Harmonic response diagram

Fig. 6.2 Harmonic response diagram. Substituting equation (6.6) into (6.3)...

For a given value of lu, equation (6.9) represents a point in complex space P(lu). When LU is varied from zero to infinity, a locus will be generated in the complex space. This locus, shown in Figure 6.2, is in effect a polar plot, and is sometimes called a harmonic response diagram. An important feature of such a diagram is that its shape is uniquely related to the dynamic characteristics of the system. 

Hence equation (6.14) can be plotted in the complex space (Argand Diagram) to produce a harmonic response diagram as shown in Figure 6.3. 

Fig. 10 (a) Schematic diagram of fabrication process for electrospun PANI/CA actuators, (b) harmonic responses, (c) hysteresis responses of electrospun actuators under sinusoidal electrical inputs with peak voltage of 3 V and excitation frequency of 0.1 Hz (Hong et aL 2013. 2013 Elsevier Ltd. Reproduced by permission of Elsevier BV. All rights reserved.)... 

In order to calculate the response function of the Feynman diagram R3, it is further assumed that the transition frequency co 12 is anharmonically shifted with respect to the ground states transition frequency so that, u>n = >oi -A. Another assumption that can be made (see later for a discussion of these assumptions) is that the fluctuations between both level pairs are strictly correlated <5co12 = <5 j0i. This implies that only the harmonic part of the potential surface is perturbed by the bath fluctuations and the anharmonicity of the vibrator is unaffected. We then obtain for R3 ... 

Full characterisation of a magnetic material with nonlinear response requires measurements of harmonic susceptibilities beyond the fundamental susceptibility Xi These are useful, for example, in the identification of multiple phase transitions in high Tc superconductors and in the determination of magnetic phase diagrams. 

The frequency response (although called transfer function) is a conmum function in signal analysis and control engineering when the dynamic behavior of a system must be analyzed. Therefore, the input and output parameters of the system will be compared as a function of frequency. For example, when the system is stimulated with a harmonic input signal of a certain frequency, the system will answer with the same frequency, but with attenuated amplitude and a shifted phase. Since the amplitude attenuation and the phase shift are both functiOTis of the stimulation frequency, it is common to plot them in Bode diagrams, where the amplirnde response and the phase response are displayed separately over the frequency. 

Fig. 7.4 Phase diagram. The numbers indicate the harmonic periods appearing in the respective regions in the unit of forced period. A limit cycle of nonintegral period appears in the shaded region Q, and a chaotic response is found in the region indicated by x ...

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