PSI - Issue 54

Alessandro Zanarini et al. / Procedia Structural Integrity 54 (2024) 107–114 A. Zanarini / Structural Integrity Procedia 00 (2023) 000–000

109

3

F f ( ω ), of the receptance FRFs H d n q f ( ω ) of size N q × N f - being N q the number of the outputs and N f of the inputs - andof the excitation signatures F f ( ω ). Eq.1 can be therefore rewritten in terms of a sum of discrete contributions, by means of a discretisation of the vibrating surface domain S ≈ q ∆ S q that scatters the sound pressure: p ( a a ,ω ) ≈− 2 ω 2 ρ 0 N q q H d n qf ( ω ) F f ( ω ) G aq ( r aq ,ω ) ∆ S q ∈ C , (2) with H d n q f ( ω ), F f ( ω ) and G aq ( r aq ,ω ) as complex-valued discrete quantities, r aq = r aq = a a − q q . Being G aq ( r aq ,ω ) and ∆ S q function of the locations of the N a discrete points in the acoustic domain and of the N q points on the structure, they can be grouped in a complex-valued collocation matrix T aq ( ω ), sized N a × N q , of element T aq ( ω ) = − 2 ω 2 ρ 0 G aq ( r aq ,ω ) ∆ S q , to transform Eq.2 into:

p ( a a ,ω ) ≈ T a q ( ω ) H d n q f ( ω ) F f ( ω ) ∈ C .

(3)

If, similarly to the acoustic transfer vectors in Ge´rard et al. (2002); Citarella et al. (2007), an acoustic transfer matrix V af ( ω ), sized N a × N f , is defined as:

V af ( ω ) = T aq ( ω ) · H d n qf ( ω ) ∈ C ,

(4)

Eq.3 can be easily rewritten as:

p ( a a ,ω ) ≈ V af ( ω ) F f ( ω ) ∈ C ,

(5)

useful in the cases where the structural response and acoustic domains are kept unchanged, while varying only the excitation signature to map the responses on the acoustic pressure field.

2.1. Notes: indirect excitation force retrieval from sound pressure fields

By reversing Eq.5, with the use of the pseudo-inverse of the acoustic transfer matrix V af ( ω ) of Eq.4, the forces induced on the structure at the excitation / shaker head by a known complex-valued pressure field can be retrieved:

ˆ F f ( ω ) ≈ V

+ fa ( ω ) p ( a ,ω ) ∈ C .

(6)

with the pseudo-inverse of the acoustic transfer matrix V af ( ω ), sized N f × N a and callable V +

fa ( ω ), precisely as:

H fa ( ω ) V af ( ω )] −

1 V H

V +

fa ( ω ) = [ V

fa ( ω ) ∈ C .

(7)

The matrix V H

fa ( ω ) V af ( ω ), to be inverted at each angular frequency ω , is a complex-valued square matrix of size

N f × N f , but this time N f is very small or unity, simplifying the inversion.