Daniel L. Sutliff

Glenn Research Center, Cleveland, Ohio

Rotating Rake Turbofan Duct Mode

Measurement System

NASA/TM—2005-213828

October 2005
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Daniel L. Sutliff

Glenn Research Center, Cleveland, Ohio

Rotating Rake Turbofan Duct Mode

Measurement System

NASA/TM—2005-213828

October 2005

National Aeronautics and

Space Administration

Glenn Research Center

Prepared for

NOISE-CON 2005

sponsored by the Institute of Noise Control Engineering

Minneapolis, Minnesota, October 17–19, 2005
Acknowledgments

The concept of inducing a Doppler shift to separate the circumferential modes was first put forth by Tom Sofrin

while at Pratt and Whitney. The first implementation of the continuously rotating rake was at NASA Glenn by a

team led by Larry Heidelberg. Kevin Konno was the primary mechanical system designer. Dave Hall developed the

original data acquisition and processing routines. The Rotating Rake won the NASA Glenn Steven V. Szabo Award

for Excellence in Engineering in 2004.

Available from

NASA Center for Aerospace Information

7121 Standard Drive

Hanover, MD 21076

National Technical Information Service

5285 Port Royal Road

Springfield, VA 22100

Trade names or manufacturers’ names are used in this report for

identification only. This usage does not constitute an official

endorsement, either expressed or implied, by the National

Aeronautics and Space Administration.

Available electronically at http://gltrs.grc.nasa.gov
NASA/TM—2005-213828 1 

Rotating Rake Turbofan Duct Mode Measurement System 

Daniel L. Sutliff 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

Abstract 

An experimental measurement system was developed and implemented by the NASA Glenn Research Center in 

the 1990s to measure turbofan duct acoustic modes. The system is a continuously rotating radial microphone rake 

that is inserted into the duct. This Rotating Rake provides a complete map of the acoustic duct modes present in a 

ducted fan and has been used on a variety of test articles: from a low-speed, concept test rig, to a full-scale 

production turbofan engine. The Rotating Rake has been critical in developing and evaluating a number of noise 

reduction concepts as well as providing experimental databases for verification of several aero-acoustic codes. More 

detailed derivation of the unique Rotating Rake equations are presented in the appendix. 

Symbols 

a,b constants 

A duct area 

B number of fan blades 

C normalizing coefficient 

D duct diameter 

E duct profile function 

f frequency 

h fan fundamental harmonic 

J Bessel function - 1st kind 

k wave number 

M Mach number 

m circumferential mode order 

N number of pulses per fan rev 

n radial mode order 

p acoustic pressure 

P modal pressure amplitude 

ˆ

Ρ modal power amplitude 

Q weighting function 

R duct radius 

r radial co-ordinate 

t temporal co-ordinate 

s fan shaft harmonic 

V number of vanes 

x axial co-ordinate 

Y Bessel function - 2nd kind

Greek 

Â admittance 

η reduced frequency

φ mode propagation angle 

κ eigenvalue 

ρ density 

σ duct hub-to-tip ratio

θ circumferential co-ordinate

ω fan shaft rotational speed 
NASA/TM—2005-213828 2 

Ω rake rotational speed 

υ fan/rake speed ratio 

ν acoustic particle velocity 

ζ cut-off ratio 

Ζ Impedance 

Introduction 

The NASA Glenn and Langley Research Centers have been involved in several programs (Advanced Subsonic 

Technology – AST, Quiet Aircraft Technology – QAT) whose goals were the reduction in transport aircraft noise 

attributed to the engine. A component of engine noise is fan tone noise caused by rotor-stator, and other interaction, 

coupled to duct propagation. Interaction modes contribute to the farfield tone noise generated by turbofan engines 

(ref. 1). In order to reduce tone noise it is desirable to understand the generation of these interaction modes. Detailed 

knowledge of the modal structure may indicate engine modifications that can quiet the overall tone signature. It is 

difficult to determine the modal content solely from farfield directivity. Analytical modal description (refs. 2 and 3) 

of turbofan noise is well understood. Previous methods to experimentally measure duct modes used a concentric 

rake face or a reposition-able radial rake. These methods had drawbacks in that they either contaminated the 

acoustic field with their wakes or required lengthy, discontinuous data acquisition. 

In order to develop fan tone noise reduction techniques, a diagnostic tool was needed to measure the interaction 

modes of ducted fans. A concept for a continuously rotating microphone rake (refs. 4 and 5) was put forth that 

would be able to obtain all modal amplitudes in a reasonable data acquisition time. The key concept of the Rotating 

Rake is that by slowly rotating in a manner locked to the fan shaft, a Doppler shift is imparted to the duct spinning 

modes that is uniquely based on the mode physics. The resulting Rotating Rake system developed by the NASA 

Glenn Research Center has made several advances in the understanding of turbofan mode generation. In addition to 

directly evaluating a noise reduction technique, this diagnostic tool can provide an experimental database to which 

the results of aero-acoustic prediction or propagation codes can be benchmarked. This paper describes the theory 

and implementation of the Rotating Rake with historical examples. An appendix to the full paper, published as a 

NASA TM, provides derivation of equations unique to the Rotating Rake. 

Duct Mode Theory 

First, a summary of the duct mode theory with an application to mode propagation and generation is presented 

in order to understand the noise signature being measured. 

Cylindrical Wave Equation 

The propagation of sound in a circular duct with uniform flow is governed by the convective wave equation. In 

cylindrical co-ordinates this equation is written in non-dimensional form as: 

∇2p(r,Θ, x,t) = ∂

∂t

+ M ∂

∂x

⎛ 

⎝ ⎜ ⎞ ⎠ ⎟ 

2

p(r,Θ, x,t) (1) 

with a general solution of 

P(θ,r, x,t) = pmnf ∗ Emn (kmn r)e

i(2πft +mθ ±kmnx)

 (2) 

where 

Emn (κ mn r) = Cmn[Jm (κ mn r) + QmnYm (κ mn r)] (3) 

is a solution to Bessel’s equation. The index m is defined as the circumferential order defining the number of 

pressure cycles in the circumferential direction. The index n defines the radial order, the number of pressure nodes 

in the radial direction. The nature of the radial rake indicates that of primary interest is the radial solution. 
NASA/TM—2005-213828 3 

The constant, Cmn, is a weighting factor that normalizes the arbitrary Bessel function profile to a desired 

physical property. (As an aside the selection of Cmn will affect the calculated mode pressure amplitude but the mode 

power level will be a fixed value regardless of the choice.) The normalizing physical property chosen for Rotating 

Rake analysis due to a desire to compare results to a predictive tool is (ref. 6): 

2

area

p dA A ≡ ∫ which results in: 

[ ]

[ ]

2 2

2 2

2

2 2

2

1 1 1 () () 2

() ()

m mn mn m mn

mn mn

m mn mn m mn

mn

m J QY

C

m J QY

⎧⎡ ⎤ ⎪ = − κ+ κ ⎨⎢ ⎥

⎪⎢ ⎥ κ ⎩⎣ ⎦

⎡ ⎤ ⎫

⎪ −σ − κ σ+ κ σ ⎢ ⎥ ⎬ ⎢ ⎥ ⎣ ⎦ κ ⎪⎭

 (4) 

For selected physical cases (hardwall condition or no-flow) this result arises from the principle of orthogonality. 

For the general case, the E-functions based on the eigenvalue solution are not orthogonal. However the above 

identity for Cmn is also used for convenience in these cases. 

The eigenvalues are determined from the boundary conditions; in the standard case it is the hardwall condition 

that the acoustic pressure be normal to the duct inner and outer wall (a consequence of zero acoustic velocity at the 

wall). 

dP

dr σ

1

= 0; E ′ mn (κ mn1) = ′ E mn (κ mnσ) = 0 (5) 

The case with a duct with passive treatment (softwall) where 

p

v = Z

ρc

; ˆ A = ρc

Z ; η = ωR

c

 (6) 

outer wall :κ mn

Em−1,n (κ mn )

Emn (κ mn ) − m = −iηˆ A ow (1− M kmn

η

)

2

;

inner wall :κ mn

Em−1,n (κ mnσ)

Emn (κ mnσ) − m

σ = iηˆ A iw (1− M kmnη

)

2

 (7) 

Note that while we are focusing on the radial solution, it is coupled to the circumferential and axial solutions 

through the mode order m and the eigenvalue κmn. 

The above boundary conditions show that real eigenvalues occur in the hardwall case and complex eigenvalues 

result from the softwall case; this is a concept that propagates throughout the analysis. The eigenvalues are 

determined from the boundary conditions. It is convenient to normalize the radial co-ordinate by the duct wall outer 

radius and define the hub-to-tip ratio as the inner wall radius normalized by the duct radius. Also, as convention the 

1st eigenvalue is called the zero radial, designated by n = 0. This allows for a short physical definition of a mode as 

the number of zero crossings of the pressure profile. The mode m = 0, n = 0, is uniform in pressure across the duct 

and is defined as the plane wave. Figure 1 shows a graphical representation of these modes with the regions of 

light/dark representing positive/negative acoustic pressure. Mathematically, an infinite number of radial modes 

(eigenvalue solutions) for each circumferential mode (also an infinite set) exist. 
NASA/TM—2005-213828 4 

circumferential mode, m

01 2

0

1

 Figure 1.—Duct mode graphical representation. Figure 2.—Wave propagation in cylindrical duct. 

