analytical study on nonlinear vibration of carbon ...€¦ · vibration, timoshenko beam theory. 1....

Vol. 1(1), Apr. 2015, PP. 01-10

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Article History: JKBEI DOI: 649123/11010 Received Date: 05 Dec. 2014 Accepted Date: 23 Feb. 2015 Available Online: 06 Mar. 2015

Analytical Study on Nonlinear Vibration of Carbon Nanotubes Using

Nonlocal Timoshenko Beam Theory S.E.Ghasemi1*, S.Gouran2 and J.Vahidi3

1 Department of Mechanical Engineering, Qaemshahr Branch, Islamic Azad University, Qaemshahr, Iran 2 Department of Mechanical Engineering, Babol University of Technology, Babol, Iran.

3 Department of Applied Mathematics, Iran University of Science and Technology, Tehran, Iran *Corresponding Author's E-mail: [email protected]

Abstract n this paper, variational iteration (VIM) and energy balance (EBM) methods have been used to investigate non-linear vibration of Timoshenko beam theory . The proposed methods do not require small parameter in the equation which is difficult to be found for nonlinear problems.

Comparison of VIM and EBM with Runge-Kutta 4th leads to highly accurate solutions.

Keywords: Variational Iteration Method (VIM), Energy balance Method (EBM),Galerkin method, non-linear vibration, Timoshenko beam theory. 1. Introduction

The demand for engineering structures is continuously increasing. Aerospace vehicles, bridges, and automobiles are examples of these structures. Many aspects have to be taken into consideration in the design of these structures to improve their performance and extend their life. One aspect of the design process is the dynamic response of structures. The dynamics of distributed-parameter and continuous systems, like beams, were governed by linear and nonlinear partial-differential equations in space and time. It was difficult to find the exact or closed-form solutions for nonlinear problems. Consequently, researchers were used two classes of approximate solutions of initial boundary-value problems: numerical techniques [28, 31], and approximate analytical methods [2, 26]. For strongly non-linear partial-differential, direct techniques, such as perturbation methods, were not utilized to solve directly the non-linear partial-differential equations and associated boundary conditions. Therefore first partial-differential equations are discretized into a set of non-linear ordinary-differential equations using the Galerkin approach and the governing problems are then solved analytically in time domain. Approximate methods for studying non-linear vibrations of beams are important for investigating and designing purposes. In recent years, some promising approximate analytical solutions have been proposed, such as Frequency Amplitude Formulation [13], Variational Iteration [5, 6, 14, 17], Homotopy-Perturbation [3, 4, 7, 24], Parametrized-Perturbation [18], Max-Min [15, 19, 29], Differential Transform Method [16], Adomian Decomposition Method [22], Energy Balance [23, 30], etc.

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S.E. Ghasemi et al. / Vol. 1(1), Apr. 2015, pp. 01-10 JKBEI DOI: 649123/11010

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ISSN: 2413-6794 (Online)

2. Nonlinear vibration analysis of nonlocal SWCNTs

Figure. 1: A single wall carbon nanotube (SWCNT) modeled as a nonlocal Timoshenko

nanobeam. Fig. 1 shows a SWCNT modeled as a Timoshenko nanobeam with length L, radius r, and effective tube thickness h. It is assumed that the SWCNTs vibrate only in the x–z plane. Based on Timoshenko beam theory ,the displacements of an arbitrary point in the beam along the x- and z-axes, denoted by 𝑈𝑈�(𝑥𝑥, 𝑧𝑧, 𝑡𝑡) and 𝑊𝑊� (𝑥𝑥, 𝑧𝑧, 𝑡𝑡), respectively ,are 𝑈𝑈�(𝑥𝑥, 𝑧𝑧, 𝑡𝑡) = 𝑈𝑈(𝑥𝑥, 𝑡𝑡) + 𝑧𝑧𝑧𝑧(𝑥𝑥, 𝑡𝑡), 𝑊𝑊� (𝑥𝑥, 𝑧𝑧, 𝑡𝑡) = 𝑊𝑊(𝑥𝑥, 𝑡𝑡) Where 𝑈𝑈(𝑥𝑥, 𝑡𝑡) and 𝑊𝑊(𝑥𝑥, 𝑡𝑡) are displacement components in the midplane, ϕ is the rotation of beam cross-section and 𝑡𝑡 is time. The nonlinear equations of motion for the nonlocal SWCNTs modeled as a Timoshenko nanobeam is given by:

𝐸𝐸𝐸𝐸 �𝜕𝜕2𝑈𝑈𝜕𝜕𝑥𝑥2

+ �𝜕𝜕𝜕𝜕𝜕𝜕𝑥𝑥� �𝜕𝜕

2𝜕𝜕𝜕𝜕𝑥𝑥2

�� = 𝜌𝜌𝐸𝐸 𝜕𝜕2

𝜕𝜕𝜕𝜕2(𝑈𝑈 − (𝑒𝑒0𝑎𝑎)2 �

𝜕𝜕2𝑈𝑈𝜕𝜕𝑥𝑥2

�) (1)

𝐾𝐾𝑠𝑠𝐺𝐺𝐸𝐸 �𝜕𝜕2𝜕𝜕𝜕𝜕𝑥𝑥2

+ 𝜕𝜕𝜕𝜕𝜕𝜕𝑥𝑥� + 𝛴𝛴1 − (𝑒𝑒0𝑎𝑎)2𝛴𝛴2 = 𝜌𝜌𝐸𝐸

𝜕𝜕2

𝜕𝜕𝜕𝜕2�𝑊𝑊 − (𝑒𝑒0𝑎𝑎)2 �

𝜕𝜕2𝜕𝜕𝜕𝜕𝑥𝑥2

�� −

𝜌𝜌𝐸𝐸(𝑒𝑒0𝑎𝑎)2𝜕𝜕2

𝜕𝜕𝜕𝜕2(𝜕𝜕

2𝑈𝑈𝜕𝜕𝑥𝑥2

− (𝑒𝑒0𝑎𝑎)2 �𝜕𝜕4𝑈𝑈𝜕𝜕𝑥𝑥4

�) (2)

𝐸𝐸𝐸𝐸 𝜕𝜕2𝜕𝜕𝜕𝜕𝑥𝑥2

− 𝐾𝐾𝑠𝑠𝐺𝐺𝐸𝐸 �𝜕𝜕𝜕𝜕𝜕𝜕𝑥𝑥

+ 𝑧𝑧� = 𝜌𝜌𝐸𝐸 𝜕𝜕2

𝜕𝜕𝜕𝜕2(𝑧𝑧 − (𝑒𝑒0𝑎𝑎)2 �

𝜕𝜕2𝜕𝜕𝜕𝜕𝑥𝑥2

� (3) Where

𝛴𝛴1 = 𝐸𝐸𝐸𝐸(�𝜕𝜕2𝑈𝑈𝜕𝜕𝑥𝑥2

� �𝜕𝜕𝜕𝜕𝜕𝜕𝑥𝑥� + 3

2�𝜕𝜕𝜕𝜕𝜕𝜕𝑥𝑥�2�𝜕𝜕

2𝜕𝜕𝜕𝜕𝑥𝑥2

� + �𝜕𝜕𝑈𝑈𝜕𝜕𝑥𝑥� �𝜕𝜕

2𝜕𝜕𝜕𝜕𝑥𝑥2

�)

