Mechanical Systems and Signal Processing 30 (2012) 330–342
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A similitude for structural acoustic enclosures S. De Rosa a,n, F. Franco a, X. Li b, T. Polito a a
PastaLab, Vibrations and Acoustics Laboratory, Department of Aerospace Engineering, Universita degli Studi di Napoli ‘‘Federico II’’, Via Claudio 21, 80125 Napoli, Italy b Key Lab of Environmental Noise and Vibration, Beijing Municipal Institute of Labor Protection, 55 Taoranting Road, Beijing 100054, China
a r t i c l e in f o
abstract
Article history: Received 30 December 2010 Received in revised form 28 October 2011 Accepted 17 January 2012 Available online 14 February 2012
A similitude is proposed for the analysis of the dynamic response of acousto-elastic assemblies. It is defined by invoking the energy distribution approach which allows the exact representation of all the fundamental parameters in terms of modal coordinates. The similitude laws are then defined by imposing that the dynamic responses in the original and parent models are the same. Herein, two test cases are discussed: the first is represented by the finite element modelling of a flexural plate coupled to an acoustic room; the second is purely analytical and concerns an infinite cylinder containing a fluid. If the original damping values are kept, it is shown that a complete similitude can be defined: it allows working with parent configurations of increased or reduced sizes. Even if the proposed schemes are very simple, the results are very promising. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Structural–acoustic similitude Energy distribution approach Scaling laws
1. Introduction From the beginning of engineering and physical sciences, there has been a continuous search for the possibility to transform the same engineering problem on different scales. Regarding the linear response of structural and structural– acoustic systems, these scales can offer several evident advantages in terms of geometries and materials, allocation of sensors and actuators, management of the tests, etc. This is true for both, very small systems studied on dimensions greater than the original ones and large systems reduced at small sizes. It is worth to highlight that in the similitude, the analyst translates the same problem on different scales, whereas with the analogies, a given class of problems is solved by looking for similar equations. For these simple reasons, the theory of the models and the analysis of the possible similitudes and analogies are a very large branch of the engineering literature and cannot be replicated here. An interesting summary of the similarity conditions between a full-scale model and one scaled by using the modal approach is given in [1], with specific reference to the dynamic response. More general views of both the problems of similitude and analogies are in [2,3], even if the main textbooks about this subject are the work in [4,5]. The approach of the similitude among models is largely used in aeronautics. In fact, during wind tunnel measurements, the scaled aircraft component is designed to represent the same natural frequencies or flutter speed and/or the windtunnel data have to be correlated with the flight-test ones [6]: all these data are aimed at producing aero-elastic information. In the present work, the idea is to investigate the possibility to define a complete similitude and the related scale laws for structural–acoustic systems. The Energy Distribution Approach (EDA) allows this kind of investigation via the modes
n
Corresponding author. Tel.: þ 39 0817683581; fax: þ 39 081624609. E-mail address:
[email protected] (S. De Rosa).
0888-3270/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.ymssp.2012.01.018
S. De Rosa et al. / Mechanical Systems and Signal Processing 30 (2012) 330–342
Nomenclature
331
x
spatial coordinate vector
aj
jth modal frequency response cross-modal power mobility between jth and kth mode frequency step Dirac d-function damping loss factor cylinder coordinate ratio of the structural and acoustic natural frequencies modal overlap factor Poisson module acoustic pressure mass density jth eigenvector modal spatial operator between jth and kth mode generic frequency interval radian excitation frequency jth eigenvalue
List of symbols AEIC A AEIC rs ca E E f ðP,tÞ h iu Ji Lx Ly Lz n NM NS Pinput PðqÞ IN Q Rc r Sf T ðrÞ t u v w
