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Enhanced He’s frequency-amplitude formulation for nonlinear oscillators
Journal Pre-proofs Microarticle Enhanced He’s frequency-amplitude formulation for nonlinear oscillators Alex Elías-Zú ñiga, Luis Manuel Palacios-Pineda, Isaac H. Jiménez-Cedeño, Oscar Martínez-Romero, Daniel Olvera Trejo PII: DOI: Reference:
S2211-3797(20)32059-3 https://doi.org/10.1016/j.rinp.2020.103626 RINP 103626
To appear in:
Results in Physics
Received Date: Revised Date: Accepted Date:
27 September 2020 9 November 2020 14 November 2020
Please cite this article as: Elías-Zú ñiga, A., Manuel Palacios-Pineda, L., Jiménez-Cedeño, I.H., MartínezRomero, O., Olvera Trejo, D., Enhanced He’s frequency-amplitude formulation for nonlinear oscillators, Results in Physics (2020), doi: https://doi.org/10.1016/j.rinp.2020.103626
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2020 Published by Elsevier B.V.
Results in Physics ## (2020) #### Contents lists available at ScienceDirect
Results in Physics journal homepage: www.elsevier.com/locate/rinp
Microarticle
Enhanced He’s frequency-amplitude formulation for nonlinear oscillators Alex Elías-Zúñigaa*, Luis Manuel Palacios-Pinedab, Isaac H. Jiménez-Cedeñoa, Oscar Martínez-Romeroa, and Daniel Olvera Trejoa aMechanical
Engineering and Advanced Materials Department, School of Engineering and Science, Tecnologico de Monterrey, Ave. Eugenio Garza Sada 2501, Sur, Monterrey 64849, NL, Mexico. bTecnológico
Nacional de México/Instituto Tecnológico de Pachuca, Carr. México-Pachuca, km 87.5, Pachuca, Hidalgo, Código Postal 42080, Mexico.
ABSTRACT In this article a trial residual Jacobi elliptic weighted function is introduced to obtain the frequency-amplitude expression of nonlinear oscillators using He´s formulation. Simulation results show an improvement in the accuracy of the frequency values when compare to numerical integration simulations
——— * Corresponding author. Tel.: +52 81; e-mail: [email protected].
A. Elías-Zúñiga, et. al.
Results in Physics ## (2020) #####
𝑘21)(((3 + 𝑚)(5 + 𝑚)(7 + 𝑚)𝛼₀ + 𝐴22(2 + 𝑚)((7 + 𝑚)((5 + 𝑚)𝛼₁ + 𝐴22 (4 + 𝑚)𝛼₂) + 𝐴42(4 + 𝑚)(6 + 𝑚)𝛼₃) ― (5 + 𝑚)(7 + 𝑚)(3 ― 2𝑘₂² + 𝑚) 𝜔22)𝛤[1 + 𝑚/2]𝛤[2 + 𝑚/2] + 8𝐴₂𝛼₂₂𝛤[(3 + 𝑚)/2]𝛤[(9 + 𝑚)/2])).
1. Introduction For nonlinear oscillators of the form 𝑢 + 𝛼0𝑢 + 𝛼1𝑢3 + 𝛼2𝑢5 + 𝛼3𝑢7 + 𝛼22𝑢2 = 0, with 𝑢(0) = 𝐴,𝑢(0) = 0 the frequency-amplitude relation can be determined using the ancient Chinese method and Jacobi elliptic functions 𝑦𝑖 = 𝐴𝑖𝑐𝑛(ω𝑖𝑡,𝑘2𝑖 ) with frequency ω𝑖 and modulus 𝑘2𝑖 [1]. This approach requires to find two trial residual functions 𝑅1(𝑡) and 𝑅2(𝑡). Then, following the steps of the approach introduced in [2]-[5] and assuming that ui Ai cn(it , ki2 ), the corresponding trial residual functions are obtained:
(1)
2. Results To study the accuracy achieved by Eq. (4), the Duffing-type oscillator (1) is considered [8, 9]. Moreover, the frequency-amplitude solution 2 8 16 2 RE 0 1 A2 2 A4 3 A6 22 A, (7) 3 15 35 4
1
𝑅𝑖(𝑡) = 64 derived by Ren and Hu in [6], and slightly modified in [10] to include 2 3 5 7 2 2 )cosquadratic (32𝐴2𝑖 α22 + (64𝐴𝑖α0 + 48𝐴3𝑖 α1 + 40𝐴5𝑖 α2 + 35𝐴7𝑖 α3 + 32𝐴𝑖( ―2 + 𝑘2𝑖 )ω2𝑖the ϕ𝑖 + 32𝐴 α1 +and 20𝐴 𝑖 α22cos 2ϕ 𝑖 +of(16𝐴 𝑖 αfrequency-amplitude 2 + 21𝐴𝑖 α3 ― 32𝐴𝑖𝑘𝑖 ω𝑖 )cos 3 nonlinear term Eq. 𝑖(1), the relation 𝜔2𝐸𝑍 = ( ― 2𝜔211(105𝐴22𝛼0 +70𝐴322𝛼1 +56𝐴522𝛼2 +48𝐴722𝛼3 +35(3𝐵22 ― 3𝐴22 +(2𝐴22 ― 𝐵22)𝑘222)𝜔222)𝐾(𝑘211 ) + (2(105𝐴11𝛼0 +70𝐴311𝛼1 +56𝐴511𝛼2 +48𝐴711𝛼3 +35(3 in which the elliptic Jacobi amplitude ϕ𝑖 = 𝑎𝑚(ω𝑖𝑡,𝑘2𝑖 ) and the 𝐵11 ― 3𝐴11 +(2𝐴11 ― 𝐵11)𝑘211)𝜔211)𝜔222) + 105𝛼22(𝐴11𝜔22 2 2 identities cos ϕ𝑖 = 𝑐𝑛(ω𝑖𝑡,𝑘𝑖 ) and sin ϕ𝑖 = 𝑠𝑛(ω𝑖𝑡,𝑘𝑖 ) where applied ― 𝐴22𝜔11)(𝐴22𝜔11 + 𝐴11𝜔22)𝐾(𝑘211))𝐾(𝑘222))/( ― 2(105 (8) [7]. Then, we use He’s frequency-amplitude formulation and 𝐴22𝛼0 +70𝐴322𝛼1 +56𝐴522𝛼2 +48𝐴722𝛼3 +35(3𝐵22 ― 3𝐴22 introduced a power-form weighted functions 𝑊𝑖(𝑡) so that the trial +(2𝐴22 ― 𝐵22)𝑘222)𝜔222)𝐾(𝑘211) + (210𝐴11𝛼0 +140𝐴311𝛼1 +112𝐴511𝛼2 +96𝐴711𝛼3 +70(3𝐵11 ― 3𝐴11 +(2𝐴11 ― 𝐵11) residual functions are now defined as [8, 9] 4 𝑇/4 𝑘211)𝜔211) + 105(𝐴11 ― 𝐴22)(𝐴11 + 𝐴22)𝛼22𝐾(𝑘211))𝐾(𝑘222)), 𝑅𝑖(𝑡)(𝑊𝑖(𝑡))𝑚𝑑𝑡, 𝑅𝑖 = (3) 𝑇 0 proposed by Elías-Zuñiga et al. in [10] are used for comparison purposes. The percentage of relative error attained from Eq. (4) when where it is assumed that the residual weighted functions have the form compare to the exact frequency-amplitude value of Eq. (1) are listed 𝑊𝑖(𝑡) = cos ϕ𝑖, 𝑖 = 1, 2, where 𝑚 is a constant that can be determined in Table 1, for the system parameter values of α0 = 1, 𝐴 = 1, 𝐴1 = 𝐴2 so that the approximate frequency-amplitude expression for the = 𝐴11 = 𝐴22 = 1, 𝐵11 = 𝐵22 = ―0.3, ω1 = ω11 = 1, ω2 = ω22 = 2, nonlinear oscillator (1) 𝑘21 = 𝑘22 = 0.0618, and 𝑘211 = 𝑘222 = 0.395 with 𝑚 = 0.023. It can be ω21𝑅2 ― ω22𝑅1 seen from Table 1, that predicted values for ω𝐴𝐸 follow well the exact ω2 = , ones. In fact, for all cases listed in Table 1, the average error value of (4) 𝑅2 ― 𝑅1 𝜔𝐴𝐸 does not exceed of 0.783% when compared to the values of ω𝑅𝐸𝑁 [6] and ω𝐸𝑍 [10]. is found [2]-[5]. Thus, substitution of Eq. (2) into Eq. (3), yields
∫
𝑅𝑖(𝑡) = (√𝜋((8𝐴2𝑖 𝛼₂₂𝛤[(3 + 𝑚)/2])/𝛤[2 + 𝑚/2] + 1/𝛤[(9 + 𝑚)/2](𝐴𝑖 (7 + 𝑚)((3 + 𝑚)(5 + 𝑚)𝛼₀ + 𝐴2𝑖 (2 + 𝑚)((5 + 𝑚)𝛼₁ + 𝐴2𝑖 (4 + 𝑚)𝛼₂)) + 𝐴7𝑖 (2 + 𝑚)(4 + 𝑚)(6 + 𝑚)𝛼₃ ― 𝐴𝑖 (5 + 𝑚)(7 + 𝑚)(3 ― 2𝑘2𝑖 +𝑚)ω2𝑖 )𝛤[1 + 𝑚/2]))/(16𝐾(𝑘2𝑖 )),
Table 1. Exact and estimated frequency values for Duffing-type nonlinear oscillators.
