Crash Analysis and Dynamical Behaviour of Light Road and Rail Vehicles

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

Crash analysis and dynamical behaviour of light road 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 other road or rail users. The crash analysis of vehicles involves structural impact and occupant biomechanics. The traditional approaches to crashworthiness not only do not take into account the full vehicle dynamics, 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 analysis or 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 occupant 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 occupant biomechanical modelling are revised. Afterwards, it is shown how multibody models of vehicles and occupants are used in crash analysis. The more traditional aspects of vehicle dynamics are then introduced in the vehicle models in order to appraise their importance in the treatment of certain types of impact scenarios for which the crash outcome is sensitive to the relative orientation and alignment between vehicles. Through applications to the crashworthiness of road and of rail vehicles, selected problems are discussed and the need for coupled models of vehicle structures, suspension subsystems and 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 comfort of occupants. The active safety of vehicles includes the manoeuvrability and stability and the study of all systems that address their improvement such as suspensions, active control and electromechanical subsystems for driving support, among others. Passive safety of vehicles addresses the protection of occupants and cargo from the moment that an accident starts until the vehicle stops. Vehicle system dynamics addresses mostly the active safety aspects of the vehicle design, whereas vehicle crashworthiness deals with the structural impact, the occupant biomechanics and all subsystems aimed to provide a better protection during a crash.

*Corresponding author. Email: [email protected] Vehicle System Dynamics ISSN 0042-3114 print/ISSN 1744-5159 online © 2005 Taylor & Francis Group Ltd http://www.tandf.co.uk/journals DOI: 10.1080/00423110500151788 386 J. Ambrósio

