Transient Characteristics of a Hydraulically Interconnected Suspension System.pdf

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SAE paper 2007-01-0582. Copyright 2007 SAE International. This

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07AC-86

Transient Characteristics of a Hydraulically InterconnectedSuspension System

J. M. Jeyakumaran, W. A. Smith and N. Zhang

Mechatronics and Intelligent Systems, University of Technology, Sydney, Australia

Copyright 2007 SAE International

ABSTRACT

This paper describes vehicle dynamic models thatcapture the large amplitude transient characteristics of apassive Hydraulically Interconnected Suspension (HIS)

system. Accurate mathematical models are developed torepresent pressure-flow characteristics, fluid properties,damper valves, accumulators and nonlinear couplingbetween mechanical and fluid systems. The vehicle ismodeled as a lumped mass system with half- and full-car configurations. The transient performance isdemonstrated by numerical integration of the second-order nonlinear differential equations. The stiffness anddamping characteristics corresponding to vehiclebounce, roll and pitch motions are extracted from thetransient simulation. Simulation results clearlydemonstrate the superiority of the HIS system duringvehicle handling and stability by providing additional roll

stiffness and reduced articulation stiffness.

INTRODUCTION

The design of vehicle suspension systems involves acompromise between ride comfort and vehicle handling[1-3]. A softer suspension yields good ride performancewith the expense of poor stability and directional control.

Alternatively, a stiffer suspension enhances handlingand minimizes the occurrence of vehicle rollover, yet itprovides poor ride comfort. In recent decades, severalsuspension systems varying from passive to semi-activeto fully active have been proposed to break the

compromise between ride comfort and handling [4-10].With the rapid development of sensing technologies, therecent trends are largely towards active, semi-active andslow-active suspension systems, and the superiority ofthese systems has been established in luxury passengervehicles [11-13]. In principle, active and semi-activesuspension systems have the greatest capability toachieve optimum performance, however, the inherentdrawbacks are the weight, cost, uncertain reliability,increased power consumption and complexity. Becauseof challenges associated with active controls withminimum hardware, the passive suspensions that

enhance vehicle handling and stability are still of greatinterest for vehicle manufacturers.

Suspension researchers have paid attention to moreeffective passive systems that interconnect suspension

actions between left and right, front and rear wheel-stations to provide optimal stiffness in bounce, pitch androll motions. There are several interconnected vehiclesuspensions acquainted in the automotive market [1415]. The anti-roll bars are the simplest and moscommonly used interconnected system. Early examplesof interconnections also include Citroen interconnectedcoil springs and Moultons hydro-elastic suspensionsPublished results show that the interconnected systemsare very effective in reducing vehicle bounce, pitch oroll, as compared to independent suspension unitsHowever, only limited systematic investigations havebeen attempted on the dynamics of vehicles fitted withinterconnected systems.

In mid 1990s, Rakheja et al. [16] and Liu et al. [17, 18investigated a Hydraulically Interconnected Suspension(HIS) system to predict vehicle ride and handlingperformance. They developed fluid dynamics models toinclude the effects of fluid compressibility, valvecharacteristics and adiabatic process of confined gasand integrated them in a four-degree-of-freedom vehiclemodel. They demonstrated that enhanced roll stiffnessand damping characteristics of the HIS provide improvedroll stability and ride performance characteristicsHowever, complex fluid properties such as the nonlinearmechanical and fluid couplings, fluid inertia and wave

propagation effects are not included in theimathematical models. Recently, more novel HISsystems have been reported [19-22] and the superiorityof these systems was demonstrated experimentallyusing the US National Highway Traffic Safety

Administrations (NHTSA) Fishhook Maneuver [23, 24]Vehicle testing is used as the main tool for improving theHIS dynamic characteristics, however, this requiresmuch effort and time during the product developmenprocess.

