Crash Energy Management of Vehicle Front-end Structures Considering Multiple Conditions: Modelling and Solution Method

: For the vehicle frontal crash development, matching the stiffness of the front end structures reasonably, i.e., impact energy 7 management, can effectively improve the safety of vehicle. A multi-condition analytical model for vehicle frontal crash is construct by three 8 dimensional decomposition theory. In the analytical model, the spring is used to express the equivalent stiffness of the local energy 9 absorption space at the front-end structure. Then based on the analytical model, the dynamic responses and evaluation indexes of the vehicle 10 in MPDB and SOB conditions are derived with input of crash pulse decomposition scheme. Comparing the actual vehicle crash data and the 11 finite element simulation results with the calculation results of the proposed solution method, the error is less than 15%, which verifies 12 validity of the modeling and the accuracy of the solution. Finally, based on the solution method in the MPDB and the SOB conditions, the 13 sensitivities of crash pulse decomposition scheme to evaluation indexes are analyzed to obtain qualitative rules which guide crash energy 14 management. This research reveals the energy absorption principle of front-end structure during the frontal impact process, and provides an 15 effective tool to manage the vehicle crash energy considering multi-condition.


18
The crash process of automobile includes three parts: barriers, vehicles, and occupants, and is a complex dynamics 19 system. According to the annual statistical report of road traffic accidents from the Chinese government, frontal impact of 20 automobile accidents account for the highest proportion among all accident forms [1,2] . In the frontal crash, the crash energy is 21 absorbed by the front-end structures of vehicle [3] . With the development of crash safety regulations, today's vehicle front-end 22 structural design needs to meet the requirements of multiple conditions such as FRB (frontal rigid barrier), MPDB (mobile 23 progressive deformable barrier), and SOB (small overlap barrier) [4,5] . This greatly increases the difficulty of vehicle safety 24 design. From the perspective of dynamics, revealing the crash energy dissipation mechanism and constructing a dynamic 25 model of the front-end structures, which solve energy management in the early design stage, is great significance to improving 26 the safety of vehicle [6] . 27 The ultimate goal of vehicle frontal crash safety research is to protect the safety of occupants. In the crash process, the 28 initial kinetic energy of occupant is dissipated through two ways: the deformation of the vehicle structure and the action of the 29 restraint system [7] . According to statistics about existing vehicles, the structure absorbs more than 60% of the kinetic energy of 30 the occupants and is the most important way to absorb energy [8,9] . Crash energy management is to control the dissipation of 31 energy by designing the rigidity and deformation of vehicle structure, so as to protect the safety of the occupants. In the 32 vehicle frontal crash process, the energy dissipation is related to the topology of auto-body and the transmission path of the 33 impact force [10][11][12] . 34 In the theory of impact mechanics, the crash energy absorbed by vehicle is approximately equal to the product of the 35 impact force and the deformation of structures. Assuming that the vehicle mass is constant during the crash process, the 36 impact force is approximately equal to the product of the acceleration and mass (constant). Define crash energy per unit mass 37 as energy density. Then, the energy density absorbed by the vehicle during the impact is the product of acceleration and 38 structural deformation [8,13] . When vehicle hits FRB, almost all of the crash energy is absorbed by the structural deformation. 39 In this process, the vehicle acceleration is also called crash pulse. The sum of the deformations of all structures of vehicle in 40 the longitudinal direction (the driving direction) is the maximum dynamic crushing. The maximum dynamic crushing is 41 obtained by the quadratic integration of crash pulse [12,14] . That is to say, the crash energy is the integral of the crash pulse in the 42 displacement domain. Therefore, from the perspective of mechanics, the frontal crash energy management of vehicle is 43 actually the design and the decomposition problems of crash pulse [8] . 44 The design problem of crash pulse has been extensively studied through theories, simulations, and experiments [15][16][17] . At 45 present, there are relatively mature methods and qualitative conclusions for guiding engineering design of crash pulse [18,19] . In  46  terms of engineering conclusions, without considering the engine layout, the high-low-height crash pulse can effectively  47  reduce occupant injuries; the double-step crash pulse is suitable for vehicles considering the engine layout, and the higher first  48 step, the lower second step are the better for occupant safety [20] . In terms of theoretical method, our research group has 49 established an automatic optimization method for crash pulse considering multiple impact conditions [21] . As a continuation of 50 the paper on multi-condition optimization of crash pulse, this article focuses on the decomposition of crash pulse. 51 In this paper, the crash pulse in displacement domain as the total energy target during the impact process is decomposed 52 into absorption energies of sub-structures based on the topology and the load path of vehicle. To achieve this, it is necessary to 53 find the correlation between the target crash pulse and the structural performance. From 2006 to 2011, a magic cube approach 54 is proposed to decompose the crash energy into sub-structure design goals from the three dimensions of time, space, and size, 55 and design the load of the front longitudinal beams through the dynamic topology optimization method [22,23] . In 2011, the 56 energy-absorbing space of the front-end structure is divided into four layer in the vertical direction according to the 57 transmission path of impact force, and used to derive the design goal of energy absorption by the simplified model [24] . In 2016, 58 the vehicle longitudinal energy management method and the lateral energy management method are summarized to divide the 59 front-end structure longitudinally, and then decompose the energies of each area according to the horizontal arrangement of 60 the energy absorbing structure [7,25] . In 2018, the energy absorption space of front-end structures is vertically layered and 61 longitudinally segmented by the deformation mode of sub-structures to obtain the decomposing energy as the design targets of 62 anti-collision beam, the energy absorbing box, and the longitudinal beam [12,13,26] . 63 At present, it is mainly focused on the research of crash energy management methods of vehicle front-end structure based 64 on engineering experience in the FRB impact condition [27][28][29] . Considering the impact force transmission characteristics of 65 FRB condition, the front-end structure of the vehicle is divided into space, and the crash pulse is decomposed according to the 66 sub-space as the energy absorption target [12] . In addition to FRB condition, the vehicle also needs to absorb energy through the 67 deformation of the front-end structure in other frontal impact conditions, such as MPDB impact condition which mainly test 68 crash compatibility and SOB impact condition which mainly test the safety of the passenger compartment [30][31][32] . Therefore, 69 how to manage the energy absorption of front-end structures to meet the performance requirements and evaluation indicators 70 of multiple frontal crash conditions is an urgent problem in the vehicle design industry, and it is also the research purpose of 71 this paper. 