Vcrash Multibody Model

8/10/2019 Vcrash Multibody Model

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Figure 14.Components of a towing with a steered-trailer

Multibody-model

Figure 15.Pedestrian accident

General

In reality, a vehicle does not get significant damages dur-ing pedestrian accident because of its body stiffness. Thedifference between a vehicles and a pedestrians weight is

large, thus the bigger damage and the higher change ofvelocity are suffered by the pedestrian. This is the reason,


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why Virtual Crash regards vehicles as rigid bodies and thedeformation of the vehicle is not taken into account is caseof a pedestrian accident.

Pedestrians have to be treated as multibody systems, be-

cause body elements can not move on independently. Ve-hicle pedestrian impacts can be simulated by means ofthis multibody-model almost as easily as vehicle - vehicleimpacts are.

The model owns the regular properties that pedestrianshave such as size and weight. These parameters are varia-ble according to the participant of an accident. Twelve dif-

ferent poses are available in Virtual Crash for the mostrealistic reconstruction of pedestrian accidents. These arethe followings: standing (default), walking left or right legis in front, sitting in two poses, lying in two poses, leaning,cyclist, motorcyclist, cat-pose, prayer-pose. Virtual Crashtakes the posture of the pedestrian into account as thegeometric shape of different vehicles.

Figure 16.Poses in Virtual Crash (walking leftwith raster, mo-torcyclist, lying pose)

Model Description

Multibody is a system of rigid bodies interconnected byjoints corresponding to humans. The parts of the body

(head, torso, arms, legs, etc.) are made of multi-degree-hyper-ellipsoids. These are special ellipsoids with differ-


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ent size-parameters and different degrees according to theparts of the body. The degrees do not need to be integer(whole number). The body elements are interconnected withjoints.

The general equation of the multi-degree-ellipsoid is thefollowing:

1

nnn

c

z

b

y

a

x (60.)

a; b; c - the semi axes of the ellipsoid

x; y; z - variablesn - the exponent (degree)

Note:In case of the exponent is n=2, the ellipsoid is asphere. If n=30, the figure is almost a cube with roundededges (Figure 17.).

n=2 n=30

Figure 17.The shapes of a hyper-ellipsoidin case of different degree

For the most appropriate setting of a collision, there aredifferent variable properties such as static and dynamicparameters that can be set before and during a simulation.

Static parameters are height, weight and posture parame-ters (pose, angular offset, height of the centre of gravity)


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dynamic properties are coefficients of friction (between thebody and the ground; between the body and the causingvehicle), coefficient of impact (k).

The dummy in the program includes 15 body elements

and 16 joints corresponding to human body. These arehead, neck, torso, waist, femurs, knees, lower legs, feet,upper and lower arms, Joints are between parts accordingto the reality.

Virtual Crash regards the human-body-model as a systemof these mentioned realistic shaped ellipsoids covered by asurface that is made of triangles. These triangles canchange their size, according to interaction of the body ele-ments (without any power impulse on one another).

The multibody system has a zero condition, when thegravity force and the internal joint forces are in balance.This is a state of equilibrium that lasts until the first externalimpact. When a vehicle just push the dummy, this balance

does not last any longer and the external forces will onlyaffect the movement of the body elements.

Coordinate system

Two additional coordinate systems are applied in case ofa simulation involving a multibody model. One coordinate

system is the inertial system in with the dummy moves andthe second coordinate system is the body fixed coordinatesystem that determines the position of the body parts. Thissystem is defined by the semi-axes of the ellipsoids. Both

coordinate systems are right hand coordinate systems.(Fig-

ure 18.)


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Figure 18.Coordinate systems

about the multibody model

Body Properties

The following properties are generally and independent-ly applied for the parts of the multibody system.

Geometry: Each element of the dummy is

represented by an ellipsoid and each has differ-ent exponents and axes corresponding to the di-mensions of a human.

Weight and moments of inertia:The moments of in-ertia about the ellipsoids three axes and its massmust be specified. The dimensions and the weightare prescribed and can be variable and the mo-

ments of inertia are calculated from these para-meters.