Propagation 

Although an infinite number of solutions exist, physical conditions allow only for a fixed subset of modes to 

propagate. The separated solution in the x-direction was given as: 

k = η

1− M 2 −M ± 1− (1− M 2

) κ

η

⎛ 

⎝ ⎜ 

⎞ 

⎠ ⎟ 

⎡ 2

⎣ 

⎢ 

⎢ 

⎤ 

⎦ 

⎥ 

⎥ ; η = 2πfR

c for M = 0 → k = ± 2πfRc

⎛ 

⎝ ⎜ ⎞ ⎠ ⎟ 

2

− β2κ2 ; β = 1− M 2 ( ) (8) 

The axial wave number is kmn. Examination of this equation gives three possible solutions: 

i) 2πfR /c > βκ mn; kmn = a → eikx = eiax ⇒ cyclic propgation in x − direction

ii) 2πfR /c = βκ mn; kmn = 0

iii) 2πfR /c < βκ mn; kmn = ib → eikx =ei(ib)x =e−bx ⇒ exponential decay in x − direction

 (9) 

A real term when incorporated into the exponential term results in cyclical variation in the combined term. Thus 

a real wave number results in propagation in the x-direction. An imaginary wave number results in a pure real in the 

exponent, resulting in exponential decay. The special case of kmn= 0 defines the cut-off frequency. Physically, the 

3D waves have a helix propagation angle. Highly cut-on modes approximate a plane wave. Cut-on modes have an 

axial wavelength that is longer than would occur in free space at that frequency (shorter wave-number). Cut-off 

modes have ‘infinite’ axial wavelength (zero wave-number - decay). Figure 2 illustrates this propagation. The 

propagation angle (Ref. 7) is given by: 

 

cos(φ) =

−M ± 1− 1

ζ2

1 ∓ M 1− 1

ζ2

; = ± 1− 1

ζ2 for M = 0; where ζ = 2πfR

κ mnc 1− M 2

; = 2πfR /κ mnc for M = 0; (10) 

It is worth repeating the concept that a given (m,n) mode will propagate only if the frequency is above the 

cut-off frequency. For frequencies only somewhat below cut-off, it can be shown that a 50 dB attenuation per duct 

diameter occurs. The eigenvalue, κmn, increases with m or n, that is, the cut-off frequency increases with higher 

modes. This results in a sequential cut-on of modes from low order to high order as the frequency (or RPM) 

increases. For a given duct, the modes that propagate are from 0 to mmax; the highest number of radial modes, from

n = 0 to n = nmax, (0,0) to (0, nmax) will propagate at m=0; each successive circumferential order will have fewer 

radial modes until at mmax only the first radial mode will propagate (mmax,0). 
NASA/TM—2005-213828 5 

Rotor-Stator Interaction 

The number of possible modes, while reduced from infinity, can still be substantial. A typical high-speed fan 

(shown later) may have up to 50 m-orders with up to 12 radial modes resulting in 600 modes. Fortunately, the 

number of modes that need to be considered is a subset of these propagate-able modes. 

The classic paper by Tyler and Sofrin (ref. 8) presents the theory of fan-duct mode generation and propagation. This 

is seen by the kinematics between the rotor and stator. Figure 3 illustrates the progression of the pulse (which can be 

extrapolated to a continuous, cyclic profile). An illustrative rotor with B = 8, and V = 6 stators demonstrates that an 

interaction between a select rotor blade and stator vane progresses in the circumferential direction as the rotor turns. 

This pulse can be seen to travel in one complete revolution when the rotor has turned only 1/4 revolution. 

The rotor locked mode (k = 0, m = B) spins at the shaft rotation speed. This mode, and higher order modes, can 

only propagate in a narrow annular duct if the blade tip speed corresponding to its spin rate is above Mach = 1.0. 

Lower order modes, which spin faster, may propagate if their spin rate results in a sonic blade tip speed. The critical 

tip Mach number is greater than 1.0 for non-narrow annular ducts. Sofrin demonstrated that generation of, and the 

spin rate of, circumferential spinning modes is governed by the following equations: 

 generation: m = sB ± kV ; spin rate: Ωm = sBΩ /m (11) 

Each circumferential mode, m, can have one or more radial modes, n. Mode propagation is dependent on the 

cut-off frequency, which is unique to each (m,n) mode. This frequency is dependent on geometric parameters and 

the eigenvalue of a combined Bessel function that is a solution to the cylindrical wave equation. Below the cut-off 

frequency the mode will decay exponentially. Above cut-off propagation occurs and acoustic power is transmitted 

down the duct and into the farfield. The Bessel function eigenvalue incorporates the duct geometry effects. The 

cut-off frequency, and cut-off ratio with Mach number effects is given by (ref. 9): 

fcut−off = cκ mn

πD ζ = πfD

R (1− M 2

) cos(2φ)

; k = R eiφ (12) 

The cut-off ratio reduces to the familiar form when the eigenvalue is real (hardwall). The cut-off ratio rewritten 

in traditional turbomachinery form gives the ratio of the mode frequency to its cut-off frequency (for M = 0, 

hardwall case): 

ζ = πsBΩD

60κ mnc = sB(Ω /60)

fcut−off

(13) 

Ω = 0 , 45

m = 0 ,360

Ω = 30

m = 120

Ω = 15

m = 60

Ω = 75

m = 300

Ω = 60

m = 240

Ω = 45

m = 180

1,7 2 3

5 6 4

Figure 3.—Interaction of rotating pressure pulse stationary object. 
NASA/TM—2005-213828 6 

-• POSSIBLE GENERATING MECHANISMS +•

rotor

only discrete

disturbances

rotor

stator interaction

spinning modes

decay duct propagation decay

OBSERVED EXTERNAL ACOUSTICS

Figure 4.—Duct mode generation and with propagation constraints. 

The primary method of generation of these spinning modes is the periodic interaction of the rotor wake 

impinging on the stator vanes as described. The stator vanes primary purpose is recovery of the rotational energy 

generated from the rotor (swirl). Other primary sources are inlet or exhaust duct struts, installed for structural 

support. These are typically the primary modes of interest in turbomachinery. Secondary methods of generation are 

caused by a change in rotor blade loading due to aerodynamic disturbances upstream of the rotor. If these 

disturbances are periodic with respect to the rotor, spinning modes may be generated. Discontinuities in the duct 

wall can generate unwanted modes. These modes, while generally undesirable, can contain significant acoustic 

levels. A summary of the preceding sections is given in figure 4. 

Rotating Rake Implementation 

Operating Principle 

The underlying principle of the Rotating Rake is the concept from the previous section that every 

circumferential mode spins at a different, discrete rate. The rake, rotating slowly, imparts a Doppler shift on the 

fan fundamental that is unique to each circumferential mode. This frequency separation is a linear function of the 

mode number and the rake rotation speed; the higher the mode number the greater the frequency shift with m = 0 

un-shifted (remaining at the harmonic frequency). This relationship, (derivation shown in the appendix) is: 

let Ω

ω= 1υ

, freq →= ω(B − m

υ

) (14) 

 

Figure 5 shows a spectrum from a single microphone with the rake stationary, compared to a spectrum for the 

same microphone with the rake rotating. The separation of the modes about the fundamental shaft order 16 is clearly 

seen. Cut-on background modes are 20 dB above the cut-off modes. The dominant rotor-stator interaction is 15 dB 

above these extraneous modes. 

∞ ∞
NASA/TM—2005-213828 7 

Figure 5.—Illustration of spectral Doppler shift due to rake rotation. 

Implementation 

The rake acquires unsteady pressure data as it spins using a series of radially distributed microphones or 

pressure transducers pointing at the fan. The rake is used in two locations—the inlet and the exhaust. The inlet rake 

rotates around the duct centerline inside the nacelle at the throat. At the nozzle, the rake rotates at the exit plane of 

the bypass duct. The rotational speed of the rake was required to be a specific whole number fraction, and in phase 

with the shaft. This means that when the fan has rotated υ times, the Rotating Rake will have completed exactly one 

revolution within ±0.2°. The speed ratio, υ, has been from 100 to 250. This speed and phase synchronization is 

necessary in order to resolve the Doppler induced frequency shift of the spinning acoustic modes. Consequently, the 

rake requires a very accurate drive and control system. This accuracy must also be repeated over the entire test 

period (5 to 10 min.) without build-up of phase error. The rake must follow the fan precisely even though the fan 

will wander in speed (5 to 10 rpm or so) throughout the test period. Consequently, the rake drive system must be 

slaved directly to the fan. 

The general method used for this precise following requirement is a programmable closed loop stepper or servo 

motor drive system. Both have been used in previous Rotating Rake applications. Commercial programmable motor 

control/drive systems are well suited for this type of application, easily meeting the design requirements if carefully 

selected. The rake motor drive system is an internal closed loop drive utilizing a programmed step/direction 

controller output driving a brushless servo or stepper motor; externally the motor drive system is an open loop 

system. The fan velocity is determined from a multi-per-rev physical sensor mounted on the shaft. A once-per-rev 

signal sets the fan position. Variations of this sensor have been an indexer mounted on the shaft, a gear tooth 

pick-up, or a blade-proximity sensor. The higher the number of signals per revolution, the better the rake system can 

track fan speed changes smoothly. This signal is fed into the manufacturer’s software for the motor control. A 

timing signal from the rake rotational body is used solely for position verification by monitoring on an oscilloscope. 

If the rake speed sensor resolution (pulses-per-rev) is equal to the speed ratio υ, then when the rake is synchronized, 

the traces of the fan-one-per-rev and the rake υ-per-rev will line up exactly, with no drift in time. Finally a rake 

body one-per-rev signal is used to provide information on the rake absolute position, which is important for 

determining absolute mode phase. 

The transducers’ time history signals must be brought across the rotating boundary. The method used is to bring 

the conditioned signal across by VHF/UHF telemetry. Again, carefully selected commercial units with custom 

design modifications have been found to be adequate to these requirements. The signal conditioning and telemetry 

transmitters must be mounted in the Rotating Rake drive housing. Power to supply these electronics can either be 

supplied by batteries in the rotating platform, or brought across by rails. The telemetry receivers must be diversity 

(two channels per frequency) type as the rotation will cause dropouts in the signal that must be picked up by a 

second channel. It has been found that close mounting of the antenna pairs is required. 

The output of the telemetry receivers is fed into a high-speed multi-channel data acquisition system. A primary 

requirement of the data acquisition system is the ability to synchronously sample the analog signal using the fan 

multi-per-rev signal, N. For analysis purposes, the ideal choice for N is 8B. This is because typically the 3rd 

harmonic (3B) is the highest ‘frequency’ typically analyzed. To satisfy the Nyquist criteria, a sampling rate more 
NASA/TM—2005-213828 8 

than twice that must be used. Further, since FFTs are more efficient when using a power-of-2, 8B fits nicely. The 

signal must be synchronously acquired to the external fan shaft signal due to the time domain averaging employed to 

separate the tone signals from the broadband and flow noise. Think of the whole system as a strobe light. If the 

strobe light is set to a fixed (internal) rate, the rotating object may appear to drift. Connecting the strobe light to a 

timing signal causes the rotating object to ‘freeze’. Experience has shown that if any one of these synchronizing 

components is in error (drift, sample drop-out, noise) the data will be invalid. Best wiring practices are essential. 