72 Based on the frontal multi-condition crash pulse design method studied in the previous stage, this paper proposes a 73 multi-condition crash energy management method, that is, the decomposition method of crash pulse. The main contributions 74 of this paper include the following three aspects: 1) three dimensional analytical model of vehicle front-end structure is 75 constructed to describe the energy absorption space, impact load path and structural stiffness; 2) based on the proposed 76 analytical model, the dynamic responses and evaluation indexes of vehicle and barrier in the MPDB and SOB conditions are 77 derived to realize the crash pulse decomposition considering multiple conditions; 3) based on the analytical model, the 78 sensitivity of the crash pulse decomposition scheme to the evaluation indexes is analyzed to obtain the qualitative crash pulse 79 energy management strategy. 80 The paper is organized as follows: Section 2 introduces a mechanical analytical model of vehicle front-end structure. In 81 Section 3, we derive the solution of analytical model in MPDB and SOB conditions. An empirical case and a simulation 82 results of existing vehicles that have been in running in China are applied to verify this constructed model and solution method 83 in Section 3. Analysis and discussion are conducted in Section 4. Finally, the conclusions of this paper are drawn in Section 5.  Lateral decomposition: On both sides of the body symmetry plane, the energy absorption structure of the vehicle is 95 almost exactly the same, and the contact area between the vehicle and the wall barrier in MPDB impact condition is also 50% 96 of the width of the vehicle. Thus, the total energy absorption space can be divided horizontally into two equal regions. To 97 ensure a better safety level of the vehicle in the condition of 25% small bias impact, energy absorption structure should be set 98 within 25% wide range on both sides of the vehicle. If the requirements of meeting various impact conditions are considered at 99 the same time, four regions can be divided horizontally. The width of each region is 25% car width. 100 Longitudinal decomposition: According to the impact force transfer path of the vehicle in the frontal impact, the total 101 energy absorption space can be vertically divided into two, three and four tiers longitudinally. Take a typical passenger 102 vehicle as an example, which includes four tiers of the total energy-absorbing apace, the first tier is the engine cover, the 103 second one is the front finger beam, the third one is the anti-impact beam, energy absorption box and longitudinal beam, and 104 the fourth one is the sub-frame structure. The energy absorption is corresponding to the impact force. The energy absorption of 105 each tier accounts for about 10%, 20%, 50% and 20% of the total energy absorption respectively [24] . The third tier includes the 106 main energy absorption area of the vehicle. To improve the lightweight effect of vehicles, the engine hood is designed to be 107 thinner and absorb less energy, so the energy absorption space is mainly divided into three tiers, that is, the first tier contains 108 the engine hood, front finger beam and other structures. In addition, the sub-frame of some vehicles is removed in 109 consideration of economic and lightweight factors, so that the longitudinal space is two tiers. 110 Vertical decomposition: The engine of traditional automobile and the motor of pure electric vehicle almost do not 111 deform during the impact, which can be regarded as rigid structure. Therefore, according to the layout position of rigid 112 components such as engine or motor, the total space of energy absorption can be divided longitudinally. For traditional 113 gasoline cars and hybrid cars, rigid components such as engine need to be placed in the middle of the front end firewall 114 considering the connection between engine and drive shaft and maintenance problems. Thus, the two sections from the front 115 end of the engine to the anti-impact beam and from the back end of the engine to the firewall are set as impact energy 116 absorption spaces. For pure electric vehicles, the motor can be considered close to the firewall layout, because there is no drive 117 shaft on the vehicle. In this way, the whole from the anti-impact beam to the front end of the motor can be used as a section of 118 energy absorption space. 119 In general, the three-dimensional decomposition of the total space of energy absorption at the front end of vehicle is to 120 take into account the impact condition, the transmission path of impact force and the arrangement of the engine and motor, etc., 121 and conduct horizontal division, vertical stratification and longitudinal segmentation successively. The decomposed fore 122 cabin energy absorption space becomes the accumulation of energy absorption subspace. As a sample, the total space of 123 energy absorption at the front end of vehicle is decomposed 4×3×2 sections as shown in Fig. 2. 124 In the FRB impact condition, the wall is rigid, and the overlap rate between the vehicle and the wall is 100%. This 128 indicates that almost all the energy absorption structures in the front end of vehicle are involved in deformation energy 129 absorption during the impact, and the acceleration response of the vehicle, namely the crash pulse, is also the result of the 130 comprehensive action of the energy absorption structure. Thus, we take the crash pulse of FRB condition as the overall energy 131 absorption objective of the vehicle's front end structure, and gradually decompose it into each energy absorption subspace and 132 the design target of the energy absorption structure in space, so as to achieve the forward design of the front-end structure. 133 The horizontal coordinate of the crash pulse in the displacement domain is the deformation, which corresponds to the 134 longitudinal direction of the energy absorption space, but there is no difference between the lateral direction and the vertical 135 direction. Therefore, the three dimensional decomposition method proposed in this section is mainly to decompose the crash 136 pulse into each energy-absorbing subspace in accordance with a given proportion according to the three-dimensional 137 decomposition scheme of the total energy-absorbing space. If the longitudinal part of the total energy absorbing space is 138 divided into two parts, the crash pulse is divided into two parts at the deformation corresponding to the longitudinal subspace. 139 Assuming that the front energy absorption space of the vehicle is divided into N tiers vertically and M zones horizontally, 140 the decomposition scheme Q of the crash pulse can be obtained as shown in Fig. 3. Note that the percentage of the absorbed 141 energy of the sub-absorbent space in the total absorbed energy is represented by nm q (n=1, 2, ..., N; m=1, 2, ..., M).  (2) 146 In Eqs. (1) and (2), Q represents the total absorbed energy; Ym Q represents the absorbed energy of mth region, m=1, 2, ..., 147 M; Ym Q represents the absorbed energy of nth tier, n=1, 2, ..., N. 148

