Critically damped response

A proportional plus integral controller is used to control the level in the reflux accumulator of a distillation column by regulating the top product flowrate. At time t = 0, the desired value of the flow controller which is controlling the reflux is increased by 3 x 10-4 m3/s. If the integral action time of the level controller is half the value which would give a critically damped response and the proportional band is 50 per cent, obtain an expression for the resulting change in level. 

MyjSp as a manipulated input3 and requesting a second-order critically damped response ... 

Figure 15.5 shows the three general types of dynamic behavior of a second-order process, which can also be nsed to describe the dynamic behavior of feedback systems overdamped, critically damped, and nnderdamped. Overdamped behavior is characterized by a monotonic approach to steady state. Underdamped behavior is characterized by an oscillatory approach to steady state. A critically damped response marks the boundary between overdamped and underdamped behavior. 

Figure 11.3 shows the plot of overshoot versus given by eq. (11.11). We notice that the overshoot increases with decreasing , while as approaches 1 the overshoot approaches zero (critically damped 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.

The foregoing discussion on complementary feedback was based primarily on critically damped response. With a pure dead-time process, this was the best obtainable. But with less difficult processes, lower damping will enhance recovery from load disturbances due to greater controller gain. 

As can be seen, the response of two first-order systems in series without feedback is a critically damped response. This result could be expected, a disturbance at the entrance propagates through the system without damping and without amplification. 

An RRS is normally constructed for several levels of critical dampings as illustrated in Figure 14.13. The most appropriate of these is then chosen for the purpose of testing. Any of the above response spectra can be developed into a time history of the earthquake, similar to that in Figure 14,12(b),... 

This is an important equation that defines the behaviour of a vibrating body under different conditions of applied force or motion F y From this it can be inferred that the response or movement of object x will depend upon t) and 7 is termed the fraction of critical damping and w , the angular natural frequency of the system. With the help of these equations, the response characteristics of an object to a force can be determined. 

By calculating the energy to heat it is possible to determine the vibration levels to which the structure can be exposed and still exhibit critical damping. There is one area that must be evaluated. Plastics exhibit a spectrum of response to stress and there are certain straining rates that the material will react to almost elastically. If this characteristic response corresponds to a frequency to which the structure is exposed the damping effect is minimal and the structure may be destroyed. In order to avoid the possibility of this occurring, it is desirable to have a curve of energy absorption vs. frequency for the material that will be used. 

In situations where absorption of the incident radiation by the transducing gas is troublesome a piezoelectric transducer (made from barium titanate, for example) can be attached to the sample (or sample cuvette in the case of liquids) to detect the thermal wave generated in the sample by the modulated light (8,9). The low frequency, critically damped thermal wave bends the sample and transducer thus producing the piezoelectric response. The piezoelectric transducer will also respond to a sound wave in the solid or liquid but only efficiently at a resonant frequency of the transducer typically of the order of 10 to 100 KHz (see Figure 4). Thus neither in the case of microphonic nor piezoelectric detection is the PA effect strictly an acoustic phenomenon but rather a thermal diffusion phenomenon, and the term "photoacoustic" is a now well established misnomer. 

The response is still blunted but any faster response would involve overshoot of the baseline. Critical damping is still too much for a rapidly responding measurement device. 

There are a number of criteria by which the desired performance of a closedloop system can be spedlied in the time domain. For example, we could specify that the closedloop system be critically damped so that there is no overshoot or oscillation. We must then select the type of controller and set its tuning constants so that it will give, when coupled with the process, the desired closedloop response. Naturally the control specification must be physically attainable. We cannot make a Boeing 747 jumbo jet airplane behave like an F-IS fighter. We cannot... 

Thus the closedloop root is located at the origin. This corresponds to a critically damped closedloop system (C = 1). The specified response in the output was for no overshoot, so this damping coefficient is to be expiected. 

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.)...

Exercise. The response of a critically damped ballistic galvanometer to a current pulse at t = 0 is u(t) = ct e-yt. Find the spectral density of the response to a stationary stream of independent random pulses. 

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. 

We used the model of the fast dynamics of the system in Equation (4.36) to design a nonlinear input-output linearizing output feedback controller with integral action (Daoutidis and Kravaris 1992) for x. The controller was designed to produce the critically damped second-order response... 

Using a low-frequency critically damped bal-listocardiograph, cardiac output was estimated in 25 subjects given 5 pg kg-1 of sarin by oral inhalation (which caused a mean 37% decrease in erythrocyte cholinesterase activity). There was a small increase in heart rate, which the investigator felt was a cortical response to the sensation... 

Figure 3.6 c and d illustrate amplitude and phase responses of oscillators having different damping coefficients. The step response of a sensor is usually determined by the time constant as well as by the typical rise and response times of the system. Figure 3.6 b shows the response of a critical damped system to a steplike change in the input signal 0 The time constant r (as defined for an exponential response), the 10% to 90% rise time t(o.i/o.9) and the 95% response time t(0 95) are marked. 

The response is also shown in Figure 11.1a. We notice that a second-order system with critical damping approaches its ultimate value faster than does an overdamped system. 

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

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