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Numerical and experimental investigations on structural intensity in plates
Composite Structures 140 (2016) 94–105
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Composite Structures journal homepage: www.elsevier.com/locate/compstruct
Numerical and experimental investigations on structural intensity in plates G. Petrone ⇑, M. De Vendittis, S. De Rosa, F. Franco Department of Industrial Engineering – Aerospace Section, University of Naples Federico II, Italy
a r t i c l e
i n f o
Article history: Available online 5 January 2016 Keywords: Structural intensity Composites Panels
a b s t r a c t The need of reducing the vibration levels is one of the most important points in many engineering fields. Therefore, the knowledge of the vibrational field can be a useful and efficient tool in problems regarding acoustic and mechanical insulation. This paper is focused on the experimental measurements and numerical predictions of the structural intensity fields in rectangular plates. The method is applied by simulating finite plates and the effects of constraints, load conditions, damping, thickness and fibres orientation are investigated. Experimental measurements are carried out on both aluminium sandwich and composite (orthotropic) panels; the results are compared through a mixed numerical/experimental approach with purely numerical predictions. The mixed approach based on both experimental and numerical information is proposed since it can give good indications of the structural intensity fields, can offer information about the energy transmission paths and can fruitfully use the numerical generated information. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Aircraft, ships, cars and all other existing vehicles are always subjected to external dynamic loads, which excite the structure in several frequency ranges depending on the sources of vibration. Vibrational energy can derive from engine operations, fluid– structure interaction, periodic contact between mechanical parts, etc. Therefore, these structures can be subjected to excitation forces at frequencies close to the structural resonances, hence exceeding the permissible vibration levels; this may result in fatigue failure, destruction of electronic and mechanical parts or very high noise levels (vibro-acoustic problem). The plates are among the most common structural configurations in the transportation engineering and thus in structural design the knowledge of their dynamic behaviour becomes of foremost importance as well as the transmission mechanisms and the vibrational fields. One of the most effective tools for predicting and measuring energy distribution is the structural intensity (SI). It is a measurement of the energy flow that propagates within a structure due to the waves and it can be considered as the analogous of the power flow (PF) [1]. The interest in evaluating the SI has become attractive thanks to its capability to offer valuable informations on the strength and location of sources and transmission paths of structures-borne sound energy by plotting a vector map. SI is able ⇑ Corresponding author. E-mail address: [email protected] (G. Petrone). http://dx.doi.org/10.1016/j.compstruct.2015.12.034 0263-8223/Ó 2015 Elsevier Ltd. All rights reserved.
to localise dampers in structures in an easy way and, for this reason, it can be a key for solving structure borne noise problems. SI has been used for several years as a tool for the design of vibration and noise control systems and there is a good literature on this topic. In 1970 Noiseux and McDevitt measured the freefield flexural intensity with two accelerometers: one supplied the normal acceleration and the other supplied the rotational velocity at the point measured by positioning the accelerometer mounted on its side [1,2]. Pavic used a new approximation technique, called the finite difference technique, allowing the reconstruction of the SI field both in the free-field and near-field, also addressing flexural vibrations: briefly, spatial derivatives of the normal displacements, directly related to the forces in the classical Euler–Bernoulli plate theory, are obtained by a grid of accelerometers, [3]. The number of the accelerometers and their positioning is changed in order to improve the accuracy of that method, but in any case this contact method has several sources of error; the first is related to the direct attachment on the structure of the accelerometers, shakers, dampers, etc. For this reason, the researchers, in the ’80s, began to study non-contact methods for the measurable quantities, such as displacement, velocity and acceleration. These quantities are found by near-field acoustic holography (NAH) or by an optical method, with laser beams [4]. Although more accurate, also these methods have some problems: (i) the misalignment of the experimental set-up, (ii) the need of phase and magnitude matching and the cost of the instruments.
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Thanks to the continuous development of computers, it has been possible to build the intensity field only by FEM prediction [5,6]: in this case, the numerical model provides quickly all necessary data with low computational costs. These predictions can be made for isotropic, but also for composite materials. In particular, the latter have highly customizable properties and therefore they can be adapted to the type of loads to which they are subjected. Other numerical methods, exploited the wavenumber processing technique; this can be used to determine SI from both contact and optical measurements, [7,8]. In these cases, the velocities and the wave-numbers are required instead of the displacements and the forces per unit length: the higher-order spatial derivatives, necessary to determine the SI, are computed from the wavenumber–frequency domain through a 2D Spatial Fourier Transform (SFT) of the velocities. SI is a vector quantity consisting of magnitude and phase, its changes may be tracked to determine the relative health of structures. As consequence, in the recent years, structural intensity received a great emphasis also for the damage detection techniques [9,10]. In this work, the evaluation of SI of different panels is investigated numerically and experimentally. Firstly, an overview of the structural intensity with emphasis on the different techniques is reported. Section 2 is dedicated to illustrate the theoretical background and the equations used to calculate the SI field. In Section 3, numerical investigations of several panels are performed, (i) to validate the technique already presented by Gavric and Pavic [5,6], and (ii) to evaluate the effect of the main parameters, such as boundary conditions, materials, damper and structural damping coefficients, etc. on SI field. In Section 4 the numerical and experimental studies of the SI field on two different classes of panels, Aluminium Foam Sandwich panels and flax-PE panels, are presented. In this section, a mixed approach, based on numerical and experimental data, is proposed to enhance the technique presented by Gavric and Pavic [5,6]. Finally, in Section 5 some concluding remarks are given.
