This paper is concerned with the vibration of disks particularly the form known as the ‘stationary wave’ which can develop in a rotating disk by the application of a stationary axial force at the periphery. The forms of vibration involving n nodal diameters are considered, and phenomena associated with linear and non-linear free and forced vibration are discussed in detail. Thereafter, attention is focused on the behaviour of real disks which, owing to inevitable imperfection, possess two independent modes of vibration, and thus two natural frequencies, for each value of n. It is shown that the stationary wave is greatly influenced by the degree of imperfection present. Experimental studies with stationary and rotating disks are presented covering forced vibration in the linear and nonlinear zones. The results show that a non-linear stationary wave may exist in a rotating disk over a wide speed range depending on the magnitude of the applied force. As speed increases this wave finally collapses and in doing so becomes a travelling wave which accelerates slowly in the direction of disk rotation as its amplitude subsides. Experiments conducted with large artificial imperfection show that the stationary wave may be transformed into two modes of vibration which are fixed in position in the disk. These modes resonate separately at their respective frequencies and apparently suffer a high degree of aerodynamic damping due to rotation. The magnitude of disk vibration is thus greatly reduced by the introduction of imperfection. Short mathematical appendices are included which contain the main theoretical results.

The flexural vibrations of the walls of thin cylinders are considered. In this type of vibration many forms of nodal pattern may exist owing to the combination of circumferential and axial nodes. Theoretical expressions are developed for the natural frequencies of cylinders with freely-supported and fixed ends and a comparison is made with the frequencies obtained experimentally. In practice, the ends of cylinders are subjected to a certain degree of fixing by end-plates, flanges, etc., and the natural frequencies thus lie between the corresponding values for freely-supported and fixed ends. To make possible the estimation of such frequencies, a method is devised in which an equivalent wavelength factor is used. This factor represents the wavelength of the freely-supported cylinder that would have the same frequency as the cylinder under consideration when vibrating in the same mode. The results of experimental investigations with various end thicknesses and flange dimensions are recorded, and from these the equivalent factors are derived. Sets of curves calculated for cylinders with freely-supported ends and covering a range of cylinder thicknesses are given. From these it is possible to obtain close approximation to the frequencies of cylinders under other end conditions by the use of an appropriate factor. An example is given of frequency calculations for a large air-receiver for which two frequencies were identified by experiment.

The paper deals with the general equations for the vibration of thin cylinders and a theoretical and experimental investigation is made of the type of vibration usually associated with bells. The cylinders are supported in such a manner that the ends remain circular without directional restraint being imposed. It is found that the complexity of the mode of vibration bears little relation to the natural frequency; for example, cylinders of very small thicknessdiameter ratio, with length about equal to or less than the diameter, may have many of their higher frequencies associated with the simpler modes of vibration. The frequency equation which is derived by the energy method is based on strain relations given by Timoshenko. In this approach, displacement equations are evolved which are comparable to those of Love and Flugge, though differences are evident due to the strain expressions used by each author. Results are given for cylinders of various lengths, each with the same thickness-diameter ratio, and also for a very thin cylinder in which the simpler modes of vibration occur in the higher frequency range. It is shown that there are three possible natural frequencies for a particular nodal pattern, two of these normally occurring beyond the aural range.

The report deals with some fundamental investigations of vibration in cutting tools. It is shown that this phenomenon may be the result of both self-induced and forced vibration. The former is a function of the cutting properties of the metal and the sharpness of the tool, while the latter depends on the interference of the tool with the surface cut during previous revolutions. Certain limitations of speed, tool frequency, and tool sharpness exist, vibration being rarely observed at low cutting speeds, at high tool frequencies, or with a freshly lapped tool. The amplitude of vibration appears to be independent of the depth of cut and seems to be limited to a value slightly in excess of that which makes the maximum vibrational velocity at the tool point equal to the speed of the work. It is governed to some extent, however, by the internal damping characteristics of the tool shank. The self-inductive influence originates from the decrease in the cutting force which occurs as speed is increased, and becomes more powerful as the wear of the tool progresses. Failure of carbide tools under vibratory cutting conditions occurs in the form of a fatigue crack which spreads inwards parallel to the top face of the tool. Photographs and profile records are given of surfaces cut under various conditions of vibration, together with a theoretical analysis of the manner in which the surfaces are formed. With small feeds, forced vibration occurs and phasing normally results. This is demonstrated by the resulting patterns on the surface, which show an orderly arrangement of the waves. A few practical examples are described.

The unusual noise emanating from what is known as a “singing” propeller is the direct result of the vibration of the blades. These are disturbed by the irregular hydrodynamic actions of cavitation and vortex shedding and induced by the extreme variations in the wake. It is demonstrated experimentally that a blade with a sweeping plan form and sharp edge, suppresses cavitation and vortex action within the working range of the propellers. Sharpening the blade edges round the periphery, in a wedge-shaped manner, has been applied successfully as a cure to many singing propellers. In addition to suppressing the disturbing forces, experiments show that the sharp edge provides considerable increase in hydrodynamic damping. The response of the blade to tip and edge excitation is considered. Blade flutter is examined, but it is concluded that such a condition is unlikely. Notes emitted by singing propellers suggest that many modes of flexural, torsional, and lateral vibration are present. It is shown that the lateral types respond easily and their importance is indicated by a number of noisy propellers which have exhibited fatigue cracks conforming to the lateral modes of vibration. It is recommended that the process of sharpening the blade edges be adopted, as this suppresses the exciting forces and damps the response of the blade.