Component Mode Type Quarter Wavelength (L/4) Additional Data String Lateral ( T A)/(4f )* U T=tension, A=area, U= mass density Rod Axial ( E )/(4f )* U E = elastic modulus Rod Torsion ( G )/(4f )* U G = shear modulus Beam Bending * 1/ 4 ( /2)(EI/ A) / 2 fS U S EI = flexural stiffness Membrane Lateral ( N h)/(4f )* U N = stress resultant Plate Bending * 1/ 4 ( /2)(D/ h) / 2 fS U S D=flexural stiffness, h=thickness 3-D Elastic Dilational ( E )/(4f )* U 3-D Elastic Shear ( G )/(4f )* U Acoustic Dilational ( B )/(4f )* U B=Bulk Modulus Table 1: Grid Spacing Guidelines for Typical Structural Components 3.2 STRESS FIDELITY REQUIREMENTS Accurate modeling of stress concentrations, for a unified dynamic-stress model, may be addressed by employment of adaptive mesh refinement[2], or by utilization of documented stress concentration formulae[5]. If mesh refinement is utilized, the number of model degrees of freedom may increase substantially, imposing undue computational resource penalties on a model that had been appropriately designed for study of dynamic response and loads. If localized stress concentrations do not affect a model’s dynamics fidelity, it is prudent to recover detailed stresses by employing (a) stress concentration formulae and/or (b) separate, detailed local or global “stress” finite element models (designed using adaptive mesh refinement). 3.3 ILLUSTRATIVE EXAMPLE SHELL STRUCTURE [3] The shell structure, shown below in Figure 1, serves as an illustrative example system throughout this paper. It consists of five substructures, namely (1) a lower cylindrical skirt (fully fixed at its base), (2) a lower hemispherical bulkhead, (3) lower cylindrical section, (4) upper cylindrical section, and (5) upper hemispherical bulkhead. The overall dimensions of the aluminum structure are length, L=100 inches, radius, R=20 inches, and wall thickness, h=0.4 inches. It should be noted that this illustrative example structure does not represent a realistic design. The rather high thickness-to-radius ratio, h/R=1/50, was selected to produce less shell breathing modes in the base frequency band (f < 2000 Hz) than typical aerospace systems, while including modes of sufficient complexity to illustrate key aspects of quantitative normal mode metrics. Shell with Applied Forces Shell Structure Component Breakdown FRFs and Modal Density Distribution Figure 1: Illustrative Example Shell Structure (Modal Density, N = number of modes per 1/3 octave band) 369
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