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Free vibration analysis of functionally graded carbon nanotubes with variable thickness by differential quadrature method
Physica E 43 (2011) 1602–1604
Contents lists available at ScienceDirect
Physica E journal homepage: www.elsevier.com/locate/physe
Free vibration analysis of functionally graded carbon nanotubes with variable thickness by differential quadrature method Maziar Janghorban a,n, Amin Zare b a b
Department of Mechanical Engineering, Shiraz Branch, Islamic Azad University, Shiraz, Iran Young Researchers club, Shiraz Branch, Islamic Azad University, Shiraz, Iran
a r t i c l e i n f o
a b s t r a c t
Article history: Received 27 January 2011 Received in revised form 24 April 2011 Accepted 2 May 2011 Available online 1 June 2011
Free vibration analysis of functionally graded carbon nanotube with variable thickness based on the Timoshenko beam theory is investigated. The material properties are assumed to be graded in the longitudinal direction, which vary according to a simple power law distribution. The differential quadrature method (DQM) is adopted to solve the equations of motion. It can be mentioned that results for free vibration analyses of the functionally graded carbon nanotube with variable thickness by differential quadrature method are not available yet and the results may be used as benchmark for future works. & 2011 Elsevier B.V. All rights reserved.
1. Introduction The differential quadrature (DQ) method is an efficient and useful numerical method for the rapid solutions of linear and non-linear partial differential equations. This method developed by Bellman and Casti [1] is an alternative discrete approach to directly solve the governing equations of various engineering problems. A comprehensive review of the differential quadrature method has been given by Bert and Malik [2]. Due to its efficiency and accuracy, DQM has been widely employed in many areas of industry and mathematics. Several researchers have addressed the linear and nonlinear static and dynamic problems of plates by the DQ method [3–8]. For beams, Feng and Bert [9] investigated the non-linear vibrations of beams using the conventional DQM. Combining DQM with Hamilton’s principle, Hsu [10] studied the electromechanical behavior of piezoelectric laminated composite beams. For micro- and nano-scale beams and tubes, Civalek et al. [11] applied the differential quadrature method to the equations of motion and bending of an Euler–Bernoulli beam using the nonlocal elasticity theory for cantilever microtubules. Civalek and ¨ [12] investigated the bending analysis of carbon nanotubes Akgoz using the nonlocal Bernoulli–Euler beam theory by differential quadrature method. Ait Atmane et al. [13] studied the thermal effect on wave propagation in double-walled carbon nanotubes embedded in a polymer matrix using nonlocal elasticity. Ansari and Hemmatnezhad [14] proposed the nonlinear vibrations of embedded multi-walled carbon nanotubes using a variational
n
Corresponding author. E-mail addresses: [email protected], [email protected] (M. Janghorban). 1386-9477/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2011.05.002
approach. Hashemnia et al. [15] investigated the dynamical analysis of carbon nanotubes conveying water considering carbon–water bond potential energy and nonlocal effects. In this paper, free vibration analysis of functionally graded carbon nanotubes (Fig. 1) with variable thickness is investigated. The governing equations are based on the Timoshenko beam theory. The differential quadrature method as an accurate numerical tool is adopted to solve governing equations. In this study the material properties are assumed to vary according to a power law distribution in terms of the volume fractions of the constituents along the longitudinal direction.
