WAVES AND FINITE ELEMENTS

Brian R Mace

Institute of Sound and Vibration Research University of Southampton

Highfield, Southampton SO17 1BJ, UK E-mail: brm@isvr.soton.ac.uk

Keywords: waves, finite elements, dispersion equation, laminates, cylinders.

ABSTRACT

Vibrations can be described in terms of waves propagating through a structure. This is particularly appealing at higher frequencies, when the size of the structure is large compared to the wavelength. Determining the wave characteristics – wavenumber, group velocity, reflection and transmission coefficients, etc – is straightforward for simple cases such as thin isotropic beams and plates. For more complex structure there are difficulties, however. In this paper a wave finite element (WFE) method for the analysis of the wave behaviour is presented. It involves conventional finite element analysis of just a small segment of the structure. Periodicity conditions are then applied, resulting in an eigenproblem, the solutions to which yield wavenumbers, wave modes and so on. The WFE approach for 1-dimensional waveguides and 2-dimensional structures is described. Examples of the free and forced vibrations of a beam, an orthotropic laminated panel and a vehicle tyre are presented. Complicated dispersion phenomena are observed. The approach gives accurate predictions at small computational cost.

1. INTRODUCTION

The propagation of waves in structures is of interest for many applications. Examples include the transmission of structure-borne sound, shock response, damage detection, statistical energy analysis, acoustic radiation and non-destructive testing. Wave approaches are particularly useful at higher frequencies, when the size and computational cost of finite element (FE) analysis of the structure as a whole becomes impractically large. For homogeneous or piecewise-homogeneous structures, knowledge of the dispersion relations, group velocity, reflection and transmission characteristics etc enables predictions to be made of the response, disturbance propagation, energy transport and so on. For simple cases analytical expressions for the dispersion relations can be found straightforwardly [1,2]. Examples include uniform 1-dimensional structures such as thin rods and beams, and 2-dimensional structures such as thin plates. For structures comprising 1-dimensional

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waveguides (beam networks, space frames etc) the dynamics can be analysed by assembling models of the individual components, either in the wave domain [3,4] or by assembling spectral elements [5].

Analyses of wave motion generally involves assumptions and approximations concerning the stress, strain and displacement states of the structure. As frequency increases (as the wavelength starts to become comparable to the cross-section dimensions) more refined models are required, and the analysis becomes increasingly complicated. For a beam in bending, for example, depending on the frequency range involved one might use Euler-Bernoulli, Rayleigh or Timoshenko theories or a solution based on the equations of 3-dimensional elasticity [1]. For more complicated structures such as cylinders [6], thick orthotropic plates, constrained layer damping treatments [7], laminates [8] (for which higher order, layer-wise theories might be required) or fibre-reinforced structures the analysis becomes far from straightforward. Aside from the need to make assumptions and approximations in the modelling, the resulting dispersion equation is usually transcendental and may be of very high order. Finding all the real, imaginary and complex solutions can be a very challenging task. Occasionally symbolic manipulation is employed to make the development tractable: for example, in deriving the dynamic stiffness matrix for a twisted Timoshenko beam Banerjee [9] derived a set of partial differential equations of 8th order and used a symbolic manipulator to generate the solutions. In studies of laminated sandwich cylinders (an application considered in section 4) Heron [10] and Ghinet and Atalla [11] developed approximate theories resulting in sets of equations of 47th and 42nd order respectively. In both cases numerical solutions were then found. Neither study is a trivial exercise.

Thus for structures of complex construction, such as those shown in Figure 1, numerical approaches are potentially of benefit. In this paper an FE based approach for the analysis of wave motion is described. The structure is homogeneous in the direction (or directions, for 2-dimensional structures) of wave propagation, but the properties can vary in an arbitrary manner over the cross-section of the structure. In this wave/finite element (WFE) method a very small segment of the structure is modelled using conventional FE methods (typically by using a commercial package) and the mass and stiffness matrices found. They are post-

(a)

(c)

(b)

Figure 1. Wave-bearing structures: 1-dimensional waveguides (a) beam (arbitrary cross-section) and (b) smooth tyre; (c) orthotropic laminate sandwich panel.

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processed by applying periodicity conditions for the propagation of a time-harmonic disturbance through the structure, resulting in various eigenproblems whose solutions yield the dispersion relations. Thus periodic structure theory is employed, although the spatial periodicity is arbitrary. Since conventional FEA is used, the full power of existing FE packages and their extensive element libraries can be utilised.

There have been various applications of FEA to spatially periodic structures. Orris and Petyt [12,13] developed methods for free and forced vibration of 1-dimensionally periodic structures. Specific applications include railway tracks [14] and stiffened cylinders [15]. Other authors have applied periodic structure theory and FE to 2-dimensional structures. Abdel-Rahman [16] developed general methods and applied them to beam-stiffened plates. Other applications include Ruzzene et al [17], who considered cellular cored structures, and Pany and Parthan [18], who investigated stiffened curved panels.

