Variational Iteration Method for the Solution of Differential Equation of Motion of the Mathematical Pendulum and Duffing-Harmonic Oscillator

In this work, the differential equation of motion of the undamped mathematical pendulum and Duffing-harmonic oscillator are discussed by using the variational iteration method. Additionally, common problems of pendulum are classified and Lagrange multipliers are obtained for each type of problem. Examples are given for illustration.


Introduction
Vibration of dynamical systems can be divided into two main classes like discrete and distributed.The variables in discrete systems depend on time only, whereas in distributed systems such as beams, plates, etc. variables depend on time and space.Therefore, equations of motion of discrete systems are described by ordinary differential equations, while equations of motion of distributed systems are described by partial differential equations [1].

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The procedure presented in this paper can be simply extended to solve more complex vibration problems; such as aeroelasticity, random vibrations etc.

Variational Iteration Method
In order to illustrate the basic concepts of VIM, the following nonlinear partial differential equation can be considered where R is a linear operator which has partial derivatives with respect to is a nonlinear term and ( ) According to VIM, the following iteration formula can be constructed.
where λ is the general Lagrange multiplier which can be identified optimally via variational theory, In this example, Mathematical Pendulum that was studied by He [15,16] is considered.
The differential equation of motion of the undamped mathematical pendulum is given by .0 sin The initial conditions for this problem are as follows: ( ) The sin y term in Eq. ( 3) is a nonlinear term and it can be expanded as .6 Substituting Eq. ( 4) into Eq.( 5) gives A more detailed form of this mathematical pendulum was investigated by He [15,16].
The Lagrange multiplier of this problem is Hence the iteration formula is The complementary solution of this problem that is used as an initial approximation is given by where α is an unknown constant.
The period can be expressed as follows .In this example, the problem that was studied by Nayfeh and Mook [17] is considered.
The differential equation of motion is given by, .0 The initial conditions for this problem are as follows: ( ) The Lagrange multiplier of this problem is The iteration formula is given by 105 The complementary solution of this problem that is used as an initial approximation is given by where α is an unknown constant.
Substituting the initial approximation given by Eq. ( 18), the following residual is obtained as follows ( ) ).
The coefficient of the ( ) The frequency that is obtained by Nayfeh and Mook [17] using the perturbation method is .8 Note that Eq. ( 23) is valid only for small ε values.However, the frequency expression given by Eq. ( 22) is valid for all ε values and takes the following form for small ε 106 values .128 27 8 Example 3.
In this example, the Duffing-harmonic oscillator that was studied by Mickens [18] and Lim and Wu [19] is considered.
The differential equation of motion is given by, .0 1 The initial conditions for this problem are as follows: ( ) For small y values, Eq. (25) reduces to .0 3 The following form of Eq. ( 27) is going to be studied in this example ( ) .0 1 where α is an unknown constant.
Substituting the initial approximation into Eq.(28), the following residual is obtained ( ) ( ) In order to discard the seculer terms, the coefficient of ( ) which is the same with the one found by Mickens [18].
The iteration formula is given by with ω defined in Eq. (30).
hand, for large y values, Eq. (25) reduces to .(26a) and (26b) respectively, it is noticed that for small y values, Eq. (27) reduces to the equation of motion of the Duffing-type nonlinear oscillator while for large y values, it reduces to the equation of motion of a linear harmonic oscillator.Therefore, Eq. (27) is called as Duffing-harmonic oscillator equation of motion.