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A mathematical theory of nonlinear chatter is developed. In this, the structure is represented by an equivalent single degree of freedom system with nonlinear stiffness characteristics and the cutting force by a third degree polynomial of the chip thickness. This model leads to a second order differential equation with nonlinear stiffness and nonlinear time delay terms from which the conditions of steady state chatter are derived. These are then discussed by applying them to an equivalent system derived from experimental data pertaining to a face milling process. The theory provides an explanation for the stages in which chatter develops and also for the “finite amplitude instability” phenomenon.

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.

Previous investigations of the vibrations of circular disks confined themselves to small amplitudes when it can be assumed that the strain energy of the disk is due to pure bending only. At large amplitudes this assumption ceases to be valid and the stretching of the middle surface has to be taken into consideration. As a result, the equations of motion become non-linear. The investigation is concerned with the vibration of ‘real’ disks, that is, disks which contain small imperfections. It is shown that, in general, the imperfections eliminate the indeterminancy of the angular position of the radial nodal lines which is characteristic of perfect disks. For those modes which contain at least one radial nodal diameter, there are two fixed nodal configurations which possess slightly different natural frequencies. At small amplitudes these nodal configurations are normal modes of vibration which vibrate independently of each other. At large amplitudes, when the equations of motion become non-linear, these configurations become coupled. The coupling is unsymmetrical in the sense that the configuration with the higher natural frequency can vibrate on its own at all amplitudes. On the other hand, above a certain amplitude, the configuration with the smaller natural frequency cannot vibrate on its own since it also excites, by parametric excitation, the other configuration. Under such conditions the phase difference between the two configurations is fixed to 90 deg. and as a result the motion of the disk corresponds to a beating travelling wave.

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The forced vibrations of modes consisting of at least one nodal diameter and an arbitrary number of nodal circles are investigated. The case when the position of the exciting force coincides with a node of one of the preferential configurations is examined. It is shown that a stationary harmonic force may excite a stationary vibration with a fixed nodal pattern, or a ‘beating’ travelling wave, depending on the conditions of excitation. The stability of vibration of modes consisting of fixed nodal patterns is investigated.