Crash analysis and dynamical behaviour of light road and ... · Crash analysis and dynamical...

Vehicle System DynamicsVol. 43, No. 6–7, June–July 2005, 385–411

Crash analysis and dynamical behaviour of lightroad and rail vehicles

JORGE AMBRÓSIO*

Instituto de Engenharia Mecânica, Instituto Superior Técnico, Av. Rovisco Pais,1049-001 Lisboa, Portugal

The main goal of crashworthiness is to ensure that vehicles are safer for occupants, cargo and otherroad or rail users. The crash analysis of vehicles involves structural impact and occupant biomechan-ics. The traditional approaches to crashworthiness not only do not take into account the full vehicledynamics, but also uncouple the structural impact and the occupant biomechanics in the crash study.The most common strategy is to obtain an acceleration pulse from a vehicle structural impact analysisor experimental test, very often without taking into account the effect of suspensions in its dynamics,and afterwards feed this pulse into a rigid occupant compartment that contains models of passengers.Multibody dynamics is the most common methodology to build and analyse vehicle models for occu-pant biomechanics, vehicle dynamics and, with ever increasing popularity, structural crash analysis.In this work, the aspects of multibody modelling relevant to road and rail vehicles and to occupantbiomechanical modelling are revised. Afterwards, it is shown how multibody models of vehicles andoccupants are used in crash analysis. The more traditional aspects of vehicle dynamics are then intro-duced in the vehicle models in order to appraise their importance in the treatment of certain types ofimpact scenarios for which the crash outcome is sensitive to the relative orientation and alignmentbetween vehicles. Through applications to the crashworthiness of road and of rail vehicles, selectedproblems are discussed and the need for coupled models of vehicle structures, suspension subsystemsand occupants is emphasized.

Keywords: Energy absorption; Occupant biomechanics; Vehicle dynamics; Multibody dynamics;Impact

AMS Subject Classification: 70E55; 92C10; 74M20

1. Introduction

The complete design of road and rail vehicles includes active and passive safety and the comfortof occupants. The active safety of vehicles includes the manoeuvrability and stability and thestudy of all systems that address their improvement such as suspensions, active control andelectromechanical subsystems for driving support, among others. Passive safety of vehiclesaddresses the protection of occupants and cargo from the moment that an accident startsuntil the vehicle stops. Vehicle system dynamics addresses mostly the active safety aspectsof the vehicle design, whereas vehicle crashworthiness deals with the structural impact, theoccupant biomechanics and all subsystems aimed to provide a better protection during a crash.

*Corresponding author. Email: [email protected]

Vehicle System DynamicsISSN 0042-3114 print/ISSN 1744-5159 online © 2005 Taylor & Francis Group Ltd

http://www.tandf.co.uk/journalsDOI: 10.1080/00423110500151788

386 J. Ambrósio

Though the most common numerical tools used to address both areas of vehicle dynamics arebased on multibody methodologies, traditionally these two aspects of vehicle design havebeen addressed separately or with little interaction. It is an objective of this work to showthat not only the two aspects of vehicle dynamics can be addressed simultaneously, but alsothat in many studies that address the passive safety of vehicles it is fundamental to includedetailed models of the vehicle active safety subsystems. Moreover, the common trend in actualvehicles is to have active driving support and intelligent safety systems, both requiring the useof sensors to provide the data required for the decision making processes. Also from this pointof view, a common approach to active and passive safety of road and rail vehicles is of majorinterest.

Until the early 1970s, crash studies relied almost exclusively on experimental testing whichfocused mostly in full scale testing and in the development of relevant testing scenarios [1]. Ina review work, Tidbury [2] stressed the facts that not only the costs associated with experimen-tal testing are very high, but also the results cannot be generalized to other impact situationsneither can such methods be used during the design of new vehicles. Therefore, the needfor accurate numerical procedures became apparent at this time. The earlier numerical meth-ods used for vehicle crashworthiness were based on the use of lumped masses and nonlinearsprings. The models built with these methods, known as lumped parameter models, use lumpedmasses to represent parts of the vehicle, such as the engine block or the passenger compart-ment, considered rigid during the analysis, and springs to represent the structural elementsresponsible for the deformation energy management [3]. The lumped parameter models aremostly one- or two-dimensional and they include at the most very simplified representationsof the suspension elements of the vehicles.

The first numerical model for the simulation of the vehicle occupant biomechanics proposedby McHenry in 1963 [4] was a two-dimensional model that included four articulated rigidbodies and the occupant restraint system. Later, Lobdell [5] proposed a lumped parameterapproach with similar numerical characteristics to that used in vehicle structural impact for thesimulation of human thorax loading. The developments of lumped parameter biomechanicalmodels continued through the years of 1970 and 1980 in parallel with the development ofmodels for the structural impact of vehicles but without the synergies associated to the use ofa common numerical framework.

The generalization of the vehicle structural and occupant biomechanical models to becomefully three-dimensional required another type of numerical methodologies. The multibodydynamics methods provided the framework in which more advanced developments could beundertaken. The plastic hinge approach, proposed by Nikravesh et al. [6] to represent thestructural dynamics of road vehicles in crash situations, uses a general multibody represen-tation where the full vehicle is described and the structural systems are divided into severalrigid bodies connected by kinematic joints to which nonlinear springs, representing the struc-tural deformation, are associated. This approach has been further developed by several authors[7, 8] and applied to the crashworthiness of road and rail vehicles proving to be a versatileand efficient methodology to deal with structural crashworthiness, especially in the initialphases of the vehicle design, when extensive re-analysis is required. Still with little or no con-tact to the developments in structural crashworthiness, the multibody methodologies beganto be extensively applied in the development of biomechanical models of vehicle occupants.Starting in the initial work of McHenry [4] and going through the developments of Robbinset al. [9] and Fleck et al. [10], the use of multibody dynamics methodologies for occupantbiomechanics reached a level of maturity that allowed it to become the most popular numericalapproach used today in impact biomechanics. The acceptance of the multibody-based modelsby the vehicle industry led to the successful development of commercial occupant simulationcomputer codes, such as SOMLA [11] and MADYMO [12].

