Analytical solution for free vibrations of rotating cylindrical shells having free boundary conditions

Abstract

a b s t r a c tIn this paper free vibrations of rotating cylindrical shells with both ends free are studied. The model used also allows for considering a flexible foundation supporting the shell in the sense of a radial and circumferential distributed stiffness. Furthermore, a circumferential tension (hoop stress) which may be due to pressurisation or centrifugal forces is taken into account. Natural frequencies and mode shapes are determined exactly for both stationary shells and for shells rotating with a constant angular speed around the cylinder axis. Trigonometric functions are assumed for the circumferential mode shape profiles, and a sum of eight weighted exponential functions is assumed for the axial mode shape profiles. The functional form of the axial profiles is shown to greatly vary with the roots of a characteristic bi-quartic polynomial that occurs in the process of satisfying the equations of motion. In the previously published work it has been very often assumed that the roots are two real, two imaginary, and two pairs of complex conjugates. In the present study, a total of eight types of roots are shown to determine the whole set of mode shapes, either for stationary or for rotating shells. The results using the developed analytical model are compared with results of experimental studies and very good agreement is obtained. Also, a parametric study is carried out where effects of the elastic foundation stiffnesses and the rotation speed are examined.

A finite strip for the vibration analysis of rotating cylindrical shells

“…The application of the analytical solution was illustrated in the case of a cylindrical shell with free-free boundary conditions, and excellent agreement with the experimentally obtained results was confirmed. The principal advantage of the analytical procedure [39] is very high accuracy confirmed by experiments.…”

“…(18), (19) and (20) has been solved in a general form in [39], assuming the displacement functions in the exponential form (21) where α is a complex parameter. Substituting (21) into Eqs.…”

“…Closed-form solutions are now difficult to obtain. A number of investigations have been undertaken to tackle this problem [33][34][35][36][37][38][39]. One of the solutions was obtained by assuming the shell displacement field as a product of Fourier series in the axial direction, and trigonometric functions in the circumferential direction [36].…”

“…A complete analytical solution for free vibrations of a rotating cylindrical shell with arbitrary boundary conditions has recently been offered in [39]. The equations of motion in [39] are based on the strain-displacement relationships of Hermann and Armenakas [23].…”

“…The equations of motion in [39] are based on the strain-displacement relationships of Hermann and Armenakas [23]. Circumferential tension due to internal pressure or centrifugal forces, as well as an elastic foundation in both radial and circumferential directions, is taken into account.…”