Underdamped response

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

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. 

Now, go to the LTI Viewer window and select Import under the File pull-down menu. A dialog box will pop out to help import the transfer function objects. By default, a unit step response will be generated. Click on the axis with the right mouse button to retrieve a popup menu that will provide options for other plot types, for toggling the object to be plotted, and other features. With a step response plot, the Characteristics feature of the pop-up menu can identify the peak time, rise time, and settling time of an underdamped response. 

The hrsl-order system considered in the previous section yields well-behaved exponential responses. Second-order systems can be much more exciting since they can give an oscillatory or underdamped response. 

Notice the very significant result that the damping coefficient is less than one on the negative real axis. This means that in sampled-data systems a real root can give underdamped response. This can never happen in a continuous system the roots must be complex to give underdamped response. 

Advantages Moderate parts count Disadvantages Underdamped response... 

Fignre 15.6 shows the key features of an underdamped response. The rise time is the time reqnired to first cross the final steady-state value. The response time or settling time is the time required for the response to settle to within 5% of the steady-state change ( 5% D). The decay ratio is the ratio C/B, which indicates how fast the oscillations damp out. 

Case C Underdamped response, when < 1. The inversion of eq. (11.4) in this case yields... 

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

Although the underdamped response is initially faster and reaches its ultimate value quickly, it does not stay there, but it starts oscillating with progressively decreasing amplitude. This oscillatory behavior makes an underdamped response quite distinct from all previous ones. 

It must be emphasized that almost all the underdamped responses in a chemical plant are caused by the interaction of the controllers with the process units they control. Therefore, it is a type of response that we will encounter very often, and it is wise to become well acquainted with its characteristics. 

Discuss the overdamped, critically damped, and underdamped responses of a second-order system. Identify their distinguishing characteristics. 

The model for a variable capacitance pressure transducer was developed in Appendix 11A and is given by eq. (13.9). It shows that the system is inherently second-order and can exhibit underdamped response. What does this mean for the applicability of such device ... 

Let us use as reference the underdamped response shown in Figure 11.2, in order to define the terms employed to describe an underdamped response. 

Period of oscillation From eq. (11.10) we see that the radian frequency (rad/time) of the oscillations of an underdamped response is given by... 

For the selection of the best values for Kc and t/ we will use simple criteria stemming from the underdamped response of a second-order system. Select the one-quarter decay ratio criterion. From eq. (11.12) we know that... 

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.

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