auxiliary power units are some examples of external and internal sources of excitation; vibration isolation systems and structural joints are examples of energy dissipation mechanisms. In order to illustrate the capabilities of the SI and for the sake of simplicity, in this study only a pair of energy source and sink is considered in order to produce a well defined energy flow through the structure. From a general point of view, the structural power is defined as the active part associated with the vibration energy. In analytical terms, the power, at a prescribed point, is represented by product of a force with the in-phase component of the velocity in the direction of the force and can be calculated as follows:
Pðx; QÞ ¼
ð1Þ
where F is the applied force and v is the complex conjugate of the velocity; they both refer to a prescribed point, Q, in the structural domain Eq. (1); R denotes the real part [12,13]. The spatial distribution of the energy propagated within the structure is defined with the structural intensity field which, in the case of plate type structures, represents the energy propagating through the unit length. Hence, according to the principle of conservation of energy, the power injected to the plate can be deduced by integrating the normal component of the structural intensity vector (I),along a closed curve L, enclosing the vibrational source:
PðxÞ ¼
I
I n dl
ð2Þ
L
Vibrational power flow per unit cross-sectional area of a dynamically loaded plate is defined as the structural intensity, I denotes the normal vector) [14]. (n The total intensity in a thin vibrating plate is due to the combined action of shear, bending, and twisting waves (Fig. 1) and can be expressed with their orthogonal components as [15]:
Ix ¼
i W x h ~ ~ þ M x ~hy Mxy ~hx I Nx u þ Nxy v~ þ Q x w
ð3Þ
Iy ¼
i W x h ~ ~ þ Q yw ~ My ~hx þ Mxy ~hy I Ny v þ Nxy u
ð4Þ
2. Theoretical background on structural intensity Structural intensity (SI) is a vector field, described by magnitude and phase, which indicates the path and the amplitude of the mechanical energy flowing through a vibrating structural component [11]. The location, where the mechanical energy enters the structure, is generally called the energy source; the location, where the energy is extracted from the system, is called the energy sink. Multiple energy sources and sinks might coexist in the same structure. Aerodynamic loads, propulsion system, rotor and gearboxes and
1 R½Fðx; Q Þv ðx; Q Þ 2
I¼
2
2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi I2x þ I2y
m
m
ð5Þ
where N x ; N y and N xy are complex membrane and in-plane shear forces per unit width, Q y and Q z are complex transverse shear forces per unit width,
Fig. 1. Generalised forces per unit length and displacements in plate element.
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M y ; M z and M xy are complex bending and twisting moments per unit width, ~ are complex conjugate of translational ~; v ~ and w u displacements, hy and ~ hz are complex conjugate of rotational displacement ~ hx ; ~ along x, y and z directions, I denotes the imaginary part. All these quantities are dependent on the excitation frequency (x). It is also useful to note that SI in plates can be expressed as a power per unit width, [W/m].
3. Numerical models The first part of the work is devoted to the numerical models of plate structures in order to have a preliminary assessment about the SI fields and to analyse the influence of different parameters. The procedure, used in this work to simulate the SI field in a mechanical system, is illustrated as block diagram in Fig. 2. All test cases have been performed at frequencies close to the natural ones (i.e. at a frequency with higher energy levels compared to other ones), even if the method can be evaluated at any frequency. The procedure is here recalled: (1) Starting from the model sizes and properties, a FE model is created with the commercial finite element analysis code MSC/Patran and a frequency response analysis is performed in MSC/Nastran; in output from a modal approach, the sets of complex displacements and internal forces per unit length are requested to perform the SI computation. (2) These sets are obtained for a specific frequency close to a natural one and imported in a specific in-house Matlab script for SI calculation by using the Eqs. (3)–(5).
Fig. 3. Conventions and symbols for numerical plates.
Table 1 Steel properties. Material
E [GPA]
G [GPA]
m
q [kg=m3 ]
Steel
205.8
79.15
0.3
7800
(3) The results are post-processed by using two Matlab functions: the Quiver and the Imagesc. The first allows to build a vector map from the two intensity components, the second provides information about SI magnitude trough a colour bar.
Fig. 2. Basic scheme of the adopted procedure.
G. Petrone et al. / Composite Structures 140 (2016) 94–105
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Fig. 4. Effects on SI of constraint conditions.
In order to facilitate the reading of the figures, a letter is assigned to the edges of the plates, starting from letter, a of the vertical left side up to letter d clockwise. Moreover, the force is indicated as a black dot and the damper as a black triangle together with the constraints: this is illustrated in Fig. 3.
3.1. Influence of different parameters on an isotropic plate The first test structure is a rectangular steel plate, 1080 mm 1440 mm 6 mm. The material properties are reported in Table 1.
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Fig. 5. Effect on SI of load magnitude, damping ratio, structural damping and thickness.
Table 2 Composite graphite/epoxy laminated properties. Material
Ex [GPa]
Ey [GPa]
Gxy [GPa]
Gyz [GPa]
mxy
myz
q [kg=m3 ]
Graphite/epoxy
175.1
6.700
4.100
2.056
0.2542
0.4689
1520
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99
Fig. 6. Effect of the variation of fibres orientation.
A 2D finite element model using 4-nodes shell elements was built and the plate model was excited by a unit nodal force along the z direction. The coordinates of the node loaded are reported in the caption of Fig. 4. The energy sink was simulated trough a viscous damper element (CVISC) located at x ¼ 810 mm and y ¼ 180 mm. It was assigned a constant damping loss factor, g ¼ 0:007. The frequency response was calculated in the range 0–150 Hz and, except where differently expressed, all the analysis of the SI were performed at a specific frequency, f ¼ 20 Hz, close to the first natural frequency of this panel, f 1 ¼ 19:56 Hz. The first investigations concerned the influence of the boundary conditions on the SI field. Three different constraint conditions were considered along the plate sides: the simply supported conditions, in which all the translational degrees of freedom were blocked; the clamped ones, in which all the translational and rotational degrees of freedom were blocked, and the free ones. Fig. 4 contains the first results carried out for two different constraints and load conditions. It can be observed that: the transmission path is always from the excitation node to the viscous damper locations; the total intensity decreases approaching to constraints; in a plate with all the edges constrained, the intensity vectors are higher in the middle of the panel, as it can be seen from the contour plot of Fig. 4a and b and the vectors flow in this central region of the panel where the DOF are not restrained. In the case of unconstrained plates, instead, the opposite occurs (see Fig. 4c–f); in particular, it can be also noticed accumulation fringes along the free edges; finally, from these first investigations, it seems that the total intensity is related to the boundary conditions. In fact, comparing Fig. 4a and e, it is clear that for the panel in free–free
boundary conditions the intensity vectors are present in all the panel including the edges, while, for other types of boundary conditions, the restrained edges reduce such intensity vectors. The simply supported plate was imposed as reference plate. On this configuration, the effects of the variation of load magnitude, damping ratio, structural damping and thickness were analysed. In Fig. 5, comparing all figures with the first one, it can be stated that: the effect of the load magnitude, Fig. 5b, is not visible considering only the intensity vectors: the variation of the total intensity is of the order of 102 for a variation on load magnitude of the order of 10; by decreasing the value of the damper from 500 to 100 [N/(m s)], Fig. 5c, the area close to the damper is less intense; by changing the structural damping of the material by a factor 10, the SI field does not change near the excitation force, but it clearly decrease near the damper, in terms of total intensity and transmission path, Fig. 5d; the effect of the variation of thickness involves two different aspects: first, the thickness is inversely proportional to the total intensity; second, decreasing the thickness, the intensity field will be more extended, see Fig. 5e. 3.2. Influence of different parameters for an orthotropic plate The second series of tests refers to orthotropic plates. The sizes are: 6000 mm 3000 mm 7 mm. The plates were here subjected to in-plane uniform pressure excitations along their edges in view to show the different behaviour when they are subjected to a force in a direction parallel and orthogonal to the fibres. In fact, the
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Fig. 7. Effect of the variation of fibres orientation. Plates with two different stacking sequences.