2. Brief statement of the DQ method The differential quadrature (DQ) method is a relatively new numerical technique in applied mechanics. It follows that the partial derivative of a function with respect to a space variable can be approximated by a weighted linear combination of function values at some intermediate points in that variable. In order to show the mathematical representation of DQM, consider a function f(x,Z) having its field in a rectangular domain 0 r x ra and 0 r Z rb. Let, in the given domain, the function values be known or desired on a grid of sampling points. According to the DQ method, the rth derivative of a function f(x,Z) can be approximated as Nx Nx X X @r f ðx, ZÞ xðrÞ xðrÞ Aim f ðxm , Zj Þ ¼ Aim fmj ðx, ZÞ ¼ ðxi , Zj Þ ¼ r @x m¼1 m¼1 for i¼1,2,y,Nx, j ¼1,2,y,NZ and r ¼1,2,y, Nx 1
ð1Þ
M. Janghorban, A. Zare / Physica E 43 (2011) 1602–1604
1603
carbon nanotube with linearly varying thickness is studied and then quadraticaly varying thickness is investigated. Both thickness profiles can be written as follows: tðxÞ ¼ t2 þ ðt1 t2 Þðx=LÞq
ð6Þ
where t1 ¼.34 nm and t2 ¼.44 nm.
5. Governing equations Consider a beam of length L and variable thickness, as shown in Fig. 2. The beam, which is simply supported at both ends, will be called an S–S beam. Free vibration analysis of an S–S functionally graded single walled carbon nanotube (SWCNT) based on the Timoshenko beam theory is studied here. The equations of motion are derived as
Fig. 1. Structure of a carbon nanotube.
From this equation one can see that the important components of DQ approximations are weighting coefficients and the choice of sampling points. In order to determine the weighting coefficients a set of test functions should be used in Eq. (1). For polynomial basis functions DQ, a set of Lagrange polynomials are employed as the test functions. The weighting coefficients for the first-order derivatives in x-direction are thus determined as 8 1 Mðxi Þ > > for i a j > > > Lx ðxi xj ÞMðxj Þ > > < Nx X ð2Þ Axij ¼ > Axij for i ¼ j; i, j ¼ 1,2. . .,Nx > > > > j ¼ 1 > > : i aj
Nx Y
@V=@x ¼ rA@2 W=@t 2
ð7Þ
where W is the lateral deflection, y is the rotation of the normal line,M ¼ EI@y=@x and V¼ aGA(qW/qx y). Assuming the sinusoidal motion in time, Eq. (7) yields ðdðaGAÞ=dxÞðdW=dxyÞ þ aGAðd2 W=dx2 dy=dxÞ þ rAo2 W ¼ 0 ðdðEIÞ=dxÞðdy=dxÞ þEIðd2 y=dx2 Þ þ aGAðdW=dxÞaGAy þ rIo2 y ¼ 0 ð8Þ The differential quadrature method is applied to discretize the equations of motion: 0 ðdðaGAÞ=dxÞ@
where Lx is the length of domain along the x direction and Mðxi Þ ¼
@M=@x þV ¼ rI@2 y=@t 2
N X
1
0
Aij Wi yi A þ aGA@
j¼1
0
ðxi xk Þ
ðdðEIÞ=dxÞ@
k ¼ 1,i a k
N X
j¼1
ð3Þ
In a similar manner, the weighting coefficients for the other direction can be obtained. The selection of locations of the sampling points plays a significant role in the accuracy of the solution of differential equations. In numerical computations, the Chebyshev–Gauss–Lobatto quadrature points are used, that is Zj 1 xi 1 ði1Þp ðj1Þp 1cos 1cos ¼ ; ¼ 2 2 a ðNx 1Þ b ðNZ 1Þ for i ¼ 1,2,. . ., Nx and j ¼ 1,2,. . .,NZ ð4Þ
Bij Wi
j¼1
1
0
Aij yi A þ EI@
N X
1
j¼1
N X
1 Aij yi A þ rAo2 Wi ¼ 0
j¼1
0
Bij yi A þ aGA@
The weighting coefficients of second order derivative can be obtained as ½Bxij ¼ ½Axij ½Axij ¼ ½Axij 2
N X
N X
1 Aij Wi AaGAyi þ rIo2 yi ¼ 0
j¼1
ð9Þ Then, the DQ discretized form of the equations of motion and the related boundary conditions can be expressed in the matrix form as follows [16]: equations of motion € ¼ f0g ½Kdb fbg þ ½Kdd fdg þ ½Mfdg
ð10Þ
3. Material properties Material properties of the beam are assumed to vary along the longitudinal direction of the beam. The material properties of functionally graded carbon nanotubes are as follows: EðxÞ ¼ E2 þðE1 E2 ÞVf , rðxÞ ¼ r2 þ ðr1 r2 ÞVf , G ¼ E=ð2ð1 þ nÞÞ
ð5Þ
where E is the Young modulus, r is the density, n is the Poisson ratio, G is the shear modulus, Vf ¼(x/L)p is the volume fraction and p is a constant defining the material property variation along the length direction; p ¼0 corresponds to an isotropic homogeneous beam. These equations are based on the rule of mixture.
