Vehicle Crash Dynamics P9 pdf

A hybrid model is a combination of Maxwell and Kelvin models.. for Hybrid Model Using the concept of an effective mass system and an approach similar to that used in derivingthe equation

Fig 5.1 Hybrid Model #1

Fig 5.2 Hybrid Model #2

CHAPTER 5 RESPONSE PREDICTION BY NUMERICAL METHODS 5.1 INTRODUCTION

The solution to a problem with an impact or excitation model having more than two masses and/orany number of non-linear energy absorbers becomes too complex to solve in a closed form Then,numerical evaluation and integration techniques are necessary to solve for the dynamic responses.Models such as the two non-isomorphic (with different structural configuration) hybrid orstandard solid models, the combination of two hybrid models, and special cases with point masseswill be treated first in closed-form The purpose of the closed-form analysis is to investigate thedynamic responses of two dynamically equivalent hybrid models In a multi-mass model, theunloading characteristics of a spring element is as important as the loading characteristics Theunloading of one mass in a model may produce loading of the neighboring masses, thereby, affectingthe total system model responses

Power curve loading and unloading simulation with hysteresis energy loss and permanentdeformation will be covered To help solve some dynamic models quickly, a lumped-parametermodel, CRUSH II [1], coupled with animation, will be utilized The force-deflection formulas ofsome simple structures are listed for ease of determining the spring stiffness for the modeling Somelumped-parameter models for the full frontal, side, and frontal offset impacts are described The basicconcepts of splitting a simple spring-mass model for the frontal offset impact and the model validationare also presented

5.2 HYBRID MODEL — A STANDARD SOLID MODEL

There are two types of hybrid models, Hybrid #1 and #2, shown in Figs 5.1 and 5.2, respectively

A hybrid model is a combination of Maxwell and Kelvin models It has three elements: twosprings and one damper These three elements are connected in such a way that the two hybrid modelsare structurally and functionally different (non-isomorphic) In impact analysis, each hybrid modelhas two mass systems with a closing speed of V12 To simplify the two-mass system analysis, theconcept of an effective mass system is introduced and utilized in the next section

The hybrid model has been used in an impact dynamics study of the body mount in a frame vehicle [2], and the lateral impact modeling of the thorax of a driver in a car accident [3]

body-on-5.2.1 E.O.M for Hybrid Model

Using the concept of an effective mass system and an approach similar to that used in derivingthe equation of motions (E.O.M) for the Maxwell model shown in Section 4.8 of Chapter 4, theE.O.M for the hybrid model can also be expressed in terms of one second order differential equation(D.E.) and one first order D.E Similar to the Maxwell model, the E.O.M for the hybrid model can

be expressed in terms of one third order D.E The corresponding characteristic equations for the twohybrid models are shown in (1) to (3) of Eq (5.1), respectively

The two-mass system can be transformed into an one effective-mass system Fig 5.3(a) showsthe hybrid #1 effective-mass system In this model, there are two springs in contact with the rigidbarrier If one of the springs is moved to the other side of the mass while still in contact with thebarrier (or ground), a new model arrangement is obtained, as shown in Fig 5.3(b) Although the EA

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Fig 5.3 Two Effective-Mass Systems (Hybrid Model #1)

(5.2)

Fig 5.4 a vs t of Hybrid Models at Three Impact Speeds

arrangements in the two models look different, the models are functionally identical to each other.The underlying assumption is that the tension behavior of spring k1 is the same as in compression

5.2.2 Dynamic Response and Principles of Superposition

The dynamics of the Kelvin model has been described in Section 4.9 of Chapter 4 The Kelvinmodel with the spring and damper in parallel produces a non-zero deceleration at time zero.Consequently, the initial deceleration of the impactor, mass m1, deviates from that in the test However, in the Kelvin model, the initial slope on the deceleration vs displacement curve is only

a function of the component parameters T, and and is independent of the impact speed Note from(1) of Eq (5.2), the sign of the initial slope of a vs d changes when the damping factor is more than0.5 This change in slope direction has been shown in the hysteresis plot, Fig 4.92 in Section 4.9.3

In Kelvin model, the initial slope of the deceleration vs displacement curve (k, a specific stiffness)multiplied by the initial velocity (vo) is equal to the jerk (j, rate of change of deceleration) at time zero.This relationship j = vo k from Eq (5.2) has been shown in Eq (1.44) of Section 1.9.3

Shown in Figs 5.4 ! 5.6 are the a vs t, d vs t, and a vs d curves of the hybrid models for Type

F body mount at the impact speeds of 5, 10, and 15 mph Note that the initial slope for each of thethree a vs d curves in Fig 5.6 is constant It is independent of the impact speed and is solelydetermined by the component parameters expressed in (7) of Eq (5.1) However, the impact speedchanges the maximum magnitudes of the g-force and deflection, as shown in Fig 5.6

Fig 5.6 a vs d of Hybrid Models at Three Speeds Fig 5.5 d vs t of Hybrid Models at Three Impact Speeds

