178 A.R. Brink et al. and Brink, calculates the mode shapes for the nominal condition, then uses the same mode shapes for all realizations of the structure. The mass and stiffness matrices are updated to reflect the new parameter set. The second method, developed by ATA Engineering, instead retains the mass and stiffness matrices of the nominal structure and perturbs the mode shape matrix with a cross-orthogonality matrix, corresponding to the expected change in mode shapes resulting from the new parameter set. Both of these methods are developed in Sects. 19.3.1 and 19.3.2, respectively. They are then applied to a prototypical structure, and their results compared for validity to full Craig-Bampton reductions, which include recalculation of the modes for each parameter set. 19.2 A Brief Review of Craig-Bampton Reduced Order Models Craig and Bampton’s landmark work [8], first published in 1968, is a simplification of work done by Hurty [9] earlier in the same decade. This simplification involves the way in which interface degrees of freedom are handled. The procedure for generating a Craig-Bampton reduced order model is as follows [10]: 1. Generate a detailed numerical model (finite element or otherwise) of the system or component that is to be reduced. 2. Identify all interface and internal degrees of freedom. 3. Calculate the fixed-interface modes modal matrix for the system with all interface degrees of freedom fixed. Engineering judgment and convergence studies dictate how many of these modes to retain. 4. Calculate the constraint modes of the interface degrees of freedom. 5. Assemble the Craig-Bampton transformation matrix. 6. Calculate the reduced mass and stiffness matrices. 7. Assemble the reduced order model into the system numerical model. 19.2.1 Model Generation and DOF Identification Generating the numerical model to be reduced is often time consuming. The analyst must build a full model capable of capturing all of the relevant dynamics. For baseline Craig-Bampton reduced order models, the model is completely linear, but can still be quite complex. To perform UQ analysis involving geometric parameters, this model needs to be rebuilt for each new parameter set. However, if changes are small, such as those attributed to geometric tolerances, then the original model can be tweaked without major effort. Identifying the interface and internal degrees of freedom is straight forward. Any degree of freedom that couples to the main structure of interest is considered an interface degree of freedom. All remaining degrees of freedom are considered internal. 19.2.2 Fixed-Interface Modes In the Craig-Bampton formulation, fixed-interface modes are defined as the normal modes of the internal degrees of freedom with all interface degrees of freedom fixed. This requires a straight forward Eigen analysis. The modal matrix is stored as ˆ. As stated earlier, the analyst must decide the appropriate number of modes to retain to achieve an acceptable level of accuracy. 19.2.3 Constraint Modes Constraint modes are calculated by statically applying a unit displacement to one interface degree of freedom at a time, while fixing all other interface degrees of freedom. This not only helps to define the proper interface stiffness, but also helps to enforce inter-component compatibility during assembly into the complete structure. The constraint modes of the substructure, are calculated with
RkJQdWJsaXNoZXIy MTMzNzEzMQ==