4. Geometry The thickness of the beam is a function of the x coordinate. In this paper, two types of thickness variation are considered. First,
Fig. 2. Carbon nanotubes as Timoshenko beams. (a) Linearly varying thickness and (b) quadratic varying thickness.
1604
M. Janghorban, A. Zare / Physica E 43 (2011) 1602–1604
Table 1 Comparison of solutions for S–S carbon nanotube with constant thickness (Do =L ¼ :1, p ¼ 0). Modes
Present
Heireche et al. [17]
Demir et al. [18]
Reddy [19]
1 2 3 4 5 6 7 8
3.1123 6.0676 8.7784 11.229 13.441 15.449 17.285 18.975
3.0929 5.9399 8.4444 10.6260 – – – –
3.1405 6.2747 9.3963 10.7218 – – – –
3.1217 – – – – – – –
Table 2 Frequencies for S–S functionally graded carbon nanotube with linearly varying thickness (a ¼ 5=6, Do =L ¼ :1, q ¼ 1, n ¼.145). P
1 2 3 4 5 6 7 8 9
.5
1
1.5
2
2.5
4.0379 7.8879 11.397 14.566 17.425 20.018 22.388 24.570 26.3006
4.4389 8.5816 12.364 15.782 18.867 21.667 24.224 26.569 27.4660
4.6840 8.9845 12.923 16.486 19.705 22.627 25.295 27.713 28.4047
4.8486 9.2508 13.203 16.954 20.263 23.267 26.009 28.4687 29.197
4.9647 9.4413 13.558 17.289 20.663 23.727 26.522 29.0198 29.872
Table 3 Frequencies for S–S functionally graded carbon nanotube with quadraticaly varying thickness (a ¼ 5=6, Do =L ¼ :1, q ¼ 2, n ¼.145). P
1 2 3 4 5 6 7 8 9
.5
1
1.5
2
2.5
4.0260 7.8706 11.376 14.543 17.402 19.995 22.367 24.549 26.325
4.4255 8.5622 12.341 15.757 18.842 21.642 24.200 26.546 27.499
4.6697 8.9636 12.899 16.461 19.679 22.601 25.270 27.691 28.436
4.8338 9.2288 13.268 16.928 20.236 23.240 25.983 28.448 29.227
4.9496 9.4186 13.533 17.262 20.636 23.700 26.496 29.000 29.901
boundary conditions ½Kbb fbg þ ½Kbd fdg ¼ f0g
ð11Þ
The elements of the stiffness matrices [Kdi](i¼b,d) and the mass matrix [M] are obtained from equations of motion and those of the stiffness matrixes [Kii](i¼b,d) are obtained from the boundary conditions. Solving the eigenvalue system of the below equation, the natural frequencies will be obtained: ð½KKo2 ½MÞfXg ¼ f0g where [KK]¼ [Kdd] [Kdb][Kbb] 1[Kbd].
