The description of road surface roughness

Journal of Sound and Vibration (1973) 31(2), 175-183

THE DESCRIPTION OF ROAD SURFACE ROUGHNESS C. J.

DODDS AND

J. D.

ROBSON

Department 0/ Mechanical Engineering, University ofGlasgow, Glasgow G12 8QQ, Scotland (Received 15 March 1973) It is shown that typical road surfaces may be considered asrealizations of homogeneous and isotropic two-dimensional Gaussian random processes.Complete description of any suchprocessisprovided by a singleautocorrelation function evaluated from anylongitudinal track, and a single direct spectral density function therefore provides a road surface description which is sufficient for multi-track vehicle response analysis. A new road classification method is proposed whichis based on this function.

1. INTRODUCTION

Road surfaces appear amenable to representation as realizations of random processes, provided that the effects of such occasional large irregularities as potholes are removed from the analysis and treated separately: these must not of course be ignored. This being the case, description of the road surface as a realization of a stationary random process will enable the response of a vehicle traversing a given road to be determined by means of the accepted techniques of the theory of random vibration. In the particular case of stationary random excitations with a Gaussian distribution and zero mean value, it is only necessary to compute from the time histories of the excitations their second-order moments, or in effect their spectral densities, to obtain a sufficient statistical description of the excitation process. Thus a spectral description of the road, together with a knowledge of traversal velocity and of the dynamic properties of the vehicle, will provide a response analysis which will describe, precisely enough to be useful, the response of the system expressed in terms of displacement, acceleration or stress. A four-wheeled vehicle running along a road is subjected to four imposed displacementexcitations, one at each wheel. Description of the road surface must be complete enough to describe adequately the displacement imposed at each wheel (in statistical terms) and all correlations between the four displacements. There is clearly no major difficulty in providing a digital survey of the two tracks along the road which the vehicle will traverse. This sampled data can then be used to compute the statistical parameters necessary to provide the required description. It was shown by Dodds and Robson [IJ, assuming that the rear wheels follow the same profile as the front and that [l?R(n), !/RL(n), !/LR(n) and !/L(n) represent in terms of a wave number, n, the direct and cross spectral densities ofthe two tracks ~R(X) and ~L(X) (Figure I), that the response spectrum SOU) for any component in the vehicle is given by

vS°(f) = [at al + a~ a2 + a! a2e-i + at ct:3 el 1>J [fL(n) + + [et! et3 + a~ a4 + a! a4e-it/J + a~ et 3 ell/J] [fiR(n) + + [C(~etl + ata2 + ct:~a2e-iJ [fLR(n). 175

(I)

176

C. J. DODDS AND J. D. ROBSON

Here ¢ = 2nfafv; a is the wheelbase of the vehicle; v is the vehicle velocity; C1. r is the vehicle receptance, i.e., the complex response of the particular component under test due to a harmonically varying imposed displacement of unit amplitude at the rth input, with all other inputs held fixed; and the asterisks indicate complex conjugates. Further simplification of equation (1) was achieved [1] by developing certain ideas due to Parkhilovski [2], and it was found that specific determination of the cross spectra could be avoided. However, the limitations imposed on the surface description in utilising Parkhilovski's hypothesis-which implied the transverse slope to be uniform at each longitudinal position-do not permit as realistic a model of the road surface as we would desire.

r''l ,

.5{(n)

""'R

4

3

,

(x'--I-

2

Figure 1. Four-input vehicle model-layout of wheel inputs.

