Nonlinear vibration of nonlocal strain gradient nanotubes under longitudinal magnetic field

Nội dung text: Nonlinear vibration of nonlocal strain gradient nanotubes under longitudinal magnetic field

1. Vietnam Journal of Mechanics, VAST, Vol.43, No. 1 (2021), pp. 55 – 77 DOI: NONLINEAR VIBRATION OF NONLOCAL STRAIN GRADIENT NANOTUBES UNDER LONGITUDINAL MAGNETIC FIELD N. D. Anh1,2, D. V. Hieu3,∗ 1Institute of Mechanics, VAST, Hanoi, Vietnam 2VNU University of Engineering and Technology, Hanoi, Vietnam 3Thai Nguyen University of Technology, Vietnam ∗E-mail: hieudv@tnut.edu.vn Received: 20 September 2020 / Published online: 10 January 2021 Abstract. The nonlinear free vibration of embedded nanotubes under longitudinal mag- netic field is studied in this paper. The governing equation for the nanotube is formu- lated by employing Euler–Bernoulli beam model and the nonlocal strain gradient theory. The analytical expression of the nonlinear frequency of the nanotube is obtained by us- ing Galerkin method and the equivalent linearization method with the weighted averag- ing value. The accuracy of the obtained solution has been verified by comparison with the published solutions and the exact solution. The influences of the nonlocal parame- ter, material length scale parameter, aspect ratio, diameter ratio, Winkler parameter and longitudinal magnetic field on the nonlinear vibration responses of the nanotubes with pinned-pinned and clamped-clamped boundary conditions are investigated and and dis- cussed. Keywords: nonlinear vibration, carbon nanotube, nonlocal strain gradient, magnetic field, Galerkin method, equivalent linearization, weighted averaging. 1. INTRODUCTION First discovered in 1991 by Iijima [1], carbon nanotubes (CNTs) show many advan- tages compared to conventional steel tubes. With theirs advantages and small sizes, CNTs have many applications such as nanoactuator [2], nano-electromechanical systems (NEMS) [3,4], nano-devices for electronics [5,6], nano-medicine [7], or nano-devices for conveying fluid and gas storage [8,9]. Researching the mechanical behavior of CNTs is an extremely important problem and attracts many interested scientists. Unlike macro-sized tubes, the size-dependent ef- fect plays an important role in the response of CNTs. Some elasticity theories have been proposed to study the size-dependent effect on the mechanical response of micro-/nano- structures such as Eringen’s nonlocal elasticity theory [10, 11], the strain gradient theory â 2021 Vietnam Academy of Science and Technology
2. 56 N. D. Anh, D. V. Hieu (SGT) [12, 13], and the modified couple stress theory (MCST) [14]. To date, many works related to the static and dynamic responses of CNTs have been published using these higher-order elasticity theories. Nonlinear free vibration responses of the single-walled carbon nanotubes (SWCNTs) were reported by Yang et al. [15] using the Timoshenko beam theory and Eringen’s nonlocal elasticity theory. The effects of nonlocal parameter, length and radius of the SWCNTs with different boundary conditions on the nonlinear free vibration behaviors of SWCNTs were examined in this work. Narendar et al. [16] studied the wave propagation problem in the SWCNTs under longitudinal magnetic field based on Eringen’s nonlocal elasticity theory and the Euler-Bernoulli beam theory. The wave method was employed by Zhang et al. [17] to analyze the nonlinear free vibration of the fluid-conveying SWCNTs based on Eringen’s nonlocal elasticity theory. Based on the Euler-Bernoulli beam theory and Eringen’s nonlocal elasticity theory, Wang and Li [18] investigated the nonlinear free vibration of the SWCNTs with the viscous damping effect. Zhen and Fang [19] used the Lindstedt–Poincare´ method to investigate the nonlinear vi- bration of the fluid-conveying SWCTNs under harmonic excitation. Nonlinear vibration of the embedded SWCNTs was reported by Valipour et al. [20] using Eringen’s nonlocal elasticity theory and the parameterized perturbation method. Goughari et al. [21] stud- ied effects of magnetic-fluid flow on the instability of the fluid-conveying SWCNTs under the magnetic field. Free vibration of the SWCNTs resting on the exponentially varying elastic foundation was examined by Chakraverty and Jena using Eringen’s nonlocal elas- ticity theory [2]. Based on the MCST and the Euler-Bernoulli beam theory, Wang [22] investigated the size-dependent vibration responses of the fluid-conveying microtubes. Flexural size-dependent vibrations of the microtubes conveying fluid was carried out by Wang et al. [23] utilizing the MCST. Based on the MCST, Tang et al. [24] developed a non- linear model to study the size-dependent vibration of the curved microtubes conveying fluid. Xia and Wang [25] studied vibration and stability behaviors of the microscale pipes conveying fluid based on Timoshenko beam theory and the MCST. Based on the second SGT, vibration and stability behaviors of the SWCNTs conveying fluid were presented by Ghazavi et al. [26]. In 2015, the nonlocal parameter and the material length scale parameter were incor- porated into a generalized elasticity theory. Lim et al. [27] introduced the nonlocal strain gradient theory (NSGT) describing two entirely different physical characteristics of ma- terials and structures at nanoscale. Many works related to behaviors of micro-/nano- beams and micro-/nano-tubes have been published by using the NSGT. Using the NSGT and the Euler-Bernoulli beam model, Simsek investigated nonlinear free vibration re- sponse of a functionally graded (FG) nanobeam [28]. Bending, buckling, and vibration responses of viscoelastic FG curved nanobeam resting on an elastic foundation were stud- ied by Allam and Radwan using the NSGT [29]. Nonlinear vibration of the electrostatic FG nano-resonator considering the surface effects was investigated by Esfahani et al. [30] employing the Euler-Bernoulli beam theory and the NSGT. Based on the NSGT, Dang et al. [31] analyzed nonlinear vibration behavior of nanobeam under electrostatic force. A nonlocal strain gradient Timoshenko beam model was developed by Bahaadini et al. [32] to analyze the vibration and instability responses of the SWCNTs conveying nanoflow. Using the NSGT and the shear deformation beam model, Malikan et al. [33] studied the
