Effect of Winkler and Pasternak elastic foundation on the vibration of rotating functionally graded material cylindrical shell

Abstract

In this paper, vibration characteristics of rotating functionally graded cylindrical shell resting on Winkler and Pasternak elastic foundations have been investigated. These shells are fabricated from functionally graded materials. Shell dynamical equations are derived by using the Hamilton variational principle and the Langrangian functional framed from the shell strain and kinetic energy expressions. Elastic foundations, namely Winkler and Pasternak moduli are inducted in the tangential direction of the shell. The rotational motions of the shells are due to the Coriolis and centrifugal acceleration as well as the hoop tension produced in the rotating case. The wave propagation approach in standard eigenvalue form has been employed in order to derive the characteristic frequency equation describing the natural frequencies of vibration in rotating functionally graded cylindrical shell. The complex exponential functions, with the axial modal numbers that depend on the boundary conditions stated at edges of a cylindrical shell, have been used to compute the axial modal dependence. In our new investigation, frequency spectra are obtained for circumferential wave number, length-to-radius ratio, height-to-radius ratio with simply supported–simply supported and clamped–clamped boundary conditions without elastic foundation. Also, the effect of elastic foundation on the rotating cylindrical shells is examined with the simply supported–simply supported edge. To check the validity of the present method, the fundamental natural frequencies of non-rotating isotropic and functionally graded cylindrical shells are compared with the open literature. Also, a comparison is made for infinitely long rotating with the earlier published paper.

Keywords

Winkler and Pasternak elastic foundation functionally graded material rotating cylindrical shell

Get full access to this article

View all access options for this article.

References

Wang

Liu

Zhu

. B-spline based functionally graded teeth modeling. J Comput Aid Des Comput Graph 2011; 23: 676–681.

Yamanouchi M, Koizumi M, Hiraj T, et al. Proceeding of first international symposium on functionally gradient materials, Japan, 8–9 October 1990.

Koizumi

. FGM activities in Japan. Composites 1997; 28: 1–14.

Isvandzibaei

Jamaluddin

Raja Hamzah

. Effects of uniform interior pressure distribution on vibration of FGM cylindrical shell with rings support based on first-order theory subjected to ten boundary conditions. Acta Mech 2014; 225: 2085–2109.

Isvandzibaei

Jamaluddin

Raja Hamzah

. Vibration analysis of supported thick-walled cylindrical shell made of functionally graded material under pressure loading. J Vib Control 2016; 22: 1023–1036.

Love

AEH

. On the small free vibrations and deformation of thin elastic shell. Phil Trans R Soc Lond 1888; 179: 491–546.

Bryan

. On the beats in the vibration of revolving cylinder. Proc Cambridge P hilos Soc 1890; 7: 101–111.

Sanders JLJR. An improved first order theory for thin shells. Rep. NASA-TR-R24, Washington, D.C, 1959.

Gol'denveizer

. Theory of thin shells, Oxford: Pergamon Press, 1961.

10.

Vlasov VZ. General theory of shells and its applications in engineering. NASA TT F – 99, 1964.

11.

Di Taranto

Lessen

. Coriolis acceleration effect on the vibration of rotating thin-walled circular cylinder. J Appl Mech 1964; 31: 700–701.

12.

Kraus

. Thin elastic shells, New York: John Wiley and Sons, 1967.

13.

Leissa AW. Vibration of shells. NASA SP-288, Reprinted by the Acoustical Society of America through the American Institute of Physics 1993, 1973.

14.

Penzes

RLE

Kraus

. Free vibrations of pre-stresses cylindrical shells having arbitrary homogeneous boundary conditions. AIAA J 1972; 10: 1309.

15.

Padovan

. Natural frequencies of rotating prestressed cylinder. J Sound Vib 1973; 32: 469–482.

16.

Padovan

. Travelling waves vibrations and buckling of rotating anisotropic shells of revolution by finite element. Int J Solid Struct 1975; 11: 1367–1380.

17.

Zohar

Aboudi

. The free vibrations of thin circular finite rotating cylinder. Int J Mech Sci 1973; 15: 269–278.

18.

Wang

Chen

. Effects of rotation on vibrations of circular cylindrical shells. J Acoust Soc Am 1974; 55: 1340–1342.

19.

Saito

Endo

. Vibrations of finite length rotating cylindrical shell. J Sound Vib 1986; 107: 17.

20.

Chun

Bert

. Critical speed analysis of laminated composite, hollow drive shafts. Compos Eng 1993; 3: 633.

21.