Crash analysis and dynamical behaviour of vehicles 387

The continuous increase in computer power and the development of the finite elementmethod ensured the feasibility of finite element-based models for the study of the structuralcrashworthiness of vehicles. One of the first successful attempts to apply nonlinear finiteelements to the prediction of a vehicle crush was reported by Thompson [13]. Belytchko andHsieh [14] proposed the finite element developments that defined the framework of their use instructural crashworthiness. During the 1970s, several nonlinear finite element codes suitableto crashworthiness applications were released. Among these, the codes DYNA3D by Hallquist[15] and WHAMS by Belytschko and Kenedy [16] are of special importance as they servedas the backbone of the codes LS-DYNA [17] and of PAM-CRASH [18] and RADIOSS [19],respectively, which are the most common finite element codes used in crash analysis. Theapplication possibilities of this method are being expanded in order to allow for the complexmodelling of vehicle and integrated occupants, but still, the representation of the suspensionsystems of the vehicles and of systems responsible for their active safety are not yet presentin the finite element models, unless at the cost of a large computational effort. The amount ofinformation required for large systems is still too high making the application of FEM codessuitable only for the analysis phase of structural and biomechanical systems. For the earlydesign phases, the multibody models are still the primary numerical tool used in simulation.

Though the features of the different numerical methodologies based on multibody dynamicsand on finite elements include most of the ingredients required for the crash analysis ofroad and rail vehicles, the models more commonly developed neither integrate the structuralcrashworthiness with the occupant biomechanics nor take into account the overall dynamicsof the vehicle. Applications that include low and medium vehicle impact speeds or multipleimpacts or long contact periods or even for which the alignment between vehicles is importantfor the outcome of the crash are examples of cases for which the vehicle dynamics associated tothe suspensions and to the wheel road–rail contact can play an important role. Traditionally, themost common numerical techniques for vehicle dynamics modelling are based on multibodydynamics methodologies, which are the same used for vehicle impact models and for occupantbiomechanics modelling. Therefore, no reason exists for not taking the advantage of theirfeatures to develop more accurate and efficient models of road and rail vehicles.

In this work, it is shown that procedures based on multibody dynamics provide a unifiedframework for the development of design and analysis tools of crashworthy structures and ofoccupants. The models developed are computationally efficient in today’s computers to allowfor the extensive re-analysis cycles required in the optimal design of vehicles for crashwor-thiness. Furthermore, in applications for which the structural patterns of deformation are notknown beforehand or for which subcomponent finite element models are available, the multi-body dynamics methods enable the effective coupling between the lumped parameter modelsand the finite element description of the structural deformations. The application of the meth-ods presented is addressed here through the study of the development of energy absorption andanti-climber devices for railway vehicles, the development of occupant protection strategies inside impact of a road vehicle and the rollover of an off-road vehicle with occupants included.

2. Multibody dynamics

There are different coordinates and formalisms that lead to suitable descriptions of multibodysystems, each of them presenting relative advantages and drawbacks. The objective of thiswork is not to discuss what is the most efficient multibody methodology that can be appliedfor vehicle dynamics crash analysis. In this work, the methods presented are based on the useof Cartesian coordinates, which lead to a set of differential-algebraic equations that need to

388 J. Ambrósio

be solved. It is assumed that appropriate numerical procedures are used to integrate the typeof equations of motion obtained with the use of Cartesian coordinates. It is also assumed thatthe different numerical issues that arise from the use of this type of coordinates, such as theexistence of redundant constraints and the possibility of achieving singular positions, are alsosolved. For a more detailed discussion on the numeric aspects of this type of coordinates, theinterested reader is referred to [20–22].

2.1 Multibody equations of motion

A typical multibody model is defined as a collection of rigid or flexible bodies that have theirrelative motion constrained by kinematic joints and that are acted upon by external forces. Ageneric description of such a model is represented in figure 1.

Let the multibody system be made of nb bodies. The equations of motion for the system ofunconstrained bodies are

Mq = g (1)

where M is the mass matrix, which includes the masses and inertia tensors of the individualbodies, q the acceleration vector and g the vector with applied forces and gyroscopic terms.

The relative motions between the bodies of the system are constrained by kinematic joints,which are mathematically described by a set of nc algebraic equations, written as

�(q, t) = 0. (2)

The first and second time derivatives of equation (2) constitute the velocity and accelerationconstraint equations, respectively, written as

�(q, t) ≡ Dq = v

�(q, q, t) ≡ Dq = γ(3)

where D is the Jacobian matrix. For a system of constrained bodies, the effect of the kinematicjoints can be included in equation (1) by adding to their right-hand side the equivalent jointreaction forces g(c) = −DTλ leading to

Mq = g − DTλ (4)

where λ is a vector with nc unknown Lagrange multipliers. Equation (4) has nb + nc unknownsthat must be solved together with the second time derivative of the constraint equations. The

Figure 1. Generic multibody system.


Figure 2. Flowchart representing the forward dynamic analysis of a multibody system.

resulting system of differential-algebraic equations is

[M DT

D 0

] [qλ

]=

[gγ

](5)

Note that the solution of equation (5) presents numerical difficulties resulting from theneed to ensure that the kinematic constraints are not violated during the integrationprocess.

2.2 Solution of the equations of motion

The forward dynamic analysis of a multibody system requires that the initial conditions of thesystem, i.e. the position vector q0 and the velocity vector q0, are given. With this informa-tion, equation (5) is assembled and solved for the unknown accelerations, which are in turnintegrated in time together with the velocities. The process, schematically shown in figure 2,proceeds until the system response is obtained for the analysis period.

