On the use of the Lorentzian function for the evaluation of the frequency averaged input power into plates

Transcription

1 On the use of the Lorentzian function for the evaluation of the frequency averaged input power into plates R. D Amico 1,2, K. Vergote 1, R. Langley 3, W. Desmet 1 1 KU Leuven, Department of Mechanical Engineering, Celestijnenlaan 3 B, B-31, Heverlee, Belgium roberto.damico@mech.kuleuven.be 2 Università degli Studi di Firenze, Dipartimento di Meccanica e Tecnologie Industriali, Via di Santa Marta, 5139, Firenze, Italy 3 University of Cambridge, Engineering Department, Trumpington Street, Cambridge CB2 1PZ, United Kingdom Abstract Predicting the average behaviour of an ensemble of complex systems is one of the most challenging tasks of vibro-acoustics. Well-established energetic formulations exist to address the problem at high-frequencies, where a diffuse and incoherent field is hypothesised and where the input power into the structure can be evaluated by assuming spatially infinite or semi-infinite systems. These approximations may be no longer valid when dealing with mid-frequency problems, where the average response of the ensemble can still be influenced by individual modes and a more accurate evaluation of the input power can significantly improve the quality of the results. This paper proposes a new approach to compute the frequency averaged input power through a single deterministic calculation using a Lorentzian function as weighting function. Thanks to its mathematical features the frequency averaging procedure can be evaluated straightforwardly for causal systems, without the need of the response computation at several frequencies. The Lorentzian function is characterised by the parameter γ, which describes its width. In order to correlate γ to the amount of uncertainty in the system, two strategies are proposed in this paper. The first one suggests to tune γ on an analytical estimate of the natural frequency statistics, while in the second case, a mode tracking approach is used. The presented approaches are used to evaluate the average input power and energy density for Kirchhoff plates perturbed by randomly distributed masses. Finally, comparisons with Monte Carlo simulations illustrate the accuracy of the proposed approaches. 1 Introduction Most manufactured systems with nominally identical properties exhibit a significant variability of their geometric and material characteristics [1]. The production and assembly processes, as well as the operational and environmental conditions, influence the system properties. The presence of such a variability in the system characteristics can create difficulties for the prediction of the dynamic behaviour of the system ensemble. At low frequencies, variabilities have negligible effects and the nominal properties can be considered representative for the ensemble. In this case, deterministic approaches, such as the Finite Element Method (FEM) [2], the Boundary Element Method (BEM) [3] or the Wave Based Method (WBM) [4], provide an accurate prediction of the system behaviour. On the other hand, at high frequencies, the wavelength is small enough 1747

