

Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

54, 4, pp. 1297-1308, Warsaw 2016
DOI: 10.15632/jtam-pl.54.4.1297

BAND GAP PROPERTIES OF PERIODIC TAPERED BEAM STRUCTURE

USING TRAVELING WAVE METHOD

Tuanjie Li

School of Electromechanical Engineering, Xidian University, Xi’an, China

e-mail: tjli888@126.com

Xiaofei Ma

Xi’an Institute of Space Radio Technology, Xi’an, China

Qian Zhang, Zuowei Wang

School of Electromechanical Engineering, Xidian University, Xi’an, China

The wavemotion equations of a tapered beam with respect to axial, torsional and flexural
deformations are deduced including the transmission and waveguide equations. Combining
the force equilibrium and displacement coordination conditions at the junction, we obtain
the relationbetween thewavenumberand frequency, and the bandgapproperties of periodic
tapered beam structures by theBloch theorem.Themodeling accuracy and efficiency of the
traveling wave method are verified by the finite element method. The band gap properties
of periodic tampered and uniform beam structures are analyzed and compared for the same
materials and lengths as well as the same volumes.

Keywords: band gap, periodic structure, traveling wave method, Bloch theorem, tapered
beam

1. Introduction

In the field of space technology, themajority of structures are complex structures, such as truss
structures, frame structures and honeycomb sandwich plate structures. The basic elements of
these structures are rods, beams, and plates, etc. For the convenience of manufacturing, most
structures are usually built by these basic elements into periodic structures. The wave-bearing
properties of periodic structures are governed by their geometries. Wave motion in periodic
structures exhibits characteristic frequency intervals called pass and stop bands over which
wavemotion can or can not occur, respectively. Theremay be an opportunity to tailormaterials
to achieve desired band gap characteristics, such that wave propagation is prevented in specified
frequency ranges. Therefore, the research on the band gap properties of periodic structures is
of significance for vibration isolation design and control of the actual structures in practical
engineering.
The analysis of band gap properties of periodic structures dates back to the investigations

by Brillouin (1953). He deduced the relation between frequency and wavenumber to describe
band gap properties of periodic structures: electric filters and crystal lattices.While since 1960s,
the periodic structures consisting of beam-type elements have been attracting great attention in
the mechanics and engineering technologies. Many different methods were applied to study the
elasticwave propagation inbeam-typeperiodic structures, including lumpmassmethod inWang
et al. (2005), transfer matrix method in Li and Wang (2005) and Yu et al. (2012), plane wave
expansion method in Wang et al. (2007), finite element method in Denys (2009) and Liu and
Gao et al. (2007), boundary elementmethod in Li et al. (2013), spectral elementmethod inWen
et al. (2014) andWu et al. (2015) and reverberation-raymatrixmethod inGuo andFang (2013).


1298 T. Li et al.

Based on the Euler-Bernoulli theorem, Wen et al. (2005) calculated the band gap properties of
flexural waves of periodic binary straight beam structures by the plane wave expansionmethod
and the vibration attenuation spectra of a finite sample by the finite element method. A novel
vibration isolation structure is designed by using band gap properties of flexural waves. Doyle
(1989) proposed an analyticsl spectral element method to obtain accurate wave solutions in the
frequency domain based on the vibration equations.

The existing researches on band gap properties of beam-type periodic structuresmainly ha-
ve two aspects of shortcomings. One shortcoming is that the methods lack enough accuracy
or versatility. The spectral element method does not have exact wave solutions for all complex
structures. Therefore, it is generally used to address band gap properties of one-dimensional
periodic structures. The finite element method discretizes a continuous structure into a struc-
ture with limited degrees of freedom, which results in a discrete error. The displacement of each
member is described by the specified interpolation function. This simplification causes the inter-
polation error. Due to these errors, the finite element method has large errors for in dynamical
analysis of the structures. The other shortcoming is that all mentioned references are based on
uniform beams, rarely based on tapered beams. A tapered beam has changeable stiffness along
the axial direction due to variable sectional area. They can improve strength and reduce weight
for space applications. In addition, they have changeable flexural wavenumber varying with the
sectional area. This paper combines the exact traveling wave method with the Bloch theorem
to investigate the band gap properties of tapered periodic structures.

