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Research ArticleModal and Dynamic Analysis of a Vehicle withKinetic Dynamic Suspension System

Bangji Zhang,1 Jie Zhang,1 Jinhua Yi,2 Nong Zhang,1,3 and Qiutan Jin1

1State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, College of Mechanical and Vehicle Engineering,Hunan University, Changsha, Hunan 410082, China2Guangzhou Automobile Group Co., Ltd., Automotive Engineering, Guangzhou, Guangdong 511434, China3School of Electrical, Mechanical and Mechatronic Systems, Faculty of Engineering and Information Technology,University of Technology, Sydney, NSW 2007, Australia

Correspondence should be addressed to Jie Zhang; [email protected]

Received 24 February 2016; Revised 17 August 2016; Accepted 29 August 2016

Academic Editor: Emiliano Mucchi

Copyright © 2016 Bangji 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 novel kinetic dynamic suspension (KDS) system is presented for the cooperative control of the roll and warp motion modes ofoff-road vehicles. The proposed KDS system consists of two hydraulic cylinders acting on the antiroll bars. Hence, the antiroll barsare not completely replaced by the hydraulic system, but both systems are installed. In this paper, the vibration analysis in termsof natural frequencies of different motion modes in frequency domain for an off-road vehicle equipped with different configurablesuspension systems is studied by using themodal analysis method.The dynamic responses of the vehicle with different configurablesuspension systems are investigated under different road excitations andmaneuvers.The results of the modal and dynamic analysisprove that the KDS system can reduce the roll and articulationmotions of the off-road vehicle without adding extra bounce stiffnessand deteriorating the ride comfort. Furthermore, the roll stiffness is increased and thewarp stiffness is decreased by theKDS system,which could significantly enhance handing performance and off-road capability.

1. Introduction

Vehicle suspension system has always been considered asan essential component to alleviate the impact from roadexcitations to the vehicle body and to ensure the propertiesof comfortable driving, good handling, and safety of thevehicles [1–4]. Conventional suspension systems are oftenadopted by automobile manufacturers due to their costeffectiveness and reliabilities, but they are also confrontedwith the contradiction between handling performance andride comfort. As is well known, the vehicle body motioncaused by four wheels can be classified as four motion modes(bounce, pitch, roll, and warp modes) and these modes playa key role in analyzing the characteristics of the vehicles [5–7]. In general, the handling performance should possess stiffpitch and roll modes, but the soft bounce andwarpmodes arebeneficial to the ride comfort [8, 9]. Therefore, an advancedsuspension system needs to be able to have both antiroll androad-holding abilities.

To date, many engineers have made great effort toimprove the properties of vehicle suspensions. Advancedsuspensions that include active suspensions and semiactivesuspensions applied to luxury passenger cars have beenproposed to balance the contradiction between vehicle ridecomfort and handling performance. For manufactures, how-ever, they are often faced with the inconvenience associatedwith these advanced suspensions, such as high cost, uncertainreliability, huge power consumption, and inherent complexity[10]. Therefore, in order to overcome the drawbacks of theabove-mentioned conventional and advanced suspensions,novel passive suspension systems are being adopted in auto-motive industry. The interconnected suspension system isone of the most effective passive suspension systems that caneliminate the compromise between vehicle stability and rideperformance. In contrast to conventional suspensions, inter-connected suspensions not only are capable of uncoupling thefour suspension modes (bounce, pitch, roll, and warp), but

Hindawi Publishing CorporationShock and VibrationVolume 2016, Article ID 5239837, 18 pageshttp://dx.doi.org/10.1155/2016/5239837

2 Shock and Vibration

also have advantages in controlling stiffness and damping ofeach suspension mode. This kind of suspensions is achievedby the use of interconnections between the individual wheelstations and is realized through mechanical, hydraulic, orhydropneumatic means [9, 11–13], where the forces can begenerated at other wheel stations when one wheel stationof vehicles has the motion [5]. As a representative example,the passive hydraulically interconnected suspension (HIS)is realized through the use of the dynamic interconnectionbetween the hydraulic circuits and the unsprung/sprungmasses of vehicles. Furthermore, the conventional shockabsorber is replaced by a single/double-acting hydrauliccylinder which is fixed at each wheel station.The chambers ofhydraulic cylinders are interconnected by hydraulic circuits,and these elements (damper valves, hydraulic accumulators,pipelines, fittings, and flexible hoses) are the indispensablecomponents for hydraulic circuits. When the vehicles aredriven on the road, the sprung and unsprung masses musthave the relativemotion thatmakes fluid flow in the circuits ofHIS systems. Due to the flow into or out of the accumulators,the volumes of accumulators are compressed or expandedand this leads to the changes of pressure in the hydrauliccircuits. And the changes of pressure in the cylinder chambersmake the restoring forces be applied to the suspension struts(the sprung and unsprung masses). HIS systems can obtainthe desired levels of stiffness and damping by pushing flowinto certain dampers and accumulators in order to reduce themotion of one given mode.

Compared to the conventional suspension systems, HISsystems have been paid only sporadic attention in the liter-ature so far. In 1970s, the investigations on hydrolastic andhydragas systems were conducted by Moulton and Best [14].Thepossiblemechanical/hydraulic interconnections arrange-ments of four-wheel vehicle were conceptually presented byOrtiz, which could provide separate damping and springingfor roll/warp or pitch/bounce modes [15]. Zapletal con-ceptually proposed the interconnected system to decouplethe full vehicle modes, which is similar to Ortiz’s system[16]. Mace introduced network theory and system synthesismethodologies to develop a theoretical study of existing pas-sive HIS systems [17]. Furthermore, mechanical admittancematrices were employed to establish the mathematical modelof this interconnected suspension, and the optimal handlingand ride performances were achieved by adjusting the warpstiffness and damping levels of a linear full-car mode. SmithandWalker also applied this approach into a novel suspensionconcept, which could decouple the stiffness and dampingof different modes [5]. From these mentioned studies, notonly were most proposals of HIS systems purely conceptualand theoretical, but also the dynamics of the interconnectingmechanism were not investigated. Then, Behave conducted aparametric study of the influence of pneumatic pitch-planeinterconnection arrangements on vehicle ride performance[18]. This study has shown that the optimum parameters(the flow loss coefficient, air spring volume to area ratio,vehicle mass, and mass distribution) could minimize theimpact on the vibration transmissibility of the bounce andpitch modes caused by the road inputs. Compared withthe reduction of the warp mode stiffness for unconnected

system, the fully decoupled damping of HIS system couldachieve better performance. An experimental investigationof the kinetic four-wheel HIS was presented to improvethe handing performance by Wilde et al. [19]. The resultsimplied that this system could provide higher roll stiffnessthan the conventional vehicle suspension systems. In recentyears, the interconnected hydropneumatic suspensions in rolland pitch planes have been proposed by Cao [7, 20, 21].They investigated the dynamic characteristics of this systemat full-car level and the results showed that the ride andhandling performances could be enhanced by adjusting thepitch stiffness and damping independently, particularly inheavy vehicle applications. Zhang and Smith have developedthe research on the multibody system dynamics of vehiclesfitted with HIS systems [22–24]. A systematic method waspresented to investigate the mechanics of roll-plane HIS intime and frequency domain and the results were verifiedby the experiments. Ding et al. have presented a new HISsystem to study the dynamics of triaxle heavy trucks [25].The theoretical results showed that the proposed HIS systemwas able to reduce the pitch motion of sprung mass andmaintain the ride comfort simultaneously. Nowadays, manyresearchers pay more attention to the investigation of theadvantages of HIS system.