Other requirements of the data system are the ability to record and store long continuous time histories 

(~5 min.) without breaking the trace into records. This requirement arises from the 25 continuous rake revolutions 

required to obtain good separation of the tones from broadband and flow noise. Ideally, each channel must have 

‘zero’ relative phase shift, or this must be accounted for in the post-processing. 

Finally, the number of transducers chosen must be greater than the highest number of radial modes that will be 

anticipated, usually the number of radial modes predicted to be cut-on at the m = 0 at the 3rd harmonic. A uniform 

radial distribution of the transducers along the rake has been chosen based upon experience. 

Rake Effects 

An obvious concern when taking inlet measurements would be the extraneous noise caused by the ingestion of 

the rake-wake contaminating the results. Indeed a stationary rake (V = 1) would generate every mode. However, in 

the reference frame of the rake, the rotation of the rake-wake and the relative speeds of the fan/rake cause all of the 

modes generated by the rake-wake rotor interaction to occur only at a single mode, that of the m = nB (fig. 5). This 

is also the so-called rotor mode that is cut-off below sonic tip speed. Therefore, the rake-wake interference is outside 

the typical modal analysis range unless the rotor tip speed is sonic. Conceptually, in the reference frame of the rake, 

it is as if the rake is stationary and the fan rotates at (Ω-ω) so that the rake-wake modes appear at that frequency. 

The mathematical verification of this intuitive observation is shown in the appendix. 

Both rakes have windscreens covering the transducers. The oscillating vortex shedding from the inlet rake body 

occurs at the measurement frequency of interest under certain conditions. A porous aluminum foam windscreen was 

attached along the trailing edge of the inlet rake covering the microphones to eliminate the vortex shedding noise 

from the rake body (the transducers are on the trailing edge of the rake) as seen in figure 6. In the exhaust duct, the 

rotor wake potential field convects downstream with a circumferential variation in a manner similar to acoustic 

modes. This can severely contaminate the microphone signals resulting in poor data. A tri-shield arrangement for the 

exhaust rake to lower interaction with the viscous wake was developed based on the concept that the convection is at 

the duct Mach number rather than the speed of sound resulting in a steeper propagation angle. The data quality was 

greatly improved with the addition of the shields as shown in figure 7 for a typical microphone. The convected wake 

levels at the non-propagating modes (m = –12, +16) is nearly eliminated, and the contamination of the propagating 

mode m = +2 propagating mode is reduced. The resulting significant improvement in the radial profile is shown in 

figure 8. Finally, in the exhaust, the rake blockage must be taken into account. Typically for a high-pressure rise fan 

the nozzle area must be increased to account for this in order to make mode comparisons on a relevant basis. 

 Figure 6.—Reduction of strouhal-shedding Figure 7.—Reduction of wake contamination 

due to installation of inlet rake shield. due to installation of exhaust rake shields. 
NASA/TM—2005-213828 9 

0

1

2

3

4

5

6

7

8

9

10

11

12

0.5 0.6 0.7 0.8 0.9 1

Prms (Pa)

Radius (r/R)

0

1

2

3

4

5

6

7

8

9

10

11

12

0.5 0.6 0.7 0.8 0.9 1

Prms (Pa)

Radius (r/R)

w/o screen w screen computed

PROPAGATING MODE

NON-PROPAGATING MODE

Figure 8.—Improvement in radial profile due to installation of exhaust rake shields. 

Data Reduction and Analysis 

The pressure time histories are acquired and recorded for each transducer as well as the timing signals through 

the data acquisition system described earlier. Experience has determined that 25 rake revolutions of data are 

required to obtain good signal-to-noise levels in the processed data. This time history is then time domain averaged 

(TDA) by taking record lengths 10 rake revolutions long and offsetting the next block by 1 rake revolution. This will 

typically result in 15 averages. Also this block length yields 10 bins in between modes when computing the FFT. 

The FFT of these TDA blocks are then reduced into complex magnitude for each Doppler shifted circumferential 

mode at each microphone radial location. A practical limit chosen for the modal decomposition is m = ±3B, about 

each harmonic. The modal pressures are computed from the least-squares-fit of the radial Bessel functions to the 

actual pressure profile. A vector error is determined at each radial location from the difference between the actual 

and computed complex pressures divided by the maximum experimental pressure. The overall error of the solution 

is defined as the average of the radial mode vector errors. The discrete form of the pressure in the duct for a given 

frequency, m-order, at a specific radial location is: 

pf ,m,r = pf ,m,n

n= 0

∞

∑ Emn (κ mn r); and in matrix form: [Pr]= [Evr][ ] Pn (15) 

applying least-squares analysis, inverting and solving for the modal amplitudes: 

[ ] Pn = [ ] Fvn

−1

[ ] Evr [ ] Pr (16) 

where 

 

[ ] Fvn =

Em0

2 (κ m0ri)

i=1

Np

∑ ... Em0 (κ m0ri)EmNmax (κ mNmax ri)

i=1

Np

∑

# #

EmNmax (κ mNmax

ri)Em0 (κ m0ri)

i=1

Np

∑ ... EmNmax

2 (κ mmNmax ri)

i=1

Np

∑

⎡

⎣ 

⎢ 

⎢ 

⎢ 

⎢ 

⎢ 

⎢ 

⎢ 

⎤ 

⎦

⎥ 

⎥ 

⎥ 

⎥ 

⎥ 

⎥ 

⎥ 

 ; 

 

[ ] Evr =

Em0 (κ m0r

1) ... Em0 (κ m0rNp )

# #

EmNmax (κ mNmax r

1) ... EmNmax (κ mNmax rNp )

⎡

⎣ 

⎢ 

⎢ 

⎢ 

⎤

⎦

⎥ 

⎥ 

⎥ 

 (17) 

Pr is the vector measured pressures at a the microphone location and the solution vector, Pn, is the modal 

coefficients; both complex. The size of the matrix is dependent on the number of microphones, and the highest 

radial mode being analyzed. Based on experience the numerical solution requires the curve fit to Nmax+1 radial 

modes for a hardwall solution. For a softwall solution the decay rate must be taken into consideration and may 

require a larger matrix. 
NASA/TM—2005-213828 10 

Mode power is an important parameter in duct acoustics and provides a better description of the acoustic field, 

compared to mode pressure, especially when relating to the external acoustics. Power rather than pressure is the 

physical parameter that is conserved. Many wall modal techniques are unable to calculate modal power. The 

definition of power, and the final form of the computation used is: 

 

 

Power → ˆ Ρ = I x

A

∫ dA; I x = ( p

ρ0

+Uu)(ρ0u + ρU); ⇒ ˆ Ρ = ∓

πR2 (1−σ2)

ρ0c0

β4 RE

1− 1

ζ2

1± M 1− 1

ζ2

2

⎧

⎨ 

⎪ 

⎪ 

⎩ 

⎪ 

⎪ 

⎫

⎬ 

⎪ 

⎪ 

⎭ 

⎪ 

⎪ 

P 2

 (18) 

This form mathematically illustrates the effect of cut-off ratio. For the hardwall case the cut-off ratio was shown 

to be a pure real number. If the cut-off ratio is greater than one, then the term in braces is also a real number, hence 

non-zero power. If the cut-off ratio is less than one, the numerator becomes a pure imaginary number due to the 

square root and power is identically zero. For the softwall, case the cut-off ratio is a complex number and non-zero 

power exists regardless of the cut-off ratio magnitude (power is the real component of the term in the braces). 

 Hardwall: (ζ real): (i) ζ < 1; RE bi { }; ˆ Ρ = 0

(ii) ζ > 1; RE a{ }; ˆ Ρ ≠ 0

 Softwall: (ζ complex): (i) ζ < 1; RE a{ } + bi ; ˆ Ρ ≠ 0

(ii) ζ > 1; RE a{ } + bi ; ˆ Ρ ≠ 0

 (19) 

Other parameters calculated from the decomposed modal data that are presented are the pressure levels of the 

inner wall and outer wall. Cmn is a the normalizing coefficient discussed earlier. The function multiplier is the 

coefficient of the un-normalized Bessel function. The Sofrin coefficient is defined as the maximum pressure along a 

radial profile. The Power Level (PWL) is calculated, including cut-off ratio and Mach number effects, and 

summed to obtain the total mode power level. Table I shows typical output from the standard data reduction. All 

circumferential modes are analyzed in this manner and typically plotted in a 3-D ‘tombstone’ plot shown in figure 9. 

The base plane axes are m- and n-order, and the vertical value axis is the PWL in the (m,n) mode. Along the wall of 

the m-order axis, the sum of all the radial modes provides the power in that circumferential mode. The sum of all the 

modes provides the PWL in the harmonic presented. A typical modal decomposition provides information as to the 

dominant modes present, usually those from the rotor stator interaction. Of secondary interest will be other modes, 

which may be due to inflow distortions or other geometric disturbances. Phase information, while tabulated is 

generally not plotted. 

TABLE I.—TYPICAL DATA OUTPUT 

 

10

8

6

4

2

32

40

0

24

16

8

-16

-8

0

-24

-32

-40

125

120

115

110

105

100

95

PWL (dB)

PWL (dB)

Circumferential Mode (m-order) Radial Mode (n-order)

95

100

105

110

115

120

125

Figure 9.—3-D plot of all mode amplitudes. 
NASA/TM—2005-213828 11 

Achievements 

Several Rotating Rake systems have been custom built for three facilities. In order of complexity of the 

turbomachinery test article, these are (1) the Advanced Noise Control Fan, (2) various high speed scale model fan 

rigs in the NASA Glenn 9- by 15-foot wind tunnel, and (3) a full scale turbofan, the Honeywell TFE731-60. 

Advanced Noise Control Fan 

The Advanced Noise Control Fan (refs. 10 and 11) (ANCF) was designed in the early 90s and first checkouts 

occurred in 1996. This unique fan was designed as a test bed for evaluating noise reduction technologies, code 

verification/development, and to continually improve the Rotating Rake system. The ANCF is a 4-ft diameter ducted 

fan (fig. 10) with several unique features. The cantilever nature of the rotor support structure allows for the rotor 

alone to be run, eliminating the need for support struts. The 16-bladed rotor, in conjunction with variable count- and 

position stator vane sets, allows for a wide combination of modes to be generated. 

