Delft University of Technology
Circumferential and radial lamina application for natural frequencies problems
Capacia, VIctor; de Almeida, Sergio Frascino Müller; Castro, Saullo G.P. DOI
10.2514/6.2021-0570
Publication date 2021
Document Version Final published version Published in
AIAA Scitech 2021 Forum
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Capacia, VI., de Almeida, S. F. M., & Castro, S. G. P. (2021). Circumferential and radial lamina application for natural frequencies problems. In AIAA Scitech 2021 Forum: 11–15 & 19–21 January 2021Virtual/online event [AIAA 2021-0570] American Institute of Aeronautics and Astronautics Inc. (AIAA).
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Circumferential and radial lamina application for natural
frequencies problems
Victor N. Capacia∗and Sergio Frascino M. de Almeida†
Department of Mechatronics and Mechanical Systems, University of Sao Paulo Polytechnic School, 05508-010, Sao Paulo, SP, Brazil
Saullo Giovani Pereira Castro‡
Faculty of Aerospace Engineering, Delft University of Technology, 2629 HS Delft, Netherlands
There is a great demand for optimization of high performance composite structures, partic-ularly in the aerospace sector. To avoid vibrational resonance in aerospace and naval structures, laminated plates can be designed using fundamental frequency constraints. Composite panels are particularly attractive because the vibration response can be optimised by tailoring the fibre angles of different layers without incurring weight penalties. A viable alternative is to use circumferential and radial layers for optimizing fundamental frequencies. This work focuses on the application these two types of theoretical layers and investigating its efficiency, for plates with and without cutouts.
I. Nomenclature
𝐾 = global stiffness matrix 𝜆 = natural frequency squared
𝛼 = mode shape normalized for unit modal mass 𝑎 = plate length in the x direction
𝑏 = plate length in the y direction nx = number of elements in the x direction ny = number of elements in the y direction
II. Introduction
To avoid vibrational resonance in aerospace and naval structures, laminated plates are usually designed for maximum fundamental frequency constraints. Composite panels are particularly attractive because the vibration response can be optimised by tailoring the fibre angles of different layers without incurring weight penalties [1].
Lamination parameters and classical lamination theory are used in [1] to maximize natural frequencies. They also develop a discrete optimality criteria based on a new generalisation of the reciproval appoximation in order to create a convex local minimization problem. For a aspect ratio (a/b) of 1 on a simply supported plate, they were able to maximize the first natural frequency in 8%.
The concept of layerwise optimization approach (LOA) is explored in [3] in order to optimize vibration behavior for maximum natural frequency of laminate composite plates. The author explores the physical consideration that the outer layer has more stiffening effect than the inner layer in the bending of plates and is more influential in determining the natural frequency. Examples are give for different aspect ratios and boundary conditions.
A frequency analysis is performed in [4] using a close form formula for thin rectangular laminates to maximize fundamental frequency. His application of specific equation to composite plates indicated that the optimal fundamental frequency parameter is a strong function of plate’s aspect ratio and composite material.
The eigenvalue problem for vibration is defined by Eq. (1).
∗Structures Engineer, EMBRAER, PhD student at University of Sao Paulo Polytechnic School, †Department of Mechatronics and Mechanical Systems, University of Sao Paulo Polytechnic School ‡Assistant Professor at Delft University of Technology, Department of Aerospace Structures and Materials
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(𝐾 − 𝜆𝑀).𝛼 = 0 (1) where K is the global stiffness matrix, M is the global mass matrix, 𝜆 = 𝜔2where 𝜔 is the natural frequency, and a is the mode shape normalized for unit modal mass such that:
𝛼. 𝑀 .𝛼=1 (2)
The global mass and stiffness matrices are obtained by assembling element matrices.
For this investigation, this eigenvalue problem is solve by Nastran®[5] software and compared in order to develop deeper insight.
The fundamental frequency performance of a composite plate is investigated. This study makes use of circumferential and radial angled lamina as means to validate its efficiency to increase performance of two plate geometries, with and without a cut-out.
