Problem statement
In this work, a simple discretetime model of impact attenuator crash will be developed. The physical phenomenon considered for the modeling is given in Fig. 1.
The car is modeled as a rigid body with mass M, the impact attenuator is attached to the car, and the impact attenuator has spring constant k and damping factor c. During the crash, the impact attenuator hits the wall and deformed. From the simulation tools that will be developed, the output variables are the deformation of the impact attenuator and the acceleration response of the car. If the deformation of the impact attenuator is sufficiently large, then it implies that the impact attenuator can efficiently absorb the impact energy. Hence, the acceleration of the car during the crash will be hugely reduced. This will lead to minimum injury of the passenger inside the car. In other words, the acceleration response of the car during the crash can be used to evaluate the effectiveness of the impact attenuator. In the model, the crash is assumed to happen in one direction.
The impact attenuator, when colliding with the wall, is expected to have large deformation. Thus, both elastic and plastic deformation of the impact attenuator must be considered. To this end, the spring constant k in the model is not only capturing the elastic behavior but also the plastic behavior of the impact attenuator. Meanwhile, the damping factor c is assumed to be constant. Damping is a function of energy dissipation property of the material (in this case the impact attenuator). In other words, we assume that the energy dissipation rate of the impact attenuator is constant throughout. To make such that the stiffness k capture the elasticplastic behavior of the impact attenuator, the value of stiffness k must be updated according to the deformation level of the impact attenuator.
The crash incidence considered in this paper is the crash without rebound. That is, when the impact attenuator deforms plastically, we assume that the impact attenuator retains its deformation when there is no more force acting on the impact attenuator (the car loses all its kinetic energy).
Methods
In general, the proposed numerical method consists of two processes:

1.
Mathematical modeling. This process includes the derivation of the discretetime model.

2.
Obtaining the stiffnessstrain curve of the impact attenuator. In this process, a quasistatic compression analysis will be done to the impact attenuator and the analysis is done using FEA software (ANSYS). The stiffnessstrain curve is used in the discretetime model simulation in MATLAB. The stiffness obtained is the effective stiffness or the change in force divided by the change in length.

3.
Simulation of the discretetime model using MATLAB. The simulation in MATLAB is done to obtain the acceleration response of the car body for different impact attenuator designs when the impact attenuator crashes into the wall.
Mathematical model
In this work, the system in Fig. 1 is modeled as two degrees of freedom system (two masses), where the first body is the car’s body with mass M and the second body is the shaker with mass M_{s}, where M_{s} ≫ M. The displacement of the car is denoted as y and x denotes the displacement of the shaker. A force F is given to the shaker such that \( \ddot{x} \) or the acceleration of the shaker follows the halfsine pulse with peak P and pulse duration T (see Eq. (1)). Below is the formula used to construct the force F which is based on the work [24].
$$ F(t)=\left\{\begin{array}{cc}{M}_sP\;\sin \left(\frac{\pi t}{T}\right)& t<T\\ {}0& t\ge T\end{array}\right. $$
(1)
The impact attenuator is modeled as a spring and damper, and these are attached to the car’s body. The crash model and the acceleration pulse can be seen in Fig. 2. Note that the direction (vertical or horizontal) does not matter for the simulation.
The impact modeling in this work models the impact as a halfsine pulse acceleration at the end of the impact attenuator (the point in contact with the wall in the real impact model). Since we are interested only in the response of the car’s body and the shaker’s mass will be set to be very high in the simulation, the model in Fig. 2a is considered as base excitation problem. Base excitation problem as an approach to study the behavior of system under impact is commonly used such as found in the works of [25, 26].
From Fig. 2a, assuming x > y, the freebody diagram can be constructed, as shown in Fig. 3.
The equation of motion in matrix form can be expressed as
$$ \left[\begin{array}{cc}{M}_s& 0\\ {}0& M\end{array}\right]\left[\begin{array}{c}\ddot{x}\\ {}\ddot{y}\end{array}\right]+\left[\begin{array}{cc}c& c\\ {}c& c\end{array}\right]\left[\begin{array}{c}\dot{x}\\ {}\dot{y}\end{array}\right]+\left[\begin{array}{cc}k\left(\in \right)& k\left(\in \right)\\ {}k\left(\in \right)& k\left(\in \right)\end{array}\right]\left[\begin{array}{c}x\\ {}y\end{array}\right]=\left[\begin{array}{c}F(t)\\ {}0\end{array}\right] $$
(2)
where ϵ is the strain of the impact attenuator and the stiffness k(ϵ) is expressed as a function of the impact attenuator’s strain. The state space equation can be expressed as
$$ \dot{\boldsymbol{x}}(t)=A\left(\in \right)\boldsymbol{x}(t)+ Bu(t) $$
(3)
with x = [x_{1}, x_{2}, x_{3}, x_{4}]^{T}, \( {x}_1=x,{x}_2=\dot{x} \), where x is the displacement of the shaker; \( {x}_3=y,{x}_4=\dot{y} \), where y is the displacement of the car’s body, and u(t) = F(t) where F(t) is given by (1). Meanwhile, matrices A(ϵ) and B will be defined shortly.
To simulate the crash, there are two conditions considered in the model:

