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Aeronautics and Aerospace Open Access Journal

Research Article Volume 7 Issue 1

On the dynamic behaviour of carbon nanotubes conveying fluid resting on elastic foundations in a magnetic-thermal environment: effects of surface energy and initial stress

Gbeminiyi M Sobamowo,2 Olorunfemi O Isaac,1 Suraju A Oladosu,1 Rafiu O Kuku1

1Department of Mechanical Engineering, Lagos State University, Nigeria
2Department of Mechanical Engineering, University of Lagos, Nigeria

Correspondence: Gbeminiyi M Sobamowo, Department of Mechanical Engineering, University of Lagos, Nigeria

Received: March 23, 2023 | Published: April 4, 2023

Citation: Isaac OO, Oladosu SA, Kuku RO, et al. On the dynamic behaviour of carbon nanotubes conveying fluid resting on elastic foundations in a magneticthermal environment: effects of surface energy and initial stress. Aeron Aero Open Access J. 2023;7(1):26-34. DOI: 10.15406/aaoaj.2023.07.00167

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Abstract

In this article, simultaneous impacts of surface elasticity, initial stress, residual surface tension and nonlocality on the nonlinear vibration of single-walled carbon conveying nanotube resting on linear and nonlinear elastic foundation and operating in a thermo-magnetic environment are studied. The developed equation of motion is solved using Galerkin’s decomposition and Temini and Ansari method. The studies of the impacts of various parameters on the vibration problems revealed that the ratio of the nonlinear to linear frequencies increases with the negative value of the surface stress while it decreases with the positive value of the surface stress. The surface effect reduces for increasing in the length of the nanotube. Ratio of the frequencies decreases with increase in the strength of the magnetic field, nonlocal parameter and the length of the nanotube. Increase in temperature change at high temperature causes decrease in the frequency ratio. However, at room or low temperature, the frequency ratio of the hybrid nanostructure increases as the temperature change increases. The natural frequency of the nanotube gradually approaches the nonlinear Euler–Bernoulli beam limit at high values of nonlocal parameter and nanotube length. Nonlocal parameter reduces the surface effects on the ratio of the frequencies. Also, the ratio of the frequencies at low temperatures is lower than at high temperatures. It is hoped that the present work will enhance the control and design of carbon nanotubes operating in thermo-magnetic environment and resting on elastic foundations.

Keywords: surface effects, carbon nanotubes, nonlocal elasticity theory, Temini and Ansari method

Nomenclature

A, area of the nanotube; E, modulus of elasticity; EI, bending rigidity; Hs, residual surface stress; Hx, magnetic field strength; I moment of area; L, length of the nanotube; mc mass of tube per unit length; N, axial/Longitudinal force; T, change in temperature; t, time coordinate; w, transverse displacement/deflection of the nanotube; W, time-dependent parameter; x, axial coordinate; ϕ(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHvp GzcaGGOaGaamiEaiaacMcaaaa@3AA3@ , trial/comparison function; α x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacqaHXo qyjuaGdaWgaaWcbaqcLbsacaWG4baaleqaaaaa@3A75@ , coefficient of thermal expansion; η, permeability

Introduction

Carbon nanotubes have shown to be nanostructures with remarkable physical, mechanical, electrical and chemical properties. Such excellent properties have aided their various medical, industrial, electrical, thermal, electronic and mechanical applications.1–5 Due to their importance for the practical applications, their dynamic behaviours have been studied.6–13 However, the effects of the surface energy and initial stress are neglected in the studies. Indisputably, the properties of the region of the solid surface are different properties from the bulk material. Also, for classical structures, surface energy-to-bulk energy ratio is small. However, nanostructures have large surface energy-to-bulk energy ratio and high ratio of surface energies to volume, elastic modulus and mechanical strength. Consequently, the mechanical behaviours, bending deformation and elastic waves of the nanostructures are greatly influenced. Therefore, the surface energy effects cannot be neglected in the dynamic behaviour analysis of nanostructures.  Such surface energy of nanostructures is composed of the surface tension and surface modulus exerted on the surface layer of nanostructures.13–21 Consequently, different works have been presented in literature to analyze the impacts of surface energy on the dynamic response and instability of nanostructures.22–28

Due to residual stress, thermal effects, surface effects, mismatches between the material properties of CNTs and surrounding mediums, initial external loads and other physical issues, carbon nanotubes often suffer from initial stresses The effects of initial stress on the dynamic behaviour of nanotubes have been studied.29–37 However, because of their significant in practically nano-apparatus applications, there is a need for a combined on the effects of surface behaviours, initial stress and nonlocality on the physical characteristics and mechanical behaviours of carbon nanotubes.

In the above past studies, different mathematical methods have been used to analyze the problem.  However, most of these methods require high skill in mathematical analysis for their applications. As a means of overcoming the drawbacks in the other approximation analytical methods, recently, Temimi and Ansari38 introduced the semi-analytical iterative technique in 2011 for solving nonlinear problems. The new iterative method has been used to solve many differential equations, such as nonlinear differential equations.39–55 The results obtained in these studies indicate that the Temimi and Ansari method (TAM) provides excellent approximations to the solution of nonlinear equation with low computational time, high accuracy, and high order of convergence. The previous studies39–55 have shown that the TAM can solve nonlinear differential and integral equations without linearization, discretization, restrictive assumptions, closure, perturbation, approximations, discretization and round-off error that could result in massive numerical computations. This method has not been applied to solve vibration problems in nanostructures. Also, scanning through the past works and to the best of the authors’ knowledge, a study on simultaneous effects of surface energy and initial stress on the vibration characteristics of nanotubes resting of Winkler and Pasternak foundations in a thermo-magnetic environment has not been carried out. Therefore, in this work, Temini and Ansari method is applied with Galerkin’s decomposition method to study the coupled impacts of surface effects, initial stress and nonlocality on the nonlinear dynamic behaviour of single-walled carbon nanotubes resting on Winkler (Spring) and Pasternak (Shear layer) foundations in a thermal-magnetic environment. Erigen’s nonlocal elasticity, Maxwell’s relations, Hamilton’s principle, surface effect and Euler-Bernoulli beam theories are adopted to develop the systems of nonlinear equations of the dynamics behaviour of the carbon nanotube. The studies of the impacts of various parameters on the vibration problems are also carried out.

Model development

Figure 1 shows a single-walled CNT of length L and inner and outer diameters Di and Do resting on Winkler (Spring) and Pasternak (Shear layer) foundations.  The SWCNTs conveying a hot fluid and resting on elastic foundation under external applied tension, initial stress, magnetic and temperature fields as shown in the figure.

Based on Erigen’s theory, Euler-Bernoulli’s theory and Hamilton’s principle.56–58 The partial differential equation governing the dynamic behaviour is derived as

( EI+ E s I s ) 4 w x 4 +( m cn + m f ) 2 w t 2 +2u m f 2 w xt +[ EA 2L 0 L ( w x ) 2 dx ] 2 w x 2 +( m f u 2 +δA σ x o H s η H x 2 A k p + EAαΔT 12ν ) 2 w x 2 + k 1 w+ k 3 w 3 ( e o a ) 2 [ ( m cn + m f ) 4 w x 2 t 2 +2u m f 4 w x 3 t +[ EA 2L 0 L ( w x ) 2 dx ] 4 w x 4 +( m f u 2 +δA σ x o H s η H x 2 A k p + EAαΔT 12ν ) 4 w x 4 + k 1 2 w x 2 +3 k 3 w 2 2 w x 2 +6 k 3 w ( w x ) 2 ]=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaadaqada qaaiaadweacaWGjbGaey4kaSIaamyramaaBaaaleaacaWGZbaabeaa kiaadMeadaWgaaWcbaGaam4CaaqabaaakiaawIcacaGLPaaadaWcaa qaaiabgkGi2oaaCaaaleqabaGaaGinaaaakiaadEhaaeaacqGHciIT caWG4bWaaWbaaSqabeaacaaI0aaaaaaakiabgUcaRiaacIcacaWGTb WaaSbaaSqaaiaadogacaWGUbaabeaakiabgUcaRiaad2gadaWgaaWc baGaamOzaaqabaGccaGGPaWaaSaaaeaacqGHciITdaahaaWcbeqaai aaikdaaaGccaWG3baabaGaeyOaIyRaamiDamaaCaaaleqabaGaaGOm aaaaaaGccqGHRaWkcaaIYaGaamyDaiaad2gadaWgaaWcbaGaamOzaa qabaGcdaWcaaqaaiabgkGi2oaaCaaaleqabaGaaGOmaaaakiaadEha aeaacqGHciITcaWG4bGaeyOaIyRaamiDaaaacqGHRaWkdaWadaqaam aalaaabaGaamyraiaadgeaaeaacaaIYaGaamitaaaadaWdXaqaamaa bmaabaWaaSaaaeaacqGHciITcaWG3baabaGaeyOaIyRaamiEaaaaai aawIcacaGLPaaadaahaaWcbeqaaiaaikdaaaGccaWGKbGaamiEaaWc baGaaGimaaqaaiaadYeaa0Gaey4kIipaaOGaay5waiaaw2faamaala aabaGaeyOaIy7aaWbaaSqabeaacaaIYaaaaOGaam4DaaqaaiabgkGi 2kaadIhadaahaaWcbeqaaiaaikdaaaaaaaGcbaGaey4kaSYaaeWaae aacaWGTbWaaSbaaSqaaiaadAgaaeqaaOGaamyDamaaCaaaleqabaGa aGOmaaaakiabgUcaRiabes7aKjaadgeacqaHdpWCdaqhaaWcbaGaam iEaaqaaiaad+gaaaGccqGHsislcaWGibWaaSbaaSqaaiaadohaaeqa aOGaeyOeI0Iaeq4TdGMaamisamaaDaaaleaacaWG4baabaGaaGOmaa aakiaadgeacqGHsislcaWGRbWaaSbaaSqaaiaadchaaeqaaOGaey4k aSYaaSaaaeaacaWGfbGaamyqaiabeg7aHjabfs5aejaadsfaaeaaca aIXaGaeyOeI0IaaGOmaiabe27aUbaaaiaawIcacaGLPaaadaWcaaqa aiabgkGi2oaaCaaaleqabaGaaGOmaaaakiaadEhaaeaacqGHciITca WG4bWaaWbaaSqabeaacaaIYaaaaaaakiabgUcaRiaadUgadaWgaaWc baGaaGymaaqabaGccaWG3bGaey4kaSIaam4AamaaBaaaleaacaaIZa aabeaakiaadEhadaahaaWcbeqaaiaaiodaaaaakeaacqGHsisldaqa daqaaiaadwgadaWgaaWcbaGaam4BaaqabaGccaWGHbaacaGLOaGaay zkaaWaaWbaaSqabeaacaaIYaaaaOWaamWaaqaabeqaaiaacIcacaWG TbWaaSbaaSqaaiaadogacaWGUbaabeaakiabgUcaRiaad2gadaWgaa WcbaGaamOzaaqabaGccaGGPaWaaSaaaeaacqGHciITdaahaaWcbeqa aiaaisdaaaGccaWG3baabaGaeyOaIyRaamiEamaaCaaaleqabaGaaG OmaaaakiabgkGi2kaadshadaahaaWcbeqaaiaaikdaaaaaaOGaey4k aSIaaGOmaiaadwhacaWGTbWaaSbaaSqaaiaadAgaaeqaaOWaaSaaae aacqGHciITdaahaaWcbeqaaiaaisdaaaGccaWG3baabaGaeyOaIyRa amiEamaaCaaaleqabaGaaG4maaaakiabgkGi2kaadshaaaGaey4kaS YaamWaaeaadaWcaaqaaiaadweacaWGbbaabaGaaGOmaiaadYeaaaWa a8qmaeaadaqadaqaamaalaaabaGaeyOaIyRaam4DaaqaaiabgkGi2k aadIhaaaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaOGaamiz aiaadIhaaSqaaiaaicdaaeaacaWGmbaaniabgUIiYdaakiaawUfaca GLDbaadaWcaaqaaiabgkGi2oaaCaaaleqabaGaaGinaaaakiaadEha aeaacqGHciITcaWG4bWaaWbaaSqabeaacaaI0aaaaaaaaOqaaiabgU caRmaabmaabaGaamyBamaaBaaaleaacaWGMbaabeaakiaadwhadaah aaWcbeqaaiaaikdaaaGccqGHRaWkcqaH0oazcaWGbbGaeq4Wdm3aa0 baaSqaaiaadIhaaeaacaWGVbaaaOGaeyOeI0IaamisamaaBaaaleaa caWGZbaabeaakiabgkHiTiabeE7aOjaadIeadaqhaaWcbaGaamiEaa qaaiaaikdaaaGccaWGbbGaeyOeI0Iaam4AamaaBaaaleaacaWGWbaa beaakiabgUcaRmaalaaabaGaamyraiaadgeacqaHXoqycqqHuoarca WGubaabaGaaGymaiabgkHiTiaaikdacqaH9oGBaaaacaGLOaGaayzk aaWaaSaaaeaacqGHciITdaahaaWcbeqaaiaaisdaaaGccaWG3baaba GaeyOaIyRaamiEamaaCaaaleqabaGaaGinaaaaaaaakeaacqGHRaWk caWGRbWaaSbaaSqaaiaaigdaaeqaaOWaaSaaaeaacqGHciITdaahaa WcbeqaaiaaikdaaaGccaWG3baabaGaeyOaIyRaamiEamaaCaaaleqa baGaaGOmaaaaaaGccqGHRaWkcaaIZaGaam4AamaaBaaaleaacaaIZa aabeaakiaadEhadaahaaWcbeqaaiaaikdaaaGcdaWcaaqaaiabgkGi 2oaaCaaaleqabaGaaGOmaaaakiaadEhaaeaacqGHciITcaWG4bWaaW baaSqabeaacaaIYaaaaaaakiabgUcaRiaaiAdacaWGRbWaaSbaaSqa aiaaiodaaeqaaOGaam4DamaabmaabaWaaSaaaeaacqGHciITcaWG3b aabaGaeyOaIyRaamiEaaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaa ikdaaaaaaOGaay5waiaaw2faaiabg2da9iaaicdaaaaa@3CDB@   (1)