(5)
where Γ[z] is the Gamma function, and K (ki2 ) is the complete elliptic integrals of the first kind with modulus ki2 . Finally, substitution of Eq. (5) into Eq. (4), provides the following frequencyamplitude expression
𝜶𝟐𝟐
𝜶𝟏
𝜶𝟐
𝜶𝟑
𝝎𝑬𝒙𝒂𝒄𝒕
𝝎𝑨𝑬
Error (%)
𝝎𝑹𝑬
Error (%)
𝝎𝑬𝒁
Error (%)
1 10 100 1000 1 1 10 100 1000 0 0 0 0 0 0 0 0 0 0 0 0 0 20 10 0 0 0 0 0 0
1 10 100 1000 0 1 10 100 1000 1 10 100 1000 0 0 0 0 0 0 0 0 1 1 10 100 1000 0 0 0 0
1 10 100 1000 0 0 0 0 0 0 0 0 0 1 10 100 1000 0 0 0 0 1 1 10 100 1000 1 10 100 1000
1 10 100 1000 0 0 0 0 0 0 0 0 0 0 0 0 0 1 10 100 1000 0 1 10 100 1000 1 10 100 1000
1.909 9 5.204 0 16.16 41 51.02 18 1.357 7 1.604 3 4.074 4 12.52 14 39.47 94 1.317 7 2.866 7 8.533 5 26.80 4 1.264 7 2.583 6 7.542 5 23.64 06 1.230 5 2.387 9 6.836 2 21.37 22 1.523 5 1.675 3 4.305 9 13.25 11 41.78 58 1.445 0 3.358 9 10.13 27 31.88 17
1.901 6 5.175 5 16.07 75 50.74 91 1.366 1 1.602 7 4.038 5 12.39 84 39.08 74 1.320 7 2.839 8 8.442 2 26.52 05 1.267 1 2.584 3 7.577 1 23.76 44 1.235 9 2.425 1 7.031 1 22.02 24 1.519 2 1.670 91 4.305 9 13.26 79 41.84 48 1.445 5 3.393 8 10.28 63 33.38 35
0.4328 0.5514 0.5389 0.5372 0.6213 0.0992 0.8897 0.9924 1.0028 0.2223 0.9421 1.0819 1.0943 0.1934 0.0287 0.4565 0.5212 0.3829 1.5343 2.7721 2.9525 0.2882 0.2650 0.0000 0.1262 0.1410 0.0362 1.0285 1.4929 1.5495
1.8554 5.0423 15.660 6 49.432 2 1.3361
2.9378 3.2073 3.2152 3.2156 1.6098 2.4536 3.4228 3.5548 3.5684 2.0745 3.5308 3.7395 3.7597 2.1343 2.6635 2.3253 2.2708 1.9412 1.1680 0.0206 0.1497 2.7201 5 2.7766
1.8852 5.1494 16.004 5 50.521 4 1.3756
1.307 3 1.060 9 0.997 4 0.990 4 1.304 4 0.251 7 0.052 2 0.009 0 0.015 9 1.929 7 3.384 1 3.592 4 3.612 6 1.989 5 2.518 0 2.180 2 2.125 8 1.796 7 1.024 5 0.121 1 0.291 3 2.574 5 2.630 9 2.577 5 2.482 6 2.471 1 2.278 3 1.574 2 1.159 2 1.107 7
1.5659 3.9396 12.091 6 38.119 1 1.2909 2.7688 8.2259 25.839 2 1.2382 2.5166 7.3711 23.115 7 1.2071 2.3603 6.8347 21.404 3 1.4832
1.6003 4.0723 12.522 6 39.485 7 1.2928 2.7728 8.2376 25.875 9 1.2400 2.5201 8 7.3815 23.148 5 1.2088 2.3674 6.8444 21.434 7 1.4853
𝜔2𝐴𝐸 = 𝐴₁𝜔22𝐾(𝑘22)(((3 + 𝑚)(5 + 𝑚)(7 + 𝑚)𝛼₀ + 𝐴21 (2 + 𝑚)((7 + 𝑚)((5 + 𝑚)𝛼₁ + 𝐴21(4 + 𝑚)𝛼₂) + 𝐴41 (4 + 𝑚)(6 + 𝑚)𝛼₃) ― (5 + 𝑚)(7 + 𝑚)(3 ― 2𝑘21 +𝑚)𝜔21 1.6300 1.6323 )𝛤[1 + 𝑚/2]𝛤[2 + 𝑚/2] + 8𝐴₁𝛼₂₂𝛤[(3 + 𝑚)/2]𝛤[(9 + 𝑚)/2]) ― 𝐴₂𝜔 4.1918 2.7232 4.1977 12.911 12.930 2.6281 2 2 𝐾(𝑘1)(((3 + 𝑚)(5 + 𝑚)(7 + 𝑚)𝛼₀ + 𝐴2 8 1 40.720 40.778 2.6166 3 1 2 4 1.4108 2.4235 1.4128 (2 + 𝑚)((7 + 𝑚)((5 + 𝑚)𝛼₁ + 𝐴2(4 + 𝑚)𝛼₂) + 𝐴2 3.3022 1.7184 3.3069 (6) 2 2 10.002 10.016 1.3028 (4 + 𝑚)(6 + 𝑚)𝛼₃) ― (5 + 𝑚)(7 + 𝑚)(3 ― 2𝑘2 +𝑚)𝜔2 4 6 31.487 31.532 1,2513 7 4 )𝛤[1 + 𝑚/2]𝛤[2 + 𝑚/2] + 8𝐴₂𝛼₂₂𝛤[(3 + 𝑚)/2]𝛤[(9 + 𝑚)/2]))/(𝐴₁𝐾( 𝑘22)(((3 + 𝑚)(5 + 𝑚)(7 + 𝑚)𝛼₀ + 𝐴21(2 + 𝑚)((7 + 𝑚)((5 + 𝑚)𝛼₁ + 𝐴213. Conclusions (4 + 𝑚)𝛼₂) + 𝐴41(4 + 𝑚)(6 + 𝑚)𝛼₃) ― (5 + 𝑚)(7 + 𝑚)(3 ― 2𝑘21 +𝑚) He´s frequency-amplitude formulation has been enhanced by 𝜔21 𝑚 introducing Jacobi elliptic weighted functions of the form (𝑊𝑖) )𝛤[1 + 𝑚/2]𝛤[2 + 𝑚/2] + 8𝐴₁𝛼₂₂𝛤[(3 + 𝑚)/2]𝛤[(9 + 𝑚)/2]) ― 𝐴₂𝐾( 2
Results in Physics ## (2020) ####
