Li-Jie Wu·Han-Wen Song

Abstract An approach is proposed to estimate the transfer function of the periodic structure,which is known as an absorber due to its repetitive cells leading to the band gap phenomenon.The band gap is a frequency range in which vibration will be inhibited.A transfer function is usually performed to gain band gap. Previous scholars regard estimation of the transfer function as a forward problem assuming known cell mass and stiffness matrices. However, the estimation of band gap for irregular or complicated cells is hardly accurate because it is difficult to model the cell exactly. Therefore, we treat the estimation as an inverse problem by employing modal identification and curve fitting.A transfer matrix is then established by parameters identified through modal analysis. Both simulations and experiments have been performed. Some interesting conclusions about the relationship between modal parameters and band gap have been achieved.

Keywords Band gap·Modal analysis·Parameter identification·Periodic structure·Transfer matrix

1 Introduction

Periodic structures are structures constructed by a series of the same cells.Periodic structure is widely used in engineering, such as the guide ways of trains, multi span bridges,stiffened plates used in aeronautics and marine,and flexible space structures used in satellites.

Periodic structure is well known for its special dynamic characteristic. Wave or vibration can only pass through the structure in some specific frequency band, which is called the pass band.In the other frequency band,wave or vibration could not pass through,which is called the stop band or band gap. The special stop band makes periodic structure a special“filter”,which offers a new idea for designing vibration isolation.

The band gap was first observed in photonic crystals[1-3], in which some waves could not propagate. Then, band gaps in the phononic crystal were found out and drew the wide attention of scholars.Researches on periodic structures have sprung up since the 1950s. The plane wave expansion (PWE) method, wavelet-based method, finite element method(FEM),spectral element method(SEM),receptance method, transfer matrix method, and wave finite element method are widely used.

The PWE method is the most widely used method to solve band gaps in phononic crystals[4,5].The displacement and material elastic parameters are expanded in Fourier series.The generalized eigenvalue equation is then obtained and,by solving the equation,the band structures can be achieved.

The PWE method, which adopts the Fourier basis, has the disadvantage of slow convergence. A wavelet-based method [6,7] is developed to calculate elastic band gaps,which use wavelet transforms as the basis functions. The advantage is the wavelet transform involves both frequency and time domain which makes it easy to describe the discontinuity.

The FEM is widely used to solve periodic structures with complicated cells[8-11].With FEM,one can easily build the model for any cell.However,for FEM,the cell model may be built with thousands of degrees of freedom(DOFs).

A structure can be treated as one spectral element with uniform geometry size and material property [12-14]. The spectral element can be represented by interpolation functions.With SEM,the DOFs in the system can be dramatically reduced compared with the FEM.

The receptance method is also a common analytic method.Vibration receptance of periodic structure suffering the force of longitude or flexural wave is deduced through the wave equation of its cell. In this regard, Mead [15] studied the band gap of simply supported periodic beams.And the study of the attenuation constant and phase constant of periodic plates under force of torsion and flexural wave were reviewed[15].Gupta studied periodic skin-stringers[16]and periodic stiffened structures[17]using a wave approach.

To be noted is that Lin and McDaniel [18] firstly used the transfer matrix method to study periodic structures.The transfer matrix method is widely used to study onedimensional periodic structures and quasi-one-dimensional periodic structures.In addition,Ruzzene and Tsopelas[19]studied the wave propagation along honeycomb periodic sandwich beams. Moreover, Richard and Pines [20] analyzed vibration band gap in periodic drive shafts. Wang et al.[21]studied the propagation and localization of waves in disordered periodic structures, like piezoelectric phononic crystals.

The nonlinear behaviors and active control involved in phononic crystals is a challenging and interesting topic as well.In this field,Wang et al.[22,23]have studied the influence on band gap considering both of the two factors.

