The self-equilibrating nature of shell breathing in modes 3 and 25 is clearly indicated by the absence of modal effective mass. Graphical illustrations indicating the character of the fundamental (Y) bending mode and the lowest frequency shell breathing mode are provided in Figure 2. Kinetic (KE) and strain or potential energy (PE) distributions are indicated in “pie” format. Mode 1: Y-Bending Mode 3: Cylinder n=3 Breathing Figure 2: Graphical Illustration of the Character of Two System Modes The modal metrics, based on kinetic energy and strain energy distributions (in accordance with direction and component activity) and modal effective mass, have been shown to clearly and quantitatively characterize modes of the illustrative example system. Most of the system modes are dominated by shell breathing activity; a few system modes are characterized by overall body bending, axial and torsional deformation. Often only the overall body modes are of interest, since they are generally the primary contributors to critical system dynamic loads. The modal filtering metric introduced below provides the means for focusing on overall body modes. 4.6 CLASSIFICATION BY “BODY” MODE SHAPE FAMILIES The modal deformation of a dynamic finite element system may be arbitrarily segmented into selected subsystems (or bodies). In the case of the illustrative example shell structure, a “body” is designated as a ring of grid points for one of the five component shells at a particular axial location. Consider transformation, depicted in Equations 14 and 15, that relates the modal deformation of designated “bodies” to rigid body motions (<Body,i) and “residual” flexible body deformations (RBody,i). Body B n Body Body Body B n Body Body Body B Body Body Body B n Body Body R R R ° ° ¿ ° ° ¾ ½ ° ° ¯ ° ° ® ° ° ¿ ° ° ¾ ½ ° ° ¯ ° ° ® » » » » » ¼ º « « « « « ¬ ª < < < ° ° ¿ ° ° ¾ ½ ° ° ¯ ° ° ® ) ) ) , ,2 ,1 , ,2 ,1 , ,2 ,1 , ,2 ,1 ... ... 0 0 0 ... 0 ... 0 ... 0 0 ... 0 0 ... M M M (14) ^ ` > @^ ` ^ `n Bodies n Bodies Bodies n R ) < M (15) Note that the defined transformation is not a constraint; it is merely a means of discriminating classes of deformation in a system mode. By employing a mass-weighted linear least squares fit that orthogonalizes local rigid body and flexible body deformations, the coefficients of the “rigid bodies” transformation are determined and the flexible body deformations are, ^ ` > @ > @n T Bodies Bodies T Bodies n Bodies M M < < < ) 1 M , ^ ` ^ ` > @^ `n Bodies Bodies n n Bodies R M ) < . (16) The degree of purity of overall body motion in a system mode is quantified as the “body” mode coherence ^ ` > @^ `n Bodies T n Bodies n Bodies R M R COH 1 , (17) 372
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