The maxima of amplitudes of thickness and loading noise harmonics are established when the radial distribution of blade chord, thickness ratio, and lift coefficient is specified. It is first shown that only airfoils with thickness distribution and chordwise loading distributions which are symmetric with respect to midchord need be considered for finding the absolute maxima of thickness and loading noise. The resulting chordwise thickness and load distributions for these maximum noise conditions require infinite slope at some points along the chord but otherwise are uniform. It is shown that sweeping the blades reduces the thickness and loading noise, but there is no optimum sweep which generates the lowest noise

Brown, T. J.

Farassat, F.

Nystrom, P. A.

English

NASA Technical Reports Server

18 
BOUNDS ON THICKNESS AND LOADING NOISE OF ROTATING BLADES 
AND THE FAVORABLE EFFECT OF B&E SWEEP ON NOISE REDUCTION 
F. Farassat+ and Paul A. Nystrom* 
The George Washington University 
Joint Institute for Advancement of Flight Sciences 
Thomas J. Brown 
Structures Laboratory, U.S. Army R$T Laboratories (AVRADCOM) 
SUMMARY 
In this paper the maxima of amplitudes of thickness and loading noise har- 
monics are established when the radial distribution of blade chord, thickness 
ratio, and lift coefficient is specified. It is first shown that only airfoils 
with thickness distribution and chordwise loading distributions which are 
symmetric with respect to midchord need be considered for finding the absolute 
maxima of thickness and loading noise. The resulting chordwise thickness and 
load distributions for these maximum noise conditions require infinite slope at 
some points along the chord but otherwise are uniform. It is shown that sweep- 
ing the blades reduces the thickness and loading noise, but there is no optimum 
sweep which generates the lowest noise. 
INTRODUCTION 
In the design of a high tip-speed rotating blade such as a helicopter rotor 
or a propeller, one important acoustic question is: given radial (spanwise) 
load distribution, thickness ratio, and chord distribution of the blade, can 
the maximum of the level of each of the sound harmonics be established? These 
maxima, of course, correspond to the worst possible acoustic design. If these 
maximum levels are kept within acceptable limits, then neither the chordwise 
load distribution nor the airfoil shape would be of concern in the acoustic 
design. Another question which comes to mind next is whether sweeping the 
blade tips appropriately can result in the lowest possible noise. In this 
paper both of the above questions are studied and answered. 
The starting point of our analysis is the following equation. Let S be 
the mean surface of the blade, h and p be the local thickness and load distri- 
bution on this surface, respectively. Then, the acoustic pressure p'6,t) in 
the far field is given by the equation 
+Work supported by U.S. Army Research Office (Durham). 
*Work supported under NASA Research Grant NSG 1474. 
373 
6X3 
https://ntrs.nasa.gov/search.jsp?R=19780024891 2020-03-22T02:06:28+00:00Z
4rrrop'(Z,t) = p - a2y : ri 
A aLi 
O at2 
Cat 
(1-d 
‘Y(b) = Is q-g-p,, dS (1-b) 
Li&t) = - / [ s *Iret dS 
The first term in eq. (l-a) is called the thickness noise. This formulation of 
thickness noise was derived by Hanson (ref. 1) and by Farassat (ref. 2), using a 
different approach. The second term in eq. (l-a) is the loading noise. 
It is assumed that the blade system, lying in y1y2-plane, is not in motion 
as a whole. That is, only hovering rotors and static propellers are considered 
here. It is also assumed that unsteady loading noise is negligible. For high- 
speed rotating blades, this assumption is justified for observer positions 
where the sources on the blades appear noncompact. Under this assumption, the 
acoustic pressure will be periodic with fundamental frequency based on blade 
passage frequency. For simplicity one blade is considered in the analysis. 
be 
The nth Fourier component of the noise, p:(z), is found from eq. (l-a) to 
4rrropA(Z) = - pon2w2 Y,(Z) 
- ikn Gi Lin(Z) (2) 
In the following analysis, the surface integrals with respect to S, used in 
evaluation of Y,(g) and Lin(2), are written in an unconventional manner in chord- 
wise direction. Written in this form, the effect of sweep can be introduced 
easily. The bounds are obtained in two stages as follows. First, it is shown 
that if the airfoil shape is deformed in such a way that the chordwise distance 
between the points of equal thickness on the airfoil surface is not changed, 
then the thickness noise is maximum if the airfoil is made symmetric with 
respect to a radius of the blade disc at each radial position. A similar result 
holds for the chordwise loading distribution. Therefore to obtain the absolute 
maxima of thickness and loading noise, only airfoil shapes and chordwise load- 
ing with midchord symmetry should be considered. These and the related result 
concerning sweep do not apply at high frequencies due to the mathematical 
limitations of some of the inequalities used in their derivation. The range of 
applicability of these results is, however, wide, particularly in the case of 
helicopter rotors and conventional propellers. 
