Vehicle Crash Dynamics P5 pot

Using accelerometer data fromboth the impacting side rib cage and the non-impacting side spine of a thorax, the torso dynamicsystem is characterized by a set of FIR coefficients, i.e., a

CHAPTER 3 CRASH PULSE PREDICTION BY CONVOLUTION METHODS

In a study by Eppinger and Chan [1], the concept of a finite impulse response (FIR) model based onconvolution theory is used to assess thoracic injury in a side impact Using accelerometer data fromboth the impacting side (rib cage) and the non-impacting side (spine) of a thorax, the torso dynamicsystem is characterized by a set of FIR coefficients, i.e., a transfer function Then, under a differentimpact condition, the torso response in the non-impacting side can then be predicted by convolutingthe FIR coefficients with the accelerometer data for the impacting side of the thorax

The basic operation of convolution theory, the derivation of the transfer function, and analgorithm using a snow-ball effect to increase the computation efficiency are discussed Cases arepresented which include but are not limited to the (1) Use of transfer functions in assessing theoccupant response prediction using various crash pulse approximations, (2) Characterization of truckbody mounts by FIR coefficients and the prediction of body pulses with different frame pulses, (3)Evaluation of the performance of air bag and steering column restraint systems for both unbelted andbelted occupant responses, and (4) Assessment of sled test pulses and the prediction of its occupantcrash severity in a barrier test condition

In body-on-frame vehicles, two types of body mounts, using man-made or natural rubbers, areevaluated for their transient transmissibility (TT), the ability of the body mount to transfer the frameimpulse to the body Two trucks with different body mounts and restraint systems were tested in highspeed barrier crashes The dynamic properties of two body mounts are characterized by transferfunctions Similarly, two restraint systems are characterized by their respective transfer functions.The occupant response in a high speed barrier crash of one truck using the interchanged body mountand restraint system from the other can then be predicted and the performance assessed

Using a Kelvin model, in which the spring and damper are connected in parallel, a digitalconvolution formula can be derived using the Laplace transform The closed-form formulas in terms

of two model parameters, K (spring stiffness) and C (damping coefficient), describe the transferfunction The dynamic properties of the components, such as air bag and body mount systems, canthen be compared for crashworthiness evaluation Other applications of FIR transfer functionsincluding the development of RIF (response inverse filtering) are discussed

RIF is based on finite impulse response (FIR) and inverse filtering (IF) methods The accuracy

in validation and prediction via FIR transfer functions depends on the frequency content of the inputand output accelerometer data from which the transfer function is developed The prediction accuracy

is low if the output data contain higher frequency components than the input Taking advantage ofthese forward prediction properties of FIR , the method of inverse filtering is thus utilized to developthe RIF for the backward prediction

The new RIF transfer function is created by the IF operation applied to the FIR transfer function.The IF technique involves four sequential matrix operations applied to the column matrix of the FIRcoefficients These matrix operations include transpose, multiplication, inverse, and multiplication.The accuracy of RIF in predicting the high frequency output (such as frame impulse) with the lowfrequency input (such as body excitation) has been shown to be high One of the applications inpredicting the truck frame pulse based on an optimized or desired body pulse is illustrated

Fig 3.1 A Transfer Function – A Convolution Process

Fig 3.2 A Dynamic System with Multiple Transfer Functions

3.2 TRANSFER FUNCTION VIA CONVOLUTION INTEGRAL

A dynamic system can be characterized by its ability to convert a set of discrete-time data (input)into another sequence (output) Such a conversion process shown in the S domain (Laplacetransformation) and time domain (convolution integral) in Fig 3.1 is defined as a transfer function

A system with an input variable x and an output variable y is linear if the following conditionsare satisfied

Condition 1: output: ay <=== input: ax

Condition 2: output: by <=== input: bx

Condition 3: output: (a+b)y <=== input: (a+b)x

A system can consist of multiple subsystems In a frontal barrier test, the front end of a vehiclecan be modeled as two subsystems, as shown in Fig 3.2 These are subsystem #1: frame rails (m1)and body rocker panels (m2), connected by body mounts (k and c), and subsystem #2: upper front endstructure (mN) connected to the body (m2), through the dash panels (kN and cN)

Relative to the body (m2) response, the two subsystems (mN and m1) are parallel In frontal impactoccupant kinematics analysis, multiple subsystems also exist Subsystem #1 may represent the beltand air bag restraint system (and the steering column for the driver side), and subsystem #2 mayrepresent the knee bolster, as shown in Table 3.1

Table 3.1 A System with Multiple Transfer Functions [TF]

Vehicle Front Structure Upper Front End Body Mount

Occupant Restraint System Belt and Air Bag Restraint

System

Knee Bolster

(3.2)

Awareness of the existence of multiple transfer functions is essential in the analysis andcomputation of the finite impulse response coefficients of a specific system The relationship betweenthe transfer functions and the overall input and output data is described by Eq (3.1)