3. Nonlinear vibration of nonlocal strain gradient nanotubes under longitudinal magnetic field 57 damped forced vibration of the SWCNTs resting on viscoelastic foundation in thermal en- vironment. Vibration behaviors of porous nanotubes were investigated by She et al. [34] utilizing the NSGT and the refined beam model. Atashafrooz et al. [35] analyzed effects of nonlocal, strain gradient and surface stresses on vibration and instability behaviors of the SWCNTs conveying nanoflow by using the NSGT and the Euler-Bernoulli beam theory. Based on the NSGT, Li et al. [36] studied size-dependent effects on critical flow velocity of microtubes conveying fluid; both Timoshenko and Euler–Bernoulli beam models were considered in this work. Nonlinear forced vibration of the SWCNTs was examined by Ghayesh and Farajpour using the NSGT and the Euler-Bernoulli beam model [37]. The coupled nonlinear mechanical behavior of nonlocal strain gradient SWCNTs subjected to distributed load was investigated by Ghayesh and Farajpour [38]. Dynamic response of the viscoelastic SWCNTs conveying fluid with uncertain parameters and under ran- dom excitation was analyzed by Azrar et al. [39] by combining the NSGT and the Euler- Bernoulli beam theory. The problem of wave propagation in the SWCNTs also attracted many authors [40–42]. According to authors’ knowledge, until now, nonlinear vibration response of the nonlocal strain gradient SWCNTs under magnetic field has not yet announced. Thus, in this paper, authors focus on studying the effect of magnetic field on the nonlinear vi- bration response of the SWCNTs based on the NSGT. Using the equivalent linearization method with the weighted averaging value [43–49], expressions of the nonlinear frequen- cies of the SWCNTs resting on the elastic foundation and under the longitudinal magnetic field are given in the analytical forms. Effects of the nonlocal parameter, material length scale parameter, aspect ratio, diameter ratio, magnetic field and elastic foundation on the nonlinear vibration behaviors of the SWCNTs with different boundary conditions are investigated and discussed in this work. Nonlinear vibration2. of MODEL nonlocal strain AND gradient FORMULATIONS nanotubes under longitudinal magnetic field 3 Considering a SWCNT made of the homogeneous material as shown in Fig.1. The 2. MODEL AND FORMULATIONS nanotube has the length L, the inner diameter d, the outer diameter D, the mass density Considering a SWCNT made of the homogeneous material as shown in Fig. 1. The nanotube ρ and the Young’shas the length modulus L, the innerE . diameter The nanotubed, the outer diameter rests D on, the a mass linear density elastic ρ and foundationthe Young’s with a coefficient ofmoduluskw (the E. The Winkler nanotube restslayer) on a and linear is elastic subjected foundation to with a longitudinala coefficient of kw magnetic(the Winkler field. layer) and is subjected to a longitudinal magnetic field. Fig.Fig. 11:. Model Model of the ofSWCNT the SWCNTresting on an restingelastic foundation on an and elastic under longitudi foundationnal magnetic field and under longitudinal magnetic field Based on the Euler-Bernoulli beam theory and the von-Karmỏn’s nonlinear strain-displacement relationship, the strain-displacement relationship for the nanotube takes a form: 2 ảảuxt(,) 1ổử wxt (,) ả2 wxt (,) e xx =+ỗữ- z 2 , (1) ảảxx2 ốứ ả x where u(x,t) and w(x,t) are the axial and transverse displacements, respectively; and t is time. According to the NSGT proposed by Lim et al. [28], the strain energy can be expressed as: 1 U=+se s(1)ẹ e dV , (2) ũ( xx xx xx xx ) 2 V (1) where s xx and s xx are the classical and higher-order stresses, respectively, which are defined as: L sa= E x,', x e a e' (')', x dx xxũ 00( ) xx (3) 0 L sa(1)= l 2 E x,', x e a e ' (')', x dx xxũ 11( ) xx (4) 0 (1) txx= ss xx-ẹ xx (5) herein, ẹ = ả represents the differential operator; t is the total stress; a and a are two nonlocal ảx xx 0 1 kernel functions; ea0 and ea1 denote the nonlocal parameters; l is the material length scale parameter. With assuming that ea01== ea ea, the general nonlocal strain gradient constitutive equation for the nanotube can be presented as [28]:
4. 58 N. D. Anh, D. V. Hieu Based on the Euler–Bernoulli beam theory and the von-Karman’s´ nonlinear strain- displacement relationship, the strain-displacement relationship for the nanotube takes a form ∂u(x, t) 1  ∂w(x, t) 2 ∂2w(x, t) ε = + − z , (1) xx ∂x 2 ∂x ∂x2 where u(x, t) and w(x, t) are the axial and transverse displacements, respectively; and t is time. According to the NSGT proposed by Lim et al. [27], the strain energy can be ex- pressed as 1 Z  ( )  U = σ ε + σ 1 ∇ε dV, (2) 2 xx xx xx xx V (1) where σxx and σxx are the classical and higher-order stresses, respectively, which are defined as L Z 0
   0 0 0 σxx = Eα0 x, x , e0a εxx(x )dx , (3) 0 L Z (1) 2 0
   0 0 0 σxx = l Eα1 x, x , e1a εxx(x )dx , (4) 0 (1) txx = σxx − ∇σxx , (5) ∂ herein, ∇ = ∂x represents the differential operator; txx is the total stress; α0 and α1 are two nonlocal kernel functions; e0a and e1a denote the nonlocal parameters; l is the material length scale parameter. With assuming that e0a = e1a = ea, the general nonlocal strain gradient constitutive equation for the nanotube can be presented as [27] h 2 2i 2 2
   1 − (ea) ∇ txx = E 1 − l ∇ εxx , (6) 2 ∂2 where ∇ = ∂x2 is the Laplacian operator. The Hamilton’s principle is used to derive the equation of motion for the nanotube. From Eqs. (1) and (2), the virtual strain energy of the nanotube can be expressed as L ( " # ) ( " # )L Z ∂u 1  ∂w 2  ∂2w  ∂u 1  ∂w 2  ∂2w  = + − + (1) + − (1) δU Nxxδ Mxxδ 2 dx Nxx δ Mxx δ 2 ∂x 2 ∂x ∂x ∂x 2 ∂x ∂x 0 0 L L Z  L Z     L  L L ∂Nxx ∂w ∂ ∂w ∂w ∂Mxx = [Nxxδu]|0 − δudx+ Nxx δw − Nxx δwdx− Mxxδ + δw ∂x ∂x 0 ∂x ∂x ∂x 0 ∂x 0 0 0 L L Z ∂2 M    ∂u  ∂w  ∂w   ∂2w  − xx + (1) + − (1) 2 δwdx Nxx δ δ Mxx δ 2 , ∂x ∂x ∂x ∂x ∂x 0 0 (7)
5. Nonlinear vibration of nonlocal strain gradient nanotubes under longitudinal magnetic field 59 (1) where Nxx is the axial force resultant, Nxx is nonclassical axial force resultant, Mxx is the (1) bending moment resultant, and Mxx is the nonclassical moment resultant, these resul- tants are defined by Z Z Z Z (1) (1) (1) (1) Nxx = txxdA, Nxx = σxx dA, Mxx = ztxxdA, Mxx = zσxx dA. (8) A A A A The virtual kinetic energy of the nanotube is given by L Z  ∂u  ∂u  ∂w  ∂w  δK = ρA δ + δ dx, (9) e ∂t ∂t ∂t ∂t 0 where A is the cross-section area of the nanotube. The virtual external work is L Z δWe = (qδw) dx. (10) 0 In this work, the external forces are caused by the linear elastic foundation and the longitudinal magnetic field q = qe + qm, (11) where the external force caused by the linear elastic layer can be expressed as qe = −kww, (12) and the external force due to the longitudinal magnetic field is [41, 42] ∂2w q = ηAH2 , (13) m x ∂x2 in which, η is the magnetic permeability and Hx is the component of the longitudinal magnetic field vector exerted on the nanotube in the x-direction. Substituting Eqs. (12) and (13) into Eq. (10), one gets L Z  ∂2w   δW = −k w + ηAH2 δw dx. (14) e w x ∂x2 0 The Hamilton’s principle is employed to derive the equation of motion for the nan- otube, this principle states that t Z (δKe + δWe − δU)dt = 0. (15) 0 Substituting Eqs. (7), (9) and (14) into Eq. (15); then using integration by parts, and collecting the coefficients of δu and δw, one gets the equations of motion ∂N ∂2u xx = ρA , (16) ∂x ∂t2
6. 60 N. D. Anh, D. V. Hieu ∂2 M ∂  ∂w  ∂2w ∂2w xx + N − k w + ηAH2 = ρA . (17) ∂x2 ∂x xx ∂x w x ∂x2 ∂t2 and the boundary conditions at x = 0 and x = L as δu : Nxx = 0 or u = 0, (18)   ∂u ( ) ∂u δ : N 1 = 0 or = 0, (19) ∂x xx ∂x ∂M ∂w δw : xx + N = 0 or w = 0, (20) ∂x xx ∂x   ∂w ( ) ∂w ∂w δ : M − N 1 = 0 or = 0, (21) ∂x xx xx ∂x ∂x  2  2 ∂ w ( ) ∂ w δ : M 1 = 0 or = 0. (22) ∂x2 xx ∂x2 Note that the boundary conditions (18)–(22) will be satisfied in only one way for any specific support conditions of the nanotube [36, 50]. From Eqs. (1) and (6), one obtains " # ∂u 1  ∂w 2 ∂2w 1 − (ea)2∇2 t = E 1 − l2∇2