III. Circumferential and radial angled ply
Advances in manufacturing techniques of composite materials through the past decades have extended the design possibilities regarding their application in lightweight structures. Novel techniques such as the automated fiber placement (AFP) and constant tow shearing (CTS) allow the fibers to follow curvilinear paths, creating laminate properties that vary within the laminate plane. These type of laminates are known as variable stiffness laminates and new designs can be explored as higher tailorability and more design space are available due to their improved in-plane behavior [6].
In a macro scale point of view the theoretical circumferential or radial angled lamina behaves like a variable stiffness composite. However locally, it can be modeled just like a uniform stiffness lamina. Fig. (1) depicts an example of a single-ply finite element model laminate with 100 elements representing a circumferential and radial angled lamina.
Fig. 1 Theoretical circumferential and radial angled ply representation
As seen in Fig. (1), each element used to model the laminate has a unidirectional orientation. The discretization of plate directions will be more detailed depending on mesh refinement. Therefore, each element stiffness is described by the classical ABD matrix.
IV. Modeling
For the investigation of maximization of natural frequency, circumferential an radial lamina are proposed. For such, two geometries are implemented.
First a simply supported plate with aspect ratio (a/b) of 1 and without cut-out is analyzed. For this case, an 8-layered symmetrical composite laminate is analyzed within all possible combinations of 0◦, 90◦, +45◦, −45◦, C and R laminae, in a total of 1296 laminates. Where C means the circumferential ply and R, the radial.
Second, a simply supported plate with cut-out and an aspect ratio (a/b) of 1 is implemented. Thus, an 8-layered symmetrical composite laminate is analyzed within all possible combinations of 0◦, 90◦, +45◦, −45◦, C and R laminae, in a total of 1296 laminates. Where C means the circumferential ply and R, radial.
Fig. 2 Geometries and contour conditions of investigated plates
Third, the use of three types of doubler are going to be investigated. One of then near the hole (C1) with 17% of the total area, another close to de edge of the plate (C3) with 35.7% of total area and an intermediary one (C2) with 47.3% of total area, as shown in Fig. (3). For this third study, a simply supported plate with cut-out and an aspect ratio (a/b) of 1 is implemented.
Fig. 3 Doubler types and geometry
For means of comparison, first, all possible combination of 8-layered symmetric composite plate (0◦, 90◦, +45◦, −45◦, C and R) are analyzed and compared with 8-layered symmetric unidirectional composite plates with the doubler addition in every possible position, as shown in Fig. (4), in a total of 5136 laminates. Afterwards, all 10-layered unidirectional symmetric laminates are included in the comparison, forming a total of 6160 laminates.
Fig. 4 Doubler types and geometry
A. Material
The material used for the plate is a carbon-epoxy tape. The summary of its mechanical properties is listed on Table 1. The same holds values for material density and reference thickness of lamina. Finally, its allowable values of tension are depicted.
Table 1 Material properties
Material properties Values
Young’s modulus in direction 1, 𝐸1 - tension 138 GPa Young’s modulus in direction 2, 𝐸2 8.96 GPa Shear modulus, G12 4825 MPa Shear modulus, G13 4825 MPa Shear modulus, G23 4825 MPa
Poisson’s ratio, 𝜈12 0.31 Material density, 𝜌 1.6e-6 𝑘𝑔/𝑚𝑚3
Layer thickness 0.1524 mm Allowable strain in direction x - tension 10500e-6 Allowable strain in direction x - compression 10500e-6 Allowable strain in direction y - tension 5800e-6 Allowable strain in direction y - compression 23000e-6
Allowable shear strain 13100e-6
B. Mesh Convergence Analysis for plate without cut-out
A mesh refinement study is performed to ensure the finite element model discretization is able to accurately represent the simply supported plate. Since it is a square plate, only quadratic elements are used with the same number of elements in both directions. For this analysis a python routine is implemented in order to write, analyze and post-process data.