It will be shown in this paper that a more realistic analysis of the probability characteristics of the road surface may be obtained by considering it not as a pair of separate tracks but as a complete surface expressed as a random function of two variables, the longitudinal coordinate x and the transverse coordinate y. This is made possible by treating the surface as a realization of a Gaussian homogeneous and isotropic random process; it will be demonstrated that the spectral properties of such a surface can be obtained from measurements taken from a single track along the road, the statistical properties being the same in all directions. Use of this hypothesis seems justified by the close agreement between the theoretical and experimental results presented in section 3. The possibility of describing a road surface by a single spectral density makes it possible to suggest a simple standard form for road-surface description. 2. THE ROAD AS A HOMOGENEOUS AND ISOTROPIC RANDOM PROCESS If the road surface be considered as a realization of a quite general random process, then two-dimensional correlation functions will provide a statistical description. To determine these latter functions the surface profile recordings relative to a fixed plane must be measured throughout every possible longitudinal road-surface section, defining the displacement ;::(xr,Yr) at all points (x"Yr)' From these values can be obtained the general autocorrelation functions (2) The establishment of these autocorrelation functions would be an extremely laborious operation both in data acquisition and computation and it is in any case doubtful whether practically useful analysis could be based on them. Thus one is inevitably led to seek a simplified means of road surface description. Suppose it is assumed that the surface irregularities Il:(x,y), form part of a two-dimensional

DESCRIPTION OF ROAD SURFACE ROUGHNESS

homogeneous Gaussian random process can now be defined as

~(~,'tJ)= x-.co4XY lim _1_ Y-'co

{~(x,y)}.

x

The two-dimensional correlation function

y

f fAX,Y).;l;(x+~,Y+I])dXdY, -x

177

(3)

-y

which may be written as ~(~,'tJ)

= <~(x,Y)·~(x + ~,Y + 1]),

(4)

where the angular brackets denote averaging over all x, y. If it is now assumed further that this process {~(x,y)} is isotropic, i.e. the homogeneous process {~(x,y)} possesses circular symmetry, some interesting results emerge. Putting respectively I] = 0, and = 0 in equation (4) gives the autocorrelation functions in the x and Y directions:

e

~xCe) = ~(e,O),

(5)

~y('tJ) = ~(O, 'tJ).

(6)

The cross correlations between the displacements on the two tracks defined by y = b, y = -b (Figure 1) can be expressed as ~RLm = <~R(X). :l:L(X + ~)

= <~(x, -b). ~(x + =

e, b)

~(" 2b)

(7)

and

RLR@ = ~(e, -2b). But, as an autocorrelation function from equations (7) and (8) that

(8)

&lee, 1]) must be even in both eand 1], it is therefore found

~RL(e) = ~LRm = &lRL(-e);

(9)

it follows from equation (9) that the cross correlation function between displacements on two parallel tracks separated by a distance 2b is even. This implies further that the cross spectral density, 9' RL(n), must be a real valued quantity: 9'RL(n) = 9'iL(n) = 9'LR(n) = 9'x(n), say.

(10)

Moreover, because of homogeneity the autocorrelation function of the profile along any straight line must be identical in form to that taken along any other parallel straight line. This implies that (11) and the corresponding spectral relationship 9'R(n) = 9'L(n) = 9'D(n), say,

(12)

must be true. However, isotropy requires ~(e, 0) = ~(O, 11)

for

(13)

e= 11, and moreover ~(p cos

e, p sin e) =

~(P),

the latter being defined in the usual way along a line making any angle (Figure 2).

(14)

e with the x-axis

178

C. J. DODDS AND J. D. ROBSON

Equations (8) and (14) together imply that ~LRm = ~(p) = ~(V ~2 + 4b2 ) .

(15)

The one-sided direct and cross spectral densities of the profiles If.R(X) and A:L(X) of two tracks separated by a distance 2b follow by Fourier transformation: eo

Y:o(n) =

J2~(~) e-12~n~ d~,

(16)

-
J2~LR(~)e-121tn~d~= J2~(p)e-f21tn~d~, ex>

f/x(n)=

ex>

(17)

where

(18) Thus from a knowledge of the autocorrelation function ~(~) for any single track in the x direction, the form of Pl(p) in any other direction is known and can then be derived from equation (15), and hence the cross spectral density, [;Px, can be evaluated from equation (17).

