Wave propagation in one-dimensional waveguides with slowly varying random spatially correlated variability

Journal of Sound and Vibration ] (]]]]) ]]]–]]]

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Wave propagation in one-dimensional waveguides with slowly varying random spatially correlated variability Adriano T. Fabro a,n, Neil. S. Ferguson a, Tarun Jain b, Roger Halkyard b, Brian R. Mace b a b

Institute of Sound and Vibration Research, University of Southampton, Highfield Campus, Southampton SO17 1BJ, UK Department of Mechanical Engineering, University of Auckland, Auckland 1142, New Zealand

a r t i c l e i n f o

abstract

Article history: Received 16 September 2014 Received in revised form 1 January 2015 Accepted 17 January 2015 Handling Editor: A.V. Metrikine

This paper investigates structural wave propagation in waveguides with randomly varying material and geometrical properties along the axis of propagation. More specifically, it is assumed that the properties vary slowly enough such that there is no or negligible backscattering due to any changes in the propagation medium. This variability plays a significant role in the so called mid-frequency region for dynamics and vibration, but wave-based methods are typically only applicable to homogeneous and uniform waveguides. The WKB approximation is used to find a suitable generalization of the wave solutions for finite waveguides undergoing longitudinal and flexural motion. An alternative wave formulation approximation with piecewise constant properties is also derived and included, so that the internal reflections are taken into account, but this requires a discretization of the waveguide. Moreover, a Fourier like series, the Karhunen–Loeve expansion, is used to represent homogeneous and spatially correlated randomness and subsequently the wave propagation approach allows the statistics of the natural frequencies and the forced response to be derived. Experimental validation is presented using a cantilever beam whose mass per unit length is randomized by adding small discrete masses to an otherwise uniform beam. It is shown how the correlation length of the random material properties affects the natural frequency statistics and comparison with the predictions using the WKB approach is given. & 2015 Elsevier Ltd. All rights reserved.

1. Introduction Manufacturing processes often result in variability of properties compared to the nominal designed product. As the requirements for optimum design increase and include a broader frequency range for the dynamic response, it is important to improve prediction capabilities. Element based techniques, such as the Finite Element (FE) method [1,2], are the main prediction tools for structural dynamics in industrial applications and are typically used for deterministic predictions. However, the higher the frequency range under analysis using FE, the finer the mesh requirement, with increased computational cost. The pollution effect, i.e. when the accuracy of the FE solution degenerates as the wavenumber or frequency increases [3], must also be taken into account, further increasing the computational cost. Besides, even small

n Corresponding author. Current address: University of Brasília, Department of Mechanical Engineering, Brasília, DF, Brazil, 70910-900. Tel.: +55 61 31075682. E-mail address: [email protected] (A.T. Fabro).

http://dx.doi.org/10.1016/j.jsv.2015.01.013 0022-460X/& 2015 Elsevier Ltd. All rights reserved.

Please cite this article as: A.T. Fabro, et al., Wave propagation in one-dimensional waveguides with slowly varying random spatially correlated variability, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.01.013i

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A.T. Fabro et al. / Journal of Sound and Vibration ] (]]]]) ]]]–]]]

variability starts to play an important role such that a deterministic FE approach by itself is no longer able to predict the structure's behaviour accurately. It is then necessary to add some level of randomness within the description of the structural models and therefore increasing the computational cost further. In this context, three different frequency bands of analysis can be defined for the typical dynamic response of weakly dissipative structures [4,5]: the low-frequency range, where deterministic models are well suited, with low order modes and low sensitivity to variability, and FE-like approaches are applicable; the high-frequency range, where energy-like methods (for instance, Statistical Energy Analysis – SEA [6]) are well suited, with many high order modes involved; and the mid-frequency range, where individual modal behaviour is still distinguishable and affected by variability. The latter frequency range is too high in frequency to be efficiently treated using FE methods, but not high enough for energy-based methods to be applicable. Wave-based methods have been developed in order to attempt to bridge this gap in the prediction tools, by increasing computational efficiency, and therefore extend the applicability of deterministic models to higher frequencies. However, most of them assume that waveguide properties are homogeneous in the direction of the travelling wave, limiting the application of such approaches. Examples include the Wave Based Method (WBM) [5,7,8], based on the indirect Trefftz approach, the spectral element method [9,10], that uses analytical solutions for the wave propagation to assemble dynamic stiffness matrices for waveguides, the Semi-Analytical FE method [11], that uses a FE formulation for the cross section of waveguides and assumes a wave like solution in the direction of propagation of the waves, and the Wave and Finite Element (WFE) method [12–14], that applies the theory of periodic structures for homogeneous waveguides using a FE model of the cross section. Limitations in the application of these wave approaches arise because analytical solutions for non-homogeneous waveguides are only possible for very particular cases, for example acoustic horns, ducts, rods and beams – e.g. [15–17] – particularly when the spatial variation in the properties of the waveguide with respect to distance compared to the wavelength are small. Lee et al. [18] have shown that the velocity of energy propagation is different to the group velocity for a class of deterministic non-homogeneity in one-dimensional waveguides, such that no wave conversion occurs. Langley [19] has shown, using a perturbation approach, that the amplitude of a wave travelling along a non-homogeneous one dimensional waveguide changes and that power is conserved. Also, he addressed the issue of wave reflection at a cut-on section. Moreover, Scott [20] has also considered the statistics of the wave intensity and phase for a specific class of one dimensional random waveguides, also considering negligible scattering. Randomness in waveguides has also been considered in the context of SEA. Manohar and Keane [21] derived expressions for the probability density functions (PDFs) of the natural frequencies and mode shapes for a class of stochastic rods, given that the mass density is a Gaussian random field and that a specific relation with the axial stiffness exists such that there is only phase change. Subsequently the flow of energy between coupled stochastic rods was considered [22]. Moreover, a numerical approach, using the WFE method, has been proposed by Ichchou [23] to include spatially homogeneous variability in waveguides using a first order perturbation [24,25]. Material or geometrical uncertainty often exhibits spatial correlation, particularly when dealing with composite materials [26– 29], and the theory of random fields, being multidimensional random processes, can be used to model this spatially distributed variability using a probability measure [30,31]. A solution of the physical problem might typically require the variability to be incorporated within a spatial discretization, e.g. an FE model, of the physical domain [32]. A number of discretization methods are available, among them being the midpoint method. This was first introduced by Der Kiureguian [33] and is used in this study. It consists in approximating the random field in a spatial domain, previously discretized by a given mesh, by using a constant but random value within each element, or group of elements. This value is given by a sample of the random field specified at the geometric centre of the element. This approach is very appealing, because it does not require any modifications of the FE code and hence making it suitable for using with commercial FE in a framework of Monte Carlo sampling. In addition, a Karhunen–Loeve (KL) expansion [31] provides characterization of the random field in terms of deterministic eigenfunctions weighted by uncorrelated random variables. For some families of correlation functions and specific geometries, there exist analytical expressions for the KL expansion, but, in general, a numerical approach has to be used [34]. The KL expansion allows a significant reduction in the number of random variables required for a random field simulation, since the number of KL modes retained in the analysis NKL is usually much smaller than the number of elements in the mesh. Moreover the expansion is optimal in a mean square sense [31]. In this work, the WKB (after Wentzel, Kramers and Brillouin) approximation for the wave solution is used along with a generalized approach for one-dimensional waveguides in terms of propagation, reflection and transmission matrices, as described by Harland et al. [35]. This approximation requires the internal reflections to be negligible or not to occur due to any local changes in the material or geometrical properties. This approach is subsequently used herein to derive expressions for the natural frequencies and input mobility of finite length waveguides. It is also shown that random fields can be straightforwardly included in the formulation, and an analytical expression of the KL expansion is used to derive closed form approximations for the PDF of the natural frequencies of straight rods and beams. The key assumption made in the WKB approximation is that there is negligible backscattering of the propagating wave due to changes in local properties. A more general approximation can be achieved if one considers a waveguide with piecewise constant material variability, separated into a finite number of discrete sections. This is an approximate representation for the more general spatially varying system. In this way, internal reflections due to local changes in the impedance can be taken into account at the junctions of the sections. A more general analysis of this kind of system can be found in [36,37]. In the next section, the WKB approximation is reviewed and the main results for free and forced vibration response given for rod and beam examples. A formulation in terms of propagation, reflection and scattering matrices is presented in detail Please cite this article as: A.T. Fabro, et al., Wave propagation in one-dimensional waveguides with slowly varying random spatially correlated variability, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.01.013i

