Analytical and Approximate Solutions to the Fee Vibration of Strongly Nonlinear Oscillators

In this paper, we implement a new approach coupled with the iteration method. This procedure is obtained by combining He’s frequency-amplitude formulation and He’s energy balance method into a new iteration procedure such that excellent approximate analytical solutions, valid for small as well as large values of amplitude, can be determined for nonlinear oscillators. This study has clarified the motion equation of nonlinear oscillators by the iteration method to obtain the relationship between amplitude and angular frequency. We compare the approximate periods obtained by our procedure with the numerical solution and with other methods like energy balance method and variational iteration method. The results show that the approximations are of extreme accuracy.

Our main aim in this paper is to apply a new approach coupled with the iteration method by combining He's FAF and He's EBM.We compare the numerical solution with others methods like energy balance method and variational iteration method.We examine accuracy of the approximation methods.

Solution Procedure
We consider a generalized nonlinear oscillator in the form: under the initial conditions: Based on He's frequency-amplitude formulation approach [3,4].The trial function to determine the angular frequency ω is given by cos , substituting from Equation (3) into Equation ( 1), one can obtain the following residual as Introducing a new function,
Solving the above equation, the relationship between the amplitude and frequency of the oscillator can be ob-tained:

Applications
In this section, some practical examples for some strongly nonlinear vibration system are illustrated to show the applicability, accuracy and effectiveness of the present method.

Autonomous Conservative Oscillator
It is known that the free vibrations of an autonomous conservative oscillator with inertia and static type fifthorder non-linearities is expressed by [7,16,17].
The initial conditions for Equation ( 6) are given by and where A represents amplitude of the oscillation.
Motion is assumed to start from the position of maximum displacement with zero velocity  is an integer which may take values of 1   , 0 or −1, and 1 ,   and 4  are positive parameters in Table 1.
By using the following trial function to determine the angular frequency  : Substituting the above trial functions into Equation ( 6) results in, the following residual Using Equation (8) into Equation ( 5), we can easily obtain Solving the above equation, an approximate frequency  as a function of amplitude A as follow: 8 .8 4 3 Hence, the approximate solution can be readily obtained Mehdipour et al. [7] obtained the approximate frequency for Equation ( 6) by the energy balance method The values of dimensionless parameters 1  , 2  , 3  and 4  associated with each of the four calculation modes are shown in Table 2 [7].
Additionally, the comparison between these methodologies can be found in Table 2 and Figure 1.It has been shown that the results of analytical approximate solution are in good agreement with those obtained from the results of energy balance method [7], and with an accurate numerical solution using fourth-order Runge-Kutta method (R-K) as shown in Table 2 and Figure 1.

Tapered Beams
Tapered beams is an important model for engineering structures which require a variable stiffness along the length, such as moving arms and turbine blades [18][19][20].In dimensionless form, the governing differential equation corresponding to fundamental vibration mode of a tapered beam is given by [20].
Assume that the solution can be expressed as: Similarly, substituting the trial solution into Equation (13), this leads to the following residual Using Equation ( 14) into Equation ( 5) we can easily obtain

Table 2. The comparison between energy balance method [7] and analytical approximate solution for four modes (1 -4) (λ = 1).
Solving the above equation, an approximate frequency as a function of amplitude equals: which is the same as given by Hoseini et al. [20].
Hence, the approximate solution can be readily obtained To illustrate the validity of the analytical approximate solution for this example, the results are compared with the variational iteration method [21] and with an accurate numerical solution, using fourth-order Runge-Kutta method (R-K) in Figure 2.

Motion of the Particle on Arrange Parabola
The governing equation of motion and initial conditions can be expressed as [5,[22][23][24].
) Finally, the frequency amplitude relationship can be obtained as: which is the same as given by Davodi et al. [5].Hence, the approximate solution can be obtained as For this following value of parameters the numerical results are compared with EBM [5] and fourth-order Runge-Kutta method (R-K) as shown in Figure 3.

Conclusion
The methods to find analytical approximate solutions for nonlinear vibrating equations are not closed, which have important application in physical sciences and engineer- ing.In this way, we illustrated that the present method is very effective and convenient and does not require linearization or small perturbation.The obtained analytical solutions were compared with those calculated by the energy balance method and variational iteration method.The obtained results are valid for the whole solution domain with high accuracy.In comparison to fourth-order Runge-Kutta method, which is powerful numerical solution, the results show that the present method is very convenient for solving nonlinear equations and also can be used for strong nonlinear oscillators.
is assumed to be the frequency of the nonlinear oscillator.Similar to previous examples we have