Phân tích dao động Duffing bậc 3-5 sử dụng phương pháp tuyến tính hóa tương đương với trung bình có trọng số

Duong Thi Hiing vd Dig Tap chi KHOA HQC & CONG NGHE 169(09) 137-142

SOLUTION OF NONLINEAR CUBIC -QUINTIC DUFFING OSCILLATORS

USING THE EQUIVALENT LINEARIZATION METHOD

WITH A WEIGHTED AVERAGING

Duong The Hung, Dang Van Hieu

University ofTechnology - TNU

ABSTRACT

In this paper, the equivalent lineanzation method with a weighted averaging proposed by Anh N.D

[1] is applied to solve die strong nonlinear cubic-quintic Duffing oscillators The closed-form

soluUons for the cubic-qiuntic Duffing oscillator are then obtamed In order to illustrate the

effectiveness and convemence of the method, some several cases of cubic-quintic Duffing

oscillator with different parameters of a, fi and y are investigated. The obtamed solutions are

compared with the exact ones The results show that the proposed method is very convenient and

can give the most precise solutions for the small as well as the large amplitudes of oscillation

Keyword; equivalent lineanzation, -weighted averaging, cubie-quintic Duffing oscillator,

nonlinear oscillator, analytical solution.

INTRODUCTION

Basically, in different fields of science and

engineering there are few issues occurring

linear whereas a great number of problems

result in the nonlinear systems. Nonlinear

oscillations are an important fact m physical

science, mechanical structures and other

engineering problems The methods of

solving linear differential equations are

comparatively easy and well established. On

the contrary, the techniques of solving

nonlinear differential equations are less

available and, in general, can only produce

approximate solutions. With the discovery of

numerous phenomena of self- excitation of a

strongly nonlinear cubic-quintic Duffing

oscillator and in many cases of nonlinear

mechanical vibrations of special types, the

methods of small oscillations become

inadequate for their analytical treatment. The

cubic-quintic Duffing equation is a

differential equation with third- and/or fiflh￾power nonlineanty. Due to the presence of

this fifth-power nonlineanty added to the

tiiird nonlineanty of the common Duffing

equation, this oscillator is difficult to handle.

To overcoming the shortcomings, many new

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analytical methods have proposed these days,

may be mentioned as the perturbation

technique [7], the harmonic balance method

[5] the Lindstedl-Poincare method [12], the

parameter-expansion method [13], the

parameterized perturbation method [17], the

approximate energy raethod [6], the

variational iteration method [8] and

variational approach [14], the Energy

Balance Method [10], the equivalent

linearization method [1], [2], [3]

The Equivalent Linearization Method of

Kryloff and Bogoliubov [17] is generalized to

the case of nonlinear dynamic systems with

random excitation by Caughey [4]. It has

been shown that the Gaussian equivalent

linearization is presently the simplest tool

widely used for analyzing nonlinear

stochastic problems However, the major

limitation of this method is seemingly that

its accuracy decreases as the nonlinearity

increases and it can lead to unacceptable

errors in the second moment. Thus, some

extensions of equivalent linearization were

proposed by many authors [ I ], [2], [3].

However, these techniques are really

complicated.

The advantage of the linearization equivalent

method is that it is very simple and