# Reduction of Coupling Interface Degrees of Freedom in Mixed-Interface Component Mode Synthesis
## 1 Introduction  
The finite element method (FEM) is one of the most effective way of solving structural dynamic problems. With a rapid increasing of computation capability, the demand for large scale FE models with huge number of degrees-of-freedom (DOFs) has raised even more quickly, which makes structural dynamic calculations much time-consuming. Hence, it is crucial to reduce computational cost, especially when physical structures are large and complex. A variety of model reduction methods have been developed during the last decades, and the focus in model reduction is mainly on reducing computational cost without too much accuracy being lost [1; 2; 3; 4; 5; 6; 7; 8]. One of the most well known model reduction methods is component modes synthesis (CMS). CMS was firstly proposed by [PERSON], and it is an efficient method for analysing the dynamics of large, complex structures which are often described by separate interconnected components/substructures [9]. [PERSON] used three displacement modes (i.e., rigid-body, constraint and normal modes) to form a complete set of modes for each substructure, and he constructed a transformation relating component coordinates to system coordinates [10]. [PERSON] and [PERSON] improved the approach by [PERSON], and they employed two forms of generalized coordinates: one isinterface generalized coordinates that is related to constraint modes, the other is normal modes that is related to the free vibration modes of the substructures [11]. As completely restrained interfaces are imposed on each component at the connections, this approach can be classified as constrained-interface CMS (CCMS). [PERSON] provided a new modal synthesis technique, in which the interface conditions used for determining component vibration modes were free-free and both compatibility and equilibrium at boundaries were satisfied for modal coupling [12]. This approach can be classified as free-interface CMS (FCMS). [PERSON] and [PERSON] developed a comprehensive method of component modal substitution that is applicable to redundant complex structures [13]. Using this method, the natural modes of vibration for each component are determined with both free-free and constrained interface conditions (known as MCMS). In general, CMS methods have relied on a bottom-up approach, which can be described by the following steps. First, partition the original structure into several substructures with same or different interface conditions; second, carry out the eigenvalue problem in each substructure, and generalized mass, stiffness, damping matrices are determined; third, transform the component physical coordinates to the modal coordinates by reduced normal modes; forth, assemble all the components based on the displacement-compatibility or/and force-equilibrium at the connections; fifth, carry out the reduced global eigenvalue problem. As the number of the retained component modes is substantially smaller than the global physical DOFs, the computational size of modal synthesis is greatly reduced [14]. In any type of the CMS methods, the most important and difficult task is choosing an appropriate reduced component modes involving modal reduction and mode synthesis, and this continues to be an area of research [15; 16; 17; 18; 19; 20; 21; 22]. Lots of work have been done to make improvements on modal reduction of internal DOFs [23; 24; 25; 26; 27], but no order reduction is performed on the boundary. Though the internal DOFs dominate the FE models of substructures, they are greatly reduced by the process of coordinate transformations based on traditional CMS theory, as only a small number of low-frequency modes are truncated. However, if the number of substructures is increasing, or if the coupling interface is large and the FE mesh is fine, the coupling interface DOFs will rule the following works, and retaining a full dimension of the coupling interface DOFs will definitely lower the computational efficiency of mode synthesis.