Vehicle Crash Dynamics P10 ppsx

(5.35) (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 the Newton-Raphson algorithm described in Eq. (5.36) and shown in Fig. 5.25. The slope of the curve at point F i is f '(F i ). F i is the value of the independent variable at iteration i. The iteration process continues until F i and F i+1 converge to the solution of the function in which f(F i+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 initial value of F i at any given time step should be the F* from the previous time step. © 2002 by CRC Press LLC 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=kD n , and the unloading by a constant slope. In the dynamic 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), and during the unloading phase, the relative deformations of the two EA Ns and the corresponding unloading force are explicitly computed. 5.6.1 Loading Phase Simulation In general, there are two non-linear springs involved in the force-deflection computation. An interactive 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 time step, 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 Newton- Raphson algorithm. © 2002 by CRC Press LLC 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 loading phase simulation. The force and the individual deflections of the two springs depend on the total deflection and unloading stiffnesses of the two springs. The method to execute the unloading phase simulation is described in Fig. 5.27 and Eq. (5.39). © 2002 by CRC Press LLC 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-loading point is at x i , the start-unloading point is at point U, and the reloading-point is at point x r . Unloading starts at the end of loading. The unloading power curve is defined by the two parameters k N and nN. Given the specified hysteresis energy, E h , and permanent deformation, x p , the two unloading parameters are derived in closed-form and shown in Eq. (5.40). It is also shown that under the specified loading and unloading conditions, the unloading power n N is always greater than the loading power n. Variable Definitions in the Loading and Unloading Cycles Loading: Unloading: , where a x : acceleration function at x x :displacement or velocity for spring damper x i : x location of starting loading or reloading cycle x p : x location of permanent deformation or deformation rate At the point of unloading, U, a max (loading) = a' max (unloading), Let E h be the hysteresis energy in the loading/unloading cycle; then E h = E L (loading energy) ! E u (unloading energy). © 2002 by CRC Press LLC (5.40) (5.41) Reloading starts at the end of unloading. During reloading, the system may unload at any moment. The unloading curve is not only a function of k N and nN, it is also a function of x i , the x location 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 numerical simulation follows the flow charts shown in Table 5.13. There are two flow charts. The one on the top shows the numerical integration procedures. The one on the bottom computes the deceleration (force) due to loading or unloading by the spring and damper. © 2002 by CRC Press LLC Table 5.13 Power Curve Loading and Unloading Flow Chart © 2002 by CRC Press LLC 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 dynamic characteristics of a body mount on a frame vehicle. The frame pulse from a truck-to-a-fixed-barrier test at 35 mph is used to excite the model. The predicted model body (output) pulse is then compared with 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 test body crash pulse with that from the optimized models with and without damping. It is clearly shown that a model without damping is not adequate to simulate the impulsive response which occurs in the early part of the body crash pulse. The body impulsive response is the result of the frame impulsive loading 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 a k and a c , respectively. Since a k is controlled by the deflection and a c by the deflection rate, both the transient deflection, )d, and the deflection rate, )v, are shown in Fig. 5.30. It is clearly shown that the first peak of the body pulse is dominated by a c , 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 )v increases at around 20 ms. © 2002 by CRC Press LLC 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 till about 19 ms, then reloading starts. Since )v keeps decreasing, contribution of damper to the output acceleration becomes minimal. However, in the steady state period after the transient response, the spring contributes more than the damper to the output deceleration, a, due to the ever-increasing deflection. Fig. 5.31shows the test frame pulse of another 35 mph truck to barrier test which is the input excitation 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 output response of the power curve model matches quite well the test responses in both the transient and steady 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 natural frequencies. These subsystems, which generate the high frequency noise, can be excluded from system modeling. CRUSH II (Crash Reconstruction Using Static History) is a one-dimensional mathematical model used 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-parameter model), 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 was used 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 component test laboratory, a testing device called Crusher can be used to generate the quasi-static force-deflection data. The components to be crushed are placed between the reaction fixtures and load plates. As the ram 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 stress- strain curve in Fig. 5.32; A, the cross sectional area; and L, the length of the material. 