energy influence coefficient matrix surface area of the plate rsth member of the AEIC matrix acoustic wavespeed energy vector Young module excitation function of space, P, and time, t plate or cylinder thickness imaginary unit Bessel function of the first kind of ith order cavity length cavity length cavity length modal density number of modes number of subsystems power input vector power input to the qth subsystem cylinder radial excitation cylinder radius cylinder coordinate spectral density force function energy of the rth subsystem time variable displacement displacement displacement
Gjk Df d
Z y L
m n P
r fj cjk O
o oj
g ðsÞ g ðaÞ
the superscript s denotes that the item g belongs to the structural domain the superscript a denotes that the item g belongs to the acoustic domain
and natural frequencies in order to determine the power input and the energy associated to each subsystem. The similitude is defined by using relationships among mode shapes, natural frequencies, damping loss factors and energies [7]. Specifically, EDA has been already used in order to predict the original and scaled responses of linear dynamic systems [8–10], and further analyses have been performed very recently [11]. In detail, in [8] investigations on a simple plate are reported; in [9] the extension to two plates is presented and in [10] the scaling between structural components with different modal density is introduced (a coupled beam-plate system). In [11] a general scaling law is formulated using Skudrzyk mean-value [12]. In [13], some preliminary analyses aimed at the complete similitude for simple plate and assemblies of plates are presented. The present goal is to define the similitude for the acousto-elastic linear systems. The work, after these remarks, presents in Section 2 a summary of EDA. As stated before, this approach is then used in Section 3 to define a similitude with a parent model tailored to represent some specific items under observation. Section 4 contains the numerical investigations concerning the linear dynamic responses selected as test cases: a flexural plate coupled to an acoustic room and an infinite cylinder containing a fluid. Some concluding remarks are in Section 5. The results show that the similitude can be found and thus the proposed approach can open interesting scenarios for applications to both modelling issues and laboratory measurements. The complete procedure is named SAMSARA,1 Similitude and Asymptotic Models for Structural–Acoustic Research and Application. The definition of SAMSARA is far to be considered a concluded topic. This work is only a first step since only simplified models have been investigated; nevertheless the possibility of working with modified structural and structural–acoustic linear dynamic models emerges rather clearly. Laboratory applications will present some more difficulties related to real problems (junctions, local effects, etc.) but the formal scheme herein discussed provides a general framework, since it uses global modes.
2. The energy distribution approach (EDA) EDA is here recalled in order to define a similitude between original and scaled models. 1
/www.britannica.com/EBchecked/topic/520716/samsaraS.
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A generic linear structural–acoustic dynamic model can be assembled by using the mode shapes and natural frequencies. This model can be now named as original. In EDA, the original eigensolutions (natural frequencies and global mode shapes) are the basis to obtain the distribution of energy in each subsystem. A generic subsystem is defined as a spatial domain characterised by specific waves. For example, a beam can be idealized as a subsystem in which longitudinal, shear, torsional and bending waves travel in an 1D domain. Thus, the whole system can be thought as an assembly of NS subsystems in which NM modes are resonating at each excitation frequency. By using EDA, it is possible to estimate the energy unknown vector, E, for a given power input vector, Pinput ; this is done by evaluating the energy influence coefficient matrix, AEIC : EðoÞ ¼ AEIC ðoÞPinput ðoÞ
ð1Þ
with AEIC rq ðoÞ ¼
PP j
P
ðrÞ oÞcðqÞ jk cjk
k Gjk ð
Z oj Gjj ðo
j j
with
ðqÞ Þcjj
r,q 2 f1, . . . ,NSg
ð2Þ
j,k 2 f1, . . . ,NMg
The energy response is defined in terms of some global parameters which depend on the modal properties of the system. The spatial coupling parameter for the generic rth subsystem is the following: Z ðrÞ cjk ¼ rðxÞfj ðxÞfk ðxÞ dx ð3Þ x2r
