Arch Appl Mech (2007) 77: 461–471DOI 10.1007/s00419-006-0106-9

ORIGINAL

Wang Yong · Huang Qibai · Zhou Minggang · Xu Zhisheng

Transfer matrix approach of vibration isolationanalysis of periodic composite structure

Received: 23 July 2006 / Accepted: 3 November 2006 / Published online: 16 January 2007© Springer-Verlag 2007

Abstract The transmission properties of elastic waves propagating in a three-dimensional composite struc-ture embedded periodically with spherical inclusions are analyzed by the transfer matrix method in this paper.Firstly, the periodic composite structures are divided into many layers, the transfer matrix of monolayer struc-ture is deduced by the wave equations, and the transfer matrix of the entire structure is obtained in the case ofboundary conditions of displacement and stress continuity between layers. Then, the effective impedance ofthe structure is analyzed to calculate its reflectivity and transmissivity of vibration isolation. Finally, numericalsimulation is carried out; the experiment results validate the accuracy and feasibility of the method adopted inthe paper and some useful conclusions are obtained.

Keywords Periodic composite structure · Vibration isolation · Transfer matrix method · Feasibility

1 Introduction

Suppressing harmful vibrations is one of the most urgent problems needing to be solved in engineering. Vibra-tion isolation is one of the most extensive applications in vibration control. Periodic composite structures havea good application prospect because of their good effect of vibration isolation within some frequency range.Therefore, the research on elastic wave transmission properties in periodic composite structures has attainedmore and more attention. Chen et al. [1] presented a theoretical analysis of acoustic stop bands in two-dimen-sional scattering arrays. A self-consistent wave scattering theory, incorporating all orders of multiple scattering,is used to obtain the wave transmission. Psarobas et al. [2] developed a formula for the calculation of the fre-quency band structure of a phononic crystal consisting of non-overlapping elastic spheres, characterized byLame coefficients which may be complex and frequency dependent, arranged periodically in a host mediumwith different mass density and Lame coefficients. Esquivel-Sirvent and Cocoletzi [3] deduced reflectivityand dispersion relation of elastic wave propagating in periodic layered composite structures. Achenbach et al.analyzed the band gap of an elastic wave in periodic medium with all kinds of cave [4–8].

Transmission properties of the three-dimensional composite structure with scatterers arranged periodicallyin a host medium with different mass density and Lame coefficients are analyzed by transfer matrix approachin this paper. Firstly, the periodic composite structures are divided into many layers, the transfer matrix ofmonolayer structure is deduced by the wave equations, and the transfer matrix of the entire structure is obtainedin the case of boundary conditions of displacement and stress continuity between layers. Then, the effectiveimpedance of the structure is analyzed to calculate its reflectivity and transmissivity of vibration isolation.Finally, numerical simulation is carried out and some useful conclusions are obtained in the paper.

Project (No. 50075029) supported by the National Natural Science Foundation of China.

W. Yong (B) · H. Qibai · Z. Minggang · X. ZhishengSchool of Mechanical Science and Engineering, Huazhong University of Science and Technology,Wuhan, Hubei 430074, People’s Republic of ChinaE-mail: wangyongcq@hotmail.com

462 W. Yong et al.

a

b

d

B

A

........... .

r-1 r r+1

Fig. 1 Schematic plot of periodic composite structure

2 Mathematical model

2.1 Wave equation

Schematic plot of periodic composite structure studied in the paper is shown in Fig. 1. The direction of coor-dinate axis x1 and x2 have been given in the figure, and that of x3 is perpendicular to paper plane upwardly.In the figure, the composite structure is periodically divided into finite monolayer ones. A is host materialand B is scatterers arranged periodically in a host medium with different mass density and Lame coefficients.The diameter of B is d . Space between scatterers, a, is lattice constant of direction x1, b is lattice constant ofdirection x2, and h is lattice constant of direction x3. The effect of vibration isolation of periodical compositestructure only along axis x1 will be discussed in the paper. We suppose:

1. Incident wave is plane harmonic wave, and is perpendicular to periodical composite structure along axisx1.

2. Host A and spherical scatterers B are composed of uniform, isotropic and continuous material.

If coordinate origin is defined in the first spherical center of left lower corner scatterer, the coordinatesof any scatterer B in axis x1x2x3 can be written as (ma, nb, lh), where m, n, l = 0,±1,±2, . . . When theincident wave is uI (x)e−iωt , the corresponding displacement field is u(x)e−iωt , and they satisfy the waveequation as follows [9,10]

L̂i j u(x) = µui, j j + (λ+ µ)u j, j i + ρω2ui = 0 (1)

where, L̂i j is positive definite Hermitian operator, λandµ are Lame constants of the materials, ρ is the density,iand j are 1 and 2, which denote coordinate axis x1andx2, and j j, j i are two order derivative of correspondingcoordinate.

