Computing the uncertain vibrations of plates with spatially-extended random excitation using the image source method

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1 Computing the uncertain vibrations of plates with spatially-extended random excitation using the image source method J. Cuenca, M. Kurowski,2, B. Peeters Siemens Industry Software NV Interleuvenlaan 68, B-3 Leuven, Belgium 2 Institute of Fluid-Flow Machinery, Polish Academy of Sciences Fiszera 4 st., 8-23 Gdańsk, Poland Abstract Flexural vibrations of thin panels are largely responsible for vehicle interior noise. Understanding the mechanisms of generation of such vibrations in conjunction with their uncertainty due to the randomness in the source is of crucial importance. This paper provides a method for computing the mid- and high-frequency flexural vibrations of a polygonal plate in response to a spatially-extended random field. The methodology is based on the image source method, which consists in representing consecutive wave reflections at internal boundaries of a polygonal domain by means of a series of mirror sources obtained from successive symmetries of the original source with respect to the boundaries. The standard image source method provides the Green s function in the entire plate for a given point source. Thus, in order to include a spatially-extended source, an integration is performed by taking advantage of the reciprocity of the Green s function. Two approaches for implementing a spatially-extended source are investigated. The first is a phenomenological statistical model based on the spatial correlation and the probability density function of the source. Such a model allows to observe the influence of the statistical properties of the source on the uncertainty in the vibration response and its behaviour with frequency. A second approach is investigated for the study of the vibrations of a panel subjected to a turbulent flow. The latter is simulated using an unsteady computational fluid dynamics approach in a rigid-wall configuration and used as an input for the image source method. The two approaches are compared and the potential for the phenomenological model to predict a statistical ensemble containing the direct approach is investigated. Introduction Using mirror sources for describing wave reflections at boundaries within an enclosed domain is a simple and powerful concept, yet most contemporary deterministic numerical computational tools are rather based on spatial discretisation. This is due to the geometrical complexity of the modelled structures, to which the numerical tools must conform. Indeed, tools based on spatial discretisation, such as finite element and boundary element methods are applicable to arbitrarily complex geometry of the overall structure and/or its boundary. 227

2 228 PROCEEDINGS OF ISMA24 INCLUDING USD24 While such methods allow for a detailed account of geometrical details, their extension to high frequencies is difficult, even for homogeneous structures of simple geometry, due to the fact that they typically require elements per wavelength []. On a comparable degree of computational complexity, methods purely based on a description of the dynamic field as a superposition of waves subjected to reflections and transmissions at the boundaries are restricted to homogeneous polygonal or polyhedral domains but do not present an intrinsic high-frequency limit. Aeronautical and vehicle applications often involve structural components such as homogeneous polygonal panels and therefore can benefit from the inherent aspects of wave methods. The present investigation is intended to contribute to the development of efficient mid- and high-frequency alternative prediction tools. In such frequency ranges, uncertainty and variability of structural parameters have a large influence on the variability of the vibration response due to the fact that phenomena occurring at short wavelengths are more sensitive to small structural variations. Previous research on the image source method has shown that it is an efficient computational tool for the flexural vibrations of polygonal plates in the mid- and high-frequency ranges [4, 5]. The image source method makes explicit use of the representation of specular reflection at a domain boundary by means of the mirror image of the original source with respect to the boundary. Successive wave reflections are therefore represented by image sources obtained from successive symmetries of the original source with respect to the boundaries. Thus, for simple boundary geometries, a low number of terms is required, e.g. two terms in the case of a semi-infinite domain (the original source and its symmetrical image source with respect to the boundary). In the case of an enclosed domain, an infinite number of reflections takes place and therefore an infinite number of image sources is required. However, the number of image sources effectively necessary at a given frequency is finite due to the existence of damping, i.e. waves having travelled more than a certain distance or having been reflected more than a certain number of times can be neglected. The image source series can then be truncated at an arbitrary degree of precision [4]. In fact, recent progress has shown that a truncated image source series presents a minimum frequency of applicability, whereas no intrinsic higher limit exists [3]. Such property is contrary to that of finite element analysis, thereby suggesting the use of the image source method as a complementary mid- and high-frequency computational tool. In this paper, the image source method is applied to the study of the vibrations of a polygonal plate subjected to a spatially-extended source. Two complementary models of the source are used in order to study the influence of its randomness on the uncertainty in the vibration response. The first approach consists of a phenomenological model of the source, in the sense of the simplest representation of its most prominent aspects, without a quantitative description of its fundamental nature. Thus, a model of the source based on the probability density function of the source amplitude and its spatial correlation is proposed. These two properties are chosen in order to provide an insight on the influence of local and global aspects of the source. As a second model of the source, a computational fluid dynamics (CFD) simulation of the turbulent flow in the vicinity of an infinite rigid plane is used. The wall pressure is used as the source in the proposed image source approach. Although this approach is computationally expensive, it provides a physically realistic framework for analysis. The paper is organised as follows. The fundamental basis of the image source method for the flexural vibrations of plates is first recalled, including the application to a spatially-extended source. Then, the two implementations of the source are developed and finally an application example is presented. The originality of the contribution herein consists in the implementation of the phenomenological and CFD models of the source within the image source method.

