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/or any 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 or standard solid models, the combination of two hybrid models, and special cases with point masses will be treated first in closed-form. The purpose of the closed-form analysis is to investigate the dynamic responses of two dynamically equivalent hybrid models. In a multi-mass model, the unloading characteristics of a spring element is as important as the loading characteristics. The unloading of one mass in a model may produce loading of the neighboring masses, thereby, affecting the total system model responses. Power curve loading and unloading simulation with hysteresis energy loss and permanent deformation will be covered. To help solve some dynamic models quickly, a lumped-parameter model, CRUSH II [1], coupled with animation, will be utilized. The force-deflection formulas of some simple structures are listed for ease of determining the spring stiffness for the modeling. Some lumped-parameter models for the full frontal, side, and frontal offset impacts are described. The basic concepts of splitting a simple spring-mass model for the frontal offset impact and the model validation are 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. © 2002 by CRC Press LLC (5.1) A hybrid model is a combination of Maxwell and Kelvin models. It has three elements: two springs and one damper. These three elements are connected in such a way that the two hybrid models are structurally and functionally different (non-isomorphic). In impact analysis, each hybrid model has two mass systems with a closing speed of V 12 . To simplify the two-mass system analysis, the concept 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 body-on- frame vehicle [2], and the lateral impact modeling of the thorax of a driver in a car accident [3]. 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 deriving the equation of motions (E.O.M) for the Maxwell model shown in Section 4.8 of Chapter 4, the E.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 two hybrid 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) shows the hybrid #1 effective-mass system. In this model, there are two springs in contact with the rigid barrier. If one of the springs is moved to the other side of the mass while still in contact with the barrier (or ground), a new model arrangement is obtained, as shown in Fig. 5.3(b). Although the EA © 2002 by CRC Press LLC 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 k 1 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 Kelvin model with the spring and damper in parallel produces a non-zero deceleration at time zero. Consequently, the initial deceleration of the impactor, mass m 1 , 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 than 0.5. 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 (v o ) is equal to the jerk (j, rate of change of deceleration) at time zero. This relationship j = v o 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 the three a vs. d curves in Fig. 5.6 is constant. It is independent of the impact speed and is solely determined by the component parameters expressed in (7) of Eq. (5.1). However, the impact speed changes the maximum magnitudes of the g-force and deflection, as shown in Fig. 5.6. © 2002 by CRC Press LLC 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 initial slope 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. The transient displacement at 10 mph is simply equal to two times that at 5 mph as shown in Fig 5.5, while the 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), one can also combine the EA Ns (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 of superposition in combining two EA’s into one EA, described in Section 4.4.3 of Chapter 4, two hybrid models in series can be simplified by joining them into one effective hybrid model as shown in the bottom of Fig. 5.7. © 2002 by CRC Press LLC 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 the two 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 with Effective EAs k11, klb/in k21, klb/in c1, klb-s/in k12, klb/in k22, klb/in c2, klb-s/in K1, klb/in) K2, klb/in c, klb-s/in 6 12 .1 3 6 .08 2 4 .0444 The initial speed of m1 impacting on the stationary m2 is 60 mph. Note that for modeling purposes, 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 responses are 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 weight interface connected by a stiff spring to the main body. In both models, the peak accelerations of m1 and m2 are !17 and 34 g for m1 and m2, respectively. These occur at the same time, at 45 ms. © 2002 by CRC Press LLC 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 EA Ns (energy absorbers) in each of the two hybrid models shown in Figs. 5.1 and 5.2, respectively. By appending a subscript i to the existing parameters, the two sets of EA Ns become unique for each model. The EA parameters are k 1i , k 2i , and c i for hybrid model #i. Similarly, the coefficients in the characteristic equation shown in Eq. (5.4) are similarly assigned as t i , u i , and v i for model #i. To establish the dynamic equivalency between the two models, the three coefficients of hybrid model #1 are set equal to the corresponding coefficients of model #2 respectively. Since there are three constraint equations, the three EA parameters of one model can then be solved in terms of those of the other model. © 2002 by CRC Press LLC (5.5) (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) are needed for the computation. It will be shown in the next section that dynamic equivalency between the 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: k 11 = 2 klb/in, k 21 = 4 klb/in, c 1 = .0444 klb-sec/in, and w 1 = 6 klb, w 2 = 3 klb, v close = 60 mph Determine the counterpart parameters for hybrid model #2 such that the dynamic responses of the two models are identical. From Eq. (5.1), the effective weight and damping factor of hybrid model #1 are computed as shown in Eq. (5.5). Using Eq. (5.4), the parameters of hybrid model #2, k 12 , k 22 , and c 2 , 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, k 22 /k 12 = 2, is the same as that in model #1, k 21 /k 11 . The computed input data for hybrid model #2 are shown at the top of Fig. 5.9, and the original input data of model #1 are shown at the bottom of Fig. 5.9. Both models are effective mass models where the effective weight is 2 klb as computed in Eq. (5.5). The effective weight is moving at a speed of 60 mph to the left where the fixed rigid barrier is located. The total crush energy, the initial kinetic energy of the effective mass in the