Table 3 Dissipated power analysis: variation of structural damping ratio.
g
P in [mW]
P out [mW]
Diff. %
0 0.007 0.020 0.070
9.1 8.4 7.4 5.0
9.1 7.7 5.9 2.6
0 8.33 20.27 48.00
Table 4 Dissipated power analysis: g ¼ 0:007; f ¼ 20 Hz).
variation
with
the
damper
constant,
D
D [N/mm]
P in [mW]
P out [mW]
2 0.5 0.05
8.4 7.0 2.8
7.7 5.3 0.7
approach can be used also with only in-plane excitation, as shown in Eqs. (3) and (4). The material was epoxy resin reinforced with graphite fibres, which were oriented both in x and y directions, i.e. one is unidirectional along x-direction and the other one along y-direction. The main aim of this strongly directional material was to investigate the effect of the fibres orientation on the SI field. The material properties are listed in Table 2. In all plates, a uniform in-plane pressure per unit length with (at
magnitude p ¼ 103 [N/mm] was applied at a frequency of f ¼ 2 Hz, value close to the first natural one, which was f 1 ¼ 1:37 Hz. For these panels only two different constraint conditions were considered: simply supported and clamped conditions. For both the conditions the translation along the z direction was assumed restrained. The results are shown in Fig. 6. Considering the same conventions of Fig. 3, a uniform in-plane pressure have been applied along the d and c edges respectively for Fig. 6a–d.
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Fig. 8. Numerical–experimental basic scheme.
Table 5 Dimensions of AFS panels. Panel A B
Table 8 Power injected into AFS panels.
Width mm
Length mm
Total thick. mm
Skins thick. mm
Core thick. mm
656 656
476 476
10 8.6
1 0.6
8 7.4
AFS panels
P num [mW]
P exp [mW]
A B Equivalent A1
0.048 0.030 0.400
0.042 0.027 –
It is possible to make further considerations:
Table 6 AFS: mechanical properties. Type
E [GPa]
g
m
q mkg3
A Core
6.48
1103
0.31
600
B Core
3.43
1103
0.31
390
Skin
71
1104
0.33
2700
Table 7 FEA-EMA correlation for AFS panels. Mode Name
FEA-A Hz
EMA-A Hz
Diff-A %
FEA-B Hz
EMA-B Hz
Diff-B1 %
(1, (0, (1, (2, (2, (0, (1, (2, (3, (0, (3, (2, (1, (3,
122.83 145.11 288.11 288.22 357.99 427.27 543.70 588.72 771.86 801.72 853.77 889.56 922.37 1080.66
124.75 146.36 292.30 296.87 363.17 434.96 548.12 598.68 786.94 805.67 867.51 897.97 924.48 1096.71
1:53 0:86 1:43 2:91 1:43 1:77 0:81 1:66 1:92 0:49 1:58 0:94 0:23 1:46
107.92 127.51 253.16 253.29 314.58 375.49 477.78 517.34 678.37 704.57 750.36 781.78 810.62 949.81
110.99 126.96 259.47 255.11 313.31 374.99 483.06 513.87 681.99 699.08 751.02 774.01 808.62 948.18
2:76 0.44 2:43 0:71 0.40 0.13 1:09 0.68 0:53 0.78 0:09 1.00 0.25 0.17
1) 2) 2) 0) 1) 3) 3) 2) 0) 4) 1) 3) 4) 2)
the transmission path of the SI preferentially follows the direction along which the elastic modulus is greater and this is evident comparing Fig. 6a and b, or Fig. 6c and d: considering the direction of the greater stiffness, i.e. the horizontal or the vertical one, both the intensity vectors and the total intensity are more developed in that direction. applying a pressure load only along the longest edge, i.e. along the y-direction, the y-oriented fibres plate (Fig. 6b) presents a greater total intensity peak than the x-oriented one (Fig. 6a). The opposite occurs in the case of a pressure load only along the shorter edge; i.e. the x-oriented fibres plate (Fig. 6c) presents a greater total intensity peak than the y-oriented one (Fig. 6d). All these considerations are even more evident considering an idealised plate with two different stacking sequences in the same plane (i.e. the left and the right part of the panel have the fibres oriented in the x and y direction respectively), as shown in Fig. 7a. Assuming a pressure load (indicated as pn in Fig. 7, where p is the pressure load and n indicated the label of the edge) on the shortest edges, a comparison has been evaluated on two panels with reference to the variation of loading and boundary conditions. 3.3. Dissipated power analysis The power injected to the structure can be computed by using the velocity and the force measured at excitation point according to Eq. (1). This power is then dissipated in the structure.
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According to the principle of conservation of energy, the power injected to the plate can be deduced by integrating the component of the structural intensity vector, which is normal to a closed curve enclosing the vibrational source [16]. In this section, the influence of different parameters on the power was investigated and results are summarised in Tables 3 and 4. It is evident the influence of the damping on the output power (P out ) that is equal to the injected power in case of g ¼ 0. The output power is calculated as well as the input one (Eq. 1): the product of the forces of the constraint and the output velocity is evaluated at the location of the scalar grid where the damper is grounded, [16]. In all other cases, the dissipated power (P in P out ) increases with the damping. 4. Numerical and experimental investigations In this section numerical and experimental investigations on two different classes of panels, Aluminium Foam Sandwich (AFS) and flax-polyethylene (flax-PE) were performed in order to carry out information about their vibro-acoustic capabilities. The dynamic behaviour of these panels were already analysed in previous works and more information about their performance and more details about material composition and characterisation of the panels can be found in [17,18]. A brief description for each
class of panels is reported in the specific paragraphs. The numerical investigations were performed by adopting the flow chart of Fig. 2; the numerical–experimental ones, indeed, following the procedure reported in Fig. 8. This last procedure may be considered as a mixed numerical–experimental approach because forces per unit length were not experimentally measured, and so, in order to have an evaluation of the SI field, the used forces were carried out from the numerical model in which the experimental displacements were used as imposed conditions. The same experimental measurements were also used as post processing data for SI evaluations. This mixed approach tries to overcome some common experimental difficulties. The cinematic of a structure can be easily measured. On the contrary, the generalised forces per unit length cannot be carried out easily from experimental data. 4.1. Aluminium foam sandwich panel For this class of panels the effect of different percentage of porosity on structural intensity of two AFS panels, having same in-plane dimensions and average cell size of the foam bubbles, was investigated experimentally and numerically. Some more details on dimension and material properties of the investigated panels can be found in Tables 6 and 5.
