Damping, critical

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. 

The object may just reach its original position. By the time it does so, it loses all its restoring force due to damping and does not overshoot. Sueh systems do not oscillate. For critically damped systems... 

Critically damped system. If = k/m, then the expression under... 

This very special case is known as critical damping. The value of c for this case is given by ... 

For critical damping, the discriminant is zero, hence the roots become... 

Specification The CNC machine-table control system is to be critically damped with a settling time of 0.1 seconds. 

Control problem To select the controller gain K to achieve the settling time and tachogenerator constant to provide critical damping. 

Calculation of K In general, the settling time of a system with critical damping is equal to the periodic time of the undamped system, as can be seen in Figure 3.19. This can be demonstrated using equation (3.62) for critical damping... 

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. 

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. 

Two repeating poles at -1/x. This case is termed critically damped. The... 

With only open-loop poles, examples (a) to (c) can only represent systems with a proportional controller. In case (a), the system contains a first orders process, and in (b) and (c) are overdamped and critically damped second order processes. 

A third, very specific case occurs when x/m = h2/4m2. The system is then said to be critically damped. 

A control loop consists of a proportional controller, a first-order control valve of time constant rv and gain Kv and a first-order process of time constant T and gain Kx. Show that, when the system is critically damped, the controller gain is given by ... 

If the desired value is suddenly changed by, an amount A R when the controller is set to give critical damping, show that the error e will be given by ... 

For critical damping, the roots of the characteristic equation are equal, hence ... 

As the system is critically damped, the roots of the denominator must be equal, that is it factorises to give (s + a)2,... 

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. 

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. 

A value between 0 (no damping) and 1 (critical damping) which quantifies the level of damping present in a system. 

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. 

What value of will give critical damping At what value of X will the system become unstable ... 

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... 

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