A study is made of the linear theory of free vibrations of membranes in which, owing to the self-weight of the membrane, some sag is present in the static profile. The theory applies only to those membranes with relatively shallow profiles. However, because a major application of the theory relates to the use of certain types of cable networks to support the roofs of buildings of large span, and because such networks must be relatively flat if structural efficiency is to be achieved, the theory is of some practical importance. A detailed examination is made of the circular membrane and the rectangular membrane. It is found that the symmetric modes of vibration are heavily dependent on the value of a characteristic geometric and elastic parameter - a parameter which can vary by several orders of magnitude in the suspended membranes typical of those under consideration. In particular, when the parameter is very large the membrane may be considered inextensible. In a practical sense, this corresponds to a membrane of shallow, although appreciable, curvature. For certain intermediate values of the parameter, situations arise in which the natural frequency of a symmetric mode is identical to that of an antisymmetric mode. And when it is very small, the symmetric modes of the classical circular and rectangular membranes are recovered, although, in the case of the classical square membrane, the theory points to conclusions which could not have been drawn from classical membrane theory alone.
The Static Profile for a Floating Particle