PSI - Issue 24

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

533

8

The numerical model in which the points of representation are not linked to the discretization of the test, better approximates the curvatures; however concavities and trends are very consistent up to the sixth mode. The graphical comparison is supported by the estimator known as MAC (Modal Assurance Criterion), defined as Allemang (2003):

N d d ϕ n , d , m ϕ ∗ e , d , m

MAC m =

(14)

N d

d ϕ n , d , m ϕ ∗ n , d , m

N d d ϕ e , d , m ϕ ∗ e , d , m

Where ϕ n , d , m is the numerical (n) modal shape, relative to the degree of freedom (d), and to the m-th mode, similarly ϕ e , d , m is the experimental homologous. The function of the modal assurance criterion (MAC) provide a measure of consistency (degree of linearity) between estimates of a modal vector. The output is a scalar between 0 and 1, with respect to the m − th mode, indicates the degree of coherence or linearity between the considered eigenvectors. The comparison, expressed as a matrix, is shown in Fig. 8 Allemang (2003).

1

0.5

0

1

2

3

4

8

5

7

6

6

5

7

4

8

3

Numerical

2

Experimental

1

(a) 2-D MAC matrix

(b) 3-D MAC matrix

Fig. 8: Numerical vs Experimental MAC.

As already graphically observed, there is a good correspondence up to the 6th mode.

2.4. Synthesis and Validation of the Vibratory Environment

During its operational life the component will be stressed by the acceleration deriving from the flight and from the operation of the aircraft (i.e. rotating parts). Today’s certification bases, for military and non-military components, make extensive use of the MIL-STD-810-H standard as a guideline for the production of evidence useful for demon strating requirements Lalanne (2009d). Here specifically, the 514 ”Vibration” method was used: this is an accelerated qualification spectrum, that compress 2500 flight hours (FH) in just 12 hours of testing (4 hours along each axis). It is a sine on random test type, which superimposes the sinusoidal vibration of the fundamental frequencies of the rotating organs, to a random signal depending on the location of the load. By applying the standard and adapting it to the aircraft, the position and the frequencies of interest, the test vibratory levels are obtained. From what was said in Par. 2.2, starting from (11, the system response can be obtained in the frequency domain in terms of Spectral Power Density (PSD) such as Braccesi et al. (2016); Preumont (2013): G q ( ω ) = H q ( ω ) G u ( ω ) H q ( ω ) T (15) then through the modal shapes of the outputs (e.g. δ ): G δ ( r , ω ) = ϕ δ ( r ) G q ( ω ) ϕ δ ( r ) T (16) The problem that arises is how to convert a deterministic quantity, like the amplitude of a sine ( A spk ), into a PSD ( G spk ). The following equivalence, derived from literature Lalanne (2009b,c):

A spk 3 ∆ f

(17)

G spk =