374 
In all the examples in this paper a rotor blade of 5-m radius and uniform 
chord of 0.4 m is used. The number of blades is two and the tip Mach number is 
0.95. 
A 
B 
b 
b' 
bO 
b 
P' 
C 
C 
CR 
&,g(?) 
h 
hm 
191 Y 
Jn 
k 
Li'Li.n 
Mr 
Mt 
n 
"i 
P 
'rn 
P' 
P,: 
SYMBOLS 
function of n (used in eq. (12)) 
function of n (used in eq. (14)), number of blades 
chordwise variable (see fig. I), m 
mean chord in blade tip region, m 
blade chord (function of n), m 
value of bLbo where sin(nb/2n) achieves its peak, m 
coefficient used in eq. (5) 
speed of sound, m/set 
section lift coefficient (a function of n) 
arbitrary positive functions 
blade section thickness variable (see fig. l), m 
maximum thickness of blade section (.function of n), m 
integrals defined in eq. (7) 
Bessel function of first kind on nth order 
wave number, w/c 
surface integral used in calculation of loading noise, and its 
amplitude of the nth Fourier component of Li (see eq. (l-c) 
and (2)), i=1,2,3 
Mach number along radiation direction 
tip Mach number of the blade 
harmonic number 
unit normal to surface S, direction from pressure,to suction 
side of the blade. i=1,2,3 
load distribution of the blade, Pa 
peak section load (function of Al), Pa 
acoustic pressure, Pa 
amplitude of the nth Fourier component of p', Pa 
375 
'E? 
Q, & 
r 
r 
0 
r i 
R,Ri 
S 
t 
t 
Tn & 
T 
+ 
X 
-+ 
Y 
a 
B 
% 
pO 
T 
Y 
'n 
w 
function defined in eq. (10-a) 
IGI, m 
observer distance from center of rotation, m 
(xi-yi) 11:) radiation vector 
blade outer and inner radius, respectively, m 
mean surface of the blade 
observer time, set 
thickness ratio of the blade (a function of q) 
function defined in eq. (10-b) 
period of the sound, set 
observer position vector, origin at rotation center 
source position vector, origin at rotation center 
geometric angle of attack (function of rl), deg 
azimuthal angle, rad 
(8,+8,)/2 azimuthal angle of point C midway points A and B in 
fig. 1, rad 
functions of n and h or p indicating azimuthal angles of points 
A and B in fig. 1, rad 
arbitrary function (see eq. 8) 
Dirac delta function 
angle between axis of rotation and z, rad 
radial position variable, m 
variable defining the degree of blade sweep 
density of the undisturbed medium, kg/m3 
source time, set 
surface integral used in calculation of thickness noise (see 
eq. (1-b)) 
amplitude of the nth Fourier component of Y 
angular velocity of the blade, rad/sec 
376 
DERIVATION OF THE BOUNDS 
In this section, our attention will be focused on I n' The manipulations 
for L in are identical to those of 'y,. If T is the period of the sound, then 
T 
Yn(S> = $ / Y (2,t)e inwt dt 
0 
For our purpose, the volume Y(z,t) will be written as 
i 
R 
Y&t> = / 
hm '!Z 
rldn 1 
R 0 
dh/ [\l-Mrj]-l df3 
i St ret 
(3) 
(4) 
where T-I is the radial position and hm is the maximum thickness of the airfoil. 
The azimuthal angle is denoted by f3. The angles St and B, are the azimuthal 
angles of points A and B, respectively, in fig. (1). Note that hm=hm(n) 
8,=B,(n,h), and B,=B, (rl,h). The only dependence on g and t in eq. (4) comes 
through the integrand so that the time integral in eq. (3) commutes with all 
the integrals in eq. (4). 