3. Vehicle structural impact

The analysis of road or rail vehicles in crash events requires that the deformation of thestructural components during impact is properly represented and that the contact forcesare described. A proper description of the structural deformation means that not only themechanisms of deformation and the relative displacements between the different structuralcomponents are captured with accuracy, but also that the deformation energy distribution iswell represented. Methodologies based on lumped deformations, such as the finite segmentand the plastic hinge approaches, or on a continuous description of the deformation, suchas the finite element method, are presented here to describe the structural deformation ofthe vehicle. The contact force models that are used in vehicle impact should be consistentwith the vehicle model used. Continuous contact force models based on penalty formula-tions are generally preferred for rigid body impact, being a representative force model ofthis type described here. The tyre contact models used for road vehicles and the wheel–railcontact models of railway vehicles are also used in some problems of vehicle impact todescribe the ground interaction forces. However, their description falls out of the scope of thiswork.

390 J. Ambrósio

Figure 3. Slender component and its finite segment model.

3.1 Finite segment approach to flexible multibody systems

Let the flexible components of the multibody system be made of slender components such asthe connecting rods of high-speed machinery or the structural frame of buses and trucks. Inthis case, each slender component can be modelled as a collection of rigid bodies connectedby linear springs, as shown in figure 3. These springs, representing the axial, bending andtorsion properties of the beams, capture the flexibility of the whole component.

Twelve generalized displacements are associated to each finite segment, that is, three trans-lations and three rotations at each end. When the beams deform, the reference frames attachedto the rigid bodies used for their model rotate and translate with respect to each other. Forcesand moments applied to the rigid bodies can be calculated from the relative end displacementsand rotations assuming that each two adjacent bodies are connected by springs and, eventually,by dampers. For a rigid body, there are deformation elements attached to each end, as shownin figure 4. The characteristics of these springs are related with the material and geometricproperties of the system components [23].

Using the principles of structural analysis, the stiffness coefficients for these springs arecalculated. For instance, the straight extensional segment and the straight bending, representedin figures 4(a) and (b), have their stiffness coefficients, respectively, given by

kei = ke

j = 2EiAi

li(6a)

kei = ke

j = 2EiIi

li(6b)

where Ei is the Young modulus, Ai the cross-section area, Ii the cross-section moment ofinertia and li the length of the finite segment.

Note that the finite segment methodology, as described in the original work by Hustonand Wang [23], only applies to structural components made of beams that have linear elasticdeformations. However, this formulation can be extended to cases with nonlinear deformations,which is relevant to vehicles models undergoing impact scenarios.

3.2 Plastic hinges in multibody nonlinear deformations

In many impact situations, the individual structural members are overloaded, principally inbending, giving rise to plastic deformations in highly localized regions, called plastic hinges.These deformations, presented in figure 5, develop where maximum bending moments occur, atload application points, joints or in locally weak areas. Therefore, for most practical situations,

Figure 4. Finite segments and their combinations: (a) extensional straight; (b) bending straight; (c) tapered bending.


Figure 5. Localized deformations on a beam and a plastic hinge.

their location is predicted well in advance. The methodology described herein is known asconceptual modelling in some industrial areas.

The plastic hinge concept has been developed by using generalized spring elements to rep-resent the constitutive characteristics of localized plastic deformation of beams and kinematicjoints in order to control the deformation kinematics, as illustrated in figure 6. A more complexstructure, such as the B-pillar of a road vehicle or the end-underframe of a train car shownin figure 7, can be represented by a collection of rigid bodies connected by the joint-springarrangements that describe its deformation and energy absorption characteristics.

For a flexural plastic hinge, the spring stiffness is expressed as a function of the changeof the relative angle between two adjacent bodies connected by the plastic hinge, as shownin figure 8. For a bending plastic hinge, the revolute joint axis must be perpendicular to theneutral axis of the beam and to the plastic hinge bending plane simultaneously. The relativeangle between the adjacent bodies measured in the bending plane is

θij = θi − θj − θ0ij (7)

where θ0ij is the initial relative angle between the adjacent bodies. Note that for the case of

flexible adjacent bodies, the relative angular values also include information on the nodalrotational displacements.

The characteristics of the spring-damper that describes the properties of the plastic hingesare obtained by experimental component testing, finite element nonlinear analysis or simplifiedanalytical methods. For instance, the typical torque–angle constitutive relation, as in figure 8,is found based on a kinematic folding model for the case of a steel tubular cross section. This

Figure 6. Plastic hinge models for different loads: (a) one-axis bending; (b) two-axis bending; (c) torsion; (d) axial.

392 J. Ambrósio

Figure 7. Plastic hinge models for vehicle substructures: (a) B-pillar of a road vehicle; (b) end-underframe of a railvehicle; (c) door of a car.

Figure 8. Plastic hinge bending moment and its constitutive relation.

model is modified accounting for elastic–plastic material properties including strain hardeningand strain rate sensitivity of some materials. A dynamic correction factor is used to accountfor the strain rate sensitivity [24]

Pd

Ps= 1 + 0.07 V 0.82

0 . (8)

Here Pd and Ps are the dynamic and static forces, respectively, and V0 is the relative velocitybetween the adjacent bodies. The coefficients appearing in equation (8) are dependent on thetype of cross section and material [24].

3.3 Nonlinear finite elements for multibody systems

The description of the structural deformations during vehicle impact using plastic hingesrequires that the mechanisms of deformation are known beforehand, which is very often thecase in most of the practical applications. However, in cases for which the structural deforma-tions are generalized, rather than concentrated in specific points, the modelling capabilitiesof the finite element method are irreplaceable. The nonlinear finite element formulation is


Figure 9. General motion of a flexible body.

summarized here in the framework of flexible multibody dynamics to be compatible with thestandard description of the multibody formulation presented before.