Figure 2 shows the effect of flow in a channel. In the fluid-conveying carbon nanotube, the condition of slip is satisfied since in such flow, the ratio of the mean free path of the fluid molecules relative to a characteristic length of the flow geometry which is the Knudsen number is larger than 10-2. Consequently, the velocity correction factor for the slip flow velocity is proposed as59:

VCF= u avg,slip u avg,noslip =( 1+ a k Kn )[ 4( 2 σ v σ v )( Kn 1+Kn )+1 ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvaiaado eacaWGgbGaeyypa0ZaaSaaaeaacaWG1bWaaSbaaSqaaiaadggacaWG 2bGaam4zaiaacYcacaWGZbGaamiBaiaadMgacaWGWbaabeaaaOqaai aadwhadaWgaaWcbaGaamyyaiaadAhacaWGNbGaaiilaiaad6gacaWG VbGaeyOeI0Iaam4CaiaadYgacaWGPbGaamiCaaqabaaaaOGaeyypa0 ZaaeWaaeaacaaIXaGaey4kaSIaamyyamaaBaaaleaacaWGRbaabeaa kiaadUeacaWGUbaacaGLOaGaayzkaaWaamWaaeaacaaI0aWaaeWaae aadaWcaaqaaiaaikdacqGHsislcqaHdpWCdaWgaaWcbaGaamODaaqa baaakeaacqaHdpWCdaWgaaWcbaGaamODaaqabaaaaaGccaGLOaGaay zkaaWaaeWaaeaadaWcaaqaaiaadUeacaWGUbaabaGaaGymaiabgUca RiaadUeacaWGUbaaaaGaayjkaiaawMcaaiabgUcaRiaaigdaaiaawU facaGLDbaaaaa@696E@   (2)

Where Kn is the Knudsen number, σv is tangential moment accommodation coefficient which is considered to be 0.7 for most practical purposes41,42

a k = a o 2 π [ ta n 1 ( a 1 K n B ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBa aaleaacaWGRbaabeaakiabg2da9iaadggadaWgaaWcbaGaam4Baaqa baGcdaWcaaqaaiaaikdaaeaacqaHapaCaaWaamWaaeaacaWG0bGaam yyaiaad6gadaahaaWcbeqaaiabgkHiTiaaigdaaaGcdaqadaqaaiaa dggadaWgaaWcbaGaaGymaaqabaGccaWGlbGaamOBamaaCaaaleqaba GaamOqaaaaaOGaayjkaiaawMcaaaGaay5waiaaw2faaaaa@4A65@   (3)

a o = 64 3π( 1 4 b ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBa aaleaacaWGVbaabeaakiabg2da9maalaaabaGaaGOnaiaaisdaaeaa caaIZaGaeqiWda3aaeWaaeaacaaIXaGaeyOeI0YaaSaaaeaacaaI0a aabaGaamOyaaaaaiaawIcacaGLPaaaaaaaaa@41FA@   (4)

a 1 =4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBa aaleaacaaIXaaabeaakiabg2da9iaaisdaaaa@3991@   and B = 0.04 and b is the general slip coefficient (b = −1).

From Eq. (2),

u avg,slip =( 1+ a k Kn )[ 4( 2 σ v σ v )( Kn 1+Kn )+1 ] u avg,noslip MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyDamaaBa aaleaacaWGHbGaamODaiaadEgacaGGSaGaam4CaiaadYgacaWGPbGa amiCaaqabaGccqGH9aqpdaqadaqaaiaaigdacqGHRaWkcaWGHbWaaS baaSqaaiaadUgaaeqaaOGaam4saiaad6gaaiaawIcacaGLPaaadaWa daqaaiaaisdadaqadaqaamaalaaabaGaaGOmaiabgkHiTiabeo8aZn aaBaaaleaacaWG2baabeaaaOqaaiabeo8aZnaaBaaaleaacaWG2baa beaaaaaakiaawIcacaGLPaaadaqadaqaamaalaaabaGaam4saiaad6 gaaeaacaaIXaGaey4kaSIaam4saiaad6gaaaaacaGLOaGaayzkaaGa ey4kaSIaaGymaaGaay5waiaaw2faaiaadwhadaWgaaWcbaGaamyyai aadAhacaWGNbGaaiilaiaad6gacaWGVbGaeyOeI0Iaam4CaiaadYga caWGPbGaamiCaaqabaaaaa@65E0@   (5)

Therefore, Eq. (1) can be written as

( EI+ E s I s ) 4 w x 4 +( m cn + m f ) 2 w t 2 +2 m f ( 1+ a k Kn )[ 4( 2 σ v σ v )( Kn 1+Kn )+1 ] 2 w xt +[ EA 2L 0 L ( w x ) 2 dx ] 2 w x 2 +( m f [ ( 1+ a k Kn )[ 4( 2 σ v σ v )( Kn 1+Kn )+1 ] ] 2 +δA σ x o H s η H x 2 A k p + EAαΔT 12ν ) 2 w x 2 + k 1 w+ k 3 w 3 ( e o a ) 2 [ ( m cn + m f ) 4 w x 2 t 2 +2 m f ( 1+ a k Kn )[ 4( 2 σ v σ v )( Kn 1+Kn )+1 ] 4 w x 3 t +[ EA 2L 0 L ( w x ) 2 dx ] 4 w x 4 +( m f [ ( 1+ a k Kn )[ 4( 2 σ v σ v )( Kn 1+Kn )+1 ] ] 2 +δA σ x o H s η H x 2 A k p + EAαΔT 12ν ) 4 w x 4 + k 1 2 w x 2 +3 k 3 w 2 2 w x 2 +6 k 3 w ( w x ) 2 ]=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaadaqada qaaiaadweacaWGjbGaey4kaSIaamyramaaBaaaleaacaWGZbaabeaa kiaadMeadaWgaaWcbaGaam4CaaqabaaakiaawIcacaGLPaaadaWcaa qaaiabgkGi2oaaCaaaleqabaGaaGinaaaakiaadEhaaeaacqGHciIT caWG4bWaaWbaaSqabeaacaaI0aaaaaaakiabgUcaRiaacIcacaWGTb WaaSbaaSqaaiaadogacaWGUbaabeaakiabgUcaRiaad2gadaWgaaWc baGaamOzaaqabaGccaGGPaWaaSaaaeaacqGHciITdaahaaWcbeqaai aaikdaaaGccaWG3baabaGaeyOaIyRaamiDamaaCaaaleqabaGaaGOm aaaaaaGccqGHRaWkcaaIYaGaamyBamaaBaaaleaacaWGMbaabeaakm aabmaabaGaaGymaiabgUcaRiaadggadaWgaaWcbaGaam4AaaqabaGc caWGlbGaamOBaaGaayjkaiaawMcaamaadmaabaGaaGinamaabmaaba WaaSaaaeaacaaIYaGaeyOeI0Iaeq4Wdm3aaSbaaSqaaiaadAhaaeqa aaGcbaGaeq4Wdm3aaSbaaSqaaiaadAhaaeqaaaaaaOGaayjkaiaawM caamaabmaabaWaaSaaaeaacaWGlbGaamOBaaqaaiaaigdacqGHRaWk caWGlbGaamOBaaaaaiaawIcacaGLPaaacqGHRaWkcaaIXaaacaGLBb GaayzxaaWaaSaaaeaacqGHciITdaahaaWcbeqaaiaaikdaaaGccaWG 3baabaGaeyOaIyRaamiEaiabgkGi2kaadshaaaGaey4kaSYaamWaae aadaWcaaqaaiaadweacaWGbbaabaGaaGOmaiaadYeaaaWaa8qmaeaa daqadaqaamaalaaabaGaeyOaIyRaam4DaaqaaiabgkGi2kaadIhaaa aacaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaOGaamizaiaadIha aSqaaiaaicdaaeaacaWGmbaaniabgUIiYdaakiaawUfacaGLDbaada WcaaqaaiabgkGi2oaaCaaaleqabaGaaGOmaaaakiaadEhaaeaacqGH ciITcaWG4bWaaWbaaSqabeaacaaIYaaaaaaaaOqaaiabgUcaRmaabm aabaGaamyBamaaBaaaleaacaWGMbaabeaakmaadmaabaWaaeWaaeaa caaIXaGaey4kaSIaamyyamaaBaaaleaacaWGRbaabeaakiaadUeaca WGUbaacaGLOaGaayzkaaWaamWaaeaacaaI0aWaaeWaaeaadaWcaaqa aiaaikdacqGHsislcqaHdpWCdaWgaaWcbaGaamODaaqabaaakeaacq aHdpWCdaWgaaWcbaGaamODaaqabaaaaaGccaGLOaGaayzkaaWaaeWa aeaadaWcaaqaaiaadUeacaWGUbaabaGaaGymaiabgUcaRiaadUeaca WGUbaaaaGaayjkaiaawMcaaiabgUcaRiaaigdaaiaawUfacaGLDbaa aiaawUfacaGLDbaadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcqaH0o azcaWGbbGaeq4Wdm3aa0baaSqaaiaadIhaaeaacaWGVbaaaOGaeyOe I0IaamisamaaBaaaleaacaWGZbaabeaakiabgkHiTiabeE7aOjaadI eadaqhaaWcbaGaamiEaaqaaiaaikdaaaGccaWGbbGaeyOeI0Iaam4A amaaBaaaleaacaWGWbaabeaakiabgUcaRmaalaaabaGaamyraiaadg eacqaHXoqycqqHuoarcaWGubaabaGaaGymaiabgkHiTiaaikdacqaH 9oGBaaaacaGLOaGaayzkaaWaaSaaaeaacqGHciITdaahaaWcbeqaai aaikdaaaGccaWG3baabaGaeyOaIyRaamiEamaaCaaaleqabaGaaGOm aaaaaaGccqGHRaWkcaWGRbWaaSbaaSqaaiaaigdaaeqaaOGaam4Dai abgUcaRiaadUgadaWgaaWcbaGaaG4maaqabaGccaWG3bWaaWbaaSqa beaacaaIZaaaaaGcbaGaeyOeI0YaaeWaaeaacaWGLbWaaSbaaSqaai aad+gaaeqaaOGaamyyaaGaayjkaiaawMcaamaaCaaaleqabaGaaGOm aaaakmaadmaaeaqabeaacaGGOaGaamyBamaaBaaaleaacaWGJbGaam OBaaqabaGccqGHRaWkcaWGTbWaaSbaaSqaaiaadAgaaeqaaOGaaiyk amaalaaabaGaeyOaIy7aaWbaaSqabeaacaaI0aaaaOGaam4Daaqaai abgkGi2kaadIhadaahaaWcbeqaaiaaikdaaaGccqGHciITcaWG0bWa aWbaaSqabeaacaaIYaaaaaaakiabgUcaRiaaikdacaWGTbWaaSbaaS qaaiaadAgaaeqaaOWaaeWaaeaacaaIXaGaey4kaSIaamyyamaaBaaa leaacaWGRbaabeaakiaadUeacaWGUbaacaGLOaGaayzkaaWaamWaae aacaaI0aWaaeWaaeaadaWcaaqaaiaaikdacqGHsislcqaHdpWCdaWg aaWcbaGaamODaaqabaaakeaacqaHdpWCdaWgaaWcbaGaamODaaqaba aaaaGccaGLOaGaayzkaaWaaeWaaeaadaWcaaqaaiaadUeacaWGUbaa baGaaGymaiabgUcaRiaadUeacaWGUbaaaaGaayjkaiaawMcaaiabgU caRiaaigdaaiaawUfacaGLDbaadaWcaaqaaiabgkGi2oaaCaaaleqa baGaaGinaaaakiaadEhaaeaacqGHciITcaWG4bWaaWbaaSqabeaaca aIZaaaaOGaeyOaIyRaamiDaaaacqGHRaWkdaWadaqaamaalaaabaGa amyraiaadgeaaeaacaaIYaGaamitaaaadaWdXaqaamaabmaabaWaaS aaaeaacqGHciITcaWG3baabaGaeyOaIyRaamiEaaaaaiaawIcacaGL PaaadaahaaWcbeqaaiaaikdaaaGccaWGKbGaamiEaaWcbaGaaGimaa qaaiaadYeaa0Gaey4kIipaaOGaay5waiaaw2faamaalaaabaGaeyOa Iy7aaWbaaSqabeaacaaI0aaaaOGaam4DaaqaaiabgkGi2kaadIhada ahaaWcbeqaaiaaisdaaaaaaaGcbaGaey4kaSYaaeWaaeaacaWGTbWa aSbaaSqaaiaadAgaaeqaaOWaamWaaeaadaqadaqaaiaaigdacqGHRa WkcaWGHbWaaSbaaSqaaiaadUgaaeqaaOGaam4saiaad6gaaiaawIca caGLPaaadaWadaqaaiaaisdadaqadaqaamaalaaabaGaaGOmaiabgk HiTiabeo8aZnaaBaaaleaacaWG2baabeaaaOqaaiabeo8aZnaaBaaa leaacaWG2baabeaaaaaakiaawIcacaGLPaaadaqadaqaamaalaaaba Gaam4saiaad6gaaeaacaaIXaGaey4kaSIaam4saiaad6gaaaaacaGL OaGaayzkaaGaey4kaSIaaGymaaGaay5waiaaw2faaaGaay5waiaaw2 faamaaCaaaleqabaGaaGOmaaaakiabgUcaRiabes7aKjaadgeacqaH dpWCdaqhaaWcbaGaamiEaaqaaiaad+gaaaGccqGHsislcaWGibWaaS baaSqaaiaadohaaeqaaOGaeyOeI0Iaeq4TdGMaamisamaaDaaaleaa caWG4baabaGaaGOmaaaakiaadgeacqGHsislcaWGRbWaaSbaaSqaai aadchaaeqaaOGaey4kaSYaaSaaaeaacaWGfbGaamyqaiabeg7aHjab fs5aejaadsfaaeaacaaIXaGaeyOeI0IaaGOmaiabe27aUbaaaiaawI cacaGLPaaadaWcaaqaaiabgkGi2oaaCaaaleqabaGaaGinaaaakiaa dEhaaeaacqGHciITcaWG4bWaaWbaaSqabeaacaaI0aaaaaaaaOqaai abgUcaRiaadUgadaWgaaWcbaGaaGymaaqabaGcdaWcaaqaaiabgkGi 2oaaCaaaleqabaGaaGOmaaaakiaadEhaaeaacqGHciITcaWG4bWaaW baaSqabeaacaaIYaaaaaaakiabgUcaRiaaiodacaWGRbWaaSbaaSqa aiaaiodaaeqaaOGaam4DamaaCaaaleqabaGaaGOmaaaakmaalaaaba GaeyOaIy7aaWbaaSqabeaacaaIYaaaaOGaam4DaaqaaiabgkGi2kaa dIhadaahaaWcbeqaaiaaikdaaaaaaOGaey4kaSIaaGOnaiaadUgada WgaaWcbaGaaG4maaqabaGccaWG3bWaaeWaaeaadaWcaaqaaiabgkGi 2kaadEhaaeaacqGHciITcaWG4baaaaGaayjkaiaawMcaamaaCaaale qabaGaaGOmaaaaaaGccaGLBbGaayzxaaGaeyypa0JaaGimaaaaaa@A977@   (6)

