Abstract
Exact solutions are obtained for transient torsional responses of a finitely long, functionally graded hollow cylinder under three different end conditions, i.e. free–free, free–fixed and fixed–fixed. The cylinder with its external surface fixed is subjected to a dynamic shearing stress at the internal surface. The material properties are assumed to vary in the radial direction in a power law form, while keep invariant in the axial direction. With expansion in the axial direction in terms of trigonometric series, the governing equations for the unknown functions about the radial coordinate r and time t are deduced. By applying the variable substitution technique, the superposition method and the separation of variables consecutively, series-form solutions of the equations are obtained. Natural frequencies and the transient torsional responses are finally discussed for a functionally graded finite hollow cylinder.
Similar content being viewed by others
References
Bhanja N.: Torsional vibration of a solid finite cylinder having transverse isotropy. Pure Appl. Geophy. 85(1), 182–188 (1971)
Xie S., Liu K.: Transient torsional wave propagation in a transversely isotropic tube. Arch. Appl. Mech. 68(9), 589–596 (1998)
Wang X., Xia X.H., Hao W.H.: An elastodynamic solution of finite long orthotropic hollow cylinder under torsion impact. J. Sound Vib. 267(1), 67–86 (2003)
Kudlička J.: Dispersion of torsional waves in a thick-walled transversely isotropic circular cylinder of infinite length. J. Sound Vib. 294(1–2), 368–373 (2006)
Horgan C.O., Chan A.M.: The pressurized hollow cylinder or disk problem for functionally graded isotropic linearly elastic materials. J. Elast. 55(1), 43–59 (1999)
Jabbari M., Sohrabpour S., Eslami M.R.: Mechanical and thermal stresses in a functionally graded hollow cylinder due to radially symmetric loads. Int. J. Press. Ves. Pip. 79(7), 493–497 (2002)
Ding H.J., Wang H.M., Chen W.Q.: A solution of a non-homogeneous orthotropic cylindrical shell for axisymmetric plane strain dynamic thermoelastic problems. J. Sound Vib. 263(4), 815–829 (2003)
Shakeri M., Akhlaghi M., Hoseini S.M.: Vibration and radial wave propagation velocity in functionally graded thick hollow cylinder. Compos. Struct. 76(1–2), 174–181 (2006)
Ootao Y., Tanigawa Y.: Transient thermoelastic analysis for a functionally graded hollow cylinder. J. Thermal Stress. 29(11), 1031–1046 (2006)
Dai H.L., Fu Y.M., Yang J.H.: Electromegnetoelastic behaviors of functionally graded piezoelectric solid cylinder and sphere. Acta Mech. Sinica 23(1), 55–63 (2007)
Mitra A.K., Mukherji P.: On the torsional vibration of a finite circular cylinder of non-homogeneous material by a particular type of twist on one of its plane ends. Pure Appl. Geophy. 95(3), 75–79 (1972)
Batra R.C.: Torsion of a functionally graded cylinder. AIAA J. 44(6), 1363–1365 (2006)
Singh B.M., Rokne J., Dhaliwal R.S.: Torsional vibration of functionally graded finite cylinders. Mechanica 41(4), 459–470 (2006)
Zhang W.B.: Equations of Mathematical Physics (in Chinese). Tsinghua University Press, Beijing (2007)
Author information
Authors and Affiliations
Corresponding author
Additional information
This project was supported by the National Natural Science Foundation of China (10872179 and 10725210), the Zhejiang Provincial Natural Science Foundation of China (Y7080298) and Zijin Plan of Zhejiang University.
Rights and permissions
About this article
Cite this article
Wang, H.M., Liu, C.B. & Ding, H.J. Exact solution and transient behavior for torsional vibration of functionally graded finite hollow cylinders. Acta Mech Sin 25, 555–563 (2009). https://doi.org/10.1007/s10409-009-0251-9
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10409-009-0251-9