The force acting on scatterer surface is zero, i.e.,

ti (x) = λu j, j ni + µ(ui, j + u j,i )n j = 0, x ∈ Smnl (2)

where, ni , n j are unit normal vector, Smnl is the surface of the sphere, which is formulated as

(x1 − ma)2 + (x2 − nb)2 + (x3 − lh)2 =(

d

2

)2

,−∞ < m, n, l < ∞ (3)

Two presumptions are satisfied at the boundary between layers of periodic composite structure: (1) dis-placement u1 must be continuous, (2) stress tensor T1 must be continuous too.

Transfer matrix approach of vibration isolation analysis of periodic composite structure 463

2.2 Wave solution of monolayer

When plane harmonic wave uI (x)e−iωt acts vertically upon periodic composite structure shown in Fig. 1,reflected wave and forward wave will appear.

The effect of vibration isolation of periodical composite structure only along axis x1 will be discussed inthe paper. So,

uI1(x) = eikL x1, uI

2(x) = uI3(x) = 0 (4)

where, kL is wave number of elastic longitudinal wave, kL = ωcL

, and cL is wave velocity of elastic longitudinal

wave, cL =(λ+2µρ

) 12.

For convenience, time terms in Eq. (4) are ignored because all the displacement and stress fields includethe time term, such as e−iωt .

The displacement field solution of wave Eq. (1) in r th layer structure can be expressed as

ur (x) = uIr (x)+ uS

r (x) (5)

where, uIr (x) indicates an incident longitudinal wave field, and uS

r (x) denotes scattered wave field.Substituting Eq. (4) into Eq. (5), we get

ur (x) = eikLr x1 i1 + uSr (x) (6)

where, i1 is unit vector along axis x1, and kLr is wave number of r th layer structure, kLr = ωcLr

It can be obtained by periodic arrangement of spherical scatterers about axis x2 and x3 and zero appliedforce of the scatterers’ surface

uSr (x1, x2 + b, x3 + h) = uS

r (x1, x2, x3) (7)

The periodic form of uSr (x) can be represented by an exponential Fourier series in both the axis x2 and x3

coordinates. To write down a general form of this series representation, it is convenient to employ the usualdisplacement decomposition of uS

r (x) in terms of a scalar potential φ(x) and a vector potential ψ(x) [4]

uSr (x) = φ,i + ei jkψk, j (8)

where φ(x) and ψk(x) satisfy reduced wave equations with longitudinal wave number kL and transverse wavenumber kT , respectively. We can write

φ±(x) =∑

p

∑q

φpq± ei(±γ L

pq x1+αp x2+βq x3) (9)

ψ±k (x) =

∑p

∑q

ψpqk±ei(±γ T

pq x1+αp x2+βq x3) (10)

here, superscript ‘+’ denote x1 > 0, and subscript ‘−’ x1 < 0.

αp = 2pπ

b, βq = 2qπ

h(11)

γ βpq =[k2β − (

αp)2 − (

βq)2

] 12, β = L , T (12)

kT is transverse wave number, kT = ωcT

;

cT is transverse wave velocity, cT =(µρ

) 12;

and Im γβpq ≥ 0.

The terms in Eqs. (9) and (10) represent wave modes. The exponentials with +γ βpq x1 denote transmitted

waves, while the ones with −γ βpq x1 denote reflected waves. For p = q ≡ 0, γ Lpq and γ T

pq are real valued, andhence the corresponding modes in Eqs. (9) and (10) represent propagating waves.

For small damped structures, wave number is defined as: kβd = kβ(1 − j ηs

2

), β = L , T

where, ηs denotes structural dissipation factor of the material.

464 W. Yong et al.

2.3 Transfer matrix of monolayer

Transfer matrix of r th layer structure shown in Fig. 1 is deduced as follows.According to Eqs. (8)–(10), displacement field equations (6) of r th structure can be written as

ur (x1) = A+eikLr x1 + A−e−ikLr x1 (13)

where, A+ is reflected wave coefficient, and A− is transmitted wave coefficient.Stress component is defined as

Tr (x1) = ρr c2Lr∂ur (x1)

∂x1(14)

Transfer matrices of Eqs. (13) and (14) can be written as[ur (x1)Tr (x1)

]=

[1 1

i Zr −i Zr

][A+eikLr x1

A−e−ikLr x1

](15)

where, Zr = ρr c2Lr kLr .

