# Investigation of Functionality of Vehicle Crumple Zones Recovered after a Traffic Accident
## 3 Methods  
### Dynamic Models of Distributed Masses of Crumple Zone Elements  
Software developed at the Department of Transport Engineering at Kaunas University of Technology was used for verification of functionality of characteristics of the crumple zone elements and the effect of change in their properties. Dynamic models of distributed masses are the basic models for the assessment of impact loads. They were used in this study to assess the contribution of an individual crumple zone to the overall energy absorption process during impact. Although the three-mass dynamic model is one of the simplest, it provides a fairly accurate reflection of the key properties of the front crumple zone of a vehicle. The distribution of the model into three zones is conditional. Each part reflects a certain group of elements that includes the assemblies located in the respective zone. The generalized mass of the elements located within the zone is attributed to the respective zone. The methodology for determining the generalized mass is not strictly defined. Usually, it is considered to be the mass of key elements of the zone and the vehicle units/assemblies connected thereto. During the development of the research methodology, the guidelines provided in [48] were applied to the control of mass distribution. That is, the sum of the masses of the first and second zones was about 33% of the vehicle mass, and the mass of the third zone was about 66% of the vehicle mass. The characteristics of the zone, i.e., the rigidity and damping coefficients, were determined in view of the character of connection between the elements (series, parallel, or mixed). The three-mass dynamic model allowed the authors to identify the key weaknesses in the modelling. One of the most important weaknesses was the impossibility of describing zone deformation properties using linear elasticity elements at higher velocities. These elements would only be applicable before the emergence of plastic deformations, for example, when calculating the bumper zone of a US-type vehicle. For this purpose, the methodology for the determination of the element characteristics and the application specifications and efficiency of the four-mass models (Figure 12) had to be adjusted. A fourth mass--vehicle crew (\(m_{4}\))--is used in the four-mass dynamic model. The relationships in the model reflect the effect of vehicle safety systems (seats, safety belts, air bags, etc.). The relationships between the 3 rd and 4 th parts are reflected by parameters  
Figure 12: Four-mass dynamic model. \(C_{4}\), \(k_{4}\), \(C_{5}\), \(k_{5}\): \(C_{4}\) and \(k_{4}\) depend on the properties of the safety belts, while \(C_{5}\) and \(k_{4}\) are determined by the properties of the seat backrest. The behaviour of the four-mass dynamic model is described by a system of differential equations:  
\[\left\{ \begin{array}{l}m_{1}\ddot{x}_{1}+k_{1}\dot{x}_{1}+c_{1}x_{1}+k_{2}( \ddot{x}_{1}-\dot{x}_{2})+c_{2}(x_{1}-x_{2})=0\\ m_{2}\ddot{x}_{2}+k_{2}(\dot{x}_{2}-\dot{x})+c_{2}(x_{2}-x_{1})+k_{3}(\dot{x}_{2 }-\dot{x}_{3})+c_{2}(x_{2}-x_{3})=0\\ m_{3}\ddot{x}_{3}+k_{3}(\dot{x}_{3}-\dot{x}_{2})+c_{3}(x_{3}-x_{2})+k_{4}(\dot{ x}_{3}-\dot{x}_{2})+c_{4}(x_{3}-x_{4})=0\\ m_{4}\ddot{x}_{4}+k_{4}(\dot{x}_{4}-\dot{x}_{3})+c_{4}(x_{4}-x_{3})=0\end{array} \right. \tag{6}\]  
The parameters used in these models were \(C_{i}\), \(k_{i}\), \(F_{cri}\) and \(x_{\lim}\). Zone damping coefficient \(k_{i}\) was accepted as constant and independent from the value of deformations. There is no rationale behind description of the crumple zone properties using elastic elements only, as this would distort the rebound process that has considerable effect on the entire process. Hence, plastic elements were introduced into the models describing the crumple zones. The characteristics of plastic elements are described by \(F_{cri}\). Stiffness coefficients \(C_{i}\) are determined by element deformation to \(F_{cri}\). The stiffness value was adjusted where deceleration was greater than 100 g at the initial impact moment.