The motion of a flexible body, depicted in figure 9, is characterized by a continuous changeof its shape and by large displacements and rotations, associated to the gross rigid body motion.Let XYZ denote the inertial reference frame and ξηζ a body fixed coordinate frame. Let theprinciple of the virtual work be used to express the equilibrium of the flexible body in thecurrent configuration t + �t and an updated Lagrangian formulation be used to obtain theequations of motion of the flexible body [25]. The finite elements used in the discretization ofthe flexible body are assembled, leading to the equations of motion of the flexible bodyMrr Mrf Mrf

Mφr Mφφ Mφf

Mf r Mf φ Mff

r

ω′u′

=

gr

g′φ

g′f

−

sr

s′φ

s′f

−

0

0f

−

0 0 0

0 0 00 0 KL + KNL

0

0u′

(9)

where r are ω′ are, respectively, the translational and angular accelerations of the body fixedreference frame and u′ denotes the nodal accelerations measured in body fixed coordinates.The local coordinate frame ξηζ , attached to the flexible body, is used to represent the grossmotion of the body and its deformation. The right-hand side of equation (9) contains the vectorgeneralized forces applied on the deformable body g and matrices KL and KNL, which arethe linear and nonlinear stiffness matrices, respectively. Vector f denotes the equivalent nodalforces due to the state of stress.

In order to improve the numerical efficiency of the solution of equation (9), a lumped massformulation is used and the nodal accelerations u′, measured with respect to the body fixedframe, are substituted by the nodal accelerations q′

f relative to the inertial frame. Furthermore,it is assumed that the flexible body has a rigid part and a flexible part and that the body fixedcoordinate frame is attached to the center of mass of the rigid part, as shown in figure 10. Theflexible and rigid parts are attached by the boundary nodes ψ . The procedure is described inAmbrósio and Nikravesh [26] leading to the new form of the equations of motion

mI + AM∗AT −AM∗S 0

− (AM∗S

)TJ′ + STM∗S 0

0 0 Mff

r

ω′q′

f

=

fr + AC′

δ

n′ − ω′J′ω′ − STC′δ − I

TC′

θ

g′f − f − (KL + KNL)u′

(10)

where J′ is the inertia tensor, expressed in body fixed coordinates, fr the vector of the externalforces applied to the body and n′ the vector with the force transport and external moments.Vector u′ denotes the nodal displacement increments from a previous configuration to the

394 J. Ambrósio

Figure 10. Flexible body with a rigid part.

current configuration, measured in body fixed coordinates. In equation (10), M∗ is a diagonalmass matrix containing the mass of the boundary nodes and

AT = [A A · · · A]T; S = −[x′

1 x′2 · · · x′

n]T; I = [I I · · · I]T

where A is the transformation matrix from the body fixed to global coordinates and xk denotesthe position of node k. Vectors C′

δ and C′θ are the reaction forces and moment of the flexible

part of the body over the rigid part is given by

C′δ = g′

δ − Fδ − (KL + KNL)δδ δ′ − (KL + KNL)δθθ′ (11)

C′δ = g′

θ − Fθ − (KL + KNL)θδδ′ − (KL + KNL)θθθ

′. (12)

In these equations, the subscripts δ′ and θ ′ refer to the partition of the vectors and matrices withrespect to the translational and rotational nodal degrees of freedom. The underlined subscriptsrefer to nodal displacements of the nodes fixed to the rigid part.

The equations of the flexible bodies are included in the equations for the constrained multi-body system using the Lagrange multiplier method when a kinematic constraint involvingthe nodal coordinates has to be set. The mechanical joints of the vehicles, modelled usingthis formulation, are described by such kinematic constraints. For more information on thedefinition of the kinematic constraints, the interested reader is referred to Ambrósio [27].

3.4 Contact detection

The numerical procedures for contact detection in crash applications are similar to method-ologies used to detect contact in other type of applications in vehicle dynamics. Let a body ofthe system get close to a surface during the motion of the multibody system, as representedin figure 11. Without lack of generality, let the impacting surface be described by a mesh oftriangle patches. In particular, let the triangular patch, where node k of the body will impact,be defined by points i, j and l. Note that node in this context means either a point of a rigidbody or a nodal point of the finite element mesh of a flexible body. The normal to the outsidesurface of the contact patch is defined as �n = �rij × �rjl/‖�rij × �rjl‖.

Let the position of the structural node k with respect to point i of the surface be

rik = rk − ri . (13)


Figure 11. Detection of contact between a multibody component and a triangular patch.

This vector is decomposed in its tangential component, which locates point k∗ in the patchsurface, and a normal component, given, respectively, by

rtik = rik − (rT

ik n)n (14)

rnik = (rT

ik n)n. (15)

A necessary condition for contact is that node k penetrates the surface of the patch, i.e.

rTik n ≤ 0. (16)

In order to ensure that a node does not penetrate the surface through its ‘interior’ face, athickness e must be associated to the patch. The thickness penetration condition is

−rTik n ≤ e. (17)

The condition described by equation (17) prevents that penetration is detected when the flexiblebody is far away, behind the contact surface. The remaining necessary conditions for contactrequires that the node is inside of the triangular patch. These three extra conditions are

(rtik rij )

T n ≤ 0; (rtjk rj l)

T n ≤ 0 and (rtlk rli )

T n ≤ 0. (18)

Equations (13)–(18) are necessary conditions for contact. However, depending on the con-tact force model actually used, they may not be sufficient to ensure effective contact. Notethat when point k∗ is on the boundary of the triangular patch the equality in one or more ofrelations (18) hold true being the node k considered to be inside such patch.

