VARIATIONAL ITERATION METHOD FOR VIBRATION PROBLEMS

Metin O. Kaya

Istanbul Technical University, Faculty of Aeronautics and Astronautics, 34469, Maslak,

Istanbul, Turkey

Abstract

In this study, linear/nonlinear, free/forced and damped/undamped vibrations of both one

degree of freedom and continous systems are discussed by using the Variational Iteration

Method. Additionally, common vibration problems are classified and Lagrange multipliers

are derived for each type of problem.

Keywords: Variational Iteration Method, VIM, Vibration, Lagrange Multiplier, Nonlinear

Vibration

1. Introduction

Vibration of dynamical systems can be divided into two main classes like discrete and

distributed. The variables in discrete systems depend on time only, whereas in distributed

systems such as beams, plates etc. variables depend on time and space. Therefore,

equations of motion of discrete systems are described by ordinary differential equations,

while equations of motion of distributed systems are described by partial differential

equations ( Meirovitch [1] ).

1

A considerable amount of studies have been made in the area of vibration problems.

Several techniques, such as finite element method, finite difference method, perturbation

techniques, series techniques, etc. have been used to handle vibration problems.

The Variational Iteration Method, VIM, was first proposed by He [2-11] and the method

has been applied to investigate many nonlinear partial differential equations, autonomous

and singular ordinary differential equations such that solitary wave solutions, rational

solutions, compacton solutions and other types of solution were found by Abdou and

Soliman [12, 13]. Additonally, He [4, 9] used VIM to solve linear/nonlinear vibration

problems.

In this study He’s studies are extended to cover vibration problems with damping, forced

vibration and vibration of beams. Therefore, here VIM is applied to various vibration

problems including vibration of linear/nonlinear, damped/undamped, free/forced vibrations

of one degree of freedom systems and beams as an example of continuous systems. The

procedure presented in this paper can be simply extended to solve more complex vibration

problems; such as aeroelasticity, random vibrations etc.

2. Variational Iteration Method

In order to illustrate the basic concepts of VIM, the following nonlinear partial differential

equation can be considered

(1)

2

where is a linear operator which has partial derivatives with respect to , is the linear

time derivative operator, is a nonlinear term and is an inhomogeneous term.

According to VIM, the following iteration formula can be constructed.

(2)

where is the general Lagrange multiplier which can be identified optimally via

variational theory, and are considered as restricted variations, i.e. ,

.

3. Different Form of Operator

According to the operator, a general classification of vibration problems can be made as

follows

3.1. Case I:

Here the following form of the operator is used

(3)

Considering Eq. (3), Eq. (1) can be expressed as follows

(4)

3

The correction functional of Eq. (4) can be written as

(5)

Making the above correction functional stationary, and noticing that , the

following iteration can be written

(6a)

(6b)

which yields the following stationary conditions:

(7a)

(7b)

(7c)

Therefore, in this case the Lagrange multiplier can be identified as follows

(8)

4

3.2. Case II:

Here the following form of the operator is used

(9)

Considering Eq. (9), Eq. (1) can be expressed as follows

(10)

Here Eq. (10) denotes vibration with damping where is the damping coefficient and is

the mass.

The correction functional of Eq. (10) can be written as

(11)

Making the above correction functional stationary, and noticing that , the

following iteration can be written

(12a)

(12b)

which yields the following stationary conditions

5

(13a)

(13b)

(13c)

Combining Eqs. (13b) and (13c), Eqs. (13a)-(13c) can be rewritten as follows

(14a)

(14b)

(14c)

Therefore, in this case the Lagrange multiplier can be identified as follows

(15)

Approximate Lagrange multiplier can be obtained simply by expanding Eq. (15) as follows

(16)

Hence,

(17)

6

3.3. Case III:

Here the following form of the operator is used

(18)

Considering Eq. (18), Eq. (1) can be expressed as follows

(19)

whre is the spring coefficient.

