resulting in the modified, frequency dependent reduced mass and stiffness matrices and applied force vector, > @ > @ > @> @ > @ na nn nn an aa aa P P M M 1 2 2 ( ) ~ , Z O Z Z , > @ > @ aa aa K K ( ) ~ Z , (11) ^ ` ^ ` > @> @ ^ `n nn an nn a a Q F P F 1 2 2 ( ) ~ , Z O Z Z . (12) The above frequency dependent reduced mass matrix suggests that any physically consistent enhancement to Guyan Reduction cannot be expressed as a constant matrix, although the IMS method[9] does provide an approximation of that effect. Whenever a conventional TAM mass matrix is required (especially in support of a modal test), the most effective strategy for determination of an appropriate analysis set is described in Reference 8 or closely related methodologies. 4. NEW REDUCTION STRATEGY CIRCUMVENTING GUYAN REDUCTION LIMITATIONS 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” Rayleigh-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 test-analysis 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[10]), producing Rayleigh-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, is inappropriate. 4.1 LOAD PATCH STRATEGY Prior to the introduction of the finite element method, investigators defined Rayleigh-Ritz shape functions as any admissible linearly independent shape functions. A common practice was utilization of deflection patterns based on deflections due to distributed static load patterns. The alternative reduction strategy, presented herein, borrows from this earlier common practice and formalizes a systematic procedure[11] that has advantages to be demonstrated. Consider the general distribution of static loads described by the matrix equation, ^ ` > @^ `\F F F F F k kk < ° ° ¿ ° ° ¾ ½ ° ° ¯ ° ° ® » » » » ¼ º « « « « ¬ ª < < < ... ... 0 0 ... ... ... ... ... 0 0 ... 0 0 2 1 22 11 , (13) where the individual load patches, [<ii], represent unit load patterns (over each specific geometric patch) that are mathematically the transposes of rigid body deflection patterns. The static displacement due to the above defined loading is ^ ` > @ ^ ` > @^ `\ U K F K F < 1 1 . (14) Pre-multiplication of this result by the transpose of unit loadings yields, ^ ` > @ > @ ^ ` > @^ `\ \ K F K F U T T < < < 1 1 , ^ ` > @ ^ ` \ \F K U T 1 1 < < . (15) The above generalized displacements are mathematically the average 6-DOF displacements at the geometric centroid of each load patch. Substitution of this result into equation 14 yields the (load patch based) “Guyan Reduction” transformation, 362
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