where the transverse area and the bending rigidity are given as

A=πdh MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiabg2 da9iabec8aWjaadsgacaWGObaaaa@3B55@

EI= π d 3 h 8 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyraiaadM eacqGH9aqpdaWcaaqaaiabec8aWjaadsgadaahaaWcbeqaaiaaioda aaGccaWGObaabaGaaGioaaaaaaa@3DED@

and

E s I s = π E s h( d o 3 + d i 3 ) 8 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyramaaBa aaleaacaWGZbaabeaakiaadMeadaWgaaWcbaGaam4CaaqabaGccqGH 9aqpdaWcaaqaaiabec8aWjaadweadaWgaaWcbaGaam4CaaqabaGcca WGObGaaiikaiaadsgadaqhaaWcbaGaam4BaaqaaiaaiodaaaGccqGH RaWkcaWGKbWaa0baaSqaaiaadMgaaeaacaaIZaaaaOGaaiykaaqaai aaiIdaaaaaaa@483B@

H s =2 τ s ( d o + d i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisamaaBa aaleaacaWGZbaabeaakiabg2da9iaaikdacqaHepaDdaWgaaWcbaGa am4CaaqabaGcdaqadaqaaiaadsgadaWgaaWcbaGaam4BaaqabaGccq GHRaWkcaWGKbWaaSbaaSqaaiaadMgaaeqaaaGccaGLOaGaayzkaaaa aa@4331@

The symbol Hs is the parameter induced by the residual surface stress. τ is the residual surface tension, d and h are the nanotube internal diameter and thickness, respectively. It should be noted that the diameter of the nanotube can be derived from chirality indices (n, m)

d i = a 3 π n 2 +mn+ m 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizamaaBa aaleaacaWGPbaabeaakiabg2da9maalaaabaGaamyyamaakaaabaGa aG4maaWcbeaaaOqaaiabec8aWbaadaGcaaqaaiaad6gadaahaaWcbe qaaiaaikdaaaGccqGHRaWkcaWGTbGaamOBaiabgUcaRiaad2gadaah aaWcbeqaaiaaikdaaaaabeaaaaa@4418@   (7)

where a 3 =0.246nm MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaka aabaGaaG4maaWcbeaakiabg2da9iaaicdacaGGUaGaaGOmaiaaisda caaI2aGaaGPaVlaad6gacaWGTbaaaa@3FDA@ . ”a” represents the length of the carbon-carbon bond. d is the inner diameter of the nanotube.

Analytical solutions of nonlinear model of free vibration of the nanotube

It is difficult to solve Eq. (6) exactly because of the nonlinear term. Therefore, an approximate analytical method is used to solve the nonlinear model. In order to develop analytical solutions for the developed nonlinear model, the partial differential equation is converted to ordinary differential equation using the Galerkin’s decomposition procedure to decompose the spatial and temporal parts of the lateral displacement functions as

w(x,t)=ϕ(x)u(t) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DaiaacI cacaWG4bGaaiilaiaadshacaGGPaGaeyypa0Jaeqy1dyMaaiikaiaa dIhacaGGPaGaamyDaiaacIcacaWG0bGaaiykaaaa@4361@   (8)

Where u(t) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyDaiaacI cacaWG0bGaaiykaaaa@3942@ the generalized coordinate of the system and ϕ(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dyMaai ikaiaadIhacaGGPaaaaa@3A14@ is a trial/comparison function that will satisfy both the geometric and natural boundary conditions.

Applying one-parameter Galerkin’s solution given in Eq. (8) to Eq. (6)

0 L R( x,t )ϕ( x )dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qmaeaaca WGsbWaaeWaaeaacaWG4bGaaiilaiaadshaaiaawIcacaGLPaaacqaH vpGzdaqadaqaaiaadIhaaiaawIcacaGLPaaacaWGKbGaamiEaaWcba GaaGimaaqaaiaadYeaa0Gaey4kIipaaaa@44E4@   (9)

where

R( x,t )=( EI+ E s I s ) 4 w x 4 +( m cn + m f ) 2 w t 2 +2 m f ( 1+ a k Kn )[ 4( 2 σ v σ v )( Kn 1+Kn )+1 ] 2 w xt +[ EA 2L 0 L ( w x ) 2 dx ] 2 w x 2 +( m f [ ( 1+ a k Kn )[ 4( 2 σ v σ v )( Kn 1+Kn )+1 ] ] 2 +δA σ x o H s η H x 2 A k p + EAαΔT 12ν ) 2 w x 2 + k 1 w+ k 3 w 3 ( e o a ) 2 [ ( m cn + m f ) 4 w x 2 t 2 +2 m f ( 1+ a k Kn )[ 4( 2 σ v σ v )( Kn 1+Kn )+1 ] 4 w x 3 t +[ EA 2L 0 L ( w x ) 2 dx ] 4 w x 4 +( m f [ ( 1+ a k Kn )[ 4( 2 σ v σ v )( Kn 1+Kn )+1 ] ] 2 +δA σ x o H s η H x 2 A k p + EAαΔT 12ν ) 4 w x 4 + k 1 2 w x 2 +3 k 3 w 2 2 w x 2 +6 k 3 w ( w x ) 2 ]=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGsb WaaeWaaeaacaWG4bGaaiilaiaadshaaiaawIcacaGLPaaacqGH9aqp daqadaqaaiaadweacaWGjbGaey4kaSIaamyramaaBaaaleaacaWGZb aabeaakiaadMeadaWgaaWcbaGaam4CaaqabaaakiaawIcacaGLPaaa daWcaaqaaiabgkGi2oaaCaaaleqabaGaaGinaaaakiaadEhaaeaacq GHciITcaWG4bWaaWbaaSqabeaacaaI0aaaaaaakiabgUcaRiaacIca caWGTbWaaSbaaSqaaiaadogacaWGUbaabeaakiabgUcaRiaad2gada WgaaWcbaGaamOzaaqabaGccaGGPaWaaSaaaeaacqGHciITdaahaaWc beqaaiaaikdaaaGccaWG3baabaGaeyOaIyRaamiDamaaCaaaleqaba GaaGOmaaaaaaGccqGHRaWkcaaIYaGaamyBamaaBaaaleaacaWGMbaa beaakmaabmaabaGaaGymaiabgUcaRiaadggadaWgaaWcbaGaam4Aaa qabaGccaWGlbGaamOBaaGaayjkaiaawMcaamaadmaabaGaaGinamaa bmaabaWaaSaaaeaacaaIYaGaeyOeI0Iaeq4Wdm3aaSbaaSqaaiaadA haaeqaaaGcbaGaeq4Wdm3aaSbaaSqaaiaadAhaaeqaaaaaaOGaayjk aiaawMcaamaabmaabaWaaSaaaeaacaWGlbGaamOBaaqaaiaaigdacq GHRaWkcaWGlbGaamOBaaaaaiaawIcacaGLPaaacqGHRaWkcaaIXaaa caGLBbGaayzxaaWaaSaaaeaacqGHciITdaahaaWcbeqaaiaaikdaaa GccaWG3baabaGaeyOaIyRaamiEaiabgkGi2kaadshaaaGaey4kaSYa amWaaeaadaWcaaqaaiaadweacaWGbbaabaGaaGOmaiaadYeaaaWaa8 qmaeaadaqadaqaamaalaaabaGaeyOaIyRaam4DaaqaaiabgkGi2kaa dIhaaaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaOGaamizai aadIhaaSqaaiaaicdaaeaacaWGmbaaniabgUIiYdaakiaawUfacaGL DbaadaWcaaqaaiabgkGi2oaaCaaaleqabaGaaGOmaaaakiaadEhaae aacqGHciITcaWG4bWaaWbaaSqabeaacaaIYaaaaaaaaOqaaiabgUca RmaabmaabaGaamyBamaaBaaaleaacaWGMbaabeaakmaadmaabaWaae WaaeaacaaIXaGaey4kaSIaamyyamaaBaaaleaacaWGRbaabeaakiaa dUeacaWGUbaacaGLOaGaayzkaaWaamWaaeaacaaI0aWaaeWaaeaada WcaaqaaiaaikdacqGHsislcqaHdpWCdaWgaaWcbaGaamODaaqabaaa keaacqaHdpWCdaWgaaWcbaGaamODaaqabaaaaaGccaGLOaGaayzkaa WaaeWaaeaadaWcaaqaaiaadUeacaWGUbaabaGaaGymaiabgUcaRiaa dUeacaWGUbaaaaGaayjkaiaawMcaaiabgUcaRiaaigdaaiaawUfaca GLDbaaaiaawUfacaGLDbaadaahaaWcbeqaaiaaikdaaaGccqGHRaWk cqaH0oazcaWGbbGaeq4Wdm3aa0baaSqaaiaadIhaaeaacaWGVbaaaO GaeyOeI0IaamisamaaBaaaleaacaWGZbaabeaakiabgkHiTiabeE7a OjaadIeadaqhaaWcbaGaamiEaaqaaiaaikdaaaGccaWGbbGaeyOeI0 Iaam4AamaaBaaaleaacaWGWbaabeaakiabgUcaRmaalaaabaGaamyr aiaadgeacqaHXoqycqqHuoarcaWGubaabaGaaGymaiabgkHiTiaaik dacqaH9oGBaaaacaGLOaGaayzkaaWaaSaaaeaacqGHciITdaahaaWc beqaaiaaikdaaaGccaWG3baabaGaeyOaIyRaamiEamaaCaaaleqaba GaaGOmaaaaaaGccqGHRaWkcaWGRbWaaSbaaSqaaiaaigdaaeqaaOGa am4DaiabgUcaRiaadUgadaWgaaWcbaGaaG4maaqabaGccaWG3bWaaW baaSqabeaacaaIZaaaaaGcbaGaeyOeI0YaaeWaaeaacaWGLbWaaSba aSqaaiaad+gaaeqaaOGaamyyaaGaayjkaiaawMcaamaaCaaaleqaba GaaGOmaaaakmaadmaaeaqabeaacaGGOaGaamyBamaaBaaaleaacaWG JbGaamOBaaqabaGccqGHRaWkcaWGTbWaaSbaaSqaaiaadAgaaeqaaO GaaiykamaalaaabaGaeyOaIy7aaWbaaSqabeaacaaI0aaaaOGaam4D aaqaaiabgkGi2kaadIhadaahaaWcbeqaaiaaikdaaaGccqGHciITca WG0bWaaWbaaSqabeaacaaIYaaaaaaakiabgUcaRiaaikdacaWGTbWa aSbaaSqaaiaadAgaaeqaaOWaaeWaaeaacaaIXaGaey4kaSIaamyyam aaBaaaleaacaWGRbaabeaakiaadUeacaWGUbaacaGLOaGaayzkaaWa amWaaeaacaaI0aWaaeWaaeaadaWcaaqaaiaaikdacqGHsislcqaHdp WCdaWgaaWcbaGaamODaaqabaaakeaacqaHdpWCdaWgaaWcbaGaamOD aaqabaaaaaGccaGLOaGaayzkaaWaaeWaaeaadaWcaaqaaiaadUeaca WGUbaabaGaaGymaiabgUcaRiaadUeacaWGUbaaaaGaayjkaiaawMca aiabgUcaRiaaigdaaiaawUfacaGLDbaadaWcaaqaaiabgkGi2oaaCa aaleqabaGaaGinaaaakiaadEhaaeaacqGHciITcaWG4bWaaWbaaSqa beaacaaIZaaaaOGaeyOaIyRaamiDaaaacqGHRaWkdaWadaqaamaala aabaGaamyraiaadgeaaeaacaaIYaGaamitaaaadaWdXaqaamaabmaa baWaaSaaaeaacqGHciITcaWG3baabaGaeyOaIyRaamiEaaaaaiaawI cacaGLPaaadaahaaWcbeqaaiaaikdaaaGccaWGKbGaamiEaaWcbaGa aGimaaqaaiaadYeaa0Gaey4kIipaaOGaay5waiaaw2faamaalaaaba GaeyOaIy7aaWbaaSqabeaacaaI0aaaaOGaam4DaaqaaiabgkGi2kaa dIhadaahaaWcbeqaaiaaisdaaaaaaaGcbaGaey4kaSYaaeWaaeaaca WGTbWaaSbaaSqaaiaadAgaaeqaaOWaamWaaeaadaqadaqaaiaaigda cqGHRaWkcaWGHbWaaSbaaSqaaiaadUgaaeqaaOGaam4saiaad6gaai aawIcacaGLPaaadaWadaqaaiaaisdadaqadaqaamaalaaabaGaaGOm aiabgkHiTiabeo8aZnaaBaaaleaacaWG2baabeaaaOqaaiabeo8aZn aaBaaaleaacaWG2baabeaaaaaakiaawIcacaGLPaaadaqadaqaamaa laaabaGaam4saiaad6gaaeaacaaIXaGaey4kaSIaam4saiaad6gaaa aacaGLOaGaayzkaaGaey4kaSIaaGymaaGaay5waiaaw2faaaGaay5w aiaaw2faamaaCaaaleqabaGaaGOmaaaakiabgUcaRiabes7aKjaadg eacqaHdpWCdaqhaaWcbaGaamiEaaqaaiaad+gaaaGccqGHsislcaWG ibWaaSbaaSqaaiaadohaaeqaaOGaeyOeI0Iaeq4TdGMaamisamaaDa aaleaacaWG4baabaGaaGOmaaaakiaadgeacqGHsislcaWGRbWaaSba aSqaaiaadchaaeqaaOGaey4kaSYaaSaaaeaacaWGfbGaamyqaiabeg 7aHjabfs5aejaadsfaaeaacaaIXaGaeyOeI0IaaGOmaiabe27aUbaa aiaawIcacaGLPaaadaWcaaqaaiabgkGi2oaaCaaaleqabaGaaGinaa aakiaadEhaaeaacqGHciITcaWG4bWaaWbaaSqabeaacaaI0aaaaaaa aOqaaiabgUcaRiaadUgadaWgaaWcbaGaaGymaaqabaGcdaWcaaqaai abgkGi2oaaCaaaleqabaGaaGOmaaaakiaadEhaaeaacqGHciITcaWG 4bWaaWbaaSqabeaacaaIYaaaaaaakiabgUcaRiaaiodacaWGRbWaaS baaSqaaiaaiodaaeqaaOGaam4DamaaCaaaleqabaGaaGOmaaaakmaa laaabaGaeyOaIy7aaWbaaSqabeaacaaIYaaaaOGaam4Daaqaaiabgk Gi2kaadIhadaahaaWcbeqaaiaaikdaaaaaaOGaey4kaSIaaGOnaiaa dUgadaWgaaWcbaGaaG4maaqabaGccaWG3bWaaeWaaeaadaWcaaqaai abgkGi2kaadEhaaeaacqGHciITcaWG4baaaaGaayjkaiaawMcaamaa CaaaleqabaGaaGOmaaaaaaGccaGLBbGaayzxaaGaeyypa0JaaGimaa aaaa@AF83@

