# Evaluation of Accelerator Pedal Strength under Critical Loads Using the Finite Element Method
## 2 Materials and Methods
### Shock Load Task  
Before proceeding to the study of shock loads, the concepts of implicit and explicit types of calculations used in Ansys will be presented, and the theory of calculations used in the finite element method (FEM) will be investigated. In the general case, any static FEM analysis (for example, Static Structural in Ansys) is determined by a strict linear Equation (1), in which the concept of time dependence is absent (static process, but not transient) [20]:  
\[[K]\{x\}=\{F\}. \tag{1}\]  
A so-called transient or modal analysis (dynamic case) follows a more complex basis equation in the following form [20]:  
\[[K]\{x\}+[C]\{\dot{x}\}+[M]\{\ddot{x}\}=\{F\}, \tag{2}\]  
where:  
\([M]\), \([C]\), \([K]\)--mass matrix, damping matrix, and stiffness matrix, respectively;  
\(\{\ddot{x}\}\), \(\{\dot{x}\}\), \(\{x\}\)--acceleration, velocity, and displacement, respectively;  
\(\{F\}\)--load vector. In the aim to find \(\{x\}\), the matrix \([K]\) needed to be inverted, which was not easy. The matrix \([K]\) was unknown due to displacements of adjacent nodes; the convergence must be sought via iterations. In most cases, the Newton-Raphson method is used as the most suitable one because of its powerful potential for solving equations in a numerical way. The  
Figure 7: Load for the hybrid task (red arrows). core idea is based on the approach suggesting that a continuous and differentiable function can be approximated by a straight-line tangent to it. The Newton-Raphson method is a root-finding algorithm aimed at finding out the value of \(x\) for which a function \(f(x)=0\). From a geometrical point of view, it can be treated as the value of \(x\) where the function being sought crosses the X axis. The Newton-Raphson method is a technique used to find the roots of nonlinear algebraic equations. The basic formulation of the method is [21]:  
\[x_{n+1}=x_{n}-\frac{f(x_{n})}{f^{\prime}(x_{n})}. \tag{3}\]  
where: \(x_{n}\)--a current estimation, \(x_{n+1}\)--a next estimation. Such an approach could be used to describe the displacements of adjacent nodes of the FEA model to find their position at any moment of time with the movement according to the nonlinear equation \(f(x)\). An implicit solution includes a \"step by step\" approach: current values at a current time step are provided by the values calculated at a previous time step. This case is also called the Euler time integration scheme. Opposite to the implicit approach, in the explicit analysis, the vector \(\left\{\tilde{x}\right\}\) is sought instead of \(\left\{x\right\}\), and the complex stiffness matrix inversion is bypassed in this way, so it is only needed to invert the mass matrix \(\left[M\right]\). The explicit scheme has a higher conditional stability than the implicit scheme, which is unconditionally stable for large time steps. However, an explicit calculation is stable if the time step size is smaller than the critical time step size of the investigated model; otherwise, the final results reached in the solution may be misleading. Integration schemes in the explicit approach use the central difference quotient to find out the velocities and accelerations at the current time step \(t_{n}\), and then find out the unknown displacements at the next time steps \(t_{n+1}\):  
\[\dot{x}=\frac{\mathrm{x}(t+\Delta t)-\mathrm{x}(t-\Delta t)}{2\Delta t};\ \tilde{x}=\frac{ \mathrm{x}(t+\Delta t)-2\mathrm{x}(\mathrm{t})+\mathrm{x}(t-\Delta t)}{2 \Delta t^{2}}. \tag{4}\]  
where: \(\left\{\tilde{x}\right\}\), \(\left\{\dot{x}\right\}\), \(\left\{x\right\}\)--acceleration, velocity, and displacement, respectively; \(\Delta t\)--time step. The time step for explicit calculations is based on the appropriate characteristic length of the finite element \(L_{e}\) and the wave propagation speed \(c\) in the medium or material.