We now introduce the source time T in a manner used by Hawkings and Lowson 
(ref. 3). Since dt=[ll-Mrllret dr, the Doppler singularity in eq. (4) is can- 
celled. Writing t=r+r/c and using the well-known integration with respect to 
-C which results in a Bessel function of first kind (ref. 3), we get 
R h m % 
Yn(Z) = c / rldrl J dh / esinB 
Ri 0 % 
Jn(nknsinc)dB (5) 
where C=(i)n e 
iknr 
Ois a constant. Let Sc=(St+BR)/2 and use BR-Bt=b/n where 
b=b(n,h) is the chordwise distance between points A and B in fig. (1). Then 
integrating eq. (5) with respect to B results in 
(6) 
We note that the angle B,=S,(n,h). We will show that the maximum of Yn 
when only 8, varies corresponds to Bc=constant, that is at each radial station 
and for all O<h<h - --, points A and B should be located symmetrically with respect 
377 
to one and the same radius of the blade disk. We need the following result. 
If g(y)>0 and ~(3) is an arbitrary function, then in any region D - 
Iy=]l eiy g d;( 5 / gd; =I 
D D 
To prove this, we note that 
I2 - IIYj2 =/ / 2sin2$ [&-&‘I l&s(?)&?l. 0 
D D 
(7) 
(8) 
In fact, strict inequality I <I holds in most cases since the integrand in eq. 
Y 
(8) has to be non-zero only in a small region in DxD. 
constant,1 IYJ= I. 
Note that if v(s)= 
In eq. (6), Jn(nknsinE)zO even for transonic and low supersonic speeds, 
since nkqsine<nMt where M t is the tip Mach number. If sin@)>O, then we can - 27-l - 
apply the result proved above. This would require nb/2nLr or n<2rn/b. Since 
for high-speed blades, the tip region is responsible for the generation of the 
noise, a reasonable value for n is nL2rz*7R/bf=4.4R/b' where b' is the mean 
chord in the blade tip region. For blades with B blades; we must have 
nBL4.4R/b'. For two-bladed helicopter rotor blades the following result typi- 
cally holds up to twenty fifth harmonics of the blade passage frequency. Apply- 
ing eq. (7) to eq. (6), we get 
Iy,&l< 2 rR rhm - 
Ri 0 
nJn(nknsine)sin(g) dhdn 
In exactly similar fashion, we can show that 
1 Q,(g) 1 = I’1Lln+‘2L2n _ k R 
pm 
I< 2 / / Jn(nknsine)sin(e)sina(n)dpdn , (10-a) 
Ri 0 
(9) 
and 
1 T,(z)] = ~~3L3n~~2~~~~e~ JR rpm nJn(nknsinc)sin($)coso(n)dpdn (10-b) 
Ri 0 
where p,=p,(n) is the peak chordwise loading and a(q) is the geometric angle of 
attack. Equations (10) and (11) describe bounds on torque and lift or thrust 
noise, respectively, when they are used in eq. (2). 
378 
We have shown above that, if we are only allowed to deform the airfoil 
shape or the chordwise loading in such a way that the chordwise distance 
between points of equal thickness or equal loading is kept fixed, the 
maximum thickness and loading noise correspond to symmetrical positioning gf 
such points with respect to the same radius. Incidentally, in this case the 
thickness noise and loading noise are 90 degrees out of phase. 
To find the absolute maxima of thickness and loading noise, eqs. (9) and 
(10) will be used. It is assumed that the airfoil thickness and load distribu- 
tion functions are monotonic with respect to variable b. This assumption is 
satisfied in most cases of interest. To be specific, thickness noise will be 
considered first. The right side of eq. (9) can be written as 
rR r 
hm 
Ri 0 
nJn(nknsinc)sin(%) dhdn 
bO 
= - rR r 
Ri 0 
nJn(nknsinc)sin($)$ dbdn 
where bo=bo(n) is the blade chord. To maximize the last integral, take 
dh --- = 
db - AG(b-bp) 
(11) 
(12) 
where A is a function determined by the maximum thickness of the airfoil and 
bp=bp(n) is the value of bzb o where sin(nb/2n) achieves its peak. If "t(n) is 
the thickness ratio of the airfoil, then A(n)=bo(n)t(n). Using eqs. (9), (11) 
and (12), we find that 
R nb 
Iy,& I =2r nbo i Jn(nknsine)sin($-) dn 
max Ri 
Similarly, to maximize the integrals in eqs. (10-a) and (lo-b), take 
* = - BG(b-b db P 
) 
(13) 
(14) 
where B is a function determined by spanwise loading of the blade. 
section lift coefficient of the blade, then B=pobo n2w2cQ/2b 
If cp is 
P' 
Equations 
(10-a) and (10-b) then give 
379 
(R(;)I = !g JR + 3) 
max Ri P 
Jn(nknsinc)sin( 2,, si.na(n)dn 
. 
