An investigation of impulsive rotor noise of a model rotor

Journal of Sound and Vibration (1973) 28(1), 55-71

AN INVESTIGATION OF IMPULSIVE ROTOR NOISE OF A MODEL ROTOR~ J. W. LEVERTON,+. AND C. B. AMOR

Westland Helicopters Limited, Yeor'il, Somerset, England (Received 27 November 1972) The investigation of gust induced impulsive rotor noise was made by using a three bladed, 9 ft diameter, model rotor. The gust was produced by a series of air-jets placed under the rotor disc and the noise characteristics were determined for a range of gust lengths and amplitudes. The main emphasis was placed on experimental measurements and theoretical prediction of discrete noise. The theoretical estimation of discrete noise was made by using a simple point dipole theory and a more complex rotational noise theory. The theoretical results have been compared with measurements and show good agreement both in amplitude and characteristics over the full range of gust profiles used in the experimental programme. Broadband noise characteristics have also been reviewed in relation to possible blade stall. 1. INTRODUCTION On a helicopter the most common source of impulsive noise is that associated with blade slap, but other aspects such as blade stall, blade/fuselage interaction and transient gust loading give rise to similar characteristics. Blade slap resulting from the unsteady lift fluctuations on a blade due to interaction with the vortex from another blade has previously been examined in detail [1] and the noise generation mechanisms are fairly well understood. To date experimental studies of other impulsive or transient rotor noise mechanisms have, in general, been limited to adhoc tests connected with blade/vortex interaction and although many theoretical investigations have been conducted, there is little or no experimental verification of the results. The position regarding transient stall is also very confused, with some authors using it to explain blade slap, while others state that it cannot occur in the traditional sense on a helicopter rotor blade. It was considered desirable, therefore, to conduct a detailed investigation into the mechanisms associated with impulsive noise. Some preliminary work [2] on the effects of gust induced blade stall had given encouraging results and it was decided to use the approach developed in this work for a more detailed study of blade/gust interaction effects. 2. SCOPE OF THE INVESTIGATION The main aim of the test programme was placed on a study of the effects of transient or impulsive loading on discrete frequency noise and as a secondary investigation the possibility of gust induced blade stall was examined. The impulsive or transient aspects were investigated t An abridged version of this paper was presented at the American Helicopter Society, Mideast Region, Symposium on "Status of Testing and Modeling Techniques", Philadelphia, U.S.A. 26-28 October 1972. :~Visiting Lecturer, Institute of Sound and Vibration Research, Universityof Southampton, Southampton SO9 5NH, England. 55

56

J. "~V. LEVERTON A N D C. B. AMOR

both experimentally and theoretically, with existing rotational noise theories [3, 4, 5, 6, 7] being adapted to meet the particular requirements of this investigation. There did not appear to be any theory applicable to "'blade stall" and thus this part of the investigation was limited to a review of the experimental data. In this acoustic-aerodynamic study a model "hover" rotor rig was used and the impulsive noise was generated by subjecting a discrete area of a rotating rotor blade to a "gust" of known size and strength. The air-jet arrangement used for producing the gust is shown in Plate 1 and the test rig set up is illustrated diagrammatically in Figure I. Microphone. ~ /

J

9 ft (2/:?)

o,o~

-

"\ 9e

Air-Jet arrangement Figure 1. Test arrangement. As the.blade passed through the gust it was subjected to a rapid change in lift, which in turn applied a fluctuating force and thus generated noise. For these tests the rotor was run with the blades set at zero incidence (and hence at zero lift). Thus the only lift on the blades was that generated by their passage through the gust. 3. TEST RIG The rotor rig was fitted with a 9 ft diameter, three bladed, rigid, rotor head. The blades were of a rectangular planform without twist and the aerofoil section was the standard NACA 0012 with a four inch chord. The gust profiles used in the experimental part of the investigation were produced by a series of air jets placed 8 in below the rotor at 0-9 radius (Plate 1). The length of gust was achieved by using a number of nozzles, up to 24, mounted on a circumferential line. The variation in amplitude (gust velocity) was obtained by varying the air pressure. The air-jet velocities were measured by using a hot-wire anemometer and the jet velocity ripple at a distance of 8 in was found to be no greater than 5 ~o of the mean. Representative gust profiles used for this investigation are shown in Figure 2. These gust velocity profiles show that there is a difference of approximately 20 ft/s between the maximum air velocity from one nozzle and three nozzles, while the addition of more nozzles merely increases the length of the gust profile without increasing the amplitude. Since it was desirable

