A solution for free vibration of rotating hybrid multiscale nanocomposite conical shell

Abstract

In the present investigation, first-order shear deformation theory (FSDT) is employed to conduct free vibration analysis of hybrid CNT-fiber nanocomposite conical shells using finite element approach. The effective material properties of this three phase nanocomposite is computed using Halpin- Tsai and Micro-mechanics model approach. The Lagrange’s equation of motion is used to derive the equilibrium equations of the rotating carbon nanotube – fibre nanocomposite conical shell wherein the nonlinearity due to rotation is also introduced. The Coriolis effect is neglected because of the moderate rotational speed of the shell. Finite element code is developed using eight noded isoparametric shell element and the same is validated with some literature to analyse the effect of twist angle, length to thickness ratio, aspect ratio, weight fraction of CNTs, and rotational speed on the fundamental frequency. The effect of such parameter on the mode shapes of the three phase nanocomposite shell are also presented here.

Keywords

Three phase hybrid composite FEM conical shell CNT

Introduction

Recently an emerging material in nano science has emerged called Carbon nanotubes (CNTs). It has excellent mechanical properties like low weight to high strength ratio, high modulus and high strength (around - 1 TPa).^1–3 Owing to the further revolution in additive manufacturing process and reinforcement techniques, CNTs have attained a huge interest as an ideal material among the scientist and engineer’s community to be used as reinforcement material with carbon fiber or in epoxy matrix. With further enormous development of nano technology, CNT-fiber is reinforced to form multiscale hybrid CNT –fiber nanocomposites. It is considered as one of most advanced form of composites that have wide range of applications^1–3 like especially in jet engine structures, advanced naval warships and aerospace applications. This new form of nanocomposite offers a great level of stiffness and wear resistance under adverse working conditions. In order to ensure safe running operation in applications like jet propel engine, both free and forced vibration analyses are required to avoid unscheduled shut down of the machineries. Some of the well-known early researches on CNT nanocomposite along with some essential recent works of this new breed- CNT composite are discussed in the following section.

A very recent literature review of the vibration on CNTRC material was carried out by Liew et al.^1,2 According to the open source literature, most of the work on hybrid CNT-fiber nanocomposite was initially based on the characterization of CNT and its parent composite matrix.^3–6 A very limited number of established works were carried out on the vibration aspect of hybrid CNT-fiber nanocomposite material. Rafiee et al.⁷ conducted a nonlinear free vibration analysis of hybrid CNT nanocomposite plate with piezoelectric layers on both sides, where carbon nanotubes are uniformly and randomly distributed in the epoxy resin matrix. Kamarian et al.⁸ used Mori-Tanaka method to compute the effective material properties and performed the vibration study with the help of generalized differential quadrature (GDQ) methods. Seidi and Kamarian⁹ performed similar study on non-uniform hybrid CNT nanocomposite beams and examined the effect of CNT agglomerations, different laminate lay-up sequence and the volume fractions of CNT on the fundamental frequency. While Ebrahimi and his research group^10–14 performed stability analysis on the hybrid CNT nanocomposite structures. They performed vibration analysis on hybrid CNT nanocomposite beam by varying the temperature¹⁰ and concluded from their studies that the multiscale hybrid composite beams are stiffer than carbon-fiber or carbon nanotube reinforced composite (CNTRC) materials. Later on, in another investigation by the same group, Halpin-Tsai homogenization theory is used by them to conduct similar analysis on the multi scale CNT nanocomposite plate.¹¹ Apart from this, the same group also reported the free vibration studies considering the agglomeration of CNT particle on hybrid CNT nanocomposite plates,¹² stability investigation on the hybrid CNT nanocomposite plates¹³ and on beams.¹⁴ However, considering the agglomeration effect of CNT on the fundamental frequency, several investigations were reported.^{3–6,15–16} Ghasemi et al.¹⁵ scrutinized the effect of agglomeration of CNTs on the multiscale hybrid CNT nanocomposite cylindrical shell. Similar analyses were also carried out on truncated conical shell made up of multiscale hybrid CNT nanocomposite material by amir et al.¹⁶ Furjan et al.¹⁷ developed a three dimensional model of circular and annular plate composed of hybrid CNT multiscale composite and analyzed the vibration response using state-space based differential quadrature theory. A related work was also reported by Zhou et al.¹⁸ considering a sandwich structure consisting of circular and annular plates. Rezaiee et al.¹⁹ examined the vibration response of conical shell made up of multi scale hybrid CNT composite using multi scale methodology, wherein the authors investigated the effects of the nanofillers aggregation. Rafie et al.²⁰ performed the nonlinear stress analysis of a piezolaminated hybrid CNT nanocomposite plate subjected to electrical and mechanical loading. However, Tornabene et al.²¹ used Eshelby–Mori–Tanaka scheme to consider agglomeration of the nanoparticles in the matrix of the three phase hybrid CNT nanocomposite and investigated the frequency response of annular plate, cylindrical and conical shell using GDQ method. Safarpour et al.²² carried out similar investigation on annular plate and concluded that by including more CNTs at the bottom surface of the annular plate results more stiffness in the structures. Gholami et al.²³ studied the nonlinear frequency response of hybrid CNT nanocomposite rectangular plate by employing time periodic discretization rule and the pseudo-arc length continuation methodology. Lee ²⁴ measured the dynamic behaviour of multi scale three phase nanocomposite skew plate considering delamination. The buckling behaviour of a hybrid CNTs multiscale nanocomposite rod was reported by Ahmadi et al.²⁵