For 1-dimensional structures there have been applications of the WFE method for free [19] and forced vibration [20], to rail structures [21] (and, in [14], using periodic structure theory for a track section), laminates [19,22], thin-walled structures [23] and fluid-filled [24,25] pipes and tyres [26]. Mencik and Ichchou [27] applied the method to calculate wave transmission through a joint. The method has been developed for 2-dimensional structures in [28] and applied to axisymmetric structures [29] and to constrained layer damping treatments [30]. Duhamel [31] presents an approach for forced vibration of a 2-dimensional structure. He enforces a harmonic motion of the form exp(-iky) in one direction, so that the equation of motion reduces to that for a 1-dimensional structure and is subsequently solved using the WFE methods in [19,20]. The Green function is then found by evaluating an integral over the wavenumber k.

For 1-dimensional waveguides the WFE approach is somewhat similar to the spectral finite element method (e.g. [32]), although in the latter case new, spectral stiffness matrices must be derived on a case-by-case basis, and this can be a not-insignificant task. The WFE approach typically exploits the power of conventional, commercial FE packages, and damping and complex stiffness matrices can be included straightforwardly.

In the next two sections the WFE method for free wave propagation in 1-dimensional waveguides and for 2-dimensional structures is described. The dispersion relations follow from the eigenvalues of various eigenproblems, while the eigenvectors give the wave modes – the internal degrees of freedom (DOFs) and forces under the passage of the wave. The wave modes give matrices that allow for transformations to be made between the physical domain (in terms of DOFs and forces) and the wave domain (wave amplitudes), which can subsequently be used to predict free wave propagation, forced response, spectral elements, reflection coefficients and so on, as discussed in section 4. In section 5 the illustrative example of a beam in bending is considered, together with applications to an orthotropic laminated cylindrical sandwich panel and the forced response of an automotive tyre, for which solutions using alternative methods are difficult at best.

2. FREE WAVE PROPAGATION IN 1-DIMENSIONAL WAVEGUIDES

Consider a structural waveguide which is homogeneous along its axis x but whose properties may vary through the cross-section in the y and z directions. Examples include a uniform beam of arbitrary cross-section or the tyre in Figure 1. Under the passage of a time harmonic wave of frequency ω any response variable ( ), , ,w x y z t varies as ( ) ( ) ( ), , , , i t kxw x y z t W y z e ω −= (1)

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where k is the wavenumber and W(y,z) is some function over the cross-section of the solid that describes the wave mode in terms of the response variable w. The wavenumber might be real (for propagating waves in the absence of damping), pure imaginary (for evanescent waves) or complex (for oscillating, decaying waves).

A segment of length is cut from the structure and meshed using conventional FEA. Typically a commercial package might be used so that existing element libraries can be exploited. The only additional constraint is that the nodes, DOFs and their ordering on the left and right hand cross-sections are identical. Typically the segment is one element long, although it is straightforward to include internal DOFs [19]. The vectors of nodal DOFs u and forces f are indicated in Fig. 2, and external excitations f

L

ext may act on the nodes. (Strictly, perhaps, these should be referred to as hypernodes formed by concatenating all the element nodes over the cross-section of the segment.) The equation of motion of the segment is 2

, ,; ; ; ;T T T T T T T T Text L R L R ext ext L ext Rω ⎡ ⎤ ⎡ ⎤ ⎡= + = − = = = ⎤⎣ ⎦ ⎣ ⎦ ⎣Dq f f D K M q q q f f f f f f ⎦ (2)

where K, M and D are the stiffness, mass and dynamic stiffness matrices (DSM), T denotes the transpose and the subscripts L and R denote the left and right hand nodes. Damping can be included by a damping matrix C or a complex stiffness matrix.

2.1 Eigenvalue problems

Free wave propagation (fext = 0) of the form of Eq. (1) is determined by applying the periodicity and equilibrium conditions

( ); 0; expR L L R ikLλ λ λ= + = = −q q f f (3) There are various approaches to estimating the eigenvalues λ and the resulting wave mode shapes. The solutions come in pairs, λ and 1λ − , representing the same wave propagating in opposite directions. If there are n DOFs per node the matrices K and M are and there are n pairs of eigenvectors.

n n×

The most straightforward approach is to substitute the periodicity condition into Eq. (2) and premultiply by . This yields, in terms of the partitions of D, the eigenvalue problem

1λ−⎡⎣I ⎤⎦I

) (4) ( 1 0LL RR LR RL Lλ λ−+ + + =D D D D q For the undamped case the propagating waves (real-valued k) can be found by specifying k and solving for ω. Pure-imaginary solutions for k can be found in a similar way.

