PSI - Issue 24

Filippo Cianetti et al. / Procedia Structural Integrity 24 (2019) 526–540 Author name / Structural Integrity Procedia 00 (2019) 000–000

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Fig. 4: CAD 3D & FEM model of the frame.

was modeled as a single body assuming that there is no loss of preload Langer et al. (2017); Lyndon (1998); Oskouei et al. (2009). The constraint with the helicopter, made with a bolted coupling in the orange holes of Fig. 4 (a) was modeled through rigid connections Zaman et al. (2013). From the FEMmodel, through a free-free modal analysis, natural frequencies and modal shapes are obtained. These represent the elementary building blocks of the dynamic model of the component,Braccesi et al. (2015); Crandall, S. H. And Mark (1963); Døssing (1988b); Hatch (2000). In fact if: [ m ] { ¨ x } + [ c ] { ˙ x } + [ k ] { x } = 0 (6) describes the law of motion for a vibrating body, then, under the assumptions of linearity, stability, causality and time invariance, it is possible to decouple the system into a summation of one degree of freedom harmonic oscillators: [ I ] { ¨ q ( t ) } + [2 σ ] { ˙ q ( t ) } + [ ω 2 ] { q ( t ) } = 0 (7) remembering the relation between physical x ( r , t ) and generalized q ( t ) coordinate: { x ( r , t ) } = [ ϕ ( r )] { q ( t ) } (8) where ϕ ( r ) are the modal shape. It is convenient to rearrange the (7) in State Space system Luis et al. (1996): { z } = [ q ( t ) , ˙ q ( t )] (9) obtaining a system of two first-order di ff erential equations:   { ˙ z ( t ) } = [ A ] { z ( t ) } + [ B ] u ( t ) { ˙ y ( t ) } = [ C ] { z ( t ) } + [ D ] u ( t ) (10) this is a smart, flexible and e ffi cient method to obtain the Frequency Response Function (FRF) outside the environment of the FEM code, using only the modal representation of the system Crandall, S. H. And Mark (1963); Preumont (1994, 2013) : H q = C ( j ω I − A ) − 1 B (11)