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CAS E F I LEfGUrs

NATIONAL ADVISORY COMMITTEE

FOR AERONAUTICS

TECHNICAL NOTE 3305

hrOV15 J954

SOME MEASUREMENTS AND POWER SPECTRA

OF RUNWAY ROUGHNESS

By _ames H. Walls, _ohn C. Houbolt, and Harry Press

Langley Aeronautical Laboratory

Langley Field, Va.

[

Washington

November 1954

https://ntrs.nasa.gov/search.jsp?R=19930084048 2018-06-17T17:30:49+00:00Z

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NATIONAL ADVISORY COMM_ FOR AERONAUTICS

TECHNICAL NOTE 3309

SOME MEASUREMENTS AND POWER SPECTRA

OF RUNWAY ROUGHNESS

By James H. Walls, John C. Houbolt, and Harry Press

SLI_4ARY

Measurements of actual runway roughness obtained by a profile-survey

method (engineer's level) are presented. Data were obtained from a survey

of a relatively rough runway and a smooth runway. The results of this

study are presented as roughness profiles of the runways surveyed and inthe form of power spectra.

INTRODUCTION

The frequency of occurrence of large load applications in routine

ground airplane operations has caused a growing concern in regard to theroughness of landing and taxiing surfaces. In order to obtain informa-

tion on this problem, it was thought desirable to make detailed measure-

ments of the roughness characteristics of actual runway surfaces. As

one of the initial steps in this study, measurements were made of two

runways available to the National Advisory Committee for Aeronautics at

Langley Field, Va. The two runways selected were known to be of very

different degrees of roughness; one runway was considered relatively

smooth whereas the other was considered rather rough - possibly rough

enough to preclude active use. The measurements made are presented here

directly as elevation profiles. In addition, the power spectra of the

runway elevations were also determined and are presented in order to

permit a description of the frequency characteristics of the runwayroughness.

SYMBOLS

D

L

m

distance over which moving average is taken

wavelength, ft

number of uniformly spaced points over the frequency range at

which power estimates are desired

2 NACA TN 3305

n

x

X

_x

y(x)

R(×)

number of equally spaced elevations taken over the runway

distance, ft

arbitrary value of x, ft

space interval, ft

random function of distance (runway height)

autocorrelation function

X

fl

power-spectral density function, defined by equation (1),

ft 2

radian/ft

displacement distance, ft

reduced frequency, 2_/L, radians/ft

root-mean-square value of y; _--_

SURVEY OF VARIATIONS IN RUNWAY KEIGHT

A diagram of the landing and parking strips that presently exist

at Langley Field is presented in figure 1 and shows the extent of the

two surveys made. Runway 17-35 was chosen because it is considered

representative of a satisfactorily smooth runway; the other runway chosen,

12-50, is considered rough and is used only for parking. Both runways

are of standard concrete construction.

The roughness measurements were made by means of a surveyor's level,

rod, and tape. This means was selected because it could be applied

directly without the delays attendant on the development of special instru-

mentation. In using this technique it was necessary to select an interval

at which elevations would be obtained. It was thought that the frequency

range between 0.5 and 35 cycles per second would be the region of prin-

cipal concern for most airplanes. At a landing speed of I00 miles per

hour this would correspond to wavelengths between about 300 and 4 feet.

As a consequence, a reading interval of 2 feet was selected. This choice

was dictated by two considerations: first, it was expected that there

would be little variation in runway height at wavelengths less than 4 feet

and, second, the communications sampling theorem (ref. I), which states

that sampling a disturbance at intervals of one-half the shortest wave-

length present completely specifies the disturbance.

NACATN 3305 3

0nly 1,400 feet at the south end of runway 3_ was surveyed so asnot to interfere with ground traffic from runway 7. Visual observationsuggested that except for the initial 100 feet, which is a sloping macadamoverrun, this 1,400 feet is fairly representative of the remainder of therunway. Three thousand feet or nearly all of runway 12 was covered. Thetwo surveys were conducted with greater ease than was originally expected.Approximately 6 hours was spent on runway 35 and slightly more than twicethat time was spent on runway 12. This meansthat on an average aboutll5 readings per hour were obtained. The pace was steady but not hurried.Detailed runway elevations are given in table I, and figure 2 shows therunway profiles. These elevations are plotted about a zero arithmeticmean.