In the damped case, or if the full dispersion characteristics for given ω are of interest, the solutions for k are complex. In principle they can be found by solving the quadratic eigenvalue problem in λ in Eq. (4). However, this is prone to numerical problems for structures with a large number of nodal DOFs, typically for 2-dimensional cross-sections. In such cases it is better to express the eigenvalue problem in terms of the transfer matrix T of the segment and

qL, fL

L Figure 2. FE of segment

of waveguide.

qR, fR

fext,L fext,R

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then reformulate it in terms of ( 1 )λ λ+ as proposed by Gry [14] and Zhong and Williams [33]. This yields

( ) ( )1

1 ( ) ( )RL LL RR LR RLL L

LR LR RL LL RRL Lλ λλλ

− + − −⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢⎢ ⎥ ⎢ ⎥− − +⎛ ⎞⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦+⎜ ⎟

⎝ ⎠

D O D D D Dq qO D D D D Dq q ⎥ (5)

This has only double eigenvalues and those with the largest magnitudes are the ones of most interest. Further details can be found in [19].

The jth eigenvalue jλ is associated with an eigenvector jφ which can be partitioned into elements and ,q jφ ,f jφ related to the DOFs q and nodal forces f respectively. These vectors form the jth columns of the matrices qΦ and fΦ . In an analogous manner left eigenvectors

and ,q jψ ,f jψ and matrices and qΨ fΨ can be found such that T = IΨ Φ . Henceforth the jth eigenvalue of T is defined such that

{ }H1; Re i 0 if 1j L Lλ ω≤ <f q jλ =

)

(6)

The eigensolutions therefore come in two sets whose eigenvalues and eigenvectors are ( ,j jλ +φ and (1 , )j jλ −φ and which represent n positive-going and n negative-going wave types respectively. Equations (6) imply that either the amplitude of a wave decreases in the direction of propagation or that, if the amplitude remains constant, there is time average power transmission in the direction of propagation.

Propagation over a distance L0 is such that, for motion in the jth wave mode, the nodal DOFs and forces at locations x and x + L0 are related by ( ) ( ) ( ) ( ) ( ) ( )0

0 0;L L L Lj

0jx L diag x x L diag xλ+ = + =q q f λ f (7)

where diag(.) indicates a diagonal matrix. In general the motion is a sum of wave components of amplitudes ja± so that

(8) ( ) ( ), , , ,1 1

;n n

q j j q j j q q f j j f j j f fj j

a a a a+ + − − + + − − + + − − + + − −

= =

= + = + = + = +∑ ∑q a a fφ φ Φ Φ φ φ Φ Φa a

2.2 Numerical issues

There are various other numerical issues besides the conditioning of the eigenvalue problem. First, as with conventional FEA, FE discretisation errors become significant if the size of the element is too large – the rule-of-thumb that there should be at least 6 or so elements per wavelength applies. Secondly, as 0ω → round-off errors in the DSM occur if 2ω M K .

The structure could be remeshed using a segment with a larger length xL or else two or more segments can be concatenated and interior DOFs condensed using one of the schemes described in [34]. Finally the discretised FE model is a spatially periodic lumped mass-spring system, while the original structure is a homogeneous continuum. The FE model thus shows

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periodic structure effects at high frequencies, where half the wavelength is comparable to the element length. These are the so-called optical branches. The effects include pass and stop bands and aliasing effects. Since the wavelengths are so short the FE description breaks down, so the issue is one of determining which solutions are artefacts of the spatial discretisation and which are valid estimates of wavenumbers in the continuum. This can be achieved by determining the sensitivity of the estimated wavenumber to the dimensions of the element, since the bounds of the pass and stop bands depend sensitively on the natural frequencies of the element with certain boundary conditions [35-36]. Wavenumber estimates which are periodic artifacts are thus sensitive to the dimensions of the element whereas those which provide estimates of wavenumbers in the continuum are insensitive to such changes. The sensitivities can be found by simple re-meshing or can be evaluated from the derivatives of K and M with respect to element length.

Finally, calculation of the higher order wave modes is prone to poor conditioning. These modes decay exponentially and very rapidly with distance so that they are not important in determining the free or forced response of the structure. Hence in practice the wave basis might be reduced, only the first m wave modes calculated and the sums in equation (8) truncated at m, so that the matrices Φ are of dimension n m× .