Fig. 5 Plate without cut-out mesh possibilities example
When modeling circumferential and radial lamina plies, the mesh refinement needs to be taken into consideration. More elements means more accurate representation of the actual angle distribution for those specific plies. A courser mesh leads to inconsistent values, at the same time that, a refined mesh may lead to a huge computational effort when analyzing hundreds of models.
Fig. 6 Convergence of Fundamental Frequency
A mesh convergence analysis is implemented in order to balance precision, computational effort and define the number of elements in the final model.
The first relation to consider is computational time against total number of elements. Fig. (6) illustrates this relation for an increasing number of mesh size.
Second, the convergence of the fundamental frequency is analyzed and Fig. (6) also depicts the results.
For this analysis, a symmetric quasi-isotropic laminate of 8 plies was considered. Taking into consideration the convergence of fundamental frequency and the required computational time, a mesh with nx = 40 and ny = 40 provides an reasonalbly accurate stiffness estimate while keeping a relatively low computational time.
C. Mesh Convergence Analysis for plate with cut-out
A similar study is implemented for the plate with cut-out. For this analysis, a python routine is also implemented in order to model nodes, elements geometry, analyze and post-process data. For this, a few parameters were established as variables. Table 2 resumes all model variables.
Table 2 Material properties
Variable name Meaning
a Length in x direction of plate b Length in y direction of plate
r Radius of hole
nx Number of elements in each plate edge c Number of circular rows of elements
cr Radius of circular elements
nf Number of elements between edge of plate and circular rows of elements
Fig. (7) shows an example of plate mesh and geometry with all variables depicted. For this example a=b=300mm and r=30mm.
Fig. 7 Plate and mesh variables example
Also, Fig. (8) depicts a few mesh and geometry possibilities that are going to be explored.
Fig. 8 Plate with cut-out mesh possibilities example
For this study, plate geometry will be fixed at a=b=300mm and r=30mm, while the convergence analysis will take place with all other variables.
First, initial values are established and a convergence analysis done by varying nx. Afterwards, three percentages of the area occupied by the circular elements are fixed (10%, 25% and 50%) and c is varied in another round of convergence check.
Fig. (9) shows an example with c=4, nf=2, nx=16 and varying cr in order to obtain the percentages above.
Fig. 9 Example of percentages of the area occupied by the circular elements
1. Nx variation
The initial values adopted are c=8, cr=10, nf=3. By varying nx while keeping other variables constant, Fig. (10) is obtained.
Fig. 10 Simulation time variation with relation to nx
For this analysis a symmetric quasi-isotropic laminate of 8 plies was considered. Taking into consideration the convergence of the fundamental frequency and the required computational time, a mesh with nx varying from 2 to 28 is analyzed.
Fig.(10) shows the simulation time varying from around 2.5 and 3 seconds, while the number of elements on each edge of plate is increased.
Further on, for the fundamental frequency convergence, it is possible to see that a nx around 15 can provide reasonable accuracy while keeping the simulation time relatively low.
For the next analysis it was decided to proceed with the value of nx=16.
2. Percentages of area analysis
For this analysis three percentages of circular elements areas are going to be fixed as shown in Fig.(9). Afterwards, c is varied consequently changing cr value. Nf is adjusted in order to keep element size as homogeneous as possible.
• 50%
Fig. (11) summarizes the variation of variable c causing an increase of simulation time. At the same time, the fundamental frequency also increases and stabilizes around 1.98Hz.
Fig. 11 Convergence analysis for 50% of circular elements area fixed
• 25%
Likewise for the 50% analysis, the simulation time responds to the increase of c in a lower rate in comparison to 25% analysis. Fig. (12) illustrates the results.
Fig. 12 Convergence analysis for 25% of circular elements area fixed
• 10%
Finally, fixing the circular elements onto 10% of total area, the variation of c does not affect the analysis in the same rate as it affects the simulation time. Fig. (13) depicts these results.
Fig. 13 Convergence analysis for 10% of circular elements area fixed
Therefore, focusing on optimizing computational cost the final values chosen for the plate and mesh are nx=16, c=4, cr=22.5mm, nf=3, a=300mm and r=30mm. Fig. (14) illustrates the final plate geometry and mesh.