Recently, Nong et al. [25] has developed an efficienhalf-car model for the vibration analysis of a vehiclefitted with a HIS system. The most obvious limitation o


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HIS system

mwl

Mv, Iv

mwr

ctl ktr

ksrsl

ktl ctr

that study was its linearity, i.e., the nonlinearitiesassociated with the cylinder volume coupling, theaccumulator air-spring effect, and damper valvecharacteristics were not included in the modelformulation. In linear models, the parameters of thehydraulic oil are held constant, and therefore thepressure- and temperature-dependence of the fluid bulkmodulus, viscosity and density are neglected. Thecoupling between mechanical and fluid systems, dampervalve characteristics and variations in fluid properties aresignificant nonlinearities of the HIS and ride evaluationrequires accurate modelling of these nonlinearities [26,27]. In addition, the hydraulic elements feature high fluidvelocities, pressures and accelerations during fasttransients, and therefore the analysis method requiresnumerical integration.

Figure 1: Schematic diagram of the HIS system in roll-plane

In this paper, a mathematical model that captures thelarge amplitude transient characteristics of a HIS ispresented. The transient responses of the vehicle,measured by its roll, bounce and pitch motions areinvestigated numerically using specified impulsive forcesand moments. Simulation of the fluidexchanges/dynamics is carried out from the set ofsecond-order differential equations describing the HISsystem. In the next section, the roll-plane models of themechanical and hydraulic system are presented.

Similarly, the full vehicle is modelled as a lumped masssystem with seven-degrees-of-freedom, i.e., the pitch,roll, and bounce motions of the vehicle and verticalmovements of the four wheel-stations. Because of thecommercial sensitivity of the full-car HIS system, adetailed derivation of the seven-degrees-of-freedommodel is not presented. The simulation results arepresented for specific impulsive loads applied at thecentre of gravity of the sprung mass. The effectivestiffness and damping coefficients corresponding tovehicle roll, bounce and pitch motions are extracted fromthe transient simulation.

MODEL FORMULATION

MECHANICAL SYSTEM MODEL

The schematic diagram of a half-car model fitted with aHIS system in roll-plane is shown in Figure 1. Theconventional shock absorbers and anti-roll barassemblies between sprung vehicle body and unsprungwheel-stations are replaced with a HIS system. The

vehicle is represented with a lumped four-degree-of-freedom system to include vehicle roll and bouncemotions and vertical movements of two wheel-stationsThe lateral and yaw motions of the vehicle are noincluded in the formulation. The stiffness and dampingof the tires linked to the wheel-stations are includedThe cylinder component is assumed as rigid in thehorizontal directions. Figure 2 shows the Free BodyDiagram (FBD) of the half-car model.

lb rb

lm

rm

tlF trF

slF hlF srFhrF

Gly Gry

, ,& &&wl wl wl y y y , ,& &&wr wr

y y y

, ,& &&vr vr vr

y y y, ,& &&vl vl vl y y y

, ,& &&v v vy y y

, M I , , & &&

Figure 2: Free body diagram of the half-car model

The equations of motion of the half-car mass and twowheel masses are derived from Newtons second law omotion. From the FBD of the left and right wheels, wehave:

( ) ( ) ( )

( )

= +

&& & &

& &

l wl tl Gl wl tl Gl wl sl wl vl

sl wl vl kl

m y k y y c y y k y y

c y y F

(1)

( ) ( ) (

( )

= +

&& & &

& &

r wr tr Gr wr tr Gr wr sr wr vr

sr wr vr kr

m y k y y c y y k y y

c y y F (2)

From the FBD of the vehicle body shown in Figure 2:

( ) ( ) ( )

( )

= + +

+ + +

&& & &

& &

v sl wl vl sl wl vl sr wr vr

sr wr vr kl kr

y k y y c y y k y y

c y y F F

(3)


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( ) ( ) ( )

( )

= +

+ +

&& & &

& &

l sl wl vl l sl wl vl r sr wr vr

r sr wr vr l kl r kr

I b k y y b c y y b k y y

b c y y b F b F

(4)

The equation of motion for the integrated model is:

[ ] { } [ ] { } [ ] { } { }+ + =&& &M Y C Y K Y F (5)

where { }Y ,{ }&

Y and{ }&&

Y are the displacement, velocity,acceleration vectors and are given by

{ } { } { }, , = , ,

Y Y Y

& &&

& &&& &&

& &&

& &&

wl wl wl

wr wr wr

v v v

y y y

y y

y y y

In Eq. (5), mass, damping and stiffness matrices, [M],[C] and [K] are 4x4 matrices with constant coefficientsand are given by

[ ]