In the present literature survey and to the best of author’s knowledge, none of the investigators have reported the free vibration response of rotating structure made of multiscale hybrid CNT nanocomposite material. In general, the blades of the jet propulsion engine are subjected to vibrate under the effect of rotation. The study of vibration response of rotating turbomachinery blade made of such a new material is of prior importance. The blade made up of this multiscale hybrid composite can be idealized as a twisted plate/cylindrical/conical shell. But it may be seen from the literature that twisted conical shell is the best choice among the researchers for the said purpose. Also, the works on vibration of the conical shell structure composed of hybrid CNT nanocomposite are also very limited. Hence, the authors have categorically reported the free vibration response of rotating hybrid CNT nanocomposite pretwisted conical shell in the present article. The most well-known early research on the vibration of isotropic plates considering rotation effect was carried out by Sreenivasamurthy and Ramamurti.²⁶ Here the rotating effects of the cantilever isotropic plates on the fundamental frequency were studied, neglecting the Coriolis effect. This study was important as far as the vibration of rotating plates was concerned wherein it is reported that the effect of Coriolis component is negligible at moderate rotational speed. While vibration study on twisted plates was carried out by Leissa et al.²⁷ However, the vibration behaviour of the pretwisted rotating composite shell considering the delamination effect was reported by Karmakar and Kishimoto.²⁸ A similar analysis for a rotating pretwisted cantilever plates was presented by Karmakar and Sinha.²⁹ However, Liew et al.³⁰ conducted the vibration analysis of pretwisted cantilever conical composite shells. While, using three dimensional modelling of composite blades, the free vibration study was reported by McGee and Chu³¹ considering centrifugal effect of the blade. Rout et al. also did several investigations on the free vibration of different rotating structures; such as composite stiffened shallow shells considering delamination,³² pretwisted stiffened cylindrical shell³³ and stiffened plate.³⁴ Recently, Maji et al.³⁵ conducted the free vibration analysis of rotating functionally graded carbon nanotube reinforced composite conical shell under varying thermal loading conditions employing finite element method. Rout et al.³⁶ also reported the time variant response of the carbon nanotube reinforced composites (CNTRC) plate subjected to low velocity impact and studied the time variant response of the material under various boundary conditions.

In the present study, both twisted and untwisted conical shell geometries are selected to carry out the free vibration analysis of hybrid CNT nanocomposite material under rotating conditions. Here, the first-order shear deformation theory (FSDT) is used to compute the strains and using eight noded isoparametric elements, finite element code is developed. Using Lagrange’s equation of motion, dynamic equations of the present problem were formulated, neglecting the Coriolis effect. The present investigation on the fundamental frequency is carried out by varying various parameters like twist angle (Ø), aspect ratio (s/h) and cone length to thickness (L/s) ratio along with weight percentage of CNTs with rotation effect. The results have been obtained and shown in the form of graphs and tables. The mode shapes of the shell are also presented.

Theoretical formulation

In Figure 1 shows the geometrical configuration of both twisted and untwisted conical shell. The structural parameters like the twist angle (Ø), thickness (h), vertex angle (θ_v) and base angle (θ₀), minor radius (β₀), length (L), major radius (a₀), cone length (s), and the base width (b₀) are illustrated in the diagram.

Figure 1.

The schematic diagram of conical shell: A untwisted conical shell configuration.