At first,a transfer matrix should be deduced from the wave equation.However,it is pretty hard to achieve an analytical transfer matrix for complicated periodic structures.With the development of finite element technology, it is possible to attain a dynamic stiffness matrix from commercial software.Combining with the transfer matrix method,the wave finite element method is a popular way to solve dynamic problems of periodic structures [24-26]. A transfer matrix is easy to achieve from a dynamic stiffness matrix.The dynamic stiffness matrix is large for complicated structures,which leads to poor conditions for a numerical transfer matrix.Some efforts have been made on solving the poor conditions.In this regard,Zhong et al. [27,28] have made a great contribution to this problem.

Now,FEM which only depends on the theoretical model is commonly used.Once a blueprint is confirmed,static characteristics, stress analysis, and dynamic characteristics are available.However,damping is not accurate in finite element software. Usually, damping needs to validate experimental results. What is more, a blueprint could hardly predict the manufacturing factors.The simulation of boundary and joint could not fit exactly.These facts lead to errors in analysis.A transfer matrix method based on modal analysis with the vector fitting(VF)method and experimental frequency response functions (FRFs) will be studied in this paper to overcome these problems.

Fig.1 Periodic structure constructed by identical cells

A modal test is already a mature method in vibration tests.Through modal tests of cells,modal parameters could be figured out.FRFs can be expressed by these parameters,making the analytic expression of FRFs and analytic transfer matrix expression available. In this paper, a periodic structure is designed.Experimental FRFs are measured by the commercial software: the SO Analyzer system, a product of m + p international. Vector fitting, a new method of curve fitting,is performed to obtain the modal parameters and the reconstructed FRFs.

The rest of this paper is organized as follows.In Sect.2,modal tests and curve fitting are introduced. In Sect. 3, an analytic transfer matrix composed by experimental FRFs is deduced. In Sect. 4, both simulations and experiments are performed to study one-dimensional periodic structures using the transfer matrix method based on experimental FRFs.Comparison of different methods is made,and some discussions about cell modal parameters and band gap are given out.

2 Cell FRFs reconstruction

In this section, the theory of modal analysis and parameter identification is introduced.Curve fitting is used to identify modal parameters of FRFs. The analytical function of cell FRFs is expressed by these modal parameters.

2.1 Cell constructions and FRF expressions

Infinite cells linked together as shown in Fig.1,construct the periodic structures.Each cell is exactly the same.The FRFs for free cells can be expressed in terms of modal parameters.Modal analysis of a single cell can be performed to obtain these modal parameters. According to Floquet's principle,the features of the cell can forecast the dynamic characteristics of the whole periodic structures.

The common expression of FRFs for a linear system with viscous damping is shown in Eq.(1)

Here Hij(jω)refers to the FRF with input at point i and output at point j,ω refers to frequency band,Ar,ij,are the residue,are poles,*means conjugate complex.

In a continuous system, the number of natural frequencies are indefinite,while some high frequencies have a low impact on the response. Similarly, for a complicated cell,there exists a huge number of natural frequencies.However,only modes that contain the most part of the energy should be retained for efficient calculations.Thus,frequency truncation should be performed to change a continuous model into a finite discrete model and to fit the expression expressed in Eq.(1),while performing curve fitting,only main modes are retained.Ignoring some small contribution,modes may lead to a certain amount of error.Due to the energy distribution,the modes ignored are those that have little influence,so the error would be acceptable.The VF algorithm for curve fitting is an algorithm that identifies modes by energy distribution.Here,VF is performed to attain modal parameters to establish cell FRFs expressions and reconstruct transfer matrix in this paper.

2.2 VF algorithms

Model parameters to express FRFs should be identified and curve fitting is used.The accuracy of the calculated band gap depends on the modal parameters identified by curve fitting.

In traditional polynomial algorithms,which are the most widely used in modal analysis, the best modal order is always defined with the consideration of the existence of the noise modes,which may lead to pretty high order.With the increasing of modal order,the numerical instability and the computation difficulty double and the demand of dividing the frequency range into several sub-bands comes up. The VF algorithm[29-31]is introduced to decompose the rational functions by taking a common set of partial fractions as basic functions.The mode order acquired is arranged according to the modal energy,to ensure the fast convergence of the VF algorithm. The parametric stability of the partial fractions and the numerical stability could be attained with the increase of the modal order.