   + − z . (23) xx ∂x 2 ∂x ∂x2 Considering Eq. (8), Eq. (23) leads to " #  ∂2   ∂2  ∂u 1  ∂w 2 1 − (ea)2 N = EA 1 − l2 + , (24) ∂x2 xx ∂x2 ∂x 2 ∂x  ∂2   ∂2  ∂2w 1 − (ea)2 M = −EI 1 − l2 . (25) ∂x2 xx ∂x2 ∂x2 Z where I = z2dA is the moment of inertia of the nanotube. Using Eqs. (16), (17), (24) A and (25), the expressions of the axial force and bending moment resultants can be get as " #  ∂2  ∂u 1  ∂w 2  ∂3u  N = EA 1 − l2 + + (ea)2 ρA , (26) xx ∂x2 ∂x 2 ∂x ∂x∂t2  ∂2  ∂2w  ∂2w ∂  ∂w  ∂2w  M = −EI 1−l2 +(ea)2 ρA − N +k w−ηAH2 . (27) xx ∂x2 ∂x2 ∂t2 ∂x xx ∂x w x ∂x2 Substituting Eqs. (26) and (27) into Eqs. (16) and (17), one gets the differential equa- tions of motion for the nanotube as ( " #) ∂  ∂2  ∂u 1  ∂w 2 ∂2  ∂2w  EA 1 − l2 + − ρA 1 − (ea)2 = 0, (28) ∂x ∂x2 ∂x 2 ∂x ∂t2 ∂x2
7. Nonlinear vibration of nonlocal strain gradient nanotubes under longitudinal magnetic field 61  ∂2  ∂4w ∂  ∂w  ∂3  ∂w  ∂2  ∂2w  −EI 1 − l2 + N − (ea)2 N − ρA w − (ea)2 ∂x2 ∂x4 ∂x xx ∂x ∂x3 xx ∂x ∂t2 ∂x2  ∂2w  ∂2  ∂2w  −k w − (ea)2 + ηAH2 w − (ea)2 = 0. w ∂x2 x ∂x2 ∂x2 (29) With assuming that the axial inertia term in Eq. (28) is neglected, the axial force resultant can be achieved as [28, 37] L L " 2# EA Z  ∂w 2 EA Z ∂w ∂3w  ∂2w  N = dx − l2 + dx. (30) xx 2L ∂x L ∂x ∂x3 ∂x2 0 0 Now, substituting Eq. (30) into Eq. (29), one obtains the equation of motion for the nanotube under the longitudinal magnetic field based on the NSGT   L L" 2#  ∂2  ∂4w  EA Z  ∂w 2 EA Z ∂w ∂3w  ∂2w    ∂2w ∂4w  − − 2 + − 2 + −( )2 EI 1 l 2 dx l 3 2 dx 2 ea ∂x ∂x4  2L ∂x L ∂x ∂x ∂x  ∂x ∂x4 0 0 2  2   2  2  2  ∂ 2 ∂ w 2 ∂ w 2 ∂ 2 ∂ w −ρA w − (ea) − kw w − (ea) + ηAH w − (ea) = 0. ∂t2 ∂x2 ∂x2 x ∂x2 ∂x2 (31) Eq. (31) shows that the motion of the nanotube is governed by the nonlinear partial differential equation. This equation satisfies the following kinematic boundary condi- tions: + For the pinned-pinned (P-P) nanotube ∂2w(0, t) ∂2w(L, t) w(0, t) = w(L, t) = = = 0 (32) ∂x2 ∂x2 + For the clamped-clamped (C-C) nanotube ∂w(0, t) ∂w(L, t) w(0, t) = w(L, t) = = = 0 (33) ∂x ∂x For convenience, the following dimensionless variables are defined s x w ea l EI = = = = ¯ = x¯ , w¯ , α , β , t t 4 , L L L L ρAL (34) L d δ2 k L4 ηAH2L2 δ = , h = , Γ = , K = w , H = x . D D 1 + h2 W EI EI Using Eq. (34), the equation of motion for the nanotube (31) is rewritten in the di- mensionless form  1   1  ∂6w¯ ∂4w¯ 1 Z  ∂w¯ 2  ∂2w¯ ∂4w¯  Z  ∂w¯ ∂3w¯   ∂2w¯ ∂4w¯  β2 − +16Γ dx¯ −α2 −16Γβ2  dx¯ −α2 ∂x¯6 ∂x¯4 2 ∂x¯ ∂x¯2 ∂x¯4 ∂x¯ ∂x¯3 ∂x¯2 ∂x¯4 0 0  1 2  Z  ∂2w¯  ∂2w¯ ∂4w¯  ∂2w¯ ∂2w¯ ∂4w¯  ∂4w¯ ∂2w¯ −16Γβ2 dx¯ −α2 −K w¯ −α2 +H −α2 + α2 − =0. ∂x¯2 ∂x¯2 ∂x¯4 W ∂x¯2 ∂x¯2 ∂x¯4 ∂x¯2∂t¯2 ∂t¯2 0 (35)
8. 62 N. D. Anh, D. V. Hieu Therefore, the kinematic boundary conditions (32) and (33) become: + For the pinned-pinned nanotube ∂2w¯ (0, t¯) ∂2w¯ (L, t¯) w¯ (0, t¯) = w¯ (L, t¯) = = = 0. (36) ∂x¯2 ∂x¯2 + For the clamped-clamped nanotube ∂w¯ (0, t¯) ∂w¯ (L, t¯) w¯ (0, t¯) = w¯ (L, t¯) = = = 0. (37) ∂x¯ ∂x¯ It is very difficult to find the exact solution of Eq. (35), the approximate method is an effective tool to find the solution of this equation. First, the Galerkin technique is used to convert equation (35) into the ordinary differential one. To apply the Galerkin technique, the solution of Eq. (35), w¯ (x¯, t¯), is assumed to have the following form w¯ (x¯, t¯) = Q(t¯) ã φ(x¯), (38) where Q(t¯) is the time-dependent function must be determined and φ(x¯) is the shape function satisfying kinematic boundary conditions of the nanotube. The shape functions, φ(x¯), for pinned-pinned and clamped-clamped nanotubes can be chosen as in Tab.1. Table 1. The shape functions for Pinned-Pinned and Clamped-Clamped nanotubes Boundary condition Shape function Pinned-Pinned φ(x¯) = sin πx¯ 1 Clamped-Clamped φ(x¯) = (1 − cos 2πx¯) 2 Applying the Galerkin technique, Eq. (35) is reduced to the following nonlinear or- dinary differential equation 3 Qă (t¯) + γ1Q(t¯) + γ2Q (t¯) = 0, (39) where the coefficients γ1 and γ2 are determined by  1 1  1 1   1 1   Z Z Z Z Z Z  2 (6) (4) 2 2 (2) (2) 2 (4) β ϕ ϕdx¯− ϕ ϕdx¯ − KW  ϕ dx¯ − α ϕ ϕdx¯ + H  ϕ ϕdx¯ − α ϕ ϕdx¯   γ = 0 0 0 0 0 0 , 1 1 1 Z Z α2 ϕ(2) ϕdx¯ − ϕ2dx¯ 0 0 (40)     Z1 Z1 Z1 Z1 Z1 1 0 0 ( ) ( ) ( ) ( ) 16Γ  (ϕ )2dx¯ − β2 ϕ ϕ 3 dx¯−β2 (ϕ 2 )2dx¯  ϕ 2 ϕdx¯ − α2 ϕ 4 ϕdx¯ 2 0 0 0 0 0 γ2 = . Z1 Z1 α2 ϕ(2) ϕdx¯ − ϕ2dx¯ 0 0 (41)
9. Nonlinear vibration of nonlocal strain gradient nanotubes under longitudinal magnetic field 63 The nanotube is assumed to satisfy the following initial conditions Q(0) = Q0, Q˙ (0) = 0. (42) where Q0 = w¯ max(0.5) is the dimensionless maximum vibration amplitude of the nan- otube. 3. SOLUTION PROCEDURE In this section, one will find the approximate solution of Eq. (39) with the initial conditions (42). It can see that Eq. (39) is cubic Duffing equation, there are many ap- proximate methods to solve this equation [51]. In this work, the equivalent linearization method with the weighted averaging value [43–49] is employed to find the approximate solution of Eq. (39). First, the equivalent linear form of the nonlinear equation (39) is introduced in the following form Qă (t¯) + ω2Q(t¯) = 0, (43) where ω is known as the approximate frequency of the nanotube, which can be deter- mined by the mean square error criterion 2 D 3 2
   2E e (Q) = γ1Q + γ2Q − ω Q → min . (44) ω2 ∂ e2(Q) Thus, from the condition = 0, leads to ∂ω2 s hQ4i ω = γ + γ . (45) 1 2 hQ2i In Eq. (45), the symbol hãi denotes the time-averaging operator which can be calcu- lated using the definition of the weighted averaging value [43] + Z ∞ h f (t)i = h(t) f (t)dt, (46) 0 with h(t) is the weighted coefficient function which satisfies the following condition + Z ∞ h(t)dt = 1. (47) 0 In this work, a specific form of the weighted coefficient function is used [43] h(t) = s2ω t e−sωt, s > 0. (48) The solution of the linearized equation (43) has a form Q(t¯) = Q0 cos(ωt¯). (49)
10. 64 N. D. Anh, D. V. Hieu With the periodic solution of linearized equation (43) given in Eq. (49), the averaging D E values Q2 and Q4 in Eq. (45) can be calculated by using Eq. (46) with the weighted coefficient function given in Eq. (48) and Laplace transform as follows + Z ∞ 2 2 2 2 2 −sωt¯ 2 Q = Q0 cos (ωt) = Q0s ωt¯ e cos (ωt¯)dt¯ 0 (50) +∞ Z s4 + 2s2 + 8 = Q2s2τ e−sτ cos2(τ)dτ = Q2 . 0 0 (s2 + 4)2 0 + Z ∞ D 4E D 4 4 E 4 2 −sωt¯ 4 Q = Q0 cos (ωt) = Q0s ωt¯ e cos (ωt¯)dt¯ w 0 (51) +∞ Z s8 + 28s6 + 248s4 + 416s2 + 1536 = Q4s2τ e−sτ cos4(τ)dτ = Q4 . 0 0 (s2 + 4)2(s2 + 16)2 0 Substituting the averaging values in Eqs. (50) and (51) into Eq. (45), the approximate frequency can be get as s 248s4 + 416s2 + 1536 + 28s6 + s8 ω = γ + γ Q2. (52) NL 1 2 (s4 + 2s2 + 8)(s2 + 16)2 0 It can be seen that the approximate frequency of the nanotube depends not only on the initial amplitude Q0 but also on the parameter s. With s = 2, the obtained results show the accuracy [44–49]. Thus, when s = 2, one obtains q 2 ωNL = γ1 + 0.72γ2Q0. (53) The approximate solution of Eq. (39) can be get as follows q  ¯ 2 ¯ Q(t) = α cos γ1 + 0.72γ2Q0t . (54) By using the shape functions in Tab.1 and integrals in Eqs. (40) and (41), one obtains the approximate nonlinear frequencies of the nanotubes: + For pinned-pinned nanotube s 1 + β2π2 ω = π4 + K + Hπ2 + 2.88Γπ4Q2. (55) NL 1 + α2π2 W 0 + For clamped-clamped nanotube s 1 + 4β2π2 1 + 4α2π2 1 + 4α2π2 ω = 16π4 + K + 4Hπ2 + 11.52Γπ4 Q2. (56) NL 3 + 4α2π2 W 3 + 4α2π2 3 + 4α2π2 0