POWERSPECTRAOFRUNWAYHEIGHT

Definition of the Power Spectrum

In addition to the actual runway-height profiles, it appeared desir-able, because of the randomcharacter of the height fluctuations, to deter-mine the power spectra of runway height. These power spectra would pro-vide a description of the frequency content of the runway height variationsand be directly applicable to the calculation of airplane responses in thefrequency plane.

Since the runway roughness under consideration here is a spacedisturbance rather than a disturbance in time, it is desirable to definethe power-spectral density function in terms of the frequency argumentin radlans per foot rather than the conventional argument _ in radiansper second. In terms of_uenc_ argument, the power-spectraldensity function of the disturbance y(x) is de'fined in the followingmanner:

where the bars indicate the modulus of the complex quantity. Equation (i)may be used to evaluate the power-spectral density function from observeddata, but, in practice, the power-spectral density function maybe deter-mined more conveniently and less tediously through use of a related func-tion, the autocorrelation function R(X), defined by

.....R(X) = lim (2)

X-"-_ _ 2X

4 NACA TN 3309

The autocorrelation function has the symmetrical property R(X) = R(-X)

and is reciprocally related to the power-spectral density function by

the Fourier cosine transformation in the following manner:

2f0-¢(_) = _- R(×)cos _× d×

_0 _

R(x) = o(n)cos ax _m

(3)

From the foregoing it is seen that

R(0)= = .2= ¢(n)_ (4)

where G is the root-mean-square value (or standard deviation) of the

disturbance and is a convenient measure for a comparison of the overall

roughness of the two runways surveyed.

Evaluation of Power Spectra

The actual evaluations of the power spectra were made by means of

the numerical-calculation procedure derived by Tukey in reference 2.

This procedure is also described and discussed in reference 3.

As a preliminary to these power-spectrum calculations, it appeared

desirable to make some modifications to the actual measured profiles.

Examination of these profiles (fig. 2) indicated that the runway height

exhibited large changes in elevation at very long wavelengths. These

large changes at low frequencies have a tendency to complicate and distort

power estimates at the higher frequencies because of the effective filter

characteristics of the numerical estimators. Since, in the present study,

there was little interest in the longer wavelengths, it was decided to

avoid these adverse affects by removing the longer wavelengths. One

simple and convenient way of prefiltering a disturbance is by the use

of moving averages. This means was used in the present study to filter

out some of the longer wavelengths. A moving average of the runway height

for a 300-foot distance, as defined by

NACATN 3309 9

K=75

Ym(X) = 1 _ y(x + K 2_x)

15o K=-75

was determined for each of the runways and is shown as the dashed curve

in figure 2. The variations of runway height about this moving average

were then determined and are shown in figure 3. As can be seen roughly

from a comparison of figures 2 and 3, the main effect of this operation

is the removal of the low-frequency components of the height variations.

The actual filtering effects of this operation on the spectrum

estimates are derived in detail in the appendix. The attenuation func-

tion for the spectrum introduced by the moving average is given by

iwhere D is the distance over which the average is taken. The attenua-

tion function is shown as a function of _D/2 in figure 4. A separate

scale for wavelength L is also shown for the present value of

D = 300 feet. It can be seen from the figure that the principal effect

of the moving average in the present case is to attenuate the effects of

the longer wavelengths, with the attenuation factor decreasing from 1

at 300 feet to 0.490 at 400 feet and 0.132 at 600 feet.

The actual steps involved in the numerical estimation of the spectra

(ref. 2) are as follows:

i. The autocorrelation coefficients were determined from the succes-

sive values of elevation Yl, Y2, • Yn according to the following

numerical form of equation (2):

n-r

Rr = __in - r = -_-_r(r = O, i, 2, . . . m) (9)

where Rr = R(X)} X = r _x.

2. Initial or "raw" estimates of the power density were then deter-

mined by use of the following numerical form of equation (3):

6 NACATN 3305

ml+2 Rqcos qr +RmCOSq=l m

(6)

where L r = L(_); _ = --.r_ These estimates have an effective filtermax

which has the undesirable character of appreciable side-band areas and

thus permits a wide diffusion of power.