Though the most common numerical tools used to address both areas of vehicle dynamics are based on multibody methodologies, traditionally these two aspects of vehicle design have been addressed separately or with little interaction. It is an objective of this work to show that not only the two aspects of vehicle dynamics can be addressed simultaneously, but also that in many studies that address the passive safety of vehicles it is fundamental to include detailed models of the vehicle active safety subsystems. Moreover, the common trend in actual vehicles is to have active driving support and intelligent safety systems, both requiring the use of sensors to provide the data required for the decision making processes. Also from this point of view, a common approach to active and passive safety of road and rail vehicles is of major interest. Until the early 1970s, crash studies relied almost exclusively on experimental testing which focused mostly in full scale testing and in the development of relevant testing scenarios [1]. In a review work, Tidbury [2] stressed the facts that not only the costs associated with experimental testing are very high, but also the results cannot be generalized to other impact situations neither can such methods be used during the design of new vehicles. Therefore, the need for accurate numerical procedures became apparent at this time. The earlier numerical methods used for vehicle crashworthiness were based on the use of lumped masses and nonlinear springs. The models built with these methods, known as lumped parameter models, use lumped masses to represent parts of the vehicle, such as the engine block or the passenger compartment, considered rigid during the analysis, and springs to represent the structural elements responsible for the deformation energy management [3]. The lumped parameter models are mostly one- or two-dimensional and they include at the most very simplified representations of the suspension elements of the vehicles. The first numerical model for the simulation of the vehicle occupant biomechanics proposed by McHenry in 1963 [4] was a two-dimensional model that included four articulated rigid bodies and the occupant restraint system. Later, Lobdell [5] proposed a lumped parameter approach with similar numerical characteristics to that used in vehicle structural impact for the simulation of human thorax loading. The developments of lumped parameter biomechanical models continued through the years of 1970 and 1980 in parallel with the development of models for the structural impact of vehicles but without the synergies associated to the use of a common numerical framework. The generalization of the vehicle structural and occupant biomechanical models to become fully three-dimensional required another type of numerical methodologies. The multibody dynamics methods provided the framework in which more advanced developments could be undertaken. The plastic hinge approach, proposed by Nikravesh et al. [6] to represent the structural dynamics of road vehicles in crash situations, uses a general multibody representation where the full vehicle is described and the structural systems are divided into several rigid bodies connected by kinematic joints to which nonlinear springs, representing the structural 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 versatile and efficient methodology to deal with structural crashworthiness, especially in the initial phases of the vehicle design, when extensive re-analysis is required. Still with little or no contact to the developments in structural crashworthiness, the multibody methodologies began to 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 Robbins et al. [9] and Fleck et al. [10], the use of multibody dynamics methodologies for occupant biomechanics reached a level of maturity that allowed it to become the most popular numerical approach used today in impact biomechanics. The acceptance of the multibody-based models by the vehicle industry led to the successful development of commercial occupant simulation computer 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 element method ensured the feasibility of finite element-based models for the study of the structural crashworthiness of vehicles. One of the first successful attempts to apply nonlinear finite elements to the prediction of a vehicle crush was reported by Thompson [13]. Belytchko and Hsieh [14] proposed the finite element developments that defined the framework of their use in structural crashworthiness. During the 1970s, several nonlinear finite element codes suitable to 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 served as 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. The application possibilities of this method are being expanded in order to allow for the complex modelling of vehicle and integrated occupants, but still, the representation of the suspension systems of the vehicles and of systems responsible for their active safety are not yet present in the finite element models, unless at the cost of a large computational effort. The amount of information required for large systems is still too high making the application of FEM codes suitable only for the analysis phase of structural and biomechanical systems. For the early design phases, the multibody models are still the primary numerical tool used in simulation. Though the features of the different numerical methodologies based on multibody dynamics and on finite elements include most of the ingredients required for the crash analysis of road and rail vehicles, the models more commonly developed neither integrate the structural crashworthiness with the occupant biomechanics nor take into account the overall dynamics of the vehicle. Applications that include low and medium vehicle impact speeds or multiple impacts or long contact periods or even for which the alignment between vehicles is important for the outcome of the crash are examples of cases for which the vehicle dynamics associated to the suspensions and to the wheel road–rail contact can play an important role. Traditionally, the most common numerical techniques for vehicle dynamics modelling are based on multibody dynamics methodologies, which are the same used for vehicle impact models and for occupant biomechanics modelling. Therefore, no reason exists for not taking the advantage of their features 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 unified framework for the development of design and analysis tools of crashworthy structures and of occupants. The models developed are computationally efficient in today’s computers to allow for the extensive re-analysis cycles required in the optimal design of vehicles for crashworthiness. Furthermore, in applications for which the structural patterns of deformation are not known beforehand or for which subcomponent finite element models are available, the multibody dynamics methods enable the effective coupling between the lumped parameter models and the finite element description of the structural deformations. The application of the methods presented is addressed here through the study of the development of energy absorption and anti-climber devices for railway vehicles, the development of occupant protection strategies in side 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 multibody systems, each of them presenting relative advantages and drawbacks. The objective of this work is not to discuss what is the most efﬁcient multibody methodology that can be applied for vehicle dynamics crash analysis. In this work, the methods presented are based on the use of 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 type of equations of motion obtained with the use of Cartesian coordinates. It is also assumed that the different numerical issues that arise from the use of this type of coordinates, such as the existence of redundant constraints and the possibility of achieving singular positions, are also solved. For a more detailed discussion on the numeric aspects of this type of coordinates, the interested 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 their relative motion constrained by kinematic joints and that are acted upon by external forces. A generic 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 of unconstrained bodies are Mq¨ = g (1) where M is the mass matrix, which includes the masses and inertia tensors of the individual bodies, 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 acceleration constraint equations, respectively, written as ˙ (q,t)≡ Dq˙ = v (3) ¨ (q, q˙,t)≡ Dq¨ = γ where D is the Jacobian matrix. For a system of constrained bodies, the effect of the kinematic joints can be included in equation (1) by adding to their right-hand side the equivalent joint reaction 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 unknowns that must be solved together with the second time derivative of the constraint equations. The

Figure 1. Generic multibody system. Crash analysis and dynamical behaviour of vehicles 389

Figure 2. Flowchart representing the forward dynamic analysis of a multibody system. resulting system of differential-algebraic equations is MDT q¨ g = (5) D0 λ γ