Suppose Ar =[

1 1i Zr −i Zr

], Eq. (15) will be changed as

[ur (x1)Tr (x1)

]= Ar

[A+eikLr x1

A−e−ikLr x1

](16)

The displacement and stress of left boundary x Lr , and right boundary x R

r = x Lr + a of r th layer structure

are linearly dependent [ur (x1)Tr (x1)

]x R

r

= Mr

[ur (x1)Tr (x1)

]x L

r

(17)

here,

Mr = Ar Tr (a)A−1r (18)

Tr (a) =[

eikLr a 00 e−ikLr a

](19)

So, Mr can be calculated by Ar and Tr (a):

Mr =[

1 1i Zr −i Zr

][eikLr a 0

0 e−ikLr a

][1 1

i Zr −i Zr

]−1

=[

cos(kLr a) sin(kLr a)Zr−Zr sin(kLr a) cos(kLr a)

](20)

Equation (20) is transfer matrix formula of r th layer structure.

2.4 Transfer matrix of periodic composite structure

Transfer matrix formula, Mr , of r th layer structure is deduced above, which is applicable to any layer structure.Transfer matrix formula of periodic composite structure will be deduced as follows.

The displacement and stress of right boundary of nth layer structure can be gotten by those of left boundaryof the first layer structure, Eq. (17) and boundary conditions between layers,[

un(x1)Tn(x1)

]x R

n

= Mn Mn−1 · · · M1

[u1(x1)T1(x1)

]x L

1

(21)

The thickness of periodic composite structure is equal to na.

Transfer matrix approach of vibration isolation analysis of periodic composite structure 465

If

M = Mn Mn−1 · · · M1 (22)

Equation (21) will be changed as [un(x1)Tn(x1)

]x R

n

= M

[u1(x1)T1(x1)

]x L

1

(23)

M is transfer matrix of periodic composite structure.

3 Transmission properties of periodic composite structure

Halevi and Fuchs adopted the surface impedance method to study optical properties of solids in 1984. Wepresent formalism similar to analyze the effective impedance of periodic composite structure in order to exploreits vibration isolation performance.

Considering that the structure is on the region x1 ≥ 0 and is contact with air of density ρ0 and velocity c0,we define the ratio [11]:

Z (0)p = T(0)n

u(0)n

(24)

as the surface impedance. When incident waves hit the surface they are reflected and transmitted. The dis-placement field u1 for the incident and reflected waves is

u1(x1) = Ai eik0

L x1 + Ar e−ik0L x1 (25)

here, k0L is incident wave number, superscript (0) denotes the corresponding parameters with zero order wave-

form, subscript i denotes incident wave, and subscript r denotes reflected wave.The reflectivity of periodic composite structure can be obtained by Eq. (25)

R = |Ar |2|Ai |2

(26)

The impedance of air is

Z0 = ik0Lρ0c2

0 (27)

Substituting Eqs. (24) and (27) into Eq. (26), we get

R =∣∣∣Z0 − Z (0)p

∣∣∣2

∣∣∣Z0 + Z (0)p

∣∣∣2 (28)

We use Eqs. (21), (22) and (24) to calculate the effective impedance,

Z (0)p = M21 − M11 Z p(x Rn )

M12 Z p(x Rn )− M22

(29)

where, Mi j are the elements of transfer matrix M of the periodic composite structure, i, j = 1, 2, Z p(x Rn ) is

the impedance of the last (nth) layer structure.When elastic waves transmit in periodic composite structure, the balance of rates of energies yields the

equation

|T |2 + |R|2 = 1 (30)

The transmissivity, T can be solved by Eqs. (26) and (30), which characterizes transmission properties ofvibration isolation of periodic composite structure.

466 W. Yong et al.

Table 1 Parameters for materials Al, Cu and Silastic

Density ρ(kg/m3) Lame coefficients Young’s modulus E(Pa) Poisson’s ratio ν

λ (Pa) µ (Pa)

Al 2,799 5.8955 × 1010 2.6812 × 1010 6.8 × 1010 0.32Cu 8,356 1.726 × 1010 7.527 × 1010 1.08 × 1011 0.31Silastic 1,300 6 × 105 4 × 104 7.84 × 106 0.47

0 20 40 60 80 100 120 140 160 180 200200-30

-20

-10

0

10

20

30

Frequency f/Hz Frequency f/Hz

Tra

nsm

issi

vity

T/d

B

Tra

nsm

issi

vity

T/d

B

0 20 40 60 80 100 120 140 160 180 200200-30

-20

-10

0

10

20

30(a) (b)

