■■■-jjiWt^

FBR _31Ü0 AFFDL-TM-75-37-FBE

AIR FORCE FLIGHT DYNAMICS LABORATORY DIRECTOR OF SCIENCE & TECHNOLOGY

AIR FORCE SYSTEMS COMMAND

WRIGHT-PATTERSON AIR FORCE BASE OHIO

STUDY OF EXCEEDANCE.CURVE SPREAD BY AIRCRAFT TAIL NUMBER BY MISSION

TYPE & BASE FOR F-4 AND A-37B FLIGHT DATA

DAVID L. BANASZAK

FEBRUARY 1975

TECHNICAL MEMORANDUM 75-37-FBE

Reproduced From Best Available Copy

APPROVED FOR PUBLIC RELEASE; DISTRIBUTION UNLIMITED

RETURN TO: AEROSPACE STRUCTURES INFORMATION AND ANALYSIS CENTER

AFFDL/FBR WPAFB, OHIO 45433

J

fr?- — ■&-

*:- ..*

FOREWARD

This report was prepared by Mr. David L. Banaszak of the

Structural Integrity Branch, Structures Division, Air Force Flight

Dynamics Laboratory. This effort was performed under Project 1367,

"Structural Integrity for Military Aerospace Vehicles," Task 136701,

"Structural Flight Loads."

This exercise was initiated largely in response to determina-

tions reached in a series of meetings with personnel in ASD/ENF;

namely Messrs. W. J. Crichlow, C. W. Luchsinger, and Troy King.

These meetings were convened to discuss methods and techniques for

analyzing and utilizing flight loads data in modifying and updating

MIL-A-8866B specification.

Appreciation is extended to Messrs. W. J. Crichlow, C. W.

Luchsinger, and Troy King for their guidance and suggestions in these

data analyses.

This technical memorandum report has been reviewed and is

approved.

{^fä**j.tJQ*{jL DENNIS J. GOLDEN, Major, USAF Chief, Structural Integrity Branch Structures Division AF Flight Dynamics Laboratory

SUMMARY

This report presents the results of the analysis of exceedance

curves by aircraft tail number for the F-4 and A-37B aircraft.

The aim of this report is to provide a statistical bound for

the different exceedance curves that are obtained from various

aircraft tail numbers on a given aircraft type. The F-4 and A-37B

aircraft were selected only because load factor (nz) exceedance data

by tail number were available in table formats in References 1 and 2.

Results of this report indicate that reasonable bounds on

exceedance curves may be obtained by tolerance limits which assume

that for a fixed load factor (nz), the log of the exceedances per

thousand flight hours has a normal distribution. Also, tolerance

limits could be obtained using nonparametric techniques when the

quantity of tail numbers is sufficiently large.

Finally, curves were fitted to the composite data. The designer

may extrapolate these curves to determine design limit load factor

and change the shape of the curves by varying values of coefficients.

ii

TABLE OF CONTENTS

Page

ii

viii

SUMMARY

DEFINITION OF SYMBOLS

SECTION I - Introduction ' 1

SECTION II - Discussion 2

1 Flight Data 2

2 Computer Programs g

3 Data Presentation 7

3.1 Exceedance Curves by Aircraft Tail Numbers 7

3.2 Confidence Intervals for the Median 7

3.3 Maximum and Minimum Values 9

3.4 Tolerance Limits Assuming Normal Distribution 10

3.5 Tolerance Limits Assuming a Log Normal 11 Distribution

3.6 Composite Exceedance Curve Fitting 13

SECTION III - Conclusions 16

SECTION IV - Recommendations 17

Appendix I - Data Results 19

Appendix II - Computer Programs Used 57

REFERENCES 69

111

LIST OF FIGUBES

FIGURES PAGE

1 Nz Exceedance Curves by Tail Number for 20

A-37B at England AFB (1969)

2 N Exceedance Curves by Tail Number for 21 z A-37B at Bien Hoa AB (1970)

3 N Exceedance Curves by Tail Number for 22

A-37B at Bihh Thuy AB (1971)

4 N Exceedance Curves by Tail Number for 23

A-37B at England AFB (1971)

5 Nz Exceedance Curves by Tail Number for 24

F-4 Air-Ground

6 Nz Exceedance Curves by Tail Number for 25

F-4 Air-Air

7 N Exceedance Curves by Tail Number for 26 z F-4 Inst. & Nav.

8 N Exceedance Curves by Tail Number for 27 z F-4 Reconnaissance

9 N Exceedance Curves by Tail Number for 28 z F-4 Test

10 Nonparametric Bounds for A-37B at 29 England AFB (1969) Exceedances

11 Nonparametric Bounds for A-37B at 30 Bien floa AB (1970) Exceedances

12 Nonparametric Bounds for A-37B at 31 Binh Thuy AB (1971) Exceedances

13 Nonparametric Bounds for A-37B at 32 England AFB (1971) Exceedances

iv

flip!., i. Ill -VjW«

LIST OF FIGURES (Cont'd)

FIGURES PAGE

14 * Nonparametric Bounds for F-4 33 Air-Ground Exceedances

15 Nonparametric Bounds for F-4 34 Air-Air Exceedances

16 Nonparametric Bounds for F-4 35 Inst. & Nav. Exceedances

17 Nonparametric Bounds for F-4 36 Reconnaissance Exceedances

18 Nonparametric Bounds for F-4 37 Test Exceedances

19 90% Tolerance Limits with 90% 38 Confidence for A-37B at England AFB (1969)

20 90% Tolerance Limits with 90% 39 Confidence for A-37B at Bien Hoa AB (1970)

21 90% Tolerance Limits with 90% 40 Confidence for A-37B at Binh Thuy AB (1971)

22 90% Tolerance Limits with 90% 41 Confidence for A-37B at England AFB (1971)

23 90% Tolerance Limits with 90% 42 Confidence for F-4 Air-Ground

24 90% Tolerance Limits with 90% 43 Confidence for F-4 Air-Air

25 90% Tolerance Limits with 90% 44 Confidence For F-4 Inst. and Nav.

26 90% Tolerance Limits with 90% 45 Confidence for F-4 Reconnaissance

LIST OF FIGURES (Cont'd)

FIGURES ' PAGE

27 90% Tolerance Limits with 90% 46 Confidence for F-4 Test

28 Composite Data Curve Fit for A-37B at 48 England AFB (1969)

29 Composite Data Curve Fit for A-37B at 49 Bien Hoa AB (1970)

30 Composite Data Curve Fit for A-37B at 50 Binh Thuy AB (1971)

31 Composite Data Curve Fit for A-37B at 51 England AFB (1971)

32 Composite Data Curve Fit for F-4 Air- 52 Ground

33 Composite Data Curve Fit for F-4 Air-Air 53

34 Composite Data Curve Fit for F-4 Inst. 54 and llav.

35 Composite Data Curve Fit for F-4 55 Reconnaissance

36 Composite Data Curve Fit for F-4 Test 56

37 Program to Store Data on A Tape Cassette 58

38 Program to Read Data on the Tape Cassette 59

39 Program to Analyze the Data by Tail Number 60

40 Revised HP Plot Pac Program 65

vi

^^^^^^^^^^F"

m-

LIST OF TABLES

Tables

I

II

III

Title

Summary of A-37B Data Used

Summary of F-4 Data Used

Curve Fit Coefficients

Page

3

3

47

vii

DEFINITION OF SYMBOLS

nz

MED (nz)

Ei(nz)

LCI(nz)

UCI(nz)

C

MAX(nz)

MIN(nz)

M(nz)

s(nz)

vertical load factor in g's.

the median value of exceedances per 1,000 hours for ä fixed value of nz.

the number of exceedances per 1,000 hours for a fixed value of nz for the ith aircraft tail number.

the lower bound of the 90% confidence Interval for MED(nz).

the upper bound of the 90% confidence interval for MED(nz).

the composite exceedance curve.

the maximum number of exceedances per 1,000 hours for a fixed value of nz.

the minimum number of exceedances per 1,000 hours for a fixed value of nz.

the percent of tail numbers that lie between 2 given limits.

confidence, i.e. the probability that a given statement is true.

mean of exceedances per 1,000 hours for fixed nz.

standard deviation of exceedances per 1,000 hours for fixed nz.

NorUL(nz)

NorLL(nz)

M3=log10(C(nz)) used as the mean for log normal assumptions

upper bound of 90% two-sided tolerance limit with 90% confidence assuming normal distribution.

lower bound Of 90% two-sided tolerance limit with 90% confidence assuming normal distribution.

viii

DEFINITION OF SYMBOLS (cont'd)

s'(nz)

LNUL

LNLL

LgNorUL

LgNorLL

F

b(nz)

M w

standard deviation of the logs of the Ei(nz) for a fixed nz.

