Influence of inhomogeneous damping distribution on sound radiation properties of complex vibration modes in rectangular plates

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

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Influence of inhomogeneous damping distribution on sound radiation properties of complex vibration modes in rectangular plates Oliver Unruh a,b,n a German Aerospace Center (DLR), Institute of Composite Structures and Adaptive Systems, Lilienthalplatz 7, 38108 Braunschweig, Germany b Technische Universitaet Braunschweig, Institute of Adaptronic and Functional Integration, Langer Kamp 6, 38106 Braunschweig, Germany

a r t i c l e i n f o

abstract

Article history: Received 20 August 2015 Received in revised form 2 May 2016 Accepted 4 May 2016 Handling Editor: D. Juve

In order to reduce noise emitted by vibrating structures additional damping treatments such as constraint layer damping or embedded elastomer layers can be used. To save weight and cost, the additional damping is often placed at some critical locations of the structure, what leads to spatially inhomogeneous distribution of damping. This inhomogeneous distribution of structural damping leads to an occurrence of complex vibration modes, which are no longer dominated by pure standing waves, but by a superposition of travelling and standing waves. The existence of complex vibration modes raises the question about their influence on sound radiation. Previous studies on the sound radiation of complex modes of rectangular plates reveal, that, depending on the direction of travelling waves, the radiation efficiency of structural modes can slightly decrease or significantly increase. These observations have been made using a rectangular plate with a simple inhomogeneous damping configuration which includes a single plate boundary with a higher structural damping ratio. In order to answer the question about the influence of other possible damping configurations on the sound radiation properties, this paper addresses the self- and mutual-radiation efficiencies of the resulting complex vibration modes. Numerical simulations are used for the calculation of complex structural modes of different inhomogeneous damping configurations with varying geometrical form and symmetry. The evaluation of self- and mutual-radiation efficiencies reveals that primarily the symmetry properties of the inhomogeneous damping distribution affect the sound radiation characteristics. Especially the asymmetric distributions of inhomogeneous damping show a high influence on the investigated acoustic metrics. The presented study also reveals that the acoustic crosscoupling between structural modes, which is described by the mutual-radiation efficiencies, generally increases with the presence of travelling waves. & 2016 Elsevier Ltd. All rights reserved.

Keywords: Structural acoustics Complex vibration modes Sound radiation Inhomogeneous damping

n Correspondence address: German Aerospace Center (DLR), Insitute of Composite Structures and Adaptive Systems, Lilienthalplatz 7, 38108 Braunschweig, Germany. E-mail address: [email protected]

http://dx.doi.org/10.1016/j.jsv.2016.05.009 0022-460X/& 2016 Elsevier Ltd. All rights reserved.

Please cite this article as: O. Unruh, Influence of inhomogeneous damping distribution on sound radiation properties of complex vibration modes in rectangular plates, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j. jsv.2016.05.009i

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1. Introduction The reduction of structural weight is one of the key goals on the way to environmentally friendly and cost efficient transportation. It is well known that there is a contradiction between a consequent structural lightweight design and noise and vibration requirements. Due to the lower mass and higher stiffness, high performance lightweight structures such as carbon fibre reinforced plastics (CFRP) exhibit increased vibration amplitudes and noise levels. A possible solution is the application of additional damping treatments such as constraint layer damping (CLD) or embedded elastomer layers, which increases dissipation rates of vibration energy and improve vibro acoustic behaviour of the structure [1,2]. In order to save weight and cost, the placement of these damping treatments can be derived from an optimisation process, which is based on the modal strain energy [3] or structural intensity [4] consideration. The optimisation leads to a very local application of damping treatments, focussed on some critical locations of the vibrating structure [5–7]. In this case the spatial distribution of structural damping becomes inhomogeneous and it leads to an occurrence of complex vibration modes. Compared to real modes, complex modes are no longer dominated by pure standing waves but by a superposition of travelling and standing waves [8]. The influence of homogeneous damping on the sound radiation of vibrating plates is discussed by Xie et al. [9] or Fahy and Gardonio [10]. With increasing homogeneous damping the radiation efficiency of the plate also increases in the frequency range of edge and corner radiators. This occurs due to a strong coupling of neighbouring resonances with resulting piston-like vibration shapes and reduced acoustic short circuit phenomena. Wodtke and Lamancusa [11] investigated the influence of inhomogeneous damping distribution and stated that it has a structural (loss factors) and an acoustic effect (radiation efficiency) on sound radiation. In this study modified vibration shapes, that occur due to inhomogeneous damping distribution, are mentioned as a reason for changes in radiation efficiency. Nevertheless, complex vibration modes with travelling wave components are not investigated here. Marburg [12] considered complex normal modes of a fluid in external acoustics using an example of an open cavity. Torres et al. [13] observed complex vibration modes in a top plate of a classic guitar. This is one of the very few publications that link the structural dynamic behaviour of complex vibration modes to the sound radiation. However, this study is focussed only on the methodology of recognising complex modes using frequency response functions calculated by the FE simulation. The acoustic relevance of complex vibration modes, and their influence on the radiation characteristics of the guitar is not investigated in this work. Consideration of complex vibration modes regarding their acoustic properties is addressed in some recent publications by Unruh et al. [14,15]. In these studies, the acoustic metrics, such as self-radiation efficiency, spatial distribution of acoustic intensity and directivity of the radiated sound field are investigated and it is shown that travelling wave components of complex vibration modes can significantly influence the sound radiation of rectangular plates. Both publications investigate the case, where bending waves travel along one dominant direction of the plate. This configuration of travelling waves occurs, when one of the plates boundaries features an increased structural damping. In [14] it is shown that structural modes with even order in the main direction of travelling waves significantly increase their radiation efficiency below the coincidence frequency, due to a better coupling in the first radiation mode. In the case of structural modes with odd order, this coupling slightly decreases, which results in a lower radiation efficiency. The coupling in the first radiation mode correlates to the volume velocity of the mode shape, which is zero for real modes of even–even or even–odd order. For example, when an even–odd order mode is affected by travelling waves in the direction of the even order, the volume velocity becomes non-zero and the coupling in the first radiation mode increases. The presence of inhomogeneous damping in plates not only leads to an occurrence of travelling waves but also to the redistribution of the vibration maxima and minima, which can affect the sound radiation. In order to separate these two mechanisms, a simplified analytical model is introduced in [15]. This model allows the calculation of complex vibration patterns with a variable amount of travelling waves without amplitude redistribution. The results show that this redistribution has a minor importance for the acoustic metrics and the presence of travelling waves dominates the sound radiation phenomena. As it has already been mentioned, these two initial studies only address a simple case, where bending waves travel along one dominant direction of the vibrating plate. This simple approach is suitable for the basic understanding of the phenomena, which affects the sound radiation of complex vibration modes. Nevertheless, in real applications the spatial distribution of inhomogeneous damping and accordingly the configuration of travelling bending waves can be much more complicated. For this reason, the present paper addresses the important question about the influence of other possible distributions of inhomogeneous damping on the acoustic properties of complex vibration modes. The presented research is focussed on the identification of relevant geometrical properties of additional damping, which influence the sound radiation. The first part of the paper describes some basic damping configurations, which can be probably found in real applications. Therefore, the geometrical forms as well as symmetry properties of additional damping are varied. In the next part, the resulting complex vibration modes are characterised regarding their self-radiation efficiency using the elemental radiator approach [10]. This allows the identification of damping configuration with the biggest impact on sound radiation, which are investigated in more detail. The self-radiation efficiencies of complex vibration modes are considered and it is shown how the geometrical properties of additional damping determine the influence on sound radiation below the Please cite this article as: O. Unruh, Influence of inhomogeneous damping distribution on sound radiation properties of complex vibration modes in rectangular plates, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j. jsv.2016.05.009i