11. Nonlinear vibration of nonlocal strain gradient nanotubes under longitudinal magnetic field 65 4. RESULTS AND DISCUSSIONS It can be seen from Eqs. (55) and (56) that the material length scale parameter (β = l/L), the Winkler parameter (KW ), the magnetic field (H), the aspect ratio (δ = L/D) and the initial amplitude (Q0) lead to an increase of the nonlinear frequency (ωNL) of the nanotubes, while the nonlocal parameter (α = ea/L) and the diameter ratio (h = d/D) lead to a decrease of the nonlinear frequency (ωNL) of the nanotubes. To study nonlinear vibration responses of the nanotubes, one introduces the scale ratio β l c = = , (57) α ea and the frequency ratio (the ratio of the nonlinear frequency to the linear frequency) ωNL ωratio = . (58) ωL Note that the linear frequency ωL can be achieved from the nonlinear one by letting Q0 = 0. 4.1. Validation of model The exact frequency of Eq. (39) is given as [51] 2π ωexact = . (59) π/2 √ Z dθ 4 2 q 2 2 2 0 2γ1 + γ2Q0 + γ2Q0 cos θ Comparison of the approximate frequency with the exact frequency for KW = 10, H = 20, L/D = 20, d/D = 0.8 and several different values of the initial amplitude Q0, the material length scale parameter β and the nonlocal parameter α is shown in Tabs.2 Table 2. Comparison of the approximate frequency with the exact frequency for P-P nanotube Q0 β α ωpresent ωexact 0.1 0.1 17.6615 17.6613 0.01 0.2 0.1 18.3896 18.3894 0.1 0.2 17.0660 17.0658 0.1 0.1 21.9770 21.9143 0.05 0.2 0.1 22.5663 22.5084 0.1 0.2 21.5014 21.4344 0.1 0.1 31.8990 31.5664 0.1 0.2 0.1 32.3979 32.9880 0.1 0.2 31.5732 31.2299
12. 66 N. D. Anh, D. V. Hieu and3 corresponding to P-P nanotube and C-C nanotube, respectively. The accuracy of analytical solutions can be observed from these tables. Table 3. Comparison of the approximate frequency with the exact frequency for C-C nanotube Q0 β α ωpresent ωexact 0.1 0.1 31.4079 31.4078 0.01 0.2 0.1 39.1177 39.1176 0.1 0.2 30.7489 30.7487 0.1 0.1 35.6032 35.5635 0.05 0.2 0.1 42.5596 42.5364 0.1 0.2 36.4813 36.4119 0.1 0.1 46.3261 46.0351 0.1 0.2 0.1 51.8637 51.6572 0.1 0.2 50.3510 49.9232 Based on the nonlocal elasticity theory, Chang [52] examined the nonlinear vibration of nanobeams under magnetic field. The frequency ratios of the nanobeams obtained in this work and those obtained by Chang [52] are compared and shown in Tab.4. Note that the results for the nanobeams can be get from the obtained results for nanotubes by letting the inner diameter equal to zero (d = 0). A good agreement can be seen between the frequency ratios obtained in this work and the frequency ratios obtained by Chang [52]. Table 4. Comparison of the frequency ratios of nanobeams with H = 50 and L/h = 20 P-P nanobeam C-C nanobeam Q0 α Chang [52] Present Chang [52] Present 0 1.0098 1.0095 1.0066 1.0063 0.1 1.0100 1.0096 1.0075 1.0072 0.01 0.2 1.0103 1.0099 1.0090 1.0087 0.3 1.0107 1.0102 1.0100 1.0096 0 1.7258 1.7027 1.5243 1.5069 0.1 1.7343 1.7110 1.5851 1.5659 0.1 0.2 1.7536 1.7298 1.6774 1.6556 0.3 1.7733 1.7489 1.7374 1.7140
13. Nonlinear vibration of nonlocal strain gradient nanotubes under longitudinal magnetic field 67 Furthermore, for P-P nonlocal nanotube (β = 0) and without the magnetic field (H = 0), the linear frequency derived from Eq. (45) is the same as the linear frequency achieved by Wang and Li [18]. 4.2. Effects of the nonlocal parameter and the material length scale parameter Effects of the nonlocal and material length scale parameters on the nonlinear vibra- tion response of the nanotubes are shown in Figs.2–4. Fig.2 presents the variation of the frequency ratios ωratio of the nanotubes to the scale ratio c for L/D = 20, d/D = 0.8, KW = 50, H = 50, Q0 = 0.1 and some different values of the nonlocal parameter. And for case of L/D = 20, d/D = 0.8, KW = 50, H = 50, Q0 = 0.1 and some different values of the scale ratio, the variations of the frequency ratios to the nonlocal parameter and the material length scale parameter are plotted in Figs.3 and4, respectively. It can be seen that the frequency ratios of the classical nanotubes (ea/L = 0, c = 0) are equal to the frequency ratios of the SGT nanotubes with c = 1. From Fig.2, it can see that the frequency ratios decrease when the scale ratio increases. When c 1 (i.e., l > ea), the frequency ratios of the nanotube decrease as the nonlocal parameter ea/L increases. Fig.3 reveals that the frequency ratios of the NSGT nanotubes are always smaller than the ones of the nonlocal nanotubes (c = 0). From Figs.3 and4, it can observe that when c 1, the frequency ratios of the nanotubes decrease as the nonlocal and length scale parameters increase. It means that the nanotubes exert a softening effect when c 1. (a) P-PFig. nanotube 2: The variation of the frequency ratios to the scale ratio(b) C-C for some nanotube values of the nonlocal Fig. 2: The variation of theparameter; frequency (a) ratiosP-P nanotube to the scale and (b)ratio C -forC nanotube some values of the nonlocal parameter; (a) P-P nanotube and (b) C-C nanotube Fig.Effects 2. The of variation the nonlocal of the and frequency material length ratios scale to the parameters scale ratio on for the some nonlinear values vibration response Effects of the nonlocal and materialof the length nonlocal scale parameter parameters on the nonlinear vibration response of the nanotubes are shown in Figs. 2-4. Fig. 2 presents the variation of the frequency ratios wratio of of the nanotubes are shown in Figs. 2-4. Fig. 2 presents the variation of the frequency ratios wratio of the nanotubes to the scale ratio c for LD/= 20 , dD/= 0.8 , KW = 50 , H = 50 , Q0 = 0.1 and some the nanotubes to the scale ratio c for LD/= 20 , dD/= 0.8 , KW = 50 , H = 50 , Q0 = 0.1 and some different values of the nonlocal parameter. And for case of LD/= 20 , dD/= 0.8 , KW = 50 , different values of the nonlocal parameter. And for case of LD/= 20 , dD/= 0.8 , KW = 50 , H = 50 , Q0 = 0.1 and some different values of the scale ratio, the variations of the frequency ratios to H = 50 ,the Q0 nonlocal= 0.1 and parameter some different and thevalues material of the lengthscale ratio, scale the parameter variations are of plottedthe frequency in Fig sratios. 3 and to 4, the nonlocalrespectively. parameter It can and be see then materialthat the frequency length scal ratiose parameter of the classical are plottednanotubes in (Figea/0,0s. L ==3 and c 4,) are respectively.equal It to can the be frequency seen that ratiosthe frequency of the SGT ratios nanotubes of the classical with c =nanotubes1. From Fig.( ea 2/0,0, L it== can c see) thatare the equal tofrequency the frequency ratios decrease ratios of when the theSGT scale nanotubes ratio increases. with c When=1. Fromc 1 (i.e., of the nanotubelea> ), the increase frequency as ratios the nonlocal of the nanotube parameter decrease ea/ L as theincreases. nonlocal However, parameter whenea/ L cincreases.>1 (i.e., Fig. lea> ), 3the reveals frequency that theratios frequency of the nanotube ratios of decrease the NSGT as nanotubesthe nonlocal are parameter always smaller ea/ L thanincreases. the ones Fig. of the 3 revealsnonlocal that the nanotubes frequency ( c ratios= 0 ). From of the Fig NSGTs. 3 and nanotubes 4, it can observe are always that when smallerc 1, the ratios frequency of the nanotubesratios ofincrease the nanotubes as the nonlocal decrease a asnd the length nonlocal scale and parameters length scale increase; parameters while increase. c >1, the It frequencymeans that the ratios of nanotubesthe nanotubes exert decrease a softening as effectthe nonlocal when c and means1. that the nanotubes exert a softening effect when c 1.
