Systems critically damped

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... 

For a critically-damped system, the roots of characteristic equation are equal, or ... 

At the origin (where z = 0) the damping coefficient is unity. Solving Eq. (19.18) for the value of controller gain that give this critically damped system... 

This value of controller gain gives a critically damped system. For larger controller gains, the system is underdamped. The complex roots are... 

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. 

Substituting A and B into Eq. (f) gives (b) Critically Damped Systems, =1... 

The value assumed for the equivalent viscous damping ratio is Co = c/Ccrit = 0.02, being Ccrit = 2 /to the viscous damping coefficient for a critically damped system. Even though quite small, this value of Co is reasonable for an undamaged structure within the linear-elastic range. 

The response of the system will depend mainly on the damping coefficient f. When f < 1, the system is underdamped and has an oscillatory response. The smaller the value of f, the greater the overshoot. If f = 1, the system is termed critically damped and has no oscillation. A critically damped system provides the fastest approach to the final value without the overshoot of an underdamped system. Finally, if f > 1, the system is overdamped. An overdamped system is similar to a critically damped system, in that the response never overshoots the final value. However, the approach for an overdamped system is much slower and varies depending upon the value of f. These typical responses are illustrated in Figure 3.27. 

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. 

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

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... 

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 ... 

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

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

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... 

If all tbe roots lie on the negative real axis, we know the system is overdamped or critically damped (all real roots). 

For Kc between zero and 4, the two roots are real and lie on the negative real axis. Tbe dosedloop system is critically damped (the dosedloop dampring coeffident is 1) at Xc = I since the roots are equal. For values of gain greater than the roots will be complex. 

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. 

Thus the system is critically damped as expected for minimal-prototype design. 

Note the overshoot at i = 7, despite the fact that the system is critically damped. This is due to the location of the zero in the controller. 

There is no more oscillation the system has the critical damping. 

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.)...

Vibration isolation 237—250 critical damping 239 pneumatic systems 250 quality factor, Q 239 resonance excitation 241 stacked plate-elastomer system 249 transfer function 240 Virus 341 Viton 250, 270, 272 Voltage-dependent imaging 16, 17 Si(lOO) 17 Si(lll)-2X1 16 Volterra equation 310 Vortex 334 W... 

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. 

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