wave propagation in periodic structuresit facilitates an efficient solution of the inverse problem,...

WAVE PROPAGATION IN PERIODIC STRUCTURES

Lecture Notes for Day 4

S. V. SOROKIN Department of Mechanical and Manufacturing Engineering,

Aalborg University

2

1 Flexural wave in a periodic infinitely long plate An infinitely long plate is composed by the set of periodically repeated (alternating) ‘strip’ elements of two types shown in Figure 1. Each set of elements (j=1,2) is specified by the following parameters: the thickness jh , the length jl , Young’s module jE , the density j , Poisson coefficient j .

Figure 1. A periodic plate

A propagation of the stationary plane flexural wave at a circular frequency is considered, so that dynamics of a plate is, in effect, reduced to the dynamics of a beam. It

is convenient to introduce the non-dimensional parameters

1

2211

21E

h ,

1

1

hl

, 1

2

ll

, 1

2

hh

, 1

2

, 1

2

EE

(hereafter 3.021 ). Then wave

motion in the plates is governed by equations formulated for the lateral displacements

21 , ww in z-direction (1

dim1

1 hw

w ,1

dim2

2 hw

w ,1

dim

hxx ), see Figure 1

012 124

1 ww , 11 nxn , n=0,1,2,… (1a) 012 2

12242

ww , 11 nxn , n=0,1,2,…. (1b)

General solutions of these equations are sought as ( hkk njnjdim )

4

1

expn

njnjj xikAxw , j=1,2 (2)

The wave numbers in (2) are explicitly expressed via the frequency parameter as 4 2

11 12k , 4 2

21 12k , 4 2

31 12 ik , 4 2

41 12 ik

4 12212 12

k , 4 12222 12 k ,

4 12232 12

ik , 4 12242 12 ik (3)

3

Matching conditions at the interfaces 0x and x (see Figure 1) are formulated as

the continuity of displacements w , slopes w , bending moments wEhM

22

112 and

transverse forces wEhQ

22

112 in the non-dimensional form

00 21 ww 00 21 ww

00 231 ww 00 231 ww (4a) 21 ww 21 ww

231 ww 231 ww (4b) The same conditions are also applied at all other junction points 1nx , n=0,1,2,… Floquet theory is used to formulate the periodicity conditions (see Figure 1)

11 exp wiw B , 11 exp wiw B , 11 exp wiw B , 11 exp wiw B (5)

These conditions are substituted in equations (4b) and a system of eight homogeneous equations with respect to eight modal amplitudes grouped in two sets, njA , 4,3,2,1n ,

2,1j is derived. This system is rather cumbersome and therefore not presented here. It has a non-trivial solution when its determinant vanishes. This condition formulates a transcendent equation with respect to Bloch parameter B , which is a standard variable in the subject, measuring a phase shift in a periodic wave guide. As follows from Floquet theory, if all roots of the characteristic equation are complex valued then at a given frequency no propagating waves exist in a periodic plate. In this simple case, it does not present any difficulties to set up the dispersion relation explicitly as a transcendent function of the frequency parameter and the Bloch parameter B . The characteristic equation is obtained by equating to zero this dispersion relation,

0, BD . (6) Then one may formulate either a direct problem of solving the characteristic equation (6) – to find the Bloch parameter for a given , or an inverse problem – to find the frequency parameters for a given B . As follows from Floquet theory, wave propagation in a periodic structure is possible only when the characteristic equation has at least one purely real root B for a given . Thus, only these roots should be determined and it is sufficient to consider the interval B0 due to the periodicity of the function

Biexp .

4

In the framework of the direct formulation, branches of characteristic curve B are plotted one by one in frequency parameter ‘sweeping’ and solving the transcendent in

B equation, which yields from equation (6) for any given . A use of the inverse formulation makes it possible to plot all branches B simultaneously. In the general case, the characteristic equation (6) involves exponents of purely real or purely imaginary argument, which contains frequency parameter , see formulas for wave numbers (3) and the inverse formulation is as complicated as the direct formulation. However, at relatively low frequencies, 1 (i..e., when Kirchhoff plate theory is applicable) these exponents of small or moderate argument may conveniently be expanded into power series in , so that the function B is reduced to a polynomial form in for a given B . It facilitates an efficient solution of the inverse problem, because a polynomial equation is easily solved by a use of standard software, e.g., Mathematica and the accuracy of this approximate solution is easily assessed by comparison with results obtained in more time-consuming solving a direct problem.

Figure2. Charasteristic curves

In Figure 2, a typical set of characteristic curves Bp is presented. The non-

dimensional frequency parameter is 211

41

21

22 11212

hElp . In this example,

the Kirchhoff theory is equally applicable to describe wave propagation in both the ‘long’ and the ‘short’ elements. As is marked by the horizontal straight lines in Figure 2, there are two band gaps located between the first and the second and between the second and the third branches, 09.909.3 p , 8.354.22 p . These curves are plotted by a use of the inverse formulation with 24 terms retained in power series expansions of the exponents and they do not differ at all from the curves obtained in the direct formulation in frequency parameter ‘sweeping’, which involves computations of transcendent functions of wave numbers. All roots B of equation (6) are complex valued in a band gap and all waves are attenuated.

5

3. Waves in periodic curved pipes In this Section, we consider three types of structures. In each case, however, a structure contains the same repeated spatial segments, which, in turn are composed of curved and straight pipe elements. To apply the Floquet theory and find propagation constants, the idealized model of an infinitely long structure is considered first. This modelling defines location of stop bands in the frequency domain. Next, we consider infinite structures composed of exactly the same segments as in the previous case, but the number of these repeated segments is fairly small. The ‘outer’ part of a piping system is then modelled as a simple straight pipe. The power flow analysis in the case of forced vibration is performed by means of the boundary integral equations method in the frequency range that covers pass bands and stop bands predicted by the Floquet theory. We are particularly concerned with the comparison of attenuation levels achieved with adding periodicity cell one by one. Finally, we consider forced vibrations of a structure of a finite length with no material losses by means of boundary integral equation method and by standard FE package ANSYS. At this stage, the point is to compare the shapes of standing waves, when the excitation frequency lies in the pass band or in the stop-bands predicted by the Floquet theory for an unbounded structure.

3.1 Boundary integral equation method and Floquet theory – travelling wave analysis A periodic structure consisting of curved and straight pipe segments shown in Figure 9 is considered. The frequency dependence of the parameters |λ| is shown in Figure 10. In accordance with the Floquet theory, the location of the frequency stop bands is defined by the condition |λ| ≠ 1 held for all twelve roots of the characteristic equation. These predictions are compared with the power flow analysis, when a finite number of repeated substructures are inserted into an infinitely long structure.

1.0m0.6m

R0.15m R0.15mR0.15m

Figure 9 The pipe segments used as the periodic repeated substructure, Pipe diameter 30mm and wall thickness 2mm, ρs = 7800 kg/m3, ρf = 1000 kg/m3, E = 210GPa and ν = 0.3.

6

|λ|

Figure 10 The magnitude of |λ| versus excitation frequency, stop-bands are identified by condition |λ| ≠ 1 The power flow in the structure shown in Figure 11 versus excitation frequency is presented in Figure 1 for variable number of the ‘periodicity cells’ inserted into a straight pipe. The stop band filter consists of two to eight chosen repeated substructures. The reference solution is for the following examples provided by the energy flux through a single periodic cell inserted in an otherwise uniform infinitely long water filled pipe. The power flow analysis is here performed for frequencies in between 1 and 955Hz.