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

2.2 Solution of the equations of motion

The forward dynamic analysis of a multibody system requires that the initial conditions of the system, i.e. the position vector q0 and the velocity vector q˙ 0, are given. With this information, equation (5) is assembled and solved for the unknown accelerations, which are in turn integrated 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 the structural components during impact is properly represented and that the contact forces are described. A proper description of the structural deformation means that not only the mechanisms of deformation and the relative displacements between the different structural components are captured with accuracy, but also that the deformation energy distribution is well represented. Methodologies based on lumped deformations, such as the ﬁnite segment and the plastic hinge approaches, or on a continuous description of the deformation, such as the ﬁnite element method, are presented here to describe the structural deformation of the vehicle. The contact force models that are used in vehicle impact should be consistent with 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 of this type described here. The tyre contact models used for road vehicles and the wheel–rail contact models of railway vehicles are also used in some problems of vehicle impact to describe the ground interaction forces. However, their description falls out of the scope of this work. 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 as the connecting rods of high-speed machinery or the structural frame of buses and trucks. In this case, each slender component can be modelled as a collection of rigid bodies connected by linear springs, as shown in figure 3. These springs, representing the axial, bending and torsion 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 attached to the rigid bodies used for their model rotate and translate with respect to each other. Forces and moments applied to the rigid bodies can be calculated from the relative end displacements and 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 shown in figure 4. The characteristics of these springs are related with the material and geometric properties of the system components [23]. Using the principles of structural analysis, the stiffness coefficients for these springs are calculated. For instance, the straight extensional segment and the straight bending, represented in figures 4(a) and (b), have their stiffness coefficients, respectively, given by

E A e = e = i i ki kj 2 (6a) li E I e = e = i i ki kj 2 (6b) li where Ei is the Young modulus, Ai the cross-section area, Ii the cross-section moment of inertia and li the length of the ﬁnite segment. Note that the ﬁnite segment methodology, as described in the original work by Huston and Wang [23], only applies to structural components made of beams that have linear elastic deformations. 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 in bending, giving rise to plastic deformations in highly localized regions, called plastic hinges. These deformations, presented in ﬁgure 5, develop where maximum bending moments occur, at load 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. Crash analysis and dynamical behaviour of vehicles 391

Figure 5. Localized deformations on a beam and a plastic hinge. their location is predicted well in advance. The methodology described herein is known as conceptual modelling in some industrial areas. The plastic hinge concept has been developed by using generalized spring elements to represent the constitutive characteristics of localized plastic deformation of beams and kinematic joints in order to control the deformation kinematics, as illustrated in figure 6.A more complex structure, such as the B-pillar of a road vehicle or the end-underframe of a train car shown in figure 7, can be represented by a collection of rigid bodies connected by the joint-spring arrangements that describe its deformation and energy absorption characteristics. For a flexural plastic hinge, the spring stiffness is expressed as a function of the change of the relative angle between two adjacent bodies connected by the plastic hinge, as shown in figure 8. For a bending plastic hinge, the revolute joint axis must be perpendicular to the neutral axis of the beam and to the plastic hinge bending plane simultaneously. The relative angle between the adjacent bodies measured in the bending plane is

= − − 0 θij θi θj θij (7)

0 where θij 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 nodal rotational displacements. The characteristics of the spring-damper that describes the properties of the plastic hinges are obtained by experimental component testing, finite element nonlinear analysis or simplified analytical 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 rail vehicle; (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 hardening and strain rate sensitivity of some materials. A dynamic correction factor is used to account for the strain rate sensitivity [24]

P d = + 0.82 1 0.07 V0 . (8) Ps

Here Pd and Ps are the dynamic and static forces, respectively, and V0 is the relative velocity between the adjacent bodies. The coefﬁcients appearing in equation (8) are dependent on the type 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 hinges requires that the mechanisms of deformation are known beforehand, which is very often the case in most of the practical applications. However, in cases for which the structural deformations are generalized, rather than concentrated in specific points, the modelling capabilities of the finite element method are irreplaceable. The nonlinear finite element formulation is Crash analysis and dynamical behaviour of vehicles 393

