# Nonlinear System Analysis by Austin Blaquiere (Auth.)

By Austin Blaquiere (Auth.)

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PENDULUM WITH FORCING FUNCTION. 6. , of oscillations whose frequencies are related to t h a t of t h e driving function by w h e r e ω is t h e frequency of t h e driving function, ω t h e frequency of a s u b h a r m o n i c oscillation, a n d η an integer: 2, 3 , 4, ... T h i s p h e n o m e n o n , which is also called frequency demultiplication, was discovered by H e l m h o l t z a n d described in his t h e o r y of physiological acoustics [2]. H e l m h o l t z pointed out t h a t s o m e t i m e s t h e ear is influenced by s o u n d s whose frequencies are not contained in t h e i n c o m i n g acoustic wave b u t are related to its frequency b y relation (94), a n d he explained this p h e n o m e n o n b y nonlinearity of t h e t y m p a n i c m e m b r a n e .

Feedback Loop of a Self-Oscillator T o simplify t h e analysis of self-oscillatory circuits, it is convenient, from a theoretical point of view, t o separate active a n d passive elements (G) χ y (F) F I G . 6. Feedback loop. into t w o c o m p o n e n t s (Fig. 6): (a) A n active network (G), w h i c h will be characterized b y its gain without feedback Γ. (b) A passive network (F), whose feedback coefficient will b e called Φ. T h e p r o d u c t ΓΦ is t h e loop transmission. It will play a very i m p o r t a n t role in t h e t h e o r y of t h e generalized N y q u i s t diagram.

Accordingly, to each given value of ω is associated either one or t w o stable s o l u t i o n s — t h e t h i r d solution, w h e n it exists inside t h e s h a d e d d o m a i n , being an instable one. FIG. 23. Linear resonance; d a m p e d p e n d u l u m . 3. The Jump Phenomenon N o w a s s u m e that, for a specified F, ω is given continuously increasing values, from ω = 0 t o infinity, a n d let ω a n d ω" b e angular frequencies which c o r r e s p o n d to points Β a n d D w h e r e t h e t a n g e n t to t h e resonance curve has infinite slope.