Fig. 14 Final plate geometry and mesh
V. Results for simply supported plate without cut-out
For the simply supported plate shown in Fig. (2a), as previously discussed, an 8-layered symmetrical composite laminate is analyzed within all possible combinations of 0◦, 90◦, +45◦, −45◦, C and R laminae, in a total of 1296 laminates. For the results visualization a ranking is established based on the highest fundamental frequency obtained. Results and comparisons can be seen in Table 3.
Table 3 Frequency results for plate without cut-out Ranking Layup 𝜔1[Hz] 𝜔2[Hz] 𝜔3[Hz] 1 [𝐶, 45.0, −45.0, −45.0]𝑠 2.661083 4.971989 5.48676 2 [𝐶, −45.0, 45.0, 45.0]𝑠 2.661083 4.971989 5.48676 3 [𝐶, 45.0, −45.0, 𝑅]𝑠 2.657744 4.959167 5.516246 4 [𝐶, −45.0, 45.0, 𝑅]𝑠 2.657744 4.959167 5.516246 5 [𝐶, 45.0, −45.0, 𝐶]𝑠 2.655574 4.908529 5.479628 6 [𝐶, −45.0, 45.0, 𝐶]𝑠 2.655574 4.908529 5.479628 7 [𝐶, 45.0, −45.0, 45.0]𝑠 2.654764 4.908491 5.519374 8 [𝐶, −45.0, 45.0, −45.0]𝑠 2.654764 4.908491 5.519374 9 [𝐶, 45.0, −45.0, 0.0]𝑠 2.649537 4.930252 5.498998 10 [𝐶, 45.0, −45.0, 90.0]𝑠 2.649537 4.930252 5.498998 11 [𝐶, −45.0, 45.0, 0.0]𝑠 2.649537 4.930252 5.498998 12 [𝐶, −45.0, 45.0, 90.0]𝑠 2.649537 4.930252 5.498998 13 [𝐶, 45.0, 𝑅, −45.0]𝑠 2.627037 4.863311 5.667103 14 [𝐶, −45.0, 𝑅, 45.0]𝑠 2.627037 4.863311 5.667103 15 [𝐶, 45.0, 𝑅, 𝐶]𝑠 2.621314 4.795891 5.658865 16 [𝐶, −45.0, 𝑅, 𝐶]𝑠 2.621314 4.795891 5.658865 17 [45.0, −45.0, 𝐶, −45.0]𝑠 2.620819 5.59665 6.124774 18 [ − 45.0, 45.0, 𝐶, 45.0]𝑠 2.620819 5.59665 6.124774 19 [45.0, −45.0, 𝐶, 𝐶]𝑠 2.620789 5.550661 6.118843 20 [ − 45.0, 45.0, 𝐶, 𝐶]𝑠 2.620789 5.550661 6.118843 43 [45.0, −45.0, −45.0, −45.0]𝑠 2.61038 5.882147 6.159503 208 [45.0, −45.0, 0.0, 90.0]𝑠 2.52042 5.613099 6.237986
The layup that shows the highest fundamental frequency is a soft laminate with a circumferential lamina at the top and bottom. It represented an increase of 5.58% in relation to the best quasi-isotropic laminate, located in position 208. On position 43 comes the first unidirectional laminate with an increase of 3.56% in relation to the quasi-isotropic laminate. This same laminate was found by Narita (2003) as his best result for a plate with a/b=1 on a simply supported plate.
VI. Results for simply supported plate with cut-out
For the simply supported plate shown in Fig. (2b), as previously discussed, an 8-layered symmetrical composite laminate is analyzed within all possible combinations of 0◦, 90◦, +45◦, −45◦, C and R laminae, in a total of 1296 laminates. Results and comparisons can be seen in Table 4.
