In this paper, nonlinear dynamics of a particle on a rotating parabola arising in the mechanical engineering is proposed. Two efficient mathematical techniques called variational iteration method and variational approach are used to solve the governing differential equation of motion. To assess the accuracy of solutions, we compare the results with the Runge-Kutta 4th order. An excellent agreement of solutions with the numerical results and published results has been demonstrated. The results reveal that the present methods are very effective and convenient in predicting the solution of such problems and find a wide application in new engineering problems

A. Nikkar

H. Jodeiri

Sh. Ahari Abesh Ahmad

English

Journal of Vibroengineering

  © JVE INTERNATIONAL LTD. JOURNAL OF VIBROENGINEERING. NOV 2015, VOLUME 17, ISSUE 7. ISSN 1392-8716 3967 
1811. Study of nonlinear dynamics of a particle on a 
rotating parabola by using efficient mathematical 
techniques 
A. Nikkar1, H. Jodeiri2, Sh. Ahari Abesh Ahmad3 
1Young Researchers and Elite Club, Ahar Branch, Islamic Azad University, Ahar, Iran 
2Young Researchers and Elite Club, Tabriz Branch, Islamic Azad University, Tabriz, Iran 
3Faculty of Civil Engineering, University of Tabriz, Tabriz, Iran 
2Corresponding author 
E-mail: 1ali.nikkar@yahoo.com, 2hjodeiri@yahoo.com, 3shaahari69@gmail.com 
(Received 13 March 2015; received in revised form 8 July 2015; accepted 15 July 2015) 
Abstract. In this paper, nonlinear dynamics of a particle on a rotating parabola arising in the 
mechanical engineering is proposed. Two efficient mathematical techniques called variational 
iteration method and variational approach are used to solve the governing differential equation of 
motion. To assess the accuracy of solutions, we compare the results with the Runge-Kutta 4th 
order. An excellent agreement of solutions with the numerical results and published results has 
been demonstrated. The results reveal that the present methods are very effective and convenient 
in predicting the solution of such problems and find a wide application in new engineering 
problems. 
Keywords: variational iteration method, variational approach, nonlinear problems, rotating 
parabola. 
1. Introduction 
Many phenomena exist in the environment where a particle's motion can be observed on them, 
such as centrifugation, centrifugal filters, industrial hopper, etc. Motion surface has different 
shapes especially for rotating application as it can be circular, parabolic or conical. On the other 
hand, the governing equations in most phenomena in applied sciences and engineering are 
nonlinear. In general, finding an exact or close-form solution for nonlinear problems is very 
difficult. So, with the rapid development of nonlinear science, there appears an ever-increasing 
interest of scientists in the analytical asymptotic techniques for nonlinear problems, and several 
analytical approximation methods have been developed; such as: homotopy perturbation method 
[1], variational iteration method [2], max-min approach [3], variational approach [4], Hamiltonian 
approach [5] and Laplace iteration method [6] .In recent years, these methods successfully applied 
arising in the applied sciences and engineering. For example; the vibrational behavior of quintic 
nonlinear in extensional beam on two-parameter elastic substrate based on the three mode 
assumptions is investigated by Sedighi [7]. He employed parameter expansion method to obtain 
the approximate expressions of nonlinear frequency-amplitude relationship for the first, second, 
and third modes of vibrations. A novel approach is applied to the analytic treatment of nonlinear 
fifth-order Equations by Saravi and Nikkar [8]. Ghaffarzadeh and Nikkar [9] applied a new 
analytical method called the variational iteration method-II (VIM-II) for the differential equation 
of the large deformation of a cantilever beam under point load at the free tip. Askari et al. [10] 