Figure 9. General motion of a flexible body. summarized here in the framework of flexible multibody dynamics to be compatible with the standard description of the multibody formulation presented before. The motion of a flexible body, depicted in figure 9, is characterized by a continuous change of 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 the principle of the virtual work be used to express the equilibrium of the flexible body in the current configuration t + t and an updated Lagrangian formulation be used to obtain the equations of motion of the flexible body [25]. The finite elements used in the discretization of the flexible body are assembled, leading to the equations of motion of the flexible body               Mrr Mrf Mrf r¨ gr sr 0 00 0 0               Mφr Mφφ Mφf ω˙ = gφ − sφ − 0 − 00 0 0 (9) ¨ + Mfr Mfφ Mff u gf sf f 00KL KNL u where r¨ are ω˙ are, respectively, the translational and angular accelerations of the body fixed reference 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 gross motion of the body and its deformation. The right-hand side of equation (9) contains the vector generalized forces applied on the deformable body g and matrices KL and KNL, which are the linear and nonlinear stiffness matrices, respectively. Vector f denotes the equivalent nodal forces due to the state of stress. In order to improve the numerical efficiency of the solution of equation (9), a lumped mass formulation is used and the nodal accelerations u¨ , measured with respect to the body fixed ¨ frame, are substituted by the nodal accelerations qf 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 fixed coordinate frame is attached to the center of mass of the rigid part, as shown in figure 10. The flexible and rigid parts are attached by the boundary nodes ψ. The procedure is described in Ambrósio and Nikravesh [26] leading to the new form of the equations of motion       ∗ T ∗ + mI + AM A −AM S0 r¨ fr ACδ   T  ∗ T T ∗  ω˙  =  − ˜ − T −  − AM S J + S M S0 n ω J ω S Cδ I Cθ (10) q¨ − − + 00Mff f gf f (KL KNL)u

where J is the inertia tensor, expressed in body ﬁxed coordinates, fr the vector of the external forces 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 diagonal mass matrix containing the mass of the boundary nodes and

T =[ ··· ]T; =−[˜ ˜ ··· ˜ ]T; =[ ··· ]T A AA A S x1 x2 xn I II I where A is the transformation matrix from the body ﬁxed to global coordinates and xk denotes the position of node k. Vectors Cδ and Cθ are the reaction forces and moment of the ﬂexible part of the body over the rigid part is given by

= − − + − + Cδ gδ Fδ (KL KNL)δδ δ (KL KNL)δθ θ (11) = − − + − + Cδ gθ 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 with respect to the translational and rotational nodal degrees of freedom. The underlined subscripts refer 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 multibody system using the Lagrange multiplier method when a kinematic constraint involving the nodal coordinates has to be set. The mechanical joints of the vehicles, modelled using this formulation, are described by such kinematic constraints. For more information on the definition 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 methodologies used to detect contact in other type of applications in vehicle dynamics. Let a body of the system get close to a surface during the motion of the multibody system, as represented in figure 11. Without lack of generality, let the impacting surface be described by a mesh of triangle 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 rigid body or a nodal point of the finite element mesh of a flexible body. The normal to the outside surface 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) Crash analysis and dynamical behaviour of vehicles 395

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 patch surface, and a normal component, given, respectively, by t = − T rik rik (rik n)n (14) n = T rik (rik n)n. (15) A necessary condition for contact is that node k penetrates the surface of the patch, i.e. T ≤ rik n 0. (16) In order to ensure that a node does not penetrate the surface through its ‘interior’ face, a thickness e must be associated to the patch. The thickness penetration condition is − T ≤ rik n e. (17) The condition described by equation (17) prevents that penetration is detected when the ﬂexible body is far away, behind the contact surface. The remaining necessary conditions for contact requires that the node is inside of the triangular patch. These three extra conditions are ˜t T ≤ ; ˜t T ≤ ˜t T ≤ (rik rij ) n 0 (rjk rjl) n 0 and (rlk rli) n 0. (18) Equations (13)–(18) are necessary conditions for contact. However, depending on the contact force model actually used, they may not be sufﬁcient to ensure effective contact. Note that when point k∗ is on the boundary of the triangular patch the equality in one or more of relations (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 the surfaces, 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, Lankarani et al. [28] propose a continuous force contact model that accounts for energy dissipation during impact. 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 be a function of the pseudo-penetration δ and pseudo-velocity of penetration δ˙ given by