rigid barrier test, has been computed to be 240 klb-ft, as shown in Eq. (5.5). © 2002 by CRC Press LLC 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 the same 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 k11 and 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 for the two models in that period. In addition to acceleration, the two hybrid models are also dynamically equivalent in terms of transient velocity and displacement beyond the deformation phase, as shown in Fig. 5.10. The dynamic 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 also equivalent (identical) in terms of transient total crush energy. The total crush energy is the sum of the three individual crush energies due to springs #1 and #2 and the damper. Figs. 5.11 and 5.12 show the total and individual crush energies vs. time of the hybrid models #1 and #2, respectively. In both hybrid 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 © 2002 by CRC Press LLC (5.7) Fig. 5.11 Total and Individual Crush Energies vs. Time for Hybrid Model #1 Fig. 5.12 Total and Individual Crush Energies vs. Time of Hybrid Model #2 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 the formula 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 m 1 and m 2 represent the frame rail (chassis) and passenger compartment masses, respectively. In other cases, m 1 may represent the vehicle structure with energy absorbers (spring and damper), and m 2 , the torso with a restraint system of spring, k 2 , and damper, c 2 . © 2002 by CRC Press LLC © 2002 by CRC Press LLC 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, e st , returns to the same form in all derivatives, it therefore satisfies the differential equations (1) and (2) of Eq. (5.8). The solution for x 1 and x 2 is shown by (3) of Eq. (5.9). Note that s is a complex root in general, as are C i , R i , S i , as shown by (4) of Eq. (5.9). 5.3.1 Solutions of the Characteristic Equation After substituting R 1 , R 2 , S 1 , and S 2 into (3) of Eq. (5.9), a characteristic equation (a 4 th order polynomial) is obtained as shown in Eq. (5.10). [...]... spring-mass model in an impact yields a sinusoidal response Such a model is useful in modeling a vehicle compartment response in a fixed barrier impact However, it may not be suitable for modeling non-perpendicular barrier crashes such as vehicle pole (or tree) and offset tests In these softer crashes, the crash pulse looks more like a haversine, having a softer pulse in the beginning followed by a stiffer... investigate the intrinsic parametric relationships in the two-mass system, the natural frequency relationship used in the vehicle suspension and handling dynamics [5] is utilized In this vehicle dynamic study, components, such as an unsprung mass (control arm and suspension) or a sprung mass (vehicle chassis), are evaluated for ride comfort in terms of their natural frequencies If the two-mass model (shown... and damping coefficients in the two mass-spring-damper model can be estimated for crash analysis Case A: In a 31 mph fixed barrier impact, the maximum displacements of the fore-frame and aftframe of a vehicle are specified to be 10 and 24 inches, respectively, and the corresponding timings are 50 and 60 ms The total vehicle weight is 3 klbs with the fore-frame weighing 1 klb and the aftframe 2 klbs... also has to be increased (from 6.77 to 8.91 klb/in) The computed structural parameters for both Cases A and B are considered to be reasonable 5.3.4 Application in Post -Crash Structural Analysis Given two sets of accelerometer data in a crash test, the structural parameters defining the components between the two accelerometer locations can be computed In a frontal impact, for example, the accelerometer... corresponding times are 78.4 and 75.6 ms, respectively The total vehicle weight is 5 klbs The fore-frame has a weight of 3 klbs and the aft-frame 2 klbs Estimate the structural parameters in terms of the stiffness and damping coefficients for the two structural members Note that the body mount at the A-pillar stays intact during the crash Using the imbedded random searching technique for this problem,... the masses in the system resonates due to a change in excitation condition, it vibrates strongly about its own natural frequency In a vehicle- to-rigid object impact, the structural response is higher and impact duration is longer than those in a soft impact such as vehicle offset or side impact In soft impact, the main component © 2002 by CRC Press LLC is characterized by a lower response magnitude... Displacement Components Fig 5.15 Mass 2 Displacement Components © 2002 by CRC Press LLC Fig 5.16 Total Displacements of Mass 1 and Mass 2 5.3.3 Application in Pre-Program Vehicle Structural Analysis In the early development stage of a prototype, the vehicle crush requirements, such as dynamic crush and timing, are defined Using the derived closed-form displacement equations and a searching technique called “Imbedded... C14 in four equations C21, C22, C23, and C24 are related to C11, C12, C13, and C14, respectively, by relationships similar to those shown in the derivation for Case 2 5.3.2 Vehicle Displacement Responses in Fixed Barrier Impact A vehicle in a 31 mph fixed barrier impact is modeled by m1= 1 klb (kilo-pound), m2 = 2 klb, k1 = 6.28 klb/in, k2 = 3.96 klb/in, c1 = 076 klb-sec/in, c2 = 056 klb-sec/in The... their natural frequencies If the two-mass model (shown in Fig 5.13) were placed in an upright position with m1 representing the mass of the tire at the bottom, and the sprung mass, m2 representing the vehicle body, on the top, k1 would represent the tire spring rate and k2 the suspension compliance 5.4.1 Formulas for the Natural Frequencies Formulas (1) and (2) of Eq 5.23 show the natural frequencies... Output Responses i mi, klb xi, in ti, ms ki, klb/in ci, klb-s/in Model xi, in Model ti, ms 1 1 10 50 8.91 045 9.7 62 2 2 20 60 7.67 070 20.0 55 Fig 5.17 Fore- and Aft-Frame Displacement in a Pre-Program Vehicle The stiffnesses of the fore-frame and aft-frames in Case B are larger than those in Case A The damping coefficients in Case B are not much different from those in Case A In order to decrease the . used in the vehicle suspension and handling dynamics [5] is utilized. In this vehicle dynamic study, components, such as an unsprung mass (control arm and suspension) or a sprung mass (vehicle chassis),. 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 m 1 and m 2 . effective hybrid model as shown in the bottom of Fig. 5.7. © 2002 by CRC Press LLC Fig. 5.7 Vehicle- to -Vehicle Impact Model: A Two- Hybrid Model (5.3) The EA parameters in the combined model are