2.3 Energy, power and group velocity

Energy-related quantities follow straightforwardly once the eigenproperties (wavenumbers and wave modes) are known. For example, the time average kinetic and potential energy densities and (i.e. the energies per unit length) for the jth wave mode are kE pE

{ } {2

H, , , ,

1Re ; Re4 4k q j q j p q j q jE E }H

L Lω

= − =Mφ φ φ φK (9)

where the superscript H denotes the Hermitian. The total energy density is then

. As the wave propagates, so the time average power transmitted through the cross-section of the waveguide is

tot k pE E E= +

{ H, ,Im

2}f j q jP ω

= − φ φ (10)

The group velocity of the jth wave mode, which by definition is the speed at which energy is propagated, is therefore given by

gj to

Pck E t

ω∂= =

∂ (11)

This is occasionally prone to numerical errors and an alternative is to estimate the group velocity by a finite difference method [23]. A further alternative, differentiating the eigenproblem, often suffers from severe numerical issues. The dissipated power follows from the imaginary part of K and/or the damping matrix C.

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3. FREE WAVE PROPAGATION IN 2-DIMENSIONAL STRUCTURES

Now consider a solid which is homogeneous in both the x and y directions, but whose properties may vary through its thickness in the z direction. An example is the laminate shown in Figure 3(a), in which each layer is uniform. Under the passage of a time harmonic wave any response variable varies as ( , , ,w x y z t ) ( ) ( ) ( ), , , x yi t k x k yw x y z t W z e ω − −= (12) where cosxk k θ= and sinyk k θ= are the components of the wavenumber k in the x and y directions and where θ indicates the direction of propagation.

The WFE analysis proceeds along similar lines to the case of 1-dimensional waveguides. A small rectangular segment of sides xL and is taken, as shown in Figure 3(b), and is meshed through the thickness, typically using 4- or 8-noded rectangular finite elements, although the use of other shaped elements or elements with internal or mid-side nodes is straightforward [28]. The vector of DOFs and nodal forces of the segment shown in Figure 3(c) are

yL

(13) 1 2 3 4 1 2 3 4;

T TT T T T T T T T⎡ ⎤ ⎡= =⎣ ⎦ ⎣q q q q q f f f f f ⎤⎦

1y

The mass and stiffness matrices of the segment are found using conventional FE methods. The equation of motion is given in equation (2).

Now consider a wave propagating through the continuum. The periodicity conditions relating the nodal DOFs are now

2 1 3 1 4; ;x y xλ λ λ= = =q q q q q λ q (14) where ; ; ;yx ii

x y x x x ye e k Lμμλ λ μ μ−−= = = = y yk L (15) Here xμ and yμ are the propagation constants. Thus

y

Lx

Ly x q1

q3

q2

q4

x

y

Figure 3. (a) 2-dimensional homogeneous solid, (b) rectangular segment and (c) definition of nodes.

(b) (c) (a)

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1;T

R R x y x yλ λ λ λ⎡ ⎤= = ⎣ ⎦q q I I I IΛ Λ (16) In the absence of external excitation, equilibrium at node 1 implies that 1 1 1 1;L L x y xλ λ λ λ− − − −

y⎡ ⎤= = ⎣ ⎦f 0 I I I IΛ Λ (17) Substituting equation (16) into the second of equation (2) and premultiplying by gives LΛ ( ) ( ) ( ) ( )2

1; , ; ; , , ,x y x y x y x yω λ λ ω λ λ λ λ ω λ λ= = −D q 0 D K M (18) where ;L R LK K M M= Λ Λ = Λ ΛR (19) are the reduced stiffness and mass matrices, i.e. the segment matrices projected onto the DOFs of node 1 under the assumption of disturbance propagation as in equation (12). In terms of the partitions of the DSM D of the segment, the eigenproblem (18) becomes

( ) ( ) ( ) ( )

( )

111 22 33 44 12 34 21 43 13 24

1 1 1 1 131 42 14 41 32 23 1

[

] 0x x

y x y x y x y x y

yλ λ λ

λ λ λ λ λ λ λ λ λ

−

− − − − −

+ + + + + + + + +

+ + + + + + =

D D D D D D D D D D

D D D D D D q (20)

If there are n DOFs per node, the nodal displacement and force vectors qj and fj are , the segment mass and stiffness matrices are

1n×4 4n n× while the reduced matrices are . n n×

The equations above are equally applicable to axisymmetric structures [29], where the azimuthal angle α replaces the coordinate x and the radial coordinate r replaces the coordinate z. A prismatic segment subtended by an angle Lα and of length is modelled using FEA. It is common to approximate curved structures as being piecewise-flat. Curvature is then included by a rotation of the coordinates by an angle

yL

Lα before applying the periodicity conditions (14).

