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Applied Bionics and Biomechanics

Research Article Volume 7 Issue 1

Nonlinear nonlocal damped free and forced vibrations of piezoelectric SWCNTs under longitudinal magnetic field due to surface effects using a two steps perturbation method

Saeed Shahsavari, S M Ali Boutorabi

School of Metallurgy and Material Engineering, Iran University of Science and Technology, Iran

Correspondence: Saeed Shahsavari, School of Metallurgy and Material Engineering, Iran University of Science and Technology, Tehran, Iran, Tel 00989376081167

Received: June 19, 2023 | Published: June 28, 2023

Citation: Shahsavari S, Boutorabi SMA. Nonlinear nonlocal damped free and forced vibrations of piezoelectric SWCNTs under longitudinal magnetic field due to surface effects using a two steps perturbation method. MOJ App Bio Biomech. 2023;7(1):88-99. DOI: 10.15406/mojabb.2023.07.00180

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Abstract

In the present work, damped free and forced vibrations of single-walled piezoelectric carbon nanotubes under longitudinal magnetic field due to surface effects surrounded on a non-linear viscoelastic medium using the nonlocal Euler-Bernoulli beam theory and multiple time scales method are investigated. Lorentz force equation is used to obtain the vertical force due to the applied voltage to the system. The surface effects as well as a combinational non-linear viscoelastic foundation are considered, and finally, the dynamic equilibrium equations are used, and non-linear equations of motion are extracted. In the following, the Galerkin and multiple time scales methods are used, and finally, analytical solutions are extracted as the non-linear free and forced vibrational responses of the system. The relevant coefficients of the extracted analytical solutions are discovered for two both simple support (S-S) and clamped (C-C) boundary conditions. In the following, , and the effects of the different parameters such as non-local parameter as well as electric-magnetic fields, effect of hardness-linear damping parameters of nonlinear considered viscoelastic foundation, applied magnetic field, base modes for different forms considering surface effects, and etc. will be studied. As one the results of this study, the presence of a non-local parameter has increased the curvature deviation to the right and the stiffening effect. In other words, the non-local parameter is a factor to increase the nonlinear effect of the system. Also, it is predictable that as the load affect position moves away from the center of the single-walled piezoelectric carbon nanotube toward the supports, the amplitude of the dynamic response decreases significantly, and this relative reduction is greater for the C-C boundary condition than for the S-S boundary condition. It is also important to note that the location of the load has no effect on the rate of deviation of the curve peak, and the degree of nonlinearity of the vibrational response of the system.

Keywords: nonlocal euler-bernoulli beam theory, non-linear non-local damped forced vibration, viscoelastic piezoelectric carbon nanotubes, magnetic field, non-linear viscoelastic foundation, surface effects

Introduction

Having unique physical and chemical properties such as extremely high tensile strength and rigidity, combined with very low density, makes CNTs an essential part of all industrial fields. Nanotubes are critical members of the design and fabrication of nanoelectromechanical systems.1 Microelectromechanical and nanoelectromechanical systems are the essential applications of piezoelectric generators.2 Surface effects are other phenomena that occur due to the static balance of atoms on the surface.3 These effects on elastic bodies are divided into three categories: surface tension, residual surface stress, and surface density.4 Indeed, surface effects will play a serious role when the surface-to-volume ratio increases.5–6 Finally, surface effects may have an important role in the vibration analysis of nanotubes.7–8

The damped forced vibration of SWCNTs was analyzed using a new shear deformation beam theory.9 In this new shear deformation beam theory, there was no need to use any shear correction factor, and also the number of unknown variables was the only one that was similar to the Euler-Bernoulli beam hypothesis. The torque effect of an axial magnetic field on a functionally graded (FG) nano-rod has been studied to capture size effects under magnetic field using Maxwell’s relation.10 As an important result of this study could be mentioned that an FG nano-rod model based on the nonlocal elasticity theory behaves softer and has a smaller natural frequency. Investigation of free vibration of viscoelastic nanotube under longitudinal magnetic field also was studied in recent years.11 According to a deep investigation on the natural frequencies and effect of different parameters such as the nonlocal parameter, structural damping coefficient, material length scale parameter, and the longitudinal magnetic field, the results of this research may be helpful for understanding the potential applications of nanotubes in Nano-Electromechanical System. Many studies in the literature are focused on the investigation of the vibration analysis of nanostructures in an elastic medium. The vibrational response of a SWCNT member considered in an elastic medium to transport a viscous fluid,12 Investigating the vibration of a DWCNT embedded in an elastic medium despite initial axial forces,13,14 Investigating the vibration of nanotubes in conditions where the surrounding medium is considered elastic,15 investigating the nonlinear free vibration of a DWCNT member considering the von Karman assumption in order to apply nonlinear effects and nonlinear,16 nonlinear vibration of a DWCNT by C-C boundary condition in an elastic medium using the nonlinear van der Waals forces, and also the von Kármán geometric to consider nonlinearity,17 Investigating the forced vibration of a DWCNT with the potential to carry a moving nanoparticle,18 investigation of the nonlinear vibration of an MWCNT in thermal environments,19 vibration analysis of an MWCNT by thermal effects, and also considering the size effects on the large amplitude,20 studying free vibration of SWCNT with elastic effects for different boundary conditions,21 vibration of SWCNT with an elastic medium due thermal conditions,22 thermal-mechanical effects consideration to study vibration and buckling instability of a SWCNT that carry out fluid, and also is rested on an elastic foundation,23 electro-thermo-mechanical effects consideration to study vibration of boron nitride nanorod that is in elastic medium by non-uniform and non-homogeneous properties,24 buckling analysis of SWCNT on a viscoelastic foundation for different boundary conditions,25 buckling investigation of SWCNT under thermal effects that is in an elastic medium with one elasticity parameter,26 and buckling behavior of single-walled CNT considering thermal effect in an elastic medium27 were studied using non-local theory of elasticity. another theory that can be considered as size-dependent continuum theories, and also can be used to study the electro-thermal transverse vibration of CNT in an elastic medium was presented based on the non-local shell theory as well as piezoelasticity properties of system.28 Continuum shell theory by considering nonlocal effects was used to study free vibration of single-walled carbon nanotubes by non-homogenous elastic medium.29 Using multiple elastic models of beams as well as continuum mechanics theory, non-linear free vibration of multi-walled carbon nanotubes was studied.30 Nonlinear vibration as well as thermal stability and of pre/post buckling due to temperature effects, and also size-dependent FG beams designed on an elastic foundation was studied based on the modified couple stress theory.31 The multiple time scales method, as a perturbation method, is known as an efficient technique in nonlinear differential equations analysis. Multiple scale method was used to study free and forced vibration of beams as well as DWCNT on an elastic medium by considering geometric nonlinearity and S-S boundary condition.32–34 Non-local elasticity theory was used to study tensioned nanobeam nonlinear vibration with considering various boundary conditions.35,36 The non-local Euler–Bernoulli theory of elasticity37 can be considered as a theory to modify nanoscale as well as nanostructers models that greatly has been used in modelling carbon nanotubes due to the size-effects in nanoscale behavior. Several references in the literatures can be found on the basis of the nonlocal theory of elasticity.38,39 In some of the recent relevant papers, the nonlinear frequency response of single-walled carbon nanotubes to primary resonance has been studied based on the nonlocal Euler-Bernoulli beam theory.40–48 The nonlocal Euler–Bernoulli elasticity theory also has been greatly used to study Nonlocal instability of cantilever piezoelectric carbon nanotubes by considering surface effects subjected to axial flow,49 Linear free vibration analysis of piezoelectric SWCNTs that the results are extracts considering linear part of a new general nonlinear viscoelastic foundation without considering external force,50 Vibration analysis of piezoelectric nanowires with surface and small scale effects,51 Flexoelectric and surface effects on size-dependent flow-induced vibration and instability analysis of fluid-conveying nanotubes based on flexoelectricity beam model‏,52 and Size-dependent nonlinear vibration of functionally graded composite micro-beams reinforced by carbon nanotubes with piezoelectric layers in thermal environments.53 Consequently, due to lots of the presented works in the literature, the objective of the present work is to study damped free and forced vibration of single-walled piezoelectric carbon nanotubes under longitudinal magnetic field considering surface effects resting on a non-linear viscoelastic foundation based on the nonlocal Euler-Bernoulli beam theory as well as multiple time scales and Galerkin methods. A theoretical and numerical study on non-linear nonlocal free and forced vibration responses are performed. This study, considering surface effects, develops a theoretical response for the non-linear nonlocal vibration of piezoelectric carbon nanotubes located on a non-linear viscoelastic foundation, and attenuation factors due to the external magnetic field and piezoelectric voltage Using Galerkin and multiple time scales methods. Surface effects, which harden the surface of nanotubes, are created by the formation of homogeneous masses by Van forces in the waltz. In the following, considering different boundary conditions of the system (S-S and C-C), the relevant coefficients of the developed analytical response are determined. Finally, the vibrational response parameters such as Amplitude-frequency response curves of nonlinear forced vibration, and etc. will be generally investigated for different boundary conditions, and the effects of the different parameters such as non-local parameter and electric-magnetic fields, Effect of hardness-linear damping parameters of viscoelastic foundation, applied magnetic field, base modes for different forms considering surface effects, and etc. will be studied.

Mathematical modelling and formulations

Scheme of the considered system

Figure 1 shows structural schematics of a piezoelectric single-walled carbon nanotube with an inner diameter , outer diameter , thickness , length , mass per unit length , and elastic modulus , subjected to voltage and a magnetic field of magnitude . This nanotube, subjected to a harmonic external point load, resting on a nonlinear viscoelastic foundation. Figure 2 shows discrete model of the single walled carbon nanotube by coordinate system.

Figure 1 Scheme of the single-walled piezoelectric carbon nanotubes under longitudinal magnetic field subjected to voltage and resting on nonlinear viscoelastic foundation with surface layer.

Figure 2 Scheme of the discrete model of the single walled carbon nanotube by coordinate system.

Non-local effects

Generally, nanoscale or microstructure analysis methods can be classified into three categories: molecular simulation, non-classical continuous environment, and multi-scale.54 The molecular simulation methods, which simulated atoms and their bonds, investigate nanostructures' behavior more profound than other methods. The high computational volume of this method makes it possible to simulate only small nanostructured dimensions using this method.55,56 Molecular methods are divided into quantum mechanics, molecular dynamics, and molecular mechanics.57 Complex energy functions are used to explain the motion of elementary particles in quantum mechanics. However, this method is associated with computational limitations due to the complexity of the equations used. In the molecular dynamics method, the motion of atoms is studied concerning their adjacent bonds. Atoms are known as rigid particles that are affected by the potential field of neighbouring atoms. The last one, the molecular mechanical method, is based on the displacement of interatomic bonds with beam and spring elements. Potential energy can express the field of interatomic forces in atomic structures. Equalizing the beam and spring elements' potential energy and strain energy, the mechanical properties are obtained equally. In classical theory, the stress state at a particular point can be calculated if the strain state is known at that point. Whereas, in non-classical theory, such as non-local tension theory, the strain state must be known throughout the range to obtain the stress state at a particular point. All other proposed theories based on non-classical continuum mechanics, including strain gradients and coupled stress theories, relate to two-dimensional elasticity.58–60 Therefore, nonlocal stress tensor can written as follows:61,62

σ( X )= V k( | X X |, τ )T( X )dV MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi abeo8aZnaabmaapaqaa8qacaWGybaacaGLOaGaayzkaaGaeyypa0Za aybCaeqal8aabaWdbiaadAfaa8aabaaaneaapeGaey4kIipaaOGaam 4Aamaabmaapaqaa8qadaabdaWdaeaapeGabmiwa8aagaqba8qacqGH sislcaWGybaacaGLhWUaayjcSdGaaiilaiaacckacqaHepaDaiaawI cacaGLPaaacaWGubWaaeWaa8aabaWdbiqadIfapaGbauaaa8qacaGL OaGaayzkaaGaamizaiaadAfaaaa@5257@   (1)

T( X )=C( X ):ε( X ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadsfadaqadaWdaeaapeGaamiwaaGaayjkaiaawMcaaiabg2da9iaa doeadaqadaWdaeaapeGaamiwaaGaayjkaiaawMcaaiaacQdacqaH1o qzdaqadaWdaeaapeGaamiwaaGaayjkaiaawMcaaaaa@447F@   (2)

That σ( X ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi abeo8aZnaabmaapaqaa8qacaWGybaacaGLOaGaayzkaaaaaa@3C2C@ is nonlocal stress tensor at point X MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadIfaaaa@38C1@ , ( | X X |, τ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbm aabmaapaqaa8qadaabdaWdaeaapeGabmiwa8aagaqba8qacqGHsisl caWGybaacaGLhWUaayjcSdGaaiilaiaacckacqaHepaDaiaawIcaca GLPaaaaaa@4338@  is nonlocal modulus, is a material constant, T( X ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadsfadaqadaWdaeaapeGaamiwaaGaayjkaiaawMcaaaaa@3B42@ is classical stress tensor, is ε( X ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi abew7aLnaabmaapaqaa8qacaWGybaacaGLOaGaayzkaaaaaa@3C10@ strain tensor and C( X ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadoeadaqadaWdaeaapeGaamiwaaGaayjkaiaawMcaaaaa@3B31@ is elasticity tensor. Because of that solving of the integral constitutive Equation (1) is complicated, a simplified differential for is greatly used as follows:

T=( 1μ 2 )σ, μ= τ 2 l 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadsfacqGH9aqpdaqadaWdaeaapeGaaGymaiabgkHiTiabeY7aTjab gEGir=aadaahaaWcbeqaa8qacaaIYaaaaaGccaGLOaGaayzkaaGaeq 4WdmNaaiilaiaacckacqaH8oqBcqGH9aqpcqaHepaDpaWaaWbaaSqa beaapeGaaGOmaaaakiaadYgapaWaaWbaaSqabeaapeGaaGOmaaaaaa a@4C84@   (3)

Where μ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi abeY7aTbaa@399A@ is non-local coefficient and is 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi abgEGir=aadaahaaWcbeqaa8qacaaIYaaaaaaa@3A72@  Laplacian operator. Also, τ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi abes8a0baa@39A9@ and l MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadYgaaaa@38D5@ are material constant.61 Finally, nonlocal stress tensor for a beam must satisfy equation (4):

σ( X ) ( τl ) 2 2 σ( X ) X 2 =Eε( X ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi abeo8aZnaabmaapaqaa8qacaWGybaacaGLOaGaayzkaaGaeyOeI0Ya aeWaa8aabaWdbiabes8a0jaadYgaaiaawIcacaGLPaaapaWaaWbaaS qabeaapeGaaGOmaaaakmaalaaapaqaa8qacqGHciITpaWaaWbaaSqa beaapeGaaGOmaaaakiabeo8aZnaabmaapaqaa8qacaWGybaacaGLOa Gaayzkaaaapaqaa8qacqGHciITcaWGybWdamaaCaaaleqabaWdbiaa ikdaaaaaaOGaeyypa0Jaamyraiabew7aLnaabmaapaqaa8qacaWGyb aacaGLOaGaayzkaaaaaa@52E8@   (4)

Where is a constant that known as the classical elasticity modulus.

Now, using the existing stress relations in the non-local theory, the relationship between the local bending torque M l ( x,t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aad2eapaWaaWbaaSqabeaapeGaamiBaaaakmaabmaapaqaa8qacaWG 4bGaaiilaiaadshaaiaawIcacaGLPaaaaaa@3E4B@  and the non-local torque M nl ( x,t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aad2eapaWaaWbaaSqabeaapeGaamOBaiaadYgaaaGcdaqadaWdaeaa peGaamiEaiaacYcacaWG0baacaGLOaGaayzkaaaaaa@3F3E@  at the cross section of the nanotube is written as follows63:

( 1 ( τl ) 2 2 x 2 ) M nl ( x,t )= M l ( x,t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbm aabmaapaqaa8qacaaIXaGaeyOeI0YaaeWaa8aabaWdbiabes8a0jaa dYgaaiaawIcacaGLPaaapaWaaWbaaSqabeaapeGaaGOmaaaakmaala aapaqaa8qacqGHciITpaWaaWbaaSqabeaapeGaaGOmaaaaaOWdaeaa peGaeyOaIyRaamiEa8aadaahaaWcbeqaa8qacaaIYaaaaaaaaOGaay jkaiaawMcaaiaad2eapaWaaWbaaSqabeaapeGaamOBaiaadYgaaaGc daqadaWdaeaapeGaamiEaiaacYcacaWG0baacaGLOaGaayzkaaGaey ypa0Jaamyta8aadaahaaWcbeqaa8qacaWGSbaaaOWaaeWaa8aabaWd biaadIhacaGGSaGaamiDaaGaayjkaiaawMcaaaaa@55A6@   (5)

In equation (5), μ  MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aabY7acaqGGcaaaa@3A49@ is the parameter of the nanotube size scale, which is also known as the non-local parameter.

External force related to Lorentz force and the effect of the magnetic field

Assume U =( u,v,w ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi qadwfapaGbaSaapeGaeyypa0ZaaeWaa8aabaWdbiaadwhacaGGSaGa amODaiaacYcacaWG3baacaGLOaGaayzkaaaaaa@3FEE@  as the displacement field of the nanostructure; the Lorentz force f l MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieWaqaaaaaaaaa Wdbiaa=zgapaWaaSbaaSqaa8qacaWGSbaapaqabaaaaa@3A22@  is equal to64:

f l =η ( J × H ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWhcaqaaabaaa aaaaaapeGaamOza8aadaWgaaWcbaWdbiaadYgaa8aabeaaaOGaay51 GaWdbiabg2da9iabeE7aOjaabckadaqadaWdaeaapeGabmOsa8aaga Wca8qacqGHxdaTceWGibWdayaalaaapeGaayjkaiaawMcaaaaa@457A@   (6)

Where J stands for the current density and H is the magnetic field strength vector in the environment. Also, in equation (6):

J =× h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi qadQeapaGbaSaapeGaeyypa0Jaey4bIeTaey41aqRabmiAa8aagaWc aaaa@3E95@   (7)

That:

h =×( U × H ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi qadIgapaGbaSaapeGaeyypa0Jaey4bIeTaey41aq7aaeWaa8aabaWd biqadwfapaGbaSaapeGaey41aqRabmisa8aagaWcaaWdbiaawIcaca GLPaaaaaa@436D@   (8)

Therefore, by considering longitudinal magnetic field:

h = H x ( v y + w z ) i + H x v x j + H x w x k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi qadIgapaGbaSaapeGaeyypa0JaeyOeI0Iaamisa8aadaWgaaWcbaWd biaadIhaa8aabeaak8qadaqadaWdaeaapeWaaSaaa8aabaWdbiabgk Gi2kaadAhaa8aabaWdbiabgkGi2kaadMhaaaGaey4kaSYaaSaaa8aa baWdbiabgkGi2kaadEhaa8aabaWdbiabgkGi2kaadQhaaaaacaGLOa GaayzkaaGabmyAa8aagaWca8qacqGHRaWkcaWGibWdamaaBaaaleaa peGaamiEaaWdaeqaaOWdbmaalaaapaqaa8qacqGHciITcaWG2baapa qaa8qacqGHciITcaWG4baaaiqadQgapaGbaSaapeGaey4kaSIaamis a8aadaWgaaWcbaWdbiaadIhaa8aabeaak8qadaWcaaWdaeaapeGaey OaIyRaam4DaaWdaeaapeGaeyOaIyRaamiEaaaaceWGRbWdayaalaaa aa@5D9A@  (9)

