Performance evaluation of vehicle front structure in crash energy management using lumped mass spring system
- Sunday M Ofochebe^{1}Email author,
- Chigbogu G Ozoegwu^{1} and
- Samuel O Enibe^{2}
https://doi.org/10.1186/s40323-015-0020-1
© Ofochebe et al.; licensee Springer. 2015
Received: 6 October 2014
Accepted: 2 January 2015
Published: 1 April 2015
Abstract
Background
The design for vehicle structural crashworthiness which ensures that components of desired crash performance characteristics are used in product manufacturing essentially involves the evaluation of the energy absorption potentials of the structures using suitable computation method. Due to unresolved difficulties in achieving detailed results through the existing methods researchers seek for more alternative computation methods. Although previous efforts in this regard are quite significant, yet some concerns still exist on accuracy or computational efficiency achievable through the conventional methods.
Method
The lumped mass spring (LMS) method is applied in the present study. Some new steps were introduced in the basic procedure to improve the accuracy and computational efficiency of the method. A new dynamic stiffness formula is written in terms of the specific energy absorption indices of the structural components. The new procedure allowed for standard state-space formulation of the crash problem.
Results
The performance of the new simulation approach is tested for a typical vehicle structure in two possible orientations called normal mode and reversed mode. The results obtained for the impact problem in normal structural mode show that a desirable energy absorption pattern of 45%, 25% and 20% of the total impact energy could be achieved through plastic deformation of the front frame, sheet metal and torque box respectively. Testing the impact system in reversed structural mode results in a rather poor energy absorption pattern in which 2.5%, 50% and 43% of the total impact energy were absorbed through deformation of the front frame, sheet metal and torque box respectively, showing that unreasonably high percentage of the total impact energy is transmitted to the interior structures.
Conclusion
The effort to quantify the energy absorbed by major vehicle front structure in both desirable and undesirable crash responses, and the computational efficiency achieved through the present method could help to enhance decision process during assessment of the components or prototype. It is found that good crash performance may be guaranteed by ensuring sufficiently high (up to 65%) contribution to the energy absorption scheme through the deformation of the foremost structures which includes the front frame and the sheet metal.
Keywords
Background
Mechanical systems can be formulated into analytical models by integrating the inertial, stiffness and energy absorption/dissipation properties of the real system in an equivalent arrangement of solid mass(es), spring(s) and/or damper(s) known as lumped mass spring (LMS) system. The development of such an analytical method has been of great importance to engineering and science owing to its relative simplicity in resolving certain research problems about the real world system it represent. Of particular importance is its significant success in vehicle crash modeling which currently undermines any other alternative analytical approach. The viability of LMS system approach in resolving nonlinear crash problems is established by the possibility of updating the input variables in the dynamic states using time-stepping numerical integration technique which usually leads to a valid approximate solution. Lumped mass spring system approach provides holistic and handy information on the dynamics of the vehicle impact based on the underlying physics of the impact system; assuming strong and un-deformable components to be rigid with concentrated masses contributing significantly to the distribution of inertia forces and transmission of impact energy, while the compliant (deformable) structural components are considered to have uniformly distributed masses contributing substantially to the dynamic resistance and energy absorption sequence. Interpreting such concrete information during prototype or component assessment is usually straightforward and requires no further averaging or integration of any sub-critical (nodal) information. However, existing reports show that achieving an acceptable accuracy via LMS approach in vehicle impact problems involves rigorous characterization of the complex elastic–plastic motion observed during structural deformation under dynamic impact condition. The necessary tasks regrettably present some unique challenges, ranging from the cost of achieving a reliable input data to the governing equation of motion to the computational difficulties in describing such input data in the dynamic state. These facts pose a perpetual hindrance to the application of the highly rated LMS system approach in precise evaluation of energy absorption potential of vehicle structure during impact. Nevertheless very few successful efforts to obtain a reliable estimate of vehicle component crush behavior using well organized impact experiment reported in literature have been of great help to researchers in expanding the scope research in this important subject. For instance, Balike successfully utilized the static crush test data (i.e. force-deformation characteristics obtained at quasi-static condition) recorded for various components of a typical framed car [1] for evaluating the performance of under-ride guard in idealized collision involving a light-weight passenger car and a heavy truck [2]. In other related works, the results of the crash calibration test of a standard Ford Fiesta [3] have been used to test the performance of linear visco-elastic models like the Maxwell model [4], Kelvin model [4,5], and auto-regressive models [5] in correlating real vehicle crash responses. The idea of assuming a perfect rigid mass in appropriate sections of the vehicle system has been exploited extensively in formulating crash simulation models in various attempts to simplify the modeling procedure and computation process [6-9].