To define a transfer function, the load paths need to be identified As an example, to accuratelycompute the body mount transfer function, the input frame (m1) and output body (m2) data associatedwith the load path through the body mount should be processed first before the upper structure startsaffecting the body response This is because, in a vehicle frontal impact, the bumper comes in contactwith the barrier followed by the deformation of the front rail Subsequently, the upper structure, such

as the shotgun and fender, may start interacting with the barrier The upper front end structure thusprovides a separate load path to the occupant compartment (body)

The loadings acting on the upper and lower structures can be obtained from the crash test data,such as the accelerometer data or the barrier load cell data In computing the FIR coefficients of acomponent such as the body mounts on a frame vehicle, the accelerometer data of the frame and body

in the first 20 ms after impact should be utilized This is due to the fact that after 20 ms, the upperstructure contributes a portion of the deceleration to the body

3.2.1 Convolution Method and Applications

The response of a linear system to a time varying input can be defined by the convolution integralshown in Eq (3.2)

Similarly to the analytical simulation using the spring-damper-mass model described in Chapters

4 and 5, a model is also linear if the spring and damper forcing functions have linear relationshipswith the deformation and deformation rates, respectively A linear system can be described by a set

of finite impulse response (FIR) coefficients obtained by digital convolution theory The relationshipsbetween the FIR coefficients and the two model parameters, spring stiffness (or natural frequency) anddamping coefficient (or damping factor), are described in the following sections

The FIR coefficients are useful in describing the dynamic characteristics of a system and inpredicting the system response under a different input condition As an example, in a frontal barriercrash, the chest deceleration and vehicle deceleration can be processed to obtain the set of FIRcoefficients which describes the dynamic characteristics of the restraint system The set of FIRcoefficients is therefore the transfer function between the vehicle and occupant systems and representsthe dynamic characteristics of the restraint system The transfer function can then be used to predictoccupant responses in a vehicle with a new or modified vehicle structures

In describing the body mount dynamic behavior in light truck barrier crashes, the transfer function

of the body mount between the frame and the body (or cab) can be obtained from the accelerometerdata at the frame and body of the truck Consequently, the body response can be predicted once given

a set of new frame accelerometer data

The method using input and output discrete data points to obtain the FIR coefficients is described

in the following Since the computation of the FIR coefficients is numerically intensive, an efficientalgorithm based on matrix symmetry and a technique called the “snow-ball effect” are presented Thevariables for the input, the actual and predicted outputs, and the FIR coefficients are defined asfollows:

x(n!m): system input at a discrete point n!m N: total number of discrete

h(m): FIR coefficient at point m

Given a set of FIR coefficients, the predicted output is then expressed by the digital convolutionformula shown in Eq (3.3):

Fig 3.3 shows the convolution process where a set of M FIR coefficients overlaps the samenumber of input discrete data points

The FIR coefficients are numbered from 1 to M in reverse order compared to the input discretedata points, which are numbered from (n-m) to (n-1), with a total of M discrete points Taking thesums of each pair of the input data and FIR coefficient yields a predicted output, y^ (n), at point n,where the corresponding input data is x(n)

At the beginning and the end of computation, where the number of overlapping data points is lessthan the number of FIR coefficients, the values of those input data points outside the range from 1 to

N are assigned a value of zero

3.2.2 Solution by the Least Square Error Method

Using the least square error method shown in Eq (3.4), the steps needed to create a set of finiteimpulse response (FIR) coefficients are presented below The set of the FIR coefficients, h(m), forthe given input and output pulses can then be solved from a set of simultaneous linear equations shown

in (3) of Eq (3.5)

(3.6)

Aij is an auto-correlation matrix, which provides a comparison of a signal (x) with itself as afunction of time shift, while Bi is a cross-correlation matrix, which provides a comparison of twosignals (x and y) as a function of time shift

The time needed to compute the matrix elements of Aij and Bi can be extremely long if a straightforward (with repetitive computation of each of the matrix elements) method is used However, anefficient method based on the symmetry of the matrix and a “snow-ball technique” have been devised.The method has been tested and is about 25 times faster than the repetitive computation approach.The snow-ball method is described in the following section

3.2.3 Matrix Properties and Snow-Ball Effect

The two sets of matrices derived from the least square error method possess certain repetitiveproperties which may be computation-intensive In Eq (3.6), the properties related to the numericaloperation are described, and the snow-balling technique is utilized to perform the computation of thematrix elements Subsequently, the solutions of the FIR coefficients can be solved efficiently on acomputer

Special Properties of Matrices A and B

(3.8)

The composition of each of the elements in Aij has been analyzed for the number sequencerepetition This repetition can be eliminated from the computation to shorten computation time Thetechnique used to eliminate this repetitiveness, as shown in Eq (3.7), uses the snow-ball effect Theprocess begins by computing the elements of the last column (j = m) in matrix A; then, the snow-balling starts from the elements in the last column and generates the rest of the elements in the upperhalf of the matrix Due to the symmetry of matrix [A], the lower half of the matrix is simply: Aji =

Aij The snow-ball and straightforward techniques have been tested on a personal computer, and thecomputation time difference was determined Given N (number of data points) = 250 and M (number

of FIR coefficients) = 150, the snow-ball technique was found to be 25 times faster than the forward method on the same computer

straight-Matrix Element Computation Methods

Matrix Element Computation Methods (Continued)

