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400 Commonwealth Drive, Warrendale, PA 15096-0001 U.S.A. Tel: (724) 776-4841 Fax: (724) 776-5760 Web: www.sae.or
2005-01-0398
Influence of Chassis Characteristics on
Sustained Roll, Heave and Yaw Oscillations
in Dynamic Rollover Testing
Aleksander Hac
Delphi Corporation
Reprinted From: Vehicle Dynamics and Simulation 2005(SP-1916)
2005 SAE World CongressDetroit, MichiganApril 11-14, 2005
SAE TECHNICAL
PAPER SERIES
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Influence of Chassis Characteristics on Sustained Roll, Heaveand Yaw Oscillations in Dynamic Rollover Testing
Aleksander HacDelphi Corporation
Copyright 2005 SAE International
ABSTRACT
In dynamic rollover tests many vehicles experiencesustained body roll oscillations during a portion of roadedge recovery maneuver, in which constant steeringangle is maintained. In this paper, qualitative explanationof this phenomenon is given and it is analyzed using
simplified models. It is found that the primary root causeof these oscillations is coupling occurring between thevehicle roll, heave and subsequently yaw modesresulting from suspension jacking forces. These forcescause vertical (heave) motions of vehicle body, which inturn affect tire normal and subsequently lateral forces,influencing yaw response of vehicle. As a result,sustained roll, heave and yaw oscillations occur duringessentially a steady-state portion of maneuver. Analysisand simulations are used to assess the influence ofseveral chassis characteristics on the self-excitedoscillations. The results provide important insights, whichmay influence suspension design.
INTRODUCTION
Dynamic rollover testing has received a lot of attention inthe past several years. In an effort to build safer vehiclesand to reduce the number of fatalities resulting fromrollover accidents, vehicle manufacturers and supplierspursue design changes, which could lead to improvedrollover resistance. These goals cannot be achievedwithout adequate testing procedures. Concurrently, theneed to evaluate and rank rollover resistance of vehiclesand pass this information to consumers led to recent
introduction by National Highway Traffic SafetyAdministration (NHTSA) of a dynamic rollover test aspart of a New Car Assessment Program (NCAP). In thistest, vehicle driven on a level, dry surface is steered by arobot at high rates and large amplitudes of steering anglefirst in one direction, then the other in order to emulatedriver steering input in an emergency road edge recoverymaneuver. The vehicle fails the test when it experiencesa Two Wheel Lift Off (TWLO) of 2 inches (about 5 cm) ormore. After the second turn, the steering angle ismaintained for about four seconds. During this portion ofmaneuver, many vehicles experience sustained, andsometimes even increasing, body roll oscillations. In
some instances, these oscillations are so severe thatTWLO occurs not in transient, but during steady statepart of the maneuver. Concurrently, generally weakeoscillations occur in lateral acceleration and yaw rateresponses of vehicle. Similar behavior is observed insevere J-turn maneuvers. To the authors besknowledge no satisfactory explanation of this
phenomenon has been offered.
In this paper an explanation of these oscillations is givenand a quantitative description is presented usingsimplified models of vehicle dynamics. It is found that theprimary root cause of these sustained vibrations iscoupling occurring between the vehicle roll, heave andsubsequently yaw modes resulting from suspension
jacking forces. These forces cause significant vertica(heave) motions of vehicle body during heavy corneringThese vertical motions result in fluctuations of tire normaforces. Since in the steady-state portion of the dynamicrollover test the vehicle is at the limit of adhesion, where
lateral tire forces are approximately proportional tonormal forces, changes in normal forces exert direcinfluence on the lateral forces, and subsequently yawresponse of vehicle. As a result, sustained roll, heaveand yaw oscillations occur during essentially a steady-state portion of maneuver. These vibrations are analyzedin this paper in more detail.
The paper is organized as follows. First, the studiedphenomenon is described quantitatively and illustratedusing vehicle test data and then reproduced insimulations using a validated vehicle model. It is shownthat in addition to roll oscillations, significant body heaveoscillations occur. The effect of several parameters onamplitude of oscillations is then examined. It is found thareducing the jacking effects in suspension can nearlyeliminate persistent body roll and heave oscillationsUsing simplified vehicle models, local stability of thevehicle system in the neighborhood of the operating poinis analyzed based on system equations linearized abouthat point. Influence of several chassis characteristics onthe self-excited oscillations is examined. The resultsprovide important insights, which may influencesuspension design.
