Numerical study of the eigenfrequencies of dry and submerged
rectangular plates
Andreas S¨
ather
Abstract
This study is an investigation of how accurately fluid-structure interaction can be modeled using a solid-acoustic model, letting an acoustic medium represent the fluid. The modeling results are compared with test results from a prevouis study on natural frequencies of submerged plates. 20 plate specimens are modeled that has the same material and physical properties as the tested plates. These plates are made of CFRP, GFRP, steel and aluminum. The results show very good agreement between numerical and experimental results with a difference of 0-3.1 % and a average difference of 0.9 %, for the first natural frequency. The results show that this type of modeling can serve as a refined analysis method for design work performed within the maritime industry.
1
Introduction
When designing components or structures that are subjected to dynamic loads it is of critical impor-tance that the natural frequencies are considered. In marine applications this means that one has to include the effect that water has on said frequency. The added mass of the water will lower the frequen-cies of any structure and it is important to know the size of this frequency reduction, so that it can be accounted for during design and modeling. It has been shown in a previous study that the method to compensate for this, by doubling the mass of the material used, is reasonably accurate when design-ing with steel but is a poor approximation when us-ing other materials, especially composites [1]. This method also fails to take geometry into considera-tion. The interest in composites, within the mar-itime industry, are increasing meaning that there is a greater need to account for their properties and behavior in the design work [2] [3].
The aim of this study is to see how well the
nat-ural frequencies of plates submerged in water, can be calculated, if the water is simplified as an acous-tic medium instead of a fluid. This will be done by simulations where a model stepwise is calibrated so that the difference between modeled and measured values can, in large, be attributed to the simplifi-cation of water as an acoustic medium. The results are compared to experimental data from a study on eigenfrequencies of dry and submerged plates under forced vibrations [1].
If successful, this approach give designers a re-fined analysis method and improve accuracy with-out the need to model fluid solid interaction by the use of CFD + FEM. Instead of calculations that take days using HPC (High Performance Comput-ing) the analysis can be done in a matter of minutes using a laptop or desktop computer.
ter-minology is used do describe what states they are meaning to represent or be compared to. (This is done for the purpose of making the report easier to understand.)
2
Method
The presented work consists mainly of a series of simulations made to produce results that can be compared with experimental data from the bench-mark study [1]. The structure of this work is out-lined in the flow chart, figure 1. In this chart the left hand side represents the physical tests or mea-surements and the right hand side represents the simulations performed to attain the results for com-parison. On the physical side, left, the majority of the work has been performed and presented in the previous paper [1], and the focus of this paper is mainly on the simulation part of the work. This chart also show the flow of different properties be-tween the different parts of the work.
The simulations are performed using COMSOL Multiphysics R 4.4. All models are meshed using
tetrahedral solid elements with second order shape functions.
Figure 2: Model of baseline plate used to evaluate
Young’s modulus.
The work can be divided into three distinct steps. The first is to evaluate the stiffness modulus for the
individual specimens. This is done, through free vibration tests and using FE-models of the plates, see figure 2, to ensure that the bending stiffness of the plates are correct in the following steps.
The second step is to calibrate the rotational in-ertia of the second model so that it can reproduce the results from the dry vibration tests presented in the previous paper. This is done by varying the density of the modeled clamp, thereby altering it’s rotational inertia. This model, see figure 3, is built to mimic the test specimens, and the moving part of the rig that works as a clamp and holds the spec-imens, in the experimental study, see figure 4.
The third, and final, step is to take the calibrated model and add an acoustic domain representing the surrounding water, figure 5. This calculation makes use of the stiffness modulus and clamp den-sity calculated in the previous steps and the results are compared to the measured values from the sub-merged tests in the benchmark study.[1]
Figure 3: Model of the test clamp and plate specimen.
The dimensions, density and material of the indi-vidual specimens are obtained from the preceding test study [1]. They are presented in appendix A.
2.1
Assumption and simplifications
in the simulations
Specimen measurements*
Model
Impulse test
Evalua6on of
Young’s modulus
Dry vibra6on test*
Dry natural frequency
with op6miza6on
Wet vibra6on test*
Wet natural frequency
L,b,t,m
f
impulseE
f
dryρ
AlComparison
f
wet/f
dryf
wet/f
dryWater proper6es measurements ρwater vsound,water
Figure 4: The actual test rig used in the physical test-ing.
Figure 5: Model of the clamp-plate submerged in water i.e. surrounded by the acoustic domain. Clamp-plate encircled in white.