The fact that the impact speed does not change the initial slope on the a vs d (g-force vs.deflection) curve does not suggest that the body mount is free of damping This is because the initialslope has already been determined by the natural frequency and damping factor of the body mount.The Hybrid model is a linear system; therefore, the principle of superposition applies Thetransient displacement at 10 mph is simply equal to two times that at 5 mph as shown in Fig 5.5, whilethe transient displacement at 15 mph is equal to the sum of the displacements at 10 mph and 5 mph

5.2.3 Combination of Two Hybrid Models

Similar to combining two Kelvin models for a vehicle-to-vehicle impact (see Section 4.4.3), onecan also combine the EANs (energy absorbers) of two hybrid models as shown in the upper figure in

Fig 5.7 The interface in the model, a point mass, has negligible weight Based on the principle ofsuperposition in combining two EA’s into one EA, described in Section 4.4.3 of Chapter 4, two hybridmodels in series can be simplified by joining them into one effective hybrid model as shown in thebottom of Fig 5.7

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Fig 5.7 Vehicle-to-Vehicle Impact Model: A

Two-Hybrid Model

(5.3)

The EA parameters in the combined model are the effective spring and effective damper shown

in Eq (5.3)

Case Study: The weights of vehicles (masses) m1 and m2 are 6 and 3 klbs (kilo-pounds), respectively.

The values of the individual and combined (or effective) parameters of the springs and damper in thetwo models are shown in Table 5.1

Table 5.1 Parameters of the Two-Hybrid and One-Hybrid Models

Vehicle m2 structure

(top left in Fig 5.7)

Vehicle m1 structure(top right in Fig 5.7)

Hybrid Model withEffective EAs

k12,klb/in

k22,klb/in

c2,klb-s/in

K1,klb/in)

K2,klb/in

c,klb-s/in

The initial speed of m1 impacting on the stationary m2 is 60 mph Note that for modelingpurposes, the interfaces between two EAs are modeled by point masses with negligible weight (e.g.,five pounds) The dynamic analysis is done by CRUSH II model simulation, and the output responsesare animated and plotted Shown in Fig 5.8 are the decelerations of the two masses in two- and one-hybrid models The deceleration curves of the two masses in the two-hybrid model are of the saw-tooth type The saw-tooth responses are the high frequency noises, generated by the light weightinterface connected by a stiff spring to the main body In both models, the peak accelerations of m1and m2 are !17 and 34 g for m1 and m2, respectively These occur at the same time, at 45 ms

Fig 5.8 Acceleration Responses of the Two Masses in the

One- and Two-Hybrid Models

(5.4)

5.2.4 Dynamic Equivalency between Two Non-Isomorphic Hybrid Models

There are three EANs (energy absorbers) in each of the two hybrid models shown in Figs 5.1 and5.2, respectively By appending a subscript i to the existing parameters, the two sets of EANs becomeunique for each model The EA parameters are k1i, k2i, and ci for hybrid model #i Similarly, thecoefficients in the characteristic equation shown in Eq (5.4) are similarly assigned as ti, ui, and vi formodel #i

To establish the dynamic equivalency between the two models, the three coefficients of hybridmodel #1 are set equal to the corresponding coefficients of model #2 respectively Since there arethree constraint equations, the three EA parameters of one model can then be solved in terms of those

of the other model

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(5.6)

Therefore, given a set of values for the two springs and one damper parameters for hybrid model

#1, one can compute the values of those for hybrid model #2 Formulas (4) to (6) of Eq (5.4) areneeded for the computation It will be shown in the next section that dynamic equivalency betweenthe two models applies to the kinematic, crush, and energy responses of the masses only There are

no equivalencies among the respective energy absorbers

Case Study: Given the values of the following parameters of hybrid model #1:

k11 = 2 klb/in, k21 = 4 klb/in, c1 = 0444 klb-sec/in, and

Using Eq (5.4), the parameters of hybrid model #2, k12, k22, and c2, can be computed from those

of hybrid model #1 and are shown in Eq (5.6) Note that the spring stiffness ratio in the hybrid model

#2, k22/k12 = 2, is the same as that in model #1, k21 /k11

The computed input data for hybrid model #2 are shown at the top of Fig 5.9, and the originalinput data of model #1 are shown at the bottom of Fig 5.9 Both models are effective mass modelswhere the effective weight is 2 klb as computed in Eq (5.5) The effective weight is moving at aspeed of 60 mph to the left where the fixed rigid barrier is located The total crush energy, the initialkinetic energy of the effective mass in the rigid barrier test, has been computed to be 240 klb-ft, asshown in Eq (5.5)

Fig 5.9 Dynamic Equivalency of Hybrid #1 (bottom) and

#2 (top) Models

Fig 5.10 Transient Kinematics of Hybrid Models #1 and #2

5.2.4.1 Dynamic Equivalency in Transient Kinematics and Crush Energy

The effective mass deceleration vs time curves of both hybrid models #1 and #2 are shown in

