Document Type : Research Paper
Authors
Composite Materials and Technology Center, MalekAshtar University of Technology, Tehran, Iran
Abstract
Keywords

Mechanics of Advanced Composite Structures 2 (2015) 6172


Semnan University 
Mechanics of Advanced Composite Structures journal homepage: http://macs.journals.semnan.ac.ir 
Free Vibration of Lattice Cylindrical Composite Shell Reinforced with Carbon Nanotubes
J. Emami^{*}, J.E. Jam, M.R. Zamani, A. Davar
Composite Materials and Technology Center, MalekAshtar University of Technology, Tehran, Iran
Paper INFO 

ABSTRACT 
Paper history: Received 31 October 2015 Received in revised form 22 November 2015 Accepted 24 November 2015 
The free vibration of the lattice cylindrical composite shell reinforced with Carbon Nanotubes (CNTs) was studied in this study. The theoretical formulations are based on the Firstorder Shear Deformation Theory (FSDT) and then by enforcing the Galerkin method, natural frequencies are obtained. In order to estimate the material properties of the reinforced polymer with nanotubes, the modified HalpinTsai equations were used and the results were checked with an experimental investigation. Also, the smeared method is employed to superimpose the stiffness contribution of the stiffeners with those of the shell in order to obtain the equivalent stiffness of the whole structure. The effect of the weight fraction of the CNTs and also the ribs angle on the natural frequency of the structure is investigated in two types of length to diameter ratios in the current study. Finally, the results which are obtained from the analytical solution are checked with the FEM method using ABAQUS CAE software, and a good agreement has been seen between the FEM and the analytical results. 



Keywords: Carbon nanotubes Lattice structures Modified HalpinTsai equations Free vibration



© 2015 Published by Semnan University Press. All rights reserved. 
The composite lattice structures have better efficiency and higher strength to weight ratio in comparison with the conventional metal structures. The lattice structures are usually made in the form of the thinwalled cylindrical or conical shells and consist of a system of (with respect to the shell axis) helical and circumferential ribs [1]. The lattice structures are based on the idea of the load carrying skin and buckling. Khalili et al. [2] studied the transient dynamic response of the initially stressed composite circular cylindrical shells under the radial impulse load. They find out that if the axial compressive load increases up to the critical buckling load, the natural frequencies decrease. Hemmatnezhad et al. [3] studied the free vibrations of the gridstiffened composite cylindrical shells. They worked in both the theoretical and numerical method and used firstorder shear deformation theory for analytical solution. Zhang et al. [4] studied the free vibration behaviors of the carbon fiber reinforced latticecore sandwich cylinder with lattice cores and uniaxial compression, and the free vibration experiments were carried out. The natural frequencies and vibration modes of CFRC LSC were revealed by the experiments for the first time. They find out that the circumferential vibration modes turning from oval, triangle to rectangle lobar mode are dominant in the low primary frequencies, and the beam bending mode appears at higher order.
In the last two decades, many investigations have been conducted into the Carbon Nanotubes (CNTs). They are known to improve the mechanical stiffness and strength. However, they also should be evaluated in terms of the free vibration of a structure. Zhu et al. [5] carried out the bending and free vibration analyses of the thintomoderately thick composite plates reinforced by the single walled carbon nanotubes. The finite element method was used. Four types of the distributions of the uniaxialaligned reinforcement material are considered. Their results revealed the influences of the volume fractions of the carbon nanotubes and the edgetothickness ratios on the bending responses, natural frequencies and mode shapes of FGCNTRC plates. Besides, they find the CNTs distributed close to the top and bottom surfaces are more efficient than those distributed near the midplane for increasing the stiffness of the plates. Ramgopal Reddy et al. [6] investigated on the free vibration analysis of the carbon nanotube reinforced laminated composite panels. Three types of panels such as flat, concave and convex are considered and the influence of the boundary conditions on the natural frequency of the CNT reinforced composite panels was analyzed. Aragh et al. [7] worked on the natural frequency analysis of the continuously graded carbon nanotubereinforced cylindrical shells based on the thirdorder shear deformation theory. The interesting finding of their study is that the graded CNT volume fractions with symmetric distribution through the shell thickness have high capabilities to reduce or increase the natural frequency in comparison with the uniformly and asymmetric CNT distribution.
In this study, a new method is introduced for estimating the material property of the composites reinforced with randomly CNTs. Then, the free vibration in both the analytical and FEM analysis is calculated for the lattice cylindrical shell. Eventually, the results of the FEM and analytical method are compared with each other and a good agreement is observed between them.
When the CNTs are added to the composite, the elastic properties change. Thus, one way to estimate the properties is using modified HalpinTsai equations. The Cox model is used to modify these equations. is the orientation factor in Eqs. (1) and (2) [8]:
(1) 

