Received 2• November 1970
7.7
Noise Reduction by Applyinl Modulation Principles
DONALD EWALD, ARMIN PAVLOVIC, AND JOHN G. BOLLINGER
The University of Wisconsin, Department of Mechanical Engineering, Madison, Wisconsin 53705
This paper presents techniques for determining nonperiodic spacing of events for the purpose of reshaping
noise-frequency spectra. The object of the application is to provide reduced noise levels and the redistribu-
tion of the frequencies at which there is noise energy so as to generate fewer perceptible sounds. Three arm-
lytical techniques for predicting the noise spectrum resulting from nonuniform spacing are presented.
Emphasis is focused on a semigraphical design technique using Bessel functions, which has proven to be of
great value in the actual selection of modulated event spacing. Fourier analysis of an impulse approximation
and a sinusoidal wave approximation are two other alternative techniques presented. The techniques out-
lined are applied to the problem of reshaping the noise spectrum of a 22-blade fan in a 5-hp induction motor.
A comparison of predicted results provides an evaluation of the alternative approaches.
INTRODUCTION
A problem of noise reduction in machinery generally
involves the reduction, shifting, or redistribution of
pure-tone effects which restfit from cyclic events in the
operation of the equipment.
K. D. Kryter and K. S. Pearsons x showed that band-
limited random noise is perceived to be less noisy than
sound composed of pure-tone superimposed on band-
limited random noise, if the over-all sound-pressure
levels (SPLs) are the same. They also showed that
sounds of larger tone-to-noise ratios are judged to be
noisier, for equal SPL.
It might be concluded that sound consisting of a pure
tone superimposed on random background noise can be
made to sound less noisy by dispersing the energy of
the tone over a number of discrete frequencies. Thus,
a system design should aim to minimize pure-tone
effects as well as to reduce the over-all broad-band
noise level.
This paper is primarily concerned with presenting
analytical methods that may be used to determine
event spacing to control the spectral distribution of
pure tones in the over-all noise spectrum generated by
the equipment components.
As an example of an approach to reduction of noise
generation, this paper will consider the air-cooling
system in a 5-hp induction motor. In the over-all
analysis, it is necessary to include a careful study of fan
characteristics and temperature distribution as well as
air-path geometry and over-all system cost. A boundary
condition might be that any design must not allow the
over-all winding temperature rise to exceed a specified
limit.
This problem may be approached by using the
following design procedures to minimize the air noise
in two broad areas:
(1) Optimize the air-flow path configuration to
maximize the cooling effect of the air.
(2) Select the number of blades to give maximum
air flow for a given fan diameter.
(3) Reduce the diameter of the fan as much as
possible to reduce the tip velocity.
(4) Space the fan blades unevenly to control pure-
tone effects.
To implement the control of pure-tone effects de-
scribed in procedure 4, several analytical approaches
will be demonstrated.
The centrifugal fan is a device which produces a
number of disturbances, or events, per operating cycle.
Its cycle is one-shaft revolution. It produces a pressure
disturbance for each blade which passes a fixed reference
point. The fundamental frequency for the disturbances
is the blade passing frequency, i.e., the product of
shaft rotational frequency and number of blades.
A straightforward approach to selecting the non-
uniform spacing is to choose a modulation principle.
For example, the events may be sinusoidally phase
modulated geometrically. The result in the noise spec-
trum is to generate side bands of a fundamental pure
tone, thereby reducing the magnitude of the funda-
The Journal of the Acoustical Society of America 1381
EWALD, PAVLOVIC, AND BOLLINGER
(a)
o.,?• S•NS•
FIG. 1. Sinusoidal modulation of a 22-blade fan. (a) Evenly
spaced fan. (b) Modulated fan.
mental. The resulting tones, if properly reduced in
amplitude and located in the proper bandwidth, may
be buried in the background noise.
Sinusoidal modulation of a fan-blade spacing involves
the modification of the position of equally spaced blades
about their nominal positions in a sinusoidal amplitude
pattern.
Modulated positions are described by:
Oi' = Oi+ ,xO sin (mO•), (1)
where 0• is the ith blade position in an evenly spaced
fan arrangement, O/ is the ith blade position after
rearranging the blades, /•0 is some maximum blade-
angle change (the modulation amplitude), and m is the
number of times the modulation cycle is repeated in one
revolution of the fan.
