By Routh E.J.
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Extra info for A treatise on the dynamics of a system of rigid bodies
T times Adding the voltage of the original impulse and the first :PI round trips we have a total of The total voltage at the point in question at the time t is given by the limiting value which (17j approaches as n is increased indefinitely ' l l r . Carson has called my attention to the fact that this series can also I)e derived fro111'l'heoreni IS. 1). 40, of his Flectric C'ircuit 'l'heory. ntion woultl be analogous to that in equations (j')-(11). REGENERA TION THEORY We shaII nest prove that the limit s(t) exists for all finite values of t.
REGENERATION THEORY We shall next evaluate the integral for a very large value of t. It will suffice to take the C integral since the I integral approaches zero. Assume originally that 1 - w does not have a root on the imaginary axis and that F(z) has the special value w'(z). The integral may be written Changing variables it becomes where z is a function of w and D is the curve in the w plane which corresponds to the curve C in the z plane. Afore specifically the imaginary asis becomes the locus s = 0 and the semicircle becomes a small curve which spirals around the origin.
In the present discussion this difficulty will be avoided by the use of a strictly linear amplifier, which implies an amplifier of unlimited power carrying capacity. The attention will then be centered on whether an initial impulse dies out or results in a runaway condition. If a runaway condition takes place in such an amplifier, it folloivs that a non-linear amplifier having the same gain for small current and decreasing gain with increasing current will be unstable as well. Tech. , vol. I I.
A treatise on the dynamics of a system of rigid bodies by Routh E.J.