3.5 Continuous contact force model

A model for the contact force must consider the material and geometric properties of thesurfaces, contribute to a stable integration and account for some level of energy dissipation.

396 J. Ambrósio

On the basis of the Hertzian description of the contact forces between two solids, Lankaraniet al. [28] propose a continuous force contact model that accounts for energy dissipation duringimpact. The procedure is used for both rigid body and nodal contact.

Let the contact force between two bodies or a system component and an external object bea function of the pseudo-penetration δ and pseudo-velocity of penetration δ given by

fs,i = (Kδn + Dδ) u (19)

where K is the equivalent stiffness, D a damping coefficient and u, in this context, a unitvector normal to the impacting surfaces. The hysteresis dissipation is introduced in equation(19) by Dδ, being the damping coefficient written as

D = 3K(1 − e2)

4δ(−)δn. (20)

This coefficient is a function of the impact velocity δ(−), stiffness of the contacting surfacesand restitution coefficient e. For a fully elastic contact e = 1, whereas for a fully plastic contacte = 0. The generalized stiffness coefficient K depends on the geometry material properties ofthe surfaces in contact. For the contact between a sphere and a flat surface, the stiffness is [28]

K = 0.424√

r

(1 − ν2

i

πEi

+ 1 − ν2j

πEj

)−1

(21)

where νl and El are the Poisson’s ratio and the Young’s modulus associated to each surface,respectively, and r is the radius of the impacting sphere.

The nonlinear contact force is obtained substituting equation (20) into equation (19)

fs,i = K δn

[1 + 3(1 − e2)

4

δ

δ(−)

]u (22)

This equation is valid for impact conditions, in which the contacting velocities are much lowerthan the propagation speed of elastic waves, i.e. δ− ≤ 10−5√E/ρ.

The contact forces between the node and the surface include friction forces modelled usingthe Coulomb friction model. The dynamic friction forces in the presence of sliding are

f friction = −µd fd

( |fns,i |

|qk|)

qk (23)

where µd is the dynamic friction coefficient and qk is the velocity of point k. The dynamiccorrection coefficient fd is expressed as

fd =

0 if |qk| ≤ v0(|qk| − v0)

(v1 − v0)if v0 ≤ |qk| ≤ v1

1 if |qk| ≥ v1.

(24)

The dynamic correction factor prevents the friction force from changing direction for almostnull values of the nodal tangential velocity, which would be perceived by the integrationalgorithm as a response with high frequency contents, forcing it to dramatically reduce the timestep size. The friction model represented by equation (24) does not account for the adherencebetween the node and the contact surface. The interested reader is referred to the work of Wuet al. [29] for a comprehensive discussion on the topics of friction and sliding in multibodydynamics.


4. Applications to crash analysis of road and rail vehicles

The applications used in this work to demonstrate the procedures proposed address bothstructural and biomechanical crashworthiness of road and rail vehicles in crash scenariosfor which the general dynamics of the vehicles, associated to the suspension systems and tothe wheel–rail or road interaction, play an important role on the system response. The firstapplication, to the design of the interface between rail vehicles of the same train, shows theimportance of having an accurate model for the vehicle pitching. In the second application, tothe side impact of a road vehicle, it is the roll and side displacement of the car that influencesthe outcome of the crash. In the third and final application, the study of the rollover of an all-terrain vehicle, not only the complex interaction between the vehicle suspension systems andthe ground is emphasized but also the occupants’ behaviour during ejection is characterized.

4.1 Railway crashworthiness

The design of railway vehicles for crashworthiness requires that not only their ends are ableto deform in a controlled manner, therefore absorbing energy during a crash, but also that allthe devices that connect the different cars in the same train also deform and absorb energy byplastic deformations. However, in order for the structural components and connection devicesto work properly, it is necessary that the vehicles remain aligned during the crash. The anti-climbers are the devices that, being located at the ends of each car, ensure that such alignment ismaintained. In their design, it is necessary to know the shear forces that they have to withstandduring the train crash. The methodology described in this work is demonstrated in the designof these rail vehicle components.

A typical arrangement of a train set with eight individual car-bodies is presented in table 1.The length and the mass of each individual car are also shown in table 1. The model of eachindividual car, shown in figure 12, is composed of five rigid bodies, B1 to B5, which representthe passenger compartment, bogie chassis and deformable end extremities. The relative motionbetween the multibody components is restricted by two revolute joints, R1 and R2, and by twotranslation joints, T1 and T2. The vehicles are assumed to be stiff and, therefore, no bendingflexibility is included in the models. The inertia and mechanical properties of the systemcomponents are described in Milho et al. [30].

The first simulation scenarios are characterized by a moving train, travelling from right toleft, which collides with another train parked with brakes applied. The trains are guided alongthe same rails, the collision velocities being 30, 40 and 55 km/h for each simulation. Of specialimportance to the anti-climber design are the simulation results for the contact forces and therelative displacements between car-body extremities [30].

The vertical relative displacement between the points of the contact surfaces defining theanti-climber devices is described by the distance g measured along the contact surface, betweenpoints A and B, which are initially leveled, as shown in figure 13. This displacement iscalculated when contact between the end extremities of the car-bodies occurs. The verticalrelative displacement obtained in the interfaces between car-bodies is illustrated in figure 14.

Table 1. Train set configuration.

Length (m) 20 26 26 26 26 26 26 26 26 26Mass (103 kg) 68 51 34 34 34 34 34 34 34 51

398 J. Ambrósio

Figure 12. Car-body model for a single car.

Figure 13. Anti-climber device contact geometry.