Figure 11. An infinitely long pipe with the eight ‘periodicity cells’ inserted; Pipe diameter 30mm and wall thickness 2mm, ρs = 7800 kg/m3, ρf = 1000 kg/m3, E = 210GPa and ν = 0.3. The wavy arrows designate infinitely long segments

7

0 100 200 300 400 500 600 700 800 900

0

20

40

60

80

100

120

Hz

IL [d

B]

2cells

3cells

4cells

5cells

6cells

7cells

8cells

Figure 12. Insertion losses in a pipe with variable number of inserted ‘periodicity cells’. The grey strips indicate Floquet-predicted stop bands. As it can be seen in the graphs in Figure 1, the performance of the stop-band filter does not much improved when the number of ‘periodicity cells becomes larger than five (Insertion loss is plotted in log scale). On the other hand, it is notably to see that only two repeated substructures already gives a noticeable attenuation effect in several frequency stop bands. The stop bands predicted by the Floquet theory are marked as grey strips in Figure 1 and Figure 13. When looking at the graphs crossing the by gray bands marked stop-bands at Figure 1, are the by Floquet theory idealized exponential decay rate per repeated ‘periodic cell’ clearly indicated, even for a relatively low number of repeated ‘periodic cells’.

-10

0

10

20

30

40

50

60

70

η =10-2η =10-3η =10-4η =10-5

η =10-1

η = 0

Hz

IL [d

B]

0 100 200 300 400 500 600 700 800 900

Figure 13. Insertion losses in a pipe with five inserted ‘periodicity cells’ and variable material loss factor η. The grey strips indicate Floquet-predicted stop bands. To illustrate the effect of structural damping on stop bands predicted by the Floquet theory, the power flow in several periodic pipes which have different magnitudes of the

8

material losses is calculated. In these calculations, the standard model of materials losses as a complex Young’s modulus defined by the loss factor, η is employed. The analysis is performed for the pipe with five ‘periodicity cells’. Material losses are disregarded in the ‘outer’ infinitely long straight segments of the pipeline. As it can be seen in Figure 13, for relatively small rates of structural damping, η < 10-3, there is no significant change in the obtained stop-band behaviour. It can also be seen that when the damping ratio of η becomes large, the stop-band effect is masked by the overall damping. The effect of repeated ‘periodicity cells’ in generation of frequency stop bands in an infinitely long structure is explained by the phenomenon of interference of waves reflected by these cells. In the case of forced response of a periodic infinite structure, this interference dramatically reduces its admittance. It results in the reduction of the energy input into the system and, therefore, in the suppression of travelling waves. However, the periodicity effect is also observed in the case of forced vibrations of a finite structure with no material losses.

3.2 Floquet theory and FE method – standing wave analysis Forced vibrations of the structure shown in Figure 14 are analysed by the Floquet theory and by the finite element code ANSYS11 using its beam44 elements where the shear stress factor is chosen to ks = 0.56. As already mentioned, the formulation of the Floquet theory involves the boundary integral equation method. The simply supported beam is loaded by a point harmonic force at the distance 3m from its left end. The force has the 10N amplitude of components in each direction of the coordinate system. For the power flow in an infinitely long periodic structure, the Floquet theory predicts stop bands in the frequency ranges shown in Figure 15. The shapes of forced vibrations (the shape of standing waves) for the excitation frequencies inside and outside stop bands obtained by the finite element method are shown in Figure 16. As is seen, the excitation inside the stop bands generates the standing waves strongly localized between the left edge and the periodic inclusion. If the excitation frequency is not in the stop band, then the forced vibrations are such that the whole pipe is involved in the harmonic motion. The element length is here set to 0.02m. These standing wave shapes are calculated for two excitation frequencies inside the predicted stop-bands 580 and 810Hz and two outside 450 and 650Hz, see Figure 15.

Figure 14. The simple supported empty pipe with five periodicity cells inserted

9

0 100 200 300 400 500 700 800 9000

0.5

1

1.5

2

2.5

3

3.5

4

Hz600

Figure 15. The magnitude of |λ| versus excitation frequency, stop-bands are identified by condition |λ| ≠ 1

XYZ

XYZ

XYZ

XYZ

XYZ

XYZ

XYZ

XYZ

Figure 16. Shapes of standing waves in the simple supported beam illustrated on Figure 14, seen from side and top view, modelled by help of ANSYS11 with beam44 elements

3.3 Parametric study As discussed in the previous sub-sections, insertion of several ‘periodicity cells’ in a straight pipe produces strong effect of suppression of wave propagation in an infinitely long structure and strong localisation effect in the shape of forced vibrations of a structure of finite length. This ‘periodicity cell’ has several geometry parameters, which may readily be tuned to produce the attenuation effect in prescribed frequency ranges. To illustrate possibility to tailor the location of the frequency stop bands, their boundaries, defined by the condition |λ| = 1 are plotted in Figure for a pipe with and without water inside. In this Figure, the pairs of the frequency and the angle of the opening of the segment, which fall into a stop band, are non-shadowed. In opposite, the combinations of these two parameters, which do not produce a stop band effect, are located in the shadowed zones.

10

Figure 17. The pipe segment employed as a periodicity cell. Radius of curvature is kept constant while angle of curvature θ are varied. Pipe diameter 30mm and wall thickness 2mm, ρs = 7800 kg/m3, E = 210GPa and ν = 0.3.

Figure 18. Pass band |λ| = 1 (shadowed) and stop bands |λ| 1 (non-shadowed) for an infinitely long empty pipe As it is seen, tailoring of stop bands to the prescribed excitation frequencies is feasible by changing the geometry of the implemented sub-structure. Therefore, the stop-band effect can be used as a tool of passive control of vibrations in the piping system. It should be noted that the stop-band behaviour is not too sensitive to small changes in angle of curvature of the curved segments.

Exercise For a periodic plate shown in Figure 1, derive Floquet equation and find pass- and stop-bands for plane dilatation waves (Section 1.4). Consider a forcing problem for a semi-infinite plate, which has 2, 4 and 6 pe4riodicity cells. Compare insertion losses in these cases. Use data from exercise 1.11 and lengths 0.4m and 0.1m.

Theoretical and experimental analysis of the stop-band behaviorof elastic springs with periodically discontinuous of curvature

Alf S�e-Knudsen,a) Radoslav Darula, and Sergey SorokinDepartment of Mechanical and Manufacturing Engineering, Aalborg University, Fibigerstraede 16,9220 Aalborg East, Denmark

(Received 21 December 2011; revised 23 June 2012; accepted 2 July 2012)

The advanced design of springs with tailored transmission characteristics is proposed. The

existence of frequency stop-bands in the transmission characteristics is theoretically predicted and

experimentally validated. The underlying mechanism of generating stop-band behavior is the

periodic discontinuity of the curvature. The theoretical analysis is performed in the framework of

the Floquet theory for an infinitely long spring. Experimental results are obtained for the springs

consisting of a small number of repeated periodicity cells.VC 2012 Acoustical Society of America. [http://dx.doi.org/10.1121/1.4740480]

PACS number(s): 43.40.At, 43.40.Cw, 43.40.Tm [KML] Pages: 1378–1383

I. INTRODUCTION

Helical springs have been used in vibration isolation for

decades. Although several alternative geometries are being

employed in technical applications (for example, conical

springs), the overwhelming majority of modern suspension

systems are equipped with cylindrical springs. These springs

usually have a constant cross-section and a constant pitch

angle except for a few laps at the ends, which are winded to

stick together to withstand mounting loads. Their transmis-

sion characteristics may be specified by the static stiffness

only in a very low frequency range, well beyond the first

eigenfrequency. At higher frequencies, the dynamic stiffness

should be employed, and it is strongly frequency-dependent.