Construction of analytical model
149 After the three-dimensional decomposition of the total energy absorption space, each subspace corresponds to a 150 decomposed crash pulse, that is sub-pulse. It is the same as the original pulse in shape, but the amplitude is different, which is 151 the product of the original pulse and the proportion of energy absorbed by each subspace. The crash pulse in the displacement 152 domain can be regarded as the equivalent specific stiffness of the front-end structure of the vehicle; similarly, the sub-pulse 153 decomposed to each energy-absorbing subspace can be regarded as the equivalent specific stiffness of the space. 154 small during the impact process, structural deformation is the most important way to absorb energy [21] . The simplified model 171 of MPDB condition can be constructed ignoring the energy converted to rotation as shown in Fig. 5. The motion response of the vehicle and the barrier can be obtained as follows: 189 where, m represents the number of forward compartment subspaces that absorb energy 204

205
In the small overlap condition, the vehicle decelerates along the longitudinal direction and rotates around its contact point 206 with the rigid barrier after contacting with the barrier. If one of the following two point occurs, the impact process of small 207 overlap condition is considered to be over: 1) when the speed of the vehicle decreases to 0, the structures of vehicle no longer 208 deforms longitudinally; 2) if the vehicle displacement in the Y direction is ≥ 25% of the vehicle width, the vehicle is detached 209 with the barrier with out structural deformation. Therefore, the vehicle motion responses in the impact process of small 210 overlap condition can be considered from two aspects, i.e. the deceleration motion and rotation motion of the vehicle.

