Harmonic input

It is interesting to note that the response of a system to a harmonic input is itself harmonic at the same frequency under the twin conditions of linearity and time invariance of the system properties for stable systems. For instability and receptivity problems, there is no general proof of the same due to the nonlinear nature of the dispersion relation, despite the fact that one is studying linearized Navier- Stokes equation. Thus it can at best be an assumption that is adopted in many analyses of this problem, except in Sengupta et al. (1994, 2006, 2006a) where the full time-dependent problem is solved as a transient problem by considering Bromwich contours in a— and u>- planes simultaneously. 

Figure 1.17 shows the variation of the main and satellite drop sizes versus the wave number of the first harmonic where the initial amphtudes of the sinusoidal disturbances are kept constant at = 0.01 and 2 = 0.05. Three sets of results are presented in this figure (a) first harmonic, only (b) added second harmonic with 0 = 0 and (c) added second harmonic with 6 = 180°. Two different behaviors are observed. For k < 0.5, when the added second harmonic is unstable, the breakup is highly dependent on the initial phase of the second-harmonic input. For no phase difference, the initial positive amplitude of the unstable second harmonic leads to satellites much larger than when no second harmonic is added. For very small wave numbers, the satellite drop becomes larger than the main drops. For 6 = 180°, which is equivalent to an initial negative amphtude of the second harmonic, the satellite drop sizes are significantly reduced. 

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. 

Let us demonstrate some results of the calculations made in [163] for an harmonic input field with a spatial frequency cjg (Table 5.2). Such a type of field could be formed, e.g., in photosensitive liquid crystal cells, when a photoconductor is illuminated by two coherent light beams which interfere with each other [163]. The harmonic distribution of light, as a result of interference, creates the corresponding harmonic dependence of the controlling field potential on one of the substrates of the liquid crystal cell (Fig. 5.25). We can determine the intensity of diffiraction in the first maximum, which correlates with the averaged square of the detector profile (0 ) (5.89), and we can calculate the Relative Modulation Characteristic (RMC) of the layer... 

Two types of force input were chosen. The first was single harmonic input (sine) and the second was a band-limited pseudo-random signal (Figure 5). The frequency range of study was from 0 to 50 Hz. The force levels chosen were representative of typical operating loads. 

Harmonic input was chosen for two reasons. The first is that a single steady state harmonic oscillation permits a simpler evaluation of system nonlinearities. There are many studies in the related literature which utilize this approach [10-14]. The second reason for choosing a harmonic input is that there exist road inputs such as long road undulations which are similar in nature. 

As in the case of single harmonic input, eight seconds of time domain integration were performed with a time step of 512 points per second. Seven different RMS force levels were utilized. For the three dampers studied this gave a total of 21 integration runs. 

Figure 7) Normalized acceleration spectra of the hub/wheel/strut assembly when subjected to 3 Hz single harmonic input.

The solution by Wood (1973) commonly used for critical facilities are, in fact, based on static 1 g loading of the soil-wall system and does not include the wave propagation and amplification of motion. On the other hand. Wood s solution is mathematically complicated to apply in engineering practice and is limited to harmonic input motions. Employing the finite element technique, in a simplified method proposed by Ostadan and White (1997) and Ostadan (2004) which incorporates the main parameters affecting the seismic soil pressure for buildings, the lateral seismic soil pressure on these structures... 

If A(co) is completely known, then the amplitude of the harmonic input signal X(co) can be calculated from the measured signal as... 

---