Numerical modelling of impact noise. Cristina D´ıaz-Cereceda, Jordi Poblet-Puig
and. Antonio Rodr´ıguez-Ferran. July 2009. Laboratori de C`alcul Num`eric ...
Numerical modelling of impact noise Cristina D´ıaz-Cereceda, Jordi Poblet-Puig and Antonio Rodr´ıguez-Ferran July 2009 Laboratori de C` alcul Num` eric (LaC` aN) Universitat Polit` ecnica de Catalunya (Spain) http://www-lacan.upc.es
July, 2009 - 1
Outline Motivation Approach to the problem Modelling the impact force Computing the displacement field and the radiated power Obtaining the outputs of interest
Developed models Infinite plates Modal analysis for finite plates Coupled plates
Conclusions July, 2009 - 2
Outline Motivation Approach to the problem Modelling the impact force Computing the displacement field and the radiated power Obtaining the outputs of interest
Developed models Infinite plates Modal analysis for finite plates Coupled plates
Conclusions July, 2009 - 3
Motivation Impact noise: noise level in rooms due to the impact of an object hitting the building structure.
Situation: Current trend of the regulations is restrictive (i.e. C´odigo T´ecnico de la Edificaci´on). 88 dB (NBE-CA-88) −→ 65 dB (April 2009) July, 2009 - 4
Outline Motivation Approach to the problem Modelling the impact force Computing the displacement field and the radiated power Obtaining the outputs of interest
Developed models Infinite plates Modal analysis for finite plates Coupled plates
Conclusions July, 2009 - 5
Approach to the problem Three main methods: Energy average thinking (SEA) Simplified methods Solving vibroacustic equations by means of numerical methods Modelling the impact force Computing the displacement field Obtaining the radiated power Computing the outputs of interest
July, 2009 - 6
Modelling the impact force Excitation: point force (tapping machine). Modelling: influence of the floor characteristics. Formulation: proposed by Brunskog.
July, 2009 - 7
Computing the displacement field and the radiated power Displacement field: governing equation D∆2 uˆ − ω 2 ρs uˆ = q where uˆ = uˆ(x, ω).
Radiated power: generic expression Z
Πrad
1 = I · ds = Re 2 s
Z
∗
pˆ v ds s
where vˆ = jωˆ u. July, 2009 - 8
Obtaining the outputs of interest Normalised impact noise pressure level (Ln ) ΠRad 4ρc Ln = 10 log p2 A0 dB. ref Adjusted normalised impact noise pressure level (Ln,w ).
July, 2009 - 9
Outline Motivation Approach to the problem Modelling the impact force Computing the displacement field and the radiated power Obtaining the outputs of interest
Developed models Infinite plates Modal analysis for finite plates Coupled plates
Conclusions July, 2009 - 10
Models for a single plate Infinite plate radiating into an infinite half-space Widely used. Detailed in the bibliography. Implies spatial Fourier transformation of the displacement. u˜ =
Πrad
F0 e i(αx0 +β0 )
D (α2 + β 2 )2 − ρs ω 2 Z Z ω 2 |˜ u (α, β)|2 kρc p = dα dβ 8π 2 k 2 − α2 − β 2 α2 +β 2 ≤k 2
July, 2009 - 11
Models for a single plate Finite plate simply supported along its edges uˆ(x, y ) =
X
apq Ψpq (x, y )
The eigenfunctions of the plate can be found anallytically 2
∆ Ψpq −
4 kpq Ψpq
=0
⇒
Ψpq = sin
pπ qπ x sin y Lx Ly
Modal contributions are uncoupled apq
4 F0 Ψpq (x0 , y0 ) = ; 4 − ω2 ρ L L D kpq s x y
Πrad
ωρ = 4π
Z Z
vˆ (x 0 , y 0 )ˆ v ∗ (x, y )
sin(kr ) dΩ dΩ0 r July, 2009 - 12
Comparison with experimental measurements
July, 2009 - 13
Parametric analysis Thickness
h (m) 0.05 0.1 0.2 0.4
Ln,w (dB) 88 80 72 62
July, 2009 - 14
Parametric analysis Loss factor
η 0% 1.5 % 3% 4.5 %
Ln,w (dB) 97 80 77 75
July, 2009 - 15
Models for multiple plates Linking: elastic joint M = kθ (θ1 − θ2 ). The same basis of functions is used. The weak form is developed: the bending moments are imposed. "Z Z y =Ly Z y =Ly ω 2 ρs 1 4 klm − u ˆ v dΩ+ Mx ∇n v dy + Mx ∇n v dy + D D Ω y =0 y =0 x=0 x=Lx # Z x=Lx Z x=Lx Z 1 + My ∇n v dx + My ∇n v dx = q v dΩ D Ω x=0 x=0 y =0 y =Ly
July, 2009 - 16
Models for multiple plates Coupled system: D1 + kθ C kθ E a f1 = kθ G D2 + kθ H b 0
a = [apq ] b = [bpq ]
Cases of interest: T-shaped structure Four plates
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Two plates Dependence on kθ
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Four plates Dependence on kθ
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T-shaped structure Dependence on kθ floor-wall. Continuous floor.
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T-shaped structure Dependence on kθ floor-wall. Non-continuous floor.
July, 2009 - 21
Outline Motivation Approach to the problem Modelling the impact force Computing the displacement field and the radiated power Obtaining the outputs of interest
Developed models Infinite plates Modal analysis for finite plates Coupled plates
Conclusions July, 2009 - 22
Conclusions Both the infinite plate model and the modal analysis model are good aproximations to the real behaviour. The thickness and loss factor of a floor are important parameters for the impact noise level. Elastic joints through the floor lower the impact noise in adjacent rooms. Floor-wall flanking transmission becomes important when floor-floor transmission is low.
July, 2009 - 23
Conclusions Both the infinite plate model and the modal analysis model are good aproximations to the real behaviour. The thickness and loss factor of a floor are important parameters for the impact noise level. Elastic joints through the floor lower the impact noise in adjacent rooms. Floor-wall flanking transmission becomes important when floor-floor transmission is low.
July, 2009 - 24
Conclusions Both the infinite plate model and the modal analysis model are good aproximations to the real behaviour. The thickness and loss factor of a floor are important parameters for the impact noise level. Elastic joints through the floor lower the impact noise in adjacent rooms. Floor-wall flanking transmission becomes important when floor-floor transmission is low.
July, 2009 - 25
Conclusions Both the infinite plate model and the modal analysis model are good aproximations to the real behaviour. The thickness and loss factor of a floor are important parameters for the impact noise level. Elastic joints through the floor lower the impact noise in adjacent rooms. Floor-wall flanking transmission becomes important when floor-floor transmission is low.
July, 2009 - 26
