Journal of Applied Mathematics
Volume 2018 (2018), Article ID 7487851, 15 pages
https://doi.org/10.1155/2018/7487851

Research Article

The Equivalent Linearization Method with a Weighted Averaging for Solving Undamped Nonlinear Oscillators

D. V. Hieu,¹ N. Q. Hai,² and D. T. Hung¹

¹Thai Nguyen University of Technology, Thai Nguyen, Vietnam
²Hanoi Architectural University, Ha Noi, Vietnam

Correspondence should be addressed to D. V. Hieu; hieudv@tnut.edu.vn

Received 27 June 2017; Accepted 5 March 2018; Published 18 April 2018

Academic Editor: Peter G. L. Leach

Copyright © 2018 D. V. Hieu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The Equivalent Linearization Method (ELM) with a weighted averaging is applied to analyze five undamped oscillator systems with nonlinearities. The results obtained via this method are compared with the ones achieved by Parameterized Perturbation Method (PPM), Min–Max Approach (MMA), Variational Iteration Method (VIM), Homotopy Perturbation Method (HPM), Energy Balance Method (EBM), Harmonic Balance Method (HBM), 4th-Order Runge-Kutta Method, and the exact ones. The obtained results demonstrate that this method is very convenient for solving nonlinear equations and also can be successfully applied to a lot of practical engineering and physical problems.

1. Introduction

Nonlinear oscillation problem is very important in the physical science, mechanical structures, and other kinds of mathematical sciences. Most of real systems are modeled by nonlinear differential equations which are important issues in mechanical structures, mathematical physics, and engineering. In most cases, it is difficult to solve such equations, especially analytically; and in addition, the most important information, such as the natural circular frequency of a nonlinear oscillation which depends on the initial conditions, will be lost during the procedure of numerical simulation.

Recently, nonlinear oscillator models have been widely considered in physics and engineering. It is obvious that there are many nonlinear equations in the study of different branches of science which do not have analytical solutions. Due to the limitation of existing exact solutions, many analytical approaches have been investigated. Many researchers have been working on various analytical methods for solving nonlinear oscillation systems in the last decades, such as Homotopy Perturbation Method (HPM) [1–5], Max–Min Approach (MMA) [5–10], Variational Iteration Method (VIM) [2, 5], Energy Balance Method (EBM) [5, 11–16], Amplitude-Frequency Formulation (AFF) [5–7, 15, 17, 18], Improved Amplitude-Frequency Formulation (IAFF) [5], Parameter Expansion Method (PEM) [5, 7, 18–20], Homotopy Analysis Method (HAM) [5, 21], Modified Homotopy Perturbation Method (MHPM) [5, 6], Modified Lindstedt-Poincare Method [22], Harmonic Balance Method [23, 24], and combined Newton’s Method with the Harmonic Balance Method [25].

The Equivalent Linearization Method is one of the common approaches to approximate analysis of dynamical systems. The original linearization for deterministic systems was proposed by Krylov and Bogoliuboff [26]. Then Caughey expanded the method for stochastic systems [27]. Thenceforward, there have been some extended versions of the Equivalent Linearization Method [28–32]. It has been shown that the Equivalent Linearization Method is presently the simplest tool widely used for analyzing nonlinear stochastic problems. Nevertheless, the accuracy of the Equivalent Linearization Method with conventional averaging normally reduces for middle or strong nonlinear systems. A reason is that some terms will vanish in the averaging process; for example, the averaging value of the functions and over one period is equal to zero. Recently, Anh proposed a new way for determining averaging values: instead of using conventional averaging process the author introduced weighted coefficient functions [31]; by this manner, the averaging value is calculated in a new way called the weighted averaging value. This proposed method has been applied effectively to analyze some strongly nonlinear oscillations such as the nonlinear Duffing oscillator with third, fifth, and seventh powers of the amplitude, the strongly nonlinear oscillators in forms and , and the cubic Duffing with discontinuity [33].

