Torsional vibration of functionally graded carbon nanotube reinforced conical shells

1 Introduction

A novel class of materials, known as carbon nanotubes (CNTs) have attracted increasing attention in recent years. The CNTs have exceptional mechanical, thermal, and electrical properties [1]. It is widely known that addition of CNTs in a matrix enhances the thermal and mechanical properties of composites [1]. Also, graded distribution of CNTs across the thickness of a structural element may enhance the thermo-mechanical characteristics of the structure where the total volume fraction of CNTs in both uniform and graded distributions are the same [1]. This is the main reason behind vast investigations carried out on the subject of functionally graded carbon nanotube reinforced composites (FG-CNTRC).

Free vibration characteristics of FG-CNTRC beams, plates, and shells have been the subject of many researches in the past 5 years. An overview of researches on the subject of free vibration of FG-CNTRC beams, plates, and shells is mentioned in the next paragraphs.

Yas and Samadi [2] analyzed the free vibration and buckling of FG-CNTRC Timoshenko beams using the generalized differential quadrature method. Composite beam is in contact with a Pasternak elastic foundation in this research. Alibeigloo and Liew [3] analyzed the free vibration of thick FG-CNTRC beams integrated with two piezoelectric layers at the top and bottom using the elasticity theory. To analyze the influences of graded CNTs in the face sheets of a sandwich beam with stiff core on vibration characteristics, Wu et al. [4] formulated the sandwich beam based on the Timoshenko beam theory and solved the eigenvalue problem using the differential quadrature method. It is shown that the ratio of face sheet thickness to core thickness is influential on the natural frequencies of the beam. Ke et al. [5] investigated the linear and nonlinear free vibration analyses of FG-CNTRC beams. The polynomial-based Ritz method is used in this study to discretize the homogeneous nonlinear motion equations. It is shown that nonlinear frequency ratios of both simply supported-simply supported and clamped-simply supported FG-CNTRC beams are dependent on the sign of the vibration amplitudes, i.e. their nonlinear frequency ratio versus amplitude curves are unsymmetrical. Lin and Xiang [6] applied both the first- and third-order shear deformation beam theories to analyze the free vibration characteristics of the beam. In this study, the Ritz method with polynomial basis functions is used to discretize the motion equations. Numerical results of this study reveal that for hard-clamped and soft-clamped supports substantial differences between frequency parameters are observed based on the first-order and third-order beam theories.

Malekzadeh and Heydarpour [7] applied a mixed solution based on the Navier and layerwise techniques to analyze the free vibration characteristics of thick rectangular plates. In this research, all edges of the plate are assumed to be simply supported. In a study by Zhang et al. [8] element-free IMLS-Ritz method is implemented to study the free vibration and associated mode shapes of an FG-CNTRC triangular plate. In another study, Zhang et al. [9] studied the free vibration and mode shapes of skew plates reinforced by FG-CNTRC. The influence of elastic foundation on the natural frequencies of a thick plate is analyzed by Zhang et al. [10] by using the element-free IMLS-Ritz method. Based on the finite element method and using the first-order shear deformation theory (FSDT), static and free vibration responses of FG-CNTRC plates are investigated by Zhu et al. [11]. Malekzadeh and Zarei [12] analyzed the free vibration of arbitrary quadrilateral laminated plates containing single or multiple FG-CNTRC layers using a two-dimensional differential quadrature method. Kamarian et al. [13] applied the Eshelby-Mori-Tanaka approach to a CNTRC plate to obtain the equivalent properties of the plate. After that free vibration response of the plate resting on a Winkler elastic foundation is analyzed using the three-dimensional theory of elasticity and Navier solution. Based on a two-step perturbation technique suitable for FG-CNTRC plates and sandwich plates with FG-CNTRC face sheets, Wang and Shen [14, 15] analyzed the linear and nonlinear free vibration responses of rectangular plates with all edges simply supported. Temperature rise influences are also taken into consideration. Malekzadeh et al. [16] investigated the influences of graded pattern of CNTs and volume fraction of CNTs on free and forced vibration responses of FG-CNTRC plates subjected to the action of a moving load. Lei et al. [17] and Zhang et al. [18, 19] investigated the free vibration of FG-CNTRC rectangular plates with various boundary conditions using mesh-free methods. Selim et al. [20] studied the free vibration of FG-CNTRC plates based on the third-order shear deformation plate theory which takes into account the nonuniform variation of shear strains across the thickness of the plate.