ðrÞ
This term depends on the global mode shapes acting within the rth subsystem: specifically, the cjk term measures the generalized work inside the rth subsystem due to the interaction between the jth and the kth modes. The global mode shapes are considered mass-normalized: the cjk terms are dimensionless. Thanks to the well-known orthogonality property, for a single system the only non-zero terms are the cjj . The frequency dependent members are here recalled: Z 1 Gjk ðOÞ ¼ o2 Re½aj ðoÞak ðoÞ do ð4Þ
O
o 2O
where the modal frequency dependent function a is given by
aj ðoÞ ¼
1
ð5Þ
o2j o2 þ iu Zj o2j
The term O represents a generic frequency interval in which the system response is analysed for a given excitation. The generic cross-modal term, Gjk , is a frequency integral whose magnitude depends primarily on the natural frequencies and the bandwidths of the jth and the kth modes. This term is small unless both modes are resonant. If the modes overlap, i.e., they lie within each others’ half-power bandwidths such that ðoj ok Þ r 12ðZj oj þ Zk ok Þ, it is particularly large. If the damping is small, then the modal terms can be approximated [7]; these involve both, the auto terms:
Gjj ðOÞ ffi
1 p 8O Zj oj
ð6Þ
as well as the cross terms:
Gjk ðOÞ ffi
p
ðoj þ ok Þ2 ðZj oj þ Zk ok Þ
16O ðo2 o2 Þ2 þ ðZj o2 þ Zk o2 Þ2 j
k
j
ð7Þ
k
The analysis of the cross terms is rather complicated and two further approximations can be used that allow separating the cross terms in large and small terms. The large terms represent the situation in which two modes well overlap (o2j ffi o2k ); the jth and the kth natural frequencies are very close one each other. The small terms refer to the case in which two modes do not well overlap, ðo2j o2k Þ2 b ðZj o2j þ Zk o2k Þ2 ; the jth and the kth natural frequencies are far one each other, even considering the effect of the relative damping values. These approximations are explicitly expressed here:
GðlargeÞ ðOÞ ffi jk
p ðoj þ ok Þ2 ðZj oj þ Zk ok Þ 16O ðZj o2j þ Zk o2k Þ2
ð8Þ
GðsmallÞ ðOÞ ffi jk
p ðoj þ ok Þ2 ðZj oj þ Zk ok Þ 16O ðo2j o2k Þ2
ð9Þ
It has to be further noted that in EDA, the loading is assumed to be proportional to mass density with zero crossspectral density, Sf in [7]. Further, it has to be noted that ½Sf ¼ ½F LT 1 .
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The expressions of the input power, PIN, to the qth subsystem and the kinetic energy, T, for the rth subsystem complete ðqÞ ðrÞ the dynamic set for the analysis, AEIC rq ðoÞ ¼ P IN ðoÞ=T ðoÞ: X Zj oj Gjj ðoÞcðqÞ ð10Þ PðqÞ IN ðoÞ ¼ 2Sf jj j
T ðrÞ ðoÞ ¼ 2Sf
XX
ðrÞ Gjk ðoÞcðqÞ jk cjk
j
ð11Þ
k
3. Introducing the similitude The definition of a useful similitude, able to represent the dynamic response, is here discussed by using few key parameters. It is expected to find the scale laws for these parameters so that a parent model can reproduce the response of the original one: the parent model will have some geometric parameters reduced or increased. These are the main hypotheses:
The models under investigations are represented by thin bending plates coupled with acoustic cavities. The plates and the fluid materials do not change in the parent models: any material variation can be interpreted as a
modification in the distribution of the natural frequencies. The boundary conditions remain the same in the parent models. The plates are excited by concentrated harmonic forces, f ðP,tÞ ¼ FðoÞdðPÞeiuot . In the parent models, the excitation acts at the same dimensionless locations. The damping values are such that the system response can be obtained using the undamped natural frequencies and the real mode shapes: more complicated models based on the complex mode shapes do not add further contributions to the present developments and results. The global modes are represented by the uncoupled structural and acoustic modal bases; any coupling effect is here neglected and will be discussed in one of the future works. It is well known that in some conditions the uncoupled modes cannot be used to describe correctly the system (aero- and hydro-elasticity, for example), but this choice is ðsÞ pursued here for the sake of simplicity. As a consequence, the jth global mode is only structural f ¼ ½f ,0 or only ðaÞ acoustic f ¼ ½0, f . In order to clearly highlight and identify the items belonging to the structural or acoustic domain, the superscripts s and a have been used, respectively.