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The motion response of the vehicle in this condition can be considered from two aspects, i.e., the deceleration motion and 217 the rotation motion of the vehicle. The 25% overlap area between the vehicle and the barrier is mainly the energy-absorbing 218 area during the impact, its equivalent stiffness (k25%) can be calculated as follows: 219 The vehicle makes a single-degree-of-freedom free vibration in the X-axis direction. The vibration equation is as 221 follows: 222 where, x(t) is the displacement of the vehicle, and ( )  The force analysis of the vehicle under the condition of small overlap condition is shown in Fig. 7. The vehicle will rotate 229 around point O under the action of impact reaction F, and the distance between the vehicle's center of mass and the center of 230 rotation is the radius of rotation r. When the vehicle is decelerating, its longitudinal displacement is xv, and the longitudinal 231 distance between the center of mass and the center of rotation is 1/2L-xv. When the vehicle is rotating, the transverse 232 displacement of the center of mass is y, and the transverse distance from the center of mass to the center of rotation is 25%B+y. 233 234 Fig. 7. The simplified model of small overlap condition.

235
The radius of rotation (r) and the rotation angle (θ) of the vehicle can be calculated as follows: 236 The torque of the vehicle (Me), the angular acceleration of the vehicle rotation (β) and the lateral displacement of the 239 vehicle (y) can be obtained as follows: 240  250

1) Occupant load criterion (OLC) 251
The OLC is calculated by the velocity-time curve of the barrier as Fig. 8. The smaller OLC, the better [33] . At the time t1 in 252 the Fig. 8, the virtual occupant makes free movement relative to the barrier, and the displacement is S1 = 0.065m; from the 253 time t1 to t2, the relative displacement between virtual occupant and barrier is S2 = 0.235m. If there is an intrusion depth of 0.63m in an area larger than 40mm × 40mm on the barrier, the barrier is considered to be 258 bottoming out [31] . At the time, deduction of two points is MD=2, otherwise MD=0. 259

3) Standard deviation (SD) 260
The SD of barrier intrusion is obtained by the homogeneity of footprint based on scans of barrier [31] . The smaller SD, the 261 better. The Barrier deformation uniformity factor h is calculated by SD as follows: 262  267 The data about three quality grade vehicles in FRB and MPDB tests [30] , are used to verify the accuracy and reliability of 268 solution method. In this condition, to research the influence of pulse parameters and vehicle quality on the evaluation index, 269

Verification of MPDB condition
including OLC, MD, SD and PM, three kinds of vehicles, i.e., V1, V2, V3, with the masses of 1700kg, 1400kg, 1100kg, 270 respectively, are applied. The crash pulses of V1, V2, V3 are shown in Fig. 9. The values of OLC, MD, SD and PM can be obtained as Table 1. The errors between the calculation results and the  280 existing data are all about 10%, which indicate that the accuracy of the solution method is acceptable. 281  It is known that the energy absorption decomposition scheme of the front-end structure of the car body is shown in Eq. 28. 291