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Fig. 1. Basic damping distributions with different shape and symmetrical properties (S – symmetric, B – bilateral symmetric, C – central symmetric, A – asymmetric).

coincidence frequency. As a complement to the self-radiation efficiencies, the mutual-radiation efficiencies [10,16] are investigated. Finally, the presented results are discussed and an outlook to future research is given.

2. Basic damping distributions in rectangular plates In rectangular plates or in more complex structures, which can be subdivided in plate elements, different configurations of inhomogeneous damping can occur. Despite the seemingly infinite number of variations it is possible to define a limited number of fundamentally different configurations. These mainly differ in their geometrical shape and the symmetry of the areas with higher damping. For example, one or more edges of the plate can have an increased damping. In real applications, this can be observed in skin panels of stiffened structures, where some stiffeners can be riveted or manufactured integrally and therefore exhibit a different amount of damping. Edges with higher damping can be distributed symmetrically or asymmetrically with respect to the plates centre. For example, two adjacent or opposite, or even all four edges can have an increased damping. It is also conceivable that some areas of the plate, for example the centre or the corners feature higher damping. This can be the case when some additional damping treatments are applied at certain areas of the structure, or the variation of damping results from different materials (e.g. different core material in a sandwich). Based on these considerations, a total of twelve configurations of inhomogeneous damping, shown in Fig. 1 can be derived. The additional damping is marked by a dark colour. The notation of the presented configurations refers to different symmetry properties of additional damping and gives S1–S5 symmetrical, B1–B3 bilateral symmetrical, C1–C2 central symmetrical and A1–A2 asymmetrical distributions. In order to ensure the comparability of different configurations, the same total area of additional damping is defined. Of course many other configurations of inhomogeneous damping are possible. However, in the author's opinion, the presented distributions cover all important properties in terms of symmetry and the related configurations of travelling waves in the resulting complex mode shapes. For instance B1 represents the case, where bending waves travel towards one of the plates boundaries. In S1 and S2, travelling waves start at the centre of the plate and propagate towards respectively two or four boundaries. The configuration S3 addresses the reversed case, where the waves are travelling from the boundaries to the centre of the plate. The asymmetric configurations A1 and A2 cause bending waves travelling in diagonal direction. Other cases have either direct similarity to the configurations shown here, or can be considered as a combination thereof. For example, a plate with three damped boundaries has partially the properties of the plate S1 and B1.

Please cite this article as: O. Unruh, Influence of inhomogeneous damping distribution on sound radiation properties of complex vibration modes in rectangular plates, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j. jsv.2016.05.009i

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Fig. 2. Calculated 4
4 structural modes of different damping configurations.

