## versão On-line ISSN 1679-7825

### Lat. Am. j. solids struct. vol.11 no.6 Rio de Janeiro nov. 2014

#### http://dx.doi.org/10.1590/S1679-78252014000600009

Application of iteration perturbation method and Hamiltonian approach for nonlinear vibration of Euler-Bernoulli beams

Morteza AhmadiI,*; Gholamreza HashemiI; Ali AsghariII

IDepartment of Civil Engineering, Aryan Institute of Science and Tech-nology, Babol, Iran
IIDepartment of Civil Engineering, Semnan University, Semnan, Iran

ABSTRACT

This paper is devoted to the new classes of analytical techniques called the Iteration Perturbation Method (IPM)and Hamiltonian Approach(HA) for solving the equation of motion governing the nonlinear vibration of Euler-Bernoulli beams subjected to the axial loads. It has been found that theIPMand HAare very prolific, rapid, functional and do not demand small perturbation and are also sufficiently accurate to both linear and nonlinear problems in engineering. Comparison of the results of these methods with together and with the results of numerical solution reveals that the IPM and HA are very effective and convenient, and can be easily extended to other nonlinear systems so that can be found widely applicable in engineering and other sciences.

Keywords: Nonlinear vibration, Euler-Bernoulli beam, Iteration Perturbation Method, Hamiltonian Approach.

1 INTRODUCTION

Vibration analysis of the beams is an important issue in structural engineering applications such as long span bridges, aerospace vehicles, automobiles and many other industrial usages. The dynamics of continuous or distributed parameter systems, such as beams, plates, and shells, are governed by nonlinear partial-differential equations in space and time. These partial differential equations and associated boundary conditions form an initial boundary-value problem (Bayat et al., 2011a).

In general it is hard to find exact or closed-form solutions for this class of problems. Consequently, approximate solutions of the original problem were sought. The study of nonlinear vibration equations solution has been applied by many researchers and various methods of solution have been used (Bayat et al., 2011b). In recent years, much attention has been devoted to the newly developed methods to construct an analytic solution for nonlinear vibration such as He's Homotopy Perturbation Method (HPM) (He, 2012; He, 2004; Bayat et al., 2012; Ganji, 2006; Beléndez et al., 2007; Barari et al., 2008), Homotopy Analysis Method (HAM) (Bayat et al., 2012; Liao, 2003; Ghotbi et al., 2009; Sohouli et al., 2010; Kimiaeifar et al., 2009), He's Parameter-Expansion Method (He, 2012; Bayat et al., 2012; He and Shou, 2007; Xu, 2007; Kimiaeifar et al., 2010a; Kimiaeifar et al., 2010b), He's Variational Iteration Method (VIM) (He, 2012; Bayat et al., 2012; He, 1997; He, 2000; Ganji et al., 2007; Barari et al., 2011; Rafei et al., 2007), He's Energy Balance Method (EBM) (He, 2002; Bayat et al., 2012; Jamshidi and Ganji, 2010; Ganji et al., 2009; Fuet al., 2011; Afrouzi et al., 2011; Sfahani et al., 2011; Momeni et al., 2011; Zhang et al., 2009a), He's Amplitude Frequency FormulationMethod (HAFF) (He, 2012; Bayat et al., 2012; He, 2008; Ganji et al., 2010; Ren et al., 2009; Zhang, 2009b), IterationPerturbation Method (IPM) (Bayat et al., 2012; He, 2001; Ozis and Yildirim, 2009; Bayat et al., 2011b), Hamiltonian Approach(HA) (He, 2012; Bayat et al., 2012; Bayat and Pakar, 2011c; Khan et al., 2010; Xu, 2010; Yazdi et al., 2010; Yildirim et al., 2011a; Yildirim et al., 2011b), Exp-Function Method (He, 2012; Wu and He, 2007; Davodi et al., 2010), Adomian's Decomposition Method (Sadighi and Ganji, 2007; Safari et al., 2009; Biazar et al., 2010), Differential Transformation Method (DTM) (Borhanifar and Abazari, 2010; Ghafoori et al., 2011; Joneidi et al., 2009), Variational Approach (VA) (He, 2007; He, 2012; Bayat et al., 2012; Amani et al., 2011; Ganji et al., 2008; Shou, 2009; Zhang, 2007), Harmonic Balance Method (HBM) (Beléndez et al., 2009), Parameterized Perturbation Method (PPM) (He, 1999; He, 2006; He, 2012; Bayat et al., 2012; Ding and Zhang, 2009; Jalaal, et al., 2011).

Among these methods, Iteration Perturbation Method (IPM)and Hamiltonian Approach(HA)are considered to solve the nonlinear vibration of Euler-Bernoulli beams subjected to the axial loads in this paper. The paper has been collocated as follows: first, the mathematical formulation of the problem is considered, and then the basic concept of IPM and HA is described. Subsequently, IPM and HA are studied to demonstrate the applicability and preciseness of these methods; this is followed by a presentation of some comparisons between analytical and numerical solutions. Eventually, it is showed that IPM and HA can converge to a precise cyclic solution for nonlinear systems.

2 Mathematical Formulation

Consider a straight beam of length , a cross-section , a mass per unit length , moment of inertia , and modulus of elasticity that subjected to an axial force of magnitude , as shown in Fig. 1. The equation of motion including of effects of mid-plane stretching is given by (Lestari and Hanagud; 2001; Lacarbonara, 1997):

Where is the viscous damping coefficient, and is a distributed load in the transverse direction.

