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# gelb_ch2_ocr - 7 SINUSOIDAL-INPUT DESCRIBING FUNCTION(DF...

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7 SINUSOIDAL-INPUT DESCRIBING FUNCTION (DF) 2.0 INTRODUCTION Of the various describing functions discussed throughout this text, the sinusoidal-input describing function is by far the most widely known and used. In the following discussion the abbreviation DF is reserved for reference to this describing function. As the name implies, the DF is a linearization of a nonlinear element subjected to a sinusoidal input. In the next two chapters we see that different interpretations regarding the origin of this sinusoidal input allow the study of totally unrelated modes of behavior occurring in nonlinear systems. However, in each such study the representation for the nonlinearity is its DF. According to the theory of optimum quasi-linearization developed in the preceding chapter, the describing function for a nonlinearity driven by an input consisting of just a single sinusoidal component is given by a

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42 SINUSOIDAL-INPUT DESCRIBING FUNCTION (DF) specialization of Eqs. (1.5-36). These expressions are repeated here for convenient reference. NA = n, + jn, (2.0-la) 2 n, = - y(0) sin 8 A The nonlinearity input is in this case x(t) = A sin (wt + 8) where the amplitude and frequency, A and w, are deterministic quantities, 8 is the random phase angle which is uniformly distributed over 2~ radians. in Eqs. (2.0-1) is the output of the nonlinearity at an arbitrary time called zero. The output of a nonlinearity is a function of its input, and in some instances the relation is quite complex. To emphasize the fact that y(t) depends on in some way, we shall indicate in the notation a depend- ence on the current value of and its first derivative. This is not a restriction on the form of the nonlinearity, however. Any functional relation is acceptable. It is required only that the steady-state output history corresponding to a sinusoidal input be defined. In view of Eqs. (2.0-2) (2.0-3), = y(A sin 8, Aw cos (2.0 4) In general, the expectations indicated in Eqs. (2.0-1) are over all random parameters required to define y(0). For the single-sinusoidal-input case, the only random parameter in the problem is the phase angle 8. The expectation is then a single integral over the range of 8, which can be taken as 0, 27~. The probability density function for this uniformly distributed variable is 11257 in that interval. Equations (2.0-1) are thus specialized in this case to 1 2n n = - sin 8, Ao cos sin 8 dB (2.0-5a) " TrA o sin (2.0-5b) Historically, the basis for the DF was established by Krylov's and Bogoliubov's continuation in the field of nonlinear mechanics of an earlier work by Van der Pol. We shall briefly develop and study this basis to see
ASYMPTOTIC METHODS FOR THE STUDY OF NONLINEAR OSCILLATIONS 43 how it relates to the DF of current usage, derived from a quite different point of view. Once having established the significant relationships involved, the remainder of this chapter is then devoted to DF calculations for a variety of nonlinear elements. Static and dynamic, memoryless and with memory, implicit and explicit nonlinearities are treated.

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gelb_ch2_ocr - 7 SINUSOIDAL-INPUT DESCRIBING FUNCTION(DF...

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