3. Numerical simulation of inhomogeneously damped plates 3.1. Calculated complex vibration modes For every configuration shown in Fig. 1, as well as for the reference plate without additional damping, a finite-element model is created in the FE-Software ANSYS. All plates have a dimension of 0:9
0:6 m. The thickness and material of the panel have no importance for the consideration of complex vibration modes because they only affect the eigenfrequencies, but in order to obtain an eigenvalue solution, aluminium material properties (E ¼72 GPa, ν ¼ 0:3, ρ ¼ 2700 kg=m3 ) and thickness of 0.005 m are defined. Nevertheless, all results presented in this paper assume that complex vibration modes can occur at any possible frequency below the coincidence. In accordance with the FE-model presented in [14], the reference Rayleigh damping of the unaffected plate material is defined by damping ratios of ξ1  0:01. For the areas affected by additional damping treatments, ratio of ξ2  0:2 is defined. This high damping ratio is needed in order to achieve a sufficient complexity of the resulting complex mode shapes. Nevertheless, this value is not unrealistic and can be expected by an intensive use (e.g. multiple elastomer layers in the composite layup of damping materials [17]). Clamped boundary conditions are defined for plates with all investigated damping configurations. Preliminary studies also addressed plates with simply supported boundaries and it is noticed that the type of boundary condition has no fundamental effect on investigated phenomena. The type of boundary condition only affects the absolute values of e.g. radiation efficiency of a mode. The DAMP-Solver is used in the ANSYS-Software for the solution of the eigenvalue problem for the inhomogeneously damped system. Depending on the damping configuration, 22 to 24 Eigenvalues and Eigenvectors are calculated in the frequency range up to 1000 Hz. Fig. 2 shows the real and imaginary parts as well as amplitude and phase distribution of the even–even 4
4 mode shape for the reference plate and for one example from every symmetry group. It is important to Please cite this article as: O. Unruh, Influence of inhomogeneous damping distribution on sound radiation properties of complex vibration modes in rectangular plates, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j. jsv.2016.05.009i

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notice that complex modes are only determined up to a arbitrary normalisation factor. This means that the picture shown in Fig. 2 is not uniquely determined and real and imaginary parts can change depending on the used normalisation factor. The 4
4 mode of the reference plate is dominated by pure standing waves, what can be seen on 01 and 1801 angles in the phase distribution. In case of the damping distributions with inhomogeneous damping the mode shape includes bending waves, which travel towards the higher damped area. This can be seen by phase ramps with angles aside of 01 and 1801. Travelling waves can also be noticed by blurred nodal lines in the amplitude distribution. 3.2. Determination of the acoustic metrics In this section, a short description of the acoustic modelling is provided. The calculation of acoustic metrics, such as selfand mutual-radiation efficiencies is conducted by the elemental radiator approach [10]. The total sound power W radiated by a plate into the far-field is calculated using an array of normal velocities vn , sound pressures p and the elemental radiator area Se as follows: W¼

Se
H  Re vn p ; 2

(1)

where the superposed H denotes the Hermitian transpose (complex conjugate transpose). For the sound power evaluation harmonic vibrations are assumed. This means that normal velocities vn and sound pressures p are space dependant and the time dependence is provided by eiωt . The sound pressure can be derived from the structural velocities vn and the frequency-dependent radiation impedance matrix Z. The off-diagonal elements of the radiation impedance matrix are expressed by: Z ij ¼

jωρ0 Se e  jkrij ; 2π r ij

(2)

where k ¼ ω=c0 is the acoustic wavenumber and rij is the distance between element i and element j. In order to estimate the sound power radiated by a complex mode shape, the field point sound pressure is calculated on the vibrating surface. The values for air density ρ0 ¼ 1:225 kg=m3 and speed of sound c0 ¼ 340 m=s are used in the simulation. The diagonal elements of the impedance matrix Zii are singular because of r ii ¼ 0 and are replaced by the formulation which corresponds to the impedance of a baffled piston [18]:  pffiffiffiffiffiffiffi  Z ii ¼ ρ0 c0 1  e  jk Se =π : (3) Using the acoustic impedance and the structural velocities, the radiated sound power can be formulated as [19]: W¼

 Se
H Re vn Zvn ¼ vH n Rv n : 2

(4)

The matrix R ¼ S2e ReðZÞ is called radiation resistance matrix and represents the free field related part of the radiation

Fig. 3. First four radiation modes of the investigated rectangular plate.

Please cite this article as: O. Unruh, Influence of inhomogeneous damping distribution on sound radiation properties of complex vibration modes in rectangular plates, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j. jsv.2016.05.009i

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impedance matrix Z. The analytical formulation of matrix R for N elemental radiators is given by: 2 3 sin ðkr 12 Þ 1N Þ 1 ⋯ sinkrðkr kr 12 1N 6 7 sin ðkr Þ 1 ⋮ 7 ω2 ρ0 S2e 6 6 kr2121 7 R¼ 6 7: 4π c0 6 ⋮ ⋱ ⋮ 7 4 5 sin ðkr N1 Þ ⋯ ⋯ 1 kr N1

(5)

The self-radiation efficiencies of the vibration modes, which are addressed in this paper can be calculated by the following equation:

σ¼

W W ; ¼ W 0 ρ0 c0 S〈 vn 2〉

(6)

where W is the sound power of the complex vibration mode according to Eq. (4), when structural velocities vn are substituted by vibration modes ψ n . W0 is the sound power radiated by a rigid piston with the same averaged velocity 〈vn 2 〉 and area S. An eigenvalue decomposition can be applied to the radiation resistance matrix R in order to calculate the so-called radiation modes Q and their efficiencies Λ: R ¼ Q T ΛQ :

(7)

In contrast to structural modes, radiation modes are frequency dependant. Fig. 3 exemplarily shows the first four radiation modes of the investigated rectangular plate for the frequency of 70 Hz. According to Eq. (4), the sound power can also be expressed with the use of radiation modes and their efficiencies as follows [10]: H T W ¼ vH n Rv n ¼ vn Q ΛQv n :