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 its stress-strain curve. The last point on the curve is the fracture point. © 2002 by CRC Press LLC 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 deflection formula. 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 deflection due to axial and shear forces are neglected. In the formulas, E denotes Young’s modulus, G, the modulus 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 perform its task. Fig. 5.35 is a sketch of the push bumper. A properly designed push bumper should not affect air bag crash sensor performance in a frontal collision. The proper design of push bumper needs to consider the effective stiffness of the push bumper and the attached front end of the vehicle, and its effect on the occupant kinematics and sensor performance in a test. © 2002 by CRC Press LLC 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 fixed ends, 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. The computation is shown in Eq. (5.42). © 2002 by CRC Press LLC [...]... the front end of the vehicle, and M is the mass of the vehicle The stiffness and mass distribution of the vehicle in the lateral direction is assumed to be that shown in Fig 5.49 The outer springs on both sides represent the frame rails, which are stiffer than those in the inner position, 0.3 k versus 0.2 k Fig 5.48 A Frontal Distributed Vehicle Model Fig 5.49 An Offset Distributed Vehicle Model In a... 5.56 Vehicle Deceleration and Intrusion in Full and Offset Barrier Tests (35mph) The selection of the optimal vehicle structure therefore is based on the set of vehicle responses, the peak deceleration in the full barrier test, and the intrusion in the offset test that provides the best overall occupant responses 5.8.2.5 An Offset Lumped-Mass Model To better model the intrusion response and vehicle crash. .. Thereafter, the door moves together with the rest of the vehicle structural components at a common velocity as shown in Fig 5.47 Meantime, the occupant separates from the door contact at a velocity higher than the common velocity of the door and vehicle Fig 5.47 Velocity Profiles of Impactor, Side Struck Vehicle, Crushed Door, and Torso Fig 5.47 shows the typical vehicle and occupant velocity profiles in a T-type... set forth by the FMVSS 208 for the occupant protection Therefore, an optimal vehicle structure needs to consider the intrusion and the vehicle peak deceleration in both full frontal and offset testing Using the vehicle weight and stiffness data shown in Section 5.8.2.3 for both the frontal and offset tests, the intrusion and vehicle peak deceleration can be computed for the impact speed of 35 mph © 2002... in the full barrier test yields an intrusion of 4 inches and vehicle peak deceleration of 31 g Shown in the plot, the vehicle peak deceleration in the full barrier test is higher than that in the offset test, and the intrusion in the offset test is always higher than that in the full barrier test The worst-case vehicle as far as the peak vehicle deceleration is the one that is too stiff, and the worst... mph) © 2002 by CRC Press LLC 5.8.2 Frontal Offset Impact In a vehicle- to -vehicle (or fixed rigid or fixed deformable barrier) frontal offset impact, due to the smaller contact area, the effective stiffness of the vehicle front end structure is reduced from that in a full frontal impact The effective mass is also less than the mass of the full vehicle The basic concept of splitting a spring mass model... the 50% offset test, 50% of the frontal area of the vehicle contacted a fixed rigid barrier In the following section, the test results of full barrier and offset barrier tests are analyzed, followed by the modeling of the full barrier and offset impacts The strategy of obtaining the optimal vehicle structure to meet the design targets of minimum vehicle intrusion and peak deceleration magnitude models... moves along with the major vehicle structure as one big mass, it impacts on the occupant with an impulsive loading Thereafter, the door moves together with the rest of the vehicle structural components at a common velocity as shown in Fig 5.47 Meantime, the occupant rebounds after contacting the door at a peak velocity (about 25 mph) higher than the common velocity of the door and vehicle (about 15 mph)... the simple spring-mass model, the vehicle peak deceleration and dynamic crush can be computed as shown in Eq (5.47) Assuming the vehicle has an available crush space of 28 inches, the intrusion is then equal to dynamic crush minus the available crush space Note that there is no intrusion if the dynamic crush is less than or equal to the available crush space (5.47) The vehicle peak deceleration and intrusion... to Sill (Side Panel to Sill in L-Type) M4: Struck Vehicle EA4: Door Pad (Bolster) M5: Rib Cage EA5: Sill to Struck Vehicle Compartment M6: Torso EA6: Torso to Rib Cage *4: Torso to Pad Clearance The impactor, after crushing the door panel, picks up the door sill, floor pan, rocker panel and/or B-pillar Before the crushed door moves along with the major vehicle structure as one big mass, it impacts on . 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. excluded from system modeling. CRUSH II (Crash Reconstruction Using Static History) is a one-dimensional mathematical model used to simulate the impact dynamics of objects or masses connected. affect air bag crash sensor performance in a frontal collision. The proper design of push bumper needs to consider the effective stiffness of the push bumper and the attached front end of the vehicle,