Skip to main content
SpringerLink
Log in

Nonlinear primary resonance of imperfect spiral stiffened functionally graded cylindrical shells surrounded by damping and nonlinear elastic foundation

  • Original Article
  • Published:
Engineering with Computers Aims and scope Submit manuscript

Abstract

In this paper, an analytical method is used to study the nonlinear primary resonance of imperfect spiral stiffened functionally graded (SSFG) cylindrical shells with internal stiffeners. The SSFG cylindrical shell is surrounded by linear and nonlinear elastic foundation and the effect of structural damping on the system response is also considered. The material properties of the shell and stiffeners are assumed to be continuously graded in the thickness direction. Three-parameter nonlinear elastic foundation model is consists of two-parameter linear elastic foundation (Winkler and Pasternak) and one hardening/softening cubic nonlinearity parameter. Based on the von Kármán nonlinear equations and the classical plate theory of shells, the strain–displacement relations are derived. The smeared stiffener technique is used to the model of the internal stiffeners. Using the Galerkin method, the partial differential equations of motion are discretized. The nonlinear primary resonance is analyzed by means of the multiple scales method. The effects of various geometrical characteristics, material parameters and elastic foundation coefficients are investigated on the nonlinear primary resonance.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16

Similar content being viewed by others

References

  1. Lajimi SAM, Heppler GR, Abdel-Rahman EM (2015) Primary resonance of an amplitude-/frequency-modulation beam-rigid body microgyroscope. Int J Non-Linear Mech 77:364–375

    Article  Google Scholar 

  2. Khadem SE, Shahgholi M, Hosseini SAA (2010) Primary resonances of a nonlinear in-extensional rotating shaft. Mech Mach Theory 45(8):1067–1081

    Article  MATH  Google Scholar 

  3. Caruntu DI, Oyervides R (2017) Frequency response reduced order model of primary resonance of electrostatically actuated MEMS circular plate resonators. Commun Nonlinear Sci Numer Simul 43:261–270

    Article  Google Scholar 

  4. Srirangarajan HR (1992) Superharmonic resonance of circular plates. J Sound Vib 157:549–552

    Article  MATH  Google Scholar 

  5. Wang YQ, Yang Z (2017) Nonlinear vibrations of moving functionally graded plates containing porosities and contacting with liquid: internal resonance. Nonlinear Dyn 90(2):1461–1480

    Article  Google Scholar 

  6. Wang YQ, Zu JW (2017) Nonlinear dynamic thermoelastic response of rectangular FGM plates with longitudinal velocity. Compos Part B Eng 117:74–88

    Article  Google Scholar 

  7. Wang YQ, Zu JW (2017) Vibration behaviors of functionally graded rectangular plates with porosities and moving in thermal environment. Aerosp Sci Technol 69:550–562

    Article  Google Scholar 

  8. Wang YQ, Zu JW (2017) Nonlinear steady-state responses of longitudinally traveling functionally graded material plates in contact with liquid. Compos Struct 164:130–144

    Article  Google Scholar 

  9. Wang YQ, Zu JW (2018) Nonlinear dynamics of a translational FGM plate with strong mode interaction. Int J Struct Stab Dyn 18(03):1850031

    Article  MathSciNet  Google Scholar 

  10. Wang YQ (2018) Electro-mechanical vibration analysis of functionally graded piezoelectric porous plates in the translation state. Acta Astronaut 143:263–271

    Article  Google Scholar 

  11. Wang YQ, Wan YH, Zhang YF (2017) Vibrations of longitudinally traveling functionally graded material plates with porosities. Eur J Mech-A/Solids 66:55–68

    Article  MathSciNet  MATH  Google Scholar 

  12. Alijani F, Amabili M (2014) Nonlinear vibrations and multiple resonances of fluid filled arbitrary laminated circular cylindrical shells. Compos Struct 108:951–962

    Article  Google Scholar 

  13. Wang YQ, Liang L, Guo XH (2013) Internal resonance of axially moving laminated circular cylindrical shells. J Sound Vib 332(24):6434–6450

    Article  Google Scholar 

  14. Mahmoudkhani S, Navazi HM, Haddadpour H (2011) An analytical study of the non-linear vibrations of cylindrical shells. Int J Non-Linear Mech 46(10):1361–1372