As shown by the computation sequence from “Order” 1 to 2 in Eq (3.9), the seed element A55,generated in Eq (3.8), is snow-balled into the computation of A44 In the operation, only onemultiplication of a pair of numbers is executed and the product is summed up with the seed to get A44.Similarly, A44 is snow-balled into the computation of A33 as shown in “Order” 3 By repeating thesame procedure for the other seeds, the upper half of the matrix is completed Due to the symmetry

of the matrix, the element computation for the lower half of the matrix is not required A Fortrancomputer program based on the computation algorithm presented is listed in Table 3.2 It includes thesnow-ball technique and a subroutine using a Gaussian method [2] to solve the set of simultaneousequations for the FIR coefficients

gin = -15 ! input data

gout = gin*(1.-cos(w * t)) ! output haversine

j = m j1 = n-j+1 a(i,j) = 0

aij = a(i,j)

c generating Seed, aij

do while (i1.gt.0 and j1.gt.0) a(i,j) = a(i,j) + xy(i1,1)*xy(j1,1) aij = a(i,j)

i1 = i1 - 1 j1 = j1 - 1 end do end do

do i = 2,m i1 = i-1 j1 = m-1

c snow-balling from Seed Aim

do while (i1.gt.0)

c copy upper right hand elements to the lower c

left hand due to matrix symmetry

end do

do i = kk,n a(i,k) = 0

end do end do h(n) = a(n,m)/a(n,n)

do nn = 1, l sum = 0

i = n-nn

ii = i+1

do j = ii,n sum = sum+a(i,j)*h(j) end do

h(i) = (a(i,m)-sum)/a(i,i) end do

return end

3.2.4 Case Studies: Computing Transfer Functions

The procedures described above are applied to an example where the transfer function converts

an input of square wave into an output of haversine pulse The data sets for the input (x) and output(y) decelerations versus time (t) are shown in Table 3.3 Shown in Fig 3.4, the input is a constantdeceleration of -15 g and the output is a haversine wave with a duration of 100 ms and a peakmagnitude of -30 g The total number of data points, N, is nine The number of FIR coefficients, M,

is set to either five or nine

Table 3.3 Input and Output Data Sets and Generation of Matrices A ij and B ij

Y(i), g 0 -4.39 -15 -25.61 -30 -25.61 -15 -4.39 0

For M (number of FIR coefficients) = 5

Generation of Matrix Elements, A ij :

(i) Generating Seeds:

Table 3.3, as are the matrix elements of Bi

M = 9: For the case where the number of FIR coefficients is equal to nine, the matrix elements of

Aij and Bi are also shown in Eqs (3.10) and (3.11) Comparing the two sets of matrices, for M = 5 and

M = 9, it is noted that the set of matrix elements for the case of M = 5 is only a subset of the casewhere M = 9 The first 5 by 5 subset matrix in the 9 by 9 matrix A is the same as that matrix A for M

= 5 The subset observation also applies to the B matrix elements

(3.11)

FIR Coefficients and Predicted Output with M =5 and 9, and N = 9

The two sets of FIR coefficients and the predicted outputs for M (number of FIR coefficients)equal to five and nine are plotted and shown in Figs 3.4 and 3.5, respectively For the case M= 5,the predicted output matches the actual output for the first four data points For the case where thenumber of FIR coefficients is equal to the number of data points, M = N = 9, the predicted outputmatches completely the actual output as shown in Fig 3.5

The plot of FIR coefficients for M = 9 has a pattern of a sine wave as shown in Fig 3.5, whilethat for M = 5 has a pattern of a halfsine wave, as shown in Fig 3.4 Note that the magnitudes of theFIR coefficients are relative They depend on the time step of the data points In this example, thetime step is 12.5 ms, and the peak magnitude of the FIR coefficients shown in Fig 3.6 is 0.7 It will

be shown in the next section that the magnitudes of the FIR coefficients are proportional to the datatime step

Fig 3.4 FIR Prediction with M=5 and Constant Input

Fig 3.5 FIR Prediction with M=9 and Constant Input

Fig 3.6 FIR Coefficient Comparison with M=5 and 9

()t=12.5 ms)

3.3 TRANSFER FUNCTION AND A SPRING-DAMPER MODEL

The spring-damper model shown in Fig 3.7 when subjected to an input excitation, p(t), yields anoutput response of q(t) The process of the model transforming the input to an output in either the timedomain (Fig 3.7) or in the S domain (Fig 3.8) is defined as a transfer function as shown in thefollowing

Fig 3.7 A spring-damper model subjected to an input

excitation, p(t)

(3.12)

Fig 3.8 Input P(s), Output Q(s), and

Transfer Function H(s) in S Domain

(3.13)

3.3.1 FIR Coefficients and K-C Parameters of a Spring-Damper Model

Given an input excitation to a Kelvin model, consisting of a spring (K) and damper (C) in parallel,the output response of the model can be obtained By solving the second order differential equation

of the spring-damper model using the Laplace transformation, the finite impulse response (FIR)