(8)

The product between the radiation modes Q and the structural velocities vn or vibration modes ψ n delivers coupling factors yn and yr : yn ¼ Qvn

and

yr ¼ Q ψ n :

(9)

These factors describe the coupling of the structural velocity patterns or vibration modes with the radiation modes. Especially the high coupling in the first radiation modes indicates a good radiation efficiency of the structural velocity patterns or vibration modes. Further in this paper, these coupling factors will be used in order to investigate the increased sound radiation of complex modes. It is important in the consideration of the radiation behaviour of structural modes that they do not radiate sound independently from each other [10]. To show this, the structural velocities vn are transformed in modal domain using the matrix of eigenvectors Ψn and modal coordinates q: vn ¼ Ψn q:

(10)

Substituting this equation in the formulation of the sound power in Eq. (4) delivers: W ¼ qH Ψn RΨn q ¼ qH Πq; H

with Π ¼ Ψn RΨn : H

(11)

Considering the matrix Π it can be noticed that the diagonal elements Πii correspond to self-radiation efficiencies. The acoustic interaction between different structural modes is specified by off-diagonal elements of the matrix Π. The mutualradiation efficiencies are finally given by a normalisation of the off-diagonal elements Πij as follows [10]:

σ ij ¼

8Π ij : ρ0 c 0 S

(12)

3.3. Quantification of the modal complexity In order to ensure the comparability of complex vibration modes regarding their impact on the sound radiation, the modal complexity of the calculated modes must be quantified. Modal complexity can be understood as a metric which quantifies the non-proportionality of structural damping using the properties of real and imaginary parts of complex vibration modes. It is important to notice that there are several different metrics available for the quantification of modal complexity [20,21]. These metrics use different properties of the complex mode, for example mean phase (MP) or mean phase deviation (MPD) in order to quantify modal complexity. Besides mathematical properties of a complex mode, also physical measures such as strain energy can be used for the quantification of modal complexity [22]. The modal collinearity index (MCI) is one of the metrics that can be used in order to quantify the level of modal complexity [21]. It is based on the Please cite this article as: O. Unruh, Influence of inhomogeneous damping distribution on sound radiation properties of complex vibration modes in rectangular plates, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j. jsv.2016.05.009i

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Fig. 4. MCI values of the calculated complex modes.

Fig. 5. Modal radiation efficiencies of complex vibration modes.

correlation between real and imaginary parts of the eigenvector. It is defined as: jReðψ~ n ÞT Imðψ~ n Þj ffi; MCI ¼ 1  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½Reðψ~ n ÞT Reðψ~ n Þ½Imðψ~ n ÞT Imðψ~ n Þ

(13)

with

ψ~ n ¼

ψn

maxðjψ n jÞ

ejπ =4 ;

(14)

where ψ~ n is the vector that results from the 45° rotation and normalisation of the eigenvector ψ n [21]. The MCI indicates real mode shapes with values of zero and mode shapes with maximum complexity with values of one. Please cite this article as: O. Unruh, Influence of inhomogeneous damping distribution on sound radiation properties of complex vibration modes in rectangular plates, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j. jsv.2016.05.009i

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Fig. 4 shows the MCI values of the complex vibration modes for the investigated damping configurations. First of all, the MCI values indicate the increasing complexity of the higher order modes, due to increasing damping ratios defined by the Rayleigh damping model [14]. Furthermore it can be noticed that configurations with the biggest continuous damping surface such as S3, B1, B2, A1 and A2 show the highest MCI values of up to 0.93. Other configurations such as S2, S5 or B3 have much lower complexities of vibration modes with MCI o 0:2. This large variation of MCI values of complex modes makes the direct comparison of different configurations difficult. In order to simplify this, the Complex Mode Radiation Index (CMRI) is defined in Section 4.3 of this paper. It allows an identification of damping distributions with the highest influence on sound radiation, normalised by the value of complexity. But first, the self-radiation efficiencies of complex vibration modes are considered in the next section.