𝛴𝛴2 = 𝐸𝐸𝐸𝐸��𝜕𝜕4𝑈𝑈𝜕𝜕𝑥𝑥4

� �𝜕𝜕𝑊𝑊𝜕𝜕𝑥𝑥

� + 3�𝜕𝜕3𝑈𝑈𝜕𝜕𝑥𝑥3

��𝜕𝜕2𝑊𝑊𝜕𝜕𝑥𝑥2

� + 3�𝜕𝜕2𝑈𝑈𝜕𝜕𝑥𝑥2

��𝜕𝜕3𝑊𝑊𝜕𝜕𝑥𝑥3

� + �𝜕𝜕𝑈𝑈𝜕𝜕𝑥𝑥� �𝜕𝜕4𝑊𝑊𝜕𝜕𝑥𝑥4

�� + 𝐸𝐸𝐸𝐸(3�𝜕𝜕2𝑊𝑊𝜕𝜕𝑥𝑥2

�3

+ 9 �𝜕𝜕𝜕𝜕

𝜕𝜕𝑥𝑥� �𝜕𝜕

2𝜕𝜕𝜕𝜕𝑥𝑥2

� �𝜕𝜕3𝜕𝜕𝜕𝜕𝑥𝑥3

� + 32

(𝜕𝜕𝜕𝜕𝜕𝜕𝑥𝑥

)2 �𝜕𝜕4𝜕𝜕𝜕𝜕𝑥𝑥4

�) where A is the cross-sectional area of the beam, I is the second moment of area and ρ is the mass density of beam material, E and G are Young’s modulus and shear modulus, respectively. The constitutive relations in classical elasticity theories can be recovered by setting the nonlocal parameter 𝑒𝑒0𝑎𝑎 = 0 and 𝐾𝐾𝑠𝑠 is the shear correction factor depending on the shape of the cross-section of the beam. Introducing the following dimensionless quantities: 𝜁𝜁 = 𝑥𝑥

𝐿𝐿 , (𝑢𝑢,𝑤𝑤) = (𝑈𝑈,𝜕𝜕)

𝑟𝑟 , 𝑧𝑧 = 𝜓𝜓 , (𝐸𝐸1, 𝐸𝐸3) = (

𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝜌

, 𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝜌𝑟𝑟2

) , 𝜂𝜂 = 𝐿𝐿𝑟𝑟 ,

𝜇𝜇 = 𝑒𝑒0𝑎𝑎𝐿𝐿

, (𝑎𝑎11 ,𝑎𝑎55 ,𝑑𝑑11) = (𝐸𝐸𝜌𝜌𝐸𝐸𝜌𝜌

, 𝐾𝐾𝑠𝑠𝐺𝐺𝜌𝜌𝐸𝐸𝜌𝜌

, 𝐸𝐸𝜌𝜌𝐸𝐸𝜌𝜌𝑟𝑟2

), 𝜏𝜏 = 𝜕𝜕𝐿𝐿 �

𝐸𝐸𝜌𝜌

(4)

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Eq. (1),(2),(3) can be expressed in dimensionless form as:

𝑎𝑎11 �𝜕𝜕2𝑢𝑢𝜕𝜕𝜁𝜁2

+ 1𝜂𝜂�𝜕𝜕𝜕𝜕𝜕𝜕𝜁𝜁� �𝜕𝜕

2𝜕𝜕𝜕𝜕𝜁𝜁2

�� = 𝐸𝐸1𝜕𝜕2

𝜕𝜕𝜕𝜕2(𝑢𝑢 − µ2 �𝜕𝜕

2𝑢𝑢𝜕𝜕𝜁𝜁2

�) (5)

𝑎𝑎55 �𝜕𝜕2𝜕𝜕𝜕𝜕𝜁𝜁2

+ 𝜂𝜂 𝜕𝜕𝜕𝜕𝜕𝜕𝜁𝜁� + 𝛴𝛴1�� − µ2𝛴𝛴2�� = 𝐸𝐸1

𝜕𝜕2

𝜕𝜕𝜕𝜕2�𝑤𝑤 − µ2 �𝜕𝜕

2𝜕𝜕𝜕𝜕𝜁𝜁2

�� − 𝐸𝐸1𝜕𝜕2

𝜕𝜕𝜕𝜕2(µ2 𝜕𝜕

2𝑢𝑢𝜕𝜕𝜁𝜁2

− µ4 �𝜕𝜕4𝑢𝑢

𝜕𝜕𝑥𝑥4�)

(6)