Therefore the transfer matrix between force/kinematic input

at one end and the output at another end has both dips and

maxima in arbitrarily chosen frequency bands. Modeling of

dynamics of springs of finite length, as well as analysis of

their dispersion characteristics is a well-established subject,

see Refs. 1–4 and references in these publications.

If a spring is exposed to a broad-band excitation, then it

may be necessary to achieve the best possible transmission

attenuation in one or several prescribed frequency ranges

rather than at individual frequencies. In this case, the dis-

cussed above frequency dependence of transfer matrices is a

serious disadvantage of the classical design of helical

springs, so that its modification might be most helpful. This

paper is concerned with the design of advanced springs,

which employs the periodicity effects to create stop-bands.

In the static limit, however, this spring design recovers the

conventional static stiffness.

The periodicity phenomenon is well known and well

understood.5,6 In the literature, it is typically referred to as

the Floquet theory. According to this theory, a wave guide

composed as an infinite chain of repeated elements with

alternating properties has so-called stop-bands, where the

wave propagation is impossible despite that each individual

component of the chain supports traveling waves within the

stop-band. This phenomenon also exists in the cases when a

homogeneous wave guide is equipped with equally spaced

supports or inertial elements. As is also well known, the de-

sirable attenuation effect is achievable, when a moderate

number of “periodicity cells” designed to fit into the fre-

quency stop-bands are employed.

In recent publication,7 it has been shown that periodi-

cally repeated discontinuities in centerline curvature can be

used to generate the stop-band structures. However, to the

best of the authors’ knowledge, it has not yet been estab-

lished whether compact stop-band helical spring components

can be manufactured and how well the predicted stop-band

behavior fit with experimental data. The focus of this paper

has therefore been given to these tasks.

It should be noted that alternative means may be used to

produce stop-band helical springs. An example of such an al-

ternative is the homogeneous helical spring bearing periodi-

cally repeated point masses. This idea for slender structures

with stop-band properties was formulated in, e.g., Ref. 6. Its

application for design of stop-band springs is theoretically

shown in Ref. 8.

In this paper, we validate the theoretical predictions

obtained by means of the classical Floquet theory for an

infinitely long spring with periodically discontinuous curva-

ture of the centerline in experiments performed with these

springs of finite length. The paper is structured as follows.

First, the two generic configurations of a periodicity cell are

presented, and the governing equations of the Timoshenko

theory of spatially curved beams are given. Then the meth-

odology of combining the boundary integral equations

method with the Floquet theory is briefly outlined, and the

stop-bands for infinite “stop-band” structures are found.

Finally, the comparison between theoretical predictions and

experimental results is presented and possibilities of the

application of the stop-band springs are discussed as well as

the limitations of the proposed design concept.

II. THE “ADVANCED SPRING” CONFIGURATIONS

As discussed in the Introduction, the aim of this paper is

to propose the design of a spring, which is capable to exhibit

a)Author to whom correspondence should be addressed. Electronic mail:

[email protected]

1378 J. Acoust. Soc. Am. 132 (3), September 2012 0001-4966/2012/132(3)/1378/6/$30.00 VC 2012 Acoustical Society of America

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the stop-band behavior in prescribed frequency ranges. The

shape of the “double spring” shown in Fig. 1(a) is chosen as

the main example. In addition, the design of a curved rod,

shown in Fig. 1(b) is also investigated. The underlying

mechanism of the stop-band behavior in both cases is the

periodically repeated discontinuities of centerline curvature

clearly seen in both structures. The double spring in Fig. 1(a)

is called hereafter the “spiral-eight” spring, and the structure

shown in Fig. 1(b) is called the “flat-bent” spring. As is seen,

the periodicity cells employed in these two designs are seg-

ments of a conventional helical spring. In the first case, the

spring element has a non-zero pitch angle, in the second

case, the ring segments are used.

III. GOVERNING EQUATIONS

For modeling of linear dynamics of a spatially curved

rod, the standard Timoshenko beam theory, which accounts

for the constant curvature of a rod, is used. The governing

equations can be found in Ref. 9. In-plane and out-of-plane

dynamics of a plane circular rod can be described by the

same set of governing equations with the pitch angle w beingequal to zero. Furthermore, by letting the radius of curvature

R tends to infinity, the governing equations for the straightrod are also recovered:

qA@2u

@t2¼ @Ru

@sþ cos

2wR

Rw �sin w cos w

RRv; (1a)

qA@2v

@t2¼ @Rv

@sþ sin w cos w

RRu; (1b)

qA@2w

@t2¼ @Rw

@s� cos

2wR

Ru; (1c)

qJx@2a@t2¼ @Ma

@sþ cos

2wR

Mc�sinw cosw

RMb�Rv; (1d)

qJy@2b@t2¼ @Mb

@sþ sin w cos w

RMa þ Ru; (1e)

qJp@2c@t2¼ @Mc

@s� cos

2wR

Ma; (1f)

where

Ma ¼ EJx@a@sþ cos

2wR

c� sin w cos wR

b

� �; (2a)

Mb ¼ EJy@b@sþ sin w cos w

Ra

� �; (2b)

Mc ¼ GJp@c@s� cos

2wR

a

� �; (2c)

Ru ¼ jGA@u

@s� sin w cos w

Rvþ cos

2wR

w� b� �

; (2d)

Rv ¼ jGA@v

@sþ sin w cos w

Ruþ a

� �; (2e)

Rz ¼ EA@w

@s� cos

2wR

u

� �: (2f)

In Eqs. (1) and (2), s is the natural centerline coordinate andt is time. The material parameters are: Young’s modulus E,the shear modulus G, the shear coefficient j, and the massdensity q. The cross section parameters are given as: thecross sectional area A, the second order area moment aroundx axis Jx, the second order area moment around y axis Jy, thepolar second order area moment Jp. The centerline parame-ters for the described helix are: the coil radius R, and thepitch angle w. The coordinate system is shown in Fig. 2. Thedisplacements u, v, and w are counted along the coordinatesx, y and z. The rotations a, b, and c are counted around the x,y, and z axes, in this order.

IV. STOP-BAND PREDICTION: DYNAMICALSTIFFNESS MATRIX AND FLOQUET THEORY

Stop-band analysis of an infinitely long chain of

repeated substructures is performed in the framework of the

classical Floquet theory for their linear time-harmonic

response. The periodicity cells for the springs shown in

Fig. 1 are presented in Fig. 3. Here the front cross sections of

the substructures are designated by “a” and the rear cross

sections are designated by “b.” The notation “c” is used to

indicate the inner node linking the two segments.