On the other hand, the antiroll bar is the most ubiq-uitous and simple interconnection arrangement in currentapplication for conventional suspension systems.The antirollbar provides higher stiffness to the roll and warp motionmodes through a mechanical interconnection between leftand right wheel-pairs, but the higher articulation stiffnessis disadvantageous to ground holding performance. Thus,this antioppositional mechanism is unable to balance thecontradiction between the roll andwarpmotions for vehicles.In addition, although the higher spring stiffness is beneficialto the handling performance, it will deteriorate the ridecomfort. Therefore, conventional suspension systems withantiroll bars are incapable of solving the contradiction amongride comfort, handling performance, and ground holdingperformance. The semiactive or active suspension systemsmay have the ability to accomplish these functions, but it willbe realized by high cost and huge power consumption. Con-sidering the above-mentioned issues for antiroll bars and theadvantages provided by HIS system, a novel kinetic dynamicsuspension (KDS) system,which is a passive hydraulic systemwithout consuming power, is proposed in this study to solvethe contradiction among different performance requirementsfor off-road vehicles. The KDS system is realized throughthe hydraulic system and a particular mechanism. It isnoteworthy that the KDS system consists of two hydrauliccylinders acting on the antiroll bars, and the antiroll barsare interconnected through the hydraulic system. Therefore,the antiroll bars are not completely replaced by the hydraulicsystem, but both systems are installed in order to coordinatethe bounce, roll, and warp motion modes. To validate theeffectiveness of the proposed KDS system, this paper presentsthe study about the characteristics of off-road vehicles fittedwith KDS system in both frequency and time domainsin terms of system modal analysis and dynamic responsesunder different conditions. It is seen from the numerical

Shock and Vibration 3

Coil springs

Hydrauliccircuit

Hydraulicactuator

Antiroll barWheel assembly

Hydraulicaccumulator

(a)

Without displacement

Off-road

On-road

Front actuator Rear actuator

With displacement

(b)

Figure 1: (a) A schematic layout of a two-axle vehicle with KDS system. (b) The working condition of a two-axle vehicle with KDS system.

results that the proposed KDS system can deliver higher rollmode stiffness than the conventional antiroll bar to improvehandling performance and can reduce the warp mode stiff-ness to enhance ground holding performance comparedto the conventional suspension system with antiroll bars.Furthermore, the KDS system does not add extra bouncestiffness and is able to maintain the ride comfort.

The remainder of the paper is organized as follows. Sec-tion 2 presents the system description of the KDS-equippedvehicle. The modelling of KDS-equipped vehicle is describedin Section 3. The characteristic analysis of KDS system ispresented in Section 4. Section 5 presents the comparativeanalysis of vehicles with the conventional suspension systemand the proposed KDS system. Section 6 describes thediscussions and future work, and the conclusions are drawnin Section 7.

2. System Description of theKDS-Equipped Vehicle

The schematic diagram of a two-axle vehicle with the KDSsystem is shown in Figure 1(a). The KDS-equipped vehi-cle system includes two subsystems: conventional vehiclesystem and hydraulic system (KDS system). The vehiclesystem consists of wheel assemblies, suspension systems (coilsprings), and vehicle chassis. The car body is supportedby the four-wheel assemblies. The coil springs are used toconnect the vehicle chassis with wheel assemblies. Further-more, coil springs can also support the chassis and alleviatethe impact from road excitations to ensure ride comfort.The hydraulic system called KDS system comprises somehydraulic elements including two hydraulic actuators thatcan directly provide the forces between the vehicle body andwheel assemblies andhydraulic accumulators that can controlthe pressure of the hydraulic circuits, pipelines, hydraulicfittings, and flexible hoses. The KDS system together withantiroll bars and the hydraulically interconnected system are

mounted at the front and rear parts of the vehicle. Antirollbars are the simplest interconnection arrangement to reducethe roll motion and are fixed on the front and rear axles ofvehicles. In Figure 1(a), antiroll bars are interconnected bymeans of the double-acting hydraulic actuator assemblies,which are different from the conventional antiroll bars.Each hydraulic actuator has a cylinder and a piston, andthe cylinder is divided into the top chamber and bottomchamber. The cylinder is mounted at the end of one antirollbar pair, while the piston is fixed at the end of the otherantiroll bar pair.The chambers of the two actuator assembliesare interconnected by hydraulic circuits to provide the flowcommunication between chambers [26]. According to theaforementioned system description, the hydraulic cylindersare actually installed between the vehicle body and wheelassemblies through the antiroll bars. Furthermore, throughthe antiroll bars, the acting forces induced by the KDSsystem are imposed on them. Thus, the antiroll bars arenot completely replaced by the hydraulic system but workedtogether through the hydraulic system.

According to Figure 1(a), the KDS system is a passivesystem without requiring external sensors to detect vehiclemotions. During the cornering, the roll motion can resultin an increase in pressure difference between the pistonassemblies on one side of the vehicle and this can preventthe transfer of fluid between the connected fluid chambersand hold the cylinder pistons in place. Thus it can suppressthe suspension stroke and ensure the suspension rigidityto reduce the roll motion of the vehicle. When the vehicleis driving off-road, the locations of the points where fourwheels contact the ground are not all in the same plane. Thisleads to the compressing of the suspension of two diagonallyopposedwheels, but the extending of the other two diagonallyopposed wheels. The mode of suspension movement calledwarp motion can minimize the change of pressure differencebetween the piston assemblies to allow the flowing of fluidin the connected hydraulic circuits and to move the cylinder


piston upwards and downwards. This is beneficial to extendthe suspension stroke and a long suspension stroke helps thewheels contact with the ground. As shown in Figure 1(b), theKDS system can ensure the excellent performance for bothon-road and off-road vehicles. However, modern vehicleswith high center of gravity are normally equipped with thestiff antiroll bars in order to prevent the roll motion of thevehicle. A disadvantageous consequence of providing antirollbars is that the ride quality is uncomfortable for passengerssince two wheels of an axle are functionally interconnected incertain level, so the single wheel’s inputs are not independentwhen the single wheel of vehicles travels on the bump or slopeblocks.

3. Modelling of KDS-Equipped Vehicle

In this paper, a seven-degree-of-freedom (seven-DOF) full-car model, which is often employed in the literature [27], isconsidered as the vehicle with conventional suspension sys-tem (VCS). The seven-DOF full-car model with KDS systemshown in Figure 2 is defined as the vehicle with KDS system(VKDS) where the lumped sprung mass is 𝑚

𝑠and unsprung

masses are𝑚𝑢𝑓and𝑚

𝑢𝑟.The lumped sprungmass of the vehi-

cle model is considered to have three DOFs (bounce motion𝑧𝑠in the vertical direction, pitch motion 𝜃, and roll motion 𝜙

around the CG center (𝑜) of the sprungmass), and eachwheelhas one DOF (bounce motion 𝑧

𝑢𝑓/𝑧𝑢𝑟

in the vertical direc-tion).The front and rear suspension strut forces are applied ata fixed distance (𝑎 and 𝑏) from the 𝑦-axis and a fixed distance(𝑡𝑓and 𝑡

𝑟) from the 𝑥-axis. The suspensions are assumed

to have the linear spring stiffness (𝑘𝑠𝑓/𝑘

𝑠𝑟) and damping

coefficient (𝑐𝑠𝑓/𝑐𝑠𝑟), and the vertical stiffness (𝑘

𝑡𝑓/𝑘

𝑡𝑟) of the

tyre is considered to be linear. In order to obtain the linearphysical parameters for a two-axle vehicle, a new physicalparameter identification method was proposed by Zhenget al., [28]. Based on the assumption of the linearity, theyintroduced the state variable method (SVM) to identify themodal parameter of the vehicle. Then, a known mass matrixwas designed to add to the vehicle in order tomake it possibleto determine the mass, damping, and stiffness matrices ofthe vehicle. The purpose of this procedure is to increase thenumber of equations to ensure that the number of equationsis more than one of the number of the unknown parameters.Consequently, the physical parameters of the two-axle vehi-cle, such as inertial parameters, stiffness, and damping, canbe calculated by using least square method. According to thesystem description in Section 2, the only difference betweenthe KDS-equipped vehicle system and the conventionalsuspension system is that the hydraulic system is installed onthe conventional vehicle through the disconnected antirollbars. Hence, the vehicle physical parameters of the KDS-equipped vehicle system are the same as the vehicle withconventional suspension system and are given in Section 5.

According to Newton’s second law and free-body dia-grams of rigid bodies, the equations of the motion for theintegratedmechanical-hydraulic system of the full-car modelcan be derived as

𝑀

𝑍 + 𝐶

𝑍 + 𝐾𝑍 = 𝐹𝜃+ 𝐹

𝑒(𝑡) , (1)

where 𝑀, 𝐶, and 𝐾 are the mass, damping, and stiffnessmatrices; 𝑍 is the displacement vector; 𝐹

𝑒(𝑡) is the road

excitation force inputmatrix;𝐹𝜃= 𝐷

1𝐾𝐴𝑅𝐵

𝜃𝐴𝑅𝐵

describes theforces in the antiroll bars due to their torsion angles, where𝐾𝐴𝑅𝐵

is the torsional stiffness matrix of antiroll bars; 𝜃𝐴𝑅𝐵

is the torsion angle of antiroll bars. The antiroll bar is a keycomponent to transmit forces between the wheels/tyres andthe chassis due to the torsion.𝐹

𝑖(𝑖 = 𝐴, 𝐵, 𝐶,𝐷) are the forces

of four wheels caused by antiroll bars, and they are expressedin the following form:

𝐹𝑖=

𝐾𝑖𝜃𝑖

𝑙1

(𝑖 = 𝐴, 𝐵) ,

𝐹𝑚

= −

𝐾𝑚𝜃𝑚

𝑙1

(𝑚 = 𝐶,𝐷) ,

𝑍 = [𝑧𝑆

𝜃 𝜙 𝑧𝑢𝐴

𝑧𝑢𝐵

𝑧𝑢𝐶

𝑧𝑢𝐷

]

𝑇

,

𝜃𝐴𝑅𝐵

= [𝜃𝐴

𝜃𝐵

𝜃𝐶

𝜃𝐷]

𝑇

,

𝐾𝐴𝑅𝐵

= diag [𝐾𝐴

𝐾𝐵

𝐾𝐶

𝐾𝐷] ,

(2)

where 𝑙1is the length of the outer arm of the front and rear

antiroll bars; 𝐾𝑖(𝑖 = 𝐴, 𝐵, 𝐶,𝐷) are the torsional stiffness

of the front and rear antiroll bars; 𝜃𝑖(𝑖 = 𝐴, 𝐵, 𝐶,𝐷) are

the angles of the front and rear antiroll bars. Accordingto Newton’s third law, 𝐹

𝜃is the vector of the antiroll bar

transmitting to the vehicle body;𝐷1is the coefficient matrix.