n ˙ fs,i = (Kδ + Dδ) u (19) where K is the equivalent stiffness, D a damping coefficient and u, in this context, a unit vector normal to the impacting surfaces. The hysteresis dissipation is introduced in equation (19) by Dδ˙, being the damping coefficient written as 3K(1 − e2) D = δn. (20) 4δ˙(−) This coefficient is a function of the impact velocity δ˙(−), stiffness of the contacting surfaces and restitution coefficient e. For a fully elastic contact e = 1, whereas for a fully plastic contact e = 0. The generalized stiffness coefficient K depends on the geometry material properties of the surfaces in contact. For the contact between a sphere and a flat surface, the stiffness is [28]

−1 √ 1 − ν2 1 − ν2 K = 0.424 r i + j (21) πEi πEj 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) − 2 ˙ n 3(1 e ) δ fs,i = Kδ 1 + u (22) 4 δ˙(−)

This equation is valid for impact conditions, in which the contacting√ velocities are much lower than 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 using the Coulomb friction model. The dynamic friction forces in the presence of sliding are | n | friction fs,i f =−µd fd q˙ k (23) |q˙ k| where µd is the dynamic friction coefﬁcient and q˙ k is the velocity of point k. The dynamic correction coefﬁcient fd is expressed as   |˙ |≤ 0ifqk v0 (|q˙ |−v ) f = k 0 v ≤|q˙ |≤v d  − if 0 k 1 (24)  (v1 v0) 1if|q˙ k|≥v1. The dynamic correction factor prevents the friction force from changing direction for almost null values of the nodal tangential velocity, which would be perceived by the integration algorithm as a response with high frequency contents, forcing it to dramatically reduce the time step size. The friction model represented by equation (24) does not account for the adherence between the node and the contact surface. The interested reader is referred to the work of Wu et al. [29] for a comprehensive discussion on the topics of friction and sliding in multibody dynamics. Crash analysis and dynamical behaviour of vehicles 397

4. Applications to crash analysis of road and rail vehicles

The applications used in this work to demonstrate the procedures proposed address both structural and biomechanical crashworthiness of road and rail vehicles in crash scenarios for which the general dynamics of the vehicles, associated to the suspension systems and to the wheel–rail or road interaction, play an important role on the system response. The first application, to the design of the interface between rail vehicles of the same train, shows the importance of having an accurate model for the vehicle pitching. In the second application, to the side impact of a road vehicle, it is the roll and side displacement of the car that influences the 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 and the 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 able to deform in a controlled manner, therefore absorbing energy during a crash, but also that all the devices that connect the different cars in the same train also deform and absorb energy by plastic deformations. However, in order for the structural components and connection devices to 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 is maintained. In their design, it is necessary to know the shear forces that they have to withstand during the train crash. The methodology described in this work is demonstrated in the design of 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 each individual car, shown in figure 12, is composed of five rigid bodies, B1 to B5, which represent the passenger compartment, bogie chassis and deformable end extremities. The relative motion between the multibody components is restricted by two revolute joints, R1 and R2, and by two translation joints, T1 and T2. The vehicles are assumed to be stiff and, therefore, no bending flexibility is included in the models. The inertia and mechanical properties of the system components are described in Milho et al. [30]. The first simulation scenarios are characterized by a moving train, travelling from right to left, which collides with another train parked with brakes applied. The trains are guided along the same rails, the collision velocities being 30, 40 and 55 km/h for each simulation. Of special importance to the anti-climber design are the simulation results for the contact forces and the relative displacements between car-body extremities [30]. The vertical relative displacement between the points of the contact surfaces defining the anti-climber devices is described by the distance g measured along the contact surface, between points A and B, which are initially leveled, as shown in figure 13. This displacement is calculated when contact between the end extremities of the car-bodies occurs. The vertical relative 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 26 Mass (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, tends to increase for the interfaces away from the high-energy interface, reaching maximum levels in the colliding train. High-energy zones (HE) mean the extremities of the train set in the frontal zone of the motor car-body and in the opposing back zone of the last car-body. The HE are potential impact extremities between two train sets. The low-energy zones (LE) are located in the remaining extremities of the train car-bodies and correspond to regions of contact between cars of the same train set. The tangential force in the anti-climber device is described as the tangential component of the contact force between the end extremities of the car-bodies. The maximum values of the tangential 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. Crash analysis and dynamical behaviour of vehicles 399

