Vehicle Crash Dynamics P10 ppsx

5.26 shows the loading and unloading curves for the struck #1 and striking #2 vehicles.The loading phase is defined by a power curve, F=kDn, and the unloading by a constant slope.. In th

(5.36)

Fig 5.25 Newton-Raphson Iterative Method

(5.37)

and the final resolution of each variable is

where d: number of design variables in space (or d-dimensional hypercube)

5.5.2 Newton-Raphson Search Algorithm

The solution of F, F*, which satisfies a function f(F) = 0 can be obtained by a method using theNewton-Raphson algorithm described in Eq (5.36) and shown in Fig 5.25 The slope of the curve

at point Fi is f '(Fi) Fi is the value of the independent variable at iteration i The iteration processcontinues until Fi and Fi+1 converge to the solution of the function in which f(Fi+1) approaches zero.The error function , shown in Eq (5.37) can also be used to check for the numerical convergence

Note that the computation of F* usually takes no more than three iterations if the relative error,,, is set at 0.001 The algorithm converges rapidly during the loading phase simulation, and the initialvalue of Fi at any given time step should be the F* from the previous time step

Fig 5.26 Loading and Unloading Phases of Two Vehicles

(5.38)

5.6 LOADING AND UNLOADING SIMULATION

Fig 5.26 shows the loading and unloading curves for the struck (#1) and striking (#2) vehicles.The loading phase is defined by a power curve, F=kDn, and the unloading by a constant slope In thedynamic simulation during the loading phase, an efficient numerical technique using the Newton-Raphson method is used to compute the individual deflections of the two non-linear springs (EA), andduring the unloading phase, the relative deformations of the two EANs and the correspondingunloading force are explicitly computed

5.6.1 Loading Phase Simulation

In general, there are two non-linear springs involved in the force-deflection computation Aninteractive scheme using the Newton-Raphson algorithm is used This is described as follows.Given a total deflection of the two springs obtained from the numerical integration at each timestep, the method computes the force and individual deflection, as shown in Eq (5.38)

The solution of F, F*, which satisfies (2) of Eq (5.38) can be obtained by applying the Raphson algorithm

Newton-Fig 5.27 Force-Deflection Computation in the Unloading

Phase

(5.39)

5.6.2 Unloading Phase Simulation

The initial conditions of the unloading phase simulation are the final conditions of the loadingphase simulation The force and the individual deflections of the two springs depend on the totaldeflection and unloading stiffnesses of the two springs The method to execute the unloading phasesimulation is described in Fig 5.27 and Eq (5.39)

Fig 5.28 Parametric Relationships in Loading/Unloading Cycles

5.6.3 Model with Power Curve Loading and Unloading

Fig 5.28 shows the general loading, reloading, and unloading power curves The start-loadingpoint is at xi, the start-unloading point is at point U, and the reloading-point is at point xr Unloadingstarts at the end of loading The unloading power curve is defined by the two parameters kN and nN.Given the specified hysteresis energy, Eh, and permanent deformation, xp, the two unloadingparameters are derived in closed-form and shown in Eq (5.40) It is also shown that under thespecified loading and unloading conditions, the unloading power nN is always greater than the loadingpower n

Variable Definitions in the Loading and Unloading Cycles

Loading:

ax : acceleration function at x

x :displacement or velocity for spring damper

xi : x location of starting loading or reloading cycle

xp : x location of permanent deformation or deformation rate

At the point of unloading, U, amax (loading) = a'max (unloading),Let Eh be the hysteresis energy in the loading/unloading cycle; then

Eh = EL (loading energy) ! Eu (unloading energy)

(5.41)

Reloading starts at the end of unloading During reloading, the system may unload at anymoment The unloading curve is not only a function of kN and nN, it is also a function of xi, the xlocation of starting loading or reloading cycle, as shown in Fig 5.28

5.6.3.1 Unloading Parameters k', n', and x i in Reloading Cycle

The derivation of the unloading parameters are described in Eq (5.41)

For a spring-damper model with power curve loading and unloading characteristics, the numericalsimulation follows the flow charts shown in Table 5.13 There are two flow charts The one on thetop shows the numerical integration procedures The one on the bottom computes the deceleration(force) due to loading or unloading by the spring and damper

Table 5.13 Power Curve Loading and Unloading Flow Chart

Fig 5.29 Test Body and Optimal Model Responses w/ and

w/o Damping in a 35 mph Test

Fig 5.30 Spring and Damper Contributions in Model Body

Response

A Kelvin model with non-linear stiffness and damping is used to simulate the dynamiccharacteristics of a body mount on a frame vehicle The frame pulse from a truck-to-a-fixed-barriertest at 35 mph is used to excite the model The predicted model body (output) pulse is then comparedwith the body accelerometer data from the test By optimizing the stiffness and damping, a ‘best’model is then found, where the output pulse matches that from the test Fig 5.29 compares the testbody crash pulse with that from the optimized models with and without damping It is clearly shownthat a model without damping is not adequate to simulate the impulsive response which occurs in theearly part of the body crash pulse The body impulsive response is the result of the frame impulsiveloading being transmitted by the body mount