They are defined as the following form:

𝐹𝜃= 𝐷

1𝐾𝐴𝑅𝐵

𝜃𝐴𝑅𝐵

=

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

−𝐹𝐴− 𝐹

𝐵− 𝐹

𝐶− 𝐹

𝐷

− (𝐹𝐶+ 𝐹

𝐷) ∗ 𝑏 + (𝐹

𝐴+ 𝐹

𝐵) ∗ 𝑎

− (𝐹𝐴

− 𝐹𝐵) ∗ 𝑡

𝑓− (𝐹

𝑐− 𝐹

𝐷) ∗ 𝑡

𝑟

𝐹𝐴

𝐹𝐵

𝐹𝐶

𝐹𝐷

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

,

𝐷1=

1

𝑙1

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

−1 −1 1 1

𝑎 𝑎 𝑏 𝑏

−𝑡𝑓

𝑡𝑓

𝑡𝑟

−𝑡𝑟

1 0 0 0

0 1 0 0

0 0 −1 0

0 0 0 −1

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

.

(3)

3.1. Model of Antiroll Bars. Antiroll bars connect the left andright wheels, which are commonly used for the suspensionsystem to increase roll stiffness and to reduce the roll motionof the vehicle. Due to the fact that the antiroll bars only workin roll motion of vehicles, the equations ofmotion of the front


y

x

o

AB

CD

a

b

KDSS

tr tr

Zs

Iyyms

Ixx𝜙

tftf

ksf csf ksf csf

ktf

muf muf

ktf

ktr

ksrcsr

mur

𝜃

Figure 2: Schematic of a 7-DOF two-axle vehicle model with KDS system.

and rear antiroll bar models are described in the followingform based on Newton’s second law:

𝐼𝐴𝑅𝐵

𝜃𝐴𝑅𝐵

+ 𝐾𝐴𝑅𝐵

𝜃𝐴𝑅𝐵

= 𝐹𝑃+ 𝐹

𝑇, (4)

where 𝐼𝐴𝑅𝐵

and 𝐾𝐴𝑅𝐵

are the inertia and stiffness matricesof the antiroll bar, respectively; 𝐹

𝑇= −𝐷

𝑇

1𝐾𝐴𝑅𝐵

𝑋 describesthe force matrix of the antiroll bar induced by the suspensionsystem. The force matrix 𝐹

𝑃of the antiroll bar due to the

pressure in the cylinder chambers is written as𝐹𝑃= 𝐷

2𝐴𝑃(𝑡),

where the area matrix 𝐴 and pressure vector 𝑃 related tothe corresponding cylinder chambers are defined as 𝑃(𝑡) =

[𝑃𝑓𝑡

𝑃𝑓𝑏

𝑃𝑟𝑡

𝑃𝑟𝑏]

𝑇 and 𝐴 = diag [𝐴𝑓𝑡

𝐴𝑓𝑏

𝐴𝑟𝑡

𝐴𝑟𝑏], and

the coefficient matrix 𝐷2is defined as the following form:

𝐷2=

[

[

[

[

[

[

−𝑙2

𝑙2

0 0

𝑙2

−𝑙2

0 0

0 0 −𝑙2

𝑙2

0 0 𝑙2

−𝑙2

]

]

]

]

]

]

, (5)

where 𝑙2is the length of the inner arm of the antiroll bars

acting on hydraulic actuators. Combining (1) and (4), theequations of motion for the completely integrated vehicle-KDS system can be given by

�� + �� + ��𝑌 = ��𝑃+ ��

𝑒, (6)

where

�� = [

𝑀 𝑂

𝑂 𝐼𝐴𝑅𝐵

]

11×11

,

�� = [

𝐶 𝑂7×4

𝑂4×7

𝑂4×4

]

11×11

,

�� = [

𝐾 −𝐷1𝐾𝐴𝑅𝐵

𝐷𝑇

1𝐾𝐴𝑅𝐵

𝐾𝐴𝑅𝐵

]

11×11

,

Front cylinder Rear cylinder

Pft

Pfb

qf qft

qfb

qrt

qrb

qrPrt

Prb

Figure 3: Mechanical-fluid system boundary conditions.

��𝑃= 𝐷

3𝐴𝑃 (𝑡) = [

07×4

𝐷2

]𝐴𝑃 (𝑡) ,

��𝑒= [

𝐹𝑒(𝑡)

𝑂4×1

]

11×1

.

(7)

The state vector of the integrated system is defined as𝑌 = [𝑍

𝑇

𝜃𝑇

𝐴𝑅𝐵]

𝑇

. Furthermore, the original mass, damping,and stiffness matrices from (1) are modified into ��, ��,and �� by adding the effects relating to the antiroll bars,and ��

𝑃describes the nonlinear coupling effect between the

mechanical and hydraulic subsystems.

3.2.Model of theMechanical-Fluid System. Figure 3 shows thedouble-acting hydraulic cylinders fixed on the antiroll barsin the new KDS-equipped vehicle system. From Figure 3, thepiston divides the cylinder into the top and bottom chambersto form the boundary within the mechanical-fluid system.The relative velocity between the ends of the piston rod resultsin a boundary fluid flow towards/away from the hydrauliclines. Thus, the volume and pressure in the top and bottomchambers change correspondingly. Based on the well-knownrelationship of fluid compressibility, the fluid compressibility


in the top and bottom chambers 𝑞comp(𝑡) and 𝑞comp(𝑏) isdescribed as

𝑞comp(𝑓𝑡) = ��𝑠𝑢𝑓

𝐴𝑓𝑡

− 𝑞𝑓− 𝑞

𝑓𝑡(𝑡)

=

𝑉0

𝑓𝑡− 𝑧

𝑠𝑢𝑓𝐴𝑓𝑡

𝐸

��𝑓𝑡

(𝑡) ,

𝑞comp(𝑓𝑏) = 𝑞𝑓𝑏

(𝑡) − (��𝑠𝑢𝑓

𝐴𝑓𝑡

− 𝑞𝑓)

=

𝑉0

𝑓𝑏+ 𝑧

𝑠𝑢𝑓𝐴𝑓𝑏

𝐸

��𝑓𝑏

(𝑡) ,

𝑞comp(𝑟𝑡) = ��𝑠𝑢𝑟

𝐴𝑟𝑡

− 𝑞𝑟− 𝑞

𝑟𝑡(𝑡)

=

𝑉0

𝑟𝑡− 𝑧

𝑠𝑢𝑟𝐴𝑟𝑡

𝐸

��𝑟𝑡(𝑡) ,

𝑞comp(𝑟𝑏) = 𝑞𝑟𝑏

(𝑡) − (��𝑠𝑢𝑟

𝐴𝑟𝑡

− 𝑞𝑟)

=

𝑉0

𝑟𝑏+ 𝑧

𝑠𝑢𝑟𝐴𝑟𝑏

𝐸

��𝑟𝑏

(𝑡) ,

(8)

where 𝐸 is the bulk modulus of the cylinder chamber; 𝑞𝑖(𝑖 =

𝑓𝑡, 𝑟𝑡) and 𝑞𝑗(𝑗 = 𝑓𝑏, 𝑟𝑏) denote the outflow and inflow of

the top and bottom chambers of the front and rear hydrauliccylinders, respectively; 𝑉0

𝑖and 𝑉

0

𝑗are the original volumes

of the top and bottom chambers. 𝑞𝑓and 𝑞

𝑟are the cross-line

leakage flows due to the pressure differential in each pistonand are given by the leakage coefficients, 𝐻

𝑓and 𝐻

𝑟. They

are defined as the following forms:

𝑞𝑓

=

(𝑃𝑓𝑡

− 𝑃𝑓𝑏

)

𝐻𝑓

,

𝑞𝑟=

(𝑃𝑟𝑡

− 𝑃𝑟𝑏)

𝐻𝑟

.

