The usual purpose of a frequency response test is to measure the linear dynamic response of the system. The tester usually wishes to make the input and output signals as large as possible in order to maximize the signal-to — noise ratio. However, the allowable maximum signal amplitude is limited by two considerations: the limits imposed by operational restrictions (maximum temperature, pressure, etc.) and the possible influence of nonlinear effects. In most nuclear reactor applications, the first restriction will dominate, but the nonlinear contamination problem may influence some tests. This section outlines current knowledge on nonlinear effects and how to minimize their influence.

It has been shown (6) that the output of a wide class of nonlinear systems may be given by the Volterra functional expansion:

oo

= J h^z) dl(t — z) dz + JJ h2(zl, z2) SI(t — z0 dl(t — z2) dzl dz2

00

+ЯІ m m i,‘+-

0 (2.13.1)

The first term on the right is the linear part is the impulse response) and all other terms represent nonlinear effects. The kernels hl, h2,h3,… consti­tute a complete representation of the system dynamics.

Equation (2.13.1) may be Fourier transformed to determine the influence of nonlinearities in a frequency response test. The Fourier transform of the first nonlinear term is

t 00

(1/T)J e~ja’ jj h^z^zjdlit — zjdllt — z2)dzldz2dt (2.13.2)

0 0

Interchange the order of integration to give

(l/T)jjh2(zl, z2) J e~Jo>,3I(t — zjdllt — z2)dtdzy dz2

0 0

The term within the square brackets may be written as follows:

<5/(0=

Then we may write

Г e“J“’ dl(t — tO 5I(t — z2) dt= e"J“’ <5/[f — — (T/2)]

J 772 J Г/2

or, letting p = t — T/2,

Г T r. T/2

e-Jat dl(t — T,) dl(t — T2) dt = e-JaTI2 e~Jap dl(p — T,)

•I T/2 Jo

xSI(p-T2)dp (2.13.7)

This may be substituted into Eq. (2.13.4) to give

r . cT’[5]

e-jat SI(t — t,) SI(t — r2) dt = (1 + e-jaTI2) е~іш SI(t — t,)

Jo Jo

x dl(t — r2) dt (2.13.8)

The factor 1 + e~j<oT/2 may be written

1 + cos(wT/2) — ;sin(wT/2)

Since o)T/2 = (2kn/T)(T/2) = kn and к is odd for an antisymmetric signal, we obtain

1 + e J“r/2 = 1 + cos kn — j sin kn = 0

Therefore, the first nonlinear term is identically zero if the input signal is antisymmetric. A similar analysis shows that all nonlinear terms contain­ing even-numbered kernels are identically zero and that the terms contain­ing odd-numbered kernels are not identically zero. Thus, half of the nonlinear terms can be eliminated by using an antisymmetric signal for frequency response testing. This will result in decreased nonlinear contamination unless the selected antisymmetric signal causes the remaining (odd-numbered) nonlinear terms to increase enough to overcome the reduction in nonlinear effects caused by elimination of the even-numbered terms. Limited practical experience (5) gave results in which the use of antisymmetric signals gave reduced nonlinear contamination, indicating that the elimination of the even-numbered terms was sufficient in that case.