Fig. 2 Performance curve of vibration isolation of periodic composite structure with the scatterer’s diameter d = 0.02 m

0 20 40 60 80 100 120 140 160 180 200200-30

-20

-10

0

10

Frequency f/Hz

Tra

nsm

issi

vity

T/d

B

0 20 40 60 80 100 120 140 160 180 200-20

-10

0

10

20

30

40

Frequency f/Hz

Tra

nsm

issi

vity

T/d

B

(a) (b)

Fig. 3 Performance curve of vibration isolation of periodic composite structure with the scatterer’s diameter d = 0.03 m

0 20 40 60 80 100 120 140 160 180 200200-60

-40

-20

0

20

40

Frequency f/Hz

Tra

nsm

issi

vity

T/d

B

0 20 40 60 80 100 120 140 160 180 200-40

-30

-20

-10

0

10

20

Frequency f/Hz

Tra

nsm

issi

vity

T/d

B

(a) (b)

Fig. 4 Performance curve of vibration isolation of periodic composite structure with the scatterer’s diameter d = 0.04 m

4 Numerical simulation

In this section we present the numerical examples of three-layer periodic composite structure, i.e., , n = 3.The materials we use belong to Al, Cu and Silastic, and their parameters are listed in Table 1.In calculation, we think of Silastic as host material and Al or Cu as scatterers. Numerical results are pre-

sented in Figs. 2–4. These results are computed for lattice constant a = 0.05 m, b = 0.05 m, h = 0.05 m. Andwhen the structure is excited with harmonic force, the coordinates of the excited point is at (0, 0.075, 0.075),the amplitude of excited displacement is 0.001 m, and the coordinates of displacement of output point is at(0.15, 0.075, 0.075).

Transfer matrix approach of vibration isolation analysis of periodic composite structure 467

Piezoelectricity accelerometer

Periodic composite structure

B&K 3560C front-end

Foundation

Machine

Piezoelectricity

accelerometer

Pulse

hammer

B&K Pulse 10.0

Fig. 5 The block diagram of the experiment of Periodic composite structure

Figure 2 shows vibration isolation transmissivity T of periodic composite structure versus excited fre-quency f when the diameter of scatterers d is 0.02 m. Figures 3 and 4 show vibration isolation effects whenthe diameter of scatterers d is 0.03 m and 0.04 m, respectively. Here, Figure (a) and (b) denote the scatterersas Cu and Al, respectively. Solid line denotes no damping, short dotted line shows that damping coefficient ofSilastic is equal to 0.0001, and long dotted line shows that damping coefficient of Silastic is equal to 0.0005.

It is shown from Figs. 2 to 4 that fundamental frequency of periodic composite structure varies with thediameter of scatterers, but change little with scatterer’s material. When the diameter of scatterers increases from0.02 to 0.04 m, the extremum of vibration isolation transmissivity decreases gradually from −17 to −28 dB,i.e., vibration attenuates quick with the increase of the diameter of scatterers. When the diameter of scatterersis equal to 0.02 m, the effective frequency range of vibration isolation is within 46 and 190 Hz under the lowfrequency. And when the diameter of scatterers increase to 0.04 m, the range changes within 39–200 Hz.

5 Experimental results

The experiment on the vibration isolation performance of periodic composite structure has been carried out byutilizing PULSE analyzers of the Danish B&K Company, and spectral analyses of the test result have been dealtwith in this paper. The main purpose of the experiment and analyses is to validate the accuracy and validity ofthe numerical algorithmic adopted above, and to offer a certain theoretical foundation for the application ofperiodic composite structure.

5.1 Experimental apparatus and makeup

The experiment was carried out using the front-end 3560 C and PULSE data acquisition and analytical system.The force signal was sent out and gathered by PCB pulse hammer, the acceleration signal of excited point andresponse point were gathered and input for PULSE front-end 3560 C by two piezoelectricity accelerometers,respectively, and the data were dealt with by PULSE analytical software. The block diagram of the experimentis shown in Fig. 5, the practical picture of every part of the test system is shown in Fig. 6, and the real sketchof the periodic composite structure is shown in Fig. 7.

The test samples of the periodic composite structure are listed in Table 2.

468 W. Yong et al.

Fig. 6 The practical picture of every part of the test system

Periodic composite structure

Section plot of periodic

composite structure

Fig. 7 The real sketch of the periodic composite structure

Table 2 The test samples of the periodic composite structure

Sample no. Host material Diameter of the scatterer (mm)

1 2, 3, 6 periods: A—rubber with thickness of 12 mm 102 2, 3, 6 periods: A—Silastic with thickness of 12 mm 10

5.2 Experimental contents and analysis method

The vibration isolation performance of periodic composite structure is tested and analyzed with the differentperiodic numbers of the scatterer and different material of the host in this paper.