The formulation for obtaining the varying radius of curvature (Ry) of the conical shell in the co-ordinate system is explained in Maji et al.³⁵ The present investigation is carried out on both the pretwisted and untwisted conical shell geometry. The z-coordinate of the conical shallow shell can be defined as

z = - \frac{1}{2} {2 \frac{x y}{R x y} + \frac{y^{2}}{R y}}

(1)

The relation between the twist angle (Ø), length of the shell (L) and the twisted radius of curvature (R_XY) can be defined as

\tan \emptyset = - \frac{L}{R_{X Y}}

(2)

Hybrid CNT nanocomposite

This new breed of multiscale three phase composite has evolved by mixing two phases in it. One phase is the continuous fiber and another is the matrix phase. In the fiber phase, it is E−glass fibers and matrix phase consists of two materials, one is epoxy-resin and the other is carbon nanotube (CNT). In order to understand the process of obtaining the hybrid composite, a schematic diagram is illustrated in Figure 2. This new material is formed by applying the principle of dispersion at nano-inclusion level in the continuous phase, which results into enhancement of mechanical properties. To predict the elastic equations of this newly breed material, both Halpin- Tsai and Micro-mechanics model approach are used.^7,39 In order to satisfy the isotropic property rule of mechanics of composite, it has been assumed that the CNTs are uniformly distributed in the matrix phase and the bonding between the two phases (fiber-matrix phase) should be perfect. The CNTs are assumed to be isotropic in nature and it is assumed that the newly formed multiphase hybrid CNT fiber nanocomposite is a void free structure. This newly formed three phase multiscale CNT fiber nanocomposite forms a laminated structure. Each laminates are oriented in different angle and are stacked upon each other along the thickness direction.^7,37,38

E_{m C N T} = \frac{E_{m}}{8} (5 (\frac{1 + 2 β_{d d} V_{C N T}}{1 - β_{d d} V_{C N T}}) + 3 (\frac{1 + 2 (\frac{l_{C N T}}{d_{C N T}}) β_{d l} V_{C N T}}{1 - β_{d l} V_{C N T}}))

(3)

where ‘E_mCNT’represents the Young’s modulus of the nano inclusive composites, E_m stands for the Young’s modulus of the metal and Where β_dl and β_dd can be evaluated from the following relations

β_{d l} = \frac{(\frac{E_{C N T}}{E_{m}}) - (\frac{d_{C N T}}{4 t_{C N T}})}{(\frac{E_{C N T}}{E_{m}}) + (\frac{l_{C N T}}{2 t_{C N T}})}; β_{d d} = \frac{(\frac{E_{C N T}}{E_{m}}) - (\frac{d_{C N T}}{4 t_{C N}})}{(\frac{E_{C N T}}{E_{m}}) + (\frac{d_{C N T}}{2 t_{C N T}})}

(4)

Figure 2.

A schematic representation of modeling three phase multiscale nanocomposite material.

‘E_CNT’ represents the Young’s modulus of carbon nanotubes, ‘V_CNT’ is the volume fraction of the carbon nanotubes, whereas l_CNT, d_CNT, t_CNT are the length, outer diameter and the thickness of the CNTs, respectively. The volume fraction of the carbon nanotube ‘V_CNT’ can be expressed in terms of weight fraction ‘w_CNT’, density of the carbon nanotube ‘ρ_CNT’ and the density of the matrix ‘ρ_m’ in the following relation.

V_{C N T} = \frac{W_{C N T}}{W_{C N T} + (\frac{ρ_{C N T}}{ρ_{m}}) - (\frac{ρ_{C N T}}{ρ_{m}}) W_{C N T}}

(5)

The relation of the Poisson’s ratio and the density of the hybrid nanocomposite can also be written from the rule of mixture³⁹ as

υ_{m C N T} = υ_{C N T} . V_{C N T} + υ_{m} . (1 - V_{C N T})

(6)

ρ_{m C N T} = ρ_{C N T} . V_{C N T} + ρ_{m} . (1 - V_{C N T})

(7)

where ‘υ’ are the effective property of the Hybrid CNT metal-ceramic material and can be ascertained by considering the Vogit rule of mixture and Mori-Tanaka approach. The shear modulus of the matrix phase and the Poisson’s ratio can be co-related as

G_{m} = \frac{E^{m}}{2 (1 + υ^{m})},

(8)

υ_{12}^{m} = υ_{m},

(9)

where in the term ‘υ’ is the Poisson ratio. The subscript ‘m’ represents the matrix phase of the hybrid multiscale nanocomposite. Following the micromechanics model approach, the elastic equations of the hybrid CNT nanocomposite plate can be obtained as³⁹

E_{11} = E_{11}^{f} V_{f} + E^{m C N T} V^{m C N T}

(10)

\frac{1}{E_{22}} = \frac{V^{f}}{E_{22}^{f}} + \frac{V^{m C N T}}{E^{m C N T}} - V^{f} V^{m C N T} (\frac{\frac{{(υ^{f})}^{2} E^{m C N T}}{E_{22}^{f}} + \frac{{(υ^{m C N T})}^{2} E_{22}^{f}}{E^{m C N T}} - 2 υ^{f} υ^{m C N T}}{υ^{f} E_{22}^{f} + υ^{m C N T} E^{m C N T}})

(11)

\frac{1}{G_{12}} = \frac{V^{f}}{G_{12}^{f}} + \frac{V^{m C N T}}{G^{m C N T}}

(12)

