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International Applied Mechanics

, Volume 54, Issue 1, pp 104–119 | Cite as

Geometric Nonlinear Vibration Analysis for Pretensioned Rectangular Orthotropic Membrane

  • C. J. Liu
  • Z. L. Zheng
  • X. Y. Yang
  • J. J. Guo
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The geometric nonlinear vibrations of pretensioned orthotropic membrane with four edges fixed, which is commonly applied in building membrane structure, are studied. The nonlinear partial differential governing equations are derived by von Kármán’s large deflection theory and D’Alembert’s principle. Because of the strong nonlinearity of governing equations, the homotopy perturbation method (HPM) to solve them is applied. The approximate analytical solution of the vibration frequency and displacement function is obtained. In the computational example, the frequency, vibration mode and displacement as well as the time curve of each feature point are analyzed. It is proved that HPM is an effective, simple and high-precision method to solve the geometric nonlinear vibration problem of membrane structures. These results provide some valuable computational basis for the vibration control and dynamic design of building and other analogous membrane structures.

Keywords

nonlinear vibration orthotropic membrane perturbation method 
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References

  1. 1.
    C. G. Wang, Y. L. Li, X.W. Du, et al, “Simulation analysis of vibration characteristics of wrinkled membrane space structure,” Int. J. Space Struct., 22, No. 4, 239–246 (2007).CrossRefGoogle Scholar
  2. 2.
    C. Jenkins and U. A. Korde, “Membrane vibration experiments: An historical review and recent results,” J. Sound Vibr., 295, No. 3–5, 602–613 (2006).ADSCrossRefGoogle Scholar
  3. 3.
    Y. L. Li, M. Y. Lu, H. F. Tan, et al, “A study on wrinkling characteristics and dynamic mechanical behavior of membrane,” Acta Mechanica Sinica/Lixue Xuebao, 28, No. 1, 201–210 (2012).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    S. W. Kang and J. M. Lee, “Free vibration analysis of composite rectangular membranes with an oblique interface,” J. Sound Vibr., 251, No. 3, 505–517 (2002).ADSCrossRefGoogle Scholar
  5. 5.
    S. W. Kang, “Free vibration analysis of composite rectangular membranes with a bent interface,” J. Sound Vibr., 272, No. 1–2, 39–53 (2004).Google Scholar
  6. 6.
    C. Y. Wang and C. M. Wang, “Exact solutions for vibrating rectangular membranes placed in a vertical plane,” Int. J. Appl. Mech., 3, No. 3, 625–631 (2011).CrossRefGoogle Scholar
  7. 7.
    R. M. Soares and P. B. Gonalves, “Nonlinear vibrations and instabilities of a stretched hyperelastic annular membrane,” Int. J. Solids Struct., 49, No. 3–4, 514–526 (2006).Google Scholar
  8. 8.
    P. B. Goncalves, R. M. Soares, and D. Pamplona, “Nonlinear vibrations of a radially stretched circular hyperelastic membrane,” J. Sound Vibr., 327, No. 1–2, 231–248 (2009).Google Scholar
  9. 9.
    Z. L. Zheng, W. J. Song, C. J. Liu, et al., “Study on dynamic response of rectangular orthotropic membranes under impact loading,” J. Adhesion Sci. Technol., 26, No. 10–11, 1467–1479 (2012).Google Scholar
  10. 10.
    C. Shin, J. T. Chung, and W. Kim, “Dynamic characteristics of the out-of-plane vibration for an axially moving membrane,” J. Sound Vibr., 286, No. 4–5, 1019–1031 (2005).Google Scholar
  11. 11.
    H. Zhang and J. Shan, “Initial form finding and free vibration properties study of membrane,” in: Proc. 2006 Xi’an Int. Conf. of Architecture and Technology, Proceedings-Architecture in Harmony, Xian, China (2006), pp. 316–320.Google Scholar
  12. 12.
    Y. L. Li, C. G. Wang, and H. F. Tan, “Research on free vibration of wrinkled membranes,” in: Proc. 5th Int. Conf. on Nonlinear Mechanics, Shanghai, China (2007), pp. 649–654.Google Scholar
  13. 13.
    J. J. Pan and M. Gu, “Geometric nonlinear effect to square tensioned membrane’s free vibration,” J. Tongji University (Natural Science), 35, No. 11, 1450–1454 (2007).Google Scholar
  14. 14.
    S. Y. Reutskiy, “Vibration analysis of arbitrarily shaped membranes,” CMES-Computer Modeling in Engineering & Science, 51, No. 2, 115–142 (2009).MathSciNetzbMATHGoogle Scholar
  15. 15.
    F. Formosa, “Nonlinear dynamics analysis of a membrane Stirling engine: starting and stable operation,” J. Sound Vibr., 326, No. 3–5, 794–808 (2009).Google Scholar
  16. 16.
