Aircraft
Noise
Theoretical
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PAGE
I_
QUALIT_
Addendum
NASA
Part
to
Technical
Memorandum
l
Aircraft
Noise Prediction
Theoretical
Manual
This addendum
document.
Please
83199
add
adds
the
three
enclosed
Program
new
sections
sections
2.4,
to the original
2.5,
and
2.6
Chapter
2 and replace the Contents
page with the enclosed
revised
Contents
page in your copy of NASA
Technical
Memorandum
83199, Part 1.
revised
11-93
in
NASA Technical
Part 1
Memorandum
Aircraft Noise Prediction
Theoretical
Manual
William
E. Zorumski
Langley
Research
Hampton,
83199
Program
Center
Virginia
National Aeronautics
and Space Administration
Scientific and Technical
Information Branch
1982
I
P
I.
Report No.
NASA
TH-83199,
2. Government
Part
l
L
Accession No.
3. Recipilmt%
CitJlo9
S. Report
4. Titleand Subtitle
Date
February
AIRCRAFT
NOISE
THEORETICAL
PREDICTION
No.
.
1982
PROGRAM
6. Pe_orming
MANUAL
Orgenizlti_Code
505-32-03-01
7. Author(s)
8. Performi_
Or_mzation
Report
No.
L-14805
William
E.
Zorumski
10. Work
9. Performing
NASA
Organization
Langley
Hampton,
Unit
Research
VA
1'i.
Center
Contract
Agency
National
Name and Addr_$
Aeronautics
Washington,
15
_pplementar¥
16.
Abstract
The
in
DC
NASA
this
and
Aircraft
Space
Noise
Administration
the
and
source
prediction
are
airframe
and
turbine
ply
with
aircraft
noise,
noise
17. Key Words(Suggested
Aircraft
Part
14. Sponsoring
Program
1
deals
propagation.
noise
for
combustion
Part
International
2
Civil
gives
Aviation
the
data
and
specific
noise,
also
(ANOPP)
with
These
parameters,
methods
noise.
the
Prediction
manual.
generation
detailed
and Period Covered
Memorandum
Agency
Code
fan
theoretical
methods
prediction
include
the
of
the
noise
noise,
single
modifications
to
Organization
are
which
aircraft
propagation
aircraft
data
flight
effects.
Part
sources.
and
the
These
dual
NASA
(ICAO)
given
affects
stream
methods
standard
2
gives
sources
jet
which
method
noise,
comfor
prediction.
by Auth,(s))
18. Distr{_tion
noise
Statement
-
Unclassified
Unlimited
prediction
noise
Turbomechanical
19. Security
of Report
Technical
20546
two-part
dynamics,
Jet
No'.
Notes
noise
Noise
or Grant
23665
13. Type
12. S_nsoring
No.
Name and Addreu
Oauif.(ofthisre_rt)
Unclassified
noise
Subject
_.SacurityClauif.(ofthispage)
21.
No. of Pages
Unclassified
193
22.
Category
_ice
A09
ForsalebytheNationalTechnicalInfo(ma/i0nService,Springfield,
Virginia2216!
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NASA-Linglex,
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Contents
Part
1
1. Introduction ...............................
I-I
2. AircraftFlight Dynamics
2.1 Atmospheric Module
2.2 Geometry
Module
2.3 Flight Dynamics
.........................
2.I-I
..........................
Module
2.2-I
.......................
2.3-I
2.4 Jet Takeoff (JTO) Module
......................
John Rawls, Jr.,Lockheed Engineering & Sciences Company
2.4-I
2.5 Jet Landing (JLD) Module
......................
Mark Wilson, Lockheed Engineering & Sciences Company
2.5-I
2.6 Steady Flyover (SFO) Module
................
John Rawls, Jr.,Lockheed Engineering & Sciences Company
.....
2.6-I
3. Propagation Effects
3.1 Atmospheric Absorption Module
3.2 Ground
...................
Reflectionand Attenuation Module
3.1-I
...............
3.2-I
4. Source Noise Parameters
4.1 Fan Noise Parameters Module
.....................
4.2 Core Noise Parameters Module
.....................
4.2-1
...................
4.3-I
4.3 Turbine Noise Parameters Module
4.4 Jet Noise Parameters'Module
4.I-I
.....................
4.5 Airframe Noise Parameters Module
4.4-I
..................
4.5-I
5. Propagation
5.1 Propagation Module
.........................
5.2 General Suppression Module
6. Received
5.1-1
.....................
5.2-1
Noise
6.1
Noise
Levels
6.2
Effective
Module
Noise
.........................
Module
6.1-1
........................
6.2-1
7. Utilities
7.1
Thermodynamic
Utilities
.......................
iii
7.1-1
rev.
11-93
Part
8.
2*
Noise
Sourc_
8.1
Fan
8.2
Combustion
8.3
Turbine
8.4
Singlc
8.5
Circular
8.6
Stone
8.7
Double
8.8
Airfrmm,
8.9
Smith
9.
Prediction
9.1
ICAO
Noig,'
Module
...........
Noi,,_
Noise
Jet
Module
Module
Stream
Circular
Shock
.Xlodul(,
Stream
(!oammlar
Nois(,, Moduk,
11-93
Noise
.
Mo(hd(,
_; 3-
.
Turtfim,
.................
.....
_..I" ............
......................
.hq
_.5_._i-
N()is_, Mo(hd¢,
..............
.......................
Noise
8.18.2-
,h,t N_)is_, Modul(
_.7-1
S._-1
.,Moduh,
.
...............
_.!)-I
Procedures
Reference
Prediction
"Chapters 8 attd 9 are published
rev.
......................
.......................
Ceil
.let Noist'
mM Bush(,ll
- ...............
Proc(,dures
under separate
(1978)
cover.
iv
.............
9.1-1
1.
INTRODUCTION
The
to
purpose
predict
of
noise
characteristics,
approach
of
i.
an
The
emit
tribution
noise
A
a
The
number
of
called
and
by
the
predicts
tion.
Level
time.
In
effects
data.
which
diction
and
2
same
prepare
contains
used
ure
Three
3.
The
Atmospheric
function
of
used
as
a
trol
function,
control
the
result
find
the
of
an
best
The
and
and
may
Source
be
as
a
function
stage
of
computation.
pared
by
the
Atmospheric
which
III
Data
modules.
in
standard
for
i-i
many
source
take
of
state
the
time
to
are
then
data
atmospheric
Module.
is
A
con-
second
These
or
may
table
to
to
for
the
Dynamics
state
the
effects
is
cri-
power
Flight
engine
passed
be
analyzed
number,
the
absorption
This
two-degree-
parameters
give
a
performance
Mach
from
table
the
module.
are
aircraft
data
variable
tables
flights
given
prepare
modules
These
Absorption
a
this
of
as
a
setting
procedure
where
of
variables
dependence.
in
fig-
processing
represents
used
of
tables
output
power
time
also
principal
properties
Module
pre-
noise.
diagram.
related
the
noise
preparing
the
the
diagram
this
and
and
definite
The
IV
the
describes
functional
for
wherein
functions
1
predicted
Dynamics
engine
in
source
modules
Modules
detail
more
the
shown
for
Level
with
Part
prediction
from
measures
for
the
and
the
them.
process
in
loca-
noise
required
to
I
generates
III.
are
is
III,
but
establishes
These
time.
and
Level
procedure
the
of
II
source
are
a
Parameters
predicted
Levels
process
angle-of-attack
ables
the
are
humidity,
taken
modules.
interpolate
to
attack,
optimization
Level
addition
Flight
of
operational
prediction
setting,
angle
methods.
2.
processed
in
has
with
figure
is
observer
analysis
function
functions
the
The
dynamic
in
which
the
the
density,
altitude.
flight
prediction
prediction
Module
pressure,
schematic
data
on
depicted
by
categories,
depends
used
stage
interpolated
prediction
four
which
computation
preparation
The
of-freedom
noise
of
direct
surface.
level
which
are
vicinity
the
observer
are
noise
III
stages
atmosphere:
terion.
source
the
general
the
of
dis-
source
the
with
data
which
Level
later
modules.
control
the
in
input
are
the
modules
to
from
ground
the
spectral
This
on
Level
deal
by
on
depends
noise
as
and
into
amount
the
sources
which
In
the
this
ANOPP
in
presence
noise
effects
time.
to
documents
the
modules
noise
information
present
modules
which
and
as
for
in
the
in
of
frequency
observer
spectral
The
measure
local
The
the
time.
signal
the
depicted
by
approximation
III,
subdivided
are
in
attenuated,
noise
is
aircraft
depicted
noise
on
by
available
which
a
being
divided
defined
predicts
Level
the
Part
of
II
predicts
are
as
directional,
the
the
atmosphere.
path
depend
(ANOPP)
of
basis,
operation,
may
reflected
are
are
effective
of
be
ray
the
flight
power,
which
receives
levels,
degree
of
Program
effects
and
this
defined
approaches
levels
an
During
atmosphere,
a
the
fundamental
arbitrary
the
approaches
functional
functional
a
observer
from
These
on
an
all
through
signal
The
Module
with
observer.
problem.
may
radiation
Prediction
for
operations,
been
follows
ground.
propagates
plus
its
the
No_se
accounting
has
characteristics,
the
ray
on
Aircrafb
engines,
problem
aircraft
observer
aircraft
of
this
NASA
aircraft,
its
to
figure
the
from
used
vari-
second
are
in
prethe
Propagation Module in the second stage of computation. The Geometry Module takes data on the aircraft position and orientation from the Flight
DynamicsModule and evaluates the vectors from the source to each ground
observer as functions of time. Each noise source will be given in a specified axis system such as engine 1 axes or aircraft wind axes so that each
observer vector will be expressed in several source coordinate
systems
at
the
same
time.
Noise
attached
wind
predictions
to
axis
the
system
directivity
in
in
figure
tance
r
which
are
of
by
this
data,
using
angle
spaced.
wind
the
positions.
to
effects
by
Absorption
noise
the
system.
the
The
engine
components,
engine
observer
axis.
process.
positions.
virtual
be
made
before
of
Output
on
functions
The
of
output
frequency,
time,
dependence
as
ceived
this
A-level
Noise
An
prediction
and
Level,
module.
evaluate
Time
variables
ANOPP
of
modules
these
Level
four
by
a
coordinate
system,
reduce
these
by
to
III
as
spectra
into
reduce
measure
made
Effective
noise
i/3-octave-band
Noise
further
are
of
in
fan,
at
virtual
at
the
noise,
is
Noise.
band
1-2
is
set
noise
may
saves
exces-
involves
are
observer.
summations
Module
noise
of
also
the
levels
data.
Per-
computed
Noise
are
and
removes
by
Module
characterized
centers
same
Module.
and
amount
Effective
module
observer
the
predictions
weighted
the
combustor,
the
taking
the
The
through
illustrates
for
Levels
Perceived
prediction
noise.
data
The
source
process,
source
to
virtual
5
of
prediction
axis
symmetry,
summation
These
converting
integrals
of
wing
used
noise
Propagation
observer.
nonlinear
such
stage
the
the
is
of
expressed
This
in
the
place.
at
summation
observers.
variables.
D-level
modules
due
noise
wind
processed
predictions
direct
3
Figure
is
to
the
axis
summed
system,
make
procedures
third
added
figure
are
and
the
Other
3 contain
they
actual
observers
in
common
these
the
from
takes
observers.
that
the
at
frequency
such
so
the
a
prediction
noise
processing
to
noise
figure
conveniently
axis
the
be
in
and
absorption
Module.
summation
with
actual
predicted
over
in
at
may
the
After
at
air
expressed
before
called
to
Module
before
by
table
Noise
depicted
noise,
given
t
axis
since
observers
adds
noise
distimes
interpolates
prepared
actual
shown
the
8.
predictions
then
Aircraft
process
to
computational
integrals
give
are
positions
processing,
operations
Module
engine
are
propagation
the
unattenuated
systems
component
observer
use
the
noise
Each
of
sive
jet
vectors
depicted
taken
Within
and
observer
Noise
sources
virtual
as
has
the
dipole
of
modules
Module
in
fixed
sequence
coordinate
few
conveniently
fan
these
and
a
observer
and
Noise
Module
turbine,
jet
in
for
a
which
Propagation
the
are
a
prediction
noise
to
systems
as
locations
Propagation
this
axis
such
the
at
a
axis
at
are
made
has
dipole
system,
single
Airframe
the
axis
a previously
looping
to
sytem
the
from
they
the
noise
of
noise
the
which
are
observers
only
Propagation
the
if
Otherwise,
propagate
the
in
being
this
use
true
Module
source
Wing
virtual
wind
free-field
component
components
at
varying
give
in
the
interpolating
to
noise
wing
Z-axis
systems
predictions
Module.
the
by
system,
The
Atmospheric
wing
for
axis
rapidly
Module,
source
minimum
coordinate
noise
for
8
located
predictions
the
the
made
Using
for
are
Geometry
the
with
prescribed
observer
ground
Noise
and
the
in
Aircraft
system
allows
several
airframe
are
widely
times
the
in
The
Predictions
observers
making
made
4.
symmetry,
virtual
are
aircraft.
based
by
on
to
the
observer frequencies and are independent of time. All other inputs to the
prediction modules are time dependent. The vectors from the source to the
observer are naturally dependent on time and observer so that the output
from a source module is a function of frequency, time, and observer, in
that order.
The prediction of i/3-octave-band noise is a serious limitation which
should not be passed over quickly.
Someof the most important noise
sources are actually tones, for example, from the fan rotor of a bypasstype engine. In the prediction modules, these tones are assigned to a
i/3-octave band and subsequently treated as broadband noise. This will
cause errors in the prediction of atmospheric attenuation, ground effects,
and even noise levels.
Nevertheless, the added complexity of carrying a
separate procedure for tones suggests that this is not an appropriate task
for ANOPPLevel III and this type of analysis has been relegated to the
Level IV manual.
Input and output of the functional modules may be a combination of
dimensional and dimensionless variables.
Two systems of dimensions are
used, the preferred SI system and alternate U.S. Customary system. One
principal distinction must be madeabout the use of the U.S. Customary
system.
The
the
popular
if
less
the
unit
mass
of
pound-mass
unit
is
of
expressed
in
used
within
at constant
the program.
pressure,
are
calories
BTU.
and
or
the
Within
accounts
for
have
been
dimensional
not
scaling,
the
different
in
are
often
(or
some
of
for
to
tabulated
power
dimensionless
to
a
setting
data
as
dimensionless
two-dimensional
use
if
function
arrays,
a
empirical
of
units.
=
by
ma
energy
only
are
the
specific
units
rather
of
heat
than
equations
altitude,
great
they
only
in
and
converted
many
this
would
it
increase
engine
data
engine
necessary
to
speed
tabulate
setting.
three-dimensional
computation
to
cases,
and
power
and
dependent
dimensional
is
the
savings
be
In
velocity,
it
using
base,
the
may
since
number
reduces
by
variables,
data
desired.
units
Mach
made
When
example,
whereas
of
thus
are
an
is
For
of
function
groups
F
heat
extraneous
form,
this
data.
variable),
a
of
dimensional
and
as
computations
dimensionless
convert
replaced
balancing
eliminates
systems
output
computation
be
that
such
as
Cp,
in mechanical
dimensional
essential
permits
of
is
groups.
approach
generated
desirable
amount
of
the
must
this
involving
variables,
expressed
the
and
for
Variables
This
variables
the
use
use
These
always
permitted
reason
slugs.
facilitates
module
the
not
The
dimensionless
groups.
facilitates
is
of
each
dimensionless
data
This
formation
is
slugs.
The
arrays
and
storage
to
of
data.
The
diagram
sections
in
stating
figure
its
method
or
technical
have
in
In
purpose
and
like
it
sections.
dimensions
parentheses.
is
All
are
its
the
is
naturally
results.
necessary
to
variables
shown
with
Variables
SI
input
methods
1-3
have
an
the
from
are
units
which
described
been
lines
module
followed
converted
the
appropriate
The
module
section
since
to
naturally
the
briefly
and
the
developed.
hand,
of
by
data,
to
independent
other
one
not
first
output
given
are
as
On
refer
the
is
and
are
written
which
along
a module
References
which
module,
organized
section,
function,
from
the
duplication
manual
each
process.
literature
interface,
put
this
3.
computation
description,
some
of
another
in
dimensionless
by
U.S.
to
so
that
modules
Customary
dimensionless
must
the
out-
will
units
groups through division
ventional
dimensional
reference
would
variable.
be
As
described
in
the
lists
example,
absolute
symbols
are
the
by
re
and
the
dimensionless
list,
temperature,
Ta
absolute
T a
Symbol
an
followed
group
group
for
conthe
(T/T a)
as
T*
and,
by a reference variable are given with their
description
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each
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re
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1-11
2. AIRCRAFT
FLIGHT
DYNAMICS
2.1
ATMOSPHERIC
MODULE
INTRODUCTION
Selection
aircraft
of
noise.
craft,
the
gation
of
noise
this
during
only
representative
Four
that
level
on
=
that
second
of
The
altitude.
the
are
used
to
gravity
is
primary
used
of
ambient
Dimensionless
equations
and
are
to
noise
in
the
sea
of
used
facilitate
working
level
from
of
water
the
value
(scale:
altitude
is
the maximum
approximation
only.
All
references
is
at
sea
aircraft
effect
altitude
to
in
the
of
Consequently,
assumptions
The
fourth
Module
are
is
approximation
geopotential
parameters
parameters
air-
propa-
signifi-
model
assigned
air.
taken
Atmospheric
atmospheric
the
the
most
analysis
neglect
these
km.
are
is
first
is
the
that
functions
of
and
this
The
standard
of
i0
predicting
km.
of
is
model
the
Since
and
made
the
For
each
10 -3 at
this
engines,
landing),
viscosity
as
its
model.
to
in
performance
constant
is
and
step
the
I0
this
is
approximation
in
purpose
of
and
usually
taken
third
first
and
below
in
is
properties
constants
arrays
atmosphere
altitude.
of the
order
atmospheric
aircraft
approximation
air
the
affect
atmosphere.
density,
The
geometric
error
is
and
sionless
for
The
is
(take-off
assumption
pressure,
weight
the
the
the
due
This
the
equal
to
relative
is
of
12.0000).
tions
through
approximations
molecular
C 12
by
operations
acceleration
performance.
model
properties
generated
terminal
value.
vapor
atmospheric
noise
cant
is
an
Atmospheric
to
equa-
1
generate
and
equal
dimen-
increments
provide
simplified
different
systems
of
forms
of
units.
SYMBOLS
C
speed
g
acceleration
H
altitude
AH
output
h
absolute
H20
of
sound,
m/s
due
(ft/s)
to
gravity,
(geometric
altitude
assumed
molecules
percent
pressure
and
k
coefficient
of
M
molecular
n
number
P
atmospheric
of
equal
increment,
humidity,
1.178
m/s 2
m
percent
to
total
at
70
(ft/s
to
2)
geopotential),
m
number
percent
fraction
of
(ratio
molecules
relative
in
humidity
of
number
mixture,
and
standard
temperature)
thermal
weight
of
output
values
conductivity,
air
pressure,
2.1-1
Pa
(ft)
(ft)
mole
(ib/ft
2)
W/m-K
(BTU/ft-s-°R)
2.
of
universal
gas
constant,
rh
relative
S
Sutherland's
constant,
T
temperature,
K
y
=
Y
ratio
humidity,
Mg r(H
-
of
density,
P
kg/m
(slug-ft2/°R-s
2)
110.4
K
(198.72°R)
(OR)
r
specific
coefficient
2
percent
HI)/RT
of
kg-m2/K-s
heats
viscosity,
3
kg/m-s
(slugs/ft
(ib/ft-s)
3)
Subscripts:
i
input
array
j
output
r
standard
1
ground
values
array
values
sea
level
level
value
value
Superscripts:
*
dimensionless
value
average
INPUT
Input
desired
and
for
altitude
pressure;
altitude.
and
model.
spheric
model
input
the
generated
the
table
of
one
default
at
of
generated
ground
equal
values
atmospheric
of
H1
relative
more
not
one
level
altitude
the
to
the
be
must
increments.
for
to
set
Table
input
I
ground
level
AH
output
altitude
Pl
pressure
at
altitude,
m
increment,
ground
preset
at
supplied.
the
level,
Constants
(ft)
m
Pa
(ft)
(Ib/ft
2.1-2
in
in
2)
the
altitude
functions
equal
of
consisting
of
a
atmo-
constant
a hydrostatic
altitude
parameters.
Input
H1
results
AH,
level
as
level
results
gives
of
ground
humidities
supplied
be
the
ground
humidity
than
have
at
consists
PI'
and
set
according
do
model
and
temperatures
atmospheric
and
Input
altitudes
the
increment;
temperature,
spheric
ever,
a
Input
altitude,
The
constructing
output
increment
increments;
Output
recommended
altitudes
ranges
atmoAH.
howare
and
Atmosphere Input Table
H
altitude,
T(X)
temperature, K (OR)
rh (H)
relative
m (ft)
humidity, percent
OUTPUT
The output is a table of dimensionless pressures, densities, temperatures, humidities, sound speeds, average sound speeds, coefficients of
viscosity and thermal conductivity, and characteristic
acoustic impedances as a function of altitude.
Atmospheric
Y
altitude,
Mgr(H
P (Y)
pressure,
re
p* (y)
density,
T (y)
temperature,
h(y)
humidity,
c (y)
sound
re
-
Properties
HI)/RT
Table
r
Pr
Pr
re
Tr
percent
speed,
mole
re
fraction
cr
--*
c
k
p
(y)
average
sound
speed,
re
cr
(Y)
coefficient
of
viscosity,
(y)
coefficient
of
thermal
conductivity,
acoustic
impedance,
c
(y)
characteristic
re
_r
re
re
k
r
PrCr
METHOD
The
be
generation
established
atmosphere.
These
tions
the
within
equations
in
the
Perfect
of
for
formed
p
atmospheric
primary
primary
constants
atmospheric
with
computational
gas
an
certain
these
sequence
model
primary
model
requires
constants
are
module
that
relevant
given
in
are
based
constants.
to
table
The
on
numerical
the
II.
values
Earth's
All
computa-
dimensionless
basic
equations
used
are:
law:
=
ORT/M
(la)
2.1-3
which in dimensionless form becomes
(ib)
p* = O'T*
Equation for speed of sound:
(2a)
c2 = yRT/M
which in dimensionless form becomes
(2b)
(c*) 2 = T*
Hydrostatic equation:
(3a)
dp/p = -Msr dH/RT
which in dimensionless form becomes
(3b)
dp*/p* = -dy/T*
where
(4)
y = Mgr(H - HI)/RTr
Figure
1
is
Using
basic
tude
a
graphic
the
atmospheric
equations,
vector
representation
the
yj
is
Ay
=
Mg r
yj
=
is
the
of
variables
atmospheric
defined
AH/RT
atmospheric
supplied
model
for
the
is
as
input,
computed
incremental
coordinates.
along
as
altitude
with
follows.
changes
An
AH
the
altiby
(5)
r
with
where
n
The
(j -
with
values
respect
number
for
temperature
integration
2,
....
(6)
n)
altitudes.
is
computed
by
(7)
r
temperature
Tj
pressures
is
may
output
I,
are
then
found
by
linear
interpolation
y.
Dimensionless
If
of
temperature
T(Hi)/T
to
(j =
Ay
dimensionless
Ti =
Output
i)
be
assumed
carried
are
to
vary
out
computed
by
linearly
exactly
integrating
between
to
yield
2.1-4
equation
y.
]
the
and
recurrence
y
(3b).
j+l_,
the
formula:
(Tj
_ =___(_/___1
-_IC_-_-_)
_
Tj_I;
j =
2,
3,
...,
(8a)
n)
or
_ylT
_
T_=Tjl,
(8b)
pj
= Pj_l
Pl
=
e
j =
2,
3,
...,
n
where
Once
the
other
the
Pl/Pr
(8c)
dimensionless
required
temperature
dimensionless
and
atmospheric
pressure
vectors
vectors
are
are
prepared,
computed
as
follows:
Density:
Sound
speed:
cj
, (_)_,'_
(i0)
=
Coefficient
of
viscosity:
(ii)
3
Coefficient
T. +
3
of
0.38313
thermal
,
k. =
conductivity:
i- 77385(T3
)3/2
(12)
3
-0.0416/T*
T*
3
Characteristic
+
0.8516
x
i0
J
impedance:
(13)
Q
The
constant
constant.
cj
= pj
0.38313
T
is
the
ratio
2.1-5
of
S/Tr,
where
S
is
Sutherland's
Average sound speed is defined by
c(y) =
Y
f0 y
where
the
denominator
sion.