The vertical relative displacement, which results from the pitch motion of adjacent cars, tendsto increase for the interfaces away from the high-energy interface, reaching maximum levels inthe colliding train. High-energy zones (HE) mean the extremities of the train set in the frontalzone of the motor car-body and in the opposing back zone of the last car-body. The HE arepotential impact extremities between two train sets. The low-energy zones (LE) are located inthe remaining extremities of the train car-bodies and correspond to regions of contact betweencars of the same train set.

The tangential force in the anti-climber device is described as the tangential component ofthe contact force between the end extremities of the car-bodies. The maximum values of thetangential force are lower than half of the weight of the passenger compartment. The higher

Figure 14. Vertical relative displacement between car-bodies at the interfaces.


Figure 15. Maximum tangential force along the interfaces.

levels for the tangential force occur at the interfaces away from the HE, where the vertical gapbetween adjacent cars reaches higher levels and are located predominantly in the collidingtrain, as illustrated in figure 15. It is observed that the tangential force at the interfaces tendsto increase both in magnitude and in frequency in the final stage of the trains impact.

In a second design stage, the multibody railway vehicle is simulated in a train crash scenariosimilar to that of an experimental test performed to validate a low-energy end design developedwithin the framework of the Brite/Euram III project SAFETRAIN [31]. The experimental testconsists of a vehicle moving with a velocity of 54 km/h toward a composition with two vehiclesstopped on the railroad, as depicted in figure 16. The two stationary vehicles are equippedwith low-energy ends and connected by a coupler device. See Milho et al. [32] for a moredetailed description of the model.

The force–time history of the buffers of wagon A is displayed in figure 17 for both thesimulation and the experimental test. Note that the experimental test results are plotted fora single buffer, whereas the expected force resulting from the simulation is shown for thecumulative effects of the two buffers of wagon A.

The velocities of the three cars during the simulation are plotted in figure 18. It can beobserved that the velocities predicted by the model are very similar to those observed in theexperimental test.

The contact between wagons A and C is predicted to happen with no initial vertical gapbetween the buffers, as shown in figure 19. However, as the buffers approach each other, thevertical gap between the wagons increases, reaching a maximum of 15 mm. The vertical forcesthat the buffers have to support in order to prevent overriding, presented in figure 19, oscillatewith a maximum peak of 15 ton.

Table 2 presents the amount of energy dissipated by the different components of the energyends and the energy breakdown for the experimental test. Note that the energies predicted inthe simulation and in the test have a difference of 10%, mainly due to the honeycomb. Another

Figure 16. Collision scenario used in the numerical simulations and in the experimental test.

400 J. Ambrósio

Figure 17. Force–time history of the buffers of wagon A. Half of the force magnitude on the buffers is comparedwith the force measured in the left buffer of wagon A.

relevant observation is that the total energy absorbed by the buffers and coupler is similar forthe test and for the simulation. However, its breakdown is completely different. This is dueto the force displacement curves used for the coupler and buffers in the model. Some of theresistance of the coupler is included in the buffers curve instead of these two systems beingmodelled independently.

4.2 Side impact of a road vehicle

The appraisal of new strategies for protection of occupants of road vehicles during side impactis the aim of the study described here, which is part of the work developed in projectAPROSYS-SP6 [33]. A multibody model of a vehicle Chrysler Neon with two US-DOT SID dummiesinside, based on the original model developed by TNO Automotive, uses the methodologiesdescribed in this work. The analysis of a crash scenario described in the norm FMVSS 214 isshown in figure 20. The dummy and impact barrier models correspond to the MADYMO 5.2

Figure 18. Time history for the wagons in the simulation and experimental test.


Figure 19. Time history of the gap and vertical contact forces in the buffers.

database US-DOT SID dummy and FMVSS 214 barrier. For the simulations carried out andreported in this document, the multibody dynamics simulation code MADYMO [12] is used.

The model of the Chrysler Neon for side impact is made of eight subsystems represented infigure 20. Besides the side structure of the vehicle and the seats all remaining structural partsare considered rigid, as they are supposed to play no role in the side impact crash scenario.Each subsystem is made by rigid bodies constrained by kinematic joints. For each part of theside structure of the vehicle, the bodies that make it up are presented in figure 21.

Table 2. Energy dissipation distribution in the components of the train.

Component Absorption (kJ) Remarks Test (kJ)

Buffers Wagon C: 624 This result does not include structural 280Wagon A: 373 deformation behind the buffersTotal: 997 No structural deformation occurs in the simulation

Coupler 300 Test data includes structural deformationbehind the coupler 835

Low energy end 1 297 Test data includes structural deformationbehind the coupler and buffers 1 435

Front honeycomb 2 780 3 016Total energy absorption 4 077 4 451

Figure 20. Multibody model for the Chrysler Neon in the side impact test defined by the norm FMVSS 214.

402 J. Ambrósio

Figure 21. Rigid bodies defining the side structure of the vehicle.

The kinematic joints represented in figure 22 are set according to the expected mechanisms ofdeformation that the vehicle can experience for the side crash tests. Caution must be exercisedif the model is to be used in any other type of side impact test besides the ones prescribed inthe FMVSS or ECE regulations.

In order for the plastic hinge definition of the side structure to be complete, it is necessary todefine the constitutive relations that relate the moments developed in the kinematic joints andthe angles associated to each degree-of-freedom. Such constitutive relations, for a selectednumber of plastic hinges, are illustrated in figure 23.

Figure 22. Kinematic joints for the side structure representation.


Figure 23. Moment–angle relations for plastic hinges of the side structure.

The energy absorption of the vehicle side structure in the test configuration can only berealized when it is included in the complete vehicle. Because all structural plastic deformationsare contained in the domain of the side structure, the remaining of the chassis can be consideredrigid. The attachments between the rigid bodies of the side structure and the rigid body ofthe chassis are realized by the crushable elements. The deformation of these elements mustbe within prescribed limits. The plastic deformation energy that these elements account for isdue to the deformation of the structure surrounding the roof, sill, A-pillar and C-pillar. If thedeformation of these crushable elements exceeds a given limit, it can be argued that the plasticdeformations of the side structure exceed the modelled region of the vehicle and, therefore,the validity of the model can be questioned.

