half-car body center 𝑎g;(b) pitch angular acceleration 𝑎𝜃; (c) acceleration of front quarter-car suspension 𝑎sf ; (d) acceleration of rear quarter-car suspension 𝑎sr; (e)travel displacement of front quarter-car suspension 𝑥df ; and (f) travel displacement of rear quarter-car suspension 𝑥dr.


xsr

ksr

xur

ktr

xir

csr

Mur

b xg

J𝜃

𝜃

Mg

Fdr

Fur

xy

c1 Fd

k1

k0c0

Bouc-Wen model

Bouc-Wen

a

ksfcsf

Fdf

Muf

xuf

xsf

ktf Fuf

xif

MRD

Figure 9: Test sketch of decoupling control of the half car with semiactive damper.

1

2 10 9

11

4

5

3

7

8

6

Id

Fd

Fd

xrxs − xu

xs − xu

Figure 10: Schematic diagram of the HIL experiment platform (1-personal computer (PC), 2-industrial PC (IPC) with a data acquisition card(DAQC), 3-the load cell, 4-shaker controller, 5-power amplifier, 6-electromagnetic vibration shaker, 7-linear variable differential transformer,8-MRD, 9-high precision programable power source, 10-interface circuit, and 11-mixed domain digitizing oscilloscope).


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Coupling suspensionDecoupled suspensionSemidecoupled suspension

(a) 𝑎g

2

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Acce

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f(m

·s−2)

0 1 2 3 4

Time t (s)


(c) 𝑎sf

0.8

0.4

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−0.8

Acce

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tion

ofa s

r(m

·s−2)

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1

0

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−3

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amic

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fxdf

(cm

)

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−0.2

−0.4

Dyn

amic

trav

el o

fxdr

(cm

)

0 1 2 3 4

Time t (s)


(f) 𝑥dr

Figure 11: Time domain response comparisons of coupling, decoupled, and semidecoupled half-car suspensions: (a) acceleration of half-car body center 𝑎g; (b) pitch angular acceleration 𝑎𝜃; (c) acceleration of front quarter-car suspension 𝑎sf ; (d) acceleration of rear quarter-carsuspension 𝑎sr; (e) travel displacement of front quarter-car suspension 𝑥df ; and (f) travel displacement of rear quarter-car suspension 𝑥dr.


suspension with semiactive damper such as MRD exhibitsa better performance than coupling suspension by partlydecoupling.

6. Conclusions

Consider the following:

(1) The coupling mechanism between the half-car frontand rear quarter-car suspensions has been originallyrevealed and quantitatively formulated by the coupleddamping force (CDF), which is a linear function ofthe pitch angular acceleration with a constant ratioof the defined coupling coefficient. The half-car pitchmotion is the main cause leading to the couplingcharacteristic.

(2) A novel dual dampers-based controllable quarter-car suspension structure was originally proposed byinstalling an unsprung mass controllable damperbetween the conventional quarter-car suspensionlower control arm and connecting rod, which servesfunction of regulating the half-car pitch motion pos-ture in a smooth and steady operation condition andmerits clear physics meaning and easy implementa-tion.

(3) A novel half-car suspension full-structure decou-pled control model, composed of independent frontand rear standard quarter-car models, was originallyproposed, on the basis of the proposed novel dualdampers-based controllable quarter-car suspensionstructure. A controller is synthesized to decouple thehalf-car suspension into independent front and rearquarter-car suspensions, by controlling the yieldedunsprung mass damper forces in response to theonline measured pitch angular acceleration.

(4) The proposed half-car suspension full-structuredecoupled control model will put forth a greatlysimplified control scheme for realizing the intelligentfull-car suspension. The suggested unsprung masscontrollable dampers are used to control the half-carpitch motion suspension performance and meantimeachieve the half-car suspension full-structuredecoupled control model, while the original sprungmass controllable dampers are used to control thedecoupled front and rear quarter-car vertical motionsuspension performances, respectively.

(5) The selection of the unsprung mass damper is dis-cussed.The active damper can guarantee the supply ofthe desired force for full decoupling. More generally,the semidecoupled suspension with semiactive MRDas unsprung mass damper is analyzed. The HILexperiment is conducted, and the results show thatthe semidecoupled suspension also exhibits a gooddecoupling performance by partly compensating theCDF.

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

The research leading to these results has received fundingfrom the National Science Foundation of China (Grant nos.51475246, 51277098, and 51075215), the Research InnovationProgram for College Graduates of Jiangsu Province (Grantno. KYLX15 0725), and the Natural Science Foundation ofJiangsu Province of China (Grant no. BK20131402).

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Research Article Coupling Mechanism and Decoupled ...downloads.hindawi.com/journals/mpe/2016/1932107.pdf · e half-car coupling mechanism between the front and rear quarter-car suspensions