barcode reader project in c#.net Response of a second-order system to a unit-impulse forcing function. in Software

Create ANSI/AIM Code 128 in Software Response of a second-order system to a unit-impulse forcing function.

Response of a second-order system to a unit-impulse forcing function.
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HIGHER-ORDER SYSTEMS: SECOND-ORDER AND TRANSPORTATION LAG
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A simple way to obtain Eq. (8.31) from the step response of Eq. (8.17) is to take the derivative of Eq. (8.17). Comparison of Eqs. (8.13) and (8.30) shows that (8.32) Y(S)limpulse = SY(S)lstep The presence of s on the right side of Eq. (8.32) implies differentiation with respect to t in the time response. In other words, the inverse transform of Eq. (8.32) is (8.33) Application of Eq. (8.33) to Eq. (8.17) yields Eq. (8.31). This principle also yields the results for the next two cases.
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CASE II IMPULSE RESPONSE FOR C = 1. For the critically damped case, the
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response is given by Y(t) = ;tfe which is plotted in Fig. 8.5.
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CASE III IMPULSE RESPONSE FOR 5 > 1. For the overdamped case, the
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(8.34)
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response is given by Y(t) = 5 J&eit sinh Jmb (8.35)
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which is plotted in Fig. 8.5. To summarize, the impulse-response curves of Fig. 8.5 show the same general behavior as the step-response curves of Fig. 8.2. However, the impulse response always returns to zero. Terms such as decay ratio, period of oscillation, etc., may also be used to describe the impulse response. Many control systems exhibit transient responses such as those of Fig. 8.5. This is illustrated by Fig. 1.7 for the stirred-tank heat exchanger.
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Sinusoidal
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Response
X(t) = Asin wt
If the forcing function applied to the second-order system is sinusoidal, then it follows from Eqs. (8.11) and (5.20) that AU y(s) = (9 + oJ*)(r*s* + 257s + 1) The inversion of Eq. (8.36) may be accomplished by first factoring the two quadratic terms to give AdT2 (8.37) Y(s) = (s - jo)(s + jw)(s - sl)(s - ~2)
LINEAR OPEN-LOOP SYSTEMS
Here ~1 and s2 are the roots of the denominator of the transfer function and are given by Eqs. (8.14) and (8.15). For the case of an underdamped system (t < l), the roots of the denominator of Eq. (8.37) are a pair of pure imaginary roots (+jw, - jw) contributed by the forcin function and a pair of complex roots (--l/r + j ,/m/r, --c/r - j J-9 1 - f2/r). We may write the form of the response Y(t) by referring to Fig. 3.1 and Table 3.1; thus Y(t) = Cl cos wt + C2 sin wt + f~-~~ ( C3 cos J-5 +
sin A ;) (8.38)
The constants are evaluated by partial fractions. Notice in Eq. (8.38) that, as t + m, only the first two terms do not become zero. These remaining terms are the ultimate periodic solution; thus ut>l f--*m = Clcos wt + C2sin wt (8.39)
The reader should verify that Eq. (8.39) is also true for 5 L 1. From this little
effort, we see already that the response of the second-order system to a sinusoidal driving function is ultimately sinusoidal and has the same frequency as the driving function. If the constants Cr and C2 are evaluated, we get from Eqs. (5.23) and (8.39) A (8.40) Y(t) = sin (ot + 4) [I - (wT>*]* + (2&07)2 where (#) = -tan- 2J-wr 1 - (wT)2
By comparing Eq. (8.40) with the forcing function X(t) = Asin ot it is seen that: 1. The ratio of the output amplitude to the input amplitude is 1 J[l - (or)212 + (256JT)2 It will be shown in Chap. 16 that this may be greater or less than 1, depending on 5 and or. This is in direct contrast to the sinusoidal response of the firstorder system, where the ratio of the output amplitude to the input amplitude is always less than 1. 2. The output lags the input by phase angle 1$ I. It can be seen from Eq. (8.40), and will be shown in Chap. 16, that 14 Iapp roaches 180 asymptotically as o increases. The phase lag of the first-order system, on the other hand, can never exceed 90 . Discussion of other characteristics of the sinusoidal response will be deferred until Chap. 16.
HIGHER-ORDER SYSTEMS: SECOND-ORDER AND TRANSPORTATION LAG
We now have at our disposal considerable information about the dynamic behavior of the second-order system. It happens that many control systems that are not truly second-order exhibit step responses very similar to those of Fig. 8.2. Such systems are often characterized by second-order equations for approximate mathematical analysis. Hence, the second-order system is quite important in control theory, and frequent use will be made of the material in this chapter.
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