Overdamped response

Number and weight average molecular weight transients are siammarized in Figure 7. The more viscous conditions of Run 4 resulted in an overdamped response whereas, the less viscous conditions of Run 5 resulted in overshoot. The simulation was more damped and delayed than the experimental response. 

This form is unnecessarily complicated. When we have an overdamped response, we typically use the simple exponential form with the exp(-t/xi) and exp(-t/t2) terms. (You ll get to try this in the Review Problems.)... 

The response has been plotted in Figure 11.1a for various values of C, > 1. It is known as overdamped response and resembles a little the response of a first-order system to a unit step input. But when compared to a first-order response we notice that the system initially delays to respond and then its response is rather sluggish. It becomes more sluggish as increases (i.e., as the system becomes more heavily over-... 

The underdamped response is initially faster than the critically damped or overdamped responses, which are characterized as sluggish. 

Finally, sjKcXt/A > 2. Then > 1 and we have an overdamped response. In Figure 11.7b we can see the dynamic response of the liquid level to a step change in the inlet flow rate, with and without control. 

Example 11.4 demonstrates very clearly how the simple first-order dynamic behavior of a tank can change to that of a second-order when a proportional-integral controller is added to the process. Also, it indicates that the control parameters Kc and r can have a very profound effect on the dynamic behavior of the system, which can range from an underdamped to an overdamped response. 

Depending on the value of the damping factor for the uncontrolled second-order system, eq. (14.23b) shows that 1. If > 1, the overdamped response of the closed-loop system is very sluggish. Therefore, we prefer to increase the value of Ke and make < 1. Then the closed-loop response reacts faster but it becomes oscillatory. Also, by increasing Kc, the offset decreases. 

Operating system, computer, 567 Operational constraints, 26 Optimal control, batch reactor, 10-11 Optimal regulatory controller, 650-53 Optimization in process operation, 3, 10-11, 26, 532-34 Optimizing control, 532-34 Orifice plate, 250 Output variables, 12, 13, 14 Overdamped response, 188-89 sampling an, 572-73 Override control, 403-5 Overshoot, 191 Overspecified systems, 88... 

Show that the following systems exhibit second-order overdamped response. 

Figure 7.12. Transient response. Typical amplifier response to an applied voltage step. (1) Overdamped response, long response time to reach final value. (2) Critically damped response, minimum response time to final value without overshoot. (3) Underdamped response, minimum response time to final value, but oscillation ( ringing ) occurs with overshooting of final value.

Figure Al.6.25. Modulus squared of tire rephasing, (a), and non-rephasing, R., (b), response fiinetions versus final time ifor a near-eritieally overdamped Brownian oseillator model M(i). The time delay between the seeond and third pulse, T, is varied as follows (a) from top to bottom, J= 0, 20, 40, 60, 80, 100,...

We do not need to carry the algebra further. The points that we want to make are clear. First, even the first vessel has a second order transfer function it arises from the interaction with the second tank. Second, if we expand Eq. (3-46), we should see that the interaction introduces an extra term in the characteristic polynomial, but the poles should remain real and negative.1 That is, the tank responses remain overdamped. Finally, we may be afraid( ) that the algebra might become hopelessly tangled with more complex models. Indeed, we d prefer to use state space representation based on Eqs. (3-41) and (3-42). After Chapters 4 and 9, you can try this problem in Homework Problem 11.39. 

Based onEq. (3-51), the time response y(t) should be strictly overdamped. However, this is not necessarily the case if the zero is positive (or xz < 0). We can show with algebra how various ranges of K and X may lead to different zeros (—l/xz) and time responses. However, we will not do that. (We ll use MATLAB to take a closer look in the Review Problems, though.) The key, once again, is to appreciate the principle of superposition with linear models. Thus we should get a rough idea of the time response simply based on the form in (3-50). 

With respect to the overdamped solution of a second order equation in (3-21), derive the step response y(t) in terms of the more familiar exp(-t/xi) and exp(-t/X2). This is much easier than... 