J = H x ( 2 v xz + 2 w xy ) i H x ( 2 v yz + 2 w x 2 + 2 w z 2 ) j + H x ( 2 v x 2 + 2 v y 2 + 2 w yz ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi qadQeapaGbaSaapeGaeyypa0Jaamisa8aadaWgaaWcbaWdbiaadIha a8aabeaak8qadaqadaWdaeaapeGaeyOeI0YaaSaaa8aabaWdbiabgk Gi2+aadaahaaWcbeqaa8qacaaIYaaaaOGaamODaaWdaeaapeGaeyOa IyRaamiEaiabgkGi2kaadQhaaaGaey4kaSYaaSaaa8aabaWdbiabgk Gi2+aadaahaaWcbeqaa8qacaaIYaaaaOGaam4DaaWdaeaapeGaeyOa IyRaamiEaiabgkGi2kaadMhaaaaacaGLOaGaayzkaaGabmyAa8aaga Wca8qacqGHsislcaWGibWdamaaBaaaleaapeGaamiEaaWdaeqaaOWd bmaabmaapaqaa8qadaWcaaWdaeaapeGaeyOaIy7damaaCaaaleqaba WdbiaaikdaaaGccaWG2baapaqaa8qacqGHciITcaWG5bGaeyOaIyRa amOEaaaacqGHRaWkdaWcaaWdaeaapeGaeyOaIy7damaaCaaaleqaba WdbiaaikdaaaGccaWG3baapaqaa8qacqGHciITcaWG4bWdamaaCaaa leqabaWdbiaaikdaaaaaaOGaey4kaSYaaSaaa8aabaWdbiabgkGi2+ aadaahaaWcbeqaa8qacaaIYaaaaOGaam4DaaWdaeaapeGaeyOaIyRa amOEa8aadaahaaWcbeqaa8qacaaIYaaaaaaaaOGaayjkaiaawMcaai qadQgapaGbaSaapeGaey4kaSIaamisa8aadaWgaaWcbaWdbiaadIha a8aabeaak8qadaqadaWdaeaapeWaaSaaa8aabaWdbiabgkGi2+aada ahaaWcbeqaa8qacaaIYaaaaOGaamODaaWdaeaapeGaeyOaIyRaamiE a8aadaahaaWcbeqaa8qacaaIYaaaaaaakiabgUcaRmaalaaapaqaa8 qacqGHciITpaWaaWbaaSqabeaapeGaaGOmaaaakiaadAhaa8aabaWd biabgkGi2kaadMhapaWaaWbaaSqabeaapeGaaGOmaaaaaaGccqGHRa WkdaWcaaWdaeaapeGaeyOaIy7damaaCaaaleqabaWdbiaaikdaaaGc caWG3baapaqaa8qacqGHciITcaWG5bGaeyOaIyRaamOEaaaaaiaawI cacaGLPaaaaaa@8E05@   (10)

Finally, the Lorentz force is:

f l = f x i + f y j + f z k =η[ 0 i + H x 2 ( 2 v x 2 + 2 v y 2 + 2 w yz ) j + H x 2 ( 2 w x 2 + 2 w y 2 + 2 v yz ) ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWhcaqaaabaaa aaaaaapeGaamOza8aadaWgaaWcbaWdbiaadYgaa8aabeaaaOGaay51 GaWdbiabg2da9iaadAgapaWaaSbaaSqaa8qacaWG4baapaqabaGcpe GabmyAa8aagaWca8qacqGHRaWkcaWGMbWdamaaBaaaleaapeGaamyE aaWdaeqaaOWdbiqadQgapaGbaSaapeGaey4kaSIaamOza8aadaWgaa WcbaWdbiaadQhaa8aabeaak8qaceWGRbWdayaalaWdbiabg2da9iab eE7aOnaadmaapaqaa8qacaaIWaGabmyAa8aagaWca8qacqGHRaWkca WGibWdamaaBaaaleaapeGaamiEaaWdaeqaaOWaaWbaaSqabeaapeGa aGOmaaaakmaabmaapaqaa8qadaWcaaWdaeaapeGaeyOaIy7damaaCa aaleqabaWdbiaaikdaaaGccaWG2baapaqaa8qacqGHciITcaWG4bWd amaaCaaaleqabaWdbiaaikdaaaaaaOGaey4kaSYaaSaaa8aabaWdbi abgkGi2+aadaahaaWcbeqaa8qacaaIYaaaaOGaamODaaWdaeaapeGa eyOaIyRaamyEa8aadaahaaWcbeqaa8qacaaIYaaaaaaakiabgUcaRm aalaaapaqaa8qacqGHciITpaWaaWbaaSqabeaapeGaaGOmaaaakiaa dEhaa8aabaWdbiabgkGi2kaadMhacqGHciITcaWG6baaaaGaayjkai aawMcaaiqadQgapaGbaSaapeGaey4kaSIaamisa8aadaWgaaWcbaWd biaadIhaa8aabeaakmaaCaaaleqabaWdbiaaikdaaaGcdaqadaWdae aapeWaaSaaa8aabaWdbiabgkGi2+aadaahaaWcbeqaa8qacaaIYaaa aOGaam4DaaWdaeaapeGaeyOaIyRaamiEa8aadaahaaWcbeqaa8qaca aIYaaaaaaakiabgUcaRmaalaaapaqaa8qacqGHciITpaWaaWbaaSqa beaapeGaaGOmaaaakiaadEhaa8aabaWdbiabgkGi2kaadMhapaWaaW baaSqabeaapeGaaGOmaaaaaaGccqGHRaWkdaWcaaWdaeaapeGaeyOa Iy7damaaCaaaleqabaWdbiaaikdaaaGccaWG2baapaqaa8qacqGHci ITcaWG5bGaeyOaIyRaamOEaaaaaiaawIcacaGLPaaaaiaawUfacaGL Dbaaaaa@8DD4@   (11)

Therefore, along the x,y,z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadIhacaGGSaGaamyEaiaacYcacaWG6baaaa@3C3E@ directions:

f x =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadAgapaWaaSbaaSqaa8qacaWG4baapaqabaGcpeGaeyypa0JaaGim aaaa@3C00@

f y = H x 2 ( 2 v x 2 + 2 v y 2 + 2 w yz ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadAgapaWaaSbaaSqaa8qacaWG5baapaqabaGcpeGaeyypa0Jaamis a8aadaWgaaWcbaWdbiaadIhaa8aabeaakmaaCaaaleqabaWdbiaaik daaaGcdaqadaWdaeaapeWaaSaaa8aabaWdbiabgkGi2+aadaahaaWc beqaa8qacaaIYaaaaOGaamODaaWdaeaapeGaeyOaIyRaamiEa8aada ahaaWcbeqaa8qacaaIYaaaaaaakiabgUcaRmaalaaapaqaa8qacqGH ciITpaWaaWbaaSqabeaapeGaaGOmaaaakiaadAhaa8aabaWdbiabgk Gi2kaadMhapaWaaWbaaSqabeaapeGaaGOmaaaaaaGccqGHRaWkdaWc aaWdaeaapeGaeyOaIy7damaaCaaaleqabaWdbiaaikdaaaGccaWG3b aapaqaa8qacqGHciITcaWG5bGaeyOaIyRaamOEaaaaaiaawIcacaGL Paaaaaa@58DC@

f z = H x 2 ( 2 w x 2 + 2 w y 2 + 2 v yz ), MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadAgapaWaaSbaaSqaa8qacaWG6baapaqabaGcpeGaeyypa0Jaamis a8aadaWgaaWcbaWdbiaadIhaa8aabeaakmaaCaaaleqabaWdbiaaik daaaGcdaqadaWdaeaapeWaaSaaa8aabaWdbiabgkGi2+aadaahaaWc beqaa8qacaaIYaaaaOGaam4DaaWdaeaapeGaeyOaIyRaamiEa8aada ahaaWcbeqaa8qacaaIYaaaaaaakiabgUcaRmaalaaapaqaa8qacqGH ciITpaWaaWbaaSqabeaapeGaaGOmaaaakiaadEhaa8aabaWdbiabgk Gi2kaadMhapaWaaWbaaSqabeaapeGaaGOmaaaaaaGccqGHRaWkdaWc aaWdaeaapeGaeyOaIy7damaaCaaaleqabaWdbiaaikdaaaGccaWG2b aapaqaa8qacqGHciITcaWG5bGaeyOaIyRaamOEaaaaaiaawIcacaGL PaaacaGGSaaaaa@598E@   (12)

Therefore, considering w( x,y,z,t )=w( x,t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadEhadaqadaWdaeaapeGaamiEaiaacYcacaWG5bGaaiilaiaadQha caGGSaGaamiDaaGaayjkaiaawMcaaiabg2da9iaadEhadaqadaWdae aapeGaamiEaiaacYcacaWG0baacaGLOaGaayzkaaaaaa@46DB@ , the vertical force affecting the lateral vibration per unit length of the Euler-Bernoulli nanotube is according to the following equation:

F lz = A f z dA=( ηA  H x 2 ) 2 w x 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadAeapaWaaSbaaSqaa8qacaWGSbGaamOEaaWdaeqaaOWdbiabg2da 9maawahabeWcpaqaa8qacaWGbbaapaqaaaqdbaWdbiabgUIiYdaaki aadAgapaWaaSbaaSqaa8qacaWG6baapaqabaGcpeGaamizaiaadgea cqGH9aqpdaqadaWdaeaapeGaeq4TdGMaamyqaiaacckacaWGibWdam aaBaaaleaapeGaamiEaaWdaeqaaOWaaWbaaSqabeaapeGaaGOmaaaa aOGaayjkaiaawMcaamaalaaapaqaa8qacqGHciITpaWaaWbaaSqabe aapeGaaGOmaaaakiaadEhaa8aabaWdbiabgkGi2kaadIhapaWaaWba aSqabeaapeGaaGOmaaaaaaaaaa@5447@   (13)

Applied force from nonlinear viscoelastic foundation

In this study, vibration analysis of SWCNTs embedded in viscoelastic medium is presented. In this case, it is assumed that the chemical bonds of SWCNT to be formed generally between the external surface of the carbon nanotube and the viscoelastic medium (see Figures 1 and 2). In this study, the present viscoelastic foundation has linear Winkler stiffness, nonlinear stiffness, linear viscosity damping, and the attenuation is nonlinear. Therefore, in the present work, the force coming from the nonlinear viscoelastic medium could be considered in the general form as follows:

F medium = F linear + F nonlinear = MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadAeapaWaaSbaaSqaa8qacaWGTbGaamyzaiaadsgacaWGPbGaamyD aiaad2gaa8aabeaak8qacqGH9aqpcaWGgbWdamaaBaaaleaapeGaam iBaiaadMgacaWGUbGaamyzaiaadggacaWGYbaapaqabaGcpeGaey4k aSIaamOra8aadaWgaaWcbaWdbiaad6gacaWGVbGaamOBaiaadYgaca WGPbGaamOBaiaadwgacaWGHbGaamOCaaWdaeqaaOWdbiabg2da9aaa @5239@

( F linea r k + F nonlinea r k )+( F linea r c + F nonlinea r c )= MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbm aabmaapaqaa8qacaWGgbWdamaaBaaaleaapeGaamiBaiaadMgacaWG UbGaamyzaiaadggacaWGYbWdamaaBaaameaapeGaam4AaaWdaeqaaa Wcbeaak8qacqGHRaWkcaWGgbWdamaaBaaaleaapeGaamOBaiaad+ga caWGUbGaamiBaiaadMgacaWGUbGaamyzaiaadggacaWGYbWdamaaBa aameaapeGaam4AaaWdaeqaaaWcbeaaaOWdbiaawIcacaGLPaaacqGH RaWkdaqadaWdaeaapeGaamOra8aadaWgaaWcbaWdbiaadYgacaWGPb GaamOBaiaadwgacaWGHbGaamOCa8aadaWgaaadbaWdbiaadogaa8aa beaaaSqabaGcpeGaey4kaSIaamOra8aadaWgaaWcbaWdbiaad6gaca WGVbGaamOBaiaadYgacaWGPbGaamOBaiaadwgacaWGHbGaamOCa8aa daWgaaadbaWdbiaadogaa8aabeaaaSqabaaak8qacaGLOaGaayzkaa Gaeyypa0daaa@64FF@

[ ( k 1 w( x,t )+( k 3 w 3 ( x,t ) ) ]+( c 0 + c 2 w 2 ( x,t ) ) w( x,t ) t ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbm aadmaapaqaa8qadaqadaWdaeaapeGaam4Aa8aadaWgaaWcbaWdbiaa igdaa8aabeaak8qacaWG3bWaaeWaa8aabaWdbiaadIhacaGGSaGaam iDaaGaayjkaiaawMcaaiabgUcaRiaacIcacaWGRbWdamaaBaaaleaa peGaaG4maaWdaeqaaOWdbiaadEhapaWaaWbaaSqabeaapeGaaG4maa aakmaabmaapaqaa8qacaWG4bGaaiilaiaadshaaiaawIcacaGLPaaa aiaawIcacaGLPaaaaiaawUfacaGLDbaacqGHRaWkdaqadaWdaeaape Gaam4ya8aadaWgaaWcbaWdbiaaicdaa8aabeaak8qacqGHRaWkcaWG JbWdamaaBaaaleaapeGaaGOmaaWdaeqaaOWdbiaadEhapaWaaWbaaS qabeaapeGaaGOmaaaakmaabmaapaqaa8qacaWG4bGaaiilaiaadsha aiaawIcacaGLPaaaaiaawIcacaGLPaaadaWcaaWdaeaapeGaeyOaIy Raam4Damaabmaapaqaa8qacaWG4bGaaiilaiaadshaaiaawIcacaGL Paaaa8aabaWdbiabgkGi2kaadshaaaGaaiykaiaacYcaaaa@65C1@   (14)

That includes spring element k 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadUgapaWaaSbaaSqaa8qacaaIXaaapaqabaaaaa@39E8@ , nonlinear spring k 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadUgapaWaaSbaaSqaa8qacaaIZaaapaqabaaaaa@39EA@ , linear damper c 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadogapaWaaSbaaSqaa8qacaaIWaaapaqabaaaaa@39DF@ , and nonlinear damper c 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadogapaWaaSbaaSqaa8qacaaIYaaapaqabaaaaa@39E1@ .

Piezoelectric equations considering surface effects

One of the characteristics that distinguish nanomaterials is their surface effects. There are two essential and mechanically distinct surface effects in nanostructures: surface stress and surface elasticity. Atoms on the surface have different bonds than atoms in the balk, and surface stresses lead to these atoms being at the minimum energy level. Also, in nanostructures, surface elasticity occurs due to the lack of bonded neighbors in surface atoms.65,66

According to Euler-Bernoulli beam theory, the axial and transverse displacement fields can be considered as follows:

u(x,y,z,t)= U 0 z w x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadwhacaGGOaGaamiEaiaacYcacaWG5bGaaiilaiaadQhacaGGSaGa amiDaiaacMcacqGH9aqpcaWGvbWdamaaCaaaleqabaWdbiaaicdaaa GccqGHsislcaWG6bWaaSaaaeaacqGHciITcaWG3baabaGaeyOaIyRa amiEaaaaaaa@49EA@

w( x,y,z,t )=w( x,t ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadEhadaqadaWdaeaapeGaamiEaiaacYcacaWG5bGaaiilaiaadQha caGGSaGaamiDaaGaayjkaiaawMcaaiabg2da9iaadEhadaqadaWdae aapeGaamiEaiaacYcacaWG0baacaGLOaGaayzkaaGaaiilaaaa@478A@   (15)

Therefore, the only nonzero (axial) strain of this beam at any point is expressed by the following equation:

ϵ xx = U 0 x z 2 w( x,t ) x 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAK1uy0H wmaeHbfv3ySLgzG0uy0Hgip5wzaGqbaabaaaaaaaaapeGae8x9di=d amaaBaaaleaapeGaaeiEaiaabIhaa8aabeaak8qacqGH9aqpdaWcaa WdaeaapeGaeyOaIyRaaeyva8aadaahaaWcbeqaa8qacaaIWaaaaaGc paqaa8qacqGHciITcaqG4baaaiabgkHiTiaabQhadaWcaaWdaeaape GaeyOaIy7damaaCaaaleqabaWdbiaaikdaaaGccaqG3bWaaeWaa8aa baWdbiaabIhacaGGSaGaaeiDaaGaayjkaiaawMcaaaWdaeaapeGaey OaIyRaaeiEa8aadaahaaWcbeqaa8qacaaIYaaaaaaaaaa@5AA6@   (16)

Where ϵ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaatuuDJXwAK1uy0H wmaeHbfv3ySLgzG0uy0Hgip5wzaGqbaabaaaaaaaaapeGae8x9di=d amaaBaaaleaapeGaaGimaaWdaeqaaaaa@44F0@  is the axial strain applied to the structure and ( x, t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aabEhacaqGGcWaaeWaa8aabaWdbiaabIhacaGGSaGaaeiOaiaabsha aiaawIcacaGLPaaaaaa@3F6D@ is also the transverse displacement (bending) at the desired location x on the center plate of the nanotube. Let's suppose E z MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aabweapaWaaSbaaSqaa8qacaqG6baapaqabaaaaa@3A02@  is the electric field created by the electric potential Φ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaiiaacqWFMoGraa a@3941@ assuming that this field exists only in the direction of the z-axis where Z indicates the distance from the center of the arrow, then the relationship between the electric current created and the existing electric potential is:67

E z = Φ z MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aabweadaWgaaWcbaGaaeOEaaqabaGccqGH9aqpcqGHsisldaWcaaqa aiabgkGi2kabgA6agbqaaiabgkGi2kaabQhaaaaaaa@4125@   (17)

For the surface piezoelectric model, the surface-related structural relationships are different from the bulk volume equations. If the direction of polarization of the piezoelectric medium is assumed in the direction of the z-axis, the stress-strain structural equations related to the surface of the piezoelectric carbon nanotube-based on the relationships presented are written as follows:3–5

σ x s = σ x 0 + c 11 s x e 31 s E z MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi abgo8aZnaaDaaaleaacaqG4baabaGaae4Caaaakiabg2da9iabgo8a ZnaaDaaaleaacaqG4baabaGaaGimaaaakiabgUcaRiaabogadaqhaa WcbaGaaGymaiaaigdaaeaacaqGZbaaaGGacOGae8hcI48aaSbaaSqa aiaabIhaaeqaaOGaeyOeI0IaaeyzamaaDaaaleaacaaIZaGaaGymaa qaaiaabohaaaGccaqGfbWaaSbaaSqaaiaabQhaaeqaaaaa@4E17@   (18)

D x s = D x 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aabseapaWaa0baaSqaa8qacaqG4baapaqaa8qacaqGZbaaaOGaeyyp a0Jaaeira8aadaqhaaWcbaWdbiaabIhaa8aabaWdbiaaicdaaaaaaa@3EFD@   (19)

In equations (18) and (19), σ x s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi abgo8aZnaaDaaaleaacaqG4baabaGaae4Caaaaaaa@3BC6@  and D x s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aabseapaWaa0baaSqaa8qacaqG4baapaqaa8qacaqGZbaaaaaa@3B07@  are axial surface stress and electric surface displacement, respectively. The parameters σ x 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi abgo8aZnaaDaaaleaacaqG4baabaGaaGimaaaaaaa@3B8A@ , D x 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aabseapaWaa0baaSqaa8qacaqG4baapaqaa8qacaaIWaaaaaaa@3ACB@ , e 31 s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aabwgapaWaa0baaSqaa8qacaaIZaGaaGymaaWdaeaapeGaae4Caaaa aaa@3BA5@  and c 11 s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aabogapaWaa0baaSqaa8qacaaIXaGaaGymaaWdaeaapeGaae4Caaaa aaa@3BA1@  indicate the residual surface stress, the residual electrical displacement, the surface piezoelectric constant, and the elastic surface constant, respectively. On the other hand, in piezoelectric carbon nanotubes, the structural relationship between bulk stress and bulk strain would be obtained as follows:68

σ x = C 11 x   e 31   E z MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHdpWCdaWgaa WcbaGaamiEaaqabaGccqGH9aqpcaWGdbWaaSbaaSqaaiaaigdacaaI XaaabeaaiiGakiab=HGiopaaBaaaleaacaWG4baabeaakiabgkHiTa baaaaaaaaapeGaaiiOa8aacaWGLbWaaSbaaSqaaiaaiodacaaIXaaa beaak8qacaGGGcGaamyramaaBaaaleaacaWG6baabeaaaaa@48F0@

σ x =  C 11 x   e 31   E z MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aabo8adaWgaaWcbaGaaeiEaaqabaGccaqG9aGaaiiOaiaaboeadaWg aaWcbaGaaeymaiaabgdaaeqaaGGacOGae8hcI48aaSbaaSqaaiaabI haaeqaaOGaeyOeI0IaaiiOaiaabwgadaWgaaWcbaGaae4maiaabgda aeqaaOGaaiiOaiaabweadaWgaaWcbaGaaeOEaaqabaaaaa@490D@   (20)

D z = e 31 ϵ x + k 33 E z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aabseapaWaaSbaaSqaa8qacaqG6baapaqabaGcpeGaeyypa0Jaaeyz a8aadaWgaaWcbaWdbiaaiodacaaIXaaapaqabaWefv3ySLgznfgDOf daryqr1ngBPrginfgDObYtUvgaiuaak8qacqWF1pG8paWaaSbaaSqa a8qacaqG4baapaqabaGcpeGaey4kaSIaae4Aa8aadaWgaaWcbaWdbi aaiodacaaIZaaapaqabaGcpeGaaeyra8aadaWgaaWcbaWdbiaabQha a8aabeaaaaa@513B@   (21)

That σ x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi abeo8aZnaaBaaaleaacaWG4baabeaaaaa@3ACF@   σ x MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aabo8adaWgaaWcbaGaaeiEaaqabaaaaa@3A53@  and D z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aabseapaWaaSbaaSqaa8qacaqG6baapaqabaaaaa@3A02@ represent the classical stress tensor component and the electrical displacement, respectively. Also, C 11 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aaboeapaWaaSbaaSqaa8qacaaIXaGaaGymaaWdaeqaaaaa@3A7A@ , e 31 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aabwgapaWaaSbaaSqaa8qacaaIZaGaaGymaaWdaeqaaaaa@3A9E@ , and k 33 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aabUgapaWaaSbaaSqaa8qacaaIZaGaaG4maaWdaeqaaaaa@3AA6@  are the elastic constant, piezoelectric constant, and dielectric constant of nanotubes, respectively.

