1. INTRODUCTION Component mode synthesis (CMS) has been a fundamental tool for the structural analysis of large complex systems for years. Instead of the system being modeled as a whole, it is broken up into substructures that are then modeled and reduced. This approach is often a necessity due to shear model size, and individual system components are often constructed by different companies. This is especially true in the aerospace community. The Craig-Bampton substructure representation [1] has become the most popular and efficient approach within the aerospace industry. In recent years, there has been a renewed interest in combining analytical based and experimental based substructures using CMS and the imposition of constraints. The direct approach would be to enforce compatibility in the physical connection degrees of freedom between two substructures, but this proves difficult in many problems for several reasons. Recently, a new approach has been introduced, called Modal Constraints for Fixture and Subsystem (MCFS) [2]. This method reduces ill-conditioning by imposing constraints on substructure modal coordinates instead of the physical interface coordinates. The experimental substructure is tested in a free-free configuration, and the interface is exercised by attaching a flexible fixture, which was dubbed a “transmission simulator” in subsequent works [3]. An analytical representation of the transmission simulator is then used to subtract its effects to produce the desired experimental model of the substructure. This process produces a substructure model that is typically much more accurate than a simple free-free model would be, because of the mass-loading effect of the transmission simulator. However, it has been observed that indefinite mass and stiffness matrices can be obtained for the experimental substructure if the analyst is not careful (e.g. the system has negative mass or stiffness). Similar problems were encountered by other researchers when removing rigid masses from a structure [4]. This paper derives simple metrics that can be used by the analyst to determine which of the systems’ modes contribute most to offending negative eigenvalues of either the mass or stiffness matrices. The metrics reveal problems with the subcomponent models that can sometimes be addressed by removing problematic modes or by refining the subcomponent models. Two examples are presented illustrating the metrics and the physics that they reveal. 2. THEORY 2.1. Application of Modal Constraint The MCFS component mode synthesis approach uses free-free substructure representations, since free-free modal tests are typically more convenient and accurate than the alternatives. In order to properly exercise a substructure during a free-free vibration test, the interface is connected to the transmission simulator [3]. Ultimately this transmission simulator must be subtracted in order to have an experimental representation of the desired substructure. A simple beam example, shown in Fig. 1, will be used to illustrate the process. Component C is the system tested to obtain an experimental model, component A is a finite element model (FEM) representation of the transmission simulator, and component B is the substructure for which an experimental representation is desired. Component C is the Combination (hence the letter C) of the Base system B and the Added transmission simulator A, so C = A+B. We wish to infer the properties of B from the measurements that were acquired on the assembly C. The transmission simulator mass-loads the left end of beam C, so the model that is obtained for B will form a good basis for B when it is subsequently assembled to some other structure at the same point (its left end). 112
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