𝑑𝑑11𝜕𝜕2𝜕𝜕𝜕𝜕𝜁𝜁2

− 𝑎𝑎55𝜂𝜂 �𝜕𝜕𝜕𝜕𝜕𝜕𝜁𝜁

+ 𝜂𝜂𝜓𝜓� = 𝐸𝐸3𝜕𝜕2

𝜕𝜕𝜕𝜕2(𝜓𝜓 − µ2 �𝜕𝜕

2𝜕𝜕𝜕𝜕𝜁𝜁2

� (7) Where

𝛴𝛴1�� =𝑎𝑎11𝜂𝜂

(�𝜕𝜕2𝑢𝑢𝜕𝜕𝜁𝜁2

� �𝜕𝜕𝜕𝜕𝜕𝜕𝜁𝜁� + 3

2𝜂𝜂�𝜕𝜕𝜕𝜕𝜕𝜕𝜁𝜁�2�𝜕𝜕

2𝜕𝜕𝜕𝜕𝜁𝜁2

� + �𝜕𝜕𝑢𝑢𝜕𝜕𝜁𝜁� �𝜕𝜕

2𝜕𝜕𝜕𝜕𝜁𝜁2

�)

𝛴𝛴2�� =𝑎𝑎11𝜂𝜂��𝜕𝜕4𝑢𝑢𝜕𝜕𝜁𝜁4

� �𝜕𝜕𝑤𝑤𝜕𝜕𝜁𝜁� + 3�

𝜕𝜕3𝑢𝑢𝜕𝜕𝜁𝜁3

��𝜕𝜕2𝑤𝑤𝜕𝜕𝜁𝜁2

� + 3�𝜕𝜕2𝑢𝑢𝜕𝜕𝜁𝜁2

��𝜕𝜕3𝑤𝑤𝜕𝜕𝜁𝜁3

� + �𝜕𝜕𝑢𝑢𝜕𝜕𝜁𝜁� �𝜕𝜕4𝑤𝑤𝜕𝜕𝜁𝜁

�� +𝑎𝑎11𝜂𝜂2

(3�𝜕𝜕2𝑤𝑤𝜕𝜕𝜁𝜁2

�3

+

9 �𝜕𝜕𝜕𝜕𝜕𝜕𝜁𝜁� �𝜕𝜕

2𝜕𝜕𝜕𝜕𝜁𝜁2

� �𝜕𝜕3𝜕𝜕𝜕𝜕𝜁𝜁3

� + 32

(𝜕𝜕𝜕𝜕𝜕𝜕𝜁𝜁

)2 �𝜕𝜕4𝜕𝜕𝜕𝜕𝜁𝜁4

�) Assuming 𝜓𝜓(𝜁𝜁 , 𝜏𝜏) = 𝑢𝑢(𝜁𝜁)𝑣𝑣(𝜏𝜏) and 𝑤𝑤(𝜁𝜁 , 𝜏𝜏) = 𝑓𝑓(𝜁𝜁)𝑔𝑔(𝜏𝜏) with ignore U, where 𝑢𝑢(𝜁𝜁) , 𝑓𝑓(𝜁𝜁) is the first eigenmode of the beam [32] and applying the Galerkin method, the equation of motion is obtained as follows: �̈�𝑔(𝜏𝜏) + 𝛼𝛼1𝑔𝑔(𝜏𝜏) + 𝛼𝛼2𝑣𝑣(𝜏𝜏) + 𝛼𝛼3𝑔𝑔3(𝜏𝜏) = 0 (8) �̈�𝑣(𝜏𝜏) + 𝛽𝛽1𝑣𝑣(𝜏𝜏) + 𝛽𝛽2𝑔𝑔(𝜏𝜏) = 0 (9) Where 𝛼𝛼1 , 𝛼𝛼2 , 𝛼𝛼3 , 𝛽𝛽1 and 𝛽𝛽2 are as follows:

𝛼𝛼1 = −𝑎𝑎55∫ 𝑓𝑓(𝜁𝜁)𝑓𝑓´´(𝜁𝜁)𝑑𝑑𝜁𝜁10∫ 𝑓𝑓2(𝜁𝜁)𝑑𝑑10 𝜁𝜁

, 𝛼𝛼2 = −𝑎𝑎55𝜂𝜂∫ 𝑢𝑢´(𝜁𝜁)𝑓𝑓(𝜁𝜁)𝑑𝑑𝜁𝜁10∫ 𝑓𝑓2(𝜁𝜁)𝑑𝑑10 𝜁𝜁

, 𝛼𝛼3 = −3𝑎𝑎112𝜂𝜂2

∫ 𝑓𝑓(𝜁𝜁)𝑓𝑓´2(𝜁𝜁)𝑓𝑓´´(𝜁𝜁)𝑑𝑑𝜁𝜁10∫ 𝑓𝑓2(𝜁𝜁)𝑑𝑑10 𝜁𝜁

,

𝛽𝛽1 = −𝑑𝑑11𝜌𝜌3

∫ 𝑢𝑢(𝜁𝜁)𝑢𝑢´´(𝜁𝜁)𝑑𝑑𝜁𝜁10∫ 𝑢𝑢2(𝜁𝜁)𝑑𝑑10 𝜁𝜁

+ 𝑎𝑎55𝜂𝜂2

𝜌𝜌3 , 𝛽𝛽2 =

𝑎𝑎55𝜂𝜂𝜌𝜌3

∫ 𝑓𝑓´(𝜁𝜁)𝑢𝑢(𝜁𝜁)𝑑𝑑𝜁𝜁10∫ 𝑢𝑢2(𝜁𝜁)𝑑𝑑10 𝜁𝜁

(10)

The Eq. (8),(9) is the governing non-linear vibration of Timoshenko beams. The center of the beam subjected to the following initial conditions: 𝑔𝑔(0) = 𝐸𝐸, �̇�𝑔(0) = 0, 𝑣𝑣(0) = 𝐵𝐵, �̇�𝑣(0) = 0 (11) where A,B denotes the non-dimensional maximum amplitude of oscillation. 3. BASIC IDEA OF VARIATIONAL ITERATION METHOD

To illustrate the basic concepts of the VIM, we consider the following differential equation: 𝐿𝐿𝑢𝑢 + 𝑁𝑁𝑢𝑢 = 𝑔𝑔(𝑡𝑡) (12) Where L is a linear operator, N a nonlinear operator and g(t) an inhomogeneous term. According to VIM, we can write down a correction functional as follows: 𝑢𝑢𝑛𝑛+1(𝑡𝑡) = 𝑢𝑢𝑛𝑛(𝑡𝑡) + ∫ 𝜆𝜆�𝐿𝐿𝑢𝑢𝑛𝑛(𝜂𝜂) + 𝑁𝑁𝑢𝑢�𝑛𝑛(𝜂𝜂) − 𝑔𝑔(𝜂𝜂)�𝑑𝑑𝜂𝜂

𝜕𝜕0 (13)



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Where λ is a general Lagrange multiplier which can be identified optimally via the variational theory [17]. The subscript n indicates the nth approximation and n is considered as a restricted variation [17], i.e. δ n = 0. 4. APPLICATION OF VARIATIONAL ITERATION METHOD

To solve Eq. (8),(9) by means of VIM, we start with an arbitrary initial approximation:

𝑔𝑔0(𝜏𝜏) = 𝐸𝐸𝐴𝐴𝐴𝐴𝐴𝐴(𝜔𝜔𝜏𝜏) , 𝑣𝑣0(𝜏𝜏) = 𝐵𝐵𝐴𝐴𝐴𝐴𝐴𝐴(𝜔𝜔𝜏𝜏) (14)

From Eq. (8), we have:

�̈�𝑔(𝜏𝜏) = −𝛼𝛼1𝑔𝑔(𝜏𝜏) − 𝛼𝛼2𝑣𝑣(𝜏𝜏) − 𝛼𝛼3𝑔𝑔3(𝜏𝜏) ⇒

�̈�𝑔(𝜏𝜏) = −𝛼𝛼1𝐸𝐸𝐴𝐴𝐴𝐴𝐴𝐴(𝜔𝜔𝜏𝜏) − 𝛼𝛼2𝐵𝐵𝐴𝐴𝐴𝐴𝐴𝐴(𝜔𝜔𝜏𝜏) − 𝛼𝛼3𝐸𝐸3𝐴𝐴𝐴𝐴𝐴𝐴3(𝜔𝜔𝜏𝜏) (15)

Integrating twice yields:

𝑔𝑔1(𝜏𝜏) =𝛼𝛼1𝜌𝜌𝐴𝐴𝐴𝐴𝑠𝑠(𝜔𝜔𝜕𝜕)