4. Acoustic characterisation 4.1. Self radiation efficiencies In this part of the paper, self-radiation efficiencies of selected configurations from every symmetry group are considered. From the preliminary studies presented in [14,15] it is known that complexity of vibration modes influences their sound radiation well below the coincidence frequency. It is demonstrated that the order of the mode (even or odd) and the corresponding direction of travelling waves have the biggest importance for self-radiation efficiencies. For that reason four examples (3
2, 2
3, 4
4, 7
1) are selected in order to represent odd–even, even–odd, even–even and odd–odd order groups of modes. Considering, that the compared configurations should have similar MCI values of the investigated modes (see Fig. 4), four examples S3, B1, A1, C1 are selected from every symmetry group. Fig. 5 shows the self-radiation efficiencies of the selected vibration modes for a reference case without additional damping and for these four configurations. In [14] it is shown that bending waves travelling in the direction of the even order of the complex mode significantly increase its radiation efficiency at frequencies well below the coincidence by 20–50 dB. This increase can be observed even at relatively low complexities of MCIo 0:2. Odd order modes slightly reduce their radiation efficiency in the entire frequency range below the coincidence by a couple of dB. In this case much higher complexities of MCI 4 0:4 are needed for an observable effect. Considering the odd–even order mode 3
2 in Fig. 5 it can be noticed that, besides the configuration A1 with asymmetric damping distribution, only a negligible influence of travelling waves on radiation efficiency occurs. Regarding the configuration B1 this observation correlates to the outcome of the previous study [14]. According to these results, bending waves which are travelling in the direction of the odd order of three do not increase the radiation efficiency of the complex mode. In this case, the volume velocity of the mode and therefore the coupling into the first radiation mode remains zero. In contrast to that, the configuration A1 increases the radiation efficiency, due to the fact that bending waves travel diagonally towards the areas with increased damping and affect the even order of two. This configuration of travelling waves leads to higher efficiencies due to an increase of volume velocity and coupling in the first radiation mode. These relations are discussed in detail in Section 4.2, but first other complex modes in Fig. 5 are considered. The complex even–odd order 2
3 mode shows a similar impact on the sound radiation as the previously considered 3
2 mode. Also for the 2
3 mode, the symmetric and central symmetric configuration shows a negligible influence on self-radiation efficiencies. Nevertheless, the biggest difference is the fact that besides the asymmetric, also the bilateral symmetric configuration increases its radiation efficiency below the coincidence frequency. This can be explained by the fact that in comparison to the 3
2 mode, the waves are now travelling along the even order of two and therefore the coupling in the first radiation mode is increased. The highest impact on the sound radiation can be observed at the even–even order 4
4 mode. In this case, the B1, A1 and C1 configuration shows a significant increase of the radiation efficiencies at low frequencies. This can also be explained by an increased coupling in lower radiation modes. Remarkable is the fact that only the 4
4 mode of the symmetric configuration S3 remains nearly unaffected in terms of self-radiation efficiency. Finally, the odd–odd order 7
1 mode is considered. This mode is the most efficient radiator under the presented series of modes. It can be noticed that none of the examined configurations indicate a significant impact on the self-radiation efficiencies. The asymmetric configuration A1 reveals, due to reduced coupling in the first radiation mode, the highest decrease of sound radiation capability by 2 dB. In order to summarise the presented results it can be stated that the discussed phenomena are of general nature and valid for all configurations specified by the symmetry properties and order of the vibration mode. All symmetric configurations S1–S5 reveal a negligible influence on the sound radiation of all orders of complex vibration modes. Following from the main direction of travelling waves, the damping distributions with bilateral symmetry B1–B3 increase the radiation efficiency of even order modes and slightly decrease in the case of odd order modes. Central symmetric configurations C1– C2 only increase the radiation capability of even–even order modes. Finally, the asymmetric configurations A1–A2 affect the highest number of vibration modes. Please cite this article as: O. Unruh, Influence of inhomogeneous damping distribution on sound radiation properties of complex vibration modes in rectangular plates, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j. jsv.2016.05.009i

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Fig. 6. Coupling factors yr in the first radiation mode.

4.2. Coupling of structural modes in radiation modes In order to improve the understanding of the results in the previous section, the coupling factors defined in Eq. (9) are considered in detail for the same group of complex vibration modes. It is well known from the literature [10,23] that at lower frequencies well below the coincidence the radiated sound power is dominated by a handful of radiation modes. The first radiation mode, already considered in Fig. 3, is the most efficient of them and corresponds to an in-phase oscillation of elemental radiators which results in a monopole character of the radiated sound field. Odd–odd order modes radiate well due to a high coupling in this first radiation mode. The second and third radiation modes are less efficient and represent the dipole character with 01 and 1801 oscillation of two opposite regions of the vibrating structure. For instance, even–odd and odd–even mode shapes of a plate have a high coupling coefficient in this group of radiation modes. Finally the fourth radiation mode corresponds to a much less efficient quadrupole radiation and is coupled with even–even order structural modes. To explain the variation of the radiation efficiencies, Fig. 6 shows the coupling factors yr as a function of dimensionless frequency kLx with acoustic wavenumber k and plate dimension in x-direction Lx. In this case the same group of modes as presented in Fig. 5 is considered. Considering the coupling factors of the odd–even 3
2 mode it can be noticed that in correlation to the results of the self-efficiencies only the A1 asymmetric configuration reveals an increased coupling in the first radiation mode. The coupling of other damping configurations remains zero. A similar coupling increase can be observed for the even–odd 2
3 mode. Besides the A1 also a B1 configuration increases the volume velocity and therefore the coupling in the first radiation mode. It can be explained by bending waves that are travelling along the even order of the complex mode. The almost negligible reduction of the radiation efficiencies of the odd–odd 7
1 mode can also be observed using the coupling coefficients. Considering the A1 configuration it can be seen that the reduction of the coupling factors results in the mentioned slight reduction of the self-radiation efficiencies by 2 dB. Interesting observations can be made on the even–even 4
4 mode. According to the results of the self-radiation efficiencies, the A1 and C1 configurations increase their radiation due to increased coupling in the first radiation mode. The self-radiation efficiency of the configuration B1 also increases but in contrast to configurations A1 and C1 the coupling in the Please cite this article as: O. Unruh, Influence of inhomogeneous damping distribution on sound radiation properties of complex vibration modes in rectangular plates, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j. jsv.2016.05.009i

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Fig. 7. Coupling factors yr of the 4
4 in the second and fourth radiation mode.

Fig. 8. CMRI values of different damping configurations.

first radiation mode remains zero. This can be explained considering the coupling in the second and fourth radiation mode in Fig. 7. The reason for the increased radiation of the B1 configuration is the improved coupling in the second radiation mode. This radiation mode is less efficient than the first, which explains the lower increase of the self-radiation efficiencies of the B1 configuration compared to A1 and C1, seen in Fig. 5. In case of the fourth radiation mode it can be noticed that, except the symmetric S3 configuration, all other configurations reduce their coupling. Regarding the coupling of complex vibration modes in the lower radiation modes and the resulting increased radiation efficiency it can be summarised:

 Complex modes which result from symmetric damping distributions have a negligible effect on the coupling into the radiation modes.