    Z. L. Zheng, C. J. Liu, X. T. He, et al., “Free vibration analysis of rectangular orthotropic membranes in large deflection,” Math. Probl. Eng., Article ID 634362 (2009).Google Scholar
  17. 17.
    C. J. Liu, Z. L. Zheng, X. T. He, et al., “L–P perturbation solution of nonlinear free vibration of prestressed orthotropic membrane in large amplitude,” Math. Probl. Eng., Article ID 561364 (2010).Google Scholar
  18. 18.
    J. H. He, “Homotopy perturbation technique,” Comp. Meth. Appl. Mech. Eng., 178, No. 3, 257–262 (1999).MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    J. H. He, “Homotopy perturbation method: A new nonlinear analytical technique,” Appl. Math. Comp., 135, No. 1, 73–79 (2003).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    J. H. He, “Comparison of homotopy perturbation method and homotopy analysis method,” Appl. Math. Comp., 156, No. 2, 527–539 (2004).MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    J. H. He, “A coupling method of a homotopy technique and a perturbation technique for non-linear problems,” Int. J. Non-Linear Mech., 35, No. 1, 37–43 (2000).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    J. H. He, “The homotopy perturbation method for nonlinear oscillators with discontinuities,” Appl. Math. Comp., 151, No. 1, 287–292 (2004).MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    L. Cveticanin, “Homotopy-perturbation method for pure nonlinear differential equation,” Chaos, Solitons and Fractals, 30, No. 5, 1221–1230 (2006).ADSCrossRefzbMATHGoogle Scholar
  24. 24.
    A. Yildirim, “Application of the homotopy perturbation method for the Fokker–Planck equation,” Int. J. Numer. Meth. Biomedical Eng., 26, No. 9, 1144–1154 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    A. M. A. El-Sayed, A. Elsaid, I. L. El-Kalla, et al., “A homotopy perturbation technique for solving partial differential equations of fractional order in finite domains,” Appl. Math. Comp., 218, No. 17, 8329–8340 (2012).MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    A. Golbabai and M. Javidi, “Application of He’s homotopy perturbation method for nth-order integro-differential equations,” Appl. Math. Comp., 190, No. 2, 1409–1416 (2007).CrossRefzbMATHGoogle Scholar
  27. 27.
    B. Ozturk and S. B. Coskun, “The homotopy perturbation method for free vibration analysis of beam on elastic foundation,” Struct. Eng. Mech., 37, No. 4, 415–425 (2011).CrossRefGoogle Scholar
  28. 28.
    H. Saffari, I. Mansouri, M. H. Bagheripour, et al., “Elasto-plastic analysis of steel plane frames using homotopy perturbation method,” J. Const. Steel Res., 70, 350–357 (2012).CrossRefGoogle Scholar
  29. 29.
    A. R. Ghotbi, A. Barari, and D. D. Ganji, “Solving ratio-dependent predator-prey system with constant effort harvesting using homotopy perturbation method,” Math. Probl. Eng., Article ID 945420 (2008).Google Scholar
  30. 30.
    K. Reck, E. V. Thomsen, and O. Hansen, “Solving the Helmholtz equation in conformal mapped ARROW structures using homotopy perturbation method,” Optics Express, 19, No. 3, 1808–1823 (2011).ADSCrossRefGoogle Scholar
  31. 31.
    I. V. Andrianov, J. Awrejcewicz, and V. Chernetskyy, “Analysis of natural in-plane vibration of rectangular plates using homotopy perturbation approach,” Math. Probl. Eng., Article ID 20598 (2006).Google Scholar
  32. 32.
    G. Domairry and A. Aziz, “Approximate analysis of MHD squeeze flow between two parallel disks with suction or injection by homotopy perturbation method,” Math. Probl. Eng., Article ID 603916 (2009).Google Scholar
  33. 33.
    F. Shakeri and M. Dehghan, “Solution of delay differential equations via a homotopy perturbation method,” Math. Comp. Model., 48, No. 3–4, 486–498 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    A. Yildirim, “Application of He’s homotopy perturbation method for solving the Cauchy reaction-diffusion problem,” Comp. Math. Appl., 57, No. 4, 612–618 (2009).MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • C. J. Liu
    • 1
    • 2
  • Z. L. Zheng
    • 3
  • X. Y. Yang
    • 4
  • J. J. Guo
    • 5
  1. 1.State Key Laboratory of Geohazard Prevention and Geoenvironment ProtectionChengdu University of TechnologyChengduChina
  2. 2.College of Environment and Civil EngineeringChengdu University of TechnologyChengduChina
  3. 3.College of Civil EngineeringChongqing UniversityChongqingChina
  4. 4.College of nuclear technology and automation EngineeringChengdu University of TechnologyChengduChina
  5. 5.Chongqing Water Resources and Electric Engineering CollegeChongqingChina

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