3. Final or "smoothed" estimates of power density, which are estimates

based on a more desirable and sharper filter, were then determined from

$0 = 0.94L 0 + 0.46L 1

@r = 0.23Lr_ 1 + 0.94L r + 0.23Lr+ 1

Sm = 0.46Lm_ 1 + 0.94L m

> (7)

The values of Sr obtained in the foregoing manner can roughly be con-

sidered to be estimates of the average power over the frequency interval

(r - 1)_ I< 2 < (r + 1)_. The distances of 1,400 and 3,000 feet coveredmZ_x max

in the survey and the interval AX of 2 feet lead to values of n of

701 and 1,901, respectively, for the two runways. Evaluation was madewith m = 40. The autocorrelation coefficients obtained are shown in

figure 5 and the resulting power estimates are plotted as a function of

the reduced frequency _ in figure 6. Each point in the present case

represents the average power in a frequency interval ± _ about the

value plotted. For clarity in presentation_ some of the power estimates

at the higher frequency are not shown but were used in obtaining the

faired curves. A scale of wavelength L is also shown in figure 6.

The root mean square of runway height o, which is a convenient measure

of the average roughness power, is listed in figure 6 for each of the

rauways in the figure.

It may be recalled that the spectra of figure 6 represent the spectra

of the runway height variations about the moving average. The effects

of the movingiaverage operation on the derived spectra have been shown

in the appendix to be equivalent to multiplying the actual spectra by the

NACATN 3305 7

I si4 -/atten_tionfunction _. _principle, then, itwo_dappear

that the effects of the moving-averse operation co_d be removed by

dividing the spectra of fig_e 6 by this attenuation function. However,

in practice this cannot be done explicit_, since the spectr_ estimates

are aver_es of the power over finite band widths. A _od approximation

of the effect may be obtained, however, by dividing t_ faired spectrum

estimates of figure 6 by the attenuation function and then averagi_ the

res_tant spectra over each of the b_d _hs. It may be seen that these

operations wo_d have a negligible effect for the range of frequencies

covered in fig_e 6.

DISCUSSION

Examination of figures 2 and 3 indicates that the runway heights

fluctuate in a random manner, with variations of as much as several inches

for each runway. The profile for runway 12 on both figures shows wider

fluctuations, particularly at the longer wavelengths. The overall varia-

tions in height at the wavelengths of principal interest are best compared

in figure 3, where the effects of the very long wavelengths (greater

than 300 feet) have been largely removed by the moving average. It is

clear from this figure that runway 12 is appreciably rougher than runway 55,

as was to be expected. The height profiles of figures 2 and 3, representing

what may be considered as a satisfactory and an unsatisfactory runway,

provide representative runway inputs for response calculations.

A more detailed comparison of the characteristics of the runway height

variations is possible from the spectra of figure 6. The overall height

fluctuations as represented by the root-mean-square values of 0.0_7 feet

for runway 12 and 0.021 feet for runway 35 indicate that runway ]2 is

almost three times as rough as runway 35. By comparing the spectra it

can be seen that runway 12 has lO times the power of runway 35 at the

longer wavelengths (300 to 500 feet) and about twice the power at the

shorter wavelengths (below _0 feet). The rapid decrease of power with

increasing frequency displayed by both spectra is perhaps generally

typical of runway height spectra and provides a guide to a representa-

tive spectrum shape. Inasmuch as the spectra of figure 6 represent

satisfactory and unsatisfactory levels of runway roughness, the heights

of the spectra provide an initial guide toward the establishment of

criteria for runway roughness.

8 NACATN 330}

CONCLUDINGR_WARKS

As an initial step in the study of runway roughness, data wereobtained from a survey of a relatively smooth runway and a runway whichis considered rough. A surveyor's level, rod, and tape were used to

obtain these data. In general, this method was found to be quick and

relatively inexpensive. The results obtained are presented as profiles

of runway height. The power spectra of runway height were also deter-

mined and are presented. These results provide an initial guide toward

establishment of criteria for runway roughnesses and are suitable for

airplane response calculations.

Langley Aeronautical Laboratory,

National Advisory Committee for Aeronautics,

Langley Field, Va., August 13, 1954.

2S

NACA TN 3305 9

APPENDIX

EFFECT OF MOVING AVERAGES ON THE POWER SPECTRL_4 OF A RANDOM FUNCTION

Consider a random function f(x). The centered moving average of

f(x) over the interval D is defined by

D

= i jx__° f(Xl)_XIg(x) D2

(_)

The deviation of the disturbance about its moving average is then given

by

h(x)= f(x)- g(x)= f(x)

D

- g Jx__o f(Xl)_Xl2

(A2)

The spectral properties of h(x) as compared with those of f(x) are

of concern; specifically, the effects on the spectrum of f(x) intro-

duced by the operation of equation (A2).

