Underdamping

Undei damped behavior. Measurement devices with mechanical components often have a natural harmonic and can exhibit underdamped behavior. The displacer type of level measurement device is capable of such behavior. 

Testing of metal-enclosed switchgear assemblies 14/447 For underdamped systems. 0 < rj < I... 

It is possible to identify the mathematieal model of an underdamped seeond-order system from its step response funetion. 

Consider a unity-gain K = ) seeond-order underdamped system responding to an input of the form... 

General case for an underdamped second-order system... 

There are different conditions of damping critical, overdamping, and under-damping. Critical damping occurs when 11, = Cl). Over-damping occurs when ji, > o). Underdamping occurs when ji, < ai. 

The only condition that results in oscillatory motion and, therefore, represents a mechanical vibration is underdamping. The other two conditions result in periodic motions. When damping is less than critical ji, < cP), then the following equation applies ... 

Barcilon, V, Singular Perturbation Analysis of the Fokker-Planck Equation Kramer s Underdamped Problem, SIAM Journal of Applied Mathematics 56, 446, 1996. 

C < 1 Two complex conjugate poles. This situation is considered underdamped. 

Consider a step input, x(t) = Mu(t) with X(s) = M/s, and the different cases with respect to the value of . We can derive the output response y(t) for the different cases. We rarely use these results. They are provided for reference. In the case of the underdamped solution, it is used to derive the characteristic features in the next section. 

Time-domain features of underdamped step response... 

From the solution of the underdamped step response (0 < f < 1), we can derive the following characteristics (Fig. 3.2). They are useful in two respects (1) fitting experimental data in the measurements of natural period and damping factor, and (2) making control system design specifications with respect to the dynamic response. 

Figure 3.2. Key features in an underdamped response. See text for equations.

The real part of a complex pole in (3-19) is -Zjx, meaning that the exponential function forcing the oscillation to decay to zero is e- x as in Eq. (3-23). If we draw an analogy to a first order transfer function, the time constant of an underdamped second order function is x/t,. Thus to settle within 5% of the final value, we can choose the settling time as 1... 

The elimination of the offset is usually at the expense of a more underdamped system response. The oscillatory response may have a short rise time, but is penalized by excessive overshoot or exceedingly long settling time. 1... 

Because of the inherent underdamped behavior, we must be careful with the choice of the proportional gain. In fact, we usually lower the proportional gain (or detune the controller) when we add integral control. 

In this section, we will derive the closed-loop transfer functions for a few simple cases. The scope is limited by how much sense we can make out of the algebra. Nevertheless, the steps that we go through are necessary to learn how to set up problems properly. The analysis also helps us to better understand why a system may have a faster response, why a system may become underdamped, and when there is an offset. When the algebra is clean enough, we can also make observations as to how controller settings may affect the closed-loop system response. The results generally reaffirm the qualitative statements that we ve made concerning the characteristics of different controllers. 

The key is to recognize that the system may exhibit underdamped behavior even though the open-loop process is overdamped. The closed-loop characteristic polynomial can have either real or complex roots, depending on our choice of Kc. (This is much easier to see when we work with... 

With the expectation that the second order system may exhibit underdamped behavior, we rewrite the closed-loop function as... 

If we assume that an oscillatory system response can be fitted to a second order underdamped function. With Eq. (3-29), we can calculate that with a decay ratio of 0.25, the damping ratio f is 0.215, and the maximum percent overshoot is 50%, which is not insignificant. (These values came from Revew Problem 4 back in Chapter 5.)... 

One reason why this approximation works is that process unit operations are generally open-loop stable, and many are multi-capacity in nature. Reminder Underdamped response of the system is due to the controller, which is taken out in the open-loop step test. 

All tuning relations provide different results. Generally, the Cohen and Coon relation has the largest proportional gain and the dynamic response tends to be the most underdamped. The Ciancone-Marlin relation provides the most conservative setting, and it uses a very small derivative time constant and a relatively large integral time constant. In a way, their correlation reflects a common industrial preference for PI controllers. 

In contrast to Eq. (6-22), we can dictate a second order underdamped system response ... 

Example 6.4 Derive the controller function for a system with a second order overdamped process but an underdamped system response as dictated by Eq. (6-27). 

With the IMC tuning setting in Example 5.7B, the resulting time response plot is (very nicely) slightly underdamped even though the derivation in Example 6.4 predicates on a system response without oscillations. Part of the reason lies in the approximation of the dead time function, and part of the reason is due to how the system time constant was chosen. Generally, it is important to double check our IMC settings with simulations. 

If we increase further the value of Kg, the closed-loop poles will branch off (or breakaway) from the real axis and become two complex conjugates (Fig. E7.5). No matter how large Kc becomes, these two complex conjugates always have the same real part as given by the repeated root. Thus what we find are two vertical loci extending toward positive and negative infinity. In this analysis, we also see how as we increase Kc, the system changes from overdamped to become underdamped, but it is always stable. 

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