If
Yj+I'
the
the
(14)
[c(Y)_-i
dY
of
equation
temperature
is
again
denominator
of
equation
following
recurrence
I
=
I
j
(14)
assumed
(14)
is
the
time
for
to
vary
linearly
can
be
integrated
vertical
transmis-
between
exactly
yj
and
to
yield
formula:
2 (Ay)
+
j-i
,
1/2
(15a)
,
1/2
where
T
with
the
following
Yl
Equation
=
(14)
The
input
terms
total
in
0
as
number
terms
of
(15b)
Ij
I1
then
(yj)
computational
=
condition:
is
c
lute
dy
=
=
computed
0
in
dimensionless
for
altitude
yj
is
expressed
it
is
mole
ratio
molecules
temperature,
more
(in
in
as
relative
convenient
percent)
of
a mixture.
pressure,
The
and
humidity
to
H20
percent.
humidity
molecules
absolute
relative
in
express
relative
humidity
humidity
For
in
is
absoto
rhj
Once
is
the
dimensional
computed
atmospheric
values
for
by
the
defined
by
hj = (rhj/pj)10
where
by
(16)
humidity
the
form
yj/Ij
purposes,
of
(15c)
(17)
linearly
values
printed
interpolating
are
output
computed
are
with
in
computed
respect
dimensionless
using
the
to
y.
form,
following
equations:
(18)
Hj
=
(RTr/Mgr)y
Pj
= prPj
j +
H1
(19)
2.1-6
Tj = TrT j
(20)
Pj = pr p
(21)
(22)
cj = CrCj
_j = CrCj
-*
(23)
Pj = _r_j
(24)
kj = krk j
(25)
pCj = PrCrPCj
(26)
REFERENCES
1. U.S. Standard Atmosphere, 1976. NOAA,NASA,and U.S. Air Force,
Oct. 1976.
2. Sutherland, Louis C.: Review of Experimental Data in Support of a
Proposed NewMethod for Computing Atmospheric Absorption Losses.
DOT-TST-75-87,U.S. Dep. Transp., May 1975.
2.1-7
TABLEI.-
RANGE
ANDDEFAULT
VALUESOF INPUTPARAMETERS
Input
parameter
Minimum
AH, m .....
PI'
T,
1
90
N/m2
H,H 1 , m
K
rh,
I01
II.-
Constant
"
°
"
"
"
"
"
S
.
.
.
.
.
o
.
.
.
.
.
o
.
.
Y
,
.
.
Pr
.......
Pr
.......
Tr
.......
.
.
.
PRIMARY
Units
9.806
gr
r
000
300
i00
CONSTANTS
U.S.
65
Customary
m/s 2
Units
32.1741
ft/s
28.9644
8314.32
2
49
718.96
ft2/°R-s
1.40
.
1.013
25
×
105
2
1.40
Pa
kg/m
1.225
2
28.9644
m2/K-s
2116.22
3
288.15
340.294
C
i0
70
STORED
SI
000
288.15
0
TABLE
ii0
0
200
.....
i00
325
-300
....
Maximum
i00
000
.....
%
Default
0.002
377
K
ib/ft
slug/ft
518.67
m/s
2
1116.45
3
oR
ft/s
.......
_r
.......
kr
.......
RTr/Mg
r
1.7894
....
x
10-5
kg/m-s
6.0530
×
l0 -6
8.434
515
6
x
3.737
4.0674
W/m-K
l03
m
2.1-8
×
10 -7
x
10 -6
2.767
210
slug/ft-s
BTU/ft-s-OR
65
x
104
ft
Altitude
Y=Yi
(H i )
I
Pi'
Ti'
Pi'
ci'
Pi'
ki'
hi
All
Altitude
Y=Yi-1
(Hi. I )
Pi-1'
Ti-l'
Pi-1'
Ci-l'
ui-1'
ki-1'
hi-1
Y
I
H
Ground
Level
(H=H I )
//I///
///
HI
Standard
Sea Level
(H-O)
Pr'
Figure
i.-
Tr'
Pr'
Cr'
Ur'
Dimensional
and dimensionless
for the atmospheric
model.
2.1-9
kr
altitudes
y--O
2.2
GEOMETRY
MODULE
INTRODUCTION
The
calculation
temporal
definition
tionship
between
defined
for
given,
Earth-fixed
tation
in
of
from
Euler
in
equations,
so
that
dimensional
tion
of
flight
Observer
fixed
from
express
the
aircraft.
all
times
It
release
is
are
not
desirable
during
the
will
observer
have
more
than
cases.
20
dB
below
Consequently,
the
times
in advance.
The
for
noise
observer.
Since
reception
noise
time,
variable
and
a
position
and
orientation
observer.
small
of
to
successive
This
module
directivity
observer
This
table
propagation
as
azimuthal
a
is used
effects.
function
by
the
six
of
degrees
case
of
or
are
not
of
Propagation
to
at
observers
at
noise
of
of
three-
by
addi-
each
observer
point
than
observer
an
which
in
results
are
evenly
are
determined.
emission
most
when
eliminated
to
reach
spaced
an
these
the
values
of
independent
The
times
selected
noise
at
the
interest
time
as
with
brake
at
finite
are
of
times
negligible
in
aircraft.
for
computation
at
the
the
to
the
in
treated
find
and
moves
little
a
Earth-
source
be
points
at
aircraft
for
each
sufficiently
characteristics
large.
the
reception
2.2-1
are
changes
directivity
of
is
times
time
table
an
is
to
desired
found
are
in
Module
Noise
takes
times
the
program
different
to
aircraft
that
a
at
savings
then
points
produces
in
described
two-degree-of-
path
system
close
certain
reception
ensure
orien-
aerodynamics
for
the
the
track.
emission
are
to
value
reception
flight
time
angle,
an
the
predictions
noise
maximum
is
to
coordinate
flight
predictions
Intermediate
increments
between
each
set
an
system
are
general
Geometry
observer
of
the
noise
the
valid
more
aircraft
noise
An
from
be
because
significant
the
emitted
a
time
make
the
a
which
on
uses
is
in
aircraft
three-dimensional
the
the
of
of
down
texts
the
input
of
on
values
miles
as
task
terms
to
will
a
The
coordinate
Module
to
(FLI),
conventions
standard
be
position
Module
time.
rela-
must
aircraft
Earth-fixed
Dynamics
given
flight.
large
several
other
of
The
and
geometric
module.
source
in
observers
the
system
Module
The
functions
and
of
the
spatial
The
Dynamics
function
to
input
are
Flight
extended
by
positions
vectors
vectors
in
be
system.
these
These
the
paths
noise
a
complete
paths.
flight.
coordinate
Geometry
may
flight
the
the
Flight
general
coordinate
vectors
is
the
aircraft
as
and
the
ANOPP
a more
from
1
a
components
respect
Although
freedom
or
These
reference
freedom
the
system
angles.
airplane.
of
with
requires
propagation
source
input
given
noise
noise
noise
duration
also
detail
the
aircraft
the
coordinate
is
terms
the
the
either
of
of
emission
angle,
time
Module
time,
and
distance,
elevation
and
source
(PRO)
for
polar
angle
coordinate
the
computation
to
system.
of
SYMBOLS
%
reference
c
speed
AdB
area
of
of
aircraft,
sound,
m/s
limiting
noise
level,
gr
standard
acceleration
h
observer
height,
i,j,k
base
m 2
(ft 2)
from
peak
(ft/s)
down
due
m
to
level
gravity,
m/s
2
(ft)
vectors
constant
M
molecular
weight
initial
mass
of
of
air
aircraft
m o
(nl, n2,n
3)
direction
cosines
O
observer
index
R
universal
gas
r
distance,
m
S
source
T
transformation
Tr
standard
t
time,
At
reception
(x,y,z)
position
Y
altitude,
re
Y
elevation
angle,
8
polar
Ae
polar
P
density,
¢
azimuthal
Euler
constant,
m2/K-s
2
(ft2/°R-s
(ft)
coordinate
system
name
matrix
sea
level
temperature,
K
s
time
increment,
coordinates,
s
m
(ft)
directivity
angle,
deg
directivity
angle
limit,
kg/m
RTr/Mg
3
deg
(slugs/ft
directivity
angles,
r
deg
3)
angle,
deg
deg
2.2-2
(OR)
2)
(ft/s
2)
Subscripts:
a
aircraft
or Earth-fixed
flight
i
time index
m
local minimum
min
global minimum
o
observer or Earth-fixed
r
standard sea level
s
source coordinate system
w
wind axis
axes
observer axes
Superscripts:
*
dimensionless value
-
average
INPUT
Input to the Geometry Module consists of the aircraft body axis position and orientation vector.
It is understood that these data are given
with respect to the Earth-fixed flight axis system of figure i. The
orientation vector is taken to be the set of three Euler angles described
in reference i. The average speed of sound as a function of altitude is
obtained from the atmospheric properties table. There may be several
noise source axis systems located within the aircraft body system. The
orientation of these systems
are given
as input
with
respect
to the body
axis
system.
origin.
shown
values
All
source
Observer
in
figure
for
the
coordinate
positions
i.
Table
input
are
systems
given
I gives
the
reference
limiting
k
constant
mo
reference
t1
initial
tn
final
are
the
assumed
Earth-fixed
recommended
ranges
parameters.
Input
AdB
in
area
of
noise
level,
mass
time,
time,
aircraft,
of
down
aircraft,
s
s
2.2-3
Constants
m 2
from
kg
(ft 2)
peak
(slugs)
level
to
have
observer
and
the
the
same
system
default
At
reception
A0
maximum
time
increment,
polar
s
directivity
angle
Flight
t
flight
time,
limit,
Dynamics
deg
Table
s
atl]
a(t)
aircraft
body
coordinates,
aircraft
body
Euler
aircraft
wind
axis
m
(ft)
I
a(t)J
e (t)l
a(t)
angles,
deg
I
a(tU
Qw(t)
Euler
angles,
deg
I
w(t)J
Observer
o
observer
index
observer
coordinates,
m
Table
(ft)
o (°)1
o(O)J
Source
s
Coordinate
source
system
name
source
Euler
angles,
relative
System
to
body
s(S) I
s(S)J
2.2-4
Table
axis,
deg
Atmospheric
Y
c
altitude,
(y)
re
average
RTr/Mg
speed
of
Properties
Table
r
sound,
re
cr
OUTPUT
Output
the
from
observer.
the
Geometry
These
coordinate
system.
the
is
source
are
In
Module
as
addition,
provided
is
expressed
for
the
use
in
reception
O
observer
S
source
t e (to,O,
the
angle
Propagation
source
in
from
the
the
to
source
observer
to
Module.
Table
s
coordinate
polar
system
time,
distance,
s)
from
components
index
emission
S)
r(to,O,S)
@ (to,O,
time,
vectors
elevation
the
Geometry
to
the
spherical
m
name
s
(ft)
directivity
angle,
_(to,O,S)
azimuthal
directivity
Y(to,O,S)
elevation
angle,
deg
angle,
deg
deg
METHOD
Each
observer
noise
dominates
(or segments)
observer.
source
for
magnitude,
greater
an
This
estimation
each
observer
and
than
operation
has
a
will
computation
only.
associated
over
other
times
is to be
estimated
of
can
as
then
eliminating
fixed
constant,
a
be
First,
time
than
all
tn
the
the
times
those
say
define
a
limited
noise
for
that
are
flight
less
than
discarded.
characteristic
done
function
from
the
initial
In
time
2.2-5
which
by
scanning
the
of
time,
for
may
which
the
be
made
Data
the
flight
dynamics
and
the
points
are
Atf
given
increment
magnitude
that
greater
are
than
closer
as
is
This
and
table
tI
the
minimum
value.
within
Excess
if
the
emitted
segment
the
from
its
observer
of
time
vector
minimum
each
the
time
to
recording
for
slice
addition,
flight
during
This
distance
times
observer
times
segment
the
flight.
on
relative
times
i0,
time
Elimination
and
time
during
based
further
time
slice
examined
the
final
together
kmo
Atf = PrCrAw
(1)
The constant
the extra points are discarded.
user parameter.
Supplementing
Input
time
the
to
points.
input
the
Geometry
If
the
interval
Module
time
points
limit
angles,
and
produce
intermediate
wind
axis
At,
The
computations
by
the
axis
may
are
the
consist
Euler
angles
are
spaced
at
now
proceed
until
all
transformed
reflection
of
of
at
aircraft
data
time
body
at
widely
intervals
interpolated
with
less
body
a
than
spaced
greater
coordinates,
intervals
Minimum
cubic
than
Euler
spline
to
At.
Distance
separately
observers
into
the
in equation (i) is a
Data
spaced
values
Location
process
continues
coordinates
are
Sparse
k
for
have
been
Earth-fixed
each
observer,
and
the
completed.
The
observer
flight
coordinate
system
transformation:
Izi]
I!
°IxI :]
Yo
=
0
The
position
of
(2)
-i
the
-
observer
relative
to
the
aircraft
is
then
(3)
oa Yo Ya
oa (t)J
Let
the
flight
to
I and
let
ri
times
the
=
z
be
Za(t) J
designated+by
observer
iXoa(t
i)
vector
+
jyoa(ti)
an
r(t i)
+
index+
=
ri
kZoa(ti)
2.2-6
which
i
ranges
from
1
be
(i
=
i,
2,
...,
I)
(4)
In addition
define a discrete path vector by
si =
These
the
vectors,
To
find
the
minimum
the
segment.
mum
at
the
test
end
of
are
is
=
ti
x
a
the
the
set
the
for
maximum
reduction
approach.
condition
basis
It
point
of
the
x
=
in
P
is
in
distance
checked
figure
to
to
2(b),
direction
of
is no minimum
order
minimum
see
rm
lies
that
r
and
distance
the
case
For
each
segment
and
time
is
-
of
which
computed
s.
If
within
a minipasses
by
(8)
ti)
no
_
is
tl
the
global
of
interest
added
data
examined
dlstance
In
of
to
rmin(O)
event
that
so
Time
Segments
given
that
the
aircraft
posilinear
interpola-
select
and
there
minimum
is
to
by
is
the
the
time
global
tmin(O
no
)
local
program
issues
a
by
10_dB/20rmln
spherical
if
(7)
_s,i)(ti+l
minimum
be
(5)
s,i
observer.
can
I-l)
by
in the
there
included
values
global
...,
a minimum
segment
time
point
tm, i
is
and
observer
vector
distance
that
of
is,
find
2,
occurs.
expressed
Determination
on
first
acceptable.
i
minimum,
a
orientation,
rma x
the
i,
(6)
local
ri
+ (nr,
to
minimum
=
1
be
(6),
tm,i
then
there
message.
The
used
the
that
is
will
ns, i
Then
minimum,
warning
equality
point
=
recorded
s i,
This
+
<
ns, i -
rm, i
Finally,
minimum.
are
distance,
s i.
equation
For
each
local
tion,
aircraft
tion
on time.
2,
which
ns
are
unit
vectors
is not
satisfied,
then
The
the
at
on
of
< -_
- nr, i
nr
and
condition
figure
time
r i
length
0
where
this
in
the
of
the
(i
ri+ 1
and
projection
within
-
shown
observer
the
ri
(9)
spreading
_dB
in
comparison
is
desired
to
find
2.2-7
to
alone
the
those
noise
points
will
give
at
ri
the
a
relative
closest
which
point
satisfy
noise
of
the
rmi n -< r i -< rma x
and
their
process
associated
is
one
and
the
as
depicted
the
figure
5,
solutions.
In
in
6,
r(tf)
condition
is
magnitudes
depicted
ri
to
in
find
figure
the
3.
starting
The
time
=
rma x
>
ri+ 1
(10a)
=
rma x
<
-
rj+ 1
(10b)
Note
that
the
at
least
figure
4.
distance
vector
order
order
examined
figure
<
in
the
time
in
minimum
This
examining
r(ts)
ending
rj
times.
of
ri _
(i0)
to
both
find
turn
the
for
to
the
find
equation
laisi
-
and
equation
for
ril
=
it
the
and
values
other
and
a
point
must
include
as
depicted
to
times,
limiting
can
ri
vectors,
(10b)
finishing
contains
end
of
two
(10a)
starting
if
set
be
have
possible
each
segment
point
Z.
written
As
in
is
shown
in
as
(ii)
rma x
where
tZ -
ti
(i
a°
l
and
t%
ti+ 1
may
be
-
=
i,
2 .....
I-l)
(12)
ti
either
a
start
or
_I + ÷
_(siri)
\2
an
end
time.
Equation
(ii)
gives
i
+ +
(siri)
+
-
a. l
s2{r 2
i, i-
2
rmax)
(13)
2
S
i
For
as
the
the
first
depicted
vector
starting
mediate
segment,
the
+in figure
sI
time
or
is
<
its
before
segments,
0
it
a.
6(a),
<
is
condition
that
extension
tI
as
required
the
is
behind
shown
added
circle
time
by
that
with
t I.
equation
aI
<
radius
If
This
rma x
a
(12).
i.
< 0,
For
means,
intersects
then
all
the
inter-
that
i
(i =
1
2.2-8
2,
3,
...,
I-2)
(14)
as shown in figure 6(b). It is possible that both roots will occur and
define a complete time segment as shown in figure 6(a); however, it will
generally turn out that only one root of equation (13) satisfies the
limits in equation (14). For the final time, the condition 0 < a allows
the final time to lie on the extension of the last path vector si_ I.
Limit on Directivity
The directivity
vector r must be
The
angle between the path vector
limited
directivity
@i
Variation
to
angle
= Arcsin
a
change
A@
s
within
and the observer
each
time
step.
is
Inr,i
x
(15)
ns ,il
and
A81 • >
If
ASi
=
8i+i
A8
then
-
(16)
@i
intermediate
time
Output
The
direction
fixed
flight
observer
r
are
given
cosines
coordinate
was
computed
from
points
must
be
added.
Computation
the
source
system
are
now
by equation
to
the
computed.
(3).
Then
observer
in
The
position
the
direction
the
Earth-
of
the
cosines
by
r'l
=
r/r
(17)
r'2 I
r,3J
from
a
The
average
the
Atmospheric
function
speed
of
the
y(t)
=
of
sound
is
Properties
flight
Mgr[r
time
known
Table.
by
as
This
a
function
variable
of
is
altitude
y
converted
to
be
setting
nr, 3 (t)_
(18)
RT r
and
c(t)
=
(19)
CrC(Y)*
2.2-9
where nr,
3
is
the
direction
value
of
reception
now be
computed
by
to
and
a
are
naturally
table
=
tf
of
The
intervals.
The
that
computed
corresponds
by
equation
to
each
(17).
value
The
of
tf
can
(20)
a
function
into
reception
times
angle
of
segments
y
found
from
to
is
by
the
by
this
the
created.
time
process
observer
The
segments
to
will
the
reception
times
previously
naturally
source
can
occur
now
be
as
y
where
as
elevation
cosine
to
r/_(t)
tf
at
computed
+
grouped
defined.
uneven
time
= Arcsin
n
is
n
the
(21)
r,3
direction
cosine.
r,3
For
equation
computation
(17)
transformation
coordinate
of
must
is
system
r,2
be
directivity
to
accomplished
as
I
the
rotated
defined
=
with
in
[T(_s)]_T(Gs)
angles,
the
source
the
figure
_
use
the
direction
coordinate
of
Euler
cosines
system.
angles
of
This
for
each
7.
[T(_s)_
nr,
r, 3Js
r,
1
(22)
a
where
T(_)
=
(23)
sin
cos
T
T
0
T(0) =
cos
sin _
0
1
co_
isin
t_
G
Q
0
0
ii
0
-sin
cos
(24)
ii
2.2-10
T(¢) =
Then,
the
polar
o
cos
_
sin
-sin
_
cos
directivity
angle
(25)
@
is
given
by
(26)
@(to,O,S)
and
the
azimuthal
= Arccos
(nr,l)
directivity
angle
s
_
is
given
by
(27)
_(to,O,S)
The
preceding
reception
the
are
always
time
geometry
systems
used
Arctan
(nr,2/nr,3)s
procedure
times,
and
nate
=
table
for
may
be
is
segments,
is
source
provided
repeated
until
observers,
complete.
coordinate
by
the
The
and
the
aircraft
systems.
outputs
sources
body
for
have
been
axes
and
Additional
all
computed
wind
source
axes
coordi-
user.
REFERENCE
i.
Etkin,
Inc.,
Bernard:
Dynamics
of
Atmospheric
c.1972.
2.2-11
Flight.
John
Wiley
&
Sons,
TABLEI.-
RANGE
ANDDEFAULT
VALUESOF INPUTPARAMETERS
Input
parameter
Minimum
Aw, m2 ......
AdB
k
.......
.
.
.
.
.
.
.
.
Maximum
1
1
i0
20
30
0.i
1
I0
m o , kg
......
1
tI , s
......
0
tn , s
......
At,
s
......
A@,
deg
.....
Default
416.8
1
x
104
1
x
106
1
x 105
0
1
0.i
5
2.2-12
x
105
0.5
2.0
i0
20
0
0
4_
-.4
m
4J
0
.,.4
_
0
•,4
_
N
m
_
m
_
m
0
0
_
4.1 _
.,.4
_
0
0
o
I
2.2-13
0
m
ri
OBSERVER
(a)
Discrete
representation
of
flight
path.
S
p
S
S
(b)
Conditions
Figure
for
2.-
minimum
Location
distance
of
minimum
2.2-14
in
path
distance.
segment.
rmin
OBSERVER
Figure
3.-
Determination
2.2-15
of
time
segments
for
observer.
(a)
(b)
Figure
4.-
Limit
Starting
Ending
points
time.
time.
on
time
2.2-16
segments.
rl
Figure
5.-
5
Minimum
set
of
of
three
single
time
2.2-17
vectors
segment.
for
determination
a<l
t_
(a)
i
2
First
-.
segment.
.- O<
a<
1
"_' tf
r
(b)
Intermediate
(c)
Figure
6.-
Final
Solution
times
for
for
time
segment.
segment.
starting
segments.
2.2-18
and
finishing
XI
X2
Y1
I ZI,Z 2
(a) Azimuth.
x3
X2
Y2 'Y3
\
\
Z2
_
Z3
(b) Elevation.
\
z3
(c) Bank.
Figure
7.-
Definition
2.2-19
of
Euler
angles.
2.3
FLIGHT
DYNAMICS
MODULE
INTRODUCTION
Accurate
requires
time.