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QUALITATIVE DESCRIPTION
NHTSA dynamic rollover test, known as a road edgerecovery maneuver or fishhook test is designed tomeasure vehicle propensity to tip up or rollover during anemergency maneuver. The result of the test, along withthe Static Stability Factor (SSF), are combined to yieldthe rollover stability star ranking of the vehicle, althoughthe contribution of the dynamic test to the overall rankingis relatively small. In the test, vehicles steered by a robotare swerved in rapid succession in two directions onsmooth, level and dry pavement. Following the secondturn, the steering angle is kept constant for about fourseconds. Details of this test, as well as results obtainedfor several vehicles have been published by NHTSA(Forkenbrock et al., 2002). One unexpected result ofthese tests was that, rather than approaching a steady-state condition during the portion of the test with constantsteering angle, most vehicles experienced quite severe,sustained body roll oscillations. These were observedregardless of whether the ESC systems were availableor enabled on tested vehicles or not. In some cases, theroll oscillations were so severe that the two-wheel lift off
(TWLO), as defined by NHTSA, was achieved during thesteady-state portion of the maneuver. In rare instancesthe roll oscillations were increasing.
In order to study this phenomenon, vehicle tests wereperformed using a large pick up truck. The test vehicle isshown in Figure 1.
Figure 1. Test Vehicle
The load rack above the truck bed was used to vary theheight of vehicle center of mass. The weight of the
vehicle equipped with all safety and test equipmentapproached the gross weight. A representative test resultin a fishhook maneuver performed at an entry speed of40 mph (about 18 m/s) is shown in Figure 2. The sameentry speed is used throughout this section.
The measured steering angle, lateral acceleration atvehicle center of gravity, yaw rate and roll angle areshown. It is seen that vehicle experiences significantbody roll oscillations and some oscillations in lateralacceleration and yaw rate responses during the portionof the maneuver when the steering angle is heldconstant. The results of simulation for the same
maneuver obtained using a validated vehicle model arealso shown. The results of simulation match the test dataquite well; more importantly, the model accuratelypredicts the roll angle oscillations. It should beemphasized that in the simulations the road wasassumed to be perfectly smooth and the surfacecoefficient of adhesion was uniform. Thus the resultingoscillations cannot be explained by variations in roadsurface, or by road roughness. It is also noted that manyvehicles tested by NHTSA (Forkenbrock et al., 2002)
experienced much more severe body roll oscillationsthan those observed in Figure 2.
0 2 4 6 8-400
-200
0
200
400
HandWheelAngle[deg]
0 2 4 6-10
-5
0
5
10
Meas.
Lat.Accel.[m/s2]
measured
simulated
0 2 4 6 8-40
-20
0
20
40
Time [s]
YawRate[deg/s]
0 2 4 6-10
-5
0
5
10
Time [s]
RollAngle[deg]
Figure 2. Measured and Simulated responses of Vehiclein a Fishhook Test Performed at 40 mph
In order to obtain further insight, vehicle roll angle andvertical position of the body in the same maneuver areshown in Figure 3.
0 1 2 3 4 5 6 7-10
-5
0
5
10
RollAngle[deg]
measuredsimulated
0 1 2 3 4 5 6 7-8
-6
-4
-2
0
2
Time [s]
BodyHeave[cm]
Figure 3. Roll and Heave Motions Of Vehicle BodyDuring Fishhook Maneuver
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According to SAE sign convention, positive verticalposition corresponds to the body being down relative tothe static equilibrium. In the test data, the body roll angleand the vertical position were determined from threeheight sensors placed in three corners of the body. Thus,the measured vertical position at the body center ofgravity is a derived value and may be inaccurate; hencethe discrepancy between the traces of body heaveobtained from test data and simulation. During a portionof the maneuver performed at a constant steering angle,
the vehicle body is subjected to approximately periodicoscillations in both roll and heave of approximately thesame frequency of about 1.6 Hz. Both types ofoscillations appear to be coupled; furthermore, verticaloscillations appear to be coupled with the lateralacceleration (not shown in Figure 3), as expected. Apossible explanation is that the vertical body motionscause changes in vertical tire forces, which inducefluctuations in lateral forces that in turn affect lateralacceleration and subsequently sustain body rolloscillations.
In order to investigate the effects of vehicle suspension
parameters on the studied phenomenon, several of themwere varied in simulations. To illustrate, the results forvehicle with nominal damping, two thirds of nominaldamping and one and a half of the nominal damping atall four corners are shown in Figure 4.
0 2 4 6 8-10
-5
0
5
10
Meas.