(9)

Assume the small roll motion of the vehicle and therelative velocity of the piston rods of the front and rearcylinders ��

𝑠𝑢𝑓and ��

𝑠𝑢𝑟are given by

��𝑠𝑢𝑓

= 𝑙2(−

𝜃2−

𝜃1) +

𝑙2

𝑙1[��

𝑢𝐴− ��

𝑠𝐴− (��

𝑢𝐵− ��

𝑠𝐵)]

,

��𝑠𝑢𝑟

= 𝑙2(−

𝜃3−

𝜃4) +

𝑙2

𝑙1[��

𝑢𝐶− ��

𝑠𝐶− (��

𝑢𝐷− ��

𝑠𝐷)]

.

(10)

When (9) and (10) are applied to (8), the flow vector𝑄(𝑡) = [𝑞

𝑓𝑡𝑞𝑓𝑏

𝑞𝑟𝑡

𝑞𝑟𝑏]

𝑇 can be written in the followingform:

𝑄 (𝑡) = 𝐷4𝐴�� (𝑡) + 𝐻𝑃 (𝑡) + �� (𝑡) �� (𝑡) , (11)

where𝐷4is a constant linear transformation matrix;𝐻 is the

leakage flow resistance matrix; ��(𝑡) is a time-variant matrixof cylinder volume and bulk modulus terms.

In order to investigate the motion properties of themechanical-fluid coupled system in (6), the fluid systemequation 𝑄 = 𝑓(𝑃) describing the relationship between the

flow and the pressure is employed. The hydraulic impedancemethod is proposed for the frequency domainmodelling dueto its computational advantages in this paper. Moreover, thismethod has been used to study the modal characteristics fortriaxle trucks with HIS by Ding et al., [25]. The equation ofthe fluid system is given as follows:

𝑄 (𝑠) = 𝑅 (𝑠)−1

𝑃 (𝑠) , (12)

where 𝑅(𝑠) is the impedance matrix of the fluid systemelements. Furthermore, based on (12), the Laplace transformof (11) with zero initial conditions is written as

𝑠𝑃 (𝑠) = �� (𝑠)−1

(𝑅 (𝑠)−1

− 𝐻)𝑃 (𝑠)

− 𝑠�� (𝑠)−1

𝐷4𝐴𝑌 (𝑠) .

(13)

Equation (6) describes the motion for the integratedmechanical-hydraulic system. According to the Laplacetransform (13), (6) can be rewritten as

𝑠2

��𝑌 (𝑠) + 𝑠��𝑌 (𝑠) + ��𝑌 (𝑠) = 𝐷3𝐴𝑃 (𝑠) + ��

𝑒(𝑠) . (14)

In this section, 𝑢 = [𝑌(𝑡) 𝑃(𝑡)]

𝑇 is defined as the coupledsystem state vector. According to the state space method, themotion equation can be given by combining (13) with (14):

𝑠𝑈 (𝑠) = �� (𝑠) 𝑈 (𝑠) + 𝐵𝑁 (𝑠) , (15)

where 𝑈(𝑠) is the Laplace transform of the state vector 𝑢 and𝑁 is the Laplace transform of the vector of applied nonsus-pension forces; ��(𝑠) is the system characteristic matrix, and𝐵 is the road disturbance input matrix. They are described inthe following form:

�� (𝑠)

=

[

[

[

[

[

𝑂 𝐼 𝑂

��

−1

��

−1

��

−1

𝐷3𝐴

𝑂 ��

−1

(𝑠)𝐷4𝐴 ��

−1

(𝑠) (𝑅−1

(𝑠) − 𝐻)

]

]

]

]

]

,

𝐵 =

[

[

[

[

𝑂

��

−1

𝑂

]

]

]

]

,

𝑁 (𝑠) = ��𝑒(𝑠) .

(16)

Equation (15) represents the general equation of motionfor the completely integrated system in frequency domain. Inorder to study the modal characteristics, this paper needs toobtain the solutions of the aforementioned equations throughnumerical iterative approach. Before solving the equationsof the coupled system, it can be seen that the calculationof the system characteristic matrix ��(𝑠) is not an easy taskdue to the impedance matrix 𝑅(𝑠). The impedance matrixdepends on the precise arrangement of the hydraulicallyinterconnected system which includes the fluid circuits andthe individual components.The details for the determinationof the impedance matrix have been shown in the literature[22].


3.3. Dynamic Equations of Mechanical-Fluid System. Thissection mainly describes the dynamic model which includesthe state space representation of the integrated full-car systemfor the transient analysis. The details of the derivation ofdynamic equations of the coupled vehicle multibody systemand hydraulic circuits by the state space method for transientanalysis can be found in the literature [23, 29]. According tothe description of the coupled system in the aforementionedsections, the state vector that describes the motion of thesprung and unsprung lumped multibody model is defined asthe following form:

𝑋𝑀

= [𝑍𝑇

𝑍

𝑇

𝜃𝑇

𝐴𝑅𝐵

𝜃

𝑇

𝐴𝑅𝐵

]

𝑇

. (17)

The state vector that describes the dynamic states of thehydraulic subsystem is given by the following form:

𝑋𝐻

= [𝑃𝑓𝑡

𝑃𝑓𝑟

𝑃𝑟𝑡

𝑃𝑟𝑏

𝑃𝑎1

𝑃𝑎2

𝑞𝑓𝑡

𝑞𝑓𝑟

𝑞𝑟𝑡

𝑞𝑟𝑟]

𝑇

,

(18)

where 𝑃𝑎1, 𝑃

𝑎2describe the pressures of the two hydraulic

accumulators. Furthermore, 𝑋 = [𝑋𝑇

𝑀𝑋𝑇

𝐻]

𝑇

is defined asthe state vector of the coupled mechanical-fluid system byintegrating the two state vectors 𝑋

𝑀and 𝑋

𝐻together.

Based on the free-body diagram approach and Newton’ssecond law, the state space equation of the full-car system isderived as the following form:

𝑇�� = 𝐺𝑋 + 𝐹. (19)

Thus, (19) can be written in the following form:

[

[

[

[

[

[

[

[

[

𝐼7

0 0 0 0

0 𝑀7

0 0 0

0 0 𝐼4

0 0

0 0 0 𝐼𝐴𝑅𝐵

0

0 0 0 0 𝑇𝐻

]

]

]

]

]

]

]

]

]32×32

[

[

[

[

[

[

[

[

[

[

𝑍

𝑍

𝜃𝐴𝑅𝐵

𝜃𝐴𝑅𝐵

��𝐻

]

]

]

]

]

]

]

]

]

]32×1

=

[

[

[

[

[

[

[

[

[

0 𝐼7

0 0 0

−𝐾 −𝐶 0 0 0

0 0 0 𝐼4

0

0 0 𝐺𝐿

0 0

0 0 0 0 𝑆𝐻

]

]

]

]

]

]

]

]

]32×32

[

[

[

[

[

[

[

[

[

𝑍

𝑍

𝜃𝐴𝑅𝐵

𝜃𝐴𝑅𝐵

𝑋𝐻

]

]

]

]

]

]

]

]

]32×1

+

[

[

[

[

[

[

[

[

[

0

𝐹𝑉

0

𝐹𝐿

𝐹𝐻

]

]

]

]

]

]

]

]

]32×1

.

(20)

According to the aforementioned study of the coupledsystem, it is noted that the external forces 𝐹

𝑉, 𝐹

𝐿, and

𝐹𝐻

include interactive elements between mechanical andhydraulic systems. They can be described as

𝐹𝑉

= 𝐷1𝐾𝐴𝑅𝐵

𝜃𝐴𝑅𝐵

+ 𝐹𝑒(𝑡) ,

𝐹𝐿= 𝐺

𝐻𝐿𝑋𝐻

− 𝐷𝑇

1𝐾𝐴𝑅𝐵

𝑍,

𝐹𝐻

= 𝐾𝐻𝑍

𝑍 + 𝐶𝐻𝑍

𝑍 + 𝐹

ℎ,

(21)

where 𝐺𝐿, 𝐺

𝐻𝐿, 𝐾

𝐻𝑍, and 𝐶

𝐻𝑍are the linear coefficients

matrices relating to themotions of themechanical system anddynamics of the suspension fluid circuits; 𝑇

𝐻and 𝑆

𝐻are the

linear coefficient matrices of the fluid circuits; 𝐹ℎdescribes

the nonlinear effects on the fluid system.In order to study the dynamic behaviors of the vehicle fit-

ted with the KDS system, the dynamic state of the multibodysystem and the fluid system can be determined simultane-ously by using the first 22 equations of the multibody systemand the last 10 equations of the fluid system. Furthermore, afourth-order Runge–Kutta numerical integration method isused to obtain the solutions of differential equation (19).