Coefficient of friction: Two coefficients of frictioncan be specified. One is used for ellipsoid to ve-hicle contacts; the other is used for ellipsoid to el-lipsoid and ellipsoid to ground contacts. Thesecoefficients of friction are assumed to be inde-pendent of the amount of penetration and usedfor the calculation of contact forces


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Stiffness coefficients: There is one coefficients ofstiffness fot all of the parts of the multiboby sys-tem. It can be prescribed and variable during asimulation as the friction coefficient is.

Equations of motion

There are some different external forces (gravity, contactforces, frictional forces and joint forces) affecting the ele-ments of the mulibody system in case of a simulation of apedestrian accident. These are calculated during each timestep. After the external forces are determined, the move-ment of each body is calculated independently by solvingthe equations of motion.

Steps during the simulation of the multibody system

1. the calculations of contacts such as positions ornormal vectors

2. the calculation of Jacobi-determinant

3. the determination of conditions and the calculationof forces

4. the application of forces to multibody system

5. the solution of the equations of motion

The parts of the Jakobi-determinant are the following:

Each condition has three rows within the determinant. Thefirst row contains the parameters of the contact planes di-rection while the second and third rows express the direc-tion of tangential forces.

There is a relation between tangential and normal forcesthat is used as a condition of the Jacobi-determinant:

NT FF (61.)


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The following equations are needed for the calculation ofthe positions and forces about the multibody system:

CvJ (62.)

dtCJMJ T )(

1 (63.)

TJF (64.)

where the parameters are:

J - the Jakobi-determinant

vi - the velocity of the multybody partsc - constants

F - the effecting forces

- the result

M - the mass matrix

t - the integration step

Equations of motion

The equations of motion consider the balance of forces,the assumption of the conservation of angular momentumand the influence of external forces and moments:

iiii

xmF (65.)

iiiiiiiM (66.)

zzyzxz

yzyyxy

xzxyxx

i

III

III

III

(67.)

mi - the weight of body part i


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ix - the acceleration of the centre of gravity of

body part i in the inertial coordinate system

i

F - the external forces on body part i in the

inertial coordinate system

iM - the external moments on body part i in

the body coordinate system

i - the mass tensor for body part i in the

body fixed coordinate systemi - the angular acceleration of body part i in

the body fixed coordinate system

i - the angular velocity of body part i in the

body fixed coordinate system

Typically a symmetric mass tensor is used for each bodyto represent an ellipsoid. For the integration of these equa-tions a Fourth order Runge-Kutta method is used.


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Definition of JointsThree types of 3D joints are utilized to construct the mul-

tibody model corresponding to the constraints and permit-

ted motions of the human body.

Revolute joint

This joint constrains relative linear motion and allows onlythe rotational motions around the common axis. Body ele-ments can rotate around the axis, but these cannot be se-parated and gone away. The origins of the relative coor-dinate systems are identical on the common axis. The num-ber of degrees of freedom of a revolute joint equals one.The joint velocity degree of freedom is the angles first timederivative that is between the main axes of the local coor-dinate systems.

Figure 19.The model of the revolute joint

This type of joints is used for the knees and elbows.

Spherical joint

This joint also constrains relative linear motion and allowsonly the rotational motions but there are more degrees of

freedom. In this case, connected body elements have fourdegrees of freedom. These are all rotations about the axes


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of a local coordinate system, that is connected to the oneelement and one rotational degree is about one axis of theothers coordinate system. The number of joint velocity de-grees of freedom equals three. This type of joint is used for

the hip femur connections and for the head torso con-nection.

Figure 20.The model of the spherical joint

Universal joint

This joint does also not allowed relative linear motions

and the rotational motions are less for this joint type thanfor spherical joint. This has only two degrees of freedom.The local coordinate systems origins are identical in themiddle of the cardan-cross. The joint velocity degrees offreedom are their first time derivatives.

