Frequency analysis of nonlinear oscillations via the global error minimization
-
M Kalami Yazdi
und
P Hosseini Tehrani
Abstract
The capacity and effectiveness of a modified variational approach, namely global error minimization (GEM) is illustrated in this study. For this purpose, the free oscillations of a rod rocking on a cylindrical surface and the Duffing-harmonic oscillator are treated. In order to validate and exhibit the merit of the method, the obtained result is compared with both of the exact frequency and the outcome of other well-known analytical methods. The corollary reveals that the first order approximation leads to an acceptable relative error, specially for large initial conditions. The procedure can be promisingly exerted to the conservative nonlinear problems.
1 Introduction
The accurate prediction of nonlinear oscillations in many areas of physics, applied mathematics and structural dynamics has been a significant subject. Surveys of the literature expose that there are different approximate analytical techniques for dealing with the nonlinear problems. Among them, one may allude to the variational iteration method [1, 2], the energy balance method [3–5], the harmonic balance method [6, 7], the parameter expansion method [8, 9], the multiple scales method [10], the homotopy analysis method [11, 12], Max–Min approach [13, 14], Hamiltonian approach [15–17], the iteration perturbation method [18-21], the variational approach [22, 23], the homotopy perturbation method [23, 24], the frequency– amplitude formulation [25–28] and so on [29, 30].
This study intends to extend the reliability and applicability of the global error minimization [31–33] by considering the governing equation of a uniform rod rocking on the cylindrical surface without slipping [34] and the Duffing-harmonic oscillator [35–37]. The algorithm transforms the nonlinear differential equation into an equivalent optimization problem. After substitution of the trial function into the functional, unknown parameters of it is acquired using a Ritz–like method. It should be mentioned that the construction of the functional in this method is similar to the least squares approach. More details about the technique can be found in the literature [31]. The rest of the manuscript is organized as follows. The outline of the method is presented in section 2. The approach is applied to the governing equation of the uniform rod rocking on the cylindrical surface in section 3. The relationship between the frequency and the initial amplitude of the Duffing-harmonic oscillator is provided for either first- and second-order approximations in section 4. Section 5 ends this study with a brief conclusion.
2 The global error minimization
This section gives the basic idea of the global error minimization. Consider a general nonlinear oscillator as follows:
By defining a functional as follows:
and assuming F (u) is an odd function. One may utilize an approximate trial function in the form of
The unknown parameters (i.e., a(2n+1) & ω) can be find through the following conditions:
To demonstrate the practicality and effectiveness of the aforementioned method, the governing equation of a uniform rod rocking on the cylindrical surface and the Duffing-harmonic oscillator are taken into consideration in the present study. The results are illustrated in next sections.
3 Case 1
Fig. 1 depicts the schematic of the uniform bar rocking on a cylindrical surface. The general equation of the motion is [34]:
where parameters l, r and g are the rod’s length, the radius of cylindrical surface and the acceleration of gravity, respectively. Eq. (5) can be rewritten as:
where
![Fig. 1 Thin uniform bar rocking on a cylindrical surface [34].](/document/doi/10.1515/nleng-2015-0036/asset/graphic/j_nleng-2015-0036_fig_001.jpg)
Thin uniform bar rocking on a cylindrical surface [34].
In the following, the global error minimization is applied to the Eq. (6). As can be seen, a good agreement with exact ones is achieved for the first-order approximation.
Based on the the basic idea of the algorithm, the minimization problem of Eq. (6) is:
To determine the approximate frequency, the following trial function is employed:
Substituting Eq. (8) into Eq. (7) yields:
where Jα(z) is the Bessel function of the first kind. by applying
To illustrate the validity and accuracy of the global error minimization, with assumption r = l/4 the Eq. (10) is reduced to:
where y = g/l . For the first order approximation, Wu et al. [6] acquired:
where the exact frequency for this condition is given as:
.
Table 1 compares the approximate frequencies with respect to the exact ones for different initial conditions when y = 1. For more convenience, the result is presented in Fig. 2. Moreover, the relative error of both methods is demonstrated in Fig. 3. In contrast to the linearized harmonic balance method (the combination of the linearization of the governing equation with the method of harmonic balance), the relative error of the global error minimization does not increase continuously, and its rate is variable in the domain. As can be seen, the accuracy of the global error minimization is better in some sections, and Eq. (12) gives a lower relative error for small amplitudes.