Fig. 9. Numerical–experimental comparison for AFS A – panels.
Fig. 10. Numerical–experimental comparison for AFS B – panels.
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G. Petrone et al. / Composite Structures 140 (2016) 94–105
Fig. 11. Comparison AFS A – equivalent panel.
Table 9 flax-PE: mechanical properties of basic lamina. h
Ex [GPa]
Ey ¼ Ez [GPa]
Gxy ¼ Gxz [GPa]
Gyz [GPa]
mxy ¼ mxz ¼ myz
q
9.50
1.30
0.55
0.40
0.40
1025
Table 10 Power injected into flax-PE panels. Flax-PE panels
P num [mW]
P exp [mW]
Unidirectional Quasi-isotropic Equivalent flax-PE
0.016 0.002 0.200
0.014 0.001 –
Numerical predictions and experimental measurements of the modal parameters of AFS panels were performed in previous works [17]. Concerning the experimental measurements, an impact hammer was used to excite the structure and a set of accelerometers was used to measure the response of the panels that were suspended by means of bunjee chord in order to simulate free–free
kg m3
i
boundary conditions. Once the frequency response data was recorded using an acquisition system, the data were postprocessed by using a Matlab script. On the other hand, from the numerical point of view, a 3-D model was realised, modelling the face sheets using 4-nodes quadrilateral (CQUAD4) elements and the core using 8-nodes solid (CHEXA) elements. The accuracy of the numerical model was hence validated comparing the numerical results, in terms of natural frequencies and mode shapes, with the experimental ones [17]. In order to apply Eqs. 3 and 4, a second numerical model consisting of all shell layers was created:. In particular, the foam core was modelled as a layer with CQUAD4 elements and then the ‘‘full shell” layer model was validated in the same way, by comparing the natural frequencies and mode shapes with experimental ones. In Table 7 the comparison between the natural frequencies of the ‘‘full 2D” model and experimental ones is reported.
Fig. 12. Numerical–experimental comparison for unidirectional flax-PE panels.
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G. Petrone et al. / Composite Structures 140 (2016) 94–105
Fig. 13. Numerical–experimental comparison for quasi-isotropic flax-PE panels.
In the present analysis, before comparing the SI behaviour of different panels, the power injected into the plates was calculated and the results are shown in Table 8. For an evaluation of structural intensity, according to the procedure described in (Fig. 8), the complex displacements were carried out from experimental measurements while the forces per unit length were obtained by FE model and both were replaced in the Eq. 5, implemented in a Matlab code. Numerical and experimental results for both AFS panels (A and B), in terms of intensity vectors and total intensity were carried out for a frequency, f ¼ 100 Hz, close to the first natural one and they are depicted in Figs. 9 and 10. Results show a good correlation, and it can be noted that the total intensity of panel A is almost 10 times lower than that of panel B. This could be due by the lower elastic core properties and lower total thickness of the panel B. A further numerical analysis was performed on the equivalent (i.e. same weight) isotropic panel of the panel A, with the same properties of AFS skins (Fig. 6). From the results reported in Fig. 11, the total maximum structural intensity of the equivalent panel is almost 10 times higher than panel A. This is mainly due to the presence of the foam material that, as known, is a good absorption material. 4.2. Flax-PE panel Also for the flax-PE, two different panels, having same dimension 706 mm 492 mm 3.6 mm, but with different stacking sequences, respectively unidirectional ½0 8 and quasi-isotropic ½0 =90 =45 = 45 s panels, were performed. The lamina properties are listed in Table 9. Also for this class of panels the dynamic behaviour, in terms of modal parameters, were already investigated in previous works [18] and the accuracy of the numerical model was already validated comparing the numerical results, in terms of natural frequencies and mode shapes, with the experimental ones. Furthermore, as for the AFS panels, the power injected into the plates was calculated and they are reported in Table 10. The same procedure, made in the previous analyses for the AFS panels, was adopted and the structural intensity behaviour of these panels was carried out for a frequency (f ¼ 7:5 Hz) close to the first natural one. Comparing Figs. 12 and 13, the quasi-isotropic panel has a maximum structural intensity higher than that of the unidirectional one. In addition, as carried out in Section 3, it can be noted that the SI transmission path follows the direction along which the elastic modulus is greater.
5. Conclusions In this paper a technique, based on the finite element method, for estimating the structural intensity of plates has been presented. First, in order to validate the technique, numerical analysis are performed and results are compared with some reference ones, already present in literature [15]. Moreover, in view to get more information, a sensitivity analysis on the influence of different parameters (boundary conditions, load and damper magnitudes, structural damping, thickness and fibre orientation) on the structural intensity field of a panel is performed. The results of the test problems indicate that the method is a valid tool for predicting the structural intensity field of a dynamically excited system at relatively low frequencies. In this paper, the results for the test problems are discussed at resonances and they are very accurate. The use of FEA, to calculate structural intensity, is accurate and economical for the first modes of a mechanical system. This accuracy is strongly dependant on the mesh model size, as expected. This technique is applied to two different classes of panels: AFS and flax-PE panels and the obtained results show a good correlation between numerical and numerical/experimental data: the experimental measurements of the forces were not available and hence the numerical ones were used. Moreover, the technique offers information of energy transmission paths and positions of sources and sinks of mechanical energy that can be suitable for designer to localise the position of application of a damper where the level of vibrations exceeds.
References [1] Mc Devitt T. Development and evaluation of laser doppler technique for measurement of flexural and longitudinal structural intensity. Technical Report No. TR 91-013. The Pennsylvania State University; 1991. [2] Noiseux D. Measurement of power flow in uniform beams and plates. J Acoust Soc Am 1970;47:238–47. [3] Pavic G. Measurement of structure borne wave intensity. J Sound Vib 1976;49:221–30. [4] Bouchard G, Bogy D. Experimental measurement of scattered surface waves using a laser-Doppler probe. J Acoust Soc Am 1985;77(3). [5] Gavric L, Pavic G. A finite element method for computation of structural intensity by the normal mode approach. J Sound Vib 1993;164:29–43. [6] Gavric L, Pavic G, editors. Computation of structural intensity in beam-plate structures by numerical modal analysis using FEM. Third international conference on intensity techniques, Senlis; 1991. p. 207–14. [7] Arruda J, Mas P. Localizing energy sources and sinks in plates using power ow maps computed from laser vibrometer measurements. J Shock Vib 1998;5:235–53.