during the calculation of the two trial residuals. The trial residuals provide an easy to use frequency-amplitude expression to calculate with great accuracy, the frequency of strong nonlinear Duffing-type oscillators. Therefore, this paper provides evidence of the versatility of Prof. He’s formulation to study nonlinear systems. Acknowledgments The authors would like to thank financial support from Tecnológico de Monterrey-Campus Monterrey, through the Research Group in Nanotechnology and Devices Design. Funding Tecnológico de Monterrey funded this research through the Research Group of Nanotechnology for Devices Design, and by the Consejo Nacional de Ciencia y Tecnología de México (Conacyt), Project Numbers 242269, 255837, 296176, and National Lab in Additive Manufacturing, 3D Digitizing and Computed Tomography (MADiT) LN299129. Conflict of Interest The authors declare that they have no conflict of interest. Author contributions: Alex Elias-Zúñiga: Conceptualization, Formal analysis, Funding acquisition, Investigation, Project administration, Writing - original draft, review & editing. Luis M. Palacios-Pineda: Formal analysis, Investigation, Software, Visualization, Writing - review. Isaac H. Jiménez-Cedeño: Formal analysis, Investigation, Software, Visualization. Oscar MartínezRomero: Formal analysis, Investigation, Software, Visualization. Daniel Olvera Trejo: Formal analysis, Investigation, Software, Visualization. References [1] Elías-Zúñiga A. Exact solution of the cubic-quintic Duffing oscillator. Applied Mathematical Modelling 2013; 37(4): 2574-2579. [2] He JH. Ancient Chinese algorithm: the Ying Buzu Shu (method of surplus and deficiency) vs. Newton iteration method, Appl. Math. Mech. (English Ed.) 2002; 23: 1407—1412. [3] He JH. An improved amplitude-frequency formulation for nonlinear oscillators. Int J Nonlinear Sci Numer Simul 2008; 9: 211—212. [4] He JH. Amplitude-frequency relationship for conservative nonlinear oscillators with odd nonlinearities. Int J Appl Comput Math 2017; 3: 1557—1560. [5] Wu Y, and Liu YP. Residual calculation in He’s frequency—amplitude formulation. J Low Freq Noise VA 2020: doi: 10.1177/1461348420913662. [6] Ren ZF Ren and Hu GF. He’s frequency—amplitude formulation with average residuals for nonlinear oscillators. J Low Freq Noise VA 2019; 38(3—4): 10501059. [7] Byrd P F, Friedman M D, Handbook of Elliptic Integrals for Engineers and Physicists. Springer-Verlag, 1953. [8] Elías-Zúñiga A and Martínez-Romero O. Accurate Solutions of Conservative Nonlinear Oscillators by the Enhanced Cubication Method. Math Probl Eng 2013; 2013: 842423. [9] Elías-Zúñiga A, Palacios-Pineda LM, Jiménez-Cedeño IH et al. Equivalent power-form transformation for fractal Bratu’s equation. Fractals 2020: doi.org/10.1142/S0218348X21500195. [10] Elías-Zúñiga A, Palacios-Pineda LM, Jiménez-Cedeño IH et al. He´s frequency-amplitude formulation for nonlinear oscillators using Jacobi elliptic functions. Submitted to the J Low Freq Noise V A. 2020.