Chen

Zhao

Shin

. Vibration of high speed rotating shells with calculation for cylindrical shells. J Sound Vib 1993; 160: 137.

22.

Hussain

Naeem

Shahzad

et al.

Vibrational behavior of single-walled carbon nanotubes based on cylindrical shell model using wave propagation approach. AIP Adv 2017; 7: 045114.

23.

Goncalves

Batista

. Non-linear vibration analysis of fluid-filled cylindrical shells. J Sound Vib 1988; 127: 133–143.

24.

Kim

. Effect of partial elastic foundation on free vibration of fluid-filled functionally graded cylindrical shells. Acta Mech Sin 2015; 31: 920.

25.

Sofiyev

Kuruoglu

. On the vibration analysis of FGM truncated and complete conical shells under various boundary conditions and resting on elastic foundations. J Eng Math 2012; 77: 131–145.

26.

Sofiyev

Schnack

. The vibration analysis of FGM truncated conical shells resting on two-parameter elastic foundations. Mech Adv Mater Struct 2012; 19: 241–249.

27.

Sofiyev

Hui

Najafov

et al.

Influences of shear stresses and rotary inertia on the vibration of functionally graded coated sandwich cylindrical shells resting on the Pasternak elastic foundation. J Sandwich Struct Mater 2015; 17: 691–720.

28.

Tornabene

Fantuzzi

Bacciocchi

et al.

A numerical investigation on the natural frequencies of FGM sandwich shells with variable thickness by the local generalized differential quadrature method. Appl Sci 2017; 7: 131.

29.

Fantuzzi

Brischetto

Tornabene

et al.

2D and 3D shell models for the free vibration investigation of functionally graded cylindrical and spherical panels. Compos Struct 2016; 154: 573–590.

30.

Tornabene

Fantuzzi

Viola

et al.

Stress and strain recovery for functionally graded free-form and doubly-curved sandwich shells using higher-order equivalent single layer theory. Compos Struct 2015; 119: 67–89.

31.

Tornabene

Fantuzzi

Bacciocchi

. Free vibrations of free-form doubly-curved shells made of functionally graded materials using higher-order equivalent single layer theories. Compos Part B 2014; 67: 490–509.

32.

Tornabene

. Free vibration analysis of functionally graded conical, cylindrical shell and annular plate structures with a four-parameter power-law distribution. Comput Meth Appl Mech Eng 2009; 198: 2911–2935.

33.

Tornabene

Viola

. Free vibration analysis of functionally graded panels and shells of revolution. Meccanica 2009; 44: 255–281.

34.

Tornabene

Viola

. Free vibrations of four-parameter functionally graded parabolic panels and shells of revolution. Eur J Mech 2009; 28: 991–1013.

35.

Tornabene

Fantuzzi

Bacciocchi

et al.

A posteriori stress and strain recovery procedure for the static analysis of laminated shells resting on nonlinear elastic foundation. Compos Part B 2017; 126: 162–191.

36.

Tornabene

Fantuzzi

Viola

et al.

Winkler–Pasternak foundation effect on the static and dynamic analyses of laminated doubly-curved and degenerate shells and panels. Compos Part B 2014; 57: 269–296.

37.

Tornabene

Reddy

. FGM and laminated doubly-curved and degenerate shells resting on nonlinear elastic foundations: a GDQ solution for static analysis with a posteriori stress and strain recovery. J Ind Inst Sci 2013; 93: 635–688.

38.

Tornabene

. Free vibrations of anisotropic doubly-curved shells and panels of revolution with a free-form meridian resting on Winkler–Pasternak elastic foundations. Compos Struct 2011; 94: 186–206.

39.

Daneshjou

Talebitooti

. Free vibration and critical speed of moderately thick rotating laminated composite conical shell using generalized differential quadrature method. Appl Math Mech 2013; 34: 437–456.

40.

Civalek

. An efficient method for free vibration analysis of rotating truncated conical shells. Int J Pressure Vessels Piping 2006; 83: 1–12.

41.

Toulokian

. Thermo physical properties of high temperature solid materials, New York: Macmillan, 1967.

42.

Naeem

Sharma

. Prediction of natural frequencies for thin circular cylindrical shells. Proc IMechE, Part C: J Mechanical Engineering Science 2000; 214: 1313–1328.

43.

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

44.

Loy

Lam

Shu

. Analysis of cylindrical shells using generalized differential quadrature method. Shock Vib 1997; 4: 193–198.

45.

Loy

Lam

Reddy

. Vibration of functionally graded cylindrical shells. Int J Mech Sci 1999; 41: 309–324.