In the absence of free electric charge D z z =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbm aalaaapaqaa8qacqGHciITcaqGebWdamaaBaaaleaapeGaaeOEaaWd aeqaaaGcbaWdbiabgkGi2kaabQhaaaGaeyypa0JaaGimaaaa@3FD4@ , and the potential energy boundary conditions related to the voltage applied to the external environment of the nanotube is obtained from the following equation:

Φ ( D 2 )=V,  Φ (D/2) = 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaiiaaqaaaaaaaaa Wdbiab=z6agjaacckacaGGOaWaaSaaaeaacaqGebaabaGaaeOmaaaa caGGPaGaeyypa0JaaiOvaiaacYcacaGGGcGaaiiOaiab=z6agjaacc kacaGGOaGaeyOeI0Iaaiiraiaac+cacaaIYaGaaiykaiaacckacqGH 9aqpcaGGGcGaaGimaaaa@4D5F@   (22)

By placing the relation (16) in equation (18), and then placing them in the constitutive equations (20) and (21) as well as using Gauss's law, the axial stress equations (in the local area) on the surface and volume of bulk nanotubes in the local space will be written as follows:

σ x = C 11 ϵ 0 e 31 V D ( C 11 +  e 31 2 k 33 ) 2 w( x,t ) x 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaiiaaqaaaaaaaaa Wdbiab=n8aZnaaBaaaleaacaqG4baabeaakiabg2da9iaaboeapaWa aSbaaSqaa8qacaaIXaGaaGymaaWdaeqaamrr1ngBPrwtHrhAXaqegu uDJXwAKbstHrhAG8KBLbacfaGcpeGae4x9di=damaaBaaaleaapeGa aGimaaWdaeqaaOWdbiabgkHiTiaabwgapaWaaSbaaSqaa8qacaaIZa GaaGymaaWdaeqaaOWdbmaalaaapaqaa8qacaqGwbaapaqaa8qacaqG ebaaaiabgkHiTmaabmaapaqaa8qacaqGdbWdamaaBaaaleaapeGaaG ymaiaaigdaa8aabeaak8qacqGHRaWkdaWcaaWdaeaapeGaaeiOaiaa bwgapaWaaSbaaSqaa8qacaaIZaGaaGymaaWdaeqaaOWaaWbaaSqabe aapeGaaGOmaaaaaOWdaeaapeGaae4Aa8aadaWgaaWcbaWdbiaaioda caaIZaaapaqabaaaaaGcpeGaayjkaiaawMcaaiaabQhacaqGGcWaaS aaa8aabaWdbiabgkGi2+aadaahaaWcbeqaa8qacaaIYaaaaOGaae4D amaabmaapaqaa8qacaqG4bGaaiilaiaabshaaiaawIcacaGLPaaaa8 aabaWdbiabgkGi2kaabIhapaWaaWbaaSqabeaapeGaaGOmaaaaaaaa aa@6D35@   (23)

σ x s = σ x 0 + c 11 s 0   e 31 s V D ( C 11 s + e 31 s  e 31 k 33 ) 2 w( x,t ) x 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi abgo8aZnaaDaaaleaacaqG4baabaGaae4Caaaakiabg2da9iabgo8a ZnaaDaaaleaacaqG4baabaGaaGimaaaakiabgUcaRiaabogadaqhaa WcbaGaaGymaiaaigdaaeaacaqGZbaaaGGacOGae8hcI48aaSbaaSqa aiaabcdaaeqaaOGaeyOeI0IaaiiOaiaabwgapaWaa0baaSqaa8qaca aIZaGaaGymaaWdaeaapeGaae4Caaaakmaalaaapaqaa8qacaqGwbaa paqaa8qacaqGebaaaiabgkHiTmaabmaapaqaa8qacaqGdbWdamaaDa aaleaapeGaaGymaiaaigdaa8aabaWdbiaabohaaaGccqGHRaWkdaWc aaWdaeaapeGaaeyza8aadaqhaaWcbaWdbiaaiodacaaIXaaapaqaa8 qacaqGZbaaaOGaaeiOaiaabwgapaWaaSbaaSqaa8qacaaIZaGaaGym aaWdaeqaaaGcbaWdbiaabUgapaWaaSbaaSqaa8qacaaIZaGaaG4maa WdaeqaaaaaaOWdbiaawIcacaGLPaaacaqG6bGaaeiOamaalaaapaqa a8qacqGHciITpaWaaWbaaSqabeaapeGaaGOmaaaakiaabEhadaqada WdaeaapeGaaeiEaiaacYcacaqG0baacaGLOaGaayzkaaaapaqaa8qa cqGHciITcaqG4bWdamaaCaaaleqabaWdbiaaikdaaaaaaaaa@6EAC@   (24)

Nonlocal governing differential equation of motion for equivalent continuum structure of embedded SWCNT

The equations of motion for the considered SWCNT, as shown in Figure 1, can be extracted as follows:

[( V( x,t ) )( V( x,t )+ V( x,t ) x dx )+ F medium dx+ F ext dx=( ρAdx ) 2 w( x,t ) t 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aacUfadaqadaWdaeaapeGaaeOvamaabmaapaqaa8qacaqG4bGaaiil aiaabshaaiaawIcacaGLPaaaaiaawIcacaGLPaaacqGHsisldaqada WdaeaapeGaaeOvamaabmaapaqaa8qacaqG4bGaaiilaiaabshaaiaa wIcacaGLPaaacqGHRaWkdaWcaaWdaeaapeGaeyOaIyRaaeOvamaabm aapaqaa8qacaqG4bGaaiilaiaabshaaiaawIcacaGLPaaaa8aabaWd biabgkGi2kaabIhaaaGaamizaiaadIhaaiaawIcacaGLPaaacqGHRa WkcaWGgbWdamaaBaaaleaapeGaamyBaiaadwgacaWGKbGaamyAaiaa dwhacaWGTbaapaqabaGcpeGaamizaiaadIhacqGHRaWkcaWGgbWdam aaBaaaleaapeGaamyzaiaadIhacaWG0baapaqabaGcpeGaamizaiaa dIhacqGH9aqpdaqadaWdaeaapeGaeqyWdiNaamyqaiaadsgacaWG4b aacaGLOaGaayzkaaWaaSaaa8aabaWdbiabgkGi2+aadaahaaWcbeqa a8qacaaIYaaaaOGaam4Damaabmaapaqaa8qacaWG4bGaaiilaiaads haaiaawIcacaGLPaaaa8aabaWdbiabgkGi2kaadshapaWaaWbaaSqa beaapeGaaGOmaaaaaaaaaa@7664@

V( x,t ) x + F medium + F ext =( ρA ) 2 w( x,t ) t 2 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi abgkHiTmaalaaapaqaa8qacqGHciITcaWGwbWaaeWaa8aabaWdbiaa dIhacaGGSaGaamiDaaGaayjkaiaawMcaaaWdaeaapeGaeyOaIyRaam iEaaaacqGHRaWkcaWGgbWdamaaBaaaleaapeGaamyBaiaadwgacaWG KbGaamyAaiaadwhacaWGTbaapaqabaGcpeGaey4kaSIaamOra8aada WgaaWcbaWdbiaadwgacaWG4bGaamiDaaWdaeqaaOWdbiabg2da9maa bmaapaqaa8qacqaHbpGCcaWGbbaacaGLOaGaayzkaaWaaSaaa8aaba WdbiabgkGi2+aadaahaaWcbeqaa8qacaaIYaaaaOGaam4Damaabmaa paqaa8qacaWG4bGaaiilaiaadshaaiaawIcacaGLPaaaa8aabaWdbi abgkGi2kaadshapaWaaWbaaSqabeaapeGaaGOmaaaaaaGccaGGSaaa aa@6037@   (25)

And also for the momentum components:

F v a eff w( x,t ) x +V( x,t ) M( x,t ) x =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadAeapaWaaSbaaSqaa8qacaWG2bGaamyya8aadaWgaaadbaWdbiaa dwgacaWGMbGaamOzaaWdaeqaaaWcbeaak8qadaWcaaWdaeaapeGaey OaIyRaam4Damaabmaapaqaa8qacaWG4bGaaiilaiaadshaaiaawIca caGLPaaaa8aabaWdbiabgkGi2kaadIhaaaGaey4kaSIaamOvamaabm aapaqaa8qacaWG4bGaaiilaiaadshaaiaawIcacaGLPaaacqGHsisl daWcaaWdaeaapeGaeyOaIyRaamytamaabmaapaqaa8qacaWG4bGaai ilaiaadshaaiaawIcacaGLPaaaa8aabaWdbiabgkGi2kaadIhaaaGa eyypa0JaaGimaaaa@596B@   (26)

Where F v a eff = P eff + F medium MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadAeapaWaaSbaaSqaa8qacaWG2bGaamyya8aadaWgaaadbaWdbiaa dwgacaWGMbGaamOzaaWdaeqaaaWcbeaak8qacqGH9aqpcaWGqbWdam aaBaaaleaapeGaamyzaiaadAgacaWGMbaapaqabaGcpeGaey4kaSIa amOra8aadaWgaaWcbaWdbiaad2gacaWGLbGaamizaiaadMgacaWG1b GaamyBaaWdaeqaaaaa@4AD0@ ; that P eff MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadcfapaWaaSbaaSqaa8qacaWGLbGaamOzaiaadAgaa8aabeaaaaa@3BD3@  is the equivalent axial force which includes the force induced in the system due to the application of axial strain, surface electric charge and the electric field created in the whole nanotube structure, V ( x, t ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadAfacaqGGcWaaeWaa8aabaWdbiaadIhacaGGSaGaaeiOaiaadsha aiaawIcacaGLPaaaaaa@3F52@ is the shear force at the cross section and M (x, t) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aad2eacaqGGcGaaiikaiaadIhacaGGSaGaaeiOaiaadshacaGGPaaa aa@3EFB@ is the bending moment at the cross section. Also, F medium MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadAeapaWaaSbaaSqaa8qacaWGTbGaamyzaiaadsgacaWGPbGaamyD aiaad2gaa8aabeaaaaa@3EA8@ is the force applied to the system by the viscoelastic foundation and F Lz MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadAeapaWaaSbaaSqaa8qacaWGmbGaamOEaaWdaeqaaaaa@3AD9@  is the force applied in the nanotube by the magnetic field to the nanotube per unit length of the nanotube. m is also the mass per unit nanotube length. Based on volume and surface stresses of SWCN, the resultant bending moment can be written as follows for a beam model:

M= A σ x zdA+ S σ x zdS=E I eff 2 w( x,t ) x 2 = MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aad2eacqGH9aqpdaGfWbqabSWdaeaapeGaamyqaaWdaeaaa0qaa8qa cqGHRiI8aaGccqaHdpWCpaWaaSbaaSqaa8qacaWG4baapaqabaGcpe GaamOEaiaadsgacaWGbbGaey4kaSYaaybCaeqal8aabaWdbiaadofa a8aabaaaneaapeGaey4kIipaaOGaeq4Wdm3damaaBaaaleaapeGaam iEaaWdaeqaaOWdbiaadQhacaWGKbGaam4uaiabg2da9iabgkHiTiaa dweacaWGjbWdamaaBaaaleaapeGaamyzaiaadAgacaWGMbaapaqaba GcpeWaaSaaa8aabaWdbiabgkGi2+aadaahaaWcbeqaa8qacaaIYaaa aOGaam4Damaabmaapaqaa8qacaWG4bGaaiilaiaadshaaiaawIcaca GLPaaaa8aabaWdbiabgkGi2kaadIhapaWaaWbaaSqabeaapeGaaGOm aaaaaaGccqGH9aqpaaa@60CE@

[ π D 4 64 ( E+ e 31 2 k 33 )+ π D 3 8 ( c 11 s + e 31 s e 31 k 33 ) ] 2 w( x,t ) x 2 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbm aadmaapaqaa8qadaWcaaWdaeaapeGaeqiWdaNaamira8aadaahaaWc beqaa8qacaaI0aaaaaGcpaqaa8qacaaI2aGaaGinaaaadaqadaWdae aapeGaamyraiabgUcaRmaalaaapaqaa8qacaWGLbWdamaaBaaaleaa peGaaG4maiaaigdaa8aabeaakmaaCaaaleqabaWdbiaaikdaaaaak8 aabaWdbiaadUgapaWaaSbaaSqaa8qacaaIZaGaaG4maaWdaeqaaaaa aOWdbiaawIcacaGLPaaacqGHRaWkdaWcaaWdaeaapeGaeqiWdaNaam ira8aadaahaaWcbeqaa8qacaaIZaaaaaGcpaqaa8qacaaI4aaaamaa bmaapaqaa8qacaWGJbWdamaaDaaaleaapeGaaGymaiaaigdaa8aaba WdbiaadohaaaGccqGHRaWkcaWGLbWdamaaDaaaleaapeGaaG4maiaa igdaa8aabaWdbiaadohaaaGcdaWcaaWdaeaapeGaamyza8aadaWgaa WcbaWdbiaaiodacaaIXaaapaqabaaakeaapeGaam4Aa8aadaWgaaWc baWdbiaaiodacaaIZaaapaqabaaaaaGcpeGaayjkaiaawMcaaaGaay 5waiaaw2faamaalaaapaqaa8qacqGHciITpaWaaWbaaSqabeaapeGa aGOmaaaakiaadEhadaqadaWdaeaapeGaamiEaiaacYcacaWG0baaca GLOaGaayzkaaaapaqaa8qacqGHciITcaWG4bWdamaaCaaaleqabaWd biaaikdaaaaaaOGaaiilaaaa@6B6B@   (27)

where A,S are the circumference and surface area of the circular point of the nanotube, respectively, and E I eff MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadweacaWGjbWdamaaBaaaleaapeGaamyzaiaadAgacaWGMbaapaqa baaaaa@3C96@ is the equivalent bending stiffness of the whole piezoelectric nanotube. The equivalent axial force to the nanotube due to the application of strain and electric charge on the volume and surface of the nanotube can be considered as follows:3–5

P eff = π D 2 4 ( E 0 + e 31 V D )+2( D( σ x 0 + c 11 s x )+ e 31 s V) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadcfapaWaaSbaaSqaa8qacaWGLbGaamOzaiaadAgaa8aabeaak8qa cqGH9aqpdaWcaaWdaeaapeGaeqiWdaNaamira8aadaahaaWcbeqaa8 qacaaIYaaaaaGcpaqaa8qacaaI0aaaamaabmaapaqaa8qacaWGfbac ciGae8hcI48aaSbaaSqaaiaabcdaaeqaaOGaey4kaSIaamyza8aada WgaaWcbaWdbiaaiodacaaIXaaapaqabaGcpeWaaSaaa8aabaWdbiaa dAfaa8aabaWdbiaadseaaaaacaGLOaGaayzkaaGaey4kaSIaaGOmam aabmaapaqaa8qacaWGebGaaiikaiabgo8aZnaaDaaaleaacaqG4baa baGaaGimaaaakiabgUcaRiaabogadaqhaaWcbaGaaGymaiaaigdaae aacaqGZbaaaOGae8hcI48aaSbaaSqaaiaabIhaaeqaaaGccaGLOaGa ayzkaaGaey4kaSIaaeyza8aadaqhaaWcbaWdbiaaiodacaaIXaaapa qaa8qacaqGZbaaaOGaamOvaiaacMcaaaa@61DE@   (28)

So, the equation of motion in local theory is developed as follows:

2 M( x,t ) x 2 P eff 2 w( x,t ) x 2 F medium ( ηA  H x 2 ) 2 w x 2 =( ρA ) 2 w( x,t ) t 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbm aalaaapaqaa8qacqGHciITpaWaaWbaaSqabeaapeGaaGOmaaaakiaa d2eadaqadaWdaeaapeGaamiEaiaacYcacaWG0baacaGLOaGaayzkaa aapaqaa8qacqGHciITcaWG4bWdamaaCaaaleqabaWdbiaaikdaaaaa aOGaeyOeI0Iaamiua8aadaWgaaWcbaWdbiaadwgacaWGMbGaamOzaa WdaeqaaOWdbmaalaaapaqaa8qacqGHciITpaWaaWbaaSqabeaapeGa aGOmaaaakiaadEhadaqadaWdaeaapeGaamiEaiaacYcacaWG0baaca GLOaGaayzkaaaapaqaa8qacqGHciITcaWG4bWdamaaCaaaleqabaWd biaaikdaaaaaaOGaeyOeI0IaamOra8aadaWgaaWcbaWdbiaad2gaca WGLbGaamizaiaadMgacaWG1bGaamyBaaWdaeqaaOWdbiabgkHiTmaa bmaapaqaa8qacqaH3oaAcaWGbbGaaeiOaiaadIeapaWaaSbaaSqaa8 qacaWG4baapaqabaGcdaahaaWcbeqaa8qacaaIYaaaaaGccaGLOaGa ayzkaaWaaSaaa8aabaWdbiabgkGi2+aadaahaaWcbeqaa8qacaaIYa aaaOGaam4DaaWdaeaapeGaeyOaIyRaamiEa8aadaahaaWcbeqaa8qa caaIYaaaaaaakiabg2da9iabgkHiTmaabmaapaqaa8qacqaHbpGCca WGbbaacaGLOaGaayzkaaWaaSaaa8aabaWdbiabgkGi2+aadaahaaWc beqaa8qacaaIYaaaaOGaam4Damaabmaapaqaa8qacaWG4bGaaiilai aadshaaiaawIcacaGLPaaaa8aabaWdbiabgkGi2kaadshapaWaaWba aSqabeaapeGaaGOmaaaaaaaaaa@7DAF@   (29)

To apply the non-local effects, the equation (5) is used. First, equation (5) is rewritten as follows:

2 M( x,t ) x 2 = 2 x 2 ( 1 ( τl ) 2 2 x 2 ) M nl ( x,t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbm aalaaapaqaa8qacqGHciITpaWaaWbaaSqabeaapeGaaGOmaaaakiaa d2eadaqadaWdaeaapeGaamiEaiaacYcacaWG0baacaGLOaGaayzkaa aapaqaa8qacqGHciITcaWG4bWdamaaCaaaleqabaWdbiaaikdaaaaa aOGaeyypa0ZaaSaaa8aabaWdbiabgkGi2+aadaahaaWcbeqaa8qaca aIYaaaaaGcpaqaa8qacqGHciITcaWG4bWdamaaCaaaleqabaWdbiaa ikdaaaaaaOWaaeWaa8aabaWdbiaaigdacqGHsisldaqadaWdaeaape GaeqiXdqNaamiBaaGaayjkaiaawMcaa8aadaahaaWcbeqaa8qacaaI YaaaaOWaaSaaa8aabaWdbiabgkGi2+aadaahaaWcbeqaa8qacaaIYa aaaaGcpaqaa8qacqGHciITcaWG4bWdamaaCaaaleqabaWdbiaaikda aaaaaaGccaGLOaGaayzkaaGaamyta8aadaahaaWcbeqaa8qacaWGUb GaamiBaaaakmaabmaapaqaa8qacaWG4bGaaiilaiaadshaaiaawIca caGLPaaaaaa@60D5@   (30)

So:

2 M( x,t ) x 2 = ( τl ) 2 4 M nl ( x,t ) x 4 + 2 M nl ( x,t ) x 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbm aalaaapaqaa8qacqGHciITpaWaaWbaaSqabeaapeGaaGOmaaaakiaa d2eadaqadaWdaeaapeGaamiEaiaacYcacaWG0baacaGLOaGaayzkaa aapaqaa8qacqGHciITcaWG4bWdamaaCaaaleqabaWdbiaaikdaaaaa aOGaeyypa0JaeyOeI0YaaeWaa8aabaWdbiabes8a0jaadYgaaiaawI cacaGLPaaapaWaaWbaaSqabeaapeGaaGOmaaaakmaalaaapaqaa8qa cqGHciITpaWaaWbaaSqabeaapeGaaGinaaaakiaad2eapaWaaWbaaS qabeaapeGaamOBaiaadYgaaaGcdaqadaWdaeaapeGaamiEaiaacYca caWG0baacaGLOaGaayzkaaaapaqaa8qacqGHciITcaWG4bWdamaaCa aaleqabaWdbiaaisdaaaaaaOGaey4kaSYaaSaaa8aabaWdbiabgkGi 2+aadaahaaWcbeqaa8qacaaIYaaaaOGaamyta8aadaahaaWcbeqaa8 qacaWGUbGaamiBaaaakmaabmaapaqaa8qacaWG4bGaaiilaiaadsha aiaawIcacaGLPaaaa8aabaWdbiabgkGi2kaadIhapaWaaWbaaSqabe aapeGaaGOmaaaaaaaaaa@66A8@   (31)

Therefore, equation (29) can be rewritten as follows:

( τl ) 2 4 M nl ( x,t ) x 4 + 2 M nl ( x,t ) x 2 P eff 2 w( x,t ) x 2 {[ ( k 1 w( x,t )+( k 3 w 3 ( x,t ) ) ]+ ( c 0 + c 2 w 2 ( x,t ) ) w( x,t ) t )}( ηA  H x 2 ) 2 w x 2 =( ρA ) 2 w( x,t ) t 2 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakqaabeqaaabaaaaaaa aapeGaeyOeI0YaaeWaa8aabaWdbiabes8a0jaadYgaaiaawIcacaGL PaaapaWaaWbaaSqabeaapeGaaGOmaaaakmaalaaapaqaa8qacqGHci ITpaWaaWbaaSqabeaapeGaaGinaaaakiaad2eapaWaaWbaaSqabeaa peGaamOBaiaadYgaaaGcdaqadaWdaeaapeGaamiEaiaacYcacaWG0b aacaGLOaGaayzkaaaapaqaa8qacqGHciITcaWG4bWdamaaCaaaleqa baWdbiaaisdaaaaaaOGaey4kaSYaaSaaa8aabaWdbiabgkGi2+aada ahaaWcbeqaa8qacaaIYaaaaOGaamyta8aadaahaaWcbeqaa8qacaWG UbGaamiBaaaakmaabmaapaqaa8qacaWG4bGaaiilaiaadshaaiaawI cacaGLPaaaa8aabaWdbiabgkGi2kaadIhapaWaaWbaaSqabeaapeGa aGOmaaaaaaGccqGHsislcaWGqbWdamaaBaaaleaapeGaamyzaiaadA gacaWGMbaapaqabaGcpeWaaSaaa8aabaWdbiabgkGi2+aadaahaaWc beqaa8qacaaIYaaaaOGaam4Damaabmaapaqaa8qacaWG4bGaaiilai aadshaaiaawIcacaGLPaaaa8aabaWdbiabgkGi2kaadIhapaWaaWba aSqabeaapeGaaGOmaaaaaaGccqGHsislcaGG7bWaamWaa8aabaWdbm aabmaapaqaa8qacaWGRbWdamaaBaaaleaapeGaaGymaaWdaeqaaOWd biaadEhadaqadaWdaeaapeGaamiEaiaacYcacaWG0baacaGLOaGaay zkaaGaey4kaSIaaiikaiaadUgapaWaaSbaaSqaa8qacaaIZaaapaqa baGcpeGaam4Da8aadaahaaWcbeqaa8qacaaIZaaaaOWaaeWaa8aaba WdbiaadIhacaGGSaGaamiDaaGaayjkaiaawMcaaaGaayjkaiaawMca aaGaay5waiaaw2faaiabgUcaRaqaamaabmaapaqaa8qacaWGJbWdam aaBaaaleaapeGaaGimaaWdaeqaaOWdbiabgUcaRiaadogapaWaaSba aSqaa8qacaaIYaaapaqabaGcpeGaam4Da8aadaahaaWcbeqaa8qaca aIYaaaaOWaaeWaa8aabaWdbiaadIhacaGGSaGaamiDaaGaayjkaiaa wMcaaaGaayjkaiaawMcaamaalaaapaqaa8qacqGHciITcaWG3bWaae Waa8aabaWdbiaadIhacaGGSaGaamiDaaGaayjkaiaawMcaaaWdaeaa peGaeyOaIyRaamiDaaaacaGGPaGaaiyFaiabgkHiTmaabmaapaqaa8 qacqaH3oaAcaWGbbGaaeiOaiaadIeapaWaaSbaaSqaa8qacaWG4baa paqabaGcdaahaaWcbeqaa8qacaaIYaaaaaGccaGLOaGaayzkaaWaaS aaa8aabaWdbiabgkGi2+aadaahaaWcbeqaa8qacaaIYaaaaOGaam4D aaWdaeaapeGaeyOaIyRaamiEa8aadaahaaWcbeqaa8qacaaIYaaaaa aakiabg2da9iabgkHiTmaabmaapaqaa8qacqaHbpGCcaWGbbaacaGL OaGaayzkaaWaaSaaa8aabaWdbiabgkGi2+aadaahaaWcbeqaa8qaca aIYaaaaOGaam4Damaabmaapaqaa8qacaWG4bGaaiilaiaadshaaiaa wIcacaGLPaaaa8aabaWdbiabgkGi2kaadshapaWaaWbaaSqabeaape GaaGOmaaaaaaGccaGGSaaaaaa@BDD2@   (32)

Finally, based on the equations (25) and (26), and for a flexural moment and cross-sectional shear force in non-local theory space, the differential equation of motion of the piezoelectric single-walled carbon nanotube considering non-local effects is developed as follows:

E I eff 4 w x 4 +[ ( k 1 w+ k 3 w 3 )+( c 0 + c 2 w 2 ) w t ] ( ηA  H x 2 ) 2 w x 2 P eff 2 w x 2 +( ρA ) 2 w t 2 F ext ( τl ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakqaabeqaaabaaaaaaa aapeGaamyraiaadMeapaWaaSbaaSqaa8qacaWGLbGaamOzaiaadAga a8aabeaak8qadaWcaaWdaeaapeGaeyOaIy7damaaCaaaleqabaWdbi aaisdaaaGccaWG3baapaqaa8qacqGHciITcaWG4bWdamaaCaaaleqa baWdbiaaisdaaaaaaOGaey4kaSYaamWaa8aabaWdbmaabmaapaqaa8 qacaWGRbWdamaaBaaaleaapeGaaGymaaWdaeqaaOWdbiaadEhacqGH RaWkcaWGRbWdamaaBaaaleaapeGaaG4maaWdaeqaaOWdbiaadEhapa WaaWbaaSqabeaapeGaaG4maaaaaOGaayjkaiaawMcaaiabgUcaRmaa bmaapaqaa8qacaWGJbWdamaaBaaaleaapeGaaGimaaWdaeqaaOWdbi abgUcaRiaadogapaWaaSbaaSqaa8qacaaIYaaapaqabaGcpeGaam4D a8aadaahaaWcbeqaa8qacaaIYaaaaaGccaGLOaGaayzkaaWaaSaaa8 aabaWdbiabgkGi2kaadEhaa8aabaWdbiabgkGi2kaadshaaaaacaGL BbGaayzxaaGaeyOeI0cabaWaaeWaa8aabaWdbiabeE7aOjaadgeaca qGGcGaamisa8aadaWgaaWcbaWdbiaadIhaa8aabeaakmaaCaaaleqa baWdbiaaikdaaaaakiaawIcacaGLPaaadaWcaaWdaeaapeGaeyOaIy 7damaaCaaaleqabaWdbiaaikdaaaGccaWG3baapaqaa8qacqGHciIT caWG4bWdamaaCaaaleqabaWdbiaaikdaaaaaaOGaeyOeI0Iaamiua8 aadaWgaaWcbaWdbiaadwgacaWGMbGaamOzaaWdaeqaaOWdbmaalaaa paqaa8qacqGHciITpaWaaWbaaSqabeaapeGaaGOmaaaakiaadEhaa8 aabaWdbiabgkGi2kaadIhapaWaaWbaaSqabeaapeGaaGOmaaaaaaGc cqGHRaWkdaqadaWdaeaapeGaeqyWdiNaamyqaaGaayjkaiaawMcaam aalaaapaqaa8qacqGHciITpaWaaWbaaSqabeaapeGaaGOmaaaakiaa dEhaa8aabaWdbiabgkGi2kaadshapaWaaWbaaSqabeaapeGaaGOmaa aaaaGccqGHsislcaWGgbWdamaaBaaaleaapeGaamyzaiaadIhacaWG 0baapaqabaGcpeGaeyOeI0YaaeWaa8aabaWdbiabes8a0jaadYgaai aawIcacaGLPaaapaWaaWbaaSqabeaapeGaaGOmaaaaaaaa@93D2@
{[ ( k 1 2 w x 2 + k 3 2 x 2 w 3 )+( c 0 3 w x 2 t + c 2 2 x 2 ( w t w 2 ) ) ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aacUhadaWadaWdaeaapeWaaeWaa8aabaWdbiaadUgapaWaaSbaaSqa a8qacaaIXaaapaqabaGcpeWaaSaaa8aabaWdbiabgkGi2+aadaahaa Wcbeqaa8qacaaIYaaaaOGaam4DaaWdaeaapeGaeyOaIyRaamiEa8aa daahaaWcbeqaa8qacaaIYaaaaaaakiabgUcaRiaadUgapaWaaSbaaS qaa8qacaaIZaaapaqabaGcpeWaaSaaa8aabaWdbiabgkGi2+aadaah aaWcbeqaa8qacaaIYaaaaaGcpaqaa8qacqGHciITcaWG4bWdamaaCa aaleqabaWdbiaaikdaaaaaaOGaam4Da8aadaahaaWcbeqaa8qacaaI ZaaaaaGccaGLOaGaayzkaaGaey4kaSYaaeWaa8aabaWdbiaadogapa WaaSbaaSqaa8qacaaIWaaapaqabaGcpeWaaSaaa8aabaWdbiabgkGi 2+aadaahaaWcbeqaa8qacaaIZaaaaOGaam4DaaWdaeaapeGaeyOaIy RaamiEa8aadaahaaWcbeqaa8qacaaIYaaaaOGaeyOaIyRaamiDaaaa cqGHRaWkcaWGJbWdamaaBaaaleaapeGaaGOmaaWdaeqaaOWdbmaala aapaqaa8qacqGHciITpaWaaWbaaSqabeaapeGaaGOmaaaaaOWdaeaa peGaeyOaIyRaamiEa8aadaahaaWcbeqaa8qacaaIYaaaaaaakmaabm aapaqaa8qadaWcaaWdaeaapeGaeyOaIyRaam4DaaWdaeaapeGaeyOa IyRaamiDaaaacaWG3bWdamaaCaaaleqabaWdbiaaikdaaaaakiaawI cacaGLPaaaaiaawIcacaGLPaaaaiaawUfacaGLDbaacqGHsislaaa@725D@
( ηA  H x 2 ) 4 w x 4 P eff 4 w x 4 +( ρA ) 4 w x 2 t 2 2 x 2 F ext }=0, MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbm aabmaapaqaa8qacqaH3oaAcaWGbbGaaeiOaiaadIeapaWaaSbaaSqa a8qacaWG4baapaqabaGcdaahaaWcbeqaa8qacaaIYaaaaaGccaGLOa GaayzkaaWaaSaaa8aabaWdbiabgkGi2+aadaahaaWcbeqaa8qacaaI 0aaaaOGaam4DaaWdaeaapeGaeyOaIyRaamiEa8aadaahaaWcbeqaa8 qacaaI0aaaaaaakiabgkHiTiaadcfapaWaaSbaaSqaa8qacaWGLbGa amOzaiaadAgaa8aabeaak8qadaWcaaWdaeaapeGaeyOaIy7damaaCa aaleqabaWdbiaaisdaaaGccaWG3baapaqaa8qacqGHciITcaWG4bWd amaaCaaaleqabaWdbiaaisdaaaaaaOGaey4kaSYaaeWaa8aabaWdbi abeg8aYjaadgeaaiaawIcacaGLPaaadaWcaaWdaeaapeGaeyOaIy7d amaaCaaaleqabaWdbiaaisdaaaGccaWG3baapaqaa8qacqGHciITca WG4bWdamaaCaaaleqabaWdbiaaikdaaaGccqGHciITcaWG0bWdamaa CaaaleqabaWdbiaaikdaaaaaaOGaeyOeI0YaaSaaa8aabaWdbiabgk Gi2+aadaahaaWcbeqaa8qacaaIYaaaaaGcpaqaa8qacqGHciITcaWG 4bWdamaaCaaaleqabaWdbiaaikdaaaaaaOGaamOra8aadaWgaaWcba WdbiaadwgacaWG4bGaamiDaaWdaeqaaOGaaiyFaiabg2da9iaaicda caGGSaaaaa@7220@   (33)

The boundary conditions of nanotubes are also expressed according to the following relations, as shown in figures 3:

For both side simple support:

x=0,L; w=0, 2 w x 2 =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadIhacqGH9aqpcaaIWaGaaiilaiaadYeacaGG7aGaaeiOaiaadEha cqGH9aqpcaaIWaGaaiilamaalaaapaqaa8qacqGHciITpaWaaWbaaS qabeaapeGaaGOmaaaakiaadEhaa8aabaWdbiabgkGi2kaadIhapaWa aWbaaSqabeaapeGaaGOmaaaaaaGccqGH9aqpcaaIWaaaaa@4A67@   (34)

For both side clamp support:

x = 0 , L ;   w = 0 , w x = 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadIhacqGH9aqpcaaIWaGaaiilaiaadYeacaGG7aGaaeiOaiaadEha cqGH9aqpcaaIWaGaaiilamaalaaapaqaa8qacqGHciITcaWG3baapa qaa8qacqGHciITcaWG4baaaiabg2da9iaaicdaaaa@4843@   (35)

Next, the equation (33) is investigated by nonlinear equation solving methods, and an analytical solution for nonlinear free and force vibration responses are extracted extracted.

Figure 3 Boundary conditions of the system:
(a) S-S boundary condition
(b) C-C boundary condition.

Analytical solution

Reducing the order of the equations of motion using Galerkin method

The Galerkin method is used to reduce the differential equations obtained. Therefore, the general form of vibrational response is considered as follows: 

w( x,t )= i=1 i=N φ i ( x ) q i ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadEhadaqadaWdaeaapeGaamiEaiaacYcacaWG0baacaGLOaGaayzk aaGaeyypa0ZaaybCaeqal8aabaWdbiaadMgacqGH9aqpcaaIXaaapa qaa8qacaWGPbGaeyypa0JaamOtaaqdpaqaa8qacqGHris5aaGccqaH gpGApaWaaSbaaSqaa8qacaWGPbaapaqabaGcpeWaaeWaa8aabaWdbi aadIhaaiaawIcacaGLPaaacaWGXbWdamaaBaaaleaapeGaamyAaaWd aeqaaOWdbmaabmaapaqaa8qacaWG0baacaGLOaGaayzkaaaaaa@5118@   (36)

Where φ i ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi abeA8aQ9aadaWgaaWcbaWdbiaadMgaa8aabeaak8qadaqadaWdaeaa peGaamiEaaGaayjkaiaawMcaaaaa@3DA8@  is a mode shape function that satisfies the geometric boundary conditions, and q i ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadghapaWaaSbaaSqaa8qacaWGPbaapaqabaGcpeWaaeWaa8aabaWd biaadshaaiaawIcacaGLPaaaaaa@3CDD@  is the generalized time coordinates of the system.

By placing equation (36) in equation (33), and applying the Galerkin method, equation (37) is extracted:

[ m ¯ d 2 q i d t 2 + β 0 d q i dt + α 1 q i ]+[ β 2 q i 2 d q i dt + α 3 q i 3 ]= f ¯ e iΩt MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbm aadmaapaqaa8qaceWGTbWdayaaraWdbmaalaaapaqaa8qacaWGKbWd amaaCaaaleqabaWdbiaaikdaaaGccaWGXbWdamaaBaaaleaapeGaam yAaaWdaeqaaaGcbaWdbiaadsgacaWG0bWdamaaCaaaleqabaWdbiaa ikdaaaaaaOGaey4kaSIaeqOSdi2damaaBaaaleaapeGaaGimaaWdae qaaOWdbmaalaaapaqaa8qacaWGKbGaamyCa8aadaWgaaWcbaWdbiaa dMgaa8aabeaaaOqaa8qacaWGKbGaamiDaaaacqGHRaWkcqaHXoqypa WaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaamyCa8aadaWgaaWcbaWd biaadMgaa8aabeaaaOWdbiaawUfacaGLDbaacqGHRaWkdaWadaWdae aapeGaeqOSdi2damaaBaaaleaapeGaaGOmaaWdaeqaaOWdbiaadgha paWaaSbaaSqaa8qacaWGPbaapaqabaGcdaahaaWcbeqaa8qacaaIYa aaaOWaaSaaa8aabaWdbiaadsgacaWGXbWdamaaBaaaleaapeGaamyA aaWdaeqaaaGcbaWdbiaadsgacaWG0baaaiabgUcaRiabeg7aH9aada WgaaWcbaWdbiaaiodaa8aabeaak8qacaWGXbWdamaaBaaaleaapeGa amyAaaWdaeqaaOWaaWbaaSqabeaapeGaaG4maaaaaOGaay5waiaaw2 faaiabg2da9iqadAgapaGbaebapeGaamyza8aadaahaaWcbeqaa8qa caWGPbGaeuyQdCLaamiDaaaaaaa@6DD1@   (37)

Where:

α 1 =E I eff 0 L φ i φ i ( 4 ) dx+ k 1 0 L φ i 2 dx( F vertica l total ) 0 L φ i φ i ( 2 ) dxμ k 1 0 L φ i φ i ( 2 ) dx+μ( F vertica l total ) 0 L φ i φ i ( 4 ) dx MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakqaabeqaaabaaaaaaa aapeGaeqySde2damaaBaaaleaapeGaaGymaaWdaeqaaOWdbiabg2da 9iaadweacaWGjbWdamaaBaaaleaapeGaamyzaiaadAgacaWGMbaapa qabaGcpeWaaybCaeqal8aabaWdbiaaicdaa8aabaWdbiaadYeaa0Wd aeaapeGaey4kIipaaOGaeqOXdO2damaaBaaaleaapeGaamyAaaWdae qaaOWdbiabeA8aQ9aadaWgaaWcbaWdbiaadMgaa8aabeaakmaaCaaa leqabaWdbmaabmaapaqaa8qacaaI0aaacaGLOaGaayzkaaaaaOGaam izaiaadIhacqGHRaWkcaWGRbWdamaaBaaaleaapeGaaGymaaWdaeqa aOWdbmaawahabeWcpaqaa8qacaaIWaaapaqaa8qacaWGmbaan8aaba WdbiabgUIiYdaakiabeA8aQ9aadaWgaaWcbaWdbiaadMgaa8aabeaa kmaaCaaaleqabaWdbiaaikdaaaGccaWGKbGaamiEaiabgkHiTmaabm aapaqaa8qacaWGgbWdamaaBaaaleaapeGaamODaiaadwgacaWGYbGa amiDaiaadMgacaWGJbGaamyyaiaadYgapaWaaSbaaWqaa8qacaWG0b Gaam4BaiaadshacaWGHbGaamiBaaWdaeqaaaWcbeaaaOWdbiaawIca caGLPaaaaeaadaGfWbqabSWdaeaapeGaaGimaaWdaeaapeGaamitaa qdpaqaa8qacqGHRiI8aaGccqaHgpGApaWaaSbaaSqaa8qacaWGPbaa paqabaGcpeGaeqOXdO2damaaBaaaleaapeGaamyAaaWdaeqaaOWaaW baaSqabeaapeWaaeWaa8aabaWdbiaaikdaaiaawIcacaGLPaaaaaGc caWGKbGaamiEaiabgkHiTiabeY7aTjaadUgapaWaaSbaaSqaa8qaca aIXaaapaqabaGcpeWaaybCaeqal8aabaWdbiaaicdaa8aabaWdbiaa dYeaa0WdaeaapeGaey4kIipaaOGaeqOXdO2damaaBaaaleaapeGaam yAaaWdaeqaaOWdbiabeA8aQ9aadaWgaaWcbaWdbiaadMgaa8aabeaa kmaaCaaaleqabaWdbmaabmaapaqaa8qacaaIYaaacaGLOaGaayzkaa aaaOGaamizaiaadIhacqGHRaWkcqaH8oqBdaqadaWdaeaapeGaamOr a8aadaWgaaWcbaWdbiaadAhacaWGLbGaamOCaiaadshacaWGPbGaam 4yaiaadggacaWGSbWdamaaBaaameaapeGaamiDaiaad+gacaWG0bGa amyyaiaadYgaa8aabeaaaSqabaaak8qacaGLOaGaayzkaaWaaybCae qal8aabaWdbiaaicdaa8aabaWdbiaadYeaa0WdaeaapeGaey4kIipa aOGaeqOXdO2damaaBaaaleaapeGaamyAaaWdaeqaaOWdbiabeA8aQ9 aadaWgaaWcbaWdbiaadMgaa8aabeaakmaaCaaaleqabaWdbmaabmaa paqaa8qacaaI0aaacaGLOaGaayzkaaaaaOGaamizaiaadIhaaaaa@B1B3@