Yas et al. [21] investigated the free vibration response of FG-CNTRC cylindrical panels using the three-dimensional theory of elasticity. Solution method of this research is suitable for panels with all edges simply supported. After applying the Navier solution to the three-dimensional motion equations, the resulting ones are ordinary differential equations which are solved using the generalized differential quadrature. Shen and Xiang [22] analyzed the linear and nonlinear free vibrations of cylindrical panel resting on a two parameter Pasternak elastic foundation. In this study the effects of thermal environment are also taken into account. Two-step perturbation technique is used to obtain a closed-form solution for the linear and nonlinear natural frequencies of the panels with all the edges simply supported. Zhang et al. [23] analyzed the free vibration and static response of an FG-CNTRC panel using the first-order shell theory. Eshelby-Mori-Tanaka approach is used to obtain the properties of the composite media. Solution method of this research is suitable for almost arbitrary type of edge supports. Mirzaei and Kiani [24] studied the free vibration characteristics of FG-CNTRC cylindrical panels with arbitrary boundary conditions using a Chebyshev-Ritz formulation.

Shen and Xiang [25] investigated the linear and nonlinear free vibrations of a composite cylindrical shell reinforced with FG-CNT fibers. Solution method of this research is suitable for shells with both the edges simply supported. Heydarpour et al. [26] examined the free vibration of a shear deformable rotating composite conical shell reinforced with either uniformly distributed or graded CNTs. A static analysis is carried out to obtain the in-plane forces induced due to rotation. Afterwards, an eigenvalue problem is established to extract the natural frequencies of the shell. Combinations of simply supported and clamped conditions are considered in this work.

Similar to free vibration, dynamic stability of cylindrical panels made of FG-CNTRC [27], large amplitude nonlinear bending of FG-CNTRC skew [28], rectangular [29] and arbitrary quadrilateral [30] plates, dynamic response of FG-CNTRC rectangular plates subjected to uniform [31] and moving loads [16], postbuckling of FG-CNTRC plates with various boundary conditions subjected to in-plane unixial and biaxial compressive loads [32, 33], aerothermoelastic characteristics of FG-CNTRC cylindrical panel in supersonic flow [34], buckling of FG-CNTRC skew plates under in-plane compression [35], thermal buckling of rectangular FG-CNTRC plate with arbitrary boundary conditions under uniform thermal field [36], thermal and mechanical buckling of FG-CNTRC conical shells [37, 38], and stress analysis of laminated plates made of FG-CNTRC layers [39] are also observed through the open literature.

The above literature survey accepts the fact that most of the researches on the subject of free vibration of FG-CNTRCs are focused on beams, plates, and cylindrical panels and only a limited number deals with the closed cylindrical or conical shells. In particular, research carried out by Heydarpour et al. [26] is the only available work on the free vibration analysis of FG-CNTRC conical shells. The present research aims to extend the investigations on FG-CNTRC conical shells. Torsional vibration of a conical shell made of a polymeric matrix reinforced by CNTs is considered. In torsional vibration analysis, when particular conditions are met, only one component of displacement field is present. Using the first-order shell theory and Donnell kinematics, two coupled equations are established by means of the Hamilton principle. These equations are discretized by means of the generalized differential quadrature method and solved as an eigenvalue problem. After validating the solution procedure for the case of a conical shell or an annular plate made of isotropic homogeneous materials, some numerical results are given for FG-CNTRC conical shells. It is shown that shell geometry, distribution profile of CNTs, and volume fraction of CNTs are important factors with regard to torsional frequencies of the FG-CNTRC conical shells; however, grading profile is not significantly influential.