Table 4 Frequency results for plate with cut-out Ranking Layup 𝜔1[Hz] 𝜔2[Hz] 𝜔3[Hz] 1 [45.0, −45.0, −45.0, 𝑅]𝑠 2.573469 5.844511 6.130792 2 [ − 45.0, 45.0, 45.0, 𝑅]𝑠 2.573469 5.844511 6.130792 3 [45.0, −45.0, −45.0, −45.0]𝑠 2.572617 5.850642 6.110728 4 [ − 45.0, 45.0, 45.0, 45.0]𝑠 2.572617 5.850642 6.110728 5 [45.0, −45.0, −45.0, 45.0]𝑠 2.56777 5.81487 6.127191 6 [ − 45.0, 45.0, 45.0, −45.0]𝑠 2.56777 5.81487 6.127191 7 [45.0, −45.0, −45.0, 𝐶]𝑠 2.564774 5.821459 6.107454 8 [ − 45.0, 45.0, 45.0, 𝐶]𝑠 2.564774 5.821459 6.107454 9 [45.0, −45.0, 𝑅, −45.0]𝑠 2.56396 5.782715 6.230247 10 [ − 45.0, 45.0, 𝑅, 45.0]𝑠 2.56396 5.782715 6.230247 11 [45.0, −45.0, −45.0, 0.0]𝑠 2.562556 5.825659 6.117507 12 [45.0, −45.0, −45.0, 90.0]𝑠 2.562556 5.825659 6.117507 13 [ − 45.0, 45.0, 45.0, 0.0]𝑠 2.562556 5.825659 6.117507 14 [ − 45.0, 45.0, 45.0, 90.0]𝑠 2.562556 5.825659 6.117507 15 [45.0, −45.0, 𝑅, 𝑅]𝑠 2.560473 5.7685 6.243949 16 [ − 45.0, 45.0, 𝑅, 𝑅]𝑠 2.560473 5.7685 6.243949 17 [45.0, −45.0, 𝑅, 𝐶]𝑠 2.556333 5.753535 6.227134 18 [ − 45.0, 45.0, 𝑅, 𝐶]𝑠 2.556333 5.753535 6.227134 19 [45.0, −45.0, 𝑅, 45.0]𝑠 2.556095 5.74067 6.241439 20 [ − 45.0, 45.0, 𝑅, −45.0]𝑠 2.556095 5.74067 6.241439 67 [45.0, −45.0, 0.0, 90.0]𝑠 2.476461 5.591131 6.176574
For the plate with cut-out, the highest fundamental frequency was found in a soft laminate with a radial ply in the middle of the laminate, with an increase of 3.91% in relation to the best quasi-isotropic laminate.
Followed by a few soft unidirectional laminates and a soft laminate with a circumferential ply in the middle of the laminate.
VII. Results for simply suported plate with cut-out and circumferential doubler
In this section, at first, laminates with doublers are compared with all possible combination of a 8-layered symmetrical composites plate with the angles 0◦, 90◦, +45◦, −45◦, C and R, in a total of 5136 laminates.
Then, all 10-layered unidirectional symmetric laminates are included in the comparison, forming a total of 6160 laminates.
A. Doubler results compared with 8-layered unidirectional laminates
Table 5 depicts all doubler results in comparison to 8-layered symmetrical composite plates.
