Nonlinear Dynamics

, Volume 98, Issue 3, pp 1853–1876 | Cite as

Self-sustained vibrations of functionally graded carbon nanotubes-reinforced composite cylindrical shells in supersonic flow

  • K. V. AvramovEmail author
  • M. Chernobryvko
  • B. Uspensky
  • K. K. Seitkazenova
  • D. Myrzaliyev
Original paper


Dynamic model of geometrical nonlinear deformations of functionally graded carbon nanotubes-reinforced composite cylindrical shell is obtained. Reddy higher-order shear deformation theory is used to derive this model. The finite-degree-of-freedom nonlinear system, which describes the structure nonlinear self-sustained vibrations, is obtained using the assumed-mode method. The linear piston theory is used to describe the supersonic flow. The loss of the cylindrical shell dynamic stability owing to the Hopf bifurcations is analyzed. The self-sustained vibrations, which describe the circumferential traveling waves flutter, occur due to this bifurcation. The harmonic balance method is applied to analyze these self-sustained vibrations. The properties of the circumferential traveling waves are analyzed.


Functionally graded carbon nanotubes-reinforced material Cylindrical shell in supersonic flow Higher-order shear deformation theory Dynamic instability Self-sustained vibration Circumferential traveling waves flutter 

List of symbols


Young’s modulus of CNTs


Young’s module of matrix

\(G_{12}^{\mathrm{CNT}} \)

Shear module of CNTs


Shear module of matrix


Identity matrix


Stiffness matrix


Length of cylindrical shell


Mass matrix

\(M_X \)

Axial stress moment resultant per unit length


Mach number

\(N_X \)

Axial stress resultant per unit length


Vector of generalized forces


Radius of cylindrical shell


Kinetic energy of cylindrical shell

\(V_{\mathrm{CNT}} (z)\)

Part of volume, which is occupied by CNTs


Part of volume, which is occupied by uniform distribution of CNTs

\(a_\infty \)

Free stream speed of sound


Constant thickness of cylindrical shell


Number of circumference waves


Number of longitudinal half-waves


Radial aerodynamic pressure

\(p_\infty \)

Free stream static pressure


Vector of generalized coordinates


Three projections of the middle surface displacements

\(u_x \left( {x,\theta ,t,z} \right) ,u_\theta \left( {x,\theta ,t,z} \right) , u_z \left( {x,\theta ,t,z} \right) \)

Projections of shell points displacements, which are placed on the z distance from middle surface

\(x,\theta ,z\)

Curvilinear coordinate system


Vector of phase coordinates

\(\prod \)

Potential energy of shell

\(\Omega \)

Frequency of self-sustained vibration

\(\alpha _{\nu j}^{\left( i \right) },\beta _{\nu jj_1 }^{\left( i \right) } \)

Numerical values, described structure geometrically nonlinear deformation

\(\gamma \)

Adiabatic exponent

\(\delta A\)

Virtual work of pressure

\(\delta w\)

Variation of radial displacements

\(\varepsilon _{XX},\varepsilon _{\theta \theta },\gamma _{\theta Z},\gamma _{XZ},\gamma _{X\theta } \)

Elements of the Green’s strain tensor

\(\eta _1,\eta _2,\eta _3\)

CNT/matrix efficiency parameters

\(\mu _{12}^{\mathrm{CNT}} \)

Poisson’s ratio of CNTs

\(\xi \)

Characteristic exponent

\(\rho ^{\mathrm{CNT}}\)

Density of CNTs

\(\rho ^{m}\)

Density of composite matrix

\(\sigma _{XZ},\sigma _{\theta Z} \)

Shear stresses

\(\sigma _{XX},\sigma _{\theta \theta },\sigma _{X\theta } \)

Elements of stress tensor

\(\phi _1,\,\phi _1 \)