𝜔𝜔2+ 𝛼𝛼2𝐵𝐵𝐴𝐴𝐴𝐴𝑠𝑠(𝜔𝜔𝜕𝜕)

𝜔𝜔2+ 𝛼𝛼3𝜌𝜌

3

4 (cos (3𝜔𝜔𝜕𝜕)

9𝜔𝜔2+ 3cos (𝜔𝜔𝜕𝜕)

𝜔𝜔2) (16)

Equating the coefficients of 𝐴𝐴𝐴𝐴𝐴𝐴(𝜔𝜔𝜏𝜏) in 𝑔𝑔0 and 𝑔𝑔1 , we have:

𝜔𝜔𝑉𝑉𝜌𝜌𝑉𝑉 = �𝛼𝛼1 +𝛼𝛼2𝐵𝐵𝜌𝜌

+ 0.75𝛼𝛼3𝐸𝐸2 (17)

And therefore,

𝑔𝑔0(𝜏𝜏) = 𝐸𝐸𝐴𝐴𝐴𝐴𝐴𝐴(�𝛼𝛼1 +𝛼𝛼2𝐵𝐵𝜌𝜌

+ 0.75𝛼𝛼3𝐸𝐸2 𝜏𝜏) (18)

Where δ n = 0 is considered as restricted variation.

𝑔𝑔𝑛𝑛+1(𝜏𝜏) = 𝑔𝑔𝑛𝑛(𝜏𝜏) + ∫ 𝜆𝜆 �𝑑𝑑2𝑔𝑔𝑛𝑛𝑑𝑑𝜂𝜂2

+ 𝛼𝛼1𝑔𝑔𝑛𝑛 + 𝛼𝛼2𝑣𝑣𝑛𝑛 + 𝛼𝛼3𝑔𝑔𝑛𝑛3� 𝑑𝑑𝜂𝜂𝜕𝜕0 (19)

𝑣𝑣𝑛𝑛+1(𝜏𝜏) = 𝑣𝑣𝑛𝑛(𝜏𝜏) + ∫ 𝜆𝜆 �𝑑𝑑2𝑣𝑣𝑛𝑛𝑑𝑑𝜂𝜂2

+ 𝛽𝛽1𝑣𝑣𝑛𝑛 + 𝛽𝛽2𝑔𝑔𝑛𝑛� 𝑑𝑑𝜂𝜂𝜕𝜕0 (20)

Its stationary conditions can be obtained as follows:

1 − 𝜆𝜆´|𝜂𝜂=𝜕𝜕 = 0 (21)

𝜆𝜆|𝜂𝜂=𝜕𝜕 = 0 (22)

𝜆𝜆´´ + 𝜔𝜔2𝜆𝜆 = 0 (23)

Therefore, the multiplier, can be identified as

𝜆𝜆 = 1𝜔𝜔𝐴𝐴𝑠𝑠𝑠𝑠𝜔𝜔(𝜂𝜂 − 𝜏𝜏) (24)

As a result, we obtain the following iteration formula:

𝑔𝑔𝑛𝑛+1(𝜏𝜏) = 𝑔𝑔𝑛𝑛(𝜏𝜏) + ∫ (1𝜔𝜔𝐴𝐴𝑠𝑠𝑠𝑠𝜔𝜔(𝜂𝜂 − 𝜏𝜏)) �𝑑𝑑

2𝑔𝑔𝑛𝑛𝑑𝑑𝜂𝜂2

+ 𝛼𝛼1𝑔𝑔𝑛𝑛 + 𝛼𝛼2𝑣𝑣𝑛𝑛 + 𝛼𝛼3𝑔𝑔𝑛𝑛3� 𝑑𝑑𝜂𝜂𝜕𝜕0

(25)

𝑣𝑣𝑛𝑛+1(𝜏𝜏) = 𝑣𝑣𝑛𝑛(𝜏𝜏) + ∫ (1𝜔𝜔𝐴𝐴𝑠𝑠𝑠𝑠𝜔𝜔(𝜂𝜂 − 𝜏𝜏)) �𝑑𝑑

2𝑣𝑣𝑛𝑛𝑑𝑑𝜂𝜂2

+ 𝛽𝛽1𝑣𝑣𝑛𝑛 + 𝛽𝛽2𝑔𝑔𝑛𝑛� 𝑑𝑑𝜂𝜂𝜕𝜕0 (26)