Lat.Accel.[m/s2]
2/3 nominalnominal
1.5 nominal
0 2 4 6 8-40
-20
0
20
40
YawRate[deg/s]
0 2 4 6 8-10
-5
0
5
10
RollAngle[deg]
Time [s]
0 2 4 6 8-15
-10
-5
0
5
Time [s]
BodyHeave[cm]
Figure 4. Effect of Suspension Damping on VehicleResponse in Fishhook Maneuver
As expected, reducing the suspension dampingincreases the amplitude and time duration of oscillations,while increasing it has an opposite effect. Simulationsperformed for wider range of damping variations showedsimilar trend, with only minimal and quickly dampedoscillations when damping is increased to double thenominal value. However, for the vehicle considered here,oscillations cannot be entirely eliminated by variation ofsuspension damping within the range, which is normally
judged acceptable for good body isolation (ride)
performance. Among other parameters considered, theheight of roll centers and, to a lesser extent, nonlinearityin suspension stiffness characteristics proved to have thelargest influence on vehicle behavior. Within the typicarange of values, reducing the height of roll centers andmaking the stiffness characteristic more linear bothreduced the amplitude of body oscillations, although theyincreased the steady-state roll angle, about which theoscillations occur. An illustrative example is shown inFigure 5. Here the results for nominal vehicle, vehicle
with linear suspension stiffness characteristic (versusprogressive characteristic for nominal suspension), andvehicle with nominal stiffness characteristic and rolcenters at the road level are shown.
0 2 4 6 8-10
-5
0
5
10
Meas.
Lat.Accel.[m/s2]
nominal
linearzero rollc
0 2 4 6-40
-20
0
20
40
YawRate[deg/s]
0 2 4 6 8-20
-10
0
10
20
RollAngle[deg]
Time [s]
0 2 4 6-10
-5
0
5
Time [s]
BodyHeave[cm]
Figure 5. Effects of Roll Centers Heights and Nonlinearity
in Suspension Stiffness Characteristics on VehicleBehavior in Fishhook
Reducing the heights of roll centers increases the peakand steady-state values of roll angle, but it alsosignificantly reduces amplitude of body heave vibrationsand roll oscillations at steady-state. The effect ofsuspension nonlinearity is subtler, but making thesuspension stiffness characteristic linear also reducesthe roll and heave oscillations and makes them appeabetter damped. This is not surprising, since reducingsuspension stiffness at the operating point, while keepingdamping unchanged, increases the damping ratio
Reducing suspension nonlinearity and the heights of rolcenters are both factors contributing to reduction in thesuspension jacking forces. These are the unbalancedvertical components of the suspension forces, whichoccur during cornering on a smooth road surface andtend to lift the vehicle body above the static equilibriumReducing the height of the roll centers reduces thevertical components of the resultant forces in the rigidsuspension arms during cornering. Reducing nonlinearityin the suspension stiffness characteristic makes thecompression of the outside suspension closer to theextension of the inside suspension, thus reducing the
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lifting of body center. This is explained in more detail inthe next section.
In maneuvers performed on smooth roads, jackingforces in the suspension constitute a primary couplingmechanism between the body roll and heave modes.Nonlinearities in suspension damping can also contributean unbalanced vertical force during transient maneuvers,but this effect is generally smaller and occurs only duringtransients.
The effects of other parameters of the vehicle wereconsidered in simulations. For example, the influence offront steer compliance is illustrated in Figure 6.Performance of a vehicle with nominal steer complianceis compared to one with one half and double the nominalcompliance. Within a reasonable range of values theeffect of steer compliance is small. However, even whenthe front steer compliance is twice the nominal value, thebody roll and heave oscillation do not decrease, eventhough the steady-state values of lateral acceleration androll angle are reduced. This is somewhat surprising,since the oscillations generally decrease as severity of
maneuver is reduced. Additional simulationsdemonstrated that in maneuvers performed at higherspeeds, increasing front steer compliance may even leadto increased body roll oscillations in the steady-stateportion of maneuver. This effect is further analyzed in thenext section.
0 2 4 6 8-10
-5
0
5
10
Meas.
Lat.Accel.[m/s2]
1/2 st.com.nominal
2 st.com.