Figure 1 shows the blade spacing of a 22-blade fan
before and after it was modulated by angular spacing
of the form
0i'= 0iq- (0.175 rad)-sin (&). (2)
Figure l(a) shows the angular locations of evenly
spaced blades before modulation. Figure l(b) shows
how the blades are spaced after sinusoidal modulation
according to Eq. 2.
The noise resulting from sinusoidal modulation of
the fundamental blade passing tone may be expressed
by the classical sinusoidal phase-modulation equation
f(t) =A0 sin(2•rF0tq-Aq0 sin2•rvt), (3)
where A 0 is amplitude of the fundamental blade passing
tone; Fo=If, blade passing frequency; I, number of
blades; fs, shaft rotational frequency; v=mfs, the
modulation frequency; and/x• = IAO, phase-modulation
amplitude.
A• refers to an angle which goes from zero to 2•r
during each nominal blade spacing, and A0 is an angle
which goes from zero to 2a- for each revolution of the
shaft. This means that Aq• will go from zero to 2•' I
times for every time that A0 goes from zero to 2•-.
In the case of the fan in Fig. l(b), this modula-
tion cycle repeats itself once (m=l) for each cycle
of the fan.
By using the trigonometric relations
sinp cosq = «[-sin (p q- q) q- sin (p-- q) •,
sin (pq-q) = sinp cosqq-cosp sinq
and the relations between the Bessel and trigonometric
functions
cos(p sinq)=J0(p)q-2 • [-J2,•(p) cos(2nq)-],
sin(p sinq) =2 •'. [-J2,-,(p) sin(2n- 1)q•,
where J,•(p) is the Bessel function of the first kind,
order n, argument p, it can be shown that
f ( l) = A o{ Jo(A½) sin(2•rFot)
+ •'. J•(a•) sin[-2•r(Fo+nv)t3
q- Z (-1)•J•(aq0) sin[-2•r(Fo--nv)t•}. (4)
Equation 4 illustrates that the frequency spectrum
will consist of a center frequency at F0 with an ampli-
tude of A 0J0(Aq0) and a number of side bands at integer
multiples of • from the center frequency, with ampli-
tudes symmetric about the center frequency.
The values of J•(A•) may be found in many mathe-
matical handbooks and are shown graphically in
Fig. 2(a).
An example of how the frequency spectrum may be
determined for a given/•4• is shown in Fig. 2.
A trial value is chosen for/•. Then a vertical line is
drawn through the trial value of •p. The intersection
of this line with the J•(/•) curves indicates the relative
amplitudes of the resulting components at frequencies
Foq-nv. The resulting frequency spectrum, Fig. 2(b), is
given to the right of the graph. Note that absolute
values are plotted on the frequency spectrum shown
in Fig. 2 (b). The dashed lines in Fig. 2 (b) indicate the
normalized amplitude of the fundamental blade passing
frequency tone for a fan with evenly spaced blades.
The example in Fig. 2 is for the modulated blade
spacing shown in Fig. l(b). In this example, 1=22,
re=l, f•=30 Hz, v=30 I-I.z, and Fo=660 Hz.
1382 Volume 49 Number 5 (Part 1) 1971
NOISE REDUCTION BY MODULATION
Fzo. 2. (•) Bessel coefficients
versus maximum phase deviation
(A•). (b) Frequency spectrum for
the phase-modulated fan with
A•=3.85.
o i 2 3
The Bessel series is for a continuous phase-modulated
function, while the actual frequency spectrum of the
fan is produced by a number of more nearly discrete
events. The amplitudes in the frequency spectrum
obtained from the Bessel series will therefore differ
somewhat from those obtained from the fan. The Bessel
series, however, will more dosely approximate the actual
fan spectrum when the number of blades is large.
For fans which have a small number of blades, a more
realistic result may be estimated by Fourier analysis of
the pressure waveform produced by the fan blades.
I. FOURIER ANALYSIS
The Fourier series analysis assumes that the function
is periodic over one revolution of the fan or one cycle
of a group of events. The function is represented by an
infinite orthogonal trigonometric series such as
fit) =/(t)+ •. Bn sin(mot)+ •. Cn cos(noa), (5)
where
(,0 f2•r/•
B, =• Jo fit) sin(nt)dt (6)
and
fit) cos(•t)dt. (7)
Since the fan traveling at an•lar velocity •
w• be anMy•d in te•s of the an•ar positions of the
b•des, B• and C• •n be written in te•s of the
an•ar position of •e fan, 0, by using the substitutions
O=•t and dO=•t in Eqs. 6 and 7 to yield
](o) sh(no)ao
and
1
fj-' f(0) cos(n0)a0.
