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Unformatted text preview: L'H6pital's rule to this t erm yields
lim y (t) = A te At €~O (2.75) Clearly, t he response does n ot go to infinity as E ---> 0, b ut i t acquires a factor t,
which approaches 0 0 a s t ---> 0 0. I f A has a negative real p art (so t hat i t lies in LHP),
e At decays faster t han t a nd y (t) ---> 0 a s t ---> 0 0_ T he resonance phenomenon in
this c ase is present, b ut i ts manifestation is a borted by t he signal's own exponential
This discussion shows t hat resonance is a cumulative phenomenon, n ot instantaneous. I t builds up Iinearly:j: with t. W hen t he mode decays exponentially, t he
signal decays a t a r ate t oo fast for resonance t o c ounteract t he decay; a s a result,
the signal vanishes before resonance has a chance to build it up. However, if the
mode were to decay a t a r ate less t han l it, we should see t he resonance phenomenon
clearly. This specific condition would be possible if Re A :::: O. For instance, when
Re A = 0, so t hat A lies on t he i maginary axis of t he complex plane, a nd therefore t For convenience we o mit m ultiplying f (t) a nd h (t) by u(t)_ T hroughout t his discussion, we
a ssume t hat t hey a re causa\.
+ I f the characteristic root in question repeats r times, resonance effect increases as t r - 1 . However)
t '-le At ---> 0 as t ---> 0 0 for any value of T , p rovided Re A < 0 (A in L HP). 160 2 Time-Domain Analysis of Continuous-Time Systems >. = jw
a nd Eq. (2.75) becomes y(t) = A te jwt (2.76) Here, t he r esponse does go t o infinity linearly with t.
For a real system, if >. = j w is a r oot, >. * = - jw must also be a root; t he
impulse response is of t he form A e jwt + A e- jwt = 2A cos wt. T he response of
this system to i nput A cos wt is 2A cos wt * cos wt. T he reader can show t hat t his
convolution contains a term of t he form A t cos wt. T he resonance phenomenon is
clearly visible. T he s ystem response to its characteristic mode increases linearly
with time, eventually reaching 0 0, as indicated in Fig. 2.21.
y ( I) t- 161 2.8 Determining t he Impulse Response a mounts t o applying a periodic force t o t he bridge. I f t he frequency of this input
force happens t o be nearer to a characteristic root of t he bridge, t he bridge may
respond (vibrate) violently a nd collapse, even though it would have been strong
enough t o c arry many soldiers marching o ut of step. A case in point is t he T acoma
Narrow Bridge failure of 1940. T his bridge was opened t o traffic in July 1940.
W ithin four months of opening (on 7 November 1940), i t collapsed in a mild gale,
n ot because of t he w ind's b rute force b ut because t he frequency of wind-generated
vortices, which matched t he n atural frequency (characteristic roots) of t he bridge,
Because of t he g reat damage which may occur, mechanical resonance is generally something to be avoided, especially in structures or vibrating mechanisms. I f
an engine with periodic force (such as piston motion) is m ounted on a platform, t he
p latform with its mass a nd springs sho...
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