υ_{12 h} = V_{F} υ^{F} + V_{m C N T} υ^{m C N T}

(13)

ρ_{h} = V_{F} ρ^{F} + V_{m C N T} ρ^{m C N T}

(14)

where ‘ $E_{11}^{f}$ ’ and ‘ $E_{22}^{f}$ ‘are the Young’s modulus in longitudinal and the transverse directions of the fiber. Here the superscript ‘f’ and ‘mCNT’ represents the material property for the glass fiber and matrix- CNT phase respectively. ‘G’, ‘υ’ and ‘V’ represents the shear modulus, Poison’s ratio and the volume fraction respectively. The material property for the Single walled carbon nanotube (SWCNT) used in the present study is given in Table 1.

Table 1.

CNT material physical properties.

CNT	E_CNT (GPa)	v_CNT	l_CNT (μm)	d_CNT (nm)	t_CNT (nm)	ρ_CNT (kg/m²)
SWCNT	640	0.33	25	1.4	0.34	1350

Finite element formulations

In the present analysis, the hybrid nanocomposite conical shell is discretized with eight noded isoparametric shell elements having five degree of freedom per node. Of which three are translational (u,v,w) and two are rotational ( $θ x$ , $θ y$ ). The shape function which has been derived from its polynomial function and expressed as⁴⁰

N_{i} = (1 + ξ ξ_{i}) (1 + η η_{i}) \frac{(ξ ξ_{i} + η η_{i} - 1)}{4} (f o r i = 1, 2, 3, 4)

(15)

N_{i} = \frac{(1 + ξ ξ_{i}) (1 + η^{2})}{2} (f o r i = 5, 7)

(16)

N_{i} = \frac{(1 + η η_{i}) (1 - ξ^{2})}{2} (f o r i = 6, 8)

(17)

where the displacement field (u,v,w) in the rectangular co-ordinate system are evaluated with respect to its mid plane considering the first-order shear deformation theory (FSDT). The expression of the displacement field is given by

u (x, y, z) = u^{0} (x, y) + z θ x (x, y)

(18)

v (x, y, z) = v^{0} (x, y) + z θ y (x, y)

(19)

w (x, y, z) = w^{0} (x, y)

(20)

where ‘u₀’, ‘v₀’ and ‘w₀’ are the mid plane displacements, whereas ‘θ_x’ and ‘θ_y’ are the rotations along x-axis and y-axis respectively. The strain-displacement relation can be obtained from the displacement field as

{ε} = {ε}^{0} + z {k}

(21)

where ‘ε⁰ ‘are the mid plane strains and ‘k’ represents the curvatures of the conical shell. The stress-strain relation can be ascertained from the elastic constant matrix [Q] and strain equation as

{σ} = {Q} {ε}

(22)

Considering the hybrid CNT-fiber nanocomposite laminate to behave like an isotropic material. The elasticity constants [Q] matrix can be represented by the following expression as

[Q] = [\begin{matrix} Q_{11} & Q_{12} & 0 & 0 & 0 \\ Q_{12} & Q_{22} & 0 & 0 & 0 \\ 0 & 0 & Q_{66} & 0 & 0 \\ 0 & 0 & 0 & k . Q_{44} & 0 \\ 0 & 0 & 0 & 0 & k . Q_{55} \end{matrix}]

(23)

The total linear strain energy of the conical shell can be expressed as

U = \frac{1}{2} \int {ε}^{T} {σ} d v

(24)

where ‘ε’ is the strain vector, ‘σ’ is the stress obtained from the strain vector. On solving equation (24) by putting the values of stress and strain

U = \frac{1}{2} {δ_{e}}^{T} [K_{e}] {δ_{e}}

(25)

where $[K_{e}]$ is the elemental stiffness matrix and ${δ_{e}}$ is the elemental displacement vector. The elemental stiffness matrix derived from the strain energy is expressed as

[K_{e}] = \int_{A} {[B]}^{T} [D] [B] d A

(26)

where [B] is the strain displacement matrix and $[D]$ represents the elasticity matrix. The [D] matrix can be expressed in the following equation as

[D] = [\begin{matrix} A_{i j} & B_{i j} & 0 \\ B_{i j} & D_{i j} & 0 \\ 0 & 0 & S_{i j} \end{matrix}]

(27)

(A_{i j}, B_{i j}, D_{i j}) = \sum_{k = 1}^{n l} \int_{z_{k - 1}}^{z_{k}} {(\bar{Q_{i j}})}_{k} (1, z, z^{2}) d z, i, j = 1, 2, 6

(28)

(S_{i j}) = k_{s} \sum_{k = 1}^{n l} \int_{z_{k - 1}}^{z_{k}} {({\bar{Q}}_{i j})}_{k} d z, i, j = 4, 5