α 3 = k 3 0 L φ i 4 dxμ k 3 ( 6 0 L φ i ' 2 φ i 2 dx+3 0 L φ i 3 φ i ( 2 ) dx ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi abeg7aH9aadaWgaaWcbaWdbiaaiodaa8aabeaak8qacqGH9aqpcaWG RbWdamaaBaaaleaapeGaaG4maaWdaeqaaOWdbmaawahabeWcpaqaa8 qacaaIWaaapaqaa8qacaWGmbaan8aabaWdbiabgUIiYdaakiabeA8a Q9aadaWgaaWcbaWdbiaadMgaa8aabeaakmaaCaaaleqabaWdbiaais daaaGccaWGKbGaamiEaiabgkHiTiabeY7aTjaadUgapaWaaSbaaSqa a8qacaaIZaaapaqabaGcpeWaaeWaa8aabaWdbiaaiAdadaGfWbqabS WdaeaapeGaaGimaaWdaeaapeGaamitaaqdpaqaa8qacqGHRiI8aaGc cqaHgpGApaWaaSbaaSqaa8qacaWGPbaapaqabaGcdaahaaWcbeqaa8 qacaqGNaaaaOWdamaaCaaaleqabaWdbiaaikdaaaGccqaHgpGApaWa aSbaaSqaa8qacaWGPbaapaqabaGcdaahaaWcbeqaa8qacaaIYaaaaO GaamizaiaadIhacqGHRaWkcaaIZaWaaybCaeqal8aabaWdbiaaicda a8aabaWdbiaadYeaa0WdaeaapeGaey4kIipaaOGaeqOXdO2damaaBa aaleaapeGaamyAaaWdaeqaaOWaaWbaaSqabeaapeGaaG4maaaakiab eA8aQ9aadaWgaaWcbaWdbiaadMgaa8aabeaakmaaCaaaleqabaWdbm aabmaapaqaa8qacaaIYaaacaGLOaGaayzkaaaaaOGaamizaiaadIha aiaawIcacaGLPaaaaaa@7086@

β 0 = c 0 0 L φ i 2 dxμ c 0 0 L φ i φ i ( 2 ) dx MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi abek7aI9aadaWgaaWcbaWdbiaaicdaa8aabeaak8qacqGH9aqpcaWG JbWdamaaBaaaleaapeGaaGimaaWdaeqaaOWdbmaawahabeWcpaqaa8 qacaaIWaaapaqaa8qacaWGmbaan8aabaWdbiabgUIiYdaakiabeA8a Q9aadaWgaaWcbaWdbiaadMgaa8aabeaakmaaCaaaleqabaWdbiaaik daaaGccaWGKbGaamiEaiabgkHiTiabeY7aTjaadogapaWaaSbaaSqa a8qacaaIWaaapaqabaGcpeWaaybCaeqal8aabaWdbiaaicdaa8aaba WdbiaadYeaa0WdaeaapeGaey4kIipaaOGaeqOXdO2damaaBaaaleaa peGaamyAaaWdaeqaaOWdbiabeA8aQ9aadaWgaaWcbaWdbiaadMgaa8 aabeaakmaaCaaaleqabaWdbmaabmaapaqaa8qacaaIYaaacaGLOaGa ayzkaaaaaOGaamizaiaadIhaaaa@5C05@

β 2 = c 2 0 L φ i 3 dxμ c 2 ( 6 0 L φ i ' 2 φ i 2 dx+3 0 L φ i 3 φ i ( 2 ) dx ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi abek7aI9aadaWgaaWcbaWdbiaaikdaa8aabeaak8qacqGH9aqpcaWG JbWdamaaBaaaleaapeGaaGOmaaWdaeqaaOWdbmaawahabeWcpaqaa8 qacaaIWaaapaqaa8qacaWGmbaan8aabaWdbiabgUIiYdaakiabeA8a Q9aadaWgaaWcbaWdbiaadMgaa8aabeaakmaaCaaaleqabaWdbiaaio daaaGccaWGKbGaamiEaiabgkHiTiabeY7aTjaadogapaWaaSbaaSqa a8qacaaIYaaapaqabaGcpeWaaeWaa8aabaWdbiaaiAdadaGfWbqabS WdaeaapeGaaGimaaWdaeaapeGaamitaaqdpaqaa8qacqGHRiI8aaGc cqaHgpGApaWaaSbaaSqaa8qacaWGPbaapaqabaGcdaahaaWcbeqaa8 qacaqGNaaaaOWdamaaCaaaleqabaWdbiaaikdaaaGccqaHgpGApaWa aSbaaSqaa8qacaWGPbaapaqabaGcdaahaaWcbeqaa8qacaaIYaaaaO GaamizaiaadIhacqGHRaWkcaaIZaWaaybCaeqal8aabaWdbiaaicda a8aabaWdbiaadYeaa0WdaeaapeGaey4kIipaaOGaeqOXdO2damaaBa aaleaapeGaamyAaaWdaeqaaOWaaWbaaSqabeaapeGaaG4maaaakiab eA8aQ9aadaWgaaWcbaWdbiaadMgaa8aabeaakmaaCaaaleqabaWdbm aabmaapaqaa8qacaaIYaaacaGLOaGaayzkaaaaaOGaamizaiaadIha aiaawIcacaGLPaaaaaa@7074@

m ¯ =( ρA ) 0 L φ i 2 dxμ( ρA ) 0 L φ i φ i ( 2 ) dx MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi qad2gapaGbaebapeGaeyypa0ZaaeWaa8aabaWdbiabeg8aYjaadgea aiaawIcacaGLPaaadaGfWbqabSWdaeaapeGaaGimaaWdaeaapeGaam itaaqdpaqaa8qacqGHRiI8aaGccqaHgpGApaWaaSbaaSqaa8qacaWG PbaapaqabaGcdaahaaWcbeqaa8qacaaIYaaaaOGaamizaiaadIhacq GHsislcqaH8oqBdaqadaWdaeaapeGaeqyWdiNaamyqaaGaayjkaiaa wMcaamaawahabeWcpaqaa8qacaaIWaaapaqaa8qacaWGmbaan8aaba WdbiabgUIiYdaakiabeA8aQ9aadaWgaaWcbaWdbiaadMgaa8aabeaa k8qacqaHgpGApaWaaSbaaSqaa8qacaWGPbaapaqabaGcdaahaaWcbe qaa8qadaqadaWdaeaapeGaaGOmaaGaayjkaiaawMcaaaaakiaadsga caWG4baaaa@5E8F@ ,  (38)

Solving nonlinear differential equation (the multiple time scales method)

The multiple time scales method, as a perturbation method, is known as an efficient technique in nonlinear differential equations analysis. In this part, the multiple time scales method is used to develop the nonlinear vibrational response of the system.

  1. Free vibration analysis

To be able to apply this method, equation (37) is rewritten in the following form:

[ d 2 q i d t 2 +ε( β 0 + β 2 q i 2 ) d q i dt + ω n 2 q i ]+[ ε( α 3 ) q i 3 ]=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbm aadmaapaqaa8qadaWcaaWdaeaapeGaamiza8aadaahaaWcbeqaa8qa caaIYaaaaOGaamyCa8aadaWgaaWcbaWdbiaadMgaa8aabeaaaOqaa8 qacaWGKbGaamiDa8aadaahaaWcbeqaa8qacaaIYaaaaaaakiabgUca Riabew7aLnaabmaapaqaa8qacqaHYoGypaWaaSbaaSqaa8qacaaIWa aapaqabaGcpeGaey4kaSIaeqOSdi2damaaBaaaleaapeGaaGOmaaWd aeqaaOWdbiaadghapaWaaSbaaSqaa8qacaWGPbaapaqabaGcdaahaa Wcbeqaa8qacaaIYaaaaaGccaGLOaGaayzkaaWaaSaaa8aabaWdbiaa dsgacaWGXbWdamaaBaaaleaapeGaamyAaaWdaeqaaaGcbaWdbiaads gacaWG0baaaiabgUcaRiabeM8a39aadaqhaaWcbaWdbiaad6gaa8aa baWdbiaaikdaaaGccaWGXbWdamaaBaaaleaapeGaamyAaaWdaeqaaa GcpeGaay5waiaaw2faaiabgUcaRmaadmaapaqaa8qacqaH1oqzdaqa daWdaeaapeGaeqySde2damaaBaaaleaapeGaaG4maaWdaeqaaaGcpe GaayjkaiaawMcaaiaadghapaWaaSbaaSqaa8qacaWGPbaapaqabaGc daahaaWcbeqaa8qacaaIZaaaaaGccaGLBbGaayzxaaGaeyypa0JaaG imaaaa@6A04@   (39)

In equation (39), ε MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi abew7aLbaa@398A@   ε MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi abew7aLbaa@398A@  is a small and dimensionless parameter. In the multiple time scales method, we first define the following parameters and equations:

T n =   n t ,  T n = D n  , n=0,1,2, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadsfapaWaaSbaaSqaa8qacaWGUbaapaqabaGcpeGaeyypa0JaaiiO aiaacckaiiGacqWFiiIZdaahaaWcbeqaaiaad6gaaaGccaWG0bGaae iOaiaacYcacaqGGcWaaSaaa8aabaWdbiabgkGi2cWdaeaapeGaeyOa IyRaamiva8aadaWgaaWcbaWdbiaad6gaa8aabeaaaaGcpeGaeyypa0 Jaamira8aadaWgaaWcbaWdbiaad6gaa8aabeaak8qacaqGGcGaaiil aiaabckacaWGUbGaeyypa0JaaGimaiaacYcacaaIXaGaaiilaiaaik dacaGGSaGaeyOjGWlaaa@5761@

d dt =( D 0 +ε D 1 + ε 2 D 2 + ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbm aalaaapaqaa8qacaWGKbaapaqaa8qacaWGKbGaamiDaaaacqGH9aqp daqadaWdaeaapeGaamira8aadaWgaaWcbaWdbiaaicdaa8aabeaak8 qacqGHRaWkcqaH1oqzcaWGebWdamaaBaaaleaapeGaaGymaaWdaeqa aOWdbiabgUcaRiabew7aL9aadaahaaWcbeqaa8qacaaIYaaaaOGaam ira8aadaWgaaWcbaWdbiaaikdaa8aabeaak8qacqGHRaWkcqGHMacV aiaawIcacaGLPaaaaaa@4C27@

d 2 d t 2 =( D 0 2 +2ε D 0 D 1 + ε 2 D 1 2 + ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbm aalaaapaqaa8qacaWGKbWdamaaCaaaleqabaWdbiaaikdaaaaak8aa baWdbiaadsgacaWG0bWdamaaCaaaleqabaWdbiaaikdaaaaaaOGaey ypa0ZaaeWaa8aabaWdbiaadseapaWaaSbaaSqaa8qacaaIWaaapaqa baGcdaahaaWcbeqaa8qacaaIYaaaaOGaey4kaSIaaGOmaiabew7aLj aadseapaWaaSbaaSqaa8qacaaIWaaapaqabaGcpeGaamira8aadaWg aaWcbaWdbiaaigdaa8aabeaak8qacqGHRaWkcqaH1oqzpaWaaWbaaS qabeaapeGaaGOmaaaakiaadseapaWaaSbaaSqaa8qacaaIXaaapaqa baGcdaahaaWcbeqaa8qacaaIYaaaaOGaey4kaSIaeyOjGWlacaGLOa Gaayzkaaaaaa@52E3@   (40)

The general form of the response is also expressed as follows:

q( t )= q 0 ( T 0 , T 1 , T 2 , )+ε q 1 ( T 0 , T 1 , T 2 , )+ ε 2 q 2 ( T 0 , T 1 , T 2 , ) + =  q 0 ( T 0 , T 1 , )+ε q 1 ( T 0 , T 1 , )+Ο( ε 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakqaabeqaaabaaaaaaa aapeGaamyCamaabmaapaqaa8qacaWG0baacaGLOaGaayzkaaGaeyyp a0JaamyCa8aadaWgaaWcbaWdbiaaicdaa8aabeaak8qadaqadaWdae aapeGaamiva8aadaWgaaWcbaWdbiaaicdaa8aabeaak8qacaGGSaGa amiva8aadaWgaaWcbaWdbiaaigdaa8aabeaak8qacaGGSaGaamiva8 aadaWgaaWcbaWdbiaaikdaa8aabeaak8qacaGGSaGaeyOjGWlacaGL OaGaayzkaaGaey4kaSIaeqyTduMaamyCa8aadaWgaaWcbaWdbiaaig daa8aabeaak8qadaqadaWdaeaapeGaamiva8aadaWgaaWcbaWdbiaa icdaa8aabeaak8qacaGGSaGaamiva8aadaWgaaWcbaWdbiaaigdaa8 aabeaak8qacaGGSaGaamiva8aadaWgaaWcbaWdbiaaikdaa8aabeaa k8qacaGGSaGaeyOjGWlacaGLOaGaayzkaaGaey4kaSIaeqyTdu2dam aaCaaaleqabaWdbiaaikdaaaGccaWGXbWdamaaBaaaleaapeGaaGOm aaWdaeqaaOWdbmaabmaapaqaa8qacaWGubWdamaaBaaaleaapeGaaG imaaWdaeqaaOWdbiaacYcacaWGubWdamaaBaaaleaapeGaaGymaaWd aeqaaOWdbiaacYcacaWGubWdamaaBaaaleaapeGaaGOmaaWdaeqaaO WdbiaacYcacqGHMacVaiaawIcacaGLPaaaaeaacqGHRaWkcqGHMacV caqGGcGaeyypa0JaaeiOaiaadghapaWaaSbaaSqaa8qacaaIWaaapa qabaGcpeWaaeWaa8aabaWdbiaadsfapaWaaSbaaSqaa8qacaaIWaaa paqabaGcpeGaaiilaiaadsfapaWaaSbaaSqaa8qacaaIXaaapaqaba GcpeGaaiilaiabgAci8cGaayjkaiaawMcaaiabgUcaRiabew7aLjaa dghapaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeWaaeWaa8aabaWdbi aadsfapaWaaSbaaSqaa8qacaaIWaaapaqabaGcpeGaaiilaiaadsfa paWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaaiilaiabgAci8cGaay jkaiaawMcaaiabgUcaRiabf+5apnaabmaapaqaa8qacqaH1oqzpaWa aWbaaSqabeaapeGaaGOmaaaaaOGaayjkaiaawMcaaaaaaa@8FCB@   (41)

Due to the attenuation in the equations of motion, the amplitude is expanded as the following equation:

a( ε )= a 0 +ε a 1 ++ ε 2 a 2 + MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadggadaqadaWdaeaapeGaeqyTdugacaGLOaGaayzkaaGaeyypa0Ja amyya8aadaWgaaWcbaWdbiaaicdaa8aabeaak8qacqGHRaWkcqaH1o qzcaWGHbWdamaaBaaaleaapeGaaGymaaWdaeqaaOWdbiabgUcaRiab gUcaRiabew7aL9aadaahaaWcbeqaa8qacaaIYaaaaOGaamyya8aada WgaaWcbaWdbiaaikdaa8aabeaak8qacqGHRaWkcqGHMacVaaa@4CD4@   (42)

Therefore, the nonlinear equation of the system is rewritten as follows:

[ d 2 q 1 d t 2 +ε( β 0 + β 2 a 2 ( ε ) q 1 2 ) d q 1 dt + ω n 2 q 1 ] +ε [( α 3 a 2 ( ε ) ) q 1 3 ]=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbm aadmaapaqaa8qadaWcaaWdaeaapeGaamiza8aadaahaaWcbeqaa8qa caaIYaaaaOGaamyCa8aadaWgaaWcbaWdbiaaigdaa8aabeaaaOqaa8 qacaWGKbGaamiDa8aadaahaaWcbeqaa8qacaaIYaaaaaaakiabgUca Riabew7aLnaabmaapaqaa8qacqaHYoGypaWaaSbaaSqaa8qacaaIWa aapaqabaGcpeGaey4kaSIaeqOSdi2damaaBaaaleaapeGaaGOmaaWd aeqaaOWdbiaadggapaWaaWbaaSqabeaapeGaaGOmaaaakmaabmaapa qaa8qacqaH1oqzaiaawIcacaGLPaaacaWGXbWdamaaBaaaleaapeGa aGymaaWdaeqaaOWaaWbaaSqabeaapeGaaGOmaaaaaOGaayjkaiaawM caamaalaaapaqaa8qacaqGKbGaamyCa8aadaWgaaWcbaWdbiaaigda a8aabeaaaOqaa8qacaWGKbGaamiDaaaacqGHRaWkcqaHjpWDpaWaa0 baaSqaa8qacaWGUbaapaqaa8qacaaIYaaaaOGaamyCa8aadaWgaaWc baWdbiaaigdaa8aabeaak8qadaqcJaWdaeaapeGaey4kaSIaeqyTdu gacaGLDbGaay5waaWaaeWaa8aabaWdbiabeg7aH9aadaWgaaWcbaWd biaaiodaa8aabeaak8qacaWGHbWdamaaCaaaleqabaWdbiaaikdaaa GcdaqadaWdaeaapeGaeqyTdugacaGLOaGaayzkaaaacaGLOaGaayzk aaGaamyCa8aadaWgaaWcbaWdbiaaigdaa8aabeaakmaaCaaaleqaba WdbiaaiodaaaaakiaawUfacaGLDbaacqGH9aqpcaaIWaaaaa@73B7@   (43)

Now, by placing equation (41) in the equation (43) and separating by different powers on both sides of the resultant equation, a set of differential equations will be obtained as follows:

Ο( ε 0 ):[ 2 T 0 2 + ω n 2 ] q 0 ( T 0 , T 1 , T 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi abf+5apnaabmaapaqaa8qacqaH1oqzpaWaaWbaaSqabeaapeGaaGim aaaaaOGaayjkaiaawMcaaiaacQdadaWadaWdaeaapeWaaSaaa8aaba WdbiabgkGi2+aadaahaaWcbeqaa8qacaaIYaaaaaGcpaqaa8qacqGH ciITcaWGubWdamaaBaaaleaapeGaaGimaaWdaeqaaOWaaWbaaSqabe aapeGaaGOmaaaaaaGccqGHRaWkcqaHjpWDpaWaa0baaSqaa8qacaWG Ubaapaqaa8qacaaIYaaaaaGccaGLBbGaayzxaaGaamyCa8aadaWgaa WcbaWdbiaaicdaa8aabeaak8qadaqadaWdaeaapeGaamiva8aadaWg aaWcbaWdbiaaicdaa8aabeaak8qacaGGSaGaamiva8aadaWgaaWcba Wdbiaaigdaa8aabeaak8qacaGGSaGaamiva8aadaWgaaWcbaWdbiaa ikdaa8aabeaaaOWdbiaawIcacaGLPaaaaaa@57CB@   (44)

Therefore:

q 0 =cos( ω n T 0 +γ( T 1 , T 2 ) ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadghapaWaaSbaaSqaa8qacaaIWaaapaqabaGcpeGaeyypa0Jaam4y aiaad+gacaWGZbWaaeWaa8aabaWdbiabeM8a39aadaWgaaWcbaWdbi aad6gaa8aabeaak8qacaWGubWdamaaBaaaleaapeGaaGimaaWdaeqa aOWdbiabgUcaRiabeo7aNnaabmaapaqaa8qacaWGubWdamaaBaaale aapeGaaGymaaWdaeqaaOWdbiaacYcacaWGubWdamaaBaaaleaapeGa aGOmaaWdaeqaaaGcpeGaayjkaiaawMcaaaGaayjkaiaawMcaaaaa@4DB7@   (45)