3.1 Forms of the eigenvalue problem

Equations (18) and (20) give eigenproblems relating the three parameters ,x yλ λ and ω for the discretised structure. First, it should be noted that the DSM D is symmetric (in the absence of gyroscopic terms). By taking the transpose of equation (20) it follows that, for real ω, if ( , )x yλ λ is a solution to equation (20) then so too is any combination of ( ),1x xλ λ and

( ),1y yλ λ . These four solutions represent the same disturbance travelling in the four directions ,θ π θ± ± . The eigenproblems take various forms depending on the physical nature of the solution being sought as described below. 3.1.1 Wavenumbers known: free wave propagation in an undamped structure. To calculate the dispersion relations for free wave propagation in an undamped structure, the real propagation constants xμ and yμ are given and the corresponding frequencies of free wave propagation are to be found. Equation (18) then becomes a standard, linear, algebraic

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eigenvalue problem in 2ω to which there are thus n real, positive solutions for 2ω . The eigenvectors define the wave modes. Although there are n solutions, not all represent wave motion in the continuous structure, some being artifacts of the spatial discretisation and periodicity. The group velocity

gx y

dd k kω ω ω∂ ∂

= = +∂ ∂

c ik

j (21)

where i and j are unit vectors in the x and y directions respectively. The derivatives can be found from the eigenvalue problem of equation (18) since, for a given eigenvalue 2

jω with mass-normalised left and right eigenvectors jψ and jφ , [37]

2

2jj j

x x xk k kω

ω∂

j

⎡ ⎤∂ ∂= −⎢ ⎥∂ ∂ ∂⎣ ⎦

K Mψ φ (22)

with a similar expression for 2

j kω∂ ∂ y . In practice the derivatives in equation (22) might be estimated numerically using a finite difference approach. 3.1.2 Frequency and one wavenumber known: quadratic eigenvalue problem In the second class of eigenproblem the frequency ω and one wavenumber, say xk , are given. This might physically represent the situation where a known wave is incident on a straight boundary so that the (typically real) trace wavenumber along the boundary is given and all possible solutions for are sought, real, imaginary or complex. Wave propagation in a closed cylindrical shell is a second example, where the wavenumber around the circumference can only take certain discrete values. In this case, equation

yk

(20) becomes a quadratic in yλ and a quadratic eigenproblem results, for which there are 2n solutions. 3.1.3 Frequency and direction of wave propagation known: polynomial and transcendental eigenvalue problems In the third type of eigenproblem the frequency ω and the direction of propagation θ are specified. Hence xλ and yλ are of the form

, ;yx i y yix y

x x

Le e

Lμμ tan

μλ λ

μ−−= = = θ (23)

where xμ and yμ might be complex, but their ratio is real and given. The nature of the eigenproblem depends on whether this ratio is rational or irrational [28,38]. If the ratio

y 2x m m1μ μ = is rational, and being integers with no common divisor, then the

eigenvalue problem can be written as a polynomial eigenvalue problem of order . On the other hand, if

1m 2m

( )1 22 m m+

y xμ μ is irrational, then a transcendental eigenvalue problem arises. In any event, unless + is small, it is more straightforward to find solutions for k to 1m 2m

( ); , 0k ω θ =D using a root-finding approach.

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3.2 Numerical issues

The eigenvalue problems for 2-dimensional WFE analysis are not prone to numerical errors because the segment matrices are typically small. The considerations regarding FE discretisation errors and round-off errors in the DSM as 0ω → apply as for the 1-dimensional case and so, too, do the effects of the spatial periodicity of the FE model. Wavenumber estimates that are artefacts of the periodicity can be identified once again by determining the sensitivity with respect to element size, i.e. by evaluating the derivatives in equation (22) with respect to ,x yL rather than ,x yk .

4 FORCED RESPONSE OF WAVEGUIDE STRUCTURES

Equation (8), together with the left eigenvectors Ψ, define transformations between the physical domain (q, f) and the wave domain (a+, a-). This enables waveguide structures to be analysed straightforwardly, using methods based on spectral element [5] or wave-based [3,4] analyses for example. In the former, the equations of motion of the structure are assembled in terms of physical DOFs and forces, which are related by dynamic stiffness matrices found from the wave analysis. In the latter, the assembly is in terms of wave amplitudes at various locations, which are related by propagation, reflection and transmission matrices, with excitation injecting waves into the structure. In WFE analysis the only difference is that the transformation matrices are found from FEA of a small segment of the structure rather than from analytical models. Previous studies include those of Duhamel [20] (forced response of finite structures), Mencik and Ichchou [27] (calculation of reflection and transmission properties of discontinuities) and Waki et al [26] (forced response of tyres). Duhamel [31] also used the WFE method to calculate the Green function of a 2-dimensional structure by superposing solutions with given wavenumber in one direction.

Numerical calculation of the higher order wave modes is prone to poor numerical conditioning. However, these modes contribute negligible amounts to the response of the structure. Hence in practice only the first m wave modes for which jλ is largest might be calculated, giving a reduced wave basis of order m. Appropriate use of the left eigenvector matrix Ψ can then remove numerical difficulties.