The layout of the paper is as follows. First, the traveling wave model of a tapered beam is
established. Then, based on the model and Bloch theorem, the relation between wavenumber
and frequency is deduced. Finally, simulations to verify the traveling wave model and analysis
of the band gap properties of periodic tapered and uniform beam structures are presented.

2. Traveling wave model of the tapered beam

Thevibration of structures can be considered as superposition of different frequencies andmodes
of elastic waves. TheFourier transformof eachmode of elastic waves is called awavemode. The
different wave modes are related to different frequencies and coupled by both the transmission
relation of amember and the scattering relation of a junction. The dynamical characteristics of
the overall structures can be described by assembling all wave modes of members.

2.1. Traveling wave model of a member

We define the coordinate systems for a member as shown in Fig. 1. x′y′z′ is the global
coordinate system, and xyz is the local coordinate system.

Fig. 1. Coordinate systems of a member

In Fig. 1, wl and wr are the left and right traveling wavemodes of themember, respectively.
In the local coordinate system, the right traveling waves propagate along the x-axis in the
positive direction. x1 and x2 are position coordinates of the endpoints of the tapered beam and


Band gap properties of periodic tapered beam structure... 1299

the subscripts 1 and 2 indicate the number of endpoints of the junction.N is the axial force, T is
the torsionwith respect to x-axis, andQ is the transverse force,F(ω) is the external stimulation.
The displacement and force of themember in the global coordinate system are expressed by

P=

{
U

F

}

=Mt

{
u

f

}

=MtY(x,ω)

{
wl
wr

}

(2.1)

where U and F are the displacement and force vectors in the global coordinate system, and u
and f are the displacement and force vectors in the local coordinate system, respectively. Mt is
the coordinate transformationmatrix between the global and local coordinate systems,Y(x,ω)
is the state transfer matrix and ω is the frequency of the external stimulation.
The wave modes of the endpoints of the tapered beam are related by a transmissionmatrix
{
wl
wr

}

|x2

= τ(x2,x1,ω)

{
wl
wr

}

|x1

(2.2)

where τ(x2,x1,ω) is the transmissionmatrix characterizing the variations of the amplitudes and
phases of each traveling wave. Thewaveguide and transmission equations (Eqs. (2.1) and (2.2))
are together called the traveling wave model of the member.
Based on the Euler-Bernoulli beam theory in Riedel and Kang (2006), the specific forms of

above equations of the tapered beam are derived in the following.

2.2. Waveguide equations of the tapered beam

Generally, thebeamstructure contains three types ofwavemodes: axialwavemode, torsional
wave mode and flexural wave mode. The thin and straight tapered beam shown in Fig. 1 is
assumed to be the ideal elastomer. Then, the length of the beam is l =x2−x1, and the sectional
area of a tapered beam A(x) is a function of x. This function is assumed to be A(x)= A0x

2/a2,
where a is the variation factor of the sectional area, A0 is the standard sectional area of the
referenced uniform beam in Riedel andKang (2006).

2.2.1. Axial waveguide equation

The force analysis of the tapered beamwith respect to stretching and compression deforma-
tions is shown in Fig. 2. The force equilibrium equation of an infinitesimal unit is

ρ
1

2
[A(x+dx)+A(x)] dx

∂2u

∂t2
=
(
N +

∂N

∂x
dx
)
−N = E∂A(x)u

∂x
dx (2.3)

where ρ is density, and u(x,t) is the axial displacement.