0 2 4 6 8-40
-20
0
20
40
YawRate[deg/s]
0 2 4 6 8-10
-5
0
5
10
RollAngle[deg]
Time [s]
0 2 4 6 8-8
-6
-4
-2
0
2
BodyHeave[cm]
Time [s]
Figure 6. The Effect of Front Steer Compliance onVehicle Response in Fishhook Maneuver
It is concluded that the most likely primary cause ofsustained body roll oscillations in steady-state portion ofthe maneuver is coupling between the vehicle roll, heaveand subsequently yaw modes resulting from suspension
jacking forces. These forces cause vertical (heave)motions of vehicle body, which in turn affect tire normaland subsequently lateral forces, influencing yawresponse of the vehicle. As a result, sustained roll, heaveand yaw oscillations occur during essentially a steady-
state portion of maneuver. In order to verify thisconjecture, provide further insights, and establishquantitative relationships between vehicle parametersand the investigated phenomenon, stability analysis ovehicle during steady-state limit cornering wasconducted. It is described in the next section.
ANALYSIS
In this section a simplified analytical model is developedwhich describes the coupling between the roll and heavebody modes and subsequently the yaw plane motionThe model needs to be simple enough to facilitatestudies of the system stability and to describe explicitlythe effects of important design parameters on vehiclebody roll oscillations in fishhook tests. At the same timethe model should capture the important aspects ovehicle behavior in a steady-state turn at the limit lateraacceleration, as the vehicle experiences in the secondphase of fishhook maneuver. In particular, the effects ofsuspension jacking forces, which couple the roll andheave modes must be included in the model.
As described earlier, during steady-state corneringjacking forces arise primarily due to two sources: verticacomponents of forces transmitted by suspension rigidlinks and nonlinearities in suspension stiffnesscharacteristics. The first effect is illustrated in Figure 7.
R
hroll
hrollc
C
F
Fz
tw/2
Fy
Figure 7. Jacking Force Exerted by Suspension Links
Lateral forces generated during cornering maneuvers aretransmitted between the body and the wheels throughrelatively rigid suspension links. In general, these
members are not parallel to the ground; therefore thereaction forces in these elements have verticacomponents, which usually do not cancel out duringcornering, resulting in a vertical net force, which pushesthe body up. It is known (Gillespie, 1993; Reimpell andStoll, 1996) that forces transmitted between the vehiclebody and a wheel through lateral arms are dynamicallyequivalent to a single force, which reacts along the linefrom the tire contact patch to the roll center osuspension. The roll center is by definition the point inthe transverse vertical plane at which lateral forcesapplied to the sprung mass do not produce suspensionroll. This is illustrated in Figure 7 for a double A arm
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suspension. If the tire lateral force is Fy, then the jackingforce, Fz, is
tanyz FF = (1)
where is the inclination angle of the line connecting tirecontact patch with the roll center to the horizontal. Notethat in SAE sign convention the force lifting the body upis negative.
The second jacking effect is due to non-linear springcharacteristics. Suspension stiffness characteristics areusually progressive; that is, stiffness increases withsuspension deflection in order to maintain good rideproperties with a full load and to minimize bottoming ofsuspension. During cornering maneuvers, a progressivecharacteristic of suspension permits smaller deflection incompression of the outside suspension than deflection inextension of the inside suspension. As a result, height ofvehicle center of gravity increases. This effect can beparticularly significant for fully loaded vehicle, for whichthe suspension can become fully compressed during
heavy cornering, thus entering the region of highnonlinearity. This effect is illustrated in Figure 8.
Fz
zO
Fz
Fz
A
zextzcomp
Figure 8. Jacking Effect Resulting from Non-linearity inSuspension Stiffness Characteristic
The nominal operating point, corresponding to staticequilibrium of lightly loaded vehicle is the point O, andthe operating point with full load is A. In Figure 8suspension extension is positive and compressionnegative. During cornering, part of the roll moment,which is not balanced by a roll bar, Mrolls, is balanced bythe suspension springs, resulting in a vertical load
transfer of Fz = Mrolls/ts, where ts is either the lateral
distance between the suspension springs (for a rigidaxle) or a track width multiplied by suspension linkageratio. Due to nonlinear spring characteristics, this changein spring load brings about extension of the inside
suspension by zext, which is larger in magnitude than
compression of the inside suspension, zcomp. As a
result, in addition to body roll, the center of the body willexperience a vertical displacement of
2
compext zzz
= (2)
Thus for any value of roll moment (in static conditionsthe corresponding values of roll angle and vertical rise ofbody center can be determined. This verticadisplacement can be expressed in terms of suspensionforce at the operating point. Carrying out thesecalculations for front and rear suspensions, the jacking
effect at vehicle center of gravity can be determined andexpressed in terms of vertical force in response to bodyroll angle:
( )zzs kF = (3)
Here Fzs is the jacking force due to the non-linearity in
suspension stiffness and kz denotes the coefficien
relating the jacking force to roll angle. Note that since thejacking force is always negative in SAE sign convention
kz is negative for positive roll angles.
