## 3 Results
Fifteen finite element analyses were performed, where linear stiffness of the shock absorber was adapted and changed every 10 N/mm. The time of a single dynamic simulation was 3 s. A total of 44,970 reference data samples were generated. It was assumed that in hybrid simulation, the shock absorber was mounted to four actuators, imposing displacements along the directions of the global coordinate system, as presented in Figure 11. In the presented case, there were four nonzero entries in the vectors \(\textbf{u^{1}}\) and \(\textbf{r^{P}}\) (corresponding to horizontal and vertical displacements and forces of points A and B--Figure 12). The exciting force vector **F** could be represented only by force **F2** as **F1** was always proportional. Therefore, the ANN got 17 inputs and four outputs. Two-layer ANN was trained, with a sigmoid activation function in the hidden layer and linear activation function in the output layer. A total of 70% of the reference data were used as training data, 15% as validation data, and 15% as test data. To train the ANN, Levenberg-Marquardt algorithm was utilized [34] with the random initial solution (normalized using the method of [PERSON] and [PERSON] [35]). When building the ANN according to the algorithm presented in Figure 3, the following parameters were set: \(N_{min}=1\), \(n=20\), \(\text{MSE}_{\text{max}}=0.5\) mm\({}^{2}\) (sum of average error of all four outputs from two simulations presented below). The ANN got 27 neurons in the hidden layer. Its architecture and a plot of training record error values against the number of training epochs are presented in Figure 8:0 The time of execution of the algorithm presented in Figure 3 was over 3 days. To verify the obtained ANN, the nonlinear elasticity of the shock absorber was described by the value of normal force generated in the element 17 \(N_{17}\) (N) in the function of its elongation \(\Delta I_{17}\) (mm). Two alternative functions were taken into consideration:  
Figure 8:1  
and  
Figure 8:2  
Substantiation about the selection of functions (12) and (13) is presented in Figure 13 with the lowest and highest stiffness of the shock absorber used to generate the highlighted reference data. Nonlinear odd functions in the presented range of stiffness were chosen. Figure 8:3 ANN obtained in example 2: (**a**) architecture, (**b**) performance of training. Figure 8:4 ANN obtained in example 2: (**a**) architecture, (**b**) performance of training. Similarly, as in the previous example, PC simulation times were measured. The average time from 20 simulations (10 for each shock absorber function) was 0.238 s for ANN and 1.296 s for FE analysis (Newmark scheme). Experimental Verification of Implementing ANN in Hybrid Simulation on the Example of Bicycle Frame with Nonlinear Shock Absorber  
Real-time hybrid simulation of the mechanical system presented in the previous example (3.2) was performed using ANNs. All the computations during hybrid simulation were executed on microcontroller National Instruments myRIO-1900 with the real-time operating system. The experiment was performed using dynamic materials testing machine Instron ElectroPuls E10000, which was controlled in a closed-loop by the microcontroller. Considering the capabilities of the testing machine, only axial elongation of the shock absorber was applied in the experiment, and its normal forces were measured. The inertia forces perpendicular to the axis of the shock absorber were transferred to the analytical subsystem. Figure 16 presents a photograph taken during the experiment. It was assumed that in hybrid simulation, the time courses of the values of **F1** and **F2** were described as follows:  
\[\textbf{{F2}}(t)=\left\{ \begin{array}{c}\frac{A_{1}}{t_{1}}t,\ \ \text{for}\ t<t_{1}\\ A_{1}-A_{2}[\cos(2\pi f(t-t_{1}))-1],\ \ \text{for}\ t\geq t_{1}\end{array} \right.