0 0 0

0 0 0

0 0 0

0 0 0

=

l

r

m

m

M

I

M

[ ]

=

tl sl sl l sl

tr sr sr r sr

sl sr sl sr l sl r sr

2 2l sl r sr l sl r sr l sl r sr

c +c 0 -c b c

0 c +c -c -b c

-c -c c +c -b c +b c

b c -b c -b c +b c b c +b c

C

[ ]

=

tl sl sl l sl

tr sr sr r sr

sl sr sl sr l sl r sr

2 2

l sl r sr l sl r sr l sl r sr

k + k 0 -k b k

0 k + k -k -b k

-k -k k + k -b k +b k

b k -b k -b k +b k b k +b k

K

The force vector, {F}, is dependent on the hydraulic

system couplings and specified vertical rise/fall of theroad profile to the individual wheels. In particular,

{ } { } { }= +

= +

&

&

kltl Gl tl Gl

krtr Gr tr Gr

kl kr

l kl r kr

-Fc y + k y

-Fc y +k y

F + F0

-b F +b F0

G KF F F

(6)

It is assumed that the mechanical suspension is linearand tire consists of a linear spring-damper combination.

HYDRAULIC SYSTEM MODEL

The hydraulic schematic diagram of the half-car HISsystem is shown in Figure 3. This is a completelyinterconnected system and it replaces the conventionashock absorbers and anti-roll bar assemblies withinterconnected hydraulic cylinders, accumulatorsdamper valves and fluid pipelines. The cylinders arekinetically linked to the two wheel-stations and thechassis. The hydraulic elements are arranged such thavehicle bounce, pitch and roll motions operate with fluidflow through selected accumulators and dampers. Thevelocities and the forces applied to the pistons from thewheel-stations cause changes in the pressures and flowrates in the fluid circuits. The vertical velocities and theforces applied to the pistons at the four wheel-stationsare defined as the physical states (inputs) and thepressures and flow rates in the cylinders as the systemstates (outputs). The interconnections between thesesystem physical states are determined by applying thekinetic principles to the motions of pistons and cylindersand the motion of the fluids.

ptrptl

blp pbr

1 2 3 4

5 6

8

7

Double-actingcylinder

Nitrogen-filledaccumulators

Cylinderpiston/rod

Compressiondamper valve

Hydraulic line 1Hydraulic line 2

Rebounddamper valve

Roll dampervalves

Figure 3: Hydraulic Schematic diagram of the half-ca

HIS system

The hydraulic system shown in Figure 3 is complex andeach hydraulic element requires careful mathematicatreatment to represent the dynamic characteristics of theoverall system. Accurate models are developed torepresent the dynamics of pistons, damper valves andupper and lower cylinder chambers. The oil aeration is asignificant nonlinearity in models of hydraulic circuits andit is an important parameter for the dynamic analysisThe fluid bulk modulus is modified to account for the airin the oil, and flexibility of hoses and pipelines. Thedifferential equations that describe the pressure changesin terms of various system flow quantities can be writtenin the following generalized form

( )= &

pp Q

V(7)

where Q represents the net flow to the fluid volumeThe value (p) is a linear function of pressure. The flowrates passing through the bleed holes, damper valvesfrestrictions in fluid lines and accumulators are describedby standard turbulent orifice flow equations, i.e.,

0 1 2- )(= dQ c A p p (8)


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Flow rate

Pressure

difference

valve fully closedtransition

regionvalve open

li

Ai

pi1 pi2pi

Qi1 Qi2Qi

The pressure drop and flow resistance in the annularflows are dominated by the viscosity effect rather thanthe orifice flow, thus the pressure drop due to viscosityeffects is incorporated in fluid lines connecting hydraulicelements.

Damper Valve Characteristics

The damper valves play an important role in the HISsystem performance. The relationship between flow rate

and pressure drop of the dampers is highly nonlinear,and simulation of such element requires modelling theinteractions between fluid flow and cylinder-piston stroke[27, 28]. The damper valve provides high dampingassociated with bleed flows corresponding to lower strutvelocities and low damping associated with flowsthrough valves corresponding to high velocity. Thevalves are designed to provide greater resistance in onedirection of the piston moment than the other. A typicalvalve characteristics is defined by the relationshipbetween the volumetric flow rate and pressure differenceas shown in Figure 4. The valve characteristics aredivided into three regions: a valve closed region,

transient region and valve open region. During thesteady state conditions, the valve is fully closed and flowis allowed through small bleed holes. During fasttransients, the valve is fully open and the flowcharacteristics are determined from the valve dynamicsand stiffness properties of the valve springs. In thetransient region, the valve is partially open to providesmooth transition in flow characteristics and dampingforce.