Also:

Ο( ε 1 ):[ 2 T 0 2 + ω n 2 ] q 1 ( T 0 , T 1 , T 2 )= MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi abf+5apnaabmaapaqaa8qacqaH1oqzpaWaaWbaaSqabeaapeGaaGym aaaaaOGaayjkaiaawMcaaiaacQdadaWadaWdaeaapeWaaSaaa8aaba WdbiabgkGi2+aadaahaaWcbeqaa8qacaaIYaaaaaGcpaqaa8qacqGH ciITcaWGubWdamaaBaaaleaapeGaaGimaaWdaeqaaOWaaWbaaSqabe aapeGaaGOmaaaaaaGccqGHRaWkcqaHjpWDpaWaa0baaSqaa8qacaWG Ubaapaqaa8qacaaIYaaaaaGccaGLBbGaayzxaaGaamyCa8aadaWgaa WcbaWdbiaaigdaa8aabeaak8qadaqadaWdaeaapeGaamiva8aadaWg aaWcbaWdbiaaicdaa8aabeaak8qacaGGSaGaamiva8aadaWgaaWcba Wdbiaaigdaa8aabeaak8qacaGGSaGaamiva8aadaWgaaWcbaWdbiaa ikdaa8aabeaaaOWdbiaawIcacaGLPaaacqGH9aqpaaa@58D3@

[ β 0 T 0 2 2 T 0 T 1 ] q 0 ( T 0 , T 1 , T 2 )( a 0 2 q 0 2 ( T 0 , T 1 , T 2 ) ) [ α 3 q 0 ( T 0 , T 1 , T 2 )+ β 2 T 0 q 0 ( T 0 , T 1 , T 2 ) ], MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakqaabeqaaabaaaaaaa aapeWaamWaa8aabaWdbiabgkHiTiabek7aI9aadaWgaaWcbaWdbiaa icdaa8aabeaak8qadaWcaaWdaeaapeGaeyOaIylapaqaa8qacqGHci ITcaWGubWdamaaBaaaleaapeGaaGimaaWdaeqaaaaak8qacqGHsisl caaIYaWaaSaaa8aabaWdbiabgkGi2+aadaahaaWcbeqaa8qacaaIYa aaaaGcpaqaa8qacqGHciITcaWGubWdamaaBaaaleaapeGaaGimaaWd aeqaaOWdbiabgkGi2kaadsfapaWaaSbaaSqaa8qacaaIXaaapaqaba aaaaGcpeGaay5waiaaw2faaiaadghapaWaaSbaaSqaa8qacaaIWaaa paqabaGcpeWaaeWaa8aabaWdbiaadsfapaWaaSbaaSqaa8qacaaIWa aapaqabaGcpeGaaiilaiaadsfapaWaaSbaaSqaa8qacaaIXaaapaqa baGcpeGaaiilaiaadsfapaWaaSbaaSqaa8qacaaIYaaapaqabaaak8 qacaGLOaGaayzkaaGaeyOeI0YaaeWaa8aabaWdbiaadggapaWaaSba aSqaa8qacaaIWaaapaqabaGcdaahaaWcbeqaa8qacaaIYaaaaOGaam yCa8aadaWgaaWcbaWdbiaaicdaa8aabeaakmaaCaaaleqabaWdbiaa ikdaaaGcdaqadaWdaeaapeGaamiva8aadaWgaaWcbaWdbiaaicdaa8 aabeaak8qacaGGSaGaamiva8aadaWgaaWcbaWdbiaaigdaa8aabeaa k8qacaGGSaGaamiva8aadaWgaaWcbaWdbiaaikdaa8aabeaaaOWdbi aawIcacaGLPaaaaiaawIcacaGLPaaaaeaadaWadaWdaeaapeGaeyOe I0IaeqySde2damaaBaaaleaapeGaaG4maaWdaeqaaOWdbiaadghapa WaaSbaaSqaa8qacaaIWaaapaqabaGcpeWaaeWaa8aabaWdbiaadsfa paWaaSbaaSqaa8qacaaIWaaapaqabaGcpeGaaiilaiaadsfapaWaaS baaSqaa8qacaaIXaaapaqabaGcpeGaaiilaiaadsfapaWaaSbaaSqa a8qacaaIYaaapaqabaaak8qacaGLOaGaayzkaaGaey4kaSIaeqOSdi 2damaaBaaaleaapeGaaGOmaaWdaeqaaOWdbmaalaaapaqaa8qacqGH ciITa8aabaWdbiabgkGi2kaadsfapaWaaSbaaSqaa8qacaaIWaaapa qabaaaaOWdbiaadghapaWaaSbaaSqaa8qacaaIWaaapaqabaGcpeWa aeWaa8aabaWdbiaadsfapaWaaSbaaSqaa8qacaaIWaaapaqabaGcpe GaaiilaiaadsfapaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaaiil aiaadsfapaWaaSbaaSqaa8qacaaIYaaapaqabaaak8qacaGLOaGaay zkaaaacaGLBbGaayzxaaGaaiilaaaaaa@9114@   (46)

By placing equation (46) in equation (43):

( D 0 2 + ω n 2 ) q 1 = 1 4 a 0 2 α 3 cos( 3 ω n T 0 +3γ( T 1 , T 2 ) )+ 1 4 ω n a 0 2 β 2 sin( 3 ω n T 0 +3γ( T 1 , T 2 ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbm aabmaapaqaa8qacaWGebWdamaaBaaaleaapeGaaGimaaWdaeqaaOWa aWbaaSqabeaapeGaaGOmaaaakiabgUcaRiabeM8a39aadaqhaaWcba Wdbiaad6gaa8aabaWdbiaaikdaaaaakiaawIcacaGLPaaacaWGXbWd amaaBaaaleaapeGaaGymaaWdaeqaaOWdbiabg2da9iabgkHiTmaala aapaqaa8qacaaIXaaapaqaa8qacaaI0aaaaiaadggapaWaaSbaaSqa a8qacaaIWaaapaqabaGcdaahaaWcbeqaa8qacaaIYaaaaOGaeqySde 2damaaBaaaleaapeGaaG4maaWdaeqaaOWdbiGacogacaGGVbGaai4C amaabmaapaqaa8qacaaIZaGaeqyYdC3damaaBaaaleaapeGaamOBaa WdaeqaaOWdbiaadsfapaWaaSbaaSqaa8qacaaIWaaapaqabaGcpeGa ey4kaSIaaG4maiabeo7aNnaabmaapaqaa8qacaWGubWdamaaBaaale aapeGaaGymaaWdaeqaaOWdbiaacYcacaWGubWdamaaBaaaleaapeGa aGOmaaWdaeqaaaGcpeGaayjkaiaawMcaaaGaayjkaiaawMcaaiabgU caRmaalaaapaqaa8qacaaIXaaapaqaa8qacaaI0aaaaiabeM8a39aa daWgaaWcbaWdbiaad6gaa8aabeaak8qacaWGHbWdamaaBaaaleaape GaaGimaaWdaeqaaOWaaWbaaSqabeaapeGaaGOmaaaakiabek7aI9aa daWgaaWcbaWdbiaaikdaa8aabeaak8qaciGGZbGaaiyAaiaac6gada qadaWdaeaapeGaaG4maiabeM8a39aadaWgaaWcbaWdbiaad6gaa8aa beaak8qacaWGubWdamaaBaaaleaapeGaaGimaaWdaeqaaOWdbiabgU caRiaaiodacqaHZoWzdaqadaWdaeaapeGaamiva8aadaWgaaWcbaWd biaaigdaa8aabeaak8qacaGGSaGaamiva8aadaWgaaWcbaWdbiaaik daa8aabeaaaOWdbiaawIcacaGLPaaaaiaawIcacaGLPaaaaaa@80FC@ + 8 4 ( T 1 γ( T 1 , T 2 ) ) ω n cos( ω n T 0 +γ( T 1 , T 2 ) ) 3 4 a 0 2 α 3 cos( ω n T 0 +γ( T 1 , T 2 ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi abgUcaRmaalaaapaqaa8qacaaI4aaapaqaa8qacaaI0aaaamaabmaa paqaa8qadaWcaaWdaeaapeGaeyOaIylapaqaa8qacqGHciITcaWGub WdamaaBaaaleaapeGaaGymaaWdaeqaaaaak8qacqaHZoWzdaqadaWd aeaapeGaamiva8aadaWgaaWcbaWdbiaaigdaa8aabeaak8qacaGGSa Gaamiva8aadaWgaaWcbaWdbiaaikdaa8aabeaaaOWdbiaawIcacaGL PaaaaiaawIcacaGLPaaacqaHjpWDpaWaaSbaaSqaa8qacaWGUbaapa qabaGcpeGaci4yaiaac+gacaGGZbWaaeWaa8aabaWdbiabeM8a39aa daWgaaWcbaWdbiaad6gaa8aabeaak8qacaWGubWdamaaBaaaleaape GaaGimaaWdaeqaaOWdbiabgUcaRiabeo7aNnaabmaapaqaa8qacaWG ubWdamaaBaaaleaapeGaaGymaaWdaeqaaOWdbiaacYcacaWGubWdam aaBaaaleaapeGaaGOmaaWdaeqaaaGcpeGaayjkaiaawMcaaaGaayjk aiaawMcaaiabgkHiTmaalaaapaqaa8qacaaIZaaapaqaa8qacaaI0a aaaiaadggapaWaaSbaaSqaa8qacaaIWaaapaqabaGcdaahaaWcbeqa a8qacaaIYaaaaOGaeqySde2damaaBaaaleaapeGaaG4maaWdaeqaaO WdbiGacogacaGGVbGaai4Camaabmaapaqaa8qacqaHjpWDpaWaaSba aSqaa8qacaWGUbaapaqabaGcpeGaamiva8aadaWgaaWcbaWdbiaaic daa8aabeaak8qacqGHRaWkcqaHZoWzdaqadaWdaeaapeGaamiva8aa daWgaaWcbaWdbiaaigdaa8aabeaak8qacaGGSaGaamiva8aadaWgaa WcbaWdbiaaikdaa8aabeaaaOWdbiaawIcacaGLPaaaaiaawIcacaGL Paaaaaa@7A7E@ +( β 0 + 1 4 a 0 2 β 2 ) ω n sin( ω n T 0 +γ( T 1 , T 2 ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi abgUcaRiaacIcacqaHYoGypaWaaSbaaSqaa8qacaaIWaaapaqabaGc peGaey4kaSYaaSaaa8aabaWdbiaaigdaa8aabaWdbiaaisdaaaGaam yya8aadaWgaaWcbaWdbiaaicdaa8aabeaakmaaCaaaleqabaWdbiaa ikdaaaGccqaHYoGypaWaaSbaaSqaa8qacaaIYaaapaqabaGcpeGaai ykaiabeM8a39aadaWgaaWcbaWdbiaad6gaa8aabeaak8qaciGGZbGa aiyAaiaac6gadaqadaWdaeaapeGaeqyYdC3damaaBaaaleaapeGaam OBaaWdaeqaaOWdbiaadsfapaWaaSbaaSqaa8qacaaIWaaapaqabaGc peGaey4kaSIaeq4SdC2aaeWaa8aabaWdbiaadsfapaWaaSbaaSqaa8 qacaaIXaaapaqabaGcpeGaaiilaiaadsfapaWaaSbaaSqaa8qacaaI Yaaapaqabaaak8qacaGLOaGaayzkaaaacaGLOaGaayzkaaaaaa@5B4F@   (47)

The condition for the solvability of the high differential equation is to prevent the formation of very extensive terms in the time response. Therefore, the coefficients of the terms sin( ω n T 0 +γ( T 1 , T 2 ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi GacohacaGGPbGaaiOBamaabmaapaqaa8qacqaHjpWDpaWaaSbaaSqa a8qacaWGUbaapaqabaGcpeGaamiva8aadaWgaaWcbaWdbiaaicdaa8 aabeaak8qacqGHRaWkcqaHZoWzdaqadaWdaeaapeGaamiva8aadaWg aaWcbaWdbiaaigdaa8aabeaak8qacaGGSaGaamiva8aadaWgaaWcba Wdbiaaikdaa8aabeaaaOWdbiaawIcacaGLPaaaaiaawIcacaGLPaaa aaa@4A90@  and cos( ω n T 0 +γ( T 1 , T 2 ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi GacogacaGGVbGaai4Camaabmaapaqaa8qacqaHjpWDpaWaaSbaaSqa a8qacaWGUbaapaqabaGcpeGaamiva8aadaWgaaWcbaWdbiaaicdaa8 aabeaak8qacqGHRaWkcqaHZoWzdaqadaWdaeaapeGaamiva8aadaWg aaWcbaWdbiaaigdaa8aabeaak8qacaGGSaGaamiva8aadaWgaaWcba Wdbiaaikdaa8aabeaaaOWdbiaawIcacaGLPaaaaiaawIcacaGLPaaa aaa@4A8B@  in equation (47) should be considered equal to zero. This prevents the following terms from appearing in the system response, and does not take the vibrational response of the system to infinity over a long period time.

Therefore, the equations of the system are obtained as follows:

8( T 1 γ( T 1 , T 2 ) ) ω n 3 a 0 2 α 3 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aaiIdadaqadaWdaeaapeWaaSaaa8aabaWdbiabgkGi2cWdaeaapeGa eyOaIyRaamiva8aadaWgaaWcbaWdbiaaigdaa8aabeaaaaGcpeGaeq 4SdC2aaeWaa8aabaWdbiaadsfapaWaaSbaaSqaa8qacaaIXaaapaqa baGcpeGaaiilaiaadsfapaWaaSbaaSqaa8qacaaIYaaapaqabaaak8 qacaGLOaGaayzkaaaacaGLOaGaayzkaaGaeqyYdC3damaaBaaaleaa peGaamOBaaWdaeqaaOWdbiabgkHiTiaaiodacaWGHbWdamaaBaaale aapeGaaGimaaWdaeqaaOWaaWbaaSqabeaapeGaaGOmaaaakiabeg7a H9aadaWgaaWcbaWdbiaaiodaa8aabeaak8qacqGH9aqpcaaIWaaaaa@53F4@   (48)

4 β 0 + a 0 2 β 2 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aaisdacqaHYoGypaWaaSbaaSqaa8qacaaIWaaapaqabaGcpeGaey4k aSIaamyya8aadaWgaaWcbaWdbiaaicdaa8aabeaakmaaCaaaleqaba WdbiaaikdaaaGccqaHYoGypaWaaSbaaSqaa8qacaaIYaaapaqabaGc peGaeyypa0JaaGimaaaa@43EA@   (49)

By solving the equations (48) and (49) together, the unknown variables are obtained in the following relation:

γ( T 1 , T 2 )= 12 α 3 a 0 2 32 ω n T 1 + γ 1 ( T 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi abeo7aNnaabmaapaqaa8qacaWGubWdamaaBaaaleaapeGaaGymaaWd aeqaaOWdbiaacYcacaWGubWdamaaBaaaleaapeGaaGOmaaWdaeqaaa GcpeGaayjkaiaawMcaaiabg2da9maalaaapaqaa8qacaaIXaGaaGOm aiabeg7aH9aadaWgaaWcbaWdbiaaiodaa8aabeaak8qacaWGHbWdam aaBaaaleaapeGaaGimaaWdaeqaaOWaaWbaaSqabeaapeGaaGOmaaaa aOWdaeaapeGaaG4maiaaikdacqaHjpWDpaWaaSbaaSqaa8qacaWGUb aapaqabaaaaOWdbiaadsfapaWaaSbaaSqaa8qacaaIXaaapaqabaGc peGaey4kaSIaeq4SdC2damaaBaaaleaapeGaaGymaaWdaeqaaOWdbm aabmaapaqaa8qacaWGubWdamaaBaaaleaapeGaaGOmaaWdaeqaaaGc peGaayjkaiaawMcaaaaa@56B3@   (50)

So, the second-order estimation of the response will be obtained as follows:

q 1 = a 0 2 α 3 32 ω n 2 cos( 3 ω n T 0 +3γ( T 1 , T 2 ) ) a 0 2 β 2 32 ω n sin( 3 ω n T 0 +3γ( T 1 , T 2 ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadghapaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaeyypa0ZaaSaa a8aabaWdbiaadggapaWaaSbaaSqaa8qacaaIWaaapaqabaGcdaahaa Wcbeqaa8qacaaIYaaaaOGaeqySde2damaaBaaaleaapeGaaG4maaWd aeqaaaGcbaWdbiaaiodacaaIYaGaeqyYdC3damaaDaaaleaapeGaam OBaaWdaeaapeGaaGOmaaaaaaGcciGGJbGaai4BaiaacohadaqadaWd aeaapeGaaG4maiabeM8a39aadaWgaaWcbaWdbiaad6gaa8aabeaak8 qacaWGubWdamaaBaaaleaapeGaaGimaaWdaeqaaOWdbiabgUcaRiaa iodacqaHZoWzdaqadaWdaeaapeGaamiva8aadaWgaaWcbaWdbiaaig daa8aabeaak8qacaGGSaGaamiva8aadaWgaaWcbaWdbiaaikdaa8aa beaaaOWdbiaawIcacaGLPaaaaiaawIcacaGLPaaacqGHsisldaWcaa WdaeaapeGaamyya8aadaWgaaWcbaWdbiaaicdaa8aabeaakmaaCaaa leqabaWdbiaaikdaaaGccqaHYoGypaWaaSbaaSqaa8qacaaIYaaapa qabaaakeaapeGaaG4maiaaikdacqaHjpWDpaWaaSbaaSqaa8qacaWG UbaapaqabaaaaOWdbiGacohacaGGPbGaaiOBamaabmaapaqaa8qaca aIZaGaeqyYdC3damaaBaaaleaapeGaamOBaaWdaeqaaOWdbiaadsfa paWaaSbaaSqaa8qacaaIWaaapaqabaGcpeGaey4kaSIaaG4maiabeo 7aNnaabmaapaqaa8qacaWGubWdamaaBaaaleaapeGaaGymaaWdaeqa aOWdbiaacYcacaWGubWdamaaBaaaleaapeGaaGOmaaWdaeqaaaGcpe GaayjkaiaawMcaaaGaayjkaiaawMcaaaaa@7A68@   (51)

Higher estimates can be obtained similarly.

  1. Forced vibration analysis (initial resonance)

To investigate forced vibrational response of the system, equation (37) is rewritten as follows:

d 2 q d t 2 +ε( β 0 + β 2 q 2 ) dq dt + ω n 2 q+ε α 3 q 3 =ε f ¯ e iΩt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbm aalaaapaqaa8qacaWGKbWdamaaCaaaleqabaWdbiaaikdaaaGccaWG Xbaapaqaa8qacaWGKbGaamiDa8aadaahaaWcbeqaa8qacaaIYaaaaa aakiabgUcaRiabew7aLnaabmaapaqaa8qacqaHYoGypaWaaSbaaSqa a8qacaaIWaaapaqabaGcpeGaey4kaSIaeqOSdi2damaaBaaaleaape GaaGOmaaWdaeqaaOWdbiaadghapaWaaWbaaSqabeaapeGaaGOmaaaa aOGaayjkaiaawMcaamaalaaapaqaa8qacaWGKbGaamyCaaWdaeaape GaamizaiaadshaaaGaey4kaSIaeqyYdC3damaaDaaaleaapeGaamOB aaWdaeaapeGaaGOmaaaakiaadghacqGHRaWkcqaH1oqzcqaHXoqypa WaaSbaaSqaa8qacaaIZaaapaqabaGcpeGaamyCa8aadaahaaWcbeqa a8qacaaIZaaaaOGaeyypa0JaeqyTduMabmOza8aagaqea8qacaWGLb WdamaaCaaaleqabaWdbiaadMgacqqHPoWvcaWG0baaaaaa@6485@   (52)

In equation (52), the excitation frequency for the initial resonance is according to the relation Ω=εσ+ ω n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi abfM6axjabg2da9iabew7aLjabeo8aZjabgUcaRiabeM8a39aadaWg aaWcbaWdbiaad6gaa8aabeaaaaa@41DD@ , where σ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi abeo8aZbaa@39A6@ is the parameter of deviation from resonance.