We have the nonlinear vibration equation of the pipe as

M u ¨ (t)+G u ˙ (t)+(K+C)u(t)+V u 3 (t)=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytaiqadw hagaWaaiaacIcacaWG0bGaaiykaiabgUcaRiaadEeaceWG1bGbaiaa caGGOaGaamiDaiaacMcacqGHRaWkcaGGOaGaam4saiabgUcaRiaado eacaGGPaGaamyDaiaacIcacaWG0bGaaiykaiabgUcaRiaadAfacaWG 1bWaaWbaaSqabeaacaaIZaaaaOGaaiikaiaadshacaGGPaGaeyypa0 JaaGimaaaa@4EDF@   (10)

where

M=( m p + m f )[ 0 L ϕ 2 (x) dx ( e o a ) 2 0 L ϕ 2 (x) d 2 ϕ d x 2 dx ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytaiabg2 da9iaacIcacaWGTbWaaSbaaSqaaiaadchaaeqaaOGaey4kaSIaamyB amaaBaaaleaacaWGMbaabeaakiaacMcadaWadaqaamaapedabaGaeq y1dy2aaWbaaSqabeaacaaIYaaaaOGaaiikaiaadIhacaGGPaaaleaa caaIWaaabaGaamitaaqdcqGHRiI8aOGaamizaiaadIhacqGHsislda qadaqaaiaadwgadaWgaaWcbaGaam4BaaqabaGccaWGHbaacaGLOaGa ayzkaaWaaWbaaSqabeaacaaIYaaaaOWaa8qmaeaacqaHvpGzdaahaa WcbeqaaiaaikdaaaGccaGGOaGaamiEaiaacMcaaSqaaiaaicdaaeaa caWGmbaaniabgUIiYdGcdaWcaaqaaiaadsgadaahaaWcbeqaaiaaik daaaGccqaHvpGzaeaacaWGKbGaamiEamaaCaaaleqabaGaaGOmaaaa aaGccaWGKbGaamiEaaGaay5waiaaw2faaaaa@6285@

G=[ 2 m f ( 1+ a k Kn )[ 4( 2 σ v σ v )( Kn 1+Kn )+1 ] ] 0 L [ ϕ(x)( dϕ dx )dx ( e o a ) 2 0 L ϕ(x) d 3 ϕ d x 3 dx ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4raiabg2 da9maadmaabaGaaGOmaiaad2gadaWgaaWcbaGaamOzaaqabaGcdaqa daqaaiaaigdacqGHRaWkcaWGHbWaaSbaaSqaaiaadUgaaeqaaOGaam 4saiaad6gaaiaawIcacaGLPaaadaWadaqaaiaaisdadaqadaqaamaa laaabaGaaGOmaiabgkHiTiabeo8aZnaaBaaaleaacaWG2baabeaaaO qaaiabeo8aZnaaBaaaleaacaWG2baabeaaaaaakiaawIcacaGLPaaa daqadaqaamaalaaabaGaam4saiaad6gaaeaacaaIXaGaey4kaSIaam 4saiaad6gaaaaacaGLOaGaayzkaaGaey4kaSIaaGymaaGaay5waiaa w2faaaGaay5waiaaw2faamaapedabaWaamWaaeaacqaHvpGzcaGGOa GaamiEaiaacMcadaqadaqaamaalaaabaGaamizaiabew9aMbqaaiaa dsgacaWG4baaaaGaayjkaiaawMcaaiaadsgacaWG4bGaeyOeI0Yaae WaaeaacaWGLbWaaSbaaSqaaiaad+gaaeqaaOGaamyyaaGaayjkaiaa wMcaamaaCaaaleqabaGaaGOmaaaakmaapedabaGaeqy1dyMaaiikai aadIhacaGGPaaaleaacaaIWaaabaGaamitaaqdcqGHRiI8aOWaaSaa aeaacaWGKbWaaWbaaSqabeaacaaIZaaaaOGaeqy1dygabaGaamizai aadIhadaahaaWcbeqaaiaaiodaaaaaaOGaamizaiaadIhaaiaawUfa caGLDbaaaSqaaiaaicdaaeaacaWGmbaaniabgUIiYdaaaa@803F@

K= 0 L ( EI+ E s I s )ϕ(x) d 4 ϕ d x 4 dx+ k 1 [ 0 L ϕ 2 (x)dx ( e o a ) 2 0 L ϕ(x) d 2 ϕ d x 2 dx ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4saiabg2 da9maapedabaWaaeWaaeaacaWGfbGaamysaiabgUcaRiaadweadaWg aaWcbaGaam4CaaqabaGccaWGjbWaaSbaaSqaaiaadohaaeqaaaGcca GLOaGaayzkaaGaeqy1dyMaaiikaiaadIhacaGGPaWaaSaaaeaacaWG KbWaaWbaaSqabeaacaaI0aaaaOGaeqy1dygabaGaamizaiaadIhada ahaaWcbeqaaiaaisdaaaaaaOGaamizaiaadIhacqGHRaWkcaWGRbWa aSbaaSqaaiaaigdaaeqaaaqaaiaaicdaaeaacaWGmbaaniabgUIiYd GcdaWadaqaamaapedabaGaeqy1dy2aaWbaaSqabeaacaaIYaaaaOGa aiikaiaadIhacaGGPaGaamizaiaadIhaaSqaaiaaicdaaeaacaWGmb aaniabgUIiYdGccqGHsisldaqadaqaaiaadwgadaWgaaWcbaGaam4B aaqabaGccaWGHbaacaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaO Waa8qmaeaacqaHvpGzcaGGOaGaamiEaiaacMcaaSqaaiaaicdaaeaa caWGmbaaniabgUIiYdGcdaWcaaqaaiaadsgadaahaaWcbeqaaiaaik daaaGccqaHvpGzaeaacaWGKbGaamiEamaaCaaaleqabaGaaGOmaaaa aaGccaWGKbGaamiEaaGaay5waiaaw2faaaaa@761D@

C=( m f [ ( 1+ a k Kn )[ 4( 2 σ v σ v )( Kn 1+Kn )+1 ] ] 2 +δA σ x o H s η H x 2 A k p + EAαΔT 12ν )[ 0 L ϕ(x) d 2 ϕ d x 2 dx ( e o a ) 2 0 L ϕ(x) d 4 ϕ d x 4 dx ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qaiabg2 da9maabmaaeaqabeaacaWGTbWaaSbaaSqaaiaadAgaaeqaaOWaamWa aeaadaqadaqaaiaaigdacqGHRaWkcaWGHbWaaSbaaSqaaiaadUgaae qaaOGaam4saiaad6gaaiaawIcacaGLPaaadaWadaqaaiaaisdadaqa daqaamaalaaabaGaaGOmaiabgkHiTiabeo8aZnaaBaaaleaacaWG2b aabeaaaOqaaiabeo8aZnaaBaaaleaacaWG2baabeaaaaaakiaawIca caGLPaaadaqadaqaamaalaaabaGaam4saiaad6gaaeaacaaIXaGaey 4kaSIaam4saiaad6gaaaaacaGLOaGaayzkaaGaey4kaSIaaGymaaGa ay5waiaaw2faaaGaay5waiaaw2faamaaCaaaleqabaGaaGOmaaaaaO qaaiabgUcaRiabes7aKjaadgeacqaHdpWCdaqhaaWcbaGaamiEaaqa aiaad+gaaaGccqGHsislcaWGibWaaSbaaSqaaiaadohaaeqaaOGaey OeI0Iaeq4TdGMaamisamaaDaaaleaacaWG4baabaGaaGOmaaaakiaa dgeacqGHsislcaWGRbWaaSbaaSqaaiaadchaaeqaaOGaey4kaSYaaS aaaeaacaWGfbGaamyqaiabeg7aHjabfs5aejaadsfaaeaacaaIXaGa eyOeI0IaaGOmaiabe27aUbaaaaGaayjkaiaawMcaamaadmaabaWaa8 qmaeaacqaHvpGzcaGGOaGaamiEaiaacMcadaWcaaqaaiaadsgadaah aaWcbeqaaiaaikdaaaGccqaHvpGzaeaacaWGKbGaamiEamaaCaaale qabaGaaGOmaaaaaaGccaWGKbGaamiEaiabgkHiTmaabmaabaGaamyz amaaBaaaleaacaWGVbaabeaakiaadggaaiaawIcacaGLPaaadaahaa WcbeqaaiaaikdaaaGcdaWdXaqaaiabew9aMjaacIcacaWG4bGaaiyk aaWcbaGaaGimaaqaaiaadYeaa0Gaey4kIipakmaalaaabaGaamizam aaCaaaleqabaGaaGinaaaakiabew9aMbqaaiaadsgacaWG4bWaaWba aSqabeaacaaI0aaaaaaakiaadsgacaWG4baaleaacaaIWaaabaGaam itaaqdcqGHRiI8aaGccaGLBbGaayzxaaaaaa@A02B@

