Axial Half Wave Research Articles

On the stability of thin pipes with an internal flow

In this paper the stability behaviour of a thin, clamped ended pipe with a high velocity internal flow is considered. The dynamic fluid loading is developed by using potential theory for an incompressible, inviscid fluid and the motion of the pipe is represented by the Flugge-Kempner shell equation. The solution is obtained by using Fourier integral theory and the method of Galerkin. For the limiting case of a relatively long thin pipe a particularly simple expression is found for the critical velocity. When both ends of a pipe are fixed, static divergence always precedes the onset of flutter and, hence, a relatively simple approach is sufficient for predicting the critical velocity. Furthermore, the mode shape at divergence instability always corresponds to that of the lowest natural frequency of the pipe, being comprised of one axial half-wave and a number of circumferential waves depending on the length and thickness ratios. The theoretical results are shown to compare well with experiments and previous work developed by different methods.

On the Parametric Resonance of a Cylindrical Shell Subjected to a Periodic Axial Load

The dynamic stability of thin-walled circular cylindrical shell is studied for the case of periodic axial load. In computing the internal membrane stress induced by the axial pressures, the cylinder is assumed to act as a longitudinal bar. The internal membrane stress is computed by considering the shell to be a longitudinal rod with axial inertia terms included. Hence, the internal axial coordinate as well as with time and to include the resonances of the cylinder acting as a longitudinal rod. The problem under study is to determine the stability of the flexural motions of the shell oscillating about this inextensional mode. The shell motion is represented by Donnell's equations. A study of the solutions of these equations reveals parametric resonance of the well-known type and a second parametric resonance which appears to be new. The latter includes the combination resonance between two modes having the same modal pattern, and also between two modes having different modal patterns. In particular, this includes combination resonance between two transverse modes having a different number of axial half waves. This result is believed to be new and is considered the principal result of the study being described. It is only obtained when axial inertia is included in computing the axial membrane stress. It is found that combination resonance does not exist between modes having a different number of circumferential waves. The two modes must have the same number of axial waves. An estimate is given for the width of the unstable regions, and numerical results are presented. The extension to a shell having the effects of rotatory inertia and shear deformation is briefly discussed.

An investigation of slot radiators in rectangular metal plates

The radiation from slots cut in conducting surfaces of limited extent is discussed. Equatorial plane patterns of an axial half-wave slot in a rectangular metal plate are measured in the X band. The experimental results compare favourably with the calculated patterns on the assumption that the plate can be represented by a thin elliptic cylinder or ribbon of infinite length. It is observed that, if the length of the plate is equal to or greater than its width, the pattern is within a few per cent of the corresponding theoretical pattern for a plate of infinite length.The admittance of the slot in the plate was also measured and compared with the computed conductance. The agreement is seen to be quite good.