(29)

where ‘nl’ represents the number of layers and ‘k_s’ represents the shear correction factor and is taken as 5/6. The kinetic energy of the shell element can be expressed as

V = \frac{1}{2} \int ρ {\bar{V}}^{2} d v

(30)

On putting the value of velocity, the kinetic energy equation becomes

V = \frac{1}{2} {δ_{e}}^{T} [M_{e}] {δ_{e}} + {δ_{e}}^{T} [F_{c e}]

(31)

Here [ $[M_{e}]$ ] and [ $[F_{c e}]$ ] are the elemental mass matrix and centrifugal force vector due to rotation of the shell,²⁶ respectively. The elemental mass matrix of the shell element is expressed as

[M_{e}] = \int_{A} {[N]}^{T} [m] [N] d A

(32)

where [m] is the inertia matrix and [N] is the shape function matrix.

The elemental centrifugal force vector can be obtained as

F_{c e} = \int ρ {[N]}^{T} [A_{x}] {\begin{matrix} x \\ y \\ z \end{matrix}} d v

(33)

The matrix [A_x] contains the angular velocity components

[A_{x}] = [\begin{matrix} Ω_{z}^{' 2} & 0 & 0 \\ 0 & Ω_{z}^{' 2} & 0 \\ 0 & 0 & 0 \end{matrix}]

(34)

where $Ω_{z}^{'}$ is the rotational component in z-coordinate. The extra strain energy stored due to the initial stresses developed for rotation can be expressed as²⁶

U_{0} = \frac{1}{2} {δ_{e}}^{T} (K_{G}) {δ_{e}}

(35)

where ‘K_G’ represents the geometric stiffness matrix due to rotation effect. In the equation (35) Green Langrange nonlinear strains are considered for computation of extra strain energy due to rotation. The geometric stiffness matrix can be given as

[K_{G}] = \int {[G]}^{T} [M_{σ R}] [G] d A

(36)

where $[M_{σ R}]$ is the matrix of in plane stress resultants developed during rotation. By applying the Lagrange’s equation of motion for moderate rotation speed, neglecting Coriolis effect, the dynamic equilibrium equation can be obtained as

[M] {\overset{\cdot\cdot}{δ}} + (K^{,}) {δ} = {F_{C}}

(37)

where [K’] is the effective stiffness matrix and can be expressed as

[K] + [K_{G}] = [K^{,}]

(38)

In equation (37), the matrices [M], [K], [K_G] and {F_c} represent the global mass matrix, global elastic stiffness matrix, global geometric stiffness matrix and global centrifugal force vector, respectively.

The above equation (37) is solved iteratively to find the natural frequency of the system.^32–35 It is observed that after three iterations the result converges. The natural frequencies ( $ω_{n}$ ) are computed by using standard Eigen value solver algorithm.

Results and discussions

Based on the discussed finite element methodology, the free vibration behavior of hybrid CNT nanocomposite conical shell under both rotating and static conditions has been investigated. Numerical results are obtained and shown here in both tabular and graphical form. Convergence study is also carried out under various mesh size as shown in Figures 3(a)–(d). In the convergence study, the results are obtained for mesh size 6x6 to 12 x 12 with increment of one element in each direction in each case. It may be seen in Figures 3(a) that the percentage difference between the results of mesh size 6x6 and 12x12 is less than one percent. On adding the CNT content in Table 2, the difference in the value of frequency on increasing the mesh size is negligible. The convergence results for second and third mode frequency shows the same characteristics in Figures 3(c) and (d), i.e. less than 1 percentage difference in the value from 7x7 mesh size onwards. So to save the computation time, 8x8 mesh size has been selected to carry out the present analysis. It consists of 64 elements and 225 numbers of nodes. The geometrical parameters of the conical shell made of hybrid nanocomposite considered for the entire analysis are furnished below:

Figure 3.

Mesh convergence results of the three phase multiscale CNT fiber nanocomposite conical shell having L = 1.0 m, w = 1.0 m, s/h = 10, l/s = 0.7, ø = 0° for (a) W_CNT = 0.0 (b) W_CNT = 0.01 (c) W_CNT = 0.0, 2^nd mode frequency (d) W_CNT = 0.0, 3^rd mode frequency.

Table 2.

Fundamental natural frequencies (rad/s) of a square [0/90]s symmetric cross ply simply supported CNTFPC plates corresponding to CNT weight fraction (WCNT) and length-thickness ratios (L/h), h = 100 mm.