3 MEDIUM AND HIGH FREQUENCY TECHNIQUES Image source model of the response of a polygonal plate subjected to a spatially-extended source 2. Green s function This section recalls the fundamental aspects of the image source method further used in this paper. The interested reader may find all relevant details in the original publications [3 7]. The method provides the response of a plate Ω with boundaries Ω to a normal harmonic point source located inside the plate, as shown in Fig.. Ω Ω Figure : Polygonal plate with source and image sources computed within a truncation radius. The Green s problem for flexural waves in a thin plate of finite dimensions can be written as [4, 5, 8] { ( D 4 kf 4 ) GΩ (r, r ; k f ) = δ(r r ) r Ω, (a) Boundary conditions r Ω, (b) where r and r respectively denote the source and observation points, D = Eh 3 2( ν 2 ) (2) is the flexural stiffness of the plate, with E its Young modulus, ν its Poisson ratio and h its thickness. The wavenumber of flexural waves is given by k f = ( ω 2 ρh D ) /4. (3) Using time dependence e iωt, structural damping is introduced by considering a complex Young s modulus in the form E = E ( iη), (4)

4 222 PROCEEDINGS OF ISMA24 INCLUDING USD24 where η is the structural damping ratio. The Green s function for a polygonal plate with arbitrary boundary conditions is obtained as [5] G Ω (r, r ; k f ) i 8πk 2 f D N v + + s= s=n v+ + e ik ξ(ξ ξ ) ei iv (r, r s ) 8πk 2 f D iv (r, r s ) 8πk 2 f D k 2 f k2 ξ µ µ k 2 f k2 ξ + e ik ξ(ξ ξ s) ei kf k f + i e kf 2 k2 ξ (µ µ(s) b ) e kf 2+k2 ξ (µ µ(s) b ) kf 2+k2 ξ µ µ k 2 f + k2 ξ T e ik ξ(ξ ξ s) A (s) pp (k ξ, k f ) ei dk ξ R(s) pp (k ξ, k f ) R (s) R (s) pe (k ξ, k f ) R (s) k 2 f k2 ξ (µ µs) k 2 f k2 ξ ep (k ξ, k f ) ee (k ξ, k f ) e i k f 2 k2 ξ (µ(s) b µs) kf 2 k2 ξ k 2 i e f +k2 ξ (µ(s) b µs) kf 2+k2 ξ dk ξ dk ξ. (5) In this expression, the first term is the direct field from the source, the second term denotes the sum of contributions from the image sources of first order, i.e. representing the first reflection at each of the N v edges of the plate, and the third term denotes the sum of contributions from the image sources of higher order, i.e. representing waves that have been reflected two or more times on the boundaries. The coordinates (ξ, µ) are local to the edge where each individual reflection takes place. Each image source contribution is a superposition of propagating and evanescent waves. In the direct contribution from the source, these two types of waves travel freely to the observation point. The contributions from image sources of first order show a coupling between both wave types at the plate boundaries via the corresponding reflection matrix. The latter contains four terms of the form R ab which are the reflection coefficients coupling incident wave a to reflected wave b, where a and b are either propagating (p) or evanescent (e). In the contributions from image sources of higher order, the evanescent components are neglected, which allows to define a cumulative reflection coefficient A pp. Furthermore, V (r, s ) is a function taking values or whether image source s is visible or not from the receiver. All details on the derivation of Eq. (5) can be found in Ref. [5]. Eq. (5) is an approximation due to the neglected evanescent components in higher order reflections and due to the fact that the image source method does not consider diffraction at plate vertices. Waves in enclosed domains are subjected to an infinite number of reflections, which are impossible to account for for a plate of arbitrary geometry [4]. Nevertheless, the presence of structural damping allows to truncate the infinite sum in Eq. (5) by eliminating all image sources that account for waves having travelled more than a predefined distance. A truncation criterion was established in [3, 4] by setting a precision parameter P such that the amplitude of the farthest image source relates to the amplitude of the original source as e ik f r r P, (6) e ik f r r t where r r and r r t are the total distances travelled by a propagating wave from the original source to the observer, and from the truncation distance to the observer, respectively. For a given precision value P and at a given frequency, the distance from the original source to the truncation is given by the inequality r t r 4 ln(p ) η ω ( E h 2 ) /4 2ρ( ν 2. (7) ) Alternatively, the frequency range of validity of the truncated image source solution is given by ( ) 4 ln(p ) 2 E h ω 2 η r t r 2ρ( ν 2 ). (8)

5 MEDIUM AND HIGH FREQUENCY TECHNIQUES 222 These inequalities show that the image source method presents a minimum frequency of validity for a given truncation distance or that there is a minimum truncation distance for a desired observation frequency. A notable aspect is that the precision of the approximation can be chosen to be arbitrarily high. 2.2 Response to a spatially-extended source A preliminary study of the vibrations of a polygonal plate subjected to a spatially-extended excitation over a finite area was carried out in Ref. [3] in the case where such excitation is characterised by its probability density function. This paragraph summarises the implementation of a spatially-extended source in the image source method in the general case. The image source method provides an approximation of the Green s function of a polygonal plate [4, 5]. In order to include a source of spatially-distributed nature, the reciprocity of the Green s function is used as follows. The source is considered as an imposed pressure on a finite subarea Ω of the plate. For further convenience, it is defined in the entire plate as p(r; k f ) r Ω, p(r; k f ) : p(r; k f ) = r Ω \ Ω, (9) Ω Ω, where the excitation may occupy a subarea or the entire plate. Following Eq. (), the response of the plate to the spatially-extended source p is the solution of the set of equations { ( D 4 kf 4 ) w(r; kf ) = p(r; k f ) r Ω, (a) Boundary conditions r Ω, (b) and is thus obtained as the convolution product D ( 4 kf 4 ) GΩ (r, r ; k f )p(r ; k f )dr = Ω Ω δ(r r )p(r ; k f )dr. () ( ) Owing to the linearity of operator D 4 kf 4 and to the defining property of the Dirac distribution, Eqs. () and (a) yield the response of the plate in the form w(r; k f ) = G Ω (r, r ; k f )p(r ; k f )dr, r Ω. (2) Ω The image source method provides the response of the plate at any point r to an excitation at a fixed point r. Thus, the integral in Eq. (2) is here evaluated by means of the reciprocity relation of the Green s function. The Green s function provided by the image source method is an approximation which partially neglects diffraction and near-field effects. However, it is constructed as a linear series of symmetrical processes, i.e. reflections, and is thus reciprocal. The reciprocity relation of the Green s function states that [9] G Ω (r, r; k f ) = G Ω (r, r ; k f ). (3) Thus, a convenient writing of Eq. (2) is w(r ; k f ) = G Ω (r, r; k f )p(r; k f )dr, Ω r Ω. (4) It is worth noting that the left-hand-side expression of Eq. (3) may as well be obtained directly by deploying a series of images of the observation point for one source point.

