CHAPTER 3 CRASH PULSE PREDICTION BY CONVOLUTION METHODS 3.1 INTRODUCTION A crash pulse is the time history of the response of a vehicle system subjected to an impact or excitation. The dynamic characteristics of the system can be described by using a “hardware” or a “software” model. A “hardware” model is a system consisting of masses interconnected by energy absorbers (springs and dampers). This will be presented in Chapters 4 and 5. The present chapter covers the use of a “software” model utilizing digital convolution theory for crash pulse prediction. In a study by Eppinger and Chan [1], the concept of a finite impulse response (FIR) model based on convolution theory is used to assess thoracic injury in a side impact. Using accelerometer data from both the impacting side (rib cage) and the non-impacting side (spine) of a thorax, the torso dynamic system is characterized by a set of FIR coefficients, i.e., a transfer function. Then, under a different impact condition, the torso response in the non-impacting side can then be predicted by convoluting the 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 an algorithm using a snow-ball effect to increase the computation efficiency are discussed. Cases are presented which include but are not limited to the (1) Use of transfer functions in assessing the occupant response prediction using various crash pulse approximations, (2) Characterization of truck body 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 and belted occupant responses, and (4) Assessment of sled test pulses and the prediction of its occupant crash severity in a barrier test condition. In body-on-frame vehicles, two types of body mounts, using man-made or natural rubbers, are evaluated for their transient transmissibility (TT), the ability of the body mount to transfer the frame impulse to the body. Two trucks with different body mounts and restraint systems were tested in high speed barrier crashes. The dynamic properties of two body mounts are characterized by transfer functions. 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 mount and 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 digital convolution 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 transfer function. The dynamic properties of the components, such as air bag and body mount systems, can then be compared for crashworthiness evaluation. Other applications of FIR transfer functions including 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 input and 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 of these forward prediction properties of FIR , the method of inverse filtering is thus utilized to develop the 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 FIR coefficients. 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 low frequency input (such as body excitation) has been shown to be high. One of the applications in predicting the truck frame pulse based on an optimized or desired body pulse is illustrated. © 2002 by CRC Press LLC 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 (Laplace transformation) 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 conditions are 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 vehicle can be modeled as two subsystems, as shown in Fig. 3.2 . These are subsystem #1: frame rails (m 1 ) and body rocker panels (m 2 ), connected by body mounts (k and c), and subsystem #2: upper front end structure (m N) connected to the body (m 2 ), through the dash panels (kN and cN). Relative to the body (m 2 ) response, the two subsystems (mN and m 1 ) are parallel. In frontal impact occupant kinematics analysis, multiple subsystems also exist. Subsystem #1 may represent the belt and air bag restraint system (and the steering column for the driver side), and subsystem #2 may represent the knee bolster, as shown in Table 3.1. Table 3.1 A System with Multiple Transfer Functions [TF] System [TF] N: kN and cN [TF]: k and c Vehicle Front Structure Upper Front End Body Mount Occupant Restraint System Belt and Air Bag Restraint System Knee Bolster © 2002 by CRC Press LLC (3.1) (3.2) Awareness of the existence of multiple transfer functions is essential in the analysis and computation of the finite impulse response coefficients of a specific system. The relationship between the 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 accurately compute the body mount transfer function, the input frame (m 1 ) and output body (m 2 ) data associated with the load path through the body mount should be processed first before the upper structure starts affecting the body response. This is because, in a vehicle frontal impact, the bumper