applied He’s energy balance method and He’s variational approach to frequency analysis of 
nonlinear oscillators with rational restoring force. Sedighi and Daneshmand [11] studied nonlinear 
transversely vibrating beams by the homotopy perturbation method with an auxiliary term. Barari 
et al. [12] studied non-linear vibration behavior of geometrically non-linear Euler-Bernoulli beams 
using variational iteration method and parameter perturbation method. Sedighi and Shirazi [13] 
presented a new asymptotic procedure to predict the nonlinear vibrational behaviour of classical 
micro-beams pre-deformed by an electric field using Parameter expansion method. Bagheri et al. 
[14] studied the nonlinear responses of clamped-clamped buckled beam. They used two efficient 
mathematical techniques called He’s variational approach and Laplace iteration method in order 
1811. STUDY OF NONLINEAR DYNAMICS OF A PARTICLE ON A ROTATING PARABOLA BY USING EFFICIENT MATHEMATICAL TECHNIQUES.  
A. NIKKAR, H. JODEIRI, SH. AHARI ABESH AHMAD 
3968 © JVE INTERNATIONAL LTD. JOURNAL OF VIBROENGINEERING. NOV 2015, VOLUME 17, ISSUE 7. ISSN 1392-8716  
to obtain the responses of the beam vibrations. Salehi et al. [15] applied two efficient methods to 
consider large deformation of cantilever beams under point load. Younesian et al. [16] studied 
free oscillations of beams on nonlinear elastic foundations by VIM. Askari et al. [17] applied 
higher order Hamilton approach to nonlinear vibrating systems, and many other problems solved 
by these methods [18-21]. Recently, nonlinear dynamics of a particle on a rotating parabola are 
analyzed by means of the analytic and semi-analytic approaches by Mirzabeigy et al. [22]. 
The Variational Approach which is also called He’s variational approach, as well as the 
variational iteration method, was utilized here to obtain the analytical expression for the following 
model of nonlinear dynamics of a particle on a rotating parabola which introduced by Nayfeh and 
Mook [23]: 
ݑ′′ + ߝଶݑ + 4ݍଶݑݑ′ଶ + 4ݍଶݑଶݑ" = 0, ݑ(0) = ܣ,  ݑ′(0) = 0. (1)
For validation of methods, we examine the advantages and disadvantages of applied methods 
by comparison of results with fourth-order Runge-Kutta solutions. 
2. Basic concept of He’s variational approach  
The variational approach to nonlinear oscillators was first proposed by Chinese mathemacian, 
Ji-Huan He [4]. We give a brief introduction of the method. To illustrate the method precisely, 
consider the following generalized nonlinear oscillations: 
ݑ′′ + ݂(ݑ) = 0,    ݑ(0) = ܣ, ݑ′(0) = 0, (2)
where ݂(ݑ) is a nonlinear function of ݑ, ݑ′ and ݑ′′. 
For simplicity, if function ݂ depends on ݑ only, its variational functional can be obtained as: 
ܬ(ݑ) = න ൜− ݑ
ᇱ
2 + ܨ(ݑ)ൠ
்
଴
݀ݐ, (3)
where ܨ is the potential, ݀ܨ ݀ݑ⁄ = ݂ and ܶ is the period of the nonlinear oscillator. 
We assume that its approximate solution can be expressed as: 
ݑ = ܣcos(߱ݐ), (4)
where ܣ and ߱ are the amplitude and frequency of the oscillator, respectively. Substituting Eq. (4) 
into Eq. (3), and setting ݀ܬ ݀ܣ⁄ = 0, we can obtain an inexplicit amplitude-frequency relationship 
of Eq. (1). 
3. Implementation of first order He’s variational approach 
In this section, we solve Eq. (1) via variational approach. We can, now, easily obtain the 
following variational formulation: 
ܬ(ݑ) = න ቊ− ݑሶ2 +
ߝଶ
2 ݑ
ଶ − 2ݍଶݑଶݑሶ ଶቋ
்
ସ
଴
݀ݐ. (5)
Assume that its approximate solution can be expressed as: 
ݑ(ݐ) = ܣcos(߱ݐ), (6)
where ߱ is the frequency to be determined and ܣ is the amplitude of oscillation. Substituting 
1811. STUDY OF NONLINEAR DYNAMICS OF A PARTICLE ON A ROTATING PARABOLA BY USING EFFICIENT MATHEMATICAL TECHNIQUES.  
A. NIKKAR, H. JODEIRI, SH. AHARI ABESH AHMAD 
 © JVE INTERNATIONAL LTD. JOURNAL OF VIBROENGINEERING. NOV 2015, VOLUME 17, ISSUE 7. ISSN 1392-8716 3969 
Eq. (6) into Eq. (5) results in: 
ܬ(ݑ) = − 12 (߱ܣsin߱ݐ) +
ߝଶ
2 (ܣcos߱ݐ)
ଶ − 2ݍଶ(ܣcos߱ݐ)ଶ(߱ܣsin߱ݐ)ଶ. (7)
Making ܬ stationary with respect to ܣ, according to He’s method, we obtain: 
߲ܬ
߲ܣ = −
1
2 (߱sin߱ݐ) + ܣߝ
ଶ(cos߱ݐ)ଶ − 8ݍଶ߱ଶܣଷ(cos߱ݐ)ଶ(sin߱ݐ)ଶ = 0. (8)