S2211-3797(20)32059-3 https://doi.org/10.1016/j.rinp.2020.103626 RINP 103626
To appear in:
Results in Physics
Received Date: Revised Date: Accepted Date:
27 September 2020 9 November 2020 14 November 2020
Please cite this article as: Elías-Zú ñiga, A., Manuel Palacios-Pineda, L., Jiménez-Cedeño, I.H., MartínezRomero, O., Olvera Trejo, D., Enhanced He’s frequency-amplitude formulation for nonlinear oscillators, Results in Physics (2020), doi: https://doi.org/10.1016/j.rinp.2020.103626
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2020 Published by Elsevier B.V.
Results in Physics ## (2020) #### Contents lists available at ScienceDirect
Results in Physics journal homepage: www.elsevier.com/locate/rinp
Microarticle
Enhanced He’s frequency-amplitude formulation for nonlinear oscillators Alex Elías-Zúñigaa*, Luis Manuel Palacios-Pinedab, Isaac H. Jiménez-Cedeñoa, Oscar Martínez-Romeroa, and Daniel Olvera Trejoa aMechanical
Engineering and Advanced Materials Department, School of Engineering and Science, Tecnologico de Monterrey, Ave. Eugenio Garza Sada 2501, Sur, Monterrey 64849, NL, Mexico. bTecnológico
Nacional de México/Instituto Tecnológico de Pachuca, Carr. México-Pachuca, km 87.5, Pachuca, Hidalgo, Código Postal 42080, Mexico.
ABSTRACT In this article a trial residual Jacobi elliptic weighted function is introduced to obtain the frequency-amplitude expression of nonlinear oscillators using He´s formulation. Simulation results show an improvement in the accuracy of the frequency values when compare to numerical integration simulations
——— * Corresponding author. Tel.: +52 81; e-mail: [email protected].
A. Elías-Zúñiga, et. al.
Results in Physics ## (2020) #####
𝑘21)(((3 + 𝑚)(5 + 𝑚)(7 + 𝑚)𝛼₀ + 𝐴22(2 + 𝑚)((7 + 𝑚)((5 + 𝑚)𝛼₁ + 𝐴22 (4 + 𝑚)𝛼₂) + 𝐴42(4 + 𝑚)(6 + 𝑚)𝛼₃) ― (5 + 𝑚)(7 + 𝑚)(3 ― 2𝑘₂² + 𝑚) 𝜔22)𝛤[1 + 𝑚/2]𝛤[2 + 𝑚/2] + 8𝐴₂𝛼₂₂𝛤[(3 + 𝑚)/2]𝛤[(9 + 𝑚)/2])).
1. Introduction For nonlinear oscillators of the form 𝑢 + 𝛼0𝑢 + 𝛼1𝑢3 + 𝛼2𝑢5 + 𝛼3𝑢7 + 𝛼22𝑢2 = 0, with 𝑢(0) = 𝐴,𝑢(0) = 0 the frequency-amplitude relation can be determined using the ancient Chinese method and Jacobi elliptic functions 𝑦𝑖 = 𝐴𝑖𝑐𝑛(ω𝑖𝑡,𝑘2𝑖 ) with frequency ω𝑖 and modulus 𝑘2𝑖 [1]. This approach requires to find two trial residual functions 𝑅1(𝑡) and 𝑅2(𝑡). Then, following the steps of the approach introduced in [2]-[5] and assuming that ui Ai cn(it , ki2 ), the corresponding trial residual functions are obtained:
(1)
2. Results To study the accuracy achieved by Eq. (4), the Duffing-type oscillator (1) is considered [8, 9]. Moreover, the frequency-amplitude solution 2 8 16 2 RE 0 1 A2 2 A4 3 A6 22 A, (7) 3 15 35 4