comes in contact with 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 thus provides 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 a component 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 upper structure 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 integral shown 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 relationships with 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 relationships between the FIR coefficients and the two model parameters, spring stiffness (or natural frequency) and damping coefficient (or damping factor), are described in the following sections. The FIR coefficients are useful in describing the dynamic characteristics of a system and in predicting the system response under a different input condition. As an example, in a frontal barrier crash, the chest deceleration and vehicle deceleration can be processed to obtain the set of FIR coefficients which describes the dynamic characteristics of the restraint system. The set of FIR coefficients is therefore the transfer function between the vehicle and occupant systems and represents the dynamic characteristics of the restraint system. The transfer function can then be used to predict occupant 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 accelerometer data 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 efficient algorithm based on matrix symmetry and a technique called the “snow-ball effect” are presented. The variables for the input, the actual and predicted outputs, and the FIR coefficients are defined as follows: © 2002 by CRC Press LLC (3.3) Fig. 3.3 Input Discrete Data Points and FIR Coefficients (3.4) y ^ (n): predicted output at point n M: total number of FIR coefficients y(n): actual output at point n n: a discrete point index 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 convolution formula shown in Eq. (3.3): Fig. 3.3 shows the convolution process where a set of M FIR coefficients overlaps the same number of input discrete data points. The FIR coefficients are numbered from 1 to M in reverse order compared to the input discrete data points, which are numbered from (n-m) to (n-1), with a total of M discrete points. Taking the sums 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 less than 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 finite impulse response (FIR) coefficients are presented below. The set of the FIR coefficients, h(m), for the given input and output pulses can then be solved from a set of simultaneous linear equations shown in (3) of Eq. (3.5). © 2002 by CRC Press LLC (3.5) (3.6) A ij is an auto-correlation matrix, which provides a comparison of a signal (x) with itself as a function of time shift, while B i is a cross-correlation matrix, which provides a comparison of two signals (x and y) as a function of time shift. The time needed to compute the matrix elements of A ij and B i can be extremely long if a straight forward (with repetitive computation of each of the matrix elements) method is used. However, an efficient 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 repetitive properties which may be computation-intensive. In Eq. (3.6), the properties related to the numerical operation are described, and the snow-balling technique is utilized to perform the computation of the matrix elements. Subsequently, the solutions of the FIR coefficients can be solved efficiently on a computer. Special Properties of Matrices A and B © 2002 by CRC Press LLC (3.7) (3.8) The composition of each of the elements in A ij has been analyzed for the number sequence repetition. This repetition can be eliminated from the computation to shorten computation time. The technique used to eliminate this repetitiveness, as shown in Eq. (3.7), uses the snow-ball effect. The process 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 upper half of the matrix. Due to the symmetry of matrix [A], the lower half of the matrix is simply: A ji = A ij . The snow-ball and straightforward techniques have been tested on a personal computer, and the computation 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 straight- forward method on the same computer. 