14. 68NonlinearNonlinear vibration vibration of of nonlocal nonlocal strain strain gradient N. gradient D. Anh, nanotubes D. nanotubes V. Hieu under under longitudinal longitudinal magnetic magnetic field field 13 13 Nonlinear vibration of nonlocal strain gradient nanotubes under longitudinal magnetic field 13 Nonlinear vibration of nonlocal strain gradient nanotubes under longitudinal magnetic field 13 Fig.Fig. 3: 3: The(a) The P-Pvariation variation nanotube of of the the frequency frequency ratios ratios to tothe the nonlocal nonlocal parameter parameter(b) C-Cfor forsome nanotube some values values of the of thescale scale ratio;ratio; (a) (a) P- P- nanotubeP nanotube and and (b) (b) C- C -nanotubeC nanotube Fig.Fig.Fig. 3 .3: The3: The The variation variation variation of of the the frequency frequency frequency ratios ratios to to tothe the the nonlocal nonlocal nonlocal parameter parameter parameter for for some some for some values values values ofof thethe scalescale ratio;ratio; (a) (a) P -PP- Pnanotube ofnanotube the scale and and ratio(b) (b) C C-C-C nanotube nanotube Fig. 4: The variation of the frequency ratios to the material length scale parameter for some values of Fig. 4: The variationthe scaleof the ratio; frequency (a) P-P ratios nanotube to the and material (b) C- Clength nanotube scale parameter for some values of the scale ratio; (a) P-P nanotube and (b) C-C nanotube (a) P-P nanotube (b) C-C nanotube Fig.Fig. 4: 4: The The variation variation of of the the fr frequencyequency ratios ratios to to the the material material length length scale scale parameter parameter for for some some valuesvalues ofof 4.2. Effect of the diameterthethe scaleratio scale ratio; ratio; (a) (a) P -PP-P nanotube nanotube and and (b) (b) C C-C-C nanotube nanotube 4.2.Fig. Effect 4. The of variation the diameter of the frequency ratio ratios to the material length scale parameter for some values of the scale ratio For case of a = 0.2 , LD/= 20 , KW = 50 , H = 50 , Q0 = 0.1 and some values of the For case of a = 0.2 , LD/= 20 , K = 50 , H = 50 , Q = 0.1 and some values of the 4.2.diameter4.2. Effect Effect ratio of of the (thedD diameter/ diameter), the variationratio ratio of the frequencyW ratios of the nanotubes0 to the scale ratio is presenteddiameter in ratio Fig. ( dD5./And), Fig the. variation6 shows theof the variation frequency of the ratios frequency of the ratios nanotubes of the to nanotubes the scale to ratiothe is 4.3. Effect of the diameter ratio presentedFor in caseFig. of5. Anda = 0.2Fig., 6LD shows/= 20the, variationKW = 50 o, f Hthe= frequency50 , Q0 = 0.1ratiosand of the some nanotubes values ofto the diameterFor ratio case for ofa =a0.2= 0.2, LD,/ LD=/ 20=, 20KW, =K50W , =H50=,50 H, Q=050= ,0.1 Qand0 = 0.1someand different some values values of ofthe the For case of α = 0.2, L/D = 20, KW = 50, H = 50, Q0 = 0.1 and some values of the diameterscadiameterdiameterle ratio. ratio ratioratio As ( ca dD (fordDn/ /seea),= ),from the0.2 the, variation Fig.LD variation/ 5 that= 20of theof , the K the frequencyW frequency= frequency50 , H ratios= ratios50 ratios of, Q the 0of of= nanotubes the0.1 the andnanotubes nanotubes some are alwaysdifferent to to the the smaller sca scavalueslele ratiothan ratio of theisis (d D) presentedthepresentedscadiameter onesle ratio. inof in Fig.the As ratioFig. nanobeams ca5 .n5 And.seeAnd/ fromFig , Fig( the.dD Fig 6/0 6 variationshows shows =5 that); theand thethe of variationthe variationfrequency the frequency frequency o o fratios fthe theratios frequency ratios frequencyof ofthe the of nanotubes nanotubes the ratiosratios nanotubes of ofare thedecreasethe always nanotubes nanotubes to thesmallerwhen scale thetoto than thethe diamediametertheratio onesteris ratio ratioof presented the increases.for nanobeams a = in0.2 From Fig., LD(5 dD./Fig./0 And= 6,= 20 Fig.it );,can Kand6 showssee =the50that frequency, theHthe= variation frequency50 , ratiosQ = of0.1ratiosof thetheand of frequency nanotubes somethe NSGT different decrease ratios nanotubes values of when the areof the diameter ratio for a = 0.2 , LD/= 20 , KWW= 50 , H = 50 , Q0 0= 0.1 and some different values of the alwaysnanotubes smaller to than the the diameter ones of ratio the nonlocal for α = nanotubes0.2, L/D (=c =20,0 i.K W e., =l =50,0);H and= for50, theQ0 NSGT,= 0.1 and the scascadiamele leratio. ratio.ter Asratio As ca ca increases.n nsee see from from From Fig. Fig. Fig.5 5that that 6, theit the can frequency frequency see that ratios theratios frequency of of the the nanotubes nanotubesratios of the areare NSGT always always nanotubes smallersmaller thanthan are scalealwayssome ratio smaller different leads to than a values decrease the ones of in the the of scale thefrequency nonlocal ratio. ratios Asnanotubes canof the see nanot ( fromc =ubes0 Fig.i e.,5 thatl = 0 the); and frequency for the NSGT, ratios the thethe ones ones of of the the nanobeams nanobeams ( dD( dD/0/0== );); and and the the frequency frequency ratios ratios of of the the nanotubes nanotubes decreasedecrease when when the the scale ratio leads to a decrease in the frequency ratios of the nanotubes. diamediameterter ratio ratio increases. increases. From From Fig. Fig. 6, 6, it it can can see seethatthat the the frequency frequency ratios ratios ofof thethe NSGT NSGT nanotubes nanotubes areare alwaysalways smaller smaller than than the the ones ones of of the the nonlocal nonlocal nanotubes nanotubes ( (cc==00 i.i. e., e., ll==00);); and and for for the the NSGT, NSGT, the the scalescale ratio ratio leads leads to to a adecrease decrease in in the the frequency frequency ratios ratios of of the the nanot nanotubesubes. .
