By J. A. Richards
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Additional resources for Analysis of Periodically Time-Varying Systems: Communications and Control Engineering Series
1 of the equation when transformed to its lossless counterpart using Eq. 1}
4) 0 where 9 1 (t) is the coefficient of the first derivative (loss) term, as seen in Eq. 3). 1'} < 0 if Eq. 4) applies. 2 The Meissner Equation The Meissner equation is a lossless Hill equation with a rectangular waveform coefficient. Its discrete transition matrix and solution have been treated in Sect. 4. It is perhaps the most readily handled Hill equation and is used here to illustrate stability. For a rectangular waveform coefficient as illustrated in Fig. 1, in which T is the length of the positive segment and n the period of the coefficient, the discrete state transition matrix is given by Eq.
32). Therefore periodic equations which lend themselves to an analytical determination of forced response are those whose coefficient waveforms render the homogeneous forms amenable to exact solution. Notwithstanding this, the above qualitative results are important guidelines in the interpretation ·of forced response in general. 15 Phase Space Analysis Certain applications involving second order Hill equation descriptions do not require a complete record of a solution but rather demand a knowledge only of specific properties such as its global maximum.
Analysis of Periodically Time-Varying Systems: Communications and Control Engineering Series by J. A. Richards