Table 5 Frequency results for plate with cut-out and doubler
Ranking Layup mass [kg] 𝜔1[Hz] 𝜔2[Hz] 𝜔3[Hz]
1 [45.0, −45.0, −45.0, −45.0, 𝐶3]𝑠 0.185244 2.933064 6.343534 6.645806 2 [ − 45.0, 45.0, 45.0, 45.0, 𝐶3]𝑠 0.185244 2.933064 6.343534 6.645806 3 [45.0, −45.0, −45.0, 45.0, 𝐶3]𝑠 0.185244 2.927315 6.31666 6.649718 4 [ − 45.0, 45.0, 45.0, −45.0, 𝐶3]𝑠 0.185244 2.927315 6.31666 6.649718 5 [45.0, −45.0, −45.0, 𝐶3, −45.0]𝑠 0.185244 2.923117 6.330408 6.61818 6 [ − 45.0, 45.0, 45.0, 𝐶3, 45.0]𝑠 0.185244 2.923117 6.330408 6.61818 7 [45.0, −45.0, −45.0, 𝐶3, 45.0]𝑠 0.185244 2.920267 6.296427 6.639245 8 [ − 45.0, 45.0, 45.0, 𝐶3, −45.0]𝑠 0.185244 2.920267 6.296427 6.639245 9 [45.0, −45.0, −45.0, 𝐶3, 0.0]𝑠 0.185244 2.915429 6.307725 6.627446 10 [45.0, −45.0, −45.0, 𝐶3, 90.0]𝑠 0.185244 2.915429 6.307725 6.627446 11 [ − 45.0, 45.0, 45.0, 𝐶3, 0.0]𝑠 0.185244 2.915429 6.307725 6.627446 12 [ − 45.0, 45.0, 45.0, 𝐶3, 90.0]𝑠 0.185244 2.915429 6.307725 6.627446 13 [45.0, −45.0, −45.0, 0.0, 𝐶3]𝑠 0.185244 2.899083 6.291008 6.613282 14 [45.0, −45.0, −45.0, 90.0, 𝐶3]𝑠 0.185244 2.899083 6.291008 6.613282 15 [ − 45.0, 45.0, 45.0, 0.0, 𝐶3]𝑠 0.185244 2.899083 6.291008 6.613282 16 [ − 45.0, 45.0, 45.0, 90.0, 𝐶3]𝑠 0.185244 2.899083 6.291008 6.613282 17 [45.0, −45.0, 𝐶3, −45.0, −45.0]𝑠 0.185244 2.895203 6.29831 6.545771 18 [ − 45.0, 45.0, 𝐶3, 45.0, 45.0]𝑠 0.185244 2.895203 6.29831 6.545771 19 [45.0, −45.0, 𝐶3, −45.0, 45.0]𝑠 0.185244 2.890929 6.263009 6.563891 20 [ − 45.0, 45.0, 𝐶3, 45.0, −45.0]𝑠 0.185244 2.890929 6.263009 6.563891 109 [45.0, −45.0, 0.0, 𝐶3, 90.0]𝑠 0.185244 2.782391 6.018458 6.602233 445 [45.0, −45.0, −45.0, 𝑅]𝑠 0.170058 2.573469 5.844511 6.130792 459 [45.0, −45.0, −45.0, −45.0]𝑠 0.170058 2.572617 5.850642 6.110728 719 [45.0, −45.0, 0.0, 90.0]𝑠 0.170058 2.476461 5.591131 6.176574
On the first position, a C3 doublered soft laminate presents an increase of 18.44% in relation to the quasi-isotropic laminate in position 719.
B. Doubler results compared with 8- and 10-layered unidirectional laminates
In this case, 10-layered symmetrical unidirectional laminates with angles 0◦, 90◦, +45◦, −45◦were included in the comparison of Table 6.
