PHÂN TÍCH DAO ĐỘNG DUFFING BẬC 3-5 SỬ DỤNG PHƯƠNG PHÁP TUYẾN TÍNH HÓA TƯƠNG ĐƯƠNG VỚI TRUNG BÌNH CÓ TRỌNG SỐ

In this paper, the equivalent linearization method with a weighted averaging proposed by Anh.N.D [1] is applied to solve the strong nonlinear cubic-quintic Duffing oscillators[r]

SOLUTION OF NONLINEAR CUBIC -QUINTIC DUFFING OSCILLATORS USING THE EQUIVALENT LINEARIZATION METHOD

WITH A WEIGHTED AVERAGING

Duong The Hung, Dang Van Hieu

University of Technology - TNU

ABSTRACT

In this paper, the equivalent linearization method with a weighted averaging proposed by Anh.N.D [1] is applied to solve the strong nonlinear cubic-quintic Duffing oscillators The closed-form solutions for the cubic-quintic Duffing oscillator are then obtained In order to illustrate the effectiveness and convenience of the method, some several cases of cubic-quintic Duffing

oscillator with different parameters of α, β and γ are investigated The obtained 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 linearization, weighted averaging, cubic-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 in 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

fifth-power nonlinearity Due to the presence of

this fifth-power nonlinearity added to the

third nonlinearity of the common Duffing

equation, this oscillator is difficult to handle

To overcoming the shortcomings, many new

*

Email: hungduongxd@gmail.com

analytical methods have proposed these days, may be mentioned as the perturbation technique [7], the harmonic balance method [5] the Lindstedt-Poincaré method [12], the parameter-expansion method [13], the parameterized perturbation method [17], the approximate energy method [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 [1], [2], [3] However, these techniques are really complicated

The advantage of the linearization equivalent method is that it is very simple and

convenient to apply The obtained results are

normally accepted for oscillators with weak

linerity Nevertheless, the accuracy of the

method normally reduces for middle or

strongly nonlinear systems Futhermore, some

terms will vanish in the averaging process, for

example the averaged value of the functions

sin(t) and cos(t) over one period is equal to

zero As a result, this method ignores some

effects of those elements in analyzing

procedure In this short communication

written by Anh.N.D [1], the main idea of the

dual conception is further extended to suggest

a new form for weighting coefficient and then

a new averaged value of functions This new

averaged value depends on the parameter s

and contains the classical averaged value

when s = 0 In the example of Duffing

oscillator, it is shown that the parameter s can

be chosen as s = n/(2π) and for n = 4 one gets

the solution that is much accurate than the

conventional one obtained by the classical

criterion of equivalent linearization This

method is applied by the authors [16] in the

analysis of some nonlinear oscillations

In this paper, the equivalent linearization

method with a weighted averaging proposed

by Anh N D [1] is used to obtain the

periodic solutions for the nonlinear

cubic-quintic Duffing oscillators The solution is

compared with the one given by Ganji et al

[9] using He’s Energy balance method as well

as with the exact solution The results show

the advantage of the proposed method over

the other ones

GENERAL DEFINITION OF CUBIC

-QUINTIC DUFFING OSCILLATORS

A cubic-quintic Duffing oscillator has the

general form of [9]: x f x( )0 (1)

with initial conditions: x(0)X0, x(0)0 (2)

where f x( )xx3x5, x and t are

generalised dismensionless displacement and

time variable, respectively, and x is the

function of t

The exact frequency ω e by imposing the

initial conditions is [15]:

1

1/ 2

0

( )

2 1 sin sin

e

k X













(3a)

1

(3b)

k





  (3c)

4 0

2

X k





LINEARIZATION METHOD WITH A

DUFFING OSCILLATOR

Consider the nonlinear cubic-quintic Dufing oscillator given by the equation:

0

xxx x  (4) with initial conditions:

0

The linearized equation of Eq (4) is:

2

0

An equation error between Eqs (4) and (6) is:

( )

ω 2

is determined by the mean square error criterion which requires that the mean square

of equation error be minimum:

2



From 2 e x2( ) 0





yields:

2

2

x

The harmonic solution of the linearized Eq

(6) is: xX0cost (10) Using the weighted coefficient is introduced

( ) s t, 0

h t ste s , the averaging operators in Eq (9) can be calculated, and according to Laplace’s transform, we have:

0

0

2

os

2 8 cos (

( 4) )

s t

s

X s

s









 