Upper bound on log scale of 90% two-sided tolerance limits with 90% confidence of exceedances per 1,000 hours.

löwer bound on log scale of 90% two-sided tolerance limits with 90% confidence of exceedances per 1,000 hours.

10

10

(LNUL)

(LNLL)

curve fitted to nz composite exceedance per 4,000 hours,

best fit polynomial that log F = b(nz).

Weighted mean.

IX

!'■ i"^^wi •

SECTION I

INTRODUCTION

As a result of a series of meetings with Mr. W. J. Crichlow,

ASD/ENF, it appears that some changes are necessary in the pre-

sentation of flight loads data. This is expected to improve the

development of more definitive structural design criteria. Mr.

Crichlow, who has prime responsibility for MIL-A-8866B revision,

explained the need for re-examining the statistical variations in

vertical load factor (n ) exceedance curves and this distribution's z

impact on more precise aircraft design, and fatigue, and fracture

analyses. Accordingly, AFFDL/FBE initiated a program to analyze

flight loads data, categorized by mission segment, type, and tail

number as available, to determine the load's distribution mean

exceedance curve and the statistical spread about the mean.

Load factor (n ) data from A-37B (Reference 1) and F-4 Z

(Reference 2) aircraft were segregated by tail number, base, and

mission type as available in the reports. The data from these

subdivisions were analyzed and the results plotted in the form of

exceedance curves.

These methods provide a precise procedure for criteria development

by computing exceedance curves with a mean and tolerance limits for

all possible exceedance curves in the given category. The results

appear promising in providing a more descriptive USAF military speci-

fication, which in turn, necessarily results in more refined structural

design for future aircraft.

j SECTION II

DISCUSSION

To achieve; tha objectives stated above, several methods of

analyzing the flight loads data as extracted from References 1 and

2 were employed. For each set of data by aircraft tail numbers, the

following statistical functions were computed for the vertical load

factor (n ) exceedance curves: z

a. Maximum, minimum and median exceedance curves with a 90%

confidence interval for the median.

b. 90% two-sided tolerance limits with 90% confidence assuming

the exceedances are normally distributed.

c. 95% two-sided tolerance limits with 95% confidence assuming

the log of the exceedances are normally distributed.

d. Least squares curve fits to the log of the composite data.

Plots of the above results are shown in Figures 1 through 36 in

Appendix I of this report. Detailed discussions of the flight data,

computer programs, and assumptions used to obtain the above results

are presented in the following paragraphs of this report.

1. Flight Data

The load factor (n ) peak data used for this study were obtained z

from References 1 and 2. Reference 1 contained operational flight

loads data from the A-37B aircraft while performing training flights

from May 1969 to September 1971. A total of 4,001 hours of data was

analyzed and is summarized in Table I. For a detailed description

of the data, the reader is referred to Reference 1.'

^#i

|W .1.1 i.iH#M|.»ll ™.l. II Uli. IllJIIflJ,.. ,. ,1. ,".■, I ^-^-— F

TABLE I

...

SUMMARY OF A-37B DATA USED1

Base and Year Hours Tail Numbers Hours/Tail Number

England AFB (1969) 541.47 11 49.22

Bien Hoa AB (1970) 2038.23 12 169.85

Binh Thuy AB (1971) 913.84 8 114.23

England AFB (1971) 507.87 4,001.41

4

TABLE II

126.97

SUMMARY OF F-4 DATA USED2

Mission Type Hours Tail Numbers Hours/Tail Number

Air-Ground 2308.6 46 50.19

Air-Air 149.0 23 6.48

In-Nav 481.2 40 12.03

Recon 515.2 19 27.12

Test 19.1 3,473.1

15 1.27

Reference 2 contained F-4 aircraft Southeast Asia (SEA) load

factor data that were segregated by tail number, and mission types.

These data were collected by Technology, Incorporated (TI), as part

of Aircraft Structural Integrity Program (ASIP) during the period

of 15 August 1969 through 31 December 1970. The 3,473 total hours

analyzed are summarized in Table II.

The above referenced reports were selected since the data are

in terms of aircraft tail number by mission type or base. However,

these data are not in terms of mission segment as desired. It

seems reasonable to assume that segregation by base, and mission

type would be as applicable to the analysis methods as segregation

by mission segment.

Before proceeding, some of the problems encountered in using

the above data should be considered. One problem that appeared in

Tables I and II was that both aircraft types have what appeared to

be a large total number of hours, but may have too small a number

of hours for any given tail number and category of mission or base.

For example, there is 4,001 hours of F-4 data but only an average

of 50 hours per tail number in the air-ground mission category, with

other mission categories having even less hours. This raised the

question about the validity of an exceedance curve for a given tail

number. A small number of hours for a tail number would probably

give the exceedance curves a wider spread than is actually the case;

this would tend to make any estimates about the variability to be

unnecessarily large. If the criteria, that 1,000 hours of data are

needed to plot a valid exceedanee curve is used, it would be necessary

to have at least 46(1,000) +40(1,000) + 19(1,000) + 15(1,000) =

143,000 total hours of F-4 data to obtain valid exceedanee curves

for all the aircraft tail numbers and categories listed in Table II.

This is a large data requirement.

Another question concerns the type of distribution the exceedances

have for a given value of n . It will be shown that normality should

probably be ruled out, but the log of the exceedances being normally

distributed is a plausible assumption. The assumption of a nonpara-

metric (not normal) distribution would also provide some usable

results, but more tail numbers would be required, and hence more data.

Lastly, the problem of zero exceedances at high nz values caused

a number of interrelated problems, for example zero peaks in 50 hours

implies zero peaks per 1,000 hours is not a valid assumption. That

is, the real number of peaks per 1,000 hours at some high load factor

may have really been 10 peaks implying .5 peaks in 50 hours. Since

there can be no fractional count of a peak, 0.5 peaks in 50 hours of

flight cannot occur. Even with over 1,000 hours of data, the result

is questionable since occurrences of nz peaks at high nz values

(above 7 g's) are rare. Handling of this zero occurrences problem

is discussed throughout this report.

It was decided that these data would suffice for this study,

since there did exist variability of the exceedanee curves for each

aircraft tail number. However, for better results, more data hours

per aircraft and more tail numbers should be used.

2. Computer Programs

Several computer programs (See Figures 37-40) were used on the

Hewlett-Packard (HP) 9830 calculator system to verify the validity

of some new approaches to analyzing flight loads data. The computer

programs were used to generate plots, tolerance limits, confidence

intervals and curve fits for the n exceedance data used herein. z

One computer program, which handles a maximum of 49 tail numbers

and 16 different values of n along the abscissa, used the n peak data

obtained from the references, and stored it on a file of a tape

cassette. Hence, each file of the tape cassette contained n data z

by tail number for a particular mission or base as shown in Tables

I and II, and also the composite data which includes all tail numbers.

The tape cassette was used to input the n data in ah analysis

program. The analysis program was used to analyze the data as

follows:

a. Plot n exceedance points per 1,000 hours for each tail number.

b. Plot n exceedance curves for each aircraft tail number. z

c. Compute and plot the minimum, maximum and median exceedance

per 1,000 hours for each of the 16 n values. z

d. Compute and plot a 90% confidence interval for the median of

the exceedances per 1,000 hours.

e. Compute and plot the mean and composite exceedances for each

of the 16 n values, z

f. Compute and plot tolerance limits on the exceedances per 1,000

hours for each n stratum assuming that the exceedances or the log z ■

of the exceedances are normally distributed.

A third computer program, which was a revision of a HP Plot Pac

Program, performs a least square.fit of a polynomial to the log of the

exceedance data. Composite data for all tail numbers for each category

were input through the keyboard to fit curves of the form

F-10b(V»

where F is the number of exceedances per 4,000 hours and b(nz) is

the best polynomial that estimates log F.

3. Data Presentation

Results of the analysis are presented in Figures 1 through 36 and

Table III in Appendix I. A detailed discussion of the various methods

that were studied is presented below.

3.1 Exceedance Curves by Aircraft Tail Numbers

For each of the nine categories listed in Tables I and II,

the exceedance curve for each individual tail number has been plotted

in Figures 1 through 9. These exceedance curves give a rough idea

of the type of spread that might be expected for different aircraft

tail numbers. The following sections primarily investigate the means

by which statistical statements can be made in describing this spread

in the exceedance curves.

