Modeling of Rollover Sequences

Copyright @ IFAC Mechatronic Systems, Darrnstadt, Gennany, 2000

MODELING OF ROLL OVER SEQUENCES Ralf Eger· Uwe Kiencke·

• Institute for Industrial Information Technology University of K arlsruhe HertzstrafJe 16 (06.35), D-76187 Karlsruhe, GERMANY Tel.: +49 (721) / 608-4520 Fax.: +49 (721) / 608-4500 Email: [email protected]

Abstract: Safety restraint systems greatly reduce the potential injury risk during vehicle accidents. One major type of accident still remains without adequate occupant protection: Vehicle Rollover. Since rollover experiments are difficult to perform, rollover models can help to understand this type if accident. It will be shown, that already quite simple models reveal important properties of vehicle rollover. The influence of various parameter variations can be investigated. Based on a simplified modeling approach, it is possible to develop an analytic stability boundary, which helps to detect a imminent rollover. This is the base for future occupant protection systems. A more complete and complex approach of modeling, covering e.g. half and full turns, requires different strategies. To include detailed suspension systems a multi body approach becomes necessary. As an arbitrary rollover is a fully spatial movement, special assumptions like wheel-road contact do not hold for the regarded time interval. To cope with these difficulties, the simulation requires a state space machine to cover the different vehicle conditions (road contact, free flight, one side lift off). The simulation can then switch between the different model structures. With this approach, classical multi body vehicle models can be extended by special rollover models to gain insight in the mechanisms causing vehicle rollover. Copyright ~ 2000 IFAC Keywords: Automotive Safety, Vehicle Rollover, Vehicle Model, Stability Function

1. INTRODUCTION

(Otte, 1988). These numbers show the potential danger to the passengers during rollover. The reduction of vehicle rollover propensity is an important part in providing increased occupant safety. Much effort has been put into experimental, statistical and analytical approaches to investigate rollover propensity and the cause for vehicle rollover. Since experimental rollover tests are difficult and expensive to perform, the development of suitable simulation models is a main focus in vehicle rollover research. Models to investigate rollover behavior were developed by several authors. Rollover as a result of a curb impact (Jones, 1975; Wood, 1990) and rollover due to critical driving maneuvers (Nalecz and Bare, 1989; Nalecz et al., 1989) represent the two major model categories.

Safety restraint systems greatly reduce potential injury risk during vehicle accidents. Today, the air bag is a common feature in most passenger cars. This has resulted in a decreasing number of serious injuries during vehicle crashes. In the past few years research has brought forth new systems like sidebags, roofbags (Mueller, 1997) or the inflatable curtain (Ohlund et al., 1998) to cover even more accident situations than frontal crash. One major type of accident still remains without adequate occupant protection: Vehicle Rollover. While rollover occurs only in 2.2% of all passenger car accidents, they contribute 14.8% to the number of fatal accidents in the US (TSF, 1997). In Germany still 5% of all severe accidents show rollover

121

velocity v

Considering rollover propensity factors during the vehicle design is one step to increase occupant safety, as numerous rollovers accidents might be avoided. If rollover actually occurs, occupant protection using safety restraint systems would be a major step in passenger safety. Belt tensioners could protect passengers from ejection while various air bags would protect the passenger from serious injuries. In order to trigger such safety restraint systems, a dynamic stability criteria has to be formulated , which reliably predicts and detects vehicle rollover situations. In this paper different approaches for modeling rollover are investigated. Starting from a planar model with few degrees of freedom , important dependencies between vehicle parameters and rollover propensity can be derived. More complex models requiring a multi body approach and suitable model structure switching will be discussed. The implementation of an occupant protection requires a stability criteria which relies only on few sensing inputs but still reliably predicts and detects vehicle rollover. The simplified model to derive a rollover stability boundary will be presented at the end.

..

curb Fig. 1. Planar rollover model with two rotational degrees of freedom investigation regards different impact velocities, that were modeled by variation of the impact force . Large impacts bring the vehicle to a rollover. The results ar presented in fig . 2. Shown is the roll angle of the cabin against the horizontal plane 'Px = et> + 'lj; and the angular velocity that :"ouI.d be measured by a body fixes sensor Wx = et> + 'lj;. Since the model assumes road-surface contact of the pivot point, the model is only valid for quarter turns 'Px = et> + 'lj; < 90° . Dotted angular velocities indicate this case. The most interesting simulation result from fig. 2 is the fact, that a critical near rollover situation is difficult to detect in the first time interval (t < 2s). It can also be seen, that the time to rollover varies between 0.5s and 3.0s which must be taken into account when developing rollover detection methods.