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in Appendix A, along with a piecewise constant formulation presented in Appendix B, used for comparison with FE results. In Section 3, the spatially varying properties are assumed to be random fields. An analytical solution of the KL expansion is assumed for Young's modulus of elasticity, and an approximate PDF for the natural frequencies is presented. Section 4 presents some numerical examples comparing the PDF of the natural frequencies and envelopes for the input mobility from the WKB approach with the piecewise constant formulation and stochastic FE. Experimental validation is described in Section 5. Measurements of the natural frequencies of an ensemble of cantilever beams have been taken. The mass per unit length of the beam is randomized by adding small discrete masses to an otherwise uniform beam. The methodology used to generate the random mass distribution and its approximation to a continuous random field is presented, and a simple model using assumed modes and a lumped mass approximation is introduced for comparison with the other approaches. Finally some concluding remarks are made in Section 6. 2. The WKB approximation The classical WKB approximation, a method for finding suitable modifications of plane-wave solutions for propagation in slowly varying media [38] was initially developed for solving the Schrödinger equation in quantum mechanics. It is also called the semi-classical method [39] or the geometrical optics approximation [38,40,41]. The formulation assumes that the waveguide properties vary slowly enough such that there are no or negligible reflections due to these locals changes, even if the net change is big. It has been applied in many fields of engineering, including ocean acoustics [42], acoustics [43–47], structural dynamics [38,39,48–50] and also for cochlear models [51,52]. However, the WKB approximation breaks down when the travelling wave reaches a local cut-off section where the wave mode ceases to propagate. This transition, also known as a turning point, leads to an internal reflection, breaking down the main assumption in the theory, requiring a different approximation for certain frequency bands (e.g. [40]). In addition, uniformly valid solutions, i.e. solutions also valid for the frequency band away from the cut-off frequency, have been derived using a slight modification of the WKB method for different applications, for instance in acoustics [43,46,47]. The application of the WKB method, or eikonal approximation, is relatively simple. It can though rapidly become arduous for more complicated structures, but has the advantage of preserving the interpretation of travelling waves. This allows the use of the same systematic approach used for homogenous waveguides, as developed by Mace et al. [35,53,54]. Bretherton [55] has extended the approximation for general linear systems, and shown that the variations in the amplitude of the waves along rays are governed by conservation of an adiabatic invariant. The method consists of writing the solution in terms of an asymptotic expansion in powers of a small parameter, using the local wavenumber and wave amplitude, and then matching the asymptotic expansion to a certain order, usually the first order. Pierce [38] has derived WKB solutions for Euler–Bernoulli and Timoshenko beams as well as for thin plates by using a conservation of energy approach, with less mathematical formalism. Waveguides with material or geometrical irregularities in its periodicity can present multiple internal reflections leading to confinement of vibration, even without energy dissipation due damping, i.e. localization [56–58]. Langley [36] has proposed a simple method for calculating the localization factor, a measure of the effects of the localization, for random disorder in one dimensional waveguides. He has also shown that the effects of damping and disorder on the attenuation of a propagating wave are not simply additive. Savin [59] has proposed three different regimes for wave propagation in heterogeneous media, based on the ratio of wavelength to correlation length of the random medium δ. The WKB approach is suitable to the regime in which the wavelength is comparable to the correlation length, i.e. δ
1, with some localization due to eventual reflections. For higher frequency, i.e. δ o 1, the wave scattering dominates to propagation and WKB approach is no longer applicable. In this sense, Luongo [60] has shown that the existence of localization is governed by a turning point problem of the WKB approximation, requiring the use of local solutions or uniformly valid solutions as mentioned. A formulation in terms of propagation matrices Λ and reflection matrices Γ for the WKB approach is presented in the appendix and used for the particular results of a rod and beam examples. The wave vector amplitudes are given by h þ iT h þ i  T aj bj and bj ¼ bj , where the propagating and evanescent waves are, respectively, given as aj ¼ aj h þ iT h  iT h i h  i  T þ T þ   aNj , b þ ¼ bjþ bNj aNj , a  ¼ aj bNj , for j ¼ 1; 2. For the non-dispersive rod and bj ¼ bj ajþ ¼ aj j j example, only propagating waves are present. The matrices Λ11 and Λ22 are the diagonal elements of the propagation matrix Λ. The waveguide natural frequencies can be determined by applying the wave train closure principle, tracing the round-trip propagating wave [61]. The input mobility Y ðωÞ due to a point force is given by the contributions of all of the wave components of the response at the excitation point and is calculated using the given framework. 2.1. Example: longitudinal vibration in rods For the case of longitudinal vibration, only a propagating wave is considered, Fig. 1, and the governing equation with spatially varying properties is given by   ∂ ∂uðx; t Þ ∂2 uðx; t Þ EAðxÞ þ ρAðxÞ ¼ pðx; t Þ; (1) ∂x ∂x ∂t 2 Please cite this article as: A.T. Fabro, et al., Wave propagation in one-dimensional waveguides with slowly varying random spatially correlated variability, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.01.013i

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Fig. 1. Finite waveguide undergoing free wave propagation and no internal reflections.

where EAðxÞ is the spatially varying longitudinal stiffness and ρAðxÞ is the spatially varying mass per unit length, pðx; tÞ is the excitation and uðx; t Þ is the axial displacement. It leads to

Λ11 ¼ e  iθL ð0;LÞ þ γ L ð0;LÞ ;

(2)

Λ22 ¼ eiθL ð0;LÞ þ γ L ð0;LÞ ;

(3)

where Λ11 and Λ22 are the elements of the 2  2 propagation matrix, as described in Appendix A, Eq. (A.12), and Z x2 θL ðx1 ; x2 Þ ¼ kL ðxÞdx;

(4)

x1

and 1 2

γ L ðx1 ; x2 Þ ¼ ln

  kL ðx1 Þ ; kL ðx2 Þ

(5)

are the phase and amplitude change, where x1 and x2 are two points within the waveguide, and θL ð0; LÞ and γ L ð0; LÞ are given for a wave propagating from one boundary to the other. For a homogeneous waveguide, then kL ¼ kL ðxÞ is a constant and the total phase change reduces to θL ð0; LÞ ¼ kL L, as well as there being no change in the amplitude, i.e. γ L ð0; LÞ ¼ 0. Applying the wave train closure principle, as described in Appendix A, for energy conserving boundaries and tracing the round-trip propagating wave [61], the natural frequencies correspond to the zeros of the characteristic equation [54,61]    1  detΛ22 ΓR Λ11 ΓL I ¼ 0; (6) where I is the identity matrix, and ð U Þij is the ith row and jth column block matrix of the corresponding propagation matrix. Solving Eq. (6), it is possible for conservative systems to find an expression for the natural frequencies 2nπ  ϕR  ϕL ωn ¼ R L pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ; 2 0 ðρðxÞ=EðxÞÞdx

n ¼ 1; 2; 3; …

(7)

where ϕL and ϕR are the phase changes produced by the reflections at the left and right boundaries, i.e., ΓL ¼ e  iϕL and ΓR ¼ e  iϕR , and EðxÞ and ρðxÞ are the spatially slowly varying Young's modulus and mass density. The internal forces are related to the positive going q þ and negative going q  travelling waves generated when the rod is excited by a harmonic force F. Using the force–deformation relationship for a rod under harmonic excitation, the internal forces at the right and the left of the excitation point are given by ∂U þ ðxÞ P þ ¼ EðL1 ÞA ∂x

(8)

and ∂U  ðxÞ P  ¼ EðL1 ÞA ∂x

(9) h R i x þ where, from Eq. h R (A.2), U ðxÞ ¼ C 1 exp i  i x0 kL ðxÞdx ð1=2Þln kL ðxÞ is the positive going wave amplitude and x  U ðxÞ ¼ C 1 exp i x0 kL ðxÞdx  ð1=2Þln kðxÞ is the negative going one. Their derivatives are then given by " !# ∂U þ ðxÞ 1 1 ∂kL ðxÞ ¼  kL ðxÞ i þ 2 (10) U þ ðxÞ; ∂x 2 k ðxÞ ∂x L

"

∂U  ðxÞ 1 1 ∂kL ðxÞ ¼ kL ðxÞ i þ 2 ∂x 2 k ðxÞ ∂x

!# U  ðxÞ:

(11)

L

Please cite this article as: A.T. Fabro, et al., Wave propagation in one-dimensional waveguides with slowly varying random spatially correlated variability, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.01.013i

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The second term in the expression can be neglected, from the WKB assumption (see Appendix A, Eq. (A.3)). This leads to expressions for the internal forces as P þ ¼  ikL ðxÞEðxÞAUðxÞ þ ;

(12)

P  ¼ ikL ðxÞEðxÞAU ðxÞ  :

(13)

and

Note that these expressions are analogous to the homogeneous case. Equilibrium and continuity of displacement gives the amplitudes of the positive going q þ and negative going q  travelling waves directly generated by an external point harmonic force F applied at x ¼ L1 , Fig. 2, as q ¼ qþ ¼ q ¼ 

i F: 2kL ðL1 ÞEðL1 ÞA

Also, the propagation matrices, Eq. (A.20), are given by " e  iθL ð0;L1 Þ þ γ L ð0;L1 Þ ΛL ¼ 0 "

ΛR ¼

(14)

#

0

;

eiθL ð0;L1 Þ þ γ L ð0;L1 Þ

(15)

#

e  iθL ðL1 ;LÞ þ γ L ðL1 ;LÞ

0

0

eiθL ðL1 ;LÞ þ γ L ðL1 ;LÞ

:

(16)

The input mobility Y ðωÞ ¼

ω

2kL ðL1 ÞEðL1 ÞA

1 þ ΓL e  iθL ð0;L1 Þ þ ΓR e  iθL ðL1 ;LÞ þ ΓL ΓR e  iθL ð0;LÞ ; 1  ΓL ΓR e  iθL ð0;LÞ

(17)

is given by substituting Eqs. (14)–(16) into Eq. (A.24), Appendix A, and using the components of the wave vector, along with the reflection coefficients from the right and left ends. 2.2. Example: flexural vibration in beams In the case of flexural vibration, the governing equation with spatially varying properties is given by   ∂2 ∂2 wðx; t Þ ∂2 uðx; t Þ EI ð x Þ ¼ qðx; t Þ; þ ρAðxÞ 2 2 ∂x ∂x ∂t 2

(18)

where EI ðxÞ is the spatially varying bending stiffness, q ðx; tÞ is the excitation and wðx; t Þ is the flexural displacement. Propagating and evanescent waves are therefore taken into account, and then the propagation matrices are given by " # e  iθB ð0;LÞ þ γB ð0;LÞ 0 Λ11 ¼ ; (19) 0 e  θB ð0;LÞ þ γ B ð0;LÞ "

Λ22 ¼

eiθB ð0;LÞ þ γB ð0;LÞ

0

0

eθB ð0;LÞ þ γ B ð0;LÞ

# ;

(20)

where, for a wave propagating between two arbitrary points x1 and x2 , one has Z x2 θB ðx1 ; x2 Þ ¼ kB ðxÞdx;

(21)

x1

and kB ðxÞ is the local wavenumber of the flexural wave

Fig. 2. Point excitation on finite length waveguide with no internal reflections.

Please cite this article as: A.T. Fabro, et al., Wave propagation in one-dimensional waveguides with slowly varying random spatially correlated variability, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.01.013i

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"

~ ðx Þ W

#

γ B ðx1 ; x2 Þ ¼ ln ~ 2 ; 2 W ðx1 Þ

(22)

~ ðxi Þ are the wave amplitudes as defined in Appendix A. Solving Eq. (6) will lead to the problem of finding the nth where W root of a transcendental equation θTn . The nth natural frequency ωn is then given by

θ2Tn ωn ¼ R pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 ; L 4 ðρðxÞA=EðxÞI yy Þ dx 0

n ¼ 1; 2; 3; …

(23)

Moreover, the nth derivative with respect to x of the positive and negative going propagating waves are n n ∂n W þ ðxÞ=∂xn ¼ ð  ikB ðxÞÞ W þ ðxÞ, ∂n W  ðxÞ=∂xn ¼ ðikB ðxÞÞ W  ðxÞ, and for the evanescent waves ∂n W Nþ ðxÞ=∂xn ¼ n n ð kB ðxÞÞ W Nþ ðxÞ, ∂n W N ðxÞ=∂xn ¼ kB ðxÞW N ðxÞ. Continuity of displacement and slope gives the amplitudes of the waves produced by an external point harmonic force F in an infinite beam as   1 i q ¼ qþ ¼ q ¼ F; (24) 3 4iEI yy ðL1 ÞkB ðL1 Þ  i where EðL1 Þ is Young's modulus, evaluated at x ¼ L1 , the point of excitation. Moreover, the propagation matrices, Eq. (A.20), are given by " # e  iθB ð0;L1 Þ þ γ B ð0;L1 Þ 0 ; (25) ΛL11 ¼ 0 e  θB ð0;L1 Þ þ γ B ð0;L1 Þ "