The ANCF has separate rotating rake systems in the inlet and exhaust ducts to allow for quick though not 

simultaneous measurements. The rotating rakes on the ANCF rotate at 1/100th of the nominal fan shaft speed of 

1800 rpm (i.e., 18 rpm) and contain 7 (inlet) or 6 (exhaust) microphones. A drive motor connected to a straight shaft 

and gear assembly is coupled to a ring gear. This ring gear is embedded into an Aluminum housing which forms part 

of the duct contour. These rake assemblies can be located between any of the discrete duct sections to allow 

measurements to be taken at several points along the duct axis. This has been extremely useful in validating 

propagation codes. Though the primary measurement system is the rotating rake, the ANCF has other aeroacoustic 

measurement systems (surface pressures, hot film, farfield) that allow for determination of the relationship of duct 

modes to the physics. 

The ANCF played a fundamental role in the NASA Advanced Subsonic Technology Active Noise Control 

(ANC) program. The evaluation of the modal effects of a tested ANC system provided by the rotating rake was 

crucial to understanding and improving the ANC algorithms. Several advances in the understanding of mode 

generation (ref. 12), propagation (ref. 13), and radiation (ref. 14) and verification of codes resulted from tests run on 

the ANCF. Other mode measurement systems (ref. 15) were verified on the ANCF providing confidence in those 

systems to be used in more demanding environments. The Hershel-Quinke (ref. 16) noise reduction concept was 

evaluated on the ANCF. These tests led to modifications of the H-Q devices for full-scale turbofan tests. A recent 

promising concept for fan noise reduction, wake filling through the use of trailing edge blowing (ref. 17) was 

evaluated on the ANCF and vetted for a more complex test on a high-speed fan. Advanced trailing edge blowing 

concepts are being currently investigated in the high-speed fan regime. 

 

INFLOW CONTROL DEVICE

ROTOR BLADES (16) STATOR VANES variable number & spacing

ROTATING RAKE

CANTILEVERED

CENTER BODY

inlet

locations

(σ = 0.0)

mid-duct

location

(σ = 0.35) exit plane

location

(σ = 0.5)

Figure 10.—Rotating rakes on advanced noise control fan. 
NASA/TM—2005-213828 12 

Twenty-Two-Inch Fan Drive Rig 

The 22 in. fan rig in the NASA Glenn 9- by 15-foot Anechoic Wind Tunnel is an air-turbine driven rig that can 

have a specifically designed fan installed and tested. Typically a test program will involve detailed performance and 

acoustic evaluations. The test section Mach number is generally held at 0.2. 

The first application of the rotating rake system (ref. 18) was in 1991 for the Advanced Ducted Propulser (ADP) 

fan model that was designed and built to test fan noise. This fan was an 18-bladed rotor 17.25 in. in diameter. The 

rake motor drive was connected to a large ring gear to which the rake body was mounted. The rake body/ring gear 

assembly rotated at 1/250th of the fan speed (12,000 rpm) or 48 rpm. This assembly was fastened to the tunnel floor 

and ceiling via a support arm. The complete fan and rake assembly is shown in figure 11. 

Configurations tested were a cut-on stator vane set (22 vanes) as well as a traditional cut-off set (40 vanes). 

Three inlet lengths were tested (L/Df = 0.21, 0.41, and 0.53). This test documented experimentally for the first time 

the effects of mode cut-on. A weak tendency for lower mode levels with shorter inlet length was noted. Also, it was 

discovered that very minor perturbations in the rotor casing treatment cause extraneous modes, resulting in higher 

acoustic farfield levels. This cause would have been difficult to identify from the farfield data. 

The next iteration of rotating rake mechanical design for the 22 in. fan rig was a conformal design to provide 

better mechanical stability and flow characteristics, and to allow for more flexible positioning of the measurement 

plane. The rake assembly was housed in a drum system that was mounted on parallel rails. The rake drum housing 

can be slid to various axial positions to accommodate different measurement planes; including all the way aft to 

enable exhaust duct mode measurements. The more robust structural mounting allowed for a faster rotational speed, 

1/200th of the fan. These features can be seen in figure 12. 

This system was used on the ADP in 1996. The purpose of the test (ref. 19) was to evaluate the potential for fan 

noise reduction by lowering fan tip speed. Acoustic modes were measured for two fans of equal pressure ratio. Fan 2 

had a tip speed 90 percent of Fan 1 depending on condition. Initial assumptions were that the lower tip speed fan 

would be quieter, but the mode measurements did not support this assumption. One complicating factor was that in 

order to maintain the performance Fan 2 required 51 stator vanes compared to Fan 1’s 45. The different interaction 

modes generated (m = –9 vs. m = –15) at 2BPF may make comparisons difficult as the ANCF has shown that some 

interaction modes couple better to the duct. This test was the first to document the assumption that co-rotating 

modes (negative m orders) do not propagate well through the rotor. This transmission loss has a significant impact 

on inlet noise. 

The Active Noise Control program culminated in the test of a hybrid passive/active system (ref. 20) for tone 

reduction in a high-speed fan. The measurements made by the Rotating Rake were key to the design of the system. 

The baseline, passive only, active only, and hybrid active/passive systems were tested. The mode measurements 

showed significant reductions were achieved using the active control system, as well as a possible ‘synergistic’ 

benefit in that the reduction achieved by the combined system was greater than the sum of the reduction achieved by 

the systems individually. 

 Figure 11.—Rotating rake ring Figure 12.—Rotating rake drum 

 assembly on geared advanced ducted assembly on 22 in. fan in the 

 propulsor in the 9- by 15-foot wind 9- by 15-foot wind tunnel. 

 tunnel. 
NASA/TM—2005-213828 13 

A Source Diagnostic Test (ref. 21) (SDT) was conducted in the 9- by 15-foot wind tunnel using the 22 in. fan 

drive rig by NASA Glenn as part of the AST. This fan was a 22-bladed fan that had sonic tip speed at 100 percent 

rpm. In one important phase of the test, the effect of stator sweep was investigated. The concept involved returning 

to a cut-on set of stators; sweep was introduced to mitigate the effect of cut-on BPF. The Rotating Rake 

measurements showed that compared to a standard cut-off set (26 vs. 54 Vanes), the swept vane set reduced the 

mode tone power significantly. This test also demonstrated the ability of the Rotating Rake to make valid mode 

measurements with the fan at sonic tip speed. Recall that at Mtip > 1, the m = B mode cuts on. It was shown that in 

the inlet the rake wake contaminates the m = B mode. By analyzing mode measurements taken at fan speeds below 

the rotor mode cut-on, the magnitude of the rake wake contamination was estimated. When the mode measurements 

were analyzed above sonic tip speed, then increase in m = B mode PWL was greater than 10 dB. The physical 

assumption is made that the rake wake contamination would increase only slightly with rpm, but the rotor mode 

increases significantly. Therefore, the m = B mode measurements above sonic fan tip speed in the inlet are assumed 

to be valid. 

The rotating rake was also used to evaluate the acoustic benefit to a forward swept fan (ref. 22), the Quiet High 

Speed Fan (QHSF), compared to a baseline fan representing the Honeywell TFE731-60 engine. Being a 

representative fan model, support struts were included in the exhaust duct, in addition to the stator vanes. This test 

also presented an extension of the Rotating Rake reduction methodology to the analysis Multiple Pure Tones 

(MPTs) or “buzz-saw” noise. MPTs occur at shaft orders below BPF with the interesting characteristic that the 

m-order generated at a given shaft order is numerically, the shaft order. The mode measurements showed that the 

QHSF was significantly quieter, but that the reduction was primarily from a lowering of the rotor-strut interaction 

tones rather than from a reduction in the MPTs that had been originally been the design goal. The Rotating Rake 

measurements suggested that the reduction mechanism may have been that the stator was shielding the strut from the 

wake of the rotor; though this is unconfirmed. 

Follow up tests to the SDT, comparing different rotor sets; and the QHSF, measuring the effects of vane 

clocking, have been done, but not yet reported as of the date of this publication. 

TFE-731-60 

The most challenging and perhaps most significant application (ref. 23) was on a Honeywell full-scale turbofan 

engine, the TFE731-60. The TFE731-60 engine is currently certified on the Dassault Falcon 900EX. The TFE731-

20/40/60 engine family also powers the Lear 45, IAI Astra SPX, and the Dassault Falcon 50EX. This test was 

accomplished under the Advanced Subsonic Technology program sponsorship of the Engine Validation of Noise 

Reduction Concepts (EVNRC). The basic engine parameters are listed in table II. The schematic of the fan flowpath in figure 13 shows multiple sources of interaction modes. The engine was tested at the Honeywell engine test 

stand in San Tan, Arizona. Photographs of the inlet and exhaust duct rake assemblies with the engine on the test 

stand are shown in figure 14. 

 TABLE II.—TFE731-60 ENGINE PARAMETERS 

 

 

 

Figure 13.—TFE731 flow-path. 

Basic Engine Weight 988 lb 

 Takeoff, Sea Level, Static Thrust 5000 lb 

Airflow 187 lb/sec 

Cruise Bypass Ratio 3.9 

Cycle Pressure Ratio 22 

Geared Fan Pressure Ratio 1.70 
NASA/TM—2005-213828 14 

 

Figure 14.—Rotating rake assembly on Honeywell TFE731-60. 

The rotor-strut interaction modes at BPF were identified as the dominant modes but were a minor contribution 

to overall PWL, due to the strength of the extraneous (background) modes. Rotor transmission losses were shown to 

result in lower co-rotating modes in the inlet compared to the exhaust. The 2BPF rotor-vane interaction mode was 

clearly identified in the inlet and exhaust, and observed to dominate. Repeatability of the measurements was shown 

to be excellent by comparing results from two separate test builds. An attempt to measure the rotor-locked noise was 

inconclusive due to the weakness of the rotor mode. The MPT sub-harmonics (“buzz-saw”) were seen to be 

dominated by the shaft-orders just under 1

/2BPF. 

Future Extensions 

An evaluation of the Trailing Edge Blowing concept on the 22 in. Fan Drive Rig is underway in the NASA 

Glenn 9- by 15-Foot Wind Tunnel. Currently, a Rotating Rake system is being designed for a Honeywell Tech7000 

Turbofan (for the NASA sponsored Engine Validation Noise Reduction Technologies (EVNERT) program. The 

rotating rake will be used to measure the full-scale version of the QHSF and to provide detailed design information 

for H-Q tubes. The analysis technique is currently being extended to measure and reduce duct modes over treated 

wall sections. The method has already been verified for treated inlet ducts (σ = 0), and current testing for annular 

treated sections is underway. These results will be reported in a future paper. A significant need is to develop 

analysis techniques to properly measure duct modes in sheared flows. 