G. Petrone et al. / Composite Structures 140 (2016) 94–105 [8] Pascal J, Li J, Carniel X. Wavenumber processing techniques to determine structural intensity and its divergence from optical measurements without leakage effects. J Shock Vib 2002;9:57–66. [9] Schmidt W, editor. Damage assessments of aluminium panel structures using structural intensity (SI)-based techniques. 52nd AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics and materials conference, Denver; 2011. [10] Lamberti A, Semperlotti F, editors. On the detection of closing delaminations in laminated composite plates using the structural intensity method. SPIE Smart Structures et al. San Diego; 2013. [11] Zhang Y. An experimental method for structural intensity and source location [Ph.D. thesis]. Yowa State University; 1993. [12] Pavic G, editor. Energy flow induced by structural vibrations of elastic bodies. Third international conference on intensity techniques, Senlis; 1991. p. 21–28.
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Composite Structures journal homepage: www.elsevier.com/locate/compstruct
Numerical and experimental investigations on structural intensity in plates G. Petrone ⇑, M. De Vendittis, S. De Rosa, F. Franco Department of Industrial Engineering – Aerospace Section, University of Naples Federico II, Italy
a r t i c l e
i n f o
Article history: Available online 5 January 2016 Keywords: Structural intensity Composites Panels
a b s t r a c t The need of reducing the vibration levels is one of the most important points in many engineering fields. Therefore, the knowledge of the vibrational field can be a useful and efficient tool in problems regarding acoustic and mechanical insulation. This paper is focused on the experimental measurements and numerical predictions of the structural intensity fields in rectangular plates. The method is applied by simulating finite plates and the effects of constraints, load conditions, damping, thickness and fibres orientation are investigated. Experimental measurements are carried out on both aluminium sandwich and composite (orthotropic) panels; the results are compared through a mixed numerical/experimental approach with purely numerical predictions. The mixed approach based on both experimental and numerical information is proposed since it can give good indications of the structural intensity fields, can offer information about the energy transmission paths and can fruitfully use the numerical generated information. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Aircraft, ships, cars and all other existing vehicles are always subjected to external dynamic loads, which excite the structure in several frequency ranges depending on the sources of vibration. Vibrational energy can derive from engine operations, fluid– structure interaction, periodic contact between mechanical parts, etc. Therefore, these structures can be subjected to excitation forces at frequencies close to the structural resonances, hence exceeding the permissible vibration levels; this may result in fatigue failure, destruction of electronic and mechanical parts or very high noise levels (vibro-acoustic problem). The plates are among the most common structural configurations in the transportation engineering and thus in structural design the knowledge of their dynamic behaviour becomes of foremost importance as well as the transmission mechanisms and the vibrational fields. One of the most effective tools for predicting and measuring energy distribution is the structural intensity (SI). It is a measurement of the energy flow that propagates within a structure due to the waves and it can be considered as the analogous of the power flow (PF) [1]. The interest in evaluating the SI has become attractive thanks to its capability to offer valuable informations on the strength and location of sources and transmission paths of structures-borne sound energy by plotting a vector map. SI is able ⇑ Corresponding author. E-mail address: [email protected] (G. Petrone). http://dx.doi.org/10.1016/j.compstruct.2015.12.034 0263-8223/Ó 2015 Elsevier Ltd. All rights reserved.
to localise dampers in structures in an easy way and, for this reason, it can be a key for solving structure borne noise problems. SI has been used for several years as a tool for the design of vibration and noise control systems and there is a good literature on this topic. In 1970 Noiseux and McDevitt measured the freefield flexural intensity with two accelerometers: one supplied the normal acceleration and the other supplied the rotational velocity at the point measured by positioning the accelerometer mounted on its side [1,2]. Pavic used a new approximation technique, called the finite difference technique, allowing the reconstruction of the SI field both in the free-field and near-field, also addressing flexural vibrations: briefly, spatial derivatives of the normal displacements, directly related to the forces in the classical Euler–Bernoulli plate theory, are obtained by a grid of accelerometers, [3]. The number of the accelerometers and their positioning is changed in order to improve the accuracy of that method, but in any case this contact method has several sources of error; the first is related to the direct attachment on the structure of the accelerometers, shakers, dampers, etc. For this reason, the researchers, in the ’80s, began to study non-contact methods for the measurable quantities, such as displacement, velocity and acceleration. These quantities are found by near-field acoustic holography (NAH) or by an optical method, with laser beams [4]. Although more accurate, also these methods have some problems: (i) the misalignment of the experimental set-up, (ii) the need of phase and magnitude matching and the cost of the instruments.
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Thanks to the continuous development of computers, it has been possible to build the intensity field only by FEM prediction [5,6]: in this case, the numerical model provides quickly all necessary data with low computational costs. These predictions can be made for isotropic, but also for composite materials. In particular, the latter have highly customizable properties and therefore they can be adapted to the type of loads to which they are subjected. Other numerical methods, exploited the wavenumber processing technique; this can be used to determine SI from both contact and optical measurements, [7,8]. In these cases, the velocities and the wave-numbers are required instead of the displacements and the forces per unit length: the higher-order spatial derivatives, necessary to determine the SI, are computed from the wavenumber–frequency domain through a 2D Spatial Fourier Transform (SFT) of the velocities. SI is a vector quantity consisting of magnitude and phase, its changes may be tracked to determine the relative health of structures. As consequence, in the recent years, structural intensity received a great emphasis also for the damage detection techniques [9,10]. In this work, the evaluation of SI of different panels is investigated numerically and experimentally. Firstly, an overview of the structural intensity with emphasis on the different techniques is reported. Section 2 is dedicated to illustrate the theoretical background and the equations used to calculate the SI field. In Section 3, numerical investigations of several panels are performed, (i) to validate the technique already presented by Gavric and Pavic [5,6], and (ii) to evaluate the effect of the main parameters, such as boundary conditions, materials, damper and structural damping coefficients, etc. on SI field. In Section 4 the numerical and experimental studies of the SI field on two different classes of panels, Aluminium Foam Sandwich panels and flax-PE panels, are presented. In this section, a mixed approach, based on numerical and experimental data, is proposed to enhance the technique presented by Gavric and Pavic [5,6]. Finally, in Section 5 some concluding remarks are given.