1
𝑅𝑖(𝑡) = 64 derived by Ren and Hu in [6], and slightly modified in [10] to include 2 3 5 7 2 2 )cosquadratic (32𝐴2𝑖 α22 + (64𝐴𝑖α0 + 48𝐴3𝑖 α1 + 40𝐴5𝑖 α2 + 35𝐴7𝑖 α3 + 32𝐴𝑖( ―2 + 𝑘2𝑖 )ω2𝑖the ϕ𝑖 + 32𝐴 α1 +and 20𝐴 𝑖 α22cos 2ϕ 𝑖 +of(16𝐴 𝑖 αfrequency-amplitude 2 + 21𝐴𝑖 α3 ― 32𝐴𝑖𝑘𝑖 ω𝑖 )cos 3 nonlinear term Eq. 𝑖(1), the relation 𝜔2𝐸𝑍 = ( ― 2𝜔211(105𝐴22𝛼0 +70𝐴322𝛼1 +56𝐴522𝛼2 +48𝐴722𝛼3 +35(3𝐵22 ― 3𝐴22 +(2𝐴22 ― 𝐵22)𝑘222)𝜔222)𝐾(𝑘211 ) + (2(105𝐴11𝛼0 +70𝐴311𝛼1 +56𝐴511𝛼2 +48𝐴711𝛼3 +35(3 in which the elliptic Jacobi amplitude ϕ𝑖 = 𝑎𝑚(ω𝑖𝑡,𝑘2𝑖 ) and the 𝐵11 ― 3𝐴11 +(2𝐴11 ― 𝐵11)𝑘211)𝜔211)𝜔222) + 105𝛼22(𝐴11𝜔22 2 2 identities cos ϕ𝑖 = 𝑐𝑛(ω𝑖𝑡,𝑘𝑖 ) and sin ϕ𝑖 = 𝑠𝑛(ω𝑖𝑡,𝑘𝑖 ) where applied ― 𝐴22𝜔11)(𝐴22𝜔11 + 𝐴11𝜔22)𝐾(𝑘211))𝐾(𝑘222))/( ― 2(105 (8) [7]. Then, we use He’s frequency-amplitude formulation and 𝐴22𝛼0 +70𝐴322𝛼1 +56𝐴522𝛼2 +48𝐴722𝛼3 +35(3𝐵22 ― 3𝐴22 introduced a power-form weighted functions 𝑊𝑖(𝑡) so that the trial +(2𝐴22 ― 𝐵22)𝑘222)𝜔222)𝐾(𝑘211) + (210𝐴11𝛼0 +140𝐴311𝛼1 +112𝐴511𝛼2 +96𝐴711𝛼3 +70(3𝐵11 ― 3𝐴11 +(2𝐴11 ― 𝐵11) residual functions are now defined as [8, 9] 4 𝑇/4 𝑘211)𝜔211) + 105(𝐴11 ― 𝐴22)(𝐴11 + 𝐴22)𝛼22𝐾(𝑘211))𝐾(𝑘222)), 𝑅𝑖(𝑡)(𝑊𝑖(𝑡))𝑚𝑑𝑡, 𝑅𝑖 = (3) 𝑇 0 proposed by Elías-Zuñiga et al. in [10] are used for comparison purposes. The percentage of relative error attained from Eq. (4) when where it is assumed that the residual weighted functions have the form compare to the exact frequency-amplitude value of Eq. (1) are listed 𝑊𝑖(𝑡) = cos ϕ𝑖, 𝑖 = 1, 2, where 𝑚 is a constant that can be determined in Table 1, for the system parameter values of α0 = 1, 𝐴 = 1, 𝐴1 = 𝐴2 so that the approximate frequency-amplitude expression for the = 𝐴11 = 𝐴22 = 1, 𝐵11 = 𝐵22 = ―0.3, ω1 = ω11 = 1, ω2 = ω22 = 2, nonlinear oscillator (1) 𝑘21 = 𝑘22 = 0.0618, and 𝑘211 = 𝑘222 = 0.395 with 𝑚 = 0.023. It can be ω21𝑅2 ― ω22𝑅1 seen from Table 1, that predicted values for ω𝐴𝐸 follow well the exact ω2 = , ones. In fact, for all cases listed in Table 1, the average error value of (4) 𝑅2 ― 𝑅1 𝜔𝐴𝐸 does not exceed of 0.783% when compared to the values of ω𝑅𝐸𝑁 [6] and ω𝐸𝑍 [10]. is found [2]-[5]. Thus, substitution of Eq. (2) into Eq. (3), yields
∫
𝑅𝑖(𝑡) = (√𝜋((8𝐴2𝑖 𝛼₂₂𝛤[(3 + 𝑚)/2])/𝛤[2 + 𝑚/2] + 1/𝛤[(9 + 𝑚)/2](𝐴𝑖 (7 + 𝑚)((3 + 𝑚)(5 + 𝑚)𝛼₀ + 𝐴2𝑖 (2 + 𝑚)((5 + 𝑚)𝛼₁ + 𝐴2𝑖 (4 + 𝑚)𝛼₂)) + 𝐴7𝑖 (2 + 𝑚)(4 + 𝑚)(6 + 𝑚)𝛼₃ ― 𝐴𝑖 (5 + 𝑚)(7 + 𝑚)(3 ― 2𝑘2𝑖 +𝑚)ω2𝑖 )𝛤[1 + 𝑚/2]))/(16𝐾(𝑘2𝑖 )),
Table 1. Exact and estimated frequency values for Duffing-type nonlinear oscillators.