Verification of Small overlap condition
Take the crash pulse of the vehicle as shown in Fig.12 Three variation schemes for six variables are proposed, as follows: 307 Alternative 1: setting any one of the six variables to increase from 0 to 50%, and the other variables are equal. 308 Alternative 2: setting any two of the six variables are increased from 0 to 50%, and the other variables are equal. 309 Alternative 3: setting any three of the six variables and increase them from 0 to 50%, and the other variables are equal. 310 The decomposition difference of each scheme can be calculated by Eq. (29). And the design and decomposition 311 difference (W) of the three alternatives are shown in Table 2.  312   11  12  11  21  11  22  11  31  11  32   12  21  12  22  12  31  12  32   21  22  21  31  21   In this section, the decomposition difference and the stiffness change position of the three schemes are adopted as the 315 criteria to measure the uniformity of the barrier. The greater W is, the poorer the uniformity of the decomposition scheme is. 316 At the same W, the more varied the locations, the better the uniformity of the decomposition scheme. With W as the horizontal 317 coordinate When W is 0, the local stiffnesses of vehicle are evenly distributed. As the W increases, the uniformity becomes worse. 322 Through the above analysis and the information in Fig. 13, it can be found that when W is 0, the OLC is the largest, the MD is 323 the smallest, and the SD equals 0. In the Fig. 13, as the W increases, the decomposition scheme becomes more and more 324 uneven, the OLC value gradually decreases, the MD and SD value gradually increases. Corresponding to the same value of W, 325 the positional relationship of local stiffness changes in the three stiffness decomposition schemes is: alternative 3 > alternative 326 2 > alternative 1. 327 When W is the same, the OLC values of three alternatives are basically the same, which shows that OLC is more sensitive 328 to W, but has nothing to do with the position of the stiffness distribution. When the W is the same, the increase in the stiffness 329 change position helps to reduce the MD, and increase SD. It shows that MD and SD are more sensitive to the change position 330 of local stiffness and W. In general, the worse the uniformity of the vehicle stiffness distribution, the smaller the OLC, the 331 larger the MD and SD. In order to ensure that the barrier is not penetrated, and to control the deformation uniformity index SD, 332 it is necessary to reasonably allocate the stiffness of front-end structure. In order to solve the dynamic response of vehicle in the overlap impact conditions, the equivalent stiffness of the BVO 343 model (KB and K) are decomposed into local stiffness in this paper, as shown in Fig. 5. The three dimensional analytical model 344 of vehicle front end structure after stiffness decomposition is shown in Fig. 4. The previous impact mechanics model can solve 345 the one-dimensional dynamic response. The analytical model and solution method proposed in this paper provide new 346 calculation ideas for the three-dimensional dynamic response of the impact system. 347 The vehicle data as the verification of the analytical model solved in MPDB test, are the same with data to verify the 348 accuracy of BVO model in the reference [21]. The results of BVO model are calculated by the vehicle mass and crash pulse in 349 the section 3.3.2, as shown in table 3. 350 Table 3 The results of BVO model. and vehicle is constant, that is, the OLC and accelerations of the system calculated by the BVO model and proposed analytical 357 model are the same, and the deformation standard deviation SD of the barrier is 0. If it is not uniformly decomposed, the 358 equivalent stiffness between vehicle and barrier is decreased, so that the OLC is decreased; and a part of local stiffnesses of 359 vehicle are increased to cause the local deformation of the barrier to increase, i.e. MD is increased; At the same time, the SD 360 value is larger than 0. 361 In engineering application, the main methods to improve vehicle compatibility are: reducing the quality of the entire 362 vehicle, evenly distributing structural stiffness, and designing a reasonable crash pulse [12,13] . In the process of safety 363 development, the quality level of the vehicle should be determined first, then the BVO model should be used to optimize the 364 crash pulse, and finally the proposed model in this paper should be used to decompose the stiffness of front-end structure. 365 The effect of vehicle stiffness decomposition on the dynamic responses of small offset impact conditions is not analyzed 366 in this paper. Theoretically, the greater the stiffness of the 25% area (QY1 or QY4 in the Fig.10), the smaller the deformation 367 of the vehicle front-end structure and the smaller intrusion into the passenger compartment. Therefore, in the engineering 368 design, the sum of the stiffness of the three positions in the 25% area should be increased as much as possible to improve the 369 safety of the passenger compartment. That is to say, adding an energy-absorbing structures or increasing the stiffnesses of the 370 existing structures in the 25% area can improve the safety of the vehicle in small offset impact condition effectively. 371 2) comparing the simulation data of the SOB test and the calculation results of constructed analytical model, the errors of 384 the maximum intrusion into passenger compartment is -5.88%; 385

Conclusion
3) as the W increases, the decomposition scheme becomes more and more uneven, the OLC value gradually decreases, 386 the MD and SD value gradually increases; 387 4) OLC is more sensitive to W; MD and SD are more sensitive to the change position of local stiffness and W; 388 5) the greater the stiffness of the 25% area, the smaller the deformation of the vehicle front-end structure and the smaller 389 intrusion into the passenger compartment. 390 In summary, this research reveals the energy absorption principle of front-end structure during the frontal impact process, 391 and provides an effective tool to manage the vehicle crash energy considering multi-condition. In future research, our works 392 will establish the theoretical relationship between the space energy absorption of the front-end structure and the energy 393 absorption of the sub-structure to carry out the forward design of vehicle safety. 394