(2) 
where E_{m}, G_{m}, K_{w}, E_{em}, G_{em}, E_{CNT} and G_{CNT} indicate Young’s and shear modulus of matrix, the waviness correction factor, Young’s and shear modulus of modified polymer, Young’s and shear modulus of CNTs respectively.
If the length of reinforcements (CNTs) is greater than the thickness of the specimen, the reinforcements are assumed to be randomly oriented in two dimensions and the factor is used. If the length of reinforcements is much smaller than the thickness of the specimen, the reinforcements are assumed to be randomly oriented in three dimensions and the parameter is considered [8]. According to the Eqs. (1) and (2), the elastic properties of the polymer which is reinforced with random CNTs can be calculated. Then, substituting the elastic properties of the polymer into the Eqs. (3), (4) and (5), the elastic property of the composite reinforced by CNTs and continuous fibers can be estimated as what follows:
(3) 

(4) 

(5) 
where the constant shape factor C is related to the aspect ratio of reinforcement length l and diameter d. Also, E_{r} and G_{r} are the moduli of the reinforcement and V_{r} is the volume fraction of the reinforcement respectively. In order to validate the above equations, the results are compared with reference [9] which is an experimental investigation and is shown in Table 1.
In order to calculate the free vibration of the lattice cylindrical composite shell, the equivalent stiffness should be determined. First, a unit cell of the stiffeners which is repeated in the whole of the structure is considered as shown in Fig. 1. Then, with the help of smeared stiffener method and the following refined assumptions, the equivalent stiffness can be calculated [10,11]:
1. The shear stresses in the cross stiffeners are not to be ignored.
2. The crosssection dimensions of the stiffeners are very small compared to the length.
3. A uniform stress distribution is assumed across the crosssectional area of the stiffeners.
4. The load on the stiffener/shell is transferred through shear forces between the stiffeners and shell.
Table1. The comparison of the results of the material properties of the present sturdy with Ref. [9]
Density (Kg/m^{3}) 
Shear modulus G_{12} (GPa) 
Transverse modulus E_{2} (GPa) 
Longitudinal modulus E_{1} (GPa) 
EpoxyCNT modulus (GPa) 
Type of CNT 

Ref.[9] 
Present 
Ref.[9] 
Present 
Ref.[9] 
Present 
Ref.[9] 
Present 
Ref.[9] 
Present 