Figure 21.The model of the universal joint


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Contacts with multibody

There are three types of contact about impacting the mul-tibody dummy. One is the contact between the body andthe vehicle involved in a pedestrian accident (ellipso-

idplane). Another similar is the dummys encounter withthe ground plane. The third is a parts impact with another

(ellipsoidellipsoid) within the multibody system. Thesethree can be regarded and defined as two different cate-gories because there are no significant dynamical differ-ences between the first two cases.

The contact models are based on a linear stiffness func-

tion and these are used to calculate the contact forces incase of the former three contact types. A coefficient of res-titution is specified to define the amount of elasticity duringthe contact. By means of the contact friction between thetwo body parts, the contact normal force and friction forcesare calculated. Thus different impact types can be calcu-lated, such as sliding impacts or full impacts. Other impact

parameters are automatically obtained, the location of theimpact point and the orientation of the contact plane for in-stance. User has the opportunity to check and supervise themultibody features and if necessary, the parameters canbe changed.

The equations, that using for the calculation of the contact

forces are:SFA (68.)

SFS (69.)


FA - the normal contact forces during ap-

proaching


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FS - the normal contact forces during separat-

ing

S - the stiffness coefficient

- the depth of penetration - the restitution coefficient

Ellipsoid plane contact

The vehicles are regarded as rigid bodies with a surfacemade of small triangle planes. The vehicle shape can bespecified by using detailed 3D vehicle shapes. The vehicle

deformations are neglected as it was mentioned. The con-tact point is on the surface of the vehicle and has to be in-side a triangular that helps to form the shape of the ve-

hicle. The penetration () is the distance between the far-ness point of the ellipsoid over the contacting plane and theplane. The calculation of the tangential contact force com-ponents uses the friction coefficient (preset between the el-

lipsoids and the vehicle) and the relative velocity of thecontact points. This contact force is taken into account as anexternal force for the vehicle.

The Figure 22.is representing the positions of points and

parameters during an ellipsoidplane contact.

Figure 22.Ellipsoid plane contact


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The calculation method is the following:

PE PP (70.)

PE nn

(71.)nPnE FF (72.)

EEnE nSF

(73.)

cpEcpPnEtPtE vvFFF

(74.)


Fni - the normal contact forces

Fti - the tangential contact forces

- the distance of penetration

- the contact friction

cpEcpP vv - the relative velocity between the ellipsoid

and the plane

Ellipsoid ellipsoid contact

This contact is specified for ellipsoids those have no com-mon joint. There is an assumption in case of the contact of

two ellipsoids: the contact point has to be on a line be-tween two points, where one point is on the surface of thefirst ellipsoid and the second point is on the surface of theother ellipsoid. The tangential planes for both points areparallel and the distance between these two planes is aminimum. The exact location of the contact point can befound by means of the body stiffness values.


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Figure 23.Ellipsoid ellipsoid contact

The Figure 23. is representing the relative locations ofthe points and parameters. For the calculation of the tan-gential component of the contact force, the body frictionhas to be the lower value between the specified for thetwo ellipsoids. The direction of the tangential force is de-fined by the direction of the relative velocity of the contactpoint

The equations are solved numerically for and the cal-culation method is the following:

2121.min PPwhere (75.)

21 nn (76.)

21 nn FF (77.)

);min( 21 C (78.)

1111

nSFn

(79.)


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2222 nSFn

(80.)

222111 nPnPPC

(81.)

1211 cpcpCnt vvFF

(82.)

2122 cpcpCnt vvFF

(83.)


Fni - the normal contact forces

Fti - the tangential contact forces

- the distance of penetration

C - the common contact friction

21 cpcp vv

- the relative velocity between the two ellip-

soid

Note: These mentioned contacts cannot occur betweentwo bodies connected with a common joint.

Collision model

General

Virtual Crash utilizes an impulse and momentum-basedimpact model by means of the restitution. Vehicles do notcrease in the program and the body stiffness coefficientsare not taken into account. One of the parameters aboutthe body crush is the depth of penetration (time-value

default: tP=0,03s), which is variable for suitable simula-tions. The impact forces are not calculated in each time stepat the so-called impulse point". The calculation of these

Vcrash Multibody Model

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