The value of frequency in the domain (y = 1).
Comparison between relative errors of the first order approximation for Case I.
4 Case 2
This section investigates accuracy of a the approach by the Duffing-harmonic oscillator. This nonlinear model has a rational form for the restoring force. The governing equation of motion for this type oscillator is:
.
Equation (14) is a mathematical model of a conservative system. for small and large values of u, it is a Duffing oscillator (i.e., ü + u³ ≈ 0) and a linear harmonic oscillator (i.e., ü + u≈ 0), respectively. The exact frequency of this oscillator is given as:
The first- and second-order approximations for this nonlinear model are given in the following context.
4.1 First-order approximation
Based on the section 2, the modified variational approach is exerted. The equation (14) can be rewritten as:
.
The minimization problem is:
for the first order approximation, the trial function is:
Where a1 = A, substituting Eq. (18) into Eq. (17) yields:
,
by applying
Comparison of approximate frequencies with exact ones (Case I)
| β(Degree) | ωGEM (RE %) | ωWu [6] (RE %) | ωe (Eq. 13) |
|---|---|---|---|
| 5 | 1.7271065591268644 (0.0004) | 1.7271127875434222 (0.0000003) | 1.7271127818492038 |
| 10 | 1.7123006522375672 (0.01) | 1.7123979618568987 (0.00002) | 1.7123976031210428 |
| 15 | 1.687723383986749 (0.03) | 1.6881969036559472 (0.0002) | 1.6881929232337736 |
| 20 | 1.6535547330682832 (0.08) | 1.6549710318127808 (0.001) | 1.6549494461907537 |
| 25 | 1.6100984795905333 (0.19) | 1.6133212470119473 (0.005) | 1.613242407046901 |
| 30 | 1.557812599993792 (0.38) | 1.5639492479882797 (0.01) | 1.5637253179110349 |
| 35 | 1.4973246338641835 (0.65) | 1.5076154205171215 (0.04) | 1.5070808958109516 |
| 40 | 1.4294245459508197 (1.01) | 1.4450968213435502 (0.08) | 1.4439726688902643 |
| 45 | 1.3550338337819872 (1.45) | 1.3771478484815676 (0.16) | 1.3749984571501832 |
| 50 | 1.2751562381380954 (1.96) | 1.3044649090836247 (0.29) | 1.3006439386772928 |
| 55 | 1.190820400643789 (2.49) | 1.227654936034564 (0.53) | 1.2212304561528435 |
| 60 | 1.1030273049757937 (2.97) | 1.147205997567743 (0.91) | 1.1368444002767581 |
| 65 | 1.0127162975457542 (3.29) | 1.0634562779649626 (1.55) | 1.047220857582407 |
| 70 | 0.9207659649820763 (3.23) | 0.9765547140194191 (2.63) | 0.9515157322021774 |
| 75 | 0.8280556886595868 (2.33) | 0.8864008582021395 (4.56) | 0.8477781234901954 |
| 80 | 0.7356390479943411 (0.58) | 0.7925386096868637 (8.35) | 0.7314320558826457 |
4.2 Second-order approximation
To illustrate the capacity of the approach, the secondorder approximation of the algorithm is applied to the Duffing-harmonic oscillator. Substituting (16) into (2), by using the following trial function:
, where a1 + a3 = A, gives:
.
by employing
and the condition
where
Comparison between relative errors for Case II.
Phase space diagram of the second-order approximation for Case II (A=1).
Comparison of approximate frequencies with exact ones for the second-order approximation (Case II)
| A | a3 | ω2 | ω2/ωe |
|---|---|---|---|
| 0.05 | 0.002104 | 0.042470 | 1.003571 |
| 0.1 | 0.004200 | 0.084682 | 1.003482 |
| 0.5 | 0.019571 | 0.387844 | 1.001235 |
| 1 | 0.030982 | 0.636696 | 0.999868 |
| 5 | 0.013949 | 0.971407 | 1.004583 |
| 10 | 0.007104 | 0.992643 | 1.001742 |
| 50 | 0.001428 | 0.999703 | 1.000095 |
| 100 | 0.000714 | 0.999926 | 1.000025 |
| 500 | 0.000143 | 0.999997 | 1.000001 |
The achieved results using the second-order approximation of this straightforward approach show that the technique is easy, convenient and accurate for conservative nonlinear oscillators that the restoring force has a rational form.