The behavior of each segment of a rod with the constant

curvature can be described by the frequency-dependent dy-

namical stiffness matrix SS, which provides the link betweenthe force/moment and displacement/rotation boundary state

FIG. 1. The periodic springs with discontinuous curvature of center line:

(a) “spiral-eight” spring; (b) “flat-bent” spring.

FIG. 2. The geometry of a helical spring: the system of coordinates (x, y, z),the coil radius R, the pitch angle w, and the cross section diameter d.

J. Acoust. Soc. Am., Vol. 132, No. 3, September 2012 S�e-Knudsen et al.: Stop-band behavior of elastic springs 1379

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variables at cross sections “a,” “c,” and “b” [the indices are

omitted in Eq. (3)]:

f ¼ SSu: (3)

The dynamical stiffness matrix is generated by use of the

boundary integral equations method as described in Refs. 7

and 10 and in the more detailed form in Ref. 11. The dynam-

ical stiffness matrix for a helix is available as soon as the

Green’s matrix, G(s,n), and its supplement, which formu-lates the force/moment response, F(s,n), are obtained by useof the wave numbers. In the condensed form, the dynamical

stiffness matrix is written as

SS ¼ �Z�1Y (4)

where

Z ¼�

Gð0; 0Þ GðL; 0ÞGð0; LÞ GðL; LÞ

�; Y ¼

�ð�Fð0; nÞj0þl � IÞ FðL; 0Þ�Fð0; LÞ ðFðL; nÞjL�l � IÞ

�; l! 0:

The derivation of the Green’s matrix for the governing Eqs.

(1) and (2) of the Timoshenko theory is similar to the deriva-

tion for the Bernoulli–Euler theory.10

We do not reproduce this derivation here for brevity, but

Sec. III B in this reference gives its full account. The Green’s

matrix is used to formulate the boundary integral equations as

explained in Sec. IV of the same reference. These boundary

integral equations are, in turn, re-written in the dynamical

stiffness matrix format (3), see also Ref. 7. We note that any

other method (say, the spectral element method or the finite

element method) may certainly be used to derive the dynamic

stiffness matrices for spiral-eight and flat-bent springs and to

yield the results reported in the Sec. V of this paper.

Each periodic substructure consists of the two curved

segments as illustrated on Fig. 3. Therefore the dynamical

equation system for each periodicity cell is also formulated

as an ensemble of two sets of equation systems, being

merged together at the interfacial cross section c. The related

degrees of freedom, uc, are condensed in the sub-matricesSaa, Sab, Sba, and Sbb, see Eq. (5):

fa

0

fb

8><>:

9>=>; ¼

SCaa SCac S

Cab

SCca SCcc S

Ccb

SCba SCbc S

Cbb

264

375

ua

uc

ub

8><>:

9>=>;

+fa

fb

� �¼

Saa Sab

Sba Sbb

� �

where

Saa ¼ SCaa � SCacðSCccÞ�1SCca

Sab ¼ SCab � SCacðSCccÞ�1SCcb

Sba ¼ SCba � SCbcðSCccÞ�1SCca

Sbb ¼ SCbb � SCbcðSCccÞ�1SCcb :

8>>>><>>>>:

(5)

The coupling between displacements/rotations and forces/

moments of the nodes at the end cross section a and the end

cross section b can be described as follows:5

� kJa ¼ b; where k ¼ expðiKBÞ;

a ¼fa

ua

� �; b ¼

fb

ub

� �and J ¼

I 0

0 �I

� �: (6)

Then it is possible to cast Eq. (5) into the following form

with only vector a being involved:

0 ¼��I Saa

0 Sba

�� k�

0 Sab

�I Sbb

�J

�a;

where SS ¼�

Saa Sab

Sba Sbb

�: (7)

When the generalized linear eigenvalue problem formulated

as Eq. (7) is solved, the set of eigenvalues k is found. It isappropriate to re-write the expression for k as follows:

k¼ expðiKBÞ¼ expðReðiKBÞÞ � ðcosðImðiKBÞÞþ isinðImðiKBÞÞÞ: (8)

The free wave propagation is impossible and the stop-band

effect is achieved when all eigenvalues fulfill the condition

that jkj 6¼ 1. This condition implies that no real-valuedpropagation constants KB exist at a given frequency. There-fore the role of KB in assessment of the waveguide proper-ties of periodic compound waveguides is exactly the same

as the role of wavenumbers for homogeneous waveguides.

This equivalence is featured in the wave finite element

method.

V. THEORETICAL ANALYSIS OF LOCATION OF THESTOP-BANDS

For the spiral-eight and the flat-bent structures, material

parameters have been chosen as follows: E¼ 210 GPa,v¼ 0.3, and q¼ 7500 kg/m3. The shear coefficient for thesolid circular cross section is12 j ¼ 6ð1þ �Þ=7þ 6�. Forthe chosen value of the Poisson ratio, it yields j¼ 0.87. The

FIG. 3. The periodically repeated substructures: (a) periodic cell of the

spiral-eight spring and (b) periodic cell of the flat-bent spring.

1380 J. Acoust. Soc. Am., Vol. 132, No. 3, September 2012 S�e-Knudsen et al.: Stop-band behavior of elastic springs

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results obtained by means of the Floquet theory are pre-

sented in Fig. 4, where the magnitudes of jkj are presented asa function of frequency for compound waveguides composed

as infinite chains of either spiral-eight or flat-bent periodicity

cells shown in Fig. 3. As is seen, each waveguide exhibits

the stop-band behavior in the frequency ranges, where

jkj 6¼ 1.

VI. EXPERIMENTAL SETUP

The setups shown in Figs. 5(a) and 5(b) have been used

for experimental verification of the theoretically predicted

stop-band behavior of the spiral-eight spring and the flat-

bent spring, respectively. In both cases, a shaker is connected

to a freely suspended spring (consisting of eight repeated

substructures) via a thin connection rod, exciting the system

in a uni-axial direction. The other end of the spring is fixed

into the fixture plate where the tri-axial accelerometer is

mounted. This plate is also suspended as shown in Fig. 5

(see also Fig. 6) with 1.5 m long soft rubber strips. The appa-

ratus used is listed in Table I.

The uni-axial shaker excitation of the spiral-eight spring

is performed parallel to the used helix segments centerlines,

see Fig. 5(a). The measured accelerations in x and y direc-tions correspond to the in-plane motion of the fixture plate.

The accelerations measured in the z direction correspond to

motion in direction perpendicular to the fixture plate, i.e., in

the direction of excitation. For the flat-bent configuration

illustrated at Fig. 5(b), the shaker excitation is performed at

the angle of 47� relatively to the z-x plane (in-plane) of theplanar curved rod. Measurements are controlled and data

pre-processed in software Bruel&Kjaer PULSE LABSHOP. A

post-processing is done in MATLAB. The qualitative analysis

is of interest, therefore just the autospectra are compared.

The frequency response of the system is measured in

two excitation cases. The first one is an excitation in the

form of broad-band random noise. The second case is a

sweep-sine, i.e., a sinusoidal excitation with a frequency

being gradually varied in time. In the case of broad-band

random noise excitation, the autospectra are averaged line-

arly in spectrum domain, where 100 averages are performed.