• As described above, under section 2, the model of the clamp has a simplified geometry con-sisting of solid rectangular boxes and cylinders without any gaps or holes.
• The tested composite plates are anisotropic and heterogeneous which means that their properties have a directional dependence. For this investigation they are treated as homoge-nous and isotropic materials. The Poisson’s ratio for the specimens as well as for the clamp material are not measured but assumed to be 0.3.
• As stated in the introduction, the water is modeled as a acoustic medium which implies that fluid dynamics and viscous dissipation are ignored.
2.2
Evaluation of Young’s modulus
To determine the Young’s modulus, for the individ-ual test specimens, a free-vibration test is made. By suspending the plate from two lightweight strings the boundary conditions of free bound-aries are approximated. Fitting the plate with ac-celerometers and knocking on it with a impulse hammer, with an incorporated load cell, means that the plates response to the impulse can be recorded, as a function of time, see figure 6.
The plates vibrations will be dominated by cer-tain distinct frequencies, i.e. the natural frequen-cies. By the use of signal processing the time de-pendent impulse data can be transformed into a fre-quency dependent response from which the natural frequencies can be acquired for the given specimen, see figure 7.
The measured frequencies are used for further evaluation of the specimens stiffness.
2.2.1 Analytical evaluation
Figure 6: Impulse recorded from a vibration test
Figure 7: Frequency response, for the recorded impulse shown in figure 6, showing peaks for natural frequencies.
is approximated as a beam, equation (1) can be used to calculate the natural frequency [4].
ωn= A s
EI
µL4 (1)
Where ω is the angular frequency, E is the Young’s modulus, I is the area moment of iner-tia, L is the length of the beam, µ is the mass per unit length of the beam and A is a factor that de-pends on the boundary conditions. For ”free-free” conditions A can be seen in table 1 [4]. After re-arranging equation 1, this is used to calculate the stiffness moduli for the individual plate specimens.
Table 1: Values for factor A for the first three modes.
First mode Second mode Third mode
22,4 61,7 121
The result from these calculations are used to get a good approximation of the moduli and serves as a starting value for the FEM evaluation.
2.2.2 Evaluation using FEM
To refine the value from analytical calculation made previously, and to take into account the presence of the drilled holes, an evaluation is made with the use of FEM. The model (see figure 2) consists of a lin-ear elastic material and the geometrical properties and the material density of the individual speci-mens are taken from the reference study [1]. For this calculation only solid mechanics are considered (no acoustic medium present).
Optimization: To find the sought after stiffness modulus an optimization is performed. The objec-tive is to minimize the difference between measured frequency, fimpulse, from the free-vibration test in section 2.2, and the frequency calculated by the model, fmodel. The objective function is written as:
(fimpulse− fmodel)2 (2)
This is done by varying the normalized stiffness modulus of the material profile. The initial stiff-ness modulus, which is obtained from the analyti-cal evaluation, is multiplied with a factor n, which has a starting value of 1. Varying the stiffness is done by varying the factor n. The normaliza-tion of the modulus serves to prevent the algorithm from exiting the optimization loop prematurely, for numerical reasons. To solve the optimization the BOBYQA algorithm is used with a convergence criteria of 0.01. BOBYQA stands for ”Bound Op-timization BY Quadratic Aproximation” and is a derivative free algorithm [5].
The resulting Young’s modulus, from these cal-culations, for the individual specimens can be found in table 4.
From these results it can be seen that, for the CFRP plates, there is a thickness dependence for the modulus. This is due to the stacking sequence of the laminate. In bending, there is a sequence dependence for the stiffness. If the zero degree lay-ers are placed in the outer surfaces of the laminate the bending stiffness will be higher than if they are placed in the center. The plates used in the bench-mark study are made up of carbon fibre mats with a surface weight of 1000 g/m2. These mats consists of a number of layers in [0 -45 90 45] directions with 400 g/m2in the 0 direction and 200 g/m2in each of the other directions. Each layer, or ”Quad”, builds approximately 1 mm in thickness i.e. the 4 mm plates consists of four layers, the 6 mm of six and so on. Al these mats are stacked so that the 0-layers are closest to the center of the plate, see table 2.
Quad+ 45 200 g/m2 90 200 g/m2 -45 200 g/m2 0 400 g/m2 Quad+ 45 200 g/m2 90 200 g/m2 -45 200 g/m2 0 400 g/m2 Neutral axis Quad-0 400 g/m2 -45 200 g/m2 90 200 g/m2 45 200 g/m2 Quad-0 400 g/m2 -45 200 g/m2 90 200 g/m2 45 200 g/m2
For the thicker plates more ”Quads”, as they are referred to in table 2, are added on the positive and negative side respectively, with the same ori-entation. Thicker plate means more layers which means more 0-layers further from the neutral axis resulting in a higher stiffness. This means that the measured modulus, in the case of the composites, are only valid in bending but since the aim of this study is natural frequencies which takes place in bending this means that, for this study, it is valid and serves it’s purpose.