auxiliary power units are some examples of external and internal sources of excitation; vibration isolation systems and structural joints are examples of energy dissipation mechanisms. In order to illustrate the capabilities of the SI and for the sake of simplicity, in this study only a pair of energy source and sink is considered in order to produce a well defined energy flow through the structure. From a general point of view, the structural power is defined as the active part associated with the vibration energy. In analytical terms, the power, at a prescribed point, is represented by product of a force with the in-phase component of the velocity in the direction of the force and can be calculated as follows:
Pðx; QÞ ¼
ð1Þ
where F is the applied force and v is the complex conjugate of the velocity; they both refer to a prescribed point, Q, in the structural domain Eq. (1); R denotes the real part [12,13]. The spatial distribution of the energy propagated within the structure is defined with the structural intensity field which, in the case of plate type structures, represents the energy propagating through the unit length. Hence, according to the principle of conservation of energy, the power injected to the plate can be deduced by integrating the normal component of the structural intensity vector (I),along a closed curve L, enclosing the vibrational source:
PðxÞ ¼
I
I n dl
ð2Þ
L
Vibrational power flow per unit cross-sectional area of a dynamically loaded plate is defined as the structural intensity, I denotes the normal vector) [14]. (n The total intensity in a thin vibrating plate is due to the combined action of shear, bending, and twisting waves (Fig. 1) and can be expressed with their orthogonal components as [15]:
Ix ¼
i W x h ~ ~ þ M x ~hy Mxy ~hx I Nx u þ Nxy v~ þ Q x w
ð3Þ
Iy ¼
i W x h ~ ~ þ Q yw ~ My ~hx þ Mxy ~hy I Ny v þ Nxy u
ð4Þ
2. Theoretical background on structural intensity Structural intensity (SI) is a vector field, described by magnitude and phase, which indicates the path and the amplitude of the mechanical energy flowing through a vibrating structural component [11]. The location, where the mechanical energy enters the structure, is generally called the energy source; the location, where the energy is extracted from the system, is called the energy sink. Multiple energy sources and sinks might coexist in the same structure. Aerodynamic loads, propulsion system, rotor and gearboxes and
1 R½Fðx; Q Þv ðx; Q Þ 2
I¼
2
2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi I2x þ I2y
m
m
ð5Þ
where N x ; N y and N xy are complex membrane and in-plane shear forces per unit width, Q y and Q z are complex transverse shear forces per unit width,
Fig. 1. Generalised forces per unit length and displacements in plate element.
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M y ; M z and M xy are complex bending and twisting moments per unit width, ~ are complex conjugate of translational ~; v ~ and w u displacements, hy and ~ hz are complex conjugate of rotational displacement ~ hx ; ~ along x, y and z directions, I denotes the imaginary part. All these quantities are dependent on the excitation frequency (x). It is also useful to note that SI in plates can be expressed as a power per unit width, [W/m].
3. Numerical models The first part of the work is devoted to the numerical models of plate structures in order to have a preliminary assessment about the SI fields and to analyse the influence of different parameters. The procedure, used in this work to simulate the SI field in a mechanical system, is illustrated as block diagram in Fig. 2. All test cases have been performed at frequencies close to the natural ones (i.e. at a frequency with higher energy levels compared to other ones), even if the method can be evaluated at any frequency. The procedure is here recalled: (1) Starting from the model sizes and properties, a FE model is created with the commercial finite element analysis code MSC/Patran and a frequency response analysis is performed in MSC/Nastran; in output from a modal approach, the sets of complex displacements and internal forces per unit length are requested to perform the SI computation. (2) These sets are obtained for a specific frequency close to a natural one and imported in a specific in-house Matlab script for SI calculation by using the Eqs. (3)–(5).
Fig. 3. Conventions and symbols for numerical plates.
Table 1 Steel properties. Material
E [GPA]
G [GPA]
m
q [kg=m3 ]
Steel
205.8
79.15
0.3
7800
(3) The results are post-processed by using two Matlab functions: the Quiver and the Imagesc. The first allows to build a vector map from the two intensity components, the second provides information about SI magnitude trough a colour bar.
Fig. 2. Basic scheme of the adopted procedure.
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Fig. 4. Effects on SI of constraint conditions.
In order to facilitate the reading of the figures, a letter is assigned to the edges of the plates, starting from letter, a of the vertical left side up to letter d clockwise. Moreover, the force is indicated as a black dot and the damper as a black triangle together with the constraints: this is illustrated in Fig. 3.
3.1. Influence of different parameters on an isotropic plate The first test structure is a rectangular steel plate, 1080 mm 1440 mm 6 mm. The material properties are reported in Table 1.
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Fig. 5. Effect on SI of load magnitude, damping ratio, structural damping and thickness.
Table 2 Composite graphite/epoxy laminated properties. Material
Ex [GPa]
Ey [GPa]
Gxy [GPa]
Gyz [GPa]
mxy
myz
q [kg=m3 ]
Graphite/epoxy
175.1
6.700
4.100
2.056
0.2542
0.4689
1520
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99
Fig. 6. Effect of the variation of fibres orientation.
A 2D finite element model using 4-nodes shell elements was built and the plate model was excited by a unit nodal force along the z direction. The coordinates of the node loaded are reported in the caption of Fig. 4. The energy sink was simulated trough a viscous damper element (CVISC) located at x ¼ 810 mm and y ¼ 180 mm. It was assigned a constant damping loss factor, g ¼ 0:007. The frequency response was calculated in the range 0–150 Hz and, except where differently expressed, all the analysis of the SI were performed at a specific frequency, f ¼ 20 Hz, close to the first natural frequency of this panel, f 1 ¼ 19:56 Hz. The first investigations concerned the influence of the boundary conditions on the SI field. Three different constraint conditions were considered along the plate sides: the simply supported conditions, in which all the translational degrees of freedom were blocked; the clamped ones, in which all the translational and rotational degrees of freedom were blocked, and the free ones. Fig. 4 contains the first results carried out for two different constraints and load conditions. It can be observed that: the transmission path is always from the excitation node to the viscous damper locations; the total intensity decreases approaching to constraints; in a plate with all the edges constrained, the intensity vectors are higher in the middle of the panel, as it can be seen from the contour plot of Fig. 4a and b and the vectors flow in this central region of the panel where the DOF are not restrained. In the case of unconstrained plates, instead, the opposite occurs (see Fig. 4c–f); in particular, it can be also noticed accumulation fringes along the free edges; finally, from these first investigations, it seems that the total intensity is related to the boundary conditions. In fact, comparing Fig. 4a and e, it is clear that for the panel in free–free
boundary conditions the intensity vectors are present in all the panel including the edges, while, for other types of boundary conditions, the restrained edges reduce such intensity vectors. The simply supported plate was imposed as reference plate. On this configuration, the effects of the variation of load magnitude, damping ratio, structural damping and thickness were analysed. In Fig. 5, comparing all figures with the first one, it can be stated that: the effect of the load magnitude, Fig. 5b, is not visible considering only the intensity vectors: the variation of the total intensity is of the order of 102 for a variation on load magnitude of the order of 10; by decreasing the value of the damper from 500 to 100 [N/(m s)], Fig. 5c, the area close to the damper is less intense; by changing the structural damping of the material by a factor 10, the SI field does not change near the excitation force, but it clearly decrease near the damper, in terms of total intensity and transmission path, Fig. 5d; the effect of the variation of thickness involves two different aspects: first, the thickness is inversely proportional to the total intensity; second, decreasing the thickness, the intensity field will be more extended, see Fig. 5e. 3.2. Influence of different parameters for an orthotropic plate The second series of tests refers to orthotropic plates. The sizes are: 6000 mm 3000 mm 7 mm. The plates were here subjected to in-plane uniform pressure excitations along their edges in view to show the different behaviour when they are subjected to a force in a direction parallel and orthogonal to the fibres. In fact, the
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Fig. 7. Effect of the variation of fibres orientation. Plates with two different stacking sequences.
