## 1 Introduction  
Flexible multibody dynamics (FMBD) simulation is a popular way to consider system dynamics with elastic deformation. The FMBD approach might be powerful because it can provide stress and strain information unlike with rigid body dynamics, but it requires much heavier computational cost than multibody dynamics (MBD) does. The computational cost comes from flexible bodies that are finite element (FE) models with large degrees of freedom (DOFs). Coordinate reduction techniques of the flexible bodies are then essential to effective FMBD simulation [1; 2; 3; 4; 5; 6]. Among the various methods, component mode synthesis (CMS) [7; 8; 9; 10; 11; 12], a projection technique with an eigenbasis, may be the most popular method in linear vibration with FE analysis. In the CMS methods, the residual eigenvectors that are less important for representing the dynamic characteristics of the original FE model are truncated. The constraint modes are then used to define the interface boundary condition of the reduced model [7; 10; 12], or the attachment modes may be employed instead of the constraint modes [9; 13]. Those depend on the type of CMS method. The small number of selected eigenvectors, known as dominant eigenvectors, are only used in the projection of the original FE model, which is a key to the model reduction process. The following two questions then arise [2]: (a) Which eigenvectors are dominant? (b) How many eigenvectors should be used to reduce FE models accurately? With this motivation, many researchers have studied mode selection and error estimation methods for use with the CMS methods. The rule of thumb of mode selection may be the frequency cut-off method in the mode superposition manner, but in the CMS methods, the effective interface mass (EIM) methods by [PERSON] and [PERSON] [14, 15] may be the first trial for the mode selection method. The EIM methods were developed for the Craig-Bampton (CB) method, which is the most popular CMS method. The optimal model reduction (OMR) methods based on the Dirichlet-to-Neumann (DTN) map have been studied under the substructuring strategy [16, 17]. Those may be the first mode selections considering the coupling effect between substructures. The CMS\({}_{\chi}\) method is an extension of the OMR method using the moment-matching approach instead of the DTN map [18]. [PERSON] et al. inherited the CMS\({}_{\chi}\) method without the relatively rough assumption in the derivation process, which is called the CMS\({}_{\sigma}\) method [19]. Unlike the above approaches, the eigenvector-based modal contribution (EMC) technique is an a posteriori data-driven method that computes the substructural modal contribution by considering the relation between the original and reduced matrices [20]. Therefore, the EMC method can provide better mode selection performance than a priori methods can, but requires relatively huge computational cost. Those issues were well examined by [PERSON] et al. [20]. Ranking of the dominant modes can be arranged using the mode selection method. We can then make a selection of the dominant modes, and then, the modes could be added to the reduced matrices in the iterative way for better accuracy. However, more modes lead to larger reduced system matrices, requiring a compromise between computation cost and accuracy. The use of mode selection only may not be enough to resolve this issue. It can be treated using the error evaluation methods of the reduced matrices. Among the various methods, the error estimators of the CB method and the Guyan reduction proposed by [PERSON] et al. may be the most highly accurate [21, 22, 23, 24]. Those approaches are direct approximations of the relative eigenvalue errors of the CB and Guyan reductions. The direct eigenvalue error estimation is then available without knowing the reference eigenvalue. Combining the error estimators and the above mode selection methods can provide an automated model reduction algorithm excluding empirical judgment [25]. This work was accelerated toward that idea.