The field of investigation is restricted to the analysis of the values of length, L, thickness, h, damping, Z and force, F, since it is further assumed that FðoÞ ¼ F. The over-line denotes the items in the parent model. For example, if F denotes the force acting on the original system, F is the force acting on the parent model. Then, a similitude has been searched with a set of parameters composed by length, thickness, force and damping: rl ¼
L , L
rh ¼
h , h
rf ¼
F , F
rZ ¼
Z Z
ð12Þ
This set is the simplest combination of the key parameters involving the geometry and the excitation. Later, it is shown that, in order to keep the same distribution of natural frequencies while looking for a complete similitude, only rl and rf are necessary. In order to reduce the number of items to be scaled and controlled, it has been further assumed that:
Any variation of the damping coefficient involves both the domains, with the same law: rZ ¼ Z ðsÞ =ZðsÞ ¼ Z ðaÞ =ZðaÞ . The area variation is due to the length variation, rA ¼ r2l . Finally, it has been taken that the global mode shapes remain unaffected:
fj ¼ f j
ð13Þ
As consequence, one gets that the scaled spatial coupling parameters are equal to the original ones: Z Z ðr Þ c jk ¼ r ðxÞf j ðxÞf k ðxÞ dx ¼ rðxÞfj ðxÞfk ðxÞ dx ¼ cðrÞ jk x2r
ð14Þ
x2r
Later the natural frequencies of the parent model are discussed. It is useful to introduce here another key parameter, the ratio of the natural frequencies: ro ¼
oj oj
ð15Þ
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The scaled auto and cross-modal frequency response operators are here defined (for the sake of brevity the O dependence has been omitted)
G jj ffi
G jk ffi
p 8O Z j o j
¼
p 8ðOr o ÞðZj r Z Þðoj r o Þ
¼ Gjj
1 r Z r 2o
r 3o r Z ðoj þ ok Þ2 ðZj oj þ Zk ok Þ
p
16Or o r 4o ðo2 o2 Þ2 þ r 4o r 2Z ðZj o2 þ Zk o2 Þ2 j
k
j
ð16Þ
ð17Þ
k
These cross terms cannot be directly posed in similitude: the involved quantities are different functions of the scaling parameters defined for the natural frequencies and for the damping. The approximations before introduced are here used: they are named large and small terms. The situation in which two modes well overlap (o2j ffi o2k ) is as follows: ðlargeÞ
G jk
ffi
p ðoj þ ok Þ2 ðZj oj þ Zk ok Þ 1 16O r Z r 2o ðZj o2j þ Zk o2k Þ2
ð18Þ
whereas if the two modes do not well overlap, ðo2j o2k Þ2 b ðZj o2j þ Zk o2k Þ2 : ðsmallÞ
G jk
ffi
p ðoj þ ok Þ2 ðZj oj þ Zk ok Þ rZ 16O r 2o ðo2j o2k Þ2
ð19Þ
Then, the scaled auto and cross modal large and small terms are respectively:
G jj ¼
1 Gjj r Z r 2o
ðlargeÞ
G jk
ðsmallÞ
G jk
ð20Þ
¼
1 GðlargeÞ r Z r 2o jk
ð21Þ
¼
r Z ðsmallÞ G r 2o jk
ð22Þ
For resonant modes, the most relevant contribution for auto and cross-modal power mobility comes from the damping controlled region. In case of separate modes, large damping values help to increase the overlap, hence the cross-modal power mobility. For the well overlapped modes, instead, the contribution of an high damping value is negligible in increasing the modal overlap whereas it is important in reducing the resonant response of each mode and the cross power mobility, as consequence. Thus, it can be argued that cross-modal power mobility is proportional to the damping for separate modes, being inversely proportional for well overlapped modes. The excitation is so scaled, according to its units: S f ¼ Sf
r 2f
ð23Þ
rm
with the structural and fluid masses being 2 r ðsÞ m ¼ rl rh ;
3 r ðaÞ m ¼ rl
ð24Þ
where for the fluid it has been assumed that only the volume can be scaled. For the sake of completeness, it is useful to list all the remaining EDA parameters when approached with the scaling procedure: r 2f X 1 r 2f ðqÞ ðqÞ ðr Z r o Þ cjj ¼ PðqÞ P IN ffi 2Sf G ð25Þ IN r m j r Z r 2o jj ro rm T
ðrÞ
ffi2Sf
r 2f XX 1 r 2f ðqÞ ðrÞ ðrÞ G jk ðcjk Þðcjk Þ ¼ T 2 rm j k rZ ro r m r Z r 2o
ð26Þ
The effect of a given damping modification on the parent models can be observed from Eqs. (20)–(22). The role of the cross modal terms is not exactly reproduced if r Z a1, even for the same distribution of the natural frequencies. This has been already studied and discussed for a number of cases in which scaled finite element models have been obtained just increasing the original damping value [9], with the aim to reduce the computational efforts: that scaling procedure is named as ASMA, Asymptotic Scaled Modal Analysis. In order to look for a complete similitude, the condition r Z ¼ 1 must be preserved. Additional conditions on the distribution of the natural frequencies and further considerations are given directly within the selected test cases.