In the following section,the basic theory of the VF algorithm and the VF algorithm in modal analysis are introduced.

2.2.1 Basic theory of VF algorithm

For a curve in the form of a rational polynomial,the unknown parameters are all included in the Eq.(2).

where brare the poles；arare the zeros；urare the residue；vrare the coefficients of high order terms,m,n are order of the polynomial.α is the coefficient of numerator.

The idea of the VF algorithm is that numerator and denominator are divided by the same polynomial T(s),whose poles are manually set as shown in Eq.(3).

pris the initial poles given.Then R(s)turns to

We also have Eq.(4)in the form of Eq.(5).

A rational polynomial can be separated into a set of basic functions as shown in Eqs.(6)and(7).

Where Φr(s)is basic function,cNrand cDrare coefficient of basic function in numerator denominator,respectively,dris coefficient for higher order terms.The basic functions are in the form of Eq.(8)

Then Eq.(4)could also have the following form as

Also,in the following matrix form

By solving Eq.(10),all coefficients can be obtained andare in a certain form.

2.2.2 VF algorithm in modal analysis

Substituting jω with s,the FRF expression is illustrated by Eq.(11).

here Rr,ijis residue,λris the poles for the new FRF.In case of displacement response,cijis zero.In case of accelerated response,cijis a constant.

By solving Eq. (10), the coefficients for ?D(jω) can be acquired.And by solving ?D(jω) = 0,the natural frequencies and damping ratio can be achieved.Then Eq.(11)turns into a series of linear equations and a residue can be achieved.Certain FRF expressions for cells can be attained.

The cell of periodic structure is of free ends, so the cell contains a rigid mode.Theoretically,the amplitude or energy for a rigid mode could be infinite, so when performing the VF algorithm,the rigid mode should be separately listed in the expression first if it exists. A modified VF algorithm is performed in detail in a simulated case in Sect.4.

3 Establishment of transfer matrix based on reconstructed FRFs and curve fitting

Periodic structure is composed of a series of identical substructures or so-called cells.Band gap is a special characteristic for periodic structure,and can be obtained by calculating the characteristic equation of the transfer matrix.

FRF of cells can be reconstructed by the VF algorithm presented in Sect.2.In this section,the transfer matrix based on these reconstructed FRFs is introduced. In this paper,one-dimensional periodic structure is studied.A single-input and single-output system is considered, so the size of the transfer matrix is 2 × 2. The four elements of the transfer matrix are expressions of reconstructed FRFs,which are polynomial functions about frequency (unknown) and are the four elements that are all presented by modal parameters(known).

3.1 Main theory of composing transfer matrix based on reconstructed FRFs

The theory of the transfer matrix is introduced as follows.

A cell of periodic structure is shown as Fig.2.L represents the left side of the cell and R is the right side.

The relation of the displacement and force on the cell can be expressed as Eq.(12).

Fig.2 A cell of periodic structure

Here XL,XRare the displacements at the left and right side of the cell respectively. FL,FRare the force at the left or right side of the cell.Hij(i, j =L,R)refers to the FRF with input at point i and output at point j.

According to Floquet's principle (or Bloch theory), the relationship of the displacement and force between left side and right side of n-th cell can also be expressed by the transfer matrix as shown in Eq.(13).

where T is the transfer matrix for the cell, μ = α +iβ is a complex constant. α is called the attenuation constant. In the zone with a nonzero attenuation constant, the flexural motion(for example)decays along the structure length and one such wave on its own can transmit no energy.β is a phase constant representing a phase shift between two consecutive spans[15].λ=eμis called the propagation constant.

According to Eq.(13),we have

According to Eq.(12)

Insert Eq. (15) into Eq. (14), then the elements of the transfer matrix can be expressed by FRFs. What is worth mentioning is that these FRFs are expressed by modal experiments.

The transfer matrix is rewritten as Eq.(17).

According to Eq.(13)and λ=eμ,we have

If Eq.(18)has a nontrivial solution,then

In this paper, the one-dimensional system is considered,so the element is an expression instead of a matrix as follows

Then the characteristic equation is

Here λ = eμ= eα+iβand Hij= Hij(jω)is a function of frequency ω.