The multibody model of the vehicle Chrysler Neon is simulated in the same crash testscenario with which prototypes of the real vehicle have been experimentally tested by NHTSA.The quantities measured during the experimental tests are used as target responses that themodel of the vehicle must meet in order to be considered validated. The vehicle responsesin time used for the model validation are: velocity of the vehicle center of mass; velocityof the barrier; velocity of the front door; velocity of the rear door; velocity of the rear floor;acceleration of the rear door in the Y -direction; velocity of the sill; acceleration of the US-DOTdriver rib; acceleration of the driver dummy torso; acceleration of the driver dummy pelvis;acceleration of the passenger dummy rib; acceleration of the passenger dummy torso andacceleration of the passenger dummy pelvis. The validation procedure, carried out by TNOAutomotive, included the fine tuning of the plastic hinges data that leads to a better correlationbetween simulation and experimental responses.

Figure 24 shows the results obtained for the simulation of the FMVSS 214 side crashtest in terms of head, pelvis, thoracic vertebras and mid-rib accelerations. The US-DOT SIDinjury criteria corresponding to this simulation are presented in table 3. The value of TTI isabove the maximum acceptable value specified in FMVSS 214 and the value for the pelvislateral acceleration is also quite high, even though below the limit specified in the norm. Thevelocities of different points of the structure and dummies are similar to those obtained in theexperimental test.

The contact forces for the dummy presented in figure 25 have not been measured in theexperimental test. It is observed that the peak forces for the pelvis, thorax and head occursimultaneously. The pelvis contact force predicted by the simulation is clearly larger than theforces carried by the head and by the thorax. This is mainly due to the proximity of the seatand occupant driver pelvis to the B-pillar, which is directly struck by the barrier.

Another type of data that is not recorded in the experimental test is the kinematics of theB-pillar and of the occupant. Such kinematics is sketched in figure 26 for different instantsof time. It is clear that the B-pillar intrudes the vehicle passenger compartment, with specialincidence in its lower part. The intrusion of the B-pillar leads to contact with the driver’s seatand pushes it to the vehicle interior.

404 J. Ambrósio

Figure 24. Y -component accelerations for parts of the dummy measured in the global coordinate system.

The results of the simulation also show the importance of modelling the suspension ofthe vehicle and the contact between the tires and the ground. The vehicle roll and its lateraldisplacement play an important role in the progression of the crash event. Moreover, theamount of kinetic energy that is dissipated by the work of the suspension systems and by thefriction between tires and ground cannot be neglected for lower impact speeds.

The results reported for the side impact demonstrate the reliability of the multibody modelof the vehicle and occupants used in this crash scenario. Though not presented here, severalmodifications on the structural components of the side of the vehicle, interior furbishing andequipment arrangement have been tested. On the basis of the outcome of this simulations,described by parameters such as relative displacements between structural components andoccupant anatomical segments, accelerations of points on the dummies and on the structure,contact forces or injury indexes, it is possible to obtain vehicle designs that optimize passengerprotection. However, the extensive number of re-analysis of the complete vehicle is onlypossible when the computation time for each analysis is acceptable. In order to appraisethe computational efficiency of the methodology described here, the time required for eachsimulation of the vehicle side impact with the multibody formulation is measured in terms ofminutes, whereas for the equivalent model in finite elements, in the same test conditions andperiod of analysis, the time required is measured in terms of days.

Table 3. Injury criteria values: pelvis accelerationsand thoracic trauma index.

Simulation ID PLA (g) TTI (g)

FMVSS 214 limits 130 85Chrysler Neon performance 110.23 87.83


Figure 25. Contact forces in the pelvis region, thorax region and head.

4.3 Road vehicle rollover

With the purpose of showing the performance of the formulation in situations where multiplecontacts are important issues, such as in the case of vehicle rollover, a model for an all-terrainvehicle, a M151A2 Jeep represented in figure 27, is presented. The full vehicle, which includesa rollbar cage for occupant protection, has a total mass of 1470 kg. The location of the vehicle’scenter of mass is 1.232 m behind the front axle and 0.607 m above the ground, when parkedon a flat horizontal surface.

The utility vehicle is modelled with 13 rigid bodies, corresponding to the chassis, doubleA-arm front suspension systems and trailing arm rear suspension systems. The interestedreader will find the full set of data for the utility vehicle in Ambrósio [34]. A rollbar cage isinstalled in the truck in order to protect the vehicle’s occupants in case of rollover. This is aflexible frame mounted over the chassis, as depicted in figure 28, and is made of 1025–1030

Figure 26. Sequence of images for the side crash.

406 J. Ambrósio

Figure 27. General dimensions of the all-terrain vehicle.

steel. The cross sectional area of each bar is annular with an outside radius of 2.54 cm. Amodel of the rollbar cage with 13 beam elements is used here.

The interaction of the vehicle and/or the rollbars with the ground is described by controllingthe coordinates of six points in the rollbar cage P1 through P6 and eight points (C1 through C8)for possible ground contact. The model for the deformation of the vehicle chassis is valid onlyif all deformations occur on the rollbar. This model cannot be used, as it is, to describe thedeformation of other parts of the chassis. However, some of the energy dissipation involvedin the impact of the rigid chassis is still described by using the continuous force contactmodel.

The all-terrain vehicle model is simulated here in a rollover situation with the initialconditions described in figure 29. The initial conditions of the simulations correspond toexperimental conditions where the vehicle moves on a cart with a lateral velocity of 13.41 m/suntil the impact with a water-filled decelerator system occurs. The vehicle is ejected with a rollangle of 23◦. The initial velocity of the vehicle, when ejected, is 11.75 m/s in the Y -direction,while the angular roll velocity is 1.5 rad/s.