References 

 1. Hubbard, H. Harvey, editor, Aeroacoustics of Flight Vehicles: Theory and Practice: Volume 1, NASA 

Reference Publication 1258, Vol. 1, WRDC Technical Report 90-3052. 

 2. Sofrin, T.G. “Aircraft Turbomachinery Noise: Fan Noise,” Notes for Lectures 21 and 22 for the Course in the 

Fluid Dynamics in Turbomachinery at ASME/Iowa State University, sponsored by Pratt & Whitney Aircraft 

Division/United Aircraft Corporation. 

 3. Moore, C.J., “Measurement of Radial and Circumferential Modes in Annular and Circular Ducts,” Journal of 

Sound and Vibration,” Vol. 62, No. 2, 1979. 

 4. Cicon, D.E., Sofrin, T.G., and Mathews. D.E., “Investigation of a Continuously Traversing Microphone System 

for Mode Measurement” NASA CR–168040, PWA-5846-26, November 1982. 

 5. Cicon, C.E. and Sofrin, T.G., “Method for Extracting Forward Acoustic Wave Components from Rotating 

Rake Measurements from Rotating Microphone Measurements in the Inlets of Turbofan Engines,” NASA 

CR–195457, April 1995. 

 6. Meyer, H.D., Envia, E. “Aeroacoustic Analysis of Turbofan Noise Generation,” NASA CR–4715, March 1996. 

 7. Rice, E.J., Heidmann, M.F., and Sofrin, T.G., “ Modal Propagation Angles in a Cylindrical Duct with Flow and 

Their Relation to Sound Radiation,” AIAA 97-0183. 

 8. Tyler, J.M., and Sofrin, T.G., “Axial Flow Compressor Studies,” SAE Transactions, Vol. 70, 1962. 

 9. Rice, E.J., “Optimum Wall Impedance for Spinming Modes–A Correlation with Mode Cut-off Ratio,” Journal 

of Aircraft, Vol. 16, No. 5. 
NASA/TM—2005-213828 15 

10. Hall, D.G., Bridges, J.E., and Heidelberg, L.J., “A Unique Ducted Fan Test Bed for Active Noise Control and 

Aeroacoustics Research,” NASA TM–107213, AIAA–96–1740. 

11. Sutliff, D.L, Nallasamy, M., Heidelberg, and L.J., Elliott, D.M., “Baseline Acoustic Levels of the NASA Active 

Noise Control Fan,” NASA TM–107214, AIAA–96–1745. 

12. Sutliff, D.L., Heidelberg, L.J., and Envia, E, “Coupling of Low Speed Fan Stator Vane Unsteady Pressures to 

Duct Modes: Measured vs. Predicted”. AIAA–99–1864. 

13. Nallasamy, M., Sutliff, D.L., and Heidelberg, L.J., “Propagation of Spinning Acoustic Modes in Turbofan in 

Turbofan Exhaust Ducts,” Journal of Propulsion and Power, Vol. 16, No. 5, pp. 736–743. 

14. Heidelberg, L.J., Sutliff, D.L., and Nallasamy, M., ‘Azimuthal Directivity of Fan Tones Containing Multiple 

Modes,” NASA TM–107464, AIAA–97–1587. 

15. Lan, J.H., Premo, J.W., and Sutliff, D.L., “Inlet Mode Measurements with an Inflow Control Device 

Microphone Array,” AIAA–2002–2563. 

16. D. de la Riva, R. Burdisso and W. Ng, “Aft Fan Noise Control Using Herschel-Quincke/liner Systems” 

AIAA–2005–3071. 

17. Sutliff, D.L., Tweedt, D.L., Fite, E.B., and Envia, E. “Low-Speed Fan Noise Reduction with Trailing Edge 

Blowing,” International Journal of Aeroacoustics,” Vol. 1, No. 3. 

18. Heidelberg, L.J., and Hall, D.G., “Inlet Acoustic Mode Measurements Using a Continuously Rotating Rake,” 

Journal of Aircraft, Vol. 32, No. 4. 

19. Heidelberg, L.J. and Elliott, D.M., “A Comparison of Measured one Modes for Two Low Noise Propulsion 

Fans,” NASA/TM—2000-210231, AIAA–2000–1989. 

20. Parente, C.A., Acas, N, Walker, B.E., Hesh , A.S., and Rice, E.J., “Hybrid Active/Passive Jet Engine Noise 

Suppression System,” NASA/CR—1999-208875. 

21. Heidelberg, L.J, “Fan Noise Source Diagnostic Test-Tone Modal Structure Results,” NASA/TM—2002-

211954, AIAA–2002–2428. 

22. Heidelberg, L.J, “Comparison of Tone Mode Measurements for a Forward Swept and Baseline Rotor Fan,” 

AIAA–2003–3293. 

23. Sutliff, D.L., Konno, K.E., and Heidelberg, L.J., “Duct Mode Measurements on the TFE731-60 Full Scale 

Engine,” NASA/TM—2002-211573, AIAA–2002–2564. 

NASA/TM—2005-213828 17 

Appendix 

This appendix outlines the derivations of the equations presented in the main body of the paper. General 

assumptions are uniform Mach number in a constant area cylindrical or annular duct. 

Symbols 

a,b constants 

A duct area 

B number of fan blades 

C normalizing coefficient 

D duct diameter 

E duct profile function 

f frequency 

G mode power Mach effect 

h fan fundamental harmonic 

J Bessel function – 1st kind 

k wavenumber 

M Mach number 

m circumferential mode order 

N number of pulses per fan rev 

n radial mode order 

p acoustic pressure 

P modal pressure amplitude 

ˆ

Ρ modal power amplitude 

Q weighting function 

R duct radius 

r radial co-ordinate 

t temporal co-ordinate 

s fan shaft harmonic 

V number of vanes 

x axial co-ordinate 

Y Bessel function - 2nd kind 

Greek

Â admittance 

ε acoustic particle displacement

η reduced frequency 

φ mode propagation angle 

κ eigenvalue 

ρ density 

σ duct hub-to-tip ratio

θ circumferential co-ordinate

ω fan shaft rotational speed 

Ω rake rotational speed 

Ψ Bessel function combination 

Λ duct mode amplitude 

υ fan/rake speed ratio 

ν acoustic particle velocity 

ζ cut-off ratio 

Ζ Impedance 

A. Solution to Wave Equation with Boundary Conditions Representative in Turbofans 

The non-dimensional form of the convective wave equation in cylindrical co-ordinates with axially uniform mean 

flow in a constant area duct is written (ref. 1) as: 

∇2p(r,Θ, x,t) = ∂

∂t

+ M ∂

∂x

⎛ 

⎝ ⎜ ⎞ ⎠ ⎟ 

2

p(r,Θ, x,t);

with the non − dimensionalizations M = u*

c 0

* ; x = x*

R ; r = r*R ; r = x*R ; t = t

*

R

c 0

*

; p = p*

ρ

*

c 0

*2 ; f = f *

R

c 0

*

 A.1 

This form of the equation is known for it’s separable solutions: 

p(θ,r, x,t) = p ∗ E(κr)ei(2πft +mθ ±kx) A.2 

A cyclic temporal variation is superimposed upon the entire solution: Pt(t)= e

i2πft A.3

The form of the circumferential solution is deduced from the observation that the pressure must be single-valued 

azimuthally: 

 Pθ (θ) = Pθ (θ + 2π); results in Pθ (θ )= cos(mθ) A.4

The variable m is defined as the circumferential mode order and indicates the number of pressure cycles in the 

circumferential direction. 
NASA/TM—2005-213828 18 

The solution in the axial direction, where k is defined as the axial wave number, is given as: 

Px ( )x = e

−ikx x

;

kx

η = 1

(1− M 2

) −M ± 1− (1− M 2

)(κ

η

)

2 ⎡

⎣ 

⎢ 

⎤

⎦ 

⎥ ; η = fR /c A.5 

Since the text clearly illustrated that the circumferential solution arises from the rotation of the rake inducing a 

Doppler-shift that separates the circumferential modes. The nature of the radial rake indicates that the primary 

interest is the radial solution. The form of the radial solution is the well-known Bessel’s Equation: 

d 2P(r)

dr 2 +

1

r

dP(r)

dr

+ η2 1− M kx

η

⎛ 

⎝ ⎜ 

⎞ 

⎠ ⎟ 

2

− kx

η

⎛ 

⎝ ⎜ 

⎞ 

⎠ ⎟ 

⎡ 2

⎣ 

⎢ 

⎢ 

⎤ 

⎦ 

⎥ 

⎥ 

2

− m

2

r 2

⎛ 

⎝ 

⎜ 

⎞ 

⎠ 

⎟ 

⎧

⎨ ⎪ 

⎩ ⎪ 

⎫

⎬ ⎪ 

⎭ ⎪ 

P(r) = 0 A.6 

The solution to this ordinary differential Equation is an eigenvalue problem with an infinite number of solutions: 

 

P(r) = aJ (κ n r) +bY(κ n r); a,b arbitrary, n = 0,1,2,... A.7 

where J and Y are the Bessel’s functions of the 1st and 2nd kind, of order n. The index n is the radial mode order, 

which physically indicates the number of pressure nodes in the radial profile. The pressure profile is a superposition 

of cirumferential and radial modes. Each term in this superposition contains a radial duct profile function written as: 

 

Emn (r) = CmnΨmn (r) = Cmn Jm (κ mn r) + QmnYm (κ mn [ r)]; n = 0,1,2,... A.8 

with subscripts m for the circumferential mode order, and n for the radial mode order. 

Since the Bessel functions can vary substantially in form, by convention a=C, b=C*Q is chosen where C is a 

weighting factor related to the integration of Ψ and Q is termed as a second eigenvalue, though technically it is not. 

κ is an eignevalue found without reference to Q as shown in Eq. B.7 or Eq. B.8 solves the proper eigenfunction P(r). 

Q is from the boundary conditions and determines the proper radial shape of the eignefunction. 