6. Numerical results For numerical illustrations, let us consider a single walled carbon nanotube having length L¼36.8 nm, outer diameter Do ¼3.68 nm, thickness h¼.34 nm, Young’s modulus E¼1012 Pa, density r ¼2300 kg/m3, Poisson’s ratio n ¼ .145 and shear correction factor a ¼5/6. The moments of inertia and the areas are calculated using I ¼ :25pðc04 ci4 Þ and A ¼ pðc02 ci2 Þ: All the numerical results are shown in the dimensionless quantities defined as follows: pffiffiffiffiffiffiffiffiffiffiffiffiffi l2 ¼ oL2 rA=EI ð13Þ In Table 1, the results are compared with those of other numerical solutions for a simply supported single walled carbon nanotube. Good agreement is achieved between the results. The effects of different power law indexes are presented in Tables 2 and 3. It may be seen that increasing the power law index will cause an increase in the frequency values of nanotubes with both linearly and quadraticaly varying thickness.
7. Conclusion Natural frequencies of simply supported functionally graded carbon nanotube of variable thickness have been obtained using the Timoshenko beam theory. The differential quadrature method was employed to convert the governing differential equations to a linear system. The material properties were assumed to be graded in the longitudinal direction, which vary according to a simple power law distribution. The accuracy of present numerical results was verified by comparing the results with those of the existing solutions. Parameter studies were also performed to show the effects of different parameters such as power law index on the frequencies of the carbon nanotube.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
ð12Þ [18] [19]
R.E. Bellman, J. Casti, J. Math. Anal. Appl. 34 (1971) 235. C.W. Bert, M. Malik, Appl. Mech. Rev. 49 (1996) 1. B.E. Richards Shu, Int. J. Numer. Methods Fluids 15 (1992) 791. K.M. Liew, T.M. Teo, Comput. Methods Appl. Mech. Eng. 159 (1998) 369. K.M. Liew, T.M. Teo, J. Sound Vib. 220 (1999) 577. K.M. Liew, T.M. Teo, J.-B. Han, Int. J. Numer. Meth. Eng 45 (1999) 1831. P. Malekzadeh, G. Karami, Eng. Struct. 28 (2006) 1307. P. Malekzadeh, Thin-Walled Struct. 45 (2007) 237. Y. Feng, C.W. Bert, Nonlinear Dyn. 3 (1992) 13. M.H.J. Hsu, Mar. Sci. Technol. 13 (2005) 148. ¨ . Civalek, C. Demir, B. Akgoz, ¨ Math. Comput. Appl. 15 (2010) 289. O ¨ . Civalek, B. Akgoz, ¨ Int. J. Eng. Appl. Sci. 1 (2009) 47. O H. Ait Atmane, A. Tounsi, S.A. Meftah, H.A. Belhadj, J. Vib. Control 17 (2011) 311. R. Ansari, M. Hemmatnezhad, Math. Comput. Model. 53 (2011) 927. K. Hashemnia, M. Farid, H. Emdad, Comput. Mater. Sci. 50 (2011) 828. P. Malekzadeh, Compos. Struct. 89 (2008) 367. H. Heireche, A. Tounsi, A. Benzair, M. Maachou, E.A.D. Bedia, Physica E 40 (2008) 2791. ¨ . Civalek, B. Akgoz, ¨ Math. Comput. Appl. 15 (2010) 57. C - . Demir, O J.N. Reddy, Int. J. Eng. Sci. 45 (2007) 288.
Contents lists available at ScienceDirect
Physica E journal homepage: www.elsevier.com/locate/physe
Free vibration analysis of functionally graded carbon nanotubes with variable thickness by differential quadrature method Maziar Janghorban a,n, Amin Zare b a b
Department of Mechanical Engineering, Shiraz Branch, Islamic Azad University, Shiraz, Iran Young Researchers club, Shiraz Branch, Islamic Azad University, Shiraz, Iran
a r t i c l e i n f o
a b s t r a c t
Article history: Received 27 January 2011 Received in revised form 24 April 2011 Accepted 2 May 2011 Available online 1 June 2011
Free vibration analysis of functionally graded carbon nanotube with variable thickness based on the Timoshenko beam theory is investigated. The material properties are assumed to be graded in the longitudinal direction, which vary according to a simple power law distribution. The differential quadrature method (DQM) is adopted to solve the equations of motion. It can be mentioned that results for free vibration analyses of the functionally graded carbon nanotube with variable thickness by differential quadrature method are not available yet and the results may be used as benchmark for future works. & 2011 Elsevier B.V. All rights reserved.
