VI. The natural frequencies of systems having restricted freedom

1. Relaxation methods for determining characteristic numbers ofcontinuous systems were described in Part IV of this series (Bradfield, Christopherson and Southwell 1939). The modes were assumed to be expressible (with sufficient accuracy) by a finite series of functions chosen to satisfy the boundary conditions, and thus in effect the system was given restricted freedom. In this way continuous systems can be brought within the scope of tabular computation. When the freedom is thus restricted only in assumption, the fact that in reality the system is continuous means that suitably chosen series of continuous functions will satisfactorily represent the modes. But systems of restricted freedom are presented (e.g. in mechanical engineering) to which this treatment is not applicable. Thus it may be desired to calculate the free periods of torsional vibration for a system consisting of heavy masses connected by (relatively) fight shafting: then usually it will not happen that consecutive masses have any systematic relation to one another, and accordingly the problem must be treated, like braced frameworks, by consideration of “joint displacements”