Rotations of middle surface normal about \(\theta \) and x axes


Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


  1. 1.
    Seidel, G.D., Lagoudas, D.C.: Micromechanical analysis of the effective elastic properties of carbon nanotube reinforced composites. Mech. Mater. 38, 884–907 (2006)Google Scholar
  2. 2.
    Liu, Y.J., Chen, X.L.: Evaluations of the effective material properties of carbon nanotube-based composites using a nanoscale representative volume element. Mech. Mater. 35, 69–81 (2003)Google Scholar
  3. 3.
    Odegard, G.M., Gates, T.S., Wise, K.E., Park, C., Siochi, E.J.: Constitutive modeling of nanotube-reinforced polymer composites. Compos. Sci. Technol. 63, 1671–1687 (2003)Google Scholar
  4. 4.
    Allaoui, A., Bai, S., Cheng, H.M., Bai, J.B.: Mechanical and electrical properties of a MWNT/epoxy composite. Compos. Sci. Technol. 62, 1993–1998 (2002)Google Scholar
  5. 5.
    Ci, L., Bai, J.B.: The reinforcement role of carbon nanotubes in epoxy composites with different matrix stiffness. Compos. Sci. Technol. 66, 599–603 (2006)Google Scholar
  6. 6.
    Richard, P., Prasse, T., Cavaille, J.Y., Chazeau, L., Gauthier, C., Duchet, J.: Reinforcement of rubbery epoxy by carbon nanofibres. Mater. Sci. Eng. A 352, 344–348 (2003)Google Scholar
  7. 7.
    Kanagaraj, S., Varanda, F.R., Zhil’tsova, T.V., Oliveira, M.S.A., Simoes, J.A.O.: Mechanical properties of high density polyethylene/carbon nanotube composites. Compos. Sci. Technol. 67, 3071–3077 (2007)Google Scholar
  8. 8.
    Andrews, R., Jacques, D., Minot, M., Rantell, T.: Fabrication of carbon multiwall nanotube/polymer composites by shear mixing. Macromol. Mater. Eng. 287, 395–403 (2002)Google Scholar
  9. 9.
    Omidi, M., Rokni, H., Milani, A.S., Seethaler, R.J., Arasteh, R.: Prediction of the mechanical characteristics of multi-walled carbon nanotube/epoxy composites using a new form of the rule of mixtures. Carbon 48, 3218–3228 (2010)Google Scholar
  10. 10.
    Nejati, M., Asanjarani, A., Dimitri, R., Tornabene, F.: Static and free vibration analysis of functionally graded conical shells reinforced by carbon nanotubes. Int. J. Mech. Sci. 130, 383–398 (2017)Google Scholar
  11. 11.
    Hu, H., Onyebueke, L., Abatan, A.: Characterizing and modeling mechanical properties of nanocomposites. Review and evaluation. J. Min. Mater. Charact. Eng. 9, 275–319 (2010)Google Scholar
  12. 12.
    Avramov, K.V.: Nonlinear vibrations characteristics of single-walled carbon nanotubes by nonlocal elastic shell model. Int. J. Nonlinear Mech. 117, 149–160 (2018)Google Scholar
  13. 13.
    Mehrabadi, S.J., Aragh, B.S.: Stress analysis of functionally graded open cylindrical shell reinforced by agglomerated carbon nanotubes. Thin Wall Struct. 80, 130–141 (2014)Google Scholar
  14. 14.
    Zhang, L.W., Lei, Z.X., Liew, K.M., Yu, J.L.: Static and dynamic of carbon nanotube reinforced functionally graded cylindrical panels. Comput. Struct. 111, 205–212 (2014)Google Scholar
  15. 15.
    Song, Z.G., Zhang, L.W., Liew, K.M.: Vibration analysis of CNT-reinforced functionally graded composite cylindrical shells in thermal environments. Int. J. Mech. Sci. 115–116, 339–347 (2016)Google Scholar
  16. 16.
    Sobhaniaragh, B., Batra, R.C., Mansur, W.J., Peters, F.C.: Thermal response of ceramic matrix nanocomposite cylindrical shells using Eshelby–Mori–Tanaka homogenization scheme. Compos. Part B 118, 41–53 (2017)Google Scholar
  17. 17.