ΛL22 ¼ "

ΛR11 ¼

eiθB ð0;L1 Þ þ γ B ð0;L1 Þ

0

0

eθB ð0;L1 Þ þ γ B ð0;L1 Þ

;

0

0

e  θB ðL1 ;LÞ þ γ B ðL1 ;LÞ

eiθB ðL1 ;LÞ þ γ B ðL1 ;LÞ

0

0

eθB ðL1 ;LÞ þ γ B ðL1 ;LÞ

(26)

#

e  iθB ðL1 ;LÞ þ γ B ðL1 ;LÞ "

ΛR22 ¼

#

;

(27)

# ;

(28)

In the same way as for the rod example, the input mobility is calculated by substituting Eqs. (24)–(28) into Eq. (A.24). Again, if the waveguide is homogeneous, then kB ðxÞ ¼ kB and the terms given reduce to θB ð0; L1 Þ ¼ kB L1 and θB ðL1 ; LÞ ¼ kB L2 , where L2 ¼ L L1 , and γ B ð0; L1 Þ ¼ γ B ðL1 ; LÞ ¼ 0. The reflected waves from the boundaries at the left and the right ends are  þ given by the respective reflection matrices as a1þ ¼ ΓL a1 and b2 ¼ ΓR b2 . 3. Piecewise constant wave approach The main results from the piecewise constant wave approach are reviewed in this section. The details of the formulation for free vibration and forced response due to a point excitation in finite length waveguides are presented in Appendix B. A finite one dimensional waveguide with length L, undergoing free wave propagation, and with piecewise constant properties, is divided into N elements, such that for the jth element the positive going and the negative going waves on the left aj and on the right side bj , are related by bj ¼ Λj aj , with, Λj the propagation matrix from the jth element. Also, two consecutives elements are connected by a scattering matrix such that aj þ 1 ¼ Gj bj , derived in Section B.3. Then wave ~ a , Fig. 25, where the propagation matrix amplitude vectors from the left side a1 and the right side bN are related by bN ¼ Λ 1 is given by Eq. (B.4). The natural frequencies are found by applying the principle of wave-train closure, tracing the round-trip propagating wave, and, differently from the previous case, considering the internal reflections, by using the scattering matrix, from Eq. (B.3), and the reflections matrices boundaries ΓR and ΓL . Thus, they correspond to the zeros of the characteristic equation    ~ Γ þΓ Λ ~ ~ ~  detΓR Λ (29) R 12  Λ 21 ΓL  Λ 22  ¼ 0; 11 L where ð UÞij is the ith row and jth column block element of the partitioned matrix. It is possible to approximate the scattering matrix as an identity matrix Gj  I, if there are negligible reflections, meaning ~ , Eq. (B.5), such that it that the wave is fully transmitted. This is equivalent to the slowly varying assumption and simplifies Λ ~ ¼Λ ~ ¼ 0, where 0 is a null matrix, which, in turn leads to Eq. (29) in a much becomes a block diagonal matrix, i.e. Λ 12 21 simpler form    ~ Γ Λ ~   0: detΓR Λ (30) 11 L 22 For a point excitation, a harmonic force F applied at the Mth junction creates a positive going q þ and a negative going q  wave that propagate to the next element and then are partially transmitted and partially reflected until they reach the Please cite this article as: A.T. Fabro, et al., Wave propagation in one-dimensional waveguides with slowly varying random spatially correlated variability, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.01.013i

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boundaries and are then only reflected. The superposition principle is used to calculate the vector of wave amplitudes w, given by Eq. (B.21), at the point of excitation including the directly excited waves plus the transmitted and reflected ones arriving at the excitation point, Eq. (B.13). The input mobility is then calculated by considering all of the wave components from w. 4. Random variability In this section, the spatially varying properties are considered to be random fields [30,31,62,63] and can be used to model spatially distributed variability using a probability measure. Equally, the natural frequencies and input mobility, for example, are themselves random variables. Analytical expressions given by the KL expansion and a first order approximation are used to derive closed form solutions for the PDF of the natural frequencies. The results are derived in terms of a spatially varying Young's modulus of elasticity, but they can be readily extended to randomness in any other material or geometric property. Irrespective of what approach is used, when predicting the response of structures with variability, some statistical model is required. Random field theory provides the elements for this representation and it typically involves expressions for the PDF, together with a model for the spatial variability of the properties. This model is usually given by expressions of spatial variability defined by a given correlation function and correlation length. The most commonly used methods for representing random fields in mechanical models includes the use of series expansions, such as the KL decomposition or the Polynomial Chaos expansion [31], and also point discretization or average discretization methods [62]. Amongst the methods of generating random fields [62–64], formulations using series expansions are able to represent the field using deterministic spatial functions and random variables. The KL expansion is a special case where these deterministic spatial functions are orthogonal and derived from the covariance function. It is especially suited for strongly correlated random fields, i.e. slowly varying, because only a few terms in the series are necessary to accurately represent the field. A Gaussian homogeneous random field H ðx; pÞ with a finite, symmetric and positive definite covariance function C H ðx1 ; x2 Þ, defined over a domain D, has a spectral decomposition in a generalized series as [31] 1 qffiffiffiffi X HðxÞ ¼ H 0 ðxÞ þ λj ξj f j ðxÞ; (31) j¼1

where ξj are Gaussian uncorrelated random variables, λj and f j ðxÞ are eigenvalues and eigenfunctions, and are solutions of the Fredholm integral equation of the second kind Z C ðx1 ; x2 Þf j ðx1 Þ ¼ λj f j ðx2 Þ: (32) D

The eigenvalues and eigenfunctions can be ordered in descending order of eigenvalues and Eq. (31) then calculated with a finite number of terms N KL , chosen by the accuracy of the series in representing the covariance function [34]. As a rule of thumb, N KL can be chosen such that λNKL =λ1 o 0:1, and NKL will depend on the correlation length of the random field. In general, this problem can only be solved numerically by discretizing the covariance function, using some collocation method, and finding its eigenvalues and eigenvectors. However, for some families of correlation functions and specific geometries, there exist analytical solutions for this integral equation. One such case is the one dimensional exponentially decaying autocorrelation function, C ðx1 ; x2 Þ ¼ e  jx1  x2 j=b ;

(33)

where b is the correlation length, in the interval  L=2 r x rL=2, where L is the length of the domain and where x1 and x2 are any two points within the interval. In this case, the KL expansion, for a zero-mean random field, can be written as [31] H ðxÞ ¼

N KL h X







αj ξ1j sin w1j x þ βj ξ2j cos w2j x

i

(34)

j¼1

where ξ1j and ξ2j are Gaussian zero-mean, unity standard-deviation, independent random variables with the properties 〈ξ1j 〉 ¼ 〈ξ1i 〉 ¼ 0;

〈ξ1i ξ2j 〉 ¼ 0 ;

〈ξ1i ξ1j 〉 ¼ δij

where δij ¼ 1 for i ¼ j and δij ¼ 0 for i aj. Moreover, the coefficients are given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

αj ¼

λ1j

ðL=2Þ ð sin ðw1j LÞ=2w1j Þ

;

βj ¼

λ2j

ðL=2Þ þ ð sin ðw2j LÞ=2w2j Þ

;

λ1j ¼

(35)

2c ; w21i þ c2

λ2j ¼

2c ; w22i þc2

(36)

where c ¼ 1=b and w1i and w2i are the ith roots of the transcendental equations c tan w1 þ w1 ¼ 0 and w2 tan w2  c ¼ 0, respectively. This expansion is truncated to N KL terms according to the weight of the higher order eigenvalues in the series. Note that the longer the correlation length the smaller the size of the eigenvalues with respect to the first, meaning that fewer terms are needed to accurately represent the series. Please cite this article as: A.T. Fabro, et al., Wave propagation in one-dimensional waveguides with slowly varying random spatially correlated variability, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.01.013i

A.T. Fabro et al. / Journal of Sound and Vibration ] (]]]]) ]]]–]]]

8

The KL expansion is then used to describe Young's modulus as a random field, in the numerical examples of the following section, given by EðxÞ ¼ E0 ½1 þ σ HðxÞ

(37)

where E0 is the nominal value for Young's modulus and σ is the standard deviation, that can also be seen as a dispersion term quantifying the influence of H ðxÞ on the mean value E0 . This continuous expression is used with the WKB formulation and in the piecewise constant approach, by evaluating Eq. (37) at the geometrical centre xcj of the jth element, such that  Ej ¼ E xcj : (38) From Eqs. (1) and (18), it can be noted that the random field given by Eq. (37) should be differentiable. This is not true for a Gaussian random field with the correlation function given by Eq. (33), because it is not continuous at x1 ¼ x2 [65]. However, this condition can be achieved by using the KL expansion of the random field with NKL terms, Eq. (34). It can be shown [34] that this truncation has the effect of smoothing the autocorrelation function at the origin, therefore making EðxÞ differentiable. Applying Eqs. (34) and (37) to the natural frequency expressions for the rod Eq. (7) and beam Eq. (23), respectively, leads to sffiffiffiffiffi E0 2nπ  ϕR  ϕL (39) ωn ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ρ 2 R0L 1=ð1 þ σ HðxÞÞdx and sffiffiffiffiffiffiffiffiffiffiffi E0 I yy θ2Tn ωn ¼ R pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 : ρA L 4 1=ð1 þ σ HðxÞÞ dx 0

(40)

In both cases, the natural frequency is therefore a random variable and an approximation for its PDF is given in the following section. The input mobility also becomes a random variable when the expression of the integral for the wavenumber for the rod (Eq. (4)) and the beam (Eq. (21)) are calculated using the KL expansion. 4.1. A probability density function for the natural frequency Assuming that Young's modulus is not only slowly varying, but also has small dispersion around its mean value, i.e. jσ H ðxÞj{1, a first order approximation for the natural frequencies of a finite length waveguide undergoing either longitudinal or flexural motion, Eqs. (39) and (40) respectively, can be written as

 Z σ L ωn  ωn0 1 þ H ðxÞdx ; (41) 2L 0 where

pffiffiffiffiffiffiffiffiffiffiffi E0 =ρ=2L, frequency of a rod with homogeneous properties for the case of longitudinal ffi  the natural 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E0 I yy =ρA is the natural frequency of a beam with homogeneous properties for vibration, from Eq. (7), and ωn0 ¼ θTn =L the case of flexural vibration, from Eq. (23). Using the KL expansion, from Eq. (34), the integral can be analytically evaluated as 

ωn0 ¼ 2nπ  ϕR  ϕL

Z 0

L

H ðxÞdx ¼ 2

N kl X

ξj

j¼1

βj w2j

sin

 w2j L ; 2

(42)

and Eq. (41) can be rewritten explicitly as a function of the Gaussian uncorrelated random variables ξj from the KL expansion as 