5 Conclusions
This study scrutinizes the accuracy of the global error minimization (GEM) by examining two nonlinear equations which arise from the free oscillations of a rigid rod rocking on the cylindrical surface without slipping and the Duffing-harmonic oscillator. The reliable results are validated by the exact solutions. This applicable technique provides a satisfactory approximate frequency for the first order approximation. Higher order estimations using this method should be more accurate for other resembling nonlinear problems with odd and rational restoring forces. At last, The GEM method is a simple and powerful algorithm that can be easily implemented to similar nonlinear systems.
References
[1] J.H. He, Variational iteration method — a kind of non-linear analytical technique: Some examples, Int. J. Non-Linear Mech. 34 (1999) 699–708.10.1016/S0020-7462(98)00048-1Suche in Google Scholar
[2] S. Bagheri, A. Nikkar, Higher order explicit solutions for nonlinear dynamic model of column buckling using variational approach and variational iteration algorithm-II, J. Mech. Sci. Technol. 28 (2014) 4605–4611.10.1007/s12206-014-1028-zSuche in Google Scholar
[3] M.K. Yazdi, P.H. Tehrani, The energy balance to nonlinear oscillations via Jacobi collocation method, Alex. Eng. J. 54 (2015) 99–103. 10.1016/j.aej.2015.03.016Suche in Google Scholar
[4] T. Ozis, A. Yildirim, Determination of the frequency–amplitude relation for a Duffing-harmonic oscillator by the energy balance method, Comput. Math. Appl. 54 (2007) 1184–1187.10.1016/j.camwa.2006.12.064Suche in Google Scholar
[5] L. Cveticanin, M.K. Yazdi, H. Askari, Analytical solutions for a generalized oscillator with strong nonlinear terms, J. Eng. Math. 77 (2012) 211–223.10.1007/s10665-012-9542-4Suche in Google Scholar
[6] B.S. Wu, C.W. Lim, L.H. He, A new method for approximate analytical solutions to nonlinear oscillations of nonnatural systems, Nonlinear Dyn. 32 (2003) 1–13.10.1023/A:1024223118496Suche in Google Scholar
[7] M.A. Hosen, Approximate solutions of the equation of motion’s of the rigid rod which rocks on the circular surface without slipping, Ain Shams Eng. J. 5 (2014) 895–899.10.1016/j.asej.2014.01.005Suche in Google Scholar
[8] S.S. Ganji, G. Domairry, A.G. Davodi, H. Babazadeh, S.H.S. Ganji, Higher-order approximations of motion of a nonlinear oscillator using the parameter expansion technique, Int. J. Mod. Phys. B 23 (2009) 4779–4789.10.1142/S0217979209053886Suche in Google Scholar
[9] H.M. Sedighi, F. Daneshmand, M. Abadyan, Dynamic instability analysis of electrostatic functionally graded doublyclamped nano-actuators, Compos. Struct. 124 (2015) 55–64.10.1016/j.compstruct.2015.01.004Suche in Google Scholar
[10] A.H. Nayfeh, D.T. Mook, Nonlinear Oscillations, Wiley, New York, 1979.Suche in Google Scholar
[11] H.M. Sedighi, K.H. Shirazi, J. Zare, An analytic solution of transversal oscillation of quintic non-linear beam with homotopy analysis method, Int. J. Non-Linear Mech. 47 (2012) 777–784.10.1016/j.ijnonlinmec.2012.04.008Suche in Google Scholar
[12] A. Reza, H.M. Sedighi, Nonlinear vertical vibration of tension leg platforms with homotopy analysis method, Adv. Appl. Math. Mech. 7 (2015) 357-368. 10.4208/aamm.2013.m314Suche in Google Scholar
[13] S.S. Ganji, D.D. Ganji, A.G. Davodi, S. Karimpour, Analytical solution to nonlinear oscillation system of the motion of a rigid rod rocking back using max–min approach, Appl. Math. Model. 34 (2010) 2676–2684.10.1016/j.apm.2009.12.002Suche in Google Scholar