2 1748 PROCEEDINGS OF ISMA212-USD212 to consider the model properties sufficiently random to describe the system behaviour by means of energetic techniques. Among these approaches, the Statistical Energy Analysis (SEA) [5] is the most widely used and efficient technique for predicting high-frequency phenomena. Under the assumptions of diffuse field and uncorrelated waves, the energy flow between subcomponents is considered proportional to the difference between their main energies, which are spatial and frequency averaged quantities representative of the statistical ensemble. The input power into the system is an important quantity when modelling SEA problems and it is commonly approximated by considering structures of infinite (or semi-infinite) spatial extent [5]. This approximation leads to a reasonable accuracy when the SEA hypotheses are fully satisfied. Nevertheless, when dealing with mid-frequency problems, the ensemble response can still be dominated by individual resonances and the approximation may be inaccurate. In order to predict the mid-frequency behaviour of the system ensemble, a frequency averaging procedure is proposed in this paper. Due to its specific mathematical properties, the Lorentzian function is used as weighting function. In fact, for causal quantities, the Lorentzian-weighted frequency average does not require the evaluation of the function at several frequencies within a band, but only one deterministic simulation computed at one complex frequency. The shape of the Lorentzian function is described by the parameter γ, which defines its width. The main goal of this research is to investigate the correlation between γ and the statistics of the perturbed system, in order to predict the average input power of the ensemble of structures. In a previous work, the averaging procedure has been applied for the prediction of the input power into rods [6]. The choice of the parameter γ was related to the eigenvalue statistics, described by the Statistical Overlap Factor (SOF) [7], and frequency averaged quantities were in good agreement with ensemble mean quantities. In this paper, the Lorentzian-weighted frequency averaging is applied to evaluate the input power into Kirchhoff plates perturbed by randomly distributed masses. Two strategies to tune the parameter γ are proposed and compared. Both of them take into account eigenfrequency statistics and the fact that variability introduces perturbation in the position of the natural frequencies. According to the first approach, the γ parameter is tuned on a first-order perturbation approximation, describing the eigenfrequency statistics [8]. The parameter γ is chosen such that the width of the Lorentzian function covers the corresponding eigenvalue distribution. Nevertheless, as the frequency increases, the response statistics saturate [9, 1]. Consequently, the correspondent parameter γ becomes stationary and the ensemble average may be overestimated. The second approach attempts to relate the perturbed eigenvalue distribution to the nominal mode shapes, by means of mode tracking. As a result, the natural frequency statistics do not level off and the Lorentzianweighted frequency average, tuned on this approach, provides an accurate estimate of the ensemble average. For both cases, the Lorentzian-weighted frequency averaged input power is compared with the Monte Carlo (MC) mean prediction of plates perturbed by randomly distributed masses. Finally, the energy distributions are evaluated and compared with the classic SEA approach. The paper is organised as follows. In Sec. 2, the Kirchhoff plate theory and the numerical approaches to its modelling are shortly introduced. Section 3 discusses the theoretical aspects of the Lorentzian-weighted averaging and the proposed approaches to tune the width of the Lorentzian function. Finally, application cases are shown and discussed in Sec Out-of-plane bending vibration of plates The most commonly used theories to model the steady-state vibration of plates are the Kirchhoff theory [11] and the Reissner-Mindlin theory [12, 13]. For thin plates and not too high frequencies, the effects of rotary inertia and shear deformation can be neglected and the Kirchhoff theory provides accurate results [14]. According to the Kirchhoff theory, the out-of-plane displacements w are governed by the following partial

3 FP7 ITN MID-FREQUENCY - CAE METHODOLOGIES FOR MID-FREQUENCY ANALYSIS 1749 differential equation, where 4 = x 4 as, x 2 y 2 4 w(x, y) kb 4 w(x, y) = F δ(x f, y f ), (1) D y 4. The plate bending wavenumber, k b, and bending stiffness, D, are defined D = k b = 4 ρhω 2 D, (2) E(1 + iη)h3 12(1 ν 2 ), (3) where ρ is the material density, h is the plate thickness, ω is the angular frequency of analysis, E is the Young modulus, η is the material damping factor, ν is the Poisson ratio and i is the imaginary unit. The plate is excited by a point force F located at point (x f, y f ) and its power into the structure can be evaluated as, P (ω, x f, y f ) = 1 2 R {F v (ω, x f, y f )}, (4) where v (x f, y f ) is the complex conjugate of the displacement velocity at point (x f, y f ). When highfrequency vibrations are considered, the plate behaviour can be described by energetic approaches like SEA [5]. In this case, the power put into an infinite plate is used and is given by, P inf (ω) = F 2 16 Dρh, (5) which is constant with frequency. As the governing eq.(1) is a fourth-order partial differential equation, two boundary conditions need to be imposed. For simply supported plates, displacements and bending moments are prescribed and both equal to zero. The modal density n(ω) of a simply supported plate can be approximated by means of the following relation, n(ω) = A 4 ρh 4π D, (6) where A is the area of the plate [5]. In the following applications, the bending vibration of Kirchhoff plates is analysed by means of two numerical approaches. The FEM [2] approach is used to compute problem eigenvalues and eigenvectors. On the other hand, to evaluate the input power and the space averaged energy density as a Frequency Response Function (FRF), the WBM [4] is used. The WBM belongs to the family of Trefftz approaches [15], in which the field variables are expanded in terms of shape functions which satisfy a priori the governing partial differential equation. The main advantages of the WBM over element based approaches are the small system dimension and the fast rate of convergence. Moreover, the use of the WBM can be particularly advantageous when performing a MC simulation for plates perturbed by random masses. In fact, the matrix related to the nominal system, of which calculation takes a large part of the total solving time, needs to be assembled only once, while the terms related to the masses have to be computed for each MC realisation. This results in a considerable reduction of solving time [16]. For the sake of brevity the numerical approach is not reported here; detailed explanations can be found in [4, 16, 17]. 3 Lorentzian-weighted frequency averaging In many engineering applications, the evaluation of the average of a function s(ω) over a frequency band, is a common procedure. Typically, a window W (ω, ω ), centered in ω, is used to weigh the average as follows, s(ω ) = + W (ω, ω )s(ω) dω (7)

