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A Modified Newton–Harmonic Balance Approach to Strongly Odd Nonlinear Oscillators

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Journal of Vibration Engineering & Technologies Aims and scope Submit manuscript

Abstract

Background

Since combinations of the Newton’s method and the harmonic balance (HB) method require, at each iteration step, calculating the first or the first- and second-order derivatives of the restoring force function, and expanding the function, its first- and second-order derivatives into Fourier series, the procedural costs are high and sometimes difficult to achieve algebraically. It is thus preferable to avoid expensive re-linearization or computation of the second-order derivative.

Purpose

A new approach is proposed to construct accurate analytical approximation solutions to strongly nonlinear conservative oscillators with odd nonlinearities.

Methods

The approach is based on a combination of a modified Newton method and the HB method. For the modified Newton method, two simplified Newton steps are taken between each Newton step where only one linearization of the restoring force function is required. The resulting equations are solved by applying the HB method appropriately.

Results

Using only one modified Newton iteration step may achieve highly accurate analytical approximation solutions to the strongly nonlinear oscillators. Three examples with physical implications are used to illustrate the proposed method.

Conclusion

Through the modified Newton iteration step, the multiple cumbersome linearizations of the restoring force function are replaced by only one linearization, and the corresponding governing equations can be properly solved by the HB method. The current work is expected to extend to the study of other nonlinear oscillations.

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Acknowledgements

The work was supported by the National Natural Science Foundation of China (Grant no. 11672118), and Research and Development Plans in Key Areas of Guangdong (Grant no. 2019B090917002).

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Authors and Affiliations

  1. School of Electro-Mechanical Engineering, Guangdong University of Technology, Guangzhou, 510006, People’s Republic of China

    Baisheng Wu, Weijia Liu & Huixiang Zhong

  2. Department of Architecture and Civil Engineering, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong, People’s Republic of China

    C. W. Lim

Authors
  1. Baisheng Wu
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  2. Weijia Liu
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  3. Huixiang Zhong
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  4. C. W. Lim
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Correspondence to Baisheng Wu.

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Appendix

Appendix

The expressions of \(P_{1} , \, P_{2} , \, P_{3} , \, P_{4} , \, P_{\varOmega } , \, P_{\text{D}}\) are as follows.