Plate 1. Air-jet arrangement.

(fachlg page 56)

IMPULSIVEROTOR NOISE

57

tO have a constant " m a x i m u m " velocity irrespective o f gust length the shortest gust considered was that associated v~ith three jets: i.e., 1.25 chord lengths. The gust velocity was restricted to 125 fl/s, since at higher velocities the air-jet noise tended to mask the r o t o r noise. I

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Length (in) I 0

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Length of profile

(chords)

Figure 2. Typical gust profiles. Values of X: 125 f/Is; 105 ft/s; 77 ft/s; 45 ft/s. 4. TEST P R O G R A M M E / A N A L Y S I S Tests were conducted over a r o t o r tip spedd r a n ~ 6 f 141 to 424 ft/s with the r o t o r set at zero pitch. The gust "length" was varied between 1.25 arid 7 chords and the gust amplitudes ranged from 45 ft/s to 125 ft/s, as illustrated diagrariih~iicalJ~/iia Figure 2. These test conditions are detailed in Table 1 and the corresponding " m a x i m u m induced angles" are given TABLE 1

Test conditions~ No. of jets 1

9 12 15 18

r 0

Gust amplitude (ft/s) "~ 45 77 105

x

x

x

x

x

XO

XO

XO

XO

XO

X

X

X

X

X

xo

xo

xo

xo

xo

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x

x

x

x

xo

xo

xo

xo

xo

"~ 125

x

x

x

x

x

21

x

x

x

x

x

24

xo

xo

xo

xo

xo

Each of the above conditions were run at the following rotor speeds: rev/min 300 • 400 • 500 • 600 x~ 700 • 800 • 900 x~ Tip speed(ft/s) 141-3 188.4 235.5 282.6 329.7 376"8 423-9 t x Conditions recorded on tape. o Conditions analysed by narrow band analysis.

58

J. ~,v. LEVERTONAND C. B. A/t,IOR TABLE2

hr&tced wtgles of hwidence Speed (rev/min)

Blade velocity at centre of gust fit/s)

300

127.2

600

255.4

900

381.6

Gust velocity fit/s)

Induced angle (degrees)

45 77 105 125 45 77 105 125 45 77 105 125

19.5 31.1 39.5 44.5 10.0 16.8 22.4 26.1 6.7 11.4 15.4 18.0

in Table 2. For each of the test conditions tape recordings of the noise were made by using standard BriJel and Kj~er (B & K) equipment and a Nagra tape recorder, the microphone being positioned I rotor diameter (9 ft) from the "gust position" and at 45 ~ to the rotor disc plane, as indicated in Figure 1. Recordings were made with "air on" (impulsive noise generation) and "air off" (normal rotor noise) so that the level of the "impulse noise" could be determined. "Ambient background" and "air-jets only" recordings were also made to verify that the results examined were not influenced by these effects. The noise recordings were analysed by making tape loops, of approximately 5 ft in length. Narrowband analysis was performed by using a Spectral Dynamics analogue analyser system. The resultant frequency spectrum traces covered the ranges 20 Hz to 150 Hz and 150 Hz to 5 kHz with 2 Hz and 5 Hz bandwidth filters, respectively. From these traces the levels of the "discrete frequency" noise associated with the impulse were extracted. For the blade stall part of the investigation ~}-octave band, overall and "A-weighted overall" (i.e., dBA) noise levels were obtained by using standard B & K. instrumentation. 5. THEORETICAL STUDY The main aim was to predict the acoustic spectra of a rotor subjected to transient/impulsive loading and verify these results with measured values obtained from the experimental tests. As the blade passed through the gust it was subjected to a rapid change in lift, which in turn applied a fluctuating force on tile surrounding air and thus generated noise. The actual noise generation was considered to be dipole in nature with the dipole axis being in the same direction as the line of action of the fluctuating force. Since the rotor was run with the blades at zero pitch the dipole axis was vertical. The blade loading was calculated by using a lifting line theory which is detailed in Appendix I. The resultant blade loading was then used in the acoustic theories to predict the noise spectra. Initially a simple point dipole theory which had been developed for a "blade slap type" study [I, 9] was used to compute the acoustic spectra of the noise generated during the passage of a blade through a short gust. This theory is summarized in Appendix II. It was