15. Nonlinear vibration of nonlocal strain gradient nanotubes under longitudinal magnetic field 69 of the nanotubes are always smaller than the ones of the nanobeams (d/D = 0); and the frequency ratios of the nanotubes decrease when the diameter ratio increases. From Fig.6, it can see that the frequency ratios of the NSGT nanotubes are always smaller than the ones of the nonlocal nanotubes (c = 0 i.e., l = 0); and for the NSGT, the scale ratio 14 N. D. Anh and D. V. Hieu 14leads to a decrease in the frequency ratiosN. D. Anh of theand nanotubes.D. V. Hieu 14 N. D. Anh and D. V. Hieu 14 N. D. Anh and D. V. Hieu Fig.Fig. 5:5: The variation(a) P-P of nanotube the frequency ratios to the scale ratio for some(b) C-C values nanotube of the diameter ratio; (a) P-P nanotube and (b) C-C nanotube Fig.Fig. 5:5: TheThe variationFig.variation 5. The of of variationthe the frequency frequency of the ratios frequencyratios to to the the ratiosscale scale toratio ratio the for scalefor some some ratio values values for some of ofthe valuesthe diameter diameter ratio; ratio; (a)(a) P P-P-P nanotube nanotubeof the diameterand and (b) (b) C ratioC-C- Cnanotube nanotube Fig.Fig. 6:6: The variation of the frequency ratios to the diameter ratio for some values of the scale parameter; (a) P-P nanotube and (b) C-C nanotube Fig.Fig. 6:6: TheThe variation variation(a) P-P of of nanotubethe the frequency frequency ratios ratios to to the the diameter diameter ratio ratio for for some some(b) values C-C values nanotube of ofthe the sca scale parameter;le parameter; 4.3.4.3. EffectEffect of the aspect ratio(a) (a) P P-P-P nanotube nanotube and and (b) (b) C -CC- Cnanotube nanotube Fig. 6. The variation of the frequency ratios to the diameter ratio for some values TheThe aspect ratio LD// has an importantof the scale influence parameteron the nonlinear vibration responses of the nanotubesnanotubes as shownshown inin FigFigs 7 and 8. For a = 0.3 , dD/= 0.8 , K = 20 , H = 50 and Q = 0.1, the 4.3. EffectEffect ofof thethe aspectaspect ratio ratio W 0 variationvariationss of the frequency ratios of the nanotubes to the scale ratio and the aspect ratio are plotted in The aspect ratio LD/ has an important influence on the nonlinear vibration responses of the FigFigss The 77 and aspect 8 coratiorrespondingrresponding LD/ has to an several important different influence valueson of the the nonlinear aspect ratio vibration and therespons scalees ratio, of the nanotubes as shown in Figs. 7 and 8. For a = 0.3 , dD/= 0.8 , KW = 20 , H = 50 and Q0 = 0.1, the nanotubesrespectively.respectively. as shown It It can can in be be Fig concluded concludeds. 7 and that8. For the a frequency= 0.3 , dD ratios/= 0.8of the, K nanotubesW = 20 , H increases= 50 and as Q the0 = aspect0.1, the variationratioratio increases. increases.ssofof thethe frequencyAndAndfrequency forfor a fixedfixedratios ratios value of of the the of nanotubes thenanotubes aspect toratio, to the the thescale scale frequency ratio ratio and and ratios the the as of pectas thepectratio nanotubes ratio are are plotted decrease plotted in in Figasasss .the.the 77 scalescale and and 8ratio 8 co co rrespondingrrespondingincreases. However, to to several several the differentfrequency different values ratios values of of of the the the NSGT aspect aspect nanotubes ratio ratio and andare the always the scale scale smaller ratio, ratio, respectively.respectively.thanthan thethe onesones It It ofof can can thethe be be nonlocal concluded concluded nanotubes that that the the( c frequency= frequency0 ). ratios ratios of of the the nanotubes nanotubes increases increases as as the the aspect aspect ratioratio increases. increases. AndAnd for for a a fixed fixed value value of of the the aspect aspect ratio, ratio, the the frequency frequency ratios ratios of ofthe the nanotubes nanotubes decrease decrease as thethe scalescale ratioratio increases.increases. However, However, the the frequency frequency ratios ratios of of the the NSGT NSGT nanotubes nanotubes are are always always smaller smaller than the ones of the nonlocal nanotubes ( c = 0 ). than the ones of the nonlocal nanotubes ( c = 0 ).
16. 70 N. D. Anh, D. V. Hieu 4.4. Effect of the aspect ratio The aspect ratio L/D has an important influence on the nonlinear vibration responses of the nanotubes as shown in Figs.7 and8. For α = 0.3, d/D = 0.8, KW = 20, H = 50 and Q0 = 0.1, the variations of the frequency ratios of the nanotubes to the scale ratio and the aspect ratio are plotted in Figs.7 and8 corresponding to several different values of theNonlinearNonlinear aspect vibration ratio and of the nonlocal scale ratio,strain gradient respectively.gradient nanotubesnanotubes It can under under be concluded longitudinal longitudinal that magnetic magnetic the frequency field field 1515 NonlinearNonlinear vibration vibration of nonlocalof nonlocal strain strain gradient gradient nanotubes nanotubes under under longitudinal longitudinal magnetic magnetic field field 15 15 Fig.Fig. 7:7: TheThe(a) variation P-P nanotube of the frequency ratratiosios toto thethe scalescale ratioratio for for some some(b) values values C-C nanotubeof of the the aspect aspect ratio; ratio; (a) PP P nanotube andand (b)(b) CC CC nanotube nanotube Fig.Fig. 7: The7: The variation variation of theof thefrequency frequency rat iosrat iosto the to thescale scale ratio ratio for forsome some values values of the of theaspect aspect ratio; ratio; Fig. 7. The variation(a) of (a)P the-P P nanotube- frequencyP nanotube and and ratios(b) (b)C-C toC nanotube-C the nanotube scale ratio for some values of the aspect ratio Fig. 8: The variation of the frequency ratios to the aspect ratio for some values of the scale ratio; Fig. 8: The variation of the frequency(a) P- ratiosP nanotube to the and aspect (b) ratioC-C nanotubefor some values of the scale ratio; (a) P-P nanotube and (b) C-C nanotube Fig.Fig. 8: The8: The (a)variation variation P-P nanotube of theof thefrequency frequency ratios ratios to the to theaspect aspect ratio ratio for forsome some(b) values C-C values of nanotube the of thescale scale ratio; ratio; (a) (a)P-P P nanotube-P nanotube and and (b) (b)C-C C nanotube-C nanotube 4.4. Effect of the elastic foundation 4.4. Effect Fig.of the 8. Theelastic variation foundation of the frequency ratios to the aspect ratio for some values of the scale ratio 4.4.4.4. Effe FigEffects. ofct9 ofthe and the elastic 10 elastic show foundation foundation effect of the Winkler parameter KW on the frequency ratios of the Figs. 9 and 10 show effect of the Winkler parameter KW on the frequency ratios of the nanotubes.Fig Fig.s. 9 9 and presents 10 show the variation effect of of the the Winkler frequency parameter ratios of Kthe nanotubeson the frequency to the scale ratios ratio of for the nanotubes.Figs. Fig.9 and 9 presents 10 show the effect variation of the of Winklerthe frequency parameter ratios K ofW theWon nanotubes the frequency to the ratios scale of ratio the for a = 0.2 , dD/= 0.8 , LD/= 20 , H = 50 , Q0 = 0.1 and some values of the Winkler parameter; while nanotubes.nanotubes., Fig. Fig. 9 presents9 presents, the the variation variation, of theof, Qthe frequency= frequency0.1 and ratios some ratios of values theof thenanotubes of nanotubes the Winkler to theto theparameter;scale scale ratio ratio forwhile for aFig.