2. NOMENCLATURE Matrices and Vectors: ^ `F Force (Eq. 1) ^ ` Bodies n n R , Residual flexible body deformation (Eq. 14) > @K Stiffness (Eq. 1) ^ `n SE Modal strain energy (Eq. 11) > @ bb K Craig-Bampton boundary stiffness (Eq. 5) > @) Modal matrix (Eq. 2) ^ `n KE Modal kinetic energy (Eq. 10) > @ i Body, < Rigid body matrix for Body “i” (Eq. 14) > @ M Mass (Eq. 1) > @ ib< Craig-Bampton “constraint modes” (Eq. 4) > @ bb M Craig-Bampton boundary mass (Eq. 5) ^ `u Displacement vector (Eq. 1) > @n Meff Modal effective mass (Eq. 8) ^ `q Modal displacement vector (Eq. 2) > @P Modal participation factor (Eq. 5) ^ ` Body i n . M Body modal coefficient vector (Eq. 14) Variables: Bodies n COH , Modal coherence (Eq. 17) i Imaginary unit (Eq. 1) f Circular frequency K Structural damping coefficient (Eq. 1) ( ) h f n Modal frequency response (Eq. 6) O Eigenvalue (Eq. 2) Subscripts: B Number of “bodies” b “boundary”, or “body” RX, RY, RZ Global rotational coordinates e “external” Tot “Total” i “internal” N Number of modes n “mode” number TX, TY, TZ Global translation coordinates q “interior” modes 3. DESIGN OF RELEVANT FINITE ELEMENT MODELS Finite element models that faithfully simulate dynamic response, loads and stresses associated with anticipated dynamic environments must be designed to (a) produce accurate system modes and/or frequency response over the frequency band of interest, and (b) accurately predict redundant load paths and peak local stresses that define structural safety margins. The above two requirements may be employed to define a unified dynamic-stress model or separate dynamic and stress models (a common practice in the aerospace industry). 3.1 DYNAMIC FIDELITY REQUIREMENTS In order to develop a relevant dynamic model, general requirements should be addressed based on 1. Frequency band width 0<f<f*, and intensity (F*) of anticipated dynamic environments. 2. General characteristics of structural or mechanical components. Dynamic environments are generally (a) harmonic, (b) transient, (c) impulsive or (d) random. For all categories, the cut-off frequency (f*) is reliably determined by shock response spectrum analysis[1]. The overall intensity level of a dynamic environment is described by the peak amplitude for harmonic, transient and impulsive events, or by the statistical amplitude (e.g., mean plus a multiple of the standard deviation) for a long duration random environment. With the cut-off frequency (f*) established, the shortest relevant wavelength (L) of forced vibration for components in a structural assembly may be calculated. For finite element modeling, the quarter wavelength (L/4) is of particular interest, since it is a rough estimate of the grid spacing needed to characterize system dynamics at the cut-off frequency. Note that the actual grid spacing requirement is a function of the specific elements being used. Grid spacing (quarter wavelength) guidelines, for typical structural components, are summarized in Table 1. 368
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