PSI - Issue 24

Corrado Groth et al. / Procedia Structural Integrity 24 (2019) 875–887 C. Groth et Al. / Structural Integrity Procedia 00 (2019) 000–000

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Fig. 1: generalized displacement and velocity for a generic smoothed step function

System input needs to excite frequencies inside the chosen range k = [0 , k max ], with a maximum amplitude com puted, for each DOF, in function of the maximum modal displacement d max :

4 ǫ L a d max k max

(6)

A q =

Choosing ǫ = tan(1 deg) so to obtain a small modal velocity compared to the reference velocity, and using the dimen sionless time τ = tV ∞ / L a the smoothed step function can be written as: q i ( τ ) =   A q 2 · � 1 − cos � k q τ �� if τ < τ q A q if τ ≥ τ q (7) The resulting generalized velocity is: ˙ q i ( τ ) =   A q k q V ∞ 2 L a · � 1 − sin � k q τ �� if τ < τ q 0 if τ ≥ τ q (8) In figure 1 the generalized displacement and velocity applied using a generic smoothed step function are shown with respect to the dimensionless time τ

2.3. Radial Basis Functions mesh morphing

RBF are mathematical interpolation functions able to retrieve, on a distance basis, scalar information known at source points, i.e. a cloud of discrete points supposed to be the input of the problem. The interpolation shape between source points can be controlled by selecting an appropriate radial function and by defining the kind of support it guarantees (local or global), meaning the domain in which the function is not zero valued (De Boer et al. (2007)).