$$\begin{aligned} P_{1} & = A(4835703278458516698824704a^{18} + 64450857758204917876523008A^{2} a^{17} b \\ & \quad + 405627669046088175630417920A^{4} a^{16} b^{2} + 1601728426736958931994673152A^{6} a^{15} b^{3} \\ & \quad + 4446974127479403668997931008A^{8} a^{14} b^{4} + 9218243172947523910489866240A^{10} a^{13} b^{5} \\ & \quad + 14785841346111404889052545024A^{12} a^{12} b^{6} + 18763524784922459763638272000A^{14} a^{11} b^{7} \\ & \quad + 19097688409745225496597102592A^{16} a^{10} b^{8} + 15706504179898943441370873856A^{18} a^{9} b^{9} \\ & \quad + 10462605459329565715349897216A^{20} a^{8} b^{10} + 5631582647280319099400355840A^{22} a^{7} b^{11} \\ & \quad + 2431167864190925491283165184A^{24} a^{6} b^{12} + 830354811239768360374763520A^{26} a^{5} b^{13} \\ & \quad + 219441322192563605505425408A^{28} a^{4} b^{14} + 43298251336294969181174272A^{30} a^{3} b^{15} \\ & \quad + 6006532208423496497883712A^{32} a^{2} b^{16} + 522786540947589345532344A^{34} ab^{17} \\ & \quad + 21485079498804067272519A^{36} b^{18} ), \\ \end{aligned}$$
$$\begin{aligned} P_{2} & = 151115727451828646838272A^{3} a^{17} b + 1905474875837901843726336A^{5} a^{16} b^{2} \\ & \quad + 11309108495780967390117888A^{7} a^{15} b^{3} + 41958844166223266993668096A^{9} a^{14} b^{4} \\ & \quad + 108994977581964717830701056A^{11} a^{13} b^{5} + 210370424231974451810402304A^{13} a^{12} b^{6} \\ & \quad + 312399213313109590649339904A^{15} a^{11} b^{7} + 364578510827628201919905792A^{17} a^{10} b^{8} \\ & \quad + 338516618651629734938542080A^{19} a^{9} b^{9} + 251508624790868463311650816A^{21} a^{8} b^{10} \\ & \quad + 149524255540205269240774656A^{23} a^{7} b^{11} + 70726792708396755185565696A^{25} a^{6} b^{12} \\ & \quad + 26291888445896461967163392A^{27} a^{5} b^{13} + 7520040035850389352161280A^{29} a^{4} b^{14} \\ & \quad + 1598200207932246614587392A^{31} a^{3} b^{15} + 237821684972978253124992A^{33} a^{2} b^{16} \\ & \quad + 22124179993691481314304A^{35} ab^{17} + 968822190832147158876A^{37} b^{18} , \\ \end{aligned}$$
$$\begin{aligned} P_{3} & = 4722366482869645213696A^{5} a^{16} b^{2} + 56004315007782198706176A^{7} a^{15} b^{3} \\ & \quad + 311498637957687129669632A^{9} a^{14} b^{4} + 1078614993058995309117440A^{11} a^{13} b^{5} \\ & \quad + 2602451747004176945643520A^{13} a^{12} b^{6} + 4639338980186234088849408A^{15} a^{11} b^{7} \\ & \quad + 6321057386653510862372864A^{17} a^{10} b^{8} + 6714526431619256808374272A^{19} a^{9} b^{9} \\ & \quad + 5619858774906742100721664A^{21} a^{8} b^{10} + 3718495779890683638185984A^{23} a^{7} b^{11} \\ & \quad + 1938655501211025962696704A^{25} a^{6} b^{12} + 788025466700714743824384A^{27} a^{5} b^{13} \\ & \quad + 244828243490659897270272A^{29} a^{4} b^{14} + 56204195663850721290240A^{31} a^{3} b^{15} \\ & \quad + 8991209738990145619840A^{33} a^{2} b^{16} + 895544245281147035584A^{35} ab^{17} \\& \quad + 41838930607835396004A^{37} b^{18} , \\ \end{aligned}$$
$$\begin{aligned} P_{4} & = 147573952589676412928A^{7} a^{15} b^{3} + 1639454379550936399872A^{9} a^{14} b^{4} \\ & \quad + 8504309305950292934656A^{11} a^{13} b^{5} + 27323998716602209009664A^{13} a^{12} b^{6} \\ & \quad + 60811895858419925516288A^{15} a^{11} b^{7} + 99305437243272100577280A^{17} a^{10} b^{8} \\ & \quad + 122919731096436997095424A^{19} a^{9} b^{9} + 117436631194839132667904A^{21} a^{8} b^{10} \\ & \quad + 87312998556222355406848A^{23} a^{7} b^{11} + 50518231926325423112192A^{25} a^{6} b^{12} \\ & \quad + 22560699826636333907968A^{27} a^{5} b^{13} + 7637023311131482243072A^{29} a^{4} b^{14} \\ & \quad + 1896881805586013365760A^{31} a^{3} b^{15} + 326363394510536538304A^{33} a^{2} b^{16} \\ & \quad + 34780496959236928136A^{35} ab^{17} + 1730732174725245369A^{37} b^{18} , \\ \end{aligned}$$
$$\begin{aligned} P_{\varOmega } & = 3A^{8} b^{4} (442721857769029238784a^{15} + 5102830579389904715776A^{2} a^{14} b \\ & \quad + 27312638386542166933504A^{4} a^{13} b^{2} + 90077300039570776129536A^{6} a^{12} b^{3} \\ & \quad + 204746874013667683205120A^{8} a^{11} b^{4} + 339795427352756527038464A^{10} a^{10} b^{5} \\ & \quad + 425361925658199361323008A^{12} a^{9} b^{6} + 408978801242226020057088A^{14} a^{8} b^{7} \\& \quad + 304485598886042791837696A^{16} a^{7} b^{8} + 175505115520599790190592A^{18} a^{6} b^{9} \\ & \quad + 77662337412392581988352A^{20} a^{5} b^{10} + 25901296598274638692352A^{22} a^{4} b^{11} \\ & \quad + 6299629977693939730944A^{24} a^{3} b^{12} + 1054263188601994088768A^{26} a^{2} b^{13} \\ & \quad + 108476685824622319800A^{28} ab^{14} + 5168358617143777623A^{30} b^{15} ), \\ \end{aligned}$$

and

$$P_{\text{D}} = 134217728\left( {4a + 3A^{2} b} \right)^{12} \left( {32a + 23A^{2} b} \right)^{3} \left( {65536a^{3} + 144384A^{2} a^{2} b + 105984A^{4} ab^{2} + 25923A^{6} b^{3} } \right).$$