    Article  Google Scholar 

  15. Tesar A (1985) Nonlinear three-dimensional resonance analysis of shells. Comput Struct 21(4):797–805

    Article  Google Scholar 

  16. Tesar A (1984) Nonlinear interactions in resonance response of thin shells. Comput Struct 18(6):1047–1055

    Article  MATH  Google Scholar 

  17. Zhang W, Liu T, Xi A, Wang YN (2018) Resonant responses and chaotic dynamics of composite laminated circular cylindrical shell with membranes. J Sound Vib 423:65–99

    Article  Google Scholar 

  18. Chen J, Li Q-S (2017) Nonlinear aeroelastic flutter and dynamic response of composite laminated cylindrical shell in supersonic air flow. Compos Struct 168:474–484

    Article  Google Scholar 

  19. Abe A, Kobayashi Y, Yamada G (2007) Nonlinear dynamic behaviors of clamped laminated shallow shells with one-to-one internal resonance. J Sound Vib 304(3–5):957–968

    Article  Google Scholar 

  20. Sheng GG, Wang X (2018) Nonlinear vibrations of FG cylindrical shells subjected to parametric and external excitations. Compos Struct 191:78–88

    Article  Google Scholar 

  21. Du C, Li Y (2014) Nonlinear internal resonance of functionally graded cylindrical shells using the Hamiltonian dynamics. Acta Mech Solida Sin 27(6):635–647

    Article  Google Scholar 

  22. Li X, Du CC, Li YH (2018) Parametric resonance of a FG cylindrical thin shell with periodic rotating angular speeds in thermal environment. Appl Math Model 59:393–409

    Article  MathSciNet  Google Scholar 

  23. Gao K, Gao W, Wu B, Wu D, Song C (2018) Nonlinear primary resonance of functionally graded porous cylindrical shells using the method of multiple scales. Thin-Walled Struct 125:281–293

    Article  Google Scholar 

  24. Sheng GG, Wang X (2018) The dynamic stability and nonlinear vibration analysis of stiffened functionally graded cylindrical shells. Appl Math Model 56:389–403

    Article  MathSciNet  Google Scholar 

  25. Nam VH, Phuong NT, Van Minh K, Hieu PT (2018) Nonlinear thermo-mechanical buckling and post-buckling of multilayer FGM cylindrical shell reinforced by spiral stiffeners surrounded by elastic foundation subjected to torsional loads. Eur J Mech-A/Solids 72:393–406

    Article  MathSciNet  MATH  Google Scholar 

  26. Thi Phuong N, Thanh Luan D, Nam VH, Hieu PT (2018) Nonlinear approach on torsional buckling and postbuckling of functionally graded cylindrical shells reinforced by orthogonal and spiral stiffeners in thermal environment. Proc Inst Mech Eng Part C J Mech Eng Sci. https://doi.org/10.1177/0954406218780523

    Google Scholar 

  27. Van Dung D (2013) Nonlinear buckling and post-buckling analysis of eccentrically stiffened functionally graded circular cylindrical shells under external pressure. Thin-Walled Struct 63:117–124

    Article  Google Scholar 

  28. Jrad H, Mars J, Wali M, Dammak F (2018) Geometrically nonlinear analysis of elastoplastic behavior of functionally graded shells. Eng Comput. https://doi.org/10.1007/s00366-018-0633-3

    Google Scholar 

  29. Van Dung D, Nam VH (2014) Nonlinear dynamic analysis of eccentrically stiffened functionally graded circular cylindrical thin shells under external pressure and surrounded by an elastic medium. Eur J Mech-A/Solids 46:42–53

    Article  MathSciNet  MATH  Google Scholar 

  30. Brush DO, Almroth BO (1975) Buckling of bars, plates, and shells, vol 6. McGraw-Hill, New York

    MATH  Google Scholar 

  31. Foroutan K, Shaterzadeh A, Ahmadi H (2018) Nonlinear dynamic analysis of spiral stiffened functionally graded cylindrical shells with damping and nonlinear elastic foundation under axial compression. Struct Eng Mech 66(3):295–303

    Google Scholar 

  32. Shaterzadeh A, Foroutan K (2016) Post-buckling of cylindrical shells with spiral stiffeners under elastic foundation. Struct Eng Mech 60(4):615–631