ITJ~:) 1 2 lcosel 
R' nb 
= POW r 
bon3cQ 
Jn(nknsinc)sin(*)cosa(n) dn 
+ max Ri bP 
(14-a) 
(14-b) 
We have shown that for all blades with a given thickness ratio i(n) at 
each radial station, the rms amplitude of the nth harmonic of thickness noise 
has the following bound 
4nroIP~(31~ fi pon2w2jYn(Z)l 
max 
(15) 
Similarly for all blades with a given section lift coefficient cR(n), the rms 
amplitude of the nth harmonic ofttorque and thrust (lift) noise have the bounds 
4~roIp$)I~ fi nk IQ,(~)] 
max 
(16-a) 
4rroIp~($IJi nk/Tn(z)I (16-b) 
max 
respectively. 
Equations (12) and (14) show that the thickness function and chordwise 
load distribution function which generate maximum noise have infinite slope at 
the same two points which are symmetrically located with respect to the mid- 
chord. The corresponding thickness and chordwise load distributions are rec- 
tangular. Note also that Ip~j is maximized by different thickness and load 
distributions for different n. In general, therefore the results of equations 
(15) and (16) are expected to be too pessimistic. 
Figure 2 shows some spanwise aerodynamic data for a two-bladed helicopter 
rotor. These performance data were calculated by a strip theory-momentum 
analysis described in reference 4. The blade thickness ratio is 8 percent, 
the blade radius is 5 m and the chord is 0.4 m. The blade planform is rectangu- 
lar. The rotor rpm is 626. Figure 3 shows the calculated thickness and load- 
ing noise with the theoretical bounds obtained above. The chordwise load distri- 
bution at each radial position was obtained from the Garabedian-Kom program 
(ref. 5). The airfoil section used is NACA 0008. The observer position is 50 m 
from the rotor center and 300 below the rotor (&=1200). The bound for loading 
noise is obtained by adding the right sides of eqs. (16-a)and (16-b). It is seen 
380 
that both bounds are very coarse although the bound on loading noise is not as 
pessimistic as that of thickness. noise. For the first harmonic level of loading 
noise, the reason for the theoretical bound being lower than the calculated level 
is not known. It may be due to the fact that the drag force (skin friction and 
wave drag) obtained from the Garabedian-Korn program has a component normal to 
the chord which was used in the acoustic calcualtions. 
For B blades, substitute nB in all the equations derived above. 
THE EFFECT OF BLADE SWEEP 
We have shown that IY,(~)I is maximum when Bc(n,h)=constant. One way of 
reducing the level of the thickness and loading noise iS blade sweep. This can 
be seen from eq. (8). The question arises whether a blade sweep can be select- 
ed which generates the least noise. We will show that among the blades with 
gradually increasing sweep towards the tip, there is no optimal sweep. 
To be specific, we take S,=-nn2 where n>O. The same argument holds as 
long as aBc/an<O. From eq. (6), we have 
R 
invn2/2 dq / 
hm 
~$1 = r n Jn(nknsins) e sin(*) dh 
Ri 0 
20 
2 
= rR g(n) e in?m j2 drl 
where g(n)>0 is defined as 
hm 
s(n) = n Jn(nknsine) / 
0 
sin($) dh 
(17) 
(18) 
It is assumed, as befor:, that nb/2n<r. - We note that as u increases, so does 
the blade sweep. We have 
R n' 
= -nu r r 
Ri Ri 
(n2-n'2)g(n)g(n')sin[y(n2-n'2)]drtdnV (19) 
381 
If it is assumed that sin[y (R2-Ri2)]>0, that is n<2n/n(R2-Ri2), then 
(20) 
For all practical angles of sweep, the above restriction on n is less strict 
than previously obtained restriction nL2mn/b. The above result indicates that 
the levels of harmonics of the thickness noise decreases as the blade sweep in- 
creases. This result is also valid for loading noise. 