The "finite" Fourier transform of equation (A2) may be written

fH(2) = h(x)e -i_xdx =

X

f (X) e-i_Xdx 1 __Xf x+D- g xJx__O f(nle-_Xdn_x2

(AS)

If the order of integration of the double integral term is inter-

changed, there results

H(_) = f(x)e-i_Xdx 1 f(_)e-i_xd_ dx-D J, D

_-_ x

(A4)

lO NACATN 3305

The x-integral of the second term maynow be evaluated to give

_X sinK(a) -- f(x) e-_X_

x a_DD2

_-x f(n)e-m_dn= - F(a)

(AS)

where F(_) is the "finite" Fourier transform of f(x)

XF(g) = f(x) e-LqXdx (A6)

X

The power spectrum of h(x) follows directly from equation (A5) in

accordance with the following equation:

1 iE(_)12 1 H(_)_(-_) (A7)Ch(N) = lim _ = lim _-_X---__ X---__

Substitution of equation (AS) into this equation yields the following

simple result:

2

_h (_) = ll sin __. _ / X--9_ 2_X2 li m _.__F(_)F(__)= ll sin_ /_f(_)_l

(A8)

The squared term on the right side of this equation is the effective

spectrum attenuation function or transfer function introduced by the

movlng-average process defined by equation (A2). It is shown plotted as

a function of gD/2 in figure 4. Also shown in the figure is a scale

of wavelength for D = 300 feet, the value used in the present study.

As a matter of added interest, an alternate derivation of equation (A8)

may be given which is slightly more involved but which is more general

in the sense that the function g(x) does not have to be functionally

related to f(x). Equation (Al) may also be written

g(x)= ! f(Xl)k(x_ Xl)_ 1D

(A9)

NACA TN 35O5 ll

where

k(x): 1 (-0/2__x __0/2)

k(x) = 0 (elsewhere)

Now, from equation (A2) the power spectra of h(x) a_d g(x) can be

shown to be related in the following manner (ref. 4):

(AI0)

where

respectively, and Cfg(_) and Cgf(_) are the cross spectra of

and g(x) defined in the following manner:

Cfg(_) = lim F(-_)G(_)X--_ 2_X

where F(n)

¢f(2) and ¢g(2) are the power spectra of f(x) and g(x),f(x)

Cgf(2) = lim F(2)G(-2)X---->_ 2_X

> (All)

is defined by equation (A6) and

__XG(G) : g(x) e-i_Xdx

X

(AI2)

The cross spectra in general have both real and imaginary terms, in con-

trast to the simple power spectra which are always real.

In order to determine the spectrum of h(x), it is necessary to

evaluate each of the spectra on the right side of equation (AI0). Con-

sidering _(_) first, it can be seen from equation (A9) that g(x) is

related to f(x) by the conventional convolution or Duhamel integral

and is thus obtained by a linear operation on f(x). Using the simple

relation between spectra of an input disturbance and an output response

for linear systems, there is immediately obtained the following relation

between g(x) and f(x):

12 _ACA TN 33o5

(A_3)

where the squared term represents the amplitude of the frequency-

response function for the operation defined by equation (A9).

The cross-spectra terms may be evaluated in the following manner.

From equations (A9), (All)_ and (A12)_ for example,

F(-2) ___i __[ f(Xl)k(x _ xl)dXl_e-i_Xdx

Cfg(g) = lim (AI4)X--_ 2_X

In the limit, this equation reduces to

sin ___DD

Cfg(2) = ¢f(2) 2 (AI_)2D

2

The cross spectrum in this case has no imaginary term reflecting the

lack of any introduction of a phase shift by the linear operation defined

by equation (A2). In the present case

_fg(2) = Sgf(_) (A__6)

Substituting the results of equations (AI3), (AIS), and (AI6) into

equation (A10) yields the same result as given by equation (A8).

NACA TN 3305 13

REFERENCES

i. Shannon, Claude E.: Communication in the Presence of Noise. Proc.

I.R.E., vol. 37, no. l, Jan. 1949, pp 10-21.

2. Tukey, John W.: The Sampling Theory of Power Spectrum Estimates.

Symposium on Applications of Autocorrelation Analysis to PhysicalProblems (Woods Hole, Mass.), June 13-14, 1949, pp. 47-67.