    Article  Google Scholar 

  33. Bich DH, Van Dung D, Nam VH, Phuong NT (2013) Nonlinear static and dynamic buckling analysis of imperfect eccentrically stiffened functionally graded circular cylindrical thin shells under axial compression. Int J Mech Sci 74:190–200

    Article  Google Scholar 

  34. Ghiasian SE, Kiani Y, Eslami MR (2013) Dynamic buckling of suddenly heated or compressed FGM beams resting on nonlinear elastic foundation. Compos Struct 106:225–234

    Article  Google Scholar 

  35. Volmir AS (1972) Non-linear dynamics of plates and shells. AS Science Edition M, USSR

  36. Bich DH, Van Dung D, Nam VH (2012) Nonlinear dynamical analysis of eccentrically stiffened functionally graded cylindrical panels. Compos Struct 94(8):2465–2473

    Article  Google Scholar 

  37. Sofiyev AH (2009) The vibration and stability behavior of freely supported FGM conical shells subjected to external pressure. Compos Struct 89(3):356–366

    Article  Google Scholar 

  38. Paliwal DN, Pandey RK, Nath T (1996) Free vibrations of circular cylindrical shell on Winkler and Pasternak foundations. Int J Press Vessels Pip 69(1):79–89

    Article  Google Scholar 

  39. Sewall JL, Naumann EC (1968) An experimental and analytical vibration study of thin cylindrical shells with and without longitudinal stiffeners. NASA TN D-4705

  40. Sewall JL, Clary RR, Leadbetter SA (1964) An experimental and analytical vibration study of a ring-stiffened cylindrical shell structure with various support conditions. NASA TN D-2398

Download references

Author information

Authors and Affiliations

  1. Faculty of Mechanical Engineering, Shahrood University of Technology, Shahrood, Iran

    Habib Ahmadi

Authors
  1. Habib Ahmadi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Habib Ahmadi.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix

Appendix

$$\begin{aligned} A & =J_{{11}}^{*}{m^4}{\pi ^4}+\left( {J_{{33}}^{*} - J_{{12}}^{*} - J_{{21}}^{*}} \right){m^2}{n^2}{\pi ^2}{\lambda ^2}+J_{{22}}^{*}{n^4}{\lambda ^4} \\ B & =J_{{21}}^{{**}}{m^4}{\pi ^4}+\left( {J_{{11}}^{*}+J_{{22}}^{*} - 2J_{{36}}^{{**}}} \right){m^2}{n^2}{\pi ^2}{\lambda ^2}+J_{{12}}^{{**}}{n^4}{\lambda ^4} - \frac{{{L^2}}}{R}{m^2}{n^2} \\ {B^*} & =A_{{21}}^{*}{m^4}{\pi ^4}+\left( {A_{{11}}^{*}+A_{{22}}^{*} - 2J_{{36}}^{{**}}} \right){m^2}{n^2}{\pi ^2}{\lambda ^2}+J_{{12}}^{{**}}{n^4}{\lambda ^4} - \frac{{{L^2}}}{R}{m^2}{n^2} \\ D & =A_{{11}}^{{**}}{m^4}{\pi ^4}+\left( {A_{{12}}^{{**}}+A_{{21}}^{{**}}+4A_{{36}}^{{**}}} \right){m^2}{n^2}{\pi ^2}{\lambda ^2}+A_{{22}}^{{**}}{n^4}{\lambda ^4} \\ G & =\left( {\frac{{{n^4}{\lambda ^4}}}{{16J_{{11}}^{*}}}+\frac{{{m^4}{\pi ^4}}}{{16J_{{22}}^{*}}}} \right) \\ \lambda & =\frac{L}{R} \\ \end{aligned}$$