Figure 15. Maximum tangential force along the interfaces. levels for the tangential force occur at the interfaces away from the HE, where the vertical gap between adjacent cars reaches higher levels and are located predominantly in the colliding train, as illustrated in figure 15. It is observed that the tangential force at the interfaces tends to 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 scenario similar to that of an experimental test performed to validate a low-energy end design developed within the framework of the Brite/Euram III project SAFETRAIN [31]. The experimental test consists of a vehicle moving with a velocity of 54 km/h toward a composition with two vehicles stopped on the railroad, as depicted in figure 16. The two stationary vehicles are equipped with low-energy ends and connected by a coupler device. See Milho et al. [32] for a more detailed description of the model. The force–time history of the buffers of wagon A is displayed in figure 17 for both the simulation and the experimental test. Note that the experimental test results are plotted for a single buffer, whereas the expected force resulting from the simulation is shown for the cumulative effects of the two buffers of wagon A. The velocities of the three cars during the simulation are plotted in figure 18. It can be observed that the velocities predicted by the model are very similar to those observed in the experimental test. The contact between wagons A and C is predicted to happen with no initial vertical gap between the buffers, as shown in figure 19. However, as the buffers approach each other, the vertical gap between the wagons increases, reaching a maximum of 15 mm. The vertical forces that the buffers have to support in order to prevent overriding, presented in figure 19, oscillate with a maximum peak of 15 ton. Table 2 presents the amount of energy dissipated by the different components of the energy ends and the energy breakdown for the experimental test. Note that the energies predicted in the 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 compared with 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 for the test and for the simulation. However, its breakdown is completely different. This is due to the force displacement curves used for the coupler and buffers in the model. Some of the resistance of the coupler is included in the buffers curve instead of these two systems being modelled independently.

4.2 Side impact of a road vehicle

The appraisal of new strategies for protection of occupants of road vehicles during side impact is 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 dummies inside, based on the original model developed by TNO Automotive, uses the methodologies described in this work. The analysis of a crash scenario described in the norm FMVSS 214 is shown 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. Crash analysis and dynamical behaviour of vehicles 401

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 and reported 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 in ﬁgure 20. Besides the side structure of the vehicle and the seats all remaining structural parts are 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 the side 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 280 Wagon A: 373 deformation behind the buffers Total: 997 No structural deformation occurs in the simulation Coupler 300 Test data includes structural deformation behind the coupler 835 Low energy end 1 297 Test data includes structural deformation behind the coupler and buffers 1 435 Front honeycomb 2 780 3 016 Total 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 of deformation that the vehicle can experience for the side crash tests. Caution must be exercised if the model is to be used in any other type of side impact test besides the ones prescribed in the FMVSS or ECE regulations. In order for the plastic hinge definition of the side structure to be complete, it is necessary to define the constitutive relations that relate the moments developed in the kinematic joints and the angles associated to each degree-of-freedom. Such constitutive relations, for a selected number of plastic hinges, are illustrated in figure 23.