    Yaser, K., Rossana, D., Francesco, T.: Free vibration of FG-CNT reinforced composite skew cylindrical shells using the Chebyshev–Ritz formulation. Compos. Part B 147, 169–177 (2018)Google Scholar
  18. 18.
    Lei, Z.X., Liew, K.M., Yu, J.L.: Free vibration analysis of functionally graded carbon nanotube-reinforced composite plates using the element-free kp-Ritz method in thermal environment. Comput. Struct. 106, 128–138 (2013)Google Scholar
  19. 19.
    Lei, Z.X., Zhang, L.W., Liew, K.M.: Elastodynamic analysis of carbon nanotube-reinforced functionally graded plates. Int. J. Mech. Sci. 99, 208–217 (2015)Google Scholar
  20. 20.
    García-Macías, E., Rodríguez-Tembleque, L., Sáez, A.: Bending and free vibration analysis of functionally graded graphene vs. carbon nanotube reinforced composite plates. Comput. Struct. 186, 123–138 (2018)Google Scholar
  21. 21.
    Wang, Q., Cui, X., Qin, B., Liang, Q.: Vibration analysis of the functionally graded carbon nanotube reinforced composite shallow shells with arbitrary boundary conditions. Comput. Struct. 182, 364–379 (2017)Google Scholar
  22. 22.
    Wang, A., Chen, H., Hao, Y., Zhang, W.: Vibration and bending behavior of functionally graded nanocomposite doubly-curved shallow shells reinforced by graphene nanoplatelets. Results Phys. 9, 550–559 (2018)Google Scholar
  23. 23.
    Moradi-Dastjerdi, R., Foroutan, M., Pourasghar, A.: Dynamic analysis of functionally graded nanocomposite cylinders reinforced by carbon nanotube by a mesh-free method. Mater. Des. 44, 256–266 (2013)zbMATHGoogle Scholar
  24. 24.
    Shen, H.S.: Nonlinear bending of functionally graded carbon nanotube-reinforced composite plates in thermal environments. Comput. Struct. 91, 9–19 (2009)Google Scholar
  25. 25.
    Shen, H.S., Xiang, Y.: Nonlinear vibration of nanotube-reinforced composite cylindrical shells in thermal environments. Comput. Methods Appl. Mech. Eng. 213–216, 196–205 (2012)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Ninh, D.G., Bich, D.H.: Characteristics of nonlinear vibration of nanocomposite cylindrical shells with piezoelectric actuators under thermo-mechanical loads. Aerosp. Sci. Technol. 77, 595–609 (2018)Google Scholar
  27. 27.
    Liew, K.M., Lei, Z.X., Yu, J.L., Zhang, L.W.: Postbuckling of carbon nanotube-reinforced functionally graded cylindrical panels under axial compression using a meshless approach. Comput. Methods Appl. Mech. Eng. 268, 1–17 (2014)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Mehri, M., Asadi, H., Kouchakzadeh, M.A.: Computationally efficient model for flow-induced instability of CNT reinforced functionally graded conical curved panels subjected to axial compression. Comput. Methods Appl. Mech. Eng. 318, 957–980 (2017)MathSciNetGoogle Scholar
  29. 29.
    Mehri, M., Asadi, H., Wang, Q.: Buckling and vibration analysis of a pressurized CNT reinforced functionally graded truncated conical shell under an axial compression using HDQ method. Comput. Methods Appl. Mech. Eng. 303, 75–100 (2016)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Mehri, M., Asadi, H., Wang, Q.: On dynamic instability of a pressurized functionally graded carbon nanotube reinforced truncated conical shell subjected to yawed supersonic airflow. Comput. Struct. 153, 938–951 (2016)Google Scholar
  31. 31.
    Frikha, A., Zghal, S., Dammak, F.: Finite rotation three and four nodes shell elements for functionally graded carbon nanotubes-reinforced thin composite shells analysis. Comput. Methods Appl. Mech. Eng. 329, 289–311 (2018)MathSciNetGoogle Scholar
  32. 32.