ωn ξ1 ; ξ2 ; …; ξNkl  ωn0 þ

N kl X

Bj ξj ;

(43)

j¼1

 where Bj ¼ ωn0 σ =L βj =w2j sin w2j L=2 are the terms included by the spatially varying Young's modulus. From this expression, a closed form expression for the PDF of the natural frequencies can be found and is given as ! ðωn  ωn0 Þ2 1 f Ωn ðωn Þ ¼ pffiffiffiffiffiffiffiffi exp  ; (44) 2σ 2Ω 2π σ Ω qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi PNKL 2 where σ Ω ¼ j ¼ 1 Bj , noting that the summation of Gaussian random variables is also a Gaussian random variable. This explicit approximation clearly shows how the statistics of the nth natural frequency depend upon the spatial variability, given by the coefficients Bj . Please cite this article as: A.T. Fabro, et al., Wave propagation in one-dimensional waveguides with slowly varying random spatially correlated variability, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.01.013i

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9

5. Numerical examples

Amplitude dB ref 1 m/(Ns)

In this section numerical examples comparing with the WKB approximation and the other methods are presented, for both cases of rods and beams. The statistics of the response are of interest, e.g. the PDF of the natural frequencies as well as envelopes for the input mobility in the frequency band. Monte Carlo simulations were performed using Ns ¼ 5000 samples. The mean rod and beam properties were assumed to be E0 ¼ 70 GPa and ρ ¼ 2700 kg/m3, but with different geometries. Structural damping was included by taking Young's modulus to be E0 ¼ E0 ð1 þiηÞ, with loss factor η ¼ 10  3 . For the rod, the cross-sectional area A ¼ 0:1  5 cm2, total length L ¼ 4 m, with free–free boundary conditions, i.e. ΓL ¼ ΓR ¼ 1 and hence ϕL ¼ ϕR ¼ 0. The frequency band under analysis was chosen to be from 1 Hz to 6 kHz, such that at least the first nine axial modes could be observed in the forced response. Moreover, the excitation point was at x ¼ L1 ¼ 0:35L. For the beam, the cross-section is rectangular with thickness 1 mm and width 30 mm, i.e. A ¼ 30 mm2, and total length L ¼ 0:5 m, with free–free boundary conditions, i.e. the reflection matrices at the left and    i 1 þi . The frequency band under analysis was chosen from 1 Hz to 1.2 kHz such that the right ends are ΓL ¼ ΓR ¼ 1i i first 10 flexural modes could be present, and the excitation point was also at x ¼ L1 ¼ 0:35L. The only spatially varying property is taken to be Young's modulus and it is described by Eqs. (34) and (37). Although this is a truncated KL expansion of a Gaussian random field with exponentially decaying correlation function, it is used here as a deterministic expression, i.e. given a correlation length, one sample of the random field is generated and then the same sample is used for all the methods in order to compare the results. They were all generated using N kl ¼ 40 terms in the KL expansion, satisfying the convergence criterion λNKL =λ1 o 0:1. The FE model was assembled with 100 elements with each element having a constant but different value for Young's modulus, given by the KL expansion evaluated at the centre of the element, i.e. using a mid-point random field discretization. The FE forced response was solved using direct inversion of the dynamic stiffness matrix, and the number of elements is such to ensure convergence for the natural frequencies and forced response. The same discretization was used in the piecewise constant approach. The integrals needed for the WKB approach were numerically evaluated using a Gauss– Legendre quadrature rule. Three different cases of Young's modulus random field were constructed, for all of the waveguide types, by changing the dispersion parameter σ and the correlation length b. Values considered are σ ¼ 0:1, b ¼ 2L; σ ¼ 0:1, b ¼ 0:8L; and σ ¼ 0:2, b ¼ 0:05L. For each one of these pairs of dispersion parameter and correlation length, one sample of Young's modulus was generated. It is possible to note that for the higher correlation lengths, the spatial distribution has a very distinguishable component of a long length scale with some small variations. It is because, from Eq. (34), sine and cosine terms with larger wavelengths (smaller wavenumbers) have much more relative importance in the series summation than the shorter ones. That relative importance decreases for smaller correlation lengths. Additionally, the input mobility was calculated for each pair b and σ , using the WKB formulation, the piecewise constant approach and the FE model using piecewise constant properties. Single realizations are shown in Figs. 3–5, for the rod, and Figs. 6–8, for the beam respectively. There is a very good agreement between the results obtained for the pairs σ ¼ 0:1 , b ¼ 2L and σ ¼ 0:1 , b ¼ 0:8L. In these cases, the Young's modulus varies slowly along the waveguide, so the WKB, the piecewise constant and the FE approaches agree very well, as expected, although the WKB method requires only a fraction of the computational cost. The results are also different from the input mobility calculated using a constant value for Young's

0 -20 -40 -60 -80 -100 -120 0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

4

4.5

5

5.5

6

Frequency, kHz 2

Phase rad

1 0 -1 -2 0.5

1

1.5

2

2.5

3

3.5

Frequency, kHz Fig. 3. Rod input mobility with nominal values (grey), WKB approximation (dash-dot), piecewise constant properties (dot) and FE approach (solid line); the dispersion parameter σ ¼ 0:1 and the correlation length b ¼ 2L.

Please cite this article as: A.T. Fabro, et al., Wave propagation in one-dimensional waveguides with slowly varying random spatially correlated variability, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.01.013i

A.T. Fabro et al. / Journal of Sound and Vibration ] (]]]]) ]]]–]]]

10

Input Mobility dB ref 1 m/(Ns)

0 -20 -40 -60 -80 -100 -120

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

2.5 3 3.5 4 Frequency, kHz

4.5

5

5.5

6

Frequency, kHz 2 Phase rad

1 0 -1 -2

0.5

1

1.5

2

Fig. 4. Rod input mobility with nominal values (grey), WKB approximation (dash-dot), piecewise constant properties (dot) and FE approach (solid line); the dispersion parameter σ ¼ 0:1 and the correlation length b ¼ 0:8L.

Input Mobility dB ref 1 m/(Ns)

0 -20 -40 -60 -80 -100 -120

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

4

4.5

5

5.5

6

Frequency, kHz 2

Phase rad

1 0 -1 -2

0.5

1

1.5

2

2.5

3

3.5

Frequency, kHz Fig. 5. Rod input mobility with nominal values (grey), WKB approximation (dash-dot), piecewise constant properties (dot) and FE approach (solid line); the dispersion parameter σ ¼ 0:2 and the correlation length b ¼ 0:05L.

Input Mobility dB ref 1 m/(Ns)

20 0 -20 -40 -60 -80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

0.8

0.9

1

1.1

Frequency, kHz 2

Phase rad

1 0 -1 -2

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Frequency, kHz Fig. 6. Beam input mobility with nominal values (grey), WKB approximation (dash-dot), piecewise constant properties (dot) and FE approach (solid line); the dispersion parameter σ ¼ 0:1 and the correlation length b ¼ 2L.

Please cite this article as: A.T. Fabro, et al., Wave propagation in one-dimensional waveguides with slowly varying random spatially correlated variability, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.01.013i

Input Mobility dB ref 1 m/(Ns)

A.T. Fabro et al. / Journal of Sound and Vibration ] (]]]]) ]]]–]]]

11

20 0 -20 -40 -60 -80

0.1

0.2

0.3

0.4

0.5 0.6 0.7 0.8 Frequency, kHz

0.9

1

1.1

0.1

0.2

0.3

0.4

0.5 0.6 0.7 0.8 Frequency, kHz

0.9

1

1.1

2

Phase rad

1 0 -1 -2

Fig. 7. Beam Input mobility nominal values (grey), WKB approximation (dash-dot), piecewise constant properties (dot) and FE approach (solid line); the dispersion parameter σ ¼ 0:1 and the correlation length b ¼ 0:8L.

Input Mobility dB ref 1 m/(Ns)

20 0 -20 -40 -60 -80

0.1

0.2

0.3

0.4

0.5 0.6 0.7 0.8 Frequency, kHz

0.9

1

1.1

0.1

0.2

0.3

0.4

0.5 0.6 0.7 0.8 Frequency, kHz

0.9

1

1.1

2 Phase rad

1 0 -1 -2

Fig. 8. Beam input mobility with nominal values (grey), WKB approximation (dash-dot), piecewise constant properties (dot) and FE approach (solid line); the dispersion parameter σ ¼ 0:2 and the correlation length b ¼ 0:05L.

modulus. For the case of σ ¼ 0:2 , b ¼ 0:05L , the FE and piecewise constant approaches do not agree with the WKB approach, showing that internal reflections are no longer negligible, as in the previous cases. The normalized PDF of the first, third, sixth and eighth natural frequencies for the three pairs of values for σ and b are shown in Figs. 9–11, for the rod and in Figs. 12–14, for the beam, calculated using the closed form expression from the WKB approximation, Eq. (44), and also by MC sampling the expression from the WKB approach, Eq. (41), the piecewise constant approximation, Eqs. (B.9) and (B.12), in addition to the FE model. It is possible to note a very good agreement, as expected, for σ ¼ 0:1 , b ¼ 2L and σ ¼ 0:1 , b ¼ 0:8L due to the slowly varying Young's modulus variation. This agreement, however, does not occur for the other pair σ ¼ 0:2 , b ¼ 0:05L where, for all of the natural frequencies, the closed form approximation does not agree with the other methods. Moreover, it is possible to note that the wave approaches, i.e. the WKB and the piecewise constant, solved by MC sampling, are in better agreement with the FE for the higher natural frequencies, as expected from this kind of approximation. Figs. 15–17 and Figs. 18–20 show the 5% and 95% percentile of the input mobility for the rod and beam, respectively. They agree well, in the same way as shown by the PDF of the natural frequencies.