By solving Eq.(21),the propagation constant is obtained.

The attenuation constant and phase constant are expressed as Eqs.(23)and(24).

In the case of α = 0,β = 0,π, the wave or vibration is inhibited in the structure [15]. Band gap or stop band corresponds to the frequency band where these conditions are satisfied. On the other hand, the frequency band where|α| = 0 and β ∈(0,π) is called the pass band in which waves or vibration can pass through.

It should be emphasized that the transfer matrix is expressed by FRFs that are polynomial functions expressed by modal parameters.So the propagation constant,attenuation constant, and phase constant are also related to modal parameters.In the next section,relationship between modal parameters and band gap will be discussed.

The transfer matrix method based on reconstructed FRF is a semi-analytic method.Because the transfer matrix here is a functional matrix instead of a numerical matrix, so the poor conditioning problem can be avoided.

Experimental data are more reliable with fine measure.With the fine stability of the VF algorithm,the transfer matrix based on experimental FRF can be well established,and the band gap can be solved more correctly.

3.2 Dynamic analysis based on transfer matrix composed by reconstructed FRFs

In Sect.3.1,FRF expressed by modal parameters is achieved.In this part,some characteristics about one-dimensional periodic structures are studied.To find the band gap is the main problem when studying periodic structures.In the following section,width and location of band gap for infinite periodic structures and the depth of band gap for finite periodic structures are studied.The differences between infinite and finite periodic structures are discussed as well. The influence of modal parameters is considered.

3.2.1 Band gap location and width

First,the most important characteristic for periodic structures is the location of the band gap.For the start and end point of the band gap or the boundary of the band gap,the attenuation constant and phase constant have the condition of λ = ±1 and the following Eq.(25).

Thus,the eigenvalue expressed by FRFs are illustrated by Eq.(21).

Substituting λ = ±1 with the eigenvalue expression and we have

FRF based on experiment could be expressed as follows.

So the boundary frequency ωγshould fit the following Eq.(30).

BysolvingEq.(30),theboundaryfrequenciesareachieved.

3.2.2 Band gap depth

The attenuation constant is used to describe the depth of the band gap,in other words,the attenuation constant represents the effect of the mechanical‘filter'of periodic structure.

In dynamics, a transfer ratio is always used to show dynamic features. As we all know, transfer ratio is also an important quota for designing and evaluating absorbers. In the following discussion, we are able to know the relationship between the band gap of infinite periodic structure and the transfer ratio for finite periodic structure.

The response of two sides of a cell(multiple DOF system)under single point excitation of the left side can be expressed by Eq.(31)

Thus,we have

Here, the transfer ratio Tdis defined as Eq. (33), which represents the amplitude change of displacement on the two sides of a cell.

According to Eq.(32),the displacement transfer ratio can also be expressed as follows.

For a system constructed by n cells,we have

It can also be reformed as

Here,

where the subscript n ofnH represents there are n cells to composethewholeperiodicstructure,andthetransferisfrom the left side of first cell to the right side of n-th cell.

Considering the condition of single input, F1= F,Fn=0,we have

where n H12means the element of first row and second column ofnH.

Similarly,

wheren-1H12means the element of first row and second column ofn-1H.

Then the transfer ratio is

The two functions n H12andn-1H12can also be reformed by the transfer matrix. The transfer matrix can be reconstructed to a diagonal matrix and the process is deduced below.

Here,

HLL,HRR,HLRare the FRFs of cell.

Then we have

Here

Thus,we have

And we have

Fig.3 Different distributions of infinite periodic structures

Fig.4 Construction of cell

The amplitude of frequency in band gap area decreases with the increasing of periodicity and the decreasing ratio approaches to the value λ when n →∞. Here, λ is the eigenvalue of the transfer matrix of the cell.

Above all, the transfer ratio of two sides of the free cell with finite periodicities presents the effect of inhibition of vibration. When the periodic structure goes to infinite, the transfer ratio equals the propagation constant in the band gap.