3.2 Confidence Intervals for the Median

The median of the exceedances per 1,000 hours (MED(nz>) for

each given value of n have been plotted in Figures 10 through 18

for each of the nine cases listed in Tables I and II. Straight

line segments were also drawn connecting the MED(nr) values. The

median was calculated so that one can say that about 1/2 of the

exceedances per 1,000 hours for each n (E. (n ) where i is the tail z 1 z

number) lie above and below the MED(n ) value. z

Next, a 90% two-sided confidence interval (CI) for the

median was found using a distribution free procedure based on the

sign test as outlined in Reference 3. The procedure was applied to

each E.(n ) for a given n so that the 90% confidence interval for

the median corresponds to the statement: the probability

LCI(n ) <_ MED(n ) <_ UCI(n ) = 0.90 z z z

where LCI(n ) is the lower limit of the CI and UCI(n ) is the upper

limit of the CI (i.e. the probability that the true median lies

between LCI(n ) and UCI(n ) is 0.90). z z

All E.(n ) = 0 cases were included for the above computation.

In addition, the composite was also plotted and found within the 90%

CI for the median for the lower values of n . Figures 10 through 18

showed that a difference exists between the median and the composite

(C) for the F-4 data whereas they are very similar for the A-37 data.

In fact, for the F-4 data the median usually lies below the composite.

This is probably due to the greater frequency of E.(n ) = 0, for the

F~4 data which was caused by an inadequate number of hours per tail

number as shown in table II and discussed earlier in data presenta-

tion. It should be noted that 1 peak in 100 hours of data would

imply 10 peaks pet thousand and may overestimate a theoretical value

of 1 peak per thousand hours by quite a margin. Intuitively, it then

seems that for the F-4, the composite would usually lie below any

exceedarice per tail number due to the overestimation of the exceedance

for a limited number of hours.

3.3 Maximum and Minimum Values

The maximum and minimum values-of ^(n ) are also plotted in

Figures 10 through 18. The maximum (MAX(nz)) and minimum values

(MIN(n )) are plotted for each n value and a line drawn connecting

the points. These curves may then be used to make distribution free

two-sided tolerance limit statements which means that with some confi-

dence gamma (y), a percentage of the tail numbers (P) lies between the

two limit values. By using Table A-32 in Reference 4, values of y

are obtained so that 90% of the tail numbers lie between the maximum

and minimum values of n exceedances per 1,000 hours. z

In Figures 10 through 18, the values of y have been included

on the plots. Note that gamma was higher as the number of tail numbers

increased. For example, Y - -30 for 11 tail numbers in Figure 10

and Y = 0.95 for 46 tail numbers in Figure 14. Hence, we can use the

maximum and minimum value of exceedances per 1,000 hours as statistical

bounds. This method would give a very conservative estimate of the

bounds on the exceedance curves since there are no assumptions made

about the underlying distribution. For Figures 10 through 14, the

bounds determined in this manner appear to be relatively narrow.

However, in Figures 15 through 18, the minimum is zero too often

since these cases have a small average hours per tail number.

3.4 Tolerance Limits Assuming Normal Distribution

In Figures 19 through 27, a plot of a 90% two-sided toler-

ance limit with 90% confidence is shown with the assumption that for

a given nz> the exceedances per 1,000 hours have a normal distribu-

tion.

The tolerance limit was computed by considering the E (n )s i z

for each given n^. The mean M(n ) was calculated using

N M(n ) = Z E (n )

Z i=l 1 z

N

where E (n ) is the exceedances per 1,000 hours for a fixed n and N -1- z z

is the number of aircraft tail numbers. Then, the sample standard

deviation(s) were computed by

/

s(nz) =

n E E (n )2 - ( E E. (n )?\1/2

i-1 x z i=i x z I

n(n-l)

which was obtained from Reference 1 on page 1-10.

The upper NorUL and lower limits NorLL were

NorUL(n ) = M(n ) + K-s (n )

10

and

NorLL = M(nz) - K>S (nz)

where K is obtained from Table A-6 of Reference 4 with y = .90,

P - 90% and N'- number of tail numbers. Before proceeding, it should

be noted that all zero exceedances per 1,000 hours were included in

the computations.

the lower limit did not appear in most of the figures,

because the assumption, that the exceedances are normally distributed,

allows the lower limit to take on negative values. This is not

possible. Another factor to consider concerns plotting of limits

on a log scale. Since

| log (M + KS) - log M | < | log (M-KS)-log M |

the upper limits would be closer to the mean than the lower limits

on thu log scale.

As was the case for the median, the mean and composite are

approximately equal at low nz values but differ at higher nz values.

This is again probably due to errors in exceedances per 1,000 hours

that occur at the higher n values. z

It was concluded from the above discussions and Figures 19

through 27 that the assumption of a normal distribution was not valid.

3.5 Tolerance Limits Assuming a Log Normal Distribution

Plots of 90% two-sided tolerance limits with 90% confidence

are shown in Figures 19 through 27 with the assumptions that the log

11

of the exceedances per 1,000 hours for each n has a normal distri-

bution. The mean M3 was computed as the common logarithm of the

composite (C(n )) at each n . Note, the composite is the same as z z

the weighted mean (M ) defined by

? E.(n ) M = £ .1 z w 1=1-^7- which equals the total number of occurrences greater than n divided z

by total time. This result is the composite. Next, a standard

deviation s'(n ) is defined as z

s'(n ) = I (log W - log C(n )2 1 1/2 Z I . - z

I1"1 N

Since, the log of zero is undefined, zero values of E (n ) were not X z

included in the solution for s'(n ). N was decreased by 1 each time

a zero was excluded for a fixed value of n . z

Then, as in the previous section, upper (LNUL) and lower (LNLL)

tolerance limits were computed by formulas:

LNUL(n ) = log C(n ) = K.S'Cn ) z z z

and

LNLL(n ) = log C(n ) - K-s'(n ). Z z *

These upper and lower tolerance limit (LNUL, LNLL) curves are then

equidistant from the composite when plotted on semilog paper as is

evident from Figures 19 through 27. For most cases, the log normal

assumption gave some reasonable bounds but deviations do occur. For

12

instance, at the extreme n values, the limits go to infinity and ■ , z

zero for the upper and lower bounds respectively. This is because

at the extreme n values there are many zeroes which make N small, ■ z

and K which was obtained from Table A-6 becomes very large. Hence,

LNUL and LNLL become very large and small respectively as can be seen

clearly in Figure 26. The spike at nz - 6 g's on Figure 22 resulted

from one aircraft tail number exceedance curve being farther from

the mean than the other three; thus, making a higher than probable

standard deviation. It was decided, that overall, the log normal

assumption did provide reasonable bounds for nz exceedance curves.

3.6 Composite Exceedance Curve Fitting

The final computer program provided for the fitting of

polynomial equations to the log of the exceedances per 4,000 hours.

Four thousand hours was chosen because it is the design life for most

fighter-type aircraft. The fitted curves that were obtained are

presented in Figures 28 through 36 and summarized in Table III.

In most instances, a third degree polynomial provided the

best fit, but in instances where this was not a good fit, (e.g. the

curve goes to plus infinity as n goes to infinity) the best straight

line was fitted through the data points. In most cases when the

straight line fit was used it proved to be adequate.

These fitted curves to the n exceedances per 4,000 Uours z

can be used as a uniform method of extrapolation to one exceedance

per 4,000 hours (i.e. the aircraft life to obtain an estimate of a

realistic design limit load factor).

13

In addition, these equations may be used to change the

magnitude and slope of the exceedance curves for test purposes.

The equation has the form

F(x) = P.10B(X),X

where F is the curve function, x corresponds to n and B is a function 2

of x of the form

B(x) - b2x2 + bjX + bQ

For the case of the cubic equation for positive n in Figure 32, the

equation for the exceedance curve for F-4 air-ground data was found

to be

w v 1A(6.9742-1.4829x + ,2914x2-O.0249x3) F(x) - 10

(See Table II in Appendix I)

This equation can be reduced to

w/ x 1„6.9742lrtx(-.0249x + .2914x- 0.0249x3)

F(x) = 10 10

This means we can place

and

P = 106-9742= 9,423,234

B(x) - (-.0249x2 + .2914x-1.4829)

P will define the point at which the curve intercepts the n = 0

axis and B(x) defines an instantaneous slope to the exceedance curve

on the log scale. This takes into account the changes in slope which

14

is typical for fighter nz exceedance curves. As long as t>2 is a

negative coefficient (-.0249 for the example cited above), we know

that the fitted curve will approach zero as nz increases without

bounds.

With these equations, ä series of curves could be obtained

by varying the magnitude P and coefficients of the slope B(x) such

that they do not exceed the bounds determined by using one of the

previous techniques. The effect of varying the P and the B(x) has

not been studied in this report, but might be an approach that may

prove effective.

Extrapolation of these curves to design limit load factor

is now a simple procedure. Since F is the exceedances per 4,000

hours, find the value of nz such that the log F = log 1 = 0. For

example in Figure 28 for A-37B, England AFB, 1969, we would find

a design limit load factor of 6.8 g's.