Example 6.2 Derive the controller function for a system with a second order overdamped process and system response as dictated by Eq. (6-22). 

From the characteristic polynomial, it is probable that we ll get either overdamped or underdamped system response, depending on how we design the controller. The choice is not clear from the algebra, and this is where the root locus plot comes in handy. From the perspective of a root-locus plot, we can immediately make the decision that no matter what, both Zq and p0 should be larger than the value of l/xp in Gp. That s how we may "steer" the closed-loop poles away from the imaginary axis for better system response. (If we know our root locus, we should know that this system is always stable.)... 

The above equation is called the characteristic equation of the system. It is the system s most important dynamic feature. The values of s that satisfy Eq. (6.67) are called the roots of the characteristic equation (they are also called the eigenvalues of the system). Their values, as we will shortly show, will dictate if the system is fast or slow, stable or unstable, overdamped or underdamped. Dynamic analysis and controller design consists of finding out the values of the roots of the characteristic equation of the system and changing their values to give the desired response. The rest of this book is devoted to looking at roots of characteristic equations. They are an extremely important concept that you should fully understand. ... 

Fig. 11.7. Transient response of the STM feedback system. Three different values of the loop gain G give different results. The response is overdamped with a gain of 100, critically damped with 200, and underdamped with 1000. (After Kuk and Silverman, 1989.)...

The effect of the value of the damping coefficient f on the response is shown in Fig. 7.28. For (< 1 the response is seen to be oscillatory or underdamped when ( >1 it is sluggish or overdamped and when (= 1 it is said to be critically damped, i.e. the final value is approached with the greatest speed without overshooting the Final value. When f = 0 there is no damping and the system output oscillates continuously with constant amplitude. 

The variability of the process parameters with flow causes variability in load response, as shown in Fig. 8-50. The PID controller was tuned for optimum (minimum-IAE) load response at 50 percent flow. Each curve represents the response of exit temperature to a 10 percent step in liquid flow, culminating at the stated flow. The 60 percent curve is overdamped and the 40 percent curve is underdamped. The differences in gain are reflected in the amplitude of the deviation, and the differences in dynamics are reflected in the period of oscillation. 

To demonstrate the potential of two-dimensional nonresonant Raman spectroscopy to elucidate microscopic details that are lost in the ensemble averaging inherent in one-dimensional spectroscopy, we will use the Brownian oscillator model and simulate the one- and two-dimensional responses. The Brownian oscillator model provides a qualitative description for vibrational modes coupled to a harmonic bath. With the oscillators ranging continuously from overdamped to underdamped, the model has the flexibility to describe both collective intermolecular motions and well-defined intramolecular vibrations (1). The response function of a single Brownian oscillator is given as,... 

To provide an example of the two-dimensional response from a system containing well-defined intramolecular vibrations, we will use simulations based on the polarized one-dimensional Raman spectrum of CCI4. Due to the continuous distribution of frequencies in the intermolecular region of the spectrum, there was no obvious advantage to presenting the simulated responses of the previous section in the frequency domain. However, for well-defined intramolecular vibrations the frequency domain tends to provide a clearer presentation of the responses. Therefore, in this section we will present the simulations as Fourier transformations of the time domain responses. Figure 4 shows the Fourier transformed one-dimensional Raman spectrum of CCI4. The spectrum contains three intramolecular vibrational modes — v2 at 218 cm, v4 at 314 cm, and vi at 460 cm 1 — and a broad contribution from intermolecular motions peaked around 40 cm-1. We have simulated these modes with three underdamped and one overdamped Brownian oscillators, and the simulation is shown over the data in Fig. 4. 

The general aspect of this curve with an inflexion point is typical of the dynamic behaviour of a second-order overdamped system in response to a step perturbation, i.e. Heaviside function [4] A property of this curve is that the concentration reached at the plateau is equal to the concentration of the fluid entering R2. Data of these runs are summarized in table 2 and presented in Figures 3 and 4. 

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