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4. The test cases 4.1. An elastic plate coupled with an acoustic cavity The first test case analysed consists of a single plate undergoing bending vibrations which radiates noise in an acoustic cavity. The plate is rectangular with simply supported edges and made of an homogeneous material; it is located in the yz-plane of generic Oxyz reference system, Fig. 1. The plate displacements are u, v, and w along the x, y and z axes, respectively. The analytical structural out-of-plane velocity response at the generic point P 1 ðy1 ,z1 Þ is here recalled for a mechanical excitation acting at P 2 ðy2 ,z2 Þ, in terms of modal components: ðsÞ
uðo; y1 ,z1 ,y2 ,z2 Þ ¼
ðsÞ
FðoÞ X fj ðy1 ,z1 Þfj ðy2 ,z2 Þ mgen j ðo2j o2 Þ þ iuZo2j
being mgen the generalised mass. The uncoupled natural radian frequencies can be obtained from the relation: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi" 2 2 # 2 pjy Eh pjz ðsÞ oj ¼ þ Ly Lz 12rðsÞ ð1n2 Þ and the uncoupled mode shapes are given by y z fjðsÞ ðy,zÞ ¼ sin jy p sin jz p Ly Lz
ð27Þ
ð28Þ
ð29Þ
The jth mode is mapped into the wavenumber space with the two associated integers jy and jz. The natural frequencies of the parent model are
oj ðsÞ ¼ ojðsÞ
rh ; r 2l
2 r ðsÞ o ¼ rl rh
ð30Þ
The fluid in the parallelepiped cavity is represented by an acoustic wavespeed, ca, and a given mass density, ra . The uncoupled acoustic natural radian frequencies can be obtained from the relation sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pkx 2 pky 2 pkz 2 ðaÞ ð31Þ ok ¼ pca þ þ Lx Ly Lz and the uncoupled mode shapes (rigid walls) are given by x y z fðaÞ ðx,y,zÞ ¼ cos k p p p cos k cos k x y z k Lx Ly Lz
ð32Þ
The kth mode is mapped into the wavenumber space with three associated integers kx, ky, kz. The uncoupled acoustic natural radian frequencies in the parent model are such that
o kðaÞ ¼
oðaÞ k rl
;
1 r ðaÞ o ¼ rl
ð33Þ
Fig. 1. Original configuration: dark grey, plate; light grey, cavity.
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4.1.1. Refining the similitude The ratio of the structural and acoustic natural frequencies, here defined as L, has to be kept in order to not alter their relative distribution, Eqs. (30) and (33):
L¼
ojðsÞ ; okðaÞ
L ¼ L-r h ¼ r l
ð34Þ
Thus, it is not possible to treat the thickness and the area of the plate independently, as it is in [13]; in fact, the natural structural and acoustic frequencies must scale with the same law. Table 1 reports the most important parameters for the acousto-elastic response and the related scaling laws, according with the three variables: rl, r Z , rf. It has to be noted that in the expression of the acoustic modal density, the area and perimeter terms are neglected.
4.1.2. The test data, the finite element models and the results The plate-cavity coupled system refers to the coordinate system in Fig. 1. The plate is made by aluminium: E ¼ 70 GPa, n ¼ 0:33, rðsÞ ¼ 2750 kg m3 ; the sizes are Ly ¼ 0:30 m, Lz ¼ 0:48 m, h ¼ 0:003 m. A 1 N mechanical point force acts in a direction normal to the plate at the point x¼0, y ¼ 0:12 m, z ¼ 0:18 m. The cavity is filled with air: cðaÞ ¼ 343 m=s, rðaÞ ¼ 1:21 kg m3 ; its dimensions are Lx ¼ 0:72 m, Ly ¼ 0:30 m, Lz ¼ 0:48 m. The FE model has been built by using 2D four-nodes elements for the plate and 3D eight-nodes elements for the acoustic part. Each element has 0.02 m long edge for a total number of nodes and elements set to 14 800 and 72 216, respectively, in the original models. There are 25 structural and 120 acoustic modes in 0–2000 Hz. The parent finite element model has been assembled with r l ¼ r h ¼ 1=3, r Z ¼ 1, r f ¼ ðr l Þ2 . The total number of nodes and elements are the same as those of the original model. As a result, the same number of structural and acoustic modes, in the parent model, resonate in a frequency range wider than the original one. The results of the coupled system are illustrated in Figs. 2–5. They refer to a 0–1000 Hz frequency range by using a frequency step Df ¼ 0:11 Hz and a modal basis containing the same number of modes for both the original and parent models. In the results concerning the parent model, the mechanical source position has been scaled in accordance to the similitude laws applied: in both models the load acts at the same dimensionless coordinates. Figs. 2 and 3 present the structural response in terms of velocity module at the driving point and mean velocity response over the plate. Each graph presents three curves: the original system, the parent model and the remodulated responses, respectively. The parent model response has been built by imposing the given triplet r l ,r Z ,r f : the results are directly compared with the original ones without further correction. The remodulated parent model responses have been achieved scaling the frequency axis by or o , in order to compensate the frequency shift introduced by the similitude. The response axis is remodulated according to the coefficients of Table 1. Two numerical analyses are needed for the original and parent models: the agreement is perfect. In fact, having defined a similitude according to the content of Table 1, it is always possible to switch from a representation to another according to the scaling laws of the involved parameters. Figs. 4 and 5 present the acoustic response in terms of pressure under the driving point and the mean pressure response over the cavity. Again, each curve involves the original system, the parent model and the remodulated responses, respectively. It is evident how the parent remodulated model response replicates exactly the original one.