Considering the unique severity posed by frontal crash which is neither opposed by the current interest in the demand and manufacture of highly economical cars of significantly reduced weight intended to meet certain requirements on environmental impact and drive energy economy, nor minimized by the persistent dependence on heavy vehicles for increased transportation needs; the study of impact energy absorption capacity and general crash performance of the front components of light-weight vehicles continues to represent an important area of research. A comprehensive review of relevant literatures reveals that the crash performance of vehicle structure in frontal impacts is largely dependent on the mechanical properties and the geometry of the front components. It is further noted that one elegant way to achieve a substantial theoretical report on this subject is to study the impact mechanics through appropriate LMS system capable of capturing the detailed nonlinear dynamics of the system. An objective review of the application of LMS method in vehicle crash simulation problems shows that the most reliable results could be traced to those in which the formulation of the dynamic resisting forces were based on the components’ crush signature (force-deformation behavior). Such data are usually recorded for every major structural component under quasi-static condition provided by a low speed crusher [2]. However, the economic cost of implementing crush test is considerably high and discourages necessary investigation of the impact system in other comparable structural configuration(s) that may serve as a useful guide to structural designers and analysts in judging a good performance through such a method. In this regards detailed numerical model may be a good choice, though with relatively high demand on computation time. Consequently, illustration of desirable crash energy management system and other extended studies in vehicle impact problems that could be conveniently achieved through the LMS system approach or other reduced order dynamic models seems to rely on pure rudimentary procedures and unrealistic data capable of giving cursory assessments of the problem [7,10]. Considering the importance of the subject under study and the computational difficulty in achieving accurate results via the standard methods, researcher seek for possible ways to utilize the simplicity of LMS computation procedure to enhance illustration of the basic concept of crash energy absorption through structural deformation. Hamza [11] suggests a unique simulation method called the equivalent mechanism model which approximates the entire vehicle structure to a continuous chain of short rigid masses connected to each other via prismatic joints with nonlinear axial springs and revolute joints with torsional springs, and subsequently compared the results of the new method to that of an equivalent coarse finite element model. Other researchers attempted to simulate vehicle impact system via simplified linear visco-elastic one degree-of-freedom models [4,5]. Some others seek for further simplifications via equivalent square wave method [10], and multi-body model [12,13].
The observed trends stated above do not allow for proper appreciation of what may be considered good energy management system in a holistic sense due to lack of specific information on the effects of impact energy on the major energy absorbing members. The present study recognized the eight-component model structure first suggested by Kamal in his pioneering work [14] visualized in 4 DOF LMS system as a good framework capable of providing a comprehensive report on the crash performance of vehicle front structure. Critical assessment of such a simulation model reveals that the major energy absorbing components could be considered in reversed orientation in the system such that any valid data may be utilized in other structural configuration for comparative study without constituting any additional experimental cost. This idea is introduced in the present report for frontal impacts by observing the side rail in both normal orientation and reversed mode, assuming the other components to maintain their normal positions and geometry in the system. The expectation is to see the extent to which such a simple modification of structural configuration in the selected component could distinguish the good and the poor energy management system. The effort led to some important deductions which help in judging the impact system as observed in normal orientation as a better energy management system and emphasizes the need for proper characterization of front structures to ensure improved crash performance in light-weight vehicle crashworthiness design.
Methods
- i.
Collision type is full-lap frontal impact against a rigid barrier.
- ii.
Sufficient rigidity is assumed in the passenger compartment to shield the occupant hence the body mass is lumped as m _{1}.
- iii.
The engine/transmission system or drive train (which include the gearbox, clutch system and drive shaft) and the cross-members/suspension system known to be structurally stronger than other sections are assumed un-deformable and collectively lumped as engine mass m _{2}.