6 2222 PROCEEDINGS OF ISMA24 INCLUDING USD24 3 Models of spatially-extended random sources In a previous study [3], the source was modelled as a two-dimensional random process characterised by its amplitude probability density function. Such a simplified model allows to observe the presence of uncertainty in the source amplitude on the vibration response. However, vibration sources of random nature present varying degrees of spatial correlation, which was not previously accounted for. A phenomenological model including spatial correlation of the source area is proposed in the first paragraph of this section. The second paragraph addresses the direct implementation of a computational fluid dynamics (CFD) simulation of a turbulent flow as the source of vibrations within the image source method. Despite its computationally demanding nature, this approach presents the advantage of providing a realistic example whose results can be compared with those of the phenomenological approach. 3. Statistical model A phenomenological model of a simplified two-dimensional random source is here proposed. The source is characterised by the statistical properties of its amplitude and its spatial dependence. The source area is defined as f p (p), χ pp (r; k f ) r Ω, p(r; k f ) : p(r; k f ) = r Ω \ Ω, (5) Ω Ω, where f p (p) is the probability density function of the source amplitude and χ pp (r; k f ) is its spatial correlation. The source is defined in an area Ω contained in, or equal to, the plate area. The spatial correlation is controlled by imposing a wavenumber energy spectrum to the source, in the form U(k f ) = p(r; k f ) 2 dr = C k q f, (6) Ω where C is a constant and q is an arbitrary real number. Figure 2 shows an example of a source area simulated with a wavenumber spectrum for q =.. It is worth noting that the wavenumber dependence of the source is purely geometrical and accordingly the frequency spectrum is considered flat. 3.2 Turbulent flow as a spatially-extended source of vibration The turbulent flow in the vicinity of a plane surface is here simulated using a CFD approach []. The simulation consists of an unsteady three-dimensional incompressible Navier-Stokes scheme in a parallelepipedic domain where the plane z = is rigid. A homogeneous input flow velocity is imposed at face y =. In order to ensure the existence of turbulent flow, an obstacle of dimensions (l, l/, l/) is positioned in contact with surface z =. Fig. 3 shows the spatial domain used for the CFD simulations. The spatial grid used for the CFD simulations is refined in the vicinity of the obstacle and thus is irregular. Therefore, a grid matching procedure is applied in order to interpolate the wall pressure extracted by the CFD simulation onto the regular grid required in the image source method [5].

7 MEDIUM AND HIGH FREQUENCY TECHNIQUES U(kf ) k f (m ) (a).6.8 (b) χpp x, (c) fp(p) p (Pa) (d) (e) (f) (g) (h) Figure 2: Example of simulated source for q =.. (a) Wavenumber spectrum, (b) corresponding simulated source, (c) spatial correlation in x and y directions, (d) probability density function, (e-h) selected spectrum components. 4 Application to a winglet 4. Properties of the structure In this section, the methodology presented above is applied to a simplified an airplane winglet. The structure consists of a polygonal uniform plate of thickness 2 cm whose vertices are located at points (x, y) = (, ), (2, ), (3, 2), (2, 2), as depicted in Fig. 4. The material of the plate is steel, with Young s modulus E = 2 GPa, Poisson s ratio ν =.3, density ρ = 785 kg m 3. The boundary conditions consist of a clamped edge at the bottom and free edges elsewhere. A turbulent flow is considered in the lower portion of the plate, in a 5 cm-wide region. 4.2 Computation of the Green s function of the winglet A high frequency range such that f [2, 5] khz is chosen for the computation, where the modal overlap factor [2] increases from 4.8 to 2. Using a precision parameter of P = in Eq. (7) results in a truncation distance for the image source method evolving from 9.9 m to 5.8 m. This is summarised in Tab..