59

IMPULSIVE ROTOR NOISE

realized that this simple point dipole theory would not be applicable to the "long gust" case. Hence a version of an advanced computer program for the prediction of rotational noise due to fluctuating loading on rotor blades, developed by Tanna [4], was modified for use in this study. A brief summary of this theory is given in Appendix III. To calculate the acoustic spectra, the blade loading was simply fed into the appropriate program. When using the rotational noise program it was found, however, that an "effective" chord rather than the actual blade chord had to be used : this is discussed in reference [8].

6. COMPARISON OF THE SIMPLE POINT DIPOLE AND ROTATIONAL NOISE THEORIES A range of "gust lengths" were considered and the acoustic spectra computed by using both the simple point dipole and the detailed rotational noise theories.

6" ~0 7[ ~0 ~1 1 l l l l l l l l l l l ~ ( O )

90 80

(/3 70 60 50 (b)

40

I 2

I 4

I 6

I 8

I I I I t I I I I0 12 14 16 18 20 22 24 26

28

5o

Harmonic order

Figure 3. Comparison of the two acoustic theories. (a) Gust length 0"75 chord (1 jet); gust amplitude 103 ft/s; rotor tip speed 282 ft/s (600 rcv/min). (b) Gust length 7 chords (24 jets); gust amplitude 125"ft/s; rotor tip speed 282 ft/s (600 rev/min). , Simple point dipole theory; - - 9--, rotational noise theory. The results for the two extreme cases (i.e., a gust length of 0-75 chords and 7 chords) are shown in Figure 3. It was found that for the short gust (upper spectrum) the difference between the two sets of data was less than 3 dB over the majority of the harmonic range. This difference is due to the fact that whereas in the "simple point dipole" method it is assumed that the lift fluctuation takes place at a fixed point and a simple correction term is applied for the spanwise effects, in the rotational noise theory one integrates over the "gust" in both azimuth and spanwise directions. In the case of the long gust shown in Figure 3 (lower spectrum) there is a difference of at least 5 dB between the " p e a k s " and a phase change, although the "fall-off" of the " p e a k " levels show the same trend. The simple point dipole program becomes, as expected, increasingly inaccurate as the length of the gust profile is increased, the reason being that this simple theory is based on a