- iv.
The resistances offered by the structural members forward of the engine mass against the barrier and that rearward of the engine mass against the body mass during the impact denoted F _{ j } correspond to the measured force deformation characteristics of the components.
- v.
The contributions to the resistance network due to highly flexible or fragile non-structural members like cables, glasses, conduits, plastics etc. are considered negligible.
- vi.
The possible contribution of the structural masses to the inertia force vector is ignored.
Equation of motion
Where M = m _{ ij } is the matrix of the lumped masses m _{ i } and X = x _{ i } represents their position vector, K = k _{ ij } is the assembly of structural stiffnesses corresponding to elastic motion and F(x, t) = f _{ j }(ϕ) in this construct, represents the vector of all internally generated forces in the spring set that sustains the plastic flow: i = 1, 2, …, n; j = 1, 2…m for an n × m mass-spring system.
The state variables φ and \( \dot{\varphi} \) could be evaluated if the initial condition, stiffness matrix K = k _{ ij } and the vector of state dependent forces f(ϕ) are known or sufficiently characterized in the dynamic state.
Where f _{ i,r}(ϕ) represents the contribution to the total dynamic resistance about a specific mass m _{ i } due to a state dependent force generated in spring r found in plastic state.
Load zone criterion
Displacements found in zones Z1, Z2 and Z3 as indicated by the displacement–load path (Figure 2b) correspond to structural deformation and contribute substantially to energy absorption scheme; such zones are regarded in this paper as active load zones while zone Z0, and zone 4 of the deformation load-path which lead to either insignificant energy absorption or total transmission of impact load are classified as idle (or passive) load zone. The transition from idle zone to active zone back to idle zone in addition to the switching of resistance formula and all other observed behaviors of the nonlinear springs were considered in arriving at the detailed governing equations of motion.
Further description of α(F _{ p }), β(δ, F _{ s }), and \( \psi \left({F}_s,\ \dot{\delta}\right) \) which characterize the dynamic resistance at the corresponding load zones is given in Appendix section.
Force deformation analysis
The solution of the system response via the proposed method requires that the hysteretic parameter pair (k_{ j,} F _{ s,j }) which describes the load path of resistance must be quantified in the dynamic state. This implies that the contribution of every individual spring to the dynamic energy absorption sequence (E _{ j }) is known preferably as fraction of the total absorbable energy of the system λ _{ j }. Hence, the proposed method adopts an approach in which all such contributions are matched in the dynamic state such that the solution of the system converges. The success of this approach lies on proper characterization of E _{ j } upon which the spring parameters k_{ j } and F _{ s,j } are estimated. In the reviewed literatures [1,2,7] the typical deformation behavior of the main energy members of front vehicle structures is illustrated by the generalized load-deformation curve of Figure 2a. In view of the complications and unmerited rise in computation time associated with tracing the details of the load path in the overall solution of E _{ j } in the proposed method, the reports are rather considered in a linearized form illustrated by the approximating force displacement diagram of Figure 2b for developing the solution algorithm; assuming that a sufficient estimate of the force-deformation behavior and the energy absorption sequence could be achieved in the active load zones (Z1, Z2 and Z3) via the approximate displacement model. This consideration enables detailed programing of the structural deformation sequence with minimized cases of iterative switches in the solution steps that essentially grants computational efficiency.
Energy dissipation in form of heat, sound and vibration denoted by ε(T) is usually assumed negligible so that all observed energy absorption in the system is credited to work done during structural deformation.
To evaluate the spring parameter F _{ s,j } and k _{ j }, the force-deformation characteristics of a given structural member (recorded either through static crush experiment or via equivalent numerical simulation) is first visualized in form of the approximate force-displacement behavior illustrated in Figure 2b. The spring geometric properties s _{ j } = F _{ p,j }/F _{ s,j }; p _{ j } = δ _{ p,j }/L _{ c,j } and q _{ j } = δ _{ s,j }/L _{ c,j }. (which characterize the contribution of individual component to the energy absorption scheme) are then evaluated.
The parameter γ is a tolerance factor (or system adjustment variable) which could be used to tune the system to the best energy absorption performance during component design.
