Summary We study numerically the frequency band-gap phenomenon for bending waves in an infinite periodic beam. The outcome of the analysis is then subjected to an optimization problem in order to maximize these band-gaps. The band-gap maximization may be performed with respect to material parameters and cross-sectional geometry

Ole Sigmund

Søren Halkjaer

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XXI ICTAM, 15–21 August 2004, Warsaw, Poland
OPTIMIZATION OF BEAM PROPERTIES WITH RESPECT TO MAXIMUM BAND-GAP
Søren Halkjær, Ole Sigmund
Department of Mechanical Engineering, Section for Solid Mechanics
Technical University of Denmark
Nils Koppels Allé, Building 404, 2800 Kgs. Lyngby, Denmark
Summary We study numerically the frequency band-gap phenomenon for bending waves in an infinite periodic beam. The outcome
of the analysis is then subjected to an optimization problem in order to maximize these band-gaps. The band-gap maximization may be
performed with respect to material parameters and cross-sectional geometry.
INTRODUCTION
The study of frequency band-gaps in certain material structures subjected to wave propagation has received a lot of at-
tention during the last years. Such materials hinder wave propagation with frequencies in certain intervals (band-gaps).
The phenomenon occurs for both acoustic, elastic and electromagnetic waves. Examples among numerous work include
[1] which analyzes band-gaps for electromagnetic waves in infinite 2D structures, [2] carries out both an analysis and
optimization for the elastic 2D structural problem (infinite and finite), [3] considers infinite periodic plates and cylindrical
shells with and without loading. Such studies are relevant in structural design regarding e. g. vibration damping and filters.
In this article, we first perform a numerical analysis of frequency band-gaps for time-harmonic bending waves in an
infinite periodic beam. We then formulate an optimization problem for maximizing these band-gaps.
THEORY
Analysis
Bending waves in an infinite periodic beam has been analyzed theoretically using Floquet theory in [4]. Here we adopt
the finite element method based on Floquet theory described in [1] to derive the eigenvalue problem for studying the
frequency band-gaps.
Bending waves in a beam are described by the fourth order beam equation. We restrict our attention to waves with a
time-harmonic behavioreiωt. The differential equation describing the transverse displacementw(x) then becomes
d2
dx2
(E(x)I(x)
d2w(x)
dx2
) = ω2ρ(x)A(x)w(x) (1)
whereE, I, ρ andA are Young’s modulus, moment of inertia, mass density and cross-sectional area respectively. Here
we consider a periodically segmented beam of infinite length as shown in Fig. 1. The beam consists of an infinite number
of identical base cells of lengthd. Each cell is made up of two different materials with possibly varying cross-section as
indicated. Thus,E(x) = E(x + d) and analogously forρ, I, andA. In order to study the problem, we assume that the
d−l
Mat 1 Mat 2
Base cell
l
d
Figure 1. Periodic infinite beam with identical base cells consisting of two materials (Mat 1 and Mat 2) with possibly varying cross-
sections.
transverse displacement can be written in terms of Bloch functions
w(x) = eikxu(x) (2)
wherek is the wave number andu(x) = u(x + d). Inserting this assumption and integrating over the length of the base
cell we obtain
−
∫ d
0
E(x)I(x)(
d
dx
+ ik)2u(x)(
d
dx
+ ik)2v(x) dx = ω2
∫ d
0
ρ(x)A(x)u(x)v(x) dx
wherev(x) is an arbitrary periodic function on the base cell. We use a standard shape function assumption based on a
third order polynomialφi(x) on each element, such thatu(x) =
∑
i uiφi(x). We then arrive at the following complex
eigenvalue problem
K(k)u = λMu, λ = ω2. (3)
Mechanics of 21st Century - ICTAM04 Proceedings
where
Kij = −
∫ d
0
E(x)I(x)(
d
dx
+ ik)2φi(x)(
d
dx
+ ik)2φj(x) dx, Mij =
∫ d
0
ρ(x)A(x)φi(x)φj(x) dx
The solution to this problem results in the dispersion relation
ωj = ωj(k), j = 1, . . . , d.o.f (4)
whereωi denotes thei’th frequency band and d.o.f. denotes the number of independent variables.
Optimization
The optimization problem for the band-gaps can now be written as a max-min objective using the dispersion relation [2]
max
ρ,E,A
: c(ρ,E, A) =
∆ω2(ρ,E,A)
ω20(ρ,E,A)
= 2
mink ω2j+1(k, ρ, E, A)−maxk ω2j (k, ρ, E, A)
mink ω2j+1(k, ρ, E, A) + maxk ω
2
j (k, ρ, E, A)
(5)
RESULTS
Here we present results for an infinite beam with constant cross-section areaA = 7.74× 10−4 m2, I = 4.16× 10−8 m4,
d = 0.254 m andl = d/2 = 0.127 m. Dispersion plots including the first five frequency bands are shown in Fig. 2 for two
material combinations shown in Table 1 (row 1 and 2 respectively). In both cases Mat 1 is Zinc, while Mat 2 is chosen to
Mat 1 E1 (GPa) ρ1 (kg/m3) Mat 2 E2 (GPa) ρ2 (kg/m3)
Zinc 98 7130 Silicon 100 2330
Zinc 98 7130 Chromium 25 7200
Table 1. Material parameters (rows) for two different base cells .
obtain differences in eitherE or ρ. It is seen that both material combinations result in relative band-gaps approximately
the same size but at different frequencies. This is to be studied more thoroughly using (5).
0 0.5 1 1.5 2 2.5 3
0
20
40
60
80
100
120
140
k
ω
 [k
H
z]
ω
5
ω
4
ω
3
ω
2
ω
1
0 0.5 1 1.5 2 2.5 3
0
10
20
30
40
50
60
70
80
k
ω
 [k
H
z]
ω
5
ω
4
ω
3
ω
2
ω
1
Figure 2. Dispersion plots including the first five frequency bands showing the band-gaps for two base cells with different material
combinations. Left: Base cell consisting of Zinc-Silicon. Right: Base cell consisting of Zinc-Chromium.
ACKNOWLEDGMENTS
This work was supported by Denmark’s Technical Research Council (the "Phonon project").
References
[1] Axmann W., Kuchment P.: An Efficient Finite Element Method for Computing Spectra of Photonic and Acoustic Band-Gap Materials.J. Comp.
Physics150:468–481, 1999.
[2] Sigmund O., Jensen J. S.: Systematic design for phononic band-gap materials and structures by topology optimization.Ph l. Tra s. R. Soc. Lond.
A 361:1001–1019, 2003.
[3] Sorokin S. V., Ershova O. A.: Frequency band gaps in periodic plates and cylindrical shells. To appear inJournal of Sound and Vibration.
[4] Lee S. L., Ke H. Y, Kao M. J.: Flexural Waves in a Periodic Beam.Journal of Applied Mechanics57:779–783, 1990.
Mechanics of 21st Century - ICTAM04 Proceedings
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OPTIMIZATION OF BEAM PROPERTIES WITH RESPECT TO MAXIMUM BAND-GAP

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OPTIMIZATION OF BEAM PROPERTIES WITH RESPECT TO MAXIMUM BAND-GAP

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