So:

( D 0 2 + ω n 2 ) q 0 =0  q 0 =A( T 1 ) e i T 0 ω n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbm aabmaapaqaa8qacaWGebWdamaaBaaaleaapeGaaGimaaWdaeqaaOWa aWbaaSqabeaapeGaaGOmaaaakiabgUcaRiabeM8a39aadaqhaaWcba Wdbiaad6gaa8aabaWdbiaaikdaaaaakiaawIcacaGLPaaacaWGXbWd amaaBaaaleaapeGaaGimaaWdaeqaaOWdbiabg2da9iaaicdacaqGGc GaeyOKH4QaamyCa8aadaWgaaWcbaWdbiaaicdaa8aabeaak8qacqGH 9aqpcaWGbbWaaeWaa8aabaWdbiaadsfapaWaaSbaaSqaa8qacaaIXa aapaqabaaak8qacaGLOaGaayzkaaGaamyza8aadaahaaWcbeqaa8qa caWGPbGaamiva8aadaWgaaadbaWdbiaaicdaa8aabeaal8qacqaHjp WDpaWaaSbaaWqaa8qacaWGUbaapaqabaaaaaaa@5724@   (53)

( D 0 2 + ω n 2 ) q 1 = β 0 D 0 q 0 2 D 0 D 1 q 0 β 2 q 0 2 D 0 q 0 α 3 q 0 3 + f ¯ e i( σ T 1 + ω n T 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbm aabmaapaqaa8qacaWGebWdamaaBaaaleaapeGaaGimaaWdaeqaaOWa aWbaaSqabeaapeGaaGOmaaaakiabgUcaRiabeM8a39aadaqhaaWcba Wdbiaad6gaa8aabaWdbiaaikdaaaaakiaawIcacaGLPaaacaWGXbWd amaaBaaaleaapeGaaGymaaWdaeqaaOWdbiabg2da9iabgkHiTiabek 7aI9aadaWgaaWcbaWdbiaaicdaa8aabeaak8qacaWGebWdamaaBaaa leaapeGaaGimaaWdaeqaaOWdbiaadghapaWaaSbaaSqaa8qacaaIWa aapaqabaGcpeGaeyOeI0IaaGOmaiaadseapaWaaSbaaSqaa8qacaaI WaaapaqabaGcpeGaamira8aadaWgaaWcbaWdbiaaigdaa8aabeaak8 qacaWGXbWdamaaBaaaleaapeGaaGimaaWdaeqaaOWdbiabgkHiTiab ek7aI9aadaWgaaWcbaWdbiaaikdaa8aabeaak8qacaWGXbWdamaaBa aaleaapeGaaGimaaWdaeqaaOWaaWbaaSqabeaapeGaaGOmaaaakiaa dseapaWaaSbaaSqaa8qacaaIWaaapaqabaGcpeGaamyCa8aadaWgaa WcbaWdbiaaicdaa8aabeaak8qacqGHsislcqaHXoqypaWaaSbaaSqa a8qacaaIZaaapaqabaGcpeGaamyCa8aadaWgaaWcbaWdbiaaicdaa8 aabeaakmaaCaaaleqabaWdbiaaiodaaaGccqGHRaWkceWGMbWdayaa raWdbiaadwgapaWaaWbaaSqabeaapeGaamyAamaabmaapaqaa8qacq aHdpWCcaWGubWdamaaBaaameaapeGaaGymaaWdaeqaaSWdbiabgUca RiabeM8a39aadaWgaaadbaWdbiaad6gaa8aabeaal8qacaWGubWdam aaBaaameaapeGaaGimaaWdaeqaaaWcpeGaayjkaiaawMcaaaaaaaa@7589@   (54)

By placing equation (53) in equation (54), and to prevent significant time response terms, the coefficient of a term e i T 0 ω n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadwgapaWaaWbaaSqabeaapeGaamyAaiaadsfapaWaaSbaaWqaa8qa caaIWaaapaqabaWcpeGaeqyYdC3damaaBaaameaapeGaamOBaaWdae qaaaaaaaa@3F2B@  must be set equal to zero, resulting is the following equation:

2i ω n A T 1 [ A ¯ ( α 3 +i β 2 ω n )A+i β 0 ω n ]A+ f ¯ e i T 1 σ =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi abgkHiTiaaikdacaWGPbGaeqyYdC3damaaBaaaleaapeGaamOBaaWd aeqaaOWdbmaalaaapaqaa8qacqGHciITcaWGbbaapaqaa8qacqGHci ITcaWGubWdamaaBaaaleaapeGaaGymaaWdaeqaaaaak8qacqGHsisl daWadaWdaeaapeGabmyqa8aagaqea8qadaqadaWdaeaapeGaeqySde 2damaaBaaaleaapeGaaG4maaWdaeqaaOWdbiabgUcaRiaadMgacqaH YoGypaWaaSbaaSqaa8qacaaIYaaapaqabaGcpeGaeqyYdC3damaaBa aaleaapeGaamOBaaWdaeqaaaGcpeGaayjkaiaawMcaaiaadgeacqGH RaWkcaWGPbGaeqOSdi2damaaBaaaleaapeGaaGimaaWdaeqaaOWdbi abeM8a39aadaWgaaWcbaWdbiaad6gaa8aabeaaaOWdbiaawUfacaGL DbaacaWGbbGaey4kaSIabmOza8aagaqea8qacaWGLbWdamaaCaaale qabaWdbiaadMgacaWGubWdamaaBaaameaapeGaaGymaaWdaeqaaSWd biabeo8aZbaakiabg2da9iaaicdaaaa@66FC@   (55)

Now, assuming the polar form A= a 2 e iγ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadgeacqGH9aqpdaWcaaWdaeaapeGaamyyaaWdaeaapeGaaGOmaaaa caWGLbWdamaaCaaaleqabaWdbiaadMgacqaHZoWzaaaaaa@3F6A@ , and placing it in the equation (55), the following set of equations will be obtained:

β 0 ω n a+ 1 4 β 2 ω n a 3 2 f ¯ sin( σ T 1 γ )=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi abek7aI9aadaWgaaWcbaWdbiaaicdaa8aabeaak8qacqaHjpWDpaWa aSbaaSqaa8qacaWGUbaapaqabaGcpeGaamyyaiabgUcaRmaalaaapa qaa8qacaaIXaaapaqaa8qacaaI0aaaaiabek7aI9aadaWgaaWcbaWd biaaikdaa8aabeaak8qacqaHjpWDpaWaaSbaaSqaa8qacaWGUbaapa qabaGcpeGaamyya8aadaahaaWcbeqaa8qacaaIZaaaaOGaeyOeI0Ia aGOmaiqadAgapaGbaebapeGaci4CaiaacMgacaGGUbWaaeWaa8aaba Wdbiabeo8aZjaadsfapaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGa eyOeI0Iaeq4SdCgacaGLOaGaayzkaaGaeyypa0JaaGimaaaa@58DD@   (56)

1 8 α 3 a 3 f ¯ cos( σ T 1 γ )=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbm aalaaapaqaa8qacaaIXaaapaqaa8qacaaI4aaaaiabeg7aH9aadaWg aaWcbaWdbiaaiodaa8aabeaak8qacaWGHbWdamaaCaaaleqabaWdbi aaiodaaaGccqGHsislceWGMbWdayaaraWdbiGacogacaGGVbGaai4C amaabmaapaqaa8qacqaHdpWCcaWGubWdamaaBaaaleaapeGaaGymaa WdaeqaaOWdbiabgkHiTiabeo7aNbGaayjkaiaawMcaaiabg2da9iaa icdaaaa@4D20@   (57)

The frequency equation response of the system with the definition of θ=σ T 1 γ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi abeI7aXjabg2da9iabeo8aZjaadsfapaWaaSbaaSqaa8qacaaIXaaa paqabaGcpeGaeyOeI0Iaeq4SdCgaaa@40FE@ , and considering that in the case of uniform response, the changes of a and θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi abeI7aXbaa@3999@ will be zero. Finally, the nonlinear vibrational response equation is presented to the following final form in the form of a closed-form equation:

4 ω n 2 [ (   1 16 β 2 a 3 + 1 4 β 0 a ) 2 + ( 1 16 α 3 a 3 ω n ) 2 ] f ¯ 2 =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aaisdacqaHjpWDpaWaaSbaaSqaa8qacaWGUbaapaqabaGcdaahaaWc beqaa8qacaaIYaaaaOWaamWaa8aabaWdbmaabmaapaqaa8qacaqGGc WaaSaaa8aabaWdbiaaigdaa8aabaWdbiaaigdacaaI2aaaaiabek7a I9aadaWgaaWcbaWdbiaaikdaa8aabeaak8qacaWGHbWdamaaCaaale qabaWdbiaaiodaaaGccqGHRaWkdaWcaaWdaeaapeGaaGymaaWdaeaa peGaaGinaaaacqaHYoGypaWaaSbaaSqaa8qacaaIWaaapaqabaGcpe GaamyyaaGaayjkaiaawMcaa8aadaahaaWcbeqaa8qacaaIYaaaaOGa ey4kaSIaaiikamaalaaapaqaa8qacaaIXaaapaqaa8qacaaIXaGaaG OnaaaadaWcaaWdaeaapeGaeqySde2damaaBaaaleaapeGaaG4maaWd aeqaaOWdbiaadggapaWaaWbaaSqabeaapeGaaG4maaaaaOWdaeaape GaeqyYdC3damaaBaaaleaapeGaamOBaaWdaeqaaaaak8qacaGGPaWd amaaCaaaleqabaWdbiaaikdaaaaakiaawUfacaGLDbaacqGHsislce WGMbWdayaaraWaaWbaaSqabeaapeGaaGOmaaaakiabg2da9iaaicda aaa@6340@   (58)

Numerical results, validations and discussions

Dimensionless form of input and output parameters are considered by the following equations:

x ¯ = x L  ,  w ¯ = w L  ,  f ¯ exc = f exc EI MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi qadIhapaGbaebapeGaeyypa0ZaaSaaa8aabaWdbiaadIhaa8aabaWd biaadYeaaaGaaeiOaiaacYcacaqGGcGabm4Da8aagaqea8qacqGH9a qpdaWcaaWdaeaapeGaam4DaaWdaeaapeGaamitaaaacaqGGcGaaiil aiaabckaceWGMbWdayaaraWaaSbaaSqaa8qacaWGLbGaamiEaiaado gaa8aabeaak8qacqGH9aqpdaWcaaWdaeaapeGaamOza8aadaWgaaWc baWdbiaadwgacaWG4bGaam4yaaWdaeqaaaGcbaWdbiaadweacaWGjb aaaaaa@51BA@

EI ¯ eff = E I eff EI  ,  P ¯ eff = P eff L 2 EI  ,  F ¯ lz = F lz L 2 EI   MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaadaqdaaqaaabaaa aaaaaapeGaamyraiaadMeaaaWdamaaBaaaleaapeGaamyzaiaadAga caWGMbaapaqabaGcpeGaeyypa0ZaaSaaa8aabaWdbiaadweacaWGjb WdamaaBaaaleaapeGaamyzaiaadAgacaWGMbaapaqabaaakeaapeGa amyraiaadMeaaaGaaeiOaiaacYcacaqGGcGabmiua8aagaqeamaaBa aaleaapeGaamyzaiaadAgacaWGMbaapaqabaGcpeGaeyypa0ZaaSaa a8aabaWdbiaadcfapaWaaSbaaSqaa8qacaWGLbGaamOzaiaadAgaa8 aabeaak8qacaWGmbWdamaaCaaaleqabaWdbiaaikdaaaaak8aabaWd biaadweacaWGjbaaaiaabckacaGGSaGaaeiOaiqadAeapaGbaebada WgaaWcbaWdbiaadYgacaWG6baapaqabaGcpeGaeyypa0ZaaSaaa8aa baWdbiaadAeapaWaaSbaaSqaa8qacaWGSbGaamOEaaWdaeqaaOWdbi aadYeapaWaaWbaaSqabeaapeGaaGOmaaaaaOWdaeaapeGaamyraiaa dMeaaaGaaeiOaaaa@63A6@

K 1 = k 1 L 4 EI  ,  K 3 = k 3 L 6 EI MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadUeapaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaeyypa0ZaaSaa a8aabaWdbiaadUgapaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGaam ita8aadaahaaWcbeqaa8qacaaI0aaaaaGcpaqaa8qacaWGfbGaamys aaaacaqGGcGaaiilaiaabckacaWGlbWdamaaBaaaleaapeGaaG4maa WdaeqaaOWdbiabg2da9maalaaapaqaa8qacaWGRbWdamaaBaaaleaa peGaaG4maaWdaeqaaOWdbiaadYeapaWaaWbaaSqabeaapeGaaGOnaa aaaOWdaeaapeGaamyraiaadMeaaaaaaa@4CBD@  (59)

C 0 = c 0 L 2 EIm  ,  C 2 = c 2 L 4 EIm MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadoeapaWaaSbaaSqaa8qacaaIWaaapaqabaGcpeGaeyypa0ZaaSaa a8aabaWdbiaadogapaWaaSbaaSqaa8qacaaIWaaapaqabaGcpeGaam ita8aadaahaaWcbeqaa8qacaaIYaaaaaGcpaqaa8qadaGcaaWdaeaa peGaamyraiaadMeacaWGTbaaleqaaaaakiaabckacaGGSaGaaeiOai aadoeapaWaaSbaaSqaa8qacaaIYaaapaqabaGcpeGaeyypa0ZaaSaa a8aabaWdbiaadogapaWaaSbaaSqaa8qacaaIYaaapaqabaGcpeGaam ita8aadaahaaWcbeqaa8qacaaI0aaaaaGcpaqaa8qadaGcaaWdaeaa peGaamyraiaadMeacaWGTbaaleqaaaaaaaa@4EF7@

μ= ( e 0 a ) 2 L 2  , T= t L 2 EI m  , Ω=ω L 2 m EI MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi abeY7aTjabg2da9maalaaapaqaa8qadaqadaWdaeaapeGaamyza8aa daWgaaWcbaWdbiaaicdaa8aabeaak8qacaWGHbaacaGLOaGaayzkaa WdamaaCaaaleqabaWdbiaaikdaaaaak8aabaWdbiaadYeapaWaaWba aSqabeaapeGaaGOmaaaaaaGccaqGGcGaaiilaiaabckacaWGubGaey ypa0ZaaSaaa8aabaWdbiaadshaa8aabaWdbiaadYeapaWaaWbaaSqa beaapeGaaGOmaaaaaaGcdaGcaaWdaeaapeWaaSaaa8aabaWdbiaadw eacaWGjbaapaqaa8qacaWGTbaaaaWcbeaakiaabckacaGGSaGaaeiO aiabfM6axjabg2da9iabeM8a3jaadYeapaWaaWbaaSqabeaapeGaaG Omaaaakmaakaaapaqaa8qadaWcaaWdaeaapeGaamyBaaWdaeaapeGa amyraiaadMeaaaaaleqaaaaa@59EF@

Validation for the results of the free vibration analysis

In this case, to validate the results with the references,69 the cross-sectional area of the nanowire is considered to be a rectangle with a small thickness and height h which its length is L. On the other hand, in this paper, using the classical local theory, the results of free vibration of a nanowire coated with piezoelectric crystals of PZT-5H for two boundary conditions, S-S and C-C, are obtained. Those properties are:

e 31 =6.5; k 33 =1.3× 10 8 ; MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadwgapaWaaSbaaSqaa8qacaaIZaGaaGymaaWdaeqaaOWdbiabg2da 9iabgkHiTiaaiAdacaGGUaGaaGynaiaacUdacaWGRbWdamaaBaaale aapeGaaG4maiaaiodaa8aabeaak8qacqGH9aqpcaaIXaGaaiOlaiaa iodacqGHxdaTcaaIXaGaaGima8aadaahaaWcbeqaa8qacqGHsislca aI4aaaaOGaai4oaaaa@4BFA@

c 11 s =7.56; e 31 s =3× 10 8 ; MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadogapaWaa0baaSqaa8qacaaIXaGaaGymaaWdaeaapeGaam4Caaaa kiabg2da9iaaiEdacaGGUaGaaGynaiaaiAdacaGG7aGaamyza8aada qhaaWcbaWdbiaaiodacaaIXaaapaqaa8qacaWGZbaaaOGaeyypa0Ja eyOeI0IaaG4maiabgEna0kaaigdacaaIWaWdamaaCaaaleqabaWdbi abgkHiTiaaiIdaaaGccaGG7aaaaa@4D34@

E=126× 10 9 ;L=20h; MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadweacqGH9aqpcaaIXaGaaGOmaiaaiAdacqGHxdaTcaaIXaGaaGim a8aadaahaaWcbeqaa8qacaaI5aaaaOGaai4oaiaadYeacqGH9aqpca aIYaGaaGimaiaadIgacaGG7aaaaa@4647@

In the following figures, the dimensionless frequency of the first mode of nanowire vibration in terms of its cross-sectional height for different values of an external voltage applied in local and non-local theory is compared with the results obtained in local formulation references.70 In the S-S boundary condition:

Figure 4 Validation for local70 and non-local (present) theory in S-S boundary conditions And also, for the C-C boundary condition:

Figure 5 Validation for local70 and non-local (present) theory in C-C boundary conditions
Furthermore, for the other forms of the surface effects:
Form 1: σ x 0 0 ,  c 11 s 0, e 31 s 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi abeo8aZ9aadaqhaaWcbaWdbiaadIhaa8aabaWdbiaaicdaaaGccqGH GjsUcaaIWaGaaeiOaiaacYcacaqGGcGaam4ya8aadaqhaaWcbaWdbi aaigdacaaIXaaapaqaa8qacaWGZbaaaOGaeyiyIKRaaGimaiaacYca caWGLbWdamaaDaaaleaapeGaaG4maiaaigdaa8aabaWdbiaadohaaa GccqGHGjsUcaaIWaaaaa@4E96@
Form 2: σ x 0 =0 ,  c 11 s =0, e 31 s =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi abeo8aZ9aadaqhaaWcbaWdbiaadIhaa8aabaWdbiaaicdaaaGccqGH 9aqpcaaIWaGaaeiOaiaacYcacaqGGcGaam4ya8aadaqhaaWcbaWdbi aaigdacaaIXaaapaqaa8qacaWGZbaaaOGaeyypa0JaaGimaiaacYca caWGLbWdamaaDaaaleaapeGaaG4maiaaigdaa8aabaWdbiaadohaaa GccqGH9aqpcaaIWaaaaa@4C53@
Form 3: σ x 0 0 ,  c 11 s =0, e 31 s =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi abeo8aZ9aadaqhaaWcbaWdbiaadIhaa8aabaWdbiaaicdaaaGccqGH GjsUcaaIWaGaaeiOaiaacYcacaqGGcGaam4ya8aadaqhaaWcbaWdbi aaigdacaaIXaaapaqaa8qacaWGZbaaaOGaeyypa0JaaGimaiaacYca caWGLbWdamaaDaaaleaapeGaaG4maiaaigdaa8aabaWdbiaadohaaa GccqGH9aqpcaaIWaaaaa@4D14@
Form 4: σ x 0 =0 ,  c 11 s 0, e 31 s =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi abeo8aZ9aadaqhaaWcbaWdbiaadIhaa8aabaWdbiaaicdaaaGccqGH 9aqpcaaIWaGaaeiOaiaacYcacaqGGcGaam4ya8aadaqhaaWcbaWdbi aaigdacaaIXaaapaqaa8qacaWGZbaaaOGaeyiyIKRaaGimaiaacYca caWGLbWdamaaDaaaleaapeGaaG4maiaaigdaa8aabaWdbiaadohaaa GccqGH9aqpcaaIWaaaaa@4D14@

Figure 6 Validation for various forms of surface effects in S-S boundary conditions

Figure 7 Validation for various forms of surface effects in C-C boundary conditions.

Results of the free vibrational analysis
  1. The effect of non-local parameter and electric-magnetic fields

In the below curves, the changes of dimensionless frequency of the free oscillations at the first mode with two different boundary conditions in terms of the applied voltage and different values of dimensionless non-local parameter, with input parameters σ x 0 =1, H x =0, C 0 =0, K 1 =10 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi abeo8aZ9aadaqhaaWcbaWdbiaadIhaa8aabaWdbiaaicdaaaGccqGH 9aqpcaaIXaGaaiilaiaadIeapaWaaSbaaSqaa8qacaWG4baapaqaba GcpeGaeyypa0JaaGimaiaacYcacaWGdbWdamaaBaaaleaapeGaaGim aaWdaeqaaOWdbiabg2da9iaaicdacaGGSaGaam4sa8aadaWgaaWcba Wdbiaaigdaa8aabeaak8qacqGH9aqpcaaIXaGaaGimaaaa@4BD2@ , are shown.

Figure 8 The dimensionless frequency change curve of the free oscillations of the first mode of the system in terms of the applied voltage for different values of the dimensionless non-local parameter of the nanotube, in SS boundary conditions.