The
plete
prediction
a detailed
dynamic
are
the
of
in
three
position
Several
of
as
two
The
equations,
first
two
force
assuming
that
all
that
moments
is
condition.
used
of
The
tings,
ties,
solved
for
of
time.
lateral
the
aircraft
aircraft
for
a
is
A
axis
is
of
engine
conditions,
position,
the
data
flight
of
mass
the
in
and
a
If
reference
area
m 2 (ft 2)
aircraft
b
wing
CD
aerodynamic
span,
wing
m
of
aircraft-engine
reference
area,
inlet-face
m 2
(ft 2)
(ft)
D
drag
coefficient,
1
QaV2Aw
2.3-1
dif-
directly
of
control
atmospheric
balance
are
a
number
cross
of
set-
proper-
equations
as
a
are
function
SYMBOLS
Ae
ordi-
trajectory
inputs
Mach
trimmed
two
vertical
inserted
two-dimensional
and
so
This
approximations
module
force
by
aircraft
solving
be
made
zero.
trajectory.
performance,
any
is
maintained
These
to
differential
are
the
attitude,
trajectory
of
aircraft
can
the
eliminating
the
two
however,
of
simplification
time.
the
From
problem;
ordinary
aircraft
computes
time.
this
moment
problem
typical
description
longitudinal
of
three
this
in
found
simultaneously
unknown
the
for
com-
ordinary
coordinates
constantly
to
required,
of
the
six
and
equation,
center
the
of
the
of
A
be
three
second
of
function
coefficients,
aircraft
to
the
the
Module
function
restrict
flight
attitude
the
moment
reduced
for
as
initial/final
the
now
computation
Dynamics
a
aerodynamic
and
the
is
one
through
can
a
problem.
unknowns
simplify
problem
and
function
complex
three
of
to
a
six
to
effects.
act
trajectory
of time.
as
is
the
equations
Flight
aircraft
if
problem
ANOPP
ferent
flight
as a function
an
valid
the
within
made
in
as
components
the
generality
aircraft
solution
theory
be
roll
forces
differential
position
and
about
The
and
the
equations
yaw
the
The
can
reduces
is
force
This
and
2.
simplification
This
of
nary
coordinates
reduces
consideration
assumption
three
an
position
aircraft
space.
approximations
dimensions.
all
the
by
aircraft
requires
the
aircraft.
references
1
simplification
flight.
any
problem
for
produced
the
three-dimensional
(Euler
angles)
textbooks
such
each
noise
of
of
the
equations
components
the
behavior
description
differential
of
knowledge
section,
CD, £g
landing-gear
CL
aerodynamic
drag
coefficient
L
lift
coefficient,
1
QaV2Aw
c
speed
D
aerodynamic
F
g
of
ground
sound,
m/s
drag,
force,
gr
acceleration
H
Heaviside
h
local
ha
absolute
N
N
gravity,
m/s 2
(ft/s
2)
function
altitude,
m
(ft)
humidity,
landing-gear
(ib)
(ib)
of
step
(ft/s)
percent
mole
fraction
position
L
aerodynamic
M
aircraft
Mach
m
aircraft
mass,
_a
engine
air
engine
mass
flow
rate,
kg/s
(slugs/s)
_f
engine
fuel
flow
rate,
kg/s
(slugs/s)
N
number
of
T
thrust,
e
Tr
lift,
N
standard
kg
flow
V/c
(slugs)
rate,
kg/s
(slugs/s)
engines
(ib)
sea
level
time,
t e
engine
V
aircraft
velocity,
W
aircraft
weight,
aircraft
position
angle
(ib)
number,
t
(x,y,z)
N
temperature,
K
(OR)
s
specific
of
attack,
thrust,
m/s
m/s
N
(ft/s)
(ft/s)
(ib)
in
Earth-fixed
coordinates,
deg
2.3-2
m
(ft)
6f
flap
E
engine
0
aircraft
0
inclination
P
dynamic
control
variable,
inclination
Euler
of
angle,
deg
angle,
deg
flight
path
viscosity,
power
deg
with
kg/m-s
respect
to
horizontal,
deg
(slugs/ft-s)
setting
P
air
Y
coefficient
density,
kg/m
of
3
(slugs/m
rolling
3)
friction
Subscripts:
a
ambient
b
body
e
engine
n
final
limit
o
break
release
r
standard
1
initial
axis
sea
level
value
Superscript:
*
dimensionless
value
INPUT
The
amount
extensive
the
aircraft
necessary
tion
in
attitude
engine
engine.
table.
of
because
the
geometry
to
a
information
of
solve
finite
and
Finally,
Default
be
A
table
values
maximum
the
atmosphere
the
is
input
increment
2.3-3
a
flight
problem.
Initial
and
and
table
aerodynamic
describe
velocity
the
variable
An
Input
AV
define
equations
control
for
to
of
provided.
setting.
the
nature
differential
time.
engine
performance
must
the
required
complex
final
to
defines
characteristics
parameters
by
the
are
the
the
and
in
an
airframe
atmospheric
given
are
solu-
aircraft
table
of
an
is
of
conditions
terminate
coefficient
described
trajectory
Description
and
properties
table
I.
Constant
for
engine
variable
table,
m/s
(ft/s)
Aircraft
Aw
Ae
aircraft
engine
b
wing
mo
fully
N
number
ho
wing
reference
inlet
span,
reference
coefficient
Ee
engine
m 2
(ft 2)
mass,
kg
(slugs)
break
of
release,
rolling
inclination
angle
mass,
kg
t1
initial
time,
s
V1
initial
velocity,
initial
distance
from
Yl
lateral
position,
m
(ft)
h1
initial
altitude,
m
(ft)
01
initial
flight-path
Vn
xn
hn
time,
final
velocity,
final
final
for
m/s
origin,
angle,
from
origin,
altitude,
m
angle
of
6f(t)
flap
attack,
setting,
m
(ft)
deg
m
(ft)
(ft)
Control
_(t)
Table
s
distance
s
engine,
(ft/s)
(ft/s)
time,
each
(slugs)
m/s
t
(ft)
Conditions
initial
final
m
friction
m1
tn
(ft 2)
area,
Initial/Final
x1
m 2
engines
at
T
Table
(ft)
loaded
altitude
area,
reference
m
of
Configuration
Variable
Table
deg
deg
2.3-4
deg
_e(t)
power setting
for each engine
tlg
landing-gear retraction
time, s
Aerodynamic
angle
of
attack,
flap
angle,
deg
h/b
wing
height
to
span
ratio
C D (_, 6f,h/b)
drag
coefficient
C L (e, 6f,h/b)
lift
coefficient
landing-gear
drag
Engine
power
M
m
(M,Z)
a
coefficient
Performance
aircraft
Mach
air
rate,
flow
fuel
te(M,_)
specific
flow
number
re
rate,
PaCaAe
re
thrust,
PaCaAe
re
ca
Atmospheric
altitude,
C
(h*)
Table
setting
mf(M,_)
h
Table
deg
_f
CD, _g (CL)
Coefficient
speed
Magrh/RT
of
sound,
re
Properties
Table
r
cr
*
p (h*)
h
a
density,
(h)
dynamic
(h*)
absolute
re
Pr
viscosity,
re
humidity,
_r
percent
mole
fraction
OUTPUT
Dynamics
Module
produces
ANOPP.
The
The
first
is
flight
trajectory
output
times
for
this
ential
equations.
This
array,
even
the
compatible
Flight
though
with
the
the
table
table
are
is
the
two
ones
expressed
two-degree-of-freedom
Geometry
Module.
2.3-5
as
used
as
tables
a
in
a
that
second
of
used
full
is
within
time.
integrating
the
The
differ-
six-degree-of-freedom
assumption
The
are
function
is
the
made,
engine
to
be
variable
table as a function of source time for use in the source parameters
modules. The output times for this table are the control variable input
times augmented, if necessary, to adequately define the data.
Flight
tf
flight
time,
Trajectory
Table
s
(x(t) ,yl, z(t))
aircraft
coordinates,
(0,0b (t) ,0)
aircraft
body
axis
Euler
angles,
(0,0 (t),0)
aircraft
wind
axis
Euler
angles
axes,
source
M(t s)
Mach
_e (ts)
engine
_f(t
flap
IZG(t
s)
s)
time,
(ft)
deg
relative
to
body
deg
Engine
ts
m
Variable
Table
s
number
power
settings
setting,
deg
landing-gear
position
Qa (ts)
ambient
density,
c a (t s )
ambient
speed
B a (t s)
ambient
dynamic
h a (t s )
absolute
kg/m
of
3
(slugs/ft
sound,
m/s
viscosity,
humidity,
3)
(ft/s)
kg/m-s
percent
mole
(slugs/ft-s)
fraction
METHOD
The
governed
along
the
two-degree-of-freedom
by
the
flight
two
flight
ordinary
flight
path
path
must
trajectory
differential
must
be
be
zero
zero.
used
equations.
and
These
the
sum
equations
of
by
this
The
sum
the
forces
module
of
are
expressed
E e)
= mV
0 -
D
the
is
forces
normal
to
as
N
Fg
+
L
-
mg r
cos
0
+
T e
sin
(_
+
_d°
_-_
(1)
e=l
and
N
-TFg
+
Te
cos
(_
+
E e)
-
mgrsin
e=l
2.3-6
=
__dV
m d-_
(2)
The definition of the force terms in equations (1) and (2) are
figure 1 for the aircraft during the ground roll and in figure
aircraft
in flight.
The term V dS/dt in equation (i) is the
acceleration normal to the flight path and the term dV/dt is
acceleration along the flight path.
shown in
2 for the
centrifugal
the
The ground force term Fg is nonzero only during the ground roll.
In addition, the centrifugal acceleration and the flight-path
angle 8
are zero during the ground roll.
Therefore, applying equation (i) during
the ground roll yields
•
N
Te sin (_ + £e)
(h = ho)
e=l
(3)
Fg =
(h > ho)
where h is the local altitude.
The coefficient of friction
T is a
function of the landing-gear and surface characteristics.
The
surface
assumed
that
it
to
the
be
friction
remains
weight
supported
total
and
throughout
force
constant
is
The
thrust
uniform
mass
results
aircraft
by
the
main
Te
for
flow
rates
ground
solely
during
thrust
the
the
from
the
the
engine
It
main
rotation.
gear,
each
roll.
is
further
landing
Since
error
is
most
gear
of
introduced
related
to
is
the
is
assumed
the
so
that
aircraft
small.
specific
by
Te
= meCat
*
e (M,_)
(4)
me
=
mf
(5)
where
ma
+
The
aircraft
coefficients
as
L=y
lift
1
QaV2Aw[CL
L
and
drag
D
is
computed
from
the
lift
and
drag
(6)
(_,6f,h/b)_
and
D=y 1 QaV2_[CD(_,6f,h/b)
In general,
function
of
Figures
and
the
lift
angle
of
3 and
figures
5
4
are
and
6
coefficient
attack
_,
examples
are
of
examples
2.3-7
(7)
+ CD,Zg(CL)_
C L
flap
the
of
and
the
setting
effect
the
of
ground
drag
_f,
flap
coefficient
and
altitude
setting
effect
on
on
CL
CD
h/b.
CL
and
are
and
C D.
a
CD
There is an additional source of drag CD,ig which is present when the
landing gear is extended. Figure 7 demonstrates the relationship between
the landing-gear drag coefficient
and
the
lift
coefficient•
The
mass
consumption
mass
as
of
of
a
the
aircraft
fuel
by
function
of
the
changes
as
engines.
time
is
mf(M,_
e)
The
given
a
function
rate
of
of
time
change
of
due
to
the
the
aircraft
by
N
d_
=
-
_
(8)
e=l
Finally,
coordinates
the
position
of
is determined
the aircraft
from
as
a
function
of
time
in
Cartesian
dx
= V cos 0
(9)
dt
and
dz
--=
-V cos 0
(I0)
dt
The
the
scales
error
(3),
numerical
differential
the
independent
terms
(8),
solution
of
equations
from
(9),
and
(I0)
equations
expressed
the
in
N
_
e=l
dt*d@ =
differential
dependent
dominating
and
Mm*
are
in
variables
solution.
and
made
form
easier
form.
prevents
Rewriting
dimensionless
{
1
m*e\<J
Ae
t*e sin
is
dimensionless
if
This
insignificant
equations
(i),
(2),
yields
M2CL
(_
+
E e)
- m*Wo*
cos
@
+ _I
(ii)
N
*
dM
dt
m
-- _
_*{Aeh
e\_J
t*
e
cos
(_
+
1
£e ) - _ M2CD
e=l
- m
WO
sin
@
-
TFgH(-z
(12)
)
N
• ./Ae_
Fg* =
m * W o* - _1
M2CL
.
sin
mel_)te
(_
+
E e)
(13)
e=l
.
N
(14)
dt---_ =
-
_)mf
e=l
2.3-8
.*
,*
m e =
.*
+ mf
m a
(15)
dx*
= M
cos
8
(16)
dt*
dz*
=
-M
sin
@
(17)
dt*
where
WS
=
m°gr
(18)
PaC_Aw
*
PaCaAw
t
=
t
(19)
m o
,
PaAw
X
------
(20)
X
m o
,
PaAw
=--(h
z
o
(21)
- h)
m o
m
and
H
is
=
m/m o
the
(22)
Heaviside
H(s)
function
=
(23)
Ii
Other
Now,
craft
symbols
all
in
the
Euler
incidence
Gb
wind
0
=
(ii)
outputs
axis
flight-path
and
equations
required
body
(s Z
(s <
_
axis
+
for
angle
angle
through
the
Qb
(17)
flight
is
have
module
related
to
been
can
the
be
angle
previously
defined.
computed.
The
of
attack
air-
and
by
(24)
@
Euler
0)
0)
angle
relative
= -e
to
the
body
axis
is
(25)
2.3-9
The Machnumber, power setting, flap setting, and landing-gear position
must be expressed in terms of the source time set, which is the input
control time set augmentedwith each time when the velocity changes
by AV. The quantity AV is a user-supplied parameter. In addition, the
atmospheric properties are expressed as a function of the source time set.
REFERENCES
i. Etkin, Bernard:
Inc., c.1972.
Dynamics of Atmospheric Flight.
John Wiley & Sons,
2. Dommasch,Daniel O.; Sherby, Sydney S.; and Connolly, ThomasF.:
Airplane Aerodynamics, Fourth ed. Pitman Pub. Corp., 1967.
2.3-10
TABLEI.-
DEFAULT
VALUESOF INPUTPARAMETERS
Input
parameter
AV,
m/s
A w,
m
..........
A e,
m 2
..........
Default
..........
30
2
b,
m
•
h o,
T
•
e e,
7/4
20
...........
m O , kg
N
i00
.
m
.
.
•
.
•
.
.
•
•
•
•
•
.
•
.
1
0.0
.
°
.
•
•
.
0.01
.
0.0
..........
m I , kg
000
.
...........
deg
tI , s
i0
..........
i0
..........
000
0.0
...........
0.0
V I,
m/s
..........
Xl,
m
...........
Yl,
m
...........
0.0
h I , m
...........
0.0
81 , deg
t n,
s
0.0
0.0
..........
i00
...........
V n , m/s
125
..........
x n,
m
.
.
. .
h n,
m
...........
.
.
2.3-11
.....
i0
000
i0
000
I!
F-
-.I
iX
N
2.3-12
\
\
\
\
.C
_m
\
,-.4
wa
D_
C
\
.,-4
\
4J
m
D
\
.,-4
0
\
\
\
r_
\
0
I
\
\
-,-I
\
N
2.3-13
..J
2
_f, deg
30
,p-
_J
°_
4W-
20 --\
10
0
t.}
0
4°e--
-5
I
i
0
5
Angle
Figure
3.-
Typical
operation
lift
with
of
l
10
Attack,
coefficient
no
I
ground
2.3-14
_,
for
effect.
15
degrees
low-speed
.4--
.3
(Sf, deg
c_
3O -_
2O
2
10 "-_
°_-
0
u
,t-
.2
O
t_
cn
L
c5
.1
I
t
-5
0
Angle
Figure
4.-
Typical
operation
drag
with
2.3-15
of
I
I
I
I
5
10
15
2O
Attack,
coefficient
no
ground
¢,
for
effect.
degrees
low-speed
.8
hlb
0.1
0.15
.7
.6
..J
c.)
2
.5
Q;
.rW--
WQ;
0
.4
W°e--
--4
.3
.2
.1
I
0
0
2
I
I
I
4
6
8
Angle
Figure
5.-
Typical
lift
of
Attack,
coefficient
a,
for
2.3-16
I
I
10
12
degrees
ground
effect.
I
14
.3 m
r_
hlb
2
°_
B
0.ii
0.15/
°_
414Q;
0
C_
1C_
I
I
2
4
Angle
Figure
6.-
Typical
I
I
6
drag
2.3-17
of
I
8
Attack,
coefficient
I
10
a,
for
12
degrees
ground
effect.
I
14
C3
O0
0
•"_
k_
0
0
0
I
0
_
'0
.,-t
I
p_
•
0
0
0
0
0
6_'G
3
'_ue_o_ao3
0
0
6eJO
=eeg-6uLpu_]
2.3-18
0
C3
•f,,,t
JTO
2.4
Jet
Takeoff
(JTO)
John Rawls.
Lockheed
Engineering
Module
Jr.
& Sciences
Company
Introduction
f
The purpose of the Jet Takeoff (JTO) Module is to calculate the position of an aircraft
during takeoff.
The basic takeoff profile consists of ground roll and climb a.s shown in
figure 1. Two optional maneuvers
may be appende(l
to tile basic takeoff profile.
One is
cutback (in engine power) which is a procedure
uscwl to reduce noise levels on the ground.
The other is a steady turn which may be initiated
after a steady-state
solution
to the
equations
of motion has been obtained.
An arbitrary
figtlt profile requires a solution to nine differential
equations:
three force
equations,
three moment equations,
and three equations
to determine
the position of the
aircraft
relative to an Earth-fixed
coordinate
system.
During
takeoff, these equations
conveniently
reduce to four first-order
nonlinear
differential
equations
which are solved
numerically
with a fourth-order
Runge-Kutta
technique.
Several assumptions
are made in
this module to simplify the analysis.
These assumptions
include zero wind, a level runway.
zero aerodynamic
ground effects, and zero weight reduction from the burning of fuel.
Symbols
ato
Co
CD.Lg
CL
aircraft
wing reference
area,
m 2 (ft 2)
aerodynamic
drag
coefficient.
aerodynamic
drag
coefficient
aerodynamic
lift coefficient,
c
speed
of sound,
m/s
D-
aerodynamic
Fy
ground
G
steady-state
gr
gravitational
H
altitude,
m (ft)
ha
absolute
humidity,
L
aerodynamic
LLg
landing
M
molecular
Mc_
aircraft
Mach
number,
m
aircraft
mass,
W/gr,
Neng
number of engines
Pw
roll rate,
drag,
•
D
½p,, v2 A_.
due to landing
gear
L
½t,,,t.2A,,'
(ft/s)
N (lb)
force, N (lb)
climb
function
constant,
9.8066
percent
m/s 2 (32.1741
ft/s 2)
mole fraction
lift, N (lb)
gear position,
weight
Up or Down
of dry air, 28.9644
V/c
kg (slugs)
deg/s
2,4-1
rev.
11-93
JTO
qw
pitch rate. deg/s
R
universal
gas constallt,
rw
yaw rate,
deg/s
T_
thrust,
T_
standard
t
time, s
ts
source
_t
incremental
V
aircraft
velocity,
m/s
W
aircraft
weight,
N (lb)
X
aircraft
longitudinal
Y
aircraft
lateral
z
aircraft
altitude
aircraft
angle of attack,
sideslip
a11gle, deg
sea level temperature.
time step,
(ft/s)
distance
inclination
from origin, In (ft)
P
air density,
T
coefficient
deg
deg
angle,
deg
kg/m-s
power
setting,
maximum
kg/m 3 (slugs/ft :l)
of rolling friction
axis Euler angles,
deg
(V_, 0_, _)
wind
axis Euler
deg
( _wb,
Euler
angles
turn
(slugs/ft-s)
percent
body
fl
m (it)
step size, s
engine
_wb)
rate,
angles,
relative
to body
axis, deg
deg/s
Subscripts:
a
ambient
b
body
bank
steady
climb
climb segment
cutback
cutback
i
initial
max
maximum
rev.
11-93
m (ft)
tolerance
II
_wb,
from origin,
above runway,
variable,
integration
Ob, _b)
K (518.67°R)
s
distance
dynamicviscosity,
(_b,
288.15
time, s
engine
error
m2/K-s 2 (49 718.96
N (lb)
flap control
_tol
8314.32
axis
turn
segment
of takeoff
of takeoff
2.4-2
thrust
ft2/°R-s
2)
JTO
min
minimum
rot
rotation
I/)
wind
axis
Superscripts:
nondimensional
derivative
with
respect
to time
iv
Input
The
JTO
Module
performance
takeoff
characteristics
procedure.
of the aircraft.
takeoff
weight
and engine
Coefficient
Table.
The
takeoff
Aircraft
set
aircraft
and
time.
basic
the
begins
takeoff
desired
climb
brake
release
from
and
brake
by setting
In order
for the
the
"desired."
If one or both of these parameters
thrust
to obtain
the anticipated
flight profile.
an
initial
after
climb
liftoff
angle
a new climb
The maximum
does
scrape
The "takeoff
The
cutback
the
Once
or set
0w to the
allows
the
climb
speed
the
the
when
rotation
define
geometry,
the
and
geometric
the
properties
maximum
Aircraft
aircraft
speed,
the
value
to attain
is attained,
the
reaches
flight
desired
climb
airborne,
the
the
angle which achieves
the maxinmm
rate
angle of attack
during
rotation
ensures
climb
there will he
reduce
0w to
establishes
velocity
Module
of climb
that the
speed
be at least
climb angle
of 2.3 °. This
JTO
a
elapsed
velocity
should
speed
and the
is chosen
too large,
Should
this occur,
default
aircraft
the
to become
the stall speed and the climb
speed.
Note that
the climb
angle
Ow which
as possible.
computes
solution.
not
climb
aircraft
to
or a designated
aircraft
are labeled
insufficient
acceptable
tables
of the aircraft
is the
of fuel is neglected.
ends
must be greater
than
greater
than the stall
minimum
the
release,
at rotation
20 percent
the
and
describe
that tile w_ight
to the burning
is defined
angle.
engines,
Parameters
distance
procedure
l)arameters
are input
through
the Aerodynamic
Lift and Drag
Drag Coefficient
Table,
and the Engine
Performance
at
a designated
of
the
Configuration
characteristics
Landing
Gear
procedure
altitude,
and
extensive
Note in this set of parameters
since weight
reduction
due
performance
Table,
the
The
an
of the
The
designated
speed,
requires
as soon
automatically
for a steady-state
tail of the aircraft
runway.
procedure
procedure
may
include
two
is implemented
optional
by setting
the
maneuvers:
cutback
CUTBACK
flag to TRUE,
and
steady
turn.
designating
the altitude
at which cutback
is to occur,
indicating
the cutback
climb angle, and indicating
the time required
for the cutback
procedure
to be completed.
To include
a turn in the takeoff
procedure,
set
the
BANK
begin,
indieate
the turn
turn will not be executed
The
final
the aircraft
Earth-fixed
therefore,
runway
noise
steps
are to establish
last
if the
the
location
origin
of the
in the noise calculation.)
an initial
angle of attack
input
Table.
indicate
rate,
and specify
the
unless a steady-state
on the runway,
and define
coordinate
system
coincides
calculations
The
to TRUE,
data
Under
are
the
most
the
altitude
new flight-path
condition
has
of the
Earth-fixed
at which
the
heading.
Note
been achieved.
that
a steady
system,
position
coordinate
the initial
conditions.
Usually,
with the location
of the aircraft
xi is zero.
The initial
altitude
and should
be greater
than
zero.
a singularity
by defining
Properties
parameter
turn
is to
the origin
of the
at brake release;
zi indicates
the height
of the aircraft
above
the
Setting
zi to zero may cause
an error
in later
aircraft
and
an observer
coincide.
The user must also configure
and an initial
flap setting.
Differential
circumstances,
2.4-3
Equation
the
Parameters
default
values
the
(This
aircraft
and
for
the
the
results
for takeoff
Atmospheric
Differential
rev.
11-93
oF
is
JTO
Equation
Parameters
are adequate.
Atmospheric
parameters
required for the calculation
of lift, drag, and sound speed are described by thc Atmospheric
Properties
Table.
Aircraft
Configuration
At/)
aircraft
wing reference
Neng
number
of engines
W
aircr_fft
weight,
engine
area,
m 2 (ft 2)
N (lb)
inclination
angle,
deg
Initial Condition
time,
ti
initial
xi
initial longitudinal
m (ft)
distance
initial
altitude
ot i
initial
angle of attack,
6i
initial
flap setting,
above
deg
Condition
time, s
maximum
distance
from origin,
maximum
altitude,
m (ft)
climb
velocity,
Oclimb
desired
flight-path
II
engine power setting,
&
rotation
amax
maximum
angle of attack
T
coefficient
of rolling
rate,
turn flag
Zbaak
turn
altitude,
fl
turn
rate,
Parameters
m/s (ft/s)
angle during
percent
climb,
deg
maximum
net thrust
deg/s
during
rotation,
deg
friction
Steady Turn
BANK
m (ft)
m/s (ft/s)
velocity,
•
Parameters
Performance
rotation
Wrot
coordinate
m (ft)
maximum
desired
of Earth-fixed
deg
Aircraft
Vclimb
from origin
runway,
Final
znlLx
Paranmters
s
zi
tmu
Parameters
Parameters
m fit)
deg/s
desired flight-path
heading after completion
of turn, Obank positive
initiates right turn and Obank negative initiates left turn, deg
rev.