V= k 3 [ 0 L ϕ 4 (x) dx ( e o a ) 2 ( 3 0 L ϕ 3 (x) d 2 ϕ d x 2 dx+6 0 L ϕ 2 (x) ( dϕ dx ) 2 dx ) ] + 0 L ϕ(x) [ EA 2L 0 L ( dϕ dx ) 2 dx ] d 2 ϕ d x 2 dx ( e o a ) 2 0 L ϕ(x) [ EA 2L 0 L ( dϕ dx ) 2 dx ] d 4 ϕ d x 4 dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGwb Gaeyypa0Jaam4AamaaBaaaleaacaaIZaaabeaakmaadmaabaWaa8qm aeaacqaHvpGzdaahaaWcbeqaaiaaisdaaaGccaGGOaGaamiEaiaacM caaSqaaiaaicdaaeaacaWGmbaaniabgUIiYdGccaWGKbGaamiEaiab gkHiTmaabmaabaGaamyzamaaBaaaleaacaWGVbaabeaakiaadggaai aawIcacaGLPaaadaahaaWcbeqaaiaaikdaaaGcdaqadaqaaiaaioda daWdXaqaaiabew9aMnaaCaaaleqabaGaaG4maaaakiaacIcacaWG4b GaaiykaaWcbaGaaGimaaqaaiaadYeaa0Gaey4kIipakmaalaaabaGa amizamaaCaaaleqabaGaaGOmaaaakiabew9aMbqaaiaadsgacaWG4b WaaWbaaSqabeaacaaIYaaaaaaakiaadsgacaWG4bGaey4kaSIaaGOn amaapedabaGaeqy1dy2aaWbaaSqabeaacaaIYaaaaOGaaiikaiaadI hacaGGPaaaleaacaaIWaaabaGaamitaaqdcqGHRiI8aOWaaeWaaeaa daWcaaqaaiaadsgacqaHvpGzaeaacaWGKbGaamiEaaaaaiaawIcaca GLPaaadaahaaWcbeqaaiaaikdaaaGccaWGKbGaamiEaaGaayjkaiaa wMcaaaGaay5waiaaw2faaaqaaiaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8Uaey4kaSYaa8qmaeaacqaHvpGzcaGG OaGaamiEaiaacMcaaSqaaiaaicdaaeaacaWGmbaaniabgUIiYdGcda WadaqaamaalaaabaGaamyraiaadgeaaeaacaaIYaGaamitaaaadaWd XaqaamaabmaabaWaaSaaaeaacaWGKbGaeqy1dygabaGaamizaiaadI haaaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaOGaamizaiaa dIhaaSqaaiaaicdaaeaacaWGmbaaniabgUIiYdaakiaawUfacaGLDb aadaWcaaqaaiaadsgadaahaaWcbeqaaiaaikdaaaGccqaHvpGzaeaa caWGKbGaamiEamaaCaaaleqabaGaaGOmaaaaaaGccaWGKbGaamiEai abgkHiTmaabmaabaGaamyzamaaBaaaleaacaWGVbaabeaakiaadgga aiaawIcacaGLPaaadaahaaWcbeqaaiaaikdaaaGcdaWdXaqaaiabew 9aMjaacIcacaWG4bGaaiykaaWcbaGaaGimaaqaaiaadYeaa0Gaey4k IipakmaadmaabaWaaSaaaeaacaWGfbGaamyqaaqaaiaaikdacaWGmb aaamaapedabaWaaeWaaeaadaWcaaqaaiaadsgacqaHvpGzaeaacaWG KbGaamiEaaaaaiaawIcacaGLPaaadaahaaWcbeqaaiaaikdaaaGcca WGKbGaamiEaaWcbaGaaGimaaqaaiaadYeaa0Gaey4kIipaaOGaay5w aiaaw2faamaalaaabaGaamizamaaCaaaleqabaGaaGinaaaakiabew 9aMbqaaiaadsgacaWG4bWaaWbaaSqabeaacaaI0aaaaaaakiaadsga caWG4baaaaa@CBDB@

The circular fundamental natural frequency gives

ω n = K+C M MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyYdC3aaS baaSqaaiaad6gaaeqaaOGaeyypa0ZaaOaaaeaadaWcaaqaaiaadUea cqGHRaWkcaWGdbaabaGaamytaaaaaSqabaaaaa@3D69@   (11)

For the simply supported pipe,

ϕ(x)=sin β n x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dyMaai ikaiaadIhacaGGPaGaeyypa0Jaam4CaiaadMgacaWGUbGaeqOSdi2a aSbaaSqaaiaad6gaaeqaaOGaamiEaaaa@41BA@   (12a)

where

sinβL=0 β n = nπ L MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CaiaadM gacaWGUbGaeqOSdiMaamitaiabg2da9iaaicdacaaMf8UaeyO0H4Ta eqOSdi2aaSbaaSqaaiaad6gaaeqaaOGaeyypa0ZaaSaaaeaacaWGUb GaeqiWdahabaGaamitaaaaaaa@484D@

Therefore,

ϕ(x)=sin nπx L MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dyMaai ikaiaadIhacaGGPaGaeyypa0Jaam4CaiaadMgacaWGUbWaaSaaaeaa caWGUbGaeqiWdaNaamiEaaqaaiaadYeaaaaaaa@4281@   (12b)

Eq. (12) can be written as

u ¨ (t)+γ u ˙ (t)+αu(t)+β u 3 (t)=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyDayaada GaaiikaiaadshacaGGPaGaey4kaSIaeq4SdCMabmyDayaacaGaaiik aiaadshacaGGPaGaey4kaSIaeqySdeMaamyDaiaacIcacaWG0bGaai ykaiabgUcaRiabek7aIjaadwhadaahaaWcbeqaaiaaiodaaaGccaGG OaGaamiDaiaacMcacqGH9aqpcaaIWaaaaa@4D7A@   (13)

where

α= (K+C) M ,β= V M ,γ= G M , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaey ypa0ZaaSaaaeaacaGGOaGaam4saiabgUcaRiaadoeacaGGPaaabaGa amytaaaacaGGSaGaaGPaVlaaykW7caaMc8UaaGPaVlabek7aIjabg2 da9maalaaabaGaamOvaaqaaiaad2eaaaGaaiilaiaaykW7caaMc8Ua aGPaVlaaykW7caaMc8Uaeq4SdCMaeyypa0ZaaSaaaeaacaWGhbaaba GaamytaaaacaGGSaaaaa@5602@

For an undamped simple-simple supported structures, where G = 0, we have

u ¨ (t)+αu(t)+β u 3 (t)=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyDayaada GaaiikaiaadshacaGGPaGaey4kaSIaeqySdeMaamyDaiaacIcacaWG 0bGaaiykaiabgUcaRiabek7aIjaadwhadaahaaWcbeqaaiaaiodaaa GccaGGOaGaamiDaiaacMcacqGH9aqpcaaIWaaaaa@479C@   (14)

Determination of natural frequencies

In order to determine the natural frequency of the vibration, we make use of the transformation, τ=ωt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiXdqNaey ypa0JaeqyYdCNaamiDaaaa@3B87@ , Eq. (14) becomes

ω 2 u ¨ (τ)+αu(τ)+β u 3 (τ)=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyYdC3aaW baaSqabeaacaaIYaaaaOGabmyDayaadaGaaiikaiabes8a0jaacMca cqGHRaWkcqaHXoqycaWG1bGaaiikaiabes8a0jaacMcacqGHRaWkcq aHYoGycaWG1bWaaWbaaSqabeaacaaIZaaaaOGaaiikaiabes8a0jaa cMcacqGH9aqpcaaIWaaaaa@4CC0@   (15)

The symbolic solution of Eq. (15) can be provided by assuming an initial approximation for zero-order deformation to be

u o (τ)=Acosτ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyDamaaBa aaleaacaWGVbaabeaakiaacIcacqaHepaDcaGGPaGaeyypa0Jaamyq aiaadogacaWGVbGaam4CaiaaykW7cqaHepaDaaa@4328@   (16)

Substitution of Eq. (16) into Eq. (15) provides

ω o 2 Acosτ+αAcosτ+β A 3 cos 3 τ=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0Iaeq yYdC3aa0baaSqaaiaad+gaaeaacaaIYaaaaOGaamyqaiaadogacaWG VbGaam4CaiaaykW7cqaHepaDcqGHRaWkcqaHXoqycaWGbbGaam4yai aad+gacaWGZbGaaGPaVlabes8a0jabgUcaRiabek7aIjaadgeadaah aaWcbeqaaiaaiodaaaGccaWGJbGaam4BaiaadohacaaMc8+aaWbaaS qabeaacaaIZaaaaOGaeqiXdqNaeyypa0JaaGimaaaa@5801@   (17)

Through trigonometry identity, we have

ω 2 Acosτ+αAcosτ+β A 3 ( 3cosτ+cos3τ 4 )=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0Iaeq yYdC3aaWbaaSqabeaacaaIYaaaaOGaamyqaiaadogacaWGVbGaam4C aiaaykW7cqaHepaDcqGHRaWkcqaHXoqycaWGbbGaam4yaiaad+gaca WGZbGaaGPaVlabes8a0jabgUcaRiabek7aIjaadgeadaahaaWcbeqa aiaaiodaaaGcdaqadaqaamaalaaabaGaaG4maiaadogacaWGVbGaam 4CaiaaykW7cqaHepaDcqGHRaWkcaWGJbGaam4BaiaadohacaaMc8Ua aG4maiabes8a0bqaaiaaisdaaaaacaGLOaGaayzkaaGaeyypa0JaaG imaaaa@60F0@   (18)

Collection of like terms gives

( αA+ 3β A 3 4 ω 2 A )cosτ+ 1 4 β A 3 cos3τ=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq aHXoqycaWGbbGaey4kaSYaaSaaaeaacaaIZaGaeqOSdiMaamyqamaa CaaaleqabaGaaG4maaaaaOqaaiaaisdaaaGaeyOeI0IaeqyYdC3aaW baaSqabeaacaaIYaaaaOGaamyqaaGaayjkaiaawMcaaiaadogacaWG VbGaam4CaiaaykW7cqaHepaDcqGHRaWkdaWcaaqaaiaaigdaaeaaca aI0aaaaiabek7aIjaadgeadaahaaWcbeqaaiaaiodaaaGccaWGJbGa am4BaiaadohacaaMc8UaaG4maiabes8a0jabg2da9iaaicdaaaa@58AA@   (19)

The elimination of secular term is produced by making

( αA+ 3β A 3 4 ω o 2 A )=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq aHXoqycaWGbbGaey4kaSYaaSaaaeaacaaIZaGaeqOSdiMaamyqamaa CaaaleqabaGaaG4maaaaaOqaaiaaisdaaaGaeyOeI0IaeqyYdC3aa0 baaSqaaiaad+gaaeaacaaIYaaaaOGaamyqaaGaayjkaiaawMcaaiab g2da9iaaicdaaaa@46D3@   (20)

Therefore, the zero-order nonlinear natural frequency becomes

ω o α+ 3β A 2 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyYdC3aaS baaSqaaiaad+gaaeqaaOGaeyisIS7aaOaaaeaacqaHXoqycqGHRaWk daWcaaqaaiaaiodacqaHYoGycaWGbbWaaWbaaSqabeaacaaIYaaaaa GcbaGaaGinaaaaaSqabaaaaa@421F@   (21)

The ratio of the zero-order nonlinear natural frequency to the linear frequency

ω o ω b α+ 3β A 2 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq aHjpWDdaWgaaWcbaGaam4BaaqabaaakeaacqaHjpWDdaWgaaWcbaGa amOyaaqabaaaaOGaeyisIS7aaOaaaeaacqaHXoqycqGHRaWkdaWcaa qaaiaaiodacqaHYoGycaWGbbWaaWbaaSqabeaacaaIYaaaaaGcbaGa aGinaaaaaSqabaaaaa@4519@   (22)

Similarly, the first-order nonlinear natural frequency is given as

ω 1 1 2 { [ α+ 3β A 2 4 ]+ [ α+ 3β A 2 4 ] 2 ( 3 β 2 A 4 32 ) } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyYdC3aaS baaSqaaiaaigdaaeqaaOGaeyisIS7aaOaaaeaadaWcaaqaaiaaigda aeaacaaIYaaaamaacmaabaWaamWaaeaacqaHXoqycqGHRaWkdaWcaa qaaiaaiodacqaHYoGycaWGbbWaaWbaaSqabeaacaaIYaaaaaGcbaGa aGinaaaaaiaawUfacaGLDbaacqGHRaWkdaGcaaqaamaadmaabaGaeq ySdeMaey4kaSYaaSaaaeaacaaIZaGaeqOSdiMaamyqamaaCaaaleqa baGaaGOmaaaaaOqaaiaaisdaaaaacaGLBbGaayzxaaWaaWbaaSqabe aacaaIYaaaaOGaeyOeI0YaaeWaaeaadaWcaaqaaiaaiodacqaHYoGy daahaaWcbeqaaiaaikdaaaGccaWGbbWaaWbaaSqabeaacaaI0aaaaa GcbaGaaG4maiaaikdaaaaacaGLOaGaayzkaaaaleqaaaGccaGL7bGa ayzFaaaaleqaaaaa@5BED@   (23)

The ratio of the first-order nonlinear natural frequency to the linear frequency

ω 1 ω b 1 2 { [ 1+ 3β A 2 4α ]+ [ 1+ 3β A 2 4α ] 2 ( 3 β 2 A 4 32 α 2 ) } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq aHjpWDdaWgaaWcbaGaaGymaaqabaaakeaacqaHjpWDdaWgaaWcbaGa amOyaaqabaaaaOGaeyisIS7aaOaaaeaadaWcaaqaaiaaigdaaeaaca aIYaaaamaacmaabaWaamWaaeaacaaIXaGaey4kaSYaaSaaaeaacaaI ZaGaeqOSdiMaamyqamaaCaaaleqabaGaaGOmaaaaaOqaaiaaisdacq aHXoqyaaaacaGLBbGaayzxaaGaey4kaSYaaOaaaeaadaWadaqaaiaa igdacqGHRaWkdaWcaaqaaiaaiodacqaHYoGycaWGbbWaaWbaaSqabe aacaaIYaaaaaGcbaGaaGinaiabeg7aHbaaaiaawUfacaGLDbaadaah aaWcbeqaaiaaikdaaaGccqGHsisldaqadaqaamaalaaabaGaaG4mai abek7aInaaCaaaleqabaGaaGOmaaaakiaadgeadaahaaWcbeqaaiaa isdaaaaakeaacaaIZaGaaGOmaiabeg7aHnaaCaaaleqabaGaaGOmaa aaaaaakiaawIcacaGLPaaaaSqabaaakiaawUhacaGL9baaaSqabaaa aa@62EF@   (24)

Approximate analytical methods of solution: Temini and Ansari method

The nonlinearity in the above Eq. (15) makes it very difficult to generate closed form solutions to the equations. Therefore; in this work, recourse is made Temini and Ansari method to provide approximate analytical solution to the problem.

