## 5 Numerical Examples
### Slider-Crank Mechanism  
Let us consider the slider-crank system shown in Figure 11 (left) with the flexible connecting rod (right). When torque is imposed on the crank, the slider reciprocates up and down inside the cylinder. The crank is constrained to the ground and the connecting rod with a revolute joint, and the connecting rod is connected to the pin. The inner cylindrical surfaces of the connecting rod are modeled using MPC (see Figure 11, middle), and the nodes on the center of the cylindrical surfaces are the connecting points of the joints. The revolute joint is also used to pin and the slider moves only in the \(x\)-direction due to the translational joint with the cylinder, where the cylinder is fixed to the ground. The dimensions of the bodies are described in Figures 11 and 12. Free-free boundary conditions are applied for the flexible connecting rod. The Young's modulus \(E\), Poisson's ratio \(\upsilon\), and density \(\rho\) of the flexible connecting rod are \(E=70\) GPa, \(v=0.33\), and \(2710\) kg/m\({}^{3}\), respectively, which are the material properties of aluminum alloy 6061-T6. The total mass of the system is measured from the CAD kernel of RecurDyn [29] based on the geometries as 5.407 kg. The number of tetrahedron elements used to model the flexible connecting rod is 4380. The average and the minimum size of elements are 0.0036 m and 0.001 m, respectively. The driving torque \(T\) shown in Figure 11 (right) is applied to the crank, which increases to 1 Nm for up to 0.1 s and remains until the simulation end time 0.5 s. The initial number of the dominant mode \(N_{init}^{i}\) is five, and the selected mode range \(N_{m}^{i}\) is 10. We select node \(A\), which represents the maximum stress in Figure 11 (middle) to compare the dynamic responses. Figure 11: The slider-crank system (left) and the flexible connecting rod (middle). The driving torque with respect to time (right) is imposed to rotate the crank. Figure 10: The error contour of the von Mises strain of the different error tolerance \(e^{i}=10\%\), \(1\%\), and \(0.1\%\) for \(t=6.8\) s. (**a**) \(e^{i}=10\%\); (**b**) \(e^{i}=1\%\); (**c**) \(e^{i}=0.1\%\). The relative eigenvalue errors from the previous [23] and the new error estimator in Equation (43) are compared in Figure 13. In this example, the cases of the number of dominant modes are 30 and 60. The results show that the new error estimator is at the upper bounds in all mode ranges in both cases. The proposed algorithm is applied for the desired mode range \(N_{m}^{i}=10\) and the error tolerance \(e^{i}=0.1\%\). For Modes 4, 5, and 6, the changes of the relative eigenvalue error and its estimation when the number of dominant modes is added are shown in Figure 14.  
Figure 12: Main dimensions of the bodies. Figure 13: Comparison of the error estimator from the previous work [23] and the new error estimator derived in Equation (43). The new error estimator gives the upper bound results for the different numbers of dominant modes \(N_{d}^{i}=30\) and \(N_{d}^{i}=60\). (**a**) \(N_{d}^{i}=30\); (**b**) \(N_{d}^{i}=60\). The time transient accelerations \(\mathbf{a}_{x}\) and \(\mathbf{a}_{y}\) of node \(A\) for the error tolerances \(e^{i}=10\%\), \(1\%\), and \(0.1\%\) are shown in Figure 15. The selected number of dominant modes (\(N_{d}^{i}\)) for the error tolerances \(10\%\), \(1\%\), and \(0.1\%\) are \(8\), \(25\), and \(105\), respectively, but the reduced flexible models with \(e^{i}=10\%\), \(1\%\), and \(0.1\%\) are identical to \(10\) by \(10\) matrices to provide a fair comparison study. The results on the left column of Figure 15 shows the absolute error \(|\delta\mathbf{a}|\) of the accelerations calculated as \(|\delta\mathbf{a}|=|\mathbf{a}^{ref}-\mathbf{a}^{e}|\). It can be seen that the errors of the model with \(e^{i}=10\%\) and \(1\%\) are bigger than in the case of \(e^{i}=10\%\), whereas in the case of \(e^{i}=0.1\%\), the error is close to zero for the overall simulation time. The von Mises stress and strain are compared in Figure 16 in the same manner.