In this paper, the Equivalent Linearization Method with a weighted averaging continues to be applied to analyze five undamped nonlinear oscillators: the Duffing oscillator with cubic nonlinearity (Example 1); the motion of simple pendulum attached to a rotating rigid frame (Example 2); the motion of a mass attached to the center of a stretched elastic wire (Example 3); the Duffing oscillator with discontinuity (Example 4); and the nonlinear oscillator with fractional elastic force (Example 5). It should be emphasized that the generalized forms of the Duffing equation (in Example 1) and the oscillator (in Example 3) have been solved by Energy Balance Method [12, 16], and the generalized form of the nonlinear oscillator with fractional elastic force (in Example 5) has been solved by Energy Balance Method and Variational Approach [14]; these solutions are achieved by Younesian et al. [12, 14, 16]. The frequency-amplitude relationship is analytically obtained by using this proposed method. Accuracy and validity of results are then presented by comparing the results with the ones obtained by the other well-known techniques and the exact and numerical ones.

2. The Equivalent Linearization Method with a Weighted Averaging

2.1. The Equivalent Linearization Method

In order to introduce the general idea of the Equivalent Linearization Method, we consider a nonlinear oscillator governed by the following equation [33]:

where is a nonlinear function only depending on two variables of velocity and displacement and and are constants. The equivalent linear oscillator is described by the equation as follows:

The equation error between the two oscillators is taken as

The coefficients of linearization in the linearized equation (2) are found from an optimal criterion. There are some criteria for determining these coefficients. The most common criterion is the mean square error criterion which requires that the mean square of equation error be minimum:

Thus,

yields

In the formulations in (4), (6a), and (6b), the symbol denotes the time-averaging operator in classical meaning [26]:

For a -frequency function , the averaging process is taken during one period ; that is,

In this technique, the importance of the attended terms is considered to be the same on time scale. In fact, their roles generally differ from time to time. That may be one of the reasons causing the classical equivalent replacement to be effective only for oscillators with weak nonlinearity but normally not being good for ones with strong nonlinearity. In order to improve this shortcoming, the averaging operation with weighting functions is proposed as in the next section. This idea is introduced by Anh [31].

2.2. The Weighted Averaging

The conventional averaging value of an integrable deterministic function on a domain : is a constant value defined by

In many cases when the function is periodic with period , the value is taken as and it leads to the averaging value of over one period:

where . Definition (9) has some deficiencies; for example, if (9) or (10) are equal to zero, the information about x(t) will be lost. For all harmonic functions such as and , this observation is true. In [31], Anh suggested replacing the constant coefficient in (9) by a weighted coefficient function . Thus one gets the so-called weighted average value:

with the condition

There are three basic weighted coefficients:

(i)Basic Optimistic Weighted Coefficients. They are increasing functions of and denoted as O(t). Examples are and , α, β > 0.(ii)Basic Pessimistic Weighted Coefficients. They are decreasing functions of and denoted as . Examples are and ; or .(iii)Neutral Weighted Coefficients. They are denoted as and are constants.

An arbitrary weighted coefficient can be obtained as summation and/or product of basic weighted coefficients. Example is

where are constant.

In this paper, a special form of weighted coefficient is introduced as follows [31, 33]:

where is constant.

The properties of the weighted function in (14) can be viewed in [31, 33].

Based on the weighted coefficient (14), a new weighted averaging value is proposed:

which is a linear operator. From Laplace transform, we get, for example,

As -periodic functions can be expanded into Fourier series; hence we can easily calculate (15) by using (16).

In this paper, for the sake of comparison, the parameter is chosen equal to 2. In the next section, some practical examples for some strongly nonlinear vibration systems are illustrated to show the applicability, accuracy, and effectiveness of the present method.

3. Some Examples and Discussions

3.1. Example 1

Consider the following Duffing equation with cubic nonlinearity:

The linearized equation of (17) is

The equation error between (17) and (18) is

where is determined by using the mean square criterion, as follows:

The periodic solution of the linearized equation (18) is

Using definition (11) and weighted coefficient (14), we can calculate averaging operators in (20):

Substituting (22) into (20), we get the approximate frequency of this oscillator:

With chosen equal to 2, we have

From (21), the approximate solution of this oscillator is

Accuracy of this method for this example is shown in Table 1 and Figures 1 and 2. Table 1 shows the comparison between the approximate frequencies obtained by this method, the ones obtained by Parameterized Perturbation Method (PPM) [5], and the exact ones ; the relative error is again obtained for small and large values of the oscillation amplitude . Table 1 shows that the maximum relative error is less than 0.4733 for this method and 2.2135 for Parameterized Perturbation Method. The numerical results obtained by three different methods are illustrated in Figures 1 and 2. Numerical results validate again accuracy of this method. As it is shown in Figures 3 and 4, validity of the solution technique is guaranteed for stronger nonlinearities. Relative error versus the oscillation amplitude is illustrated in Figure 3.