Equation 19 shows that the energy absorbing capacity of all nonlinear springs showing similar force-deformation characteristics varies according to the total crushable length L _{ c } of the components. The information may be useful at early design stage to enhance development of a workable design especially in integral body construction where vehicle front structures are usually made of intermediate columns of comparable deformation pattern.
The formulation of the dynamic steady-force F _{ s.j } and dynamic stiffness k _{ j }. based on the specific energy absorption index and the known geometric properties of the components ensures convergence of the solution.
Solution procedure
The system response is computed dynamically based on the state variable formulation (23) subject to the load zone criteria (14). In the programming, the numerical integration utilizes various forms of Equation 23 in which each form reflects a unique observation of the spring system in the load zoning system. The number of independent observations utilized in the programing was minimized based on some practical considerations. Only components that show both linear elasticity and significant plasticity in the static crush characteristics data were considered in both perspectives in the solution program. For practical details additional cases were recognized for pure dynamically compliant springs in fully compressed and totally consumed states so as to improve the accuracy of the current method at impact speed of about 50km/h where extended structural deformation is anticipated. By and large, crash modeling is usually intended to minimize fatality in a survivable crash occurring at moderate impact velocity range usually below 100km/h. At this range, transition to fully compressed or totally consumed state by the interior components of real vehicle structural rigidity is certainly not desirable.
Considering separately the two structural modes under study at full-compliant state (where γ = 1), the distinct energy absorption capacities E _{ j } of the nonlinear springs were first evaluated from Equations 20 for a specific value of energy absorption index λ _{ j }. This enables the calculation of dynamic parameters F _{ s,j } and k _{ j } from (21) and (22) respectively, substituting the known components’ geometric properties p _{ j }, q _{ j } , s _{ j }, and L _{ C,j } (i.e. measured force-deformation behavior for a typical framed car components found in [1,2] mapped as proposed in Figure 2b), and the typical mass distribution of a conventional light-vehicle given in Appendix section. The results of this first stage analysis known as spring tuning were then applied for the solution of the system response under crash condition through a computer program written to solve the governing differential equations of motion for a specific value of λ _{ j } (accounting for all observed cases of mass displacements and structural load zones) given the initial conditions ( ẋ _{1} = ẋ _{2} = ẋ _{3} = ẋ _{4} = V _{0}, x _{1} = x _{2} = x _{3} = x _{4} = 0). The solutions were completed through numerical integration employing simple logics that check the deformation states of the springs and select appropriate governing differential equation corresponding to each case such that the displacements and velocities of the masses arising from a preceding case are automatically fed as initial conditions to the new governing equation in the current case.
Results and discussion
The present research problem was solved based on state variable formulation. The resulting accelerations of the various masses were integrated iteratively using ODE45 numerical solver. The results are presented as time histories of the impact events. The necessary comparison between the results of the impact system in normal and the reversed structural modes were recorded accordingly. As anticipated some significant disparities were observed in all the compared events. The results obtained in terms of displacement, velocity and acceleration of the masses are all typical of compliant vehicle structures that show significant plastic deformation in both the foremost structures and the interior front components.
Displacement response
Velocity response
Acceleration response of the body mass
Further comparison of the system response was accomplished through the recorded peak acceleration/deceleration of the body mass. Observing the results at each iteration steps shows that operating the system in normal and reversed structural configurations results in comparable peak body mass decelerations of 31.5 g and 30.7 g respectively. This occurred in both cases at the instant when the front frame and the sheet metal were in steady force state. Both figures fall within acceptable range recommended by automobile safety regulatory bodies [10]. Hence further assessments of the relative performance of the exemplified structural plans in impact energy management were embarked upon using the measured instantaneous axial crush of components and the total energy loss history as follows.
Axial crush of components
On the other hand, the reversed system witnessed rapid consumption of the front frame within the first 20 milliseconds of the impact leading to transfer of huge amount of the impact load on the other components. As a result both the radiator and the sheet metal experienced unreasonably high peak deformation. Some of the interior structures which include; the torque box and the engine mount equally show extended deformation, suggesting increased structural intrusion and poor energy management scheme (Figure 5b). However, the firewall, the drive-line and transmission mount maintain similar deformation pattern as in normal structural mode but with slightly increased peak values (Figure 6b).