An overlap is chosen 66.67%, and a time signal is weightedusing a Hanning window. The frequency range of sweep

sine is from 20 Hz to 1.6 kHz, and the sweep rate is chosen

10 Hz/s. The autospectra are recorded each second, and a

vibration response at excitation frequency is extracted. In

FIG. 4. Magnitudes of jkj for (a) thespiral-eight spring, with w¼ 0.049 rad(stop-bands from 259 to 427 Hz and

from 595 to 950 Hz); (b) the flat-bent

spring (stop-bands from 129 to 367 Hz

and from 461 to 977 Hz).

FIG. 5. The experimental setup: (a)

with the spiral-eight spring; (b) with

the flat-bent structure.


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such a way, the frequency response over the whole excitation

band is obtained. In the case of sine-sweep measurements,

no spectral averaging is applied.

VII. THE EXPERIMENTAL RESULTS AND DISCUSSION

The autospectra measured in each direction for both

types of excitation and the two spring designs are presented

in Fig. 7. The theoretical predictions of the stop-band fre-

quency ranges are marked as the dashed vertical lines. It can

be clearly seen that for all measurements, there is a very rea-

sonable agreement between the predictions from Floquet

theory and the experimental data showing a sharp reduction

of the vibration amplitude in these frequency ranges. In the

both excitation cases, the location of frequency stop-bands is

the same.

In measurements of the response of the spiral-eight

spring to random-noise excitation, a single resonant peak in

y direction at around 900 Hz occurs [Fig. 7(a)]. However,this resonance is not observed in the response of the same

spring exposed to the sine-sweep excitation. The detailed

analysis by means of Bruel&Kjaer PULSE LABSHOP system

showed that this is a higher order non-linear effect, which is

not captured in the framework of the linear theory employed

in this paper. This experimental result hints toward the limi-

tations of the linear theory but does not undermine its quali-

tative predictions.

From the experimental results, we conclude that the

design of vibration isolators for prescribed frequency bands

as spatially curved rods with discontinuous curvature is a

meaningful alternative to existing designs of such devices.

The close agreement between predictions of the Floquet

theory and power flow analysis in periodic structures with

FIG. 6. The tri-axial accelerometer mounted on the fixture plate.

TABLE I. The equipment used for measurements in the experimental setup.

Type Device

Data acquisition Bruel&Kjaer Pulse 3506 C

Electrodynamic vibration exciter Bruel&Kjaer 4809

Power amplifier Bruel&Kjaer 2706

Tri-axial accelerometer Bruel&Kjaer 4321

Charge-to-DeltaTron converter 3 x Bruel&Kjaer 2647 A

FIG. 7. Amplitudes of acceleration of the spiral-eight spring: (a) random-noise excitation, (b) sine-sweep excitation; and of the flat-bent spring: (c) random-

noise excitation, (d) sine-sweep excitation.

1382 J. Acoust. Soc. Am., Vol. 132, No. 3, September 2012 S�e-Knudsen et al.: Stop-band behavior of elastic springs

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moderate number of periodicity cells has been demonstrated

in numerical simulations, see Ref. 7. It has been found that

the insertion loss is linearly dependent upon the number of

periodicity cells and that the material damping masks the pe-

riodicity effect. It is reasonable to anticipate that the influ-

ence of these parameters on the transmission characteristics

of spiral-eight and flat-bent springs should be qualitatively

the same. Design recommendations and specific case-

studies, where the limitations on the space available for in-

stallation of an isolator, probably, confront the requirements

on the attenuation level, lie beyond the scope of this paper.

The absence of real-valued propagation constants inside

the stop-bands means that all waves excited at the loaded

end of the spring are attenuated and their amplitudes decay

along the length of the spring. This phenomenon may be

explained as the destructive interference of the incoming

wave and the waves reflected at the discontinuities of curva-

ture. Therefore the periodicity-induced vibration isolation is

unlikely to be sensitive to the boundary and loading condi-

tions at the ends of the advanced spring. The invariance of

the structural response to the excitation conditions is

documented in reference,7 where the periodicity-induced

insertion loss has been computed for all possible types of

time-harmonic point excitation.

VIII. POSSIBLE INDUSTRIAL APPLICATIONS ANDLIMITATIONS

The experiments have been conducted with flexible

spring designs, and, therefore, small excitation forces have

produced the measured accelerations. The large-amplitude

vibrations of the spiral-eight and the flat-bent springs have

not been studied because these designs obviously cannot

qualify for immediate industrial application. However, it is

certainly possible to find radii of curvature and diameters of

the wire for the spiral-eight and flat-bent springs, which pro-

vide their sufficient stiffness in the static limit and preserve

the stop-band behavior in the specified frequency ranges.

The efficiency of the proposed design of the stop-band

springs in industrial applications may possibly be limited due

to non-stationary excitation. The response of elastic wave

guides with periodic inserts to transient impact-type excitation

requires further theoretical and experimental analysis.

IX. CONCLUSION

The periodically repeated discontinuities in the curva-

ture of elastic rods can be utilized to design the vibration

isolators with prescribed stop-band frequency ranges. The

theoretical background of such a design is constituted by the

Floquet theory in which the dynamical stiffness matrices for

the periodicity cells are employed. The dynamical stiffness

matrices within the employed Timoshenko beam model may

be obtained either approximately by means of the finite ele-

ments method, or exactly, by means of the boundary integral

equations method and Green’s matrices as has been done in

this paper.

The reported experimental results have proved that the

designs of the spiral-eight and the flat-bent springs provide

the significant vibration isolation in the stop-band frequency

ranges predicted by the Floquet theory.

The spiral-eight design of a helical spring is particularly

advantageous. It is as compact, as the conventional design,

and yet it provides the stop-band effect in practically mean-

ingful frequency ranges.

1J. E. Mottershead, “Finite elements for dynamical analysis of Helical

Rods,” Int. J. Mech. 22, 267–283 (1980).2J. Lee and D. J. Thompson, “Dynamic stiffness formulation, free vibra-

tion and wave motion of helical springs,” J. Sound Vib. 239, 297–320(2001).

3J. Lee, “Free vibration analysis of cylindrical helical springs by the pseu-

dospectral method,” J. Sound Vib. 302, 185–196 (2007).4V. Yildirim, “Investigation of parameters affecting free vibration fre-

quency of helical springs,” Int. J. Numer. Methods Eng. 39, 99–114(1996).

5L. Brillouin, Wave Propagation in Periodic Structures, 2nd ed. (DoverPublications, New York, 1953), Chap. 4.

6D. J. Mead, “Wave propagation in continuous periodic structures: research

contributions from Southampton, 1964–1995,” J. Sound Vib. 190, 495–524 (1996).

7A. S�e-Knudsen, and S. V. Sorokin, “Modelling of linear wave propaga-tion in spatial fluid filled pipe systems consisting of elastic curved and

straight elements,” J. Sound Vib. 329, 5116–5146 (2010).8E. Manconi, L. Paganini, A. S�e-Knudsen, and S. V. Sorokin, “Optimumdesign of a periodic pipe filter using waves and finite elements,” in

Proceedings, 8th International Conference on Structural DynamicsEURODYN, Leuven, Belgium (July 4–6, 2011), Paper MS21-1013,pp. 3118–3124 (available on CD-ROM from Department of Civil Engi-

neering, Katholieke Universiteit, Leuven, Belgium).9W. H. Wittrick, “On elastic wave propagation in helical springs,” Int. J.

Mech. Sci. 8, 25–47 (1966).10S. V. Sorokin, “The Green’s matrix and the boundary integral equation for

analysis of time-harmonic dynamics of elastic helical springs,” J. Acoust.