1532.0 
1539.00 
1.36 
1.3826 
4.29 
4.2909 
22.80 
23.0000 
2.00 
2.0000 
No CNT 
1540.0 
1540.32 
1.48 
1.5079 
4.67 
4.6678 
22.91 
23.1315 
2.19 
2.1878 
0.5% MWNT 
1541.5 
1541.64 
1.60 
1.6331 
5.04 
5.0419 
23.02 
23.2635 
2.37 
2.3764 
1.0% MWNT 
1543.0 
1542.97 
1.72 
1.7579 
5.41 
5.4133 
23.13 
23.3959 
2.56 
2.5656 
1.5% MWNT 
1540.0 
1540.32 
1.48 
1.5102 
4.67 
4.6745 
22.91 
23.1339 
2.19 
2.1912 
0.5% SWNT 
Figure 1. The unit cell and coordinate system for a stiffened cylindrical shell [3]
The strains on the interface of the stiffener and the shell are given by [12]:
(6) 
where are the midplane strains and curvatures of the shell and t is the shell thickness. These strain components should be transformed along the stiffener direction l and t. So the transformation matrix is as what follows [12]:
(7) 
where , and is the stiffener orientation angle. Fig. 2 shows the force free body diagram on the unit cell. The axial and shear forces in the stiffener direction are expressed as what follows [12]:
(8) 
where and are the longitudinal and shear modulus of the stiffeners, respectively.
Figure 2. The force distribution on the unit cell [12]
Resolving the axial and shear forces in the and directions, we have the following equations [12]:
(9) 
Substituting Eqs. (6) and (8) into Eq. (9) and dividing the force expressions by the corresponding edge width of the unit cell, the forces per unit length are carried out as what follow[12]:
(10) 
According to the shear forces between stiffeners and shell, the torsion and bending moments are applied to stiffeners and shell. Fig. 3 shows the moments free body diagram which is applied to unit cell. Then, the same as before, the resultant moments can be obtained as what follows [12]:
(11) 
Fig. 4 shows the transverse shear forces, and the strain components are as what follows [12]:
(12) 
where are the transverse shear strains in the stiffeners, and are the transverse shear strains in the shell. Then, the shear forces resulting from the shear strains are given as the following expressions [12]:
(13) 
Rewriting these forces in the and directions, we have the following equations [12]:
(14) 
Figure 3. The moments free body diagram on the unit cell [12]
Figure 4. The transverse shear forces [12]
The resultant shear forces per unit length can be obtained by dividing the above forces by the corresponding length are what follows [12]:
(15) 
Therefore, the matrices for the stiffeners are expressed as the following [12]:
(16) 
Also, the resultant force and moments can be written as following (due to the shell in terms of the strain components of the midplane surface of the shell) [12].
(17) 
The total force and moment on the structure are the superposition of the forces and moments due to the stiffeners and the shell according to their volume fractions as what follows [12]:
(18) 
where and are the volume fractions of the stiffener and the shell respectively.
In Firstorder Shear Deformation Theory (FSDT), it is assumed that the transverse normal does not remain perpendicular to the mid surface after the deformation, so the displacement fields are as what follows [13]:
(19) 
Based on (FSDT), the equilibrium equations for a cylindrical shell are as the following equations [3]:
(20) 
In the above equation, and are the slope in the plane of and respectively.
are defined by the following relation [3]:
(21) 
where is the density for each layer. Moreover, the stiffness of the shell is given by the following expressions [2]:
(22) 
where and are the extensional, coupling, bending and thickness shear stiffness matrices respectively and they are defined as what follows [2]:
(23) 
where is the shear correction factor and equals to [2]. The mid plane strain and curvature components are given by the following equations [2]:
(24) 
The boundary conditions for the cylindrical shell which are simply supported along its curved edges at and are considered as what follows [2]:
(25) 
The following functions are assumed to satisfy the simplysupported boundary conditions and the equations of motion:
(26) 
Now, substituting Eq. (24) into Eq. (22) and then enforcing the result into Eq. (20), the free vibration Eigenequations yield to what follows:
(27)
(28) 
where and are stiffness and mass matrices respectively and are shown in appendix.
The geometry of the structure is shown in Table 2. Two types of L/D are considered. The cylindrical shell is assumed to be [45/45/45/45/45] stacking sequence, while in the stiffener structure, the fibers are considered to be oriented in the ribs’ directions. It is supposed that the CNTs have randomly orientation for both the singlewall and multiwall CNTs. The material properties of the glass fiber, carbon nanotubes and epoxy resin are given as what follows:

The boundary conditions are applied in both edges of the cylinder in the ABAQUS software. The structure is fixed in both the circumferential and thickness directions and is free in the longitudinal direction. Also, the M_{x} is fixed, too. The number of the elements is about 475395 and their types are structured ones for the lattice cylinder and the shell sweep for the cylindrical shell. Table 3 and Table 4 show a comparison of the first five natural frequencies got via analytical solution and FEM approach in two types of the length to diameter ratio (L/D = 1 and 2) and the carbon nanotubes (MWNT and SWNT).
A good agreement has been seen among the results and the maximum error is less than 10%. As shown in Fig. 5, it is clear that with an increase in the weight fraction of the CNTs, the natural frequency of the structure increases linearly.
Figs. 6 and 7 give the first five mode shapes of the lattice cylindrical shell in L/D = 1 and 2 ratios, respectively.
Tables 5, 6 and 7 give a comparison of different rib angles. Three kinds of the rib angle are chosen (30°, 35° and 40°) to investigate the effect of the rib orientation on the natural frequency of the structure.
Fig. 8 shows the natural frequency decreases at first and then by enhancing the amount of rib angle, it increases.
Table 2. Geometrical parameters of the lattice cylindrical shell
Shell height (mm) 
For L/D = 1, L = 240 For L/D = 2, L = 480 
Diameter (mm) 
For L/D = 1, D = 240 For L/D = 2, D = 242.584 
Shell thickness (mm) 
3 
Helical rib distance, a_{k} (mm) 
60 
Stiffener orientation (ϕ°) 
±30 
Stiffener thickness (mm) 
3 
Stiffener width (mm) 
5 
(a)
(b)
Figure 5. Variations of the natural frequencies with different weight fractions of the CNTs; (a) L/D = 1 and (b) L/D = 2
Table 3. Natural frequencies of the lattice cylindrical shell reinforced with the glass fiber and MWNT
m 
n 

Analytical (Hz) 
FEM (Hz) 
Difference % 
Analytical (Hz) 
FEM (Hz) 
Difference % 

1 
1 1 1 1 1 
4 5 3 6 7 
614.05 737.56 743.94 974.71 1275.89 
656.58 793.78 801.15 1050.30 1373.40 
6.92 7.62 7.69 7.76 7.64 
636.37 758.43 776.34 1000.73 1310.39 
681.23 819.58 832.58 1084.4 1419.5 
7.05 8.06 7.24 8.36 8.33 
2 
1 1 1 2 1 
3 2 4 4 5 
273.72 366.17 397.72 606.09 614.05 
292.12 401.44 430.49 656.71 658.73 
6.72 9.63 8.24 8.35 7.28 
284.65 386.81 409.34 623.29 636.37 
304.38 421.75 445.05 679.43 683.69 
6.93 9.03 8.72 9.01 7.44 
Continued Table 4 Natural frequencies of the lattice cylindrical shell reinforced with the glass fiber and MWNT
m 
n 

Analytical (Hz) 
FEM (Hz) 
Analytical (Hz) 
FEM (Hz) 
Analytical (Hz) 
FEM (Hz) 

1 
1 1 1 1 1 
4 5 3 6 7 
656.69 778.02 804.74 1025.36 1343.12 
703.56 843.36 860.36 1116 1462 
656.69 778.02 804.74 1025.36 1343.12 
703.56 843.36 860.36 1116 1462 
656.69 778.02 804.74 1025.36 1343.12 
703.56 843.36 860.36 1116 1462 
2 
1 1 1 2 1 
3 2 4 4 5 
294.71 405.05 420.31 639.63 656.69 
315.57 439.76 458.55 700.46 706.27 
294.71 405.05 420.31 639.63 656.69 
315.57 439.76 458.55 700.46 706.27 
294.71 405.05 420.31 639.63 656.69 
315.57 439.76 458.55 700.46 706.27 
Table 5. Natural frequencies of the lattice cylindrical shell reinforced with the glass fiber and SWNT
m 
n 

Analytical (Hz) 
FEM (Hz) 
Analytical (Hz) 
FEM (Hz) 
Analytical (Hz) 
FEM (Hz) 

1 
1 1 1 1 1 
4 5 3 6 7 
614.05 737.56 743.94 974.71 1275.89 
656.58 793.78 801.15 1050.30 1373.40 
614.05 737.56 743.94 974.71 1275.89 
656.58 793.78 801.15 1050.30 1373.40 
614.05 737.56 743.94 974.71 1275.89 
656.58 793.78 801.15 1050.30 1373.40 
2 
1 1 1 2 1 
3 2 4 4 5 
273.72 366.17 397.72 606.09 614.05 
292.12 401.44 430.49 656.71 658.73 
273.72 366.17 397.72 606.09 614.05 
292.12 401.44 430.49 656.71 658.73 
273.72 366.17 397.72 606.09 614.05 
292.12 401.44 430.49 656.71 658.73 
Continued Table 6. Natural frequencies of the lattice cylindrical shell reinforced with the glass fiber and SWNT
m 
n 