where

$$\begin{aligned} \Delta & ={J_{11}}{J_{22}} - {J_{12}}{J_{21}},~~J_{{22}}^{*}=\frac{{{J_{22}}}}{\Delta },~~J_{{12}}^{*}=\frac{{{J_{12}}}}{\Delta },\,\,\,J_{{11}}^{*}=\frac{{{J_{11}}}}{\Delta },~~~J_{{21}}^{*}=\frac{{{J_{21}}}}{\Delta },~~~J_{{33}}^{*}=\frac{1}{{{J_{33}}}},~~~J_{{36}}^{*}=\frac{{{J_{36}}}}{{{J_{33}}}} \\ J_{{11}}^{{**}} & =J_{{22}}^{*}{J_{14}} - J_{{12}}^{*}{J_{24}},~~~J_{{12}}^{{**}}=J_{{22}}^{*}{J_{15}} - J_{{12}}^{*}{J_{25}},\,\,\,J_{{21}}^{{**}}=J_{{11}}^{*}{J_{24}} - J_{{21}}^{*}{J_{14}},~~~J_{{22}}^{{**}}=J_{{11}}^{*}{J_{25}} - J_{{21}}^{*}{J_{15}} \\ A_{{11}}^{*} & =J_{{22}}^{*}{J_{14}} - J_{{21}}^{*}{J_{15}},~~~A_{{21}}^{*}=J_{{11}}^{*}{J_{15}} - J_{{12}}^{*}{J_{14}},\,\,\,A_{{12}}^{*}=J_{{22}}^{*}{J_{24}} - J_{{21}}^{*}{J_{25}},~~~A_{{22}}^{*}=J_{{11}}^{*}{J_{25}} - J_{{12}}^{*}{J_{24}} \\ A_{{11}}^{{**}} & =J_{{11}}^{{**}}{J_{14}} - J_{{21}}^{{**}}{J_{15}} - {J_{41}},\,\,\,A_{{12}}^{{**}}=J_{{12}}^{{**}}{J_{14}} - J_{{22}}^{{**}}{J_{15}} - {J_{42}},\,\,\,A_{{21}}^{{**}}=J_{{11}}^{{**}}{J_{24}} - J_{{21}}^{{**}}{J_{25}} - {J_{51}} \\ A_{{22}}^{{**}} & =J_{{12}}^{{**}}{J_{24}} - J_{{22}}^{{**}}{J_{25}} - {J_{52}},\,\,\,A_{{36}}^{{**}}=J_{{36}}^{{**}}{J_{36}} - {J_{63}} \\ \end{aligned}$$

where

$$\begin{aligned} {J_{11}} & =\frac{{{E_1}}}{{1 - {\nu ^2}}}+{Z_1}{E_{1s}}\left( {{{\cos }^3}\theta +{{\cos }^3}\beta } \right),\,\,\,~{J_{12}}=\frac{{{E_1}\nu }}{{1 - {\nu ^2}}}+{Z_1}{E_{1s}}\left( {{{\sin }^2}\theta \cos \theta +{{\sin }^2}\beta \cos \beta } \right) \\ {J_{14}} & =\frac{{{E_2}}}{{1 - {\nu ^2}}}+{Z_1}{E_{2s}}\left( {{{\cos }^3}\theta +{{\cos }^3}\beta } \right),\,\,\,{J_{15}}=\frac{{{E_2}\nu }}{{1 - {\nu ^2}}}+{Z_1}{E_{2s}}\left( {{{\sin }^2}\theta \cos \theta +{{\sin }^2}\beta \cos \beta } \right) \\ {J_{21}} & =\frac{{{E_1}\nu }}{{1 - {\nu ^2}}}+{Z_2}{E_{1s}}\left( {\sin \theta {{\cos }^2}\theta +\sin \beta {{\cos }^2}\beta } \right),\,\,\,{J_{22}}=\frac{{{E_1}}}{{1 - {\nu ^2}}}+{Z_2}{E_{1s}}\left( {{{\sin }^3}\theta +{{\sin }^3}\beta } \right) \\ {J_{24}} & =\frac{{{E_2}\nu }}{{1 - {\nu ^2}}}+{Z_2}{E_{2s}}\left( {\sin \theta {{\cos }^2}\theta +\sin \beta {{\cos }^2}\beta } \right),\,\,\,{J_{25}}=\frac{{{E_2}}}{{1 - {\nu ^2}}}+{Z_2}{E_{2s}}\left( {{{\sin }^3}\theta +{{\sin }^3}\beta } \right) \\ {J_{33}} & =\frac{{{E_1}}}{{2\left( {1+\nu } \right)}}+2{Z_3}{E_{1s}}\left( {\sin \theta \cos \theta +\sin \beta \cos \beta } \right),\,\,\,{J_{36}}=\frac{{{E_2}}}{{2\left( {1+\nu } \right)}}+2{Z_3}{E_{2s}}\left( {\sin \theta \cos \theta +\sin \beta \cos \beta } \right) \\ {J_{41}} & =\frac{{{E_3}}}{{1 - {\nu ^2}}}+{Z_1}{E_{3s}}\left( {{{\cos }^3}\theta +{{\cos }^3}\beta } \right),\,\,\,{J_{42}}=\frac{{{E_3}\nu }}{{1 - {\nu ^2}}}+{Z_1}{E_{3s}}\left( {{{\sin }^2}\theta \cos \theta +{{\sin }^2}\beta \cos \beta } \right) \\ {J_{51}} & =\frac{{{E_3}\nu }}{{1 - {\nu ^2}}}+{Z_2}{E_{3s}}\left( {\sin \theta {{\cos }^2}\theta +\sin \beta {{\cos }^2}\beta } \right),\,\,\,{J_{55}}=\frac{{{E_3}}}{{1 - {\nu ^2}}}+{Z_2}{E_{3s}}\left( {{{\sin }^3}\theta +{{\sin }^3}\beta } \right) \\ {J_{63}} & =\frac{{{E_3}}}{{2\left( {1+\nu } \right)}}+2{Z_3}{E_{3s}}\left( {\sin \theta \cos \theta +\sin \beta \cos \beta } \right) \\ \end{aligned}$$