Soc. Am. 129, 1315–1323 (2011).11A. S�e-Knudsen, “On advantage of derivation of exact dynamical stiffness

matrices from boundary integral equations (L),” J. Acoust. Soc. Am. 128,551–554 (2010).

12I. H. Shames, and C. L. Dym, in Energy and Finite Element Methods inStructural Mechanics, edited by S. I. Units (Taylor and Francis Books,New York, 1991), pp. 200.


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Rapid Communications

On accuracy of the wave finite element predictions of wavenumbersand power flow: A benchmark problem

Alf Søe-Knudsen n, Sergey Sorokin

Department of Mechanical and Manufacturing Engineering, Aalborg University, Pontoppidanstraede 101, Aalborg East DK-9220, Denmark

a r t i c l e i n f o

Article history:

Received 6 December 2010

Received in revised form

9 February 2011

Accepted 21 February 2011

Handling Editor: G. Degrande

a b s t r a c t

This rapid communication is concerned with justification of the ‘rule of thumb’, which

is well known to the community of users of the finite element (FE) method in dynamics,

for the accuracy assessment of the wave finite element (WFE) method. An explicit

formula linking the size of a window in the dispersion diagram, where the WFE method

is trustworthy, with the coarseness of a FE mesh employed is derived. It is obtained by

the comparison of the exact Pochhammer–Chree solution for an elastic rod having the

circular cross-section with its WFE approximations. It is shown that the WFE power

flow predictions are also valid within this window.

& 2011 Elsevier Ltd. All rights reserved.

1. Introduction

The wave finite element method (WFE) is often used in situations, when the wave propagation in a slender structure

should be analysed, but wavenumbers cannot be found analytically (see [1–6]). Unlike analytical methods, WFE method

generates wavenumbers dependent on a type and a size of the finite elements employed. As is recognised, a few lowest

branches in the dispersion diagram obtained with WFE are genuine and accurately found, with other branches being

spurious or inaccurate.

Therefore, benchmark problems of elastic wave propagation in rods, thin plates, thin cylindrical shells, and similar have

been solved to prove the validity of WFE predictions of location of dispersion curves in Mace et al. [2] and Manconi and

Mace [5]. However, to the best of the authors’ knowledge, the benchmarking of this method by comparison of location of

dispersion curves with the classical Pochhammer–Chree solution has not yet been done. Likewise, the inaccuracies in

predicted location of dispersion curves have not yet been traced to the inaccuracy in the power flow computations. This

rapid communication fills in the gap. Section 2 contains the brief reminders of the WFE technique and the Pochhammer–

Chree solution. Sections 3 and 4 are concerned with comparison of dispersion diagrams and power flows, respectively.

2. Wavenumbers

The dynamic properties of slender structures, which may be modelled as linear elastic waveguides, are controlled by

their wavenumbers. The standard harmonic time- and space-dependence assumption, u(x,t)¼c exp(ÿ iotþ ikx), reduces asystem of governing linear differential equations in spatial and temporal coordinates to the dispersion equation. It may

have a polynomial (in elementary situations, for instance, a standard beam theory) or a transcendental form (for instance,

the Rayleigh–Lamb problem for an elastic layer). The magnitudes of purely real wavenumbers and the associated

Contents lists available at ScienceDirect

journal homepage: www.elsevier.com/locate/jsvi

Journal of Sound and Vibration

0022-460X/$ - see front matter & 2011 Elsevier Ltd. All rights reserved.

doi:10.1016/j.jsv.2011.02.022

n Corresponding author. Tel.: þ45 9940 7328; fax: þ45 9815 1675.E-mail address: [email protected] (A. Søe-Knudsen).

Journal of Sound and Vibration 330 (2011) 2694–2700

eigenmodes govern the transmission of the vibroacoustic energy in waveguides. The knowledge of the imaginary and

complex-valued wavenumbers is needed to predict the interaction between travelling waves in the boundary layers at

discontinuities and at loaded points.

2.1. The wavenumbers and modal vectors obtained by the WFE method

The wavenumbers, k, for an infinitely long rod with an arbitrarily shaped cross-section can be obtained numerically by

considering a finite element model of a ‘slice’ of this structure in the framework of a modification of the finite elements

method, known as WFE, as well described in e.g. [1]. The version of this method presented in the paper should be seen as

one of possibilities (see e.g. [2,3]). This approach works well for the relatively simple problems but other formulations

should probably be employed for more advanced geometries of cross section (see e.g. [4–6]).

The dynamical equations for a slice of a uniform infinitely long structure can be written in the classical finite element

style where only nodes at its left and right side are involved. In Eq. (1), note that ‘a’ and ‘b’ mark the left and the right edges

of the slice, and Le is its length

fa

fb

( )

¼ Ssua

ub

( )

where Ss ¼ ½KSÿo2MS (1)

To secure that the system of equations is well conditioned, it is helpful to scale Ss, ua and ub, so the largest component

in the dynamical stiffness matrix becomes equal to one, as given here

fa

fb

( )

¼ ŜSûa

ûb

( )

, where ŜS ¼ Ss1

s,

ûa

ûb

( )

¼ua

ub

( )

s and s¼maxðSsÞ (2)

Since the structure is regarded to be infinitely long, and the slice is modelled by finite elements as its periodically repeated

part (see, for instance, [2]), the coupling between the displacements-rotations and forces-moments in the nodes of the

cross sections ‘a’ and ‘b’ can be described by means of Floquet theory as given below [7,8]:

ÿLJa¼ b, where L¼ expðikLeÞ, a¼fa

ûa

� �

, b¼fb

ûb

� �

and J¼I 0

0 ÿI

� �

(3)

By help of these, it is possible to reformulate Eq. (2) into the following form, only dependent of vector a

0¼ÿI Saa0 Sba

" #

ÿL0 Sab

ÿI Sbb

" #

J

" #

a (4)

With this standard eigenvalue problem being solved, the wavenumbers, k, and associated eigenmodes are easily found

for each root L. The order of the characteristic equation in L is determined by the amount and the type of finite elements

used in the model. As discussed, some of the computed wavenumbers are authentic for the considered wave guide, but

some wavenumbers are spurious. It is necessary to distinguish between them and, furthermore, to assess the accuracy of

numerical calculations of wavenumbers. These tasks are easily accomplished in the elementary cases listed in the

Section 1, because exact dispersion equations have the polynomial form. As soon as a more advanced wave guide should

Nomenclature

A cross section area

c1 speed of dilatation wave

c2 speed of shear wave

c vector containing a set of constants

E Young’s module

f vector containing forces and moments

G shear module

J axial moment of inertia

k wavenumber

K non-dimensional wavenumber

K stiffness matrix

Le length of the mesh segment

M mass matrix

r radial coordinate

R cross section radius

Ss dynamical stiffness matrix

ŜS scaled dynamical stiffness matrix

u vector containing displacements and

rotations

û vector containing scaled displacements and

scaled rotations

x axial coordinate

z number of elements per wave length

y angular coordinate

k shear coefficientl Lamé coefficient

L periodic constant

m Lamé coefficientn Poisson ratior mass densitys scaling parametero circular frequencyO non-dimensional frequency

A. Søe-Knudsen, S. Sorokin / Journal of Sound and Vibration 330 (2011) 2694–2700 2695

be considered (for example, wave propagation in a solid elastic rod at high frequencies), the range of reliability of the WFE

predictions is challenged and requires a careful analysis.