To test the validity of the above result, figure 4 shows the thickness 
noise spectra of three blades with increasing sweep. The tip Mach number is 
0.95 and the thickness ratios of all the blades are 8 percent. The observer 
is in the rotor plane and 50 m from the rotor center. It is seen that the above 
conclusion is indeed correct and should hold up to the 22nd harmonic. In fact it 
holds for much higher harmonics. The airfoil section used in the calculations 
is NACA 0008. 
CONCLUSIONS 
In this paper, bounds are established on thickness and loading noise of ro- 
tating blades. Only steady loading noise is considered which restricts the re- 
sults to high tip speeds. It is shown that only chordwise thickness and load 
distributions with midchord symmetry need be considered to establish these 
bounds. The resulting thickness and load distributions have infinite slopes at 
two points symmetrically located with respect to the midchord. Due to the fact 
that the amplitude of each harmonic of the spectrum is maximized, the resulting 
bounds are too coarse. A more appropriate approach may be to search for chord- 
wise thickness and load distributions which maximize overall acoustic level. 
It is also shown that sweeping the blade tips is beneficial in reducing 
the radiated noise. Also for blades with sweeps that increase towards the tip, 
there is no optimal sweep for minimum noise level. 
. 
382 
REFERENCES 
1. Hanson, D. B.: Near Field Noise of High Tip Speed Propellers. AIAA Paper 
No. 76-565, July, 1976. 
2. Farassat, F.: Discontinuities in Aerodynamics and Aeroacoustics: The 
Concept and Applications of Generalized Derivatives. Jour. Sound and 
Vib.., vol. 55, no. 2, Nov. 1977, pp. 165-193. 
3. Hawkings, D. L.; Lowson, M. V.: Theory of Open Supersonic Rotor Noise. 
Jour. Sound and Vib., vol. 36, no. 1, Sep. 1974, pp. l-20. 
4. Mantay, W. R.; Shidler, P. A.; Campbell, R. L.: Some Results of the 
Testing of a Full-Scale Ogee Tip Helicopter Rotor; Acoustics, Loads, and 
Performance. AIAA Paper No. 77-1340, Oct. 1977. 
5. Bauer, F.; Garabedian, P.; Korn, D.; Jameson, A.: Supercritical Wing 
Sections II, A Handbook. Lecture Notes in Economics and Mathematical 
Systems, No. 108. Springer-Verlag, 1975. 
383 
h 
(a) Thickness. (b) Load. 
Figure 1 .- Chordwise distributions. 
14 
r 
QL 
18 I 
16 SPANWI SE 
s;;;;;;E 14 
LIFT ” 
DISTRIBUTION, 
DISTRIBUTION. 12 
kN/m 6 
kN-m/m 10 
8 
6 
4 
2 
0 
2 
0 
0 
Mt 
SPANWISE TORQUE DISTRIBUTION 
SPANWISE LIFT DISTRIBUTION 
COEFFICIENT OF LIFT 
COEFFICIENT OF DRAG 
= 0.95 1.0 
.8  
.6 
I I I I I I 
0 .2 .4 .6 .8 1.0 
FRACTION OF RADIUS. i-//R 
Figure 2.- Performance curves for two-bladed helicopter. 
384 
120- 
THEORETICAL BOUND 
110 - 
SPL,dB 
(RE 20 pPa) loo 
f 
CALCULATED 
HARMONIC NUMBER 
(a) Thickness noise. 
THEORETICAL BOUNDS 
HARMONIC NUMBER 
(b) Loading noise. 
Figure 3.- Comparison of theoretical bound with calculated thickness 
and loading noise spectra. 
A- 
0.4 m 
t SPL, dB 
- STRAIGHT 
(RE 20 pPa) 
---MEDIUM SWEEP 
----- BIG SWEEP 
I 
8oo 
I I I 
10 20 30 I 
I HARMONIC NUMBER - .a r..,rrn 
Figure 4.- Effect of blade sweep on thickness noise. 
385 


Bounds on thickness and loading noise of rotating blades and the favorable effect of blade sweep on noise reduction

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server