The generalized Vincent circle in vibration suppression

In 1972 A. H. Vincent, the then Chief Dynamicist at Westland Helicopters, discovered that when a structure excited at point p with a constant frequency is modified, for example by the addition of a spring between two points r and s, then the response at another point q   traces a circle when plotted in the complex plane as the spring stiffness is varied from minus infinity to plus infinity. This discovery, although apparently little known today, has many useful applications some of which are described in papers by various authors appearing in the 1970s and early 1980s. Vincent's discovery is in fact a particular example of the bilinear transformation due to August Ferdinand Moebius (1790–1868). In this paper, the Vincent circle method is generalized for the case of any straight-line modification in the complex plane, typically z=k+iωc–ω2mz=k+iωc–ω2m, where c=α(k-ω2m)+βc=α(k-ω2m)+β. A new method for the visualization of Vincent circle results, including the case of multiple modifications is also presented.