As an example, consider the application of excitation fext at a point as shown in Figure 4. It generates waves e+ and e- which propagate away from the excitation point. Continuity and equilibrium condition at the excitation point give

q q

extf f

+ − +

+ − −

⎡ ⎤ ⎧ ⎫ ⎧ ⎫⎢ ⎥ ⎨ ⎬ ⎨ ⎬⎢ ⎥ ⎩ ⎭⎩ ⎭⎣ ⎦

0efe

Φ Φ=

−Φ Φ (24)

Inverting the matrix on the left hand side of this equation can cause numerical difficulties, especially for waveguides with many DOFs. However, premultiplication by the left eigenvector matrix Ψ and using the orthonormality relations ΨΦ = I leads to (25) ;q ext q ext

+ + − −= =e f e fψ

e+e-

fext

Figure 4. Point excitation and excited waves.

ψ In effect, the external forces are projected onto the (reduced) wave modes.

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5 ILLUSTRATIVE EXAMPLES AND APPLICTIONS

5.1 Uniform beam in bending

Consider a uniform Euler-Bernoulli beam undergoing bending vibrations. The possible wavenumbers are where , ibk k± ± b

2

4b

AkEI

ρ ω= (26)

where EI is the bending stiffness and ρA the mass per unit length. The waves form a pair of propagating waves and a pair of evanescent, or nearfield, waves. The FE matrices for an element of length L are well known (e.g. [39]). There are two DOFs at each node, displacement and rotation, and correspondingly two nodal forces, shear and bending moment. The WFE method thus predicts two pairs of eigenvalues which are given by [19]

2 3 4 51,2

2 3 4 53,4

1 1 11 (2 6 24 ;1 11 (2 6 24

OkL

ii O

λ μ μ μ μ μμ

λ μ μ μ μ μ

= ± + ± + + )

)=

= ± − + +∓ (27)

These expression are accurate to 4th order terms in μ = kl, with errors being of order 5μ .

The dispersion curves for real μ , i.e. for free wave propagation, are shown in Figure 5. The WFE solutions are periodic functions of μ with period 2π because of the spatial periodicity. There are 3 pass bands whose cut-off frequencies, 0 and , , ,a b c dΩ , are indicated in the Figure. Also shown is the analytical solution, equation (26). The first pass band

of the WFE results gives accurate results for ( aΩ < Ω ) μ up to 3π or so. The sensitivities with respect to the element length of the solutions which truly represent motion in the continuum are insensitive to the element length while those that are artefacts of the spatial periodicity have sensitivities approximately equal to 1.

60

-3 -2 -1 0 1 2 30

20

30

40

50

μx/π

Ω

First pass band

Second pass band

Third pass band

Ωa Ωb

Ωc

Ωd

μ /π

Band 2

Band 3 Ωd

Ωc

Band 1 Ωa

Ωb

Figure 5. Dispersion curves for isotropic thin beam, propagating waves: ….. analytic solution; WFE results: ____ first, - - - second and third pass bands.

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Figure 6 shows the solutions for μ as a function of Ω. The first is real in the pass bands and becomes complex in the stop bands. The other is imaginary, representing the evanescent wave. The results become inaccurate for 3μ π> or so due to discretisation errors and break down completely for μ π> because of periodic structure phenomena.

5.2 Free wave propagation in an orthotropic laminated cylindrical sandwich panel

As a second example, and one for which other approaches are difficult, consider a cylindrical sandwich shell comprising two laminated skins and a foam core. The two skins each comprise 4 orthotropic sheets of glass/epoxy with a lay-up of [+45/-45/-45/+45] while the core material is a polymethacrylamide Rohacell foam. The radius is 1 m and the thickness-to-mean radius ratio 0.018. Further details and material properties of the orthotropic skins can be found in [38]. Very similar shells were considered by Heron in [10] and by Ghinet and Atalla [11]. Heron followed a classical theory for sandwich structures, assuming the core is in shear while the skins are thin and undergo both bending and in-plane extensional and shear motions. This resulted in a set of equations of 47th order. Ghinet and Atalla developed a discrete layer theory based on Flugge theory, yielding a 42nd order set of equations describing the motion. In both cases numerical solutions were then found. Neither study is a trivial exercise: apart from the effort in developing the equations of motion and applying inter-layer compatibility and equilibrium conditions, assumptions are made concerning the internal stress and strains distributions. Finding numerical solutions to the resulting equations is not difficult for freely propagating waves in undamped structures (real wavenumbers). Finding numerical solutions for damped structures and finding complex solutions is not straightforward, involving root-searching in the complex wavenumber domain.