[14] M.K. Yazdi, H. Ahmadian, A. Mirzabeigy, A. Yildirim, Dynamic analysis of vibrating systems with nonlinearities. Commun. Theor. Phys. 57 (2012) 183–18710.1088/0253-6102/57/2/03Suche in Google Scholar
[15] J.H. He, Hamiltonian approach to nonlinear oscillators. Phys. Lett. A 374 (2010) 2312–231410.1016/j.physleta.2010.03.064Suche in Google Scholar
[16] H. Askari, Z.S. Nia, A. Yildirim, M.K. Yazdi, Y. Khan, Application of higher order Hamiltonian approach to nonlinear vibrating systems, J. Theor. Appl. Mech. 51 (2013) 287–296.Suche in Google Scholar
[17] Y. Khan, Q. Wu, H. Askari, Z. Saadatnia, M.K. Yazdi, Nonlinear vibration analysis of a rigid rod on a circular surface via Hamiltonian approach, Math. Comput. Appl. 15 (2010) 974–977. 10.3390/mca15050974Suche in Google Scholar
[18] J.H. He, Iteration perturbation method for strongly nonlinear oscillations. J. Vib. Control 7 (2001) 631–642.10.1177/107754630100700501Suche in Google Scholar
[19] S.S. Ganji, A. Barari, S. Karimpour, G. Domairry, Motion of a rigid rod rocking back and forth and cubic-quintic Duflng oscillators. J. Theor. Appl. Mech. 50 (2012) 215–229. Suche in Google Scholar
[20] H.D. Kaliji, M. Ghadimi, M. Eftari, Investigating the dynamic behavior of two mechanical structures via analytical methods, Arab. J. Sci. Eng. 38 (2013) 2821–2829.10.1007/s13369-012-0494-9Suche in Google Scholar
[21] H.M. Sedighi, F. Daneshmand, Static and dynamic pull-in instability of multi-walled carbon nanotube probes by He’s iteration perturbation method, J. Mech. Sci. Technol. 28 (2014) 3459–3469.10.1007/s12206-014-0807-xSuche in Google Scholar
[22] J.H. He, Variational approach for nonlinear oscillators. Chaos Solitons Fractals 34 (2007) 1430–1439.10.1016/j.chaos.2006.10.026Suche in Google Scholar
[23] D. Younesian, H. Askari, Z. Saadatnia, M.K. Yazdi, Free vibration analysis of strongly nonlinear generalized Duflng oscillators using He’s variational approach & homotopy perturbation method, Nonlin. Sci. Lett. A 2 (2011) 11–16.Suche in Google Scholar
[24] S.E. Ghasemi, A. Zolfagharian, D.D. Ganji, Study on motion of rigid rod on a circular surface using MHPM, Propulsion Power Res. 3 (2014) 159–6410.1016/j.jppr.2014.07.003Suche in Google Scholar
[25] M. Ghadimi, A. Barari, H.D. Kaliji, G. Domairry, Periodic solutions for highly nonlinear oscillation systems, Arch. Civ. Mech. Eng. 12 (2012) 389–395.10.1016/j.acme.2012.06.014Suche in Google Scholar
[26] S.S. Ganji, D.D. Ganji, H. Babazadeh, N. Sadoughi, Application of amplitude–frequency formulation to nonlinear oscillation system of the motion of a rigid rod rocking back, Math. Meth. Appl. Sci. 33 (2010) 157–166.10.1002/mma.1159Suche in Google Scholar
[27] Y. Khan, M.K. Yazdi, H. Askari, Z. Saadatnia, Dynamic analysis of generalized conservative nonlinear oscillators via frequency amplitude formulation. Arab. J. Sci. Eng. 38 (2013) 175–179.10.1007/s13369-011-0035-ySuche in Google Scholar
[28] A. Fereidoon, M. Ghadimi, A. Barari, H.D. Kaliji, G. Domairry, Nonlinear vibration of oscillation systems using frequency– amplitude formulation, Shock Vib. 19 (2012) 323–332.10.1155/2012/303189Suche in Google Scholar
[29] L. Cveticanin, M.K. Yazdi, H. Askari, Z. Saadatnia, Vibration of a two-mass system with non-integer order nonlinear connection, Mech. Res. Commun. 43 (2012) 22–28.10.1016/j.mechrescom.2012.04.002Suche in Google Scholar
[30] I. Kovacic, Application of the field method to the non-linear theory of vibrations, J. Sound Vib. 264 (2003) 1073–1090. 10.1016/S0022-460X(02)01192-6Suche in Google Scholar