Table 1 Derived parameters (three variables: rl, r Z , rf). Structural natural frequencies Acoustic natural frequencies
oðsÞ p Ah j
1 r ðsÞ o ¼ rl
ðaÞ pV 1=3 k
o
1
1 r ðaÞ o ¼ rl rL ¼ 1
Ratio of the natural frequencies Structural mass
m pAh
3 r ðsÞ m ¼ rl
Fluid mass
mðaÞ pV
3 r ðaÞ m ¼ rl
Structural modal density
nðsÞ pAh nðaÞ pV
Acoustic modal density
ðsÞ j
L¼o
ðaÞ j
o
ðsÞ
1
r ðsÞ n ¼ rl 3 r ðaÞ n ¼ rl
Structural modal overlap factor mðsÞ pnðsÞ Z
r ðsÞ m ¼ rl rZ
Acoustic modal overlap factor
mðaÞ pnðaÞ Z
3 r ðaÞ m ¼ rl rZ
Frequency terms
Eqs. (20) and (21)
r G r 2l r 1 Z
Eq. (22)
r G r 2l r Z
Input power
Eq. (25)
2 r P ¼ r 2 l rf
Spectral density
Eq. (23)
3 2 r ðsÞ S ¼ rl rf
Energy
Eq. (26)
1 2 r T ¼ r1 l rZ rf
Square response
2 r 1 v2 ðor p2 ÞpT ðsÞ ðmðsÞ Þ1 r sr ¼ r 4 Z rf l
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100 original similitude
Velocity (m/s)
10−1
10−2
10−3
10−4
10−5 100
200
300
400
500
600
700
800
900
1000
Frequency (Hz) 100 original similitude
Velocity (m/s)
10−1
10−2
10−3
10−4
10−5 100
200
300
400
500
600
700
800
900
1000
Frequency (Hz) Fig. 2. Driving point velocity (m/s) vs frequency (Hz) ðr L ¼ 13Þ; (top) original and similitude responses; (bottom) original and remodulated similitude responses.
4.2. An infinite cylinder The aim of this section is to introduce the similitude through an analytical development and thus all the considerations about the analytical model in itself are left to the literature. An analytical model is available in a very simple and common configuration. This is represented by a uniform, infinitely long cylinder of thickness h, mean radius Rc, excited by a radial force acting at a given angle y (Fig. 6). The cylinder is made in aluminium and contains a fluid of density ra and wave speed ca. The solution involves trigonometric and Bessel functions, as summarised in [14,15]. The displacement bases are expressed in terms of a given number of radial components, NW. They are for the axial, tangential and radial components, respectively: uðy, oÞ ¼
NW X
U i ðoÞ cosðiyÞ
ð35Þ
V i ðoÞ sinðiyÞ
ð36Þ
i¼0
vðy, oÞ ¼
NW X i¼0
wðy, oÞ ¼
NW X i¼0
W i ðoÞ cosðiyÞ
ð37Þ
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100 original similitude
Velocity (m/s)
10−1
10−2
10−3
10−4 100
200
300
400
500
600
700
800
900
1000
Frequency (Hz) 100 original similitude
Velocity (m/s)
10−1
10−2
10−3 100
200
300
400
500
600
700
800
900
1000
Frequency (Hz) Fig. 3. Mean plate response, velocity (m/s) vs frequency (Hz) ðrL ¼ 13Þ; (top) original and similitude responses; (bottom) original and remodulated similitude responses.
