Thermally Induced Vibrations

  • Richard B. HetnarskiEmail author
  • M. Reza Eslami
Part of the Solid Mechanics and Its Applications book series (SMIA, volume 158)


This chapter presents thermally induced vibrations of isotropic and homogeneous beams and shallow arches and functionally graded beams. The vibration occurs when the heat conduction equation is of parabolic type and the first-order time derivative of temperature function is involved in the heat conduction equation. This equation may be solved with the hyperbolic equation of motion to provide thermally induced vibrations if the limit of inertia parameter is met.


  1. 1.
    Boley BA, Weiner JH (1960) Theory of thermal stresses. Wiley, New YorkzbMATHGoogle Scholar
  2. 2.
    Boley BA (1956) Thermally induced vibrations of beams. J Aeronaut Sci 23(2):179–182zbMATHGoogle Scholar
  3. 3.
    Mobley FF, Fischell RE (1966) Results from gavity-gadient sabilized stellites. In: Proceedings of the Symposium on Passive Gravity-Gradient Stabilization. NASA – SP-107, pp 237–251Google Scholar
  4. 4.
    Rimrott FPJ (1981) The frequency criterion for thermally induced vibrations in elastic beams. Ing Arch 50:281–287CrossRefGoogle Scholar
  5. 5.
    Boley BA, Barber AD (1957) Dynamic response of beams and plates to rapid heating. ASME J Appl Mech 24(3):413–416MathSciNetzbMATHGoogle Scholar
  6. 6.
    Kraus H (1966) Thermally induced vibrations of thin nonshallow spherical shells. AIAA J 4(3):500–505CrossRefGoogle Scholar
  7. 7.
    Nakajo Y, Hayashi K (1984) Response of circular plates to thermal impact. J Sound Vib 95(2):213–222CrossRefGoogle Scholar
  8. 8.
    Das S (1983) Vibrations of polygonal plates due to thermal shock. J Sound Vib 89(4):471–476CrossRefGoogle Scholar
  9. 9.
    Venkataramana J, Jana MK (1974) Thermally forced vibrations of beams. J Sound Vib 37(2):291–295CrossRefGoogle Scholar
  10. 10.
    Stroud RC, Mayers J (1970) Dynamic response of rapidly heated plate elements. AIAA J 9(1):76–83Google Scholar
  11. 11.
    Brush JC, Adalis S, Sadek IS, Sloss JM (1993) Structural control of thermoelastic beams for vibration suppression. J Therm Stress 16(3):249–263CrossRefGoogle Scholar
  12. 12.
    Nakajo Y, Hayashi K (1988) Response of simply supported and clamped circular plates to thermal impact. J Sound Vib 122(2):347–356CrossRefGoogle Scholar
  13. 13.
    Manolis GD, Beskos DE (1980) Thermally induced vibrations of beam structures. Comput Methods Appl Mech Eng 21(3):337–355MathSciNetCrossRefGoogle Scholar
  14. 14.
    Hill DL, Mazumdar J (1987) A study of the thermally induced large amplitude vibrations of viscoelastic plates and shallow shells. J Sound Vib 116(2):323–337CrossRefGoogle Scholar
  15. 15.
    Mazumdar J, Hill DL (1980) Clements, thermally induced vibrations of a viscoelastic plate. J Sound Vib 73(1):31–39CrossRefGoogle Scholar
  16. 16.
    Mazumdar J, Hill DL (1982) Clements, dynamic response of viscoelastic plates of arbitrary shape to rapid heating. Int J Solids Struct 18(11):937–945CrossRefGoogle Scholar
  17. 17.
    Hill DL, Mazumdar J (1984) Thermally induced vibrations of viscoelastic shallow shell. J Sound Vib 93(2):189–200CrossRefGoogle Scholar
  18. 18.
    Tauchert TR (1989) Thermal shock of orthotropic rectangular plates. J Therm Stress 12(2):241–258MathSciNetCrossRefGoogle Scholar
  19. 19.
    Chang JS, Wang JH, Tsai TZ (1992) Thermally induced vibrations of thin laminated plates by finite element method. Comput Struct 42(1):117–128CrossRefGoogle Scholar
  20. 20.
    Huang NN, Tauchert TR (1992) Thermally induced vibration of doubly curved cross-ply laminated panels. J Sound Vib 154(3):485–494CrossRefGoogle Scholar
  21. 21.
    Huang NN, Tauchert TR (1993) Large amplitude vibrations of graphite reinforced aluminum cylindrical panels subjected to rapid heating. Compos Eng 3(6):557–566CrossRefGoogle Scholar
  22. 22.
    Khdier AA (2001) Thermally induced vibration of cross-ply laminated shallow shells. Acta Mech 151(3–4):135–147CrossRefGoogle Scholar