= 0.210 showdD/s =the 0.8 variationLD/= of 20 theH frequency= 50 0 ratios of the nanotubes to the Winkler parameter for a =a0.2= 0.2, dD, /dD/= 0.8= 0.8, LD, /LD/= 20= 20, H, =H50= ,50 Q, Q=00.1= 0.1andand some some values values of theof theWinkler Winkler parameter; parameter; while while Fig.a = 0.210 show, dD/s the= 0.8 variation, LD/ of= 20 the, H frequency= 50 , 0 Q ratios= 0.1 ofand the some nanotubes values of to the the scale Winkl ratio.er parameter As can see for Fig.Fig. 10 10 show shows thes the variation variation of of the the frequency frequency ratios0 ratios of ofthe the nanotubes nanotubes to the to the Winkl Winkler parameterer parameter for for a = 0.2 , dD/= 0.8 , LD/= 20 , H = 50 , Q0 = 0.1 and some values of the scale ratio. As can see afrom=a0.2= these0.2, ,dD / FiguresdD/= 0.8= 0.8that, LD, / theLD/= Winkler 20= 20, H, =Hparameter50= ,50 Q, Q= 0.1leads= 0.1and toand some a some decrease values values of theof the thescale frequency scale ratio. ratio. ratiosAs Ascan ofcan see the see fromnanotubes. these FiguresThe fact that that the the Winkler elastic parameter foundation0 0 leads enhances to a thedecrease stiffness of the of frequencythe nanotubes, ratios so of the the fromnanotubes.frequencyfrom these these of FiguresThe the Figures factnanotubes that thatthat the theincreases Winkler Winkler elastic byparameter foundationparameter increasing leads leads theenhances to coefficient to a decrease a the decrease of stiffness ofthe oftheelastic the frequencyof frequency thefoundation nanotubes, ratios ratios. However, of so theof the nanotubes.frequencythenanotubes. increase ofThe intheThe the fact nanotubes fact nonlinear that that the increasesthe frequency elastic elastic by foundation foundationdueincreasing to the enhances the elastic enhances coefficient foundation the the stiffness of stiffness the coefficient elastic of of the thefoundation nanotubes, is nanotubes, smaller. However, sothan so the the the frequencytheincreasefrequency increase in of the ofthein thelinear the nanotubes nanotubes nonlinear frequency, increases increases frequency thus by the by increasing due frequency increasing to the the ratios elastic the coefficient coefficient of foundationthe nanotubeof theof the elastic coefficients elastic decreases foundation foundation is when smaller. However, the. However, thanelastic the thethe increase increase in in the the nonlinear nonlinear frequency frequency due due to theto the elastic elastic foundation foundation coefficient coefficient is smaller is smaller than than the the increasefoundation in thecoefficientlinear frequency, increases. thus the frequency ratios of the nanotubes decreases when the elastic increasefoundationincrease in inthecoefficient thelinearlinear frequency, increases. frequency, thus thus the the frequency frequency ratios ratios of theof the nanotube nanotubes decreasess decreases when when the theelastic elastic foundationfoundation coefficient coefficient increases. increases.
17. Nonlinear vibration of nonlocal strain gradient nanotubes under longitudinal magnetic field 71 ratios of the nanotubes increases as the aspect ratio increases. And for a fixed value of the aspect ratio, the frequency ratios of the nanotubes decrease as the scale ratio increases. However, the frequency ratios of the NSGT nanotubes are always smaller than the ones of the nonlocal nanotubes (c = 0). 4.5. Effect of the elastic foundation 16 16 N. D. Anh andN. D. D. Anh V. Hieuand D. V. Hieu 16 16 Figs.9 and 10 show effect of theN. WinklerD.N. AnhD. Anh and parameterand D. V.D. HieuV. Hieu K W on the frequency ratios of the nanotubes. Fig.9 presents the variation of the frequency ratios of the nanotubes to Fig. 9: The Fig.variation (a)9: The P-P of variation nanotubethe frequency of the ratiosfrequency to the ratios scale to ratio the scalefor some ratio values for some of (b)the values C-CWinkler of nanotube the parameter; Winkler parameter; (a) P-P nanotube(a) P-P and nanotube (b) C- Cand nanotube (b) C-C nanotube Fig.Fig. 9: The 9: The variation variation of the of thefrequency frequency ratios ratios to the to thescale scale ratio ratio for forsome some values values of theof theWinkler Winkler parameter; parameter; Fig. 9. The variation(a) of P(a) the-P P nanotube- frequencyP nanotube and and ratios(b) (b)C-C toC nanotube- theC nanotube scale ratio for some values of the Winkler parameter Fig. 10: The variation of the frequency ratios to the Winkler parameter for some values of the scale Fig. 10: The variation of the frequencyratio; (a)ratios P-P to nanotube the Winkler and (b)parameter C-C nanotube for some values of the scale ratio; (a) P-P nanotube and (b) C-C nanotube Fig.Fig. 10: (a)10:The P-PThe variation nanotubevariation of the of thefrequency frequency ratios ratios to the to theWinkler Winkler parameter parameter(b) C-Cfor forsome nanotube some values values of theof thescale scale 4.5. Effect of the magneticratio;ratio; field (a) P(a) -P P nanotube-P nanotube and an (b)d (b)C-C C nanotube-C nanotube 4.5. Effect of the magnetic field Fig. 10.Final, The variationeffect of the of themagnetic frequency field on ratios the frequency to the Winkler ratios of parameter the nanotubes for some is investigated values in this Final, effect of the magnetic field on the offrequency the scale ratios ratio of the nanotubes is investigated in this 4.5. Effectsub-section. of the magnetic The variation field of the frequency ratios of the nanotubes to the scale ratio for a = 0.2 , sub4.5.-section. Effect of The the variation magnetic of field the frequency ratios of the nanotubes to the scale ratio for a = 0.2 , dD/= 0.8 , LD/= 20 , KW = 30 , Q0 = 0.1 and some values of the magnetic field is presented in Fig. Final,, effect ,of K the= magnetic30 , Q = field0.1 and on thesome frequency values of ratios the magne of thetic nanotubes field is presented is investigated in Fig. in this dD/=Final, 0.811.LD effectAnd/ =Fig. of 20 the12 showsWmagnetic the fieldvariation0 on the of thefrequency frequency ratios ratios of ofthe the nanotubes nanotubes is toinvestigated the magnetic in fieldthis for 11sub. Andsub-section.- Fig.section. 12 The shows The variation variation the variation of theof theof frequency the frequency frequency ratios ratios ratios of ofthe of the nanotubesthe nanotubes nanotubes to to theto the scale magnetic scale ratio ratio forfield fora for =a 0.2= 0.2, , a = 0.2 , dD/= 0.8 , LD/= 20 , KW = 30 , Q0 = 0.1 and some values of the scale ratio. It can be dD/dD/= 0.8= 0.8, LD,/ LD/= 20= , 20 K, K=W30=,30 Q, Q= 00.1= 0.1andand some some values values of theof the magne magnetic ticfield field is presented is presented in Fig. in Fig. a = 0.2 , concludeddD/= 0.8 that, LD / effectW= 20 of, theKW magne0= 30 , ticQ 0 field= 0.1 hasand the some same values as effect of the of scale the elasticratio. Itfoundation can be to the concluded11. 11And. And Fig.nonlinear that Fig. 12 effect shows12 vibration shows of the the the variationresponses magne variationtic ofof field theofthe the hasfrequencynanotubes. frequency the same ratiosAn asratiosincrease effectof theof inthe ofnanotubes the thenanotubes magnetic elastic to thetofoundation field the magnetic leadsmagnetic toto fielda the decrease field for for of nonlineara = 0.2 vibration, dD/ responses= 0.8 , LD of/ the= nanotubes. 20 , K = An30 increase, Q = 0.1 in theand magneticsome values field of leads the to scale a decrease ratio. ofIt can be a = 0.2 , the dD/ frequency= 0.8 , ratiosLD/ = of 20 the, K nanotubesW =W30 , ,Q this0 =0 0.1 observationand some is values completely of the consistentscale ratio. with It canthe be results theconcluded frequencyconcludedobtained that ratios that effect by effect of Chang of the theof nanotubes [the52 magne ]. magneAndtic, thisfortic field field a observation fixed has has the value the same is same of completely as the aseffect magnetic effect of consistent of the field, the elastic theelastic with frequencyfoundation foundationthe results ratios to to the of the the obtainednonlinearnonlinear by vibration Chang vibration [responses52 ]responses. And of for the of a the fixednanotubes. nanotubes. value An of An theincrease increase magnetic in thein field,the magnetic magnetic the frequency field field leads leads ratios to ato decrease ofa decrease the of of thethe frequency frequency ratios ratios of of the the nanotubes nanotubes, this, this observation observation is completelyis completely consistent consistent with withthethe results results obtainedobtained by by Chang Chang [52 []52. And]. And for for a fixed a fixed value value of ofthe the magnetic magnetic field, field, the the frequency frequency ratios ratios of of the the