4 175 PROCEEDINGS OF ISMA212-USD212 where represents the frequency average. If the mathematical feature of the window is complicated and no hypotheses are made on the function s, the evaluation of the integral may require a large computational effort. In case of numerical integration, the function s has to be evaluated at several frequencies in order to accurately compute the integral (7). An alternative way to evaluate the integral is to consider the real frequency axis as part of a contour line in the complex plane and make use of the complex residue theorem [18]. Under proper assumptions, the value of the contour integral of an analytic function can be evaluated by simply computing the residues at the poles of the function which are located inside the contour line. Although this procedure can simplify the calculation, it is required to know the position of the poles of the integrand, which in eq.(7) is composed by the functions s and W. The physical quantity described by s has to be causal, meaning that the change of its value is temporally delayed with respect to the perturbation which generates the change itself. As a mathematical consequence of this assumption, the poles of the function s are located in the upper half complex plane [14]. Their position is related to the time convention chosen; in this paper a positive sign convention is taken. If a negative sign convention is taken, the poles of the function s are located in the lower half plane. The second function of the integrand is the weight function W, which in this paper is taken to be a Lorentzian, L(ω, ω, γ). The Lorentzian function, illustrated in Fig. 1, is a single-peak function of mathematical form, L(ω, ω, γ) = 1 πγ 1 [ ], (8) 1 + ( ω ω γ ) 2 where ω is the location of the peak of the distribution and γ is a scale parameter which specifies the halfwidth at half-maximum. In the complex plane, the Lorentzian function has two poles located at ω ± iγ. The PDF (ω) ω Figure 1: Lorentzian function centered in ω = and with γ = 1. knowledge of the position of the poles of the function L(ω, ω, γ)s(ω) allows the choice of the contour of integration in the lower half complex plane and the result of the Lorentzian-weighted frequency averaging, using the complex residue theorem, leads to the following relation, s(ω ) = + L(ω, ω, γ)s(ω) dω = s(ω iγ). (9) Equation (9) states that the frequency average of the function s around a frequency ω, using a Lorentzian as weighting function, corresponds to the evaluation of the function s at a complex frequency ω iγ. It is worth noting that no additional computational effort is required to evaluate the average, which can be computed even if the positions of the poles of s are not known. 3.1 Evaluation of the input power When dealing with uncertainties and variabilities, the interest is mainly focused on the prediction of the ensemble behaviour. For this, the vibrational response of a structure can be described both in kinematic and