    Asadi, H.: Numerical simulation of the fluid–solid interaction for CNT reinforced functionally graded cylindrical shells in thermal environments. Acta Astron. 138, 214–224 (2017)Google Scholar
  33. 33.
    Gholami, R., Ansari, R.: Nonlinear harmonically excited vibration of third-order shear deformable functionally graded graphene platelet-reinforced composite rectangular plates. Eng. Struct. 156, 197–209 (2018)Google Scholar
  34. 34.
    Gao, K., Gao, W., Chen, D., Yang, J.: Nonlinear free vibration of functionally graded graphene platelets reinforced porous nanocomposite plates resting on elastic foundation. Comput. Struct. 204, 831–846 (2018)Google Scholar
  35. 35.
    Dowell, E.H.: Panel flutter: a review of the aeroelastic stability of plates and shells. AIAA J. 8, 385–399 (1979)Google Scholar
  36. 36.
    Evensen, D.A., Olson, M.D.: Circumferentially travelling wave flutter of a circular cylindrical shell. AIAA J. 6, 1522–1527 (1968)zbMATHGoogle Scholar
  37. 37.
    Mehri, M., Asadi, H., Kouchakzadeh, M.A.: Computationally efficient model for flow-induced instability of CNT reinforced functionally graded truncated conical curved panels subjected to axial compression. Comput. Methods Appl. Mech. Eng. 318, 957–980 (2017)MathSciNetGoogle Scholar
  38. 38.
    Ganapathi, M., Varadan, T.K., Jijen, J.: Field-consistent element applied to flutter analysis of circular cylindrical shells. J. Sound Vib. 171, 509–527 (1994)zbMATHGoogle Scholar
  39. 39.
    Amabili, M., Pellicano, F.: Nonlinear supersonic flutter of circular cylindrical shells. AIAA J. 39, 564–573 (2001)zbMATHGoogle Scholar
  40. 40.
    Amabili, M., Pellicano, F.: Multimode approach to nonlinear supersonic flutter of imperfect circular cylindrical shells. ASME J. Appl. Mech. 69, 117–129 (2002)zbMATHGoogle Scholar
  41. 41.
    Jansen, E.L.: Effect of boundary conditions on nonlinear vibration and flutter of laminated cylindrical shells. ASME J. Vib. Acoust. 130, 011003 (2008)Google Scholar
  42. 42.
    Avramov, K.V., Chernobryvko, M.V., Kazachenko, O., Batutina, T.J.: Dynamic instability of parabolic shells in supersonic gas stream. Meccanica 51, 939–950 (2016)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Chernobryvko, M.V., Avramov, K.V., Romanenko, V.N., Batutina, T.J., Suleimenov, U.S.: Dynamic instability of ring-stiffened conical thin-walled rocket fairing in supersonic gas stream. Proc. IMechE Part C J. Mech. Eng. Sci. 230, 55–68 (2016)Google Scholar
  44. 44.
    Chai, Y., Song, Z., Li, F.: Investigations on the aerothermoelastic properties of composite laminated cylindrical shells with elastic boundaries in supersonic airflow based on the Rayleigh–Ritz method. Aerosp. Sci. Technol. 82–83, 534–544 (2018)Google Scholar
  45. 45.
    Haddadpour, H., Mahmoudkhani, S., Navazi, H.M.: Supersonic flutter prediction of functionally graded cylindrical shells. Comput. Struct. 83, 391–398 (2008)Google Scholar
  46. 46.
    Song, Z.G., Li, F.M.: Aerothermoelastic analysis and active flutter control of supersonic composite laminated cylindrical shells. Comput. Struct. 106, 653–660 (2013)Google Scholar
  47. 47.
    Chen, J., Li, Q.S.: Nonlinear aeroelastic flutter and dynamic response of composite laminated cylindrical shell in supersonic air flow. Comput. Struct. 168, 474–484 (2017)Google Scholar
  48. 48.
    Lin, H., Cao, D., Shao, C.: An admissible function for vibration and flutter studies of FG cylindrical shells with arbitrary edge conditions using characteristic orthogonal polynomials. Comput. Struct. 185, 748–763 (2018)Google Scholar