60

s. x,v. LEVERTONAND C. B. AMOR

point source assumption and, although the short gusts are a fair approximation to this, the loading profile, when it is increased in size, can no longer be assumed to be at a "point". 7. IMPULSIVE ROTOR NOISE RESULTS 7. I. EXPERIMENTAL Two seis of typical narrowband analysis results are presented in Figures 4 and 5. Figure 4 shows the effect on the spectrum of increases in rotor speed for the "long gust", while the effect of changing the gust length at a fixed rotor speed is illustrated in Figure 5. From these and similar analyses the harmonic levels were extracted and plotted in a convenient form for comparison with the theoretical results, as shown in Figures 6 and 7. 7.2. THEORETICAL Due to the inaccuracies associated with the "point dipole" method, the modified rotational noise programme was used for the main part of this investigation. Although results were obtained for the complete range of gust profiles used in the experimental programme, the theoretical results presented in this paper are restricted to those illustrated in Figures 3, 6 and 7. 8. COMPARISON OF MEASURED AND COMPUTED SPECTRUM SHAPES The predicted SPL of the acoustic spectrum for harmonic orders, based on values of 3R, the blade passing frequency, o f up to 30 Hz are compared with measurements in Figure 6. It can be seen that in the case ofthe short gust (upper spectrum) that, apart from experimental scatter (typically +3 dB), the measured results agree well with the theoretical curves. It will also be observed, however, that some of the "harmonic peaks", particularly those in the harmonic range 18-26 differ by -4-8 dB from the theoretical values. This is considered to be due to cancellation and reinforcement of some of the harmonics arising from the fact that one of the blades was displaced by 2 ~ from its ideal position and that the individual blades were likely to respond in a slightly different manner to the gust impulse. In this context it is worth noting that it was found that while tracking the blades it was necessary to apply 1~ of positive pitch at the cuff of one blade to give an acceptable track over the operating speed range. This meant that although two blades were at zero pitch and hence zero lift, the third blade had small equal positive and negative forces on it. The results for the long gust (7 chord lengths) are compared in the lower spectrum of Figure 6. The measured values, although of the same amplitude and shape as the theoretical spectra, exhibit "dips" and "peaks" which are in general displaced in frequency from the predicted curves. These differences are again considered to be due to the possible nonuniformity and irregularity of blade loading "impulses". All the conditions examined showed the same order of agreement between experiment and theory with the exception of the fundamental, or first harmonic, whose level could not be predicted. 9. DISCUSSION OF RESULTS 9.1. VARIATIONOF SPL WITH GUSTLENGTHAND AMPLITUDE Frequency spectra for various gust lengths for a rotor tip speed of 282 ft/s (600 rev/min) and gust amplitude of 125 ft/s are shown in Figure 5. As can be seen, the amplitude of the fundamental frequency (30 Hz), or first harmonic, increases dramatically as the gust is

61

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63

IMPULSIVE ROTOR NOISE 80

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14

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18

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24

26

28

30

Hormontc order

Figure 6. Comparison of calculated and measured SPL. (a) Gust length 1-25 chords; gust amplitude 45 ft/s.

-, Theory; measurements: +, 300 rev/min; o, 600 rev/min; x, 900 rev/min. (b)Gust length 7 chords; gust amplitude 45 ft/s. , Theory; measurements: o, 300 rev/min; n, 900 rev/min. lengthened. Each increase in gust length also resulted in a slight increase in the number of higher harmonics. At constant rotor speed and gust length the SPL of the fundamental (or first harmonic) was found to be independent of gust amplitude, while the level of the second and higher harmonics increased dramatically. The number of harmonics detectable also significantly increased with increase in the amplitude of the gust. There was a discrepancy in all cases between the predicted and measured level of the fundamental. For the short gust the SPL value was under predicted and it is considered that the measured value was that associated with normal rotor noise. For the long gust, however, the predicted level was higher than that measured: this cannot be explained but it is of interest to note that although, in general, the level of the fundamental is independent of gust amplitude nevertheless it is dependent on the length of the gust. It appears, therefore, that the level of the fundamental is associated with the "drag" of the rotor blades which is expected to be a function of gust length. 9.2. VARIATIONor SPL ~,VITItROTORSPEED Frequency spectra are reproduced in Figure 4 and depict the variation with rotor speed for a gust length of 7 chords and amplitude of 125 ft/s. As can be seen, there is an increase

64

J. W . LEVERTON A N D C. B. AMOR 80

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700

8o0 900

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f

I 282

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I I 576 4 2 5

Rotor (rev/min) l 188

I 25,5

Tip speed (ft/s)