Figure 22. Kinematic joints for the side structure representation. Crash analysis and dynamical behaviour of vehicles 403

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 be realized when it is included in the complete vehicle. Because all structural plastic deformations are contained in the domain of the side structure, the remaining of the chassis can be considered rigid. The attachments between the rigid bodies of the side structure and the rigid body of the chassis are realized by the crushable elements. The deformation of these elements must be within prescribed limits. The plastic deformation energy that these elements account for is due to the deformation of the structure surrounding the roof, sill, A-pillar and C-pillar. If the deformation of these crushable elements exceeds a given limit, it can be argued that the plastic deformations 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 test scenario 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 the model of the vehicle must meet in order to be considered validated. The vehicle responses in time used for the model validation are: velocity of the vehicle center of mass; velocity of 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-DOT driver 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 and acceleration of the passenger dummy pelvis. The validation procedure, carried out by TNO Automotive, included the fine tuning of the plastic hinges data that leads to a better correlation between simulation and experimental responses. Figure 24 shows the results obtained for the simulation of the FMVSS 214 side crash test in terms of head, pelvis, thoracic vertebras and mid-rib accelerations. The US-DOT SID injury criteria corresponding to this simulation are presented in table 3. The value of TTI is above the maximum acceptable value specified in FMVSS 214 and the value for the pelvis lateral acceleration is also quite high, even though below the limit specified in the norm. The velocities of different points of the structure and dummies are similar to those obtained in the experimental test. The contact forces for the dummy presented in figure 25 have not been measured in the experimental test. It is observed that the peak forces for the pelvis, thorax and head occur simultaneously. The pelvis contact force predicted by the simulation is clearly larger than the forces carried by the head and by the thorax. This is mainly due to the proximity of the seat and 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 the B-pillar and of the occupant. Such kinematics is sketched in figure 26 for different instants of time. It is clear that the B-pillar intrudes the vehicle passenger compartment, with special incidence in its lower part. The intrusion of the B-pillar leads to contact with the driver’s seat and 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 of the vehicle and the contact between the tires and the ground. The vehicle roll and its lateral displacement play an important role in the progression of the crash event. Moreover, the amount of kinetic energy that is dissipated by the work of the suspension systems and by the friction between tires and ground cannot be neglected for lower impact speeds. The results reported for the side impact demonstrate the reliability of the multibody model of the vehicle and occupants used in this crash scenario. Though not presented here, several modifications on the structural components of the side of the vehicle, interior furbishing and equipment arrangement have been tested. On the basis of the outcome of this simulations, described by parameters such as relative displacements between structural components and occupant 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 passenger protection. However, the extensive number of re-analysis of the complete vehicle is only possible when the computation time for each analysis is acceptable. In order to appraise the computational efficiency of the methodology described here, the time required for each simulation of the vehicle side impact with the multibody formulation is measured in terms of minutes, whereas for the equivalent model in finite elements, in the same test conditions and period of analysis, the time required is measured in terms of days.

Table 3. Injury criteria values: pelvis accelerations and thoracic trauma index.

Simulation ID PLA (g) TTI (g)

FMVSS 214 limits 130 85 Chrysler Neon performance 110.23 87.83 Crash analysis and dynamical behaviour of vehicles 405

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 multiple contacts are important issues, such as in the case of vehicle rollover, a model for an all-terrain vehicle, a M151A2 Jeep represented in figure 27, is presented. The full vehicle, which includes a rollbar cage for occupant protection, has a total mass of 1470 kg. The location of the vehicle’s center of mass is 1.232 m behind the front axle and 0.607 m above the ground, when parked on a flat horizontal surface. The utility vehicle is modelled with 13 rigid bodies, corresponding to the chassis, double A-arm front suspension systems and trailing arm rear suspension systems. The interested reader will find the full set of data for the utility vehicle in Ambrósio [34]. A rollbar cage is installed in the truck in order to protect the vehicle’s occupants in case of rollover. This is a flexible 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. A model 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 controlling the 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 only if all deformations occur on the rollbar. This model cannot be used, as it is, to describe the deformation of other parts of the chassis. However, some of the energy dissipation involved in the impact of the rigid chassis is still described by using the continuous force contact model. The all-terrain vehicle model is simulated here in a rollover situation with the initial conditions described in ﬁgure 29. The initial conditions of the simulations correspond to experimental conditions where the vehicle moves on a cart with a lateral velocity of 13.41 m/s until the impact with a water-ﬁlled decelerator system occurs. The vehicle is ejected with a roll angle 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. The two occupants in the front of the vehicle have shoulder and lap seatbelts, whereas the occupant seated in the back of the vehicle has no seatbelt. The initial positions of the occupants correspond 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 of the vehicle with three Hybrid III dummies that have been carried at the Transportation Research

