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Optimum core design to improve noise attenuation performance and stiffness of sandwich panels used for high-speed railway vehicles

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Abstract

In this study, we optimally design the core of a metal sandwich panel used in high-speed railway vehicles to minimize the amount of metal solid in the core and subsequently reduce its weight. The optimum core must satisfy constraints regarding sound transmission class (STC) and compliance. Because the solid-void layout in the core strongly affects the acoustic and static characteristics of sandwich panels, the core layout should be carefully designed when reducing the amount of metal solid. To this end, three topology-optimization problems and one size-optimization problem are formulated and sequentially solved. A single unit cell is periodically repeated in the core. Thus, structural and acoustic topology-optimization problems are first formulated and solved for the unit cell. Based on the optimal topologies obtained for the unit cell, a moderate initial solid-void layout in the unit cell is determined for the size-optimization problem to optimally design core. The effectiveness of the current design approach for the core of the sandwich panel is validated by comparing the STC and compliance values of the optimal core design and a reference design.

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Abbreviations

\( \underset{n=1}{\overset{N}{\mathbf{A}}} \) :

finite element assembly operator

B :

matrix relating the element strain vector to its nodal displacement vector

C:

mean compliance

C ini :

initial mean compliance

c :

speed of sound in air

D :

constitutive matrix

d 1 :

thickness of thin flat plates at both ends in a sandwich panel

d 2 :

thickness of the bar inside a sandwich panel

dis(r,m):

distance between the centers of the r th and m th elements

E r (χ r ):

Young’s modulus of the r th element

F :

uniform force applied to the left-end panel in static analysis

\( {\overrightarrow{\mathbf{F}}}^s \) :

equivalent global force vector in static analysis

\( {\overset{\rightharpoonup }{\mathbf{F}}}^a \) :

nodal vector of the applied equivalent force in acoustical analysis

f :

frequency

\( {\overrightarrow{\mathbf{f}}}^a \) :

element force vector

f t :

target frequency

Ĥ m :

convolution operator in a filter used in topology optimization

I :

total number of design variables

K a :

global stiffness matrix in acoustical analysis

K s :

global stiffness matrix in static analysis

k a n :

element stiffness matrix in acoustical analysis

k s n :

element stiffness matrix in static analysis

L i :

objective function (i = 1, ⋯, 4)

l i :

length of each part in analysis models (i = 0, 1, ⋯, 5)

M a :

global mass matrix in acoustical analysis

m a n :

element mass matrix in acoustical analysis

N :

number of total finite elements

N a n :

shape function matrix of the n th element

N in :

number of elements associated with the left-end boundary in acoustical analysis

\( \overrightarrow{\mathbf{P}} \) :

nodal vector of the acoustic pressure

p i :

acoustic pressures at three points (i = 1,2,3)

q :

penalization parameter

R :

number of total design variables

S upp :

upper limit value of the total sum of the weighted design variables

s i :

design variable used to determine the core metal layout (i = 1, ⋯, 10)

STC:

sound transmission class

\( TL{\left(\overrightarrow{\chi}\right)}_{f={f}_t} \) :

TL value at the target frequency f t

TL ini :

initial value of TL

TL low :

allowable lower-limit value of TL

\( \overrightarrow{\mathbf{U}} \) :

global nodal displacement vector in static analysis

V a :

volume ratio of the solid region and the design domain

V b :

core metal volume

V ini b :

initial value of the core metal volume

V f :

volume fraction (the ratio of the allowable solid volume and the design domain)

w i :

weighting factor of each design variable

x 12 :

distance between two points in the non-design domain

α :

compliance fraction

β :

allowable volume fraction of an optimized shape and an initial shape

γ :

strain

K :

bulk modulus

π i :

acoustic power of an incident acoustic wave

π t :

acoustic power of a transmitted acoustic wave

ρ :

density

τ :

stress

\( \overrightarrow{\chi} \) :

design variable vector

χ r :

rth design variable

Ω n :

element area of the nth element in acoustical analysis

∂Ω in n :

left-end boundary in acoustical analysis model

∂Ω out n :

right-end boundary in acoustical analysis model

ω :

angular frequency

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Acknowledgments

This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No. 2013R1A1A2010158), Science, and Technology (No. 2010–0005299). The authors would like to thank Professor Seockhyun Kim at the Kangwon University in the Republic of Korea for the extremely valuable discussion regarding the core structure in sandwich panels for high-speed railway vehicles.

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  1. Division of Mechanical Engineering, Ajou University, 206 World cup-ro, Yeongtong-Gu, Suwon, 443-749, Republic of Korea

    Hong Geun Yoon & Jin Woo Lee

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  1. Hong Geun Yoon
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Correspondence to Jin Woo Lee.

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Yoon, H.G., Lee, J.W. Optimum core design to improve noise attenuation performance and stiffness of sandwich panels used for high-speed railway vehicles. Struct Multidisc Optim 55, 723–738 (2017). https://doi.org/10.1007/s00158-016-1502-6

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  • DOI: https://doi.org/10.1007/s00158-016-1502-6

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