By the iteration formula (25),(26), we can directly obtain other components as:



5



𝑔𝑔1(𝜏𝜏) = 𝐸𝐸𝐴𝐴𝐴𝐴𝐴𝐴(𝜔𝜔𝜏𝜏) +1

32𝜔𝜔2(−𝛼𝛼3𝐸𝐸3 cos(𝜔𝜔𝜏𝜏) + 16𝜔𝜔3𝐸𝐸𝜏𝜏 sin(𝜔𝜔𝜏𝜏) − 16𝜔𝜔𝛼𝛼1𝐸𝐸𝜏𝜏 sin(𝜔𝜔𝜏𝜏) −

16𝜔𝜔𝛼𝛼2𝐵𝐵𝜏𝜏 sin(𝜔𝜔𝜏𝜏) − 12𝜔𝜔𝛼𝛼3𝐸𝐸3𝜏𝜏 sin(𝜔𝜔𝜏𝜏) + 𝛼𝛼3𝐸𝐸3cos (3𝜔𝜔𝜏𝜏) (27)

Where ω is evaluated from Eq. (17). In the same manner, the rest of the components of the iteration

formula can be obtained.

5. APPLICATION OF ENERGY BALANCE METHOD

In order to asses the advantages and the accuracy of the energy balance method; we will apply this

method to the discussed system.

Equation of motion, which reads:

�̈�𝑔(𝜏𝜏) + 𝛼𝛼1𝑔𝑔(𝜏𝜏) + 𝛼𝛼2𝑣𝑣(𝜏𝜏) + 𝛼𝛼3𝑔𝑔3(𝜏𝜏) = 0 (28)

�̈�𝑣(𝜏𝜏) + 𝛽𝛽1𝑣𝑣(𝜏𝜏) + 𝛽𝛽2𝑔𝑔(𝜏𝜏) = 0 (29)

𝑔𝑔(0) = 𝐸𝐸, �̇�𝑔(0) = 0, 𝑣𝑣(0) = 𝐵𝐵, �̇�𝑣(0) = 0 (30)

Its variational formulation can be easily established:

𝐽𝐽(𝑢𝑢) = ∫ �𝑔𝑔´2

2+ 𝛼𝛼1

𝑔𝑔2

2+ 𝛼𝛼2𝑣𝑣𝑔𝑔 + 𝛼𝛼3

𝑔𝑔4

4� 𝑑𝑑𝜏𝜏𝜕𝜕0 (31)

It’s Hamiltonian, therefore, can be written in the form

𝐻𝐻 = 𝑔𝑔´2

2+ 𝛼𝛼1

𝑔𝑔2


𝑔𝑔4

4 (32)

and

𝐻𝐻𝜕𝜕=0 = 𝛼𝛼1𝜌𝜌2

2+ 𝛼𝛼2𝐸𝐸𝐵𝐵 + 𝛼𝛼3

𝜌𝜌4

4 (33)

𝐻𝐻𝜕𝜕 − 𝐻𝐻𝜕𝜕=0 =𝑔𝑔´2

2+ 𝛼𝛼1

𝑔𝑔2


𝑔𝑔4

4− 𝛼𝛼1

𝜌𝜌2

2− 𝛼𝛼2𝐸𝐸𝐵𝐵 − 𝛼𝛼3

𝜌𝜌4

4 (34)

We will use the trial function to determine the angular frequency ω, i.e., 𝑔𝑔0(𝜏𝜏) = 𝐸𝐸𝐴𝐴𝐴𝐴𝐴𝐴(𝜔𝜔𝜏𝜏) , 𝑣𝑣0(𝜏𝜏) = 𝐵𝐵𝐴𝐴𝐴𝐴𝐴𝐴(𝜔𝜔𝜏𝜏) (35) If we substitute (35) into (34), it results the following residual equation: (𝜌𝜌𝜔𝜔sin (𝜔𝜔𝜕𝜕))2