15

SECTION III

CONCLUSIONS

The plotting of n exceedance curves by aircraft tail number

to obtain statistical bounds did appear to be feasible. Finding

two-sided tolerance limits for exceedance curves with the assumption

that the log of the exceedances per 1,000 hours are normally distri-

buted appeared to give reasonable bounds. Nonparametric techniques

gave reasonable, but more conservative bounds. The assumption that

the exceedances per 1,000 hours were normally distributed appeared to

give the widest bound and hence were not as useful. Curve fitting

polynomials to the log of the exceedances per 4,000 hours for compo-

site n data provided usable equations for representing n exceedance

curves in most cases for the two aircraft types that were considered.

In addition, the polynomials provide a standard mean of extrapolating

n exceedance curves to design limit load factor.

16

SECTION IV

RECOMMENDATIONS

As noted earlier, the data used for the F-4 represented a small

amount of flight time per tail number. A further study should be

made to determine what minimum time on a tail number is required in

each category to ensure that a representative exceedance curve can

be obtained for that particular aircraft. The A-37B data indicated

that 100 hours per aircraft tail number would probably be sufficient,

but more information would have to be studied before this statement

could be made with certainty.

Separating the data into mission segments would increase the

need for a larger data base. For example, if nbnparametric techniques

are used, then at least 50 tail numbers should be used for each

mission segment. If 100 hours per tail number are required and five

mission segments for a typical aircraft type are assumed, there would

need to be at least 25,000 hours of data on the particular aircraft

type. If the data were not categorized by mission segment, 5,000

hours would suffice, and hence, it is seen that the total number of

hours needed is directly proportional to the number of categories

within the data base. Thus, it should be remembered that segregating

data into mission segment would greatly increase the amount of data

that needs to be collected.

Also, the uncertainty of the number of exceedances per 1,000 hours

at higher n values is of concern. A minimum of 100 hours per tail

number would be a minimum requirement; however, there is still a problem

17

when zero exeeedances per 1,000 hours occur, because exceedance per

1,000 hours approaches zero as n increases, but does not really

reach zero. These problems at high n values could easily form the

major topic of further study.

Lastly, the aircraft for this report were both fighter-type

aircraft. For other aircraft, like transport, other techniques may

be more applicable than those cited in this study.

18

APPENDIX I

DATA RESULTS

19

--1 - --1-- -.-.-._.... - --(■— I— — -' - ■ -ffgp---ppf 11 Tail Numbers 541,47.Total Hours

: ASD-TR-72-1 Volume I T

i 000000 -■! ■ 1 -■ Pv ;4 1- ■

j |T F hit :z(- - } ;.;', 1 - m I::: rt - — —^ -■ — — ........... — — _

b __ - xvßv ■•" . —I 1 . Ü-

1 1--|— | rv^V J,t 1 _, L: [ ^/Bfe-i Mil I \ X^ lsl--i -UuJ . oooao -■-■-:—--j— r ■ . a [_■■■!. j | k . V

k i V^. 1 *k 1 /•* i S \IN\\ """ " . :.".+:

If] 'v is' N\A I 1 I- /» L \ :U8t:::..T7i rh_zi

11 /I i J_ ■ J_ ^ \ >oj\ks. L' ■ j.[ J__.. |f| I ' t *\ i \\ \ ^ \

■ÜÜOO ■' ! 'ill '' \\ \\\! V

\ [ Yl \ \ \ \ . i" 3 ^ i'fl r~ ... \ _i__5^i vW

\ 1 Nl\\\ V li ^ JC ' n/a! I L ■ ~\ Vv\ N

8 jj/i I 1 ^ VM^!

* IM iM ' . T ^JvÄ T PU ! /fJ [ 1 1 \ > '\

1 ] Mi ' 1 1 :' M \ CO 1 / M 1 L j ^1 \ CD [ , 121 \ A»' G 1 y i j \ \ \ m 'II j \ |V; \ It

"S ' f f i i <U I j! Ali \ w / j M IN T

S i \ ' i ; 1 ■ i i f, \ l! \ J

- - t i Composite

^ 1 li II : ft J\

1 "* 1 [ r '

! li ~~l—" -i ~t t

-2 -10 12 3 4 5 6 7 8 n (g zlK'^ *

FIGURE 1. N Exceedance Curves by Tail Number for A-37B at

England AFB (1969) 20

-v- --.»*•-•<•:: • v-'^^mrn»

1000000

100000

10000

T

12 Tail Numbers 2038.23 Total Hours ASD-TR-72-IVolume I

100O —

itle

8 n (g»s)

FIGURE 2. N Exceedance Curves by Tail Number for A-37B at

Bien Hoa AB (1970) 21

.üOOOOO

100000

looqp

10QO — QJ-

1<V0

6 7

FIGURE 3. Nz Exceedance Curves by Tall Number for A-37B at

Binh Thuy AB (1971)

22

8 nz(g»s)

!:ttöv|irrFE!!^|#Ö m ititEi : . . : i . : ': ! : ' : i : ; ; ; : :

4 Tail Numbers

""* 507.87 Total Hours 1 t - - ASD-TR-72-1 Volume I

1000000 —pH—H~—r~r~~~T~ ~r:""""" «^ "' j...Li.. - " r.-:::: :z_: -~.:r|;::::::.. _-.: izzi:.:..::'.;:

::;::.:::: :p::.::::: r::..;: "JäSL:.\i J -■ -

__ ::t::::::J..h„|.l_,:^_ -: •-

looocx) 4rr.t. ..-q±. - L-p ■}.■■. :J::.-... i~zz-:

■i I

.. .. - _—<-- _ - j_- _- ;;_-_-- ___ " UT^NSTP 1 ' f1 r ^ow\ /• VVSü— i—

L_ i U- \\^ r t T ! I : [ :z:tÄ..____tT±i__.

loooe H j ■ '■ j._a^trrr.jr^pri|' ;...T_|—j—i-j- -pt-p} - • w

- vv' - — 3' ~ [f \ V^ 4-

33 'l' \'Vv 4~ \

o ff '::::::::::::».: -

<-T 1 [ ■ ■

1000 ■ 2J ~.4_ Jf«J—j—

■ (ö /li ^ 'l V V\( J <u i^ A Vv C , // ~L v V\ '

*o iJ A w. a) 1/

. ■ _ - „ »J \,

V-1- Composite

iao - -i- a- - - ^ / -ii- -t" -+1+*- t —t~ i-r-H-' ~j/r /j

/ Composite •

W ._ . - -_■ .;_ ■ " :z~~."' ."'.V. .1.. .. . _ .-■ _.— -. .... — _ - - . . — -

.- - - - ■ - ■ • .- ■ - — -

■*

-2 -1 0 1 2 3 4 5 6 7 8 n (g*

FIGURE 4. N Exceedance Curves by Tail Number for A-37B at z England AFB (1971) 23

,0 JüO<V>

1000O0

icoao

100

.11-

-' 46 Tail Numbers 2308.6 Total Hours ASD-TR-71-37

-2 "I 0 1 2 3 4 5 6 7

FIGURE 5. Nz Exceedance Curves by Tail Number for F-4 Air-Ground

24

8 n (g's) z

T' _; 23 Tail Numbers 149.0 Total Hours ASD-TR-71-37

100000,0

1000Ö0

10000

1000

1O0

8 nz(g's),

FIGURE 6. N Exceedance Curves by Tail Number for F-4 Air-Air z

25

:J.(-'|:t|4;H-h|.fr? •■ -H--H--

OQß

. or.ao "—'

100,0

10£>

.1 4- -2 -1 8 n (g's) z

FIGURE 7. Nz Exceedance Curves by Tail Number for F-4 Inst. & Nav.

26

1000000

100000

1000/) r arr

100,0 —

1Q0

8 n (g'8)

FIGURE 8. Nz Exceedance Curves by Tail Number for F-4 Reconnaissance 27

-4... -)...,.

HT

iU<.-00Op

ooooo

:l .l'.t Wr"

-t-

15 Tail Numbers 19.1 Total Hours ASD-TR-71-37

«i L LLi -2 -1

li _L

FIGURE 9. Nz Exceedance. Curves by Tail Number for F-4 Test

8 n (g's) z

28

... ::|^lrt|-i:1:^^4*t-l4--^-=l----"L--- "" -4—■■ —----- :i 11 Tail Numbers

- 541.47 Total Hours ASD-TR-72-1 Volume I

Confidence that yu* or tall numbers lie between the maximum

-- and minimum is 0.30 oooooo r\~ ■ r ~ ■"' r:—."'." 'V-' JyS. - — — ... _. — — -- — ■ ^^^\,

„_. . ' LVVCV. L ~ " " "" i\v ^V'

^Ov ^V^^. \ ^^yV)

vv\ J\ — . _ _

\ x\f^^V 100000 -H^----rtT^rjjr^r4z ^pr:"™^": '*^^N^^ i^L— —'~~

—__jf|L._jr -V. ^^.^J 1 _ f fi ^

k V vv^ In Sy ^\WV

$ \m\ to - ■ ißf * I mi a IM

IOOOO j-f33 |_|_j.._|_4_j ^-l-[— -/"/HTI- T-H'T-i-f-H- -rt-i ..„1.M...-.J..........