(11)

4 4 4 4 2 2 4 4

0

2 0

24

( 4) ( 16)

X s



0

6 2

0

2

1658880 440064 282496 45712 3168 94

( 4) ( 16) (

)

( )

36

s t

s

s









(13)

Substituting Eqs (11) – (13) into Eq (9),we obtain:

4

4 2

2

0

2

248 416 1536 28

( 16) (2 8 ) (1658880 440064 282496 45712 3168 94 )

( 16) ( 36) (2 8 )

X

X







(14)

It can be seen from Eq (14) that the

frequency of oscillation ω depends not only

on the initial amplitude X 0 but also on the

parameter s

In this paper, with s=2, from Eq (14) we get

the approximate frequency of oscillation:

0.72 X 0.575 X

Substituting Eq (15) into Eq (10), we

obtain the harmonic approximate solution of

the oscillator

For example, with     1 and

0 3

X  , we get the approximate frequency

of oscillator:

7.3522 (16) and the approximate solution:

3cos 7.3522

x t (17)

COMPARISONS The results obtained by the proposed method, the ones obtained by the energy balance method (EBM) [9] and the exact solutions are compared For this nonlinear problem, the exact frequency is stated by S.K Lai et al [15] The results are presented

in Figs 1 - 3 and Table 1 for different values

of the parameters α, β, γ and X 0.

Figures 1, 2, 3 illustrate the time history diagram of the displacement, velocity and phase phane of the oscillator for different

values of the parameters α, β, γ and X 0

Figure 1 The solution of the nonlinear cubic-quintic Duffing oscillator with α=β=γ=1 and X 0 =3

(a) (b) (c)

Figure 2 The solution of the nonlinear cubic-quintic Duffing oscillator with α=2, β=γ=1 and X 0 =50

(a) (b) (c)

Figure 3 The solution of the nonlinear cubic-quintic Duffing oscillator with α=5, β=3, γ=1 and X 0 =100

The comparisons of frequencies for different

parameters via numerical are presented in

Table 1 As can be seen from the Table, the

proposed method can give the better results

than the one obtained by the energy balance

method Specifically, for different nonlinearity degree of systems, the highest error of the method is only 1.53% compared with 2.29% of the energy balance method

Table 1 Percentage errors of approximate frequency when α=1, β=10, γ=100

X 0 ω exact ω EBM R Error (%) ω present R Error (%)

0.1 1.039700000 1.039642196 0.006 1.038147388 0.15

8 478.4630000 489.4110824 2.29 485.7795796 1.53

To show the effects of the parameters α, β

and γ on the frequency base on the

amplitudes are considered and shown in

Figures 4-5 The results are compared with the EBM solutions

Figure 4 Influence of α on frequency base on X 0 for

β=γ=1

Figure 5 Influence of β on frequency base on X 0

for α=10, γ=2

To achieve a better understanding of the

effects of the parameters α, β and γ and

amplitude X 0 on the frequency of the

systems, they are considered simultaneously via sensitivity analysis of the frequency; the results are given in Figure 6

Figure 6 Sensitivity analysis of frequency for:

(a) 0<α<100, 0<X 0 <100, β=γ=10; (b) 0<β<50, 0<X 0 <50, α=100, γ=10;

CONCLUSIONS

In this paper, the equivalent linearization

method with a weighted averaging is applied

to solve the nonlinear differential equations

governing on the Duffing oscillators Some

examples of Duffing oscillators with

differential parameters of α, β and γ were

presented Comparisons between the results

of exact solutions and those of the EBM have

been made The results show that the present

method can give the solutions in excellent

agreement with the corresponding exact ones

The maximum error is only 1.53% in

comparison with 2.29% given by the EBM

This method can be easily extended to analyse other nonlinear oscillators However,

proper values of the parameter s should be

chosen to give better and the best solution is still required further investigation.

