## 5 Numerical Examples
### Quadruple Pendulum Problem  
A quadruple pendulum model consisting of four cylinders is considered. The pendulum consists of three rigid bodies and one flexible body, and they are connected by spherical joints sequentially. Body 1 is connected to the ground, and one side of the flexible body is modeled as a free end. The model is described in Figure 2. When the simulation starts, the bodies fall freely in the \(-y\) direction due to gravitational acceleration g = 9.806 m/s\({}^{2}\), and the chaotic behavior appears. Here, the simulation end time is seven seconds, and the number of steps is 500. The length and the radius of the cylinders are 0.3 m and 0.015 m, respectively. The distance from a spherical joint to a cylinder is 0.05 m. For the flexible body modeling, the free-free boundary condition is applied. An ideal Young's modulus, which is half of steel's, is 100 GPa. [PERSON]'s ratio \(v\) is 0.285, and the density \(\rho\) is 7850 kg/m\({}^{3}\). The flexible body is modeled using 3528 tetrahedron elements. The average and the minimum size of the elements are 0.0075 m and 0.0056 m, respectively. The initial number of dominant modes \(N_{init}^{i}\) is five, and the multi-point constraint (MPC) [29] node on the right side of the flexible body is selected as the boundary node. The mode range \(N_{m}^{i}\) selected is 30, and the desired error tolerance is considered to be 0.1%. The dynamic responses of node \(A\) in Figure 2 (right), which were selected arbitrarily, are compared in the following numerical studies. Figure 3 is a relative eigenvalue error comparison of the previous [23] and the new error estimators in Equation (43). Two cases with 50 and 100 dominant modes are considered here. The previous error estimator well describes the relative eigenvalue errors with high accuracy, but it generally tends to be at the low bounds of the exact error. This may lead to excessive coordinate reduction. Otherwise, the new error estimator tends to be at the upper bounds of the exact error. In this work, in a conservative manner, the new error estimator is considered, but both error estimators could be used as the lower and upper bounds of the exact relative eigenvalue error. This is because neglecting the additional term of the new error estimator directly leads to the previous one, as shown in Equations (42) and (45). Figure 3: 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 number of dominant modes \(N_{d}^{i}=50\) and \(N_{d}^{i}=100\). **(a)**\(N_{d}^{i}=50\); **(b)**\(N_{d}^{i}=100\). Figure 2: A quadruple pendulum with a flexible body (left) and the flexible cylinder with node \(A\) (right) to compare dynamics responses. To evaluate the performance of the proposed algorithm, the estimated and exact relative eigenvalue errors change as the dominant mode is added for the \(n\)-th mode (shown in Figure 4). Among the mode range, some modes (\(n=1,6,10,15,16,17,19,26\), and \(30\)) are only considered in Figure 4 for better readability. We found that the accuracy of lower modes shows fast convergence adding dominant modes than the accuracy of higher modes in Figure 4, which follows a typical tendency of the mode superposition. To satisfy the error tolerance \(e^{i}=0.1\%\) within the mode range of \(N_{m}^{i}=30,446\) dominant modes are selected in the algorithm. It can be confirmed that the exact relative eigenvalue error of the RFlex using the error estimator satisfies the specified error tolerance for the \(n\)-th mode. The sufficiently accurate RFlex that satisfies the error tolerance \(e^{i}\) for the desired mode range can be obtained by using the newly derived error estimator at the upper bounds. Here, dynamic responses are compared for three RFlex models that have different error tolerances \(e^{i}=10\%,1\%\), and \(0.1\%\). The selected number of dominant modes (\(N_{d}^{i}\)) is \(35\) and \(58\) for the error tolerances \(e^{i}=10\%\) and \(1\%\), respectively.