Figure 7. Variation of theoretical and experimental harmonic sound pressure levels with rotor speed. Gust length 1.25 chords (5 in or 3 jets); gust amplitude 45 ft/s. Experimental Theoretical o 14th harmonic ..... x 20th harmonic - - 9- [] 24th harmonic + 30th harmonic A 34th harmonic in amplitude of each harmonic and an appearance of higher harmonics with each increase in speed. Harmonic orders of up to 30 are clearly shown and the discrete frequencies appear more distinctive at high rotor speeds. Theoretical and experimental results for a number of the blade passing harmonics for a short gust are shown in Figure 7. Theoretically the predicted SPL of the low and high frequency harmonics tend to be dependent on V 4 and V6, respectively, while for longer gusts the variation for all harmonics is typically V4. The experimental results showed similar trends in terms of velocity variation, but the absolute amplitudes differed, in general, by 4-3 dB from the theoretical values. For all the gust amplitudes tested, the measured OASPL varied as V 2 at the low rotor speeds and V 6 at the high speeds. At low speeds the OASPL showed a dramatic variation with gust length, while at the highest test speed it was relatively insensitive to changes in gust lengths. At all rotor speeds the OASPL showed an increase in level up to a gust length of 589chords and then as the gust was lengthened to 688chords the level decreased. When the gust length was extended to 7 chords the level increased again. This was considered to be a possible indication of local blade stall as discussed in the following section. 10. BLADE STALL STUDY It was anticipated that a positive indication of blade stall, or at least a significant change in the noise spectrum, would have been obtained when the blade was subjected to the longer gusts. Blade stall was expected to give rise to a change in the broadband noise characteristics, but there was no noticeable difference. The discrete frequency analysis suggested, however, that there was some change from the general pattern when the gust length was in excess of

65

IMPULSIVE ROTOR NOISE

5{- chords. This study was limited to an upper frequency of 5 kHz and, combined with the experimental scatter, this prevented any meaningful review of the data. It was decided, therefore, to investigate this aspect further in terms of 1/3-octave band and OASPL results, since it was considered that a "wider frequency band" approach would reduce some of the difficulties encountered with discrete frequency analysis and allow the high frequency broadband noise also to be taken into account. i

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Figure 8. Variation of SPL with gust length. (a) 1/3 octave band frequency spectrum (50 Hz-10 kHz); (b) SPL versus gust length for l 0 k Hz 1/3 octave band; (c) variation of OASPL with gust length. Key for (a): o, 3 jets/l.25 chords; z~, 18 jets/5"25 chords; x, 21 jets/6-25 chords. Tip speed 282 ft/s (600 rev/min); gust velocity 125 ft/s. Key for (b) and (c), gust amplitude: o, 45 ft/s; x, 77 ft/s; zx, 105 ft/s; +, 125 ft/s.

A representative set of 1/3-octave band results are shown in Figure 8(a) for gust lengths corresponding to 3, 18 and 21 jets, respectively. Intermediate values follow the general trend shown, but have been omitted for clarity. The SPL above 2 kHz reached a'maximum value at a gust length of 589chords (18 jets), decreased as the gust was lengthened to 688chords (21 jets) and then increased in level again as the length of the gust was extended to 7 chords (24jets). Similar variations occur below 150 Hz, but between 150 Hz and 2 kHz, particularly around 500 Hz, the "1-25 chord" (3 jets) condition had the highest level. This is because with the short gust some very high "peaks" were obtained and thus the "3 jets" results are a function of just a few discrete components. This has been discussed previously and is thought to be associated with the non-uniformity of the response of the blades to the gust impulses.