8 2224 PROCEEDINGS OF ISMA24 INCLUDING USD24 2l symmetry axis Ω y 2l l/2 l/ z x Figure 3: Spatial domain for the CFD simulations. The dimensions of the obstacle are magnified for the illustration. The dashed line represents the symmetry axis for the simulations and the dot-dashed line represents the subdomain Ω where the wall pressure is extracted. The arrows denote the direction of the flow. 2.5 Ω r.5 Ω Figure 4: Geometry of the winglet. Ω and Ω respectively denote the entire plate domain and the source domain. The observation point is located at r. As pointed out earlier, the truncation distance of the image source series is larger for lower frequencies than for higher frequencies. Fig. 5 shows the positions of the image sources used for the computation of the Green s function, together with the truncation radii for 2 khz and 5 khz. 4.3 Influence of the statistical parameters of the source on the vibration response The influence of the parameters of the phenomenological model is here investigated. Using a phenomenological model as the input of the approach does not allow to draw explicit conclusions as to the resulting uncertainty in the vibration response. Therefore, a comparative study is carried out here, by observing the evolution of the vibration response by varying the two statistical properties. The spatial correlation and probability distribution of the source are controlled by respectively modifying the order q of the wavenumber spectrum and the type of amplitude probability density function. The influence of such source properties on the vibration response is here investigated in a comparative study. In order to do so, it is necessary to evaluate the statistical properties of the vibration response, namely the mean value and standard deviation per frequency. This is performed by applying the method within a Monte Carlo scheme, that is, repeating the simulation a large number of times. Each run of the simulation provides one spectrum

9 MEDIUM AND HIGH FREQUENCY TECHNIQUES 2225 f (khz) MOF r t r (m) sources Table : Modal overlap factor (MOF) [2], truncation radius and number of image sources at the lower and higher bounds of the frequency range of interest, for precision parameter P = khz 4 5 khz Figure 5: Image source positions and truncation radii for 2 khz and 5 khz. and thus the ensemble of spectra is taken as the population for determining the statistical momenta. In the present case, 2 Monte Carlo samples have been used Power of the wavenumber spectrum The power q defining the wavenumber spectrum controls the spatial dependence of the source region. Indeed, a negative value of q imposes a prevalence of lower wavelengths, whereas a positive value of q favours the existence of small wavelengths. Accordingly, the spatial correlation is respectively wide and narrow in these cases. Figure 6 illustrates the cases q = 5 and q = 5. The vibration response shows that a source with a narrow spatial correlation produces a vibration response with a higher degree of randomness in its frequency dependence than a more spatially-correlated source. This can be observed in the resonant behaviour of the structure, in particular in the disappearance of the anti-resonance at 4.6 khz. Here, a normal distribution has been used as the probability density function for each component of the wavenumber spectrum. It is worth noting with respect to Fig. 6(d) that the probability density function of the compound source differs from the fundamental distribution, due to the fact that it is a superposition of the random fields at different spatial scales.

10 2226 PROCEEDINGS OF ISMA24 INCLUDING USD24 q = 5 q = 5 (a) (b) (c) (d) U(kf ) χpp fp(p) k f (m ) x, p (Pa) U(kf ) χpp fp(p) k f (m ) x, p (Pa) (e) w(r) 2 w(r) f (khz) f (khz) Figure 6: Simulation of the flexural vibrations of the winglet using a normal amplitude distribution. Left: q = 5; right: q = 5. (a) Wavenumber energy spectrum; (b) plate, source and observation point at one Monte Carlo sample; (c) spatial correlation in the x and y directions; (d) probability density function of the source amplitude; (e) modulus of the spectrum of the out-of-plane displacement at the observation point with mean, standard deviation and Monte Carlo ensemble.