—

000 rrr : - ^ ^%JV^ -: -~~

L Cv\. V \ \jC \^

® ' 1! Hi * Ml -j- -*- x" i T i. 1-4- °> /IB/1 8 / i/i

10O0 - Jjj ■=r= '"i~jf'r~~!—M M— * \ M 4' II

- 2'--:r~ 7Ull u ~+~ ___.3L Jfl-iJf—q:+t-.__—

_ a 1 IJI-1- v \ \ \ K ±tt»' i \'\ 4 ■«■ - . - ""JT"1-

5-\v 4 1 II - [[[ - -L-

1 !i loo ir-fr-?—'in 1_— "i "

\j 1 J EEEEE±EEEEEEEE-^EEEEEEEEgEEEE=E

_ J_ . ■ EEEEEEEEEEEEiEEE.EEEEEEEEEEEEiE _J -- 1 {

. 6 ■ ■ T 1,0 -. ■ "J -

6 -_±—i-i—I—L..I... 1—1—1—1,1 II—III 1 Liii—i.. — — —. ——

I! - ... 1 - Maxin um — 2 - Upper Bound of 90% CI for the Median

- V _ MoH < a ' Rr\tm/3 nf Oftrn frvr flip MpiRflrT4 _ . — _-- - ^ _ jjower

5 - Minim urn 6 - Compo

, | •'- ' i " . i M i M i i

site

-2-1 0 1 2 . 3 4 5 6 7 8 n (e

FIGURE 10. Nonparametric Bounds for A-37B at England AFB (1969) Exceedances

29

-- f 1-1- ----- -L.Iir.".: I_.X 12 Tail Numbers H i~l zUJö.^j Total Hours —'

ft - -4 . . .. ASD-TR-72-1 Volume I L

Confidence that 90% of tail numbers lie between the maximum

sooooo _::. "i " zr^';■ ~ _„ "■ "j ^_:;■..z ™~ _crr ana minimum is U.j4

J^~ | | | | | — ■ ' «l ZL_

-i ■ wi J_ i "i h 11 r m f T T

1000QO '[li—p- r r-—■ —r ^x«[Sc i - —1 i—

_t. .1 . _i_ i " \^LSST '— - —T 4 V^SSvVk

i : ' -+*^fc»k. _i "Sv§N | 1 i J-NST X I LJ

,„. _r JJI__LJ| .X-I I_.m--I 11 J _g: J-UZ-jL-JI ± _A\^ _"!_ -

i *—' (_ ' j"j~ ffji \W L ■-( m\ W\

aJ| jl is yt\ TX

,„ -isL JDT -JLJEI- J W\ ^«^fHffflfl h^ljjjffitf Hill T MIli^lzrafffflE^ffl

y //f II _l_ \\\\ _\

w /flf-il-4 J\A\ \ Zt- Ylii+ Vu J-j

iao -|——j j_l_//iLL!i—. . _ jn

ttffffi^ i ,j #n_ T r

'« ' ■ fi "eT [. . Hrmrl 1111 n IrrffH r to i 1 - Maximum

~j~T r ~"~L ~ 4 ~ Lower Bound of 90% CI for the Median __. - _i : 5 - Minimum

1 J 6 - Composite

.11, 1 _,_J ...J I I, l.ili| LU.LU i n i i i i i i i i u i i i n i i i k -2 -l 0 l 2 3 4 5 7 T1"' "fl"J L', FIGURE 11. Nonparametric Bounds for A-37B at Bien Hoa AB

(1970) Exceedances

30

&h

. TTTZTT . , TrT"™ IT *"" 1 "I ,".. _ T - .... H-H-h-H H-i-H-l 1-H-l+H-hhH-hl- H-" ::i::::::.|i'-:.::::::.:::::±.; : .f":. l4

--- 8 Tail Numbers 913.84 Total Hours ASD-TR-72-1 Volume I

Confidence that 90% of tail numbers lie between the maximum

1000000 """ ~ —..1... ... — — - ^ i , . .| ., «f^ — — - ■ ■ - - and minimum is 0.19

>

-r -I

: '%^Ä J ' ' ~T '"

mm %! flk S Ifl '

_ O If

Ö Jf u ÄW

O- Iff m Mi

100(0 - ?!'—t 1 JsP i ' ■ —— 1

a) fZm A\ n * Ilk lift

W ""' Jl

ioo 1— *-—j —« - —

Ua

.*. - J - - T mi' f[ r __ I_ J A

--I)-- _ ._ - -

_ _ _[ V ■

~ — - — .- j^ _ Maxln ium ■ Bound of 90% CI for the Median

~ 3 - Media M 4 - Lowei

_ _ _ ■' ' 5 — M-in-tn • Bound of 90% CI for the Median

6 - Compc »site

-2 -1 6 8 ' nz(g's)

FIGURE 12. Nonparametric Bounds for A-37B at Binh Thuy AB (1971) Exceedances

31

1 I i ! .| Ll. - I ! !

-T-

1

i .. i -'-'- 4

tr-

ial! Numbers '^r:

LT\ { f • j 507.87 Total Hours ASD-TR-72-1 Volume I

Confidence that 90% of tail numbers lie between the maximum and minimum is 0.05

;l ^1 i

1 1 T"H "^

1

i 1

1 |

jd--r-jE: ̂ ^"-j-f-f-r-'- -r' 5rl n yV 1~ "1

—r

—X. „

- r .._T_

SJ f i

i\

-Tt-1- ' -1-' i ! T

■—

i .•coo L i \ ■ ' '

i I

■■V ' " ■_!. r-U -L. — -—— J|_Ö__: . . -- _|_ Jo ■ I™ -H~-

i i 1 L ■ j s i | h 4^ 4- ~' M X V- I I 1

-i l j I i T ! * ! u ±fu i —|— -W

B^ffi: ̂ .-:jJ:lM -1—r 1— ... - -V Ar- -'. 1

—r—

j_| 0^ 1 \\

'S V \ Ja ._...!.. . 1 \\i u - - ■ ■ iiA i i i

to loao — a; H-

- or-t--T

1 _|_ .... —1— V —"«

i

!...

- «cx.rt "t-p-f-fj—r-":i :- -i~r —j— -L ~-\ ^YtT~-

i ■ur ■ '7 1 1 0). /// V i\\ o

— !4 /»' jjju W /I :

100 1—|—, // 3I

; 1 1

5 (l —In——.—. L

i -

r— — ^-|—h r

i fiS -r+ - —t—

4-p &J

+:_:! .'' / ' T '"i -

-j- ' i 1

N-*4- / 4-4 i 1

r* T

III •■ J ' - 1 L

4-j-, i——! -j- -)■-.+ Li~'.: 1 - maximum - -ff#"4" 1--4- 2 - Upper Bound of 90% CI for the Median :

! _j7 j J ! II'

J-LL-,_, i I _| —,-- -

- Lower Bound of 90% CI for the Median i ' ! lit-- i !-J4 -4-j- 4

! I ' ' -j- 5

J 6 - Minimum - Composite 1 "

.1 ;i L ' 1 ill ' 1 -2 -1 0 1 2 3 4 5 6 7 8-xl 7£T

FIGURE 13. Nonparametric Bounds for A-37B at England AFB (1971) Exceedances

32

_l_--l--- -J - ---±J ...fl„ z

-i-l-li I .+M-H-H-I-I-H-H-H-H4+H- ._ .._ ..:- X] _._^ f6 Tail Numbers r

»308.6 Total Hours iSD-TR-71-37

Jonfidence that 90% of tail lumbers lie between the maximum md minimum is 0.95 -

/

(. r

i ooooop ^4— ■" ■ :-ri_:|::_ . A, ::t ::

■ I w x. *ÜN - _ _[/ WT I

1/ ^V • 1 J Si

ill iil i

. 1,1 r. .. •_ V - r - - A >^_ 1 n *, ^s. 1 'H \ tl i Jr' V~ 10 V

jf'L. V- i

T^ V^^J

n u

looop - 5"" ' ' .JX_____._ «^ 1 —. — kJ ^»^*

-a-—-4- :=jjTT-EEE==-2Eq _Sf* f^-^ :.; I . ._ :fc is^L- ^ VI ^\\ o 7 1 - _ _ _ *" J .

rH If Wt - i II' \\N\

H / P.

/ 10CUO - ?l—— 1--

/// C. I TV A. " 5' " I

--»II |_. „

" .2 -■' / Tu i " \3\ si / ifff .