Assume the non-conservative forces were equal to zero. Therefore, Eq. (1) can be written as follows:

For convenience, the following non-dimensional variables are use:

Where R is the radius of gyration of the cross-section. As a result, Eq. (2) can be written as follows:

Assuming W (X,t) = φ (X)Q(t) where Q(t) is the first eigenmode of the beam (Tse et al., 1978) and applying the Galerkin method, the equation of motion is obtained as follows:

Where α = α1 + Fα2 and α1, α2 and β are as follows:

The Eq. (5) is the differential equation of motion governing non-linear vibration of Euler-Bernoulli beams. The center of the beam is subjected to the following initial conditions:

Where Qmax denotes the non-dimensional maximum amplitude of oscillation.

3. Solution Procedures

3. 1 Basic concept of Iteration Perturbation Method (IPM)

The Iteration Perturbation Method was first proposed by He (2001). To illustrate its basic solution process, we consider the following general nonlinear oscillator:

The variable y = is introduced, and then Eq. (8) can be replaced by equivalent system

With assumption that its initial approximate can be expressed as

Where ω is the angular frequency of the oscillation. Then:

Substituting Eq. (11) and Eq. (12) into the Eq. (10) yields:

Using Fourier expansion series in the right hand of Eq. (13):

Substituting Eq. (14) into Eq. (13) yields:

Integrating Eq. (15) yields:

Comparing Eq. (12) and (16):

3. 2 Basic concept of Hamiltonian Approach (HA)

Previously, He (2002) had introduced the energy balancemethod based on collocation and Hamiltonian. Recently, in 2010 it was developed into the Hamiltonian approach (He, 2010). This approach is a kind of energy method with a vast application in conservative oscillatory systems. In order to clarify this approach, the following general oscillatoris considered (He, 2010):

With initial conditions:

Oscillatory systems contain two important physical parameters, i. e. the frequency ω and the amplitude of oscillation . It is easy to establish a variational principle for Eq. (19), which reads:

Where T is period of the nonlinear oscillator = f.

In the Eq. (21), is kinetic energy and F(u) potential energy, so the Eq. (21) is the least Lagrangian action, from which we can immediately obtain its Hamiltonian, which reads

From Eq. (22):

Introducing a new function, (u), defined as:

Eq. (23) is, then, equivalent to the following one

Or

From Eq. (26) approximate frequency-amplitude relationship of a nonlinear oscillator can be obtained.

4 Runge-Kutta

The fourth RK (Runge-Kutta) method has been used for the numerical approach to verify the analytic solution. This iterative algorithm was written in the form of the following formulae for the second-order differential equation:

Where Δt is the increment of the time and h1, h2, h3 and 4 h are determined from the following formulae:

The numerical solution starts from the boundary at the initial time, where the first value of the displacement function and its first-order derivative are determined from initial condition. Then, with a small time increment Δt the displacement function and its first-order derivative at the new position can be obtained using Eq. (27). This process continues to the end of the time limit.

5 Application

In this step application of IPM and HA is going to be indicated as follows:

5. 1 Applying IPM

After introducing the variable y = , and Substituting Q = A cos(Δt) into the Eq. (5), it is obtained:

By using Fourier series expansion:

By integrating Eq. (30), and comparing with Eq. (12):

According to Eq. (31), the following approximate solution is obtained:

5. 2 Applying HA

Consider Eq. (5). Its Hamiltonian can be easily obtained, which reads:

Integrating Eq. (33) with respect to t from 0 to :

With assumption that the solution can be expressed as:

Substitute it to Eq. (34), leads to:

Setting

From Eq. (37), it is obtained:

According to Eq. (38), the following approximate solution is resulted:

6 Results and Discussion

To illustrate and verify the accuracy of these new approximate methods, for the problem, a comparison of the time history oscillatory displacement responses with the numerical solution using Runge-Kutta method is presented in Table 1. Figure 2 represents a comparison of analytical solution of Qt based on time between Iteration Perturbation Method (IPM) and Hamiltonian Approach (HA) for α = π/3,A = π/6 Figure 3 shows comparison of analytical solution of dQ/dt based on time between Iteration Perturbation Method (IPM) and Hamiltonian Approach (HA) for α = π/3,A = π/6, and Figures 4 and 5 are the comparisons of frequency corresponding to various parameters of α and A.

The effect of different parameters α and A are studied in Figures 6 and 7 simultaneously. It is evident that IPM and HA show an excellent agreement with the numerical solution and quickly convergent, and are valid for a wide range of vibration amplitudes and initial conditions. The accuracy of the results shows that the IPM and HA can be potentiality used for the analysis of strongly nonlinear oscillation problems accurately.

7 Conclusion

In this study, two new methods called Iteration Perturbation Method (IPM) and Hamiltonian Approach (HA) have been used for non-natural oscillators. IPM and HAwere utilized for analyzing the equation of the motion governing the nonlinear vibration of Euler-Bernoulli beams. It has been proved that the IPM and HAare very efficient, comfortable and sufficiently exact in engineering problems. IPM and HA can be simply extended to nonlinear equation for the analysis of nonlinear systems. The obtained results from the approximate analytical solutions are in excellent agreement with the corresponding exact solutions.

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