The WFE model was realised using 18 SOLID45 elements in ANSYS, 4 for each skin and 10 for the core, resulting in 57 DOFs after reduction of the mass and stiffness matrices. Figure 7 shows the complex dispersion curves for the circumferential orders n = 0,1 and 3, the latter

Figure 6. Dispersion curves for beam: _____analytic solution; - - - WFE results; (a) Re{μ}, (b) Im{μ}.

0 2 4 6 8 10 12 14 16 18 200

0.5

1

1.5

Imag

(μ/π

)

Ω

0 2 4 6 8 10 12 14 16 18 200

0.5

1

1.5

Ω

Real

(μ/π

)

Im μπ

⎧ ⎫⎨ ⎬⎩ ⎭

Re μπ

⎧ ⎫⎨ ⎬⎩ ⎭

Ω

Ω

(a)

(b)

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being typical of higher orders. The ring frequency is at approximately 623 Hz. Well above the ring frequency the wavenumbers tend to those for the flat panel. Below the ring frequency the dispersion curves are very complicated and the motion cannot be described simply in terms of flexural, extensional and shear motion alone. In Figure 7 real, imaginary and complex wavenumbers are shown. Some very complicated dispersion phenomena, common in the dispersion curves of complicated structures, can be seen:

• Three waves propagate from ω = 0: the two for which n = 0 correspond broadly to torsional and extensional waves, while that for n = 1 is a bending wave.

• A bending evanescent wave propagates from ω = 0 for n = 1 (Figure 7(b)). • At point A in Figure 7(a) a propagating wave cuts on: the wavenumber of this branch

changes from pure imaginary to real. This is an example of very common and familiar wave cut-on. At high frequencies this wave become predominantly extensional.

• At point B in Figure 7(b) a pair of waves with complex conjugate wavenumbers bifurcates into a pair of propagating waves with real wavenumbers. One branch, that with the lower wavenumber, has phase and group velocities of opposite signs (the wavenumber decreases with increasing frequency). The motion thus somewhat resembles a Michael Jackson “Moonwalk” [40]: it goes forward but moves backwards.

• At point C two propagating waves bifurcate into a pair of waves with complex conjugate wavenumbers, while at D they bifurcate back into real wavenumbers.

• Point E illustrates another form of bifurcation, where a pair of complex conjugate wavenumbers bifurcate into a pair with pure imaginary values. Point F shows the reverse process, where two imaginary wavenumbers bifurcate into a complex conjugate pair.

5.3 Free wave propagation and forced response of a tyre

Tyre vibration to 2kHz or so produces both interior and exterior noise, but there are major problems in modelling tyres in this frequency range using conventional FE methods. In this section the WFE method is applied to a smooth tread (195/65R15) tyre provided by Bridgestone Corporation. Further details can be found in [26]. A tyre is a complicated structure comprising several different rubbers, fillers, steel and textile fibres, but the smooth tyre can be regarded as a homogeneous waveguide in the circumferential direction (Figure 1(b)). A segment subtending an angle of 1.8º in the circumferential direction as shown in Figure 8 was modelled using ANSYS 7.1. Curvature in the circumferential direction was allowed by rotating coordinates through 1.8º when applying the periodicity condition in a manner analogous to that for axisymmetric structures described in section 3. The solid element SOLID46, which generates equivalent single material properties for a layered structure, was used. An internal pressure of 200kPa was simulated by the application of surface loads on the elements. The number of DOFs of the segment was 324 after the condensation of interior DOFs. The material properties of the rubber components were assumed to be frequency dependent.

The dispersion curves are shown for the undamped case for the sake of clarity. Figure 9 shows the circumferential wavenumber (rad/rad) below 600Hz for modes where the tread motion is predominantly in the plane of the tyre cross-section (i.e. not in the circumferential direction). The wave mode shapes are also indicated for some of the propagating branches. Results for real, imaginary and complex (conjugate) wavenumbers are shown. Complex conjugate wavenumbers are only plotted close to their cut-off frequencies for clarity. Natural frequencies occur if Re{k} = 1, 2, …. and also if Re{k} = 0 for the purely breathing modes, although such modes do not occur for this tyre in the frequency range shown.

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The anti-symmetric A1 mode cuts-on first. This involves lateral motion of the tread as illustrated in the figure, the dashed line showing the deformed shape. In the A1 mode and the symmetric S1 “tread bounce” wave mode the tread mass is vibrating on the sidewall stiffness.

Figure 7. Dispersion curves for laminated sandwich cylinder: (a) n = 0, (b) n = 1 and (c) n = 3; ……. complex-valued, …… pure real and pure imaginary wavenumbers.