11 MEDIUM AND HIGH FREQUENCY TECHNIQUES 2227 σ =.6 σ =. (a) U(kf ) U(kf ) (b) (c) (d) fp(p) χpp k f (m ) x, p (Pa) χpp fp(p) k f (m ) x, p (Pa) (e) w(r) 2 w(r) f (khz) f (khz) Figure 7: Simulation of the flexural vibrations of the winglet using a wavenumber spectrum defined by q = and a normal amplitude distribution with two values of the standard deviation σ. Left: σ =.6; right: σ =.. (a) Wavenumber energy spectrum; (b) plate, source and observation point at one Monte Carlo sample; (c) spatial correlation in the x and y directions; (d) probability density function of the source amplitude; (e) modulus of the spectrum of the out-of-plane displacement at the observation point with mean, standard deviation and Monte Carlo ensemble.

12 2228 PROCEEDINGS OF ISMA24 INCLUDING USD Amplitude probability density function In order to assess the sensitivity of the vibration response to the probability density function of the source amplitude, a comparative study is here made by controlling explicitly and exclusively the standard deviation of the source amplitude. Fig. 7 shows a simulation using a flat wavenumber spectrum and a normal distribution of the amplitude, with two different values of the standard deviation. The figure shows that by controlling the standard deviation of the source amplitude, the change in the variability of the vibration response is uniform across the entire frequency range of interest. 4.4 Vibration response to a turbulent flow A CFD simulation according to the specifications in sec. 3.2 is here performed. The source is placed on the plate such that the direction of the flow is horizontal and matches the source area Ω previously defined. The properties of the setup are as follows. The length of the obstacle is l =. m for a flow velocity of 5 m s, the total simulation time is.5 s, with a time step of 4 s, i.e. a sampling frequency of khz, which results in a maximum frequency of 5 khz. Figure 8 shows a snapshot of the time-domain two-dimensional pressure field at the surface of the plate Figure 8: Modulus of the dynamic pressure at t =.755 s in the CFD simulation. A Fourier transform is applied to the time series in order to obtain the pressure field in the frequency domain. The full two-dimensional pressure field at each frequency is then used as an input for the image source method. In doing so, the source is considered stationary and as such does not contain any variability such as in the simulations of the previous paragraph. An estimate of such variability is made by extracting the spatial correlation and probability density function from the CFD results so as to define an equivalent statistical model. Fig. 9 shows the spatial correlation and probability density function extracted from the CFD simulation, a Monte Carlo sample of the source using the equivalent model and the flexural displacement response using directly the CFD source and predicted by the statistical model. It can be observed that the statistical ensemble provided by the phenomenological model does not entirely contain the direct simulation with the CFD results. This is here interpreted as a failure of the statistical model to correctly represent the CFD results, due to the fact that the flow has a rather deterministic behaviour, as observed in Fig. 8. Nevertheless, the response given by the two approaches share the same order of magnitude and follow comparable trends with frequency.

13 MEDIUM AND HIGH FREQUENCY TECHNIQUES 2229 (a) (c) χpp x, (b) (d) fp(p) w(r) p (Pa) f (khz) Figure 9: Simulation of the flexural vibrations of the winglet using the source obtained from the CFD simulation. (a) Spatial correlation of the source in the x and y directions; (b) probability density function of the source amplitude; (c) Plate, source and observation point at one Monte Carlo sample; (e) modulus of the spectrum of the out-of-plane displacement at the observation point: solid line, direct CFD source; dashed lines, mean and standard deviation of the equivalent phenomenological model; gray lines: Monte Carlo ensemble. 5 Conclusion In this paper, the flexural vibrations of polygonal plates with a random spatially-extended excitation have been investigated. The framework for the study is the image source method, previously developed and shown to be an efficient prediction tool in the mid- and high-frequency ranges. The image source method provides the Green s function of a plate for a point source. A spatially-extended excitation is applied by taking advantage of the reciprocity of the Green s function. A simple phenomenological model is proposed for random sources, based on a superposition of random spatial fields at different spatial scales, obeying to a wavenumber energy spectrum. The source thus simulated is characterised by two statistical parameters, namely its spatial correlation and by its probability density function. Such a phenomenological model allows to observe the influence of the statistical properties of the source on the structural vibrations. In a comparative study, it is observed that the more the source area is spatially correlated, the more the resonant behaviour of the structure is excited. Indeed, a low spatial correlation results in a loss of phase coherence in the vibrations. Additionally, it is shown that the probability density of the source, and in particular its standard deviation, has a uniform effect across the frequency range of interest. This is due to the fact that the probability density function does not carry any spatial information. A physical model of a spatially-extended random source of vibrations is also provided, in the case of a turbulent flow induced by a small obstacle on a planar surface. The turbulent flow is predicted using a CFD approach and the pressure at such surface is used as the source in the present method. Additionally, the phenomenological model is applied using statistical properties extracted from the CFD simulation. It is observed that the phenomenological model captures the overall trends of the frequency response and shares the same order of magnitude with the direct CFD source. However, the latter does not present a sufficient degree of