2

Y.1J

-~------

I I I I

b

I I

b

x,(

I _---1-

_

~

(

F igure 2. Two-dimensional road surface parameters.

It is now convenient to define a coherency function, fAn), by $l(n) =

[;Px(n)/ [;Po(n) .

(19)

Here the quantity ?2(n) is analogous to the quantity --/(/) as used elsewhere in the literature [3] but it is convenient to distinguish it from "y2(/) as ?(n) itself defines, in this special case, the inter-dependence between the two tracks. In the general case the cross spectrum may be written as [3] (20)

and the coherency function defined as 2(f) Y

2

=

ISxif) 1 SxC!) S)1(/)

(21)

It can be seen that the function '1 2( / ) is a measure of the magni tude of I Sxif) 12 in terms of the associated values of SxCf) and Sy(/) at the same value off To assess the relationship between the two signals, in the general case, the phase relationship, ()Xy(/), must also be considered. However, for the special case considered here, equation (10) implies that (22)

179

DESCRIPTION OF ROAD SURFACE ROUGHNESS

From equation (19), it is evident that the coherency function, ;9'(11), can then be obtained for any track width, 2b, by substitution in the relationships (16), (17) and (18):

;9'(n) =

f'"PA(V ~2 + 4b

e-121tn~ d~ -'" -------2

2)

(23)

eo

2

f ~(~) e-i21tn~ d~

Just as the general coherency function y2(f) satisfies 0 .;;; y2(f) .;;; 1 for all f [3], so it can be shown that 0<; ;9'(n).,;; 1,

0";;11";;;; co.

(24)

Moreover, from practical considerations it is possible to say that ;9'(n) must approach unity for small wave numbers (i.e., long wavelengths) and zero for large wave numbers (i.e., short wavelengths). Examination of equation (23) shows that as 2b ~ 0 (i.e., the two tracks corresponding to ~R(X) and ~L(X) become closer together) then f;(n)~I.

Also, since the typical road autocorrelation function 2b ~ 00

(25) PA(~)

tends to zero for large

;9'(11) -s o.

~,

then as (26)

The assumptions of homogeneity and isotropy which make it possible to describe a road surface by a simple spectral density are major steps in the analysis and must be justified by experiment before they can be adopted. The experimental evidence reported in the following section is not extensive enough to be conclusive but does give a very encouraging degree. of support for the typical roads considered. 3. EXPERIMENTAL RESULTS

In order to test the hypothesis of isotropy, geodesic surveys were carried out on several different types of road surface for which the surface undulations on two or three parallel tracks were measured at equal longitudinal increments. The resultant data was high-pass filtered to remove trends and low-frequency signals and then statistically analysed. The results are shown in Figures 3, 4, 5 and 6. Figure 3 shows spectral densities taken from pairs of tracks for each of three different road surfaces. It will be seen that the spectral density of the road surface in the longitudinal direction is independent of the track taken in each case, and this supports the assumption of homogeneity. While these plotted spectral densities are typical of the majority of profiles which will be encountered by vehicles, special treatment will of course be required for tracks in close proximity to verge and kerb where increased short-wavelength content must be expected. Figure 4 shows a typical plot of ~LR(~) and ~RL(~)' All cross-correlation functions investigated to date were found to be similarly symmetrical about the ~ = 0 axis; this implies that cross spectral densities taken across two tracks were real in each case. Figure 5 shows the coherency function {An) for a given road with three different track widths: it clearly behaves in a manner consistent with the assumption of isotropy. Moreover, Figure 6, which shows curves of ;9'(n) for a single track width for three different road surfaces, compares values computed from pairs of tracks with those computed (on the assumption of isotropy) from single tracks. Here the assumption of isotropy is strongly supported.