Fig 5.10 The peak accelerations of both models are equal, -51 g @ 46 ms, and both models have thesame deceleration in the early portion of the rebound phase The deceleration difference after 80 ms

in the rebound phase is attributed to the way the EA(s) is in contact with the rigid barrier Spring k22

in model #2 is in contact with the barrier (not rigidly attached to the barrier), while both springs k11and k21 in model #1 are in contact with the barrier Since at any given time, the spring contact forces

in the two models may not be the same, the separation time and the deceleration may be different forthe two models in that period

In addition to acceleration, the two hybrid models are also dynamically equivalent in terms oftransient velocity and displacement beyond the deformation phase, as shown in Fig 5.10 Thedynamic crush occurs at 68 ms where the velocity of the effective mass is zero

Beyond kinematics, the two hybrid models under dynamic equivalency conditions are alsoequivalent (identical) in terms of transient total crush energy The total crush energy is the sum of thethree individual crush energies due to springs #1 and #2 and the damper Figs 5.11 and 5.12 showthe total and individual crush energies vs time of the hybrid models #1 and #2, respectively In bothhybrid models, the transient individual crush energies due to two springs and one damper are different.However, the sums of the three energies of the two hybrid models are identical in the deformation

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phase and in the early portion of the rebound phase The maximum total crush energy is 240 klbs-ft

at 68 ms, the time of dynamic crush The maximum total crush energy can also be checked by theformula for the energy density, e, as shown in Eq (5.7):

5.3 TWO MASS-SPRING-DAMPER MODEL

A numerical method based on semi-closed-form solutions of a two mass-spring-damper (2-MSD)model (shown in Fig 5.13) is presented Applications of the model solutions to the vehicle pre-program and post-crash structural analyses are described The model in these applications simulates

a rigid barrier impact of a vehicle where m1 and m2 represent the frame rail (chassis) and passengercompartment masses, respectively In other cases, m1 may represent the vehicle structure with energyabsorbers (spring and damper), and m2, the torso with a restraint system of spring, k2, and damper, c2

Fig 5.13 A Two Mass-Spring-Damper Model

(5.8)

(5.9)

(5.10)

The method for solving the impact responses of the two masses is adapted from the method used

in the free vibration analysis of a two-degree of freedom damped system [4] The equations of motion(EOM) of the 2-MSD are shown in Eq (5.8)

Since an exponential function of t, est, returns to the same form in all derivatives, it thereforesatisfies the differential equations (1) and (2) of Eq (5.8) The solution for x1 and x2 is shown by (3)

of Eq (5.9) Note that s is a complex root in general, as are Ci, Ri, Si, as shown by (4) of Eq (5.9)

5.3.1 Solutions of the Characteristic Equation

After substituting R1, R2, S1, and S2 into (3) of Eq (5.9), a characteristic equation (a 4th orderpolynomial) is obtained as shown in Eq (5.10)

Case 1: Two pairs of complex conjugates,

Case 2: One pair of complex conjugate and two real and negative roots, and

Case 3: Four real and negative roots.

The solutions to the three cases are described as follows

Case 1: Two pairs of complex conjugates

The system in this case has moderate damping The rate of decay is defined by p1, the real part

of the root, and the frequency of vibration is specified by q1, the imaginary part

where p1, p2, q1, and q2 are all positive s1 and s2 are the first pair of complex conjugates, and s3 and

s4, the second pair The two roots s1 and s2 in the first pair will yield the solutions x11 and x21, wherethe first subscript refers to the mass index and the second subscript the pair number of the complexconjugate:

(I) Displacement components x 11 and x 21 due to s 1 and s 2 , respectively

The ratios of complex numbers C21/C11 and C22/C12 can be obtained by substituting s1 and s2 intothe characteristic equation, respectively The ratio of amplitudes A21/A11 can thus be obtained, and thephase angle relationship between M21 and M11 can be related as shown in Eq (5.13)

(II) Displacement components x 12 and x 22 due to s 3 and s 4 , respectively

Similarly, using the same approach as in (I), the ratio of amplitudes A22 to A12 and the phase anglerelationship between M22 and M12 can also be obtained They are shown in Eq (5.14)

(5.15)

(5.16)

(III) General Solution

A set of simultaneous equations is formulated based on the displacement and velocity expressionsand the initial conditions as shown in Eq (5.15)

Note that A21, A22, N21, and N22 are related to A11, A12, N11, and N12, respectively, by the relationshipsshown in the derivation before Thus, the general displacement equations are completely defined aftersolving the four unknowns in the four non-linear equations Note that this system has a moderatevalue of damping

Case 2: One pair of complex conjugate and two real and negative roots

The rate of decay is defined by p1, the real part of the root, and the frequency of vibration, by q1

imaginary part If the damping in the system is slight, the values of p1 and p2 will be small, and q1 and

q2 will approximate the values of T1 and T2, the frequencies of the system without damping.The solutions of the 4th order posinomial yield four roots where the coefficients p1, q1, F3, and

F4 are all positive The general displacement solutions are shown in Eq (5.16)

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