0 500 1000 1500

-10

-8

-6

-4

-2

0

2

4

6

8

10

Frequency[Hz]

Imag

(k yR)

R

eal(k

yR)

D

(a)

0 500 1000 1500

-10

-8

-6

-4

-2

0

2

4

6

8

10

Frequency[Hz]

Imag

(k yR)

R

eal(k

yR)

B

A

C

(b)

0 500 1000 1500

-10

-8

-6

-4

-2

0

2

4

6

8

10

Frequency[Hz]

Imag

(k yR)

R

eal(k

yR)

(c)

(a)

Re{kyR}

Im{kyR}

(b)

Re{kyR}

Im{kyR}

(c)

Re{kyR}

Im{kyR}

F

Frequency (Hz)

E A

C D B

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The A2, S2 and higher wave modes are cross-sectional modes. The wave modes TS,A predominantly involve shear motion of the tread. Complicated cut-off phenomena, curve veering, cut-on with a nonzero real wavenumber and waves with phase and group velocities of opposite sign are evident. At higher frequencies the dispersion curves tend to those of a flat structure.

Belt

Fixed to rim

Sidewall

Figure 8. Segment of tyre.

A point force acting on the tyre injects waves whose amplitudes can be found from equation (25). The total wave amplitudes a+ and a- immediately to the right of the excitation point are given by the superposition of the directly excited waves e+ and e- and the sum of the waves which have propagated around the circumference of the tyre. From this it follows that (28) { } { }1 1,− −+ + −= − = −a I τ e a I τ τe−

exp jik L− L where is the diagonal wave propagation matrix whose jth element is , being the circumferential length of the tyre. The displacements at any point then follow from equation

τ ( )

(8). Excitation over the contact patch is found by superposing the responses to a number of point forces. Damping is included by a frequency dependent loss factor. All wavenumbers become complex because of the damping.

Experimental measurements were also taken with the tyre excited over a disc of diameter 23.5mm centred on the centre-line of the tyre. For this excitation, only symmetric modes are excited. The effect of the finite size of excitation becomes significant only at higher frequencies.

The input mobility is shown in Figure 10. In the WFE predictions only about 70 pairs of positive- and negative-going wave modes are used to extract the numerical result. This implies that only 140 DOFs are needed to model the dynamic behaviour of the tyre model below 2kHz. The response broadly falls into 3 regions: low frequencies, below the first resonance at which the S1 mode cuts-on and where the sidewall stiffness dominates the system motion; the mid-frequency region between 100 and 400Hz or so where distinct resonances can be seen; high frequencies where the effects of damping become larger and distinct modal behaviour cannot be observed. Good agreement can be seen on the whole. Agreement between the numerical and experiment results could be improved by updating the FE model parameters.

6. CONCLUDING REMARKS

Describing the dynamics of structures in terms of waves is an appealing approach, particularly at higher frequencies. Knowledge of the dispersion relations and wave modes enables predictions to be made of disturbance propagation, structural response, transmission and so on. However, prediction of these dispersion characteristics is not always straightforward. This is particularly the case with the growing use of laminated and fibre-reinforced components. In this paper a wave finite element (WFE) approach was described. It involves post-processing mass and stiffness matrices of a small segment of the structure. These can be found using conventional FE methods and commercial FE packages, so that the

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full range of existing element libraries can be exploited. Since the size of the FE model is typically small, computational cost is not an issue.

Application to free wave propagation in 1-dimensional waveguides and 2-dimensional structures was described, as was the application to the forced response of waveguide structures. Two applications were presented – these would raise significant challenges for alternative approaches but can be analysed in a straightforward and systematic manner using WFE analysis. A variety of complicated dispersion behaviour was observed: cut-on, bifurcations between complex conjugate and real wavenumbers, bifurcations between complex conjugate and pure imaginary wavenumbers, cut-on with non-zero real wavenumber and veering. These are often seen in the dispersion behaviour of complicated structures comprising various coupled wave modes.

0 100 200 300 400 500 600-4

-2

0

2

4

6

8

10

Frequency [Hz]

- | Im

(k) |

| R

e(k)

|

[ra

d/ra

d]

A1

S1

A2

TS1

S2

A3S3

TA2 TS2

TS1

5

5

Figure 9: Dispersion curves for tyre: ― real; ···· imaginary; −·− complex conjugate wavenumbers. Anti-symmetric (Ai) and symmetric (Si) wave modes, i=1,2,3.

Ti denotes a shear wave mode.

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102 103

10-3

10-2

Frequency [Hz]

Mag

nitu

de [m

/Ns]

102 103-180

-90

0

90

180

Frequency [Hz]

Phas

e [d

egre

e]

Figure 10. Magnitude and phase of input mobility: ― WFE; ···· experiment.

ACKNOWLEDGEMENTS

The work presented here is due in no small part to Yoshiyuki Waki, Elisabetta Manconi, Lars Hinke, Denis Duhamel and Michael Brennan and I both acknowledge their contributions and express my thanks for their willingness to work with me.

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