5 FP7 ITN MID-FREQUENCY - CAE METHODOLOGIES FOR MID-FREQUENCY ANALYSIS 1751 energetic terms and the choice of the approach is strictly dependent on the aspect to investigate. Evaluating the average of a kinematic quantity, i.e. displacement, would be unsatisfactory at mid- and high-frequencies. In that case, the average will tend to the direct field within the domain. On the contrary, energetic quantities are of higher interest, since they are more representative of the average response. Nevertheless, the energy is a quadratic quantity and its poles are spread all over the complex plane, both in the upper and in the lower half-planes. Consequently, integral (7) cannot be evaluated in the efficient manner as described before. On the other hand, the calculation of the input power only requires the knowledge of the excitation, F, and the velocity field, v(x, y, ω), at the points where the system is excited. If the Lorentzian is used to weigh the input power, the frequency average can be computed as indicated in eq.(9) and, for steady-state problems, it results in the following expression, P (ω, x, y) = L(ω, ω, γ)r {F v (ω, x, y)} dω = 1 2 R {F v (ω iγ, x, y)}. (1) In the next sections, this result is applied to evaluate the frequency averaged input power into plates perturbed by additional randomly distributed masses. Since the result is assumed to be representative of the mean frequency response function, the average is not evaluated at center frequencies of wide bands, e.g. thirdoctave bands, but continuously over the frequency range of analysis. For very high frequencies and relatively simple structures, the input power can be approximated by assuming infinite or semi-infinite spatial extent of the model and analytical expressions can be found in literature [5, 14]. On the other hand, when dealing with mid-frequency problems and complex structures, using such an approximation may not be satisfactory. In this context, using a frequency averaged input power can provide more complete information to an energetic model. Using a Lorentzian weighting function allows this evaluation just by computing the response of the unperturbed, deterministic system at a complex frequency. This becomes even more advantageous when the numerical approach used to model the system does not require an increase in computational effort when handling complex quantities, i.e. the BEM and the WBM. Finally, it is assumed that the frequency average is representative of the mean ensemble behaviour. This hypothesis is usually valid for high frequency phenomena, where the SOF is sufficiently high [7, 19]. In this paper, it is hypothesised that the equivalence holds also in the mid-frequency range. 3.2 Choice of the parameter γ and eigenvalue statistics An important factor in the evaluation of eq.(1) is the width of the Lorentzian function, represented by the parameter γ. The goal of the following analysis is to formulate a relation between the frequency averaging and the effects of the variabilities on the system response. To do that, the parameter γ has to be properly tuned. In the following, two approaches are proposed. Both of them are based on the fact that variabilities produce scatter in the position of natural frequencies of the system, which can be quantified through the standard deviation σ. Thus, the parameter γ is chosen according to the value of σ(ω), considered as a continuous function of frequency. As the position of the nominal natural frequencies is modified due to the perturbation, the amplitude of the Lorentzian is taken to be large enough to encompass part of them and provide a frequency averaged value Natural frequency statistics based on first order perturbation approximation The SOF was introduced by Manohar and Keane to estimate the frequency beyond which individual modes cease influencing the response [7]. Its mathematical expression is defined as follows, S pq = 2σ pq µ pq, (11) where σ pq is the standard deviation of the pq-th natural frequency from its mean value and the mean frequency spacing, µ pq, is equal to the inverse of the modal density, which for bare plates can be computed by