2 Basic formulation

Consider an FG-CNTRC circular conical shell of thickness h, end radii R₁<R₂, length along the generator L, and vertex half angle α. The meridional, circumferential, and thickness directions of the shell are denoted by x, θ, and z, respectively. A schematic of the shell with the assigned coordinate system and geometric characteristics is shown in Figure 1.

Figure 1:

Schematic and geometric characteristics of the conical shell.

As reported by Kwon et al. [40], using the powder metallurgy process, the concept of functionally graded materials and CNTs may be achieved together. Volume fraction of CNT may vary according to a prescribed function across the thickness of a structural element. However, compatible with the available manufacturing process [40], only linear variation of CNT across the thickness of the conical shell is considered. As a result, the single walled carbon nanotube (SWCNT) reinforcement is either uniformly distributed (referred to as UD) or functionally graded in the thickness direction (referred to as FG). FG-O, FG-X, FG-V, and FG-Λ CNTRC are the functionally graded distribution of carbon nanotubes over the thickness direction of the composite conical shell [37, 38, 41, 42].

It is assumed that the CNTRC conical shell is made from a mixture of SWCNT, graded in the thickness direction, and a matrix which is assumed to be isotropic. The effective material properties of the two-phase nanocomposites, mixture of CNTs and an isotropic polymer, may be estimated according to the well-known Mori-Tanaka scheme or the rule of mixtures. Due to simplicity and convenience, in the present study the rule of mixtures is employed by introducing CNT efficiency parameter and the effective material properties of CNTRC conical shell [43]. This efficiency parameter is introduced to match the shear modulus obtained from the molecular dynamics simulations with those obtained according to the modified rule of mixtures approach. Thus the effective material properties which are involved in this research may be written as

where in the above equations, G12CN and ρ^CN are the shear modulus and mass density of SWCNTs. Besides, G^m and ρ^m indicate the shear modulus and mass density of the isotropic matrix. The coefficient η₃ is introduced to account for the scale-dependent shear modulus [43]. This constant is evaluated by matching the effective property of CNTRC obtained from the molecular dynamic simulations with those from the rule of mixtures. Furthermore, in Eq. (1), V_CN and V_m are the volume fractions of CNTs and matrix phase, respectively, which satisfy the condition [44].

Uniform and four types of functionally graded distributions of the CNTs along the thickness direction of the nanocomposite conical shell are assumed. The mathematical expression of CNTs’ volume fraction in each case of distribution is given in Table 1. In this table, VCN∗ is the overall volume fraction of the structure which may be obtained in terms of mass density of the constituents and mass fraction of the CNT, w_CN as

For type FG-V, the outer surface (z=+h/2) of the shell is CNT-rich and the inner surface (z=-h/2) is free of CNTs. For type FG-Λ, the distribution of CNT reinforcements is inverse and the inner surface (z=-h/2) of the shell is CNT-rich whereas the outer surface is free of CNTs. For type FG-X, a mid-plane symmetric graded distribution of CNT reinforcements is achieved and both outer and inner surfaces are CNT-rich. For type FG-O, a mid-plane symmetric graded distribution of CNT reinforcements is achieved and both outer and inner surfaces are free of CNT. Configurations of UD- and FG-CNTRC conical shells are shown in Figure 2. As seen from Table 1, UD- and all of FG-CNTs will have the same value of mass fraction of CNTs.

Figure 2:

Configurations of various CNTRC conical shells.

To capture the through the thickness shear deformations effects, the first-order shear deformation theory of shells is used to formulate the governing equations of the shell. Based on the FSDT, the only available component of displacement field, i.e. tangential displacement uθ may be written in terms of the tangential displacement of the mid-surface v and the rotation of a normal element in the zθ plane, φ as [45].