4. Characteristic Analysis of KDS System

The hydraulic layout of the KDS system including twoidentical fluid circuits is shown in Figure 4.The two identicaldouble-acting piston-cylinder actuators, which are fixed atthe front and rear antiroll bars, are coupled with each otherkinetically through the two hydraulic pipelines, hydraulicaccumulators and hydraulic fittings. During the vehicle cor-nering, the roll motion can result in the long displacement ofsuspensions B and D and the short displacement of suspen-sions A and C. Due to the movements of four suspensions,the hydraulic oil in circuit B flows into accumulator B whilethe hydraulic oil in circuit A flows out from accumulatorA. The gas of accumulator B is compressed and the gas ofaccumulator A is released which can result in the pressure ofhydraulic circuit B rising and the pressure of hydraulic circuitA falling. The generated hydraulic force coupled throughthe antiroll bars acts against the vehicle body roll motion.The restoring roll moments of the KDS system (𝑀KDS) andconventional suspension system (𝑀CS) can be expressed as

𝑀KDS = 𝑀spring + 𝑀ℎ,

𝑀CS = 𝑀spring + 𝑀Anti-roll,(22)

where 𝑀spring is the restoring roll moment developed dueto restoring forces of springs; 𝑀Anti-roll is the restoring rollmoment of the conventional antiroll bars;𝑀

ℎis the restoring

roll moment developed due to the restoring hydraulic forceand is written as

𝑀ℎ= 2𝑛 (𝑝

𝑓𝑡𝐴𝑓𝑡

− 𝑝𝑓𝑏

𝐴𝑓𝑏

) 𝑡𝑓

+ 2𝑛 (𝑝𝑟𝑡𝐴𝑟𝑡

− 𝑝𝑟𝑏𝐴𝑟𝑏) 𝑡

𝑟,

(23)

where 𝑛 is the lever ratio related to 𝑙1and 𝑙

2of the discon-

nected antiroll bars. Due to the nonlinear change of oil pres-sures for two hydraulic circuits,𝑀

ℎcan be significantly larger


Rear antiroll bar

Suspension D

Hydraulic circuit B

Hydraulic accumulator B

Warp motion

Suspension BRoll motion Hydraulic circuit A

Hydraulic accumulator A

Suspension C

Suspension CFront antiroll bar

Front actuator

Rear actuator

Figure 4: Three-dimensional diagram of a KDS system.

than 𝑀Anti-roll. Hence, the roll stiffness of the KDS systemis higher than the conventional suspension system, and theKDS system has the ability to enhance handing performance.

When the vehicle is driving off-road, the locations of fourwheels contacting the ground are not all in the same plane. Asis shown in Figure 4, suspensions A and D of two diagonallyopposed wheels are compressed and suspensions B and Cof the other two diagonally opposed wheels are extended.For the KDS system, the hydraulic oil in the top chamberof the front actuator flows into the top chamber of the rearactuator through circuit A and there is no oil flowing into/outaccumulator A. Furthermore, the hydraulic oil in the bottomchamber of the rear actuator flows into the rear chamber ofthe front actuator through circuit B and there is no oil flowinginto/out accumulator B. Hence, the oil pressures of the twohydraulic circuits A and B are unable to be changed. In thewarp motion, the vehicle body twisting moments of the KDSsystem (𝑇KDS) and conventional suspension system (𝑇CS) canbe expressed as

𝑇KDS = 𝑇spring + 𝑇ℎ,

𝑇CS = 𝑇spring + 𝑇Anti-roll,(24)

where 𝑇spring is the twisting moment of the vehicle body dueto restoring forces of springs; 𝑇Anti-roll is the restoring rollmoment of the vehicle body due to the conventional antirollbars; 𝑇

ℎis the twisting moment of the vehicle body due to the

restoring hydraulic force and is written as

𝑇𝐻

= (𝐹𝐻𝐴

𝑡𝑓− 𝐹

𝐻𝐵𝑡𝑓) + (𝐹

𝐻𝐷𝑡𝑟− 𝐹

𝐻𝐶𝑡𝑟) , (25)

where 𝐹𝐻𝑖

(𝑖 = 𝐴, 𝐵, 𝐶,𝐷) are the hydraulic forces acting onthe vehicle body through the disconnected antiroll bars.Theycan be expressed as

𝐹𝐻𝐴

= −𝐹𝐻𝐵

= 𝑛 (𝑝𝑓𝑡𝐴𝑓𝑡

− 𝑝𝑓𝑏

𝐴𝑓𝑏

) ,

𝐹𝐻𝐷

= −𝐹𝐻𝐶

= 𝑛 (𝑝𝑟𝑡𝐴𝑟𝑡

− 𝑝𝑟𝑏𝐴𝑟𝑏) .

(26)

Because of the invariability of oil pressures, the twistingmoment 𝑇

ℎdue to the restoring hydraulic force is very small.

But, the twisting moment 𝑇Anti-roll is usually larger due to thelimitation of conventional antiroll bars in the warp motion.Thus, the KDS system has the ability to provide smallertwisting moment than conventional suspension system. Inthe warp motion, the smaller twisting moment is beneficialto the vehicle body, but the larger twisting moment is ableto damage the vehicle body. Due to the oil free flow inhydraulic circuits, the suspension movements in the warpmotion cannot be restricted. Therefore, the KDS system hasthe ability to improve ground holding performance and keepthe balance of the body attitude in the warp motion.

5. Numerical Analysis

In this section, the comparative analysis of vehicles withthe conventional suspension system and the proposed KDSsystem is shown. The comparison is done about both themodal characteristic and the dynamic response. The physicalparameters of a seven-DOF vehicle model, which can beidentified by the physical parameter identification method[28], are shown in the Parameters. The Parameters alsolists the hydraulic parameters of the KDS system based onthe literature [23]. The solutions of the motion equationsare obtained according to the vehicle parameters and theappropriate numerical method.

5.1. Modal Analysis of Mechanical-Fluid System. Accordingto Section 3, it is obvious that the frequency-dependentcoefficient A(s) is the systemmodal characteristic matrix andthis matrix is changed with the input frequency s. Accordingto the definition of eigenvalue, the solutions of the equation|det(��(𝑠) − 𝑠𝐼)| = 0 are the eigenvalues of ��(𝑠). Since theexact eigenvalues of ��(𝑠) are difficult to obtain numerically,approximate roots are acceptable if the function 𝑊(𝑠) =

|det(��(𝑠) − 𝑠𝐼)| is approaching a minimum value in thevicinity of s. Based on this theory, the Matlab optimizationlibrary function fminsearh is employed to determine the realeigenvalues from the local minimum values.

A three-dimensional plot of the function 𝑊(𝑠) withrespect to the real and imaginary parts of 𝑠 shows clearly thatthe modulus of the determinant reaches a minimum valueat the eigenfrequency. The logarithm value 𝑊(𝑠) in Figure 5is used to evaluate the numerical solutions under the givenfrequency range. It can be seen that the numerical valueof the vertical axis can theoretically obtain the infinitesimalvalue, but this depends on the changes of parameters and thecomputer accuracy.

According to the numerical solution of modal equationsin Section 3.2, the identified eigenvalues and eigenvectors, thenatural frequencies, and the free vibration mode shapes areshown inTable 1.The results in Table 1 show that the first threemode shapes involve the bounce, roll, and pitch motions ofthe sprung mass, and the next four mode shapes are relatedto the wheel hop motions for the VKDS and VCS. Comparedwith VCS, the KDS system can effectively increase the rollmode (the 2ndmode) frequency of the sprungmass vibration


Table 1: Natural frequencies and mode shape comparison.