180

C. J. DODDS AND J. D. ROBSON Wove number(cycle/m) 10. 2

10·'

no

10-2

10- 1

(Cycle/ft)

Figure 3. Spectral densities of two parallel tracks along three different types of motor road. Route: A, motorway; B, minor road; C, pave; . _._.•, !/L(n); - - - , !/R(n).

10

20

30

(m

Figure 4. Normalized cross correlation function, minor road. 1·0

0·8

0 ·6

:s

tJ,

0 ·4

0 ·2

o

0 '0 2

0'04 I

0 ·1

006

Wave number [

0 '1

n (cycle/ft) ,

0 ·2

0 ·3

0 ·12

0 ·14

!

0 '4

( Cycle / m)

Figure S. Coherency function, g(n), as a function of track width, 2b (minor road). 2b ~ 4 ft; ..., 2b = 8 ft.

, 2b = 2 ft; - ' - ,

181

DESCRIPTION OF ROAD SURFACE ROUGHNESS

A'

1-0

0-8

0-6

~.-~~ ••• ~, \~,\:

\\ ~~'\B ":\ B\ uII

r.\ \ "\ / \J,' ~ ~I .x \ ''''. 11 \ .I \ '.', >t: {':j;\ . c ~ \ x'1-, i''i., ; ('\'~ I,

cd,.., \'1

I, I

0-4

I

«

\

\"

.... \

I

'V

I

I

'''' \

,..:v 'il I \

.

I

\'

0'02 !

I

.,

1 J \IJ,' ..,

\ . ./

o

J\

I '\

iI

0-2

\

I \ ~

0'04 0-06 0'1 Wave number, n (cycle Zf t ) I

I

0'12

0·14

I

This data therefore provides evidence that the assumption of isotropy is justified in the case of a variety of roads. While this evidence cannot be regarded as conclusive without a very extensive investigation of road surfaces there does seem to be a strong presumption that very many road surfaces are isotropic and that the very considerably simplified analysis resulting from the assumption of isotropy is a valid approach to road surface description. 4. ROAD DESCRIPTION The possibility of describing a complete road surface by a single spectral density suggests the feasibility of approximating to that spectral density by a simple analytical form, From an analysis of road spectra presented by MIRA [4; see, for example, Figure 7] one finds that these spectra can be approximated by an equation of the form

(

n)-WI

[/(no) no

Yen) =

( [/(no)

(

:0

)

l

(27)

- W2 l

where [/(no) is the "roughness coefficient" (value of the spectral density at the discontinuity frequency no). The measurements for typical road surfaces can be classed into various groups and the parameter values for each group are given in Table 1 [5]. From the spectra presented by MIRA the discontinuity between the two branches of equation (27) was found to correspond to a wavelength of approximately 6'3 m (20 ft): it is worth noting that this is one of the standard lengths used in road construction techniques. This gave to no a value of approximately 1/2rccycle/m and this was taken as the datum value. Classification of road spectra by this means is proposed by Dodds [5] and although much work is still needed to confirm the parameters in Table 1, it is felt that this could form a sound basis for future road classification. [/(no) provides a basic roughness coefficient, and since it

182

C. J. DODDS AND J. D. ROBSON

is expressed in terms of the spectral density the ratio of amplitude roughness between two roads is proportional to the square root of the ratio of their respective g(n o) values. The exponents Wi and W2 have completely contradictory significance. For a given f/(no) a high value of Wi indicates a road with increase in proportional roughness at the longer wavelengths. Conversely a road with a high W2 exponent suggests a road with a decrease in proportional roughness at shorter wavelengths. As a road's state ofrepair deteriorates, a decrease in the exponent W 2 is to be anticipated.

o-:

001

1

10

Wave number (cycle 1m)

Figure 7. The spectral density, .9(n), of a typical principal road showing the dual slope fitting characteristic [5].