6. Experimental validation In the experiment, measurements of the flexural natural frequencies of an ensemble of cantilever beams were performed. The mass per unit length of the beam was randomized by adding small discrete masses to an otherwise uniform beam. Moreover, the methodology to generate the random mass distribution and its approximation to a continuous random field is discussed below Please cite this article as: A.T. Fabro, et al., Wave propagation in one-dimensional waveguides with slowly varying random spatially correlated variability, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.01.013i

12

A.T. Fabro et al. / Journal of Sound and Vibration ] (]]]]) ]]]–]]]

Fig. 9. Normalized PDF of the (a) first, (b) third, (c) sixth and (d) eighth natural frequencies of the rod using the closed form expression (grey), the WKB (dash-dot), the piecewise constant (dot) and the FE (solid) for σ ¼ 0:1 , b ¼ 2L:

A simple model using assumed modes and a lumped mass approximation is presented and the proposed wave approach using the WKB approximation for free vibration of flexural waves is applied for validation. Some issues of representing a discrete implementation of a continuous random field are also discussed along with the results obtained using the WKB formulation. 6.1. Distribution of random masses The positions of the attached masses are evenly distributed along the beam, i.e. their positions xi always remain the same, as shown in Fig. 21, but the values of each individual mass mi vary according to a Gaussian homogeneous random field H ðxÞ with exponentially decaying correlation function. A collocation method was used to calculate a correlation matrix. This matrix was then used to solve the associated KL eigenproblem [31] into eigenvectors f i and their respective eigenvalues λi , such that the zero mean random field is given by Eq. (31). The masses were added at 10 different locations in multiples of 1 g, such that the nominal or reference beam had m0 ¼ 6 g added at each location, adding up 60 g total. The values mi ¼ m0 þ μi are calculated by individual realizations of the zero-mean random field μi , with a given correlation length, at the point xi , i.e. using the mid-point approach for the random field discretization and adding it to the baseline value of 6 g. Furthermore μi was discretized to be an integer multiple of 1 g. This random field is numerically generated using a solution of the KL expansion, discretizing the domain and solving an eigenproblem from the correlation matrix [31,62]. The PDF of this random field is given such that the values between 6 g and 6 g fall inside the 6σ region. Then they are rounded to the nearest integer, and any value sampled outside of this region is rounded to 6 g or 6 g accordingly. Fig. 22 gives the normalized histogram obtained from this procedure. The uniform beam is L ¼ 0:4 m long with rectangular cross section, width b ¼ 39:85 mm and thickness h ¼ 2 mm. It has Young's modulus E ¼ 1:91  1011 Pa, estimated using the measured natural frequencies and a mass density ρ ¼ 7:8  103 kg=m3 estimated by weighing the beam. Five different correlation lengths were used to generate the mass distributions: b ¼ 0, meaning that the values generated for the masses are statistically independent or uncorrelated; b ¼ 0:10 L, equal to the distance between two adjacent masses; b ¼ 0:25 L; b ¼ 0:60 L; and b ¼ 1, meaning that all of the masses are the same. Please cite this article as: A.T. Fabro, et al., Wave propagation in one-dimensional waveguides with slowly varying random spatially correlated variability, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.01.013i

A.T. Fabro et al. / Journal of Sound and Vibration ] (]]]]) ]]]–]]]

13

Fig. 10. Normalized PDF of the (a) first, (b) third, (c) sixth and (d) eighth natural frequencies of the rod using the closed form expression (grey), the WKB (dash-dot), the piecewise constant (dot) and the FE (solid) for σ ¼ 0:1, b ¼ 0:8L:

The mass distribution is expected to approximate a continuous mass density spatial distribution, for the flexural vibration of a straight beam, and also for a range of correlation lengths. This spatial distribution is represented in the form   ρðxÞ ¼ ρ0 1 þ ηρ þ σ HðxÞ ; (45) where ηρ represents the effects due to the mean mass loading, m0 and σ is a dispersion parameter for the random field. The offset value ηρ ¼ 0:2409 is calculated using the reference beam, i.e. the nominal properties, having 6 g at each location, and the dispersion parameter σ ¼ 0:0803 is calculated based on the 6σ requirement.

6.2. Assumed modes with the lumped mass model A simple model using the assumed modes from an analytical solution of a uniform cantilever Euler–Bernoulli beam then with added lumped masses is used to give an approximation of the natural frequencies. It assumes point masses, with no rotational inertia, which give no added stiffening. If ϕj ðxÞ represents the jth mass normalized mode of the uniform cantilever beam, i.e. for the nominal system RL RL ρ 0 Aϕi ðxÞϕj ðxÞdx ¼ δij , then the modal stiffness is kij ¼ 0 EI yy ϕi'' ðxÞϕj'' ðxÞdx. The perturbation of the natural frequencies due to changes in the added masses can be approximated by adding their kinetic energy contributions to each modal mass μij , using the mode shape of the homogeneous system, so

μij ¼

Nm X

mk ϕi ðxk Þϕj ðxk Þ;

(46)

k¼1

Please cite this article as: A.T. Fabro, et al., Wave propagation in one-dimensional waveguides with slowly varying random spatially correlated variability, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.01.013i

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A.T. Fabro et al. / Journal of Sound and Vibration ] (]]]]) ]]]–]]]

Fig. 11. Normalized PDF of the (a) first, (b) third, (c) sixth and (d) eighth natural frequencies of the rod using the closed form expression (grey), the WKB (dash-dot), the piecewise constant (dot) and the FE (solid) for σ ¼ 0:2, b ¼ 0:05L:

and mk is the kth mass at xk . Additionally, the diagonal terms can be used to estimate the natural frequencies of the cantilever beam with added masses so that approximately

ωj 

sffiffiffiffiffiffiffiffiffiffiffiffiffi kjj : 1 þ μjj

(47)

This expression is used to derive statistics of the natural frequencies by generating samples of the discrete random field for the mass distribution, in a Monte Carlo framework [66].

6.3. WKB approximation So far, the results obtained from the WKB approximation considered Young's modulus of elasticity as a random field. However, the same approach can be used in the case of varying mass density, of the form of Eq. (45). Thus, the flexural natural frequency ωn , from Eq. (23), is given by

θ2Tj ωj ¼ R pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ; L 4 dx ρ ðxÞA=EI yy 0

(48)

where θTn is the nth root of a transcendental equation related to the cantilever boundary conditions [67]. The integral in the denominator can be calculated numerically, using for example a Gauss–Legendre scheme. In this experiment, only the mass density is considered to be spatially varying. Moreover, this expression is also be used to derive statistics of the natural frequencies in a Monte Carlo framework [66], using the continuous random field approximation for the mass density and constant flexural stiffness. Please cite this article as: A.T. Fabro, et al., Wave propagation in one-dimensional waveguides with slowly varying random spatially correlated variability, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.01.013i

A.T. Fabro et al. / Journal of Sound and Vibration ] (]]]]) ]]]–]]]

15

Fig. 12. Normalized PDF of the (a) first, (b) third, (c) sixth and (d) eighth natural frequencies of the beam using the closed form expression (grey), the WKB (dash-dot), the piecewise constant (dot) and the FE (solid) for σ ¼ 0:1, b ¼ 2L:

6.4. Experimental results and discussion The natural frequencies of the 2nd to the 7th mode of the cantilever beam with added masses were measured using 20 statistically independent samples for the mass configuration, for each correlation length except for the uniform sample corresponding to b ¼ 1. A slightly different procedure was used because there are only 13 possible integer values for the masses between  6 g and 6 g. The mean value and standard deviation for this case were calculated by using the normalized discrete PDF, i.e.

ω¼

13 X

ωi pi ;

(49)

i¼1

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 13 uX σ ω ¼ t ð ωi  ω Þ 2 p i ;

(50)

i¼1

where pi is the normalized PDF for the discrete set of masses, as shown in Fig. 22. Table 1 gives the mean value and standard deviation of the measured natural frequencies from the 2nd, to the 7th modes for each value of correlation length. The 95% confidence interval for the mean value is calculated using a t-student distribution, for the cases with 20 samples. Note that for all the correlation lengths, the mean values of the natural frequencies are almost the same for each mode. That is expected for small values of σ , from Eq. (45). The sample mean ω and standard deviation σ ω for each mode and correlation length were used to calculate the Coefficient of Variation COV ¼ σ ω =ω. Figs. 23 and 24 show the COV for the 2nd, 3rd and 4th modes and for the 5th, 6th and 7th modes, respectively, as functions of the correlation length, along with the COV obtained with 5000 Monte Carlo samples using the simple added mass theory, Eq. (47), and the WKB approximation, Eq. (48). Note that, in general, the results agree well for all of the modes and correlation lengths except for the uncorrelated case, b ¼ 0, when the assumption of slowly varying properties is no longer valid and therefore the WKB approximation breaks down. Moreover, it can be shown that representing a random field using the midpoint method along with a coarse mesh, i.e. the distance between two consecutive masses is longer than the correlation length, may introduce higher variability into the Please cite this article as: A.T. Fabro, et al., Wave propagation in one-dimensional waveguides with slowly varying random spatially correlated variability, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.01.013i

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A.T. Fabro et al. / Journal of Sound and Vibration ] (]]]]) ]]]–]]]

Fig. 13. Normalized PDF of the (a) first, (b) third, (c) sixth and (d) eighth natural frequencies of the beam using the closed form expression (grey), the WKB (dash-dot), the piecewise constant (dot) and the FE (solid) for σ ¼ 0:1, b ¼ 0:8L:

stochastic response [32]. This result agrees with the much higher COV found for the experimental results and added mass theory prediction. For the case when b ¼ 1, the beam mass can be treated as a random variable, instead of a random field, i.e. the expressions for natural frequency from the assumed modes, Eq. (47), reduces to

ω0j ωj ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ;   1 þ mρ0 A 1 þ ηρ σ m ξ1

(51)

and from the WKB approximation, Eq. (48), reduces to

ω0j ffi; ωj ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ ηρ þ σ ρ ξ1

(52)

where ω0j is the natural frequency of the deterministic system. The random mass is given by m ¼ σ m ξ1 , where σ m is the standard deviation of the beam mass, and the mass density is a random variable given by ρ ¼ ρ0 1 þ ηρ þ σ ρ ξ1 , from Eq. (45), ξ1 being a Gaussian, zero-mean, unity standard deviation random variable. Using a first order approximation, it is possible to find analytically a value for the covariance for the WKB approach, given by COV ¼

σ

: 2ð1 þ ηρ Þ

(53)

and also for the assumed modes model, COV ¼

σm

N  m ; 2 ρ AL 1 þ η 0 ρ

assuming, from mode normalization, that the approximation

PNm

i¼1

(54)

ϕ2ij  ρ ALðN1mþ η Þ holds. This approximation depends on 0

ρ

the number of masses Nm distributed over the beam and on the jth mode shape ϕj , i.e. for high order mode shapes, more masses distributed over the beam length are needed for a good approximation. Results are shown in Table 2. Note that they show a good agreement with the results obtained from Monte Carlo sampling of Eqs. (47) and (48), also given in Figs. 23 and 24, and from the experimental results (Fig. 26). Please cite this article as: A.T. Fabro, et al., Wave propagation in one-dimensional waveguides with slowly varying random spatially correlated variability, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.01.013i

A.T. Fabro et al. / Journal of Sound and Vibration ] (]]]]) ]]]–]]]

17

Fig. 14. Normalized PDF of the (a) first, (b) third, (c) sixth and (d) eighth natural frequencies of the beam using the closed form expression (grey), the WKB (dash-dot), the piecewise constant (dot) and the FE (solid) for σ ¼ 0:2, b ¼ 0:05L:

Input Mobility dB ref 1 m/(Ns)

0 -20 -40 -60 -80 -100 -120

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

Frequency [kHz]

Fig. 15. 5% and 95% percentile and mean value of the input mobility of the rod using the WKB approach (dark grey dashed), the FE (black solid) and the piecewise constant (light grey dotted), σ ¼ 0:1, b ¼ 2L:

7. Conclusions In this paper, wave propagation in one dimensional waveguides with spatially varying properties was modelled using the WKB approximation, an analytical formulation considering slowly varying properties. It is assumed that when propagating over a finite distance, the total phase change of a wave is given by the integral of the spatially varying wavenumber. This is used to find analytical expressions for the natural frequencies and input mobility of a finite length waveguides undergoing either flexural or longitudinal waves. The KL expansion, a series representation of random fields, is then used to include spatially correlated material random variability in the proposed approach. It significantly reduces the computational time when compared to the standard stochastic FE method and provides a suitable framework to account for spatially correlated randomness and to calculate natural frequency and forced response statistics. Closed form expression for the PDFs of the natural frequencies for a rod and a beam example are derived, using a first order approximation. It also allows efficient approximations for small standard Please cite this article as: A.T. Fabro, et al., Wave propagation in one-dimensional waveguides with slowly varying random spatially correlated variability, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.01.013i

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18

Input Mobility dB ref 1 m/(Ns)

0 -20 -40 -60 -80 -100 -120

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

Frequency [kHz]

Fig. 16. 5% and 95% percentile and mean value of the input mobility of the rod using the WKB approach (dark grey dashed), the FE (black solid) and the piecewise constant (light grey dotted), σ ¼ 0:1, b ¼ 0:8L:

Input Mobility dB ref 1 m/(Ns)

0 -20 -40 -60 -80 -100 -120 0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

Frequency [kHz]

Fig. 17. 5% and 95% percentile and mean value of the input mobility of the rod using WKB approach (dark grey dashed), the FE (black solid) and the piecewise constant (light grey dotted), σ ¼ 0:2, b ¼ 0:05L:

Input Mobility dB ref 1 m/(Ns)

20 0 -20 -40 -60 -80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Frequency [kHz]

Fig. 18. 5% and 95% percentile and mean value of the input mobility of the beam using the WKB approach (dark grey dashed), the FE (black solid) and the piecewise constant (light grey dotted), σ ¼ 0:1, b ¼ 2L:

deviations and large correlation lengths of the given random field. Despite the fact that only a Gaussian homogeneous random field was assumed, other non-Gaussian random field distributions for the waveguide properties can be readily included in the formulation. It also can be extended for two dimensional structures and to take different wave modes into account. The randomness was also included in the piecewise constant wave formulation by discretizing the random field, sampling the KL expansion at the geometric centre of each element. This approach is more general in the sense that it considers internal reflections, and provided very good results, but it might require a fine discretization mesh, increasing the computational cost significantly. In addition, experimental validation was carried out, in which the spatial correlation was controlled a priori. The mass per unit length of an otherwise uniform cantilever beam was randomized by adding masses and the issue of implementing a random field in a structure according to an analytical model was described. Natural frequencies were measured for five different correlation lengths, including one where the masses were homogeneously distributed, i.e. all the masses along the Please cite this article as: A.T. Fabro, et al., Wave propagation in one-dimensional waveguides with slowly varying random spatially correlated variability, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.01.013i

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19

Input Mobility dB ref 1 m/(Ns)

20 0 -20 -40 -60 -80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Frequency [kHz]

Fig. 19. 5% and 95% percentile and mean value of the input mobility of the beam using the WKB approach (dark grey dashed), the FE (black solid) and the piecewise constant (light grey dotted), σ ¼ 0:1, b ¼ 0:8L:

Input Mobility dB ref 1 m/(Ns)

20 0 -20 -40 -60 -80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Frequency [kHz]

Fig. 20. 5% and 95% percentile and mean value of the input mobility of the beam using WKB approach (dark grey dashed), the FE (black solid) and the piecewise constant (light grey dotted), σ ¼ 0:2, b ¼ 0:05L:

Fig. 21. Evenly distributed masses along a cantilever beam. The mass values can vary from 0 g to 12 g added at each location.

0.25

PDF

0.2

0.15

0.1

0.05

0

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

Mass, g Fig. 22. Marginal distribution of the values for the added masses mi to the baseline value 6 g.

Please cite this article as: A.T. Fabro, et al., Wave propagation in one-dimensional waveguides with slowly varying random spatially correlated variability, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.01.013i

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20

Table 1 Mean value with 95% confidence interval and the estimated standard deviation of the measured natural frequencies, in Hz, for the 2nd to the 7th mode for each value of correlation length. Correlation Length b

0 0:10 L 0:25 L 0:60 L 1

Mode 2

Mode 3

Mode 4

Mode 5

Mode 6

Mode 7

ω

σω

ω

σω

ω

σω

ω

σω

ω

σω

ω

σω

57.4 7 0.3 57.5 7 0.4 57.7 7 0.6 57.9 7 0.7 57.6

0.6 0.8 1.3 1.5 1.6

159.6 7 0.9 160.4 7 1.0 160.1 7 1.5 161.0 7 1.9 160.1

2.1 2.2 3.2 3.9 4.3

310.5 7 1.3 310.5 7 2.1 310.7 7 3:0 311.1 7 3.5 309.9

2.8 4.3 6.2 7.4 8.1

506.7 7 1.9 506.3 7 2.6 508.4 7 4.4 509.3 7 5.7 505.6

4.2 5.4 9.1 11.8 12.7

760.0 7 2.5 759.8 7 4.3 761.1 7 6.3 764.0 7 8.5 757.9

5.6 8.9 13.0 17.8 18.6

1072.6 7 4.4 1075.0 7 7.0 1074.7 7 9.8 1077.0 7 12.0 1069.1

9.8 14.7 20.5 25.1 27.3

0.04

COV

0.03 0.02 0.01 0

0L

0.10 L

0.25 L

0.60 L

∞

Correlation length Fig. 23. Coefficient of Variation for the 2nd, 3rd and 4th mode for each correlation length from the measurements (respectively circle, þ and  marker), simple added mass theory (respectively dashed, dotted and dash-dotted line) and WKB (grey continuous line for all of the modes).

0.04

COV

0.03 0.02 0.01 0

0L

0.10 L

0.25 L

0.60 L

∞

Correlation length Fig. 24. Coefficient of Variation for the 5th, 6th and 7th mode for each correlation length from the measurements (respectively circle, þand  marker), simple added mass theory (respectively dashed, dotted and dash-dotted line) and WKB (grey continuous line for all of the modes).

Fig. 25. Free wave propagation in piecewise constant waveguide considering internal reflections.

Table 2 Coefficient of variation for the 2nd, 3rd, 4th, 5th, 6th and 7th modes natural frequencies for the infinite correlation length b ¼ 1, from the experimental results and the first order approximation on the assumed modes and WKB approach. COV

Mode 2

Mode 3

Mode 4

Mode 5

Mode 6

Mode 7

Assumed modes WKB Experimental

0.0325 0.0324 0.0269

0.0325 0.0324 0.0267

0.0325 0.0324 0.0258

0.0325 0.0324 0.0250

0.0325 0.0324 0.0244

0.0325 0.0324 0.0254

Please cite this article as: A.T. Fabro, et al., Wave propagation in one-dimensional waveguides with slowly varying random spatially correlated variability, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.01.013i

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21

Fig. 26. Point excitation of a finite length rod with piecewise constant properties and non-dissipative junctions, taking into account internal reflections.

beam had the same value, meaning an infinite correlation length, and another such that they were completely uncorrelated, i.e. statistically independent from each other, meaning zero correlation length. The experimental results were compared to a simple model with assumed modes from an analytical solution for a cantilever beam with added lumped mass, assuming point mass, no rotational inertia and no added stiffening and with the proposed WKB formulation for flexural waves, in which the discrete mass distribution is approximated to a spatially continuous mass density. The results agreed very well and it was shown that the greater the correlation length the higher the standard deviation of the natural frequencies.

Acknowledgements The authors gratefully acknowledge the financial support of the Brazilian National Council of Research (CNPq). Appendix A. The WKB approximation for longitudinal and flexural waves In this section, the WKB solutions for longitudinal and flexural waves are briefly reviewed. The free vibration and forced response due to a point excitation in finite length waveguides is formulated in terms of propagation matrices. The WKB solutions for longitudinal and flexural waves in straight rods and beams, respectively, are reviewed for sake of completeness. Consider a rod undergoing longitudinal vibration, with slowly changing material propertiesp along its length, ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi such that a propagating wave has negligible reflections due to local impedance changes. In this case, cL ðxÞ ¼ EðxÞ=ρðxÞ is the local phase velocity at position x, and uðx; t Þ the axial displacement. Assuming a time harmonic and separable solution, uðx; t Þ ¼ U ðxÞ e  iωt ; it is possible to define a local wavenumber kL ðxÞ ¼ ω=cL ðxÞ. Thus, the eikonal function SðxÞ ¼ ln U~ ðxÞ þiθðxÞ [41,42] is introduced, in order to find wave solutions of the form U ðxÞ ¼ eSðxÞ ¼ U~ ðxÞe 7 iθðxÞ :

(A.1)

Applying this solution to the governing equation and neglecting the higher order terms it is possible to find two solutions of the kind Rx Rx i kL ðxÞdx i kL ðxÞdx  1=2  1=2 x0 ðxÞe þ C 2 kL ðxÞe x0 ; (A.2) U ðxÞ ¼ C 1 kL where C 1 and C 2 are arbitrary constants. The two terms correspond respectively to positive going and negative going Rx travelling waves. Note that the complex term in the exponential θðxÞ ¼ x0 kL ðxÞdx corresponds to a phase change of a  1=2 propagating wave along the waveguide and the term U~ ðxÞ ¼ C 1;2 kL ðxÞ corresponds to an amplitude change. x0 and x correspond to two arbitrary points along the waveguide. In a homogenous rod, only the phase would change in a propagating wave, but local changes in the impedance in a spatially varying rod would also lead to an amplitude change. The  validity of the WKB approximation can be evaluated by defining a length scale LðxÞ ¼ kL ðxÞ=ðdkL ðxÞ=dxÞ and assuming that the wavelength 2π =kL ðxÞ is small compared to LðxÞ, i.e. 1=kL ðxÞ{LðxÞ. Thus, the WKB assumption is valid when [68]    1 dk ðxÞ   L (A.3)  2 {1: k ðxÞ dx  L

The phase and amplitude change of the positive and negative going waves travelling a distance L, from x ¼ 0 to x ¼ L are related by the waves propagating in the same direction at the two ends and is given by (Fig. 1) sffiffiffiffiffiffiffiffiffiffiffi R kL ð0Þ  i L kL ðxÞdx þ þ 0 b ¼ e a ; (A.4) kL ðLÞ sffiffiffiffiffiffiffiffiffiffiffi R kL ð0Þ i L kL ðxÞdx  e 0 a ; (A.5) b ¼ kL ðLÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where kL ð0Þ and kL ðLÞ are the wavenumbers at the two ends, i.e. kL ðxÞ ¼ ρðxÞ=EðxÞω evaluated at x ¼ 0 and x ¼ L respectively. 