STABILTY ANALYSIS OF ROLL AND HEAVE MOTION
As stated earlier, our objective here is to develop asimplified model, which would permit one to study theeffects of design parameters on stability of roll and heavemodes during limit cornering. According to Liapunovsindirect method, local stability of a nonlinear system canbe determined, under quite general conditions, bystudying the stability of linearized model, as long as thelinearized system is not marginally stable (Vidyasagar1978, Section 5.4). In this section we first introduce asimple non-linear model describing the lateral motion ovehicle along with the body heave and roll, which takesinto account fundamental couplings between these
modes. Subsequently, we linearize equations for thismodel around the operating point corresponding tosteady-state limit cornering; using this linearizedequations, we formulate necessary and sufficienconditions for the asymptotic stability (or instability) of thesystem. Since the parameters in the equations of motiondepend on the characteristic parameters of the vehicleand the suspension system, the effects of these designparameters on stability in steady-state limit cornering canbe examined.
The lateral, roll and heave modes can be approximatelydescribed by the following equations:
( )
( )
( )( ) ( ) ( )
cos1tan
tancos
)(
cos
+
=++
=++
+=+
gmkFF
FFzkzczm
gazhmkcI
FFzhmma
szryRRyLR
fyRFyLFzzs
yrollsxx
yryfrollsy
(4)
In the above equations mdenotes vehicle mass, ms thebody (sprung) mass, hroll is the height of body center ofmass above the roll axis at equilibrium, Ixx the body rolmoment of inertia about the roll axis, Fyf and Fyr are the
lateral forces of front and rear axle, is the front whee
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steering angle, c
and k
are the roll damping and roll
stiffness of suspension, cz and kz are the suspensiondamping and stiffness in vertical direction (in heave),
FyLF, FyRF, FyLR, and FyRR are the lateral tire forces, f and
r are the inclination angles of lines connecting the tire
contact patches to the roll centers for front and rear
suspensions, respectively, kz is the term describing the
jacking force due to suspension stiffness nonlinearity,and g is gravity acceleration. Variable ay denotes lateral
acceleration, - the body roll angle, and z verticalposition of the body with respect to static equilibrium.SAE sign convention is used in equations (4). The
stiffness parameter values kand kz include the effects of
suspension and tire compliance and they correspond tolinearized values about the operating point.
In order to simplify the terms involving the differences inlateral tire forces appearing in the last of equations (4), itis assumed that the tire lateral forces in limit corneringare proportional to the normal forces. That is, for thefront axle
zf
zRFzLF
yf
yRFyLF
FFF
FFF = (5)
where subscript z refers to vertical forces. Similarequation holds for the rear axle. In addition, the normalforce for the left front tire can be approximated by thefollowing equation
( ) gat
mhz
m
mg
L
mbF y
wf
szLF +
=
2(6)
Here L denotes vehicle wheelbase, b is the distance of
vehicle center of mass to the rear axle, h the height ofvehicle center of mass above the ground, and f is the
fraction of the total suspension roll stiffness contributedby the front suspension. In addition to the staticcomponent, this equation reflects the influence of thequasi-static load transfer due to cornering and the effectof body heave. Analogous equations can be written forthe remaining tires. Using equations (5) and (6) and theiranalogues for remaining tires, as well as the statedassumption that the lateral forces are proportional to thenormal forces, the following equations are obtained:
( ) ( )
gaaAFF
gaaAFF
yyrryRRyLR
yyffyRFyLF
=
=
tan
tancos
(7)Here Afand Arare the following coefficients:
r
w
f
rf
w
f
fgt
hmA
gt
hmA
tan
12,tan
2 == (8)
It can further be assumed that in the limit cornering, thesum of all lateral tire forces is proportional to the totalnormal load. That is
( )zmmgFF syyryf =+ cos (9)where gayy /max= and aymax is the maximum steady
state acceleration, which vehicle can sustain on dry roadThe sign of expression on the right hand side is thesame as that of ay. Substituting equations (7) and (9) intoequations (4) yields
( ) ( )( )
( ) ( )( ) ( )
cos1
)(
+
+=++
=++=+
gmk
gaaAAzkzczm
gazhmkcI
zmmgzhmma
sz
yyrfzzs
yrollsxx
syrollsy
(10)
The system of equations (10) can now be linearized