11-93
2.4-4
system,
JTO
Cutback
CUTBACK
cutbadk_ag
Zcutback
cutbad_altitude,
0cutback
cutbad_climb
angle,
_ttcutback
time ra_[uired
.;_
to complete
At
flight li_ae increment,
Ctol
error qeranee
_IIIKX
_min
1
m (ft)
deg
Differential
integr_on
Parameters
cutback,
Equation
s
Parmneters
s
step size, s
maxin_mn
integration
step size. s
mininm_n
integration
step size, s
_erodynamic
aircraft_angle
Lift and Drag
of attack,
flap comtrol variable,
Coefficient
Table
deg
deg
CL(a,, f)
aerody_amicl
lift coefficient,
CD(ct,
aerody_amic_
drag. coefficient,
½p, L2A,,.v
D
_f)
_p_V2.,l,, '
)
Landing
Gear Drag
Coefficient
Table
i
eL
CD,Lg(CL)
aerodyaamic,
lift coefficient,
L
_paV2A,, '
aerodynamic
drag
due to landing
coefficient
Atmospheric
*
altitude,
Table
(ARM)
re _gr
c*(H*)
speed qf sound,
p*(H*)
air density,
p*(H*)
dynamic
ha(H*)
Properties
gear extension
absolute
re Ca
re O-
viscosity,
re _a
humidity,
percent
Engine
II
engine
M_o
aircraft _/Iach number
T (U,
net thrtmt
power setting,
per engine,
mole fraction
Performance
percent
Table
maximum
net thrust
N (lb)
2.4-5
rev.
11-93
JTO
Output
Two output
tables are created by the JTO-module.
The Flight-Path
Table gives the
aircraft ground coordinates
and Euler angles in both the body axis and the wind axis
coordinates
systems.
The Source Variables Table is a function of source time with eight
dependent
variables
including
Mach number, engine power setting,
flap setting,
landing
gear position,
and ambient
atmospheric
conditions.
A new source time is added to the
Source Variables Table whenever one of the eight dependent
quantities
changes value.
Flight-Path
Table
t
flight time, s
(x(t), y(t) z(t))
aircraft
ground coordinates,
(_b(t),
aircraft
body
axis Euler
angles,
aircraft
wind
axis Euler
angles
Ob(t), d/b(t) )
(O, Owb(t),O)
Source
source
Variables
deg
relative
to the body
axis, deg
Table
time, s
M_(t_)
Mach
n(t )
engine
6s(t,,) •
flap setting,
LLg(t. )
landing
gear position,
ambient
air density,
ambient
speed
ambient
dynamic
absolute
humidity,
ha(t._)
m (ft)
number
power setting,
percent
maximum
thrust
deg
Up or Down
kg/m 3 (slugs/ft
of sound,
3)
m/s (ft/s)
viscos!ty,
percent
kg/m-s
(slugs/ft-s)
mole fraction
Method
The JTO Module defines the takeoff profile of an aircraft, relative to a fixed position on
Earth.
Three frames of reference
are used to describe the motion of the aircraft.
One is
fixed to the Earth with the origin placed at brake release, a.s shown in figure 1. The x axis
is parallel to the runway with positive x in the direction of takeoff. The y and z axes form
a right-handed
coordinate
system with the z axis pointing positive "downward."
The other
two reference frames are fixed to the aircraft with the origin located at the center of mass.
The two aircraft
reference
frames used in this module
are shown in figure 2. One reference
frame is the body axis coordinate
system denoted
by (x b, Yb, Zb). The positive x b a_ds
extends forward from the center of the aircraft.
The Yb and z b axes form a right-handed
coordinate
system with the z b axis pointing positive downward.
The other reference frame
is the wind axis coordinate
system denoted
by (xw, Yw, zw). In the wind axis coordinate
system, the xw axis is aligned with the aircraft velocity vector. The wind axis coordinate
system is used to solve the equations
of motion.
The orientation
Euler angles denoted
rev.
11-93
of the Earth-fixed
axes and the
aircraft
body
axes are defined
by (¢b, 0b, Cb) as shown in figure 3. In the wind axis coordinate
2.4-6
by the
system,
JTO
by (¢n,,Ow, dPw). The body and wind axis Euler anglesare
a and the sideslip angle _ as follows:
the Euler angles are denoted
related by the angle of attack
% = Cn, +
(1)
Ob = On, + a
(2)
=
In this analysis,
The
JTO
the sideslip
Module
angle
computes
(3)
L_is always
the
zero.
Earth-fixed
coordinates
(x, y, z), the
angles (¢b, Ob,.ZPb)and the wind axis Euler angles relative to the body
The wind axis Euler angles relative to the body axes are given by
On, b =
_wb
=
dPu,b =
Ob =
m--_-
axis Euler
axes (¢wb, On,b, Cn,b).
(4)
Ow
--
_bw
-
_b
=0
(5)
¢w
--
¢b
=0
(6)
The equations governing the position and velocity
system in a still atmosphere
(refs. 1 and 2) are
dV
body
--_
of the aircraft
in the wind axis coordinate
Neng
= E
Te cos(a
+ e)-rF#
- D-
(7)
WsinOn,
e----I
mVrw
= W cos On,sin 0n,
(8)
Neng
mVqw = E
Te sin (a + _) + Fg + L - W cos Ow cos ¢w
(9)
e=l
dOw
- lpn, sin 0w
Pw = dt
dOw
qn, = _
cos On, + _P,,,sin 0n,
rn,=
d_Pw
_
cosOn,cosCn,-
(10)
(11)
dOw .
dt----_smCpw
(12)
dr,
d--i = V cos On,cos _pn,
(13)
dy = V cos 0n, sin lpn,
dt
dz
d_ = -Vsin0n,
Equations (7), (8),
Yw, and zn, directions.
yaw rates, respectively.
nates as a function of
and flight-path
heading
the roll and yaw rates
and (9) represent
the
Equations
(I0), {11),
Equations (13), (14),
time. These equations
angles are zero (i.e.,
are also zero (i.e., pn,
(15)
equations
of motion of the aircraft in the xn,,
and (12) are expressions for the roll, pitch, and
and (15) give the change in the ground coordiare simplified during takeoff since the bank
Cn, = 0 and Cn, = 0). With this simplification,
= 0 and rn, = 0) and the pitch rate reduces to
qn, =
2.4-7
(14)
d0n,
d-'T
(16)
rev.
11-93
JTO
The system
equations
of equatiotm
dV
dt
now reduces
Tecos(a
to the
following
+e)
rF_
first-order
D
IVsin0w
nonlinear
differential
17)
m
dOu: =
Te sin (a + ¢) + F.q + L - II" cos 0,,. r-_
dt
Le=l
18)
da:
d-'-/= V cos 0,,
19)
dz
d-_ = - V sin 0,,,
Equations
technique.
(17) through
(20) are solved numerically
A solution is obtained at each flight time
(20)
with
,_l)CCitie([
a fourth-order
by
Runge-Kutta
tnew = told + At
An integration
step size _ is chosen
(21)
such that
<_At
(22)
_min < _ < _t,_ax
(23)
and
The integration
scheme adjusts the step size to meet the desired error tolerance.
Figures 4
and 5 show the forces acting on the aircraft during the _rouml roll and climb. The ground
force term Fy in equations
(17) and (18) represents
the resistance due to friction resulting
from contact between the aircraft wheels and the runw_w. This force is positive as long
as the wheels remain in contact with the runway and becomes zero at the point of liftoff.
An expression
for the ground force is obtained by noting that the flight-path
angle 0_v and
pitch rate _)w in equation (18) are zero during ground roll. Consequently,
b) =
WcosOw-L-
e=l
_ Tesin(a
+-)
(z=0)
(24)
N'eng
o
The rolling friction
surface characteristics
(z < o)
coefficient 7- in equation (17) is a function of the landing gear and the
of the runway. Two assumptions
are made to simplify this parameter.
The surface is assumed to be uniform during the ground roll and the ground force F.q is
assumed, to result solely from the main landing gear. This assumption
allows the friction
coefficieht to remain constant
until liftoff. The aircraft lift and drag are computed by
1
2
(25)
and
(26)
{ _paVaAw [CD (c*,61) + CD,Lg(CL)]
½PaV2AwCo (_, 6I)
rev.
11-93
(atelirnb < 3)
(Atclirab > 6)
2.4-8
JTO
In equation(26),Atclim
b
is
Atclimb-----
and
tclim b
coefficient
(27)
t -- tclim b
is the time at the start of the climb stage. The lift coefficient CL and the drag
CD are functions of the aircraft angle of attack and flap settings. An additional
source of drag CD,Lg,
which is a function of the lift coefficient, is due to the extension of
the landing gear. Landing gear drag is present during the ground roll stage and the first
3 seconds after liftoff. The landing gear drag coefficient is multiplied by a cosine term
to ensure a smooth transition in the drag force. These coefficients are obtained from the
Aerodynamic Lift and Drag Coefficient Table and the Landing Gear Drag Coefficient Table.
The net thrust Te is a function of the engine power setting and the flight Mach number.
Values for the net thrust are obtained from the Engine Performance Table.
Takeoff Procedure
The takeoff procedure is divided into four stages: ground roll, climb, cutback, and steady
turn. Ground roll and climb are the basis for all takeof profiles. Cutback and steady turn
are optional
stages.
Ground roU. Ground roll begins with the aircraft at rest. During the ground roll stage,
the climb angle Ow tuld the pitch rate 0w are zero. With these restrictions, the problem is
one-dimensional
and requires solutions to equations
(17) and (19). The initial conditions
for the ground
roll stage are
V (ti) = 0
(28)
x (ti) = _:i
(29)
The angle of attack ta, climb angle 0w, and coordinates
y and z remain constant during
ground roll; that is, t_ = cq, 0w = 0, y -- 0, and z = zi. Equations (17) and (19) are solved
iteratively until the rotation velocity is achieved. After reaching the rotation velocity, the
angle of attack
is increased
by
r_ne w =
Otoid +
(30)
¢_ At
until the maximum angle of attack has been achieved. As the nose of the aircraft rotates
"upward,"
the velocity and the angle of attack continue to increase. The velocity vector
however remains parallel to the runway. The ground roll stage terminates when the ground
force term Fg becomes
zero.
Climb.
Climb begins at the point of liftoff when the aircraft
main landing gear
leaves the ground, and 3 seconds into the climb stage, the landing gear is automatically
retracted.
During the climb stage, the aircraft accelerates
to the desired climb velocity,
and simultaneously,
the climb angle increases from zero until the desired climb angle is
obtained.
The aircraft coordinates
are obtained by solving equations
(17) through (20).
During
the solution
of these equations,
one of the following
conditions
will arise:
Condition 1 (V < Vclim b and 0w < 0climb):
Every climb stage begins with the aircraft
velocity and climb angle less than the desired values. The angle of attack is evaluated
at the beginning of the climb stage to determine if rotation has been completed.
If
rotation has not been completed, the angle of attack continues to increase until the
maximum angle of attack is achieved or until the aircraft acceleration rate becomes
negative.
If the acceleration
rate becomes negative, the aircraft has "overrotated';
in which case, the angle of attack is reduced by 1 deg/s until the acceleration
rate
becomes positive. ,Equations (17) through (20) are solved numerically until one of the
conditions described subsequently occurs or until a steady-state
solution is obtained.
2.4-9
rev.
11-93
JTO
Condition 2
(V= Vclim b and Ow < 0climb): When the aircraft
attains
the climb
velocity before attaining
the desired climb angle, the angle of attack is adjusted
so
that the aircraR no longer accelerates.
Ttie appropriate
angle of attack is computed
by setting equation
(17) to zero
[Ne__nl_
re cos (a+¢)-O-ll"
and solving for a numerically.
altitude.
If additional
energy
climb segment is automatically
sin 0,L,] --- 0
(31)
The remainin_
energy in the system is used to gain
is available after the climb angle is attained,
a second
initiated as described
by condition 4.
Condition
3 (V < Wclim b and /gw = 0climb):
The alternative
to condition
2 is when
the aircraft attains the climb angle before air,tining
the desired climb velocity.
For
most takeoff procedures,
setting a climb angle
less than the maximum
climb angle
attainable
by the aircraft is desirable.
This establishes
0clim b as an initial
climb
angle which is held constant to allow the climb velocity to be achieved as quickly as
possible. To ensure that the initial climb angle i._maintained,
equation (18) is set to
zero and solved numerically
for a:
ZTesin(a+e)+L-ll'c°sO'v
=0
(32)
c=l
The angle of attack
now controls
the climb angle, and the remaining
energy in
the system is used to accelerate
the aircraft.
If additional
energy is available
after
the climb velocity is attained,
a second climb segment is automatically
initiated
as
described
by condition
4.
Condition
4 (V = Vclim b and 0w = 0climb):
Condition
4 arises when the aircraft
is capable of a greater climb speed or climb _mgle than is specified by the input
parameters.
When this condition
occurs.
_].limb
is held constant
and the excess
energy is used to increase the rate of climb.
A new climb angle is computed
by
setting the right side of equations
(17) and (18) to zero and simultaneously
solving
for Ow. The new climb angle can be written in terms of the lift-drag ratio and the
thrust-weight
ratio as
Ow = sin -1
(
where the steady-state
1 (L/D)
+ (L/D)G 2
I
climb function
[ 1 (L/D)G
+ (L/D) 2
G is defined
+
- G2 2
1 +1 (L/D)
(33)
as
Neng
[D
(34)
Steady-state
solution: A steady-state
solution is defined
rate and the pitch rate satisfy the following criteria:
m
rev.
11-93
IdVl
-_-
_
2.4-10
Ctol
when both the acceleration
(35)
JTO
and
dO,v ¸
mV l- _
The Earth-fixed
coordinates
(36)
<_Et,,I
can now be computed
Xnew = Xold + VAt
by
cc_sBtc
(37)
sill 8u.
(38)
and
Znew
The climb stage
---- Zold
--
VAt
ends when one of the following
conditions
are met:
x >_x,,,_x
(39)
z > z,.ax
(40)
or
t >_ tmax
The climb
steady-state
stage may also end
solution is obtained
,
when tile cutback
and a steaxly turn
(41)
altitude
is obtained
or when
maneuver
is to be performed.
a
Cutback.
Cutback
is initiated
when the aircraft
reaches the user-specified
cutback
altitude.
With the aircraft
angle of attack and the climb velocity remaining
constant
throughout
the maneuver, the thrust required to obtain the cutback climb angle is computed
by
W IcosOcutbaek + (L/D) sin Ocutback]
Tcutback
-_
. (L/D)
cos (a + e) + sin (a + e)
(42)
-
Thrust
is reduced
linearly
such that
Neug
Neng
N_,ng
(43)
e=l
e----1
c=l
N¢,ng
where the thrust increment
ATe is given by
e=l
Neng
_T_ =
(44)
•
htcutback
e=l
and T_.(II) is a function of the power setting at the beginning of the cutback maneuver.
The
climb angle corresponding
to each new thrust setting is computed
by equation
(33) with
Te ---- Te,new
in equation
(34). At the end of Atcutbac k, in seconds, cutback is completed
and a steady-state
solution to equations
(17) and (18) is obtained with the new climb angle
equal to the cutback climb angle.
Steady turn.
When an aircraft executes
a steady turn, the aircraft velocity, climb
angle, and roll angle are required
to be constant.
Consequently,
a steady-state
solution
to equations
(17) and (18) is required before a turn can be initiated.
Once this condition
is met, a steady turn is initiated as soon as the user-specified
turn altitude is achieved.
2.4-11
rev.
11-93
JTO
The flight-path headingduring
the
turn varies as
Ikw,new = _Pw,old ÷ fl At
where
the turn
pilot's
perspective.
rate
fl is constant.':'
Under
A positive
these csmditions,
(45)
value for f/produces
a right turn,
from the
the yaw rate is
~
"ru, = fl cos 0_,.cos ¢bw
(46)
,.,.
and the balance
of forces in the Yr direction
mVfl
when solving
equation
_
is
O_vcos _,,, = W cos 0u, sin 0,,
(47) for Co
(47)
an angle
Vfl
_I _pw= tan -1-
was found
to be required
from the following
to execrate
the turn.
(48)
gr
The Earth-fixc(l
coordinates
are obtained
equations:
_T¢limb
The steady
turn
cos Ow [sin _w -- sin ('_b,L.-- _ At)]
Ynew
=
cos/_w
Znew
---- Zold -- V'dimb
YoIO ÷
procedure
_
is terminated
At
[cos
fl AT)
(_w
m
Xnew = Xohl + _
(49)
(50)
CO8
sin Ow
(51)
when
Y)w = '_l,ank
or when x, z, or t exceed
the maxinmm
values
(52)
given by equations
(39), (40), and (41).
References
1. Etkin,
Bernard:
2. Dommanch,
Pitman
Publ.
Dyrmmics
Daniel
Corp.,
of Atmospheric
O.; Sherby,
Sydney
Flight.
John
S.; and Connolly,
Wiley
& Sons.
Thomas
F.:
Inc.. c.1972.
.4_TTdane Aerodynamws.
Fourth
ed.
e.1967.
ONGINAL
rev.
11-93
OF
PA"JE I_
Q ALI'r
JTO
\
\
\
\
\
\
\
\
E
•-
\
\
\
F_
-!
.o
2
\
E
\
u
°\
_L
2.4-13
rev.
11-93
JTO
E
-i
rev.
11-93
2.4-14
JTO
_"
II
2X
\
I
W
o
x
I
°
II
E
C
II
i-
.E
i
fm
".o
°
x
..
fxl
•
,,4
E
°.
.
.
.o
a_
°
.o
o-
o°
2.4-15
rev.
11-93
JTO
\
\
\
\
\
\
\
\
\
\
\
..q
_b
\
C
\
-t
\
\
0
\
\
L_
\
-i
\
\
.m
k.
\
\
\
\
rev.
11-93
2.4-16
JTO
\
\
\
\
\
\
.E
\
\
e.m
t_
\
\
r
\
\
\
r,
\
\
\
_Mb
\
\
tl
2.4-1T
rev.
11-93
JLD
2.5 Jet Landing (JLD) Module
Mark-Wilson
Lockheed Engineering& SciencesCompany
Introduction
The Jet Landing (JLD) Module computes the positionof an aircraft
during an approach.
to the runway based on the aircraft
performancc characteristics.
The landingprocedurecan
be dividedintoas many as fivesegments. The basiclandingprofile
consistsof one segment
with an approach angle of 3°. During each segment, the approach speed and flight-path
angle are requiredto be constant. This constraintreduces the differential
equations of
motion to an algebraicform which is used to determine the altgleof attack and thrust
requiredto maintain the input approach speed and fight-pathangle.
Symbols
aircraft wing reference
CD
CD.LK
CL
area, m 2 (ft 2)
D
½p,,V._Aw
aerodynamic
drag
coefficient,
aerodynamic
drag coefficient
due to landing gear
aerodynamic
lift coefficient,
½Oat,2Au
L
_
c
speed of sound, m/s
(ft/s)
D
aerodynamic
F9
ground force, N (lb)
gr
gravitational
H
altitude,
ha
absolutehumidity,percentmole fraction
L
aerodynamic lift,
N (Ib)
LLg
landinggear position,
Up or Down
M
molecularweightof dry air,28.9644
Moo
aircraft
Mach number, V/c
fit
aircraft
mass, W/gr,
Neng
number
of engines
Nseg
number
of flight-path
R
universal
Te
thrust,
T,
standard
t
time, s
ts
source time, s
drag, N (lb)
constant,
9.8066 m/s 2 (32.1741
ft/s 2)
m (ft)
gas constant,
kg (slugs)
segments
8314.32
m2/K-s 2 (49 718.96
ft2/°R-s2)
N (lb)
sea level temperature,
2._-1
288.15 K (518.67°R)
rev.
11-93
JLD
At
incremental
V
aircraft velocity,
W
aircraft
weight,
aircraft
longitudinal
aircraft
lateral
aircraft
altitude
aircraft
angle
Y
z
time step, s
flap control
engine
m/s
N (lb)
distance
distance
from origin,
from origin,
above runway,
of attack,
variable,
inclination
dynamic
(ft/s)
viscosity,
m (ft)
m (ft)
m (it)
deg
deg
angle, deg
kg/m-s
(slugs/ft-s)
II
engine
power setting,
IIRv
thrust
reverse
P
air density,
r
coefficient
(Oh, 8b, Oh)
body axis Euler angles, deg
(_Pu,,Ow, Ow)
wind axis Euler angles, deg
power
percent
maximum
setting,
kg/m 3 (slugs/ft
percent
thrust
maximum
thrust
3)
of rolling friction
(_b_.b,On,b, Otvb) • Euler angles relative
to body axis, deg
Subscripts:
a
ambient
b
body axis
f
final
H
runway
i
initial
Lg
landing
n
counter
w
wind axis
1
first segment
2
second segment
threshold
gear
Superscript:
,
nondimensional
Input
A description
of the aircraft geometry is required.
The Aerodynamic
Lift and Drag
CoefficientTable and an Engine Performance Table describethe characteristics
of the
airframeand engine. The atmosphere isdescribedby the Atmospheric PropertiesTable.
Input parameters are used to definethe number of landing segments and other altitude
dependent variablessuch as the altitudeforlandinggear extension.An array of altitudes,
my.
11.93
2.5-.2
JLD
flight-pathangles,andapproachvelocitiesfor eachsegmentofthe landingprofilearedefined
in the LandingProfileTable.
Input Parameters
Nseg
number
of segments
thrust reverser
power setting,
percent
At
time increment,
vf
final aircraft velocity,
W
aircraft
ZLg
altitude
for flap and landing
ZH
altitude
for end of runway crossing,
by
landing configuration
weight,
m/s
(ft/s)
N (lb)
array of appro_h
altitudes,
Ow,i
array
of approach
flight-path
I6
array
of approach
velocities,
Aircraft
aircraft wing reference
geng
number of engines
deg
Profile
Table
m (ft)
angles,
m/s
deg
(ft/s)
Parameters
area, m 2 (ft 2)
angle, deg
of rolling friction
Aerodynamic
Lift and Drag Coefficient
or
aircraft angle of attack, deg
/i.t
flap control
CL(a,_y)
aerodynamic
Co(,-,, 61)
m (ft)
m (ft)
Configuration
mu_
engine inclination
gear extension,
flap setting,
Zi
coefficient
net thrust
s
Landing
T
maximum
variable,
Table
deg
L
lift coefficient,
_paV2A
w
D
aerodynamic
drag coefficient,
Landing
_paV2A
w
Gear Drag Coefficient
Table
L
CL
CD,Lg(CL)
aerodynamic
lift coefficient,
_paV2A
aerodynamic
drag coefficient
due to landing gear extension
2.5-3
w
rev.
11-93
JLD
Atmospheric
Properties
Table
ATM)
altitude,
il
c*(H*)
p*(H')
ha(H*)
speed I_ sound,
air de-;_ity,
re Ca
re pa
dynamic
viscosity,
re/Za
absohte
humidity,
percent
:_
Engine
mole fraction
Performance
Table
•!
II
enginel_ower
Moo
aircraf$:/vlach
T,. (n, M_)
net tllmst
setting,
percent
maxinmm
net thrust
number
-p
per engine,
N (lb)
Output
Two output tables are created by this module. The Flight-Path
Table gives the aircraft
ground coordinates
amt Euler angles in both the l)ody axis and the wind axis coordinates
systems.
The Source .Variables Table is a function of source time with eight dependent
variables including Mai:h •number, engine power setting, flap setting, landing gear position,
and ambient atmospheric
conditions.
A new source time is added to the Source Variables
Table whenever one orthe eight dependent
quantities
changes value.