$w_{c n}$	L/h
	10			20			100
	Rafiee et al [7]	Lee [24]	Present FEM	Rafiee et al [7]	Lee [24]	Present FEM	Rafiee et al [7]	Lee [24]	Present FEM
0.0	2617.85	2619.41	2567.42	680.44	682.06	662.58	27.574	27.678	26.795
1%	2954.59	2936.91	2932.19	761.80	760.92	752.39	30.783	30.825	30.367
2%	3197.95	3185.46	3195.96	822.07	822.70	818.33	33.185	33.314	33.005

Length of the shell (L) = 0.01 m, cone length (s) = 1.0 m, base width (w) = 0.01 m, θ₀ = 30°and θ_v = 15°.

Validation of results

The present investigation on hybrid CNT fiber nanocomposite conical shell is thoroughly validated in all aspect with the available literature. Table 2 shows the validation of the hybrid CNT fiber nanocomposite plate, wherein the natural frequency of a simply supported squarecarbon nanotube fiber polymer composite (CNTFPC) plate is obtained corresponding to different weight fraction of CNT and different L/h ratio. The computed results are in good agreement with the results of Rafiee et al.⁷ and Lee.²⁴ Since the author has introduced the rotation effect, henceforth the validation of the same has been performed by solving one example, which was earlier solved by Sreenivasamurthi and Ramamurthi.²⁶ Table 3 shows the results of the rotating isotropic plate and the computed results are in close agreement. Table 4 shows the results of non-dimensional fundamental frequency of pretwisted conical shell, wherein the evaluated results show good accuracy with that of Liew et al.³⁰ In all the solved examples, it has been observed that the current model shows very small deviations from the previously found results and henceforth, the present formulation can successfully compute the free vibration response of the rotating hybrid CNT nanocomposite pretwisted conical shell.

Table 3.

Non-dimensional fundamental frequency ( $ϖ = ω L^{2} \sqrt{ρ h / D}$ ) of an isotropic rotating cantilever plate. (L/b = 1, h/L = 0.12, $D = E h^{3} / 12 (1 - ν^{2})$ and ν = 0.3).

Non-dimensional rotational speed $Ω = Ω^{'} / ω_{n}$	Sreenivasamurthy and Ramamurthi [26]	Present FEM
0.0	3.43685	3.41683
0.2	3.51858	3.49704
0.4	3.75280	3.72663
0.6	4.12875	4.07865
0.8	4.56786	4.52205
1.0	5.09167	5.02989

Table 4.

Non-dimensional fundamental frequencies, $ϖ = ω b_{0}^{2} \sqrt{(ρ h / D)}$ , $D = E h^{3} / 12 (1 - ν^{2})$ for the pretwisted shallow conical shell with $ν$ = 0.3, s/h = 1000, $θ_{v}$ = 15⁰and $θ_{0}$ = 30⁰.

Ø	L/s	Liew et al [30]	Present FEM
0°	0.6	0.35997	0.36013
	0.7	0.30608	0.30619
	0.8	0.27832	0.27840
15°	0.6	0.34116	0.34150
	0.7	0.29410	0.29438
	0.8	0.26964	0.26986
30°	0.6	0.28828	0.28901
	0.7	0.25752	0.25821
	0.8	0.24179	0.24242
45°	0.6	0.21015	0.21117
	0.7	0.19603	0.19714
	0.8	0.19070	0.19185

Numerical solution

In the present investigation, free vibration behavior of rotating hybrid CNT multiscale nanocomposite pretwisted conical shell is examined under cantilever boundary conditions. The effect of changing the structural parameters on the fundamental frequencies is explored. The parameters that are changed in the present analysis are the twist angle, cone length to thickness (s/h) ratio, ply length to cone length (L/s) ratio, weight fraction of CNTs and the non-dimensional rotation speed (NRS). The boundary condition for the cantilever pretwisted shell is defined as:

At $x = 0, u = v = w = θ_{x} = θ_{y} = 0$

In the current study, four layered symmetrical cross ply (0°/90°/90°/0°) of carbon nanotube fiber-polymer composite (CNTFPC) plates have been considered. The material properties of the hybrid CNT fiber nanocomposite considered are as follows: Epoxy resin coupled with single walled carbon nanotube (SWCNT) forms the matrix phase with the property values SWCNT depicted in Table 1. Whereas the Epoxy resin material property that are used in the present study are furnished below:

E_m = 2.72 GPa, υ_m = 0.33 and ρ_m = 1200 kg/m².

On the other side, the fiber phase that is used exclusively is E-Glass fiber, whose material proporties are E_f = 69 GPa, υ_f = 0.2, ρ_f = 1200 kg/m² and the V_f = 0.8. The total thickness of the plate is 0.1 m unless otherwise specified.