about the operating point (ay0, 0, z0), which corresponds
to steady-state limit cornering. Let us denote
zzzaaa oyyy +=+=+= ,, 00 (11)
where the symbol signifies presumably smalincremental variables. Carrying out the linearizationprocedure (that is, substituting (11) into (10), cancelingequal terms on both sides of equations, and neglectinghigher order terms in incremental variables), thefollowing linear equations are obtained:
( )( )[ ]
+++
+=++
=++
=+
00
1
121
1
sin
2
gmgaAAk
aaAAzkzczm
ahmzamkcI
zmhmam
syrfz
yyrfzzs
yrollsysxx
syrollsy
(12)
The following notation is used in the above equations:
( )
002001
0101
,2
1
,
gaagaa
zhgmkkzhh
yyyy
rollsrollroll
==
==(13)
Note that since z0 < 0, hroll1 > hroll. Eliminating the
incremental lateral acceleration, ay, from equations (12
yields a system of two second-order linear differentiaequations with two unknowns. After taking the Laplacetransform on both sides of these equations, the following
fourth-order characteristic equation is obtained:
0012
23
34
4 =++++ bsbsbsbsb (14)
In order to abridge the subsequent equations, thefollowing notation is introduced:
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( )
( )
( ) ,sin
2
2,
001
11
1122
21
2
1
gmgaAAkk
m
maAAmm
hm
maAAb
m
hmII
syrfzz
syyrfss
rolls
yrfrolls
xxxx
++=
+=
+==
(15)
Now the coefficients in equation (14) are given by:
m
hmbmIbmccIb
bam
m
hmkccmkkIb
ckkcbkamkkb
rollsysxxszxx
ys
rollszyzszxx
zzzysz
12
22114113
222
12
11112
111210
,
,
,,
=+=
++=
+==
(16)
Stability of the linear system of equations (12) isdetermined by the roots of the characteristic equation(14). All of them must have negative real parts for thesystem to be asymptotically stable. The presence of
exponentially stable, but weakly damped modes can alsobe detected. Since the coefficients of equations (12) and(14) depend on the parameters of the vehicle and onsuspension design parameters, their effects on systemstability can be investigated. To illustrate, the effect oflocation of roll centers on vehicle stability is shown inFigure 9. These results were obtained for nominalvehicle parameters and lateral acceleration at theoperating point ay0= 8.5 m/s
2.
0 0.1 0.2 0.3 0.4 0.50
0.1
0.2
0.3
0.4
0.5
Front Roll Center Height [m]
RearRollCenterHeight[m]
UNSTABLE
STABLE
Figure 9. Influence of Roll Center Locations on Stabilityof Linearized System
Increasing the height of roll centers (with otherparameters unchanged) can make the vehicle unstable.The size of the stability region, however, is sensitive tothe operating point, in particular to the lateralacceleration, ay0. The line separating the stable andunstable regions is a straight line since the effects of rollcenter heights are contained in parameters hroll, Af andAr, with the first being a linear function of roll centerheights and the latter appearing only as a sum in
linearized equations (12). Instability of the linearizedmodel implies only local instability of the nonlinearvehicle model, since as the amplitude of oscillationsincreases, the assumption that the incremental variablesare small is violated. Thus, the nonlinear system may nobecome unstable, but develop a limit cycle. On the othehand, significant oscillation may occur when thelinearized system is stable, but with poorly dampedmode(s). In this case vibrations induced during thetransient phase of fishhook maneuver will not increase
but may decay very slowly.
It has been known based on quasi-static analysis (Hac2002) that in order to maximize rollover resistance, rolcenters cannot be too high, since then the positive effectof reducing steady-state roll angle may be offset or evendominated by the negative effect of increase in height othe body center of mass during cornering due to jackingforces. The analysis in this paper demonstrates that theheights of roll centers are also limited by the requiremenof stable roll response in the steady-state portion of thefishhook test. Although for the vehicle considered herethe heights of roll centers that can induce unstable
behavior are larger than typically occurring in lightvehicles, even lower values can contribute to stable, bupoorly damped response.
It should be noted that when both roll centers are at theground level and the suspension characteristic is linear
then the coefficients Af = Ar = 0 (equation 8) and kz= 0
Consequently, there is no coupling between the roll andheave modes according to equation (12) and heavemode is not excited at all; as a result, the system isasymptotically stable.