 Bilateral symmetric configurations increase the coupling of even–odd order structural modes into the first radiation  

mode. In case of even–even order modes the coupling in the second radiation mode is improved. The coupling of odd– odd order modes in the first radiation mode is slightly reduced. Central symmetric damping distributions only affect even–even order structural modes and increase their coupling into the first radiation mode. Asymmetric configurations always increase the coupling into the first radiation mode when an even order in any direction of the plate is present.

In other words, travelling bending waves usually increase the coupling of structural modes with even order into the lower radiation modes and therefore lead to increased radiation efficiency. In contrast to that, the observed reduction of radiation efficiency of odd order modes is almost negligible. 4.3. Complex mode radiation index In order to substantiate the statements of the previous sections, this part of the paper verifies the influence of complexity on sound radiation with a new evaluation index. In the previous sections it is shown that self-radiation efficiencies indicate the amount of influence with a single scalar metric. This motivates the use of self-radiation efficiencies for the definition of a Please cite this article as: O. Unruh, Influence of inhomogeneous damping distribution on sound radiation properties of complex vibration modes in rectangular plates, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j. jsv.2016.05.009i

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evaluation index which quantifies the impact on sound radiation normalised by the level of complexity in complex vibration modes. This Complex Mode Radiation Index CMRI is calculated for every damping configuration and every complex vibration mode and is introduced by the following equation: Z fi 10log10ðjσ r ðf Þ  σ n ðf ÞjÞ CMRI ¼ df ; (15) MCI 0 where σr is the radiation efficiency of the real mode of the reference plate without additional damping, the σn is the radiation efficiency of the complex vibration mode and MCI its level of complexity. The CMRI represents the integral of absolute deviations between the radiation efficiency of the real mode and the corresponding complex vibration mode for the frequencies fi below the coincidence, normalised by the MCI value. According to the results from previous studies, frequencies below 10 percent of coincidence are taken into account. In other words, the CMRI indicates with higher values a bigger deviation of complex modes from the real modes per MCI unit of complexity. Fig. 8 shows the CMRI values of complex vibration modes for the investigated damping configurations. In this analysis the lower 17 modes are taken into account, because of the noticeable amplitude redistribution of modes with higher order, as discussed in Section 3.1. Too high redistribution of vibration amplitudes, due to a very high damping, do not allow a reliable statement about the influence of pure complexity in form of travelling bending waves. First it can be noticed that even within a single damping configuration, there is a remarkable variation in evaluation coefficients depending on the number of the mode. For example the bilateral symmetric B1 configuration reveals the CMRI from 6  100 to 1:5  103 . In average the CMRI values are approximately around 102, what corresponds to deviation of radiation efficiencies of 1 to 2 dB. Regarding the acoustic impact of configurations with different symmetry it is found that the symmetric damping distributions S1–S5 never exceed values of 3  102. In contrast, the bilateral configurations B1–B3 can reach values of 9  103. The highest observable values of up to 3  104 are reached at the asymmetric A1–A2 and central symmetric C1–C2 configurations. It is remarkable that central symmetric distributions show very high CMRI values only for four even–even order modes with numbers 5, 9, 15 and 17. In contrast, configurations with asymmetric damping affect much more modes and also reveal a higher averaged level of CMRI values. To illustrate this, Fig. 9 shows the CMRI values for every damping distribution, which are average over all considered modes of the corresponding configuration. For convenience they are additionally normalised by the maximum value, found in the A1 configuration. The main conclusions, from Sections 4.1 and 4.2, can be confirmed by the average CMRI values. Symmetric damping distributions show a negligible effect of travelling waves on the self-radiation efficiencies. Plates with bilateral symmetry show three to ten times higher impact on sound radiation than symmetrical configurations. Finally, the central symmetric and asymmetric damping distributions reveal the highest influence on self-radiation efficiency of complex vibration modes. It seems that the application of inhomogeneous damping with bilateral, central symmetric and asymmetric properties can lead to an increased sound radiation of structural resonances and should be avoided at low frequencies below 10 percent of coincidence. 4.4. Mutual-radiation efficiencies As it is mentioned in Section 3.2, structural modes do not radiate sound independently of each other. The acoustic interference between structural modes is described by the mutual-radiation efficiencies, defined in Eq. (12). Especially between the resonance frequencies, the dynamic response implies a superposition of different modes and therefore the contribution of mutual-radiation efficiencies to the radiated sound power grows. It is important to notice that the crosscoupling only occurs between structural modes with similar order. This means that for example even–even order modes only influence even–even order modes. Accordingly, the mutual-radiation efficiencies of even–even with odd–odd, even– odd or odd–even order modes are zero. Fig. 10 shows the mutual-radiation efficiencies for the reference plate with real structural modes. Besides the mentioned group-wise cross-coupling between the structural modes it can be noticed that modes with similar order and therefore similar bending wavelengths such as 2
2 and 2
4 have higher mutual efficiencies than for example 2
2 and 6
8 modes. Furthermore, it can be seen that the mutual efficiencies as a function of frequency can be positive or negative. This

Fig. 9. Normalised and averaged CMRI values.

Please cite this article as: O. Unruh, Influence of inhomogeneous damping distribution on sound radiation properties of complex vibration modes in rectangular plates, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j. jsv.2016.05.009i

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Fig. 10. Mutual-radiation efficiencies of the real structural modes.