Finally collocation at ߱ݐ = ߨ 4⁄ , the frequency can be approximated as: 
߱௏஺ =
ߝ
ඥ1 + 2ݍଶܣଶ, (9)
and zero-order approximate solution from Eq. (22) is: 
ݑ = ܣcos ቆ ߝඥ1 + 2ݍଶܣଶ ݐቇ. (10)
4. Basic concept of He’s variational iteration method  
To clarify the basic ideas of He’s variational iteration method consider the following general 
differential equation: 
ܮݑ + ܰݑ = ݂(ݔ), (11)
where ܮ  is a linear and ܰ  is a nonlinear operator, ݂(ݔ)  an inhomogeneous or forcing term. 
According to the variational iteration method, we can construct a correct functional as follows: 
ݑ௡ାଵ(ݔ) = ݑ௡(ݔ) + න ߣ൫ܮݑ௡(߬) + ܰݑ෤௡(߬) − ݂(߬)൯݀߬,
௫
଴
(12)
where ߣ is a general Lagrange multiplier, which can be identified optimally via the variational 
theory, the subscript ݊ denotes the ݊th approximation, ݑ෤௡ is considered as a restricted variation, 
i.e. ߜݑ෤௡ = 0. 
For linear problems, its exact solution can be obtained by only one iteration step due to the 
fact that the Lagrange multiplier can be exactly identified. In this method, the problems are 
initially approximated with possible unknowns and it can be applied in non-linear problems 
without linearization or small parameters. The approximate solutions obtained by the proposed 
method rapidly converge to the exact solution. 
5. Implementation of He’s variational iteration method 
Now, for solving Eq. (1) by using VIM, first, we assume that the angular frequency of the 
system (1) is ߱, we have the following linearized equation: 
ݑሷ + ߱ଶݑ = 0. (13)
So we can rewrite Eq. (1) in the form: 
ݑሷ + ߱ଶݑ + ݂(ݑ) = 0, (14)
1811. STUDY OF NONLINEAR DYNAMICS OF A PARTICLE ON A ROTATING PARABOLA BY USING EFFICIENT MATHEMATICAL TECHNIQUES.  
A. NIKKAR, H. JODEIRI, SH. AHARI ABESH AHMAD 
3970 © JVE INTERNATIONAL LTD. JOURNAL OF VIBROENGINEERING. NOV 2015, VOLUME 17, ISSUE 7. ISSN 1392-8716  
where ݂(ݑ) = (ߝଶ − ߱ଶ)ݑ + 4ݍଶݑଶݑሷ + 4ݍଶݑݑሶ ଶ = 0. 
Applying the proposed method, the following iterative formula is formed as: 
ݑ௡ାଵ(ݐ) = ݑ௡(ݐ) + න ߣ൫ݑሷ ௡(߬) + ߱ଶݑ෤௡(߬) − ݂(߬)൯݀߬.
௧
଴
(15)
Its stationary conditions can be obtained as follows: 
1 − ߣᇱ|ఛୀ௧ = 0, 
ߣ|ఛୀ௧ = 0, 
ߣ′′ − ߱ଶߣ|ఛୀ௧ = 0. 
(16)
The Lagrange multiplier, therefore, can be identified as: 
ߣ = 1߱ sin߱(߬ − ݐ). (17)
Substituting the identified multiplier into Eq. (15) results in the following iteration formula: 
ݑ௡ାଵ(ݐ) = ݑ௡(ݐ) 
      + 1߱ න sin߱(߬ − ݐ)(ݑሷ ௡(߬) + ߝ
ଶݑ௡(߬) + 4ݍଶݑ௡(߬)ଶݑሷ ௡(߬) + 4ݍଶݑ௡(߬)ݑሶ ௡(߬)ଶ
௧
଴
)݀߬. (18)
Assuming its initial approximate solution has the form: 
ݑ଴(ݐ) = ܣcos(߱ݐ). (19)
And substituting Eq. (19) into Eq. (1) leads to the following residual: 
ܴ଴(ݐ) = −ܣ߱ଶcos(߱ݐ) + ߝଶܣcos(߱ݐ) + 4ݍଶܣଶcosଶ(߱ݐ)
      −4ݍଶܣଷ߱ଶcosଷ(߱ݐ) + 4ݍଶܣଶ߱ଶcosଶ(߱ݐ). (20)
By the Eq. (18), we can obtain: 
ݑଵ(ݐ) = ݑ଴(ݐ) +
1
߱ න sin߱(߬ − ݐ)൫ܴ଴(߬)൯݀߬,
௧
଴
(21)
ݑଵ(ݐ) =
1
4߱ (ܣ(4߱cos(߱ݐ) + ܣ
ଶݍଶ߱cos(߱ݐ) + 2߱ଶsin(߱ݐ)ݐ 
      −2ߝଶsin (߱ݐ)ݐ + 4ܣଶݍଶ߱ଶsin(߱ݐ)ݐ − ܣଶݍଶ߱cos (3߱ݐ))).
(22)
In the same manner, the rest of the components of the iteration formula can be obtained. In 
order to ensure that no secular terms appear in ݑଵ, resonance must be avoided. To do so, the 
coefficient of cos(߱ݐ) in Eq. (20) requires being zero, i.e.: 
߱௏ூெ =
ߝ
ඥ1 + 2ݍଶܣଶ. (23)
6. Results and discussion 
In this stage, we show the some figures to validation of the methods. In order to verify the 
precision of the presented methods, current results were compared with Runge-Kutta 4th order 
1811. STUDY OF NONLINEAR DYNAMICS OF A PARTICLE ON A ROTATING PARABOLA BY USING EFFICIENT MATHEMATICAL TECHNIQUES.  
A. NIKKAR, H. JODEIRI, SH. AHARI ABESH AHMAD 
 © JVE INTERNATIONAL LTD. JOURNAL OF VIBROENGINEERING. NOV 2015, VOLUME 17, ISSUE 7. ISSN 1392-8716 3971 
method, presented in Figs. 1, 2. It is observable that our results are in excellent agreement with 
the results provided by Runge-Kutta 4th order. Also, for show the effects of different parameters 
on the period, we presented Figs. 3, 4. We can see that period increases when value of the ߝ 
decreases or value of the ݍ increases and other parameters remain constant. 
 