3.2.3 Cell distribution

Considering the cell from A to B and the other distribution from B to A,as shown in Fig.3,infinite periodic structures composed by the two cells are exactly the same.A more intuitive model is shown in Fig.4.An infinite periodic structure could be composed by numerical kinds of cells,so how does the way of distribution effect the characteristics of periodic structures or in other words,on band gap? This is what we want to discuss in this section. The subscript O represents outside of A or B and subscript I represents the inner side of A or B.

Case A-B:

Rewritten as

Case B-A:

Similarly,the FRF is

Equation(52)could also rewritten as Eq.(53).

For the transfer matrix as shown in Eq.(21),exchanging HBBand HAA, the characteristic equation is not changed,which means that the distribution of cells has no effect on the dynamic characteristics of the whole periodic structure.

3.2.4 Invariable frequency

For different cells,the eigenvalues of the cells are different.In finite periodic structures constructed by a cell,the eigenvalues of the cell are also the eigenvalues of the whole cells.

The kinetic equation for a cell is

Dij(i, j =L,R) is the dynamic stiffness with input i at point and output at point j, f refers to force on the right or left side and q refers to the displacement on the both side of the cell.

The natural frequencies could be calculated by

For the one-dimensional system, Dijis not a matrix but a function.

Consider a structure made of two cells,

The global dynamic stiffness matrix of the two cells is thus

Rewritten as

Suppose Dijis a function,

According to Eq.(56),then we have

So, the natural frequencies of cells are also natural frequencies of the whole structure,which are called invariable frequencies in this essay.

Here comes a problem, when the periodic structures go to infinite, these periodic structures constructed by different cells should be the same.How to explain the invariable frequency? In fact, the amplitude of invariable frequency decreases with the increasing of periodicity due to the discussion about transfer ratio in the last section.

4 Simulated case

In the last section,transfer matrix expressions composed by modalparametershavebeenachievedandsomefeatureshave been discussed. In this section, some simulated cases will be studied. The results show that one-dimensional periodic structures with different kinds of cells all have these characteristics.

Fig.5 Periodic structure with single pass band

Fig.6 Cell of periodic structure studied

Fig.7 FRFs of cell shown in Fig.6.a H11.b H13.c H33

4.1 Periodic structures constructed by three masses

In this part,single pass band or stop band periodic structure is studied first.Periodic structure composed by two different masses has single pass band.The whole periodic structure is shown in Fig.5.The cell is shown in Fig.6.Here,suppose that m =3, M =1,k =10,000,and c=1.

4.1.1 Transfer matrix method based on reconstructed FRF and curve fitting

FRFs of the cell could be gained directly by Matlab,as shown in Fig.7.The common expression of FRF for linear systems with viscous damping is shown in Eq. (1). Because of the free boundary, there exists a rigid mode, which should be eliminated when performing curve fitting.

The FRF modified is shown as Eq.(63)

with mc=4.

Fig.8 FRF of H11 and curve fitting result

Fig.9 Decomposed natural frequency a H11.b Decomposed part

The FRF without rigid mode is

Set H11as example,H11and ?H11are shown in Fig.8.The result of VF matches well with the original curve.VF is also able to split H11to several single curves that contain only one natural frequency as shown in Fig.9.

The parameters to be identified are Ar,ijand sr.sris the pole of curve and includes the information of natural frequency ωrand damping ratio ξr. Using the VF algorithmintroduced in Sect.2 and the parameters,systems poles and residues could be identified,shown in Table 1.

Table 1 Poles and residues of H11(jω)

Fig.10 Comparison of simulated FRF and reconstructed FRF

Above all,FRF can be expressed by parameters gained.

The comparison of FRF gained by Matlab simulation and FRF reconstructed by modal parameters are shown in Fig.10.The results matche well between simulated FRF and reconstructed FRF.