Fig. 11. Mutual-radiation efficiencies of real and complex modes with similar orders.

means that the acoustic interaction between neighbouring resonances can be constructive (increasing sound power) or destructive (decreasing sound power). According to Eq. (11) it is important that the positive or negative contribution to the radiated sound power of the plate is also depending on the modal coordinates q, which assign the amplitude and phase of different modes to each other. Mutual-radiation efficiencies of complex modes, which result from different damping distributions, can be calculated similar to real modes using Eq. (12). Fig. 11 shows the mutual-radiation efficiencies of the odd–odd 1
1 and 3
1 modes, as well as for the even–even 2
2 and 4
2 modes. Due to the fact that the mutual-radiation efficiencies of complex vibration modes are also complex, they are presented as frequency dependant locus curves in the complex plane. The locus curves are shown for the frequencies between 10 and 2500 Hz and start at the coordinate origin. As expected, both pairs of real structural modes of the reference plate stay on the real axis. In contrast, the mutual-radiation efficiencies of inhomogeneously damped plates extend through the entire complex plane. Due to the fact that matrix Π is hermitian the mutual-

Please cite this article as: O. Unruh, Influence of inhomogeneous damping distribution on sound radiation properties of complex vibration modes in rectangular plates, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j. jsv.2016.05.009i

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radiation efficiencies σij and σji are conjugate complex. Therefore the algebraic sign of the imaginary parts is determined by the choice of either σij or σji. The complexity of the mutual efficiencies complicates the interpretation of the associated physical phenomena. In order to explain how complex mutual-radiation efficiencies contribute to the radiated sound power, Eq. (11) is considered using a simple example of a system with two modes. Knowing that the matrix Π is hermitian the sound power of a simplified system with two natural modes can be specified as follows: "  #T " #" # q1 Π 11 Π 12 q1 W¼ ; (16) q2 Π 21 Π 22 q2 where Π11, Π22 are the contributions of the self-radiation efficiencies and Π12, Π21 are conjugate values that are related to mutual-radiation efficiencies. The values q1 and q2 are the corresponding modal coordinates of the two investigated structural modes. After the multiplication of matrix and vector, the equation rearranges to: W ¼ q1 Π 11 q1 þq2 Π 22 q2 þq1 Π 12 q2 þq2 Π 21 q1 :

(17)

The first two terms in this equation correspond to the contributions of the self-efficiencies to the radiated sound power. The last two terms result from the cross-coupling of both mode shapes and represent the contribution of the mutual-radiation efficiencies. Due to the fact that both cross-coupling terms q1 Π 12 q2 and q2 Π 21 q1 are conjugate complex, the overall contribution of the mutual-radiation efficiencies results in: W m ¼ q1 Π 12 q2 þq2 Π 21 q1 ¼ 2Reðq1 Π 12 q2 Þ:

(18)

This equation shows that only the real parts of the mutual-radiation efficiencies contribute to the radiated sound power. Important is that the quantity of the real part Reðq1 Π 12 q2 Þ depends on the phase angle given by the modal coordinates q1 and q2. This means that the locus curve given in Fig. 11 rotates around the coordinate origin depending on the phase angle between two modes given by their modal coordinates. The projection of the locus curve on the real axis gives the absolute value and the algebraic sign of the contribution to the radiated sound power. For complex modes, it is required to investigate the influence of the eigenvector complexity on mutual-radiation efficiencies of pairings between modes with different orders e.g. even–odd and odd–even. As a reminder, in the case of real

Fig. 12. Mutual-radiation efficiencies of real and complex modes with different orders.

Please cite this article as: O. Unruh, Influence of inhomogeneous damping distribution on sound radiation properties of complex vibration modes in rectangular plates, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j. jsv.2016.05.009i

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Fig. 13. Off-diagonal elements Πij at 400 Hz.

mode shapes these mutual-radiation efficiencies are equal to zero. Fig. 12 shows locus curves of mutual-radiation efficiencies for some mode pairings of the reference plate and the configurations A1, B1, C1. As expected, the locus curve of the real mode shapes of the reference plate remains in the coordinate origin at a value of zero. In contrast to that, the locus curves of inhomogeneously damped configurations are non-zero. This means that travelling waves increase the mutual interaction between complex mode shapes of different order. In order to consider this effect in more detail, Fig. 13 visualises the logarithmic off-diagonal elements Πij of the matrix Π, defined in Eq. (11) for a frequency of 400 Hz. This frequency is selected in order to visualise the maximum mutual-radiation efficiencies. The dominant diagonal elements Πii, which are correlated to the self-radiation efficiencies, are set to zero in order to focus on the lower valued mutual efficiencies. In Fig. 13 it can be seen that the reference plate with real modes has a little number of spots that are not equal to zero. Compared to the reference plate, the symmetric damping distribution S3 reveals a marginal and almost negligible increase of acoustic interaction between the complex vibration modes. In contrast to that, the bilateral symmetric configuration B1 shows an improved cross-coupling between complex modes, which can be seen by an increased number of non-zero elements. A similar behaviour can also be observed for the configuration C1 with a central symmetric damping distribution. The influence of complex modes on the mutual-radiation efficiencies is especially apparent considering the asymmetric configuration A1. It can be noticed that in this case all modes interact with each other. The highest sensitivity of this asymmetric configuration regarding the acoustic properties of present complex modes has already been observed using self-radiation efficiencies and couplings into lower radiation modes in Sections 4.1 and 4.2. Due to the fact that the asymmetric damping distribution induces bending waves that are travelling diagonally, all nodal lines of the mode shape are also travelling in this direction. This travelling of the nodal lines blurres the opposite properties of even and odd order modes and increases the acoustic interaction between them, which raises the mutual-radiation efficiencies. 4.5. Discussion In this part of the paper, several important issues regarding technical relevance of the presented results are discussed. One of the questions is whether it is important to consider the complex mode shapes in acoustic simulations, which are for example conducted in order to design low-noise structures. Despite the fact that most of the observed effects address very low frequencies below the coincidence frequency, the consideration of complex vibration modes seems to be important for accurate predictions of sound relevant design parameters. This might be relevant for bilateral and central symmetries and Please cite this article as: O. Unruh, Influence of inhomogeneous damping distribution on sound radiation properties of complex vibration modes in rectangular plates, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j. jsv.2016.05.009i

