Page 44 - MSAM-1-4

Materials Science in Additive Manufacturing Acoustic performances of SC lattices fabricated by DLP

Figure 3. Discretization of the air domain of the SC-Truss unit cell for the acoustic modeling using the MMC model.

By assuming the closed tubes of square cross-section cos kl cav jZ sin( kl )
as a set of resonant materials, the transfer matrix is given 0 0 0 cav
by [25] T j 1 sin( kl ) cos kl cav , (VIII)
c
Z 0 cav 0
1 Z 0
T r , (VI)
r
c and k
0 1 Where Z 0 0 0 c are the characteristic
0
0
Where Z is the characteristic impedance of the closed impedance and the wave number of the air layer,
r
tube. Based on the works by Maa, Morse and Ingard [21,22] respectively. Thereafter, the transfer matrix of one layer of
on microperforated panel absorbers, Z can be expressed SC-Truss unit cells is given by,
r
as follows: 1 Z
T T TT r
32 t k 2 t layerx r c r 0 1
tube
Z d tube 1 32 2 1 R j 0 tube cos 0 cav jZ sin( kl )
r
s

2
kl

cav
0
r
1 0 0 cav 1 Z . (IX)
0
1
j Z sin(kl ) cos kl
0 cav
1 1 d tube , 0 0
k 2 2 t (VII)
9 tube The sound absorption coefficient is a measure of the
2 proportion of sound energy absorbed and dissipated
Where η, ρ refer to the dynamic viscosity and density by a material when a sound wave is incident on it and
0
of air at 25°C respectively, w is the angular frequency of the is commonly used to quantify the sound absorption
sound waves, j is the imaginary unit such that j = −1, ε is capability of a material . An estimate of the sound
2
[28]
the perforation ratio of the square tubes given by absorption coefficient α TMM for each frequency f for the
d 2 d multi-layered material can be determined directly for T
tube , k tube is the perforate constant, by the following expression : total
[25]
D 2
1 2
and R 2 2 is the resistive end-correction factor T cT
00 21
s
0
11
cos

1
f
that accounts for the air friction arising from the oscillatory TMM cT , (X)
00 21
viscous flow on the surface of the perforated panel. δ and T cos
11
1
δ are the resistive and reactive correction factors to be
2
determined from experimentally determined sound Where θ is the angle of incidence of the sound wave.
absorption coefficient results from this work. For normal incidence, θ = 0° and hence cos θ = 1.
For the open cavity of the SC-Truss unit cell, the layer Based on experimental data, the optimal values of the
was simply modeled as a layer of air of density ρ and speed correction factors δ and δ were determined and their
1
0
2
of sound c , similar to the air outside the lattice cells. The relations to the geometric parameters of the SC-Truss unit
0
transfer matrix of the cavity is given by . cell were proposed. Thereafter, α for each case were
[25]
TMM
Volume 1 Issue 4 (2022) 5 https://doi.org/10.18063/msam.v1i4.22