IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v39y2009i3p1308-1316.html
   My bibliography  Save this article

On methods for continuous systems with quadratic, cubic and quantic nonlinearities

Author

Listed:
  • El-Bassiouny, A.F.

Abstract

Methods for study of weakly nonlinear continuous systems are discussed. The method of multiple scales is used to analyze the nonlinear response of a relief valve under combined static and dynamic loadings. We determine a second-order approximation to the response of the system for the case of primary resonance. Second, we derive a second-order nonlinear ordinary differential equation that describes the time evolution of a single-mode, the so-called single-mode discretization. Then, we use the multiple scales method to determine second-order approximate solutions of this equation, thereby obtaining the equations describe the modulations of the amplitude and phase of the response. We show that the results of the second approach are erroneous.

Suggested Citation

  • El-Bassiouny, A.F., 2009. "On methods for continuous systems with quadratic, cubic and quantic nonlinearities," Chaos, Solitons & Fractals, Elsevier, vol. 39(3), pages 1308-1316.
    • Handle: RePEc:eee:chsofr:v:39:y:2009:i:3:p:1308-1316
    DOI: 10.1016/j.chaos.2007.06.021
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0960077907003992
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.chaos.2007.06.021?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. El-Bassiouny, A.F., 2006. "Single-mode control and chaos of cantilever beam under primary and principal parametric excitations," Chaos, Solitons & Fractals, Elsevier, vol. 30(5), pages 1098-1121.
    2. Gendelman, O.V. & Lamarque, C.H., 2005. "Dynamics of linear oscillator coupled to strongly nonlinear attachment with multiple states of equilibrium," Chaos, Solitons & Fractals, Elsevier, vol. 24(2), pages 501-509.
    3. Alasty, Aria & Salarieh, Hassan, 2007. "Nonlinear feedback control of chaotic pendulum in presence of saturation effect," Chaos, Solitons & Fractals, Elsevier, vol. 31(2), pages 292-304.
    4. Cao, Hongjun, 2005. "Primary resonant optimal control for homoclinic bifurcations in single-degree-of-freedom nonlinear oscillators," Chaos, Solitons & Fractals, Elsevier, vol. 24(5), pages 1387-1398.
    5. El-Bassiouny, A.F., 2006. "Vibration and chaos control of non-linear torsional vibrating systems," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 366(C), pages 167-186.
    6. Musielak, D.E. & Musielak, Z.E. & Benner, J.W., 2005. "Chaos and routes to chaos in coupled Duffing oscillators with multiple degrees of freedom," Chaos, Solitons & Fractals, Elsevier, vol. 24(4), pages 907-922.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. El-Bassiouny, A.F., 2006. "Single-mode control and chaos of cantilever beam under primary and principal parametric excitations," Chaos, Solitons & Fractals, Elsevier, vol. 30(5), pages 1098-1121.
    2. Amer, Y.A., 2007. "Vibration control of ultrasonic cutting via dynamic absorber," Chaos, Solitons & Fractals, Elsevier, vol. 33(5), pages 1703-1710.
    3. Layeghi, Hamed & Arjmand, Mehdi Tabe & Salarieh, Hassan & Alasty, Aria, 2008. "Stabilizing periodic orbits of chaotic systems using fuzzy adaptive sliding mode control," Chaos, Solitons & Fractals, Elsevier, vol. 37(4), pages 1125-1135.
    4. Jiang, Song & Li, Wei & Xin, Gaifang & Sheng, Lianchao & Fan, Mengbao & Yang, Xuefeng, 2020. "Analysis of torsional vibration characteristics and time delay feedback control of semi-direct drive cutting transmission system in shearer," Chaos, Solitons & Fractals, Elsevier, vol. 132(C).