The correction functional of Eq. (19) can be written as

(20)

Making the above correction functional stationary, and noticing that , the

following iteration can be written

(21a)

(21b)

which yields the following stationary conditions

(22a)

7

(22b)

(22c)

Therefore, in this case the Lagrange multiplier can be identified as follows

(23)

where the circular frequency, , is given by

. (24)

Approximate Lagrange multiplier can be obtained simply by expanding Eq. (23) as follows

(25)

Hence,

(26)

3.4. Case IV:

Here the following form of the operator is used

(27)

8

Considering Eq. (27), Eq. (1) can be expressed as follows

(28)

The correction functional of Eq. (28) can be written as

(29)

Making the above correction functional stationary, and noticing that , the

following iteration can be written

(30a)

(30b)

which yields the following stationary conditions

(31a)

(31b)

(31c)

Combining Eqs. (31b) and (31c), Eqs. (31a)-(31c) can be rewritten as follows

(32a)

9

(32b)

(32c)

Therefore, in this case the Lagrange multiplier can be identified as follows

(33)

where

(34)

Approximate Lagrange multiplier can be obtained simply by expanding Eq. (34) as follows

(35)

Hence,

(36)

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4. Illustrative Examples

EXAMPLE 1:

In this example, a simple mass-spring system that undergoes forced vibration is examined.

The differential equation of motion of this system is given by

(37)

where

Dividing both sides by , Eq.(37) can be rewritten as follows

(38)

where .

Here it is easily seen that the Lagrange multiplier of this problem is

(39)

Additionally, the iteration formula of this problem is

(40)

The complementary solution of this problem is given by

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(41)

Using Eq. (41) as an initial approximation, we get

(42)

or

(43)

Since the last term in Eq. (43) automatically satisfies the complementary equation, this term

will not be used. Thus, Eq. (43) can be simplified to

(44)

which is the general solution of Eq. (37).

In order to point out the importance of using the exact Lagrange multiplier instead of the

approximate one, the following multiplier may be considered

(45)

Considering Eq. (45), the following iteration expression can be written

(46)

12

If we use the complementary solution given by Eq.(41) as an initial approximation, we get

(47a)

or in compact form

(47b)

and

(48a)

or in compact form

(48b)

In the same way, can be written as follows

(49)

Since,

13

(50)

The expression for can be written as follows

(51)

As , the compact form of solution becomes

(52)

Note that Eq. (52) is the same as Eq. (47b).

If we use a different approximate form of the Lagrange multiplier, , such as

(53)

the following iteration expression can be written

(54)

Again using the complementary solution given by Eq.(41) as an initial approximation, we

get

(55a)

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or in compact form

(55b)

and

(56)

It can be easily seen that approximate Lagrange multiplier

converges faster than .

EXAMPLE 2:

In this example, a damped mass-spring system that undergoes forced vibration is examined.

The differential equation of motion of this system is given by

(57)

The Lagrange multiplier of this problem is

(58)

15

where

(59)

Hence using this Lagrange multiplier, the iteration formula can be written as

(60)

Dividing both sides of Eq. (57) by , the following equation of motion is obtained

(61)

where

and .

Therefore, Eq. (60) becomes

(62)

and

(63)

The complementary solution of this problem is given by

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(64)

Taking this complementary solution as an initial approximation, the following expression is

obtained

(65)

Integrating the second term of Eq. (65), we get

(66)

Since the last two terms of Eq. (66) automatically satisfy the homogeneous equation, they

will not be used. The second term of Eq. (66) can be written in a more compact form as

follows

(67)

where

(68)

Hence, the exact solution of Eq. (57) is obtained after one iteration

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(69)

EXAMPLE 3:

In this example, transverse vibration of a uniform beam with simply supported ends is

examined. The differential equation of motion of this system is given by

, , (70)

The initial and the boundary conditions for this problem are as follows:

(71a)

(71b)

(71c)

(71d)

(71e)

(71f)

,

The Lagrange multiplier of this problem is

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It is easily noticed that the Lagrange multiplier of this problem is

(72)

Using this Lagrange multiplier, the iteration formula can be written as

(73)

The complementary solution of this problem is given by

(74)

Using the iteration formula given by Eq. (73) and taking the complementary solution as an

initial approximation, we get

(75)

and

(76)

Hence the expression for can be written as follows

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(77)

Thus we have,

(78)

which is the exact solution.

EXAMPLE 4:

In this example, transverse vibration of a variable coefficient beam that was studied by

Wazwaz [14] is considered. The differential equation of motion of this system is given by

, , , (79)

Here it can be that the equation of motion of the beam is as follows:

(80)

The initial and the boundary conditions for this problem are as follows:

, (81a)

, (81b)

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. (81c)

, (81d)

, (81e)

, (81f)

The Lagrange multiplier of this problem is

and for this problem. Hence the iteration formula can be written as

(82)

The complementary solution of this problem that is used as an initial approximation is

given by

(83)

By using the iteration expression given by Eq. (82) and the initial approximation given by

Eq. (83), we get

(84a)

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or in compact form

(84b)

and

(85)

Eq. (85) can be simplified to the following expression

(86)

As it is seen, the exact solution of this problem is obtained quickly in two iteration.

