Ratio. Damping

The vane can be used for both average wind direction and the fluctuation statistic determined over hourly intervals. The vane should have a distance constant of less than 5 m and a damping ratio greater than or equal to 0.4 to have a proper response. Relative accuracy should be 1° and absolute accuracy should be 5°. In order to estimate accurately, the direction should be sampled at intervals of 1-5 sec. This can best be accom-... 

Equations (3.42) and (3.43) are the standard forms of transfer functions for a second-order system, where K = steady-state gain constant, Wn = undamped natural frequency (rad/s) and ( = damping ratio. The meaning of the parameters Wn and ( are explained in sections 3.6.4 and 3.6.3. 

The ratio of the damping eoeffieient C in a seeond-order system eompared with the value of the damping eoeffieient Q required for eritieal damping is ealled the Damping Ratio ( (Zeta). Henee... 

Consider a second-order system whose steady-state gain is K, undamped natural frequency is Wn and whose damping ratio is (, where C < 1 For a unit step input, the block diagram is as shown in Figure 3.18. From Figure 3.18... 

The resulting output Xo t) would be as shown in Figure 3.20. There are two methods for ealeulating the damping ratio. 

A system eonsists of a first-order element linked to a seeond-order element without interaetion. The first-order element has a time eonstant of 5 seeonds and a steady-state gain eonstant of 0.2. The seeond-order element has an undamped natural frequeney of 4 rad/s, a damping ratio of 0.25 and a steady-state gain eonstant of unity. 

Since the Bulk Modulus of hydraulic oil is in the order of 1.4 GPa, if m and F[ are small, a large hydraulic natural frequency is possible, resulting in a rapid response. Note that the hydraulic damping ratio is governed by Cp and A c. To control the level of damping, it is sometimes necessary to drill small holes through the piston. 

The Process Reaction Method assumes that the optimum response for the closed-loop system occurs when the ratio of successive peaks, as defined by equation (3.71), is 4 1. From equation (3.71) it can be seen that this occurs when the closed-loop damping ratio has a value of 0.21. The controller parameters, as a function of R and D, to produce this response, are given in Table 4.2. 

This corresponds to a damping ratio of 0.23. These vaiues are very ciose to the Zeigier-Niciiois optimum vaiues of 4.0 and 0.2i respectiveiy. 

Control problem For a speeifie hull, the eontrol problem is to determine the autopilot setting K ) to provide a satisfaetory transient response. In this ease, this will be when the damping ratio has a value of 0.5. Also to be determined are the rise time, settling time and pereentage overshoot. 

Part (b) From equation (5.52), line of constant damping ratio is... 

Find the asymptotes and angles of departure and hence sketch the root locus diagram. Locate a point on the complex locus that corresponds to a damping ratio of 0.25 and hence find... 

Plot line of constant damping ratio on Figure 5.16 and test trial points along it using angle criterion. 

Ship roll damping ratio ( = 0.248 Ship steady-state gain A s =0.5... 

As explained in Figure 6.12 the pure integrator asymptote will pass through OdB at l.Orad/s (for A" = 1 in equation (6.67)) and the seeond-order element has an undamped natural frequeney of 2.0rad/s and a damping ratio of 0.5. 

Design a full-order observer that has an undamped natural frequeney of 10 rad/s and a damping ratio of 0.5. 

The elosed-loop eontrol system analysed by the root-loeus method in Example 5.8 ean be represented by the bloek diagram shown in Figure 10.44. Using root-loeus, the best setting for K was found to be 11.35, representing a damping ratio of 0.5. 

Finally, for a complex pole, we can relate the damping ratio (t, < 1) with the angle that the pole makes with the real axis (Fig. 2.5). Taking the absolute values of the dimensions of the triangle, we can find... 

C = damping ratio (also called damping coefficient or factor)... 

Use MATLAB to make plots of overshoot and decay ratio as functions of the damping ratio. 

Example 5.2 Derive the closed-loop transfer function of a system with proportional control and a second order overdamped process. If the second order process has time constants 2 and 4 min and process gain 1.0 [units], what proportional gain would provide us with a system with damping ratio of 0.7 ... 

In terms of design specification, it is not uncommon to use decay ratio as the design criterion. Repeating Eq. (3-29), the decay ratio DR (or the overshoot OS) is a function of the damping ratio ... 

While we have the analytical results, it is not obvious how choices of integral time constant and proportional gain may affect the closed-loop poles or the system damping ratio. (We may get a partial picture if we consider circumstances under which KcKp 1.) Again, we ll defer the analysis... 

If we assume that an oscillatory system response can be fitted to a second order underdamped function. With Eq. (3-29), we can calculate that with a decay ratio of 0.25, the damping ratio f is 0.215, and the maximum percent overshoot is 50%, which is not insignificant. (These values came from Revew Problem 4 back in Chapter 5.)... 

By and large, a quarter decay ratio response is acceptable for disturbances but not desirable for set point changes. Theoretically, we can pick any decay ratio of our liking. Recall Section 2.7 (p. 2-17) that the position of the closed-loop pole lies on a line governed by 0 = cos C In the next chapter, we will locate the pole position on a root locus plot based on a given damping ratio. 

The integral time constant is x = xb and the term multiplying the terms in the parentheses is the proportional gain Kc. In this problem, the system damping ratio Q is the only tuning parameter. 

Transient response criteria Analytical derivation Derive closed-loop damping ratio from a second order system characteristic polynomial. Relate the damping ratio to the proportional gain of the system. 

In terms of controller design, the closed-loop poles (or now the root loci) also tell us about the system dynamics. We can extract much more information from a root locus plot than from a Routh criterion analysis or a s = jco substitution. In fact, it is common to impose, say, a time constant or a damping ratio specification on the system when we use root locus plots as a design tool. 

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