[31] Y. Farzaneh, A.A. Tootoonchi, Global error minimization method for solving strongly nonlinear oscillator differential equations, Comput. Math. Appl. 59 (2010) 2887–2895. 10.1016/j.camwa.2010.02.006Suche in Google Scholar
[32] A. Mirzabeigy, M.K. Yazdi, A. Yildirim, Analytical approximations for a conservative nonlinear singular oscillator in plasma physics. J. Egypt. Math. Soc. 20 (2012) 163–166.10.1016/j.joems.2012.05.001Suche in Google Scholar
[33] M.K. Yazdi, A. Mirzabeigy, H. Abdollahi, Nonlinear oscillators with non-polynomial and discontinuous elastic restoring forces, Nonlin. Sci. Lett. A 3 (2012) 48–53.Suche in Google Scholar
[34] E.W. Gaylord, Natural frequencies of two nonlinear systems compared with the pendulum, J. Appl. Mech. 26 (1959) 145– 146.10.1115/1.4011945Suche in Google Scholar
[35] R.E. Mickens, Mathematical and numerical study of the Duflng-harmonic oscillator, J. Sound Vib. 244 (2001) 563–567.10.1006/jsvi.2000.3502Suche in Google Scholar
[36] C.W. Lim, B.S. Wu, A new analytical approach to the Duflngharmonic oscillator. Phys. Lett. A 311 (2003) 365–373.10.1016/S0375-9601(03)00513-9Suche in Google Scholar
[37] C.W. Lim, B.S. Wu, W.P. Sun, Higher accuracy analytical approximations to the Duflng-harmonic oscillator, J. Sound Vib. 296 (2006) 1039–1045.10.1016/j.jsv.2006.02.020Suche in Google Scholar
© 2016 Walter de Gruyter GmbH, Berlin/Boston
This article is distributed under the terms of the Creative Commons Attribution Non-Commercial License, which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Artikel in diesem Heft
- Frontmatter
- Original Articles
- Flow and heat transfer of hydromagnetic Oldroyd-B fluid in a channel with stretching walls
- Research Article
- Numerical approximation of Newell-Whitehead-Segel equation of fractional order
- Research Article
- Frequency analysis of nonlinear oscillations via the global error minimization
- Research Article
- Uniform practical stability in terms of two measures with effect of delay at the time of impulses
- Research Article
- Multiple Solutions of an Unsteady Stagnation-Point Flow with Melting Heat Transfer in a Darcy–Brinkman Porous Medium
- Research Article
- On using block pulse transform to perform equivalent linearization for a nonlinear Van der Pol oscillator under stochastic excitation
- Research Article
- Sea Level Rise and Land Subsidence Contributions to the Signals from the Tide Gauges of China
- Research Article
- Analytical method for space-fractional telegraph equation by homotopy perturbation transform method
Artikel in diesem Heft
- Frontmatter
- Original Articles
- Flow and heat transfer of hydromagnetic Oldroyd-B fluid in a channel with stretching walls
- Research Article
- Numerical approximation of Newell-Whitehead-Segel equation of fractional order
- Research Article
- Frequency analysis of nonlinear oscillations via the global error minimization
- Research Article
- Uniform practical stability in terms of two measures with effect of delay at the time of impulses
- Research Article
- Multiple Solutions of an Unsteady Stagnation-Point Flow with Melting Heat Transfer in a Darcy–Brinkman Porous Medium
- Research Article
- On using block pulse transform to perform equivalent linearization for a nonlinear Van der Pol oscillator under stochastic excitation
- Research Article
- Sea Level Rise and Land Subsidence Contributions to the Signals from the Tide Gauges of China
- Research Article
- Analytical method for space-fractional telegraph equation by homotopy perturbation transform method