Mode shape1st 2nd 3rd 4th 5th 6th 7th

VCS𝑓 (Hz) 1.493 1.528 1.842 9.837 10.110 10.211 10.552𝜉 0.340 0.240 0.334 0.219 0.207 0.250 0.239

𝑍𝑠

1 0 0.560 − 0.210𝑖 −0.016 + 0.050𝑖 0 −0.024 + 0.060𝑖 0

𝜃 −0.390 + 0.110𝑖 0 1 −0.023 + 0.060𝑖 0 0.012 − 0.050𝑖 0

𝜑 1 1 0 0 0.048 − 0.060𝑖 0 0.023 + 0.090𝑖

𝑍𝜇𝐴

0.011 − 0.080𝑖 0.058 − 0.036𝑖 −0.030 + 0.060𝑖 −0.030 − 0.068𝑖 −0.030 − 0.020𝑖 1 −1

𝑍𝜇𝐵

0.011 − 0.080𝑖 −0.058 + 0.036𝑖 −0.030 + 0.060𝑖 −0.030 − 0.068𝑖 0.030 + 0.020𝑖 1 1

𝑍𝜇𝐶

0.025 − 0.040𝑖 0.052 − 0.027𝑖 0.190 − 0.130𝑖 1 −1 0.024 + 0.060𝑖 0.024 + 0.063𝑖

𝑍𝜇𝐷

0.025 − 0.040𝑖 −0.052 + 0.027𝑖 0.190 − 0.130𝑖 1 1 0.024 + 0.060𝑖 −0.024 − 0.063𝑖

VKDS𝑓 (Hz) 1.492 1.689 1.840 9.040 9.840 9.930 10.210𝜉 0.346 0.350 0.330 0.396 0.225 0.215 0.257

𝑍𝑠

1 0 −0.390 + 0.110𝑖 0.110 + 0.080𝑖 0.110 + 0.080𝑖 0.030 + 0.040𝑖 0.030 + 0.040𝑖

𝜃 0.560 − 0.210𝑖 0 1 0.002 + 0.080𝑖 −0.002 − 0.080𝑖 −0.020 − 0.080𝑖 0.020 + 0.080𝑖

𝜑 0 1 0 −0.030 − 0.070𝑖 −0.030 − 0.070𝑖 0.190 + 0.130𝑖 0.190 + 0.130𝑖

𝑍𝜇𝐴

0 0 −0.210 − 0.410𝑖 1 −1 −0.810 − 0.160𝑖 0.810 + 0.160𝑖

𝑍𝜇𝐵

−0.016 − 0.050𝑖 −0.023 − 0.060𝑖 0 −0.030 + 0.070𝑖 −0.030 + 0.070𝑖 1 1

𝑍𝜇𝐶

0 0 0.050 − 0.004𝑖 −0.870 − 0.230𝑖 0.870 + 0.230𝑖 −1 1

𝑍𝜇𝐷

−0.024 − 0.060𝑖 0.012 + 0.050𝑖 0 1 1 0.0240 − 0.060𝑖 0.024 − 0.060𝑖

Real part of complex (s)

Imaginary part of complex (s)

1.7

1.65

1.6

1.550

0−20−20−40

−40−60−60

−80 −80

log10

(abs

(det

(A−s∗I)

)

Figure 5: Three-dimensional diagram of log(abs(𝑊(𝑠))).

(from 1.528Hz to 1.689Hz) and maintain the bounce andpitch modes (the 1st and 3rd modes) frequencies of thesprung mass vibration (from 1.493 and 1.842Hz to 1.492 and1.840Hz). Furthermore, it can reduce the natural frequenciesof the warp and oppositional hops between the left and rightwheels (from 10.552 and 10.110Hz to 9.040 and 9.930Hz) andhold the natural frequencies of oppositional and synchronoushops between the front and rear wheels (from 9.837 and10.211Hz to 9.840 and 10.210Hz). The KDS system cansimultaneously increase the damping ratios corresponding tothe roll and warp modes of the integrated system.

Based on the analytical results of mode shapes, theKDS system is able to provide higher stiffness of the rollmode to improve the handing performance and also providessofter stiffness of the warp mode to improve ground holdingperformance. Moreover, the KDS system does not change thestiffness of the bounce and pitchmodes and hence guaranteesthe ride comfort.

5.2. Dynamic Response. This section introduces the time-domain response of the full-car model to validate theadvantages of the KDS system to an off-road vehicle underexcitations of lateral acceleration and road inputs. In thefollowing actual driving scenarios, the roll/warp and bouncemotions of the VKDS and VCS are compared to investigatethe vehicles performance.

5.2.1. Roll Motion. It is well known that roll motion isimportant for the handling performance and too severe rollmotion may make the vehicle out of control. The handlingperformance is defined as the percentage of the availablefriction or the maximum achievable lateral accelerationaccording to Harty [30]. When the centrifugal force that avehicle experiences during cornering exceeds the availablefriction force, sliding or yaw motions of the vehicle mayoccur. This section investigates the well-known slalom testand fishhook maneuver of the VKDS and VCS. The equiv-alent roll moments of the vehicle body induced by these twotests are applied to the full-car model. Figures 6(a) and 6(b)show these two equivalent lateral acceleration inputs as afunction of time.


0 2 4 6 8 10 12−8

−6

−4

−2

0

2

4

6

8

Time (s)

Late

ral a

ccel

erat

ion

(m/s2)

(a)

0 2 4 6 8 10 12−6

−4

−2

0

2

4

6

Time (s)

Late

ral a

ccel

erat

ion

(m/s2)

(b)

Figure 6: (a) Equivalent lateral acceleration of the fishhook maneuver. (b) Equivalent lateral acceleration of the slalom test.

0 2 4 6 8 10 12−6

−4

−2

0

2

4

6

Roll

angl

e (de

g)

Time (s)

VCSVKDS

(a)

0 2 4 6 8 10 12−25

−20

−15

−10

−5

0

5

10

15Ro

ll ve

loci

ty (d

eg/s

)

Time (s)

VCSVKDS

(b)

Figure 7: (a) Sprung mass roll angle of the VCS and VKDS under the fishhook maneuver. (b) Sprung mass roll velocity.

Figures 7(a), 7(b), 8(a), and 8(b) compare sprung massroll angles and velocity responses of the vehicle models withtwo suspension configurations (VKDS and VCS) under thesimulations of the slalom test and fishhook maneuver givenin Figures 6(a) and 6(b). In these tests, both the sprung massroll angle and velocity are used to describe the handlingperformance. The designed stiffness of the antiroll bar isdefined as the minimum target value so that the roll stiffnessof KDS system is not less than the value of antiroll bar. It canbe clearly seen from Figures 7(a), 7(b), 8(a), and 8(b) thatthe VKDS experiences a significantly smaller roll angle andvelocity. According to the formula 𝑓roll = √𝑘roll/𝐼roll/2𝜋 (𝑓rollis the roll oscillation frequency, 𝑘roll is the roll mode stiffness,and 𝐼roll is the roll moment inertia of the sprung mass), the

higher roll mode stiffness of the VKDS is also verified fromthe relatively higher oscillation frequency of the roll modecompared to the VCS (from 1.528Hz to 1.689Hz) in Table 1.Furthermore, the results show that the roll velocity responsesof the VKDS decay faster than the VCS, which is attributedto the enhanced roll mode damping properties of the KDSsystem, and the higher roll mode damping has the ability toensure the balance of the body attitude in short time.

Figures 9(a) and 9(b) illustrate the comparisons of sprungmass roll angle responses of the VKDS with different oilpressures (20 bar, 30 bar, and 40 bar) in this set of simulations.The results indicate that the sprung mass roll angle of theVKDS with the 40-bar oil pressure is smaller than the valueof the antiroll bar, but the roll angle in the 20-bar oil pressure


0 2 4 6 8 10 12−3

−2

−1

0

1

2

3Ro

ll an

gle (

deg)

Time (s)

VCSVKDS

(a)

0 2 4 6 8 10 12−8

−6

−4

−2

0

2

4

6

8

Roll

velo

city

(deg

/s)

Time (s)

VCSVKDS

(b)

Figure 8: (a) Sprung mass roll angle of the VCS and VKDS under the slalom test. (b) Sprung mass roll velocity.

0 2 4 6 8 10 12−6

−4

−2

0

2

4

6

Roll

angl

e (de

g)

Time (s)

20 bar30 bar40 bar

(a)

0 2 4 6 8 10 12−3

−2

−1

0

1

2

3

Roll

angl

e (de

g)

Time (s)

20 bar30 bar40 bar

(b)

Figure 9: (a) Sprung mass roll angle of the VKDS with different oil pressures under the fishhook maneuver. (b) Sprung mass roll angle of theVKDS with different oil pressures under the slalom test.

condition is nearly identical to the value of antiroll bar.Compared with the antiroll bars, the passive KDS systemcould automatically generate fluidic mass flows into/out ofthe actuators so that the oil pressures of the top and bottomchambers of hydraulic actuators change rapidly to generatethe reacting forces to control relative roll displacementbetween the sprung mass and the unsprung masses. It can beseen that higher oil pressure is able to provide the higher rollstiffness to prevent roll motion of the off-road vehicles.