  23. 23.
    Khdier AA (2001) Thermally induced vibration of cross-ply laminated shallow arches. J Therm Stress 24(11):1085–1096CrossRefGoogle Scholar
  24. 24.
    Adam C, Heuer R, Raue A, Ziegler F (2000) Thermally induced vibrations of composite beams with interlayer slip. J Therm Stress 23(8):747–772CrossRefGoogle Scholar
  25. 25.
    Chang JS, Shyong JW (1994) Thermally induced vibration of laminated circular cylindrical shell panels. J Therm Stress 51(3):419–427Google Scholar
  26. 26.
    Chen LW, Lee JH (1989) Vibration of thermal elastic orthotropic plates. Appl Acoust 27(4):287–304CrossRefGoogle Scholar
  27. 27.
    Raja S, Sinha PK, Prathap G, Dwarakanathan D (2001) Thermally induced vibration control of composite plates and shells with piezoelectric active damping. Smart Mater Struct 13(4):939–950CrossRefGoogle Scholar
  28. 28.
    Kumar R, Mishra BK, Jain SC (2008) Thermally induced vibration control of cylindrical shell using piezoelectric sensor and actuator. Int J Adv Manuf Technol 38(5–6):551–562CrossRefGoogle Scholar
  29. 29.
    Esfahani SE, Kiani Y, Eslami MR (2013) Non-linear thermal stability analysis of temperature dependent FGM beams supported on non-linear hardening elastic foundations. Int J Mech Sci 69(1):10–20CrossRefGoogle Scholar
  30. 30.
    Kiani Y, Eslami MR (2013) Thermomechanical buckling of temperature-dependent FGM beams. Lat Am J Solids Struct 10(2):223–246CrossRefGoogle Scholar
  31. 31.
    Kargani A, Kiani Y, Eslami MR (2013) Exact solution for nonlinear stability of piezoelectric FGM timoshenko beams under thermo-electrical loads. J Therm Stress 36(10):1056–1076CrossRefGoogle Scholar
  32. 32.
    Kiani Y, Eslami MR (2014) Geometrically non-linear rapid heating of temperature-dependent circular FGM plates. J Therm Stress 37(12):1495–1518CrossRefGoogle Scholar
  33. 33.
    Alipour SM, Kiani Y, Eslami MR (2016) Rapid heating of FGM rectangular plates. Acta Mech 227(2):421–436MathSciNetCrossRefGoogle Scholar
  34. 34.
    Pandey S, Pradyumna S (2017) A finite element formulation for thermally induced vibrations of functionally graded material sandwich plates and shell panels. Compos Struct 160:877–886CrossRefGoogle Scholar
  35. 35.
    Yaghoubi MR, Kiani Y (1397) Thermal induced vibration in beams considering the rotational inertia, transactions of ISME. J Syst Dyn Solid Mech (in Persian), appearGoogle Scholar
  36. 36.
    Ghiasian SE, Kiani Y, Eslami MR (2014) Nonlinear rapid heating of FGM beams. Int J Non-Linear Mech 67(1):74–84CrossRefGoogle Scholar
  37. 37.
    Eslami MR (2014) Finite elements methods in mechanics. Springer International Publishing, SwitzerlandCrossRefGoogle Scholar
  38. 38.
    Reddy JN, Chin CD (1998) Thermomechanical analysis of functionally graded cylinders and plates. J Therm Stress 21(6):593–626CrossRefGoogle Scholar
  39. 39.
    Ghiasian SE, Kiani Y, Eslami MR (2013) Dynamic buckling of suddenly heated or compressed FGM beams resting on nonlinear elastic foundation. Compos Struct 106(1):225–234CrossRefGoogle Scholar
  40. 40.
    Reddy JN (2003) Mechanics of laminated composite plates and shells, theory and application. CRC Press, Boca RatonGoogle Scholar
  41. 41.
    Reddy JN (2004) An introduction to nonlinear finite element analysis. Oxford University Press, OxfordCrossRefGoogle Scholar
  42. 42.
    Keibollahi A, Kiani Y, Eslami MR Nonlinear rapid heating of shallow arches, To appear, ZAMMGoogle Scholar
  43. 43.
    Eslami MR (2018) Buckling and postbuckling of beams, plates, and shells. Springer International Publishing, SwitzerlandCrossRefGoogle Scholar
  44. 44.
    Eslami MR (2010) Thermo-mechanical buckling of composite plates and shells. Amirkabir University Press, AmirkabirGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.NaplesUSA
  2. 2.Department of Mechanical EngineeringAmirkabir University of TechnologyTehranIran

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