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Dual approach to averaged values of functions: Advanced formulas

November 2012
Vietnam Journal of Mechanics 34(4)

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Averaged values play major roles in the study of dynamic processes. The useof those values allows transforming varying processes to some constant characteristicsthat are much easier to be investigated. In order to extend the use of averaged valuesone may apply the dual approach which suggests a consideration of two different aspectsof a problem in question. This short communication proposes new advanced formulas foraveraged values of functions based on the dual conception.

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Vietnam Journal of Mechanics, VAST, Vol. 34, No. 4 (2012), pp. 321 – 325

Short Communication

DUAL APPROACH TO AVERAGED VALUES

OF FUNCTIONS:

ADVANCED FORMULAS

N. D. Anh

Institute of Mechanics, VAST, Vietnam

Abstract. Averaged values play major roles in the study of dynamic processes. The use

of those values allows transforming varying processes to some constant characteristics

that are much easier to be investigated. In order to extend the use of averaged values

one may apply the dual approach which suggests a consideration of two different aspects

of a problem in question. This short communication proposes new advanced formulas for

averaged values of functions based on the dual conception.

Key words: advanced formula, averaged value, global-local averaged, dual approach,

Duffing oscillator.

1. INTRODUCTION

Averaged values play major roles in the study of dynamic processes. The use of

those values allows transforming varying processes to some constant characteristics that

are much easier to be investigated. In order to extend the use of averaged values one may

apply the dual approach recently proposed and developed in [1]-[2]. One of significant

advantages of the dual conception is its consideration of two different aspects of a problem

in question which allows investigation to be more appropriate. This advantage has been

used in [3] to propose some new global and local averaged values of functions. The main

objective of the short communication is to present a further extension of global-local

averaged values of function. New advanced formulas for averaged values of functions are

proposed based on the dual conception.

2. ADVANCED FORMULAS FOR AVERAGED VALUES OF

DETERMINISTIC FUNCTIONS

Let y(x)be an integrable deterministic function of x∈[a, b]. The following advanced

formulas for averaged value of y(x)can be considered.

322 N. D. Anh

Classical formula. Define

I0(y(x), a, b) = y(x)(1)

Classical formula for averaged value of y(x)is well-known as follows

< y(x)>=< I 0(y(x), a, b)>=1

b−aZb

I0(y(x), a, b)dx =1

b−aZb

y(x)dx. (2)

Formula I.1. Define a function of x∈[a, b]by

I1(y(x), a, b) = 1

2(1

x−a

y(σ)dσ +1

b−x

y(σ)dσ).(3)

Formula I.1 for averaged value of y(x)is defined by

< y(x)>1=< I 1(y(x), a, b)>=1

b−aZb

I1(y(x), a, b)dx. (4)

Formula I.2. Define

I2(y(x), a, b) = I1(I1(y(x), a, b), a, b) =

2(1

x−a

I1(y(σ), a, b)dσ +1

b−x

I1(y(σ), a, b)dσ).(5)

Formula I.2 for averaged value of y(x)is defined by

< y(x)>2=< I 2(y(x), a, b)) >=1

b−aZb

I2(y(x), a, b))dx. (6)

Analogously, one can define

In(y(x), a, b) = I1(In−1(y(x), a, b), a, b)(7)

and Formula I.n for averaged value of y(x)is defined by

< y(x)>n=< I n(y(x), a, b)>=1

b−aZb

In(y(x), a, b)dx. (8)

Further, according to (3) define

I1(y(r), a, x) = 1

2(1

r−a

y(σ)dσ +1

x−r

y(σ)dσ),

Dual approach to averaged values of functions: Advanced formulas 323

I1(y(s), x, b) = 1

2(1

s−x

y(σ)dσ +1

b−s

y(σ)dσ),(9)

and

A1(y(x), a, b) = 1

2(1

x−a

I1(y(r), a, x)dr +1

b−x

I1(y(s), x, b)ds).(10)

