Systems overdamped

The damping strength of the object is such that it may absorb most of its restoritig force before it reaches its original position. There are no oscillations. For overdamped systems, rj > I. 

For C > 1 (overdamped system). If the damping coefficient is greater than unity, the quantity inside the square root is positive. Then St and S2 will both be real numbers, and they will be different (called distinct roots). 

Example 6.11. The overdamped system of Example 6.8 is forced with a unit step function. 

Consider an overdamped system driven by a periodic force K(q <[>) and white noise (r), with equation of motion... 

The general aspect of this curve with an inflexion point is typical of the dynamic behaviour of a second-order overdamped system in response to a step perturbation, i.e. Heaviside function [4] A property of this curve is that the concentration reached at the plateau is equal to the concentration of the fluid entering R2. Data of these runs are summarized in table 2 and presented in Figures 3 and 4. 

Overdamped system forced by a square wave) Consider an overdamped linear oscillator (or an / C-circuit) forced by a square wave. The system can be nondimensionalized to x + x = F(f), where F(r) is a square wave of period T. To be more specific, suppose... 

Another driven overdamped system) By considering an appropriate Poincare map, prove that the system 0-i-sin0 =sinz has at least two periodic solutions. Can you say anything about their stability (Hint Regard the system as a vector field on a cylinder f=l, 0 = sinz-sin0. Sketch the nullclines and thereby infer the shape of certain key trajectories that can be used to bound the periodic solutions. For instance, sketch the trajectory that passes through (z,0) = (f,f).)... 

Before going on, we will specialize Eq. (3.24) to a high friction and weak force or strongly overdamped system. Given the arguments of Appendix Section F.9, this may be done by letting mx t) — 0 in Eq. (3.24) to yield... 

We will see that to justify Eq. (A.29) we require a high-friction, weak-force (overdamped) system. This requirement is in complete harmony with, but is even more restrictive than the assumption that, the unconstrained A s are slow variables. 

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. 

Figure 11.6 shows the qualitative features of the response, which are the same as those of an overdamped system. A comparison with the first-order response would be instructive. Thus from Figure 11.6 we notice that ... 

The rule developed in Example 27.1 for the sampling rate of a first-order response can be extended to cover a large class of overdamped systems. Figure 16.9 shows the experimental response of an overdamped process to an input step change. The S-shaped response of Figure 16.9 can be approximated by the response of a first-order plus dead time system,... 

In Chapter 11 we found that almost all second-order, open-loop processes are overdamped systems (( > 1) composed of two interacting capacities in series. Therefore, the transfer function can be written as... 

EX A M PLE 2.13 The overdamped system of Example 2.8 is now forced with a ramp input ... 

This relation above shows that in an overdamped system the total force times the displacement corresponds to dissipation. Integrated over a time interval t, we have the followings individual trajectories (Seifert, 2012)... 

For overdamped systems, the normalized amphtude ratio is attenuated (A/KA < 1) for all co. For underdamped systems, the amplitude ratio plot exhibits a maximum (for values of 0 < < V2/2) at the resonant frequency... 

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. 

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