348 



THEORY OF SEAKEEPING 



Response 



Tfme 



Frequency 

 Fig. 5 Power spectral density 



ing equipment. Consequently each sample on the tape 

 is spliced into a continuous loop before being played 

 back into the FM demodulators and the analyzers. 



The three types of analyzers through which data re- 

 corded on tape may be processed are a probability dis- 

 tribution analyzer, a power spectral density analyzer, 

 and a cross spectral density analyzer. These three 

 analyzers and the type of statistical information pro- 

 duced by each will be discussed in the following sec- 

 tions of this paper. 



Probability Distribution 



The time function shown at the top of Fig. 2 illus- 

 trates a randomly varying cjuantity of the type to be 

 considered throughout this paper. One way to ana- 

 lyze such a function is to determine its probability dis- 

 tribution. This is a measure of the proportion of total 

 time during which the amplitude of the varying quan- 

 tity exceeds given levels. 



For example, suppose the time function shown at the 

 top of the figure were a plot of the gust velocities en- 

 countered by an airplane flying through rough air. 

 The intensity of the turbulence could be shown by plot- 

 ting the probability that the gusts would exceed specific 

 velocities against gust velocity. The gusts in the at- 

 mosphere represented by the distribution shown would 

 be expected to exceed 3 or 4 fps about 95 per cent of the 

 time and to exceed 20 fps only about 1 per cent of the 

 time. 



The electronic analyzer used to determine probability 

 distributions is illustrated in Fig. 3. The data sample 

 recorded on an endless loop of magnetic tape is contin- 

 uously played into the analyzer, and its instantaneous 

 amplitude is compared to a reference voltage En. The 

 level of the reference voltage is determined by the po- 

 sition of the potentiometer slider. If the input voltage 

 is larger than the reference voltage, the amplifier will 



Transfer 



Function 



Y(f) 



Fig. 6 Input-output power spectral density relationship 



be saturated and the output of the limiter will be at 

 level B: if the input is less than the reference voltage, 

 the amplifier will be cut off and the output of the lim- 

 iter circuit will be at level .4. The percentage of time 

 that the level is at B then represents the probability of 

 the input being above the fixed level E^. By averaging 

 the output of the limiter this probability may be read 

 directly and recorded. By slowly changing the level 

 of the reference voltage and driving the paper on the re- 

 corder in synchronization, a complete plot of the prob- 

 ability distribution may be obtained. 



To illustrate the use of the analyzer, I-'ig. 4 shows a 

 distribution of wing buffeting loads obtained during a 

 gradual pull-up at high speeds. The solid line is the 

 analyzer record and the circled points are values deter- 

 mined numerically. Setup and running time on the 

 electronic analyzer was about 20 to 30 min and about 7 

 man-hours were recjuired to read the time-history rec- 

 ord, calculate the mean, and determine numericallj' the 

 circled points. 



Power Spectral Density 



The probability distribution describes the intensity 

 characteristics of the data but does not describe their 

 frequency or spectral characteristics. Nor does it fur- 

 nish information sufficient for calculating system input- 

 output relationships or transfer functions. If this sort 

 of information is required, a second type of statistical 

 analysis called the power spectral density analysis may 

 be u.sed. 



Power spectral density analysis is sometimes referred 

 to as "generalized harmonic analysis" and is similar to 

 the familiar Fourier series harmonic frequency analysis 

 except in the following re.spects: (1) Fourier series anal- 

 ysis is applicable to repetiti^'e functions while general- 

 ized harmonic analysis is applicable to stationary ran- 

 dom time functions, (2) as illustrated in Fig. 5, the .spec- 

 ti'um is repi'esented as a continuous curve rather than 

 as discrete harmonically-related frequencies, and (3) 

 the spectrum is plotted in terms of mean-scjuared am- 



