A Parallel Approach of the Enhanced Craig–Bampton Method

The enhanced Craig–Bampton (ECB) method is a novel extension of the original Craig–Bampton (CB) method, which has been widely used for component mode synthesis (CMS). The ECB method, using residual modal compensation that is neglected in the CB method, provides dramatic accuracy improvement of reduced matrices without an increasing number of eigenbasis. However, it also needs additional computational requirements to treat the residual flexibility. In this paper, an efficient parallelization of the ECB method is presented to handle this issue and accelerate the applicability for large-scale structural vibration problems. A new ECB formulation within a substructuring strategy is derived to achieve better scalability. The parallel implementation is based on OpenMP parallel architecture. METIS graph partitioning and Linear Algebra Package (LAPACK) are used to automated algebraic partitioning and computational linear algebra, respectively. Numerical examples are presented to evaluate the accuracy, scalability, and capability of the proposed parallel ECB method. Consequently, based on this work, one can expect effective computation of the ECB method as well as accuracy improvement.

Oriented by Characteristic Roots Reduced Matrices in the Class of Semiscalarly Equivalent

A set of polynomial 3 × 3 -matrices of simple structure has been singled out, for which a so-called oriented by certain characteristic roots reduced matrix is established in the class of semiscalarly equivalent. The invariants of such reduced matrices and the conditions of their semiscalar equivalence are indicated. The obtained results will also be applied to the problem of similarity of sets of numerical matrices.

GRASSMANNIAN SEMIGROUPS AND THEIR REPRESENTATIONS

The set of row reduced matrices (and of echelon form matrices) is closed under multiplication. We show that any system of representatives for the $\text{Gl}_{n}(\mathbb{K})$ action on the $n\times n$ matrices, which is closed under multiplication, is necessarily conjugate to one that is in simultaneous echelon form. We call such closed representative systems Grassmannian semigroups. We study internal properties of such Grassmannian semigroups and show that they are algebraic semigroups and admit gradings by the finite semigroup of partial order preserving permutations, with components that are naturally in one–one correspondence with the Schubert cells of the total Grassmannian. We show that there are infinitely many isomorphism types of such semigroups in general, and two such semigroups are isomorphic exactly when they are semiconjugate in $M_{n}(\mathbb{K})$. We also investigate their representation theory over an arbitrary field, and other connections with multiplicative structures on Grassmannians and Young diagrams.

Modeling Support Effects: Finite Element and Experimental Modal Methods

The effects of support/foundation dynamics are often significant in high speed turbomachinery, and can affect the stability and response to unbalance. In some cases additional critical speeds are introduced, related to resonances in the foundation or interaction with rotor resonances of foundation resonances. This paper reviews several methods for representing these effects, including (1) reduced matrices from finite element substructures (ANSYS, for example), (2) matrices generated from modal data, and (3) direct use of experimental transfer functions. These methods are implemented in a finite element rotor program in a PC-DOS environment. The application of the methods to two laboratory rotor configurations described and results presented. Situations with a foundation resonance above and near the rotor critical are included. The importance of including coupling effects between supports is shown.