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Unformatted text preview: uld be designed so t hat t heir characteristic
roots are n ot close t o t he engine's frequency of vibration. P roper design of this
platform can not only avoid resonance, b ut also a ttenuate v ibrations if t he system
roots are placed far away from t he frequency of vibration. 2 .8 Appendix 2.1: Determining t he Impulse Response We now derive t he u nit impulse response of a n LTIC s ystem S specified by t he
n th-order differential equation
Fig. 2 .21 Build up of system response in resonance. Q (D)y(t) = P (D)f(t) or
Recall t hat w hen>. = j w, t he system is marginally stable. As we have indicated, t he full effect of resonance cannot be seen for an asymptotically stable
system; only in a marginally stable system does t he resonance phenomenon boost
t he s ystem's response t o infinity when t he s ystem's input is a c haracteristic mode.
B ut even in a n a symptotically stable system, we see a manifestation of resonance if
its characteristic roots are close t o t he i maginary axis, so t hat Re >. is very small.
We c an show t hat when t he c haracteristic roots of a system are I j ± j wo, t hen, t he
system response t o t he i nput ejwot or the sinusoid cos wot is very large for small
I j. t T he r esponse drops off rapidly as t he i nput signal frequency moves away from
woo T his frequency-selective behavior can be studied more profitably after a n understanding of frequency-domain analysis is acquired. For this reason we p ostpone
full discussion of t his s ubject until C hapter 7.
Importance o f t he Resonance Phenomenon T he r esonance phenomenon is very i mportant because it allows us t o design
frequency-selective systems by choosing their characteristic roots properly. Lowpass, bandpass, highpass, a nd b andstop filters are all examples of frequency selective networks. I n mechanical systems, t he i nadvertent presence of resonance can
cause signals of s uch t remendous magnitudes t hat t he system may fall a part. A
musical note (periodic vibrations) of proper frequency can s hatter a glass if t he
frequency is m atched t o t he c haracteristic root of the glass, which acts as a mechanical system. Similarly, a company of soldiers marching in step across a bridge
t This follows d irectly from Eq. (2.74) w ith .\ = (j + j wo a nd < = (j (Dn + a n_ID n- 1 + ... + a ID + ao)y(t)
= (bnDn + bn_ID n - 1 + ... + bID + b o)f(t) (2.77a) (2.77b) In Sec. 2.3 we showed t hat t he impulse response h (t) is given by h (t) = A 0 8(t) + characteristic modes (2.78) We now show t hat in t he above equation Ao = bn where bn is t he coefficient of t he
n th-order t erm on t he r ight-hand side of Eq. (2.77b). W hen t he i nput f (t) = 8(t),
t he response y (t) = h (t). Therefore, from Eq. (2.77b), we o btain
( Dfi + a n_I Dn - 1 + ... + a ID + ao)h(t) = (bnDn + bn_ID n- 1 + ... + bID + bo)8(t) In this equation we s ubstitute h (t) from Eq. (2.78) a nd compare t he coefficients of
similar impulsive terms on b oth sides. T he highest order of t he derivative of impulse
on b oth sides is n , w ith its coefficient value as Ao o n the left-hand side a nd bn on
the right-hand side. T he two values must be...
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