66

J. W. LEVERTON AND C. B. AMOR

If this aspect is taken into account and the "'average" harmonic value in the l]3-octave band is considered, then these results tend to follow the same trends as the low and high frequency regions. In the case of the results for the longer gust lengths the variation in level between the individual discrete frequencies is small and hence the I/3-octave band values are representative of the "average" level of the discrete frequencies. The 10 kHz, l/3-octave band and OASPL values have been plotted in Figures 8(b) and 8(c) as a function of gust length. As can be seen, there was, with the exception of the low velocity (45 ft/s) case, a decrease in level at a "gust length" of 688chords (21 jets). Theoretically the 1/3-octave bands in the region of 1000/1250 Hz (and 100/125 Hz) were expected to show a 6/10 dB increase in level a s t h e gust was initially lengthened to 5 chords, followed by a reduction of I or 2 dB as the gust length was further extended to 7 chords. Although the 7 chords (24 jets) value is below that of the 589chords (18 jets) as expected (see Figure 8(a)), the additional reduction in level at 688chords (21 jets) was not predicted. The 1/3-octave band controlled by the fundamental frequency did not follow these trends. Although it cannot be confirmed at present, it is considered that this reduction at 688chords is possibly associated with some form of blade stall, but except for this there was no other evidence of it occurring.

11. CONCLUDING REMARKS Theoretical and experimental results have shown good agreement in both amplitude and frequency characteristics over the full range of gust profiles used in the investigation. This implies that the blade loading and rotational noise estimates are reliable and thus it is clear that providing precise aerodynamic information is available accurate detailed noise predictions can be made. Discrete gusts of up to 7 chords in length and producing induced angles of 45 ~ did not appear to give rise to any significant blade stall or changes in frequency characteristics which could not be predicted. In other words, a rapid rate of change of"angle of attack" did not induce stall. ACKNOWLEDGMENTS The "air-jet" arrangement and the basic test procedure used in this study were developed by Mr A. R. Whatmore at the University of Southampton Institute of Sound and Vibration Research (see reference [2]). This paper is based on the work by Amor reported in reference [8]. The investigation was carried out at the Institute of Sound and Vibration Research, University of Southampton, under a Ministry of Defence contract; with Westland Helicopters Limited providing financial support of Amor. The authors wish to thank Westland Helicopters Limited and the Ministry of Defence for permission to publish this paper, and colleagues for their help in its preparation. Views expressed are, however, those of the authors. REFERENCES 1. J. W. LEVERTON1966 M.Sc. Thesis, blst#ttte of Sound and Vibration Research, University of Southampton. Helicopter blade slap. 2. A. R. WHA'rMOrtE1969 M.Sc. Thesis, htstitute of Sotmd and Vibration Research, University of Southampton. A study of transient noise from helicopter rotor blades. 3. H. K. TANNA 1970 Ph.D. Thesis, hlstitute of Sound and Vibration Research, University of Southampton. Sound radiation from sources in circular motion with application to helicopter rotor noise.

IMPULSIVE R O T O R NOISE

67

4. H. K. TANNA1968 Institute o f Somtd and Vibration Research Technical Report No. 13. Computer program for the prediction of rotational noise due to fluctuating loading on rotor blades. 5. S.E. WRIGHT1969 hmitute of Sound and Vibration Research Technical Report No. 14. Theoretical study of rotational noise. 6. S. E. WRIGHTand H. K. TANNA 1969 hzstitute of Sotmdand Vibration Research Technical Report No. 15. A computational study of rotational noise. 7. S. E. WRIGHt 1971 Journal of Sound and Vibration 17, 437-498. Discrete radiation from rotating periodic sources. 8. C. B. AMOR 1972 lnstitttte of Sotmd attd Vibration Research Contract Report 72/23. Transient rotor noise study. 9. G. R. TAYLOR 1972 IVestland Helicopters Limited, Research Memorandum 122. Determination of the acoustic radiation from a point dipole source. 10. R. L. BISPHNGHOrr, H. AsaezY and R. L. HALrMAN 1957 Aeroelasticity. New York: Addison Wesley Publishing Co. APPENDIX I T H E O R Y FOR D E T E R M I N A T I O N OF BLADE L O A D I N G

The approach employed, originally adapted by Leverton [1 ], uses Kussner's function [10] for the determination of the loading. Kussner's function is based on the lifting line theory applied to a two-dimensional aerofoil. The rotor blade is assumed to be moving as a wing through a gust at a velocity equal to that of the blade section at the centre of the gust. The load at any point, L(s), in terms of an arbitrary gust profile, w(a), is given by L(s) = 89 Vcao (w(~r) ~b(s - tr) d~r,