?
Flight-Path
flight
time, s
aircraft
(0, Ob(t), O)
(0, Owb(t ), O)
ground coordinates,
body
axis Euler
angles,
aircraft
wind
axis Euler
angles
source
Variables
Mach
number
engine
power
6i(t,)
flap setting,
LLg(ts)
landing
gear position,
p.(t.)
ambient
air density,
ambient
speed
ambient
dynamic
absolute
humidity,
rev.
11-93
deg
relative
to body
axis, deg
Table
time, s
Moo(t.)
II(t.)
ha(G)
m (ft)
i aircraft
Source
t8
Table
setting,
deg
percent
maximum
thrust
'
Up or Down
kg/m 3 (slugs/ft
of sound, m/s
viscosity,
percent
3)
(h/s)
kg/m-s
(slugs/ft-s)
mole fraction
2.5-4
OiMO,IN,IM... PAOE m
OT POOR _ALITY
JLD
Method
The flight trajectory
horizontal direction by
generated
by this module
is based
on balancing
Neng
-rFg
+ Z
the forces in th-e
dV
Tecos(a+e)-
WsinOw-
(1)
D=m-_
e=l
and
in the vertical
direction
by
Neng
Fg + Z
Te sin (o + E)-
(2)
W cosSw + L = mV--_- t
effil
These
figure
equations
(refs. 1 and 2) are defined in the wind axis coordinate
1. The ground force term Fy is defined by
Fy =
W - L - e=l
_ :re sin (a + e)
and remains
function
zero until the moment
of landing
The aircraft
gear and surface
lift L and drag
shown
in
(3)
(z = 0)
Neng
0
system
(z > 0)
of touchdown.
The coefficient
of rolling
friction
r is a
characteristics.
D are computed
by
1
2
L = _paV Aw [CL (a,_f)]
(4)
and
D-
lpaV2Aw
[CD (o,,$f)+ CD,Lg(CL)]
(5)
z
where the
Coefficient
The
lift and drag coefficients
are obtained
from
Table and the Landing Gear Drag Coefficient
position
of the aircraft
as a function
the Aerodynamic
Table.
Lift
and
of time is given by
dx
and
Drag
-= V cos6o
dt
(6)
dz
-- = V sin Ow
dt
(7)
This module divides the landing procedure into segments defined by the parameter
Naeg.
An array of altitudes, flight-path
angles, and approach velocities are specified in the Landing
Profile Table. The arrays are defined such that the first value corresponds
to the outermost
segment.
Figure 2 shows an example of a two-segment
landing profile. Note that the flightpath angles are defined to be negative down. The origin of x is defined at the end of the
runway. The initial distance of the aircraft from the end of the runway is computed
from
the array of altitudes
and flight-path
angles by
Naeg
xi =
2.5-5
(8)
Zn -- Zn.__+_+
l
rev.
11-93
JLD
wherethe
runway;
last altitude
that
in the summation
corresponds
to the
altitude
at the end of the
is,
z,v_+l
= zH-
(9)
By maintaining
a constant
approach
speed and flight-path
angle, the derivatives
in
equations
(1) and (2) are zero. This allows the use of the following algebraic equations
(assuming
the aircraft is not on the ground; i.e., Fg = 0):
Neng
Z
T_:cos (a + _) - W sin O,,. - D = 0
(10)
Te sin (a + _) - WcosOw
(11)
e=l
and
Neng
Z
+ L = 0
To calculate the aircraft angle of attack a required to maintain
a small angle approximation
is made for a + e in equation (11).
and equation
(1 I) becomes
the desired approach speed.
Assume that sin (a + _) ._. 0
L = WcosOw
which
can be written
in terms
of the lift coefficient
(12)
as
W cos0.,
eLThe angle of attack that satisfies
Lift and Drag Coefficient Table.
The
thrust
required
equation
to maintain
(13)
_paV2A w
(13) is found
the desired
by interpolating
flight-path
the Aerodynamic
angle is obtained
from
Neng
W [cosOw + (L/D)sin
0w]
Z
Te, = (L/D) cos (a + e) + sin (c_ + _)
e=l
During the airborne
segments
of the landing
constant so that the aircraft coordinates
are
xj
=
sj_
1 +
trajectory,
V cosOw
the
approach
(14)
velocity
remains
At,
(15)
At
(16)
and
zj = zj-1
- VsinOw
where j indicates
segment
number.
After touchdown,
the aircraft
velocity
function of time due to the ground force term. The velocity then becomes
vj=
where
the reduction
in velocity
AV = _1
-rFg
is given
+ av
11-93
a
(1"0
by
- 17 - rngr sin O_ +
r_ cos (o_ + e)
e=l
rev,
becomes
_-.5,-6
At
(18)
JLD
Thrust
and
Te is calculated
aircraft
Mach
by using
number
the Engine
are used
Performance
as inputs.
After
Table
where
touchdown,
engine
_hrllst
employed
to rapidly
decelerate
the aircraft.
TILe engine
power set
thrust reverser
parameter
as input to the Enginf3 Performance
Table.
assigning
a negative
HRV = -0.5
The
aircraft
of-freedom
plane (i.e.,
angles
the
would
value
to the
provide
thrust
Euler
angles
thrust
reverser
reversing
are also
parameter
at a power
required
!_b and
following
(_b are
relation
both
zero.
between
The
angle
aircraft
body
of attack
setting
a,s output
assumption
has been made, the landing
no yaw or roll motion).
This assumption
and
HRV.
For example,
3).
half
Since
profile takes place
implies
that the
axis
Euler
flight-path
angle
.setting
may
a value
axis
Euler
angle
Ob is transformed
to the wind
of
full power.
the
two-degree-
only in the vertical
values of the Euler
0 b can
he calculated
by
angle:
9 b = a + Ot,,
The body
be
' is replaced
by the
i hrust
is reversed
by
of one
(fig.
power
reversing
(19)
axis
by using
the
following
equation:
0wb = -_
(2o)
References
1. Etkin. Benmrd:
Dynamics
of Atmo._pheT_c Flight. John Wiley & Sons, Inc.. c.1972.
2. Domma.sch, Daniel O.; Sherby, Sydney S.: and Connolly. Thoma._ F.: Airplane
Pitman Publ. Corp., c.1967.
Aervdynamics.
Fourth ed.
i
2.5-7'
r_v.
II-93
JLD
\
\
\
\
\
\
\
\
e-
\
\
\
\
\
\
o_
c_
cO
e-
\
\
\
\
\
\
\
\
rev.
11-93
2.5-8
r_
JLD
2.5-9
rev.
11-93
JLD
+
I
_D
II
,._.,,_
+-
-
+
!
._
II
.N
•.
-_
°o
.".Oo
L_
".o
Z
."
ooo
°°
o'
rev.
11-93
2.5-10
SFO
2.6 Steady Flyover (SFO) Module
JohnRawls,Jr.
LockheedEngineering
& Sciences
Company
Introduction
The Steady Flyover (SFO) Module defines the position of an aircraft based oil kinematics
(ref. 1). This approach
enables flight paths to be defined without
requiring
engine and
aircraft performance
data as input.
Aircraft
motion is restricted
to constant
velocity or
uniform acceleration
along a rectilinear
flight path. Flight paths that do not conform to
this restriction
may be divided into segments.
A segmented
flight path is constructed
by
executing
this module once for each segment.
Input to the SFO Module consists of parameters
that define a single flight-path
_gment.
Output consists of parameters
that allow repeated executions
of the module, and two tables
that are used by other ANOPP
(Aircraft
Noise Prediction
Program_
' -,dules to predict
noise produced
by a moving source.
One table definc_s the position of the aircraft as a
function of time, and the other identifies changes in ttle noise source characteristics.
Symbols
a
aircraft
acceleration,
c
speed of sound,
d
distance,
gr
gravitational
H
altitude,
m (ft)
h,,
absolute
humidity,
LLg
landing
M
molecular
m/s 2 (ft/s 2)
m/s
(ft/s)
m (ft)
constant,
9.8066
percent
gear position,
weight
m/s 2 (32.1741
ft/s 2)
mole fraction
Up or Down
of dry air, 28.9644
aircraft
Mach number,
V/c
N
number
of incremental
time steps
R
universal
gas constant,
Tr
standard
sea level temperature,
t
time,
tLg
landing
ts
source time, s
At
incremental
V
aircraft
velocity,
:r
aircraft
longitudinal
Y
aircraft
lateral
Z
aircraft
altitude
8314.32
m2/k-s 2 (49 718.96
288.15
ft2/°R-s
2)
K (518.6T'R)
s
gear reset
time, s
time step, s
m/s
(ft/s)
distance
distance
above
from origin,
from origin,
runway,
2.6-1
m (ft)
m (ft)
m (ft)
rev.
11-93
SFO
aircraft angle of attack,
(I
flap control variable,
deg
deg
_
dynamic
viscosity,
kg/m-s
(slugs/ft-s_
lI
engine
power setting,
p
air density,
( _-'b.Oh.os )
body
axis Euler
angles, deg
( u... O... o.. )
wind axis Euler
angles, deg
( u..b. 8..b.O..b)
Euler angles
percent
kg/m :_ (slugs/ft
relative
maximun_
rhru._t
3)
to body axis. lie-
Subscripts:
a
ambient
b
body
/
final
i
initial
ref
referellce
tC
wind
i
first segnteut
2
second
axis
axis
segntent
Superscript:
nondimensional
Input
A flight profile nmy consist of as many segment._ a._ tim user desires.
Each fiioht-t)ath
segment requires one execution
of the SFO .Module. To create a nmltiple segmeltt flight
profile, tile APPEND
parameter
must be set to TRUE. This causes the data created for the
Flight-Path
Table and the Source \'ariables
Table to be appended
to tile output created by
the previous execution of the SFO Module.
Therefore.
it is important
that each segment
be created in the appropriate order. If the APPEND
parameter
is FALSE. each execution
of the SFO .Module creates a new flight profile.
The Mach number update criterion parameter _X.llx identifies when the flight speed has
altered the noise source characteristics
sufficiently to warrant updating
the Source Variables
Table. If ttle aircraft velocity is not constant and .-X.llx = 0. the Source Variables
Table
contains an entry for every incremental
change in the aircraft velocity.
A large number
of noise predictions
may result that nmy not be warranted
considering
the approximations
required to define the flight oath. The user can limit the number of source noise predictious
and. as a result, reduce the total computation
time by setting the .Xlaeh number update
criterion
parameter.
The computational
Generally.
option
-X3Ix
-- 0.05 is adequate.
flag ZOPT
allows the final z position
of the
aircraft
to be
specified directly by the input parameter
z/ or to be computed
by using the inclination
angle 8u.: Sometimes
the inclination
angle is a more convenient parameter
for constructing
a flight-path
segment.
rev.
11-93
2.6-2
SFO
The position of an aircraft is calculated with the initial and final condition parameters.
The Aircraft Configuration Parameters do not affect the flight profile but are required as
output by the Source Variables Table.
Information from the Source Variables Table is
used by other ANOPP modules to predict the noise characteristics
of the engine and the
airframe.
With the exception of the landing gear, the Aircraft Configuration
Parameters
remain constant throughout a flight segment. The Earth-fixed coordinated system is defined
such that z is positive down as shown in figure 1. For convenience,
the z coordinates
(corresponding
to altitude) are input as positive values and converted to negative values
within the module. In order to compute the aircraft Mach number, the speed of sound is
provided by the Atmospheric
Properties Table.
Input Parameters
APPEND
multiple
Ji
initial step number
AM_
Mach number
At
time step increment,
zref
altitude
ZOPT
computational
option flag, for ZOPT = 1, input
for ZOPT = 2, input 0w and disregard zf
segment
flight-path
flag, True or False
update criterion
s
of runway above reference
level, m (ft)
zf and disregard
inclination angle of flight vector with respect to horizontaJ,
climb and negative for descent, deg
Initial Condition
positive
8w and
for
Parameters
ti
initial
time,s
¼
initial
velocity,
m/s (R/s)
xi
initial
longitudinal
positionfrom origin,m (ft)
initial
lateralpositionfrom origin,m (ft)
zi
initial
altitudeabove runway, m (ft)
Final Condition
tf
final time, s
v/
final velocity,
xf
final longitudinal
_f
final lateral
zf
final altitude
m/s
Parameters
(R/s)
distance
position
from origin, m (ft)
from origin, m (ft)
above runway, m (ft)
2.6-3
rev.
11-93
SFO
Aircraft
LLg,i
initial landing
tLg
landing
II
engine
61
flap setting,
Configuration
gear position,
power setting,
percent
air density,
ha(H*)
thrust
deg
Properties
Table
(ATM)
re
speed of sound,
p'(H*)
maximum
deg
Atmospheric
altitude,
Up or Down
gear reset time, s
angle of attack,
S
Parameters
re ca
re pa
dynamic
viscosity,
re #a
absolute
humidity,
percent
mole fraction
Output
The Final Condition Parameters
provide pertinent
information
execution of the Steady Flyover Module. When a multiple-segment
the final eonditior_.' are used
tables are created
by this
coordinates
and Euler angles
The Source Variables Table
necessary for a repeat
flight profile is created,
as the initial conditions
for the next segment.
Two output
module.
The Flight-Path
Table gives the aircraft
ground
in both the body axis and tile wind axis coordinates
systems.
is a function of source time with eight dependent
variables
including
Mach number,
engine power setting,
flap setting landing gear position,
and
ambient
atmospheric
conditions.
A new source time is added to the Source Variables
Table whenever one of the eight dependent
quantities
changes value.
Final Condition
If the
become
SFO
Module
the input
is executed
parameters
with
APPEND
final step number
LLg,J"
final landing
tl
final time, s
v!
actual
final velocity,
x/
actual
final longitudinal
Yl
actual
final lateral
z/
actual
final altitude
II-93
= TRUE,
for the next execution
:i
rev.
Parameters
gear position.
the
following
of the SFO Module:
Up or Down
m/s (ft/s)
position
position
above
2.6-4
from origin,
from origin,
runway,
m (ft)
m (ft)
m (ft)
parameters
SFO
Flight-Path
t
flight
(x(t),
Y(t),
z(0)
0b(t), 0)
(0,0wb(t),0)
time.
Table
s
aircraft
position
coordinates,
aircraft
body
axis
Euler
angles,
deg
aircraft
wind
axis
Euler
angles
relative
Source
t_
source
time,
M_c ( t._)
Mach
number
n(t,,)
engine
power
m (ft)
Variables
to body
deg
Table
s
setting,
percent
maximum
thru,_ _
flap setting,
deg
LLg(t.,)
landing
position,
p,,(t.,)
ambient
air density,
ambient
speed
ambient
dynamic
viscosity,
kg/m-s
(slugs/ft-s)
ambient
absolute
humidity,
percent
mole
ha(t_)
axis.
gear
Up or Down
kg/m 3 (slugs/ft
of sound,
m/s
3)
(ft/s)
fraction
Method
The
SFO
Module
computes
the
position
of
an
approach
is useful when aircraft
performance
data
profiles
are required.
Figures
2 through
4 illustrate
can be constructed
by using the SFO Module.
Figure
remain
aircraft
2 illustrates
a single-segment
constant.
A flight-path
and by the initial
and
of attack,
flight path
the flap setting,
can be defined
descending,
therefore,
accelerating,
a flight profile
The
profiles
flight
module.
The landing
approach
and ground
the initial
conditions
roll,
Figure
first
path
or decelerating.
may be created
shown
based
where
the
on
kinematics.
or when
of flight
velocity
This
simple
profiles
and
the
flight
that
altitude
segment
is defined
by the initial
and final position
of the
final velocity.
At the beginning
of each segment,
the angle
and the power
setting
between
two points,
the
in figures
are provided
as input.
As long as the
motion
of the aircraft
may be climbing,
The aircraft
motion
for a hovering
aircraft.
3 and
4 are constructed
may
by repeated
also
be
stationary;
executions
of the
profile shown
in figure 3 is divided
into two segments,
representing
roll. The final position
and velocity
for the approach
segment
becomes
for the ground roll segment.
4 illustrates
a takeoff
climb segment,
second
first climb segment,
the
the first climb segment
measured
from the
the aircraft
velocity
setting.
flight
aircraft
are not available
three examples
profile that
has been divided
climb segment,
and cutback.
aircraft
is accelerating.
by specifying
the new
The
gear
landing
position
into four segments:
ground
During
ground
roll and the
gear may
and the
be retracted
reset time
during
which is
beginning
of the first climb segment.
During
the last two segments,
is constant,
but the two segments
differ in climb angle and in power
rev.
11-93
SFO
The position
coordinates
of the aircraft
by the following
equations:
Xk+l = Xk + (Vk At + _a At2)
cosOwcosg'w
(l)
Yk+l = ?ik + (Vk At + _a At2)
cosO,,sinCw
(2)
zk+t = zkEquations
part of
are_computed
(1), (2), and
(V,
At + _u At2).si,,Ow
(3) are solved iteratively
(3)
for k = .I to N, where
N is the integer
t
g = J, + A--7
In equation
flight-path
step k is
(4)
(4), Ji is the initial step number, t is the total
segment,
and At is the incremental
time step.
time required
The aircraft
to complete the
velocity at time
vk = vk-t + ,, _t
where Vk_ 1 is the velocity from the previous
acceleration
of the aircraft given by
(5)
time step
and
a is the acceleration.
.= v]- v/'
The
(6)
2d
and
is computed
segment,
which
from the initial
and final velocities
and the distance
traveled
during
d = CAx 2 + Ay '_+ Az 2
The incremental
distances
The total time required
flight-path
heading
heading
(7)
Ax, Ay, and Az are
to complete
t =
The flight-path
Ax = z/-.,',
(8)
Ay = yf - 71i
(9)
Az = zf - zi
(10)
a segment
is
{t:_ -
(v, =(lal
v: >=O)
0)
_bw and the climb angle Ow remain
constant
(11)
during
a segment.
¢" = tan-I
Ay
X-_
An option is provided which allows either the final altitude
z¢ or the climb angle
specified as input. If z I is specified, the climb angle is computed
by
Ow ----tan_ 1 _Az
Ax
11-93
The
is given by
'
rev.
the
is
2.6-6
(12)
to be
(13)
SFO
If _),_. i_ .-pecified.
_X: i_, COmlmted
by
_X: = __.r tall H,,
,14 t
Tile aircraft
body axi,_ Euler
angle.,, _, and t'_, ar(' (alculatcd
b.v u:ing
the followizlg
relation_hil)
between
au.,,,.Ic of' _ittack. flight-l)ath
angle, told Hight-path
headiug..
See fig. 5.,
The climb angle in the ))od.v a×i.- i,,
e;, = _)-- H
and
tlm
Hight-path
hcadi.o
,15,
m tile l)o¢[v axi_ i_
t- L = _.',
The
ho(l.v axi,- Eult,l
,nab'
_;, i, tran-tLrmed
, ](j_
to tilt, wi,(l
_xi,
with
[h,
',lLJwizlg rclatioli:
H ,_,= --_)
Thi.tile
mo_hde
aircraft
Source
create,,
l)o,,itio,
Varia|)lc_
two
and
Tal)]c.
ouH)ut
Euler
which
of the ,,om'ce noi.,,e _ll'(, altered
aml)ienr
condition,,
or all'craft
entry
The
ct)rrcsl)on,
tlioht
lino
Mach
to the
mmfi)rr,
tal)le..,.
_ulglc.i(lentifie.-
One
for each
time.-
1_',
i,. rhc
(luring
l)v changes
ill engine
velocity.
The Source
start
givcn
of
the
flioht
Table.
.=Xt.
when
Tlle
the
which
(icfine.-
other
i_ tlle
characteri.-tic_.
...coment.
l)v
.l/×,
t. ) =
(18)
(,_H
=
i..,rvahm_e(l
from the aircnd't
velocity
and the ami)iem
i_ ol)taim,d
from the Atmo.-l)heric
Pl'ol)ertie_
Tahle
altitude
H" -ix'ell l)v
H" =
The parameter
_pecified
in the
i,crcmcnt
l)owt'r ,,ettillg.
aircraft
configuration.
Varial)le.Table
ahva.v.,, contain.-,
all
flew flight-path
a
Fli_ht-Path
time
,._ml)(l .-pee(].
_. a hmction
The _otmd _peed c( H" )
of the nondinmn.,,ional
_"
- :_':
RT,. My,.
:n.: i.-: the height
of tilt, runway
Atmo.-:pheric
Properties
Table.
above
(19)
the
reference
level
(u_,_ually .-.ea level)
Reference
1. Bert. Ft.nlimtnd P.: .mi .luhn,.to:l. E. Ru.-._.ll..Jr.: I'+rto,..11,,,,),..,,.
McGraw-Hill Book Co.. 1962.
for
Et*.qm_ f r.--
Starer..
{md Dy....¢.,.
OF
2.6-7
rev.
POOR
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2.6-11
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rev.
11-95
SFO
rev. 11-93
2.6-12
3. PROPAGATION
EFFECTS
3.1
ATMOSPHERIC
ABSORPTION
MODULE
INTRODUCTION
As
sound
attenuated
Module
waves
due
accounts
sphere.
which
the
sound
depends
on
altitude
expressed
as
a
a
given
The
the
molecular
loss
molecules.
of
in
altitude.
Module
due
be
the
the
module
Fourth
a
is
sound
due
to
the
table
as
then
a
of
It
of
the
molecular
speed
f
frequency,
Hz
frl
relaxation
frequency,
gr
acceleration
H
altitude,
HI
ground
h
absolute
humidity,
M
molecular
weight
P
pressure,
Pa
of
sound,
the
available
due
m
level
m/s
of
to
for
be
the
m/s 2
m
percent
used
(ft/s
of
(ib/ft
air
2)
3.1-1
(ft)
mole
to
be
is
that
causes.
nitrogen
a
level
due
and
with
fraction
2)
the
oxygen
relaxation
to
and
First
is
vibrational
the
vibra-
coefficient
coeffidimensionless
the
atmospheric-absorption
(ft)
altitude,
atmo-
ground
Second
frequency
Hz
gravity,
shown
from
atmospheric-absorption
(ft/s)
to
is
basic
loss
SYMBOLS
c
ANOPP
The
total
absorption
of the
four
effects.
function
intensities
in
effects.
to
the
coefficient
intensity.
relaxation
due
by
coefficient
value
four
viscous
are
humidity.
assumed
average
sound
loss
is
and
absorption
the
molecules.
for each
form
table
and
rotational
computes
correct
occurs
molecular
of oxygen
coefficients
dimensionless
to
they
Absorption
absorption
frequency.
the
observer,
produced
an
are
the
and
thermal
as
pressure,
is
applied
the
Atmospheric
intensity
humidity
altitude
to
molecules.
This
to
to
is
relaxation
sum
of the
This
cient
due
Third
nitrogen
tional
is the
and
absorption
loss
noise
expressed
which
can
atmospheric
classical
of
to
The
Therefore,
coefficient,
altitude,
source
temperature,
only.
function
absorption
in
is
pressure,
of
to
decrease
frequency,
temperature,
the
absorption.
attenuation
functions
mean
from
atmospheric
for
This
The
propagate
to
Propagation
effect.
is
<p2
mean-square
>
pressure,
universal
gas
r
distance,
m
T
temperature,
X
fractional
Y
dimensionless
Pa 2
constant,
m2/K-s
Mgr(H
coefficient,
l-I
dimensionless
P
reference
(ft2/°R-s
2)
concentration
altitude,
characteristic
2
(OR)
molar
0
4)
(ft)
K
absorption
(ib2/ft
nepers/m
vibrational
HI)/RT
K
coefficient,
kg/m
3
r
(nepers/ft)
temperature,
absorption
density,
-
(slugs/ft
(OR)
_Cr/f,
nepers
3)
Subscripts:
cl
classical
n
nitrogen
o
oxygen
r
standard
rot
rotational
s
source
vib
vibrational
sea
level
value
Superscript:
*
dimensionless
value
INPUT
The
two
altitude.
values
basic
independent
Center-frequency
in
inputs.
properties
the
In
Atmospheric
addition,
produced
the
by
i/3-octave-band
variables
values
the
for
Module
(ATM)
module
requires
for
the
i/3-octave
define
the
several
ATM.
center
frequency,
Hz
3.1-2
module
are
frequency
bands
and
the
altitude
independent-variable
of
the
atmospheric
and
Atmospheric-Properties
dimensionless
altitude,
h(y)
humidity,
percent
p
(y)
pressure,
re
T
(y)
temperature,
Table
Mgr(H
mole
-
HI)/RT
r
fraction
Pr
re
Tr
OUTPUT
The
a
output
function
is
of
a
table
frequency
of
and
the
dimensionless
absorption
coefficient
as
altitude.