Figure 4: A typical damper valve characteristics of theHIS system

The equation of motion of the damper valves andcylinder-pistons is primarily dictated by pressure forces,spring forces, damping forces and flow forces and canbe written in the following generalized form:

+ + + = &&p s d f s sF F F F m x (9) (9)

In the half-car configuration, the equations of motion offour cylinder-damper valves, two accumulator valvesand cylinder-pistons are derived and coupled with thefour-degree-of-freedom mechanical system model.

Pipelines

The vehicle layout and packaging constraints of the HISsystem requires relatively long flexible pipelines. Alumped parameter model is developed by dividing thefluid pipelines into several elements. The fluid pipe oconstant diameter is handled as one element. Figure 5shows a pipeline element i with a constant diameterThe mean pressure and mean flow in each element isassumed as an arithmetic mean of the pressure and flowrate at both ends of the pipe. The fluid flow is assumedas one-dimensional compressible flow to accommodatethe water hammer phenomenon.

Figure 5: A model of the pipeline element i with aconstant diameter

Assuming the pressure losses due to viscosity areproportional to the mean flow rate, and the magnitude olosses is the same as the inertia and pressure forcesthe momentum equation can be written as

1 2( - ) -

=&i i i i i i ii

lQ p p k l Q

A(10

where the viscous loss factor28 /=i ik A . The

continuity equation for the pipeline is written in terms othe mean pressure and flow difference between theends of the pipe element as

1 2

( )( - )

=& i i i

i

pp Q Q

V(11

where (p) is defined in Eq. (7). The mean pressure andmean flow rate of the pipe element is given as thearithmetic mean at both ends of the element, i.e.,

1 2 1 2

1 1( ) ; ( )

2 2

= + = +i i i i i ip p p Q Q Q (12a, b)

In general, the pipe ends are connected by a junctionpoint. If the net flows to the junction are denoted by Qi1inand Qi2, out then taking time derivative of Eq. (12a) andapplying continuity equation for the junctions, we have

11 2 1, 1

1

22 2,

2

( )2 ( )( - ) ( - )

( )( - )

=

+

ii i i in i

i i

ii i out

i

ppQ Q Q Q

V V

pQ Q

V

(13


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Eqs. (10), (12b) and (13) describes the fluid flow withunknown flow rates Qi, Qi1and Qi2. It is noted that theequation of motion of the next pipeline is function of Qi1and Qi2, and therefore a system of coupled equations isformulated for the pipeline.

Accumulator Model

As shown in Figure 3, the HIS system features gas-pressurized hydraulic accumulators to reduce shock

pressure loading due to system inputs. The accumulatorconsists of a pressure housing divided into twochambers by an elastometric diaphragm. One chamberis filled with gas and the other chamber filled withhydraulic fluid. The compressibility of the oil in theaccumulator is neglected, as the oil stiffness is muchgreater than that of the nitrogen contained in thebladder. A drop in the system pressure is accompaniedby flow from the accumulator and therefore theaccumulator needs to be sufficiently large to meet thepeak flow demands without appreciable drop in thesystem pressure. Accumulators also supply hightransient flow when required for short periods and they

determine the system pressure. The accumulator ismodeled by assuming an adiabatic process [29].

const. = =o opV p V (14)

The adiabatic gas law is used to model the accumulatorpressure as a function of gas volume at a prechargedpressure. Taking partial time derivative of Eq. (14)noting that the flow into the accumulator Q, is given

by /= Q V t, the pressure gradient of the

accumulator can be written as a nonlinear function ofpressurep

1/

=

&

o o

Q p pp

V p(15)

The numerical integration of Eq. (15) is carried out todetermine accumulator pressure.