Fig. 2. Axial deformation of the tapered beam

The coefficient of the quadratic differential term of u in Eq. (2.3) must be simplified to
obtain analytical wave solutions. We adopt Taylor series expansion of A(x+dx) to obtain the
approximation relation


1300 T. Li et al.

[A(x+dx)+A(x)]dx ≈ 2A(x)dx+ ∂A(x)
∂A(x)

d2x (2.4)

By omitting the second-order terms of dx in Eq. (2.4), we can further simplify Eq. (2.3) as

ρA(x)dx
∂2u(x,t)

∂t2
= EA(x)

∂2u(x,t)

∂x2
dx+E

∂u(x,t)

∂x

dA(x)

dx
dx (2.5)

whereE is the elasticitymodulus.With separation of variablesu(x,t) = û(x,ω)ejωt, the equation
of motion of the axial wave is given by

∂2û

∂x2
+
2

x

∂û

∂x
+
ρω2

E
û =0 (2.6)

The general solution to Eq. (2.6) is

û(x,ω) = ul+ur =
cul
kαx
exp

[
j
(
kαx−

π

2

)]
+
cur
kαx
exp

[
−j
(
kαx−

π

2

)]
(2.7)

where ul and ur are the undetermined left and right axial traveling wave modes, respectively.
The axial wavenumber kα =

√
ρω2/E depends on the density, elasticity modulus and frequency

except the sectional area.
Based on Eq. (2.7), the waveguide equation is expressed by

{
û
N

}

=




1 1

−EA
x
+jkαEA −

EA

x
− jkαEA




{
ul
ur

}

(2.8)

The corresponding transmission equation of wave modes is
{
ul
ur

}

|x=x2

= diag
(x1
x2
ejkαl,

x1
x2
e−jkαl

){ul
ur

}

|x=x1

(2.9)

2.2.2. Torsional waveguide equation

Fig. 3. Torsional deformation of the tapered beam

The torsional motion of the tapered beam is shown in Fig. 3, where ψ(x,t) indicates the
angle with respect to the x-axis. The moment equilibrium equation of the infinitesimal unit is
given as

ρIp(x)dx
∂2ψ(x,t)

∂t2
= T(x+dx)−T(x)=

∂T(x,t)

∂x
dx

T =GIp
∂ψ

∂x

(2.10)

where IP is the polar moment of inertia of the section and G is the shear modulus.


Band gap properties of periodic tapered beam structure... 1301

Setting ψ(x,t) = φ(x,ω)ejωt, we can obtain the torsional motion equationas follows

∂2φ

∂x2
+
4

x

∂φ

∂x
+
ρω2

G
φ =0 (2.11)

In a similar way, the waveguide equation of the torsional wave is

{
φ
T

}

=






1+
j

ktx
1−

j

ktx(
−
3

x
−
3j

ktx2
+jkt

)
GIP

(
−
3

x
+
3j

ktx2
− jkt

)
GIP






{
φl
φr

}

(2.12)

whereφl andφr are the undetermined left and right torsional travelingwavemodes, respectively.
The torsional wavenumber kt =

√
ρω2/G is related to the density, shearmodulus and frequency,

and is independent of the sectional area.
The transmission equation of the torsional wave is described as

{
φl
φr

}

|x=x2

= diag

(
x21
x22
ejktl,

x21
x22
e−jktl

){
φl
φr

}

|x=x1

(2.13)

2.2.3. Flexural waveguide equation

The flexural stress analysis of the tapered beam is shown in Fig. 4, where wy(x,t) is the
y-axis deflection of the beam, Iz =

∫
Ay
2 dA indicates the moment of inertia of the section with

respect to the z-axis. Mz is the bendingmoment with respect to the z-axis and the distribution
force is q = ∂2Mz/∂x

2. The force equilibrium equation of the flexuralmotion of the infinitesimal
element in the plane xy is

∂2

∂x2

(
EIz

∂2wy(x,t)

∂x2

)
= ρA

∂2wy(x,t)

∂t2
(2.14)

Fig. 4. Flexural deformation of the tapered beam

With the expression of wy(x,t)= ŵy(x,ω)e
jωt, Eq. (2.14) yields thewave equation ofmotion

x2
∂4ŵy
∂x4
+8x

∂3ŵy
∂x3
+12x2

∂2ŵy
∂x2
−
4πa2ρω2

EA0
ŵy =0 (2.15)