Mesh sensitivity study To determine that the mesh is refined enough to model the frequencies for the plates accurately a test is performed on how the number of elements effect the resulting frequencies. This is done using the baseline model. The mesh density is stepwise taken from an extremely coarse mesh to a finer resulting in a increase in the num-ber of elements used. The numnum-ber of elements and output frequencies are recorded and plotted for the first frequency. The data is presented in figure 8.
It can be observed that variations in the number of elements used have very little effect on calcu-lated frequency. For the first frequency there is a difference of 1.4 % between the first and the sec-ond calculation, going from 295 to 659 elements. After that the difference between the data points falls under 0.25 %, for the first frequency. The pre-ceding stiffness calculations are made with the finer mesh density resulting in 17142 elements (the high-est number of elements in figure 8) meaning that it is refined enough to produce good results.
2.3
Dry natural frequency
This is done to simulate the frequency that was measured by vibration in air, in the preceding study. This is strictly a solid mechanics model. The model consists of two different material pro-files. One for the part of the model that represents the clamp and another for the test specimen. Both
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 104 90 95 100 105 110 115 120 Number of elements Natural frequency [Hz]
Figure 8: Graph showing modeled frequency dependence of number of elements for the free vibration test. Green dot represents chosen mesh density.
Boundary conditions The outer surfaces of the model are set as free boundaries and initial values for displacement and structural velocity are set to zero (u = 0,∂u∂t = 0) for all the domains of the clamp and specimen model. To make sure that the model rotates around the right axis a condition of prescribed displacement is applied on the centre axis of the outer cylinders of the model (see figure 9).
Figure 9: White line in the center of the cylinder rep-resents axis of rotation.
These axis are set to have zero displacement in the x and z direction (u = 0, w = 0), making the y axis the axis of rotation. To make sure the clamp and plate in the model does not move around the ”tank” but stay fixed in one position a second con-dition of prescribed displacement is also set on the nodes depicted in figure 10. In this case the dis-placement in the y direction was set to zero (v = 0).
Calibration of dry frequency As in the previ-ous optimization, done to find the stiffness modu-lus, the goal is to minimize the difference between the measured frequency, ftest, and the modeled fre-quency, fmodel. For this optimization the objective function is written as:
(ftest− fmodel)2 (3)
Figure 10: Nodes, circled in white, with prescribed dis-placement which restricts the model from movement in the y direction
In this case this is done by varying the density of the material profile used for the clamp. The com-bined weight and volume of the aluminum profiles, the steel bolts, the washers and the part of the plate held inside the clamp will vary from test to test. If we see all these parts as one entity the overall den-sity of this will vary from test to test. The resulting density of this optimization is meant to represent this overall density. The BOBYQA algorithm is used with a convergence criteria of 0.01 to perform this optimization [5].
The resulting clamp densities from this optimiza-tion are presented in table 4.
Mesh sensitivity study An investigation of how the mesh density effects the results is per-formed to ensure the results are converging. The mesh density is taken from an extremely coarse set-ting to extra fine resulset-ting in a number of elements, for the camp and plate, going from 273 to 12323. The results are presented in figure 11.
0 0.5 1 1.5 2 2.5 x 104 75 75.5 76 76.5 77 77.5 78 78.5 79 79.5 80 Number of elements
First natural frequency [Hz]
With tank present without tank
Figure 11: Graph showing modeled frequency as a func-tion of the number of elements in the model, for the dry vibration test. Green dot represents chosen mesh den-sity.
air includes the acoustic domain of the tank. This presence has no direct influence on the calculations since only solid mechanics is considered. However, this mesh sensitivity study has shown a indirect effect of this presence. How the elements are dis-tributed in the model differs between a case where the acoustic domain are present and a case where it isn’t. In figure 12 this difference can be seen. Fig-ure 12a shows meshing done with acoustic domain present and a setting resulting in 1339 elements. Figure 12b shows meshing done without the acous-tic domain present and a setting resulting in 854 elements. It is clear to see that the meshing done without the presence of the acoustic domain results in a more evenly distributed and more refined mesh for the plate. This will produce better, or more ac-curate, results for the bending of the plate.