L_ \ _ 1 A \ m ■1

\ '

tOO T— Ju f-f \ . „1 _ _. _ \z 4. . L_ ■ \ -* / ' j 5

/__.

_ 1 - Maxd .mum : - " j_ 2 - Upp« ;r Bound of 90% CI for the Median

Lan »r Bound of 90% CI for the Median Lmum »osite

I J 1 / -.- - T - Med- 1/ £ 4 - Low«

_. fi- ■ '_ 5 - Mind 6 - Comi

..-L-- — 1 1 1 1 111 1 -2 -1 0. 1 2 3 4 5 6 7 8 n (g's)

z

FIGURE 14. Nonparametric Bounds for F-4 Air-Ground Exceedances

33

. yooao

i ÖOuC^O

1O0QO

ioao

160

.1 L i\ \ I

1 - Maximum 2 - Upper Bound of 90% CI for the Median 3 - Median 4 - Lower Bound of 90% CI for the Median 5 - Minimum 6 - Composite

-1 ' ' ' ' ' ' ' ' ' ' ' ' * '■ i i i i i i •dmMm

"2 • _1 ° X 2 3 * ■ 5 6 7 8n (g's) z FIGURE 15. Nonparametric Bounds for F-4 Air-Air Exceedances

34

«■»p»

1000000

100000 -

10000 -j

1000 ->■

1O0

1 2

FIGURE 16. Nonparametrlc Bounds for F-4 Inst. & Nav. Exceedances

35

500000

i CO 000

oooo

100,0

10,0

.1 i

FIGURE 17. Nonparametric Bounds for F-4 Reconnaissance Exceedances

36

* '*'.'••.-

■

zrrrr~—:-z-.-r-z- ----? - TTT r rmri :+-l,l-t-l.4 lrt:l4:t-]/±.4:rt:i:l±d-i:±±±j±t±:t-l l-j"

Tail Numbers .1 Total Hours D-TR-71-37

nfidence that 90% of tail numbers e between the maximum and minimum is

--*—--£ ~" ' .... ._:. 19 . _ . _ - - AS

- - - Co li

, 0. loooooo .:[•_'_.'.{.■ ...'-LLL^rrrrrTr^pp' r^ -

-it' - "" " - ■-■(- \V[ 4- -T (-• :-— ._-.).. _ - \ Vw

■ J]"" j vSS i . 1,1 IN j^

100000 rFri' rHTT'l' rt f"!>r J?4" i H"v"i FV^J o t- -----. x:::::::::-^::|:^-:::|::| -vC-V ^ ~"'' / Jffi'1- \L J r \ x > \

/ 1 3 V ' y_ ^ -^-l-S^ Sy I

/ LF , 10000 - :rrr-rTr~r J_ .-- &zp~'z-~* + ■ * X \ +s' M' - - . iMififl - 3 -t h1 ~ i _ V 5 - _ -M „ -rf : 1 i__A t o / _ O-l F- -' IH y

1000 - <u—} 4r TT i "* *" • -t - "" UHlil QJ., |_ ■ _, O , 6 - ■ _ g.__6_ —-- -

•o ■ ■ ~ .

w ioo --T-1—--*—"hit ii—'TT "i—

. ________..i — —. — —. — — — — — — — — — — — — —— — — —■ —

\ • '.

Lmum »r Bound of 90% CI for the Median Lan er Bound of 90% CI for the Median lmum )osite

_ 2 - tipp . 3 - Med

._ _ ._. K _ M-f-n

_ 6 - Com

1.LL1LJ.1.LI» Jx 'j ! z 3 * ' * h l Ä nz(g's)

FIGURE 18. Nonparametric Bounds for F-4 Test Exceedances

37

rr

.1 t-

\\r\~

< i

! I

-2 ■t

! i l I I ■ . I

i !.! i 11 Tail Numbers . 541.47 Total Hours

ASD-TR-72-1 Volume I

i -t-;-

: 1 - Upper bound of tolerance limit assuming normal. . 2 - Mean 3 - Lower bound of tolerance limit assuming normal

; 4 - Upper bound of tolerance limit assuming lognormal 5 - Composite 6 - Lower bound of tolerance limit assuming lognormal

8 n Oafs) z

FIGURE 19. 90% Tolerance Limits with 90% Confidence for A-37B at England AFB (1969).

38

10000O0

1000O0

1000O

ioao

iao

V>

.t 1

:b:.:i ■i-rr -\T\ 1 -:T

I ! > i

L-H±

! I ! i

ilTi." i; ! frr

til"

1 - Upper bound of tolerance limit assuming normal 2 - Mean 3 - Lower bound of tolerance limit assuming norial 4 - Upper bound of tolerance limit assuming logqprmal

5 - Composite 6 - Lower bound of tolerance limit assuming logi^ormal

-2 -1 T T B ting's)

FIGURE 20. 90% Tolerance Limits with 90% Confidence for A-37B at Bien Hoa.AB (1970)

39

l-.sjp

i'l

lOQO

i-l

I'l I ! I ! i

100

J i.

-;-4 i ! I j I i

I I

-2

'l-M- : t '- l I i I. i ! I ! i

■i:r : :n : I-l i

T-lt-l-t.'" ; ,. (.ui-

I ! ! I ]Y 8 Tail Numbers

i 913.84 Total Hours ASD-TR-72-1 Volume I

! i ' i

i ! I

1 - Upper bound of tolerance limit assuming normal 2 - Mean

3 - Lower bound of tolerance limit assuming normal 4 - Upper bound of tolerance limit assuming lognormal 5 - Composite 6 - Lower bound of tolerance limit assuming logjnormal

-1 0 1—5—r—T-—frrtfa) FIGURE 21. 90% Tolerance Limits with 90% Confidence for A-37B

at Binh Thuy AB (1971).

40

10000O0

1000&0

10000

1 - Upper bound of tolerance limit assuming normal 2 - Mean 3 - Lower bound of tolerance limit assuming normal 4 - Upper bound of tolerance limit assuming lognormal 5 - Composite 6 - Lower bound of tolerance limit assuming lognormal

8 n (g's) z

FIGUKE 22. 90% Tolerance Limits with 90% Confidence for A-37B

at England AFB (1971).

41

joooao

1JOOÜC

.1u I I

5 - Composite >| 6 - Lower bound of tolerance limit assuming

lognormal

LL.L

i i i i -1

FIGURE 23. 90% Tolerance Limits with 90% Confidence for F-4 Air-Ground.

8 nz(g»s)

42

looooao

lOOOCVD

100O0

ioao

-j-.i;

.j.

rt-r col Ml 3 I o as

I ol '§;

u 0) P.

CO -r CU

o § CU cu o

w

100

V>

rrtr ij.jj.

!!l

:&Ti.

,1 W—W—mtiaVl.f ..a. W>-E.lAwHlgl."g?LfJ*li.'il

1 - Upper bound of tolerance limit assuming normal 2 - Mean 3 - Lower bound of tolerance limit assuming nornjal 4 - Upper bound of tolerance limit assuming logrtormal

.r........ ^ _ compOSit;e i i i 16 - Lower bound of tolerance limit assuming logr

1 ! =»= =a=

ormal

8 nz(g's) -2 -1 0 1 2 3 4 5 6 7

FIGURE 24. 90% Tolerance Limits with 90% Confidence for F-4 Air-Air

43

100

> i

11 ~ nomIlbOUnd °f tolerance limit assuming "2 - Mean 3 ~ rormalbOUnd °f tolerance limit assuming 4 " lgln^rmard °f tol««ice limit assuming 5 - Composite 6 - Lower bound of tolerance limit assuming loga -2-101234

axmaJ

«5678 nz(g's) FIGURE 25. 90% Tolerance Limits with 90% Confidence for F-4

Inst. and Nav.

44

looocao

lOOOQO

100O0

ioao

to

: - i i.

!-i i-

iiii -2

4

-: -i i.

i___LL 1 - Upper bound of tolerance limit assuming normal | jj 2 - Mean

I : ; ; 3 - Lower bound of tolerance limit assuming noräal : i * : 4 - Upper bound of tolerance limit assuming logrormal .;..!..! 15- Composite

I■ | ! i 6 - Lower bound of tolerance limit assuming logiiormal ! I i 'I :

JL J. X -1 0 6

Ju 8 n (g*s)

FIGURE 26. 90% Tolerance Limits with 90% Confidence for F-4 Reconnaissance.

45

! 15 Tail Numbers 19.1 Total Hours ASD-TR-71-37

.1 k—

1 - Upper bound of tolerance limit assuming normjal 2 - Mean

-L,- -2

:.;..;. , 3 - Lower bound of tolerance limit assuming normal -|■■}-I- 4 - Upper bound of tolerance limit assuming lognarmal

! I 5 - Composite I j ; 6 - Lower bound of tolerance limit assuming lognLrmal

8 nz(g's) -1 0 1 2 3 4 5 6 7

FIGURE 27. 90% Tolerance Limits with 90% Confidence for F-4 Test.