While testing, periodic composite structure is installed between the machine and foundation. The piezo-electricity accelerometers are fixed on machine and foundation respectively, which is shown in Fig. 8.

In the course of testing, the sampling parameter of FFT is set up as follows: the analysis bandwidth is400 Hz, the spectral line is 400, the frequency resolution ratio is 1 Hz, sampling cycle is 1 s, and the averagetimes of the data are five.

Frequency spectrum analysis and coherence analysis are adopted to analyze and deal with the test datain order to obtain the excited and response characteristic of periodic composite structure, by which vibrationisolation transmissivity can be solved so as to analyze the effect of vibration isolation.

Transfer matrix approach of vibration isolation analysis of periodic composite structure 469

Pulse

hamme

Piezoelectricity

accelerometer

Periodic composite structure Piezoelectricity accelerometer

r

Fig. 8 Fixing sketch of the piezoelectricity accelerometer

0 40 80 120 160 200 240 280 320 360 400-30

Frequency f/Hz

-20

-10

0

10

202• •3• •6• •

2 p3 periods

eriods

eriods6 p

Tra

nsm

issi

vity

T/d

B

Fig. 9 The curves of vibration isolation transmissivity of periodic composite structure with the host of rubber

0 40 80 120 160 200 240 280 320 360 400-30

Frequency f/Hz

-20

-10

Tra

nsm

issi

vity

T/d

B

0

10

202• •3• •6• •

2 p3 periods

eriods

eriods6 p

Fig. 10 The curves of vibration isolation transmissivity of periodic composite structure with the host of Silastic

5.3 Experimental results and discuss

The curves of vibration isolation transmissivity of periodic composite structure are obtained and shown inFigs. 9 and 10 by test. Vibration isolation transmissivity in two periods, three periods and six periods when thediameter of the scatterers of periodic composite structure which is made up of rubber and Silastic is 10 mmare shown in Figs. 9 and 10, respectively.

From Figs. 9 and 10, we can see that vibration isolation effect of periodic composite structure which ismade up of a rubber and steel ball with the diameter of 10 mm is worse than that made up of Silastic and steelball with the diameter of 10 mm. Effective frequency range of vibration isolation is probably near 50–230 Hz,and the vibration isolation transmissivity of the former in two periods, three periods and six periods is higherthan that of the latter, respectively. Vibration attenuates more quickly and vibration isolation effect gets betterwith more periods.

The comparisons of the testing value with the theoretical value of periodic composite structure with threeperiods are shown in Fig. 11. The effective frequency range of vibration isolation of theoretical curve is between

470 W. Yong et al.

0 40 80 120 160 200-20

-10

0

10

20

30

Tra

nsm

issi

vity

T/d

B

3• • • • •3• • • • •Theoretical valueExperimental value

Frequency f/Hz

Fig. 11 The comparisons of the testing value with the theoretical value of periodic composite structure with three periods

about 36 and 180 Hz while the testing value between about 50 and 230 Hz, and the theoretical vibration isolationeffects is better than the experimental ones.

5.4 Error analysis

Comparing vibration isolation effect of the theoretical calculations with experimental results, it can be seenthat the errors which exist in the vibration isolation frequency range and effect mainly root in as follows:

1. Because the test model used in this paper was made up of overlapped layer structure, the connectionbetween layers was not an ideal seamless link. While in the theoretical model, its connection was con-sidered to be the ideal seamless link. So, there are certain differences between test model and theoreticalmodel, which will cause the difference between experimental result and theoretical result.

2. While setting up the vibration model of the periodic composite structure, the size of damping consideredis distinguished with that existed actually, which influences the vibration isolation performance of thestructure. This is one of the reasons for bringing the error too.

3. Geometry size of the real structure and material properties, including density, Yang’s modulus, Poisson’sratio, etc. have deviations with parameter value of theoretical model, which has brought certain errors tosetting-up of the theoretical model too.

6 Conclusion and future work

Vibration isolation performances of the periodic composite structure have been analyzed by transfer matrixapproach above and the following conclusions are drawn:

1. Fundamental frequency of periodic composite structure vary with the diameter of scatterers, but changelittle with scatterer’s material.

2. The effect of vibration isolation will get better with the increase of the diameter of scatterers.3. The more difference the density of the host and that of the scatterers has, the better the effect of vibration

isolation will get.4. The effective frequency range of vibration isolation will get wider with the increase of the diameter of

scatterers.

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