Kayran and Vinson [46] examined the conditions for independent torsional vibration motions. As mentioned by Kayran and Vinson [46] for the case when the coupling terms A_i6, B_i6 and D_i6 (i=1, 2) do not exist, the torsional motion is uncoupled from the extensional and bending motions and one may study the free torsional vibrations only by considering the tangential displacement uθ. For classical FG-CNTRC conical shells, the coupling stiffnesses A_i6, B_i6 and D_i6 (i=1, 2) are absent and, therefore, the free torsional vibration may be analyzed only by considering the tangential displacement uθ.

In FSDT, when torsional vibration is uncoupled from the bending and extensional motions, only two components of the strain filed are present, i.e. γ_xθ and γθ_z [47]. For a conical shell with the assigned coordinates system as shown in Figure 1, these two components may be written as

In the above equations, r(x)=R₁+x sin(α) stands for the radius of the shell at each point along the length.

Components of stress in terms of strains under the assumption of linear elastic material are evaluated as

where Q₄₄ and Q₆₆ are the reduced material stiffness coefficients compatible with the plane-stress conditions and are

The components of stress resultants are obtained using the components of stress field as

In the above equation, κ is the shear correction factor of FSDT. As this factor depends on the boundary conditions, material properties, geometry and loading type [48], determination of its exact value is not straightforward. However, the approximate values of κ=1, κ=5/6, or κ=π²/12 are used extensively. In this research, the correction shear factor is set equal to κ=π²/12.

Substitution of Eq. (6) into Eq. (8) with the simultaneous aid of Eq. (5) generates the stress resultants in terms of the mid-surface characteristics of the shell as

In the above equation, the constant coefficients A₄₄, A₆₆, B₆₆, and D₆₆ indicate the out-of-plane shear, in-plane shear, coupling shear-twist, and twist stiffnesses respectively, which are calculated by

The complete set of linear motion equations and boundary conditions of an FG-CNTRC conical shell in free torsional vibration regime may be obtained based on the Hamilton principle [45]. Statement of this concept reads

where in the above equation δU is the virtual strain energy of the shell which may be calculated as

And δT is the virtual kinetic energy of the shell. Considering the tangential kinetic energy of the shell, one may write

Integrating the expressions (12) and (13) with respect to z coordinate and performing the integration by part technique to relieve the virtual displacement gradients result in the expressions for the linear torsional motion equations of an FG-CNTRC conical shell as

The complete set of boundary conditions are revealed through the process of virtual displacement relieving. For the two ends of the shell, i.e. x=0 and x=L, the boundary conditions are extracted as

The motion equations (14) may be written in terms of the kinematic variables v and φ. To obtain such equations, Eq. (9) should be substituted into Eq. (14). After substituting and performing proper mathematical operations, the resulting equations are

Two types of boundary conditions may be defined. Clamped (C) boundary condition with v=φ=0 and free (F) boundary condition with N_xθ=M_xθ=0.

3 Solution method

As only free vibration motion is under investigation, we may write

where ω is the natural frequency. Upon substitution of Eq. (17) into Eq. (16) one may reach to

The above system of coupled ordinary differential equations are discretized via the generalized differential quadrature method. After applying the differential quadrature method to Eq. (18) one may reach to

where in the above equation

In the above equations, C⁽⁰⁾ is the identity matrix and C⁽¹⁾ and C⁽²⁾ are the weighting coefficient matrices associated with the first and second derivatives of a function [49]. Distribution of the points along the slanted edge of the shell is described based on the Chebyshev-Gauss-Lobatto distribtuion which reads

Here, N is the number of grid points through the length of the shell. Similar to the governing motion equation, generalized differential quadrature method should be applied to the boundary conditions. Recalling the definition of stress resultants (9), one has

In Eq. (22), i=1 is for the edge x=0 and i=N is for the edge x=L. The boundary conditions are applied directly to Eq. (19). The resulting eigenvalue problem is solved by means of the standard eigenvalue procedures.

