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5 TWO-SINUSOID-INPUT DESCRIBING FUNCTION (TSIDF) 5.0 INTRODUCTION Circumstances which lead to periodic but highly nonsinusoidal nonlinearity inputs render the DF approach invalid. In a number of such cases the nonlinearity input is well described as being composed of two additive sinusoids. These sinusoids result, in most cases of interest, from a sinusoidal input to the system, from system limit cycles, or both. The input to the nonlinearity is assumed, for the purpose of describing function calculation, to have the form x(t) = A sin (w,t + 6,) + B sin (wBt + OB) (5.0-1) in which the amplitudes A and B and frequencies w, and will be deter- mined by the nature of the system and its inputs. From the point of view of the general describing function theory developed in Chap. 1, the random variables which characterize this input are the phase angles 6, 6,. When two or more sinusoidal components are assumed at the nonlinearity input, the statistical independence of these components, which if true
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INTRODUCTION 251 simplifies describing function calculation considerably, must be established with some care. Since each sinusoidal component is characterized by an amplitude and frequency which are deterministically fixed, and by a phase angle, the independence of the input components is established if the phase angles can properly be described as independent random variables. There are a number of important situations in which these phase angles are not independent. If one sinusoid were harmonically related to the other, the periods of the sinusoids would be commensurate, and a consistent phase relation would exist between them. The nature of the nonlinearity output would depend on this relative phase, and the quasi-linear approximator should reflect that dependence. In that case, with two sinusoidal inputs, only one phase angle could be treated as a random variable with a uniform distribution over one cycle; the other phase angle would be deterministically related to the first. With the statistical approach of Chap. 1, the coupled set of integral equations indicated in Eq. (1.5-13) would have to be solved to determine the optimum quasi-linear approximator for the nonlinearity. Alternatively, the interpretation of these describing functions as the amplitude and phase relation between each input component and the harmonic com- ponent of the same frequency in the nonlinearity output can be employed for describing function calculation. This latter point of view is expressed in the statements phasor representation of output component of frequency w, NA = (5.0-2) phasor representation of input component of frequency w, o, NB = (5 .O-3) phasor representation of input component of frequency wB where the subscript on Ndenotes the input component to which the describing function applies.
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This note was uploaded on 02/03/2012 for the course AERO 16.30 taught by Professor Ericferon during the Spring '04 term at MIT.

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