Table 6 Frequency results for plate with cut-out and doubler
Ranking Layup mass [kg] 𝜔1[Hz] 𝜔2[Hz] 𝜔3[Hz]
1 [45.0, −45.0, −45.0, −45.0, −45.0]𝑠 0.212573 3.223267 7.464737 7.514569 2 [ − 45.0, 45.0, 45.0, 45.0, 45.0]𝑠 0.212573 3.223267 7.464737 7.514569 3 [45.0, −45.0, −45.0, −45.0, 45.0]𝑠 0.212573 3.223202 7.481591 7.498169 4 [ − 45.0, 45.0, 45.0, 45.0, −45.0]𝑠 0.212573 3.223202 7.481591 7.498169 5 [45.0, −45.0, −45.0, 45.0, −45.0]𝑠 0.212573 3.219241 7.390488 7.572898 6 [ − 45.0, 45.0, 45.0, −45.0, 45.0]𝑠 0.212573 3.219241 7.390488 7.572898 7 [45.0, −45.0, −45.0, −45.0, 0.0]𝑠 0.212573 3.218913 7.46335 7.511932 8 [45.0, −45.0, −45.0, −45.0, 90.0]𝑠 0.212573 3.218913 7.46335 7.511932 9 [ − 45.0, 45.0, 45.0, 45.0, 0.0]𝑠 0.212573 3.218913 7.46335 7.511932 10 [ − 45.0, 45.0, 45.0, 45.0, 90.0]𝑠 0.212573 3.218913 7.46335 7.511932 11 [45.0, −45.0, −45.0, 45.0, 45.0]𝑠 0.212573 3.217583 7.370533 7.586148 12 [ − 45.0, 45.0, 45.0, −45.0, −45.0]𝑠 0.212573 3.217583 7.370533 7.586148 13 [45.0, −45.0, −45.0, 45.0, 0.0]𝑠 0.212573 3.213743 7.376442 7.578707 14 [45.0, −45.0, −45.0, 45.0, 90.0]𝑠 0.212573 3.213743 7.376442 7.578707 15 [ − 45.0, 45.0, 45.0, −45.0, 0.0]𝑠 0.212573 3.213743 7.376442 7.578707 171 [45.0, −45.0, −45.0, −45.0, 𝐶3]𝑠 0.185244 2.933064 6.343534 6.645806 313 [45.0, −45.0, 𝐶3, 0.0, 90.0]𝑠 0.185244 2.812886 6.055425 6.619359 1129 [45.0, −45.0, −45.0, 𝑅]𝑠 0.170058 2.573469 5.844511 6.130792 1143 [45.0, −45.0, −45.0, −45.0]𝑠 0.170058 2.572617 5.850642 6.110728 1581 [45.0, −45.0, 0.0, 90.0]𝑠 0.170058 2.476461 5.591131 6.176574
Without considering the increase of mass, 10-layered laminates have an advantage in relation to 8-layered laminates with doubler.
VIII. Conclusion
For both geometries the theoretical circumferential and radial plies showed to be a good alternative for increasing the first fundamental frequency of the plate.
For the plate without a cut-out the circumferential ply was between the best placed layups with a 5.58% frequency increase in relation to the best quasi-isotropic laminate. It also showed a 1.9% increase in relation to the best unidirectional laminate found by [3].
The radial theoretical lamina also showed up between the best laminates for the plate with cut-out with a 3.9% increase in relation to the best quasi-isotropic laminate.
Also, making use of a circumferential doubler was a good strategy for increasing its fundamental frequency. When comparing the use of doubler on 8-layered plate it was possible to obtain a 18% frequency increase in relation to the best quasi-isotropic laminate. However 10-layered unidirectional plates showed better frequency results when compared to 8-layered plates with doubler. It is also worth mentioning that the 10-layered unidirectional laminates are heavier than the previous, however weight was not used as a criteria for the established ranking.
References
[1] Abdalla, M. M., Setoodeh, S., and Gurdal, Z., “Design of variable stiffness composite panels for maximum fundamental frequency using lamination parameters,” Composite Structures, Vol. 81, 2007, pp. 283–291.
[2] Capacia, V. N., Hernandes, J. A., and Castro, S. G. P., “Wing Panel Optimization using Lamination Parameters and Inverse Distance Weighting Interpolation,” 2018.
[3] Narita, Y., “Layerwise optimization for the maximum fundamental frequency of laminated composite plates,” Journal of Sound
and Vibration, Vol. 263, 2003, pp. 1005–1016.
[4] Bert, C. W., “Optimal Design of a Composite-Material Plate to Maximize its Fundamental Frequency,” Journal of Sound and
Vibration, Vol. 50, No. 2, 1977, pp. 229–237.
[5] MSC Nastran Quick Reference Guide, MSC, chapter and pages.
[6] Machado, T. G. P., Hernandes, J. A., Capacia, V. N., and Castro, S. G. P., “Design and Optimization of Wing Stiffened Panels Using Variable Stiffness Laminates with Coupled Variable Thickness,” 2020.