The general solution to Eq. (2.15) is

ŵy = wy1+wy2+wy3+wy4 (2.16)

where


1302 T. Li et al.

wy1 =
cy1
5
√
x4
exp

[
j
(
2kzx− π4

)]
wy2 =

cy2
5
√
x4
exp

(
2kzx− jπ4

)

wy3 =
cy3
5
√
x4
exp

[
−j
(
2kzx− π4

)]
wy4 =

cy4
5
√
x4
exp

(
−2kzx− jπ4

)

Differently from the wavenumber of axial and torsional wave modes, the wavenumber of the
flexural wave mode kz =

4
√
ρAω2/(EIz) not only depends on the density, frequency, elasticity

modulus andmoment of inertia of area, but also on the sectional area. Therefore, thewavenum-
ber of flexural wave mode of the tapered beam varies with the sectional area. For convenient
calculation, the position variable of the section x is separated from kz, i.e. kz = k

′
z/
√
x, where

k′z =
4
√
4πa2ρω2/(EA0) is a constant.

Then, the waveguide equation of the flexural wave mode is





ŵy
ϕz
Vy
Mz





=






1 1 1 1
ay1+jay2 ay1+ay2 ay1− jay2 ay1−ay2
m31 m32 m33 m34
m41 m42 m43 m44











wy1
wy2
wy3
wy4





(2.17)

where m31 = EIz(ay3− jay4−ay5+ja3y2), m32 = EIz(ay3−ay4+ay5−a3y2), m33 = EIz(ay3+
jay4−ay5− ja3y2), m34 = EIz(ay3+ay4+ay5+a3y2), m41 = EIz[−(9ay1/(4x))−(3jay2/x)−a2y2],
m42 = EIz[−(9ay1/(4x)) − (3ay2/x) + a2y2], m43 = EIz[−(9ay1/(4x)) + (3jay2/x) − a2y2],
m44 = EIz[−(9ay1/(4x))+(3ay2/x)+a2y2], and ϕz = ∂ŵy/∂x is the bending angle with respect
to the z-axis and Vy = EIz∂

3ŵy/∂x
3 is the shear force with respect to the y-axis. Some coeffi-

cients in Eq. (2.17) are ay1 =−5/(4x), ay2 = k′z/
√
x, ay3 =585/(64x

3), ay4 =177k
′
z/(16

√
x5),

ay5 =21k
′
z
2
/(4x2). The transmission equation of the flexural wave modes is

{
wl
wr

}

|x=x2

= diag
(
ay6e

2jk′zay7,ay6e
2k′zay7,ay6e

−2jk′zay7,ay6e
−2k′zay7

){wl
wr

}

|x=x1

(2.18)

where wl = {wy1,wy2}T, wr = {wy3,wy4}T, ay6 = 4
√
x51/x

5
2, ay7 =

√
x2−
√
x1.

Due to flexural deformation of the beam with the circular cross-section, in the plane xy
and xz there is rotational symmetry with respect to x-axis. Similarly, ky = k

′
y/
√
x, where

k′y =
4
√
4πa2ρω2/(EA0) is a constant. Thewaveguide equation and the transmission equation of

wave modes in the plane xz are obtained just replacing Iz and kz by Iy and ky.

2.2.4. Spatial beam element

As the wave motion couples the axial, torsional and flexural wave modes, the corresponding
u, f,wl andwr in Eq. (2.1) can be defined by

u=
{
û ŵy ŵz φ ϕy ϕz

}T
f =

{
N Vy Vz T My Mz

}T

wl =
{
ul wy1 wz1 φl wy2 wz2

}T
wr =

{
ur wy3 wz3 φr wy4 wz4

}T (2.19)

For the sake of simplicity, the state transfer matrix of tapered beam is described as

Y =

[
Yul Yur
Yfl Yfr

]

(2.20)

where Yul, Yur, Yfl and Yfr can be obtained by Eqs. (2.8), (2.12) and (2.17).
The transmission matrix of the spatial tapered beam is obtained as

t(x2,x1,ω)=

[
t′(x2−x1) 0
0 t′(x1−x2)

]

(2.21)

where t′(x2−x1) and t′(x1−x2) can be founf from Eqs. (2.9), (2.13) and (2.18).