Principle of Temini and Ansari method

The principle of the method is described as follows. The general system of nonlinear equation is in the form

L( u(x) )+N( u(x) )+g(x)=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGmb qcfa4aaeWaaOqaaKqzGeGaamyDaiaacIcacaWG4bGaaiykaaGccaGL OaGaayzkaaqcLbsacqGHRaWkcaWGobqcfa4aaeWaaOqaaKqzGeGaam yDaiaacIcacaWG4bGaaiykaaGccaGLOaGaayzkaaqcLbsacqGHRaWk caWGNbGaaiikaiaadIhacaGGPaGaeyypa0JaaGimaaaa@4C21@   (25)

with the boundary conditions

B( u, du dx )=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGcb qcfa4aaeWaaOqaaKqzGeGaamyDaiaacYcajuaGdaWcaaGcbaqcLbsa caWGKbGaamyDaaGcbaqcLbsacaWGKbGaamiEaaaaaOGaayjkaiaawM caaKqzGeGaeyypa0JaaGimaaaa@4398@   (26)

where x denotes the independent variable, u(x) represents an unknown function, g(x) is a known function, L is a linear operator, N is a nonlinear operator and B is a boundary operator. Since L is taken as the linear (highest order derivative) part of the DE, it is possible to take some or the remaining linear parts of the DE and add them to N as needed. The procedure of the proposed TAM is as follows.

Assuming that u(x) is an initial guess of the solution to the problem u(x) and is the solution of the equation

L( u 0 (x) )+g(x)=0,B( u, d u 0 dx )=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGmb qcfa4aaeWaaOqaaKqzGeGaamyDaKqbaoaaBaaaleaajugibiaaicda aSqabaqcLbsacaGGOaGaamiEaiaacMcaaOGaayjkaiaawMcaaKqzGe Gaey4kaSIaam4zaiaacIcacaWG4bGaaiykaiabg2da9iaaicdacaGG SaGaamOqaKqbaoaabmaakeaajugibiaadwhacaGGSaqcfa4aaSaaaO qaaKqzGeGaamizaiaadwhajuaGdaWgaaWcbaqcLbsacaaIWaaaleqa aaGcbaqcLbsacaWGKbGaamiEaaaaaOGaayjkaiaawMcaaKqzGeGaey ypa0JaaGimaaaa@5641@   (27)

In order to generate the next improvement to the solution, Eq. (28) is solved

L( u 1 (x) )+g(x)+N( u 0 (x) )=0,B( u 1 , d u 1 dx )=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGmb qcfa4aaeWaaOqaaKqzGeGaamyDaKqbaoaaBaaaleaajugibiaaigda aSqabaqcLbsacaGGOaGaamiEaiaacMcaaOGaayjkaiaawMcaaKqzGe Gaey4kaSIaam4zaiaacIcacaWG4bGaaiykaiabgUcaRiaad6eajuaG daqadaGcbaqcLbsacaWG1bqcfa4aaSbaaSqaaKqzGeGaaGimaaWcbe aajugibiaacIcacaWG4bGaaiykaaGccaGLOaGaayzkaaqcLbsacqGH 9aqpcaaIWaGaaiilaiaadkeajuaGdaqadaGcbaqcLbsacaWG1bqcfa 4aaSbaaeaajugibiaaigdaaKqbagqaaKqzGeGaaiilaKqbaoaalaaa keaajugibiaadsgacaWG1bqcfa4aaSbaaSqaaKqzGeGaaGymaaWcbe aaaOqaaKqzGeGaamizaiaadIhaaaaakiaawIcacaGLPaaajugibiab g2da9iaaicdaaaa@6444@   (28)

Following the above procedure, the Temini and Ansari method gives the possibility to write the solution of the general nonlinear equation in the iterative formula

L( u n+1 (x) )+g(x)+N( u n (x) )=0,B( u n+1 , d u n+1 dx )=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbsacaWGmb qcfa4aaeWaaOqaaKqzGeGaamyDaKqbaoaaBaaaleaajugibiaad6ga cqGHRaWkcaaIXaaaleqaaKqzGeGaaiikaiaadIhacaGGPaaakiaawI cacaGLPaaajugibiabgUcaRiaadEgacaGGOaGaamiEaiaacMcacqGH RaWkcaWGobqcfa4aaeWaaOqaaKqzGeGaamyDaKqbaoaaBaaaleaaju gibiaad6gaaSqabaqcLbsacaGGOaGaamiEaiaacMcaaOGaayjkaiaa wMcaaKqzGeGaeyypa0JaaGimaiaacYcacaWGcbqcfa4aaeWaaOqaaK qzGeGaamyDaKqbaoaaBaaabaqcLbsacaWGUbGaey4kaSIaaGymaaqc fayabaqcLbsacaGGSaqcfa4aaSaaaOqaaKqzGeGaamizaiaadwhaju aGdaWgaaWcbaqcLbsacaWGUbGaey4kaSIaaGymaaWcbeaaaOqaaKqz GeGaamizaiaadIhaaaaakiaawIcacaGLPaaajugibiabg2da9iaaic daaaa@69FC@   (29)

Application of Temini and Ansari method to the nonlinear problem

From Eq. (15), it is clear that

u ¨ (τ)+ α ω 2 u(τ)+ β ω 2 u 3 (τ)=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyDayaada Gaaiikaiabes8a0jaacMcacqGHRaWkdaWcaaqaaiabeg7aHbqaaiab eM8a3naaCaaaleqabaGaaGOmaaaaaaGccaWG1bGaaiikaiabes8a0j aacMcacqGHRaWkdaWcaaqaaiabek7aIbqaaiabeM8a3naaCaaaleqa baGaaGOmaaaaaaGccaWG1bWaaWbaaSqabeaacaaIZaaaaOGaaiikai abes8a0jaacMcacqGH9aqpcaaIWaaaaa@4FA0@   (30)

Assuming that u(t) is an initial guess of the solution to the problem and is the solution of the equation

The initial problem is

L( u 0 (τ) )=0 u ¨ 0 (τ)=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamitamaabm aabaGaamyDamaaBaaaleaacaaIWaaabeaakiaacIcacqaHepaDcaGG PaaacaGLOaGaayzkaaGaeyypa0JaaGimaiaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaeyO0H4TaaGPaVlaaykW7caaMc8UabmyDayaadaWa aSbaaSqaaiaaicdaaeqaaOGaaiikaiabes8a0jaacMcacqGH9aqpca aIWaaaaa@549F@   (31)

with initial conditions

u o (0)=Acosτ, u ˙ o (0)=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyDamaaBa aaleaacaWGVbaabeaakiaacIcacaaIWaGaaiykaiabg2da9iaadgea caWGJbGaam4BaiaadohacaaMc8UaeqiXdqNaaiilaiqadwhagaGaam aaBaaaleaacaWGVbaabeaakiaacIcacaaIWaGaaiykaiabg2da9iaa icdaaaa@48CD@   (32)

On solving Eq. (32), one arrives at

u o (τ)=Acosτ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyDamaaBa aaleaacaWGVbaabeaakiaacIcacqaHepaDcaGGPaGaeyypa0Jaamyq aiaadogacaWGVbGaam4CaiaaykW7cqaHepaDaaa@4328@   (33)

In order to generate the next improvement to the solution, Eq. (34) is solved

L( u 1 (τ) )+g(τ)+N( u 0 (τ) )=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamitamaabm aabaGaamyDamaaBaaaleaacaaIXaaabeaakiaacIcacqaHepaDcaGG PaaacaGLOaGaayzkaaGaey4kaSIaam4zaiaacIcacqaHepaDcaGGPa Gaey4kaSIaamOtamaabmaabaGaamyDamaaBaaaleaacaaIWaaabeaa kiaacIcacqaHepaDcaGGPaaacaGLOaGaayzkaaGaeyypa0JaaGimaa aa@4C4B@   (34)

with initial conditions

u 1 (0)=Acosτ, u ˙ 1 (0)=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyDamaaBa aaleaacaaIXaaabeaakiaacIcacaaIWaGaaiykaiabg2da9iaadgea caWGJbGaam4BaiaadohacaaMc8UaeqiXdqNaaiilaiqadwhagaGaam aaBaaaleaacaaIXaaabeaakiaacIcacaaIWaGaaiykaiabg2da9iaa icdaaaa@485B@   (35)

Where

L( u 1 (τ) )= u ¨ 1 (τ),N( u 0 (τ) )= α ω 2 u 0 (τ)+ β ω 2 u 0 3 (τ),g(τ)=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamitamaabm aabaGaamyDamaaBaaaleaacaaIXaaabeaakiaacIcacqaHepaDcaGG PaaacaGLOaGaayzkaaGaeyypa0JabmyDayaadaWaaSbaaSqaaiaaig daaeqaaOGaaiikaiabes8a0jaacMcacaGGSaGaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaamOtamaabmaabaGaamyDamaaBaaale aacaaIWaaabeaakiaacIcacqaHepaDcaGGPaaacaGLOaGaayzkaaGa eyypa0ZaaSaaaeaacqaHXoqyaeaacqaHjpWDdaahaaWcbeqaaiaaik daaaaaaOGaamyDamaaBaaaleaacaaIWaaabeaakiaacIcacqaHepaD caGGPaGaey4kaSYaaSaaaeaacqaHYoGyaeaacqaHjpWDdaahaaWcbe qaaiaaikdaaaaaaOGaamyDamaaDaaaleaacaaIWaaabaGaaG4maaaa kiaacIcacqaHepaDcaGGPaGaaiilaiaaykW7caaMc8UaaGPaVlaayk W7caWGNbGaaiikaiabes8a0jaacMcacqGH9aqpcaaIWaaaaa@7704@

The above equations implies that

u ¨ 1 (τ)+ α ω 2 u 0 (τ)+ β ω 2 u 0 3 (τ)=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyDayaada WaaSbaaSqaaiaaigdaaeqaaOGaaiikaiabes8a0jaacMcacqGHRaWk daWcaaqaaiabeg7aHbqaaiabeM8a3naaCaaaleqabaGaaGOmaaaaaa GccaWG1bWaaSbaaSqaaiaaicdaaeqaaOGaaiikaiabes8a0jaacMca cqGHRaWkdaWcaaqaaiabek7aIbqaaiabeM8a3naaCaaaleqabaGaaG OmaaaaaaGccaWG1bWaa0baaSqaaiaaicdaaeaacaaIZaaaaOGaaiik aiabes8a0jaacMcacqGH9aqpcaaIWaaaaa@523B@   (36)

Which can be written as

u ¨ 1 (τ)= 1 ω 2 [ α u 0 (τ)+β u 0 3 (τ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyDayaada WaaSbaaSqaaiaaigdaaeqaaOGaaiikaiabes8a0jaacMcacqGH9aqp cqGHsisldaWcaaqaaiaaigdaaeaacqaHjpWDdaahaaWcbeqaaiaaik daaaaaaOWaamWaaeaacqaHXoqycaWG1bWaaSbaaSqaaiaaicdaaeqa aOGaaiikaiabes8a0jaacMcacqGHRaWkcqaHYoGycaWG1bWaa0baaS qaaiaaicdaaeaacaaIZaaaaOGaaiikaiabes8a0jaacMcaaiaawUfa caGLDbaaaaa@5169@   (37)

Substitute Eq. (33) into Eq. (37), we have

u ¨ 1 (τ)= 1 ω 2 [ αAcosτ+β A 3 co s 3 τ ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyDayaada WaaSbaaSqaaiaaigdaaeqaaOGaaiikaiabes8a0jaacMcacqGH9aqp cqGHsisldaWcaaqaaiaaigdaaeaacqaHjpWDdaahaaWcbeqaaiaaik daaaaaaOWaamWaaeaacqaHXoqycaWGbbGaam4yaiaad+gacaWGZbGa aGPaVlabes8a0jabgUcaRiabek7aIjaadgeadaahaaWcbeqaaiaaio daaaGccaWGJbGaam4BaiaadohadaahaaWcbeqaaiaaiodaaaGccaaM c8UaeqiXdqhacaGLBbGaayzxaaaaaa@5657@   (38)

Integrating both sides twice, we have

u 1 (τ)= 1 ω 2 0 τ 0 τ ( αAcosτ+β A 3 co s 3 τ )dτdτ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyDamaaBa aaleaacaaIXaaabeaakiaacIcacqaHepaDcaGGPaGaeyypa0JaeyOe I0YaaSaaaeaacaaIXaaabaGaeqyYdC3aaWbaaSqabeaacaaIYaaaaa aakmaapehabaWaa8qCaeaadaqadaqaaiabeg7aHjaadgeacaWGJbGa am4BaiaadohacaaMc8UaeqiXdqNaey4kaSIaeqOSdiMaamyqamaaCa aaleqabaGaaG4maaaakiaadogacaWGVbGaam4CamaaCaaaleqabaGa aG4maaaakiaaykW7cqaHepaDaiaawIcacaGLPaaacaWGKbGaeqiXdq Naamizaiabes8a0bWcbaGaaGimaaqaaiabes8a0bqdcqGHRiI8aaWc baGaaGimaaqaaiabes8a0bqdcqGHRiI8aaaa@6510@   (39)