(5)
where Γ[z] is the Gamma function, and K (ki2 ) is the complete elliptic integrals of the first kind with modulus ki2 . Finally, substitution of Eq. (5) into Eq. (4), provides the following frequencyamplitude expression
𝜶𝟐𝟐
𝜶𝟏
𝜶𝟐
𝜶𝟑
𝝎𝑬𝒙𝒂𝒄𝒕
𝝎𝑨𝑬
Error (%)
𝝎𝑹𝑬
Error (%)
𝝎𝑬𝒁
Error (%)
1 10 100 1000 1 1 10 100 1000 0 0 0 0 0 0 0 0 0 0 0 0 0 20 10 0 0 0 0 0 0
1 10 100 1000 0 1 10 100 1000 1 10 100 1000 0 0 0 0 0 0 0 0 1 1 10 100 1000 0 0 0 0
1 10 100 1000 0 0 0 0 0 0 0 0 0 1 10 100 1000 0 0 0 0 1 1 10 100 1000 1 10 100 1000
1 10 100 1000 0 0 0 0 0 0 0 0 0 0 0 0 0 1 10 100 1000 0 1 10 100 1000 1 10 100 1000
1.909 9 5.204 0 16.16 41 51.02 18 1.357 7 1.604 3 4.074 4 12.52 14 39.47 94 1.317 7 2.866 7 8.533 5 26.80 4 1.264 7 2.583 6 7.542 5 23.64 06 1.230 5 2.387 9 6.836 2 21.37 22 1.523 5 1.675 3 4.305 9 13.25 11 41.78 58 1.445 0 3.358 9 10.13 27 31.88 17
1.901 6 5.175 5 16.07 75 50.74 91 1.366 1 1.602 7 4.038 5 12.39 84 39.08 74 1.320 7 2.839 8 8.442 2 26.52 05 1.267 1 2.584 3 7.577 1 23.76 44 1.235 9 2.425 1 7.031 1 22.02 24 1.519 2 1.670 91 4.305 9 13.26 79 41.84 48 1.445 5 3.393 8 10.28 63 33.38 35
0.4328 0.5514 0.5389 0.5372 0.6213 0.0992 0.8897 0.9924 1.0028 0.2223 0.9421 1.0819 1.0943 0.1934 0.0287 0.4565 0.5212 0.3829 1.5343 2.7721 2.9525 0.2882 0.2650 0.0000 0.1262 0.1410 0.0362 1.0285 1.4929 1.5495
1.8554 5.0423 15.660 6 49.432 2 1.3361
2.9378 3.2073 3.2152 3.2156 1.6098 2.4536 3.4228 3.5548 3.5684 2.0745 3.5308 3.7395 3.7597 2.1343 2.6635 2.3253 2.2708 1.9412 1.1680 0.0206 0.1497 2.7201 5 2.7766
1.8852 5.1494 16.004 5 50.521 4 1.3756
1.307 3 1.060 9 0.997 4 0.990 4 1.304 4 0.251 7 0.052 2 0.009 0 0.015 9 1.929 7 3.384 1 3.592 4 3.612 6 1.989 5 2.518 0 2.180 2 2.125 8 1.796 7 1.024 5 0.121 1 0.291 3 2.574 5 2.630 9 2.577 5 2.482 6 2.471 1 2.278 3 1.574 2 1.159 2 1.107 7
1.5659 3.9396 12.091 6 38.119 1 1.2909 2.7688 8.2259 25.839 2 1.2382 2.5166 7.3711 23.115 7 1.2071 2.3603 6.8347 21.404 3 1.4832
1.6003 4.0723 12.522 6 39.485 7 1.2928 2.7728 8.2376 25.875 9 1.2400 2.5201 8 7.3815 23.148 5 1.2088 2.3674 6.8444 21.434 7 1.4853
𝜔2𝐴𝐸 = 𝐴₁𝜔22𝐾(𝑘22)(((3 + 𝑚)(5 + 𝑚)(7 + 𝑚)𝛼₀ + 𝐴21 (2 + 𝑚)((7 + 𝑚)((5 + 𝑚)𝛼₁ + 𝐴21(4 + 𝑚)𝛼₂) + 𝐴41 (4 + 𝑚)(6 + 𝑚)𝛼₃) ― (5 + 𝑚)(7 + 𝑚)(3 ― 2𝑘21 +𝑚)𝜔21 1.6300 1.6323 )𝛤[1 + 𝑚/2]𝛤[2 + 𝑚/2] + 8𝐴₁𝛼₂₂𝛤[(3 + 𝑚)/2]𝛤[(9 + 𝑚)/2]) ― 𝐴₂𝜔 4.1918 2.7232 4.1977 12.911 12.930 2.6281 2 2 𝐾(𝑘1)(((3 + 𝑚)(5 + 𝑚)(7 + 𝑚)𝛼₀ + 𝐴2 8 1 40.720 40.778 2.6166 3 1 2 4 1.4108 2.4235 1.4128 (2 + 𝑚)((7 + 𝑚)((5 + 𝑚)𝛼₁ + 𝐴2(4 + 𝑚)𝛼₂) + 𝐴2 3.3022 1.7184 3.3069 (6) 2 2 10.002 10.016 1.3028 (4 + 𝑚)(6 + 𝑚)𝛼₃) ― (5 + 𝑚)(7 + 𝑚)(3 ― 2𝑘2 +𝑚)𝜔2 4 6 31.487 31.532 1,2513 7 4 )𝛤[1 + 𝑚/2]𝛤[2 + 𝑚/2] + 8𝐴₂𝛼₂₂𝛤[(3 + 𝑚)/2]𝛤[(9 + 𝑚)/2]))/(𝐴₁𝐾( 𝑘22)(((3 + 𝑚)(5 + 𝑚)(7 + 𝑚)𝛼₀ + 𝐴21(2 + 𝑚)((7 + 𝑚)((5 + 𝑚)𝛼₁ + 𝐴213. Conclusions (4 + 𝑚)𝛼₂) + 𝐴41(4 + 𝑚)(6 + 𝑚)𝛼₃) ― (5 + 𝑚)(7 + 𝑚)(3 ― 2𝑘21 +𝑚) He´s frequency-amplitude formulation has been enhanced by 𝜔21 𝑚 introducing Jacobi elliptic weighted functions of the form (𝑊𝑖) )𝛤[1 + 𝑚/2]𝛤[2 + 𝑚/2] + 8𝐴₁𝛼₂₂𝛤[(3 + 𝑚)/2]𝛤[(9 + 𝑚)/2]) ― 𝐴₂𝐾( 2
Results in Physics ## (2020) ####