6 1752 PROCEEDINGS OF ISMA212-USD212 using eq.(6). The SOF provides a measure of the amount of crossing and veering modes that can occur in a perturbed system. When the frequency is low, the perturbation often has negligible effects on the eigenvalue statistics and statistical overlap barely occurs between modes. When the wavelength becomes smaller, the perturbation has a major influence on the position of the natural frequencies, which can cross or veer from each other. As a result, the SOF increases until it saturates, meaning that further perturbation has negligible influence on the eigenvalue statistics [9, 1, 2, 21]. For rectangular plates perturbed by N m randomly distributed masses of mass m a, it is possible to approximate the variance of a natural frequency, as suggested in Ref. [8]. The mathematical relation is based on a first-order perturbation analysis and assumes that the pq-th natural frequency is modified according to the following relation, ωpq 2 ωpq 2 1 N m j=1 m a ψpq(x 2 mj ), (12) where ψ pq is a mass normalised shape function associated to the pq-th mode. As a result, the variance of the position of the pq-th eigenvalue can be written as, σ 2 pq =.3125 N mm 2 a M 2 ω 2 pq, (13) where M is the total mass of the bare plate. This expression is a useful guideline for the degree of statistical overlap but it becomes invalid as the SOF approaches unity, that is when the standard deviation σ is equal to half the mean frequency spacing, µ. In order to cover the eigenvalue distribution and provide a frequency averaged value, the parameter γ will be chosen as equal to the standard deviation σ, evaluated through expression (13) Natural frequency statistics based on mode tracking The frequency average performed as described in eq.(1) refers to the nominal system, meaning that natural frequencies and mode shapes also refer to nominal properties. This constitutes the basic concept for the second proposed strategy, which attempts to tune the parameter γ on the statistics of the nominal mode shapes. To evaluate the effects of mass perturbation on the nominal behaviour of the system the following procedure is applied. After having evaluated the nominal natural frequencies and mode shapes, the eigenvalues and eigenvectors of the perturbed systems are evaluated through MC simulations. For each MC realisation, perturbed mode shapes are associated to nominal ones according to the highest MAC (Modal Assurance Criterion) value [22]. Finally, perturbed eigenvalues are reorganised as function of the nominal mode shapes and the statistics of the nominal natural frequencies are evaluated. It is important to underline the conceptual limits of this procedure. At low frequencies, individual modes dominate the average response of the structure and investigating the statistics of nominal mode shapes can be indicative of their distributions. As frequency increases, the behaviour of the perturbed system becomes sufficiently random such that the exact modal statistics are no longer required to determine the average response. Furthermore, due to the complexity of mode shapes, the tracking of modes may become difficult and misleading. From a probabilistic point of view, this procedure allows to translate variability on the system properties to variability on the frequency behaviour of the system. As a consequence, for mid-frequencies the ergodic assumption can be relaxed, once the frequency PDFs associated to the nominal modes are known. 4 Application cases In this section, applications of the previous strategies are presented. Once the model properties are introduced, the analysis on the eigenvalue statistics and evaluation of frequency averaged input power are

7 FP7 ITN MID-FREQUENCY - CAE METHODOLOGIES FOR MID-FREQUENCY ANALYSIS 1753 discussed. 4.1 Problem description A plate of 1 by.6 m and 1 mm thickness is considered and shown in Fig. 2. The plate is made of Aluminum, with Young modulus 7 GPa, density 27 kg/m 3, Poisson ratio.3 and damping coefficient.1. A harmonic unit force excitation is applied at point (.7,.4)m. The plate is simply supported along all the boundaries. The system behaviour is perturbed by adding randomly distributed masses over its surface, for a total of four different cases consisting of 1, 2, 3 and 5 masses. Each mass is.2% of the total mass of the bare plate. The following analyses are performed over a frequency range between and 1 khz, in which the plate under consideration has approximately 185 modes F y [m] x [m] Figure 2: Geometry of the plate with point force excitation, F ( ). 4.2 Evaluation of natural frequency statistics Each MC simulation of the perturbed plate consists of 2 realisations. Natural frequencies and mode shapes are evaluated by means of a FE model consisting of 6165 nodes and 6 quadrilateral shell elements. The software MD.Nastran 21 is used to perform the eigenmode calculations. Figure 3 shows the standard deviation of the natural frequencies according to the strategies proposed in the previous section. The black solid curves represent the standard deviation evaluated according with the approach presented in Sec For all cases, σ increases with increasing frequency, until it saturates. The analytical approximation, described by eq.(13), is represented by the black dashed curves. When the frequency is low, σ is well-approximated and once the SOF value reaches unity, the standard deviation is taken to be equal to the half of the mean frequency spacing, which is evaluated on the MC realisations. For all cases the σ value computed by MC simulation is approximated with reasonable accuracy by the analytical model. The red solid curves represent the standard deviation evaluated by retrieving nominal modes in simulations of perturbed plates. When frequency is low the black and the red curves are superposed to each other, meaning that perturbed mode shapes are similar to the nominal ones. As frequency increases, the standard deviation increases and it does not level off as in the previous case. To approximate the behaviour of the standard deviation a linear fitting over the whole set of data is performed and it is represented by the red dashed curves. This allows not only to regularise the curves but also to homogenize σ values. For high level of