Please cite this article as: A.T. Fabro, et al., Wave propagation in one-dimensional waveguides with slowly varying random spatially correlated variability, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.01.013i

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22

The same procedure can be applied to a Euler–Bernoulli beam, where EI yy ¼ EðxÞI yy ðxÞ and ρA ¼ ρðxÞAðxÞ are the spatially varying flexural stiffness and mass per unit length, respectively, and wðx; t Þ is the transverse displacement along the beam. Assuming a time harmonic solution wðx; t Þ ¼ W ðxÞe  iωt , thus the eikonal SðxÞ is also used here in order to find wave solutions of the kind ~ ðxÞe 7 iθðxÞ : W ðxÞ ¼ W

(A.6)

Applying this solution to the governing equation, and neglecting higher order terms it is possible to find solutions of the form [38] Rx Rx Rx Rx     3 1 i kB ðxÞdx  kB ðxÞdx i kB ðxÞdx kB ðxÞdx x0 W ðxÞ ¼ ρA 8 EI yy 8 C 1 e þC 2 e x0 þ C 3 e x0 þ C 4 e x0 (A.7) where C 1 , C 2 , C 3 and C 4 are arbitrary constants and the four terms correspond to positive going and negative going Rx propagating and evanescent waves. Moreover, exponential terms θðxÞ ¼ 7 i x0 kB ðxÞdx correspond to a phase change of  38  the Rx  18 ~ the respective waves and W ðxÞ ¼ ρA EI yy and θðxÞ ¼ 7 x0 kB ðxÞdx correspond to an amplitude change because of  1=4 pffiffiffiffi changes in the properties of the beam and decay of evanescent waves. In terms of bending waves, kB ðxÞ ¼ ρA=EI yy ω is the local free bending wavenumber. The phase and amplitude change of the positive going and negative going propagating and evanescent waves travelling, from x ¼ 0 to x ¼ L, are again related to the positive and negative going waves at the two ends i.e. ~ ðLÞ  i R L k ðxÞdx W B 0 aþ ; e ~ W ð0Þ

(A.8)

~ ðLÞ i R L k ðxÞdx W a ; e 0 B ~ W ð0Þ

(A.9)

bN ¼

~ ðLÞ  R L k ðxÞdx W aNþ ; e 0 B ~ W ð0Þ

(A.10)



~ ðLÞ R L k ðxÞdx W aN ; e 0 B ~ ð0Þ W

(A.11)

b

þ

b

¼



þ

¼

bN ¼

 3 1 ~ ð0Þ and W ~ ðLÞ are given by W ~ ðxÞ ¼ ρA  8 EI yy  8 evaluated at the points x ¼ 0 and x ¼ L, respectively. where W

A.1. Free wave propagation Considering a one dimensional finite waveguide with length L, then the positive going a þ and the negative going þ  a wave amplitude vectors on the left and on hand boundary, b and b , can be related by b ¼ Λa where " # Λ11 0 Λ¼ (A.12) Λ22 0 

is a propagation matrix. The wave amplitudes are given by the vectors ( ) þ þ a b ; b ¼ : a¼  a b

(A.13)

and can include propagating and evanescent waves. The waves reflected from boundaries at the left and the right hand ends  þ are given by the respective reflection matrices as a þ ¼ Γ L a  and b ¼ Γ R b . Thus, from Eq. (A.12) þ

¼ Λ11 a þ ;

(A.14)



¼ Λ22 a  ;

(A.15)

¼ Λ11 ΓL a  ;

(A.16)

ΓR b þ ¼ Λ22 a 

.(A.17)

b b

and using the reflection relations at the boundaries, then b

þ

This is then used in the wave train closure principle, for finding the natural frequencies, as shown in Section 2. Please cite this article as: A.T. Fabro, et al., Wave propagation in one-dimensional waveguides with slowly varying random spatially correlated variability, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.01.013i

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23

A.2. Point excitation A point time harmonic excitation force F, applied at L1 , creates positive going q þ and negative going q  waves, so that the waves at this point are given by the sum of these directly excited and incident ones such that þ

a2þ ¼ b1 þq þ ;

(A.18)

and 

b1 ¼ a2 þq  ;

(A.19)

as indicated in Fig. 2. Propagation from the boundaries to the excitation point is given by b1 ¼ ΛL a1 and b2 ¼ ΛR a2 , where " # " # ΛL11 0 ΛR11 0 ΛL ¼ ; ΛR ¼ (A.20) ΛL22 ΛR22 0 0 are propagation matrices and the wave amplitudes are given by the vectors ( þ) ( þ) ( þ) a1 a2 b1 ; a ; b ; ¼ ¼ a1 ¼  2 1 a1 a2 b1

( b2 ¼

þ

b2

 b2

) :

(A.21)

The waves reflected from the boundaries at the left and right ends are given by the respective reflection matrices as  þ a1þ ¼ Γ L a1 and b2 ¼ Γ R b2 . Using Eqs. (A.18) and (A.19) together with the propagation and reflection properties, and noting that the inverses of the matrices ΛL11 and ΛR11 are simply the inverse of each of their diagonal elements, then the positive going wave vector a2þ is given by   1 1 (A.22) a2þ ¼ ΛL11 ΓL ΛL 22 ΛR 22 ΓR ΛR11 a2þ þ q  þq þ : It is related to the negative going vector a2 by 1

a2 ¼ ΛR

22

ΓR ΛR11 a2þ :

(A.23) a2þ

The wave vector at the point of excitation is given by the sum of the positive going and negative going vectors, namely     1  1 1 1 1 a2þ þ a2 ¼ I þ ΛR 22 ΓR ΛR11 I ΛL11 ΓL ΛL 22 ΛR 22 ΓR ΛR22 I þ ΛL11 ΓL ΛL 22 q:

a2

wave

(A.24)

This expression is used to calculate the input mobility Y ðωÞ from the contributions of all wave components to the response. These relationships can be used to find the wave amplitudes at any point of the waveguide. Appendix B. Finite length waveguide with piecewise constant variability In this section a piecewise constant wave formulation is briefly reviewed. The free vibration and forced response due to a point excitation in finite length waveguides is formulated in terms of propagation matrices, and particular results are given for the rod and beam examples. B.1. Free wave propagation For a finite one dimensional waveguide with length L, undergoing free wave propagation, with piecewise constant properties, divided into N elements, such that for the jth element the positive going and the negative going waves on the left aj and on the right side bj , are related by bj ¼ Λj aj , Fig. 25, with, " # Λj11 0 Λj ¼ 0 Λ (B.1) j22 where Λj11 and Λj22 are propagation matrices, and the wave amplitudes ( þ) ( þ) aj bj aj ¼ ; bj ¼ ;   aj bj þ

(B.2)



are given by the positive ajþ , bj and negative aj , bj going wave amplitudes. Assuming that there are no energy losses from a wave transmitted from one element to another, two consecutives elements are connected by a scattering matrix aj þ 1 ¼ Gj bj , which can be partitioned as ( þ ) " #( þ ) aj þ 1 bj Gj11 Gj12 ¼ ; (B.3)   aj þ 1 Gj21 Gj22 bj where Gj11 , Gj12 , Gj21 and Gj22 are blocks whose derivations are given below. Please cite this article as: A.T. Fabro, et al., Wave propagation in one-dimensional waveguides with slowly varying random spatially correlated variability, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.01.013i

24

A.T. Fabro et al. / Journal of Sound and Vibration ] (]]]]) ]]]–]]]

It follows that the wave amplitudes incident on the left of the waveguide a1 and the ones at the right hand boundary bN , ~ a , where [36] at the ðN 1Þth junction are given by bN ¼ Λ 1 N1

Λ~ ¼ ΛN ∏ GN  j ΛN  j :

(B.4)

j¼1

Natural frequencies can be found by applying the principle of wave-train closure, tracing the round-trip propagating wave, and, differently from the previous case, considering the internal reflections, using the partitioned matrix ( þ) " #( þ ) a1 bN Λ~ 11 Λ~ 12 ¼ ~ ; (B.5)  ~ a1 bN Λ 21 Λ 22 together with the reflections matrices at the boundaries ΓR and ΓL , so that they correspond to the zeros of the characteristic equation, given by Eq. (29). If the scattering matrix is approximated as an identity matrix Gj  I, i.e. internal reflections are negligible, then the natural frequencies can be found from the zeros of the characteristic equation given by Eq. (30). For the case of longitudinal vibration, the propagation matrices reduce to

Λj11 ¼ e  iθj

(B.6)

Λj22 ¼ eiθj

(B.7)

where θj ¼ kLj l is the phase change within the jth element. In addition, the scattering matrix is given by Eq. (B.42) can also be put in the form " þ  # þ  r jþ 1 t j t j r j r j Gj ¼ þ ; (B.8)  r j 1 tj where t jþ is the transmission coefficient from the jth junction from the left to the right, t j is the transmission coefficient from the right to the left, r jþ is the reflection coefficient of the propagating wave reflecting at the junction j from the left to the right and r j is the reflection coefficient for a propagating wave reflecting at the same junction but from the right to the left. For simplicity, all the elements are assumed to be the same length l ¼ L=N. Then substituting back into Eq. (30), one can then obtain the natural frequency as

ωn ¼

2nπ  ϕR  ϕL PN qffiffiffiffiffiffiffiffiffiffiffiffi; ρj =Ej j¼1

(B.9)

2L

where ϕL and ϕR are the phase changes at the left and right boundaries, i.e., ΓL ¼ e  iϕL and ΓR ¼ e  iϕR . Analogously, for the case of flexural vibration, Eq. (B.1) reduces to " # e  iθ j 0 ; (B.10) Λj11 ¼ 0 e  θj "

Λj22 ¼

eiθj

0

0

eθj

# ;

(B.11)

where θj ¼ kBj l is the phase change within the jth element. Also, the scattering matrix block elements are given by Eqs. (B.43)–(B.46). Substituting into Eq. (30) and solving for θTn , it is possible to obtain the natural frequency for flexural vibration in a beam as

ωn ¼ P

θ2Tn

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 N 4 ðρAj =EI yy j ÞLÞ ; j¼1

(B.12)

where θTn is the nth root of the transcendental Eq. (30). B.2. Point excitation The excitation is given by a harmonic force F applied at the Mth junction, at a distance L1 from the left boundary, so that it creates a positive going q þ and a negative going q  propagating wave. The waves at this point are given by the sum of these directly excited waves plus the transmitted and reflected ones such that ( þ ) " þ #(  ) ( þ ) þ aM þ 1 rM tM aM þ 1 q ¼ þ ; (B.13)  þ   tM rM bM q bM þ þ   where the reflection and transmission matrices rM , tM , tM and rM are related to the scattering matrix GM at the excitation point by Eqs. (B.38)–(B.41).