Absorption-Coefficient
Table
f
frequency,
Y
dimensionless
altitude,
Mgr(H
_ (f,Y)
dimensionless
absorption
coefficient,
Hz
-
HI)/RT
r
_Cr/f,
nepers
METHOD
The
as
an
the
sound
intensity
due
to
atmospheric
coefficient.
This
absorption
is
expressed
coefficient
is
defined
by
relation
<p2(r)>
where
=
--<p2(r)>
from
the
is
source,
source,
is
lost
atmospheric-absorption
and
expressed
_cl
is
_
is
as
a
is
the
p2(rs)>
the
the
sum
>
is
exp[-2_(r
acoustic
the
absorption
of
+ erot
the
_
mean-square
<p2(rs)
= ecl
where
<rj>t
four
+
loss
The
some
distance
at
pressure
absorption
r
the
coefficient
as
(2)
_vib,n
due
molecular-absorption
at
acoustic
coefficient.
+
(i)
rs)_
pressure
mean-square
components
evib,o
classical
-
loss
to
thermal
due
to
and
the
viscous
effects,
rotational
relaxation
of
rot
oxygen
and
nitrogen
due
to
the
vibrational
the
molecular-absorption
gen
molecules.
to
be
molecules,
Additional
_vib,o
relaxation
loss
due
sources
negligible.
3.1-3
is
of
oxygen
to
the
of
the
molecular-absorption
molecules,
vibrational
atmospheric
and
relaxation
attenuation
loss
evib,n
of
are
is
nitroassumed
As a result of extensive theoretical and experimental effort,
expressions have been developed for each of the terms in equation (2).
Sutherland (ref. i) presents the theoretical development and all existing
experimental data. Reference 2 further refines that work. All of the
empirical equations used in this module are results of the work in these
references.
An important parameter in the vibrational-relaxation
absorption
loss of a gas is the relaxation frequency. The relaxation frequency is
defined as that frequency at which the maximumvibrational absorption
loss per unit wavelength occurs. In general, the relaxation frequency
is a function of temperature, pressure, and humidity for a given gas.
Air is assumedto be composedof nitrogen and oxygen, neglecting the
absorption of the other components. Therefore, the following empirical
expressions for the relaxation frequencies are used:
frl,n
= (p/pr)(293.15/T)I/2(9
+ 350h expl-6.142[(293.15/T)i/3-
(3)
13})
and
frl,o
(4)
= (p/pr){24 + 44100h_0.05 + h)/(0.391 + h)_l
In equations (3) and (4), frl,n
is the relaxation frequency of nitrogen
in hertz, frl,o
is the relaxation frequency of oxygen in hertz, p is
the ambient pressure, T is the ambient temperature in Kelvin, and h is
the absolute humidity in percent mole fraction.
Rewriting equations (3)
and (4) in terms of dimensionless variables yields
frl,n
=
p*/
=
p*{24
(T*)
l/2(9.08
+
expl-6.1VSECT* -l/3-
340.65h
1]})
,s,
and
frl,o
where
a
T
=
function
presented
+
44100h
T/T r
of
in
figure
at
standard
oxygen.
It
is
and
value
that
of
is
and
p
temperature
humidity
frequency
_0.05
1
the
value
the
absolute
for
sea
much
of
=
the
h)/(0.391
P/Pr"
and
interesting
always
+
A
graph
humidity
at
nitrogen.
level
to
+
note
relaxation
of
the
standard
Relaxation
pressure
greater
(6)
h)_}
is
that
than
sea
value
nitrogen
frequency
humidity.
3.1-4
is
in
of
frequency
level
frequency
presented
the
the
relaxation
the
as
a
figure
as
is
function
2
oxygen
relaxation
highly
pressure
of
for
relaxation
frequency
dependent
on
the
The four terms on the right-hand side of equation (2) are now computed. The classical and rotational terms are combinedand expressed as
a function of temperature, pressure, and frequency as follows:
_cl + _rot = (1.84 × i0-ii)
Each of the two vibrational
loss terms are written
exp
evib,i
= 35 \T
8i
is
fractional
i
the
molar
= o
for
oxygen
the
term
equations
(7)
the values
relation
c
yields
the
-
characteristic
and
in
and
i
(8)
for
the
=
Cr(T*)
=
square
n
for
exp(-Si/T
2ffrl,
terms
physical
f2
+
i
f2rl,i
temperature
In
(8),
equation
nitrogen.
can
of
_ 2
vibrational
brackets
in
(f/c)
form:
(8)
concentration.
large,
(-Si/T)
in the following
/
i
where
(7)
(T/293.15) I/2 f2/(p/pr)
Since
be
replaced
dimensionless
constants
given
and
i
is
the
by
value
table
of
unity.
variables,
in
X i
defined
is
the
such
that
8i/T
is
Expressing
substituting
I,
and
using
the
I/2
(9)
following:
_cl
+
_rot
=
=
(9.555
_vib,o
x
(6.207
exp
x
x
10-9)(f/Cr)(T*)i/2(f/p
10 -4 ) (f/Cr)
[7" 771(T*
-
(I0)
*)
(T*)-5/2
I)/T_
12ffrl,o/(f2
+
(ii)
f2Irl,o
and
_vib,n
=
(1.683
x
10 -4 ) (f/Cr)
x exp[II'633(T*
The
total
(12).
units
sary
absorption
The
of
to
coefficient
absorption
nepers
multiply
-
per
the
is
coefficients
meter.
To
(T*) -5/2
I)/T*]
the
in
side
3.1-5
sum
these
convert
right-hand
12ffrl,n/(f2
to
of
of
equations
equations
decibels
each
+
equation
(i0),
are
per
(ii),
expressed
meter,
by
(12)
f2rl,n)_
8.69.
it
and
in
is
the
neces-
Figure 3 showsa typical graph of the total absorption coefficient
as a function of frequency with h = 0.2, T* = 1.0, and p* = 1.0. The
three distinct regimes for the absorption coefficient are readily apparent from the figure.
The first regime, where the frequency is less than
the relaxation frequency of nitrogen, is dominated by the vibrational
absorption of nitrogen.
The second regime includes values of frequency
between the nitrogen relaxation frequency and the oxygen relaxation frequency and is dominated by the oxygen vibrational absorption. The classical and rotational losses dominate in the third regime for frequencies
above the oxygen relaxation frequency.
The data in figure 3 include a wide range of frequencies. For aircraft noise problems, the frequency range of interest is normally limited
to less than i0 000 Hz. To further demonstrate the properties of the
absorption coefficient,
the classical and rotational,
nitrogen vibrational, oxygen vibrational,
and total coefficients are plotted as functions of frequency in figures 4 to 7. The effect of changing relaxation
frequency is shownby the lines of constant humidity in figures 5 to 7.
All data are for standard sea level temperature and pressure, and all four
figures are plotted to the samescale to allow direct comparisons between
the figures.
Finally, the relaxation frequencies which correspond to each
constant humidity line are shown in figures 5 to 7.
The total absorption-coefficient
curves on figure 7 demonstrate how
the characteristics
of the absorption coefficient dramatically change as a
result
of
changing
properties
change,
frequency
range
can
assist
the
coefficient
The
cient
_
in
_
height.
loss
in
quency.
is
the
is
altitude
y,
y
absorption
p
form,
absorption
is
a
falls
atmospheric
within
the
7 with
figure
figure
dimensional.
dimensionless
as
per
unit
function
of
Atmospheric
humidity
To
, and
3
express
absorption
a
in
- HI)/RT
coefficient
wavelength
coeffi-
is
used
in
altitude
the
is
defined
ATM,
(ATM),
only.
to
the
and
given
In
fre-
the
temperature,
The
dimensionless
as
(14)
r
h
a
absorption
conditions.
humidity,
ANOPP
of
ground
standard
pressure,
function
the
expresses
under
temperature,
Table
humidity
from
coefficient
Module
are
discussed
= Mgr(H
coefficient
absorption
Atmospheric-Properties
pressure
of
the
(13)
nepers
a
the
and
that
of
as
average
of
is
For
values
term.
coefficient
dimensionless
_
term
Comparison
dominant
dimensionless
the
_Cr/f
the
terms
As
absorption
changes.
identifying
defined
pressure,
The
dominant
interest
in
The
general,
the
of
frequency.
atmospheric-absorption
=
where
relaxation
from
as
expressed
ATM
gives
functions
as
a
of
function
3.1-6
temperature
y.
T*,
Therefore,
of
y
and
the
f.
The total change in sound intensity due to atmospheric absorption is
an integral over the length of the conical ray tube from the source to
the observer. An average absorption coefficient is defined as this integral divided by the length. Since the absorption coefficient is a function of y and f only, the average dimensionless absorption coefficient
from ground level to somevalue y is given by
ifoy
(f,y)
Equation
(15)
cient.
= Y
is
also
Equations
_cl
(O_Cr/f)
valid
(I0),
+
for
(ii),
_rot
=
=
(9.555
(15)
dy
each
and
component
(12)
are
of
then
the
absorption
rewritten
(T*)
(6.207
x
coeffi-
as
dy
10-9)_f0Y
Y
(16)
i/2/p*
2ffrl,o
_vib,o
x
exp
x
10-4)if0Yy
[7 .771(T*
(T*) -5/2
-
I)/T*J
f2
+
f2
rl,o
(17)
dy
and
_vib,n
=
(1.683
x
The
temperature
tions
of
This
altitude
y.
T
The
, pressure
total
module
produces
H
frequency
Module
absorption
effects.
_cl
p
+ _rot
corrects
a
fo
y
2ffrl,n
(T*) -5/2
-
average
:
Propagation
1
exp[ll.633(T*
_(f'Y)
and
x 10-4)
f2
I)/T*]
, and
table
f
of
for
the
3.1-7
use
sound
f2
rl,n
(18)
dy
relaxation
dimensionless
+ _vib,o
+
frequencies
absorption
are
func-
coefficient
is
(19)
+ _vib,n
_(f,y)
by
the
intensity
for
a
range
Propagation
values
of
values
Module.
for
of
The
atmospheric-
the
REFERENCES
i. Sutherland, Louis C.: Review of Experimental Data in Support of a
Proposed NewMethod for Computing
Atmospheric
Absorp£ion
Losses.
DOT-TST-75-87,
2.
American
National
Absorption
23-1978),
U.S.
of
Dep.
Standard
Sound
American
by
Natl.
Transp.,
Method
the
May
for
1975.
the
Atmosphere.
Stand.
Calculation
ANSI
Inst.,
3.1-8
Inc.,
of
SI.26-1978
June
23,
the
(ASA
1978.
TABLEI.-
Constant
STORED
PRIMARY
CONSTANTS
U.S. Customary
Units
SI Units
340.294
C
r
-
•
.
gr
.......
M
•
Pr
.
.
•
.
.
9.806
.
•
•
m/s
65
1.225
.......
kg/m
m2/K-s
288.15
r
•
o
.
•
•
•
32.1741
m/s 2
ft/s
ft/s
3
2
K
0.002
49
377
718.96
slug/ft
3
ft2/°R-s
2
518.67°R
.
Xn
.......
0.781
0.781
XO
.......
0.209
0.209
n
Oo
"
"
"
•
• "
"
.......
3.1-9
2
28.9644
28.9644
•
8314.32
T
1116.45
•
3352.0
K
2239.1
K
6033.6°R
4030.38°R
T.:
1.1//
400
T
= 1.00
3O0
=
N
85
W-
U
e-,
GJ
200
o"
CIJ
SIJc
0
K
m
p,-,
100
P* = 1.0
I
0
I
0.2
Absolute
Figure
i.-
0.4
Humidity,
Relaxation
i
I
0.6
h,
frequency
3.1-10
0,8
percent
for
I
1.0
mole
nitrogen.
fraction
I
1.2
4
2
lx10 4
N
o
8
r-q--
U
C
O"
cO
K
lx10
3
8
6
2x10 2
0
I
I
i
I
I
I
0.2
0.4
0.6
0.8
1.0
1.2
Humidity,
h,
Absolute
Figure
2.-
Relaxation
3.1-11
percent
frequency
mole
for
fractton
oxygen.
10
S4_
E
10 -1
1-
d
10-2
2
q..
qo
Oxygen
Relaxatlon
Frequency
10"3
o
so
(#1
10-4
o
F-
L
10"5
p*=
Nitrogen
Relaxatlon
Frequency
1o-6
10
T*
3.-
I
I
I
10 2
10 3
10 4
10 5
Typical
sea
- 1.0
I
Frequency,
Figure
1.0
total
level
absorption
temperature
f,
Itz
coefficient
and
pressure.
3.1-12
for
standard
I
10 6
ix10-I
8-6-4-k
2--
E
1xI0-2_
8--
c-
6-o
f,.
4_
a
+
P
U
2--
.2
C
U
Ixi0-3_
8-6--
G,I
0
4--
cO
2-f..
0
1xI0-4_
8-6--
e0
4-0
2--
lx10
U
"5_
8--
T* = 1.0
6-tO
4--
I II
2x10-6
2xlO
4
Frequency,
Figure
4.-
Classical
plus
sea
level
rotational
temperature
3.1-13
6
f,
absorption
and
8 lx10 3 2
4
6
8 lx10 4
Hz
coefficient
pressure.
for
standard
S_U
E
I/I
L
e_
e-
JO
m
2
eCP
-e-
qq.-
h=
1.20
0
(J
h " 0.60
0
q--
4_
00
(/1
.0
lx10 "4
8
h = 0.20
e,o
h = 0.08
c
o
o
h = 0
lx10-5
8
6
C) Nitrogen
Relaxation
Frequency
4
2 ,io-6
I I II
2x10
4
6
8 lx1022
Frequency,
Figure
5.-
Nitrogen
sea
4
6 8 lx103
f,
l
I
I II
2
4
6
8 lx104
Hz
vibrational
absorption
level temperature
and
3.1-14
coefficient
pressure.
for
standard
Ixi0-I_
8-i
6-4--
2--
T* • 1.0
p* = 1.0
[_ Oxygen
ReIaxatlon
Frequency
, i
2x10"62xlO
4
6
8 lx102
2
4
Frequency,
Figure
6.-
Oxygen
vibrational
sea
level
6
f,
absorption
temperature
3.1-15
8 lx103
j i_l
4
6
8 lx104
Hz
coefficient
and
2
pressure.
for
standard
lxi0 -2
8
6
f,.
4
4-*
E
ul
$.
(P
2
(P
e-
lxi0 -3
8
l
6
eGJ
4
QJ
0
2
C
0
e-I
S.
0
lx10 -4
8
6
4
O
p-
Q
2
Nitrogen
[] Oxygen
lxlO
-5
8
T*
Relaxatlon
Relaxation
Frequency
Frequency
- 1.0
6
p" = 1.0
I
2x10 -6
2xi0
I
II
I
I
4
6 8 lx10 2 2
4
Frequency,
Figure
7.sea
Total
level
absorption
temperature
I
!
,,I
6 8 lx10 3 2
4
6
III
f, Hz
coefficient
and
for
pressure.
3.1-16
standard
8 lx10 4
3.2
GROUND
REFLECTION
AND
ATTENUATION
MODULE
INTRODUCTION
The
and
noise
landing,
ground.
when
The
influence
the
of the
reflection
surface
observer
reflection
propagation
The
for
aircraft
in
most
the
attenuation
sound
are
under
change
sound
and
during
both
effects
waves
accurately
concern
aircraft
causes
a
of
the
reflection
to
of
and
of
ground
order
is
and
Earth's
surface
and
attenuation
waves.
accounted
by
the
ground
on
presence
to the
produced
close
are
these
to
a
the
the
significant
conditions.
in the
sound
waves
and
the
attenuation
predict
take-off
must
aircraft
The
spectrum
creation
due
of
be
properly
noise
at
the
observer.
Sound
may
be
waves
is
dependent
greatly
characterized
by
reflected
intensity
be
by
waves
The
for
function
and
dated
(ref.
assumes
the
source
with
the
that
The
4)
Ground
effects
of
is
is
length
difference,
distance.
the
a
data
and
which
for
in
incidence
incoherence
C
coherence
c
speed
F
spherical-wave
f
frequency,
of
is
also
sound
Finally,
of
these
(ref.
l)
is
Delany
a
and
locally
spheri-
effects
model
has
computes
coefficient
surface
form
and
waves.
as
a
the
must
Chien-
Bazley
impedance
reacting
(refs.
Module
frequency,
function
3.2-1
the
shift.
All
Scholes
and
(ft/s)
Hz
can
free-field
This
and
dimensionless
coefficient
shape
waves
diminish
atten-
surface
normal
dimensionless
angle,
m/s
is
source.
constant
sound,
the
ground
SYMBOLS
a
the
surface
of
A
sound
or
ANOPP
with
attenuation,
tabulated
to
Attenuation
a
amount
model.
Parkin
is
Earth's
The
phase-angle
influence.
point
of
the
addition
theory,
that
The
enhance
on
chosen
This
reflection,
either
complete
the
characteristics.
impedance.
ground
a
Reflection
factor,
factor
the
to
ground.
surface
can
in
model
3),
the
depending
produce
2).
ground-effects
the
to
to
(ref.
(ref.
plane
effects
due
the
and
parallel
of
acoustic
produced
ground-effects
theory
on
ground
be
nearly
absorption
directly,
waves
accounted
Soroka
the
can
acoustic
the
a complex
received
surface
cal
propagate
by
uation
be
which
attenuated
5
uniform
been
and
a
vali6).
table
of
the
incorporating
The
function
source-to-observer
groundof
path-
G
ground-effects
H
source
h
observer
i
unit
K
constant,
k
wave
factor
altitude,
m
altitude,
imaginary
(ft)
m
(ft)
nttmber
21/(6Nb)
number,
2_f/c
number
of
subbands
Nd
number
of
ground
<p2>
mean-square
R
magnitude
of
r
distance,
m
U
unit
function
argument
Ar
path-length
E
constant,
D
=
8
incidence
band
dips
Pa 2
complex
(Ib2/ft
4)
spherical-wave
reflection
coefficient
(ft)
of
complex
i/3-octave
pressure,
step
F
per
complex
plane-wave
spherical-wave
reflection
difference,
K
-
m
reflection
coefficient
coefficient
(ft)
1
2_Qf/o
complex
angle,
specific
P
density,
kg/m
O
specific
flow
T
=
deg
ground
3
admittance
(slugs/ft
3)
resistance
(kr2/2i)i/2(cos
8 +
of
ground,
_)
Subscripts:
c
center
ff
free
gr
ground
£
lower
band
value
field
effect
limit
3.2-2
kg/s-m
3
(ib/s-ft
3)
u
upper limit
1
direct
2
reflected
INPUT
The basic independent variables for the model are path-length difference, incidence angle, frequency, and source-to-observer distance.
The range of path-length differences is computedwithin the program based
on the user-specified number of ground dips. The ranges for other variables are specified by upper and lower limits.
The atmospheric properties
of density and speed of souDd and the specific flow resistance of the
ground are also required. These values are assumedto be constant
throughout the model. The numberof I/3-octave subband intervals adjusts
the predicted effect for bandwidth. The incoherence constant is an
empirical quantity which limits cancellation effects.
The range and
default values of each input are given in table I.
a
incoherence coefficient
speed of sound at the observer, m/s (ft/s)
(f£,fu)
Nd
frequency lower and upper limits,
Hz
number of subbandsper i/3-octave
band
number
of
ground
(rl,r u)
source-to-observer
(e£,8u)
incidence
P
air
a
specific
dips
at
flow
be
included
distance
angle
density
to
lower
the
lower
and
upper
observer,
resistance
upper
limits,
kg/m
of
and
the
3
m
(ft)
deg
(slugs/ft
ground,
limits,
3)
kg/s-m
3
(slugs/s-ft
3)
OUTPUT
The
tion
of
incidence
module
four
produces
a
table
dimensionless
angle,
frequency,
and
Ground
kAr
path-length
COS
cosine
of
of
the
parameters:
Effects
angle
3.2-3
path-length
source-image-to-observer
difference
incidence
ground-effects
Table
factor
as
difference,
distance.
a
func-
cosine
of
= 2_pf/O
kr 2
image distance
G(kAr,cos @,n,kr 2)
ground-effects
factor
METHOD
The ground-effects geometry is shown in figure i. A source is
located at an altitude
H over a ground plane. Sound arrives at the
receiver at a height h from the direct path r I and from a reflected
path r2, which appears to the observer to be from an image source. The
incidence angle of the reflected wave is 8. The path-length difference
Ar = r 2 - r I is the most significant parameter of ground effects.
As
shown in figure 2, the path-length difference can be approximated in
terms of the observer height and the incidence angle as
Ar _ 2h cos @
(i)
The Chien-Soroka theory is derived from a solution to the wave equation in the half space of figure i.
The derivation of the theory is presented in references 1 and 2. The resulting expression for the meansquare pressure with ground effect <p2>_
r_
is
<p2>gr
where
ence
is
<p2>ff
is
the
coefficient,
the
term
in
and
G
The
in
the
C
wave
complex
equation
defined
phase
+
is
(2)
kAr)_
pressure,
R
is
the
C
to
as
the
magnitude,
reflection
referred
is
the
coher-
and
coefficient.
The
ground-effects
(3)
f
C
relation
is
a
Gaussian
distribution
=
exp
the
+
follows:
approximation
The
(_
number,
reasonable
is
cos
spherical-wave
(2)
as
2RC
mean-square
coefficient
which
A
a
is
= <p2>gr/<p2>f
assuming
function.
is
+ R2
free-field
the
in
coherence
process.
where
of
brackets
G
energy
k
argument
factor
by
= <p2>ff[l
[- (akAr)
incoherence
incoherence
is
the
fraction
maintained
for
of
of
throughout
the
the
coherence
the
initial
the
acoustic
propagation
coefficient
is
made
form
(4)
23
constant
constant
and
is
exp
normally
3.2-4
denotes
given
the
exponential
a value
of
0.01,
which corresponds to a value of C of 0.37 at a Ar value of 16 wavelengths. After substitution of equation (4), the ground-effects factor
becomesthe following:
G : 1 + R2 + 2R exp[-(akAr)23 cos (d + kAr)
The
wave
Chien-Soroka
reflection
Re id
where
F
is
theory
(ref.
coefficient
=
the
F
+
can
(i -
complex
the
tion
F(T)
accounts
complex
(6) is
for
specific
F(y)
=
the
- _T
that
the
as
complex
spherical-
follows:
(6)
plane-wave
reflection
coefficient
given
as
8 + v)
(7)
spherical-wave
ground
1
shows
expressed
F)F(T)
F = (cos 8 - _)/(cos
and
2)
be
(5)
shape.
admittance.
The
In
function
this
equation
F(T)
in
is
equa-
C8)
W(iY)
where
y
and
W
is
=
the
W(z)
For
error
any
value
function
(kr2/2i)i/2(cos
following
is
ITl
+
complex
i f__
= _
_
of
@
>
e -t2
z_-_
dt
i0,
an
used.
This
_)
(9)
error
function:
(Im(z)
asymptotic
allows
approximation
F(T)
to
be
for
expressed
>
O)
the
as
(I0)
complex
the
following:
F(T) = -2 _U[-Re
(T)] Te
1
T2
(Ii)
2T 2
3.2-5
3
+
(2T2)
2
where U is the unit step function defined as follows:
u(s) = i
(s > 0)_
U(S)
(S
=
1/2
u(s) = 0
The
ground
remaining
where
the
9.
graph
of
For
plified
the
an
is
n
=
2
cos
assumed
(_
small.
For
standard
where
which
2
original
The
integral
the
3)
i(4.36D)-0"73)_
_
is
=
complex
specific
developed
the
following
(13)
-I
is
R =
in
(_ =
i,
and
figure
3.
