Parametric aspects of mode and stability diagrams for general periodic systems

Stability and natural modes of general linear periodically time-varying systems are investigated using the classical procedure of Hill infinite determinant analysis. It is shown that for a normally stable passive system, instability can be invoked most easily by varying any one of the system parameters at a frequency of
2 \beta/k, k
integral, where
\beta
is a natural frequency of the system. This is a generalization of a result well known for systems of second order. In addition, a conjecture propounded by Keenan [5] in 1966 regarding natural modes, is extended to its most general form. Embodied in the analysis is a new choice of parameter for the vertical axis of made and stability diagrams. Whilst the horizontal axis is retained as the amplitude of the time-varying system parameter the vertical axis is redefined in terms of the period of the variation. Examples for an analytically tractable system are presented and exhibit features predicted by the theory.