Model simulation

The transient responses of the vehicle, measured by itsmotion in roll, bounce and pitch is investigatednumerically using specified inputs under typicaloperating conditions. The simulation is performed with

Advanced Continuous Simulation Language (ACSL), anAcslXtreme product distributed by AEgis technologies.The system is solved by Runge-Kutta-Fehlberg fourthorder integration with variable time steps. In general, thenonlinear system of equations of the hydraulic system isvery stiff and requires extensive numerical computations.Time histories of displacements, velocities andaccelerations are recorded for half- and full-carconfigurations. In addition, hydraulic system parameterssuch as cylinder chamber and accumulator pressures,pressure losses in the pipe lines, junctions and dampervalves, flow demands to cylinder chamber, gas volume,etc., are recorded at each communication interval.

RESULTS AND DISCUSSION

In order to demonstrate the transient characteristics othe HIS system, the developed half- and full-car modelsare simulated with various impulsive forces. In particularlarge amplitude roll, bounce and pitch impulse inputs ofvarious magnitudes with duration of 0.01 sec arespecified at the centre of gravity of the sprung mass(vehicle body). The time history of selected vertica

displacements and typical hydraulic system pressuresare presented. The effective stiffness and dampingcoefficients during roll, bounce and pitch motions arecalculated from the vehicle displacements. The validityof dynamic properties of the half-car model is assessedagainst those of the full-car model. The mechanical andhydraulic system parameters used for the half-casimulation are presented in Tables 1 and 2. Because othe commercial sensitivity the hydraulic schematic of thefull-car HIS and system parameters of certain hydraulicelements such as cylinders, pipelines and dampevalves are not provided.

Table 1: Mechanical system parameters

Symbol Value Units

700 kg

jm 50 kg

I 250 kg/m2

jb 0.75 m

sjk 20 kN/m

tjk 200 kN/m

tjc 3000 N.s/m

kjc 100 N.s/m

Table 2: Hydraulic system parameters

Symbol Value Units

870 kg/m3 0.05 N.s/m2 1400 MPa

d 7.07 mm

pt 2.0 mm

E 210 GPa

pV 200 ml

pP 0.5 MPa

1.45 -

Figures 6 and 7 show the simulation results of the half-car model when an impulsive roll moment of 50 kNm is


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applied at the centre of gravity of the sprung mass. Thisroll moment simulates the dynamic characteristics of theHIS system during high amplitude roll motion at high rollrates. In particular, the impulsive moment simulates aninitial maximum role rate of 105 deg/sec. This roll rate isnearly twice as large as the maximum value of about 52deg/sec reported during Fishhook Maneuver vehiclerollover testing specified by NHTSA [22-24]. Figure 6shows the time histories of left wheel displacement (thin-solid line), right wheel displacement (dot line) andvehicle roll angular displacement (thick-solid line) whenthe impulsive moment is applied at t=0.1 sec for theduration of only 0.01 sec. Results show a heavilydamped roll response and wheel and roll displacementsdamp out in about 1.5 sec. The maximum rolldisplacement is about 0.090 rad (~5.2

deg), occurring at

t=0.185 sec.

Figure 6: Wheel and roll displacements of the half-carmodel due to impulse roll moment of 50 kNm.

As described in Section 2, the characteristics of cylinderand accumulator valves are highly nonlinear andtherefore the dynamic properties such as roll stiffnessand damping coefficients are highly depend on themagnitude of the roll displacement. In addition, thedamper valves have asymmetric damping properties andprovide flow resistance in only one direction. When theroll angle is very small, the force due to the pressuredifferential across the spool valves is not large enoughto overcome the spring preload to open the dampervalves and therefore the roll dynamics are governed byfluid flow through bleed holes of the valves. When the

roll angle is high, the dynamic response is dominated byrapid opening and closing of damper valvesaccompanied by large fluid motions, i.e., the operatingregion is in the second ramp of the damper valvecharacteristics shown in Figure 4.

The spool displacement profiles of the valves showwhen a large impulsive roll moment of 50 kNm isapplied, the cylinder and accumulator valves are openedfor open for about 1.0 sec. The best measure of effectiveroll stiffness is calculated from the roll response aftert=1.0 sec. The analysis of time history of rolldisplacement shows that the effective roll stiffness and

roll damping are about 240 kNm/rad and 30%respectively. This dynamic roll stiffness is significantlyhigher that the static roll stiffness of 124 kNm/radResults presented in Figure 6 also show the maximumwheel displacement of 29 mm occurring at t=0.18 sec.

