Abstract
The undamped vibration response of a single degree of freedom system with a hardening non-linear spring is investigated for the case of a centrifugal disturbing force. The restoring force characteristic is asymmetrical. The degree of asymmetry is expressed in terms of the parameter of static deflection, which is produced by the weight of the vibrating mass. The approximate method of Ritz-Galerkin is used for the theoretical analysis. On the basis of published literature this method appears to be generally superior to other analytic methods, and this is supported in this investigation by a direct comparison of theoretical results for the case of free vibration. Experimental results are obtained by means of an electronic analogue computer. Free vibration, harmonic resonance, superharmonic resonance of order 2, and subharmonic resonance of order 1/2 are investigated for several magnitudes of the static deflection and of the disturbing force amplitude. It is found that in general the static deflection, or the force of gravity, has a considerable effect on the vibration response and hence cannot be neglected in a theoretical analysis. The superharmonic resonance is practically negligible even at small magnitudes of damping. In contrast the subharmonic resonance is very pronounced, and from the practical point of view it is at least as important as the harmonic resonance. Theoretical results obtained by means of the Ritz-Galerkin method compare favourably with experimental results. The error of approximation increases slightly with the magnitude of non- linearity, but in the range of the investigation it remains within acceptable limits.