36 R.N. Coppolino ŒMff fRufgCŒKff fufgDfFfg! Maa Mao Moa Moo Rua Ruo C Kaa Kao Koa Koo ua uo D Fa Fo . (5.11) If only the “analysis” partition of the mass matrix were non-zero, and external forces were only applied to “analysis” degrees of freedom, the relationship between “analysis” and “omitted” degrees of freedom would be fufgD ua uo D Iaa K 1 ooKoa f uagDŒ‰fa c fuag. (5.12) In that situation, the reduction transformation in Eq. (5.12) would be exact. When the “omit” partitions of the mass matrix are non-zero, the reduction transformation is approximate (its columns are Ritz [3] shape functions). Application of the reduction transformation, in a symmetric manner following the Ritz method, yields the “classical” Guyan reduction TAM mass matrix, ŒMaa c DŒ‰fa T c ŒMff Œ‰fa c. (5.13) It should be noted that the reduction transformation matrix columns (in Eq. 5.12) are physically consistent with deflection shapes associated with application of individual unit “analysis” set loads. 5.4.2 Modified Guyan Reduction When the Guyan reduction method was introduced in 1965, the majority of matrix structural dynamic models were assembled using finite elements based on technical theories (e.g., beams, plates and shells). Deformation shapes for technical theory based structural models, subjected to point loads, are generally smooth resulting in “well-behaved” Ritz shape functions. As finite element technology continued to evolve, elements based on 3-D elasticity theory matured to the point that many of today’s highly refined finite element models incorporate 3-D elastic elements. Dynamic models using 3-D elastic elements are generally quite accurate and effective, except for situations in which reduced models are required (e.g., preparation of testanalysis models or TAMs). Since highly refined 3-D elastic models closely follow exact mathematical behavior, deformations associated with point loads are extreme (infinite in the limit, as in the case of the Boussinesq problem [6]), producing Ritz shape functions that do not resemble normal modes. Thus application of Guyan Reduction on dynamic models composed of 3-D elements, as well as several types of one and two dimensional elements (especially shells), is inappropriate. Consider the general distribution of static loads described by the matrix equation, fFfgDŒ fa fFag, (5.14) where [ fa], represents the collection of unit load patterns (or load patches). The static displacement shapes due to unit load patches are Œ‰fa DŒKff 1 Œ fa , whichimplies that, fufgDŒ‰fa fFag. (5.15) Pre-multiplication of this result by the transpose of unit loadings yields, fqgDŒ‰fa TŒKff 1 Œ‰fa fFag. (5.16) Substitution of this result into Eq. (5.15) yields the modified Guyan reduction transformation, fufgDŒ‰fa ‰fa TKff 1‰fa 1 fqgD ‰fq fqg. (5.17) Unlike the reduction transformation defined by classic Guyan reduction, which provides a direct relationship between “free” and “instrumented” DOFs, the above transformation requires further development. This is accomplished by first focusing on a partition that relates “instrumented” and generalized DOFs, i.e., fuagD ‰aq fqg, (5.18)
RkJQdWJsaXNoZXIy MTMzNzEzMQ==