2.2. The exact benchmark solution

The classical solution of equations of elasto-dynamics for an elastic straight rod with the circular cross-section

(Pochhammer–Chree solution) (see [9]), is briefly reproduced here for consistency. The displacement field in the

cylindrical coordinates (see Fig. 1), is presented in the form of the Lamé decomposition (see, for instance, [10])

u¼rFþr � ðixC1Þþr �r � ðixC2Þ (5)Here F, C1, C2 are potentials, which satisfy the Helmholz equations

rrFþ o2

c21F¼ 0 (6)

rrCnþ o2

c22Cn ¼ 0, where n¼ 1,2 (7)

Here c1 is the speed of a dilatation wave in unbounded elastic medium, and c2 is the speed of a shear wave.

The traction-free boundary conditions at the lateral surface of a cylinder of the radius R are formulated as follows

(l and m are Lamé coefficients, r¼R):

trr ¼ lrrFþ2m @2F

@r2þ 1

r

@2C1@r@y

ÿ 1r2

@C1@y

þ @3C2

@x@r2

!

¼ 0 (8)

trx ¼ m 2@2F

@x@rþ 1

r

@2C1@x@y

ÿ @3C2@r3

ÿ1r

@2C2@r2

þ 1r2

@C2@r

ÿ 1r2

@3C2

@r@y2þ 2

r3@2C2

@y2þ @

3C2@x2@r

!

¼ 0 (9)

try ¼ m2

r

@2F

@r@yÿ 1r2

@F

@yÿ @

2C1@r2

þ 1r

@C1@r

þ 1r2

@2C1

@y2

ÿ2r

@3C2

@x@r@yÿ 2r2

@2C2@x@y

!

¼ 0 (10)

The potentials must also be bounded at the axis r¼0, so that the general solutions of Eq. (5) acquire the simple form

Fðx,r,yÞ ¼X

1

m ¼ 0AmJmðk1rÞcosmyexpðikxÞ, where k1 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

o2

c21ÿk2

s

(11)

C1ðx,r,yÞ ¼X

1

m ¼ 0BmJmðk2rÞsinmyexpðikxÞ, where k2 ¼


o2

c22ÿk2

s

(12)

C2ðx,r,yÞ ¼X

1

m ¼ 0CmJmðk2rÞcosmyexpðikxÞ, where k2 ¼


o2

c22ÿk2

s

(13)

These formulas are substituted to the boundary conditions (8)–(10), and the determinant of three homogeneous

algebraic equations with respect to Am, Bm. Cm is equated to zero. It yields a set of dispersion equations for each individual

circumferential wavenumber m. Each of these dispersion equations is transcendental, and has infinitely many roots, which

are either real, or imaginary, or complex-valued.

3. Dispersion diagrams

The finite element model is set up by use of the linear eight-node solid elements (ANSYS11’s Solid45 element). The

number of elements and the length of the meshed segment of an infinitely long rod are varied and the location of

Fig. 1. The cylindrical coordinate system used for the Lamé decomposition.

A. Søe-Knudsen, S. Sorokin / Journal of Sound and Vibration 330 (2011) 2694–27002696

approximate dispersion curves is compared with location of the exact dispersion curves. The three meshes used for the

WFE analysis are shown in Fig. 2.

In the FE modelling of structural dynamics, it is the generally recognised ‘rule of thumb’ for relatively low frequencies

that, to secure accuracy of calculations, it is sufficient to choose the largest element length, Le, to be about 1/12 to 1/10 of

the shortest wave length of interest, kc:

Le ¼2p

zkcwhere z¼ 10. . .12 (14)

It should be observed that this simple formula is not applicable at high frequencies because of pollution errors,

associated with variation nature of the FE method formulation (see [11] for details). In the high frequency regime (which

lies beyond of the scope of the present communication), pollution errors could be removed when, e.g. the ‘refined rule of

thumb’ as given in [12, p. 186], has been applied.

In a solid elastic straight rod, the largest wavenumber (and, hence, the shortest wave length) is attributed to the

propagating flexural wave, which is very accurately captured by the Timoshenko beam theory. Its wavenumber is given by

kT ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

oJEkGðorJðEþkGÞþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

o2r2J2ðE2ÿ2EkGþk2G2Þþ4rAJEk2G2p

q

JEkGffiffiffi

2p (15)

Eq. (15) employed with shear correction factor k¼6(1þn)2/(7þ12nþ4n2) [13] is in the perfect agreement with theexact solution. This equation can be combined with the ‘rule-of-thumb’ Eq. (14) to determine the threshold magnitude of

o, up to which the WFE method employed with the given finite element type and mesh gives a reliable prediction ofwavenumbers. This task may be identified as the ‘WFE windowing’ of the dispersion diagram.

The selected size of finite elements introduces the threshold range of wavenumbers kc¼2p/(zLe) as follows fromEq. (15). Then the threshold frequencies are found from Eq. (16) used with z¼10 and 12

oc ¼

ffiffiffiffiffiffiffiffi

1

2rJ

s

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Jk2c ðEþkGÞþkGAÿZq

where Z¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

J2k4c ðE2ÿ2EkGþk2G2Þþ2JAk2c ðEkGþk2G2ÞþA2k2G2q

(16)

In the case under consideration, this formula is written in the non-dimensional form given in Eq. (17) with the non-

dimensional wavenumber K¼kR and the non-dimensional frequency parameter O¼oR/c1

Oc ¼


1ÿ2n4ð1ÿnÞ

s

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2K2c ð1þnÞþkðK2c þ4ÞÿHq

where H¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

4K4c ð1þ2nþn2Þþkð1þnÞð16K2c ÿ4K4c Þþk2ð16þ8K2c þK4c Þq

(17)

From the mathematical point of view, there is no difference between purely real and complex roots of dispersion

equation, so that the window in the dispersion diagram concerned with imaginary wavenumbers, i.e. those with Re(K)¼0,may be set up by employing the formula Im(K)¼ÿKc¼ÿkcR with z¼10 and 12.

The dispersion diagrams in Fig. 3 are plotted for the non-dimensional parameters K and O. Only purely real and purely

imaginary wavenumbers are displayed, because the accuracy of WFE in calculations of complex-valued wavenumbers is

the same. Therefore, their real and imaginary parts are not presented in this figure for clarity. Unlike the FE solution, the

solution of exact dispersion equation is obtained for each circumferential wavenumber m independently, and the exact

dispersion curves (dotted lines) belonging to different spectra are designated differently: ‘*’ for m¼0, ‘W’ for m¼1, ‘X’ form¼2, ‘J’ for m¼3 and ‘þ ’ for m¼4.

Fig. 2. Three finite element meshes for the segment of a rod with the eight-node solid elements (ANSYS11’s solid45) used: (a) 16 elements, the segment

length Le/R¼0.5, (b) 64 elements, the segment length Le/R¼0.25, and (c) 196 elements, the segment length Le/R¼0.14.


For each case illustrated in Fig. 2, Eqs. (14) and (17) are used to draw the grey ‘transition strips’, z¼10 to 12 separatingthe windows, where WFE predictions are reliable, with the rest of the plane of dispersion diagram.