Road spectra presented by other authors [6, 7] can also be classified by this method and close agreement between road types is obtained. However, although equation (27) defines road spectra by using the simplest mathematical expression possible and is ideal for computational or design purposes it does not as it stands TABLE

1

Classification ofroads [5] based on road spectra presented by MIRA [4] 1 no = 271 cycle/m Wt

Motorways Principal roads

Minor roads

t

Very good Good Very good Good Average Poor Average Poor Very poor

.9(n) measured in units of 10-6 m 3/cycle.

Wz

-"'-

'""

0·464

1'360

0·221

2·05

0·487

1·440

0·266

2'28

0·534

1·428

0·263

Standard deviation

2-8 8-32

1'945

32-128 128-512 32-128 128-512 512-2048

,..,....

Standard '"" deviation

Mean

2-8 8-32

r Mean

Range

9'(no)t

Road class

r

DESCRIPTION OF ROAD SURFACE ROUGHNESS

183

allow for any easy mathematical analysis, as 9"(11) becomes infinite at n = O. This inconvenience can be overcome in various ways, the simplest analytically being to filter the road spectrum with a high pass filter with a transfer function of the form ).i2n:n cx(in) = 1 + }j21!n ' (28) in which }. is chosen to be a length greater than any wavelength of interest. The effect of this is to reduce the spectrum to zero at zero wave number and thus to yield a finite 9"(n) at n = O. 5. CONCLUSIONS The treatment of road-surface undulations by consideration of a homogeneous and isotropic random process seems very promising. The experimental evidence obtained to date is limited, but in the case of all three of the very different road surfaces if,vestigated it strongly supports the two hypotheses of homogeneity and isotropy. Vehicle response, which is governed by equation (1), can now be reduced to the form

1 SOU)

= -

u

{[F0(f)]2

+ gvC/)[Fx(fW} 9"o(n),

(29)

where [r 0(/)]2 and [Fx(f)]2 represent the contributions of the ex's from equation (1) and as such are dependent on vehicle dynamic properties and speed. The coherency function, gv(f), is defined in terms of the spatial coherency function , fl.(n) :

(30) Thus, for response prediction, apart from the vehicle dynamic properties, a knowledge is required only of the direct spectrum, 9" o(n), of any single track along the road, from which gvC/) can be derived for any speed, v. If this new hypothesis regarding road-surface roughness is found to be generally true, then the road classification method proposed will, for all practical and design purposes, completely define the road-surface roughness. ACKNOWLEDGMENTS Thanks are due to Mr N. F . .Barter of MIRA for providing the wide range of road spectra necessary for the road classification method and to members of B,S. panel MEE/158/3/1 for their comments and discussion on the proposed method of road classification which is currently the British proposal for International Standard ISO/TC l08/WG9,

1. 2. 3. 4, 5. 6. 7.

REFERENCES C. J. DODDS andJ. D , ROBSON 1970 Proceedings ofthe 13 FlSITA Congress, Brussels, Paper 17.2D. The response of vehicle components to random road surface undulations. I. G . PARKHILOVSKI 1968 Avtomobil'naya Promyshlennost' 8, 18-22. Investigations of the probability characteristics of the surfaces of distributed types of roads. J. S . BENDAT and A. G. PIERSOL 1971 Random Data : Analysis and Measurement Procedures. New York: Wiley Interscience. R. P . LABARRE, R. T. FORBES and S. ANDREW 1969 Motor Industry Research Association Report No. 1970/5. The measurement and analysis of road surface roughness. C. J. DODDS 1972 (May) BSI Document 72/34562 (ISO/TC/I08/WG9 (MEE/158j3jl)), Generalised terrain dynamic inputs to vehicles. H. BRAUN 1966 Deutsche Kraftfahrtforschung und Strassenuerkehrstechnik 186, 1-83, Untersuchungen tiber Fahrbahnunebenheiten. 1. KANESHIGE 1969 Society of Automotive Engineers Paper 690111, Application of the probability theory to the design procedure of an endurance test track surface.