8 1754 PROCEEDINGS OF ISMA212-USD212 perturbation, the two approximations behave differently from each other. On the contrary, for a low number of masses, σ evaluated by mode tracking can be well approximated by the relation (13), considered over the whole frequency range. Figure 4 shows how the perturbed modes are distributed with respect to their nominal positions. As the perturbation level or the mode number increase, their position is spread around the diagonal. On the contrary, when frequency is low, modes are concentrated on the diagonal, meaning that the perturbation is not drastically modifying the mode shapes. For the sake of brevity, the MAC curves are not reported here, but as expected their values decrease with increasing frequency and perturbation. When the frequency is low, the MAC value is high (from.99 to.9) at the correspondent nominal eigenfrequency. For increasing frequencies the MAC value decreases (from.9 to.4, in the highest part of the frequency range) and is spread around the correspondent nominal eigenvalue. With increasing perturbation, the spread effect increases but the associated mode shapes, especially for the cases with 1 and 2 masses, are univocally detected. Finally, it is interesting to compare the Probability Distribution Functions (PDFs) of some eigenvalues with the respective Lorentzian functions, when the parameter γ is tuned on the red dashed curve. Fig. 5 shows the PDFs related to four eigenvalues of the plate perturbed by 5 masses. A good correlation between the eigenvalue distributions and the Lorentzian functions is observed, meaning that the variability is correctly captured masses 16 2 masses Standard deviation [Hz] Standard deviation [Hz] masses 16 5 masses Standard deviation [Hz] Standard deviation [Hz] Figure 3: Standard deviation of the eigenvalue distributions evaluated without tracking nominal modes (black solid curves), by tracking nominal modes (red solid curves) and respective approximations (black and red dashed curves) 4.3 Evaluation of the frequency averaged input power In the following applications, the frequency average input power is computed for all the aforementioned perturbed cases. Each MC simulation consists of 5 realisations. The Lorentzian-weighted frequency average input power is evaluated as described in eq.(1). Instead of evaluating the nominal response of the system at a real frequency ω, the value is computed at a complex

9 FP7 ITN MID-FREQUENCY - CAE METHODOLOGIES FOR MID-FREQUENCY ANALYSIS 1755 Figure 4: Distribution of perturbed eigenvalues with respect to their nominal position. Blue color indicates large dispersion; red color indicates small dispersion..8 5 random masses eig random masses eig PDF PDF random masses eig random masses eig PDF.3.2 PDF Figure 5: Probability distribution functions for four eigenvalues. Black curve, distribution evaluated by MC simulation; red curve, Lorentzian function.

10 1756 PROCEEDINGS OF ISMA212-USD212 frequency ω iγ. As previously mentioned, the parameter γ is chosen to be equal to the value of the standard deviation evaluated by using the two aforementioned strategies. Figure 6 shows the results when γ is tuned according to the first approach (black dashed curve in Fig. 3). After the SOF reaches unity, the approximation of the standard deviation is taken to be constant. Consequently, the width of the Lorentzian function does not increase with frequency. This leads to an overestimation of the average response, which is represented by the red curve. In Fig. 7 the Lorentzian function is tuned on the linear approximation of the standard deviation based on the statistics of the nominal mode shapes evaluated by mode tracking. For all cases there is a good agreement between MC mean values and Lorentzian-weighted frequency averages. It is interesting to see that as frequency increases, the average input power asymptotically tends to the input power into an infinite plate, which is represented by the thin dashed curves. This confirms that for high frequencies infinite or semi-infinite plates accurately approximate the average behaviour of the system. 1 1 masses 1 2 masses 1 3 masses 1 5 masses Figure 6: Input power evaluation. Grey curves, MC samples; red curve, MC average; black curve, Lorentzian frequency average with γ tuned on the first approach; thin, dashed black curve, input power to an infinite plate. 4.4 Improvements on the SEA prediction The following results show the benefits achieved by enhancing classic SEA simulations by using the Lorentzianweighted frequency averaged input power. Figures 8 and 9 show the space averaged energy density for the previously presented cases. Each MC sample is evaluated by averaging the WBM deterministic response over the spatial extent of the plate. As expected, the MC mean energy asymptotically tends to the SEA prediction, computed by using input power into infinite plates. Results in Fig. 8 are obtained by using the frequency averaged input power evaluated according to the first strategy. Except for low perturbation, the predictions tend to overestimate the MC average. On the contrary, the second proposed approach allows obtaining more accurate results for all cases.

