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Unformatted text preview: SecondOrder Systems The denominator of a secondorder transfer function with C < 1 can be expressed as
form 4, Table 4.3. If C 2 1, the denominator can be written as the product of two ﬁrst
order factors like form 3. A special case of a secondorder system that does not ﬁt into the previous cases occurs when C = 00. The form is = L 4537
T“) s(rs + 1) ( ) An example is a mass with a damper but no spring (k = 0). The three building blocks
are K, s, and TS + 1. Because it is in the denominator, the 8 term shifts the composite m
curve upward for a) < 1 and shifts it down for a) > 1. The composite m curve follows
that of the 5 term until u) z l/T, when the (rs + 1) term begins to have an effect. For
a) > 1/1, the composite slope is —40 db/decade. The s term contributes a constant
— 90° to the (1) curve. The result is to shift the ﬁrstorder lag curve (Figure 4.11b) down
by 90°. The results are shown in Figure 4.18. 0.1 1 10 100 1000
O.)
(a)
—90° ‘
¢
—135°
_180° L;
0.1 1 10 10 1000
0.)
(12) FIGURE 4.18 Frequency response plots for T (s) = s(1:s + 1)
with K = 1 and 1' = 0.2. TABLE 4.3 Factors Commonly Found in Transfer Functions of the Form: N N T(s) = K 1(5) 2(5)
Dl(s)D2(s) Factor N ,(s) or Dj(s)
1. Constant, K
2. s”
3. TS + l
2 2
4. 32+2Cwns+w§=[(—S—) +—£s+1:wﬁ, C<1
(Du a)". An Overdamped System
Consider the secondorder model m)? + at + kx = du(t)
Its transfer function is d _____ (4.538)
ms2 + cs + k T(s) = If the system is overdamped, both roots are real and distinct, and we can write T(s) as T(s) = __£’ﬂ‘___ = L (45—39)
m 2 c (11s +1)(12s +1)
—s + —s + 1
k k
where 11 and 12 are the time constants of the roots.
Returning to (4522) and (4523), we see that
d
m(w) = 20 log T(iw) = 20 log E‘ — 20 log lrlwi + II
— 20 log Itzwi + 1 (4.540)
d
4m) = A E — A(tlwi+1)— 25,(r2wi + 1) (4541) where K = d/k, Dl(iw) = rlwi + 1 and D2(ia)) = tzwi + 1. Thus, the magnitude ratio
plot in db consists of a constant term, 20 log Id/kl, minus the sum of the plots for two
ﬁrstorder lead terms. Assume that II > 12. Then for 1/11 < co <1/12, the slope is
approximately —20 db/decade. For a) > 1/12, the contribution of the term (Izwi + 1)
is signiﬁcant. This causes the slope to decrease by an additional 20 db/decade, to
produce a net slope of —40 db/decade for w>1/12. The rest of the plot can be
sketched as before. The result is shown in Figure 4.1% for d > k. The phase angle plot
shown in Figure 4.1% is produced in a similar manner by using (4.541). Note that if
d/k > 0, A (d/k) = 0°. m(db)
(scale for
composite
curve) m(db)
(scale for
individual
factors) 20 log I Z] ‘20 (a) qb (deg) (scale
for all terms) —90° Composite 435° 480° (1” c,.)=‘l/‘r1 o.)=1/‘r2
logo.) FIGURE 4.19 Frequency response plots for the overdamped system
d/k T“) = (as + 1)(r2s +1)‘ An Underdamped System If the transfer function given by (45—38) has complex conjugate roots, it can be
expressed as form 4 in Table 4.3. T(5)= M = S 2 M (4.5.42)
Es2+£s+1 — +2ci+1
k k a),l cu We have seen that the constant term d/k merely shifts the magnitude ratio plot up or
down by a ﬁxed amount and adds either 0° or —180° to the phase angle plot.
Therefore, for now, let us take d/k = l and consider the following quadratic factor,
obtained from (4.542) by replacing s with ice. T(ico) = —— = —_ (4.543) ' 2 2 2 2
(1—01)+—£wi+1 1(3) + {mi
w" a)” a)” a)" Man) = 20 log The magnitude ratio is = _ 10 log [(1 _ “’2 >2 + (23")? (4544)
a)" a),l The asymptotic approximations are as follows. For a) < a)”, mm); —20 log 1 =0 For m > can,
(04 w2
E —201 4 2
W») 0g (0: + C a):
4
a)
a —20 10
g a):
= —4010g£0—
a) n Thus, for low frequencies, the curve is horizontal at m = 0, while for high frequencies, it
has a slope of — 40 db/decade, just as in the overdamped case. The highfrequency and
lowfrequency asymptotes intersect at the corner frequency a) = a)". The underdamped case differs from the overdamped case in the vicinity of the
corner frequency. To see this, examine M (w). l 2 2 2 (“w—2) WC”)
wit a)” This has a maximum value when the denominator has a minimum. Setting the
derivative of the denominator with respect to a) equal to zero shows that the maximum M (w) occurs at a) = a)”. /1 — 28. This frequency is the resonant frequency
60,. The peak of M (co) exists only when the term under the radical is positive;rrthat is,
when E g 0.707. Thus, MW) = (4545) w, = com /1 — 252 0 g c g 0.707 (4546)
The value of the peak M p is found by substituting a), into M (w). This gives 1 .
—— 0 s g 0.707 (4.547)
2L / 1 — c2 L: If E > 0.707, no peak exists, and the maximum value of M occurs at a) :0 where M = 1. Note that as C —>0, c0,—>w,,, and M p—+ 00. For an undamped system, the reso
nant frequency is the natural frequency w". Mp: M(w,)= m (db) ¢(deg) FIGURE 4.20 Frequency response plots for the underdamped system
1 T(5)=‘—2——‘_
(1) ﬂ“
0)" a)" A plot of m(a)) versus log a) is shown in Figure 4.20a for several values of Z. Note
that the correction to the asymptotic approximations in the vicinity of the corner
frequency depends on the value of C. The peak value in decibels is m = mm) = —20 log (2M1 — :2) (4548) m(u),,) = — 20 log 22; (4549) The curve can be sketched more accurately by repeated evaluation of (4544) for
values of a) near can. At (0:0)", The phase angle plot is obtained in a similar manner. From the additive property
for angles (4.523), we see that for (4543), ¢<w>=All(%)z+2if’il ZCw
tan ¢(w) = — 2 (4.550)
1‘ where ¢(w) is in the 3rd or 4th quadrant. For a) < a)",
¢(w) ; —tan'10 = 0° Thus For a) > a)",
(Mao) ; —180°
At the corner frequency,
¢(w,,) = —tan‘1 oo = —90° This result is independent of C. The curve is skewsymmetric about the inﬂection point
at 45 = — 90° for all values of C. The rest of the plot can be sketched by evaluating
(4550) at various values of w. The plot is shown for several values of C in Figure 4.2%. At the resonant frequency.
_ 2
(1,00,) = _tan—1___\/1C2‘: (4.551) For our applications, the quadratic factor given by form 4 in Table 4.3 almost
always occurs in the denominator; therefore, we have developed the results assuming
this will be the case. If a quadratic factor is found in the numerator, its values of m(w)
and (15((0) are the negative of those given by (4544) and (4550). ...
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 Spring '09

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