Using trigonometric identity in Eq, (39), one arrives at

u ¨ 1 (τ)= 1 ω 2 0 τ 0 τ [ αAcosτ+β A 3 ( 3cosτ+cos3τ 4 ) ]dτdτ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiqadwhagaWaam aaBaaaleaacaaIXaaabeaakiaacIcacqaHepaDcaGGPaGaeyypa0Ja eyOeI0YaaSaaaeaacaaIXaaabaGaeqyYdC3aaWbaaSqabeaacaaIYa aaaaaakmaapehabaWaa8qCaeaadaWadaqaaiabeg7aHjaadgeacaWG JbGaam4BaiaadohacaaMc8UaeqiXdqNaey4kaSIaeqOSdiMaamyqam aaCaaaleqabaGaaG4maaaakmaabmaabaWaaSaaaeaacaaIZaGaam4y aiaad+gacaWGZbGaaGPaVlabes8a0jabgUcaRiaadogacaWGVbGaam 4CaiaaykW7caaIZaGaeqiXdqhabaGaaGinaaaaaiaawIcacaGLPaaa aiaawUfacaGLDbaacaWGKbGaeqiXdqNaamizaiabes8a0bWcbaGaaG imaaqaaiabes8a0bqdcqGHRiI8aaWcbaGaaGimaaqaaiabes8a0bqd cqGHRiI8aaaa@6F5B@   (40)

After the integration, we arrived at

u 1 (τ)=Acosτ+ 1 ω 2 [ αA( cosτ1 )+β A 3 ( 27( cosτ1 )+( cos3τ1 ) 36 ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwhadaWgaa WcbaGaaGymaaqabaGccaGGOaGaeqiXdqNaaiykaiabg2da9iaadgea caWGJbGaam4BaiaadohacaaMc8UaeqiXdqNaey4kaSYaaSaaaeaaca aIXaaabaGaeqyYdC3aaWbaaSqabeaacaaIYaaaaaaakmaadmaabaGa eqySdeMaamyqamaabmaabaGaam4yaiaad+gacaWGZbGaaGPaVlabes 8a0jabgkHiTiaaigdaaiaawIcacaGLPaaacqGHRaWkcqaHYoGycaWG bbWaaWbaaSqabeaacaaIZaaaaOWaaeWaaeaadaWcaaqaaiaaikdaca aI3aWaaeWaaeaacaWGJbGaam4BaiaadohacaaMc8UaeqiXdqNaeyOe I0IaaGymaaGaayjkaiaawMcaaiabgUcaRmaabmaabaGaam4yaiaad+ gacaWGZbGaaGPaVlaaiodacqaHepaDcqGHsislcaaIXaaacaGLOaGa ayzkaaaabaGaaG4maiaaiAdaaaaacaGLOaGaayzkaaaacaGLBbGaay zxaaaaaa@7216@   (41)

and subsequent problems can be obtained from the iterative problem generating relation by we build a correcting practical as

L( u n+1 (τ) )+g(τ)+N( u n (τ) )=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamitamaabm aabaGaamyDamaaBaaaleaacaWGUbGaey4kaSIaaGymaaqabaGccaGG OaGaeqiXdqNaaiykaaGaayjkaiaawMcaaiabgUcaRiaadEgacaGGOa GaeqiXdqNaaiykaiabgUcaRiaad6eadaqadaqaaiaadwhadaWgaaWc baGaamOBaaqabaGccaGGOaGaeqiXdqNaaiykaaGaayjkaiaawMcaai abg2da9iaaicdaaaa@4E59@   (42)

with initial conditions

u n+1 (0)=Acosτ, u ˙ n+1 (τ)=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyDamaaBa aaleaacaWGUbGaey4kaSIaaGymaaqabaGccaGGOaGaaGimaiaacMca cqGH9aqpcaWGbbGaam4yaiaad+gacaWGZbGaaGPaVlabes8a0jaacY caceWG1bGbaiaadaWgaaWcbaGaamOBaiabgUcaRiaaigdaaeqaaOGa aiikaiabes8a0jaacMcacqGH9aqpcaaIWaaaaa@4D10@   (43)

where

L( u n+1 (τ) )= u ¨ (τ),N( u n+1 (τ) )= α ω 2 u(τ)+ β ω 2 u 3 (τ),g(τ)=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamitamaabm aabaGaamyDamaaBaaaleaacaWGUbGaey4kaSIaaGymaaqabaGccaGG OaGaeqiXdqNaaiykaaGaayjkaiaawMcaaiabg2da9iqadwhagaWaai aacIcacqaHepaDcaGGPaGaaiilaiaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaad6eadaqadaqaaiaadwhadaWgaaWcbaGaamOBai abgUcaRiaaigdaaeqaaOGaaiikaiabes8a0jaacMcaaiaawIcacaGL PaaacqGH9aqpdaWcaaqaaiabeg7aHbqaaiabeM8a3naaCaaaleqaba GaaGOmaaaaaaGccaWG1bGaaiikaiabes8a0jaacMcacqGHRaWkdaWc aaqaaiabek7aIbqaaiabeM8a3naaCaaaleqabaGaaGOmaaaaaaGcca WG1bWaaWbaaSqabeaacaaIZaaaaOGaaiikaiabes8a0jaacMcacaGG SaGaaGPaVlaaykW7caaMc8UaaGPaVlaadEgacaGGOaGaeqiXdqNaai ykaiabg2da9iaaicdaaaa@7814@

The solutions of u n (τ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyDamaaBa aaleaacaWGUbaabeaakiaacIcacqaHepaDcaGGPaaaaa@3B37@ form the approximate analytical solutions of u(τ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyDaiaacI cacqaHepaDcaGGPaaaaa@3A0E@ . The analytical solutions are simulated and the results are shown below

Recall that τ=ωt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiXdqNaey ypa0JaeqyYdCNaamiDaaaa@3B87@

u(t)=Acosωt+ 1 ω 2 [ αA( cosωt1 )+β A 3 ( 27( cosωt1 )+( cos3ωt1 ) 36 )+... ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaadwhacaGGOa GaamiDaiaacMcacqGH9aqpcaWGbbGaam4yaiaad+gacaWGZbGaeqyY dCNaamiDaiabgUcaRmaalaaabaGaaGymaaqaaiabeM8a3naaCaaale qabaGaaGOmaaaaaaGcdaWadaqaaiabeg7aHjaadgeadaqadaqaaiaa dogacaWGVbGaam4CaiabeM8a3jaadshacqGHsislcaaIXaaacaGLOa GaayzkaaGaey4kaSIaeqOSdiMaamyqamaaCaaaleqabaGaaG4maaaa kmaabmaabaWaaSaaaeaacaaIYaGaaG4namaabmaabaGaam4yaiaad+ gacaWGZbGaeqyYdCNaamiDaiabgkHiTiaaigdaaiaawIcacaGLPaaa cqGHRaWkdaqadaqaaiaadogacaWGVbGaam4CaiaaiodacqaHjpWDca WG0bGaeyOeI0IaaGymaaGaayjkaiaawMcaaaqaaiaaiodacaaI2aaa aaGaayjkaiaawMcaaiabgUcaRiaac6cacaGGUaGaaiOlaaGaay5wai aaw2faaaaa@7129@   (45)

where

ω 1 2 { [ α+ 3β A 2 4 ]+ [ α+ 3β A 2 4 ] 2 ( 3 β 2 A 4 32 ) } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyYdCNaey isIS7aaOaaaeaadaWcaaqaaiaaigdaaeaacaaIYaaaamaacmaabaWa amWaaeaacqaHXoqycqGHRaWkdaWcaaqaaiaaiodacqaHYoGycaWGbb WaaWbaaSqabeaacaaIYaaaaaGcbaGaaGinaaaaaiaawUfacaGLDbaa cqGHRaWkdaGcaaqaamaadmaabaGaeqySdeMaey4kaSYaaSaaaeaaca aIZaGaeqOSdiMaamyqamaaCaaaleqabaGaaGOmaaaaaOqaaiaaisda aaaacaGLBbGaayzxaaWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0Yaae WaaeaadaWcaaqaaiaaiodacqaHYoGydaahaaWcbeqaaiaaikdaaaGc caWGbbWaaWbaaSqabeaacaaI0aaaaaGcbaGaaG4maiaaikdaaaaaca GLOaGaayzkaaaaleqaaaGccaGL7bGaayzFaaaaleqaaaaa@5AFC@

Substitute Eq. (12) and (45) into Eq. (8), we have

w(x,t)=[ Acosωt+ 1 ω 2 [ αA( cosωt1 )+β A 3 ( 27( cosωt1 )+( cos3ωt1 ) 36 ) ]+... ]{ sin nπx L } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DaiaacI cacaWG4bGaaiilaiaadshacaGGPaGaeyypa0ZaamWaaeaacaWGbbGa am4yaiaad+gacaWGZbGaeqyYdCNaamiDaiabgUcaRmaalaaabaGaaG ymaaqaaiabeM8a3naaCaaaleqabaGaaGOmaaaaaaGcdaWadaqaaiab eg7aHjaadgeadaqadaqaaiaadogacaWGVbGaam4CaiabeM8a3jaads hacqGHsislcaaIXaaacaGLOaGaayzkaaGaey4kaSIaeqOSdiMaamyq amaaCaaaleqabaGaaG4maaaakmaabmaabaWaaSaaaeaacaaIYaGaaG 4namaabmaabaGaam4yaiaad+gacaWGZbGaeqyYdCNaamiDaiabgkHi TiaaigdaaiaawIcacaGLPaaacqGHRaWkdaqadaqaaiaadogacaWGVb Gaam4CaiaaiodacqaHjpWDcaWG0bGaeyOeI0IaaGymaaGaayjkaiaa wMcaaaqaaiaaiodacaaI2aaaaaGaayjkaiaawMcaaaGaay5waiaaw2 faaiabgUcaRiaac6cacaGGUaGaaiOlaaGaay5waiaaw2faamaacmaa baGaai4CaiaadMgacaWGUbWaaSaaaeaacaWGUbGaeqiWdaNaamiEaa qaaiaadYeaaaaacaGL7bGaayzFaaaaaa@7E6C@   (46)

Results and discussion

Figure 1 shows the comparison between the results of Temimi and Ansari method (TAM) and numerical method (NM) using Fourth-order Runge-Kutta method. The obtained results of using TAM as compared with the numerical procedure are in good agreements. The high accuracy of TAM gives high confidence about validity of the method in providing solutions to the problem. Also, the effects of various parameters of the model on the dynamic response of the single-walled carbon nanotube are also presented in the figures under various subsections in the section in Figure 3.

Figure 1 Carbon nanotube conveying hot fluid resting on elastic foundation.

Figure 2 Effect of slip boundary condition on velocity profile.59

Figure 3 Comparison between the obtained results and the numerical solution for the nonlinear vibration.

The importance of surface residual stress on the vibration behaviour of the nanotube is shown in Figure 4. It is shown that the dynamic response of the nanotube different for negative and positive values of surface residual stress. This establishes that the dynamic behaviour of the fluid-conveying nanotube depends on the sign of the residual surface stress. Indisputably, as it is shown in the figure, at any given dimensional amplitude, there is an increase in the frequency ratio when the negative value of the surface stress increases while the frequency ratio decreases when the positive value of the surface stress increases. This is because, the negative values of surface stress decrease the linear stiffness of the nanostructure while the positive values of surface stress increase the linear stiffness of the carbon nanotube.

Figure 4 Effect of surface residual stress per unit length on the frequency ratio of the nanotube.

Additionally, the positive surface elasticity produces softening effect in the nanotube, while negative surface elasticity gives stiffening influence in the nanotube. Therefore, it can be stated that when the surface stress is zero, the effect of surface elasticity is not so important. Consequently, one can infer that the surface stress alone is important and effective even without consideration of the surface elasticity. However, when the surface stress is nonzero, the surface elasticity plays a significant role in the dynamic behaviour of the nanostructure.

The importance of surface stress, nonlocality and nanobeam length on the frequency ratio of the fluid-conveying nanostructure is displayed in Figure 5. The figures show that the frequency ratio decreases with increase in the length and thickness ratio of the of the nanotube. It could also be stated that nonlocal parameter reduces the influence of the surface energy and stress on the frequency ratio. The results also presented that the vibration frequency of the nanotube under the consideration of the effects of surface energy and stress is larger than vibration frequency of the nanobeam given by the classical beam theory which does not consider the surface effect. Also, the figures present a clear statement that when the nanotube length increase, the natural frequency of the nanotube gradually approaches the nonlinear Euler–Bernoulli beam limit. This is as a result of decrease in the surface effect. Therefore, high thickness ratios and long nanotube length make the impacts of surface energy and stresses on the on the frequency ratio to vanish. The impact of the initial stress on the dynamic behaviour of the nanotube is shown in Figure 6. It is depicted at any adimensional amplitude increases, there is an increase in the frequency ratio as the initial stress increases.

Figure 5 Effects of the nanotube nonlocal parameter and length on the frequency ratio.

Figure 6 Effect of initial stress on the frequency ratio of the nanotube.

The nonlocal parameter is a scaling parameter which makes the small-scale effect to be accounted in the analysis of microstructures and nanostructures. The effect of the nonlocality on the frequency ratio decrease for varying adimensional amplitude is illustrated in Figure 7. The fundamental frequency ratio of the fluid-conveying structure decreases as the nonlocal parameter increases. Also, the effect of the nonlocality on the frequency ratio decreases by increasing the amplitude ratio of the structure. The variations in the ratio of the frequencies with adimensional nonlocal parameter for different change in temperature are presented in Figure 8&9. In Figure 8, it is shown that increase in temperature change at high temperature causes decrease in the frequency ratio. However, at room or low temperature, the frequency ratio of the hybrid nanostructure increases as the temperature change increases as shown in Fig. 8. Also, the ratio of the frequencies at low temperatures is lower than at high temperatures.