The other two FRFs could be expressed by modal parameters in the same way.The transfer matrix composed by FRFs is shown in Eq.(17).In this condition,we have

The characteristic equation is shown in Eq.(21).By solving Eq. (21), the relationship between attenuation constant and frequency,the relationship between phase constant and frequency could be revealed,as shown in Fig.11(ignoring damping, the effect of damping will be discussed in Sect.4.1.3

Discussion:the frequency band where α ＞0 or β = 0,180°is where vibration could be inhibited.As shown in Fig.11,13-22 Hz and after 26 Hz,the curve matches the requirement.Usually,13-22 Hz is called the stop band or band gap.The frequency band where α = 0 or β ∈(0,180°)is the pass band,where vibration can pass through.

Fig.11 Propagation constant.a Attenuation constant.b Phase constant

4.1.2 Comparison

To certify the result, the method illustrated in Ref. [32] is performed to solve the same problem.

The kinetic equation of mass spring system is shown in Eq.(65).

According to Floquet's principle (or Bloch's principle),= AN, j =1 and= A1, j = N.

In this case, N =2,then

Here, A0= A2, j =1, A3= A1, j =2.

Equation(67)could be rewritten in matrix form.

Fig.12 Band gap for 3-masses periodic structure

The condition of Eq. (68) having nontrivial solution is Eq.(69).

The relationship between ω and γ (γ ∈(0,π))is shown in Fig.12.The band gap is about 13-22 Hz.Figure 12 shows the symmetry of band gap in the range of the square of the frequency.

4.1.3 Discussion about band gap and cell composition

In Sect.3,some dynamic characteristics are revealed and in this part,those conclusions are tested.

(a)Band gap width

In this case of single band gap, by solving Eq. (30), the frequencies of the limitation point of band gap are achieved.

These frequencies represent the several starting and ending frequencies of band gap,that are same as the propagation in Fig.11.With a precise expression of FRFs,the exact band gap width and location can be solved.

(b)Band gap depth

Figure 13 shows the transfer ratios of periodic structures(single band gap)composed by 1-8 cells.With the increasing of periodicities, the depth of transfer ratio in band gap decreases.

Fig.13 Transfer ratios of periodic structures constructed by 1-8 cells

Fig.14 Decreasing ratios at 15Hz

Fig.15 Cell with even mass distribution

Picking one-point 15 Hz in the band gap as an example,amplitudes of transfer ratios composed by 1-10 cells are calculated.The decreasing ratio at 15 Hz is shown in Fig.14.

According to Fig.11 and Eq.(48),λ|15Hz=2.30,which is printed in blue in Fig. 14. Meanwhile, decreasing ratio is directly calculated from the transfer ratio with different number of cells in red in Fig.14.According to Fig.14,the red line representing the decreasing ratio approaches while the blue line representing the propagation constant.

(c)Cell distribution

In the simulation case above,the distribution of the three masses is 2-1-1,while the mass distribution can be different.

Fig.16 Propagation constant of different cell distribution(distribution of mass in the substructure:red line 2-1-1；blue line 1.5-1-1.5).a Attenuation constant.b Phase constant

Fig.17 FRFs of periodic structures constructed by 1-8 cells

If the mass distribution is 1.5-1-1.5 as shown in Fig. 15,the relationship between propagation constant and frequency will not be changed, as shown in Fig. 16. The two lines,orange and blue lines,representing two cells are exactly the same,which means distribution of cell is irrelevant to band gap.

(d)Invariable frequency

As practical structures are not infinite but finite, invariable frequencies would exist.The FRFs of periodic structures constructed by 1-8 cells are shown in Fig.17.As shown in Fig.17,the natural frequencies are concentrated in the bands 0-13 Hz and 22-26 Hz.

Fig.18 Natural frequencies of periodic structure(mass distribution of 2-1-1)with 1-50 periodicities

Fig.19 Natural frequencies of periodic structure(mass distribution of 1.5-1-1.5)with 1-50 periodicities

Figures 18 and 19 exhibit the change of natural frequencies with 1-50 periodicities,where the invariable frequencies more clearly show.Figure 18 is the condition of mass distributionof2-1-1.Figure19istheconditionofmassdistribution of 1.5-1-1.5.Figure 20 is the condition of mass distribution 0.5-3-0.5. As we know, if the periodicity goes to infinite,periodic structures composed by the three cells would be the same.