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 A 55 , generated in Eq. (3.8), is snow-balled into the computation of A 44 . In the operation, only one multiplication of a pair of numbers is executed and the product is summed up with the seed to get A 44 . Similarly, A 44 is snow-balled into the computation of A 33 as shown in “Order” 3. By repeating the same 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 Fortran computer program based on the computation algorithm presented is listed in Table 3.2. It includes the snow-ball technique and a subroutine using a Gaussian method [2] to solve the set of simultaneous equations for the FIR coefficients. © 2002 by CRC Press LLC (3.9) Table 3.2 Gaussian Routine for Determining Finite Impulse Response Coefficients (FIR) c main routine COMMON/SOL/a(151,152),h(151) dimension xy(251,2) write(*,"(' no. of pts (t=100ms)=',$)") read(*,*) n npb = n step = 100./float(npb-1) write(*,"(' step=',f9.3)") step pi = 3.1415926 f = 10. ! hz w = 2*pi*f ! rad./sec, do i = 1, npb tms = (i-1)* step t = .001 * tms gin = -15 ! input data gout = gin*(1 cos(w * t)) ! output haversine xy(i,1) = gin xy(i,2) = gout end do 2 write(*,"(' # of fir coeff.(max=',i3,') =>',$)") n read(*,*) m if(m.gt.n) goto 2 c prepare a(i,j) matrix with order of m x m do i = m,1,-1 i1 = n-i+1 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 A im do while (i1.gt.0) © 2002 by CRC Press LLC i2 = i1+1 j2 = j1+1 i3 = n-i1+1 j3 = n-j1+1 a(i1,j1) = a(i2,j2) + xy(i3,1)*xy(j3,1) i1 = i1-1 j1 = j1-1 end do end do c copy upper right hand elements to the lower c left hand due to matrix symmetry do i = 2, m do j = 1, i-1 a(i,j) = a(j,i) end do end do c prepare Bb(i) matrix with order of m x 1 c store in a(i,k),i=1,2, ,m; k=m+1 m1 = m+1 do i = 1, m a(i,m1) = 0. do j = i, n l = j-i+1 a(i,m1) = a(i,m1) + xy(j,2)*xy(l,1) end do end do call gauss(m) c Compute the predicted response, xy(n,1) do n = 2, npb+1 ! Fir prediction n1 = n-1 xy(n1,2) = 0. xyn=xy(n1,2) do j = 1, m nj = n-j if(nj .gt. 0) then xy(n1,2) = xy(n1,2) + h(j)*xy(nj,1) xyn = xy(n1,2) end if end do end do stop end subroutine gauss(n) c to solve for simultaneous equations [2] COMMON/SOL/a(151,152),h(151) m = n+1 l = n-1 do k = 1, l kk = k+1 do i = kk,n qt = a(i,k)/a(k,k) do j = kk,m a(i,j) = a(i,j)-qt*a(k,j) end do 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 constant deceleration of -15 g and the output is a haversine wave with a duration of 100 ms and a peak magnitude 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. © 2002 by CRC Press LLC Table 3.3 Input and Output Data Sets and Generation of Matrices A ij and B ij i 1 23456789 t, ms 0 12.5 25 37.5 50 62.5 75 87.5 100 X(i), g -15 -15 -15 -15 -15 -15 -15 -15 -15 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: A 55 = X(5)*X(5)+X(4)*X(4)+X(3)*X(3)+X(2)*X(2)+X(1)*X(1) = 1125.0 A 45 = X(6)*X(5)+X(5)*X(4)+X(4)*X(3)+X(3)*X(2)+X(2)*X(1) = 1125.0 A 35 = X(7)*X(5)+X(6)*X(4)+X(5)*X(3)+X(4)*X(2)+X(3)*X(1) = 1125.0 A 25 = X(8)*X(5)+X(7)*X(4)+X(6)*X(3)+X(5)*X(2)+X(4)*X(1) = 1125.0 A 15 = X(9)*X(5)+X(8)*X(4)+X(7)*X(3)+X(6)*X(2)+X(5)*X(1) = 1125.0 (ii) Snow-balling from Seeds: A 14 = A 25 +X(9)*X(6) = 1125.0 + -15.0 * -15.0 = 1350.0 A 24 = A 35 +X(8)*X(6) = 1125.0 + -15.0 * -15.0 = 1350.0 A 13 = A 24 +X(9)*X(7) = 1350.0 + -15.0 * -15.0 = 1575.0 A 34 = A 45 +X(7)*X(6) = 1125.0 + -15.0 * -15.0 = 1350.0 A 23 = A 34 +X(8)*X(7) = 1350.0 + -15.0 * -15.0 = 1575.0 A 12 = A 23 +X(9)*X(8) = 1575.0 + -15.0 * -15.0 = 1800.0 A 44 = A 55 +X(6)*X(6) = 1125.0 + -15.0 * -15.0 = 1350.0 A 33 = A 44 +X(7)*X(7) = 1350.0 + -15.0 * -15.0 = 1575.0 A 22 = A 33 +X(8)*X(8) = 1575.0 + -15.0 * -15.0 = 1800.0 A 11 = A 22 +X(9)*X(9) = 1800.0 + -15.0 * -15.0 = 2025.0 Generation of Matrix Elements, B i : B 1 = y(1)*X(1)+y(2)*X(2)+. . .+y(9)*X(9) = 1800.0 B 2 = y(2)*X(1)+y(3)*X(2)+. . .+y(9)*X(8) = 1800.0 B 3 = y(3)*X(1)+y(4)*X(2)+. . .+y(9)*X(7) = 1734.1 B 4 = y(4)*X(1)+y(5)*X(2)+. . .+y(9)*X(6) = 1509.1 B 5 = y(5)*X(1)+y(6)*X(2)+. . .+y(9)*X(5) = 1125.0 M = 5 : The computations of matrix elements, A ij and B i , the FIR coefficients, and the predicted output, y^, are shown in the Table 3.3 and Eq. (3.11). For the case where the number of FIR coefficients is five, the matrix elements of A ij generated by the snow-balling approach are shown in Table 3.3, as are the matrix elements of B i . M = 9 : For the case where the number of FIR coefficients is equal to nine, the matrix elements of A ij and B i 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 case where 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. © 2002 by CRC Press LLC (3.10) (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 the number of FIR coefficients is equal to the number of data points, M = N = 9, the predicted output matches 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, while that for M = 5 has a pattern of a halfsine wave, as shown in Fig. 3.4. Note that the magnitudes of the FIR coefficients are relative. They depend on the time step of the data points. In this example, the time 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 data time step. © 2002 by CRC Press LLC [...]