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seems to be even more important for asymmetrical distributions of inhomogeneous damping. It is also important to notice that if the placement of additional damping is carried out by conventional methods based on modal strain energy [3] or structural intensity distribution [4], symmetrical damping configurations will be the result of this optimisation process. In this case the consideration of complex mode shapes and their acoustic characteristics is not mandatory and can be limited to real modes. Nevertheless, in the case of asymmetric damping distributions with very high damping ratios, which result in vibration modes with high complexity, the effects presented in this paper should be considered in order to optimise the acoustic performance of the vibrating structure. Due the fact that an increase of the radiation efficiency of even order modes is expected at frequencies lower than 10 percent of the coincidence frequency, especially plates with very low eigenfrequencies (high mass and low stiffness) and small scales (small bending wavelengths and large distance to the coincidence frequency) are affected by presented phenomena. This means that lightweight structures with high stiffness and low mass are not seriously affected by effect presented in this paper. The discussion of symmetry in acoustics is not a new topic. The influence of symmetry has already been investigated by Constans et al. [24] who uses an optimisation tool for the placement of point masses in vibrating shells. Supplementary to the state of the art the presented paper addresses the influence of symmetry from a different perspective by putting the complex vibration modes into account. Regarding the results of this paper, the symmetric placement of damping treatments must be aimed in order to avoid the increase of radiation efficiency of complex vibration modes. However, it must be taken into account that perfectly symmetrical configurations can hardly be produced in experiment due to uncertainties as it has already been stated e.g. by Marburg in [25].

5. Conclusions This paper investigates the influence of different spatial damping configurations on the resulting complex mode shapes and their sound radiation properties. It is shown that symmetric, bilateral symmetric, central symmetric and asymmetric distributions of inhomogeneous damping cover the majority of cases that can probably be expected in real applications. Using an example of five configurations with different damping properties, the self-radiation efficiencies of complex vibration modes are considered in detail. First, it is shown that the symmetric damping distributions do not affect the sound radiation of complex modes. For bilateral symmetric configuration it is shown that modes with even order in the direction of travelling waves significantly increase and modes with odd order slightly decrease their radiation capability below the coincidence frequency. An interesting observation is made regarding the central symmetric configuration, which affects only even–even order modes. Other groups of modes seem to remain unaffected in their radiation. The highest influence on the self-radiation efficiencies is observed for asymmetric damping distributions. According to these results, all structural modes with even order in any direction of space increase their radiation. A new Complex Mode Radiation Index (CMRI) is introduced, which allows the evaluation of the influence on selfradiation efficiencies below the coincidence frequency, normalised by complexity level, estimated by MCI. Using this metric it is shown that symmetric damping distributions have an almost negligible influence on sound radiation. Furthermore, it is noticed that bilateral and central symmetric distributions affect radiation capability of certain groups of modes. Nevertheless, the highest influence on sound radiation of the majority of complex vibration modes is observed by the asymmetric damping configuration. The consideration of the mutual-radiation efficiencies reveals that in the case of complex vibration modes this metric also becomes complex valued. It is shown that the contribution of cross-couplings to the radiated sound power is related to the real part of the complex mutual efficiencies. The quantity of the real part and therefore the amount and algebraic sign of the contribution depend on the amplitude and phase given by the modal coordinates. The most important result regarding the mutual-radiation efficiencies is that, depending on the symmetry of the damping distribution, the acoustic coupling between structural modes increases, even in the case of different orders. Especially the asymmetrically distributed inhomogeneous damping leads to an increase of all mutual-radiation efficiencies. With regard to the application of additional damping treatments the results of this paper show that, in order to avoid an increase of the self-radiation efficiencies of complex vibration modes, the damping should be placed symmetrically. Bilateral symmetric, central symmetric and especially asymmetric distributions of additional damping may improve the radiation capability of structural resonances and increase the emitted noise levels. Nevertheless this increase is expected well below the coincidence frequency for plates with small scale and very low eigenfrequencies. Future works will include the investigation of the influence of complex vibration modes on the sound power radiated by the inhomogeneously damped plates. Furthermore, an important question is the coupling of complex vibration modes in the fluid modes of an adjacent cavity. The coupling of different types of excitations such as plane acoustic waves and turbulent boundary layers can also have an importance for inhomogeneously damped plates. Please cite this article as: O. Unruh, Influence of inhomogeneous damping distribution on sound radiation properties of complex vibration modes in rectangular plates, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j. jsv.2016.05.009i

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Please cite this article as: O. Unruh, Influence of inhomogeneous damping distribution on sound radiation properties of complex vibration modes in rectangular plates, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j. jsv.2016.05.009i