11 FP7 ITN MID-FREQUENCY - CAE METHODOLOGIES FOR MID-FREQUENCY ANALYSIS masses 1 2 masses 1 3 masses 5 masses 1 Figure 7: Input power evaluation. Grey curves, MC samples; red curve, MC average; black curve, Lorentzian frequency average with γ tuned on the second approach; thin, dashed black curve, input power to an infinite plate. 1 1 masses 1 2 masses masses 1 5 masses Figure 8: Energy density evaluation. Grey curves, MC samples; red curve, MC average; black curve, SEA with Lorentzian-weighted frequency averaged input power and γ tuned by using the first approach; thin, dashed black curve, single component SEA simulation.

12 1758 PROCEEDINGS OF ISMA212-USD masses 1 2 masses masses 1 5 masses Figure 9: Energy density evaluation. Grey curves, MC samples; red curve, MC average; black curve, SEA with Lorentzian-weighted frequency averaged input power and γ tuned by using the second approach; thin, dashed black curve, single component SEA simulation. 5 Discussion At this stage, it is interesting to discuss the possible physical interpretation of the Lorentzian-weighted frequency averaging. As can be seen from eq.(9), evaluating the response at a complex frequency, ω iγ, may be interpreted as a variation of the material damping proportional to γ. As pointed out by Langley [23], there is a strong correlation between the damping effect and the strength of the reverberant field of the ensemble of systems. When the damping effect is relatively low, waves travel from the excitation points towards the boundaries, where they are reflected. This contributes to the creation of a reverberant field within the structure. When the damping is high, waves are damped out before they can interact with each other to create a reverberant field. Consequently, the system behaves as if it was infinitely extended and no reflection comes from the boundaries. Now, it is clear the reason why the Lorentzian-weighted frequency averaged input power of a finite system, asymptotically converges to the input power of an infinite structure, when the frequency and γ are high. Nevertheless, it has to be underlined that the introduction of the parameter γ does not perturb directly the properties of the system, since the original damping value of the system is unaltered. Furthermore, it allows a better control on the value of such a fictitious damping, as function of frequency. Previous results show that a deterministic approach can be used to evaluate the input power into the structure and that SEA prediction accuracy can be drastically increased. For both proposals to tune γ, the knowledge of the eigenvalue statistics is required. Nevertheless, while for simple systems this can be easily evaluated, estimating natural frequency statistics constitutes the main obstacle towards more complex application cases.