Effect of W_CNT and rotation

Table 5 shows the effect of weight fraction (W_CNT) of carbon nanotube and rotational speed on the fundamental frequencies of the pretwisted multiscale hybrid nanocomposite conical shells. The fundamental frequencies have been computed for increasing the value of weight fraction of CNT (W_CNT) for different value of twist angles viz 00,150,300 and 450, respectively. The effect of the non-dimensional rotation of speed (NRS)(NRS = Ω/ωn, where ω_n is the natural frequency of the stationary conical shell, Ω = actual speed of rotation) on the fundamental frequencies are also included here. It is evident that for increasing the twist angle of the pretwisted conical shell, fundamental frequency decreases. Henceforth, the twisted conical shell is considered to be less stiff than its untwisted case. However, on increasing the W_CNT, the fundamental frequency increases in each of the respective cases. Therefore, on increasing the carbon nanotube content the stiffness of the conical shell increases. Also, in case of rise in NRS, the value of fundamental frequency increases. This may have occurred due to the centrifugal stiffening of the conical shell. It may be observed that the percentage increase in fundamental frequency due to rotation is higher in twisted shells as compared to untwisted shells. For example, at NRS = 0.6 and W_CNT = 0.02, the percentage increase in fundamental frequency in untwisted case is 22.26 while the value in twisted case (Ø = 30⁰) is 22.41. Similar observation can be seen when the shell is enriched with maximum W_CNT, which means that at a fixed value of NRS, the percentage increase in fundamental frequency is found higher in the shell having maximum value of WCNT.

Table 5.

Variation of fundamental frequencies of hybrid CNT fiber nanocomposite conical shell having s/h = 10, L/s = 0.7 and L = 0.01 m.

Pretwist angle (Ø)	NRS	W_CNT
Pretwist angle (Ø)	NRS	0.0	0.01	0.02	0.04
0°	0.00	138.66	146.71	152.86	161.85
	0.20	142.46	150.73	157.05	166.30
	0.40	153.21	162.14	168.94	178.90
	0.60	169.44	179.35	186.89	197.91
	0.80	189.54	200.65	209.11	221.46
	1.00	212.19	224.68	234.17	248.02
15°	0.00	138.11	146.09	152.18	161.07
	0.20	141.90	150.11	156.36	165.50
	0.40	152.63	161.48	168.22	178.06
	0.60	168.81	178.64	186.11	197.02
	0.80	188.84	199.88	208.26	220.49
	1.00	211.42	223.83	233.24	246.95
30°	0.00	136.19	143.94	149.81	158.35
	0.20	139.94	147.92	153.95	162.73
	0.40	150.56	159.17	165.68	175.15
	0.60	166.57	176.14	183.37	193.87
	0.80	186.39	197.15	205.26	217.04
	1.00	208.71	220.82	229.93	243.15
45°	0.00	131.73	138.99	144.40	152.21
	0.20	135.39	142.87	148.44	156.47
	0.40	145.75	153.84	159.86	168.54
	0.60	161.35	170.37	177.07	186.72
	0.80	180.65	190.81	198.34	209.19
	1.00	202.39	213.82	222.30	234.50

Effect of L/s ratio

The effect of shell length to cone length ratio (L/s) on the fundamental frequency of the hybrid nanocomposite conical shell is illustrated in Figure 4. Here the effect of L/s ratio is predicted in two different ways. In first case, the effect is studied on the shells having different values of s/h ratio while in second case; it is investigated for the shells having different values of weight fraction of CNT. It is evident from both cases that on increasing the L/s ratio, the fundamental frequency decreases irrespective of the values of s/h and W_CNT. But the decrease in fundamental frequency is found rapid in the shell having lower value of s/h. Increasing the value of W_CNT only increases the fundament frequency but the reduction pattern of fundamental frequency due to increase in L/s ratio is following the same trend in all the four the shells having different values of W_CNT.

Figure 4.

Variation of the fundamental frequencies of hybrid multiscale CNTs composite with respect to L/s ratio. ø = 0°, L = 0.01 m and s/h = 10.

Effect of Twist Angle (Ø)- The effect of twist angle (Ø) of the conical shell on first four fundamental frequencies has been studied on four different shells having 0.0, 0.01, 0.02 and 0.04 weight fraction of CNTs respectively. The results are shown in Table 6. It may be noted from Figure 5 that on increasing the twist angle, the first fundamental frequency decreases in each shell irrespective of the weight fractions of CNT. It is also evident from Table 6 that on increasing the weight percentage of CNT, the stiffness of the pre twisted conical shell structure increases. It may be seen that on increasing the twist angle from 0° to 45° with an increment of 15°, the percentage decrease of fundamental frequencies for W_CNT = 0 are 0.40,1.79 and 5.00 corresponding to twist angles 15⁰, 30⁰ and 45⁰, respectively. The percentage decrease of fundamental frequency is calculated with respect to untwisted shell. In case of W_CNT = 0.01, the percentage of decrease of first fundamental frequencies are 0.42, 1.88 and 5.26 respectively while in case of W_CNT = 0.02, the percentage of decrease are 0.44, 2.00, 5.54. Finally, in case of W_CNT = 0.04, and the percentage decrease of first fundamental frequencies are 0.48, 2.16 and 5.95 corresponding to twist angles 15⁰, 30⁰ and 45⁰, respectively. It is evident that the percentage decrease in fundamental frequency due to twist angle is found maximum and minimum in the shells having W_CNT = 0.04 and 0.0 respectively. Hence it can be said that twist angle has striking effect on the shells reinforced with carbon nanotubes. Higher is the weight fraction of carbon nanotubes, the effect will be more striking.

