350 



THEORY OF SEAKEEPING 



0^5 



0,4 _ 



0.3 







10 20 30 40 5C 



Frequency, cps 



Fig. 8 Power spectral density of wing shear loads 



GiiCt) 



Fig. 9 Cross spectral density 



The bandwidth of the scanning filter in the XACA's 

 equipment can be adjusted to values ranging from 1/2 

 cycles per sec to 200 cycles per sec and the true-time 

 range of frequencies which can be analyzed is from 3 

 cycles per sec to 15,000 cycles per sec. 



By taking advantage of the possible changes in tape 

 speed previously mentioned, it is possible to obtain 

 equivalent filter band widths of 0.01 cycles per sec or 

 less and to handle frequencies ranging from a few hun- 

 dredths of a cycle to 50 or 60 kc. By changing tape 

 speeds where necessary, it is also possible to handle 

 record lengths ranging from a few tenths oi a second 

 to several minutes in duration 



To illustrate the use of the power spectrum analyzer, 

 Fig. 8 shows the spectrum of the shear loads on a fighter- 

 type airplane wing under buffeting conditions. The 

 continuous curve was obtained from a magnetic tape 

 record played through the analyzer and the circled points 

 were obtained by reading the time-history record and 

 numerically calculating the spectrum. 



The results from the two methods differ by a maxi- 

 mum of about 10 per cent at the low-frequency end of 

 the spectrum. This difference is due, in part at least, 

 to a large, undesii'ed, very-low-frequency component 

 in the original data sample which had to be attenuated 

 before the analysis. The electrical high-pass filter 

 u.sed to attenuate this component was not identical to 

 the equivalent numerical high-pass filter used for the 

 same purpose; therefore, some differences in the spec- 

 trum at the low-frequencj' end were to be expected. 



Some difference between the analyzer and the nu- 

 merical values also result from the fact that the shape 

 and bandwidth of the analyzer's scanning filter were 

 not identical to the equi\alent filter resulting from the 

 numerical process. 



Actually, differences of 10 per cent are not partic- 



ularly alarming. We are dealing with statistical proc- 

 esses such that, e\'en with errorless data processing 

 schemes, the results obtained from repeating the same 

 test might easily vary as much as plus or minus 25 to 

 30 per cent. jNIore will be said about this subject later. 

 Here, to read the record and determine the spectrum 

 using an automatic digital computer required about 10 

 man-hours; the time required for setting up and run- 

 ning the electronic analyzer was less than 30 min. 



Cross Spectral Density 



It has been shown that the power spectral density 

 furnishes information regarding the freciuency content 

 of fluctuating quantities which is not pro\ided by the 

 probability distribution. However, where the phe- 

 nomenon being investigated involves study of two or 

 more related fluctuating (|uantities, which is usually the 

 case, some knowledge of the time or phase correlation 

 between the two ciuantities might also be required. 

 This information is not provided by the power spectra 

 of the individual functions but may be obtained from 

 another statistical process called cross spectra analysis. 



The cross spectrimi between two time functions is 

 a \-ector quantity and two spectra are recjuired to fur- 

 nish the cf)mplete cross spectral density. These are 

 illustrated in Fig. 1). The real part is called the "co- 

 power spectrum" and indicates the product of the in- 

 phase frequency com])onents in the two functions. The 

 imaginary part is called the "c|uadrature .spectrum" and 

 indicates the product of the 90° out-of-phase freciuency 

 components in the two functions. The absolute value 

 and phase angle of the cross spectrum are deteiinined 

 by vectorially comliining tfle in-phase and quadrature 

 spectra. 



To gain an understanding of the physical significance 

 of cross spectra, consider two fluctuating time functions 



