Damping factors

The EF algoritlnn [ ] is based on the work of Cerjan and Miller [ ] and, in particular, Simons and coworkers [70,1Y. It is closely related to the augmented Hessian (rational fiinction) approach[25]. We have seen in section B3.5.2.5 that this is equivalent to addmg a constant level shift (damping factor) to the diagonal elements of the approximate Hessian H. An appropriate level shift effectively makes the Hessian positive definite, suitable for minimization. 

From Equation (17-4), one will find that the phase lag is a function of the relative rotating speed lu/lu and the damping factor (See Figure 17-1.) The force direction is not the same as the maximum amplitude. Thus, for maximum benefit, the correction weight must be applied opposite to the force direction. 

Assume that a damping factor of 0.707 or greater is good and provides a -3dB attenuation at the corner frequency and does not produce noise due to ringing. Also assume that the input line impedance is 50 ohms since the regulatory agencies use an TISN test which make the line impedance equal this value. Calculate the values needed in the common-mode inductor and Y capacitors ... 

Real world values do not allow a capacitor of this large a value. The largest value capacitor that will pass the ac leakage current test is 0.05 pF. This is 50 percent of the calculated capacitor value, so the inductor must be increased 200 percent in order to maintain the same corner frequency. The inductance then becomes 900 pH and the resultant damping factor is 2.5 which is acceptable. The resulting schematic is shown in Figure 3-77. 

Another area of concern is the Q of the filter at self-resonance. If the Q is too high, or in other words, the damping factor is too low, the filter will actually generate noise in the form of narrow-band ringing. This can be dealt with during the design. 

The minimum damping factor (Q should be no less than 0.707. Less than that would allow ringing to occur and produce less than 3dB of attenuation at the corner frequency. 

The differential-mode filter should have a lower damping factor than the common-mode because the combined damping response of the entire filter section would be too sluggish if higher damping factors were used. A damping factor of a minimum of 0.5 is acceptable. 

From the solution of the underdamped step response (0 < f < 1), we can derive the following characteristics (Fig. 3.2). They are useful in two respects (1) fitting experimental data in the measurements of natural period and damping factor, and (2) making control system design specifications with respect to the dynamic response. 

There are two very useful MATLAB features. First, we can overlay onto the root locus plot lines of constant damping factor and natural frequency. These lines help us pick the controller gain if the design specification is in terms of the frequency or the damping factor. 

Photoelectrons that experience inelastic losses will not have the appropriate wave vector to contribute to the interference process. Such losses are taken into account by an exponential damping factor,... 

Figure 10A shows the oscillation of the surface pressure. However, if we include the damping factors and increase the number of cells in the simulation, the oscillation would be smoothed to a flat plateau such as it is represented by the dotted line. 

An analysis of the calculated values of AW shows that the damping factor in the indirect interaction is dr1, as exemplified in Fig. 8.8, for Cr and Ti substrates and 77 = 1.667. Envelope curves of the form ad x are shown, where a is chosen, so that the curve passes through the data point corresponding to d = 4. The fit to d 1 is seen to improve as d increases, agreeing with the asymptotic nature of the interaction law. 

Second, the fluctuation is delayed by a time 5t which is a function of the residence time t, of the element in the reservoir. For an infinite residence time the argument of the tangent tends towards n/2 and the delay 5f towards T/4, while for a short residence time, the delay tends towards zero. As expected, reactive elements respond more rapidly than inert elements. The phase shift and the damping factor relating input to output concentrations represent the angular phase and argument of a complex function known as the transfer function of the reservoir. Such a function, however, is most conveniently introduced via Laplace and Fourier transforms. Applications of these geochemical concepts to the dynamics of volcanic sequences can be found in Albarede (1993). 

Dynamic mechanical testers apply a small sinusoidal stress or strain to a small sample of the polymer to be examined and measure resonant frequency and damping versus temperature and forced frequency. Instrument software computes dynamic storage modulus (G ), dynamic loss modulus (G") and tan delta or damping factor. Measurements over a wide range of frequency and temperature provide a fingerprint of the polymer with sensitivity highly superior to DSC. 

Note 2 Damping factor, is related to logarithmic decrement, 8, by the equation ... 

For both new and 2x new clearances, the damping factor versus separation margin between any bending natural frequency and the synchronous run line shall be within the acceptable region shown on Figme 1.2. If this condition cannot be satisfied, the damped response to imbalance shall be determined (see 1.1.4). 

Figure I-2—Damping Factor Versus Frequency Ratio...

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