Table 6.

Variations of first four fundamental frequencies of conical shell hybrid multi scale composite on twist angle (ø), taking s/h = 10, L/s = 0.7, L = 0.01 m and W_CNT = 0.04.

WCNT	phi(Ø)	ω1	ω2	ω3	ω4	% Deviation in ω1 on increasing Ø
0.0	0⁰	138.66	284.57	550.40	867.68	—
	15⁰	138.11	285.53	550.12	867.70	0.40
	30⁰	136.19	288.94	549.15	867.76	1.79
	45⁰	131.73	297.02	546.90	867.90	5.00
0.01	0⁰	146.71	292.12	586.48	1089.90	—
	15⁰	146.09	293.18	586.23	1089.91	0.42
	30⁰	143.94	296.92	585.35	1089.96	1.88
	45⁰	138.99	305.77	583.31	1090.09	5.26
0.02	0⁰	152.86	296.85	612.80	1241.80	—
	15⁰	152.18	298.00	612.54	1241.81	0.44
	30⁰	149.81	302.06	611.64	1241.86	2.00
	45⁰	144.40	311.63	609.55	1241.97	5.54
0.04	0⁰	161.85	303.49	650.47	1400.21	—
	15⁰	161.07	304.79	650.19	1400.08	0.48
	30⁰	158.35	309.36	649.20	1399.53	2.16
	45⁰	152.21	320.05	646.89	1397.58	5.95

Figure 5.

Variation of first fundamental frequency of conical shell hybrid multiscale CNT fiber nanocomposite on twist angle (ø) for both MWCNT in Figure 4a and SWCNT in Figure 4b along with their W_CNT, s/h = 10, L = 0.01 m and l/s = 0.7.

Mode shapes-The effect of the rotational speed on the first three mode shapes of twisted and untwisted conical shell are furnished in Table 7. For the untwisted conical shell, the first mode shape is span wise bending mode (1B) and the second mode shape is the second span wise bending mode (2B), while the third mode shape is the first torsion mode (1T). For the 15° twisted conical shell, the slope in nodal lines of first and second modes are observed but the effect of rotational speed is not significant. This may be due to 0.04 weight fraction of carbon nanotubes, which makes the conical shell stiff enough to resist the deformation arises on account of moderate rotational speed.

Table 7.

The mode shapes of the hybrid CNT multi scale composite conical shell plotted for both untwisted and twisted (ø = 15°) having L/s = 0.7, l/h = 10, L = 0.01 m and W_CNT = 0.04.

Twist Angle (°)	NRS	1st mode	2nd mode	3rd mode
0°	0
	0.2
	0.4
	0.6
	0.8
	1
15°	0
	0.2
	0.4
	0.6
	0.8
	1

Conclusions

This paper presents the free vibration characteristics of rotating hybrid CNT multiscale nanocomposite conical shell employing finite element method. Parametric study is presented and the important conclusions drawn are as follows:

(a) On increasing the rotational speed of the hybrid CNT multiscale nanocomposite conical shell, the fundamental frequencies increases. However, the percentage increase in fundamental frequency is found more in the shell containing more weight fraction of CNT.

(b) On increasing the twist angle, the fundamental frequencies decreases. However, the decrease in fundamental frequency is found less when weight fraction of CNT is more.

(c) On increasing the weight fraction of CNT the fundamental frequencies increase. However, the stiffness of the twisted conical shell can be enhanced by increasing the weight fraction of CNT.

(d) On increasing the aspect ratio of the hybrid CNT multiscale nanocomposite conical shell, fundamental frequencies decreases. The net drop in fundamental frequencies is more prominent in thin shell than that of thick shell.

(e) The effect of moderate rotational speed on mode shape is found insignificant. The presence of carbon nanotubes in the shell dominates the effect of moderate rotational speed on mode shapes.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Ranojit Banerjee

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A solution for free vibration of rotating hybrid multiscale nanocomposite conical shell

Abstract

Keywords

Introduction

Theoretical formulation

Hybrid CNT nanocomposite

Finite element formulations

Results and discussions

Validation of results

Numerical solution

Effect of WCNT and rotation

Effect of L/s ratio

Conclusions

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References

Effect of W_CNT and rotation