REFERENCES

1 Anh N D., (2015), “Short Communication Dual

Approach to Averaged Values of Functions: a form

for weighting coefficient”, Vietnam Journal of Mechanics ,Vol 37, No 2, pp 145 –150

2 Anh N D., Di Paola M., (1995), “Some extensions of Gaussian equivalent linearization”,

International Conference on Nonlinear Stochastic Dynamics, Hanoi, Vietnam, pp 5-16

3 Anh N D., Schiehlen W., (1997), “An

Extension to the Mean Square Criterion of

Gaussian Equivalent Linearization”, Vietnam

Journal of Mathematics, 25 (2), pp 115-123

4 Caughey T K., (1959), “Equivalent

linearization technique”, J Acoust Soc Am., 35,

pp 1706–1711

5 Chen Y M., Liu J K., (2007), “A new method

based on the harmonic balance method for

nonlinear oscillators”, Physics Letters A, 368, pp

371-378

6 D′Acunto M (2006), “Determination of limit

cycles for a modified van der Pol oscillator”

Mechanics Research Communications, 33, pp

93-100

7 Ganji D.D., Rafei M (2006), “Solitary wave

solutions for a generalized Hirota-Satsumacoupled

KdV equation by homotopy perturbation method”,

Physics Letter A, 356, pp 131-137

8 Ganji D.D., Nourollahi M., Rostamian M (2007),

“A comparison of variational iteration method with

Adomian’s Decomposition Method in some highly

nonlinear equations”, International Journal of

Science and Technology, 2(2), pp 179-188

9 Ganji D D., Gorji M., Soleimani M.,

Esmaeilpour M (2009), “Solution of nonlinear

cubic-quintic Duffing oscillators using He’s

Energy Balance Method”, Journal of Zhejiang

University, 10(9), pp 1263-1269

10 Ganji S S., Ganji D D., Ganji Z Z.,

Karimpour S (2008), “Periodic solution for

strongly nonlinear vibration systems by He’s Energy Balance Method”, Acta Applied

Mathematics, 106(1), pp 79-92

11 He J H (2000), “A review on some new recently developed nonlinear analytical

techniques”, International Journal of Nonlinear

Science and Numerical Simulation, 1, pp 51-70

12 He J H (2001), “Modified Lindstedt-Poincare, methods for some strongly nonlinear oscillations, Part III: double series expansion”,

International Journal of Non-linear Science and

Numerical Simulation, 2, pp 317-320

13 He J H (2006), “Some asymptotic methods

for strongly nonlinear equations”, International

Journal of Modern Physic B, 20, pp 1141-1199

14 He J H (2007), “Variational approach for

nonlinear oscillators”, Chaos Solitons & Fractals,

34, pp 1430-1439

15 Lai S K., Lam C.W., Wu B.S., Wang C., Zeng Q.C., He X F (2008), “Newton-harmonic balancing approach for accurate solutions to nonlinear

cubic-quintic Duffing oscillators”, Applied Mathematical

Modeling, 33(2), pp 852-866

16 N D Anh, N Q Hai, D V Hieu (2017), “The Equivalent Linearization Method with a Weighted Averaging for Analyzing of Nonlinear Vibrating

Systems”, Latin American Journal of Solids and Structures 14 (2017), pp 1723-1740

17 Krylov N., Bogoliubov N., (1943),

Introduction to nonlinear mechanics, New York:

Princenton University Press

TÓM TẮT

PHÂN TÍCH DAO ĐỘNG DUFFING BẬC 3-5 SỬ DỤNG PHƯƠNG PHÁP

TUYẾN TÍNH HÓA TƯƠNG ĐƯƠNG VỚI TRUNG BÌNH CÓ TRỌNG SỐ

Dương Thế Hùng * , Đặng Văn Hiếu

Trường Đại học Kỹ thuật Công nghiệp - ĐH Thái Nguyên

Trong bài báo này, phương pháp tuyến tính hóa tương đương với trung bình có trọng số (phương pháp được đề xuất bởi N D Anh [1]) được áp dụng để phân tích dao động phi tuyến Duffing bậc 3-5 Nghiệm giải tích dạng đóng của hệ Duffing bậc 3-5 đã thu được Để minh họa sự tin cậy và thuận tiện của phương pháp, một số trường hợp của hệ dao động Duffing bậc 3-5 với các thông số

khác nhau của α, β và γ đã được khảo sát Nghiệm thu được đã được so sánh với nghiệm chính

xác Các kết quả cho thấy phương pháp đề xuất là rất tiện lợi và có thể thu được nghiệm rất chính xác cho cả trường hợp biên độ dao động nhỏ hoặc lớn

Từ khóa: tuyến tính hóa tương đương, trung bình trọng số, dao động Duffing bậc 3-5, dao động

phi tuyến, nghiệm giải tích.

Ngày nhận bài: 12/7/2017; Ngày phản biện: 29/7/2017; Ngày duyệt đăng: 30/9/2017

*

Email: hungduongxd@gmail.com