5.6.3.2 Deceleration Contributions of Spring and Damper

To further analyze the contribution of the spring and damper to the overall model body response,the deceleration contributions of the spring and damper are denoted as ak and ac, respectively Since

ak is controlled by the deflection and ac by the deflection rate, both the transient deflection, )d, andthe deflection rate, )v, are shown in Fig 5.30 It is clearly shown that the first peak of the body pulse

is dominated by ac, which is controlled by )v Note that the first damper unloading occurs very early,

at around 10 ms, when )v decreases after reaching the peak Subsequent re-loading occurs when )vincreases at around 20 ms

Fig 5.31 Body Responses for the Truck Test and

Power-Curve Model

Note that loading occurs while )v is increasing, until it starts decreasing At 10 ms, )v reaches

a maximum velocity of 10 mph, which is the end of the damper loading Damper unloading lasts tillabout 19 ms, then reloading starts Since )v keeps decreasing, contribution of damper to the outputacceleration becomes minimal However, in the steady state period after the transient response, thespring contributes more than the damper to the output deceleration, a, due to the ever-increasingdeflection

Fig 5.31shows the test frame pulse of another 35 mph truck to barrier test which is the inputexcitation to the Kelvin model for the body mount The magnitude of the test frame impulsive loading

is about 138 g, and that of the test and model body impulsive response is about 53 g The outputresponse of the power curve model matches quite well the test responses in both the transient andsteady state domains, as shown by the body test and model curves in Fig 5.31

5.7 A LUMPED-PARAMETER MODEL — CRUSH II

A system can be modeled by a lumped mass model with an infinite number of natural frequencies.The motion of a system can be studied by the summation of many subsystems with different naturalfrequencies These subsystems, which generate the high frequency noise, can be excluded fromsystem modeling

CRUSH II (Crash Reconstruction Using Static History) is a one-dimensional mathematical modelused to simulate the impact dynamics of objects or masses connected by springs and dampers Since

it uses connecting masses, it is frequently referred to as a lumped-mass model (or lumped-parametermodel), and the spring and dampers are referred to as energy absorbers (EA) The original version

of the software was developed under a contract with the U.S Department of Transportation [1] It wasused to perform a computer simulation of collinear car-to-car and car-to-barrier collisions

5.7.1 Simple Structure Force-Deflection Table

The operation of CRUSH II requires the use of quasi-static force-deflection data In a componenttest laboratory, a testing device called Crusher can be used to generate the quasi-static force-deflectiondata The components to be crushed are placed between the reaction fixtures and load plates As theram is extended, generally in steps of 0.3 inches, force measurements are recorded

In the elastic range, stiffness (K) is the slope of the force-deflection curve K is equal to AE/L,

as shown in the load vs elongation plot of Fig 5.32 E is Young's Modulus, the slope of the strain curve in Fig 5.32; A, the cross sectional area; and L, the length of the material

stress-The stress-strain curves of ductile and brittle materials are shown in Figs 5.33 and 5.34,respectively In contrast to ductile materials, the brittle materials do not have a yield point on itsstress-strain curve The last point on the curve is the fracture point

Fig 5.32 Relationship between Force-Deflection and Stress-Strain Curves

Fig 5.33 F vs , for Structural Steel Fig 5.34 F vs , for Brittle Materials

The elastic stiffness of a structure under a specific loading can be obtained from a deflectionformula The stiffness of the structure in the direction of applied load is the ratio of the applied load

to the deflection in the loading direction The deflection of a structure at a given point can be derived

by the strain energy method using Castigliano’s Theorem [7] Fifteen simple structures are shown in

Table 5.14 and the deflection formulas for the structures are shown Note that most of the deflectiondue to axial and shear forces are neglected In the formulas, E denotes Young’s modulus, G, themodulus of rigidity, and I, the moment of inertia of the cross section of the structure member

5.7.2 Push Bumper Force-Deflection Data

A push bumper mounted on the front bumper of a police car requires a certain stiffness to performits task Fig 5.35 is a sketch of the push bumper A properly designed push bumper should not affectair bag crash sensor performance in a frontal collision The proper design of push bumper needs toconsider the effective stiffness of the push bumper and the attached front end of the vehicle, and itseffect on the occupant kinematics and sensor performance in a test

Fig 5.35 Police Car Push Bumper

Fig 5.36 Push Bumper Force vs Deflection

Fig 5.37 Beam and Spring Modeling of Push Bumper

The stiffness of the push bumper, based on the stiffness formulas of various structures shown in