Table 3 Dissipated power analysis: variation of structural damping ratio.
g
P in [mW]
P out [mW]
Diff. %
0 0.007 0.020 0.070
9.1 8.4 7.4 5.0
9.1 7.7 5.9 2.6
0 8.33 20.27 48.00
Table 4 Dissipated power analysis: g ¼ 0:007; f ¼ 20 Hz).
variation
with
the
damper
constant,
D
D [N/mm]
P in [mW]
P out [mW]
2 0.5 0.05
8.4 7.0 2.8
7.7 5.3 0.7
approach can be used also with only in-plane excitation, as shown in Eqs. (3) and (4). The material was epoxy resin reinforced with graphite fibres, which were oriented both in x and y directions, i.e. one is unidirectional along x-direction and the other one along y-direction. The main aim of this strongly directional material was to investigate the effect of the fibres orientation on the SI field. The material properties are listed in Table 2. In all plates, a uniform in-plane pressure per unit length with (at
magnitude p ¼ 103 [N/mm] was applied at a frequency of f ¼ 2 Hz, value close to the first natural one, which was f 1 ¼ 1:37 Hz. For these panels only two different constraint conditions were considered: simply supported and clamped conditions. For both the conditions the translation along the z direction was assumed restrained. The results are shown in Fig. 6. Considering the same conventions of Fig. 3, a uniform in-plane pressure have been applied along the d and c edges respectively for Fig. 6a–d.
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Fig. 8. Numerical–experimental basic scheme.
Table 5 Dimensions of AFS panels. Panel A B
Table 8 Power injected into AFS panels.
Width mm
Length mm
Total thick. mm
Skins thick. mm
Core thick. mm
656 656
476 476
10 8.6
1 0.6
8 7.4
AFS panels
P num [mW]
P exp [mW]
A B Equivalent A1
0.048 0.030 0.400
0.042 0.027 –
It is possible to make further considerations:
Table 6 AFS: mechanical properties. Type
E [GPa]
g
m
q mkg3
A Core
6.48
1103
0.31
600
B Core
3.43
1103
0.31
390
Skin
71
1104
0.33
2700
Table 7 FEA-EMA correlation for AFS panels. Mode Name
FEA-A Hz
EMA-A Hz
Diff-A %
FEA-B Hz
EMA-B Hz
Diff-B1 %
(1, (0, (1, (2, (2, (0, (1, (2, (3, (0, (3, (2, (1, (3,
122.83 145.11 288.11 288.22 357.99 427.27 543.70 588.72 771.86 801.72 853.77 889.56 922.37 1080.66
124.75 146.36 292.30 296.87 363.17 434.96 548.12 598.68 786.94 805.67 867.51 897.97 924.48 1096.71
1:53 0:86 1:43 2:91 1:43 1:77 0:81 1:66 1:92 0:49 1:58 0:94 0:23 1:46
107.92 127.51 253.16 253.29 314.58 375.49 477.78 517.34 678.37 704.57 750.36 781.78 810.62 949.81
110.99 126.96 259.47 255.11 313.31 374.99 483.06 513.87 681.99 699.08 751.02 774.01 808.62 948.18
2:76 0.44 2:43 0:71 0.40 0.13 1:09 0.68 0:53 0.78 0:09 1.00 0.25 0.17
1) 2) 2) 0) 1) 3) 3) 2) 0) 4) 1) 3) 4) 2)
the transmission path of the SI preferentially follows the direction along which the elastic modulus is greater and this is evident comparing Fig. 6a and b, or Fig. 6c and d: considering the direction of the greater stiffness, i.e. the horizontal or the vertical one, both the intensity vectors and the total intensity are more developed in that direction. applying a pressure load only along the longest edge, i.e. along the y-direction, the y-oriented fibres plate (Fig. 6b) presents a greater total intensity peak than the x-oriented one (Fig. 6a). The opposite occurs in the case of a pressure load only along the shorter edge; i.e. the x-oriented fibres plate (Fig. 6c) presents a greater total intensity peak than the y-oriented one (Fig. 6d). All these considerations are even more evident considering an idealised plate with two different stacking sequences in the same plane (i.e. the left and the right part of the panel have the fibres oriented in the x and y direction respectively), as shown in Fig. 7a. Assuming a pressure load (indicated as pn in Fig. 7, where p is the pressure load and n indicated the label of the edge) on the shortest edges, a comparison has been evaluated on two panels with reference to the variation of loading and boundary conditions. 3.3. Dissipated power analysis The power injected to the structure can be computed by using the velocity and the force measured at excitation point according to Eq. (1). This power is then dissipated in the structure.