Figures 18-20 show that there exist some invariable frequencies in each case, which are natural frequencies of the cell. No matter how many periodicities, the invariable frequency always exists in finite periodic structures. The difference of periodic structures constructed by the three conditions is also the difference of invariable frequency.

Fig.20 Natural frequencies of periodic structure(mass distribution of 0.5-3-0.5)with 1-50 periodicities

(e)Effect of damping ratio

Wave propagation along real structures will encounter material or structural change,or both.At such discontinuity,the incident wave is partially reflected which leads to the transmission of reduced energy. Damping isusually associated with conversation of the energy. For example, conservation of mechanical energy into heat may cause loss of mechanical[33].Traditionally,damping is not applied to the attenuation above.However,damping is an important part in practical engineering structures and also an important factor for FRF and modal parameters.So,here we introduced the damping into the system and study the influence of different damping ratios.

Damping ratio is a quantitative indicator for damping and is available through parameter identification.Figure 21 shows us the relationship.

When there is no damping in the system,the attenuation constantα =0inthepassband.Withtheincreaseofdamping ratio,the attenuation constant is modified.If damping ratio is comparatively small,the band gap still exists.However,with the increase of damping ratio,there will exist attenuation in the pass band as well due to the energy loss.Once damping ratio is equal to or bigger than, there would be no pass band at all.The vibration would be reduced in all frequency bands.

4.2 Complicated cell with multiple masses and springs

Fig. 21 Effect of damping ratio on band gap (damping ratio: blue:ξ =0；red:ξ =0.01；yellow:ξ =0.1；purple:ξ =black:ξ =1；green:ξ =1.1).a Attenuation constant.b Phase constant

Fig.22 Complicated cells

In the case above,a simple example was studied to verify the special characteristics of periodic structures. In this part, a more complicated cell will be studied.As shown in Fig.22,the model contains six masses and seven springs,which is not a chain model.However,it is worth to mention that the cell of the complicated model is single input and single output.Between the input and output there exist multiple masses and springs.

The FRFs needed to calculate the propagation constants and band gap are shown in Fig.23.Curve fitting and modal parameter identification are performed to achieve the propagation constants as shown in Fig.24.

(a)Band gap width and location

AccordingtoEq.(30),wehaveλ=1, fb=11.77,17.70,24.13 and λ=-1, fb=9.63,13.00,16.17,24.4.(fbis the frequency of the beginning and ending point of the band gap)(b)Band gap depth

Fig.23 FRFs of complicated cells.a H11.b H16.c H66

Similarly, band gap depth at 21 Hz is calculated as an example to show the decreasing ratio.According to Fig.24,λ|21Hz=47.2759.By computing the amplitude of transfer ratios of different number of cells (Fig. 25), the decreasing ratio could also be calculated.=47.2675,Tn,n =3,4 represents the transfer ratio of 3 or 4 cells.(c)Cell distribution

As shown in Fig.24,the propagation constants calculated by two different cells are exactly the same.

(d)Invariable frequency

Another characteristic is that there are invariable frequencies in FRFs of finite periodic structures composed of different numbers of cells as shown in Fig.26.Frequencies 0,10.31,14.64,16.17,19.27,and 25.50 Hz appear in each curve.

Fig.25 Transfer ratios of periodic structures composed by 1-4 cells

Fig.26 FRFs of periodic structures composed by 1-4 cells

4.3 Uniform cell

If all the masses and springs in the cell are identical, then there will be no band gap phenomenon. In other words, if the cell is uniform, there will be no band gap, as shown in Figs.27 and 28.Natural frequencies concentrated in the band 0-32 Hz and |λ| = 1, so there is no band gap for uniform structures.

According to the cases above,the transfer matrix method based on reconstructed FRFs can be used to solve the band gap of one-dimensional periodic structures.