Band gap properties of periodic tapered beam structure... 1303

3. Band gap properties of periodic tapered beam structure

The periodic beam structure shown in Fig. 5 is rigidly composed of tapered beam units. Each
periodic unit has two tapered beams with different materials rigidly connected together at the
junctionH.URnB andF

R
nB indicate the output displacement and force of the n-th periodic unit.

The superscript R and L denote the right and left endpoints of the tapered beam, respectively.
According to the force equilibriumand displacement coordination at the junction,we obtain the
following relation of state vectors:
— beam A and beam B

F
R
nA+F

L
nB =0 U

R
nA =U

L
nB (3.1)

— periodic units n and n−1

F
R
(n−1)B +F

L
nA =0 U

R
(n−1)B =U

L
nA (3.2)

Fig. 5. Periodic tapered beam structure; (a) periodic tapered beam structure, (b) periodic unit

According to the connection relationship of periodic units in Fig. 5, PR
(n−1)B

=

{UR
(n−1)B

,FR
(n−1)B

}T is the input state vector of the n-th unit, PRnB = {URnB,FRnB}T is the
output state vector of the n-th periodic unit.With Eqs. (3.1) and (3.2), the relationship of state
vectors between two tapered beams of the periodic unit and between two periodic units are
given by

P
R
nA = diag(I,−I)PLnB PR(n−1)B = diag(I,−I)P

L
nA (3.3)

The state vectors Pi1ni2 = {U
i1
ni2
,Fi1ni2}

T and the wave modes vector Wi1nA = {wnl,wnr}T
are defined, where the superscript i1 indicates R or L and the subscript i2 indicates A or B.
According to waveguide equation Eq. (2.1) and transmission equation Eq. (2.2), we get

P
i1
ni2
= diag(I,−I)Vni2Y

i1
ni2
W
i1
ni2

W
R
ni2
= Tni2W

L
ni2

(3.4)

where I is a 3×3 identitymatrix andVn is the coordinate transformation of the taperedbeam i2
of the n-th periodic unit.
If m is the number of periods, the relation of the input and output state vectors of the

(n+m)-th periodic unit can be obtained based on Eqs. (3.1)-(3.4)

P
R
(n+m)B =A(n+m−1)BB(n+m−1)AP

R
(n+m−1)B

A(n+m−1)B =V(n+m−1)BY
R
(n+m−1)BT

L→R
(n+m−1)B(Y

L
(n+m−1)B)

−1(V(n+m−1)B)
−1

B(n+m−1)A =V(n+m−1)AY
R
(n+m−1)AT

R→L
(n+m−1)A(Y

L
(n+m−1)A)

−1(V(n+m−1)A)
−1

(3.5)

where the subscriptL → R indicates the coordinate transformationmatrix fromthe left endpoint
to the right endpoint, similarly R → L.


1304 T. Li et al.

The relation between the input state vector of the n-th and the output state vector of
(n+m)-th periodic unit is

P
R
(n+m)B =A(n+m−1)BB(n+m−1)AP

R
(n+m−1)B =

n+1∏

j=n+m

A(j−1)BB(j−1)AP
R
nB =CP

R
nB (3.6)

where

C=
n+1∏

j=n+m

A(j−1)BB(j−1)A

The undetermined state vectors of Eq. (3.6) are much larger than the number of equations.
Therefore, Eq. (3.6) is a multiple solutions problem. More equations are needed to obtain the
analytical solutions. The state vectors of different periodic units are related by the Bloch the-
orem.The relation describing the inputandoutput state vectors between then-th and (n+m)-th
periodic unit by the Bloch theorem is given as

P
R
(n+m)B = diag(e

jkmb
I,−ejkmbI)=DPRnB (3.7)

where k is the wavenumber of the periodic unit, b is the length of the periodic unit.
Combining Eq. (3.6) with Eq. (3.7), the following expression is obtained

(C−D)PRnB =0 (3.8)

If thematrix determinant det(C−D)= 0, we can get the relation between the wavenumber
and frequency, and the band gap properties.