A second test is therefore done, where the mesh density is varied in the same way as described above, but without the acoustic domain present. The number of elements are varied from 318 to 22261 elements and the resulting frequencies are recorded and presented in figure 11. There is no fluctuations seen in these results indicating that the
source of this was indeed the element distribution in the mesh, effected by the presence of the acous-tic domain. These effects are avoided by increasing the mesh density.
(a) (b)
Figure 12: (a) describes the output mesh, for a normal setting, with the tank present. (b) describes the out-put mesh, for the extra course setting, without the tank present.
The modeling and the calibration are performed, for the 19 plate specimens, with the mesh density marked in green in figure 11 resulting in 12323 ele-ments for the clamp and plate. They are all meshed with the acoustic domain present. This is done so that the model is calibrated using the same mesh that will be used for the acoustic-solid interaction that will be performed in the in next step.
2.4
Wet natural frequency
The model depicted in figure 5 contain the same clamp-plate model, as in the ”dry” case, but now an acoustic domain is also present representing the surrounding water tank. This is a multi physics model with solid mechanics-acoustics interaction. The density, for the acoustic medium, is 1002 kg/m3and the speed of sound, is 1481 m/s. These properties was measured in the water at the time the submerged vibration test was performed. Prop-erties for the clamp-plate domain are the same as in section 2.3.
are largely the same as in the dry case except for the free boundaries. These are now replaced by an acoustic-structure boundary which connects the solid mechanics to the pressure-acoustics-domain in the surrounding tank. On the top surface of the tank (the blue side in figure 14) we have a condi-tion of zero pressure and the remaining five sides are set as sound hard boundaries (the grey sides in figure 14).
Mesh sensitivity study A test is performed to ensure that the mesh density is refined enough for the acoustic-solid simulation. The mesh density is stepwise taken from a extremely course mesh set-ting to a fine, for the acoustic domain while keep-ing the mesh density of the clamp-plate domain constant. This gives a total number of elements, for the entire model, ranging from 33434 to 85070. The number of elements and resulting frequency are recorded for each step and the results are shown in figure 13. This shows very small variations in cal-culated frequency which never exeeds 0.1 %. These small variations are due to the fact size of the ele-ments, in the acoustic domain, are governed by the mesh density of the clamp-plate domain which is kept constant. The calculations leading to the re-sults presented in table 4 is performed at the fine mesh density, with 85070 elements, marked green in figure 13.
A minimum of 8 elements per wavelength is also preferable to accurately model the frequencies in the acoustic domain. This can be evaluated using equation 4.
f = v
λ (4)
Where f is the frequency, v is the wave speed and λ is the wave length. The maximum element length in the fine setting, used for the calculation of the wet frequency, is 0.48 m. Using equation 4 shows that with this setting the model should be able to accurately calculate frequencies up to 385
Hz. As can be seen in table 4 this is higher then all the calculated frequencies.
2 3 4 5 6 7 8 x 104 35 35.5 36 36.5 37 37.5 38 38.5 39 39.5 40 Number of elements
First natural frequency [Hz]
Figure 13: Graph showing resulting number of elements and frequency for variations in mesh density. Green dot represents chosen mesh density.
3
Complex geometry
Figure 14: Picture of the FE-model. Top surface, in-dicated by the white arrow, represents the zero pres-sure boundary and the remaining five surfaces repre-sents sound hard boundaries.
Figure 15: Model of the plate with a curved geometry used for free vibration simulation.
Figure 16: Figure of the mode for the first natural fre-quency in free vibration.
3.1
Free vibration test
An evaluation of the natural frequency trough a free vibration test utilizing the same setup and sig-nal processing, already described in section 2.2, is performed. This gives that the first natural fre-quency is 120.7 Hz , the second is 281.1 Hz and the third is 352.6 Hz.
3.2
Evaluation of material
proper-ties
Using laminate theory [6] and the material prop-erties presented in table 3 the bending stiffness and average laminate properties can be calculated. The material properties, presented in table 3, come from a material with the same kind of fibre as the one used in the experimental study [7]. The bending stiffness is evaluated through the D-matix, where D11describes the bending stiffness in the x-direction and D22in the y-direction. This gives the result D11= 845 N m, D22= 776 N m. The shear modulus is also calculated to be Gxy = 14 GP a. This means that the bending stiffness, per unit width, and thereby the Young’s modulus, trough equation (5), should have a relationship where E2 is approximately 92% of E1, even though there is twice the amount of fibres in the 1-direction. This can be explained by the stacking sequence al-ready discussed in section 2.2.2. Since E for the same material, in section 2.2, showed an average of 43.21 GPa it is reasonable to expect E1 for the orthotropic material be similar in magnitude. In equation 5, D is the bending stiffness per unit width, E is the Young’s modulus and ν is the Pois-son’s ratio.