14 223 PROCEEDINGS OF ISMA24 INCLUDING USD24 randomness in order for the statistical ensemble to be representative of the phenomenon. Nevertheless, in situations where statistical properties are sufficient to describe the source from a vibration standpoint, i.e. in case of a fully developed turbulence, a phenomenological model is a viable and computationally efficient alternative to CFD. The proposed methodology is computationally efficient and the uncertainty and variability analysis here undertaken would not be possible using conventional finite element techniques. A major advantage of the approach is that it exploits the fact that the Green s function of the plate and the source are independent from each other. Therefore the Green s function only has to be computed once for a given structure. Acknowledgements The European Commission is greatly acknowledged for the financial support of this work through the EU FP7 Marie Curie IAPP project STADYWICO, Grant Agreement The authors would like to thank the Academic Computer Centre in Gdansk (TASK) for the resources allocated for the CFD simulations. The first author wishes to thank Prof. François Gautier and Prof. Laurent Simon at Université du Maine, Le Mans, France, Prof. Brian R. Mace at University of Auckland, New Zealand, and Dr. Neil S. Ferguson at University of Southampton, United Kingdom, for their valuable contribution to the work on the image source method and on uncertainty analysis. References [] J. Blazek. Computational Fluid Dynamics: Principles and Applications: Principles and Applications. Elsevier Science, 2. [2] J. Cuenca. Wave models for the flexural vibrations of thin plates. PhD thesis, Université du Maine, France, 29. [3] J. Cuenca. Computing the vibrations of polygonal panels under distributed random excitation using the image source method. In 5th European Conference for Aeronautics and Space Sciences (EUCASS), Munich, -5 July 23. [4] J. Cuenca, F. Gautier, and L. Simon. The image source method for calculating the vibrations of simply supported convex polygonal plates. Journal of Sound and Vibration, 322(4-5):48 69, 29. [5] J. Cuenca, F. Gautier, and L. Simon. Harmonic Green s functions for flexural waves in semi-infinite plates with arbitrary boundary conditions and high-frequency approximation for convex polygonal plates. Journal of Sound and Vibration, 33(6):426 44, 22. [6] J. Cuenca, B.R. Mace, and N.S. Ferguson. High-frequency vibrations of uncertain coupled beams using an image source approach. In Noise and Vibration: Emerging Methods (NOVEM), Sorrento, -4 April 22. [7] J. Cuenca, B.R. Mace, and N.S. Ferguson. Computing the response of an assembly of uncertain beams using an image source approach. In ISMA/USD, Leuven, 7-9 September 22. [8] K.J. Graff. Wave Motion in Elastic Solids. Dover, New York, 99. [9] P.M. Morse and H. Feshbach. Methods of Theoretical Physics. McGraw-Hill, New York, 953. [] O.C. Zienkiewicz and R.L. Taylor. The Finite Element Method for Solid and Structural Mechanics. Butterworth-Heinemann, 25.