4. Numerical simulations

Two numerical examples are applied to illustrate the proposed method of the tapered beam.
The first one is to analyze the vibration response of the tapered cantilever beam with flexural
deformations. The veracity and superiority of the proposed method are revealed by comparing
with the results of the finite element method. The second one is to compare the differences
between the band gap properties of the periodic tapered and uniform beam structure.

4.1. Dynamic response analysis

A transverse stimulation applied to the endpoint 1 of the cantilever tapered beam shown in
Fig. 1 is F(ω)= ejωt and the other endpoint 2 is fixed.Thematerial and geometrical parameters
are listed in Table 1.

Table 1.Material and geometrical parameters of the tapered beam

Elasticity Density Poisson’s Length Variation Standard
modulus E [Pa] ρ [kg/m3] ratio µ L [m] factor a area A0 [m

2]

2.0 ·109 7800 0.3 1 2 5.0 ·10−4

The frequency response of the transverse displacement of the free end is shown in Fig. 6.
The results of the traveling wave model are compared with those of the finite element method.
In Fig. 6, the results of the traveling wave method are displayed by the black solid line, and
the finite element results obtained by 20 elements and 5 elements permember are shown by the
grey dash dot line and black dash line, respectively.


Band gap properties of periodic tapered beam structure... 1305

Fig. 6. Frequency response of transverse displacement of the free end

As shown in Fig. 6, the results of the traveling wave model well coincide with the finite
element results obtained by 20 elements per member in the low frequency range, which verifies
the veracity of the proposed traveling wavemethod. In the frequency range of 0-1100rad/s, just
3 resonance peaks are found for finite element results obtained by 5 elements per member. As
the number of disperse elements increases, more resonance peaks appear, and the finite element
results are converge to the results of the traveling wave method.

4.2. Band gap analysis

Aperiodic unit of the periodic tapered beam structure is shown inFig. 5. The corresponding
material and geometrical parameters are listed in Table 2.

Table 2.Material and geometrical parameters of periodic unit

Items
Tapered beam Uniform beam
A B A B

Elastic modulus E [Pa] 8.43 ·109 1.10 ·1011 8.43 ·109 1.10 ·1011
Density ρ [kg/m3] 1210 2730 1210 2730

Poisson’s ratio µ [–] 0.3 0.3 0.3 0.3

Length L [m] 0.5 0.5 0.5 0.5

Variation factor a 2 2 – –

Standard area A0 [m
2] 1.59 ·10−2 1.59 ·10−2 – –

Sectional area A [m2] – – 6.28 ·10−3 6.28 ·10−3

Fig. 7. Band gap properties in three kinds of waves

The band gap properties of the periodic tapered beam structure are shown in Fig. 7. It
can be noted that no stop band is found for the axial wave and two stop bands are found
for the torsional wave: 0Hz-1353Hz and 1758Hz-3229Hz. The flexural wave has four stop


1306 T. Li et al.

bands: 152.9Hz-445.9Hz, 684.7Hz-2038Hz, 2484Hz-3908Hz and 4318Hz-6025Hz. The frequ-
ency range 0Hz-152.9Hz is the pass band for the axial wave and the flexural wave, but the
stop band for the torsional wave, which is called the partial stop band.Other partial stop bands
are 152.9Hz-445.9Hz, 684.7Hz-1758Hz, 1758Hz-2038Hz, 2484Hz-3229Hz, 3229Hz-3908Hzand
4318Hz-6025Hz. In the frequency range 152.9Hz-445.9Hz, only the axial wave contributes to
the wavenumber. Therefore, the material or structural parameters of the axial wave can be
changed to make it a stop band to prevent wave propagation.

The group velocity of waves is defined as v = dω/dk. The value of the group velocity
represents the speeds of thewave energy transmission. The group velocity curve of the torsional
wave is shown in Fig. 8. The group velocity is non-zero in the pass bands 1353Hz-1758Hz and
3229Hz-5003Hz. The group velocity equals to zero in the stop bands 0Hz-1353Hz and 1758Hz-
3229Hz,which indicates no energy propagation in the stop bands. It can be observed fromFig. 8
that the group velocity changes with the frequency in the pass band, and the fastest velocity of
the energy propagation is at the frequency 3904Hz. The velocities of the energy transmission in
high frequencies are much faster than that in low frequencies.