46

TABLE III. CURVE FIT COEFFICIENTS

3 2 F(X)=10(a3X +a2X +alX+ao)where x=n and F(X)= Exceedances/4,000 Hours

A. Coefficients for n >lg

Figure

28 -0.0592 29 -0.0305 30 -0.0954 31 -0.0254 32 -0.0249 33 -0.0177 34 -0.0525 35 36 -0.0117

R-Square Hours

.5281

.2907

.8393

.2390

.2914

.1833

.5726

.1180

-1.9186 -1.3611 -2.7681 -1.1635 -1.4829 -1.0033 -2.6019 -.6364 -.8723

7.3527 .99908 541.47 6.6859 .99796 2038.23 7.6166 .99956 913.84 6.9181 .99913 507.87 6.9742 .99681 2308.6 6.4857 .99945 149.0 7.7763 .99583 401.2 6.4918 .98916 515.2 6.3584 .99442 19.1

B. Coefficients for n <lg

Figure

28 29 30 31 32 33 34 35 36

o R-Square Hours

.6777

.2336

.5177

2.4887

.9398

1.2371

3.4216 4.2709 .99962 541.47 4.4009 4.1837 1 2038.23 1.7549 3.3475 .87189 913.84 2.4042 4.4226 .99650 507.87 2.4319 2.9379 .99968 2308.6 1.6055 3.7242 .98488 149.0 2.7188 2.7525 .99223 401.2 2.5057 3.2584 .99435 515.2 1.2208 3.5295 .99835 19.1

47

en , LH Ul LU ■ mn w *«-•

u\ T. 3

ca f ft," —.1 u. -.*-. a <ai :=»■

r^i ui trn 11,

—:^ — , t—-. 1 cc »«• r~j l"M w_{ »~ r- L.*..* *~.i « jiT.» ..—* !■-• c« UJ '-'ll fcj !— £„. « 1 C: IP fill cc —^ Ifi cc

t®3*ss

48

CSS r-

ro

3;

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APPENDIX II

COMPUTER PROGRAMS USED

57

s

FIGURE 37. Program to Store Data on Tape Cassette

'■.'■■'A\ NSC 50 3 :> NSC 50? 16 , ='.::■■ J-i TO 59

.;■: rOP K=i TO 16 PI flT J 3 = 9 uxr.hoa-o

:'■ Tr,J]==Q

50 3JY3C 16 3

ill.-.,---, ; !•-.

ROXi J MOP "ft0NL=jC0l INPUT L.JM !'0P 1 = 1 TO L

1-4 M

fi; INPUT RL I 3 ■ L FOR J = l TO M up MOP ":;(i:i)", "j" •••i, i.ni:,UT ;•■;[ I» j] IPO' NT XT J US PROP "T(" I .">=?" i:'-6 :M;PUT TI: I ] 130 PENT I HG FOR J=l TO II o.;i> aisp "v("j")="; ,00 INPUT YCJ3 HO PENT J "ON"' FÜR 0=1 TO L A-o PRINT y-A- PRINT h[J3?

:F:- FRONT XCJJK"]; ;'P0 I-if:XT K A A FTP! NT TCJ]» ;.■!!■: PFXT .J : --ii !;R1NT ;oo:. FOR I = I TO 8 POP PRINT YC I 3; POO HEXT I ■■TO PRINT ;-,-: 7 pRTHT PF;P FOR ...1=1 TO L FLO FOR K=9 TO 16

58

FIGUJB 37. Continued «WM

' • ; •IM, .-.'LJJKJ:

■'>'. ■■:'.'■:\ K

'■■■■■■ 7.7" ,.l '■■ : ■• :Ni

.•••■••• :-uk -7-9 Tu [..■ .••'^ i'-.iMT vi: i jj

1"

7-5 PRINT 71.7; i'iPP "iVNTtP 1 ■'77 ;:•:('!.:7 Tl ■■■-•ü ■!• Tl-J THEN 4 ••■'••■"" vi'r 'IPffH b7L

i_ i ? i " -i „ T i

1 ;r Ml "I •i :;■ ■ :,i , _

■■'■-> :■■'*. mr ;;cLi ü^CLI^M! h n.il i?vrm 3 'U 7770 :57Ü

:-' L-i:-i- "nLti'NC."";■'

i. 5 ',' i" '■ n:

'!" iP ÖPTR

-it..* END i:

FIGURE 38. Program to Read Data on Tape Cassette

7i. .. Nu. r ,777 7 !. :> 7 H]'7 7

■J i i! J

iff Hi ...! 7 ;'■'

7 I'O 50 .!.':!•■ 1 1-Hi!

59

■IGÜBE 39. Program to Analyze the Data by Tail Number

I" i L.i::.h:J„ !i i■■:I.iI.':!:.

7 i ' J i... .'ii

J I.M'1 ' H I

I '■ il

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FH: .1 IN L

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! ™ ] T

iu h

t... ■■.' i ü

Pi- 3"! HLi. F>3:ExIl!ii'-!CF :-liIH"

60

>^™^^^w^^pp

^ gIGURE 39. Continued

'■: '."' ..'. .■:'■■ c.ii I c.P 1 .' G I 'i"' P-0 ': NI"'U f n

: "GO:, UHN ':':! f-r-:: •OP 01 .U fliiii J

<N! \<J .1

f :l: ■ i! u '.■■'• u

- O:1 i'l.GF "IM I ER 1. 1 i'l PLOT Gi"Pi'-;P cPG .PvUl' Tl G-G IF TH?i THEN 300 GGf Ri£M PLOT COMPOSITE

GPP GO SUB 730 70S FEH ' :..0 GOTO 380

PT POP P»-l TCI M "•if- G:: Xi: i-?i ü K 3=9 THEN PPG : : PLOT" VLKJJLGT PL Rl i'l,J) ■'■■ ■ 'T KiiKl THEN '"'SO ■V-.) I EM

GTGP|i TSP OTtrEF' I. TO

:J ■ i !,'■!( ;"

i J.

"1 j; ■. ':;.. ' '.'"'.' •'.": * ■ ! M •:•:• ihiti ! ■.-0 f'1

v. l ,.:;p ■ ■■"■■! ! "f

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:, fP " '"! .' "G'O '• ■: .; ■:::• r ,::':. H P.: :::- j

-■ :. '' i'l J; ■ i Ti'; 1-

■ i i ■. ..' * '. TO 1-

NO HT. EfihiKEIi

' : i FOR -I--.! TO H- '! "OG Pi'P JI-J-U TO M ' !.;■'■ i!- R: ..! !':• i-:E J: j T ■ - E- =-_£ oo id if to ' '■: -J PPO<iPF'! ' ''0 !-:' .1 J = RiPJl T ' -'.00 Pr Jl MO

10 PERT Jl Tii'i PORT ,1

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■ I! ! h '. :.. •'". ,; TO: :'P '. J=--J.--v-= J ':":- Of 0 ■ P !•■■ i-:! I/O r■;': : :fi G; ; I •■;:,

61

FIGURE 39. Continued

! ; CO

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i i.

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■62

FIGURE 39. Continued

: ;■ I i i

'• Di:-! !"; . i i;

1.000 HEFT F 1510 lUSF1 "ENTER i TO FIND M£' .■-20 INPUT Tl . ■■'■

HSS- .'h i. HE LOG GGEOEL LI ST R1BU11 Of

:::": 0 ! E /,[ ... j !''. T-O i IE: h 3 i CV? :E:0 I'!,;:'":i...GTXL !...:'F J si-'(3 EOF J~! TO H ';38 Jr' Xi:j>K] = 6 THEE 102* 399 Hi-ti-J+j :00i '"■- .,...GTOC JJK J P'i;.-:vt-2

.918 GOTO 193.0 \i-y.Y J

.J=J+1 TO H S 40 i' I- E L J i \: J=O T H E H 10 8 0 :.. 5 u\ Q;,:-... i... G T F [ J J K J - M 2) '■• 2+Q

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: o £.'

20 v;-:-Q/Mi

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EOT ER :'. TO Flhl« I

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i ■!!■•;..11

00 _..>ri£+K5*S2 'ri,K ]=v2 :i2:. !<] = !• 1 :.r 3'K]=!''i?