Fig. 1. Comparison between the VIM, VA with the Runge-Kutta 4th order when ߝ = 1, ݍ = ܣ = 1 
 
Fig. 2. Comparison between the VIM, VA with the Runge-Kutta 4th order when ߝ = 1, ݍ = ܣ = 0.5 
 
Fig. 3. The effects of ݍ on the period of motion when ߝ = ܣ = 1 
1811. STUDY OF NONLINEAR DYNAMICS OF A PARTICLE ON A ROTATING PARABOLA BY USING EFFICIENT MATHEMATICAL TECHNIQUES.  
A. NIKKAR, H. JODEIRI, SH. AHARI ABESH AHMAD 
3972 © JVE INTERNATIONAL LTD. JOURNAL OF VIBROENGINEERING. NOV 2015, VOLUME 17, ISSUE 7. ISSN 1392-8716  
 
Fig. 4. The effects of ɛ on the period of motion when ݍ = 2 and ܣ = 1 
7. Conclusion 
Study of nonlinear dynamics of a particle on a rotating parabola is main objective of this paper. 
We successfully applied two efficient and easy mathematical techniques called He’s variational 
iteration method and He’s variational approach to indicate the frequency-amplitude relationships. 
Comparison of results with numerical solutions are investigated. The performance and capability 
of applied methods are discussed by showing some figures. 
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1811. STUDY OF NONLINEAR DYNAMICS OF A PARTICLE ON A ROTATING PARABOLA BY USING EFFICIENT MATHEMATICAL TECHNIQUES.  
A. NIKKAR, H. JODEIRI, SH. AHARI ABESH AHMAD 
 © JVE INTERNATIONAL LTD. JOURNAL OF VIBROENGINEERING. NOV 2015, VOLUME 17, ISSUE 7. ISSN 1392-8716 3973 
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Ali Nikkar received M.Sc. degree in Civil Engineering from University of Tabriz. Now 
He is Ph.D. student at Friedrich-Schiller-University Jena. He was selected as Iranian 
distinguished university student by the Ministry of Science, Research and Technology in 
2014. His research interests include earthquake engineering, application of analytical and 
numerical methods in engineering and applied science, nonlinear vibration of structural 
systems. 
 
Hadi Jodeiri received M.Sc. degree in Civil Engineering from University of Tabriz. Now 
he is Ph.D. student Islamic Azad University. His current research interests include 
nonlinear dynamic, analytical methods, numerical methods and vibrations. 
 
Shahin Ahari Abesh Ahmad received M.Sc. degree in Civil Engineering from University 
of Tabriz. Now he works at Young Researches Club at University of Tabriz. His current 
research interests include nonlinear dynamic, analytical methods, numerical methods and 
vibrations. 
 


Study of nonlinear dynamics of a particle on a rotating parabola by using efficient mathematical techniques

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Study of nonlinear dynamics of a particle on a rotating parabola by using efficient mathematical techniques

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