5.2.2. Warp Motion. For off-road vehicles, a stiff warp modeis harmful to the handling performance and chassis structure.When off-road vehicles are driven on the rugged road, fourvehicle wheels are not all in a single plane as shown inFigure 10. A stiff warp mode can worsen the uneven weightdistribution of the four wheels and constrain the relativemotion of the wheels, which may lead to the lifting up ofa wheel on the rugged road. The lift of driving wheels andsteering wheels has the negative impact on forward driving


Figure 10: The warp mode excitation.

0 2 4 6 8 10 12

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

Time (s)

Susp

ensio

n de

flect

ion

(m)

VCSVKDS

(a)

0 2 4 6 8 10 12

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

Time (s)

Susp

ensio

n de

flect

ion

(m)

VCSVKDS

(b)

Figure 11: (a) Vehicle suspension deflection of the VCS and VKDS in front left station. (b) In rear right wheel station.

torque, steering, and handling performance. Furthermore,the too large twist torque can damage the chassis due to a pairof roll moments. The directions of this pair of roll momentsare opposite to each other, which leads to twist vehicle chassisalong 𝑥-axis. Thus, this motion of the four wheels is definedas the warp mode.

The antiroll bars are normally applied to connect thewheels of the front and rear axles of off-road vehicles toincrease roll stiffness, but they can result in the stiff warpmode. Compared with the antiroll bars, the KDS systeminterconnects the front and rear wheels, and it cannot affordthe additional stiffness to the warp mode. When the leftwheels’ inputs are opposite to the right wheels, the antirollbars can generate the antiroll forces exerted on the vehiclebody to prevent roll motion. When two diagonally opposedwheels’ inputs are opposite to the other two diagonallyopposed wheels as shown in Figure 10, they could twistvehicle chassis along 𝑥-axis.

The warp mode is excited by uneven ground and it canimpact the handling performance at low frequency, and thisroad excitation is defined by the following equation:

𝑧𝑔𝑖

(𝑡) = 𝐴𝑖sin (2𝜋𝑓𝑡) , (27)

where 𝐴𝐴

= −𝐴𝐵= −𝐴

𝐶= 𝐴

𝐷= 0.03m and 𝑓 = 1Hz.

Figures 11(a) and 11(b) show the comparisons of the frontleft and the rear right suspension deflections of the vehiclemodels with two suspension configurations (VKDS andVCS)under the warp mode excitation. The off-road vehicles arenormally equipped with the front and rear antiroll bars, andthe front antiroll bar generally contributes to the major rollstiffness of the vehicles. The results show that the front leftsuspension deflection of the VCS is smaller than the VKDSdue to the installation of the conventional front antiroll bar.When the road excitation is applied to the front wheels, thefront antiroll bar could push the full vehicle body rotatingdue to the vehicle major roll stiffness induced by the front


0 2 4 6 8 10 12−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8Ro

ll an

gle (

deg)

Time (s)

VCSVKDS

(a)

0 2 4 6 8 10 12−5

−4

−3

−2

−1

0

1

2

3

4

5

Roll

velo

city

(deg

/s)

Time (s)

VCSVKDS

(b)

Figure 12: (a) Sprung mass roll angle of the VCS and VKDS under the warp mode excitation. (b) Sprung mass roll velocity.

0 2 4 6 8 10 120

0.01

0.02

0.03

0.04

0.05

0.06

Time (s)

Disp

lace

men

t (m

)

Front leftFront right

Rear leftRear right

Figure 13: The road inputs of four wheels.

antiroll bar. The vehicle body motion resists the rear wheels’motion, which can increase the rear right wheel’s suspensiondeflection of the VCS. In this case, the rear right wheel’ssuspension deflection of the VKDS is smaller than the VCSdue to the oil free flow in hydraulic circuits. The resultsalso illustrate that the twisting moment of the vehicle bodyfor the VCS is larger than the VKDS in the warp mode.Thus, this means that the conventional antiroll bars heavilyput the vehicle loads on two diagonally opposed wheelswhere the coil springs are already compressed, but the othertwo diagonally opposed wheels have the tendency to lift upin order to further diminish the normal forces. Thus, the

conventional antiroll bars could worsen the vehicle road-holding performance under the warp motion.

Figures 12(a) and 12(b) illustrate the comparisons ofsprung mass roll angle and velocity responses of the VKDSandVCS.Vehicle body rollmodemotion is also excited underwarp excitation due to the stiffness difference of the frontand rear suspensions. The results show that the vehicle bodyroll angle and velocity are noticeably lower for the vehicleequippedwithKDS systemwhich decouples the roll andwarpmodes and has no negative effect on the warp mode. FromTable 1, it can be seen that the natural frequency 9.040Hz ofVKDS is smaller than 10.552Hz of VCS in warp mode. Thismeans that the warp stiffness of VKDS is softer than VCS.These results in time and frequency domain can validate theperformance of KDS system.

In order to further test the system performance, thetypical bump road profile 𝑧

𝑔is used to simulate the warp

inputs. The ground displacement for an isolated bump in anotherwise smooth road surface can be expressed as

𝑧𝑔(𝑡) =

{{{{{

{{{{{

{

0 𝑡 < 1

𝐻

2

[1 − cos(2𝜋V

0

𝑙

𝑡)] 1 ≤ 𝑡 ≤

𝑙

V0 𝑡 >

𝑙

V,

(28)

where𝐻 and 𝑙 are the height and the length of the bump andV is vehicle forward speed. We choose 𝐻 = 0.06m, 𝑙 = 2m,and V = 3.6 km/h in this paper. The road inputs to the frontand rear wheels have same peak amplitude with a time delayof (𝑎 + 𝑏)/V shown in Figure 13.

In this case, the front and rear wheels inputs with a timedelay are applied to excite the warp motion. Figure 14 showsthe comparisons of dynamic tyre forces for the four wheels(fl, fr, rl, and rr) of the VKDS and VCS. The dynamic tyreforce is proportional to the tyre deflection and is calculated


0 2 4 6 8 10 12−1500

−1000

−500

0

500

1000

1500

Time (s)

Dyn

amic

tyre

forc

e (N

)

VCSVKDS

(a)

0 2 4 6 8 10 12−1500

−1000

−500

0

500

1000

1500

Time (s)

Dyn

amic

tyre

forc

e (N

)

VCSVKDS

(b)

0 2 4 6 8 10 12−1500

−1000

−500

0

500

1000

1500

Time (s)

Dyn

amic

tyre

forc

e (N

)

VCSVKDS

(c)

0 2 4 6 8 10 12−1500

−1000

−500

0

500

1000

1500

Time (s)

Dyn

amic

tyre

forc

e (N

)

VCSVKDS

(d)

Figure 14: (a) Dynamic tyre force of the VCS and VKDS in front left wheel. (b) In front right wheel. (c) In rear left wheel. (d) In rear rightwheel.

as 𝑘𝑡𝑖(𝑧𝑔𝑖(𝑡) − 𝑧

𝑢𝑖(𝑡)) with the tyre stiffness 𝑘

𝑡𝑖. It can be

seen from Figure 14 that the KDS system generates lesstyre dynamic force than the conventional suspension system.As mentioned in Section 4, the KDS system is unable torestrict the suspensionmovement in the warpmode, which isbeneficial to the contact between the wheels and the ground,but the conventional antiroll bar can limit the suspensionmovement due to the connection between the left and rightwheels. It leads to the small fluctuation of the dynamic tyreforce for the VKDS. Hence, the KDS system can improveground holding performance.

5.2.3. Bounce Motion. The VKDS and VCS are evaluated forrelative ride dynamic analysis under random road condition

and different vehicle speeds. In order to reasonably evaluatevehicle ride comfort, the road surface spectrums that mainlyconsist of four random roads (smooth, medium-smooth,medium-rough, and rough roads) are applied in this set ofsimulations.The vertical displacement power spectral density(PSD) characteristics of the three road profiles at a forwardspeed of 90 km/h are shown in Figures 15(a), 15(b), 15(c), and15(d), which are defined as the road inputs of the four wheelsfor the dynamic analysis. Based on the displacement PSDcharacteristics of four wheels shown in Figure 15, these fourwheels are described as the front left wheel (fl), the front rightwheel (fr), the rear leftwheel (rl), and the rear right wheel (rl).Similar displacement PSD characteristics of three roads canbe found in the literature [31].