The expressions of \(R_{1} , \, R_{2} , \, R_{3} , \, R_{\varOmega } , \, D_{\text{SN2}}\) are as follows.

$$\begin{aligned} R_{1} & = - 24576A^{2} S_{1} + 22912A^{4} S_{1} - 3840A^{6} S_{1} - 98304S_{2} + 165376A^{2} S_{2} - 84096A^{4} S_{2} \\ & \quad + 11520A^{6} S_{2} - 32768S_{3} + 28672A^{2} S_{3} + 6704A^{4} S_{3} - 6276A^{6} S_{3} + 24576A^{2} S_{1} \varOmega_{\text{SN1}} \\ & \quad - 22912A^{4} S_{1} \varOmega_{\text{SN1}} + 3840A^{6} S_{1} \varOmega_{\text{SN1}} - 24576S_{1}^{3} \varOmega_{\text{SN1}} + 22912A^{2} S_{1}^{3} \varOmega_{\text{SN1}} - 3840A^{4} S_{1}^{3} \varOmega_{\text{SN1}} \\ & \quad + 884736S_{2} \varOmega_{\text{SN1}} - 1488384A^{2} S_{2} \varOmega_{\text{SN1}} + 756864A^{4} S_{2} \varOmega_{\text{SN1}} - 103680A^{6} S_{2} \varOmega_{\text{SN1}} \\ & \quad - 630784S1^{2} S_{2} \varOmega_{\text{SN1}} + 920832A^{2} S_{1}^{2} S_{2} \varOmega_{\text{SN1}} - 381084A^{4} S_{1}^{2} S_{2} \varOmega_{\text{SN1}} + 35541A^{6} S_{1}^{2} S_{2} \varOmega_{\text{SN1}} \\ & \quad - 233472S_{1} S_{2}^{2} \varOmega_{\text{SN1}} + 7296A^{2} S_{1} S_{2}^{2} \varOmega_{\text{SN1}} + 194180A^{4} S_{1} S_{2}^{2} \varOmega_{\text{SN1}} - 45771A^{6} S_{1} S_{2}^{2} \varOmega_{\text{SN1}} \\ & \quad - 663552S_{2}^{3} \varOmega_{\text{SN1}} + 1116288A^{2} S_{2}^{3} \varOmega_{\text{SN1}} - 567648A^{4} S_{2}^{3} \varOmega_{\text{SN1}} + 77760A^{6} S_{2}^{3} \varOmega_{\text{SN1}} \\ & \quad + 819200S_{3} \varOmega_{\text{SN1}} - 716800A^{2} S_{3} \varOmega_{\text{SN1}} - 167600A^{4} S_{3} \varOmega_{\text{SN1}} \, + 156900A^{6} S_{3} \varOmega_{\text{SN1}} \\ & \quad - 1216512S_{1}^{2} S_{3} \varOmega_{\text{SN1}} + 1651968A^{2} S_{1}^{2} S_{3} \varOmega_{\text{SN1}} - 555768A^{4} S_{1}^{2} S_{3} \varOmega_{\text{SN1}} + 22194A^{6} S_{1}^{2} S_{3} \varOmega_{\text{SN1}} \\ & \quad - 2007040S_{1} S_{2} S_{3} \varOmega_{\text{SN1}} + 2849280A^{2} S_{1} S_{2} S_{3} \varOmega_{\text{SN1}} - 1274560A^{4} S_{1} S_{2} S_{3} \varOmega_{\text{SN1}} \\ & \quad + 210000A^{6} S_{1} S_{2} S_{3} \varOmega_{\text{SN1}} - 704512S_{2}^{2} S_{3} \varOmega_{\text{SN1}} + 352256A^{2} S_{2}^{2} S_{3} \varOmega_{\text{SN1}} + 390440A^{4} S_{2}^{2} S_{3} \varOmega_{\text{SN1}} \\ & \quad - 176214A^{6} S_{2}^{2} S_{3} \varOmega_{\text{SN1}} - 626688A^{2} S_{1} S_{2}^{2} \varOmega_{\text{SN1}} + 584256A^{4} S_{1} S_{3}^{2} \varOmega_{\text{SN1}} - 97920A^{6} S_{1} S_{3}^{2} \varOmega_{\text{SN1}} \\ & \quad - 3141632S_{2} S_{3}^{2} \varOmega_{\text{SN1}} + 5203328A^{2} S_{2} S_{3}^{2} \varOmega_{\text{SN1}} - 2652640A^{4} S_{2} S3_{3}^{2} \varOmega_{\text{SN1}} + 403560A^{6} S_{2} S_{3}^{2} \varOmega_{\text{SN1}} \\ & \quad - 614400S_{3}^{3} \varOmega_{\text{SN1}} + 537600A^{2} S_{3}^{3} \varOmega_{\text{SN1}} + 125700A^{4} S_{3}^{3} \varOmega_{\text{SN1}} - 117675A^{6} S_{3}^{3} \varOmega_{\text{SN1}} \\ \end{aligned}$$
