Abstract
The extinction of second order subharmonic vibration response of a single degree of freedom system is investigated for the case of damped centrifugal excitation. The degree of the restoring force asymmetry resulting from the effect of gravity is expressed in terms of the parameter of static deflection. The resonance under gravity effects is analysed theoretically for a wide range of physical conditions to determine the behavioural characteristics of the subharmonic components. The inherently coupled algebraic equations are obtained by the approximate energy method of Ritz-Galerkin and by the method of harmonic balance. These two methods are not bounded to any degree of non-linearity. As there is no exact solution for this investigation and because of the dissipative forces inevitably introducing the problem of stability, the actual existence of the approximate solution over the frequency band-width is ascertained. There are no real roots in the instability region. The algebraic polynomial expressions cannot be satisfied simultaneously because of the accumulative effect in an accompanying harmonic of the vibratory motion. The build-up oscillation occurs in the second order region, having a frequency the same as that of the main component of subharmonic motion. The stability criterion is derived from comparing the characteristic exponent of solution to the variational equation with damping coefficient of the system. The response characteristics are then investigated where the polynomial equations are simplified through justifying the approximation of the fundamental harmonic as the effective amplitude of the disturbing force. The results are of comparable accuracy for cases in which gravity effects do not increase the effective non-linearity with resonance. The approximation, however, is applicable whatever tha physical characteristic behaviour of the non-linearity in the region of the critical state of subharmonic extinction. The subharmonic motion in the process of analysis is shown to exist in two opposite phases, differing by pi radians. The resulting phase of periodic vibration depends upon initial conditions. The isocline graphical method is used to depict the transient motion. The effective non-linearity is determined to be governed by the influence of gravity effects on the equilibrium of moition. The pronounced subharmonics are of the order one half, and the extinction conditions for the resonance predominant over the higher orders ere investigated through expressing the limiting inequalities in terms of the system parameters. In the critical state, complete suppression of the subharmonics is achieved. The limiting condition is then examined where damping is fixed at a convenient minimum value and the corresponding optimum limit of gravity effect tolerable can be evaluated for which the amplitude of excitation has no influence on the effective non-linearity as regard to exciting the resonance. As inequality is also presented from which the limiting frequencies of the subharmonic vibration can be predicted with reasonable accuracy. In these investigations the limiting inequalities are not dependant on the resulting variables in the non linear phenomena. An experimental test-rig is designed to demonstrate the subharmonic response. The values recorded from it compare favourably with the approximated theoretical results and with the experimental results obtained from the electronic analogue computer.