Analytical (Hz) 
FEM (Hz) 
Analytical (Hz) 
FEM (Hz) 
Analytical (Hz) 
FEM (Hz) 

1 
1 1 1 1 1 
4 5 3 6 7 
657.39 778.70 805.69 1026.23 1344.27 
704.32 844.19 861.31 1117.10 1463.50 
657.39 778.70 805.69 1026.23 1344.27 
704.32 844.19 861.31 1117.10 1463.50 
657.39 778.70 805.69 1026.23 1344.27 
704.32 844.19 861.31 1117.10 1463.50 
2 
1 1 1 2 1 
3 2 4 4 5 
295.05 405.67 420.69 640.20 657.39 
315.95 440.37 459.02 701.19 707.04 
295.05 405.67 420.69 640.20 657.39 
315.95 440.37 459.02 701.19 707.04 
295.05 405.67 420.69 640.20 657.39 
315.95 440.37 459.02 701.19 707.04 
(b) 2^{nd} Mode (m=1,n=5)
(d) 4^{th} Mode (m=1,n=6) 
(a) 1^{st} Mode (m=1,n=4)
(c) 3^{rd} Mode (m=1,n=3) 

(e) 5^{th} Mode (m=1,n=7)



Figure 6. The mode shapes of the lattice cylindrical shell with L/D = 1
(b) 2^{nd} Mode (m=1,n=2)
(d) 4^{th} Mode (m=2,n=4) 
(a) 1^{st} Mode (m=1,n=3)
(c) 3^{rd} Mode (m=1,n=4) 
(e) 5^{th} Mode (m=1,n=5)

Figure 7. The mode shapes of the lattice cylindrical shell with L/D = 2
Table 7. Nondimension natural frequencies of the lattice cylindrical shell reinforced with the multiwall CNTs and rib angle of 30
m 
n 

Analytical (Hz) 
FEM (Hz) 
Analytical (Hz) 
FEM (Hz) 
Analytical (Hz) 
FEM (Hz) 

2 
1 1 1 2 1 
3 2 4 4 5 
0.0463 0.0620 0.0673 0.1026 0.1039 
0.0494 0.0679 0.0729 0.1112 0.1115 
6.70 9.52 8.32 8.38 7.31 
0.0481 0.0653 0.0691 0.1053 0.1075 
0.0514 0.0712 0.0752 0.1148 0.1155 
6.86 9.04 8.83 9.02 7.44 
Continued Table 8. Nondimension natural frequencies of the lattice cylindrical shell reinforced with the multiwall CNTs and rib angle of 30
m 
n 

Analytical (Hz) 
FEM (Hz) 
Analytical (Hz) 
FEM (Hz) 
Analytical (Hz) 
FEM (Hz) 

2 
1 1 1 2 1 
3 2 4 4 5 
0.0497 0.0683 0.0709 0.1078 0.1107 
0.0532 0.0741 0.0773 0.1181 0.1191 
7.04 8.49 9.03 9.55 7.59 
0.0511 0.0709 0.0725 0.1102 0.1136 
0.0548 0.0767 0.0793 0.1211 0.1223 
7.24 8.18 9.38 9.89 7.66 
Table 9. Nondimension natural frequencies of the lattice cylindrical shell reinforced with the multiwall CNTs and rib angle of 35
m 
n 

Analytical (Hz) 
FEM (Hz) 
Analytical (Hz) 
FEM (Hz) 
Analytical (Hz) 
FEM (Hz) 