$$\begin{aligned} R_{2} & = 2816A^{4} S_{1} - 1056A^{6} S_{1} + 11264A^{2} S_{2} - 12672A^{4} S_{2} + 3168A^{6} S_{2} + 32768S_{3} \\ & - 53248A^{2} S_{3} + 26368A^{4} S_{3} - 4128A^{6} S_{3} - 2816A^{4} S_{1} \varOmega_{\text{SN1}} + 1056A^{6} S_{1} \varOmega_{\text{SN1}} + 2816A^{2} S_{1}^{3} \varOmega_{\text{SN1}} \\ & - 1056A^{4} S_{1}^{3} \varOmega_{\text{SN1}} - 101376A^{2} S_{2} \varOmega_{\text{SN1}} + 114048A^{4} S_{2} \varOmega_{\text{SN1}} - 28512A^{6} S_{2} \varOmega_{\text{SN1}} + 90112S_{1}^{2} S_{2} \varOmega_{\text{SN1}} \\ & - 84480A^{2} S_{1}^{2} S_{2} \varOmega_{\text{SN1}} + 10560A^{4} S_{1}^{2} S_{2} \varOmega_{\text{SN1}} + 3168A^{6} S_{1}^{2} S_{2} \varOmega_{\text{SN1}} + 233472S_{1} S_{2}^{2} \varOmega_{\text{SN1}} \\ & - 350208A^{2} S_{1} S_{2}^{2} \varOmega_{\text{SN1}} + 170620A^{4} S_{1} S_{2}^{2} 2^{2} \varOmega_{\text{SN1}} - 23997A^{6} S_{1} S_{2}^{2} \varOmega_{\text{SN1}} + 76032A^{2} S_{2}^{3} \varOmega_{\text{SN1}} \\ & - 85536A^{4} S_{2}^{3} \varOmega_{\text{SN1}} + 21384A^{6} S_{2}^{3} \varOmega_{\text{SN1}} - 819200S_{3} \varOmega_{\text{SN1}} + 1331200A^{2} S_{3} \varOmega_{\text{SN1}} - 659200A^{4} S_{3} \varOmega_{\text{SN1}} \\ & + 103200A^{6} S_{3} \varOmega_{\text{SN1}} + 552960S_{1}^{2} S_{3} \varOmega_{\text{SN1}} - 781056A^{2} S_{1}^{2} S_{3} \varOmega_{\text{SN1}} + 296892A^{4} S_{1}^{2} S_{3} \varOmega_{\text{SN1}} \\ & - 26325A^{6} S_{1}^{2} S_{3} \varOmega_{\text{SN1}} + 286720S_{1} S_{2} S_{3} \varOmega_{\text{SN1}} - 161280A^{2} S_{1} S_{2} S_{3} \varOmega_{\text{SN1}} - 103880A^{4} S_{1} S_{2} S_{3} \varOmega_{\text{SN1}} \\ & + 57750A^{6} S_{1} S_{2} S_{3} \varOmega_{\text{SN1}} + 704512S_{2}^{2} S_{3} \varOmega_{\text{SN1}} - 1144832A^{2} S_{2}^{2} S_{3} \varOmega_{\text{SN1}} + 597184A^{4} S_{2}^{2} S_{3} \varOmega_{\text{SN1}} \\ & - 100104A^{6} S_{2}^{2} S_{3} \varOmega_{\text{SN1}} + 71808A^{4} S_{1} S_{3}^{2} \varOmega_{\text{SN1}} - 26928A^{6} S_{1} S_{3}^{2} \varOmega_{\text{SN1}} + 241664S_{2} S_{3}^{2} \varOmega_{\text{SN1}} \\ & + 30208A^{2} S_{2} S_{3}^{2} \varOmega_{\text{SN1}} - 316004A^{4} S_{2} S_{3}^{2} \varOmega_{\text{SN1}} + 110979A^{6} S_{2} S_{3}^{2} \varOmega_{\text{SN1}} + 614400S_{3}^{3} \varOmega_{\text{SN1}} \\ & - 998400A^{2} S_{3}^{3} \varOmega_{\text{SN1}} + 494400A^{4} S_{3}^{3} \varOmega_{\text{SN1}} - 77400A^{6} S_{3}^{3} \varOmega_{\text{SN1}} \\ \end{aligned}$$