It is seen that I1(y(r), a, x)is a function of r∈[a, x],I1(y(s), x, b)is a function of s∈[x, b],

and A1(y(x), a, b)is a function of x∈[a, b].Formula A.1 for averaged value of function

y(x)is defined by

< y(x)>1=< A1(y(x), a, b)>=1

b−aZb

A1(y(x), a, b)dx, (11)

Analogously, one can define

An(y(x), a, b) = A1(An−1(y(x), a, b), a, b)(12)

and Formula A.n for averaged value of y(x)is defined by

< y(x)>n=< An(y(x), a, b)>=1

b−aZb

An(y(x), a, b)dx. (13)

One can also define mixed products as

An(Im(y(x), a, b), a, b) = A1(An−1(Im(y(x), a, b), a, b), a, b)(14)

and Formula A.nI.m for averaged value of y(x)as

< y(x)>n

m=< An(Im(y(x), a, b), a, b)>=1

b−aZb

An(Im(y(x), a, b), a, b)dx. (15)

3. APPLICATION TO NONLINEAR EQUATIONS

For illustration of possible uses of the proposed advanced formulas consider the

following equation

¨z+γz3= 0, z(0) = a, ˙z(0) = 0.(16)

Let xis a solution of the linear equation

¨x+kx = 0, x(0) = a, ˙x(0) = 0.(17)

324 N. D. Anh

The equation error is to be

e(x) = γx3−kx

The value of kcan be determined from a minimum condition, for example,

L(e2(x)) →min

k(18)

where Lis a linear averaging operator. Thus, from (18) one has

ω2=k=γLx4

L(x2)(19)

- Taking

L(·) = 1

aZa

(·)dx (20)

one gets

ω2

0=k0=γ

aRa

0x4dx

aRa

0x2dx

=γa4

a2=3

5γa2,(21)

or,

ω0=r3

5γa2= 0.77460√γa. (22)

- Taking

L(·) = <·>1=1

aZa

I1(·,0, a )dx, (23)

one gets

ω2

1=k1=γ< x4>1

< x2>1

=γ

aRa

10(a4+a3x+a2x2+ax3+ 2x4)dx

aRa

6(a2+ax + 2x2)dx

=γ149a4

600

13a2=447

650γa2,

or,

ω1=r447

650γa2= 0.82927√γa. (24)

- Taking

L(·) = <·>2=1

aZa

I2(·,0, a )dx, (25)

Dual approach to averaged values of functions: Advanced formulas 325

one gets

ω2

2=k2=γ< x4>2

< x2>2

=γ

aRa

1200(209a4+ 119a3x+ 79a2x2+ 54ax3+ 48x4)dx

aRa

72(19a2+ 10ax + 8x2)dx

= 0.2649427

10γ a2= 0.71535γa2,

(26)

or,

ω2=p0.71535γa2= 0.84578√γa. (27)

Comparing the solution (27) with the exact solution ωeand the one ωlobtained by the

classical criterion of equivalent linearization, respectively,

ωe= 0.847√γa, ωl= 0.866√γa, (28)

one shows a significant accuracy of (27).

4. CONCLUSION

In this short communication the main idea of the dual conception is further extended

to suggest new advanced formulas for averaged values of functions. These advanced for-

mulas contain the classical formula of averaged value as a particular case. In the example

of Duffing oscillator it is shown that advanced formulas can give a series of approximate

solutions that are more accurate than the conventional one obtained by the classical cri-

terion of equivalent linearization.

ACKNOWLEDGEMENTS

The research reported in this paper is supported by Vietnam National Foundation

for Science and Technology Development.

REFERENCES

[1] N. D. Anh, Duality in the analysis of responses to nonlinear systems, Vietnam Journal of

Mechanics,32(4) (2010) 263-266.

[2] N. D. Anh, N. N. Hieu, N. N. Linh, A dual criterion of equivalent linearization method for non-

linear systems subjected to random excitation, Acta Mechanica, (2012) DOI 10.1007/s00707-

011-0582-z.

[3] N. D. Anh, Dual approach to averaged values of functions, Vietnam Journal of Mechanics,

34(3) (2012) 211-214.

Received April 16, 2012

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