Direct Sine-Wave Testing

Once the G(( , curves have been found, they can be used directly to examine the dynamics and stability of the system or to design controllers in the frequency domain (see Chap. 13). 

If a transfer-function model is desired, approximate transfer functions can be fitted to the experimental curves. First the log modulus Bode plot is used. The low-frequency asymptote gives the steadystate gain. The time constants can be found from the breakpoint frequency and the slope of the high-frequency asymptote. The damping coefficient can be found from the resonant peak. 

Once the log modulus curve has been adequately fitted by an approximate transfer function G(J ), the phase angle of G( a) is compared with the experimental phase-angle curve. The difference is usually the contribution of deadtime. The procedure is illustrated in Fig. 14.2. 

It is usually important to get an accurate fit of the frequency response of the model to the experimental frequency only near the critical region where the phase 

Filling approximate tranufer function to experiinenUi) frequency-response data. 

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The main disadvantage of direct sine-wave testing is that it can be very time-consuming when applied to typical large time-constant chemical process equipment. The steadystate oscillation must be established at each value of frequency. It can lake days to generate the complete frequency-response curves of a slow process. 

When the test is being conducted over this long period of time, other disturbances and changes in operating conditions can occur that can affect the results of the test. Therefore direct sine-wave testing is only rarely used to get the complete frequency response. 

One of the most useful and practical methods for obtaining experimental dynamic data from many chemical engineering processes is pulse testing. It yields reasonably accurate frequency-response curves and requires only a fraction of the time that direct sine-wave testing takes. 

J. The frequency-response data given below were obtained from direct sine-wave tests of a chemical plant. Fit an approximate transfer function to these data. 

If the input U does not provide enough excitation of the process over the important frequency range, the model fidelity is poor, particularly in processes with appreciable noise. This is why direct sine wave testing at a frequency near the ultimate frequency and relay feedback testing are such usefril methods. 

Direct sine wave testing is an extremely useful way to obtain precise dynamic data. Damping coefficients, time constants, and system order can all be quite accurately found. Direct sine wave testing is particularly useful for processes with signals that are noisy. Since you are putting in a sine wave signal with a known frequency and the output signal has this same frequency, you can easily filter out all of the... 

Direct Sine Wave Testing Pulse Testing... 

The next level of dynamic testing is with direct sine waves. The input of the plant, which is usually a control valve position or a flow controller setpoint, is varied sinusoidally at a fixed frequency o). After waiting for all transients to die out and for a steady oscillation in the output to be established, the amplitude ratio and phase angle are found by recording input and output data. The data point at this frequency is plotted on a Nyquist, Bode, or Nichols plot. See Fig. I6.3u. Then the frequency is changed to another value, and a new amplitude ratio and a new phase angle are... 

The most popular dynamic test procedure for viscoelastic behavior is the application of an oscillatory stress of small amplitude. This shear stress applied produces a corresponding strain in the material. If the material were an ideal Hookean body, the shear stress and shear strain rate waves would be in phase (Fig. 14A), whereas for an ideal Newtonian sample, there would be a phase shift of 90° (Fig. 14B), because for Newtonian bodies the shear strain is at a maximum, when a maximum of stress is present. The shear strain, when assuming an oscillating sine fimction, is at a maximum in the middle of the slope, because there is the steepest increase in shear strain due to the change in direction. For a typical viscoelastic material, the phase shift will have a value between >0° and <90° (Fig. 14C). 

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