$$\begin{aligned} R_{3} & = - 396A^{6} S_{1} - 1584A^{4} S_{2} + 1188A^{6} S_{2} - 4608A^{2} S_{3} + 5760A^{4} S_{3} - 1548A^{6} S_{3} + 396A^{6} S_{1} \varOmega_{\text{SN1}} \\ & \quad - 396A^{4} S_{1}^{3} \varOmega_{\text{SN1}} + 14256A^{4} S_{2} \varOmega_{\text{SN1}} - 10692A^{6} S_{2} \varOmega_{\text{SN1}} - 12672A^{2} S_{1}^{2} S_{2} \varOmega_{\text{SN1}} + 7128A^{4} S_{1}^{2} S_{2} \varOmega_{\text{SN1}} \\ & \quad + 1188A^{6} S_{1}^{2} S_{2} \varOmega_{\text{SN1}} - 77824S_{1} S_{2}^{2} \varOmega_{\text{SN1}} + 116736A^{2} S_{1} S_{2}^{2} \varOmega_{\text{SN1}} - 43776A^{4} S_{1} S_{2}^{2} \varOmega_{\text{SN1}} \\ & \quad - 2660A^{6} S_{1} S_{2}^{2} \varOmega_{\text{SN1}} - 10692A^{4} S_{2}^{3} \varOmega_{\text{SN1}} + 8019A^{6} S_{2}^{3} \varOmega_{\text{SN1}} + 115200A^{2} S_{3} \varOmega_{\text{SN1}} - 144000A^{4} S_{3} \varOmega_{\text{SN1}} \\ & \quad + 38700A^{6} S_{3} \varOmega_{\text{SN1}} - 110592S_{1}^{2} S_{3} \varOmega_{\text{SN1}} + 134784A^{2} S_{1}^{2} S_{3} \varOmega_{\text{SN1}} - 34020A^{4} S_{1}^{2} S_{3} \varOmega_{\text{SN1}} \\ & \quad - 864A^{6} S_{1}^{2} S_{3} \varOmega_{\text{SN1}} - 286720S_{1} S_{2} S_{3} \varOmega_{\text{SN1}} + 510720A^{2} S_{1} S_{2} S_{3} \varOmega_{\text{SN1}} - 289800A^{4} S_{1} S_{2} S_{3} \varOmega_{\text{SN1}} \\ & \quad + 45010A^{6} S_{1} S_{2} S_{3} \varOmega_{\text{SN1}} - 99072A^{2} S_{2}^{2} S_{3} \varOmega_{\text{SN1}} + 123840A^{4} S_{2}^{2} S_{3} \varOmega_{\text{SN1}} - 37539A^{6} S_{2}^{2} S_{3} \varOmega_{\text{SN1}} \\ & \quad - 10098A^{6} S_{1} S_{3}^{2} \varOmega_{\text{SN1}} - 241664S_{2} S_{3}^{2} \varOmega_{\text{SN1}} + 430464A^{2} S_{2} S_{3}^{2} \varOmega_{\text{SN1}} - 267624A^{4} S_{2} S_{3}^{2} \varOmega_{\text{SN1}} \\ & \quad + 61301A^{6} S_{2} S_{3}^{2} \varOmega_{\text{SN1}} - 86400A^{2} S_{3}^{3} \varOmega_{\text{SN1}} + 108000A^{4} S_{3}^{3} \varOmega_{\text{SN1}} - 29025A^{6} S_{3}^{3} \varOmega_{\text{SN1}} \\ \end{aligned}$$
$$\begin{aligned} R_{\varOmega } & = - 262144S_{1} + 405504A^{2} S_{1} - 166912A^{4} S_{1} + 14100A^{6} S_{1} - 40960A^{2} S_{2} + 48512A^{4} S_{2} \\ & \quad - 10272A^{6} S_{2} + 16384A^{2} S_{3} - 24576A^{4} S_{3} + 8532A^{6} S_{3} + 262144S_{1} \varOmega_{\text{SN1}} - 405504A^{2} S_{1} \varOmega_{\text{SN1}} \\ & \quad + 166912A^{4} S_{1} \varOmega_{\text{SN1}} - 14100A^{6} S_{1} \varOmega_{\text{SN1}} - 