5 Experimental verification

Fig.27 Propagation constant of uniform cell.a Attenuation constant.b Phase constant

Fig.28 Poles of transfer ratio in uniform structure

To verify the theory and simulation, an experiment is designed here.The experimental object is shown in Fig.29.The cell is designed with uniform material, 304 stainless steel. The geometry size for mass 1 is 75 mm×75 mm×20 mm,and geometry size for mass 2 and mass 3 is 75 mm×75 mm×30 mm.The geometry size for the C-shaped beam is R25 mm ×15 mm×2 mm with 20 mm×15 mm×2 mm in both two ends. Two measure points are attached on the point 1 and point 3.

The data acquisition device is the m+p SO Analyzer.The sampling frequency is 2048 Hz.The sources of excitation are Gaussian white noise from 5 Hz to 1000 Hz.

First of all, modal test is performed to gain the FRFs of the cell as shown in Fig.30.H11,H13,H33are what we need to gain the propagation constant and band gap.The FRFs are shown in Fig.31.

Fig.29 Three-mass cell

Fig.30 Experimental system

Fig.31 Experimental FRFs

Fig.32 Curve fitting results for H11

Fig.33 Curve fitting results for H13

Fig.34 Curve fitting result for H33

5.1 Curve fitting and FRF reconstruction

Fig.35 Propagation constant for three-mass cell

Fig.36 Experiment of four cells

To calculate the propagation constants, first of all, FRF expressions of H11,H13,H33should be achieved.VF is used here to gain the modal parameters and thus FRF expressions could be reconstructed.Figures 32-34 show the fitting results of H11,H13,H33,respectively.

5.2 Propagation constants

By applying the method introduced above, the propagation constants could be attained.The result is shown in Fig.35.Band gap where the attenuation constant is bigger than zero and phase constant is zero or pi.So according to Fig.35,we could find out that band gap of periodic structure constructed by the substructure is about 65-80 Hz.

5.3 FRF and transfer ratio function of periodic structure of four periodicities

To certify whether the band gap is about 69-85 Hz or not,experiments of periodic structure with two to four periodicities are carried out as shown in Fig.36.The result of FRFs and Transfer Ratio function of four cells are shown in Fig.37.The band gap of four periodicities is 58-82 Hz,a little wider than the infinite band gap calculated.These experiments also certified that the method introduced is feasible and reliable.

Fig. 37 FRFs and transfer ratio of four-cells structure. a Amplitude.b Transfer ratio

6 Conclusion and discussion

Transfer matrix method based on experimental FRFs was introduced in the paper. The elements of transfer matrix were constructed by functions of modal parameters identified through modal analysis with the VF algorithm. Both simulation and experiment were performed to verify the reliability and validity of the transfer matrix method based on experimental FRFs.

Some interesting conclusions are drawn about stop band(band gap)and pass band.First of all,the way of distribution of cells for a same periodic structure has no effect on the stop band and propagation constants,but it does have an effect on some specific frequencies of finite periodic structure(invariable frequency).Secondly,the influence of damping ratio of substructure is discussed.Periodic structure acts as a“filter”if damping ratio is small (10-2), while for large damping ratio, there is no obvious stop band. Finally, the effect of the natural frequencies of cell is studied.There will be some same natural frequencies of periodic structures constructed by different numbers of cells, which are called invariable frequencies. What is worth to mention is that the value of the invariable frequencies locating in band gap are equal to the same start or end frequency of band gap for the symmetrical substructure.Considering transfer ratio function of finite periodic structure,the band gap calculated by the transfer matrix method based on experimental FRFs accords well with the area where the transfer ratio is smaller than 1 (or smaller than zero in the log figure).The depth of transfer ratio function would increase with the periodicities.The increasing velocity is also related to the propagation constant.A transfer matrix based on experimental FRFs containing damping information is more accurate and direct,especially for complicated structure.The elements of the transfer matrix are functions about modal parameters, which avoids the problem of poor conditions in calculating eigenvalues of a numerical matrix.Qualitative and quantitative analysis based onthetransfermatrixofexperimentalFRFshelpustoachieve some special conclusion. Because the method is based on experiments,it can solve practical problems.

AcknowledgementsThis work was supported by the National Natural Science Foundation of China(Grant No.11272235).