... derived from the crash pulse alone without satisfying any test boundary conditions can be computed by Eq (2.27) in Chapter 2 In the Truck #2 rigid barrier 31 mph test, the FEW has been computed and is shown in Eq (3.18) and Fig 3.24 (3.18) The 5th order FEW, shown in Fig 3.24, exhibits a double hump wave and approximates the test vehicle crash pulse very closely To assess the effect of the FEW crash pulse... column in belted and unbelted occupant conditions plays an important role in the occupant responses In Chapter 1, crash pulse characterization methods were developed to approximate an original crash pulse by the use of only a few parameters These techniques utilize only a few parameters for crash pulse approximation and capture the major responses of a dynamic system in a test The © 2002 by CRC Press... the approximation technique in predicting the occupant response depends on the number of parameters used The crash test data, such as vehicle and occupant accelerometer data, for two trucks in rigid barrier 31 mph tests are used in this study Table 3.4 shows the summaries of the occupants and vehicles, truck #1 and truck #2 The data for the two trucks are used to obtain the transfer functions of the... Factor) = peak occupant deceleration, g / ESW, g 3.4.1 Test Vehicle and Occupant Responses Two trucks, one a pickup truck with a full-powered air bag for an unbelted occupant and the other an SUV (sport utility vehicle) with a depowered air bag for a belted occupant, were tested in 31 mph rigid barrier impacts The essential data describing the vehicle and occupant responses are shown in Table 3.4 Truck... TWA crash pulse as input The predicted occupant torso decelerations (yTWA) using ^ TWA input (xTWA) are shown in Fig 3.22 The peak deceleration magnitudes of both the predicted and the test torso responses occur during the vehicle rebounding phase and result in a relative high deceleration However, the magnitudes of the peak femur deceleration for both the predicted and test responses occur before vehicle. .. with a 3-point belt and an air bag system in a rigid barrier test at 31 mph, the vehicle left rocker and left front occupant torso decelerations are shown in Fig 3.22 The TWA (trapezoidal wave approximation), satisfying two boundary conditions of the vehicle dynamic crush and velocity change, is overlapped with the test crash pulse, as shown in the figure 3.4.3.1 Restraint Transfer Function Validation... objectives of the study in this section are two-fold One is to evaluate the crash pulse approximation techniques in predicting test occupant performance, the other, to assess the belted and unbelted left front occupant performance in air bag equipped vehicles Since the air bag module is attached to the steering wheel system, the impact dynamics of the steering column in belted and unbelted occupant conditions... occupant in the trucks subjected to TWA and FEW pulses The predicted occupant responses from these approximated pulses are compared with those from the tests Table 3.4 Vehicle and Occupant Responses in the Truck 31 mph Rigid Barrier Tests Vehicle Truck (type, restraint condition) #1: pickup truck, unbelted, w/ fullpowered air bag Torso / Femur Dy Cr*., in @ Tm, ms ESW, g t*, restraint contact time,ms... total number of data points fourteenfold The total number of data points and the number of FIR coefficients are greater than those for truck #1, due to the longer duration of the crash pulse To validate the FIR model, the test vehicle pulse (x) is convoluted with the FIR coefficients Similar to the Truck #1 FIR model validation © 2002 by CRC Press LLC (plotted in Fig 3.17), the predicted torso output... TWA (Trapezoidal Wave Approximation) crash pulse on the occupant response, a TWA is used as the input in the derived FIR model The predicted output torso deceleration with a TWA is compared to the test torso deceleration and shown in Fig 3.22 The predicted peak torso deceleration of 30 g, lower than the test torso deceleration of 36 g, occurs around 65 ms, before the vehicle comes to a stop at the rigid . CHAPTER 3 CRASH PULSE PREDICTION BY CONVOLUTION METHODS 3.1 INTRODUCTION A crash pulse is the time history of the response of a vehicle system subjected to an impact. used. The crash test data, such as vehicle and occupant accelerometer data, for two trucks in rigid barrier 31 mph tests are used in this study. Table 3.4 shows the summaries of the occupants and vehicles,. obtained from the crash test data, such as the accelerometer data or the barrier load cell data. In computing the FIR coefficients of a component such as the body mounts on a frame vehicle, the accelerometer