Similarly the effects of other vehicle design parameters
which affect parameters in equation (12), on stability ofvehicle in steady state limit cornering can beinvestigated. These parameters include suspensionstiffness and damping, roll stiffness distribution betweenfront and rear axles, vehicle mass (payload), roll momenof inertia, etc. Although the effect of tires is not directlyincluded, it is reflected to some extend in the steadystate lateral acceleration at the limit, ay0. For example, ican be shown that increasing this value by use of moreaggressive tires makes the linearized system less stable
It is noted that during Fishhook maneuvers vehicle bodyexperiences not only roll and heave, but also pitchmotion. This has primarily two causes: longitudinadeceleration of vehicle during hard cornering anddifferences in proportion between front and rear jackingforces relative to weigh distribution. The effect of bodypitch was neglected in the interest of simplicity. Taking iinto account would not provide significant new insightsbut would make close form solution too complex.
ANALYSIS OF EFFECT OF STEER COMPLIANCE
Vehicle response in the steady-state cornering at thelimit can be affected by steering compliance, whichintroduces coupling between the lateral acceleration and
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front steering angle. Considering a vehicle model thatdescribes yaw plane motion along with the roll and heavebody motions yields equations that are too complex forexplicit analysis. Fortunately, significant insightsregarding this coupling mechanism can be gained byconsidering vehicle motion in the yaw plane only. Theequations of motion are:
bFaFI
FFvvm
yryfzz
yryfxy
=
+=+
cos
cos
(17)
Here denotes vehicle yaw rate, vx and vy are thelongitudinal and lateral velocities of vehicle center ofmass, a and b are the distances of vehicle center ofmass to the front and rear axles, respectively, Izzdenotesvehicle yaw moment of inertia and the meaning of othersymbols remains the same as before. Due to front steerlateral force compliance, the front steer angle is
yfFFK= 0 (18)
where 0 is the front steer angle without compliance andKF is the lateral force steer compliance coefficient, whichis assumed constant. Assuming that the incremental
steer angle due to compliance, yfFFK , is much smaller
than 0, the function cos can be approximated by a
Taylor expansion limited to linear terms. This yields
0110 sin,coscos Fyf KKFK =+= (19)
Note that the presence of compliance reduces the
steering angle, , compared to 0, but increases cos,
thus increasing the forcing term in equation (17).
Substituting equation (19) into (17) and subsequentlylinearizing these equations about the operating point (vy0,
0), the following system of equations for incrementalvariables is obtained:
( ) bFFKaFI
FFKFvvm
yryfyfzz
yryfyfxy
+=
++=+
010
010
2cos
2cos
(20)
In the above Fyf0 is the front axle lateral force at theoperating point. Let us assume further that the corneringstiffness of both tires of front and rear axles at theoperating point are Cf0 and Cr0, respectively. The
cornering stiffness at the operating point can beinterpreted as a tangent of the inclination angle of thelateral force versus slip angle characteristic at the tireslip angle corresponding to the operating point. This isconceptually illustrated in Figure 10 for the front axle.The cornering stiffness in the linear range of the tirecharacteristic is Cfl and at the operating pointcorresponding to severe cornering it is Cf0. The latter ismuch smaller than the former.
Fyf
f
0
l
Cfl = tanl
Cf0 = tan0
Figure 10. Front Axle Lateral Force Characteristic
The incremental front axle lateral force, Fyf, is related to
the front axle incremental slip angle, f, as follows:
+== f
x
y
fffyfv
avCCF 00 (21)
Since, however, due to equation (18)
yfFf FK = (22)
equations (21) and (22) yield
x
y
Ff
f
yfv
av
KC
CF
+
+=
0
0
1(23)
Similarly for the rear axle,
==
x
y
rrryrv
bvCCF 00 (24)
Substituting equations (23) and (24) into the stateequations (20) and performing some manipulationsyields
+
=
++
+=
xzz
rf
y
xzz
fr
x
fr
xy
x
rf
y
vI
bCaCv
vI
aCbC
mv
aCbCvv
mv
CCv
20
2110
1001
(25)
In the above equations
Ff
yfF
ffKC
FKCC
0
00
0011
tan21cos
+
+= (26)
Equations (25) lead to a second order characteristicequation, from which stability conditions can be derived
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However, these equations have the same form as thosedescribing a linear bicycle model (Wong, 1993).Therefore, when Cf1 and Cr0 are both positive, thesufficient condition for asymptotic stability at all speeds isthat the value of Cr0b-Cf1a, which is directly related to theundesteer gradient, be positive. Using equation (26) it isstraightforward to show that the following two conditionsmust be satisfied in order to make the understeergradient of the linearized system negative:
0sin2 00
(27b)