where

$$\begin{aligned} {E_1} & =\mathop \int \limits_{{ - h/2}}^{{h/2}} {E_{{\text{sh}}}}\left( z \right){\text{d}}z=\left( {{E_m}+\frac{{{E_c} - {E_m}}}{{k+1}}} \right)h,\,\,\,{E_2}=\mathop \int \limits_{{ - h/2}}^{{h/2}} z{E_{{\text{sh}}}}\left( z \right){\text{d}}z=\frac{{\left( {{E_c} - {E_m}} \right)k{h^2}}}{{2\left( {k+1} \right)\left( {k+2} \right)}} \\ {E_3} & =\mathop \int \limits_{{ - h/2}}^{{h/2}} {z^2}{E_{{\text{sh}}}}\left( z \right){\text{d}}z=\left[ {\frac{{{E_m}}}{{12}}} \right.~\left. {+\left( {{E_c} - {E_m}} \right)\left( {\frac{1}{{k+3}}+\frac{1}{{k+2}}+\frac{1}{{4k+4}}} \right)} \right]{h^3} \\ {E_{1s}} & =\mathop \int \limits_{{h/2}}^{{h/2+{h_s}}} {E_s}\left( z \right){\text{d}}z=\left( {{E_c}+\frac{{{E_m} - {E_c}}}{{{k_s}+1}}} \right){h_s} \\ {E_{2s}} & =\mathop \int \limits_{{h/2}}^{{h/2+{h_s}}} z{E_s}\left( z \right)dz \\ & =\frac{{{E_c}}}{2}~h{h_s}\left( {\frac{{{h_s}}}{h}+1} \right)+\left( {{E_m} - {E_c}} \right)~h{h_s}\left( {\frac{1}{{{k_s}+2}}\frac{{{h_s}}}{h}+\frac{1}{{2{k_s}+2}}} \right) \\ {E_{3s}} & =\mathop \int \limits_{{\frac{h}{2}}}^{{\frac{h}{2}+{h_s}}} {z^2}{E_s}\left( z \right){\text{d}}z \\ & =\frac{{{E_c}}}{3}h_{s}^{3}\left( {\frac{3}{4}\frac{{{h^2}}}{{h_{s}^{2}}}+\frac{3}{2}\frac{h}{{{h_s}}}+1} \right)+\left( {{E_m} - {E_c}} \right)h_{s}^{3}\left[ {\frac{1}{{{k_s}+3}}+\frac{1}{{{k_s}+2}}\frac{h}{{{h_s}}}+\frac{1}{{4\left( {{k_s}+1} \right)}}\frac{{{h^2}}}{{h_{s}^{2}}}} \right] \\ \end{aligned}$$

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ahmadi, H. Nonlinear primary resonance of imperfect spiral stiffened functionally graded cylindrical shells surrounded by damping and nonlinear elastic foundation. Engineering with Computers 35, 1491–1505 (2019). https://doi.org/10.1007/s00366-018-0679-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00366-018-0679-2

Keywords

Search

Navigation