13 FP7 ITN MID-FREQUENCY - CAE METHODOLOGIES FOR MID-FREQUENCY ANALYSIS Conclusions In this paper, a new proposal for evaluating the frequency averaged input power into plates is presented. Using a Lorentzian function as weight of a causal variable allows computing the frequency average simply by using a complex frequency as frequency of analysis. In order to relate the frequency average response to the mean response of an ensemble of structures, the amplitude of the Lorentzian is tuned by using the statistics of the natural frequencies for the given perturbed system. Two approaches to tune the parameter γ are proposed. The first one suggests to chose γ equal to the standard deviation of the natural frequency position evaluated by means of a first order perturbation analysis. The second approach takes into account the eigenvalue statistics by tracking the nominal mode shapes in perturbed systems. Also in this case γ is taken to be equal to the computed standard deviation. Results show a good correlation between MC averages and Lorentzian-weighted frequency average quantities, especially when the second strategy is applied. 7 Acknowledgments The research of R. D Amico is funded by an Early Stage Researcher grant within the European Project MID- FREQUENCY Marie Curie Initial Training Network (GA 21499). The IWT Flanders through the ASTRA project, the Fund for Scientific Research - Flanders (F.W.O.), and the Research Fund KU Leuven are also gratefully acknowledged for their support. References [1] M. S. Kompella, R. J. Bernhard, Measurements of the statistical variation of structural-acoustic characteristics of automotive vehicles, Proceedings of the SAE Noise and Vibration Conference, Warrendale, USA (1993). [2] O. Zienkiewicz, R. Taylor, The finite element method, Butterworth-Heinemann (The three volume set, 6 th edition) (25). [3] C. Brebbia, J. Telles, L. Wrobel, Boundary element techniques: theory and applications in engineering, Springer-Verlag (1984). [4] W. Desmet, A wave based prediction technique for coupled vibro-acoustic analysis, Ph.D. Thesis, KU Leuven, Leuven (1998). [5] R. Lyon, R. DeJong, Theory and application of statistical energy analysis, Butterworth-Heinemann (2 nd edition) (1995). [6] R. D Amico, K. Vergote, W. Desmet, Lorentzian-weighted frequency averaging for the evaluation of the input power into one-dimensional structural dynamic systems, Proceedings of Inter-noise 212 Conference, New York, USA (212). [7] C. Manohar, A. Keane, Statistics of energy flows in spring-coupled one-dimensional subsystems, Philosophical Transaction of the Royal Society of America A, Vol. 346 (1994), pp [8] N. Kessisoglou, G. Lucas, Gaussian orthogonal ensemble spacing statistics and the statistical overlap factor applied to dynamic systems, Journal of Sound and Vibration, Vol. 324 (29), pp [9] N. Kessisoglou, R. Langley, Application of the statistical overlap factor to predict GOE statistics, Noise and Vibration: Emerging Methods (NOVEM25), Saint Raphael, France (25).

14 176 PROCEEDINGS OF ISMA212-USD212 [1] N. Kessisoglou, R. Langley, Natural frequency statistics of engineering structures, Eleventh International Congress on Sound and Vibration (ICSV24), St. Petersburg, Russia (24). [11] A. Leissa, Vibration of plates, Acoustical Society of America, Woodbury, New York (1993). [12] E. Reissner, The effect of transverse shear deformation on the bending of elastic plates, Journal of Applied Mechanics ASME, Vol. 12 (1945), pp. A69-A77. [13] R. Mindlin, Influence of rotary intertia and shear on flexural motions of isotropic, elastic plates, Journal of Applied Mechanics ASME, Vol. 38 (1951), pp [14] L. Cremer, M. Heckl, B.A.T. Petersson, Structure-borne sound: Structural vibrations and sound radiation at audio frequencies, Springer (3 nd edition) (25). [15] E. Trefftz, Ein Gegenstück zum ritzschen verfahren, Second International Congress on Applied Mechanics, Zurich, Switzerland (1926). [16] K. Vergote, Dynamic analysis of structural components in the mid frequency range using the wave based method: non-determinism and inhomogeneities, Ph.D. Thesis, KU Leuven, Leuven (212). [17] C. Vanmaele, Development of a wave based prediction technique for the efficient analysis of low- and mid- frequency structural vibrations, Ph.D. Thesis, KU Leuven, Leuven (27). [18] M.D. Greenberg, Advanced engineering mathematics, Pearson (2 nd edition) (1998). [19] F. J. Fahy, J. Walker, Advanced applications in acoustics, noise and vibration (Chap. 11 High-frequency structural vibration), Spon Press (24). [2] R. Langley, A. Brown, The ensemble statistics of the band-averaged energy of a random system, Journal of Sound and Vibration, Vol. 275 (24), pp [21] R. Langley, A. Brown, The ensemble statistics of the enrgy of a random system subjected to harmonic excitation, Journal of Sound and Vibration, Vol. 275 (24), pp [22] D.J. Ewins, Modal Testing: theory and practice, Research Studies Press Ltd (2 nd edition), England (21). [23] R. Langley, On the diffuse field reciprocity relationship and vibrational energy variance in a random subsystem at high frequencies, Journal of the Acoustical Society of America, Vol. 121(2) (27), pp