0),
the
theory
_
0.
The
=
is
greatly
expression
simfor
the
to
exp[-(akAr)2_
kAr
cos
(15)
(kAr)
only.
predictions
are
the
given
effect
integrated
is
shown
surface
i,
reduces
of
of
over
included
variation
the
as
for
finite-frequency
finite
bandwidth,
band,
a
but
only
variable
of
the
other
terms
is
acceptable
for
in
in
bandwidths.
the
the
the
the
ground-
variation
of
integration.
width
i/3-octave
or
of
the
It
1 band
is
is
narrower
4).
purposes
of
i/3-octave
Nb
has
F
approximation
(ref.
+
hard
0,
kAr)
the
This
bands
+
is
+
that
-0"75
admittance
approximate
factor
term
=
noise
to
is
(ref.
(14)
function
Actual
Bazley
frequency
factor
a
order
effects
(6.86_)
acoustically
G
determined
_:
ground
ground-effects
In
+
be
and
2_Qf/o
since
which
for
E1
to
Delaney
dimensionless
=
A
parameter
equation
x) =
(12)
0)
(s < 0)
admittance
empirical
=
a
analyzing
bands
of
is an odd
number.
center
frequency
i/3-octave
ratio
for
of
attenuation
and
reflection
sound
are
subdivided
Using
equal
an odd
number
to the center
into
effects,
Nb
gives
a
frequency
the
subbands,
center
subband
of the
band.
the
averaging
subband
the
limit
cosine
frequencies
term
over
following:
3.2-6
is
the
2 I/(3Nb)
subband
is
, so
the
that
the
= _i f Kfc cos (_ + kAr) df
Af jfc/K
<cos (_ + kAr)>
(16)
I/(6Nb)
where
K
of
subband.
the
band,
=
an
2
,
_f
=
Assuming
approximation
(K -
K-l)fc,
that
for
_
and
fc
remains
equation
is
the
constant
(16)
is
expressed
sin
center
frequency
throughout
the
sub-
as
(£kcAr)
(17)
<cos
where
kc
duces
(_
is
the
+
the
kAr)>
=
subband
following
cos
(_
center
final
+ kcAr)
wave
EkcA
number
expression
for
r
and
the
E
=
K
-
i.
=
1
+
R2
+
2R{expl-(akAr)_l_
cos
(_ +
pro-
factor:
sin
G
This
ground-effects
(ekAr)
(18)
kAr)
ekAr
where
For
it
an
band
is
understood
that
acoustically
hard
k
refers
surface,
to
the
the
subband
averaged
sin
plot
of
of
an
It
=
G
is
as
a
2,
...,
to
minimum
evaluated
that
it
user
can
n
+
ground
each
h/r
that
most
final
is
much
r 2 _
less
rI
G
to
be
three
source
and
Ar
presented
(19)
in
figure
ground-effects
kAr
of
caused
and
reflected
is
as
the
as
4
for
function;
the
case
If
N d.
3.2-7
are
for
which
separation
Under
cos
In
8
z
this
2hH/r
the
of
after
the
kAr
is
is
the
range
divided
the
is
commonly
points
addition,
condition
I.
value
of
of
not
G
fifth
so
met,
noise-generation
it
can
be
dip
small
the
frequency,
equally
This
at
be
interest
into
large.
a
of
are
condition
incidence-angle
ranges
and
intermediate
this
has
where
cancellation
waves
values
constant.
interest
i)_,
however,
value
observer
factor
-
the
that
at
increasing
for
(2n
by
imperative
well
for
and
2h
the
of
are
unity.
z
that
variable
of
and
than
is
(EkAr)
EkAr
nodes
define
in
conditions
the
4
It
different
These
(kAr)
direct
dip
distance,
user
inputs.
intervals.
For
the
dips.
assumed
a
kAr
values
These
ground
variation
be
of
at
adequately
provide
valid,
i.
cos
figure
between
of
the
can
from
source-to-observer
are
number.
the
surface.
occurring
as
to
5)
function
hard
intensity
=
2{exp[-(akAr)2]}
apparent
nodes
referred
(Nd
2 +
of
i,
sound
the
=
acoustically
series
n
wave
for
is
G
A
center
expression
means
assumed
are
spaced
models
that
The output of the module is a table of the ground-effects-factor
values as functions of four dimensionless variables:
kAr, cos 8, _,
and kr 2. For a hard surface, the last three variables do not affect the
ground-effects factor.
In this case, the module output is a fourdimensional
table
with
_, and
kr 2.
The
table
as
it would
In
been
and
6.
case.
in
reference
compared
Recent
that
the
with
the
the
and
are
generally
the
Chien-Soroka
of
ground
in
fair
method
smaller
by
a
for
method
from
T-38A
flyover
in
have
in
data
reference
attenuation
of
5
this
given
8
show
agrees
the
8,
this
references
agreement
measured
be
cos
result
this
good
given
the
may
and
to
amplitudes
than
method
dimensions
data
ground
The
reflection
the
the
same
in
kAr.
with
data
maximum
frequency.
cating
effects
are
of
of
predicted
flyover
frequency
measured
each
propagation
this
747
attenuation
the
in
produces
table
effects
data
of
Boeing
predicted
that
entry
ground
comparisons
7 and
the
one
ground-to-ground
predictions
reference
well
4,
to
The
only
interpolation,
logic
for
a one-dimensional
very
predicted
amplitudes,
conservative
indi-
estimate
of
attenuation.
REFERENCES
i.
Pao,
S.
Paul;
Ground
2.
3.
Chien,
C.
F.;
and
Plane.
1975,
9-20.
pp.
Delany,
M.
pp.
E.;
Parkin,
P.
and
Sound
P.
Sound
7.
H.;
a
W.:
Oncley,
NASA
Sound
& Vib.,
E.
N.:
Appl.
E.:
Paul
B.:
TP-II04,
Prediction
43,
no.
Acoustical
Along
an
Nov.
8,
i,
Properties
vol.
3,
of
1978.
Propagation
vol.
Acoust.,
Willshire,
NASA
Scholes,
Jet
Engine
no.
2,
of
Apr.
Fibrous
1970,
and
Scholes,
Jet
Engine
a
& Vib.,
William
of
William
Engined
W.
i,
no.
W.
Close
vol.
2,
L.,
Jr.:
Aircraft
of
Aircraft
Sideline
Noise
1978.
Close
H.;
TP-1747,
Willshire,
Ratio
and
vol.
From
Sound
Prediction
TM-78717,
& Vib.,
Propagation
8.
W.
Sound
Bazley,
NASA
From
Parkin,
J.
J.
William
Sound
6.
Soroka,
and
Noise.
105-116.
Zorumski,
J.
R.;
Aircraft
Materials.
Attenuation.
5.
Alan
on
Impedance
Absorbent
4.
Wenzel,
Effects
no.
E.:
The
to
i,
the
Jan.
E.:
4,
the
Oct.
Assessment
Noise:
The
Propagation
at
1964,
The
to
Horizontal
Ground,
pp.
1-13.
Horizontal
Ground,
of
T-38A
Propagation
at
1965,
L.,
Jr.:
Lateral
Noise.
NASA
pp.
353-374.
Ground
Effects
Flight
on
Experiment.
Attenuation
of
TM-81968,
1981.
3.2-8
of
Hatfield.
1980.
Aircraft
of
Radlett.
High-By-Pass
the
TABLEI.-
RECOMMENDED
RANGES
FORINPUTPARAMETERS
Input
a
c,
•
,
•
m/s
.
Minimum
°
°
•
0.001
•
......
f£,
Hz
......
fu'
Hz
......
Default
Maximum
0.010
300
0.i00
340.294
13
400
5O
2000
4000
Nb
........
1
5
9
Nd
........
2
5
i0
i0
I0
r£,
m
.
r u , km
. .
.
. .
I0
......
e£,
deg
.....
0
@u'
deg
.....
89
P,
kg/m
3
_,
kg/s-m
.....
3
....
1.0
1.0
3.2-9
x 105
i0
89
1.225
2.5
x
105
1.5
5.0
x
105
SOURCE
_
c:2
_
IMAGE
Figure
i.-
Ground-effects
geometry.
SOURCE
IMAGE
Figure
2.-
Derivation
of
path-length
3.2-10
difference.
N
m
0
0
m
_
u
e-
m
N
0
-H
0
(%1
0
C_
I1)
N
_
_D
,-4
Q
k
II
g4
d
e_
0
.,._
t,..l
0
d
-,-.I
°_
C)
,
I
I
I
,,-4
0
0
d
d
d
I
_D
0
I
0
o%
0
d
0
!
!
_o
epn_uSeN
suel.pea
3.2-11
',_
6jr
1°
o
!
m
oH
r_
2
J
0
O
u
"2
U.
u
-4
w-
e-
-6
o
,-8
,-10
,.12
1
2r
Dimensionless
Figure
4.-
Ground-effects
I
I
I
4r
path
factor
I
I
6_r
length
for
difference,
acoustically
3.2-12
8_"
k_r
hard
surface.
I
1
lO_r
4. SOURCE
NOISE
PARAMETERS
4.1
FAN
NOISE
PARAMETERS
MODULE
INTRODUCTION
Fan
jet
and
ANOPP.
The
physical
The
data
for
is
a
purpose
for
fan
of
turbojet
fan
noise
and
data
the
noise
methods
Fan
Noise
Parameters
Module
by
Heidmann's
turbofan
exit
are
then
(ref.
states
are
converted
to
The
provided
a
function
required
computed.
SYMBOLS
A
area,
m 2
(ft 2)
C
speed
of
sound,
D
fan
diameter,
aircraft
mass
flow
n
number
number
P
pressure,
R
dry-air
T
temperature,
t
time,
s
¥
ratio
of
power
setting
kg/s
speed,
of
gas
times
(lb/ft
2)
constant,
K
specific
kg/m
(slugs/s)
Hz
source
Pa
(ft/s)
(ft)
rate,
rotational
density,
m
Mach
N
p
m/s
3
m2/K-s
(OR)
heats
(slugs/ft
4.1-1
produced
are
is
to
I)
by
turbo-
included
in
generate
for
fan
the
user.
the
noise
engines.
flow
first
method
characteristics.
prediction
total
prediction
are
flight-path
of
noise
and
entrance
the
the
part
Fan
required
engine-state
from
significant
engines.
parameters
prediction
These
noise
turbofan
3)
2
(ft2/°R-s
2)
by
of
fan
time,
using
parameters
Subscripts:
e
engine
i
entrance
j
exit
t
total
ambient
Superscript:
*
dimensionless quantity
INPUT
This module provides the parameters for a typical axial-flow fan as
shown in figure i. The entrance and exit flow states for the fan are
required from the user. The engine power setting, aircraft Machnumber,
and ambient density and speed of sound are provided by the engine variable
table.
Fan
engine
aircraft
Moo
mass
power
Mach
flow
rotational
total
Tt, i (_,M)
Entrance
number
rate,
re
AeP_/R_
re
speed,
temperature,
M
aircraft
power
State
setting
_
R_/D
re
Fan
engine
Flow
T
Exit
Flow
State
setting
Mach
number
o0
"*(_,M
mj
Tt, j(_,M
)
mass
)
total
flow
rate,
re
temperature,
Engine
t
S
source
(t)
aircraft
time,
Mach
Q_AeP_/R_
re
_
T
Variable
Table
s
number
4.1-2
_(t)
engine
c_(t)
ambient
speed
p_ (t)
ambient
density,
power
setting
of
sound,
kg/m
m/s
3
(ft/s)
(slugs/ft
3)
OUTPUT
The
outputs
execution
of
to
the
this
fan
module
noise
Fan
n
number
of
t
source
time,
m. (t)
entrance
1
m
(t)
exit
3
N* (t)
AT*
source
(t)
total
Noise
time
flow
flow
rotational
ambient
M
(t)
aircraft
p_ (t)
speed,
speed
re
re
re
of
required
source
for
time.
p
c
p
c A e
Ae
c_/D
rise
of
Mach
ambient
parameters
function
Parameters
rate,
across
Ambient
(t)
physical
a
values
rate,
temperature
c
the
as
s
mass
mass
are
modules
fan,
re
T
Conditions
sound,
m/s
(ft/s)
number
density,
kg/m
3
(slugs/ft
3)
METHOD
The
fan
the engine
tables
must
entrance
to a function
as a function
the
user.
The
must
by
be
the
and
power
setting
be provided
fan
of
of
source
source
entrance
converted
to
exit
time.
time
and
the
flow
_
and
directly
as
exit
states
are
the aircraft
by
the user.
expressed
as
Mach
number
These
data
a
M
are
The
engine
provided
variable
table
gives
by the
Flight
Dynamics
mass
rate
referred
flow
variables
and
used
the
by
fan
the
Fan
function
of
.
These
converted
M
and
Module
or
rotational
Noise
speed
Module
relations
-*
m
m
(t)
=
(i)
47
4.1-3
and
.
N
N (t) =
(2)
where the ambient value of the ratio of specific heats y is 1.4.
Finally, the temperature rise across the fan AT* is the difference
total temperature,
AT*
=
Tt,j
-
(3)
Tt,i
REFERENCE
i.
Heidmann,
Source
M.
F.:
Noise.
Interim
NASA
TM
in
Prediction
X-71763,
Method
1975.
4.1-4
for
Fan
and
Compressor
Ttj
-_
mi
Tt ,i
N
j7
Figure
i.-
Schematic
diagram
4.1-5
of
a typical
axial-flow
fan.
4.2
CORE
NOISE
PARAMETERS
MODULE
INTRODUCTION
Core
noise
turboprop,
are
is
included
to
tion
data
for
a
significant
in
generate
and
ANOPP.
the
part
turbojet
The
purpose
physical
of
the
engines.
of
parameters
total
Core
the
noise
noise
Core
Noise
required
for
a
produced
by
prediction
methods
Parameters
Module
core
predic-
noise
module.
The
These
is
turbofan,
core
entrance
engine-state
from
core
and
data
the
flight-path
noise
prediction
exit
are
flow
first
states
converted
to
characteristics.
are
then
are
The
provided
a
function
required
computed.
SYMBOLS
A
area,
c
ambient
M
aircraft
mass
m 2
(ft 2)
speed
Mach
flow
number
p
pressure,
R
dry-air
T
temperature,
t
time,
Q
density,
of
source
Pa
gas
sound,
m/s
kg/s
(slugs/s)
times
(ib/ft
2)
constant,
K
(OR)
kg/m
setting
3
(slugs/ft
Subscripts:
e
engine
i
entrance
j
exit
m2/K-s
s
power
(ft/s)
number
rate,
n
engine
of
4.2-1
3)
2
(ft2/°R-s
2)
by
of
core
the
user.
time,
using
parameters
t
total
ambient
Superscript:
*
dimensionless quantity
INPUT
The entrance and exit combustor flow states are required from the
user. The engine power settings and aircraft Machnumbers in the engine
variable table are provided by the Flight DynamicsModule or the user.
Core
engine
power
aircraft
Entrance
Flow
State
setting
Mach
number
o*
m l (_,M)
mass
flow
rate,
Pt,i(rf'M
)
total
pressure,
Tt,i(_r'M
)
total
temperature,
re
AeP_/R_
re
p_
re
Core
-g
engine
M
aircraft
total
Tt, j (F,M)
power
M
source
Flow
State
setting
Mach
number
temperature,
time,
aircraft
(t)
T
Exit
re
Engine
t
_
T_
Variable
Table
s
Mach
power
number
_(t)
engine
setting
c(t)
ambient
speed
p_(t)
ambient
density,
of
sound,
kg/m
3
m/s
(ft/s)
(slugs/ft
3)
OUTPUT
The
the
outputs
execution
to
of
the
this
core
module
noise
are
the
modules
physical
as
a
4.2-2
parameters
function
of
required
source
time.
for
Core
source
Noise
time
Parameters
number
of
values
t
source
time,
m.
(t)
l
combustor
entrance
mass
Pi (t)
combustor
entrance
total
pressure,
T. (t)
combustor
entrance
total
temperature,
combustor
exit
s
flow
rate,
re
p c
re
p_
re
Ae
T
1
T. (t)
3
total
temperature,
Ambient
c
(t)
ambient
speed
M
(t)
aircraft
Mach
ambient
p_(t)
of
sound,
re
T
Conditions
m/s
(ft/s)
number
density,
kg/m
3
(slugs/ft
3)
METHOD
and
tion
These
the
the
rate
Noise
A
schematic
exit
flow
of
the
diagram
states
data
engine
is
are
power
be
Module
by
to
tables
table.
converted
the
a
typical
in
setting
converted
core
flow
state
engine
variable
must
of
shown
to
a
with
In
the
combustor
figure
_
i.
and
function
the
of
respect
addition,
referred
depicting
These
flow
aircraft
source
time
to
M (t)
and
the combustor
variables
used
the
states
Mach
by
entrance
are
a
number
funcM
.
interpolating
_(t)
entrance
by
the
values
mass
from
flow
Combustion
relation
m.
m.
where
the
(i)
(t) =
ambient
value
of
the
ratio
4.2-3
of
specific
heats
y
is
1.4.
/
/
/
/
/
/
//
ii
o
\
0
4-)
w
F-
0
u-
I
/
0
L
/
J
w
w
z
I
z
y
.,-I
I#l
wz
/
i_ t-,i
z_
DO
/
/
/
4.2-4
4.3
TURBINE
NOISE
PARAMETERS
MODULE
INTRODUCTION
Turbine
turbofan
noise
and
modules
are
Module
noise
prediction
The
data
eters
is
for
the
are
flight-path
of
the
low-power
purpose
physical
and
and
data
turbine
The
the
turbojet
entrance
part
during
ANOPP.
generate
for
turbine
significant
engines
in
to
engine-state
from
a
included
eters
These
is
turbojet
of
exit
flow
states
converted
prediction
to
area,
m 2
(ft 2)
c
speed
of
sound,
D
turbine
f
fuel-to-air
h*
specific
enthalpy,
re
ha
absolute
humidity,
percent
M
Mach
M
m/s
rotor
a
m
then
required
computed.
(ft)
ratio
RT
mole
fraction
number
aircraft
Mach
number
co
m
mass
N
rotational
n
number
P
pressure,
R
dry-air
R
gas
T
temperature,
t
time,
flow
rate,
kg/s
speed,
of
Pa
times
(lb/ft
2)
gas
constant,
constant,
m2/K-s
K
(slugs/s)
Hz
source
m2/K-s
2
(OR)
s
4.3-1
2
(ft2/°R-s
by
noise
Param-
for
provided
function
(ft/s)
diameter,
Noise
required
SYMBOLS
A
produced
Turbine
Turbine
are
The
are
noise
turbine
engines.
characteristics.
noise
the
parameters
turbofan
first
total
operations.
(ft2/°R-s
2)
2)
by
of
the
time,
turbine
user.
using
param-
ratio
¥
of
engine
P
specific
power
heats
setting
density,
kg/m
3
specific
entropy
(slugs/ft
3)
function,
re
R
Subscripts:
e
engine
i
entrance
j
exit
s
static
t
total
ambient
Superscript:
*
dimensionless
quantity
INPUT
This
as
shown
are
the
from
settings,
engine
provides
figure
required
power
by
module
in
the
i.
The
the
user
aircraft
variable
parameters
entrance
for
Mach
M
aircraft
power
Mach
exit
predicting
numbers,
typical
flow
the
and
a
Entrance
Flow
State
setting
number
0o
mass
N (_,_)
Tt, i (_ ,M
flow
rotational
)
total
rate,
re
speed,
temperature,
Turbine
engine
aircraft
power
Mach
A
re
re
p_/R_
R_/D
T
Exit
Flow
State
setting
number
4.3-2
axial-flow
states
turbine
ambient
tables.
Turbine
engine
for
and
for
noise.
densities
the
The
are
turbine
turbine
engine
provided
A.3 (_,M)
turbine
f (_,M)
fuel-to-air
exit
area,
re
A e
ratio
o*
m
3
(_,M)
Pt,j(_r'M
Tt,
)
j (_,N)
exit
mass
flow
rate,
exit
total
pressure,
exit
total
temperature,
source
M
time,
aircraft
(t)
Mach
p_
T
Variable
Table
number
engine
power
c
(t)
ambient
speed
p
(t)
ambient
density,
absolute
re
_
s
_(t)
ha(t)
AeP_/R_
re
Engine
t
re
setting
of
sound,
kg/m
humidity,
m/s
3
(ft/s)
(slugs/ft
percent
3)
mole
fraction
OUTPUT
The
the
outputs
execution
of
of
this
the
module
turbine
are
noise
Turbine
n
number
of
t
source
time,
f(t)
fuel-to-air
m
mass
(t)
N* (t)
Tt, i (t)
T
(t)
source
as
parameters
a
Parameters
values
s
rate,
rotational
exit
Noise
time
physical
ratio
flow
entrance
the
modules
re
speed,
total
static
D
re
c A e
c_/D
temperature,
re
temperature,
re
T
T
s,3
Ambient
c(t)
ambient
h a (t)
absolute
speed
of
humidity,
sound,
Conditions
m/s
percent
4.3-3
(ft/s)
mole
fraction
function
required
of
source
for
time.
aircraft
Mach
ambient
number
density,
kg/m
3
(slugs/ft
3)
METHOD
The
of
the
tables
turbine
function
of
a function
the
user.
time.
source
turbine
referred
time
*
exit
_
by
flow
and
the
The
engine
as
provided
rotational
variables
N
and
setting
directly
source
of
The
the
entrance
engine
power
are provided
speed
used
by
are
variable
by
and
the
states
the
aircraft
user.
These
table
the
mass
as
function
These
to a
M
and
Dynamics
rate
noise
a
number
M .
are
converted
gives
Flight
flow
turbine
expressed
Mach
data
must
be
modules
_
Module
by
converted
the
to
relation
N
(t)
as
or
=-
(i)
m
= --
(2)
and
•,
m
(t)
where
the
ambient
turbine
exit
modules
for
temperature
variable
value
static
of
the
ratio
temperature
the
computation
of
can
be
assuming
specific
computed
the
ratio
sionless
T*t, j'
gas
can
be
of
specific
R*
constant
ratio
thermodynamic
computed
is
heats
required
work
either
y
by
the
is
turbine
extraction.
constant
1.4.
The
specific
The
noise
exit
static
heats
or
heats.
fuel-to-air
appropriate
specific
ideal
Constant
The
of
T*
s,j
by
the
Specific
heats
at
the
=
R/R
are
f,
and
Heats
turbine
computed
absolute
utility.
exit
from
humidity
The
turbine
dimen-
temperature
ha,
using
the
Mach
number
M.
3
7t
+
V 't Mj/I\
•_'-
the
total
relation
R'fPT*
=
and
the
exit
1
_ W, * t,_
A.
] Pt,j
Yt
-
1
_
)
M2
4.3-4
Yt +I
2 7t-i
(3)
as discussed in ThermodynamicUtilities.
temperature
Ts , j
T s,j
=
As
discussed
be
computed
the
turbine
exit
static
is
Tt, j 1
+
in
from
(4)
M
Variable
can
Then,
Specific
Thermodynamic
simultaneous
Heats
Utilities,
solution
of
the
static
temperature
T
s,j
: e-(*t-*s)
(5)
Pt,j
and
m
,
3.
where
sionless
_*
0
.
Aj
Pt,j
is
the
j
=
*
2ht_
T
(6)
h
.
s, 3 Pt,j
dimensionless
entropy
enthalpy.
4.3-5
function
and
h*
is
the
dimen-
_j
Tt ,i
Pt ,J
Tt ,J
N
Figure
i.-
Schematic
diagram
of
typical
4.3-6
axial-flow
turbine.
4.4
JET
NOISE
PARAMETERS
MODULE
INTRODUCTION
Exhaust
by
are
to
jet
turbojet
included
The
from
noise
the
a
The
physical
modules
engine
flight
exit
are
first
path
prediction
purpose
turbojet
nozzle
part
of
and
flow
the
then
the
prediction
Jet
Noise
Parameters
turbofan
engines.
state
is
to
the
provided
a
The
exhaust
by
area,
m 2
c
speed
of
of
required
jet
computed.