1. Introduction The differential quadrature (DQ) method is an efficient and useful numerical method for the rapid solutions of linear and non-linear partial differential equations. This method developed by Bellman and Casti [1] is an alternative discrete approach to directly solve the governing equations of various engineering problems. A comprehensive review of the differential quadrature method has been given by Bert and Malik [2]. Due to its efficiency and accuracy, DQM has been widely employed in many areas of industry and mathematics. Several researchers have addressed the linear and nonlinear static and dynamic problems of plates by the DQ method [3–8]. For beams, Feng and Bert [9] investigated the non-linear vibrations of beams using the conventional DQM. Combining DQM with Hamilton’s principle, Hsu [10] studied the electromechanical behavior of piezoelectric laminated composite beams. For micro- and nano-scale beams and tubes, Civalek et al. [11] applied the differential quadrature method to the equations of motion and bending of an Euler–Bernoulli beam using the nonlocal elasticity theory for cantilever microtubules. Civalek and ¨ [12] investigated the bending analysis of carbon nanotubes Akgoz using the nonlocal Bernoulli–Euler beam theory by differential quadrature method. Ait Atmane et al. [13] studied the thermal effect on wave propagation in double-walled carbon nanotubes embedded in a polymer matrix using nonlocal elasticity. Ansari and Hemmatnezhad [14] proposed the nonlinear vibrations of embedded multi-walled carbon nanotubes using a variational
n
Corresponding author. E-mail addresses: [email protected], [email protected] (M. Janghorban). 1386-9477/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2011.05.002
approach. Hashemnia et al. [15] investigated the dynamical analysis of carbon nanotubes conveying water considering carbon–water bond potential energy and nonlocal effects. In this paper, free vibration analysis of functionally graded carbon nanotubes (Fig. 1) with variable thickness is investigated. The governing equations are based on the Timoshenko beam theory. The differential quadrature method as an accurate numerical tool is adopted to solve governing equations. In this study the material properties are assumed to vary according to a power law distribution in terms of the volume fractions of the constituents along the longitudinal direction.
2. Brief statement of the DQ method The differential quadrature (DQ) method is a relatively new numerical technique in applied mechanics. It follows that the partial derivative of a function with respect to a space variable can be approximated by a weighted linear combination of function values at some intermediate points in that variable. In order to show the mathematical representation of DQM, consider a function f(x,Z) having its field in a rectangular domain 0 r x ra and 0 r Z rb. Let, in the given domain, the function values be known or desired on a grid of sampling points. According to the DQ method, the rth derivative of a function f(x,Z) can be approximated as Nx Nx X X @r f ðx, ZÞ xðrÞ xðrÞ Aim f ðxm , Zj Þ ¼ Aim fmj ðx, ZÞ ¼ ðxi , Zj Þ ¼ r @x m¼1 m¼1 for i¼1,2,y,Nx, j ¼1,2,y,NZ and r ¼1,2,y, Nx 1
ð1Þ
M. Janghorban, A. Zare / Physica E 43 (2011) 1602–1604
1603
carbon nanotube with linearly varying thickness is studied and then quadraticaly varying thickness is investigated. Both thickness profiles can be written as follows: tðxÞ ¼ t2 þ ðt1 t2 Þðx=LÞq
ð6Þ
where t1 ¼.34 nm and t2 ¼.44 nm.
5. Governing equations Consider a beam of length L and variable thickness, as shown in Fig. 2. The beam, which is simply supported at both ends, will be called an S–S beam. Free vibration analysis of an S–S functionally graded single walled carbon nanotube (SWCNT) based on the Timoshenko beam theory is studied here. The equations of motion are derived as
Fig. 1. Structure of a carbon nanotube.