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According to the principle of conservation of energy, the power injected to the plate can be deduced by integrating the component of the structural intensity vector, which is normal to a closed curve enclosing the vibrational source [16]. In this section, the influence of different parameters on the power was investigated and results are summarised in Tables 3 and 4. It is evident the influence of the damping on the output power (P out ) that is equal to the injected power in case of g ¼ 0. The output power is calculated as well as the input one (Eq. 1): the product of the forces of the constraint and the output velocity is evaluated at the location of the scalar grid where the damper is grounded, [16]. In all other cases, the dissipated power (P in P out ) increases with the damping. 4. Numerical and experimental investigations In this section numerical and experimental investigations on two different classes of panels, Aluminium Foam Sandwich (AFS) and flax-polyethylene (flax-PE) were performed in order to carry out information about their vibro-acoustic capabilities. The dynamic behaviour of these panels were already analysed in previous works and more information about their performance and more details about material composition and characterisation of the panels can be found in [17,18]. A brief description for each
class of panels is reported in the specific paragraphs. The numerical investigations were performed by adopting the flow chart of Fig. 2; the numerical–experimental ones, indeed, following the procedure reported in Fig. 8. This last procedure may be considered as a mixed numerical–experimental approach because forces per unit length were not experimentally measured, and so, in order to have an evaluation of the SI field, the used forces were carried out from the numerical model in which the experimental displacements were used as imposed conditions. The same experimental measurements were also used as post processing data for SI evaluations. This mixed approach tries to overcome some common experimental difficulties. The cinematic of a structure can be easily measured. On the contrary, the generalised forces per unit length cannot be carried out easily from experimental data. 4.1. Aluminium foam sandwich panel For this class of panels the effect of different percentage of porosity on structural intensity of two AFS panels, having same in-plane dimensions and average cell size of the foam bubbles, was investigated experimentally and numerically. Some more details on dimension and material properties of the investigated panels can be found in Tables 6 and 5.
Fig. 9. Numerical–experimental comparison for AFS A – panels.
Fig. 10. Numerical–experimental comparison for AFS B – panels.
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Fig. 11. Comparison AFS A – equivalent panel.
Table 9 flax-PE: mechanical properties of basic lamina. h
Ex [GPa]
Ey ¼ Ez [GPa]
Gxy ¼ Gxz [GPa]
Gyz [GPa]
mxy ¼ mxz ¼ myz
q
9.50
1.30
0.55
0.40
0.40
1025
Table 10 Power injected into flax-PE panels. Flax-PE panels
P num [mW]
P exp [mW]
Unidirectional Quasi-isotropic Equivalent flax-PE
0.016 0.002 0.200
0.014 0.001 –
Numerical predictions and experimental measurements of the modal parameters of AFS panels were performed in previous works [17]. Concerning the experimental measurements, an impact hammer was used to excite the structure and a set of accelerometers was used to measure the response of the panels that were suspended by means of bunjee chord in order to simulate free–free
kg m3
i
boundary conditions. Once the frequency response data was recorded using an acquisition system, the data were postprocessed by using a Matlab script. On the other hand, from the numerical point of view, a 3-D model was realised, modelling the face sheets using 4-nodes quadrilateral (CQUAD4) elements and the core using 8-nodes solid (CHEXA) elements. The accuracy of the numerical model was hence validated comparing the numerical results, in terms of natural frequencies and mode shapes, with the experimental ones [17]. In order to apply Eqs. 3 and 4, a second numerical model consisting of all shell layers was created:. In particular, the foam core was modelled as a layer with CQUAD4 elements and then the ‘‘full shell” layer model was validated in the same way, by comparing the natural frequencies and mode shapes with experimental ones. In Table 7 the comparison between the natural frequencies of the ‘‘full 2D” model and experimental ones is reported.
Fig. 12. Numerical–experimental comparison for unidirectional flax-PE panels.
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Fig. 13. Numerical–experimental comparison for quasi-isotropic flax-PE panels.
In the present analysis, before comparing the SI behaviour of different panels, the power injected into the plates was calculated and the results are shown in Table 8. For an evaluation of structural intensity, according to the procedure described in (Fig. 8), the complex displacements were carried out from experimental measurements while the forces per unit length were obtained by FE model and both were replaced in the Eq. 5, implemented in a Matlab code. Numerical and experimental results for both AFS panels (A and B), in terms of intensity vectors and total intensity were carried out for a frequency, f ¼ 100 Hz, close to the first natural one and they are depicted in Figs. 9 and 10. Results show a good correlation, and it can be noted that the total intensity of panel A is almost 10 times lower than that of panel B. This could be due by the lower elastic core properties and lower total thickness of the panel B. A further numerical analysis was performed on the equivalent (i.e. same weight) isotropic panel of the panel A, with the same properties of AFS skins (Fig. 6). From the results reported in Fig. 11, the total maximum structural intensity of the equivalent panel is almost 10 times higher than panel A. This is mainly due to the presence of the foam material that, as known, is a good absorption material. 4.2. Flax-PE panel Also for the flax-PE, two different panels, having same dimension 706 mm 492 mm 3.6 mm, but with different stacking sequences, respectively unidirectional ½0 8 and quasi-isotropic ½0 =90 =45 = 45 s panels, were performed. The lamina properties are listed in Table 9. Also for this class of panels the dynamic behaviour, in terms of modal parameters, were already investigated in previous works [18] and the accuracy of the numerical model was already validated comparing the numerical results, in terms of natural frequencies and mode shapes, with the experimental ones. Furthermore, as for the AFS panels, the power injected into the plates was calculated and they are reported in Table 10. The same procedure, made in the previous analyses for the AFS panels, was adopted and the structural intensity behaviour of these panels was carried out for a frequency (f ¼ 7:5 Hz) close to the first natural one. Comparing Figs. 12 and 13, the quasi-isotropic panel has a maximum structural intensity higher than that of the unidirectional one. In addition, as carried out in Section 3, it can be noted that the SI transmission path follows the direction along which the elastic modulus is greater.
5. Conclusions In this paper a technique, based on the finite element method, for estimating the structural intensity of plates has been presented. First, in order to validate the technique, numerical analysis are performed and results are compared with some reference ones, already present in literature [15]. Moreover, in view to get more information, a sensitivity analysis on the influence of different parameters (boundary conditions, load and damper magnitudes, structural damping, thickness and fibre orientation) on the structural intensity field of a panel is performed. The results of the test problems indicate that the method is a valid tool for predicting the structural intensity field of a dynamically excited system at relatively low frequencies. In this paper, the results for the test problems are discussed at resonances and they are very accurate. The use of FEA, to calculate structural intensity, is accurate and economical for the first modes of a mechanical system. This accuracy is strongly dependant on the mesh model size, as expected. This technique is applied to two different classes of panels: AFS and flax-PE panels and the obtained results show a good correlation between numerical and numerical/experimental data: the experimental measurements of the forces were not available and hence the numerical ones were used. Moreover, the technique offers information of energy transmission paths and positions of sources and sinks of mechanical energy that can be suitable for designer to localise the position of application of a damper where the level of vibrations exceeds.
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