Table 5.14, is computed in this section The force-deflection data from the quasi-static component test

is shown in Fig 5.36 The slope of the straight line approximating the test curve is k=18 klbs/in

The push bumper is modeled as a half circular spring with a beam supported by free and fixedends, as shown in Fig 5.37 Assuming a concentrated load acting at the midpoint of the push bumper,the combined force-deflection can be estimated by formulas #8 and #15 in Table 5.14 Thecomputation is shown in Eq (5.42)

Table 5.14 Deflection and Stiffness of Structural Members

Table 5.14 Deflection and Stiffness of Structural Members (Continued)

Fig 5.38 Force-Deflection Data with Unloading Properties

The computed push bumper stiffness in the direction perpendicular to the face plate is about 19klbs/in This analytical estimate is fairly close to that obtained from the component test, 18 klbs/in

5.7.3 Basic Operation of EA Types

In addition to the use of quasi-static force-deflection data for the structural components in themodel, there are eight types of EA and five types of CV (velocity sensitive) factors to choose from foreach of the components used in the model The EA type specifies how the loading and unloading aresimulated and the CV type specifies the type of material strain rate effects [8,9] to be used for that EA.The EA Type 4 allows the user to input discrete force-deflection (F-D) data points for a structuralcomponent In addition to the loading portion of the data set, it permits the unloading and reloadingalong the same slope, ku If the unloading is driven lower, such that it intercepts the second unloadingslope, ku', the unloading then follows this new path, as shown in Fig 5.38 During the unloading alongku', if it encounters reloading, it would then create a new reloading path with the same slope as ku

Fig 5.39 Loading and Unloading Events

The EA Type 4 was created to model the material property in which the corresponding permanentdeformation and hysteresis energy can be specified This is done by the use of two unloading slopes

ku and ku', as shown in Fig 5.38

Even though ku' is specified for the unloading of a certain structural component, it does notguarantee that the deformed material will return to its initial undeformed state The material specifiedfor compression (or tension) alone will not return to its undeformed state unless it has enough elasticenergy returned in the restitution phase (area of the force-deflection under the unloading curve) and/or

is forced to return to its initial position by the neighboring component(s) connected to it

The data format used to specify the discrete force-deflection data points for the EA Type 4 isshown as follows: Given a data point 2 with (d2, F2) and a corresponding unloading force level F2', thesecond unloading slope, ku', is then equal to F2' /d2 The unloading force levels of the other data pointscan then be computed by simply multiplying ku' by the corresponding deflection

Pt d F load FNunload

1 d1 F1 F1'

2 d2 F2 F2

ku: initial unloading or subsequent reloading stiffness for the

EA Type 4 with F-D tabular input

ku': the second unloading slope, equal to F2' /d2

In a special case, ku' equals zero Then, the intercept of the unloading slope ku on the zero forcelevel is the permanent deformation, dp , as shown in Fig 5.39

stiffnesses, specified in EA Type 3F-D tabular input

5.7.4 Basic Operation of CV Factor (Velocity Sensitive Factor)

The velocity sensitivity (CV) factors provide an important link in converting the crusher staticforce-deflection data to dynamic Strain-rate has an important effect on the dynamic response ofstructural members Various equations have been formulated to express the strain-rate effect Atcrush rates of 30 mph, CV factors applied to the static force to obtain the dynamic force range from1.3 to 1.5 for heavy frame type sections The effect is less important for light elements such as sheetmetal The strengths of highly elastic rubber mounts seem to be independent of the strain-rate Thereare five CV factors built into the CRUSH II model As an example, CV Factor 3 is based on materialproperties and test strain rate effects caused by the quasi-static Crusher test[1] Eq (5.43) shows the

CV factors based on the following formula for hot-rolled steel with various percentages of carboncontent

where A: a constant, usually 1.0

KR: a material factor (See Table 5.15)

CR: the absolute value of instantaneous crush rate

SCR: the crush rate of the test specimen, usually 03333 in/sec

Table 5.15 Some KR Values for Various Materials

5.7.5 Coefficient of Restitution, Static, and Dynamic Crush Relationship

It will be shown that the coefficient of restitution is equal to the square root of the ratio of elasticrebound displacement to the dynamic crush This relationship is identical to that of a ball released atrest from a height of h Assuming that the maximum height that the ball rebounds to is h' Then hcorresponds to dynamic crush, and h' corresponds to the elastic rebound displacement

The derivation for such a relationship is shown in Eq (5.44) It is also shown that the ratio oflinear loading stiffness to unloading stiffness is equal to the coefficient of restitution squared Ifunloading is perfectly plastic, and the unloading stiffness is infinite, the coefficient of restitutionbecomes zero