The acoustic pressure, P, can be expressed through the coupling with the radial displacements Ji coa r cosðiyÞ Pðr, y, oÞ ¼ ðra ca oÞ W i ð oÞ J i coa Rc i¼0 NW X
ð38Þ
where the symbol Ji denotes the Bessel function of the first kind of ith order. The uncoupled structural solution (it is neglected the effect of the fluid over the surrounding structure) can be found by using the Donnell–Mushtari equations, LDM ðoÞ with or without some modifiers, LMOD ðoÞ, which refine the representation of a given stress–strain state. In terms of operators [14]: "
2
LDM ðoÞ þ
h
#
LMOD ðoÞ fsðoÞg ¼ fFðoÞg 12R2c
where fsgT ¼ fu,v,wgT and fFgT ¼ fO,O,Q gT .
ð39Þ
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130 original similitude
120
Pressure (dB)
110
100
90
80
70
60 100
200
300
400
500
600
700
800
900
1000
Frequency (Hz) 130 original similitude 120
Pressure (dB)
110
100
90
80
70
60 100
200
300
400
500
600
700
800
900
1000
Frequency (Hz) Fig. 4. Driving point pressure (dB) vs frequency (Hz) ðr L ¼ 13Þ; (top) original and similitude responses; (bottom) original and remodulated similitude responses.
The structural excitation, acting at yQ , is defined as follows: Q dðyyQ Þ ¼
NW X
Q i cosðiyÞ
ð40Þ
i¼0
being Q i ¼ E=2p; the parameter E ¼ 1 if i¼0, while E ¼ 2 otherwise. Figs. 7 and 8 report the results for an aluminium cylinder: h ¼ 0:8128 103 m; Rc ¼0.9144 m. It is filled with air: the material constants are the same as in the previous tests. The number of the radial components has been set to NW¼100; the structural damping is assumed constant, Z ¼ 0:01. The structural excitation is located at yQ ¼ p=3. The structural velocity response is evaluated at the driving point, Fig. 7; the acoustic pressure is calculated at y ¼ p=3 and r ¼ 0:7 Rc , Fig. 8. Again, it can be observed that the parent model perfectly replicates the response of the original configuration for both the structural and acoustic domains after having remodulated the responses. 4.3. Some considerations A list of considerations are here given:
Variations of the thickness and the area of the plate (or cylinder, respectively), such that rh arl , lead to a modification of the natural structural and acoustic frequency distributions; for such coupled acousto-elastic systems, this choice cannot
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125 120
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115 110 105 100 95 90 85 100
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Frequency (Hz) 130 original similitude
125 120
Pressure (dB)
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Frequency (Hz) Fig. 5. Mean cavity response, pressure (dB) vs frequency (Hz) ðr L ¼ 13Þ; (top) original and similitude responses; (bottom) original and remodulated similitude responses.
Fig. 6. Cylinder configuration: dark grey, shell; light grey, fluid.
be pursued since the structural and acoustic poles in the parent model would scale with different laws. This point needs to be better investigated. Having defined a complete similitude, according to the contents of Table 1, it is always possible to switch from a representation to another according to the scaling laws of the involved parameters. This step can be performed by invoking the similitude in the reversal direction and then by modifying both the axes.
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Structural velocity [m/s]
original similitude
Frequency [Hz]
Structural velocity [m/s]
original similitude
Frequency [Hz] Fig. 7. Structural response: radial velocity (m/s) vs frequency (Hz) (y ¼ p=3, rL ¼ 0.5); (top) original and similitude responses; (bottom) original and remodulated similitude responses.