I b) • J4(Aq )) -r-- J6(A• )
-- J
II
ii
Ii
ø ,,
S 6 ?
(a) (b)
Since the phase angle of the harmonic components is
generally not of interest in noise analysis, only the
amplitude, A•= (BJ-]-C•)I, needs to be calculated.
A Fourier analysis of the noise generated for modified
blade spacing requires some knowledge of the wave-
form, which in turn depends on the position of the
blades. This waveform may be approximated by an
arbitrarily chosen function. Two such functions are
impulses occurring at blade tip locations and single-
period sinusolds whose periods are equivalent to the
blade'spacings.
II. IMPULSE
The easiest waveform to analyze would result from
the assumption that an impulse of strength K, Ka(O)
is produced by each blade. In this case, the function
f(O) becomes
f(O)=K$(O), for 0=0,, i=l, ..., I,
= 0, for all other O,
where 0i is the angular position of the ith blade, and I
is the number of blades in the fan.
Since fKlf(O)dO=K, substituting this function for
frO) into the equation for B• and C, yields
Kr
B• =-- •.. sin(nO/)
•l-i 1
and
Kr
C• =-- • cos(nO/).
;ri 1
Letting K=•r/I, these equations yield
Ar=A:•=Aar ..... 1.0
for an evenly spaced fan arrangement, where A •r is the
amplitude at frequency nfj.
The main problem with the assumption of an impulse
waveform in this case is evidenced above. All harmonics
The Journal 9f the Acoustical Society of America 1383
EWALD, PAVLOVIC, AND BOLLINGER
,o ' "-I!,;:i• •" .... • '1: •:•; ;:.1.•::.........._..[ .i......i ..i..:
20
100 200 300 400 500 600 700 800
Frequency (Hz)
•4o
•o
200 300 400 500 600 700 800
Frequen.cy (Hz)
(b)
FIO. 3. (a) Frequency spectrum of an induction motor with an
evenly spaced 22-blade fan. (b) Frequency spectrum of an induc-
tion motor with blade spacing of Fig. 1 (b).
are equal in amplitude. An examination of an actual
frequency spectrum for an evenly spaced fan arrange-
ment shows that this is not true. The higher harmonics
in reality have smaller amplitudes.
III. SINUSOIDAL APPROXIMATION
In the case of unequally spaced blades, one approach
to the analysis is to assume single-period sine terms
with period equal to the blade spacing. The equation
for this function is
r 2•r(0-00
f(O)=• sin.--, for 0•<0<0•+•,
•=• 0i+•--01
where 0z+•=0•q-br rad. Letting D•= br/(0•+]--0•) and
substituting into the equations for B, and C, yields
1 {sinE(O,--n)O--O•O,']
B•=--•
2•r i=• t D--•-n
sin[- ( D i+n )O - D •O i'] l
D-•-•n I ol
I
2 •r i= • t D--•n
cos[-(Di--n)0--DiOi-] I
D--•--n I •
The values of B, and C, are defined for all n•D•.
and
Cn =
The values of the functions at (Oi+•--Oi) = 2a'/n must be
evaluated by use of L'Hospital's rule. Taking the deriva-
tive of both the numerator and the denominator with
respect to D• and evaluating them at D• = n yields
lim { sinE(D'--n)Oi+•--DiOi]--sin[-(Di--n)O'--D'O[] }
Di•n
= (01+l--0i) cos(DiO0,
and
= (0,+•-0,) sin(D,O•).
Use of the sinusoidal waveform results in more
complicated calculations than the impulse waveform.
However, it produces more realistic results in that the
harmonic content will more closely resemble that of the
hardware. This is especially important when phase
modulation causes the side bands of successive har-
monics to overlap in the frequency spectrum.
IV. EXPERIMENTAL RESULTS
These methods for analyzing and predicting a fre-
quency spectrum were used to determine a fan arrange-
ment for a 5-hp induction motor having a shaft rota-
tional frequency of 30 Hz.
The original fan on the motor had 22 evenly spaced
blades and therefore the fundamental disturbance fre-
quency, F0, was 660 Hz. This number of blades was
retained for the modulated arrangement because the
660-Hz blade passing frequency was in an area of
relatively low background-noise level [see Fig. 3(a)-].
The band of low-level background noise extends about
150 Hz below the blade passing frequency to allow five
significant side-band pairs to be produced in a fan
which has one modulation cycle for each revolution of
the fan (m = 1).
