In-plane vibrations of annular rings

Abstract

The in-plane vibrations of thin disks and narrow thin rings which already have been completely analysed are the two extremes of the annulus configuration. The general case of the annulus has now been analysed and the results confirmed by experiment. The modes of disks are characterized by two integers giving the number of nodal circles —including a possible central nodal point—and the number of nodal diameters. In the absence of nodal diameters two series of pure radial and pure tangential modes are possible. These have the characteristics of plate and shear waves respectively. Compound modes which have both radial and tangential components of displacement have been found to fall into four modal series. There are six series of narrow ring counterparts of these disk resonances, every disk mode moving to a particular ring mode as the hole is developed in the centre. Flexural modes approach zero frequency, the extensional series have finite frequencies and there is one series of compound shear modes and one of compound plate modes both of which approach infinite frequency as do the two pure modes. The eigenvalues, in the form of a dimensionless frequency constant, have been evaluated for steps of hole size from the disk case to that of an indefinitely narrow ring and for a wide range of values of Poisson's ratio. The theoretical results have been supported by measurements of the spectra of a range of annuli and by exploring the pattern of vibrations with a probe pick-up. A feature of the change from disk to thin ring is the disappearance of one or more nodal circles in certain cases. Typically all modes with one nodal circle and two or more nodal diameters lose the nodal circle to become the finite frequency extensional ring series. Compound modes having 2, 4, … circles as disks have 1, 2, … circles as shear ring modes and the 3, 5, … series become 1, 2, … plate ring modes. This creates the apparent paradox that the inner free boundary of an annulus can be a node. In fact the force at the edge arising from a radial strain is balanced by the hoop stress arising from a tangential strain. The parallel between the modes of resonance of thin rings and infinite tubes is drawn.