during the calculation of the two trial residuals. The trial residuals provide an easy to use frequency-amplitude expression to calculate with great accuracy, the frequency of strong nonlinear Duffing-type oscillators. Therefore, this paper provides evidence of the versatility of Prof. He’s formulation to study nonlinear systems. Acknowledgments The authors would like to thank financial support from Tecnológico de Monterrey-Campus Monterrey, through the Research Group in Nanotechnology and Devices Design. Funding Tecnológico de Monterrey funded this research through the Research Group of Nanotechnology for Devices Design, and by the Consejo Nacional de Ciencia y Tecnología de México (Conacyt), Project Numbers 242269, 255837, 296176, and National Lab in Additive Manufacturing, 3D Digitizing and Computed Tomography (MADiT) LN299129. Conflict of Interest The authors declare that they have no conflict of interest. Author contributions: Alex Elias-Zúñiga: Conceptualization, Formal analysis, Funding acquisition, Investigation, Project administration, Writing - original draft, review & editing. Luis M. Palacios-Pineda: Formal analysis, Investigation, Software, Visualization, Writing - review. Isaac H. Jiménez-Cedeño: Formal analysis, Investigation, Software, Visualization. Oscar MartínezRomero: Formal analysis, Investigation, Software, Visualization. Daniel Olvera Trejo: Formal analysis, Investigation, Software, Visualization. References [1] Elías-Zúñiga A. Exact solution of the cubic-quintic Duffing oscillator. Applied Mathematical Modelling 2013; 37(4): 2574-2579. [2] He JH. Ancient Chinese algorithm: the Ying Buzu Shu (method of surplus and deficiency) vs. Newton iteration method, Appl. Math. Mech. (English Ed.) 2002; 23: 1407—1412. [3] He JH. An improved amplitude-frequency formulation for nonlinear oscillators. Int J Nonlinear Sci Numer Simul 2008; 9: 211—212. [4] He JH. Amplitude-frequency relationship for conservative nonlinear oscillators with odd nonlinearities. Int J Appl Comput Math 2017; 3: 1557—1560. [5] Wu Y, and Liu YP. Residual calculation in He’s frequency—amplitude formulation. J Low Freq Noise VA 2020: doi: 10.1177/1461348420913662. [6] Ren ZF Ren and Hu GF. He’s frequency—amplitude formulation with average residuals for nonlinear oscillators. J Low Freq Noise VA 2019; 38(3—4): 10501059. [7] Byrd P F, Friedman M D, Handbook of Elliptic Integrals for Engineers and Physicists. Springer-Verlag, 1953. [8] Elías-Zúñiga A and Martínez-Romero O. Accurate Solutions of Conservative Nonlinear Oscillators by the Enhanced Cubication Method. Math Probl Eng 2013; 2013: 842423. [9] Elías-Zúñiga A, Palacios-Pineda LM, Jiménez-Cedeño IH et al. Equivalent power-form transformation for fractal Bratu’s equation. Fractals 2020: doi.org/10.1142/S0218348X21500195. [10] Elías-Zúñiga A, Palacios-Pineda LM, Jiménez-Cedeño IH et al. He´s frequency-amplitude formulation for nonlinear oscillators using Jacobi elliptic functions. Submitted to the J Low Freq Noise V A. 2020.