  49. 49.
    Avramov, K.V.: Bifurcation behavior of steady vibrations of cantilever plates with geometrical nonlinearities interacting with three-dimensional inviscid potential flow. J. Vib. Control 22, 1198–1216 (2016)MathSciNetzbMATHGoogle Scholar
  50. 50.
    Avramov, K.V., Strel’nikova, E.A., Pierre, C.: Resonant many-mode periodic and chaotic self-sustained aeroelastic vibrations of cantilever plates with geometrical non-linearities in incompressible flow. Nonlinear Dyn. 70, 1335–1354 (2012)MathSciNetGoogle Scholar
  51. 51.
    Avramov, K.V., Papazov, S.V., Breslavsky, I.D.: Dynamic instability of shallow shells in three-dimensional incompressible inviscid potential flow. J. Sound Vib. 394, 593–611 (2017)Google Scholar
  52. 52.
    Asadi, H., Beheshti, A.R.: On the nonlinear dynamic responses of FG-CNTRC beams exposed to aerothermal loads using third-order piston theory. Acta Mech. 229, 2413–2430 (2018)MathSciNetzbMATHGoogle Scholar
  53. 53.
    Keleshteri, M.M., Asadi, H., Wang, Q.: Postbuckling analysis of smart FG-CNTRC annular sector plates with surface-bonded piezoelectric layers using generalized differential quadrature method. Comput. Methods Appl. Mech. Eng. 325, 689–710 (2017)MathSciNetGoogle Scholar
  54. 54.
    Keleshteri, M.M., Asadi, H., Wang, Q.: On the snap-through instability of post-buckled FG-CNTRC rectangular plates with integrated piezoelectric layers. Comput. Methods Appl. Mech. Eng. 331, 53–71 (2018)MathSciNetGoogle Scholar
  55. 55.
    Mei, C., Abdel-Motagaly, K., Chen, R.: Review of nonlinear panel flutter at supersonic and hypersonic speeds. Appl. Mech. Rev. 52, 321–332 (1999)Google Scholar
  56. 56.
    Fidelus, J.D., Wiesel, E., Gojny, F.H., Schulte, K., Wagner, H.D.: Thermo-mechanical properties of randomly oriented carbon/epoxy nanocomposites. Compos. Part A 36, 1555–1561 (2005)Google Scholar
  57. 57.
    Wang, Q., Qin, B., Shi, D., Liang, Q.: A semi-analytical method for vibration analysis of functionally graded carbon nanotube reinforced composite doubly-curved panels and shells of revolution. Comput. Struct. 174, 87–109 (2017)Google Scholar
  58. 58.
    Reddy, J.N.: A simple higher-order theory for laminated composite plates. ASME J. Appl. Mech. 51, 745–752 (1984)zbMATHGoogle Scholar
  59. 59.
    Reddy, J.N.: A refined nonlinear theory of plates with transverse shear deformation. Int. J. Solids Struct. 20, 881–896 (1984)zbMATHGoogle Scholar
  60. 60.
    Amabili, M., Reddy, J.N.: A new non-linear higher-order shear deformation theory for large-amplitude vibrations of laminated doubly curved shells. Int. J. Nonlinear Mech. 45, 409–418 (2010)Google Scholar
  61. 61.
    Amabili, M.: Nonlinear vibrations of laminated circular cylindrical shells: comparison of different shell theories. Comput. Struct. 94, 207–220 (2011)Google Scholar
  62. 62.
    Alijani, F., Amabili, M., Bakhtiari-Nejad, F.: Thermal effects on nonlinear vibrations of functionally graded doubly curved shells using higher order shear deformation theory. Comput. Struct. 93, 2541–2553 (2011)zbMATHGoogle Scholar
  63. 63.
    Amabili, M.: Nonlinear Vibrations and Stability of Shells and Plates. Cambridge University Press, Cambridge (2008)zbMATHGoogle Scholar
  64. 64.
    Olson, M.D., Fung, Y.C.: Supersonic flutter of circular cylindrical shells subjected to internal pressure and axial compression. AIAA J. 4, 858–864 (1966)Google Scholar
  65. 65.