For realistic values of parameters these conditionscannot be satisfied. However, the handling modelbecomes less stable as a result of steer forcecompliance if the presence of compliance increases thevalue of coefficient Cf1, since in this case the value ofCr0b-Cf1a, and consequently the understeer gradient, are
reduced. According to equation (26), Cf1 = Cf0cos0when
the steer force compliance, KF = 0. Thus the presence ofcompliance increases the apparent front corneringstiffness value, Cf1, if
000 tan2 yf FC < (28)
In the linear range of tire operation this condition is notsatisfied since Cf is much larger than Fy (for example, for
the vehicle considered here, Cf 110,000 N/rad and Fy isalways less than 20,000 N). Thus in the linear range, thesteer force compliance reduces the front corneringstiffness coefficient, Cf1, improving stability. When the
vehicle is at the limit, however, as is the case in thefishhook maneuver, the front cornering stiffness can belower than Cf by an order of magnitude and may even beclose to zero, as illustrated in Figure 10. At the same
time, the value of 2Fy0tan0 can be substantial when the
front steering angle is very large. For example, for thevehicle considered here in the fishhook maneuver
2Fy0tan0 2*15,000N*tan20o 11,000 N, which can be
larger than Cf0 at the limit. Thus, the presence of frontsteer compliance can significantly reduce the stabilitymargin of vehicle in the yaw plane for large steeringangles experienced in fishhook maneuver. This isparticularly true for maneuvers performed at higher
speeds, when front tire slip angle becomes large,resulting in small values of front axle cornering stiffnessat the operating point. Therefore, yaw plane motion ofthe vehicle becomes more sensitive to changes innormal loads, thus reinforcing the coupling between thebody roll and heave and vehicle yaw modes.
The observation that the presence of steer forcecompliance may reduce vehicle stability margin byincreasing the apparent front axle cornering stiffness ofthe linearized model at large steering angles may seemsurprising. This can be heuristically explained as follows.The impact of the steer force compliance on the
apparent front cornering stiffness of the linearized modeis the result of two influences, one of which acts toincrease, the other to reduce, the cornering stiffness(and consequently the lateral force). The net resultdepends on which effect dominates. In fact, the steecompliance reduces the cornering stiffness of thelinearized system, as seen in equation (23). However
the steer compliance also reduces the steering angle,
and thus increases cos, as expressed in equation (19)
The latter effect is significant only for large steeringangles. Since the front axle lateral force in equation (17)
is a product of Fyf, which is reduced, and cos, which
increases, the net result depends on which effecdominates. For very large steering and slip angles ofront axle, the steering compliance may provide apositive feedback of lateral force, thus increasing thecoupling between vertical and yaw modes of vehicle andcontributing to instability.
CONCLUSION
In this paper sustained body roll oscillations experienced
by vehicles during a steady-state portion of road edgerecovery maneuver were explained and analyzed. It wasfound that these oscillations occurring during hardcornering arise primarily because of coupling betweenthe body roll and heave, and subsequently vehicle yawmodes resulting from suspension jacking forces. Theseforces result in vertical motions of vehicle body, whichcause fluctuations in tire normal forces and consequentlytire lateral forces. The lateral forces directly influencelateral acceleration of vehicle, which in turn affects bodyroll. The presence of front steer compliance mayreinforce the coupling between the vertical and yawmodes and make the vehicle yaw motion less stableunder the conditions analyzed here, but for realistic
parameter values this is a higher order effect.
Stability analysis of coupled roll, heave and yaw motionsdemonstrated that increasing roll center heights, whichincreases jacking forces and can be used as means oreducing steady-state roll angle, generally reducesdynamic stability of vehicle roll and heave motion in limitcornering by reinforcing the coupling between the roland heave modes, which is the primary mechanismresponsible for sustained oscillations. Thus, in addition toother considerations, the heights of roll centers arelimited by the requirement of stable roll response in thesteady-state portion of the fishhook test. Other
suspension design characteristics, which lead toincrease of jacking forces, such as extremelyprogressive suspension stiffness characteristic at theoperating point (which may be the case whensuspension bottoms) also contribute to oscillatoryresponse. As expected, increasing suspension dampingmakes vehicle response more stable, but the levels ofdamping necessary to effectively suppress theoscillations may be, for some vehicles with high center ogravity, difficult to reconcile with requirements of ridecomfort in normal conditions.
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