Figure 7: Typical pressure traces of the half-car modedue to impulse roll moment of 50 kNm.

Figure 7 shows typical hydraulic pressure profiles of thehalf-car model when an impulsive roll moment of 50kNm is applied. Only pressure profiles corresponding totop-left cylinder (thin-solid line), top-right cylinder (thicksolid line) and accumulator-2 (dot line) are presentedThe impulsive moment simulates an initial role rate o105 deg/sec and therefore the initial pressure at top-righcylinder is significantly higher than the mean operatingpressure. In particular, the maximum top-right cylindepressure is about 13.0 MPa, occurring at t=0.16 secThe maximum top-left cylinder pressure is about 6.0

MPa, occurring at t=0.36 sec. As shown in Figure 3, theaccumulator-2 is located in the hydraulic line-2 andlinked to the top-right cylinder and therefore accumulatopressure-2 closely resembles that of the top-righcylinder pressure. The initial pressure increase in thetop-right cylinder is accompanied by significant flow tothe accumulator-2 of the HIS system. The cylinder andaccumulator pressures continue to oscillate about themean pressure of 2.0 MPa until about t=1.5 sec.

Figure 8: Wheel and roll displacements of the full-carmodel due to impulse roll moment of 100 kNm.


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Simulation is also performed with the full-carconfiguration and the corresponding results for animpulsive roll moment of 100 kNm are shown in Figures8 and 9. Figure 8 shows the time histories of left wheeldisplacement (thin-solid line), right wheel displacement(dot line) and vehicle roll angular displacement (thick-solid line) when the impulsive moment is applied at t=0.1sec for the duration of 0.01 sec. For clarity, only thefront wheel displacements of the full-car model arepresented in Figure 8. However, the comparison of thefront- and back-wheel displacements (results notpresented here) indicates very similar effective stiffnessand damping characteristics. It is noted that theaccumulators of the HIS system are located near thefront wheels of the vehicle and therefore the stiffness ofthe back wheels is slightly higher than that of the frontwheels.

The analysis of roll displacement shows that theeffective roll stiffness and roll damping are about 308kNm/rad and 49%, respectively. Results also show thatthe maximum roll displacement is about 0.083 rad(~4.8

o), occurring at t=0.185 sec. The pressure profiles

of top-left cylinder (thin-solid line), top-right cylinder(thick-solid line) and accumulator-2 (dot line) are shownin Figure 9. The maximum top-right and top-left cylinderpressures are 10.9 MPa and 4.6 MPa, respectively.

Figure 9: Typical pressure traces of the full-car modeldue to impulse roll moment of 100 kNm.

The direct comparison of roll displacement of the half-and full-car models is presented in Figure 10. Resultsshow the time histories of roll displacement are similar

but the half-car model overestimates the vehicle rolldisplacement. In particular, the initial maximum rolldisplacement predicted by the half-car model is 0.090rad. This is about 8% higher than the correspondingvalue of 0.083 rad predicted by the full-car model. Thewheel and roll displacements of the full-car model exhibitfaster decay rates and the half-car modelunderestimates roll stiffness and roll dampingcoefficients. In particular, the roll stiffness and rolldamping predicted by the half-car model are 24% and38% lower than the corresponding values predicted bythe full-car model, respectively. This is also evident fromthe cylinder pressure profiles of the half- and full-car

models compared in Figure 11. The maximum top-righcylinder pressure predicted by the half-car model isabout 13.0 MPa compared to 10.9 MPa predicted by thefull-car model. The right cylinder pressure oscillationspredicted by the full-car model damp out very quickly. Ingeneral, consistent trends are noted when comparingthe roll response of the half- and full-car models of theHIS system, i.e., the half-car model underestimates theeffective stiffness and damping ratio during largeamplitude roll motion.

-0.08

-0.04

0.00

0.04

0.08

0.0 0.5 1.0 1.

Time (sec.)

Rollangulardispl.(rad.)

full-car model

half-car model

Figure 10: Comparison of roll displacements of full-andhalf-car models due to impulse roll moment.