Fig. 8. Group velocity curve of the torsional wave

Thematerial and geometrical parameters of theperiodic uniformbeamstructure are listed in
Table 2.Thecomparisons ofbandgappropertiesbetween theperiodic taperedanduniformbeam
structures with respect to axial, torsional and flexural waves are respectively shown in Fig. 9.
The axial wave has three stop bands and the torsional wave has five stop bands. Similarly, the
flexural wave has six stop bands. Among these stop bands, no waves can propagate through
structure in the frequency ranges 414Hz-675.2Hz, 1143Hz-1341Hz and 3322Hz-3723Hz, which
are called the complete stop bands. The other stop bands are the partial stop bands.

As shown in Fig. 9a, the axial wave of the periodic tapered beam structure has no stop
bandswhile that of the periodic uniformbeam structure has three stop bands: 1032Hz-2430Hz,
2975Hz-4634Hz and 5357Hz-6010Hz. It can be inferred that the filter in certain frequency
ranges that are complete pass bands for the axial wave and complete stop bands for torsional
and flexural waves can be achieved by the periodic tapered beam structure with variational
sectional area. Furthermore, the curve slope of the periodic tapered beam structure representing
the velocity of energy propagation is bigger than that of the periodic uniform beam structure
at the frequency range 0Hz-2000Hz.

Referring to Fig. 9b, the torsional wave of the periodic tapered beam structure has two stop
bands and five stop bands are for the torsional waves of the periodic uniform beam structure:
640.1Hz-1506Hz, 1844Hz-2873Hz, 3322Hz-3726Hz, 4264Hz-4841Hz and 5236Hz-6293Hz.But
the stop bands of the periodic tapered beam structure covering a large frequency range are
relatively wider than those of the periodic uniform beam structure. It means that the bearing
capacity of the torsional force of the periodic tapered beam structure is much better than that


Band gap properties of periodic tapered beam structure... 1307

Fig. 9. Comparison of band gap properties of the: (a) axial wave, (b) torsional wave, (c) flexural wave

of the periodic uniformbeam structure. Furthermore, the first stop band of the periodic tapered
beam structure covers the whole low frequencies 0Hz-1353Hz. Therefore, the periodic tapered
beam structure is very suitable for vibration isolation of torsional motion.

Referring to Fig. 9c, the frequency range of stop bands of the flexural wave of the periodic
uniform beam structure is small. The widest frequency range is just 567Hz, which is much
smaller than that of the periodic tapered beam, 1707Hz. It means that the bearing capacity of
the flexural force of the periodic tapered beam structure ismuchbetter than that of the periodic
uniform beam structure. The high frequency ranges of stop bands of the periodic tapered beam
structure are as approximately the same as those in low frequencies due to the wavenumber
changing with the sectional area.

5. Conclusions

In this paper, the traveling wave model of the tapered beam is established including the trans-
mission and waveguide equations with respect to the axial, torsional and flexural deformations.
Combining these equations with the Bloch theorem, band gap properties of the periodic tape-
red beam and the periodic uniform beam are analyzed and compared. Some conclusions are
summarized as follows.

• Compared with the conventional finite element method, the traveling wave method has
higher precision in mid and high frequencies as well as shorter calculation time and less
memory occupation. This is because the traveling wave method uses the continuous mo-
del to exactly describe the transmission relationship of the member as well as the force
equilibrium and displacement coordination conditions of the junction.

• Compared with the uniform beam, the wavenumbers of axial and torsional waves of the
tapered beam cannot change with variable sectional area, while the wavenumbers of the
flexural wave decrease with an increase in the sectional area.

• The stop bands of the periodic tapered beamstructure arewider than those of the periodic
uniform beam structure. The periodic tapered beam structure has more advantages over


1308 T. Li et al.

the periodic uniform beam structure to achieve vibration isolation and filtering in some
frequency ranges.

Acknowledgment

This project is supported by National Natural Science Foundation of China (Grant No. 51375360,

U1537213).

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Manuscript received August 24, 2015; accepted for print March 7, 2016