Energy absorption
Specific energy absorption indices of the structural components
Structural mode | λ _{ 1 } | λ _{ 2 } | λ _{ 3 } | λ _{ 4 } | λ _{ 5 } | λ _{ 6 } | λ _{ 7 } | λ _{ 8 } |
---|---|---|---|---|---|---|---|---|
Normal | 0.20 | 0.45 | 0.001 | 0.25 | 0.001 | 0.015 | 0.001 | 0.10 |
Reversed | 0.43 | 0.025 | 0.001 | 0.50 | 0.001 | 0.010 | 0.001 | 0.05 |
Future work
Considering the cost of obtaining full-scaled crash test data, the development of a workable dynamic vehicle crash model based on static crush behavior of the structural components conceived in this study represents a significant simplification in vehicle crash design capable of enhancing on-line decision during design for vehicle structural crashworthiness. The proposed method quantifies the contributions of the various component in impact energy absorption by observing the effects λ _{ j } on the system response within a useful range of values such that the known initial and of course the anticipated final conditions of the problem are substantially realized. Fast convergence of the solution is always guaranteed since the sampling of λ _{ j } is conducted within a short data range of (0–1). The solution converges when the velocity of the system (or the bumper force) approaches zero within typical impact duration assuming full-plastic collision. Further studies may be tailored to validate the proposed computation method through standard numerical method such as finite element method. Recent developments in CAD/CAE increase the possibility of obtaining reliable component static force-deformation behavior through numerical simulation once the mechanical properties and geometry of the components are specified. It then implies that the expected crash behavior of any proposed design could be readily tested during component formulation, allowing for prompt system adjustment in the case of any observed indication of unwanted performance, even before final prototype assembly using the proposed method. This represents a significant cost reduction from conducting multiple real component crush test or full-scaled crash test experiments and computation time saving from repeated detailed numerical simulation of fully assembled vehicle model needed during component formulation. With the crush characteristics of components attained through standard numerical simulation method, the necessary validation of the proposed modeling procedure could be achieved through straightforward comparison of the results of the proposed simulation of vehicle crash response via integrated LMS model employing component static force deformation behavior and those of an equivalent fully assembled system attained through detailed finite element method. Other necessary extension of the present work may be directed towards developing automated system for monitoring the solution convergence (which may require the use of graphical/numerical optimization tool such as genetic algorithm or differential evolution) that would generally enhance the application of the proposed method in crash performance evaluation.
Conclusion
The present study reveals that proper design for light-weight vehicle that shows high crashworthiness potential in frontal impact such as; minimized intrusion of structures into the passenger compartment, controlled restitution/rebound of the vehicle masses, acceptable occupants’ acceleration and maximum absorption of impact energy via structural deformation could be achieved through adequate evaluation of the vehicle front components in crash energy management system using the proposed method. The results of the present study suggest that the design criteria on frontal impact in terms of standard upper limit of acceleration of the occupants (or payload mass) of 32 g and minimized intrusion of structure into the passenger compartment, could be realized by ensuring sufficiently high ( ≥ 65%) contribution to the energy absorption scheme through the deformation of the front frame and the sheet metal. As seen from the results of the normal structural mode, this amount of energy absorption ensures that significantly reduced fractions of the total impact energy are absorbed by the interior front structures. Moreover, the studied crash energy management plan represented as the normal structural mode grants fully resisted, unidirectional (no rebound) displacement, minimal terminal restitution and fairly uniform deceleration of the vehicle masses which are all desirable crash trends. The necessary design considerations for reaching such a desired crash performance involve proper selection of structural stiffness and component geometry which determine the distribution of impact energy in the vehicle system. It is noted that the dynamic resistance of the vehicle structures and the associated energy absorption during the impact depend on such distribution of the impact energy within the structural zones. Hence, a major contribution of this work is the construction of the dynamic peak/mean resistant forces and the dynamic structural stiffness based on specific energy absorption of the main structural members which enables the evaluation of the unknown dynamic resistance and the system crash response. The demonstrated efficiency of the proposed method represents significant relief from the usual computational burden presented by the existing methods.
Declarations
Authors’ Affiliations
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