18. 72 N. D. Anh, D. V. Hieu the scale ratio for α = 0.2, d/D = 0.8, L/D = 20, H = 50, Q0 = 0.1 and some values of the Winkler parameter; while Fig. 10 shows the variation of the frequency ratios of the nanotubes to the Winkler parameter for α = 0.2, d/D = 0.8, L/D = 20, H = 50, Q0 = 0.1 and some values of the scale ratio. As can see from these Figures that the Winkler param- eter leads to a decrease of the frequency ratios of the nanotubes. The fact that the elastic foundation enhances the stiffness of the nanotubes, so the frequency of the nanotubes increases by increasing the coefficient of the elastic foundation. However, the increase in the nonlinear frequency due to the elastic foundation coefficient is smaller than the in- crease in the linear frequency, thus the frequency ratios of the nanotubes decreases when the elastic foundation coefficient increases. 4.6. Effect of the magnetic field Final, effect of the magnetic field on the frequency ratios of the nanotubes is inves- tigated in this sub-section. The variation of the frequency ratios of the nanotubes to the scale ratio for α = 0.2, d/D = 0.8, L/D = 20, KW = 30, Q0 = 0.1 and some values of the magnetic field is presented in Fig. 11. And Fig. 12 shows the variation of the fre- quency ratios of the nanotubes to the magnetic field for α = 0.2, d/D = 0.8, L/D = 20, KW = 30, Q0 = 0.1 and some values of the scale ratio. It can be concluded that effect of the magnetic field has the same as effect of the elastic foundation to the nonlinear vibra- tionNonlinearNonlinear responses vibration vibration of the of of nanotubes. nonlocal nonlocal strain strain An gradient increase gradient nanotubes in nanotubes the magnetic under under longitudinal fieldlongitudinal leads magnetic to magnetic a decrease field field of 1717 the frequency ratios of the nanotubes, this observation is completely consistent with the results obtained by Chang [52]. And for a fixed value of the magnetic field, the frequency nanotubesnanotubes decreasedecrease as as the the scale scale ratio ratio increases. increases. The The magnetic magnetic fie fieldld has has an an important important influence influenceonon the the ratios of the nanotubes decrease as the scale ratio increases. The magnetic field has an nonlinearnonlinear vibration vibration responses responses of of the the nanotub nanotubes.es. important influence on the nonlinear vibration responses of the nanotubes. Fig.Fig. 11: 11: The The variation variation(a) P-P nanotubeof of the the frequency frequency ratios ratios to to the the scale scale ratio ratio for for some some(b) values C-C values nanotube of of the the magnetic magnetic field; field; (a)(a) P P-P-P nanotube nanotube and and (b) (b) C C-C- Cnanotube nanotube Fig. 11. The variation of the frequency ratios to the scale ratio for some values of the magnetic field Fig. 12: The variation of the frequency ratios to the magnetic field for some values of the scale ratio; Fig. 12: The variation of the frequency ratios to the magnetic field for some values of the scale ratio; (a) P-P nanotube and (b) C-C nanotube (a) P-P nanotube and (b) C-C nanotube 5. CONCLUSIONS 5. CONCLUSIONS The nonlinear vibration response of the nanotubes under the magnetic field based on the The nonlinear vibration response of the nanotubes under the magnetic field based on the nonlocal strain gradient theory and the Euler-Bernoulli beam theory is investigated in this work. The nonlocal strain gradient theory and the Euler-Bernoulli beam theory is investigated in this work. The equation of motion for the nanotubes is derived by the Hamilton principle. The Glerkin technique and theequa equivalenttion of motion linearizationfor the nanotubeswith the weighted is derived averaging by the Hami valuelton are principle. employed The to findGlerkin the approximatetechnique and frethequencies equivalent of thelinearization nanotubes with with the pinned weighted-pinned averaging and clamped value -areclamped employed boundary to find conditions. the approximate The comparisonfrequencies of of the the obtained nanotubesapproximate with pinned solutions-pinned with and the clamped exact ones-clamped and the boundary published conditions. ones shows The thecomparison accuracy of thethe obtained obtained approximateapproximate solutions. solutions with the exact ones and the published ones shows the accuracy of the obtained approximate solutions.
19. 17 NonlinearNonlinear vibration vibration of nonlocal of nonlocal strain strain gradient gradient nanotubes nanotubes under under longitudinal longitudinal magnetic magnetic field field 17 nanotubesnanotubes decrease decrease as the as scale the scaleratio ratioincreases. increases. The magneticThe magnetic field fie hasld an has important an important influence influenceon theon the nonlinearnonlinear vibration vibration responses responses of the of nanotub the nanotubes. es. Fig. 11:Fig. The 11: variation The variation of the offrequency the frequency ratios ratiosto the to scale the scaleratio forratio some for some values values of the of magnetic the magnetic field; field; Nonlinear vibration of nonlocal strain gradient nanotubes under longitudinal magnetic field 73 (a) P-P(a) nanotube P-P nanotube and (b) and C -(b)C nanotube C-C nanotube Fig. 12:Fig. The 12: variation (a)The P-P variation nanotubeof the offrequency the frequency ratios ratiosto the to magnetic the magnetic field forfield some for(b) some values C-C valuesnanotube of the of scale the scaleratio; ratio; (a) P-P(a) nanotube P-P nanotube and (b) and C -(b)C nanotubeC-C nanotube Fig. 12. The variation of the frequency ratios to the magnetic field for some values of the scale ratio 5. CONCLUSIONS5. CONCLUSIONS 5. CONCLUSIONS The nonlinearThe nonlinear vibration vibration response response of the of nanotubes the nanotubes under under the magnetic the magnetic field field based based on the on the nonlocalnonlocalThe strain nonlinear straingradient gradient vibrationtheory theory and response the and Euler the Euler- ofBernoulli the-Bernoulli nanotubes beam beam theory under theory is investigated the is magneticinvestigated in this field in work.this based work. The on The equathetionequa nonlocal oftion motion of strainmotionfor the gradientfor nanotubesthe nanotubes theory is derived is and derived by the the Euler–Bernoulliby Hami the Hamilton principle.lton principle. beam The theory GlerkinThe Glerkin is techniqu investigated technique ande and thein equivalent thisthe equivalent work. linearization The linearization equation with of thewith motion weighted the weighted for averaging the averaging nanotubes value value are is derivedemployed are employed by to the find to Hamilton findthe approximatethe approximate princi- freple.quenciesfre Thequencies Glerkinof the of nanotubes the technique nanotubes with and with pinned the pinned equivalent-pinned-pinned and linearization clampedand clamped-clamped- withclamped boundary the boundaryweighted conditions. conditions. averaging The The comparisonvaluecomparison are of employed the of obtained the obtained toapproximate findapproximate the approximate solutions solutions with frequencies with the exact the exact ones of theonesand nanotubes theand publishedthe published with ones pinned- onesshows shows the accuracy of the obtained approximate solutions. thepinned accuracy and of the clamped-clamped obtained approximate boundary solutions.conditions. The comparison of the obtained approximate solutions with the exact ones and the published ones shows the accuracy of the obtained approximate solutions. Effects of the nonlocal parameter ea/L, the material length scale parameter l/L, the aspect ratio L/D, the diameter ratio d/D, the elastic foundation KW and the magnetic field H on the nonlinear vibration responses of the nanotubes are examined. It can be concluded that: - The nonlocal parameter leads to a decrease in the nonlinear frequencies of the nan- otubes; while the material length scale parameter, the aspect ratio, the diameter ratio, the elastic foundation and the magnetic field lead to an increase in the nonlinear frequencies of the nanotubes. - When l ea, the frequency ratios of the nanotube decrease as the nonlocal parameter ea/L increases. The frequency ratios of the NSGT nanotubes are always smaller than the ones of the nonlocal nanotubes (c = l/ea = 0). When c 1, the frequency ratios of the nanotubes decrease as the nonlocal and length scale parameters increase.
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