2+ 𝛼𝛼1

(𝜌𝜌𝐴𝐴𝐴𝐴𝑠𝑠(𝜔𝜔𝜕𝜕))2

2+ 𝛼𝛼2𝐸𝐸𝐵𝐵(𝐴𝐴𝐴𝐴𝐴𝐴(𝜔𝜔𝜏𝜏))2 + 𝛼𝛼3

(𝜌𝜌𝐴𝐴𝐴𝐴𝑠𝑠(𝜔𝜔𝜕𝜕))4

4− 𝛼𝛼1

𝜌𝜌2


𝜌𝜌4

4= 0

(36) If we collocate at 𝜔𝜔𝜏𝜏 = 𝜋𝜋

4, we obtain:

(𝜌𝜌𝜔𝜔)2

4+ 𝛼𝛼1

𝜌𝜌2

4+ 𝛼𝛼2𝜌𝜌𝐵𝐵

2+ 𝛼𝛼3

𝜌𝜌4

16− 𝛼𝛼1

𝜌𝜌2


𝜌𝜌4

4= 0 (37)

or

𝜔𝜔𝐸𝐸𝐵𝐵𝑉𝑉 = �𝛼𝛼1 +2𝛼𝛼2𝐵𝐵𝜌𝜌

+ 0.75𝛼𝛼3𝐸𝐸2 (38)

6. RESULTS AND DISCUSSIONS

The behavior of 𝑔𝑔(𝐸𝐸, 𝑡𝑡) obtained by VIM and EBM frequency at 𝛼𝛼1=2.910191164, 𝛼𝛼2=9.828472866, 𝛼𝛼3=0.4357628396 and B=0,0.005 is shown in Figs.2 , 3 and 4. Results of VIM and EBM frequency versus amplitude has been investigated and plotted in Figs.5, 6. Influence of B on



6



frequency (VIM & EBM) and amplitude has been depicted in Figs.7, 8. The solutions are also compared for t=0.5 in Table1. It can be observed that there is an excellent agreement between the results obtained from VIM and EBM with those of Runge-Kutta 4th order method [1].

Figure 2: VIM & EBM deflection at 𝛼𝛼1 = 2.910191164 , 𝛼𝛼2 = 9.828472866 ,

𝛼𝛼3 = 0.4357628396 for B=0

Figure 3: VIM deflection at 𝛼𝛼1 = 2.910191164 , 𝛼𝛼2 = 9.828472866 ,

𝛼𝛼3 = 0.4357628396 for B=0.005



7



Figure 4: EBM deflection at 𝛼𝛼1 = 2.910191164 , 𝛼𝛼2 = 9.828472866 ,

𝛼𝛼3 = 0.4357628396 for B=0.005

Figure 5: Results of VIM frequency versus amplitude



8



Figure 6: Results of EBM frequency versus amplitude

Figure 7: Results of VIM frequency versus amplitude

associated with influence of B



9



Figure 8: Results of EBM frequency versus amplitude

associated with influence of B

Table 1: comparison between VIM & EBM with time marching solution for motion equation (8) When 𝑡𝑡 = 0.5 (𝐴𝐴), 𝛼𝛼1 = 1, 𝛼𝛼2 = 1, 𝛼𝛼3 = 1

A VIM EBM Runge- Kutta Error(VIM) Error(EBM) 0.01 0.009945 0.009945 0.010098371 0.000153 0.000153

0.1 0.099490 0.099490 0.10008375 0.000593 0.000593 0.2 0.198956 0.198956 0.200067529 0.001111 0.001111 0.3 0.298362 0.298362 0.300051358 0.001689 0.001689 0.4 0.397678 0.397678 0.400035263 0.002357 0.002357 0.5 0.496874 0.496874 0.500019267 0.003144 0.003144

1 0.990026 0.990026 0.999941663 0.009915 0.009915 10 5.814516 5.814517 10.00264166 4.188126 4.188125

Conclusions

In this work energy balance method (EBM) and variational iteration method (VIM) are proved to be very convenient and powerful mathematical tools to solving nonlinear oscillators and the solutions obtained are in good agreement with numerical values. References

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