(i)
The first condition is when the car is still compressing the impact attenuator, and the impact attenuator is still in the elastic region.

(ii)
The second condition is when the car compresses the impact attenuator until the impact attenuator reaches the plastic region. In this region, we assume that the impact attenuator has permanent deformation.
The simulation considers that condition (i) is continued by condition (ii) and then the car stops. Therefore, the matrices A(ϵ) and B in the state space equation are:
$$ A\left(\in \right)=\left[\begin{array}{llll}\frac{\begin{array}{c}0\\ {}k\left(\in \right)\end{array}}{M_s}& \frac{\begin{array}{c}1\\ {}c\end{array}}{M_s}& \frac{\begin{array}{c}0\\ {}k\left(\in \right)\end{array}}{M_s}& \frac{\begin{array}{c}0\\ {}c\end{array}}{M_s}\\ {}\frac{\begin{array}{c}0\\ {}k\left(\in \right)\end{array}}{M}& \frac{\begin{array}{c}0\\ {}c\end{array}}{M}& \frac{\begin{array}{c}0\\ {}k\left(\in \right)\end{array}}{M}& \frac{\begin{array}{c}1\\ {}c\end{array}}{M}\end{array}\right],B=\left[\frac{\begin{array}{c}0\\ {}1\end{array}}{\begin{array}{c}{M}_s\\ {}\begin{array}{c}0\\ {}0\end{array}\end{array}}\right] $$
(4)
The matrix A is a function of ϵ since it will be updated according to the strain of the impact attenuator.
To simulate the problem in MATLAB, the continuoustime model (3), is converted into the discretetime model:
$$ \boldsymbol{x}\left(k+1\right)=\left(I+A\left(\in \right)h\right)\boldsymbol{x}(k)+ Bhu(k),k=0,1,2,\cdots $$
(5)
where k is the time step, h is the sampling time, and I is the identity matrix. To obtain the time t in seconds from the discrete time simulation, we multiply k and h or t = kh. Meanwhile, the acceleration of the car at time t, a(t), is obtained by \( a(t)=\frac{1}{h}\left({x}_2\left(k+1\right){x}_2(k)\right) \), for t ≥ h. We assume that a(0) = 0.
Quasistatic compression test analysis of the impact attenuator
In this work, the impact attenuator is a hollow box made from Aluminum 6082T6. The reason of using a hollow box as the impact attenuator model is so that we can perform a parametric study on its stiffness by changing the thickness of the shell, while the outer dimension of the box remains the same. To obtain the stiffness data to be used in the simulation, a compression test analysis is done to the hollow box and the compression is done with a strain rate of 0.1/s. The compression test analysis is done using ANSYS Mechanical APDL with the following steps:

The Aluminum 6082T6 compression stressstrain curve is first obtained from the work of [27]. The curve is obtained from compression test (experiment) on standard compression test specimen.

The stressstrain curve from the above step is keyed into the material properties of the finite element model of the hollow box.

Static analysis in ANSYS is done by giving one end of the hollow box a constant velocity such that the strain rate becomes 0.1/s. The strain rate is a function of the velocity and the length of the hollow box.

Stiffnessstrain curve can be obtained easily using postprocessing menu in ANSYS Mechanical APDL.