0.0E+00

5.0E+06

1.0E+07

0.0 0.5 1.0 1.

Time (sec)

Pressure(MPa)

full-car model (left cylinder)

full-car model (right cylinder)

half-car model (left cylinder)

half-car model (right cylinder)

Figure 11: Comparison of pressure profiles of full-andhalf-car models due to impulse roll moment.

To investigate the vehicle bounce response of the HIS

system, the simulation is performed for an impulsiveforce of 100 kN applied at the centre of gravity of thesprung mass for a duration of 0.01 sec. Because of theassumed left-right symmetry, the time history of all fouwheel displacements are of the same magnitude and theroll and pitch displacements are zero. Figure 12 showsthe time history of wheel displacement (dot line) andvehicle bounce displacement (thick-solid line). Thedisplacement profiles show a highly damped bounceresponse with an equivalent damping ratio of abou54%. The maximum bounce is about 79.2 mmoccurring at t=0.30 sec. Analysis shows that theeffective stiffness of the bounce motion is 73.5 kN/m


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Given that the actual stiffness of the four suspensionsprings is 80 kN/m (see Table 1, stiffness of eachsuspension spring is 20 kN/m), the HIS system of thefull-car model softens the bounce motion by about 8%.

Figure 12: Wheel and bounce displacements of the full-car model due to impulse bounce of 100 kN.

To illustrate the effects of hydraulic pressures on vehiclebounce response, typical pressures of bottom cylinder(thin-solid line), top cylinder (thick-solid line) andaccumulator-1 (dot line) are shown in Figure 13. Due tothe initial impulse bounce rate of about 700 mm/sec, theinitial pressure in the bottom cylinders overshoots toabout 5.0 MPa immediately after the impulsive force isapplied at t=0.1 sec. With the absence of the rollmotion, the pressure oscillations in the cylinders andaccumulators are significantly lower than the valuespresented for the vehicle roll motion (see Figure 9). Inparticular, the maximum accumulator pressure

registered is only about 2.3 MPa.

Figure 13: Typical cylinder and accumulator pressuresof the full-car model due to impulse bounceforce of 100 kN.

The direct comparison of bounce displacement of thehalf- and full-car models is presented in Figure 14. Thetime history of bounce and wheel displacements arealmost indistinguishable and the analysis shows that the

effective stiffness and damping coefficient of the vehiclebounce from the half- and full-car models are within 1%The simplified half-car model therefore captures theessential dynamics of the bounce motion.

-0.050

-0.025

0.000

0.025

0.050

0.075

0.0 0.5 1.0 1.5 2.0 2.5

Time (sec.)

Displacement(

m)

Wheel displ - full-car

vehicle bounce - full-car

Wheel displ - half-car

vehicle bounce - half-car

Figure 14: Comparison of wheel and vehicle bouncedisplacements of full- and half-car modelsdue to impulse bounce force.

The vehicle pitch motion of the HIS system is evaluatedby applying an impulsive pitch moment of 100 kNm athe centre of gravity of the sprung mass for a duration o0.01 sec. Because of the assumed left-right symmetryof the HIS system, the time history of roll displacement iszero. Figure 15 shows the back-wheel displacemen(dot line), front wheel displacement (thick-solid line) andvehicle pitch angular displacement (thin-solid line)Results also show that the maximum pitch displacemenis about 0.042 rad (~2.4

o), occurring at t=0.26 sec. The

analysis of pitch displacement shows that the initiaeffective pitch stiffness and pitch damping are about 160

kNm/rad and 60%, respectively.

Figure 15: Wheel displacements and pitch anguladisplacements of the full-car model due toimpulse pitch moment of 100 kNm.

To illustrate the effects of hydraulic pressures on vehiclepitch motion, typical pressure traces of bottom cylinde(thin-solid line), top cylinder (thick-solid line) andaccumulator-1 (dot line) are shown in Figure 16. Fo


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clarity, only the front cylinder pressures are presented.The initial pressure oscillations in the top and bottomcylinders are due to the large impulse pitch motionapplied at t=0.1sec. In general, the pressure oscillationsin the cylinders are significantly lower than the valuespresented for the vehicle roll motion (see Figure 9). Inthe absence of roll motion, there is no flow to theaccumulators and the accumulator pressure remainsconstant at the mean pressure of 2.0 MPa.