Figure 28. Computational model of the rollbar cage. Crash analysis and dynamical behaviour of vehicles 407

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 tests is shown in figure 30. The first 2 s of the simulations, presented in figure 31, show that the first contact of the wheels with the ground occurs at 0.3 s, for all simulations, causing the vehicle to bounce with an increasing roll velocity. At 0.75 s, the vehicle impacts the ground with the rollbar cage and continues its rolling motion with contact by different points of the structure. The occupants in 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 occupant is complete. The results do not seem to be very sensitive to different contact models or to the complexity of the finite element mesh applied in the rollbar, but they are extremely sensitive to the values of the friction between vehicle structure and ground. Owing to the highly nonlinear nature of the problem, after relatively large periods of analysis, the behaviours of different models 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 in the lateral direction and 20 cm in the frontal direction obtained in the simulations is similar to the permanent deformations observed in experimental tests. Models with 12 and 24 beam finite elements for the rollbar cage show similar results for the first part of the analysis. The differences between the responses in the second part of the analysis must be attributed to the nonlinearity of the problem rather than to the models used for the rollbar cage.

Figure 33. Severity index for the vehicle occupants. Crash analysis and dynamical behaviour of vehicles 409

The severity index observed for the occupants in ﬁgure 33 indicates a very high probability of fatal injuries in the conditions simulated. Note that the model has rigid seats, interior trimming for the dashboard, side and ﬂoor panels, and that the ground is also considered to be rigid. If some compliance is included in the vehicle interior, it is expected that the head accelerations are lower. The kinematics of the biomechanical models of the occupants, and in particular that of the ejected occupant, are similar to the kinematics of the crash test dummies used in the experimental tests. Several simulations of the vehicle rollover with occupants seating with different postures have been performed. These simulations show that regardless of the rear occupant seating posture the ejection and post-ejection occupant kinematics remains basically unchanged.

5. Conclusions

This work demonstrated a multibody dynamics-based formulation that effectively promotes the simultaneous analysis of the vehicle stability and manoeuvrability dynamics with structural and occupant crashworthiness. As the formalisms used are common to all disciplines that are relevant to vehicle design, including structural crashworthiness, it is possible to recycle vehicle models for different applications, to account for the different mechanisms of energy absorption for a wider range of crash velocities and to have a better appraisal for the occupant’s mechanisms of injury and injury criteria. In the process, it was also demonstrated how nonlinear ﬁnite elements can be integrated with conventional rigid multibody descriptions in order to build better general vehicle models. Though not demonstrated through the applications used here, the reliability of the prediction of certain crash mechanisms allows devising driving support systems that assist the driver in crash avoidance manoeuvres. Through the railway crashworthiness application, it was demonstrated that the design of the train systems that ensure the train alignment during the crash requires that the relative motion between the cars of the same train is well described, and therefore, that the suspension models, developed in the framework of typical vehicle dynamics applications, are used to account for the relative pitching between adjacent vehicles. The side impact of the road vehicle conﬁrmed these conclusions, now in what the vehicle roll motion is concerned. The rollover simulations showed how the vehicles can withstand multiple contacts when the simulation is being preformed. All these applications are run in a reasonable short amount of time, measured in minutes, which demonstrates 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 project 2/2.1/TPAR/2041/95, to study the vehicle rollover is gratefully acknowledged. The support of EC 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), AEA Technology/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 and through project TIP3-CT-2004-506503 (APROSYS-SP6), with the partners SIEMENS VDO (Ge), Faurecia (Ge), DaimlerChrysler (Ge), CIDAUT (Sp), Fh G (Ge), Warsaw University of Technology (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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