4 Numerical result and discussion

The procedure outlined in the previous sections is used herein to study the torsional vibration of a composite truncated conical shell made from a polymeric matrix reinforced with FG-CNTs. In the rest, the following convention is used for the edge supports. For instance, F-C indicates a conical shell which is free at x=0 and clamped at x=L. Unless otherwise stated, Poly (methyl methacrylate), referred to as PMMA, is selected for the matrix with material properties E^m=2.5 GPa, ν^m=0.34, and ρ^m=1150 kg/m³. The matrix is isotropic and its shear modulus may be obtained easily in terms of elasticity modulus and Poisson’s ratio. (10,10) armchair SWCNT are chosen as the reinforcements. In-plane shear modulus of SWCNT at room temperature is G12CN=1.9445 TPa and mass density is ρ^CN=1400 kg/m³ [25]. The CNTs efficiency parameter η₃ is calculated by Shen [43] by matching the elastic properties evaluated by molecular dynamics simulation and rule of mixtures. For three different volume fractions of CNTs, this parameter is given as: η₃=0.7154 for VCN∗=0.12,η₃=1.1382 for VCN∗=0.17, and η₃=1.1095 for VCN∗=0.28. The shear modulus G₂₃ is taken as equal to 1.2G₁₂. In the next paragraph, at first two comparison studies are carried out for the case of isotropic homogeneous shells and plates. Afterwards, numerical results are provided for FG-CNTRC conical shells. In the rest, number of grid points is chosen as N=31 after the examination of convergence of first five natural frequencies up to four digits.

4.2 Parametric studies

In this section, parametric studies are conducted to investigate the influences of various parameters on the torsional frequencies of FG-CNTRC conical shell. In the whole of this section, nondimensional frequency parameter is defined as Ω=ωR1ρm/Gm.

The first parametric study analyzes the influence of CNTs’ volume fraction on the first five torsional frequencies of CNTRC conical shells. Numerical results are provided in Table 4. Four different cases of boundary conditions and the first five natural frequencies are considered. Numerical results are evaluated for a shell with geometric characteristics α=30°, L/R₁=2, and R₁/h=50. As one may see from this table, as the volume fraction of CNTs increases, torsional frequencies of the shell also increases. Except for the zero-valued frequency of F-F shell which is not mentioned in the table (associated to rigid body motion), frequency of an F-F shell is higher than F-C shell followed by in order C-F and C-C shells.

Table 4:

Torsional frequency parameter Ω=ωR1ρm/Gm for FG-X CNTRC conical shells with α=30°, L/R₁=2, and R₁/h=50.

Boundary conditions	VCN∗	Ω1	Ω2	Ω3	Ω4	Ω5
C-C	VCN∗=0.12	1.4271	2.8180	4.2163	5.6166	7.0178
	VCN∗=0.17	1.8510	3.6553	5.4690	7.2854	9.1029
	VCN∗=0.28	1.9789	3.9077	5.8467	7.7885	9.7315
C-F	VCN∗=0.12	0.4404	2.0485	3.4739	4.8858	6.2934
	VCN*=0.17	0.5713	2.6572	4.5060	6.3374	8.1633
	VCN∗=0.28	0.6108	2.8407	4.8172	6.7750	8.7270
F-C	VCN∗=0.12	1.0149	2.2518	3.5994	4.9762	6.3641
	VCN∗=0.17	1.3165	2.9208	4.6688	6.4547	8.2549
	VCN∗=0.28	1.4074	3.1225	4.9912	6.9004	8.8249
F-F	VCN∗=0.12	1.5210	2.8696	4.2513	5.6431	7.0390
	VCN∗=0.17	1.9728	3.7221	5.5144	7.3197	9.1304
	VCN∗=0.28	2.1091	3.9792	5.8952	7.8251	9.7609