2 
1 1 1 2 1 
3 2 4 4 5 
0.0435 0.0578 0.0636 0.0972 0.0972 
0.0465 0.0623 0.0697 0.1045 0.1066 
6.90 7.79 9.59 7.51 9.67 
0.0452 0.0611 0.0654 0.0998 0.1006 
0.0483 0.0656 0.0718 0.1084 0.1100 
6.86 7.36 9.79 8.62 9.34 
Continued Table 10. Nondimension natural frequencies of the lattice cylindrical shell reinforced with the multiwall CNTs and rib angle of 35
m 
n 

Analytical (Hz) 
FEM (Hz) 
Analytical (Hz) 
FEM (Hz) 
Analytical (Hz) 
FEM (Hz) 

2 
1 1 1 2 1 
3 2 4 4 5 
0.0467 0.0639 0.0670 0.1022 0.1037 
0.0500 0.0685 0.0738 0.1118 0.1131 
7.07 7.20 10.15 9.39 9.06 
0.0481 0.0664 0.0685 0.1044 0.1065 
0.0516 0.0710 0.0756 0.1149 0.1160 
7.28 6.93 10.36 10.06 8.92 
Table 11. Nondimension natural frequencies of the lattice cylindrical shell reinforced with the multiwall CNTs and rib angle of 40
m 
n 

Analytical (Hz) 
FEM (Hz) 
Analytical (Hz) 
FEM (Hz) 
Analytical (Hz) 
FEM (Hz) 

2 
1 1 1 2 1 
3 2 4 4 5 
0.0469 0.0560 0.0701 0.1048 0.1074 
0.0504 0.0639 0.0772 0.1133 0.1182 
7.46 14.11 10.13 8.11 10.06 
0.0487 0.0633 0.0720 0.1083 0.1102 
0.0524 0.0674 0.0796 0.1172 0.1219 
7.60 6.48 10.5 8.22 10.6 
Continued Table 12. Nondimension natural frequencies of the lattice cylindrical shell reinforced with the multiwall CNTs and rib angle of 40
m 
n 

Analytical (Hz) 
FEM (Hz) 
Analytical (Hz) 
FEM (Hz) 
Analytical (Hz) 
FEM (Hz) 

2 
1 1 1 2 1 
3 2 4 4 5 
0.0503 0.0663 0.0737 0.1116 0.1128 
0.0541 0.0705 0.0817 0.1208 0.1252 
7.55 6.33 10.8 8.24 10.9 
0.0518 0.0689 0.0754 0.1145 0.1152 
0.0558 0.0732 0.0837 0.1240 0.1284 
7.72 6.24 11.01 8.30 11.46 
Since the design of the lattice cylindrical shell with different rib angles in a specific length to diameter ratio has some limitations, the effect of weight should be omitted using Eq. (27) [14]. The main reason of this behavior can only be described in changing the structure stiffness.
(27) 
Figure 8. Variations of the nondimensional natural frequency of the lattice cylindrical shell reinforced with the MWNT nanotubes in different rib angles of 30°, 35° and 40°
The free vibrations of the lattice cylindrical shell reinforced with the carbon nanotubes are investigated. The analytical method is based on Firstorder Shear Deformation Theory (FSDT). According to the smeared method, the equivalent stiffness of the structure was calculated and then it was entered into the analytical procedure to find the natural frequencies. In order to find out the material properties of a composite reinforced with CNTs, the modified HalpinTsai equations were used. The analytical results are validated by the FEM results and a good agreement has been observed. It can be understood from the results that adding CNTs increases the natural frequency linearly. Besides, by changing the rib angle, the stiffness of the structure changes, too. First, the natural frequency decreases and then it increases.
Nomenclature
Young’s and shear modulus of modified polymer reinforced with CNTs 

Waviness factor 

Orientation parameter 

Mid plane strains 

Mid plane curvatures 

Transverse shear strain 

Slope in the plane of 

Slope in the plane of 

Mass inertia 

Weight fraction of CNTs 

Shear correction factor 

Volume fraction of fiber 

Density of fiber, polymer and carbon nanotubes 

Young modulus of fiber, polymer and carbon nanotubes 

Poisson ratio of fiber and polymer 

Length of CNT 

Diameter of single wall and multi wall nanotubes 

Rib angle 
Appendix
Stiffness Coefficients:
Mass Coefficients:
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