2. ROLLOVER MODEL A first rollover model uses a two body approach, shown in fig. 1 and consists of two rotational degrees of freedom: et> angle between unsprung mass and road plane 'lj; angle between sprung and unsprung mass The model is based on the model presented by (Eger, 1998) with modifications regarding the nonlinear force elements acting on the system. The limitation of suspension travel was implemented by an exponential force law of the form Co ' X+dO . i:

F(x) =

{

Ixl - xo

eoe

"'le

X + doe

Ixl-xo "'Id

i:

IXI > Xo

Another task was to model the resulting impact forces which generally poses some difficulties in modelling (Chatterjee and Ruina, 1998) . An accurate impact force model was not the main goal of the research, since only the resulting motion of the vehicle was of interest. This lead to the assumptions, that the impact time is small compared to the rotational dynamics and that the impact forces are large compared to all other external forces . This resulted PT l model of the impact force, allowing a force build up in the compression phase. The amplitude an impact duration were adjusted to generate different impact conditions.

-100L..---'-----'-----3~--~

o

2

t[sJ

2.1 Simulation results

Fig. 2. Simulation of curb trip rollover at different impact velocities and comparison of critical near rollover situation

The following section presents some simulation results derived with the presented model. The first

Exemplary, the influence of the suspension is presented. Figure 3 shows the simulation results with

122

fixed impact conditions and variable suspension spring stiffness. As expected, lower spring constants increase the tendency for rollover. But significant parameter variations are needed to produce rollover for the given impact situation. Rollangle

,

..... .......... . : ,

,

.. .

",'"/

60

Co - 66%

;.~~~+66% . . . . ..... . .... ...- ..- -

Co

1.5

t[sl

angular velocity

3. EXTENDED ROLLOVER MODELLING The presented model has many limitations due to its simple character. For a more complex model, different modelling considerations have to be made. In a first step, the complexity of the mechanical model can be increased. The possibility of spacial movement as well as the inclusion of detailed suspension systems might be the focus of interest. This leads to a vehicle model with many interconnected bodies, resulting in complex equations of motion with many degrees of freedom . To generate these equations, a multi body formalism is needed (Roberson and Scwertassek, 1988) . In a loop free multi body system (fig. 5) the kinematics can be derived recursively in the following manner: (1)

AjO = AjiAiO, jWOj

=

jWij

+ Aji iWOi

(2) (3)

o -100

jrSj = jrC j

.. ......... .. ... .

o

0 .5

1.5

t[sJ

jVSj

=

Fig. 3. Influence of suspension spring stiffness variation on rollover behavior

+

Aji(irS.

Aji(iVS.

jaS j

=

Aji(ias,

(4)

+i WOi irC. +i rc,)

+jWij jrCj

Interesting in this context is the behavior of the suspension system and the limitation of the spring elongation. Figure 4 shows the angle between sprung and unsprung mass. It can be seen, that in the rollover case, the relative movement between both bodies is neglectable. This behavior will be used in section 4 to develop a model based rollover detection method. To cope with the different time scales in which rollover occurs, a state space plot fig . 4 is introduced.

+i rc,)

+j rC j

(5)

+i ~Oi irC. +i WOi iWOi irC.

(6)

4

Fig. 5. Adjacent bodies in a multi body system

o -4 -8

Loops require a special treatment but can be incorporated as well. With d'Alembert's principle and the use of generalized coordinates q and Jacobians J, the equations of motion can be derived in the following form

. ... ~:--:~ . ~+~~ . - -' Co

-12 - - - - -......:.::....---'-----1.... .5---.......I [sJ 0 05 t

State space plane

M(q)ij + g(q, q) = qe(q, q) -- ... _-_ ... -- ---

---

with the mass matrix M generalized coriolis forces

· ~~.:.~~.:..:66% ·

-10

and the vector g of

n

M(q) =

- . - cQ + 66%

-100

(7)

==C;;

L [J~.miJT. + J~. eS,JR'] i=l

0

n

Fig. 4. Angle between vehicle axle and body showing the limit in spring travel and state space plot of the simulation results from fig. 3

g(q,q) =

L i=l

123

[J~.mijT,q+

(8)

4. MODEL BASED ROLLOVER DETECTION

The external forces and torques are collected in n

qe(q,q)=L[J~Jt+J~Jie].