D = Eh 3
[40, 40, 15] GPa, ¯G = [14, 5, 5] GPa. These values was then varied to see how they effected the mod-eled frequencies. From this study it’s evident that G2, G3 and E3had very little effect on the results. By leaving them at starting values and focusing on E1, E2 and G1 a solution could be found. The result from this parametric study shows that set-ting ¯E = [39.4, 38, 15] GPa, ¯G = [13.7, 5, 5] GPa gives the first three frequencies as f1 = 120.7 Hz, f2= 281.0 Hz and f3= 355.7 Hz.
Material property Value
E1 107 GPa
E2 15 GPa
G12 4.3 GPa
ν12 0.3
t 0.193 mm1
Table 3: Material properties for a UD lamina used to calculate laminate properties.
3.3
Dry natural frequency
To simulate the dry vibration test made for this geometry and to calibrate the model for the next step a simulation is performed in the same manor as described in section 2.3. A model similar to the one used in that simulation is built but with a curved geometry, figure 17. The specimen ma-terial is modeled orthotropic with values from the evaluation described in 3.2.
A optimization is performed to find the overall density of the clamp and to calibrate the model. This is performed using the same objective, algo-rithm and convergence criteria described in section 2.3. The result for this optimization can is pre-sented in table 4.
1This value is calculated from the average laminate
thick-ness of the 5 first specimens devided by the number of layers in the stacking sequence.
Figure 17: Figure of the model used to simulate the dry vibration test of the sample with complex geometry.
Boundary conditions This simulation is per-formed using the same boundary conditions as in the simulation described in section 2.3.
3.4
Wet natural frequency
A solid mechanics-acoustics interaction calculation is performed to simulate the submerged test. This is performed using the same set-up as in the simula-tion described in secsimula-tion 2.4 but with the addisimula-tion off the orthotropic material in the test specimen. The result from this simulation is presented in ta-ble 4.
Boundary conditions This simulation is per-formed using the same boundary conditions as in the simulation described in section 2.4.
4
Results
0 20 40 60 80 100 120 0 10 20 30 40 50 60 70 80 90
Natural frequency dry [Hz]
Natural frequency wet [Hz]
Test Model
5
Discusion
The results from this study shows excellent agree-ment between measured and modeled results. The quality of the result depends on how well the ge-ometry and the material properties is represented in the simulation, i.e. if the dry frequency can be modeled with accuracy then so can the wet fre-quency.
Variations in the evaluated Young’s modulus and the clamp density can be seen. Theoretically, plates with the same width and thickness should have the same clamp density. The fact is that they don’t have the same dimensions. Even if plates are made of the same material and cut from the same plate there are variations. The measured thickness, for instance varies between 5.75-5.85 mm for the CFRP plates specified as 6 mm. In bending, the stiff-ness varies dramatically with thickstiff-ness and even a change of a few hundreds of a millimeter can have a big effect on the outcome of these calculations. This is also the reason why the curved plate has a lower Young’s modulus in the 1-direction then the rectan-gular plates with the same thickness. In that model the geometry is imported from a CAD file, setting the thickness to 6 mm. For the calculations made with the rectangular plates the measured thickness is used.
There is still a geometrical difference between modeled and actual clamp. This is a source of error but since the dimensions of the modeled clamp are from measurements of the actual clamp the differ-ence in effected area is very small compared to the area of the plate specimens.
The simplification that the composite materials are isotopic, done for the first 19 specimens, does not present problems, in this study. This is due to the fact that the first frequency bending mode, for these specimens, are just that, pure bending. For other bending modes, including any torsional components, the anisotropic nature of fiber com-posites would probably lead to larger differences
between modeled and measured values. This is why the curved specimen in section 3 was modeled as a orthotropic material. This is still a simplification of the plate material but a better representation then the isotropic material used for the rectangu-lar specimens. However, this study only aimed to evaluate modeling water as an acoustic medium. This means that models built for design work may need to include the directional dependence for the composite material properties to produce reliable results.
6
Conclusions
References
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[3] Russell C., Composites: long-term viabil-ity and benefits, Reinforeced plastics, 0034-3617/05, Oktober 2005
[4] Cyril M. Harris, Allan G. Piersol, Harris’ Shock and Vibration Handbook , McGraw-Hill, Fifth edition 2002
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