# Reduction of Coupling Interface Degrees of Freedom in Mixed-Interface Component Mode Synthesis
## 6 Conclusions  
The CMS method is one of the well-developed modal order reduction technique that describes the whole structure with several independent components/substructures. Modal analysis are carried out in each component and then mode synthesis are done based on the displacement-coordination or/and force-equilibrium at the interfaces of substructures. As the internal DOFs are greatly reduced by the first coordinate transformation, the number of the coupling interface DOFs involved in mode synthesis will rule the following computational efficiency and accuracy. In this paper, we focus on reducing the coupling interface DOFs to solve the problems of docking difficulties and incomplete boundary conditions. The following conclusions may be drawn:  
(a) Displacement and force interpolation functions of the coupling interface DOFs using nodal displacement interpolation formulations based on FEM are set up for the MCMS method. The corresponding interpolation transformation is done before mode synthesis, and only a small number of IBNs are kept so that the proposed method is very computationally efficient for solving eigenvalue problems.  
(b) The numerical performance of the MCMS-rid method is demonstrated by three different examples, where different structural properties and FE model size are discussed properly. The results indicate that favourable accuracy with a least number of the retained DOFs involved in modal synthesis can be obtained when compared with other MCMS methods.  
Figure 16: Modal assurance criterion between the modes by the MCMS-fid method and MCMS-rid method different interpolation schemes. (**a**) MCMS-rid:b vs. MCMS-fid; (**b**) MCMS-rid:c vs. MCMS-fid.  
Figure 17: Frequency response functions of the beam by FE model, MCMS-fid and MCMS-rids with different interpolation schemes for the first mode.  
(c) The number and distribution of IBNs chosen for mode synthesis have significant influence on the numerical precision. It is noticed that the more the IBNs involved, the higher the calculation accuracy will be obtained. And when the IBNs uniformly distribute on the coupling interface, the accuracy is the best.  
(d) The shape of sub-regions also greatly effects the computational accuracy. Dividing the interface with quadrangular interpolation sub-regions may receive a better accuracy than that with triangle ones. However, the triangle interpolation sub-regions show advantages when the coupling interface is irregular.  
Conceptualization, Y.T. and H.Q.; methodology, Y.T.; software, H.Q.; data curation, Y.T.; writing-original draft preparation, Y.T.; writing-review and editing, H.Q.; funding acquisition, Y.T. and H.Q. All authors have read and agreed to the published version of the manuscript.  
This research was funded by \"the National Key R & D Program of China (Grant No. 2017 YFC0404900)\", \"the National Natural Science Foundation of China (Grant No. 41904095 and 51979027)\", \"the Open Fund of Key Laboratory of Geotechnical and Underground Engineering of Ministry of Education at Tongji University (Grant No. KLE-TJGE-B1804)\", \"the State Key Laboratory of Mechanical Behavior and System Safety of Traffic Engineering Structures at Shijiazhuang Tiedao University (Grant No. ZZ2020-04)\", and \"the Fundamental Research Funds for the Central Universities (Grant No. DUT19 JC23/DUT19 RC(4)020)\".  
The authors declare no conflict of interest.