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8 NONOSCILLATORY TRANSIENTS IN NONLINEAR SYSTEMS 8.0 INTRODUCTION In general, describing function concepts are directed primarily at the steady- state responses of nonlinear systems. The test inputs used to develop most describing functions are therefore based upon signal forms which are expected in steady-state system operation. We have seen this to be the case in the formulation of the sinusoidal-input describing function (DF), two-sinusoid- input describing function (TSIDF), and dual-input describing function (DIDF) of previous chapters. In the latter instance the linearization em- ployed also permitted determination of the transient response of a limit cycling nonlinear system under certain conditions. When a random process was considered, its statistical properties were determined on a steady-state basis. If, on the other hand, an approximate solution for the transient response of a non-limit-cycling nonlinear system is desired, we are led to consider aperiodic nonlinearity test inputs. In this chapter we treat two describing functions based on aperiodic test inputs. One is the transient-input describing function due to Chen (Refs.
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TRANSIENT-INPUT DESCRIBING FUNCTION 439 3, 4). It is formulated in a way considerably different from that presented in earlier sections of the text. Second is the exponential-input describing function. This new describing function is based on an exponential test input and follows very closely the describing function framework established earlier in the text. Both describing functions, emphasizing simplicity in use, are intended to facilitate the design of nonlinear systems. 8.1 The philosophy of quasi-linearization employed here is adopted specifically for the purpose of studying the transient response of nonlinear systems. For convenience, in transient investigation we take the input to be a step. This assumption, however, need not exclude the consideration of another form of input, which may be regarded as generated by the step response of a certain linear network. Nonlinearities under consideration are taken to be piece- wise-linear as a result of the approximation procedure to be presented. These are assumed to be followed by linear low-pass elements, as in conventional describing function theory. At the outset, a problem of circular nature arises. In order to perform an appropriate quasi-linearization of a nonlinear element it is necessary to test that element with a signal which simulates the actual input; however, determination of the actual input depends upon the nonlinearity quasi- linearization, which has allowed the system investigation in the first place. This apparent difficulty may be resolved by recalling that, since the non- linearity under consideration is piecewise-linear, resulting in a piecewise- linear overall system, the shifting of the nonlinearity operating point during the transient can be readily determined on a "marching" basis. A rough determination of the actual transient response in the first linear range via
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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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