Please cite this article as: A.T. Fabro, et al., Wave propagation in one-dimensional waveguides with slowly varying random spatially correlated variability, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.01.013i

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25

The wave amplitudes at the left end of the waveguide a1 and the ones on the right end of the Mth element bM , are related ~ a ; also the wave amplitudes on the right end of the waveguide b and the ones on the left end of the ðM þ1Þth by bM ¼ Λ L 1 N ~ a element aM þ 1 , are related by bN ¼ Λ R M þ 1 , where M 1

Λ~ L ¼ ΛM ∏ GM  j ΛM  j ;

(B.14)

j¼1

Λ~ R ¼ ΛN

NM1

∏

j¼1

GN  j ΛN  j :

(B.15)

~ ¼Λ ~ ~ G Λ Also, these matrices are related to the free wave propagation matrix by Λ L M R . Applying the boundary conditions,  þ i.e. a1þ ¼ ΓL a1 and bN ¼ ΓR bN , to these expressions leads to ( ) " #( þ ) þ aM þ 1 bN Λ~ R11 Λ~ R12 ¼ ~ ; (B.16)  aM ΓR bNþ Λ Λ~ þ1 R21

(

þ

bM



bM

)

R22

"

Λ~ L11 Λ~ L12 ¼ ~ Λ L21 Λ~ L22 þ

#(

ΓL a1 a1

) ;

(B.17)



þ  from which it is possible to find that aM þ 1 ¼ TR aM þ 1 and bM ¼ TL bM , where    1  ~ ~ T R ¼ ΓR Λ Λ~ R21  ΓR Λ~ R11 ; R12  Λ R22

(B.18)

  1 ~ ~ TL ¼ Λ Λ~ L21 ΓL þ Λ~ L22 : L11 ΓL þ Λ L12

(B.19)

This, in turn, can be applied to Eq. (B.13) such that h i   1  i  1h þ  1  þ þ þ   aM tM TR tM TL I rM TL q þqþ : þ 1 ¼ I rM T R tM TL I  rM TL

(B.20)

The vector of waves w at the point of excitation can be calculated by using the superposition principle of the vector of þ  positive going wave aM þ 1 and a negative going waves aM þ 1 at the point of excitation, such that þ w ¼ ðI þTR ÞaM þ 1:

(B.21) þ



For the case of longitudinal vibrations, continuity of displacement leads to q ¼ q ¼ q ; and equilibrium of the forces at the excitation, i.e. F þ P þ  P  ¼ 0, where P þ ¼  ikLM þ 1 EM þ 1 Aq þ and P  ¼ ikLM EM Aq  , leads to the directly excited waves amplitudes as q¼ 

i F: ðkLM EM þkLM þ 1 EM þ 1 ÞA

(B.22)

For flexural vibrations continuity and equilibrium lead to qþ ¼ q ¼ q ¼

  1  iF   : 3 3 2I yy EM kM þ EM þ 1 kM þ 1  i

(B.23)

B.3. Scattering matrix In this section the scattering matrix used in the piecewise constant waveguide approach is derived. Assuming two connected waveguides the degrees of freedom and internal forces as grouped into two vectors, for each waveguide i and j [35], as Waj ¼ ψajþ ajþ þ ψaj aj ; þ



Wbi ¼ ψbiþ bi þ ψbi bi ; where ψ ψ ψ waveguide, and þ a ,

 a ,

þ b

and ψ

 b

þ



þ

þ





Fbi ¼ ϕbi bi þ ϕbi bi ;  aj ,

þ bi

(B.25)

are the displacement matrices for the positive and negative going waves at each respective Faj ¼ ϕaj ajþ þ ϕaj aj ;

þ aj ,

(B.24)

(B.26) (B.27)

 bi

where ϕ ϕ ϕ and ϕ are the internal forces matrices for the positive and negative going waves at each respective waveguide. Continuity and equilibrium conditions, can be written as [35] Please cite this article as: A.T. Fabro, et al., Wave propagation in one-dimensional waveguides with slowly varying random spatially correlated variability, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.01.013i

26

A.T. Fabro et al. / Journal of Sound and Vibration ] (]]]]) ]]]–]]]

Fig. 27. Waves at a waveguide junction.

Caj Waj ¼ Cbi Wbi ;

(B.28)

Eaj Faj ¼ Ebi Fbi ;

(B.29)

where Caj and Cbi are the continuity matrices and Eaj and Ebi are the equilibrium matrices. A scattering matrix Gi , relating the waves from the left side of the junction to the waves at its right side, Fig. 27, can be defined as ( þ) ( þ) aj bi ¼ Gi : (B.30)   aj bi Applying the continuity and equilibrium conditions, and assuming the matrix inversion exists, thus 2 3  1" # Caj ψajþ Caj ψaj Cbi ψbiþ Cbi ψbi 4 5 Gi ¼ þ  þ  : Eaj ϕaj Eaj ϕaj Ebi ϕbi Ebi ϕbi

(B.31)

Moreover, the partitions of Gij are

1  1  1 þ  þ Caj ψaj Eaj ϕaj  Eϕaj Caj ψajþ Caj ψaj Gi11 ¼  Ebi ϕbi Caj ψajþ

1    1 þ þCbi ψbiþ Caj ψajþ  Caj ψaj Eaj ϕaj Eaj ϕaj ;

(B.32)

1  1  1   þ Caj ψaj Eaj ϕaj  Eaj ϕaj Caj ψajþ Caj ψaj Gi12 ¼  Ebi ϕbi Caj ψajþ

1    1 þ þCbi ψbi Caj ψajþ  Caj ψaj Eaj ϕaj Eaj ϕaj ;

(B.33)

1      1 þ  1 þ Eaj ϕaj Caj ψajþ  Caj ψaj Eaj ϕaj Eaj ϕaj Gi21 ¼  Cbi ψbiþ Eaj ϕaj þ

þEbi ϕbi



1  1  þ Eaj ϕaj Eaj ϕaj Caj ψajþ Caj ψaj ;

(B.34)

1      1 þ  1 þ Eaj ϕaj Caj ψajþ  Caj ψaj Eaj ϕaj Eaj ϕaj Gi22 ¼  Cbi ψbi Eaj ϕaj 

þEbi ϕbi



1  1  þ Eaj ϕaj Eaj ϕaj Caj ψajþ Caj ψaj :

(B.35)

It is also possible to write the scattering matrix Ti , at the ith junction relating the wave incoming to the junction to the waves going out of the junction as þ  a a ¼ T : (B.36) i þ  b b Also applying the continuity and equilibrium conditions, and assuming the matrix inversion exists, 2 3  12 3 Cja ψajþ  Cbi ψbi Caj ψaj Cbi ψbiþ 5 4 5: Ti ¼ 4 þ þ þ  Eaj ϕaj Ebi ϕbi Eaj ϕaj  Ebi ϕbi

Please cite this article as: A.T. Fabro, et al., Wave propagation in one-dimensional waveguides with slowly varying random spatially correlated variability, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.01.013i

A.T. Fabro et al. / Journal of Sound and Vibration ] (]]]]) ]]]–]]]

This matrix can be partitioned in terms of reflection and transmission matrices " þ # ri tiþ Ti ¼ ; ti ri

27

(B.37)

where tiþ and riþ are the transmission and reflection matrices from the left to the right, and ti and ri are the transmission and reflection matrices from the right to the left. They are related to G by 1 riþ ¼ Gi12 Gi22 ;

(B.38)

1 tiþ ¼ Gi11  Gi12 Gi22 Gi21 ;

(B.39)

1 ti ¼ Gi22

(B.40)

1 ri ¼  Gi22 Gi21 :

(B.41)

For the coupling between two waveguides undergoing longitudinal waves, with different properties, the continuity and equilibrium matrices are Ca ¼ Cb ¼ Ea ¼ Eb ¼ 1, the displacement matrices are ψaþ ¼ ψa ¼ ψ ¼ ψb ¼ 1, and internal forces þ  þ  matrices are ϕa ¼ iEAj kLj , ϕa ¼ iEAj kLj , ϕb ¼  iEAi kLi , ϕb ¼ iEAi kLi . This leads to the scattering matrix 0 1 EA1 k1 EA1 k1 1 1 2 þ 2EA2 k2 2  2EA2 k2 @ (B.42) G ¼ 1 EA1 k1 1 EA1 k1 A: 2  2EA2 k2 2 þ 2EA2 k2 For the coupling between two waveguides undergoing flexural waves, the continuity and equilibrium matrices are ! !

 1 1 1 1 1 0 þ  ψbi ¼ ψbi ¼ Caj ¼ Cbi ¼ Eaj ¼ Ebi ¼ , , , the displacement matrices are  ikBi  kBi ikBi kBI 0 1 0 1 ! ! 3 3 1 1 1 1 ikBi kBI þ  @ A, ψajþ ¼ ik ψ ¼ ϕ ¼ EI , , and internal forces matrices are i bi 2 2 aj  kBj ikBj kBj Bj kBI kBI 0 1 0 1 0 3 1 3 3 3 3 3  ikBj kBj ikBj  kBj ikBi kBi  þ  @ A @ A @ A. , ϕaj ¼ EI j , ϕaj ¼ EI j ϕbi ¼ EIi 2 2 2 2 2 2 kBi kBi kBj kBj kBj kBj This leads to the scattering matrix

2

Gi11 ¼

1 4Z j kBj

Gi12 ¼

1 4Z j kBj



 ki þ kj Z i þZ j

 iðk1 ik2 ÞðZ 1  Z 2 Þ    ki  kj Z i þZ j iðk1 þ ik2 ÞðZ 1  Z 2 Þ

  ! i ki þ ikj Z i Z j   ; ki þkj Z i þ Z j   ! i ki  ikj Z i  Z j   ; ki kj Z i þ Z j

(B.43)

(B.44)

Gi21 ¼ Gi12 ;

(B.45)

Gi22 ¼ Gi11 ;

(B.46)

2

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Please cite this article as: A.T. Fabro, et al., Wave propagation in one-dimensional waveguides with slowly varying random spatially correlated variability, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.01.013i