Sage Journals: Discover world-class research

Abstract

Governing equations for free vibrations of thin orthotropic or isotropic helicoidal plates are formulated by expressing displacements in terms of components in a helical orthogonal coordinate system. Equations derived for isotropic helicoidal plates are applied to the vibration problem of twisted rectangular turbine blades without camber. These twisted cantilever rectangular blades are simulated to be isotropic helicoidal plates with their angles of twist approximated to be their centre-line helical angles. Natural frequencies are calculated by solving the eigenvalue problem using the finite difference method and are compared to experimental measurements reported in the literature. Reasonably good agreement has been found for flexural modes; however, some discrepancies have also been observed for higher-order modes due to the inaccuracy of modelling the blade geometry as that of a helicoidal plate and due to the approximation of assuming the angle of twist to be the helical angle.

Keywords

vibration plates shells helicoidal

Get full access to this article

View all access options for this article.

References

Reissner

Washizu

On torsional vibrations of a beam with a small amount of pretwist. J. Japan Soc. Aeronaut. Engrs, 1957, 5, 330–355.

Knowles

J. K.

Reissner

Note on stress strain relations for thin elastic shells. J. Math. Physics, 1958, 37, 269–282.

Knowles

J. K.

Reissner

Torsion and extension of helicoidal shells. Q. Appl. Math., 1959, 17, 409–422.

Reissner

On twisting and stretching of helicoidal shells. In The theory of thin elastic shells (Proceedings of IUTAM Symposium, Delft, 1959) ( Koiter

W. T.

), 1960, pp. 434–466 (North-Holland, Amsterdam).

O'Mathuna

D. P.

Rotationally symmetric deformation in helicoidal shells. J. Math. Physics, 1963, 42, 85–111.

O'Mathuna

D. P.

Reissner

A problem of shearing and transverse bending of shallow hyperbolic, paraboloidal shells. J. Appl. Mechanics, 1963, 30, 295–296.

Yaroshenko

A. R.

Resolving system of equations and boundary conditions for a shell in the form of an oblique helicoid. Soviet Appl. Mechanics, 1969, 5, 385–391.

Simmonds

J. G.

General helicoidal shells undergoing large, one-dimensional strains or large inextensional deformations. Int. J. Solids and Structs, 1984, 20, 13–30.

Leissa

A. W.

Vibration of plates. NASA SP-160, NASA, Washington, D.C., 1969, p. 245.

10.

Chia

C.Y.

Nonlinear analysis of plates, 1980 (McGraw-Hill, New York).

11.

Montoya

Coupled bending and torsional vibrations in a twisted, rotating blade. Brown Bovei Rev., 1966, 53, 216–230.

12.

MacBain

J. C.

Vibratory behavior of twisted cantilevered plates. J. Aircraft, 1975, 12, 343–349.

13.

Love

A. E. H.

A treatise on the mathematical theory of elasticity, 1944 (Dover, New York).

14.

Sokolnikoff

I. S.

Redheffer

R. M.

Mathematics of physics and modern engineering, 1966, pp. 311–315 (McGraw-Hill, New York).

15.

Olson

M. D.

Lindberg

G. M.

Dynamic analysis of shallow shells with a doubly-curved triangular finite element. J. Sound Vibr., 1971, 19, 299–318.

16.

Billington

D. P.

Thin shell concrete structures, 1965, Ch. 1 (McGraw-Hill, New York).

17.

Knowles

J. K.

Reissner

A derivation of the equations of shell theory for general orthogonal coordinates. J. Math. Physics, 1956, 35, 351–358.

18.

Kirchhoff

Vorlesungen uber Mathematisch Physik, Mechanik, 1883 (B. G. Teubner, Leipzig).

19.

Timoshenko

Woinowsky-Krieger

Theory of plates and shells, 1959, p. 84 (McGraw-Hill, New York).

20.

Gerald

C. F.

Applied numerical analysis, 1970, pp. 2–4 (Addison-Wesley, Reading, Massachusetts).

21.

Barton

M. V.

Vibration of rectangular and skew cantilever plates. J. Appl. Mechanics, 1951, 18, 129–133.

22.

Vogel

S. M.

Skinner

D. W.

Natural frequencies of transversely vibrating uniform annular plates. J. Appl. Mechanics, 1965, 32, 926–931.

Vibration Analysis of Helicoidal Plates

Abstract

Keywords

Get full access to this article

References