(A 1)

0

where s is any point measured from the start of the gust (it is a dimensionless variable given by s = Vt/b, where Vis the velocity of the wing at the centre ofthe gust and b is the half chord), cr is the point at which the load is being considered and ~, is K.ussner's function. A suitable approximation for ~ ( s - tr) is given in reference [10] and takes the form r

-

a ) _~ I -

89 - ~ 1 7 6

-

89

(A2)

From this it follows that ~,(s - a) = 0.065 e -~

+ 0-5 e -c*-~).

(A3)

Hence equation (A1) can be rewritten as $

L(s) = 89 Vcao f w(a) [0.065 e -~176

+ 0.5 e-~S-Oq dtr.

(A4)

0

F o r convenience, let A = 89 becomes

0.065, B = 89

L(s) = .4 e -~

0.5 and a = 0.13. Then equation (A4)

w(e) e ~' d e + B e -~ f w(e) e ~ d~. O

(A5)

o

Consider now a gust of arbitrary profile. While the blade is in the gust (i.e., s < Y where Y is the non-dimensional gust length), then the loading L(s) will be given by equation (A5). After the blade passes out of the gust, w(a) becomes zero and cannot directly affect the loading. This does not imply that the blade load is zero, since a distribution of shed vorticity would be left in the wake by the aerofoil as it passed through the gust. This would affect

68

s.w.

L E V E R T O N A N D C. B. A M O R

the aerofoil even when outside the original gust, and the loading would therefore take a finite time to decay. Hence the loading for the blade outside the direct effect of the gust (s/> Y) is given by u

u

L(s) = A e - ~ f w(a) e~" da + Be -~ (. w(a)e~da. 0

(g6)

o

Since Y is constant the value of the integral will be a constant, and the loading decay will take on an exponential form with increasing s. This theory was programmed such that the blade loading time history, resulting from any arbitrary gust shape, could be obtained.

APPENDIX II SIMPLE P O I N T D I P O L E T H E O R Y

The pressure at a point in the far field whose position relative to the dipole has coordinates Xl (i = 1, 2, 3) is given by the Lighthill equation [1, 9]: Xt l a F l 4nr2co at (t--r/Co),

P--Pc

(A7)

where F, is the fluctuation force, r is the distance of the observation point from the source and (t - r/co) is referred to as the "retarded time" and is the instance when a wave travelling at the speed of sound (Co) has to leave the source in order to reach the point of observation in time t. For a blade loading per unit span (Ls) perpendicular to the plane of rotation such that 6F~ -- 6F~ = Ls 3x,

where 6x is a small increment of span over which L~ acts, then 6(p - po)

X.

1 6L~

9

4~r ~ Co ~ " (t - r/co),~x.

(A8)

Integrating over the blade span region of interest (ro - rl) and taking the mean square ofpressure one obtains

(P -- Po)2

lffa/ (t)y

~'~-2--](ro

(A9)

where K is a constant dependent upon the spanwise load distribution (e.g., for a half sine wave distribution, K = 2/n), i is the observer distance from the centre of the elemental blade area, X, is the normal coordinate from the mid-point of the area under consideration and

?L;\ (TJ= \W/,,o_,,, [Note: When the elemental span length (ro - rx) is small compared with the acoustic wavelength then the "retarded time" (t - r/Co) is unimportant.] The blade loading, L, is represented by the following Fourier equation which is based on the "blade spacing interval": L = 89 + ~ an sin (nwt) + ~ b, cos (nwt), n=l

n=l

(AI0)

IMPULSIVE ROTOR NOISE

69

where a, and b, are constants, w = 2rrfwherefis the blade passing frequency and n = 1,2, 3 . . . is the harmonic number, from which is obtained, for the nth harmonic,

OL(t)n]

--if-, / =

4re2 1 .[""~"~

(All)

where C,z = a2, + b,2 and z is the blade passing interval. Hence substituting in equation (A9) gives [ 1 1X.