The general boundary conditions are determined from the relationship between the non-dimensional particle velocity 

and pressure at the duct walls: 

η2 1− M kx

η

⎛ 

⎝ ⎜ 

⎞ 

⎠ ⎟ 

2

ε = dp

dr ; piηε N

= Z

ρ 0

*

c0

* = 1

Α ˆ A.9 

Where ε is the particle displacment in the flow positive in the radial direction, and is the radial particle displacment 

at the wall, positive into the wall. When the boundary conditions are applied, this becomes: 

dP(1)

dr = −iηˆ A ow 1− M kx

η

⎛ 

⎝ ⎜ 

⎞ 

⎠ ⎟ 

2

P(1); dP(σ)

dr = iηˆ A iw 1− M kx

η

⎛ 

⎝ ⎜ 

⎞ 

⎠ ⎟ 

2

P(σ) A.10 

The boundary condition for the hardwall case is that the pressure be normal to the duct inner and outer wall (a 

consequence of the zero acoustic velocity at the wall). Or, for a hardwall duct the inner and outer wall admittances 

are 0 so Eq. A.10 reduces to: 

dP(1)

dr = dP(σ)dr = 0 A.11 
NASA/TM—2005-213828 19 

B. Eigenvalue Solutions 

The solution of the eigenvalue problem is based on the general boundary conditions from the continuity of particle 

displacement at the wall (Eq. A.10) that are repeated here: 

dP(1)

dr = −iηˆ A ow 1− M kx

η

⎛ 

⎝ ⎜ ⎞ 

⎠ ⎟ 

2

P(1); dP(σ)

dr = iηˆ A iw 1− M kx

η

⎛ 

⎝ ⎜ ⎞ 

⎠ ⎟ 

2

P(σ) B.1 

where: Pmn (r) = pmnEmn (r) = CmnΨmn (r) = Cmn Jm (κ mn r) + QmnYm (κ mn [ r)] B.2 

Taking the derivative of this equation with respect to r, and applying the recursive definition of the Bessel Function 

derivative ( mJm (κ mn ) −κ mn J ′ m (κ mn ) = κ mn Jm+1(κ mn ) ) results in: 

 

dEmn (r)

dr = d Cmn Jm (κ mn r) + QnmYm (κ mn [ ] r)

dr = Cmnκ mn J ′ m (κ mn r) + QmnY m′ (κ mn [ ] r)

= Cmnκ mn

m

κ mn r

⎛ 

⎝ ⎜ 

⎞ 

⎠ ⎟ Jm (κ mn r) − Jm+1(κ mn r)

⎡ 

⎣ 

⎢ 

⎤ 

⎦ 

⎥ + Qmn

m

κ mn r

⎛ 

⎝ ⎜ 

⎞ 

⎠ ⎟ Ym (κ mn r) − Ym+1(κ mn r)

⎡ 

⎣ 

⎢ 

⎤ 

⎦ 

⎥ 

⎧ 

⎨ ⎪ 

⎩ ⎪ 

⎫ 

⎬ ⎪ 

⎭ ⎪ 

 B.3 

Substituting Eq. B.3 into the left side of Eq. B.1, and Eq. B.2 into the right side, canceling out the coefficient Cmn

results at the outer wall: 

 

κ mn

m

κ mn

⎛ 

⎝ ⎜ ⎞ 

⎠ ⎟ Jm (κ mn ) − Jm+1(κ mn )

⎡ 

⎣ 

⎢ 

⎤ 

⎦ 

⎥ +Qmn

m

κ mn

⎛ 

⎝ ⎜ ⎞ 

⎠ ⎟ Ym (κ mn ) − Ym+1(κ mn )

⎡ 

⎣ 

⎢ 

⎤ 

⎦ 

⎥ 

⎧

⎨ ⎪ 

⎩ ⎪ 

⎫

⎬ ⎪ 

⎭ ⎪ 

= −iηˆ A ow 1− M kx

η

⎛ 

⎝ ⎜ 

⎞ 

⎠ ⎟ 

2

Jm (κ mn ) + QmnYm (κ mn [ ])

 B.4 

and at the inner wall: 

 

κ mn

m

κ mnσ

⎛ 

⎝ ⎜ 

⎞ 

⎠ ⎟ Jm (κ mnσ) − Jm+1(κ mnσ)

⎡ 

⎣ 

⎢ 

⎤ 

⎦ 

⎥ + Qmn

m

κ mnσ

⎛ 

⎝ ⎜ 

⎞ 

⎠ ⎟ Ym (κ mnσ) − Ym+1(κ mnσ)

⎡ 

⎣ 

⎢ 

⎤ 

⎦ 

⎥ 

⎧

⎨ ⎪ 

⎩ ⎪ 

⎫

⎬ ⎪ 

⎭ ⎪ 

= +iηˆ A iw 1− M kx

η

⎛ 

⎝ ⎜ 

⎞ 

⎠ ⎟ 

2

[ ] Jm (κ mnσ) + QmnYm (κ mnσ)

 B.5 

Solving both Equations for Qmn and setting the result equal to one another yields: 

 

2 2

1 1

2 2

1 1

ˆ 1 () () 1 ( ) ( )

ˆ ˆ 1 () () 1 ( ) (

x x

ow m mn mn m mn iw m mn mn m mn

x x

ow m mn mn m mn iw m mn mn m m

k k m m iA M J J iA M J J

k k m m iA M Y Y iA M Y Y

+ +

+ +

+ η − κ −κ κ − η − κ σ −κ κ σ

η ση

=

+ η − κ −κ κ − η − κ σ −κ κ

η ση

⎡⎤ ⎡⎤ ⎛⎞ ⎛⎞ ⎢⎥ ⎢⎥ ⎜⎟ ⎜⎟ ⎝⎠ ⎝⎠ ⎣⎦ ⎣⎦

⎡⎤ ⎡⎤ ⎛⎞ ⎛⎞ ⎢⎥ ⎢⎥ ⎜⎟ ⎜⎟ ⎝⎠ ⎝⎠ ⎣⎦ ⎣⎦



) nσ

 B.6 

This equation can be solved iteratively by a number of means to obtain κ , then substitute into Eq. B.4 or Eq. B.5 to 

obtain Q. For the hardwall boundary conditions the admitances, Âiw and Âow, in Eq. B.6 are set to zero: 

mJm (κ mn ) −κ mn Jm+1(κ mn )

mYm (κ mn ) −κ mnYm+1(κ mn )

=

m

σ

Jm (κ mnσ) −κ mn Jm+1(κ mnσ)

m

σ

Ym (κ mnσ) −κ mnYm+1(κ mnσ) B.7 
NASA/TM—2005-213828 20 

The profile for a cylindrical duct (σ=0) requires that the radial solution be finite at the centerline. To Q=0 to Thus 

the radial duct function, E, reduces to: 

Pmn (r) = CmnEmn (r) = CmnJm (κ mnr) B.8 

The cylindrical duct eigenvalue solutions for softwall boundary conditions are found by noting Âiw in Eq. B.1 and 

Qmn in Eq. B.4 are zero. This results in: 

dP(1)

dr = −iηˆ A ow 1− M kx

η

⎛ 

⎝ ⎜ 

⎞ 

⎠ ⎟ 

2

P(1); dP(0)

dr = 0

 B.9 

κ mn

Jm+1(κ mn )

Jm (κ mn ) − m − iηˆ A ow 1− M kx

η

⎛ 

⎝ ⎜ 

⎞ 

⎠ ⎟ 

2

= 0

 B.10 

Finally, the eigenvalue equation for the cylidrical hardwall duct is found by setting Âow in Eq. B.10 to zero: 

 

κ mn

Jm+1(κ mn )

Jm (κ mn ) − m = 0

 B.11 

C. Derivation of the Weighting Factor, Cmn

Since the profile of the Bessel functions can vary significantly with order, it is desirable to normalize the duct profile 

by a physical constant in order that meaningful comparisons of the mode amplitude can be made. The weighting 

factor can take several forms based on the desired physical reasoning. One form chosen in early modal analysis 

(ref. 2) was acoustic intensity based normalization by setting the integration across the duct to unity. 

intensity: p

2

dA = 1

A

∫ C.1 

An alternative form is to normalize by relating to power. In order to match the methodology of standard CAA codes 

(ref. 3) the normalization chosen is power based so that: 

 power: p

2

dA = A

A

∫ C.2 

From the section A, the radial profile is given by : 

 

Ψmn (r) = Jm (κ mnr) +QmnYm (κ mnr); n = 0,1,2,... C.3 

integrating, 

Cmn

2

σ

1

∫ 0

2π

∫ Ψmn

2 (κ mnr)rdrdΘ = 2π (1−σ2

)

Cmn

2 Ψmn2 (κ mnr)rdr σ

1

∫ = (1−σ2

)

 (1−σ

2

)

Cmn

2 = Ψmn

2 (κ mnr)rdr σ

1

∫ C.4 

Under certain conditions the functions are orthogonal. The orthogonality property of the Bessel Function (ref. 4) is: 

2

11 2

12 1 1 2

0,

( )( ) ; ( ),

:

(1) ( ) ( ) 0

(2) & 0 ( ) ( ) 0

b

b

j l

l a a

l

ll l

if j l

t t t dt

t t dt if j l

provided the following two conditions hold

is a real zero of h b h b

there exisits k k such that k a k a

υ υ ν

ν+ ν

ν+ ν

⎧ ≠

⎪ ξλ ξλ = ⎨ ξλ = ⎪

⎩

λ λξ λ − ξ λ =

≠ λξ λ − ξ λ =

∫ ∫

 C.5 
NASA/TM—2005-213828 21 

Applying the hardwall boundary conditions from Eq. A.11 and the recursive definition of the Bessel function shows 

that (1) & (2) are satisfied with the eigenvalues κ. Re-arranging the impedance boundary conditions Eq. A.10: 

d(Emn (κ mn r))

dr = −i ˆ A (1− M kx

η

)

2Emn (κ mn r) → κ mnE ′ mn (κ mn r) + i ˆ A (1− M kx

η

)

2Emn (κ mn r) = 0 C.6 

At the outer wall (b=1) this equation takes on the form of condition (1): 

2

1,

2

1,

ˆ ( ) ( ) (1 ) ( ) 0

ˆ ( ) (1 ) ( ) 0

x mn m n mn mn mn ow mn mn

x mn m n mn ow mn mn

k

E mE iA M E

k

E m iA M E

+

+

⎡ ⎤ −κ κ + κ + − κ = ⎣ ⎦ η

⎡ ⎤

κ κ −+ − κ = ⎢ ⎥ ⎣ ⎦ η

 C.7 

Likewise, at the inner wall (a=σ) the boundary conditions take on the form of condition (2): 

−κ mnEm+1,n (κ mnσ) + m

σ

Emn (κ mnσ) ⎡

⎣ ⎢ 

⎤

⎦ ⎥ − i ˆ A iw (1− M kx

η

)

2Emn (κ mnσ) = 0

−κ mnEm+1,n (κ mnσ) + m

σ − i ˆ A iw (1− M kxη

)

2 ⎡ 

⎣ ⎢ 

⎤ 

⎦ ⎥ Emn (κ mnσ) = 0

 C.8 

It is recognized that this is not strictly correct because for the general case the κmn is not a real eigenvalue, but may 

be complex. Nonetheless, the form of the solution will be used in order to be consistent in the normalization. Future 

work may investigate the derivation of a rigorously correct solution of the normalizing constant. 