Table 1: Comparison of the approximate frequencies with the exact frequency, Example 1.

. error (%)

. error (%)

0.1
0.5
0.1
0.7076
0.7076
0.0000
0.7076
0.0000

0.5
0.1
2
0.6800
0.6892
1.3501
0.6782
0.2647

1
2
0.5
1.5403
1.5411
0.0520
1.5362
0.2662

2
5
2
3.2958
3.3166
0.6313
3.2802
0.4733

5
2
5
9.5818
9.7852
2.1228
9.5917
0.1033

10
1
0.5
6.0722
6.2048
2.0994
6.0828
0.1746

15
0.5
2
17.9866
18.3848
2.2135
18.0139
0.1518

20
5
1
17.0977
17.4642
2.1436
17.1172
0.1141

Figure 1: Comparison of time history diagram of displacement between the present, PPM, and exact solutions at , Example 1.

Figure 2: Comparison of time history diagram of displacement between the present, PPM, and exact solutions at , Example 1.

Figure 3: Comparison of relative error between the present and PPM solutions with and , Example 1.

Figure 4: A simple pendulum attached to a rotating rigid frame.

The approximate frequency obtained by PPM [5] is the same as that obtained by EMB [12], as follows:

And the exact frequency of this oscillator is [5]

3.2. Example 2

The motion of a simple pendulum attached to a rotating rigid frame that is shown in Figure 4 has following nonlinear differential equation [6, 14]:

with the initial conditions:

in which and are generalized dimensionless displacements and time variables, and .

Equation (28) can be rewritten as follows:

Based on Taylor-Maclaurin expansion, the approximations of and are considered:

Substituting (31) into (30) yields

The linearized equation of (32) is

The equation error between (32) and (33) is

The unknown coefficient is determined from the mean square error criterion:

It yields

The periodic solution and the frequency of (33) are

Similarly to Example 1, we calculate averaging operators , , and :

Substituting averaging operators in (38) into (36), and then substituting (36) into (37), we get the approximate frequency (with chosen equal to 2):

and the approximate solution:

Comparisons between the results of the present method and the ones achieved by Jouybari and Ramzani [6] using Min–Max Approach (MMA), by Younesian et al. [14] using EBM, and using accurate numerical method are demonstrated in Tables 2 and 3, for two cases of and . A very interesting agreement between the results is observed.

Table 2: The results of present, approximate, obtained by Jouybari & Ramzani and by Yousesian et al., and numerical solution for when and , Example 2.

Numerical solution [12]
MMA solution [6]
. error
(%)
EBM solution
[14]
. error
(%)
Present solution
. error
(%)

0
1.000000000
1.000000000
0.000000000
1.000000000
0.000000000
1.000000000
0.000000000

0.5
0.939545503
0.945169971
0.598637105
0.944189348
0.494265045
0.945834709
0.669388122

1
0.769687769
0.786692549
2.20930885
0.782987048
1.727879737
0.789206596
2.535940908

1.5
0.520367974
0.541946378
4.146758655
0.534386712
2.694004762
0.547083274
5.13392471

2
0.226896964
0.237770336
4.792206916
0.226137434
0.334746656
0.245694103
8.284438306

2.5
−0.082424976
−0.092479615
12.19853434
−0.107353589
30.24400395
−0.082311253
0.137971529

3
−0.386261801
−0.412588246
6.815699852
−0.428861683
11.0287587
−0.401399783
3.919098902

3.5
−0.660436345
−0.687452426
4.090641165
−0.702499667
6.369019864
−0.677004442
2.50865918

4
−0.871990011
−0.886930535
1.713382471
−0.897723721
2.951147338
−0.879268816
0.834734906

4.5
−0.986529936
−0.989147789
0.265359712
−0.992742682
0.629757473
−0.986281489
0.025183929