In Table 5, the influence of grading profile on the torsional frequencies of conical shells are analyzed. Total volume fraction of CNTs is considered as VCN∗=0.28 which may be graded or uniformly distributed across the thickness. First five frequency parameters and four different cases of boundary conditions are taken into consideration. Geometric characteristics of the shell are α=30°, L/R₁=2, and R₁/h=50. It is observed that the influence of grading profile is almost negligible on the torsional frequencies of the shell. The differences between the frequency parameters of FG-X, FG-O, FG-V, and FG-Λ shells are almost negligible. However, FG-V has the highest frequency and FG-Λ has the lowest frequency. Frequency parameters of all graded shells are higher than the associated UD shells. However, discrepancy of frequency parameter between an FG-CNTRC and UR-CNTRC conical shell is small. This conclusion is different from the observations in flexural vibration analysis of Mirzaei and Kiani [24] for cylindrical panels and Selim et al. [20] for rectangular plates. In most of the available works on flexural vibration analysis of FG-CNTRC structures, it is mentioned that FG-X patten results in higher frequencies and also the graded pattern is an influential factor on frequencies. The reason is that Young’s modulus of the composite media is highly affected by the CNT volume fraction and, therefore, extensional and bending stiffnesses are also dependent on the graded pattern and volume fraction of CNTs. However, due to the severe difference between the shear modulus of polymeric matrix and CNTs, and according to Eq. (1) the effect of shear modulus of CNT on shear modulus of composite media almost disappears. As an example, the associated mode shapes for C-C and F-C shells are provided in Figures 3 and 4.

Table 5:

Torsional frequency parameter Ω=ωR1ρm/Gm for different CNTRC conical shells with VCN∗=0.28,α=30°, L/R₁=2, and R₁/h=50.

Boundary conditions	Grading profile	Ω1	Ω2	Ω3	Ω4	Ω5
C-C	UD	1.9261	3.8035	5.6908	7.5809	9.4721
	FG-V	1.9814	3.9127	5.8539	7.7978	9.7424
	FG-Λ	1.9762	3.9023	5.8383	7.7769	9.7164
	FG-O	1.9789	3.9077	5.8467	7.7885	9.7315
	FG-X	1.9789	3.9077	5.8466	7.7884	9.7314
C-F	UD	0.5945	2.7650	4.6887	6.5944	8.4943
	FG-V	0.6114	2.8443	4.8233	6.7833	8.7371
	FG-Λ	0.6100	2.8368	4.8104	6.7652	8.7138
	FG-O	0.6108	2.8407	4.8172	6.7750	8.7270
	FG-X	0.6107	2.8407	4.8171	6.7750	8.7269
F-C	UD	1.3699	3.0393	4.8582	6.7165	8.5897
	FG-V	1.4094	3.1267	4.9977	6.9090	8.8352
	FG-Λ	1.4053	3.1181	4.9841	6.8903	8.8115
	FG-O	1.4074	3.1225	4.9912	6.9004	8.8249
	FG-X	1.4074	3.1225	4.9912	6.9004	8.8249
F-F	UD	2.0529	3.8731	5.7380	7.6165	9.5007
	FG-V	2.1120	3.9844	5.9026	7.8345	9.7719
	FG-Λ	2.1060	3.9735	5.8867	7.8134	9.7457
	FG-O	2.1091	3.9792	5.8952	7.8251	9.7609
	FG-X	2.1091	3.9791	5.8952	7.8251	9.7608

Figure 3:

First five mode shapes of an FG-X CNTRC conical shell with C-C boundary conditions, VCN∗=0.28 and geometric characteristics, α=30°, L/R₁=2 and R₁/h=50.

Figure 4:

First five mode shapes of an FG-X CNTRC conical shell with F-C boundary conditions, VCN∗=0.28 and geometric characteristics, α=30°, L/R₁=2 and R₁/h=50.

The next parametric study analyzes the influence of shell vertex angle on the torsional frequencies of conical shells. Numerical results are tabulated in Table 6. Conical shells with FG-X distribution of CNTs and volume fraction VCN∗=0.28 are considered. Shell parameters are L/R₁=2 and R₁/h=50. Various values are considered for the semi vertex angle of the cone. It is observed that, for C-C, F-C, and F-F shells, torsional frequency increases as the semi-vertex angle increases, however, for the C-F shells trend is different. In C-F shells, torsional frequency decreases as the semi-vertex angle decreases and afterwards increases slightly with an increase in the semi-vertex angle. Such trend is observed in some other types of shell structures for instance conical panels [52].

