FREQUENCY RESPONSE OF A SECOND-ORDER SYSTEM

The transfer function of a second-order system is given in (7.3,9). The frequency-re­sponse vector is therefore

Me’*

From the modulus and argument of (7.5,11), we find that

1

{[1 – (o>/&>„)2]2 + 4£2(cu/wn)2}l/2

w (rad/s) (a)

Figure 7.18 Frequency-response functions, elevator angle input. Jet transport cruising at high altitude, (a) Load factor amplitude. (b) Load factor phase.

A representative vector plot of Me“p, for damping ratio £ = 0.4, is shown in Fig. 7.11, and families of M and cp are shown in Figs. 7.12 and 7.13. Whereas a single pair of curves serves to define the frequency response of all first-order systems (Fig. 7.10), it takes two families of curves, with the damping ratio as parameter, to display the char­acteristics of all second-order systems. The importance of the damping as a parame­ter should be noted. It is especially powerful in controlling the magnitude of the reso­nance peak which occurs near unity frequency ratio. At this frequency the phase lag is by contrast independent of £, as all the curves pass through <p = —90° there. For all
values of £, Л/ —> 1 and <p —» 0 as ш/шп 0. This shows that, whenever a system is driven by an oscillatory input whose frequency is low compared to the undamped natural frequency, the response will be quasistatic. That is, at each instant, the output will be the same as though the instantaneous value of the input were applied stati­cally.

The behavior of the output when £ is near 0.7 is interesting. For this value of £, it is seen that (p is very nearly linear with шІшп up to 1.0. Now the phase lag can be in­terpreted as a time lag, т = (<р/2тг)Т = <p/a> where T is the period. The output wave form will have its peaks retarded by т sec relative to the input. For the value of £ un­der consideration, ср/(шІшп) Ф it/2 or <p/w = ттІ2шп = Tn, where Tn = 2тг/ш, п the un­damped natural period. Hence we find that, for £ = 0.7, there is a nearly constant time lag t = Tn, independent of the input frequency, for frequencies below reso­nance.

The “chain” concept of higher-order systems is especially helpful in relation to frequency response. It is evident that the phase changes through the individual ele­ments are simply additive, so that higher-order systems tend to be characterized by greater phase lags than low-order ones. Also the individual amplitude ratios of the el­ements are multiplied to form the overall ratio. More explicitly, let

G(s) = G,(i) • G2(s) G„(s)

be the overall transfer function of n elements. Then

G(ioS) = G,(7ft>) • G2(ia>) Gn(i(o)

= (A, M, • K2M2 ■■■

= KMei4>

n

so that KM — PI KrMr

r= 1

n

r= 1

On logarithmic plots (Bode diagrams) we note that

n

log KM = ^ log KrMr (7.5,14)

r= 1

Thus the log of the overall gain is obtained as a sum of the logs of the component gains, and this fact, together with the companion result for phase angle (7.5,13) greatly facilitates graphical methods of analysis and system design.

RELATION BETWEEN IMPULSE RESPONSE AND FREQUENCY RESPONSE

We saw earlier (7.3,4), that h{t) is the inverse Fourier transform of G(iw), which we can now identify as the frequency response vector. The reciprocal Fourier transform relation then gives

G(iw)= h(t)e~iuj’ dt

j — oo

228 Chapter 7. Response to Actuation of the Controls-Open Loop

that is, the frequency response and impulsive admittance are a Fourier transform pair.

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