(10)

i=1

A Matlab implementation of the multi body formalism allows the modelling of mechanical systems. With a versatile force library including mechanical interconnections like springs or electronically controlled actuators, designed in the Matlab/Simulink environment, the modelling and simulation of mechatronic systems is possible. In the case of rollover modelling, another aspect has to be considered. If a full rollover is to be modelled, the following situations can occur • vehicle travel on road prior to rollover • lift of of one side of vehicle • airborne travel • impact with obstacle or ground • possible deformations due to impact The goal is to combine existing and verified vehicle dynamic models with the ability to cover rollover situations. This can be achieved by applying model structure switching algorithms, that selects the appropriate model for the current simulation situation. The simulation is then controlled by a state machine, shown in fig . 6. side obstacle

Next to the models describing rollover accidents more or less accurately, the development of occupant protection systems is most important. While complex models help to verify and optimize rollover detection algorithms, analytical strategies usually become impossible. Since the rollover detection has to rely on few sensing inputs like acceleration an angular velocity (Eger and Kiencke, 1999a) a simple model containing only few states is preferred. This promises the possibility to derive parameter dependent analytical criteria that can be adapted to different vehicle types. One possible method do develop rollover detection is based on the conservation of energy theorem. But the resulting stability boundary is too sensitive and might indicate false rollover (Eger and Kiencke, 1999b). Another approach is the application of Newton's 2. law of conservation of momentum. The problem of missing energy terms disappears, as the resulting friction- or damping forces still act on the system, changing its linear and angular momentum. Figure 7 shows the vehicle at an arbitrary roll angle I.{Jx < K . Defining an inertial coordinate system with origin at the pivot point P , the coordinates of the CC are expressed by

. chassi : ground : contact

'---'-;-,--, , , ,

r obst =

2 wheel Vehicle FN .whe = 0 dynamics 1--'--+'1 rolling model model

0

FN.whe

= 0

airborne tavel model

Fig. 7. Vehicle coordinates and forces on P y

= -rsin(K -

I.{Jx);

z

= -rcos(K -

I.{Jx) (11)

and the corresponding accelerations wheel ground impact

y = r sin(K z = r COS(K -

wheel ground impact

I.{Jx)cp;

+ r COS(K -

I.{Jx)
I.{Jx)cp; - r sin(K - I.{Jx)
Employing the balance of forces and moments we get

Fig. 6. State machine controlling model structure switching for an extended rollover simulation

mz

Unilateral contact conditions are used for transitions between the states. This requires an accurate model of the contact kinematics (Pfeiffer and Clocker, 1996) which contributes much to the model complexity. The general multi body approach together with model switching strategies promise a more detailed insight in vehicle rollover mechanisms. However the problem of calculating impact forces and possible structural deformations remain to be investigated. Also the ability to validate such complex model with real world experiments should be regarded critically.

= mg -

N;

my

= -F (13)

assuming the pivot point P remains on ground all times. The rollover sequence is split into two intervals. In the first interval [0 :s; t :s; tcritl the vehicles angular momentum is build up through external forces N and F generating a potential rollover. tcrit

Lcrit = / -N(t)ly(t)1 + F(t)lz(t)1 dt o

124

(14)

values of 'Px if rolling motion is allowed to switch from left to right pivot. An equivalent boundary can be found for negative values of W x · The performance of the rollover detection is verified using experimental and simulation results. Figure 8 shows the simulated trajectories using the rollover model in chapter 2. It can be seen, that the stability boundary eq.(24) used to detect rollover clearly discriminates between rollover (gray) and non rollover (dashed) situations. Also the critical near rollover situation where the vehicle turns close to the static trip angle is covered.

In the second interval tcrit < t < T, the forces supporting rollover are assumed to act only until tcrit . This only leaves the normal force, slowing down the rolling motion until the static tip angle 'Px = I'\, -+ t = T is reached. T

Ll

J

=

-N(t)ly(t)1 dt

(15)

tcrit

If the angular momentum for 'Px = I'\, remains positive, rollover occurs. Applying eq.(14) and (15) the rollover condition is

Lcrit + Ll > 0

(16)

Rollover Simulation

meaning that with no supporting lateral forces F, rollover is sure to occur after tCrit. To derive a stability boundary, eq.(13) is evaluated in the second interval. The stability border is the condition

Wx = CPx(t) :$ 0 For t 2:

excpx

tcrit

for

'Px =

(17)

I'\,

we get

= -Nlyl

= mz·

y - mg · y

(18)

.. [~r2sin2(1'\,-'Px)Jcpx -mgrsin(I'\,-'Px) 'Px= 2 ex + mr2 sin (I'\, - 'Px) To find a trajectory in the ('Px, wx)-plane the following on must find a representation

Wx = f('Px)

-+

CPx = j'('Px)f('Px)

Fig. 8. Stability boundary and simulated Rollover. The test includes the following situations: rollover (gray), non critical situation (- - -) and critical near rollover situation (--)

(19)

Applying this to eq.(18) a differential equation of Bernoulli-type can be found

!' = g('Px) . f

- h('Px) .

r

with

Q

= -1 (20)

Experimental data of a mid size car rollover test is shown in fig . 9. Here the vehicle slides sidewards with different lateral velocities. Again it is possible to discriminate rollover from non rollover events ,

Substituting

z('Px)

= f('Px)2

-+

f('Px)

= VZ('Px)

