## 3 Results
Two alternative functions were taken into consideration:  
\[\mathbf{N_{s,1}}=150u_{\text{C}}-50(u_{\text{C}})^{3} \tag{8}\]  
Figure 5: ANN obtained in example 1: (a) architecture, (b) performance of training. \[\mathbf{N}_{s,2}=50u_{\text{C}}+1000(u_{\text{C}})^{3}. \tag{9}\]  
Substantiation about the selection of functions (8) and (9) is presented in Figure 6 with the lowest and highest spring stiffness \(k\) used to generate the highlighted reference data. Nonlinear odd functions in the range of \(k\) on which neural network was trained (or near it, as ANNs have the ability to generalize) were chosen. Two alternative time courses of force \(\mathbf{F}\) were taken into consideration and are presented in Figure 7.  
Time courses of vertical displacement of point C obtained from ANN and FE model of hybrid simulation (considered as exact) are presented in Figure 8, and mean-squared errors are summarized in Table 1.  
Figure 3:0 Nonlinear functions used to simulate the physical subsystem: (**a**) function (8), (**b**) function (9). Figure 3:1 Time course of force \(\mathbf{F}\): (**a**) load case 1, (**b**) load case 2. Table 2 contains a comparison of simulation time for FE analysis (Newmark scheme) and ANN. Simulations were performed in MATLAB software on a PC (Intel Core i7-8700 3.20 GHz, 8 GB RAM). The table shows the average of 10 measurements. Although the calculations were not performed in real-time as in actual hybrid simulation, the simulation times presented in Table 2 are also a measure of the model reduction. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Function of Spring Stiffness** & **Load Case 1** & **Load Case 2** \\ \hline Equation (8) & FEM: 0.54 s & FEM: 0.55 s \\  & ANN: 0.26 s & ANN: 0.25 s \\ \hline Equation (9) & FEM: 0.56 s & FEM: 0.54 s \\  & ANN: 0.25 s & ANN: 0.25 s \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Simulation times comparison—example 1. Figure 8: Time course of vertical displacement of point C: (**a**) load case 1 and function (8), (**b**) load case 2 and function (8), (**c**) load case 1 and function (9), (**d**) load case 2 and function (9). FEM, finite element method. Numerical Verification of Implementing ANN in Hybrid Simulation on the Example of Bicycle Frame with Nonlinear Shock Absorber  
A steel (Young modulus 200 GPa, density 7800 kg/m\({}^{3}\)) mountain bicycle frame was considered as the second numerical example and is presented in Figure 9. The frame was equipped with nonlinear shock absorber, consisting of the physical subsystem in hybrid simulation. The rest of the mechanical system was linear elastic, consisting of the analytical subsystem. The finite element model consisted of 27 finite elements, 73 DOFs, and is presented in Figure 10. The element 17 represented the shock absorber and was of the rod type. The rest of the elements were of the plane frame type. The node linking elements 18 and 19 were a ball joint; therefore, they had four DOFs: vertical translation, horizontal translation, left-side rotation, and right-side rotation. Force **F1** represented the load from the handlebar, and force **F2** represented the load from the saddle. Cross-section parameters of subsequent elements are given in Table 3. The Rayleigh damping (6) coefficients were \(\alpha\) = 2 and \(\beta\) = 0.1. The time step was 1 ms.  
Figure 10: Finite element model of bicycle frame (reprinted from [23], with the permission of AIP Publishing). Figure 9: Numerical example 2—mechanical system: (**a**) photograph; (**b**) CAD model (reprinted from [23], with the permission of AIP Publishing). It was assumed that in hybrid simulation, the time courses of the values of **F1** and **F2** were described as follows:  
\[\textbf{F2}(t)=-A[\cos(2\pi f\cdot t)-1] \tag{10}\]  
and  
\[\textbf{F1}(t)=0.4\cdot\textbf{F2}(t) \tag{11}\]  
where \(A=400\) N, \(f=3\) Hz, and \(t\) denotes time. In order to generate the reference data, it was assumed that the nonlinear stiffness of the shock absorber could vary in a range from 60 to 200 N/mm.