Three occupants, with a 50th percentile, are modeled and integrated with the vehicle. Thetwo occupants in the front of the vehicle have shoulder and lap seatbelts, whereas the occupantseated in the back of the vehicle has no seatbelt. The initial positions of the occupants corre-spond to a normally seated driver, a front passenger bent to check out the ‘glove compartment’and a rear occupant with a ‘relaxed’ position.

This setup and the simulation outcome are compared with that of two experimental tests ofthe vehicle with three Hybrid III dummies that have been carried at the Transportation Research

Figure 28. Computational model of the rollbar cage.


Figure 29. Rollover simulation scenario: (a) initial conditions; (b) position of the vehicle occupants.

Figure 30. View of the experimental test for the truck rollover.

Center of Ohio [35, 36]. An overview of the footage obtained in one of the experimental testsis shown in figure 30.

The first 2 s of the simulations, presented in figure 31, show that the first contact of thewheels with the ground occurs at 0.3 s, for all simulations, causing the vehicle to bounce withan increasing roll velocity. At 0.75 s, the vehicle impacts the ground with the rollbar cage andcontinues its rolling motion with contact by different points of the structure. The occupantsin the front of the vehicle are hold in place by the seatbelts. Upon continuing its roll motion,the vehicle impacts the ground with its rollbar cage, while the ejection of the rear occupantis complete. The results do not seem to be very sensitive to different contact models or to thecomplexity of the finite element mesh applied in the rollbar, but they are extremely sensitive tothe values of the friction between vehicle structure and ground. Owing to the highly nonlinearnature of the problem, after relatively large periods of analysis, the behaviours of differentmodels diverge.

Figure 31. View of the outcome of the rollover simulation of a vehicle with three occupants.

408 J. Ambrósio

Figure 32. Deformations of the rollbar cage, represented by node P1.

In figure 32, the permanent deformations of the front nodal points of the rollbar cage,which impacted the ground first, are displayed. The permanent deformation of 13 cm inthe lateral direction and 20 cm in the frontal direction obtained in the simulations is sim-ilar to the permanent deformations observed in experimental tests. Models with 12 and24 beam finite elements for the rollbar cage show similar results for the first part of theanalysis. The differences between the responses in the second part of the analysis must beattributed to the nonlinearity of the problem rather than to the models used for the rollbarcage.

Figure 33. Severity index for the vehicle occupants.


The severity index observed for the occupants in figure 33 indicates a very high probability offatal injuries in the conditions simulated. Note that the model has rigid seats, interior trimmingfor the dashboard, side and floor panels, and that the ground is also considered to be rigid. Ifsome compliance is included in the vehicle interior, it is expected that the head accelerationsare lower.

The kinematics of the biomechanical models of the occupants, and in particular that ofthe ejected occupant, are similar to the kinematics of the crash test dummies used in theexperimental tests. Several simulations of the vehicle rollover with occupants seating withdifferent postures have been performed. These simulations show that regardless of the rearoccupant seating posture the ejection and post-ejection occupant kinematics remains basicallyunchanged.

5. Conclusions

This work demonstrated a multibody dynamics-based formulation that effectively promotes thesimultaneous analysis of the vehicle stability and manoeuvrability dynamics with structuraland occupant crashworthiness. As the formalisms used are common to all disciplines thatare relevant to vehicle design, including structural crashworthiness, it is possible to recyclevehicle models for different applications, to account for the different mechanisms of energyabsorption for a wider range of crash velocities and to have a better appraisal for the occupant’smechanisms of injury and injury criteria. In the process, it was also demonstrated how nonlinearfinite elements can be integrated with conventional rigid multibody descriptions in order tobuild better general vehicle models. Though not demonstrated through the applications usedhere, the reliability of the prediction of certain crash mechanisms allows devising drivingsupport systems that assist the driver in crash avoidance manoeuvres. Through the railwaycrashworthiness application, it was demonstrated that the design of the train systems thatensure the train alignment during the crash requires that the relative motion between the carsof the same train is well described, and therefore, that the suspension models, developed inthe framework of typical vehicle dynamics applications, are used to account for the relativepitching between adjacent vehicles. The side impact of the road vehicle confirmed theseconclusions, now in what the vehicle roll motion is concerned. The rollover simulations showedhow the vehicles can withstand multiple contacts when the simulation is being preformed. Allthese applications are run in a reasonable short amount of time, measured in minutes, whichdemonstrates the suitability of the methods for studies that require extensive re-analysis.

Acknowledgements

The support of the Portuguese Foundation for Science and Technology, FCT, through project2/2.1/TPAR/2041/95, to study the vehicle rollover is gratefully acknowledged. The support ofEC through projects BE-96-3092 (SAFETRAIN), with the partners ADTRANZ/SOREFAME(Pt), ERRI (Nl), SNCF (Fr), DB (Ge), PKP (Pl), FMH (Pt), CIC (UK), GEC/MC (UK), AEATechnology/BRR (UK), U. Valencienne (Fr), GAC/CIMT (Fr), ALS/DDF (Fr), IFS (Ge),TU Dresden (Ge), DUE (Ge), enabled to develop the studies on train crashworthiness andthrough project TIP3-CT-2004-506503 (APROSYS-SP6), with the partners SIEMENS VDO(Ge), Faurecia (Ge), DaimlerChrysler (Ge), CIDAUT (Sp), Fh G (Ge), Warsaw University ofTechnology (Pl) and TNO (Nl) provided the results of the road vehicle side impact studies.The EC support is also greatly appreciated.

410 J. Ambrósio

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