Figure 7 Effects of maximum amplitude and nonlocal parameter on ratio of the frequency ratio.

Figure 8 Effects of change in temperature on the frequency at high temperature.

Figure 9 Effects of change in temperature on the frequency ratio at low temperature.

Figure 10 shows the significance of the magnetic field strength on the frequency ratio of the nanotube. It is shown that the frequency ratio decreases when the strength of the magnetic field increases. Also, at high values of magnetic fields and amplitude of vibration, the discrepancy between the nonlinear and the linear frequencies increases. A further investigation shows that the vibration of the nanotube approaches linear vibration when the magnetic force strength increases to a certain high value. Such very high value of magnetic force strength which causes great attenuation in the beam can be adopted as a control and instability strategy for the nonlinear vibration system. 

Figure 10 Effects of magnetic field strength on the frequency ratio.

Comparison of the midpoint deflection of linear and nonlinear vibrations of the nanostructure is analyzed in Figure 11. The nonlinear term causes stretching effect in the nonlinear in the nonlinear vibration. As stretching effect increases, the stiffness of the system increases which consequently increases in the natural frequency and the critical fluid velocity.

Figure 11 Linear and nonlinear dynamic behaviour of the nanostructure.

Figure 12 presents the effect of nonlocal parameter on the vibration of the nanotube. It is depicted that increase in the nonlocal parameter leads to decrease in the frequency of vibration and decrease in the critical velocity. The significance of slip parameter on the dynamic response of the carbon nanotube is shown in Figure 13. From the figure, it is established that increase in the slip parameter causes decrease in the frequency of vibration and the critical velocity. Also, the Figures depict the critical speeds corresponding to the divergence condition for different values of the system’s parameters for the varying nonlocal and slip parameters.

Figure 12 Effects of nonlocal parameter and fluid flow velocity on the natural frequency of the nonlinear vibration.

Figure 13 Effects Slip parameter (Knudsen number) on the natural frequency of the nonlinear vibration.

Conclusion

In the current paper, Galerkin decomposition and Temini and Ansari methods have been applied to explore the simultaneous impacts of surface elasticity, initial stress, residual surface tension and nonlocality on the nonlinear vibration of single-walled carbon conveying nanotube resting on linear and nonlinear elastic foundation and operating in a thermo-magnetic environment have been analyzed. Through the parametric studies, it was revealed that the

  1. Ratio of the nonlinear to linear frequencies increases with the negative value of the surface stress while it decreases with the positive value of the surface stress  At any given value of nonlocal parameters, the surface effect reduces for increasing in the length of the nanotube.
  2. Ratio of the frequencies decreases with increase in the strength of the magnetic field, nonlocal parameter and the length of the nanotube. The natural frequency of the nanotube gradually approaches the nonlinear Euler–Bernoulli beam limit at high values of nonlocal parameter and nanotube length.
  3. Nonlocal parameter reduces the surface effects on the ratio of the frequencies.
  4. Increase in temperature change at high temperature causes decrease in the frequency ratio. However, at room or low temperature, the frequency ratio of the hybrid nanostructure increases as the temperature change increases. Also, the ratio of the frequencies at low temperatures is lower than at high temperatures.
  5. Increase in the nonlocal and slip parameters leads to decrease in the frequency of vibration and decrease in the critical velocity.

It is hoped that through this study, the control and design of carbon nanotubes operating in thermo-magnetic environment and resting on elastic foundations will be greatly enhanced.

Acknowledgments

None.

Conflicts of interest

The author declares that there is no conflict of interest.

Funding

None.

References

  1. Iijima S. Helical microtubules of graphitic carbon. Nature. 1991;354:56–58.
  2. Abgrall, Nguyen NT. Nanofluidic devices and their applications. Anal Chem. 2008;80:2326–234.
  3. Zhao D, Liu Y, Tang YG. Effects of magnetic field on size sensitivity of nonlinear vibration of embedded nanobeams. Mech Adv Mater Struct. 2018;1–9.
  4. Azrar A, Ben Said M, Azrar L, et al. Dynamic analysis of Carbon Nanotubes conveying fluid with uncertain parameters and random excitation. Mech Adv Mater Struct. 2018;1–16.
  5. Rashidi V, Mirdamadi HR, Shirani E. A novel model for vibrations of nanotubes conveying nanoflow. Comput Mater Sci. 2012;51:347–352.
  6. Reddy JN, Pang S. Nonlocal continuum theories of beams for the analysis of carbon nanotubes. J Appl Phys. 2008;103:023511.
  7. Wang L. A modified nonlocal beam model for vibration and stability of nanotubes conveying fluid. Physica E. 2011;44:25–28.
  8. Lim CW. On the truth of nanoscale for nanobeams based on nonlocal elastic stress field theory: equilibrium, governing equation and static deflection. Appl Math Mech. 2010;31:37–54.
  9. Lim CW, Yang Y. New predictions of size-dependent nanoscale based on nonlocal elasticity for wave propagation in carbon nanotubes. J Comput Theor Nanoscience. 2010;7:988–995.
  10. Bahaadini R, Hosseini M. Nonlocal divergence and flutter instability analysis of embedded fluid-conveying carbon nanotube under magnetic field. Microfluid Nanofluid. 2016;20:108.
  11. Mahinzare M, Mohammadi K, Ghadiri M, et al. Size-dependent effects on critical flow velocity of a SWCNT conveying viscous fluid based on nonlocal strain gradient cylindrical shell model. Microfluid Nanofluid. 2017;21:123.
  12. Bahaadini R, Hosseini M. Flow-induced and mechanical stability of cantilever carbon nanotubes subjected to an axial compressive load. Appl Math Modell. 2018;59:597–613.
  13. L Wang. Vibration analysis of fluid-conveying nanotubes with consideration of surface effects. Physica E. 2010;43:437–439.
  14. Zhang J, Meguid SA. Effect of surface energy on the dynamic response and instability of fluid-conveying nanobeams. Eur J Mech A/Solids. 2016;58:1–9.
  15. Hosseini M, Bahaadini R, Jamali B. Nonlocal instability of cantilever piezoelectric carbon nanotubes by considering surface effects subjected to axial flow. J Vib Control. 2016.
  16. Bahaadini R, Hosseini M, Jamalpoor A. Nonlocal and surface effects on the flutter instability of cantilevered nanotubes conveying fluid subjected to follower forces. Physica B. 2017;509:55–61.
  17. Wang GF, Feng XQ. Effects of surface elasticity and residual surface tension on the natural frequency of micro-beams. J App Phys. 2007;101:013510.
  18. GF Wang, Feng XQ. Surface effects on buckling of nanowires under uniaxial compression. Appl Phys Lett. 2009;94:1419133.
  19. Farshi B, Assadi A, Alinia ziazi A. Frequency analysis of nanotubes with consideration of surface effects. Appl Phys Lett. 2010;96:093103.
  20. Lee HL, Chang WJ. Surface effects on axial buckling of non-uniform nanowires using non-local elasticity theory. Micro & Nano Letters. 2011;6(1):19–21.
  21. Lee HL, Chang WJ. Surface effects on frequency analysis of nanotubes using nonlocal Timoshenko beam theory. J Appl Phys. 2010;108:093503.
  22. Guo JG, Zhao YP. The size dependent bending elastic properties of nanobeams with surface effects. Nanotechnology. 2007;18:295701.
  23. Feng XQ, Xia R, Li XD, et al. Surface effects on the elastic modulus of nanoporous materials. Appl Phys Lett. 2009;94:011913.
  24. He J, Lilley CM. Surface stress effect on bending resonance of nanowires with different boundary conditions. Appl Phys Lett. 2008;93:263103–8.
  25. He J, Lilley CM. Surface effect on the elastic behavior of static bending nanowires. Nano Lett. 2008;8:1798–1802.
  26. Jing GY, Duan HL, Sun XN, et al. Surface effects on elastic properties of silver nanowires: contact atomic-force microscopy. Phys Rev B. 2010;73:235406.
  27. Sharm P, Ganti S, Bhate N. Effect of surfaces on the size-dependent elastic state of nano-inhomogeneities. Appl Phys Lett. 2003;82:535–537.
  28. ZQ Wang, Zhao YP, Huang ZP. The effects of surface tension on the elastic properties of nano structures. Int J Eng Sci. 2010;48:140–150.
  29. Selim MM.Vibrational analysis of carbon nanotubes under initial compression stresses. NANO Conference. King Saud University; KSA; 2009.
  30. Zhang H, Wang X. Effects of initial stress on transverse wave propagation in carbon nanotubes based on Timoshenko laminated beam models. Nanotechnoology. 2006;17:45–53.
  31. Wang X, Cai H. Effects of initial stress on non-coaxial resonance of multi-wall carbon nanotubes. Acta Mater. 2006;54:2067–2074.
  32. Liu K, Sun C. Vibration of multi-walled carbon nanotubes with initial axial loading. Solid State Communications. 2007;143:202–207.
  33. Chen X, Wang X. Effects of initial stress on wave propagation in multi-walled carbon nanotubes. Phys Scr. 2008;78:015601.
  34. Selim MM. Torsional vibration of carbon nanotubes under initial compression stress. Brazilian J Phys. 40(3):283.
  35. Selim MM. Vibrational Analysis of Initially Stressed Carbon. Acta Physica. 2011;119.
  36. Selim MM. Vibrational analysis of initially stressed carbon nanotubes. Acta Phys Pol A. 2011;119(6):778–782.
  37. Selim MM, El Safty SA. Vibrational analysis of an irregular single-walled carbon nanotube incorporating initial stress effects. Nanotechnology Reviews. 2020;9:1481–1490.
  38. Temimi H, Ansari AR. A semi analytical iterative technique for solving nonlinear problems. Comput Math Appl. 2011;61:203–210.
  39. Temimi H, Ansari AR. A new iterative technique for solving nonlinear second order multi point boundary value problems. Appl Math Comput. 2011;218:1457–1466.
  40. Temimi H, Ansari AR. A computational iterative method for solving nonlinear ordinary differential equations. LMS J Comput Math. 2015;18:730–753.
  41. Al Jawary MA, Al Razaq SG. A semi analytical iterative technique for solving duffing equations. Int J pure app Math. 2016;108(4):871–885.
  42. Ehsani F, Hadi A, Ehsani F, et al. An iterative method for solving partial differential equations and solution of Kortewegde Vries equations for showing the capability of the iterative method. World App Program. 2013;3(8):320–327.
  43. Al Jawary MA, Raham RK. A semi-analytical iterative technique for solving chemistry problems. J King Saud Univer Sci. 2017;29(3):320–332.
  44. Al-Jawary MA. A semi-analytical iterative method for solving nonlinear thin film flow problems. Chaos, Solitons & Fractals. 2017;99:52–56.
  45. Al-Jawary MA, Radhi GH, Ravnik J. Semi-analytical method for solving Fokker-Planck’s equations. J Assoc Arab Univer Basic App Sci. 2017;24:254–262.
  46. Al-Jawary MA, Hatif S. A semi-analytical iterative method for solving differential algebraic equations. Ain Shams Eng J. 2018;9(4):2581–2586.
  47. Al-Jawary MA, Al-Qaissy HR. A reliable iterative method for solving Volterra integro-differential equations and some applications for the Lane–Emden equations of the first kind. Monthly Notices of the Royal Astronomical Society. 2015;448(4):3093–3104.
  48. Al-Jawary MA, Adwan MI, Radhi H. Three iterative methods for solving second order nonlinear ODEs arising in physics. J King Saud Univer Sci. 2020;32(1):312–323.
  49. Al-Jawary MA, Mohammed AS. A Semi-Analytical Iterative Method for Solving Linear and Nonlinear Partial Differential Equations. Int J Sci Res. 2015;6(5):978–982.
  50. Al-Jawary MA, Nabi AZA. Reliable iterative methods for solving convective straight and radial fins with temperature-dependent thermal conductivity problems. Gazi Univer J Sci. 2019;32(3):967–989.
  51. Yassein SM. Application of Iterative Method for Solving Higher Order Integro-Differential Equations. Ibn AL Haitham J Pure App Sci. 2019;32(2):51–61.
  52. Agheli B. Solving fractional Bratu’s equations using a semi analytical technique. Punjab Univer J Math. 20220;51(9).
  53. Sobamowo MG, Adeleye OA. Application of a New Iterative Method to Analysis of Kinetics of Thermal Inactivation of Enzyme. UPB Sci Bull Series B. 2019;81(1).
  54. Ehsani F, Hadi F, Ehsani R. An iterative method for solving partial differential equations and solution of Kortewegde Vries equations for showing the capability of the iterative method. World Appl Program. 2013;3(8):320–327.
  55. Eringen AC. Nonlocal polar elastic continua. Int J Eng Sci. 1972;10(1):1–16.
  56. Eringen AC. Linear theory of nonlocal elasticity and dispersion of plane waves. Int J Eng Sci. 1972;10(5):425–435.
  57. Eringen AC. On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J Appl Phys. 1983;54(9):4703–4710.
  58. Arania GA, Roudbaria MA, Amir S. Longitudinal magnetic field effect on wave propagation of fluid conveyed SWCNT using Knudsen number and surface considerations. Applied Mathematical Modelling. 2016;40:2025–2038.
  59. Bahaadini R, Hosseini M. Effects of nonlocal elasticity and slip condition on vibration and stability analysis of viscoelastic cantilever carbon nanotubes conveying fluid. Comput Mater Sci. 2016;114:151–159.
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