40 ET ;" j 0 0? 0 J=S2 r;;; rO ■;....OT 0

OiPiJT Tl . !'■ TlMl THE;-; : ■ !;,' PI :::::,.'; TO -•

:! ■',. f: i ,'' i": 1

.,- 3. <■ 0

ÜG HOE HOI... DISTR» ' 5

i l ■;" ■'»'..'! *

. ' L ■ • f t.l-: ' 11

f T 63

FIGWRE 39. Continued

i. i

■■I"ii"l!:k 1 -fi P ::!PU'i' T:i : !•" I1-:] T! !r"r]

. ,j. i:OiO 05 -JU

'■■'-■ >:. .'en !•■:'

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■ iiir:-i'i !■. . i

■:' i'U-i ■ OOfORH ' ' :

:i -JOTO S50Ö ' . f. ~i Pp i MT I

r-.;.. 1 i.H\li

• -' iH i I'' R 1 j :•:. ] i RETURN

(.

64

FIGURE 40. Revised HP Plot Pac Program

-PVlM Pi. 6'.- 'iiGI' i ■< F-OL If! J "i' TOP' 1 1 TO LI oo PL i 3--r-c i :i-6

?ij PETY t :>< PL J ]-"!

lf-ü ]),::F "f-JR::. L'EGl'LE- " ^ 1 '0 ;.OPU7 HZ iPO TF I.i^'>? THEN iPU i 20 DISP " AM I(■■•( j Ml TP: ? IHCRF1 '„ -•' >' !•■!:■ i'HPUT Y10<2,2 3 15Ö PISP "YMH-KYMfitfj IHCRM. =": i oti j Mr U i j 1 4 ■ :- 5 } ..:,

,80 ■>• Y2-Yl>/:l? YrO Y3 = Yl-2-*.J

.'-.OK1 Y £-~ Y 21J • '. ', ■.:.-'.. Ml..I::. Y 'l "~ü'£ J 'i ,,;:t^"i'' [ ? v..'i i To .:;•'« r"i.or YYj'r'i . '•■■•■-; FLO I KIJYI POP PL Of 3.:. Yi

•2-HY..0

7-Rill Y2-J.

HYP "PITTEP 1 TO PP.I'M [TPYYT P9 ;.F l-Ptrl TPPN 4:.'!:■

üi i i;

r r i H i P iY, 20.. ' i PL j

Tj T p,p ' >.:• !';:• i.:.' •"• ■■'; ' PITT ER

FOP MR i u'..!'" i' ,. 2

Y---HI-30 j . V-'-.-iPS r~ L N I ..PHY; i" \ ''i .... '2 '■; .;: ;'

■;■!;;,■;•;''

;-i o K i" •!-;! '•'

!" '.'■■'•. i. ■-Ü TO PIEP ■ PL OV [■/■ NO '! T '• K >:■ 2'. i- 2 L.0'1 i? ;'; ,. _.

or' r i

! ..:;:.L--! ■■'■■ wm

' ■■4t'«0

! vi

: 0

;. MPT

IVOH

65

FIGURE 40. Continued

10 IF W THEN 60 15 DISP "TCHOHPRV; 16 INPUT T 20 DISP "N,V" ' ' 30 INPUT BC2]JY 32 PRINT "NZ="BC2]J"CUMULATIVE PEAKS = "Y 34 Y = Y/1" 25 Y=Y*4O80 36 PRINT "NZ="BC£]) "CUMULATIVE PEAKS PER 41*00 HR.="'i

60 IF FNNi THEN 28 58'END 60 DISP "NOT ALLOWED" 70 END

10 IF N THEN SO 20 DISP "WRONG XJY="! 20 INPUT BC£]?Y 40 IF FNY(-l) THEN 20 56 END '23 DISP "NOT ALLOWED" 78 END

iy yy-SQR<<s2-Sit2/H)..--(H~i>) y b o b' ~ S Q R (.(S 4 ■• - S 312 .'• N ) / (N ■■- in : •; :::■! R q = (s 5 - S1 * S 3 / H) / (N -1 ':■ .-- R R / R 9 40 PRINT SO PRINT "NO. POINTS ="N 60 PRINT 75 PR I NT " X: riEHN=" S1 ,-'N; TRE25" ST. DEV. =" RR o y P k IN T " Y: M E A N =" S 3 / N 5 T A B 2 5" S T. D E V. =" fi 9

106 PRINT "CORR;.CnFFF. = "Rc» 110 PRINT .-■■■■" t 120 END

"'0 IF N <= D2-W THEN 268 2 8 DISP "DEG.REfc.="! 30 INPUT Dl 40 IF Dl <= D2-W THEN 70 56 DISP "MAX DEG-="5D2-W 60 END 70 IF W=0 THEN 250 SO T = 0 90 FOR 1=1 TO Dl+1 100 BE I ] = 0 110 FOR J=l TO Dl 1+2 12 0 R = < I + J -1 > * (D 2+2 - 8. 5 * < I +. n') 130 B[I]:=BEIII+CCT+J3*r:ipi 148 NEXT J ' ,,

■m ■ p

1 FIGURE 40. Continued V

■ ' * • " ■ * I

i69''NEXT I * ■■- --■ "—- ■■■•"- } 178 R1=0 | 189 FOR 1=2 TO Dl+1 f 199 Rl=Rl+CCr*<D2+<3-I)/2)]t2 \ £09 NEXT I • | 210 T0=CC<D2+l)*<D2+2>/2-3' i 226 T0=T0-CCD2+13t2 ' ' j 238 DISP "DONE" f

240 END 250 IF N>D2 THEN 288 < } 260 DISP "NOT ENOUGH POINTS" :" i 270 END * i 280 P=W=1 ' f 290 D2=D2+1 ^ I 388 FOR J=l TO D2 \ 318 CCP3=SQRCCP] 328 FOR 1=1 TO D2-J+1 338 CCP+I]=C[P+I3/CCP-] 340- NEXT I 350 R=P+I 3€d S=R ' * 378 FOR L=l TO D2-J 3S8 P=P+1 398 FOR M=l TO- D2+2-J-L . 488 C[ R+M-l 3=i::CR+f'1~l]-C[ P ]*Cf P+M-l ] 418 NEXT M- 420 R=R+M-1 " • 430 NEXT L 448 P=S 458 NEXT J 468 T=(D2+l)*(D2+2)/2 470 FOR 1 = 1 TO D2-1 488 T-T-l-I 498 CCT3=1/CIT3' 500 FOR J=l TO D2-I 510 P=D2+1-I-J 520 P=P*(D2+1-(P-1)/2>~I 530 R=P-J ■ 540 S=8 . 550 U=I+J+1 560 V=P 570 FOR K=l TO J 588 V=V+U-K 590 S=S-CCR+K]*CCV3' 680 NEXT K 618 CCP3==S/CCR3 628 NEXT J 638 NEXT I 648 CC13=1/CC13 650 GOTO 88 , ' 67

■ FIGURE 40. Continued {

■0 IF N = 0 THEN 128 ;-o -PRINT

-1 I'RIHT "COEFFICIENTS" ' O PRINT

:'0 FORMAT F3.9JF12.4 0 FOR 1=1 TO Dl+1

"n HRITE C15?50)"B<"I-t" J=:::"Bf I ]- :.f.' NEXT -I VH PRINT 100 PRINT "R SQUARE = "Pi/TO US PRINT 120 END

it' FOR N = N1 TO N2 STEP <X2 XI >/10Fi 3 0 Y = FN2N 32 IF V<Y5 OR Y>Y6 THEN fifi 3 0 PLOT KJY 00 GOTO 79 68 PEN "0 NEXT v Tj Z=FN i<@ 'Ü END

•>fc .Li rp "CHARACTER HEIGHTS)"; 20 INPUT H • ■ 30 LABEL (*:> Hi> 2i O.'i P/":') 40 LETTER 50 Z=FNK0 60 END

.0 Disp ":

20 DISP "Y(C.ALC)^"FN2R -O END

68

REFERENCES

1. Clay, Larry E. and John F. Nash; Technology, Incorporated; Flight and Taxi Data from A-37B Aircraft During Combat and Training Operations, ASD-TR-72-1, Volume I, Deputy for Engineering, Aeronau- tical Systems Division, Air Force Systems Command, Wright-Patterson Air Force Base, Ohio, January 1972.

2. Clay, Larry E,, Ronald I. Rockafellow and James A. Strohm, Technology, Incorporated; Structural Flight Loads Data from SEA and Thunderbird F-4 Aircraft, ASD-TR-71-37, Deputy for Engineering, Aeronautical Systems Division, Air Force Systems Command, Wright- Patterson Air Force Base, Ohio, April 1971.

3. Hollander, Myles and Douglas A. Wolfe, Non-parametric Statis- tical Methods. John Wiley & Sons, Inc., New York, 1973

4. AMCP706-110(AMC), Engineering Design Handbook, "Experimental Statistics," Sections 1-5, December 1969.

5. Hewlett-Packard Calculator, 9830A Plotter Pack. Hewlett-Packard Company.

6- Hewlett-Packard 9830A Calculator Operating and Programming. Hewlett-Packard Company, 1973.

69

a^_