A complete similitude is achievable only if the original damping is left unaltered, rZ ¼ 1. The similitude is approximate if rZ a1: in such a case only a mean response can be replicated, as already shown in [8,11,13]. There, the responses were not re-modulated since the investigations were aimed at reducing the 1 computational costs. The test presented in [8] is a particular one obtained with rh ¼1, r A ¼ b, r Z ¼ b , r f ¼ 1 and then r sr ¼ 1, being b a parameter chosen by the analyst. At this point, even by using these very simplified models, SAMSARA is well posed. In fact, for structural and structural–acoustic configurations if both the original damping values and the distribution of the natural frequencies are kept, a complete similitude is defined: it is thus possible to work on different scales. For structural and structural–acoustic configurations in which the original damping values are modified, only an estimation of the mean square response is achievable: a scaling is possible rather than a similitude. In this case the predictive models can have great computational advantages. 5. Concluding remarks A structural–acoustic similitude has been presented and discussed with both analytical and finite element approaches. They both allow reproducing the local response of simple coupled structural–acoustic systems for any given choice of the length parameter, rl. This led to the definition of parent models greater or smaller than the original configuration. In the parent models, the original damping values and the distribution of the natural frequencies are kept. In this way, it is always possible to switch from the original model to the parent one and to perform the reversal step, being the similitude complete. Even by using simplified models, the present results allow defining a whole procedure in which is possible to work with scaled models or parent models. This tool has been named SAMSARA, Similitude and Asymptotic Models for Structural– Acoustic Research and Application. A complete similitude is obtained if both the original damping values and the distribution of the natural frequencies are kept: the parent models will work on different scales, larger or smaller than the original ones. This can have a great importance in the laboratory experiences.
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Acoustic pressure [Pa]
original similitude
Frequency [Hz]
Acoustic pressure [Pa]
original similitude
Frequency [Hz] Fig. 8. Acoustic response: pressure (Pa) vs frequency (Hz) (y ¼ p=3, r ¼ 0:7Rc , rL ¼0.5); (top) original and similitude responses; (bottom) original and remodulated similitude responses.
A scaling procedure can be applied if the original damping values are modified. This allows building models useful to estimate the mean square response at very reduced computational costs. This can have a great importance in enhancing the numerical tools. Further work is clearly needed to fully define SAMSARA even introducing more complicated and realistic schemes, such beams, stiffened plates and shells, but the actual investigations have properly framed the opportunities offered by the similitude and scaling applied to linear problems in structural–acoustics. References [1] J. Wu, Prediction of the dynamic characteristics of an elastically supported full-size plate from those of its complete-similitude model, Comput. Struct. 84 (2006) 102–114. [2] P. Kroes, Structural analogies between physical systems, Br. J. Philos. Sci. 40 (1989) 145–154. [3] G.C. Everstine, Structural analogies for scalar field problems, Int. J. Numer. Methods Eng. 17 (March) (1981) 471–476. [4] E. Szucs, Similitude and Modelling, Elsevier Science Ltd., 1980 ISBN: 0444997806. [5] S.J. Kline, Similitude and Approximation Theory, Springer-Verlag, 1986 ISBN: 0387165185. [6] C.H. Wolowicz, J.S. Bowman Jr., W.P. Gilbert, Similitude Requirements and Scaling Relationships as Applied to Model Testing, NASA Technical Paper 14–35, 1976. [7] B.R. Mace, Statistical energy analysis, energy distribution models and system modes, J. Sound Vib. 264 (2003) 391–409. [8] S. De Rosa, F. Franco, A scaling procedure for the response of an isolated system with high modal overlap factor, Mech. Syst. Signal Process. 22 (7) (2008) 1549–1565. [9] S. De Rosa, F. Franco, On the use of the asymptotic scaled modal analysis for time-harmonic structural analysis and for the prediction of coupling loss factors for similar systems, Mech. Syst. Signal Process. 24 (2) (2010) 455–480. [10] S. De Rosa, F. Franco, T. Polito, Analysis of the short and long wavelength coupling through original and scaled models, in: NOVEM 2009, Noise and Vibration: Emerging Methods, April, Keble College, Oxford, UK, 2009 paper no. 28. [11] X. Li, A scaling approach for high-frequency mean responses, J. Acoust. Soc. Am. 127 (5), doi:10.1121/1.3397257. [12] E.J. Skudrzyk, The mean-value method of predicting the dynamic response of complex vibrators, J. Acoust. Soc. Am. 67 (1980) 1105–1135. [13] S. De Rosa, F. Franco, T. Polito, Structural similitude for the dynamic response of plates and assemblies of plates, Mech. Syst. Signal Process. 25 (3) (2011) 969–980. [14] A.W. Leissa, Vibration of Shells, NASA SP-288, 1973. [15] G. SenGupta, W.H. Weatherill, Numerical prediction of airborne noise transmission into a fuselage, in: AIAA 11th Aeroacoustics Conference, Sunnyvale, CA, 19–21 October 1987.