(21) Critical Sliding test

and writing z' = 2 . f . f' results in the following linear differential equation for z A(t)z

+ B(t)

. (22)

Solving this equation leads to the general solution (23)

+ 2mgr[cos(1'\, - 'Px.o) - cos(1'\, - 'Px)] ex + mr2 sin 2 (1'\, - 'Px) Applying the instable equilibrium values Wx,O = 0, 'Px,o = I'\, and zO('Px,o) = 0 the following stability boundary can be derived.

o

20

40

60

Fig. 9. Stability boundary and real world rollover tests. The test data includes the following situations: rollover (gray), non critical situation (- - -) and critical near rollover situation

2· mg· r[l - cos(1'\, - 'Px)] ex + m . r2 sin 2 (1'\, - 'Px)

(-)

(24) The boundary is valid for the first quadrant of the ('Px, wx )- plane but can be extended to negative

Characteristic for both results is the overshot at small roll angles. Even a driving maneuver 125

6. REFERENCES

where the roll angle remains below 10° shows this overshot. This results from the influence of the suspension system. Only at significant roll angles leading to a lift of one side of the vehicle, the simplified model assumption becomes valid. Since the suspension was not included in the model, the range of approx. l4'x I < 10° has to be excluded from the rollover prediction.

Chatterjee, A. and A. Ruina (1998). Two i~­ terpretations of rigidity in rigid-body collIsions. Journal of applied mechanics: contributions of the ASME Applied Mechanics Division 65(4),894-900. Eger, R. (1998). Model based detection .of critical driving situations. In: Advances m Vehzcle Control and Safety AVCS'98. Amiens. Eger, R. and U . Kiencke (1999a). Motion estimation in vehicle rollover. In: Proceedings of the IFAC14 th World Congress. Beijing. Eger, R. and U . Kiencke (1999b). Stability in vehicle rollover. In: Proceedings of the European Control Conference ECC'99. Karlsruhe. Jones, 1. S. (1975). The mechanics of rollover as the result of curb impact. Transactions 75046l. SAE. Mueller, H. E . (1997) . Roof airbags. Transactions 970167. SAE. Nalecz, A. G., A. C. Bindemann and H . K. Brewer (1989). Dynamic analysis of vehicle rollover. In: 12th International Technical Conference on Experimental Safety Vehicles. Goteborg. pp. 803-819. Nalecz, A. G. and C. Bare (1989) . Developement and analysis of intermediate tripped rollover model (itrs). Final report DTNH2287-D27174. National Highway Traffic Safety Administration - U.S . DoT. Washington. Ohlund, A. et al. (1998). The inflatable curtain (IC) - a new head protection system for side impacts. In: 1(fh Experimental Safety Vehicles (ESV) Conference. Windsor, Kanada. Otte, D. (1988). Charakteristika von Unfallen mit Fahrzeugiiberschlag. Verkehrsunfall und Fahrzeugtechnik 4,92-98. Pfeiffer, Friedrich and Christoph Glocker (1996) . Multibody dynamics with unilateral contacts. John Wiley and Sons. New York. Roberson, R. and R. Scwertassek (1988) . Dynamics of Multibody Systems. Springer. Berlin, Heidelberg, New-York. TSF (1997). Traffic Safety Facts 1996, National Highway Traffic Safety Administration - U.S . DoT, HS 808649, Washington. Wood (1990) . Model of vehicle rollover due to side impact collision. Proceedings of the Institution of Mechanical Engineers 204(2),83-92.

5. CONCLUSION The paper presented different modelling approaches for vehicle rollover. Already a simple two degree of freedom models provide helpful insight for rollover sequences. Especially the influence of suspension system and the different time durations for rollover can be seen. This model is the basis for further investigations on the way to develop occupant protection systems for rollover accidents. In order to get more realistic simulation results, extended models become necessary. Based on widely used multi body formalisms, a more sophisticated mechanical model can be implemented an also cover modern suspension systems in detail. Since the rollover sequence can be split into sperate parts with different system behavior, a model switching strategy is proposed. Using a state machine, the simulation can switch between the appropriate model descriptions. This allows the combination of existing vehicle dynamic models with rollover models. That increased model complexity is not always necessary was demonstrated in the last section. The third an simplest modelling approach was used to derive a dynamic rollover stability criteria to detect imminent rollover. Due to the simple model used, an analytical solution is possible. The method makes use of only one sensor input, namely the angular velocity about the vehicles roll axis. As the stability criterium includes various vehicle parameters, it can easily be adapted to different types of vehicles. One inherent problem is the effect of the suspension system. Is occurs at small roll angles, where the suspension influence cannot be neglected. As no information about the suspension is measured and implemented in the criterium, this problem remains, if no additional sensing information is provided. To cover these cases, a special treatment or the exclusion of rollover detection at small angles is necessary.

126