Figure 16: Typical cylinder and accumulator pressuresof the full-car model due impulse pitchmoment of 100 kNm.

CONCLUSION

A vehicle dynamic model that captures the transientcharacteristics of a HIS system is presented. Thevehicle is modelled as a lumped mass system with ahalf-car model with four-degrees-of-freedom and a full-car model with seven-degrees-of-freedom. The lumped

mass full-car model includes the roll, bounce and pitchmotions of the vehicle and vertical movements of thefour wheel-stations. Mathematical models aredeveloped to represent the nonlinearities associatedwith pressure-flow characteristics, fluid properties,damper valves, accumulators and coupling betweenmechanical and fluid systems. The transientperformance is demonstrated by numerical integration ofthe second-order nonlinear differential equations,utilizing the Runge-Kutta fourth order integrationscheme.

The HIS system performance is evaluated from the timehistory of displacements when the sprung mass issubjected to impulsive forces and moments. Simulationresults show that a simplified half-car model capturesthe essential dynamics of the vehicle bounce motion,however, it overestimates the displacements andhydraulic pressures during vehicle roll motion. The half-car model underestimates the effective stiffness anddamping ratio of the HIS system during large amplituderoll motion and the coupling effects of the HIS systemare best represented by the full-car model. The HISsystem provides improved roll stability by significantlyenhancing roll stiffness without altering bounce stiffnessand damping properties. The developed nonlinear

models provide insight into various hydraulic systemparameters and nonlinearities, and reveal directions fodesign improvements for vehicles fitted with similar HISsystems.

ACKNOWLEDGMENTS

Financial support for this research is provided jointly bythe Australian Research Council (ARC LP0562440) and

University of Technology, Sydney.

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CONTACT

Jeku M. Jeyakumaran

Mechatronics and Intelligent SystemsFaculty Of EngineeringUniversity of Technology, Sydney

PO Box 123, BroadwayNSW, Australia

Ph: (02) 9514 1613Fax: (02) 9514 2655email:[email protected]

NOMENCLATURE

iA

0A

jb

cd

tjc

sjc

FpFs

Fd

Ff

Fkj

Fhj

I

ik

sjk

tjk

il

jm

sm

p

&p

0p Q

&iQ

iQ

,i inQ

,i outQ

V

0V

xs , &&sx

, &Gj Gjy y

Cross sectional area of pipe element i

Effective flow area of the orifice

Lateral distance from the vehicle body

c.g. to suspension strut

Fluid discharge coefficient

Damping coefficient of tire

Damping coefficient of piston strut

Pressure force on the piston/spoolSpring force on the piston/spool

Damping force on the piston/spool

Flow force on the piston/spool

Force between vehicle body and road

wheel due suspension spring

Force between vehicle body and road

wheel due to HIS system

Sprung mass moment of inertia about roll

plane

Viscous loss factor pipe element i

Stiffness of suspension springs

Stiffness of the tire

Length of the pipe element i

Sprung mass of the vehicle body

Unsprung mass of the wheel-station

Mass of piston/spool

Pressure

Time derivative of pressure

Precharge gas pressure in accumulator

Flow rate

Time derivative of the flow rate

Mean flow through pipe element i

Input flow to the pipe junction i

Output flow from the pipe junction i

Fluid volume

Precharge gas volume in accumulator

Displacement/acceleration of the

piston/spool

Vertical displacement/velocity of tire-


12/12

, ,v v vy y y& &&

, ,vj vj vjy y y& &&

, ,wj wj wjy y y& &&

Greek letter

, , & &&

Subscripts

i

j

1

2

i1, i2

ground contact patch

Vertical displacement / velocity /

acceleration of vehicle body c.g.


acceleration of one side of vehicle body


acceleration of road wheel-station

Density of the oil

Viscosity of the oil

Bulk modulus of the oil

Ratio of specific heat for nitrogen

Angular displacement / velocity /

acceleration of vehicle body-roll plane

Pipe element iwith constant diameter

Left or right side, i.e., l or r

upstream of a flow restrictor

downstream of a flow restrictor

End 1 and 2 of the pipe element i

Transient Characteristics of a Hydraulically Interconnected Suspension System.pdf

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