(Sponsored by 0NR, Dept. Navy.)

3. Press, Harry, and Houbolt, John C.: Some Applications of Generalized

Harmonic Analysis to Gust Loads on Airplanes. Preprint No. 449,

S.M.F. Fund Preprint, Inst. Aero. Sci., Jan. 25-29, 1954.

4. James, Hubert M., Nichols, Nathaniel B., and Phillips, Ralph S.:

Theory of Servomechanisms. McGraw-Hill Book Co._ Inc., 1947.

14 NACA TN 3305

oo_ , , , , ............................. , .........

o_

r_

o

ooooo ooooo 0oooo oo00o ..........

o

--A.

d ...................... ; ; ;;i;" ;;;;; ....................

NACATN 3505 15

H

o

.._ ......... _ .._ .... _ _..I_ .... I ....... _ ..........

°i

16 NACA TN 33O5

35

NACA TN 330_ 17

o' ; _ _ _ ; ; _ _" ;;;;i' , , , , , , , , , , , , , , , , , , , , , , ' , ' ' ' ' ' " " " "" " _'_ _ _

............................... ° .......... ,,o ,,,°° ,

18 NACA TN 330_

i

o

........ o .... • ............. • , .... ° ...... , • •

4r;¢: ;_:;; :; ..... ,,, ,,,,, ,,,,, ,',', , .... ''''' ''''' ''

NACA TN 330_ 19

20 NACATN 3305

]i

..0 ...............................................

_oo o_..... _oo o_o _ooo _o oo_oo o_o_o o_oo o_. _.

4$

NACA TN 330_ 21

n

!A

ii............

_ _ _ _,d_ _._.d_._. _._._,_-

_;;;; ?;tt¢ _tttt ¢;;;," ;?tft ;;;;;

!i ................

22 NACA TN 3305

for p

Portion of runways surveyed

Figure i.- Diagram of runways at Langley Field, Va.

NACATN 3505

.2

0

.2

1,2

-.6

--8 1 1 I I I I 1 I l 1 I

.h

.2

E oo

.,-t

>

-.4-

-.6

0

__(b) Runway 12_

I l I 1 £ l t ] I I

2 4 6 8 I0

Figure 2.- I

/C-,6_ _ /

_1. I i I l L i I l ! L......... ___3

(b) Runway 12.

I I I I J l 1

4 6 8J l I J I 1 l

I0 12 14 16

Distance along runway, ft

Figure 2.- Variation of runway elevation about an a

23

I l I l i ] I I I I _ [ I18 20 22 24 26 28 30 XlO 2

ithmetic mean.

2_

o.r-,i

r_b.a)

.2-

0

-.2

(a) Runway35.

i I _ t 1 I _ l I _-_____L 1 J 1

.2 -

o-M

oil

a)

0

-.2

L I .... I _2 J J_- ! L-_I • ____j_

2 4 6 8 iO 12

Figure 3.- Va

(a) _ay 35.

l j 1 _.t I I_ I I l

_ A [ - I J L I l I I I J ...... ] I I L__6 8 i0 12 lh 16 18 20

Distance along runway, ft

Figure 3.- Variation of runway elevation about a 300-foot moving mean

6--,c=._-/_

NACA TN 3305

I L I I !

20 22

1 l L_ 1

24 26 28

I30 x 102

-foot moving mean.

_6S

NACATN 3305 25

L I l

II

I

ILL_LL L__. __-- l l

o

o

o.,o

o

o__1_

o

ao

,,o

o

8

!

II

OJ

v

0.,-4

0

0.,-I

©hO

I4

.,..4

_o

I

©

.,l-I

26 NACA TN 33O9

o

0cJ,

0

000

4 1 x 1o-3

I3100000

0000

2 -

i -

Runway0 12[] 35

000O000000

00

000000000

000

__ I 1 I I-'1o 20 40 60 80

X, ft

Figure 9.- Autocorrelation coefficients for runway height. X = r _x.

NACATN3305 27

.10

.01

•001

i .0001

• 00001

I

I

\I\ I\ I

Runway

D) 123_

u

.05737

,o:_117

I

I

I

.I•000001 --- --. 10•01 I

Reduced freqsency, II, radiana/ft

I DO _ I 1 L_ __L J50o 2 loo 50 20 10 _ 1

Wavelength, L, ft

Figure 6.- Power-spectral density functions for the two runways.

NACA-Langley - 11-10-54 - 1000