Figure 9 The dimensionless frequency change curve of the free oscillations of the first mode of the system in terms of the applied voltage for different values of the dimensionless non-local parameter of the nanotube, in CC boundary conditions.

In the below curves, the frequency changes of dimensionless free oscillations at the first mode of free vibration of the linear system for two different boundary conditions according to the magnitude of the applied magnetic field in different values of dimensionless non-local parameter, with input parameters σ x 0 =1,V=0.5, C 0 =0, K 1 =10 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi abeo8aZ9aadaqhaaWcbaWdbiaadIhaa8aabaWdbiaaicdaaaGccqGH 9aqpcaaIXaGaaiilaiaadAfacqGH9aqpcaaIWaGaaiOlaiaaiwdaca GGSaGaam4qa8aadaWgaaWcbaWdbiaaicdaa8aabeaak8qacqGH9aqp caaIWaGaaiilaiaadUeapaWaaSbaaSqaa8qacaaIXaaapaqabaGcpe Gaeyypa0JaaGymaiaaicdaaaa@4BE0@ , are shown.

Figure 10 The dimensionless frequency change curve of the free oscillations of the first mode of the system in terms of the magnitude of the magnetic field for different values of the dimensionless non-local parameter of the nanotube, in SS boundary conditions.

Figure 11 The dimensionless frequency change curve of the free oscillations of the first mode of the system in terms of the magnitude of the magnetic field for different values of the dimensionless non-local parameter of the nanotube, in CC boundary conditions.

  1. Effect of hardness-linear damping parameters of viscoelastic foundation

In the below curves, the frequency changes of the dimensionless free oscillations at the first mode of the free vibration of the linear system for two different boundary conditions according to the magnitude of the linear stiffness of the foundation in different forms considering surface effects, with input parameters σ x 0 =1, H x =1× 10 7 ,μ=0.1, C 0 =0,V=1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi abeo8aZ9aadaqhaaWcbaWdbiaadIhaa8aabaWdbiaaicdaaaGccqGH 9aqpcaaIXaGaaiilaiaadIeapaWaaSbaaSqaa8qacaWG4baapaqaba GcpeGaeyypa0JaaGymaiabgEna0kaaigdacaaIWaWdamaaCaaaleqa baWdbiaaiEdaaaGccaGGSaGaeqiVd0Maeyypa0JaaGimaiaac6caca aIXaGaaiilaiaadoeapaWaaSbaaSqaa8qacaaIWaaapaqabaGcpeGa eyypa0JaaGimaiaacYcacaWGwbGaeyypa0JaaGymaaaa@542B@ , are shown.

Figure 12 The dimensionless frequency change curve of free oscillations of the first mode of the system in terms of dimensionless linear stiffness of the substrate for different forms considering surface effects, in SS boundary conditions.

Figure 13 The dimensionless frequency change curve of free oscillations of the first mode of the system in terms of dimensionless linear stiffness of the substrate for different forms considering surface effects, in SS boundary conditions.

In the below curves, the frequency changes of the dimensionless free oscillations at the first mode of the free vibration of the linear system for two different boundary conditions according to the linear damping coefficient of the foundation in different forms considering surface effects, with input parameters σ x 0 =1, H x =1× 10 7 ,μ=0.1, K 1 =100,V=0.5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi abeo8aZ9aadaqhaaWcbaWdbiaadIhaa8aabaWdbiaaicdaaaGccqGH 9aqpcaaIXaGaaiilaiaadIeapaWaaSbaaSqaa8qacaWG4baapaqaba GcpeGaeyypa0JaaGymaiabgEna0kaaigdacaaIWaWdamaaCaaaleqa baWdbiaaiEdaaaGccaGGSaGaeqiVd0Maeyypa0JaaGimaiaac6caca aIXaGaaiilaiaadUeapaWaaSbaaSqaa8qacaaIXaaapaqabaGcpeGa eyypa0JaaGymaiaaicdacaaIWaGaaiilaiaadAfacqGH9aqpcaaIWa GaaiOlaiaaiwdaaaa@5719@ , are shown.

Figure 14 The dimensionless frequency change curve of real and imaginary parts of the special values of the first mode of the system in terms of dimensionless linear damping of the substrate for different forms considering surface effects, in SS boundary conditions.

Figure 15 The dimensionless frequency change curve of real and imaginary parts of the special values of the first mode of the system in terms of dimensionless linear damping of the substrate for different forms considering surface effects, in CC boundary conditions.

Frequency results of nonlinear vibration of the non-damped system

In the below curves, the results of the nonlinear vibration for the non-damped system are presented.

For H x =0,μ=0.1, K 3 =100,V=0.5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadIeapaWaaSbaaSqaa8qacaWG4baapaqabaGcpeGaeyypa0JaaGim aiaacYcacqaH8oqBcqGH9aqpcaaIWaGaaiOlaiaaigdacaGGSaGaam 4sa8aadaWgaaWcbaWdbiaaiodaa8aabeaak8qacqGH9aqpcaaIXaGa aGimaiaaicdacaGGSaGaamOvaiabg2da9iaaicdacaGGUaGaaGynaa aa@4C17@ :

Figure 16 Graph of Nonlinear Frequency to Linear Frequency Ratio of the Immortal System Base Mode for Different Values of the foundation Linear Hardness Parameter, Left: SS boundary condition, right: CC boundary condition.

For H x =0,μ=0.1, K 1 =100, K 3 =100 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadIeapaWaaSbaaSqaa8qacaWG4baapaqabaGcpeGaeyypa0JaaGim aiaacYcacqaH8oqBcqGH9aqpcaaIWaGaaiOlaiaaigdacaGGSaGaam 4sa8aadaWgaaWcbaWdbiaaigdaa8aabeaak8qacqGH9aqpcaaIXaGa aGimaiaaicdacaGGSaGaam4sa8aadaWgaaWcbaWdbiaaiodaa8aabe aak8qacqGH9aqpcaaIXaGaaGimaiaaicdaaaa@4D3F@ :

Figure 17 Graph of Nonlinear to Linear Frequency Ratio Changes of the Immortal System Base Mode for Different Values of Applied Voltage, Left: SS Boundary Conditions, Right: CC Boundary Conditions.

For μ=0.1, K 1 =100, K 3 =100,V=0.5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi abeY7aTjabg2da9iaaicdacaGGUaGaaGymaiaacYcacaWGlbWdamaa BaaaleaapeGaaGymaaWdaeqaaOWdbiabg2da9iaaigdacaaIWaGaaG imaiaacYcacaWGlbWdamaaBaaaleaapeGaaG4maaWdaeqaaOWdbiab g2da9iaaigdacaaIWaGaaGimaiaacYcacaWGwbGaeyypa0JaaGimai aac6cacaaI1aaaaa@4D4D@ :

Figure 18 Graph of Nonlinear Frequency to Linear Frequency Ratio of the Immortal System Base Mode for Different Values of Applied Magnetic Field, Left: SS Boundary Conditions, Right: CC Boundary Conditions.

For H x =1× 10 7 , K 1 =100, K 3 =100,V=0.5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aabIeapaWaaSbaaSqaa8qacaqG4baapaqabaGcpeGaeyypa0JaaGym aiabgEna0kaaigdacaaIWaWdamaaCaaaleqabaWdbiaaiEdaaaGcca GGSaGaae4sa8aadaWgaaWcbaWdbiaaigdaa8aabeaak8qacqGH9aqp caaIXaGaaGimaiaaicdacaGGSaGaae4sa8aadaWgaaWcbaWdbiaaio daa8aabeaak8qacqGH9aqpcaaIXaGaaGimaiaaicdacaGGSaGaaeOv aiabg2da9iaaicdacaGGUaGaaGynaaaa@5102@  

Figure 19 Graph of changes in the ratio of nonlinear to linear frequency ratio of the base mode of the immortal system for different values of the non-local parameter of the nanotube, left: SS boundary condition, right: CC boundary condition.

In the following two diagrams for zero voltage and negative voltage, nonlinear to linear frequency ratio diagrams are drawn for SS boundary conditions:

Figure 20 Graph of Nonlinear Frequency to Linear Frequency Ratio of Immortal System Base Mode for Different Forms Considering Surface Effects without Voltage Applied to SS Boundary Conditions.

Figure 21 Graph of changes in the ratio of nonlinear to linear frequency ratio of the base mode of the immortal system for different forms Considering surface effects with applied voltage V = -0.5, for SS boundary conditions.

Amplitude-frequency response curves of nonlinear forced vibration

For V=0, H x =0, K 1 =0, K 3 =50, C 0 =0.15, C 2 =0, x 0 ¯ =0.5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aabAfacqGH9aqpcaaIWaGaaiilaiaabIeapaWaaSbaaSqaa8qacaqG 4baapaqabaGcpeGaeyypa0JaaGimaiaacYcacaqGlbWdamaaBaaale aapeGaaGymaaWdaeqaaOWdbiabg2da9iaaicdacaGGSaGaae4sa8aa daWgaaWcbaWdbiaaiodaa8aabeaak8qacqGH9aqpcaaI1aGaaGimai aacYcacaqGdbWdamaaBaaaleaapeGaaGimaaWdaeqaaOWdbiabg2da 9iaaicdacaGGUaGaaGymaiaaiwdacaGGSaGaae4qa8aadaWgaaWcba Wdbiaaikdaa8aabeaak8qacqGH9aqpcaaIWaGaaiila8aadaqdaaqa a8qacaqG4bWdamaaBaaaleaapeGaaGimaaWdaeqaaaaak8qacqGH9a qpcaaIWaGaaiOlaiaaiwdaaaa@59F2@ :

Figure 22 Amplitude-frequency vibration curves for different values of μ (μ = 0 simple line, μ = 0.2 bold fold line); Figure above: SS boundary conditions, bottom figure: CC boundary conditions.

For V=0, H x =0, K 1 =0, K 3 =50, C 0 =0.15,μ=0.1, x 0 ¯ =0.5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aabAfacqGH9aqpcaaIWaGaaiilaiaabIeapaWaaSbaaSqaa8qacaqG 4baapaqabaGcpeGaeyypa0JaaGimaiaacYcacaqGlbWdamaaBaaale aapeGaaGymaaWdaeqaaOWdbiabg2da9iaaicdacaGGSaGaae4sa8aa daWgaaWcbaWdbiaaiodaa8aabeaak8qacqGH9aqpcaaI1aGaaGimai aacYcacaqGdbWdamaaBaaaleaapeGaaGimaaWdaeqaaOWdbiabg2da 9iaaicdacaGGUaGaaGymaiaaiwdacaGGSaGaaeiVdiabg2da9iaaic dacaGGUaGaaGymaiaacYcapaWaa0aaaeaapeGaaeiEa8aadaWgaaWc baWdbiaaicdaa8aabeaaaaGcpeGaeyypa0JaaGimaiaac6cacaaI1a aaaa@5AAA@ :

V=0,  H x =0,  K 1 =0, K 3 =50,  C 0 =0.15, μ=0.1,  x ¯ 0 =0.5: MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadAfacqGH9aqpcaaIWaGaaiilaiaacckacaWGibWaaSbaaSqaaiaa dIhaaeqaaOGaeyypa0JaaGimaiaacYcacaGGGcGaam4samaaBaaale aacaaIXaaabeaakiabg2da9iaaicdacaGGSaGaam4samaaBaaaleaa caaIZaaabeaakiabg2da9iaaiwdacaaIWaGaaiilaiaacckacaWGdb WaaSbaaSqaaiaaicdaaeqaaOGaeyypa0JaaGimaiaac6cacaaIXaGa aGynaiaacYcacaGGGcGaeqiVd0Maeyypa0JaaGimaiaac6cacaaIXa GaaiilaiaacckaceWG4bGbaebadaWgaaWcbaGaaGimaaqabaGccqGH 9aqpcaaIWaGaaiOlaiaaiwdacaGG6aaaaa@6050@

Figure 23 Amplitude-frequency vibration curves for different values of C_2, (C_2 = 1 simple line, C_2 = 3 bold fold lines), Figure above: SS boundary conditions, bottom figure: CC boundary conditions.

For V=0, H x =0, K 1 =0, K 3 =50, C 2 =1,μ=0.1, x 0 ¯ =0.5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aabAfacqGH9aqpcaaIWaGaaiilaiaabIeapaWaaSbaaSqaa8qacaqG 4baapaqabaGcpeGaeyypa0JaaGimaiaacYcacaqGlbWdamaaBaaale aapeGaaGymaaWdaeqaaOWdbiabg2da9iaaicdacaGGSaGaae4sa8aa daWgaaWcbaWdbiaaiodaa8aabeaak8qacqGH9aqpcaaI1aGaaGimai aacYcacaqGdbWdamaaBaaaleaapeGaaGOmaaWdaeqaaOWdbiabg2da 9iaaigdacaGGSaGaaeiVdiabg2da9iaaicdacaGGUaGaaGymaiaacY capaWaa0aaaeaapeGaaeiEa8aadaWgaaWcbaWdbiaaicdaa8aabeaa aaGcpeGaeyypa0JaaGimaiaac6cacaaI1aaaaa@5881@ :  

V=0,  H x =0,  K 1 =0, K 3 =50,  C 2 =1, μ=0.1,  x ¯ 0 =0.5: MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadAfacqGH9aqpcaaIWaGaaiilaiaacckacaWGibWaaSbaaSqaaiaa dIhaaeqaaOGaeyypa0JaaGimaiaacYcacaGGGcGaam4samaaBaaale aacaaIXaaabeaakiabg2da9iaaicdacaGGSaGaam4samaaBaaaleaa caaIZaaabeaakiabg2da9iaaiwdacaaIWaGaaiilaiaacckacaWGdb WaaSbaaSqaaiaaikdaaeqaaOGaeyypa0JaaGymaiaacYcacaGGGcGa eqiVd0Maeyypa0JaaGimaiaac6cacaaIXaGaaiilaiaacckaceWG4b GbaebadaWgaaWcbaGaaGimaaqabaGccqGH9aqpcaaIWaGaaiOlaiaa iwdacaGG6aaaaa@5E27@

Figure 24 Amplitude-frequency vibration curves for different values of C_0, (C_0 = 0.1 simple line, C_0 = 0.3 bold fold line), high: SS boundary conditions, low: CC boundary conditions.

For H x =0, K 1 =100, K 3 =50, C 0 =0.1, C 2 =1, μ=0.1, x 0 ¯ =0.5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aabIeapaWaaSbaaSqaa8qacaqG4baapaqabaGcpeGaeyypa0JaaGim aiaacYcacaqGlbWdamaaBaaaleaapeGaaGymaaWdaeqaaOWdbiabg2 da9iaaigdacaaIWaGaaGimaiaacYcacaqGlbWdamaaBaaaleaapeGa aG4maaWdaeqaaOWdbiabg2da9iaaiwdacaaIWaGaaiilaiaaboeapa WaaSbaaSqaa8qacaaIWaaapaqabaGcpeGaeyypa0JaaGimaiaac6ca caaIXaGaaiilaiaaboeapaWaaSbaaSqaa8qacaaIYaaapaqabaGcca qG9aGaaeymaiaabYcapeGaaiiOaiaabY7acqGH9aqpcaaIWaGaaiOl aiaaigdacaGGSaWdamaanaaabaWdbiaabIhapaWaaSbaaSqaa8qaca aIWaaapaqabaaaaOWdbiabg2da9iaaicdacaGGUaGaaGynaaaa@5D54@ :

H x =0,  K 1 =100, K 3 =50,  C 0 =0.1,  C 2 =1, μ=0.1,  x ¯ 0 =0.5: MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqkY=MjYJH8sqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaaWdbi aadIeadaWgaaWcbaGaamiEaaqabaGccqGH9aqpcaaIWaGaaiilaiaa cckacaWGlbWaaSbaaSqaaiaaigdaaeqaaOGaeyypa0JaaGymaiaaic dacaaIWaGaaiilaiaadUeadaWgaaWcbaGaaG4maaqabaGccqGH9aqp caaI1aGaaGimaiaacYcacaGGGcGaam4qamaaBaaaleaacaaIWaaabe aakiabg2da9iaaicdacaGGUaGaaGymaiaacYcacaGGGcGaam4qamaa BaaaleaacaaIYaaabeaakiabg2da9iaaigdacaGGSaGaaiiOaiabeY 7aTjabg2da9iaaicdacaGGUaGaaGymaiaacYcacaGGGcGabmiEayaa raWaaSbaaSqaaiaaicdaaeqaaOGaeyypa0JaaGimaiaac6cacaaI1a GaaiOoaaaa@61E6@

Figure 25 Amplitude-frequency vibration curves for different values of V, (V = -0.5 purple line, V = 0 blue fold line, V = 0.5 bold black), high: SS boundary conditions, low: CC boundary conditions.

According to the presented diagrams, the presence of a non-local parameter has increased the curvature deviation to the right and the stiffening effect. In other words, the non-local parameter is a factor to increase the nonlinear effect of the system. On the other hand, the presence of this factor reduces the maximum range of fluctuations.

Conclusion

By comparing the graphs related to two different boundary conditions, the following results are inferred:

  1. At the C-C boundary conditions, the diagram is collected and the amount of deviation to the right is reduced. For example, the deviation parameters for the maximum dynamic response for the S-S and C-C boundary condition are 0.065 and 0.0045, respectively. In addition, the dynamic response decreases with less degrees of freedom in boundary conditions. For example, in these diagrams, the maximum dynamic range for S-S and C-C are 0.79 and 0.26, respectively. Also, the damping of the system has a greater effect in the boundary conditions of C-C and has significantly reduced the sharpness of the diagram. Therefore, for the C-C boundary conditions, the effect of substrate damping in reducing the dynamic response at excitation frequencies close to the resonant state of the system was greater.
  2. The higher the foundation attenuation, the lower the amplitude of the amplitude-frequency curve, and consequently the lower the maximum dynamic response. Of course, it should be noted that the amplitude-excitation frequency curve for a system with C-C boundary conditions is more sensitive to the damping changes of the foundation than the amplitude-frequency curve of the system excitation of the S-S boundary conditions.
  3. As the nonlinear hardness coefficient of the substrate increases, the amplitude-frequency curve deviates to the right, but the maximum dynamic response amplitude remains constant. By looking more closely at the two diagrams, it could be found that the effect of this parameter on deviating of the diagram is greater for S-S boundary conditions due to more flexibility, so the dynamic response in the CC boundary condition is less sensitive to the increasing nonlinear foundation stiffness parameter.
  4. According to the obtained results, increasing the linear stiffness of the substrate has led to a decrease in the dynamic response as well as the peak deviation of the curve. Increasing the linear stiffness of the system reduces the nonlinear effects of the system. In addition, in the C-C boundary conditions, where the overall stiffness of the system is higher, this factor is less effective.
  5. The presence of a magnetic factor, in addition to reducing the amplitude of the dynamic response, has led to a decrease in the amplitude of the peak and also a reduction in its curvature to the right. However, these effects have attracted more attention for S-S boundary conditions, which have a more flexible system than for C-C boundary conditions.
  6. According to the curves, it is quite clear that by changing the applied voltage from zero to a positive value, the response amplitude has increased sharply and the deviation to the right has increased as well. However, by changing the voltage from zero to a negative value, both the dynamic response amplitude and the peak deviation of the graph are reduced as well as nonlinear effects of the system. It is also important to note that under the same conditions these changes for C-C boundary conditions are much less than for S-S boundary conditions and in general it could be said that the sensitivity of the dynamic response curve of the system with C-C boundary condition to most investigated changes in the system input parameters such as foundation stiffness, voltage, and magnetic factor, non-local factor and surface effects are less than the sensitivity of the system with SS boundary conditions.
  7. It could be seen that by considering the parameter of residual surface stresses for the system, the response amplitude, and the amount of curvature deviation to the right are reduced.
  8. It is predictable that as the load affect position moves away from the center of the wire toward the supports, the amplitude of the dynamic response decreases significantly, and this relative reduction is greater for the C-C boundary condition than for the S-S boundary condition. It is also important to note that the location of the load has no effect on the rate of deviation of the curve peak and the degree of nonlinearity of the dynamic response of the system.
  9. Given that the voltage applied to the system is positive for the last two numerical diagrams. It is expected that the graph with the highest deviation and the lowest peak amplitude is related to Form 4 and the graph with the lowest deviation and the lowest peak amplitude is also related to Form 2. In addition, in this case, by comparing the two curves of Form 1 and the main form, it can be concluded that by considering the surface parameters (Form 1), the maximum peak amplitude and the deviation of the graph have increased.

Acknowledgments

None.

Funding

None.

Conflicts of interest

The authors declare that they have no competing interests.

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