EXAMPLE 5:

In this example, Mathematical Pendulum that was studied by He [4, 9] is considered.

The differential equation of motion of the undamped mathematical pendulum is given by,

(87)

The initial conditions for this problem are as follows:

(88a)

(88b)

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The term in Eq. (87) is a nonlinear term and it can be expanded as

(89)

Substituting Eq. (89) into Eq. (87) gives

(90)

A more detailed form of this mathematical pendulum was investigated by He [5, 7].

The Lagrange multiplier of this problem is

(91)

Hence the iteration formula is

(92)

The complementary solution of this problem that is used as an initial approximation is

given by

(93)

where is an unknown constant.

Substituting the initial approximation into Eq. (90), the following residual is obtained

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(94)

The coefficient of the term is set to zero in order to eliminate the secular term

which may occur in the next iteration. Doing so, the expression of is found as follows

(95)

Hence,

(96)

with defined in Eq. (95).

The period can be expressed as follows

(97)

If , then . On the other hand He’s [4, 9] approximation gives ,

while the exact period is , where .

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EXAMPLE 6:

In this example, the problem that was studied by Nayfeh and Mook [15] is considered.

The differential equation of motion is given by,

(98)

The initial conditions for this problem are as follows:

(99a)

(99b)

The Lagrange multiplier of this problem is

(100)

The iteration formula is given by

(101)

The complementary solution of this problem that is used as an initial approximation is

given by

(102)

where is an unknown constant.

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Substituting the initial approximation given by Eq. (102), the following residual is obtained

as follows

(103)

The coefficient of the term is set to zero in order to eliminate the secular term

which may occur in the next iteration. Doing so, the expression for is obtained as follows

(104)

Hence,

(105)

with defined in Eq. (104).

The new frequency is defined as follows

(106)

The frequency that is obtained by Nayfeh and Mook [15] using the perturbation method is

(107)

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Note that Eq. (107) is valid only for small values. However, the frequency expression

given by Eq. (106) is valid for all values and takes the following form for small values

(108)

EXAMPLE 7:

In this example, the Duffing-harmonic oscillator that was studied by Lim and Wu [16] and

Mickens [17] is considered.

The differential equation of motion is given by,

(109)

The initial conditions for this problem are as follows:

(110a)

(110b)

with initial conditions and .

For small values, Eq. (109) reduces to

(111a)

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On the other hand, for large values, Eq. (109) reduces to

(111b)

Considering Eqs. (111a) and (111b) respectively, it is noticed that for small values, Eq.

(109) reduces to the equation of motion of the Duffing-type nonlinear oscillator while for

large values, it reduces to the equation of motion of a linear harmonic oscillator.

Therefore, Eq. (109) is called as Duffing-harmonic oscillator equation of motion.

The following form of Eq. (109) is going to be studied in this example

(112)

He’s technique is going to be used to overcome seculer terms that appear in the iterations.

The initial approximation is,

(113)

where is an unknown constant.

Substituting the initial approximation into Eq. (112), the following residual is obtained

(114)

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In oreder to discard the seculer terms, the coefficient of is set to zero which gives

the expression of as follows

(115)

Hence the new frequency is defined as follows

(116)

which is the same with the one found by Lim and Wu [16] and Mickens [17].

The iteration formula is given by

(117)

Hence,

(118)

with defined in Eq. (116).

For small values of amplitude A, the frequency expression given in Eq. (116) is expressed

as follows

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(119a)

Additionally, for large values of amplitude A, the frequency expression given in Eq. (116)

is expressed as follows

(119b)

which agree with the approximations made for the equations of motion given in Eqs. (111a)

and (111b).

5. Conclusion

In thispaper, which extends He’s studies , various types of vibration problems including

vibration of linear/nonlinear, damped/undamped, free/forced vibrations of one degree of

freedom systems and beams as an example of continuous systems are solved using the

Variational Iteration Method. Lagrange multipliers which arise from different types of

vibration problems are presented and the solutions are made in a detailed way.

Additionally, the procedure presented in this paper can be simply extended to solve more

complex vibration problems; such as aeroelasticity, random vibrations etc.

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