The general solution to the integral is then: 

() ()

1 1 2

2 2 22 1

2 2 2 m mn mn mn mn mn () 1

mn

m r rdr r r r

r σ σ

⎡⎛ ⎞ ⎤

ψ κ = − Ψ κ +Ψ κ ⎢⎜ ⎟ ′ ⎥ ⎜ ⎟ ⎢ κ ⎥ ⎣⎝ ⎠ ⎦ ∫ C.9 

() () () ()

2 2 22

22 2 2

2 2 22

(1 ) 1 1 1

2 2 mn mn mn mn mn mn mn mn

mn mn mn

m m

C

− σ ⎡ ⎤⎡ ⎤ ⎛⎞ ⎛ ⎞ σ = − Ψ κ +Ψ κ − − Ψ κ σ +Ψ κ σ ⎢ ⎥⎢ ⎥ ⎜⎟ ⎜ ⎟ ′ ′

⎝⎠ ⎝ ⎠ κ κσ ⎣ ⎦⎣ ⎦

 C.10 

Applying the general annular duct, non-zero impedance boundary conditions: 

dP(1)

dr = −iηˆ A ow 1− M kx

η

⎛

⎝ ⎜ ⎞⎠ ⎟ P(1); dP(σ)

dr = iηˆ A iw 1− M kx

η

⎛

⎝ ⎜ ⎞⎠ ⎟ P(σ); C.11 

dΨm (κ mnr)

dr = −iηˆ A 1− M kx

η

⎛

⎝ ⎜ ⎞⎠ ⎟ Ψm (κ mn r) C.12 

κ mnΨ ′ m (κ mn ) = −iηˆ A ow 1− M kx

η

⎛

⎝ ⎜ ⎞⎠ ⎟ Ψm (κ mn ); κ mnΨ ′ m (κ mnσ) = iηˆ A iw 1− M kx

η

⎛

⎝ ⎜ ⎞⎠ ⎟ Ψm (κ mnσ) C.13 

Ψ m′ (κ mn ) = − iηˆ A ow

κ mn

1− M kx

η

⎛ 

⎝ ⎜ 

⎞ 

⎠ ⎟ Ψm (κ mn ); Ψ m′ (κ mnσ) = iηˆ A iw

κ mn

1− M kx

η

⎛ 

⎝ ⎜ 

⎞ 

⎠ ⎟ Ψm (κ mnσ) C.14 
NASA/TM—2005-213828 22 

Substituting the re-arranged boundary conditions equations into the definition of Cmn derivative: 

( )

( )

2 2 2

2 2

2 22

2 2 2

22 2 2

2

1 1 1 1 ()

2(1 )

1 ()

ow x mn mn m mn

mn mn mn

iw x mn mn m mn

mn mn

m A k

M

C

m A k

M

⎧⎡ ⎤ ⎛ ⎞ ⎪ ⎛ ⎞ η ⎛ ⎞ = − Ψ κ− − Ψκ ⎨⎢ ⎥ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟

−σ κ ⎪⎢ ⎥ ⎜ ⎟ ⎝ ⎠ κ η ⎝ ⎠ ⎣ ⎦ ⎝ ⎠ ⎩

⎡ ⎤⎫ ⎛ ⎞ ⎛ ⎞ η ⎛ ⎞ ⎪ − σ − Ψ κ σ −σ − Ψ κ σ ⎢ ⎥ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎬ ⎢ ⎥ ⎝ ⎠ ⎜ ⎟ κ ⎝ ⎠ κ η ⎝ ⎠ ⎪ ⎣ ⎦⎭

 C.15 

For the hardwall condition A=0; the definition of Cmn reduces to: 

1

Cmn

2 = 1

2(1−σ2

)

1− m2

κ mn

2

⎛ 

⎝ 

⎜ ⎞ 

⎠ 

⎟ Ψmn

2 ( ) κ mn − σ2 − m2

κ mn

2

⎛ 

⎝ 

⎜ ⎞ 

⎠ 

⎟ Ψmn

2 ( ) κ mnσ

⎡

⎣ 

⎢ ⎢ 

⎤ 

⎦

⎥ ⎥ C.16 

For a cylindrical duct, equation C.15 simplifies to (impedance condition): 

with σ = 0; Ψmn (κ mn ) = Jm (κ mn ); Ψmn (0) = Jm (0) = 0; m ≠ 0 C.17 

( )

2 2 2

2 2

2 2

ˆ 1 1 1 1 () 2

ow x mn mn m mn

mn mn mn

m A k

J MJ

C

⎡ ⎤ ⎛ ⎞ ⎛ ⎞ η ⎛ ⎞ ⎢ ⎥ = − κ− − κ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎢ ⎥ ⎜ ⎟ κ ⎜ ⎟ κ η ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎣ ⎦

 C.18 

and for the hardwall condition, this further simplifies to: 1

Cmn

2 = 1

2

(1− m2

κ mn

2 )Jm

2 (κ mn ) C.19 

D. Doppler-Shift of Duct Modes in Rotating Reference Frame 

The pressure at a fixed radial and axial location in the duct from all modes is: p(Θ,t) = Am

s e−i(sωt−mθ )

m=−M

m= +M

∑ s=1

∞

∑ D.1 

If the location is slowly rotating in the circumferential direction: Θ=Ωt; p(Θ,t) = Am

s e−i(sω−mΩ)t

m=−M

m= +M

∑ s=1

∞

∑ D.2 

For a typical application the rake rotates at a fraction of the fan speed: Ω = 1

υ.

ω; s = B D.3 

Substituting Eq. D.3 into Eq. D.2 it is seen that the cyclical part becomes: 

( ) ( () () ) ( ) sm Bm B ω− Ω = ω− =ω − ω m

υ υ

 D.4 

The mode frequency is then: m ( ( )) f B =ω − m

υ

 D.5 
NASA/TM—2005-213828 23 

Thus the frequency is shifted from the fundamental by an amount proportional to the mode number, and inversely 

proportional to the speed ratio. Note that for a rake that rotates in the same direction as the rotor, co-rotating modes 

(+m) will be reduced in frequency and counter-rotating modes (–m) will be shifted higher in frequency. The plane 

wave (m=0) is at the fan harmonic. 

E. Non-Interference of Rake Wake 

The wake of the rake is ingested by the rotor and contaminates the acoustic field. Fortunately, due to the rotating 

frame of reference all of the contamination occurs at a single Doppler-shifted frequency. A physical explanation was 

given in the text; here is the analytical proof: 

The pressure due to a fixed distortion: p(Θ,t) = Am

s e

−i(sωt−(s±l)θ )

 E.1 

If this distortion is slowly rotating: p(Θ,t) = Am

s e

−i s[ (ω±lΩ)t−(s±l)θ ]

 E.2 

The cyclical portion is: 

(sω ± lΩ)t − (s ± l)θ = (sω ± lΩ)t − (s ± l)Ωt = sωt ± lΩt − sΩt ∓ lΩt = s(ω − Ω)t E.3 

so the frequency (with s=B) is: frake = s(ω − Ω) = ωs(1− Ω /ω) = ωB(1− 1

υ

) E.4 

from the previous section note that for m=B: ( ) ( ) 1 fB B m B ( ) 1( ) m = =ω − =ω −

υ υ

 E.5 

In an unpublished letter, dated 20 November 1981, (this was to be an appendix to an earlier report (ref. 5), Sofrin 

demonstrated the non-interference this way (original text and notation): 

“Consider a single mode generated by the rotor cutting the wake of a radial probe positioned at θ=0: 

pmn (0;θ,t) = Re Amn expi(mθ − nBΩt) (1) 

If the wake is moved to θ=β, then the pressure at (θ+β) and time (t+ β/Ω) will be the same as given by (1), or 

pmn (β;θ + β,t + β

Ω) = pmn (0;θ,t) (2) 

Consequently, the pressure at θ and t due to the wake at θ=β is 

p t p t A i m nB t mn mn mn ( ; , ) (0; , ) Re exp ( ) ( ) β β ⎡ ⎤ β θ = θ−β − = θ−β − Ω − Ω ⎢⎣ Ω ⎥⎦

 (3) 

The complete pressure, due to all modes and frequencies is then 

pmn (β;θ,t) = p(β;θ,t)

m,n

∑ = Re Amn expi m[ ] (θ − β) − nBΩt + nBβ)

m,n

∑ (4) 

But the microphone senses the pressure only at the position θ=β, (or θ=β+const) so the microphone signal pmic is: 

pmic = pmn (β;β,t) = Re Amn expi nB [ ] β − nBΩt

m,n

∑ (5) 
NASA/TM—2005-213828 24 

Since the exponential does not depend on m, this can be written 

pmic = Re Amn

−∞

∞

∑

⎛

⎝ 

⎜ ⎜ 

⎞

⎠ 

⎟ ⎟ expi nB ( ) β − nBΩt

n=1

∞

∑

or, with Amn

−∞

∞

∑ = An ,

pmic = Re An expi nB( ) β − Ωt

n=1

∞

∑

 (6) 

From Eq. (6) it is seen that the microphone signal, at any frequency nBΩ behaves exactly as if it were sensing the 

direct rotor field which has nB lobes, spinning at Ω. This signal will undergo a 2π phase shift when the probe angle 

β changes by 2π/β. 

It should now become clear on reflection that no matter how complicated the wake-generated acoustic field, that 

after the probe has turned through 1 blade gap, the pressure sensed by the microphone is indistinguishable in 

amplitude and phase from its initial state, thus providing intuitive support to the analytical results.” 