There are no other purely real wavenumbers except those within the windows in the windowed frequency ranges, so

that the energy transmission phenomena should be captured with sufficient accuracy by means of WFE method up to the

threshold frequency. However, just a few evanescent waves are adequately captured by this methodology. It suggests that

the boundary layer effects at the loaded zones and at the discontinuities in geometry or in material properties are

described by the WFE method in an approximate manner.

For the three examples given in Fig. 3a–c the relation between Kc, Oc, requested length of the slice and the requested

mesh fineness are summarised in Table 1. The reported values should be seen as qualified ‘rule-of-thumb’ guesses

estimating validity range of the WFE method.

4. Power flow

As shown in the previous section, the WFE method correctly predicts the wavenumbers within the window in the

dispersion diagram. Therefore, the power flow calculations by means of the WFE method should be trustworthy within the

Fig. 3. The dispersion diagrams for a solid rod obtained by use of WFE versus exact solution. The numerical solution is marked by ‘’’ (a) the mesh shown

in Fig. 2a is used, (b) the mesh shown in Fig. 2b is used, (c) the mesh shown in Fig. 2c is used. Designation of branches belonging to the exact solution is

made by use of following symbols; ‘*’ for m¼0, ‘W’ for m¼1, ‘X’ for m¼2, ‘J’ for m¼3 and ‘þ ’ for m¼4.

Table 1

Summary of results illustrated in Fig. 3a–c.

Threshold scaled wavenumber Threshold scaled frequency Requested length of the ‘slice’ Requested fineness,

number of elements

KcE1.05–1.25 3 OcE0.34–0.44 3 Le/R¼0.50 and 4�4¼16KcE2.09–2.50 3 OcE0.90–1.12 3 Le/R¼0.25 and 8�8¼64KcE3.74–4.50 3 OcE1.79–2.20 3 Le/R¼0.14 and 14�14¼196


same frequency range. Calculation of the vibro-acoustic power flow are done from Eq. (18) (see [14]). Here ‘‘n’’ indicates

the complex conjugate function

~EfluxðsÞ ¼1

2ReððÿioÞ � fðsÞTuðsÞ�Þ (18)

As an example, the energy flux through cross sections ‘a’ or ‘b’ of the infinitely long rod illustrated on Fig. 4 is calculated.

The following dimensional quantities are used: E¼210 GPa, n¼0.3, r¼7800 kg/m3, R¼0.005 m. For the used numericalSimpson’s rule of integration (see e.g. [15]), step length is 0.015 m. The calculations are performed by using the

wavenumbers obtained numerically as specified in Section 3, for the used wave formulation. The distributed load is

applied equally for all degrees of freedom in the used FE models, so that the force per unit length in the loaded area will be

2 N/m along each coordinate axis.

In Fig. 5a and b, the power flows predicted with the coarser FE meshes shown in Fig. 2a and b are compared with the

power flow calculated with the fine mesh shown in Fig. 2c. In the frequency ranges considered, the latter should be

regarded as the exact solution. It should be noticed that the width of the frequency range displayed in Fig. 5b is the same as

in Fig. 5a, but it is shifted towards higher frequencies. The range of power is the same in both figures.

The ‘transition zones’ reported in Table 1 are shown in Fig. 5 in the same way as in Fig. 3. As expected, the inaccuracies

in calculations of the wavenumbers corrupt accuracy of the power flow predictions. The significant differences between

the results obtained using the 4�4 mesh and the 14�14 mesh emerge in the range of O from 0.34 to 0.44, as illustrated inFig. 5a. The significant differences between the results obtained using the 8�8 mesh and the 14�14 mesh emerge in therange of O from 0.90 to 1.12, as illustrated in Fig. 5b.

5. Conclusion

Dispersion diagrams for an elastic rod of the circular cross-section, obtained by WFE method with the use of the eight-

node linear solid elements are compared with the exact analytical solution. It is shown that the size of a window in the

dispersion diagram, in which the WFE predictions are trustworthy for the chosen FE mesh, can be found from a simple

formula. The power flow predictions obtained by the WFE method are accurate in the same frequency range as the WFE

predictions of wavenumbers.

References

[1] D. Duhamel, B.R. Mace, M.J. Brennan, Finite element analysis of the vibrations of waveguides and periodic structures, Journal of Sound and Vibration294 (2006) 205–220.

Fig. 4. Infinitely long rod with the distributed load; the wavy arrows indicate boundaries at infinity.

Fig. 5. The power flow in cross-sections ‘a’ and ‘b’ (see Fig. 4), calculated using WFE and the three finite element meshes given in Fig. 2: (a) the 16

element meshes given in Fig. 2a (full line) and the 196 elements mesh at Fig. 2c (dashed line). (b) The 64 elements mesh from Fig. 2b (full line) and again

the 196 elements mesh from Fig. 2c (dashed line). The vertical grey band indicates values given in Table 1.


[2] B.R. Mace, D. Duhamel, M.J. Brennan, L. Hinke, Finite element prediction of wave motion in structural waveguides, Journal of the Acoustical Society ofAmerica 117 (2005) 2835–2843.

[3] Y. Waki, B.R. Mace, M.J. Brennan, Numerical issues concerning the wave and finite element method for free and forced vibrations of waveguides,Journal of Sound and Vibration 327 (2009) 92–108.

[4] Y. Waki, B.R. Mace, M.J. Brennan, Free and forced vibrations of a tyre using a wave/finite element approach, Journal of Sound and Vibration 323 (2009)737–756.

[5] E. Manconi, B.R. Mace, Wave characterization of cylindrical and curved panels using a finite element method, Journal of the Acoustical Society ofAmerica 125 (2009) 154–163.

[6] J. Ryue, A waveguide finite element and boundary element approach applied on submerged fluid-filled cylindrical shells, in: Proceedings of theX International Conference on Recent Advances in Structural Dynamics, Sorthampton, United Kingdom, July 2010, paper 121.

[7] L. Brillouin, Wave Propagation in Periodic Structures, second ed, Dover Publications, New York, 1953.[8] M.G. Floquet, Sur les equations differentielles linéairea á coefficents périodiques (On linear differential equations with periodic coefficients), Annales

Scientifiques de l’École Normale Supérieure 2 (1883) 47–88.[9] C. Chree, The equations of an isotropic elastic solid in polar and cylindrical coordinates, their solutions and applications, Transactions of the

Cambridge Philosophical Society 14 (1889) 250–309.[10] Y.-H. Pao, Elastic waves in solids, Transactions of the ASME 50 (1983) 1152–1164.[11] M.J. Crocker, Handbook of Noise and Vibration Control, John Wiley & Sons, New Jersey, 2007, pp. 107.[12] F. Ihlenburg, Finite Element Analysis of Acoustic Scattering, Springer-Verlag, New York, 1998, pp. 186.[13] J.R. Hutchinson, Shear coefficients for Timoshenko beam theory, Journal of Applied Mechanics 67 (2001) 87–92.[14] L. Cremer, M. Heckl, B.A.T. Petersson, Structure-borne Sound: Structural Vibrations and Sound Radiation at Audio Frequencies, third ed, Springer-Verlag,

Berlin, 2005, pp. 337.[15] E. Kreyszig, Advanced Engineering Mathematics, eighth ed, John Wiley & Sons Inc., New York, 1999, pp. 872–874.


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