The modulation amplitude, A0, of 0.175 rad was
chosen to ensure low amplitudes at all frequencies
while limiting the number of significant side-band pairs
to four. Note that Aq•=AOXI=0.175X22=3.85 rad,
indicated on Fig. 2(a). The half bandwidth in which
significant side-band amplitudes occur • is «BW=v,•4•
=mXf•XA•= 1X30X3.85•115 I-Iz.
V. COMPARISON
The frequency spectrum of the induction motor with
evenly spaced blades is shown in Fig. 3 (a). The spec-
trum for the modulated 22-blade fan is shown in Fig.
3 (b). The amplitudes which are evident above the back-
ground level in this spectrum are shown in Fig. 4(d).
Figs. 4(a), 4(b), and 4(c) show the[frequency spectra
predicted using the methods of analysis discussed
earlier in this paper.
All three methods produced good estimates of the
frequency spectrum which would be produced by the
modulated fan. This fact is explained by the large
number of blades and the small blade-angle changes
involved in the modulated fan. The large number of
blades made the system approximate a continuous
system. The small degree of modulation prevented the
higher harmonics in the Fourier analysis of the impulse
train from appearing in the low-frequency ranges. The
1384 Volume 49 Number 5 (Port 1) 1971
NOISE REDUCTION BY MODULATION
higher harmonics were ignored, since it was known that
their amplitudes were insignificant.
The Bessel series analysis is useful because it permits
the designer to tailor the frequency spectrum by direct
visual inspection of Fig. 2. This analysis is best for large
numbers of blades and small modulation amplitudes.
The Fourier analysis of an impulse for each blade
gives good results in the special cases of small modu-
lation amplitudes (a) where side bands from the higher
harmonics do not overlap with those of the funda-
mental, or (b) where the higher harmonics of the fan
are nearly equal in amplitude to the fundamental
frequency.
The Fourier analysis of a sine wave between blades
requires more calculations, but the predicted spectrum
is the most accurate when the fundamental frequency
is much greater in amplitude than the higher harmonics.
VL CONCLUSION
Nonuniform event spacing in multi-event cydic
proceases results in reduction of the amplitudes of
tonal noise components. In the illustrative example,
sinusoldM phase modulation resulted in a series of tones
with amplitudes at least 8 dB below that of the un-
modulated tone, while reasonable blade spacing was
maintained. The use of Fig. 2 permits rapid and con-
venient selection of event spacing. In cases where the
number of events per cycle is small (say 20 or less),
Fourier analysis may be done, using assumed wave-
forms, as a check on the predictions of Fig. 2.
In the application of the induction motor fan cited,
the 660-Hz blade passing tone was distinctly perceptible
in the original design. Following modification of the
fan by application of the principles of phase modulation
discussed in this paper, the 600-Hz blade passing tone
became audibly imperceptible. The one-fold modulation
(m= 1) chosen for this application did not result in an
undesirable mechanical unbalance or aerodynamic loss.
However, in other applications, this aspect should be
critically evaluated. It is interesting to note that the
mechanical balance, a/ter modulation, is perfect in all
cases where m is an integer greater than 1 and where
the number of blades, I, is an integer multiple of m.
The frequency spectrum in Fig. 3(b) may be further
(a)
%
05Aø t
o
400 600 800
FREQUENCY {H;)
(b)
0.SA o
0
FREQUENCY {Hz)
Iooo
(c)
o
400 600 8OO
(d)
0.SA o
0 200
400 600 •OO I000
FREQUENCY (Hz)
FIO. 4. Frequency spectra for the sinusoidally modulated
22-blade fan shown in Fig. 3. (a) Spectrum predicted using Fourier
analysis of an impulse for each blade. (b) Spectrum predicted
using Fourier analysis of a sinusoidal waveform between blades.
(c) Spectrum predicted using Bessel series coefficients. (d) Actual
spectrum shown in Fig. 3(b), amplitudes measured above back-
ground noise.
altered by other geometric modification of the fan,
such as the reduction of fan diameter. The result
effected is the further burying of the tonal noise in the
broad-band spectrum.
l K. D. Kryter and K. S. Pearsons, "Judged Noisiness of a
Band of Random Noise Containing an Audible Pure Tone,"
J. Acoust. Soc. Amer. 38, 106-112 (1965).
s S. Goldman, Frequem:y Analysis, Modulation and Noise
(Dover, New York, 1948).
The Journal of the Acoustical Society of America 1385