F. Least Squares Fit 

The modal solution is a least-squares curve fit to the measured pressure using the analytical radial pressure profile 

solutions as the basis functions. If the pressure has been separated by frequency and circumferential mode order, the 

pressure for at given radial and axial location is then: 

pmrf = pmnf

n= 0

∞

∑ Emnf (κ mnf r) F.1 

Limiting to the maximum number of radial modes: pmrf = pmnf

n= 0

N max

∑ Emnf (κ mnf r) F.2 

The difference between the measured and calculated pressure: δ = pmnf

n= 0

N max

∑ Emnf (κ mnf r) − pmrf F.3 

A weighting function is introduced to allow for weighting of errors based on area or other physical means. (In 

practice the weighting function Wi has the effect of bad coding a microphone by setting Wi = 0 for a suspicious 

channel, 1 for all others.) 

δ i = Wi pmnf

n= 0

N max

∑ Emnf (κ mnf ri) − pmri f

⎡

⎣ 

⎢ 

⎢ 

⎤ 

⎦

⎥ 

⎥ F.4 

The square of the error is: δ i

2 = Wi pmnf

n= 0

N max

∑ Emnf (κ mnf ri) − pmri f

⎡ 

⎣ 

⎢ 

⎢ 

⎤ 

⎦ 

⎥ 

⎥ 

⎧ 

⎨ ⎪ 

⎩ ⎪ 

⎫ 

⎬ ⎪ 

⎭ ⎪ 

2

 F.5 

Summing over the number of points p: δ2 = δ i

2

i=1

Np

∑ = Wi pmnf

n= 0

N max

∑ Emnf (κ mnf ri) − pmri f

⎡ 

⎣ 

⎢ 

⎢ 

⎤ 

⎦ 

⎥ 

⎥ 

⎧ 

⎨ ⎪ 

⎩ ⎪ 

⎫ 

⎬ ⎪ 

⎭ ⎪ 

2

i=1

Np

∑ F.6 
NASA/TM—2005-213828 25 

The concept of least-square method is to minimize the error squared, by taking the differential with respect to each 

Pmnf and setting the result to zero. Which taking the series terms out of the summation and extending to the radial 

modes: 

( ) ( )

( ) ( )

1 1

1

2 2

2

2 max 2

1 0

2 max 2

1 0

2 ( ) () ,

2 ( ) () ,

i

i

Np N

i mn f mn f i mnf mnf mnf i mr f

mn f i n

Np N

i mn f mn f i mnf mnf mnf i mr f

mn f i n

WE r p E r p

p

WE r p E r p

p

= =

= =

∂δ ⎡ ⎤

= κ κ− ⎢ ⎥ ∂ ⎢⎣ ⎥⎦

∂δ ⎡ ⎤

= κ κ− ⎢ ⎥ ∂ ⎢⎣ ⎥⎦

∑ ∑

∑ ∑

...

 F.7 

Setting the derivative to zero and splitting the summation: 

() ()( )

( )( )

max 2 2

10 1

max 2 2

01 1

0 () () ()

( )( ) ( )

i

i

Np Np N

i mvf mvf i mnf mvf mnf i i mvf mvf i mr f

in i

N Np Np

mnf i mvf mvf i mnf mnf i i mvf mvf i mr f

ni i

WE r p E r WE r p

p WE rE r W E r p

== =

== =

= κ κ− κ

κ κ= κ

∑∑ ∑

∑∑ ∑

 F.8 

Which can be written in matrix form for a fixed frequency: 

[ ] Fvn [ ] Pn = [Evr][Pr] [ ] Pn = [ ] Fvn

−1

[ ] Evr [ ] Pr F.9 

where: [ ]

max max

max max max max

22 2

100 00

1 1

2 22

1 00

1 1

() () ( )

( )( ) ( )

p p

p

p p

mN p

N N

m m i m m i mN mN i N

i i

vn

N N

mN i m m i mmN i N mN

i i

WE r W E rE r

F

WE rE r W E r

= =

= =

⎡ ⎤ ⎢ κ κκ ⎥ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ κκ κ ⎢ ⎥ ⎣ ⎦

∑ ∑

∑ ∑

...

# #

...

 F.10 

 

[ ] Evr =

W1

2

Em0 (κ m0r

1) ... WNp

2 Em0 (κ m0rNp )

# #

W1

2

EmNmax (κ mNmax r

1) ... WNp

2 EmNmax (κ mNmax rNp )

⎡

⎣ 

⎢ 

⎢ 

⎢ 

⎤ 

⎦

⎥ 

⎥ 

⎥ 

measured pressures: [ ] Pr modal amplitudes (solution): [Pn ]

note that the matrix Equation dimensions are: [Nmax] ~ [Nmax, Nmax][Nmax,Nmeas][Nmeasx1] 
NASA/TM—2005-213828 26 

G. Derivation of Power 

The basic definition of power is the integration of the acoustic flux across a radial plane of the duct: 

ˆ Ρ = I x

A

∫ dA; G.1 

For uniform axial flow it is straight-forward to show (* represents the complex conjugate.): 

I x = ( p

ρ 0

+Uu)(ρ 0u + ρU); p = c0

2

ρ; M = U /c0 → I x = (1+ M 2) pu* +

M

ρ 0c0

pp* + ρ 0c0Muu* G.2 

Inserting Eq. G.2 into Eq. G.1 results in: 

2 1 2 2

2

0 1

2 2* 2

2 22

0

(1 ) ˆ ;

(1 )

; ()

mns ax mns mns o m ns

mns mns mns

mns mns mns mn

mns

R P I rdrd G P

U

where G M M

sB and

sB c

U

π ∞ ∞∞

σ =−∞ = =−∞

π −σ = θ=

ρ

⎡ ⎤ = − + Λ +Λ ⎢ ⎥ ⎣ ⎦

γ Ω Λ = γ = −β κ Ω

+ γ

∫ ∫ ∑∑∑

 G.3 

The derivation of this is shown in detail in reference 3. It is recognized that some error is introduced since the above 

result relies on orthogonality to eliminate cross terms. Summation of the +/- terms yields: 

Gmns = M 2 − (1+ M 2)Λ mns

* + Λ mnsΛ mns*

mns

G−mn(−s) = M 2 − (1+ M 2)Λ−mn(−s)

* + Λ−mn(−s)

Λ−mn(−s)

*

Λ−mn(−s) = Λ mns

*

Gmn(−s) = M 2 − (1+ M 2)(Λ mns

* )

* + Λ mns

* (Λ mns* )

*

 G.4 

Gmns + G−mn(−s) = 2M 2 − (1+ M 2)(Λ mns

* + Λ mns) + 2Λ mnsΛ mns* G.5 

Substituting the expression for Λmns into Eq. G.5, after manipulation: 

 

Gmns = Re ∓M 2

β4 (sBΩ /U)kmns

sBΩ /c ( ) o ± Mkmns

2

⎧

⎨ ⎪ 

⎩ ⎪ 

⎫ 

⎬

⎪ 

⎭

⎪ G.6 

Traditionally, this has been reduced assuming either pure real or pure imaginary eigenvalues to the form: 

 

Gmns = ∓M 2

β4 (sBΩ /U)kmns

sBΩ /c ( ) o ± Mkmns

2 ; for a real eigenvalue Gmns = 0; for an imaginary eigenvalue. G.5 

 

A slightly different form will arise if the assumption that complex eigenvalues can occur: 
NASA/TM—2005-213828 27 

 

Gmns =

M 2

β4 Re ω

U (∓kmns) ⎧

⎨ ⎩ 

⎫

⎬ ⎭ 

ω

c0

± Mkmns

2 G.6 

The final version for the rotating rake data analysis is: 

 

Ρ = ∓

πR2 (1−σ2)

ρ 0c0

β4 RE

1− 1

ζ2

1± M D 1− 1

ζ2

2

⎧

⎨ 

⎪ 

⎪ 

⎩ 

⎪ 

⎪ 

⎫

⎬ 

⎪ 

⎪ 

⎭ 

⎪ 

⎪ 

P 2

 G.7 

F. Comment 

Derivation of the equations unique to the Rotating Rake system have been presented as used for several 

applications. The data reduction based on these equations has been validated by experience and comparison to other 

methodologies. In order to keep the solutions manageable a significant simplification assumption was made, that the 

orthogonality solutions for various integrals applies. For certain boundary conditions (namely flow in treated ducts) 

this is not rigorously correct, as it assumes product terms involving non-identical modes due not contribute. 

However, this assumption provides a solution that can be meaningfully analyzed. Future extensions of this analysis 

should include the solution complete with cross terms that were not included herein. 

G. References 

1. Hubbard, H. Harvey, editor, Aeroacoustics of Flight Vehicles: Theory and Practice: Volume 1, NASA Reference 

Publication 1258, Vol. 1, WRDC Technical Report 90-3052. 

2. Moore, C.J., “Measurement of Radial and Circumferential Modes in Annular and Circular Ducts,” Journal of 

Sound and Vibration,” Vol. 62, No. 2, 1979. 

3. Meyer, H.D., Envia, E. “Aeroacoustic Analysis of Turbofan Noise Generation,” NASA CR–4715, March 1996. 

4. Ambramowitz, M., and Stegun, I.A., Handbook of Mathematical Functions, National Bureau of Standards 

Applied Mathmatics Series, No. 55, U.S. Government Printing Office, Washington, DC, 1964. 

5. Cicon, D.E., Sofrin, T.G., and Mathews. D.E., “Investigation of a Continuously Traversing Microphone System 

for Mode Measurement” NASA CR–168040, PWA-5846-26, November 1982 
This publication is available from the NASA Center for AeroSpace Information, 301–621–0390.

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32

Rotating Rake Turbofan Duct Mode Measurement System

Daniel L. Sutliff

Turbofan; Duct modes; Aeroacoustics

Unclassified -Unlimited

Subject Categories: 07 and 71

Prepared for NOISE-CON 2005 sponsored by the Institute of Noise Control Engineering, Minneapolis, Minnesota,

October 17–19, 2005. Responsible person, Daniel L. Sutliff, e-mail: Daniel.L.Sutliff@nasa.gov, organization code RTA,

216–433–6290.

An experimental measurement system was developed and implemented by the NASA Glenn Research Center in the 1990s

to measure turbofan duct acoustic modes. The system is a continuously rotating radial microphone rake that is inserted

into the duct. This Rotating Rake provides a complete map of the acoustic duct modes present in a ducted fan and has

been used on a variety of test articles: from a low-speed, concept test rig, to a full-scale production turbofan engine. The

Rotating Rake has been critical in developing and evaluating a number of noise reduction concepts as well as providing

experimental databases for verification of several aero-acoustic codes. More detailed derivation of the unique Rotating

Rake equations are presented in the appendix.