5
−0.982752773
−0.982895042
0.014476581
−0.970950401
1.200950262
−0.986449715
0.376182302

5.5
−0.861382758
−0.868857967
0.867815025
−0.852109657
1.07653664
−0.879755271
2.132909305

6
−0.644890958
−0.659541879
2.271844692
−0.632155313
1.974852468
−0.677756428
5.09628327

6.5
−0.368002797
−0.377900390
2.689542873
−0.341638967
7.164029789
−0.402335838
9.329559797

7
−0.063204827
−0.054818323
13.26877139
−0.012988435
79.45024832
−0.083329972
31.84115194

7.5
0.245785975
0.274275125
11.59103972
0.317111883
29.01951912
0.244703077
0.440585758

8
0.537446333
0.573291546
6.669542762
0.611815759
13.8375539
0.546227301
1.633831596

8.5
0.78289188
0.809440784
3.391132885
0.838227903
7.068156461
0.788578403
0.726348445

9
0.946701653
0.956826699
1.069507586
0.971676667
2.638108207
0.945502349
0.12668236

9.5
0.999764591
0.999286944
0.047775947
0.995531392
0.423419577
0.999999478
0.023494231

10
0.931963943
0.932165326
0.021608454
0.908764206
2.489338474
0.946166081
1.523893505

Table 3: The results of present, approximate obtained by Jouybari & Ramzani, and numerical solution for when and , Example 2.

Numerical solution [2]
MMA solution
[6]
. error
(%)
EBM solution
[14]
. error
(%)
Present solution
. error
(%)

0
1.000000000
1.000000000
0.000000000
1.000000000
0.000000000
1.000000000
0.000000000

0.5
0.953968489
0.963119576
0.959265123
0.961786066
0.819479583
0.964200044
1.072525468

1
0.825442672
0.855198634
3.604849011
0.850064873
2.982908666
0.859363450
4.109404463

1.5
0.638888746
0.684197515
7.091808908
0.673375035
5.397855138
0.692996509
8.469043059

2
0.422287711
0.462729406
9.576810773
0.445220578
5.430626183
0.477011079
12.95878771

2.5
0.195637881
0.207129984
5.874170657
0.183038862
6.439969057
0.226871698
15.96511721

3
−0.032545617
−0.063747522
95.87129659
−0.093132124
186.1587291
−0.039511677
21.40398813

3.5
−0.260619945
−0.329922957
26.5915995
−0.362185221
38.97064593
−0.303066019
16.28657929

4
−0.485739543
−0.571762994
17.70978959
−0.603557273
24.25533019
−0.544920861
12.18375544

4.5
−0.696309233
−0.771429307
10.78832083
−0.79880073
14.71925003
−0.747759417
7.388984888

5
−0.869266232
−0.914194339
5.16850941
−0.93299355
7.331162267
−0.897058465
3.197206101

5.5
−0.976272484
−0.989527622
1.357729345
−0.995879662
2.008371466
−0.982128207
0.599804163

6
−0.996192580
−0.991872507
0.43365842
−0.982652815
1.35915136
−0.996877655
0.068769334

6.5
−0.924854298
−0.921056034
0.410687825
−0.894323908
3.301102678
−0.940250751
1.664743629

7
−0.776876524
−0.782301685
0.698329893
−0.737643731
5.050068034
−0.816301976
5.074867213

7.5
−0.579081909
−0.585844101
1.167743612
−0.524587017
9.410567167
−0.633906052
9.467424582

8
−0.358097172
−0.346174158
3.329547098
−0.271437236
24.20011739
−0.406122511
13.41125894

8.5
−0.130576292
−0.080970115
37.99018661
0.002457915
101.8823593
−0.149260634
14.30913814

9
0.097635936
0.190206352
94.8118283
0.276165213
182.8520157
0.118288292
21.15241257

9.5
0.325420035
0.447353037
37.46942071
0.528765792
62.48716586
0.377367786
15.96329218

10
0.548087739
0.671502582
22.51735155
0.740953929
35.1889262
0.609427780
11.19164627

To illustrate and verify the accuracy of the current analytical approach, the Equivalent Linearization Method with a weighted averaging, in comparison with other well-known techniques, we performed a comparison between the results obtained by this method and outcomes achieved by Jouybari and Ramzani [6], by Younesian et al. [15], and by using the Runge-Kutta 4th-Order Method. Variations of for different are illustrated in Figures 5–7.