    Olson, M.D., Fung, Y.C.: Comparing theory and experiment for the supersonic flutter of circular cylindrical shells. AIAA J. 5, 1849–1856 (1967)Google Scholar
  66. 66.
    Barr, G.W., Stearman, R.O.: Influence of a supersonic flow field on the elastic stability of cylindrical shells. AIAA J. 8, 993–1000 (1970)Google Scholar
  67. 67.
    Man, Y., Li, Z., Zhang, Z.: Interfacial friction damping characteristics in MWNT-filled polycarbonate composites. Front. Mater. Sci. China 3, 266–272 (2009)Google Scholar
  68. 68.
    Khan, S., Li, C., Siddiqui, N., Kim, J.: Vibration damping characteristics of carbon fiber-reinforced composites containing multi-walled carbon nanotubes. Compos. Sci. Technol. 71, 1486–1494 (2011)Google Scholar
  69. 69.
    Zhou, X., Shin, E., Wang, K.W., Bakis, C.E.: Interfacial dampingcharacteristics of carbon nanotube-based composites. Comput. Sci. Technol. 64, 2425–2437 (2004)Google Scholar
  70. 70.
    Messina, A., Soldatos, K.P.: Ritz-type dynamic analysis of cross-ply laminated circular cylinders subjected to different boundary conditions. J. Sound Vib. 227, 749–768 (1999)zbMATHGoogle Scholar
  71. 71.
    Timarci, T., Sodatos, K.P.: Comparative dynamic stidies for symmetric cross-ply circular cylindrical shells on the basis of a unified shear deformable shell theory. J. Sound Vib. 187, 609–624 (1995)zbMATHGoogle Scholar
  72. 72.
    Khdeir, A.A., Reddy, J.N., Frederick, D.: A study of bending, vibration and buckling of cross-ply circular cylindrical shells with various shell theories. J. Eng. Sci. 27, 1337–1351 (1989)zbMATHGoogle Scholar
  73. 73.
    Leissa, A.W.: Vibration of Shells. NASA SP-288. Government Printing Office, Washington, DC (1993)Google Scholar
  74. 74.
    Kochurov, R., Avramov, K.V.: Nonlinear modes and traveling waves of parametrically excited cylindrical shells. J. Sound Vib. 329, 2193–2204 (2010)Google Scholar
  75. 75.
    Kochurov, E., Avramov, K.V.: On effect of initial imperfections on parametric vibrations of cylindrical shells with geometrical non-linearity. Int. J. Solids Struct. 49, 537–545 (2012)Google Scholar
  76. 76.
    Amabili, M.: A comparison of shell theories for large-amplitude vibrations of circular cylindrical shells: Lagrangian approach. J. Sound Vib. 264, 1091–1125 (2003)Google Scholar
  77. 77.
    Bogolubov, N.N., Mitropolsky, Y.A.: Asymptotic Methods in the Theory of Nonlinear Oscillations. Gordon and Beach, New York (1961)Google Scholar
  78. 78.
    Hayashi, C.: Nonlinear Oscillations in Physical Systems. McGraw-Hill, New York (1964)zbMATHGoogle Scholar
  79. 79.
    Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer, Berlin (1990)zbMATHGoogle Scholar
  80. 80.
    Guckenheimer, J., Holmes, P.J.: Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer, New York (1983)zbMATHGoogle Scholar
  81. 81.
    Galishin, A.: Calculations of plates and shells according to improved theory. Research on Plates and Shells Theory, USSR, Kazan, pp 66–92 (1967) (in Russian)Google Scholar
  82. 82.
    Duc, N.D., Cong, P.H., Tuan, N.D., Tran, P., Thanh, N.V.: Thermal and mechanical stability of functionally graded carbon nanotubes (FG CNT)-reinforced composite truncated conical shells surrounded by the elastic foundations. Thin Wall Struct. 115, 300–331 (2017)Google Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Vibrations, Podgorny Institute for Mechanical EngineeringNational Academy of Science of UkraineKharkivUkraine
  2. 2.Department of Mechanics and EngineeringM. Auezov South Kazakhstan State UniversityShymkentRepublic of Kazakhstan

Personalised recommendations