    5. Tang, Yun & Yang, Fenghong & Chen, Guanrong & Zhou, Tianshou, 2006. "Classification of homoclinic tangencies for periodically perturbed systems," Chaos, Solitons & Fractals, Elsevier, vol. 28(1), pages 76-89.
    6. Salarieh, Hassan & Alasty, Aria, 2009. "Chaos control in an economic model via minimum entropy strategy," Chaos, Solitons & Fractals, Elsevier, vol. 40(2), pages 839-847.
    7. Yang, Jianping & Jing, Zhujun, 2009. "Controlling chaos in a pendulum equation with ultra-subharmonic resonances," Chaos, Solitons & Fractals, Elsevier, vol. 42(2), pages 1214-1226.
    8. Ture Savadkoohi, Alireza & Manevitch, Leonid I. & Lamarque, Claude-Henri, 2011. "Analysis of the transient behavior in a two dof nonlinear system," Chaos, Solitons & Fractals, Elsevier, vol. 44(6), pages 450-463.
    9. Yang, Jianping & Jing, Zhujun, 2008. "Inhibition of chaos in a pendulum equation," Chaos, Solitons & Fractals, Elsevier, vol. 35(4), pages 726-737.
    10. Bonakdar, Mohammad & Samadi, Mostafa & Salarieh, Hassan & Alasty, Aria, 2008. "Stabilizing periodic orbits of chaotic systems using fuzzy control of Poincaré map," Chaos, Solitons & Fractals, Elsevier, vol. 36(3), pages 682-693.
    11. Salarieh, Hassan & Alasty, Aria, 2008. "Stabilizing unstable fixed points of chaotic maps via minimum entropy control," Chaos, Solitons & Fractals, Elsevier, vol. 37(3), pages 763-769.
    12. Yu, Xiang & Zhu, Shi-Jian & Liu, Shu-Yong, 2008. "Bifurcation and chaos in multi-degree-of-freedom nonlinear vibration isolation system," Chaos, Solitons & Fractals, Elsevier, vol. 38(5), pages 1498-1504.
    13. Manevitch, L.I. & Gourdon, E. & Lamarque, C.H., 2007. "Parameters optimization for energy pumping in strongly nonhomogeneous 2 dof system," Chaos, Solitons & Fractals, Elsevier, vol. 31(4), pages 900-911.
    14. Ario, Ichiro, 2013. "Multiple duffing problem in a folding structure with hill-top bifurcation," Chaos, Solitons & Fractals, Elsevier, vol. 51(C), pages 52-63.
    15. Salarieh, Hassan & Alasty, Aria, 2009. "Chaos control in uncertain dynamical systems using nonlinear delayed feedback," Chaos, Solitons & Fractals, Elsevier, vol. 41(1), pages 67-71.
    16. Zhang, Yan & Chen, Shihua & Zhou, Hong, 2009. "Synchronizing the noise-perturbed Lü chaotic system," Chaos, Solitons & Fractals, Elsevier, vol. 40(5), pages 2475-2482.
    17. Reis, Eduardo V.M. & Savi, Marcelo A., 2022. "Spatiotemporal chaos in a conservative Duffing-type system," Chaos, Solitons & Fractals, Elsevier, vol. 165(P1).
    18. Litak, Grzegorz & Borowiec, Marek & Syta, Arkadiusz & Szabelski, Kazimierz, 2009. "Transition to chaos in the self-excited system with a cubic double well potential and parametric forcing," Chaos, Solitons & Fractals, Elsevier, vol. 40(5), pages 2414-2429.
    19. Litak, Grzegorz & Syta, Arkadiusz & Borowiec, Marek, 2007. "Suppression of chaos by weak resonant excitations in a non-linear oscillator with a non-symmetric potential," Chaos, Solitons & Fractals, Elsevier, vol. 32(2), pages 694-701.
    20. Salarieh, Hassan & Alasty, Aria, 2008. "Delayed feedback control via minimum entropy strategy in an economic model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(4), pages 851-860.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:chsofr:v:39:y:2009:i:3:p:1308-1316. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Thayer, Thomas R. (email available below). General contact details of provider: https://www.journals.elsevier.com/chaos-solitons-and-fractals .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.