Table 6:

Torsional frequency parameter Ω=ωR1ρm/Gm for different semi-vertex angles of FG-X CNTRC conical shells with VCN∗=0.28,L/R₁=2, and R₁/h=50.

Boundary conditions	α[°]	Ω1	Ω2	Ω3	Ω4	Ω5
C-C	0°	1.9448	3.8897	5.8345	7.7793	9.7242
	10°	1.9513	3.8929	5.8367	7.7810	9.7255
	20°	1.9644	3.8998	5.8413	7.7844	9.7282
	30°	1.9789	3.9077	5.8467	7.7885	9.7315
	40°	1.9924	3.9155	5.8520	7.7925	9.7347
	50°	2.0040	3.9224	5.8568	7.7962	9.7377
	60°	2.0131	3.9281	5.8608	7.7993	9.7402
C-F	0°	0.9724	2.9172	4.8621	6.8069	8.7517
	10°	0.8044	2.8701	4.8341	6.7870	8.7362
	20°	0.6907	2.8491	4.8219	6.7783	8.7295
	30°	0.6108	2.8407	4.8172	6.7750	8.7270
	40°	0.5533	2.8384	4.8161	6.7744	8.7265
	50°	0.5120	2.8390	4.8169	6.7750	8.7270
	60°	0.4829	2.8407	4.8183	6.7760	8.7279
F-C	0°	0.9724	2.9172	4.8621	6.8069	8.7517
	10°	1.1551	2.9883	4.9054	6.8380	8.7759
	20°	1.2973	3.0583	4.9494	6.8699	8.8009
	30°	1.4074	3.1225	4.9912	6.9004	8.8249
	40°	1.4915	3.1785	5.0288	6.9282	8.8468
	50°	1.5547	3.2249	5.0608	6.9521	8.8658
	60°	1.6004	3.2612	5.0865	6.9715	8.8812
F-F	0°	1.9373	3.9031	5.8573	7.8537	9.8959
	10°	1.9772	3.9061	5.8455	7.7876	9.7307
	20°	2.0409	3.9401	5.8684	7.8049	9.7446
	30°	2.1091	3.9792	5.8952	7.8251	9.7609
	40°	2.1710	4.0172	5.9217	7.8453	9.7771
	50°	2.2225	4.0510	5.9456	7.8636	9.7919
	60°	2.2624	4.0785	5.9654	7.8789	9.8043

5 Conclusion

Free torsional oscillations of an FG-CNTRC conical shell with combinations of free and clamped boundary conditions are considered in this research. Five different profiles are considered for the distribution of CNTs across the thickness. The properties of the composite reinforced media are obtained by means of a modified rule of mixtures approach. First-order shell theory and Donnell kinematic assumptions are incorporated with the Hamilton principle to construct the free motion equations of the conical shell and the associated boundary conditions. The obtained system of differential equations are solved using the generalized differential quadrature method. After examination of the efficiency and accuracy of the proposed formulation for isotropic homogeneous conical shells and annular plates, parametric studies are conducted for FG-CNTRC conical shells. It is shown that volume fraction of CNTs is an important factor with regard to natural frequency parameters. Generally higher volume fraction of fibers enhances the natural torsional frequency parameters. Furthermore, edge supports are also influential on the mode shapes and frequencies of the shell. Generally, F-F conical shell has the highest torsional frequency followed by, in order, F-C, C-F, and C-C conical shells. Vertex half angle of the cone is also influential on torsional frequency of the conical shell. However, influence of grading profile, as shown, is almost negligible. Generally FG-CNTRC conical shells have higher torsional frequencies in comparison to UD-CNTRC conical shells. The discrepancy between the frequency parameters of UD- and FG-CNTRC conical shells seems to be negligible. Unlike the flexural vibration modes, where FG-X pattern results in higher frequencies, in torsional modes, influences of graded patten is almost negligible; however, FG-V pattern results in higher frequencies.