Frequency (Hz)

flfr

rlrr

Disp

lace

men

t PSD

(m2 /H

z)10−2

10−3

10−4

10−5

10−6

10−7

10−8

10−9

10−2 10−1 100 101

(a)

flfr

rlrr

Frequency (Hz)

Disp

lace

men

t PSD

(m2 /H

z)

10−2

10−1

10−3

10−4

10−5

10−6

10−7

10−8

10−9

10−2 10−1 100 101

(b)

Disp

lace

men

t PSD

(m2 /H

z)

10−1

10−2

10−3

10−4

10−5

10−6

10−7

10−8

Frequency (Hz)

flfr

rlrr

10−2 10−1 100 101

(c)

Disp

lace

men

t PSD

(m2 /H

z)

10−1

10−2

10−3

10−4

10−5

10−6

10−7

10−8

100

flfr

rlrr

Frequency (Hz)10−2 10−1 100 101

(d)

Figure 15: Vertical displacement temporal PSD characteristics of four road profiles at a speed of 90 km/h: (a) Smooth. (b) Medium-smooth.(c) Medium-rough. (d) Rough.

Furthermore, the root mean square (RMS) vertical accel-eration of vehicle body is used to quantify ride comfort. Thefrequency-weighted RMS acceleration levels of the humanseat interface are defined as the evaluation of human per-ception of ride comfort according to ISO-2631-1 [32], but theunweighted RMS vertical acceleration response of the sprungmass can also assess the vehicle ride vibration in significantmeasure. Figures 16(a), 16(b), 16(c), and 16(d) illustrate thecomparisons of RMS vertical acceleration of the sprung masswith VKDS and VCS under different vehicle speeds (50, 60,70, 80, and 90 km/h). The results show that the VKDS andVCS experience comparable RMS vertical acceleration of thesprungmass undermost of road surfaces and different vehiclespeeds. The comparable vertical ride responses of the VKDS

and VCS are also verified from similar oscillation frequencyof the bounce mode compared to the VCS (from 1.492Hz to1.493Hz) in Table 1. The results suggest that the KDS systemdoes not provide the additional stiffness of bounce mode andhas no negative effect on the vehicle ride comfort.

6. Discussions and Future Work

The study of this paper mainly focuses on the performanceanalysis of the newly proposed KDS system. According tothe system description, the KDS system is installed on theconventional vehicle through the disconnected antiroll bars,so the basic physical parameters of the VCS and VKDSare the same. Because of the sameness, the characteristics


50 60 70 80 900

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Vehicle speed (km/h)

VCSVKDS

Vert

ical

acce

lera

tion

(m/s

2)

(a)

50 60 70 80 900

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


VCSVKDS

Vert

ical

acce

lera

tion

(m/s

2)

(b)

VCSVKDS

50 60 70 80 900

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2


Vert

ical

acce

lera

tion

(m/s

2)

(c)

VCSVKDS

50 60 70 80 900

0.5

1

1.5

2

2.5

3

3.5

4


Vert

ical

acce

lera

tion

(m/s

2)

(d)Figure 16: Comparisons of the sprung mass vertical ride responses of the VCS and VKDS: (a) smooth; (b) medium-smooth; (c) medium-rough; (d) rough.

of the suspension system and tyre assembly are definedto be linear ones for simplifying the analysis of the KDSsystem. In addition, a linear suspension model may beapplicable in some small-to-moderate amplitude applications[24]. Under general conditions, many light vehicles may nothave the larger body roll angle and the longer suspensiondisplacements for the rollmotion. Inwarpmotion, the vehicleis driven on the rough road at the lower velocity and the lowerexaction frequency. Although the wheel and tyre assemblyhave the important role of attenuating high frequency vibra-tions, the dynamic characteristic of wheels may be assumedto be linear in lower frequency vibrations in order to analyzethe vertical tyre dynamic force. Furthermore, in this paper,the frequency-dependent factors in the mechanical system,such as tyre damping, are neglected in the model without

adding further complexity to the solution. Hrovat suggeststhat this full vehicle model generally predicts the accelerationresponse up to around 10 or 12Hz on the rough road, withdeclining accuracy above that [33]. The hydraulic impedancemethod may be applicable for any conceivable analyticalvehicle model in small amplitude applications, in whichnonlinearities in the fluid system may be negligible [25].Considering the above-mentioned reasons, the linear vehiclemodel is also introduced to analyze the modal characteristicsof the VCS and VKDS.

Nevertheless, it can be seen that the most obviouslimitation of the adopted analysis method is its linearityin this paper. In practical application, we often encountermany nonlinearities in the suspension hydraulic system andtyre assembly, such as conventional damper characteristics,


the nonlinearities in the fluid system, and the nonlinearcharacteristics of wheels. Hence, an extension of the proposedmethodology to a full nonlinear vehicle model needs to beconsidered in future works. This extension would be moreapplicable conceptually and would be essential for capturingdynamic response between all the vehicle rigid-body modes.Meanwhile, the experiment will be conducted to confirm theadvantage of the KDS system.Mikhlin has used the nonlinearnormalmodes (NNMs)method to obtain the bouncemotionproperty of a seven-DOF vehicle model with the front andrear springs of the nonlinear elastic characteristic [34]. Thus,nonlinear normal modes will be applied to analyze themodalproperty of the nonlinear vehicle model in future works,which can offer a solid theoretical and mathematical toolfor interpreting a wide class of nonlinear modal phenom-ena. The other limitation is that the pitch motion has notbeen deeply explored in this paper. It can be seen that anadaptive vibration control strategy has been proposed for thenonlinear uncertain suspension systems to stabilize both thebounce and pitch motions of the car in the literature [35, 36].Based on these theories, the active control strategy would beconsidered as future works in order to improve ride comfortand control the pitch motion of the passive VKDS.

7. Conclusions

In this paper, a novel KDS system is proposed to improvevehicle roll and warp motion performance without affectingride comfort. A multidisciplinary approach is applied todevelop a hydromechanical coupled model for the vehiclewith KDS system. The characteristics of the KDS-equippedvehicle about vehicle bounce, roll, and warp motions inthe frequency domain and time domain are presented. Thenatural frequencies and modal shapes of the system havebeen obtained in the frequency domain, and the dynamicresponses of the bounce, roll, and warp motions have beenanalyzed under various road excitations in time domain.

In order to illustrate the potential benefits of the pro-posed suspension, the comparison analysis of the vehiclesequippedwith the conventional suspension and the proposedKDS system has been carried out in this study. From thecomparison, the results demonstrated that (1) the proposedKDS system could decouple the roll and warp modes, (2) theKDS system could provide higher roll stiffness to enhancehandling performance, (3) theKDS systemcould reducewarpstiffness to improve wheels ground holding performance, and(4) the KDS system could hardly provide additional stiffnessto bounce and pitch modes so that the VKDS has similar ridecomfort as the VCS.

Parameters

(i) The Physical Parameters of a Seven-DOF Vehicle Model

𝑚𝑠: Vehicle sprung mass, 1558 kg

𝐼𝑥𝑥: Pitch moment inertia of the sprungmass, 2381 kg⋅m2

𝐼𝑦𝑦: Roll moment inertia of the sprungmass, 600 kg⋅m2

𝑎: Distance from the sprung mass CG tothe front axle, 1.25m

𝑏: Distance from the sprung mass CG tothe rear axle, 1.53m

𝑡𝑓: Half width of the front axle, 0.6m

𝑡𝑟: Half width of the rear axle, 0.5m

𝑚𝑢𝑓: Front unsprung masses, 96.4 kg

𝑚𝑢𝑟: Rear unsprung masses, 105.6 kg

𝑘𝑠𝑓: Spring rate of the front suspension,

36.6N/mm𝑘𝑠𝑟: Spring rate of the rear suspension,

42.7N/mm𝑐𝑠𝑓: Damping coefficient of the front suspen-

sion, 3000N∗m/s𝑐𝑠𝑟: Damping coefficient of the rear suspen-

sion, 2700N∗m/s𝑘𝑡𝑓/𝑘

𝑡𝑟: Vertical stiffness of tyres, 381N/mm

𝑘𝑎𝑟𝑏𝑓

: Front antiroll bar stiffness, 45N/mm𝑘𝑎𝑟𝑏𝑟

: Rear antiroll bar stiffness, 8N/mm.

(ii) Hydraulic Parameters

𝜌: Hydraulic fluid density, 870 kg/m3

𝐸: Bulk modulus, 1400MPa𝑉𝑝: Accumulator precharge gas volume, 0.65 L

𝑃𝑝: Accumulator precharge pressure, 20 bar

𝐷: Diameter of cylinder piston, 0.05m𝑑: Diameter of cylinder piston rod, 0.03m𝑃0: Mean system pressure, 40 bar.

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this article.

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Grant no. 51175157) and the project ofState Key Laboratory of Advanced Design and Manufacturefor Vehicle Body (Grant no. 71575005). The authors wish togratefully acknowledge the help of Professor Haiping Du inthe final language editing of this paper.

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