196608S_{1}^{3} \varOmega_{\text{SN1}} + 293888A^{2} S_{1}^{3} \varOmega_{\text{SN1}} - 113056A^{4} S_{1}^{3} \varOmega_{\text{SN1}} \\ & \quad + 8007A^{6} S_{1}^{3} \varOmega_{\text{SN1}} + 368640A^{2} S_{2} \varOmega_{\text{SN1}} - 436608A^{4} S_{2} \varOmega_{\text{SN1}} + 92448A^{6} S_{2} \varOmega_{\text{SN1}} - 720896S_{1}^{2} S_{2} \varOmega_{\text{SN1}} \\ & \quad + 934912A^{2} S_{1}^{2} S_{2} \varOmega_{\text{SN1}} - 259776A^{4} S_{1}^{2} S_{2} \varOmega_{\text{SN1}} + 5742A^{6} S_{1}^{2} S_{2} \varOmega_{\text{SN1}} - 2490368S_{1} S_{2}^{2} \varOmega_{\text{SN1}} \\ & \quad + 3969024A^{2} S_{1} S_{2}^{2} \varOmega_{\text{SN1}} - 1718816A^{4} S_{1} S_{2}^{2} \varOmega_{\text{SN1}} + 156180A^{6} S_{1} S_{2}^{2} \varOmega_{\text{SN1}} - 276480A^{2} S_{2}^{3} \varOmega_{\text{SN1}} \\ & \quad + 327456A^{4} S_{2}^{3} \varOmega_{\text{SN1}} - 69336A^{6} S_{2}^{3} \varOmega_{\text{SN1}} - 409600A^{2} S_{3} \varOmega_{\text{SN1}} + 614400A^{4} S_{3} \varOmega_{\text{SN1}} - 213300A^{6} S_{3} \varOmega_{\text{SN1}} \\ & \quad - 27648A^{4} S_{1}^{2} S_{3} \varOmega_{\text{SN1}} + 19845A^{6} S_{1}^{2} S_{3} \varOmega_{\text{SN1}} - 4587520S_{1} S_{2} S_{3} \varOmega_{\text{SN1}} + 6522880A^{2} S_{1} S_{2} S_{3} \varOmega_{\text{SN1}} \\ & \quad - 2132480A^{4} S_{1} S_{2} S_{3} \varOmega_{\text{SN1}} - 420A^{6} S_{1} S_{2} S_{3} \varOmega_{\text{SN1}} - 2818048S_{2}^{2} S_{3} \varOmega_{\text{SN1}} + 4711424A^{2} S_{2}^{2} S_{3} \varOmega_{\text{SN1}} \\ & \quad - 2322688A^{4} S_{2}^{2} S_{3} \varOmega_{\text{SN1}} + 335013A^{6} S_{2}^{2} S_{3} \varOmega_{\text{SN1}} - 6684672S_{1} S_{3}^{2} \varOmega_{\text{SN1}} + 10340352A^{2} S_{1} S_{3}^{2} \varOmega_{\text{SN1}} \\ & \quad - 4256256A^{4} S_{1} S_{3}^{2} \varOmega_{\text{SN1}} + 359550A^{6} S_{1} S_{3}^{2} \varOmega_{\text{SN1}} - 1087488A^{2} S_{2} S_{3}^{2} \varOmega_{\text{SN1}} + 1380128A^{4} S_{2} S_{3}^{2} \varOmega_{\text{SN1}} \\ & \quad - 359841A^{6} S_{2} S_{3}^{2} \varOmega_{\text{SN1}} + 307200A^{2} S_{3}^{3} \varOmega_{\text{SN1}} - 460800A^{4} S_{3}^{3} \varOmega_{\text{SN1}} + 159975A^{6} S_{3}^{3} \varOmega_{\text{SN1}} \\ \end{aligned}$$
$$D_{\text{SN2}} = 262144 - 602112A^{2} + 460800A^{4} - 127156A^{6} + 8007A^{8}$$

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Wu, B., Liu, W., Zhong, H. et al. A Modified Newton–Harmonic Balance Approach to Strongly Odd Nonlinear Oscillators. J. Vib. Eng. Technol. 8, 721–736 (2020). https://doi.org/10.1007/s42417-019-00176-3

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