From this equation one can see that the important components of DQ approximations are weighting coefficients and the choice of sampling points. In order to determine the weighting coefficients a set of test functions should be used in Eq. (1). For polynomial basis functions DQ, a set of Lagrange polynomials are employed as the test functions. The weighting coefficients for the first-order derivatives in x-direction are thus determined as 8 1 Mðxi Þ > > for i a j > > > Lx ðxi xj ÞMðxj Þ > > < Nx X ð2Þ Axij ¼ > Axij for i ¼ j; i, j ¼ 1,2. . .,Nx > > > > j ¼ 1 > > : i aj
Nx Y
@V=@x ¼ rA@2 W=@t 2
ð7Þ
where W is the lateral deflection, y is the rotation of the normal line,M ¼ EI@y=@x and V¼ aGA(qW/qx y). Assuming the sinusoidal motion in time, Eq. (7) yields ðdðaGAÞ=dxÞðdW=dxyÞ þ aGAðd2 W=dx2 dy=dxÞ þ rAo2 W ¼ 0 ðdðEIÞ=dxÞðdy=dxÞ þEIðd2 y=dx2 Þ þ aGAðdW=dxÞaGAy þ rIo2 y ¼ 0 ð8Þ The differential quadrature method is applied to discretize the equations of motion: 0 ðdðaGAÞ=dxÞ@
where Lx is the length of domain along the x direction and Mðxi Þ ¼
@M=@x þV ¼ rI@2 y=@t 2
N X
1
0
Aij Wi yi A þ aGA@
j¼1
0
ðxi xk Þ
ðdðEIÞ=dxÞ@
k ¼ 1,i a k
N X
j¼1
ð3Þ
In a similar manner, the weighting coefficients for the other direction can be obtained. The selection of locations of the sampling points plays a significant role in the accuracy of the solution of differential equations. In numerical computations, the Chebyshev–Gauss–Lobatto quadrature points are used, that is Zj 1 xi 1 ði1Þp ðj1Þp 1cos 1cos ¼ ; ¼ 2 2 a ðNx 1Þ b ðNZ 1Þ for i ¼ 1,2,. . ., Nx and j ¼ 1,2,. . .,NZ ð4Þ
Bij Wi
j¼1
1
0
Aij yi A þ EI@
N X
1
j¼1
N X
1 Aij yi A þ rAo2 Wi ¼ 0
j¼1
0
Bij yi A þ aGA@
The weighting coefficients of second order derivative can be obtained as ½Bxij ¼ ½Axij ½Axij ¼ ½Axij 2
N X
N X
1 Aij Wi AaGAyi þ rIo2 yi ¼ 0
j¼1
ð9Þ Then, the DQ discretized form of the equations of motion and the related boundary conditions can be expressed in the matrix form as follows [16]: equations of motion € ¼ f0g ½Kdb fbg þ ½Kdd fdg þ ½Mfdg
ð10Þ
3. Material properties Material properties of the beam are assumed to vary along the longitudinal direction of the beam. The material properties of functionally graded carbon nanotubes are as follows: EðxÞ ¼ E2 þðE1 E2 ÞVf , rðxÞ ¼ r2 þ ðr1 r2 ÞVf , G ¼ E=ð2ð1 þ nÞÞ
ð5Þ
where E is the Young modulus, r is the density, n is the Poisson ratio, G is the shear modulus, Vf ¼(x/L)p is the volume fraction and p is a constant defining the material property variation along the length direction; p ¼0 corresponds to an isotropic homogeneous beam. These equations are based on the rule of mixture.
4. Geometry The thickness of the beam is a function of the x coordinate. In this paper, two types of thickness variation are considered. First,
Fig. 2. Carbon nanotubes as Timoshenko beams. (a) Linearly varying thickness and (b) quadratic varying thickness.