Figure 5: A comparison between the approximate and numerical solutions with and , Example 2.

Figure 6: A comparison between the approximate and numerical solutions with and , Example 2.

Figure 7: A comparison between the approximate and numerical solutions with and , Example 2.

The approximate frequency obtained by using MMA [6] is

The approximate frequency obtained by using EBM [15] is

3.3. Example 3

The equation of motion of a mass attached to the center of a stretched elastic wire in dimensionless form is as follows [5, 16, 18]:

where is a parameter, .

Equation (43) can be written as follows:

The linearized equation of (44) is

The equation error between (44) and (45) is

where is determined by using the mean square criterion, as follows:

The periodic solution of the linearized equation (45) is

Using (48), we have

Thus

Substituting (49) and (50) into (47), we get

Using definition (11) and weighted coefficient (14), we calculate averaging operators in (51):

With chosen equal to 2, substituting (52) into (51), we get the approximate frequency of this oscillator:

and the approximate solution of this oscillator:

where

Thus, the approximate period of this oscillator is

For , , , and , comparison of the approximate periods in (56), the approximate Variational Iteration Method (VIM) periods [5] in (57), and the Energy Balance Method (EBM) periods [16] in (58) with exact periods in (59) is tabulated in Table 4. From Table 4, the accuracy of this proposed method can be observed.

Table 4: Comparison of the approximate periods with the exact periods, Example 3.

[18]
[5]
. error (%)

. error (%)
[16]
. error (%)

0.1
0.1
6.62168
6.621237
0.00669
6.62174
0.00091
6.62169
0.00011

1
0.1
6.537508
6.517854
0.30063
6.53806
0.00844
6.53573
0.02723

10
0.1
6.322938
6.314678
0.13064
6.32097
0.03112
6.32006
0.04558

100
0.1
6.287182
6.286329
0.01357
6.28696
0.00353
6.28687
0.00498

0.1
0.5
8.869257
8.863794
0.06159
8.86990
0.00725
8.86925
0.0001

1
0.5
7.992133
7.814722
2.21982
7.99408
0.02436
7.97282
0.2416

10
0.5
6.490208
6.445572
0.68774
6.47925
0.16884
6.47431
0.24498

100
0.5
6.303280
6.298952
0.06866
6.30214
0.01809
6.30167
0.02557

0.1
0.75
12.49673
12.47385
0.18309
12.49943
0.02161
12.4967
0.00041

1
0.75
9.625404
9.168186
4.75012
9.61936
0.06279
9.56406
0.63733

10
0.75
6.602092
6.531632
1.06724
6.58442
0.26767
6.57665
0.38536

100
0.75
6.313404
6.306879
0.10335
6.31168
0.02731
6.31097
0.03854

0.1
0.95
27.15679
26.86136
1.08787
27.18975
0.12137
27.1540
0.01043

1
0.95
12.075277
10.96676
9.18005
12.00887
0.54994
11.8737
1.66932

10
0.95
6.696117
6.60302
1.39031
6.67235
0.35494
6.66211
0.50781

100
0.95
6.321539
6.313243
0.13123
6.31934
0.03479
6.31844
0.04897

The approximate period obtained by Variational Iteration Method is as follows [5]:

The approximate period obtained by Energy Balance Method is as follows [16]:

The exact period of this oscillator is as follows [18]:

Comparisons of the approximate solutions obtained by two analytical methods with the exact ones is presented in Figures 8 and 9. And the relative error versus the oscillation amplitude is illustrated in Figure 10.

Figure 8: Comparison of time history diagram of displacement between the present, VIM, and exact solutions at , Example 3.

Figure 9: Comparison of time history diagram of displacement between the present, VIM, and exact solutions at , Example 3.

Figure 10: Comparison of relative error between the present, VIM, and EBM solutions with , Example 3.