"(p -- po)Z = [.2Q-~r r l(ro -- rl) K -C,n]Z T- ] 9

(A12)

Therefore the SPL for the nth harmonic is given by [ SPL,=201og

1

lcos0r

C,n]

[2Q-27 Co SK-~-'z ] - 2OI~

(A13)

where S = ro - ra and cos0 = AO,ff, where 0 is the angle subtended between the dipole axis and the source-observer line.

APPENDIX III ROTATIONAL NOISE THEORY

1. List of symbols B rotor blade number blade chord (in) speed of sound fit/s) D distance from source point on rotor disc to field point (in) L(r, V) total blade section loading at point (r, V) on the rotor disc (lbf/in) harmonic number m P~e real component of sound pressure (lbf/in 2) imaginary component of sound pressure (ibf/in 2) distance from centre of rotor head to source point on the rotor disc (in) r tip radius (in) rT R distance from centre of rotor head to field point (in) SPmB root-mean square sound pressure at a field point (lbf/in 2) rotor rotational speed (radian/s) blade pitch angle (deg) V azimuth angle in rotor plane (deg) (0 ~ at tail, positive in direction of rotation) 0 field point azimuth angle (deg) a angle between rotor plane and field point (deg) (positive upwards) C

Co

2. Rotational noise program A modified version of the computer program for the prediction of rotational noise due to fluctuating loading on rotor blades, developed by Tanna [4], was used to calculate the rotational noise. This method is based on the work of Wright and Tanna whichis described in more detail in references [4], [5], [6] and [7].

3. Summary of rotational noise program With reference to the rotor and field point geometry shown in Figure AI, the r.m.s, value of the ruth harmonic of sound pressure, SP,,B, at afield point (R,O,a) due to a B-bladed

70

J. "W. LEVERTON AND C, B. AMOR

Directionof ftig~! RotoraxiZ[ / = 1 8 0 ~ Rotordisc

I

/ . . . . . . .

-a" /I

]ll~ 1

~,=270~

J//= Oj0~ )

y

/

',

,

tlIIField

point

Y

x @-o0 Figure A I , R o t o r geometry and field point location. (a is positive if field point is above the r o t o r disc.)

rotor rotating at s radians per second is given by R 2V'2~ 2 C [(P/e)2 + (Pfm)Z]ll2'

SPins

where p:~=

(mcB)[mBf2sinU+ l[sinflcosasin(tp_O)+

ffZ.(r,O) mD 2 sin~[ o

co

o

(A15)

+ cosflsina]rdrd~9

and p~,. =

(AI4)

; f ',r,, o o

~

roD2

sm ~

(2r) [ Co

cos U -

[sin fl cos a sin (~ - 0) +

+ cos fl sin a] r dr d t9,

(AI6)

for which

[---

]

D -= [R 2 + r z - 2Rr cos a cos (0 - 0)] 1'2.

(A17) (A18)

The method, used repeatedly, evaluated the basic sound pressure equations (AI4), (A15) and (AI6). This includes a double integration. One integration is around the rotor disc with the sample points (azimuth angles) chosen at constant intervals. The other integration is along the radius where sample radial stations are unevenly spaced. A subroutine called mMCOR is included to perform the integration by the trapezoidal rule.

IMPULSIVE ROTOR NOISE

71

4. Computational procedure The original program developed by Tanna [4] required the "steady" loading value and the Fourier components and phase angle of the blade loading harmonics. This was not very suitable to this investigation since the blade loading was calculated as a function of time. The program was therefore modified to enable the lift on the blade to be read in as direct loading in terms of Ib/ft at various azimuth stations. The hover rig was run with the blades at zero pitch (and hence at zero lift) so that the only lift obtained was when the blade passed through the gust. Thus the majority of the azimuth loading was zero. Rather than have to input a large number of zeros the program was further modified so that it was only required to read in the blade loading values over the range associated with the gust.