1604
M. Janghorban, A. Zare / Physica E 43 (2011) 1602–1604
Table 1 Comparison of solutions for S–S carbon nanotube with constant thickness (Do =L ¼ :1, p ¼ 0). Modes
Present
Heireche et al. [17]
Demir et al. [18]
Reddy [19]
1 2 3 4 5 6 7 8
3.1123 6.0676 8.7784 11.229 13.441 15.449 17.285 18.975
3.0929 5.9399 8.4444 10.6260 – – – –
3.1405 6.2747 9.3963 10.7218 – – – –
3.1217 – – – – – – –
Table 2 Frequencies for S–S functionally graded carbon nanotube with linearly varying thickness (a ¼ 5=6, Do =L ¼ :1, q ¼ 1, n ¼.145). P
1 2 3 4 5 6 7 8 9
.5
1
1.5
2
2.5
4.0379 7.8879 11.397 14.566 17.425 20.018 22.388 24.570 26.3006
4.4389 8.5816 12.364 15.782 18.867 21.667 24.224 26.569 27.4660
4.6840 8.9845 12.923 16.486 19.705 22.627 25.295 27.713 28.4047
4.8486 9.2508 13.203 16.954 20.263 23.267 26.009 28.4687 29.197
4.9647 9.4413 13.558 17.289 20.663 23.727 26.522 29.0198 29.872
Table 3 Frequencies for S–S functionally graded carbon nanotube with quadraticaly varying thickness (a ¼ 5=6, Do =L ¼ :1, q ¼ 2, n ¼.145). P
1 2 3 4 5 6 7 8 9
.5
1
1.5
2
2.5
4.0260 7.8706 11.376 14.543 17.402 19.995 22.367 24.549 26.325
4.4255 8.5622 12.341 15.757 18.842 21.642 24.200 26.546 27.499
4.6697 8.9636 12.899 16.461 19.679 22.601 25.270 27.691 28.436
4.8338 9.2288 13.268 16.928 20.236 23.240 25.983 28.448 29.227
4.9496 9.4186 13.533 17.262 20.636 23.700 26.496 29.000 29.901
boundary conditions ½Kbb fbg þ ½Kbd fdg ¼ f0g
ð11Þ
The elements of the stiffness matrices [Kdi](i¼b,d) and the mass matrix [M] are obtained from equations of motion and those of the stiffness matrixes [Kii](i¼b,d) are obtained from the boundary conditions. Solving the eigenvalue system of the below equation, the natural frequencies will be obtained: ð½KKo2 ½MÞfXg ¼ f0g where [KK]¼ [Kdd] [Kdb][Kbb] 1[Kbd].
6. Numerical results For numerical illustrations, let us consider a single walled carbon nanotube having length L¼36.8 nm, outer diameter Do ¼3.68 nm, thickness h¼.34 nm, Young’s modulus E¼1012 Pa, density r ¼2300 kg/m3, Poisson’s ratio n ¼ .145 and shear correction factor a ¼5/6. The moments of inertia and the areas are calculated using I ¼ :25pðc04 ci4 Þ and A ¼ pðc02 ci2 Þ: All the numerical results are shown in the dimensionless quantities defined as follows: pffiffiffiffiffiffiffiffiffiffiffiffiffi l2 ¼ oL2 rA=EI ð13Þ In Table 1, the results are compared with those of other numerical solutions for a simply supported single walled carbon nanotube. Good agreement is achieved between the results. The effects of different power law indexes are presented in Tables 2 and 3. It may be seen that increasing the power law index will cause an increase in the frequency values of nanotubes with both linearly and quadraticaly varying thickness.
7. Conclusion Natural frequencies of simply supported functionally graded carbon nanotube of variable thickness have been obtained using the Timoshenko beam theory. The differential quadrature method was employed to convert the governing differential equations to a linear system. The material properties were assumed to be graded in the longitudinal direction, which vary according to a simple power law distribution. The accuracy of present numerical results was verified by comparing the results with those of the existing solutions. Parameter studies were also performed to show the effects of different parameters such as power law index on the frequencies of the carbon nanotube.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
ð12Þ [18] [19]
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