3.4. Example 4

Consider the Duffing oscillator with discontinuity [3]:

The linearized equation of (60) is

The equation error between (60) and (61) is

The unknown coefficient is determined from the mean square error criterion, as follows:

The periodic solution of (61) is

Similar to Examples 1–3, we calculate averaging operators and ; then, substituting these operators into (63) with note that the parameter is chosen equal to 2, we have the approximate frequency of this oscillator:

Thus, the approximate solution is

Accuracy of the approach for this example is shown in Figures 11 and 12. We performed a comparison between the results obtained by this method, the ones obtained by He [3] using the Homotopy Perturbation Method, and outcomes achieved using the Runge-Kutta 4th-Order Method for different values of , , and .

Figure 11: Comparison of time history diagram of displacement between the present, HPM, and numerical solutions at , Example 4.

Figure 12: Comparison of time history diagram of displacement between the Present, HPM, and numerical solutions at , Example 4.

3.5. Example 5

Consider the nonlinear oscillator equation of the form [23, 24, 34]

where is a positive integer.

The linearized equation of (67) is

The equation error between (67) and (68) is

The unknown coefficient is determined from the mean square error criterion, as follows:

The periodic solution of (68) is

Using definition (11) and weighted coefficient (14), we calculate averaging operators in (70):

With chosen equal to 2, from (72) and (73), we have

Substituting (71) and (72) into (70), we get the approximate frequency of this oscillator:

where

For (the harmonic oscillator case) it follows from (77) and (76) that is equal to the well-known value 1. And, for , is equal to . For other values of , the integral in (77) can be calculated numerically by using the formula manipulation package Maple. For some values of approximations of the frequency are given in Table 5. Table 5 presents the comparison of the approximate frequencies in (76), the approximate frequencies obtained by van Horssen [34], and the approximate frequencies obtained by Mickens [23]. As can be seen from this table the fractional errors of the approximations obtained in [34] for range from approximately 2 to approximately 11 percent, while the fractional errors of the approximations obtained in this paper range from 2 to 10 percent. And, from Table 5, when increases, the relative error of the approximate frequencies obtained by van Horssen increases, while the relative error of the approximate frequencies in this paper decreases.

Table 5: Comparison of the approximate frequencies, Example 5.

[23]
[34]
. error (%)

. error (%)

0
1.00000
1.000000
0.00000
1.000000
0.00000

1

2.03313

10.10847

2

3.62649

8.680304

3

4.69039

7.72281

4

5.45295

7.037311

5

6.03179

6.517881

6

6.48627

6.110074

7

6.85704

5.779148

8

7.16298

5.504079

9

7.42243

5.272273

10

7.64509

5.07269

50

9.93668

3.028171

500

10.8938

2.17776

11.07200

2.02100

Comparison of the approximate solutions obtained by three methods is present in Figures 13 and 14.

Figure 13: Comparison of time history diagram of displacement between the present, HBM [23], and App [34] solutions with , Example 5.

Figure 14: Comparison of time history diagram of displacement between the Present, HBM [23] and App [34] solutions with , Example 5.

4. Conclusions

In this paper, the Equivalent Linearization Method with a weighted averaging is applied to analyze five undamped nonlinear oscillation systems. This method is proposed by Anh in 2015. Instead of using the classical averaging process in (9) and (10), a special form of weighted coefficient is introduced in (14), whereby a new weighted average value is computed by (15). The proposed weighted function is a combination of two basic coefficient functions (the optimistic weighted coefficient and the pessimistic weighted coefficient ). The frequency-amplitude relationship was analytically obtained by this method. The frequency depends not only on the initial amplitude of the oscillator but also on the parameter in the expression of weighted coefficient . Accuracy of this method is investigated by five nonlinear oscillation systems. The results show that this method is useful in obtaining analytical solutions for oscillators and vibration problems with nonlinearities. The approximate solutions obtained by this method are valid not only for small parameters but also for very large parameters. And the results indicate that the solution procedure is easy and provides a remarkable accuracy. In this paper, is chosen equal to 2 for comparison and convenient calculating purposes. However, the value of the parameter in the expression of the weighted coefficient should be chosen to give better results and the best solution still requires further investigation.

Conflicts of Interest

The authors declare that no conflicts of interest took place during the preparation of the manuscript.

Acknowledgments

The research is supported by Thai Nguyen University of Technology grant for a scientific project (no. “T2018-B27”).

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