De
equivalent
Dh
hydraulic
Dp
plug
f
fuel-to-air
ha
absolute
M
Mach
M
co
(ft 2)
sound,
m/s
circular
diameter,
diameter,
(ft/s)
nozzle
m
m
diameter,
m
(ft)
(ft)
(ft)
ratio
humidity,
percent
mole
fraction
number
aircraft
mass
Mach
flow
number
rate,
n
number
P
pressure,
R
dry-air
R
gas
T
temperature,
t
time,
of
kg/s
source
Pa
2)
gas
constant,
constant,
m2/K-s
K
(slugs/s)
times
(ib/ft
m2/K-s
2
(OR)
s
4.4-1
2
(ft2/°R-s
(ft2/°R-s
2)
the
function
SYMBOLS
A
noise
noise
for
converted
total
jet
required
characteristics.
are
of
Several
parameters
for
data
significant
engines.
ANOPP.
the
engine-state
is
turbofan
in
generate
prediction
noise
and
2)
produced
methods
Module
jet
user.
time,
using
parameters
is
noise
These
data
for
jet
V
velocity,
Y
ratio
m/s
of
engine
(ft/s)
specific
power
heats
setting
P
density,
kg/m
0"
specific
entropy
3
(slugs/ft
3)
function,
re
R
Subscripts:
a
aircraft
e
engine
fe
fully
p
plug
s
static
t
total
1
primary
2
secondary
expanded
stream
stream
ambient
Superscript:
*
dimensionless
quantity
INPUT
This
tion
is
of
module
jet
required
primary
such
for
and
as
on
are
since
addition
area
must
be
It
is
inner
since
that
of
conditions
neglected
density
of
without
and
are
computed
the
not
or
be
input
the
outer
outer
-
density,
static
needed
the
of
speed
if
SPL
only
is
on
and
their
areas
of
of
the
be
the
shown
and
error
if
it
used
in
computing
normalized
corrected
is
not
mean-squared
to
standard
4.4-2
included
as
setting.
have
reference
a
is
equal
-
humidity
may
available.
decibel
to
A e
it.
the
atmo-
are
used
in
be
The
local
levels;
pressure
conditions.
to
Local
humidity
plug
area
relative
i.
jets
state
power
may
nozzle
figure
the
state
engine
engine
however,
state
Both
dual-stream
nozzle
inner
predic-
flow
associated
specified
in
sound,
variables;
for
of
primary
the
engines.
are
functions
the
for
nozzle
turbojet
required
usually
as
used
are
and
diameter
nozzle
are
primary
The
be
A 2,
will
flow
sound
i.
specified,
data
the
jets
may
and
significant
speed
if
A1
are
The
as
states
figure
sizes
must
such
These
in
areas
which
diameter
evaluation
they
the
Ap,
flow
nozzle
which
noise.
jets
engines.
the
to
given
parameters
cell
nozzle
illustrated
assumed
spheric
the
shock
single-stream
turbofan
variables
with
and
secondary
variables
In
provides
mixing
is
however,
to
Table
be
I
gives the recommendedranges and the default values for the input
parameters.
Ae
engine reference area, m2 (ft 2)
A*
P
primary nozzle plug area, re Ae
Primary-Nozzle
engine
M
power
aircraft
Flow
State
setting
Mach
number
co
nozzle
area,
re
fuel-to-air
fl (_ 'Mco)
A e
ratio
.*
m 1 (g,
mass
Mco)
Pt,l(_'Mco
)
Tt,l(g,Mco)
flow
rate,
total
pressure,
total
temperature,
re
AePco/R_
re
p_
re
Secondary-Nozzle
engine
M
power
aircraft
_
T co
Flow
State
(Optional)
setting
Mach
number
o0
A 2 (_)
nozzle
area,
re
fuel-to-air
f2 (_, Mco)
Ae
ratio
m 2 (_,M)
mass
Pt,
total
pressure,
total
temperature,
2 ('n" ,N_)
Tt, 2 (_' M
)
flow
rate,
re
AePco/R_
re
pco
re
Engine
t
source
_(t)
aircraft
7T(t)
engine
Cco (t)
ambient
speed
pco (t)
ambient
density,
h a (t)
absolute
time,
T
Variable
Table
s
Mach
power
_
number
setting
of
humidity,
sound,
kg/m
3
m/s
(slugs/ft
percent
4.4-3
(ft/s)
mole
3)
fraction
OUTPUT
The outputs to this
execution of the exhaust
Unless otherwise stated,
fully expanded jet which
module are the physical parameters required for
jet noise modules as a function of source time.
all parameters are computed for a hypothetical
has a static pressure equal to ambient pressure.
Primary
number
of
source
source
time,
Jet
time
Parameters
values
s
Afe, 1(t)
primary
jet
De,1 (t)
actual
primary
jet
equivalent
Dh,
actual
primary
jet
hydraulic
M I (t)
primary
jet
Mach
T 1 (t)
primary
jet
total
primary
jet
velocity,
primary
jet
density,
1 (t)
area,
re
Ae
diameter,
re
diameter,
re
_e
_e
number
temperature,
re
T
*
Vl(t)
re
c
*
Pl (t)
ratio
Y1 (t)
of
re
specific
heats
Secondary
Jet
area,
for
primary
jet
Parameters
Afe,2(t)
secondary
D*
actual
secondary
jet
equivalent
actual
secondary
jet
hydraulic
(t)
jet
p_
re
(Optional)
A e
diameter,
re
_e
e,2
Dh,2(t)
M 2 (t)
secondary
jet
Mach
T2(t)
secondary
jet
total
V2(t)
secondary
jet
velocity,
P2 (t)
secondary
jet
density,
Y2 (t)
ratio
of
specific
diameter,
re
number
temperature,
heats
re
re
re
T
c
Q_
for
secondary
4.4-4
jet
_e
Ambient
c(t)
ambient
speed
Ma(t)
aircraft
D_(t)
ambient
of
Mach
Conditions
sound,
m/s
(ft/s)
number
density,
kg/m
3
(slugs/ft
3)
METHOD
The
is
the
stream
of
user
has
or
static
heats
state
The
for
option
jet.
or
The
variable
the
of
method
of
nozzle
the
tables
must
a
flow
engine
be
verted
to
and
_
Module
as a function
or the
user.
second
ratio
is
of
from
function
states
power
of
the
shown
_
directly
source
of
as
setting
provided
The
ratio
of
the
total
temperature
be
ber
heats
specific
is
by
The
either
of
heats
for
as
the
Tt, I,
a
first
single-
constant
ratio
calculation
of
heats.-
The
from
the
are
expressed
Mach
These
variable
by
primary
the
the
primary
total
jet
fuel-to-air
as
number
data
are
table
a
M a.
con-
gives
Flight
Y1
is
ratio
f,
Thermodynamic
jet
specific
jet
user.
engine
provided
the
in
constant
1
aircraft
Ma
Dynamics
Jet
the
discussed
assuming
computed
for
figure
the
the
The
time
heats
h a , as
estimated
M1
module.
for
selection
in
and
by
time.
source
specific
humidity
manual.
Constant
may
this
data
specific
Single-Stream
absolute
of
this
of
computing
variables.
input
function
options
stated
dual-stream
specific
These
two
previously
pressure
The
and
and
Utilities
velocity
heats.
computed
the
portion
and
jet
density
jet
Mach
num-
ratio
of
specific
relation
--¥i / (YI -I )
Ps,l
where
*
Ps,I
The
number
=
= Pt,l
1
for
primary
and
jet
specific
s,l
= Tt,l
+
a
2
fully
static
heat
expanded
temperature
ratio
+
(i)
M
jet.
is
computed
from
the
jet
Mach
as
2
4.4-5
M
(2)
as discussed in ThermodynamicsUtilities.
•
The jet density is then
Ps,l
Pl(t)
=
(3)
R'T*
s,l
where
R*
=
velocity
R/R
is
is
given
the
I
_,
•
Finally,
the
ratio
jet
of
Pt,
(4)
specific
heats.-
static
Ps,I
--
As
heats
y_
discussed
temperature
is
is
in
1.4.
Thermodynamic
evaluated
from
Utilities,
the
relation
- (*t-*s)
=
(5)
e
1
=
where
$*
fully
jet
7
specific
primary
constant.
\1/2
_
ambient
Variable
the
gas
l_sK Ts,i_
vI=MI_
the
jet
by
,
where
primary
is
the
expanded
dimensionless
jet.
The
entropy
jet
I R*T*
Mach
number
function
is
and
then
1
for
a
Ps,l
given
by
"*
(6)
MI
where
Ys
=
is
Q
_s'l
the
, ml,
AlPs,
1
ratio,
of
specific
heats
evaluated
at
T*s,l.
Finally,
*
the
jet
equations
Jet
density
(3)
Pl(t)
and
The
equivalent
and
values
may
slightly
diameter
the
jet
velocity
V 1
are
computed
from
respectively.
The
=
geometric
occur
(4),
geometry.-
A*
fe,l
and
area
of
the
fully
expanded
primary
jet
is
"'1 * *
ml/PlVl
(7)
hydraulic
for
the
away
diameters,
nozzles
from
the
De
instead
of
nozzle
exit
and
the
D
fully
plane.
, are
based
expanded
The
on
area,
actual
which
equivalent
is
D
=
z
(8)
e,l
4.4-6
and the hydraulic diameter is
Dh,l
j
=
Dp*2
--+
-
(9)
D*P
Secondary
Computations
primary,
outer
the
for
except
diameter
secondary
that
of
jet
*
the
secondary
secondary
inner
stream
plug
nozzle.
are
identical
diameter
is
Therefore,
the
is
,I 4
=
the
the
Jet
•
1
•
A2
D 2
p
4.4-7
!,I4
•
_ A1
D 2
p
taken
to
to
hydraulic
those
be
for
the
diameter
the
first
of
TABLEI.-
RANGE
ANDDEFAULT
VALUESOF INPUTPARAMETERS
Input
parameter
2
Ae, m ......
Ap ........
Minimum
Default
0.01
_/4
0
i0
0
4.4-8
Maximum
1
/
/
/
/
/_
PRIMARY-NOZZLE
FLOW STATE
r
AI'
z
/
/
/
F
Single-stream
circular
]
SECONDARY- NOZZLE
FLOW STATE
A2'
_'
i
/
/
/
f2' m2'
PRIMARY-NOZZLE
FLOW STATE
r
/
fl' ml'
PI'
T1
x
/
(b)
Figure
i.nozzle
TI
nozzle.
v
/
PI'
_
(a)
/
fl' ml'
Dual-stream
Schematic
depicting
coannular
diagram
of
the
appropriate
4.4-9
a
nozzle.
circular
flow
and
coannular
states.
P2'
T2
4.5
AIRFRAME
NOISE
PARAMETERS
MODULE
INTRODUCTION
Airframe
craft
in
ANOPP.
parameters
values
is
The
have
a
the
of
their
Noise
for
by
form
a
significant
part
operations.
Airframe
needed
the
is
low-power
output
in
must
noise
during
the
input
Parameters
execution
module
data
are
member
as
of
Airframe
the
Module
of
the
to
in
whereas
the
noise
modules
of
are
generates
airframe
identical
ANOPP,
total
noise
the
modules.
input;
the
the
air-
included
three
The
however,
prediction
time
physical
the
input
modules
parameters.
SYMBOLS
ambient
Co0
speed
of
landing-gear
I£g
M
aircraft
sound,
m/s
(ft/s)
position
Mach
number
o0
n
number
t
time,
6f
flap
P_
of
source
times
s
setting,
deg
ambient
dynamic
viscosity,
ambient
density,
kg/m
kg/m-s
3
(slugs/ft
(slugs/ft-s)
3)
INPUT
Input
Flight
to
this
Dynamics
module
Module
is
or
the
the
Engine
t
I£g (t)
source
time,
landing-gear
M_(t)
aircraft
6f(t)
flap
s
position
Mach
setting,
engine
variable
user.
number
deg
4.5-1
Variable
Table
table
as
provided
by
the
C
(t)
ambient
speed
of
p_ (t)
ambient
density,
p_(t)
ambient
dynamic
sound,
kg/m
m/s
3
(ft/s)
(slugs/ft
viscosity,
3)
kg/m-s
(slugs/ft-s)
OUTPUT
The
execute
outputs
the
to
this
airframe
module
noise
are
modules
Airframe
n
number
of
t
source
time,
IZg
M
(t)
(t)
_f(t)
source
landing-gear
aircraft
flap
time
time
function
parameters
of
source
required
time.
Parameters
values
s
position
Mach
setting,
ambient
speed
p_(t)
ambient
density,
_(t)
ambient
dynamic
(t)
three
a
Noise
number
deg
Ambient
c
the
as
of
sound,
kg/m
3
Conditions
m/s
(ft/s)
(slugs/ft
viscosity,
3)
kg/m-s
(slugs/ft-s)
METHOD
The
parameters
engine
are
variable
generated.
table
is
read
and
the
4.5-2
appropriate
output
to
5.
PROPAGATION
5.1
PROPAGATION
MODULE
INTRODUCTION
The
the
Propagation
noise
appropriate
ence.
a
sum
of
system,
propagation
absorption,
data
some
by
(ABS),
the
and
There
of
to
more
the
sum
is
the
Atmospheric
are
six
performed
i.
Interpolate
the
and
input
2.
Apply
spherical
effects
3.
Divide
the
accurate
the
ground
6.
Combine
output
position
a
is
subband
c
speed
f
frequency,
gr
acceleration
H
altitude,
amplitude
of
sound,
into
as
attenuation
m
(ft)
5.1-1
the
observer.
function
of
emission
change
for
(ref.
more
I)
effect
i/3-octave
frequency,
gravity,
a
subbands
bands
acoustic
reception
factor
(ft/s)
to
to
modeling
Hz
due
sound
impedance
into
effects
mean-square
of
adjusting
m/s
pre-
Module
effect
and
the
data
SYMBOLS
A
The
previously
Absorption
the
bands
absorption
function
time.
been
characteristic
ground
subbands
table
as
propagate
and
reflection
frequency
computations
atmospheric
retarded
have
by
the
refer-
angles
and
Apply
same
required
Atmospheric
noise
frequency
5.
resulting
source
i/3-octave
atmospheric
of
the
of
order:
absorption
the
observer
observer.
to
spreading
Apply
and
all
frame
in
spreading,
effects
directivity
4.
observer
The
generated
(ATM).
following
observer
are
applies
desired
spherical
the
necessary
the
is
attenuation,
(GEO),
which
and
the
first.
of
Module
steps
in
to
sources
propagation
Module
data
reference
them
performed
and
various
noise
of
noise
effects
reflection
the
takes
frame
transfer
or
the
are
(PRO)
source
two
Geometry
time
The
the
include
ground
for
pared
They
in
computations
If
coordinate
for
Module
modules
m/s 2
(ft/s
2)
pressure
time,
at
and
the
h
observer
height,
k
wave
M
molecular
number,
m
2_f/c
weight
number
of
o
observer
<p2>
mean-square
(ft)
a
of
air
subbands
per
i/3-octave
band
index
acoustic
universal
gas
r
distance,
m
Ar
path-length
SPL
sound
T
temperature,
t
time,
utv
spectrum
w
ratio
Y
dimensionless
Y
elevation
pressure,
constant,
Pa 2
m2/K-s
2
(ib2/ft
(ft2/°R-s
4)
2)
(ft)
difference,
pressure
level,
K
m
(ft)
dB
(OR)
s
slopes
of
n
=
o
incidence
0
polar
subband
center
frequencies
altitude,
angle,
deg
angle,
deg
Mgr(H
-
HI)/RT
r
2_Qaf/U
directivity
average
absorption
p
density,
kg/m
(J
specific
flow
azimuthal
solid
angle,
angle
3
deg
coefficient,
nepers
ambient
e
emission
o
observer
wavelength
(slugs/ft3)
resistance
directivity
for
a
of
angle,
ray
cone,
the
ground,
deg
sr
Subscripts:
a
per
5.1-2
kg/s-m
3
(slugs/s-ft
3)
r
standard sea-level reference value
s
source
Superscript:
*
dimensionless quantity
INPUT
The input to this module consists of one or more noise data tables
computed for the samesource coordinate system on the aircraft.
Whentwo
or more source noise data tables are input, they are summedprior to being
propagated to the observer. Additional tables are required which incorporate the various propagation effects.
The range and default values of
the required input parameters are given in table I.
Nb
number of subbandsper i/3-octave
rs
source radius, m (ft)
specific
flow resistance of the ground, kg/s-m3 (slugs/s-ft 3)
Source
f
frequency,
e
polar
emission
te
Noise
Data
Table
Hz
directivity
azimuthal
band
angle,
directivity
time,
deg
angle,
deg
s
24
<p2
(f, @,_,te)>*
mean-square
acoustic
Geometry
t
reception
time,
o
observer
r (t,o)
distance,
t e (t,o)
emission
8(t,o)
polar
_(t,o)
azimuthal
directivity
7(t,O)
elevation
angle,
h(o)
observer
pressure,
Table
s
index
m
(ft)
time,
s
directivity
height,
angle,
angle,
deg
m
deg
(ft)
5.1-3
deg
re
QaCa
Atmospheric
Y
altitude,
c (y)
speed
pc
characteristic
(y)
P
(Y)
Mgr(H
of
-
sound,
density,
Properties
H1)/RT
re
r
cr
impedance,
re
Table
re
PrCr
Pr
Absorption
f
frequency,
Hz
Y
altitude,
Mgr(H
_(f,Y)
average
-
Coefficient
HI)/RT
absorption
Table
r
coefficient,
nepers
per
wavelength
OUTPUT
This
the
module
observer
produces
as
a
a
function
table
of
Received
f
frequency,
Hz
t
reception
time,
O
observer
*(o)
speed
Ca
air
Pa (o)
<p2
mean-square
acoustic
time,
Noise
Data
and
pressures
at
observer.
Table
s
index
of
sound
density
(f, t,o)>*
of
frequency,
at
at
the
the
observer,
re
observer,
mean-square
re
acoustic
cr
Pr
pressure,
2 4
PaCa
re
METHOD
Noise
If
produce
terms
two
or
one
data
of
more
noise
data
table.
mean-square
pressure,
addition.
source
and
increments
the
observer,
corresponding
directivity
values
(pseudo
they
to
angles
observers).
be
reception
to
the
are
tables
noise
the
noise
data
summation
noise
data
table
polar
Before
time,
the
input
and
polar
data
are
for
data
the
directivity
observers.
This
5.1-4
are
summed
expressed
is
is
azimuthal
these
interpolated
actual
input,
the
This
constant
must
Interpolation
tables
Since
element-by-element
time
Data
a
a
in
simple
function
of
directivity
can
be
emission
angles,
is
angle
propagated
time
and
accomplished
to
values
azimuthal
by
to
using the polar directivity
angle, azimuthal directivity
emission time values from the geometry table. That is,
_(t,o), and te(t,o),
the input table
<p2(f,8,_,te)>*
angle, and
using 8(t,o),
is
converted
to
of
the
<p2(f,t,o)>*.
scale
Figure
1
is
a
graphic
representation
time
mapping.
Spherical
Spreading
and
Characteristic
Impedance
Effects
*
The
basis
time
and
table
the
is
remainder
now
of
processed
the
of
Figure
angle
conservation
2 is
d_
a
a
of
the
intensity
source
equation
radius
form
stant
from
a
expressed
in
<P2
acoustic
schematic
from
product
source
(rs)>
for
pc s
an
rs
as
<P2
d_
(ro) >,
=
to
in
column-by-column
apply
These
an
(ro)>
value
conical
a
conical
ray
tube
acoustic
observer
of
spherical
are
derived
cross-sectional
ray
of
power,
area,
radius
r o.
received
spreading
from
the
tube.
solid
which
is
be
con-
must
This
is
2
pc o
(i)
a
a
each
within
The
r
d_
(i)
o
dimensionless
*
<p2
the
=
s
equation
of
observer.
and
to
power
diagram
to
2
r
Expressing
of
on
module
and observer
index.
The
next
step
is
the characteristic
impedance
correction.
condition
the
<p2(f,t,o)>
throughout
form
yields
2
pc (yo)
rs
pc (ys)*
r2
<p2
(rs) > ,
(2)
O
where
for
less
pc o
the
and
observer
height
at
pc s
are
height
the
Yo
=
Ys
= Mgr(r
determined
and
source
from
source
and
the
altitude,
observer
are
atmospheric
properties
respectively.
given
The
table
dimension-
by
Mgrh/RTr
(3)
and
The
observer
geometry
term
and
distance
table.
In
pc(Yo)*/pC(Ys)
sin
y
r
+ h)/RT
and
equation
(4)
observer
(2),
is
r
the
5.1-5
height
2/ r o
2
rs
is
characteristic
h
the
are
found
from
the
spherical
spreading
impedance
correction.
SubbandDivision
2
The
of
the
mean-square
frequency
corresponding
in
acoustic
the
to
f.
The
the
ANSI
reference,
ground
the
effects
N b = 2m
default
+ 1
value
subbands,
of
m
=
W
table
standard
the
where
2.
m
The
*
>,
is
expressed
>
has
values
i/3-octave-band
application
that
is
<p
<p2(f,t,o)
accurate
requires
fj+l
pressure
of
i/3-octave
as
of
a
frequencies.
atmospheric
bands
be
function
frequency
As
shown
absorption
divided
is an integer
greater
ratio
of subband
center
and
into
than
0.
The
frequencies
is
101/10Nb
=
(5)
f.
]
where
j
index
is
the
number
i/3-octave
each
band
is
center
j =
Then,
index
j
of
to
-
I)N b
subband
+
k
subband
The
by
center
number
the
frequency
is
the
original
relation
h
center
The
frequencies.
of
expressed
(h
=
I,
in
terms
2 .....
N b)
of
a
2,
....
(6)
i/3-octave-
as
fj
= w
is
the
i
total
frequencies
determined
subband
index
i
h-m-lf
where
the
the
frequencies
(i
frequency
number
related
value
for
of
from
number
the
the
i/3-octave
center
half
of
the band
m
=
(h
=
of
I/3-octave
2
are
mean-square
slopes
of
frequency
is
i,
2,
...,
fi,
i =
bands.
given
in
acoustic
the
Nb;
table
pressure
i/3-octave-band
the
slope
of
i,
The
i/3-octave
(7)
and
II.
for
each
subband
spectrum.
the
k)
spectrum
For
in
the
is
each
lower
<p2>
i
(8)
ui=
and
the
slope
vi
The
term
2
*
<P >i-i
for
=
the
upper
half
is
(9)
ui+ I
<p2>
is
the
value
and
vk
of
the
mean-square
are
not
defined
acoustic
pressure
for
l
fi"
The
terms
u I
5.1-6
by
equations
(8)
and
(9);
therefore, the end slopes are established by setting
uI = vI and
vk = uk. Then the value of the mean-square pressure for each subband
<p2>j
is given by
i<p2>:/Ai) ui h-m-I
(h =
i,
2,
...,
m)
(h=m+
(<p2>; ffi)
The
subband
adjusting
vih-m-
factor
(h =
1
A i
is
m
+
defined
2,
i)
...,
such
(m
> 0)
(i0)
N b)
that
Nb
<p2
(ii)
= <p2>i
h=l
Substituting
equation
(i0)
into
equation
(ll)
and
solving
for
Ai
yields
m
A i
=
1
+
_,
/ i
ku
h-m-i
+
vi
h)
(12)
h=l
Defining
the
the
sum
total
of
subband
the
adjusting
mean-square
mean-square
acoustic
factor
A i
acoustic
pressure
pressure
of
the
Atmospheric
The
the
atmospheric
subband
function
of
Absorption
source
data.
frequency
Module
to
absorption
The
the
_(f'Ys'Y°)
f
(ABS).
observer
= Ys
1
altitude
average
Yo
fyYO
applied
Yo
the
and
Ys
mean-square
are given
pressure
ensures
subbands
that
equals
the
band.
ground
y
effects
are
coefficient
as
computed
absorption
now
_
by
coefficient
applied
is
the
to
a
Atmospheric
from
the
as
_(y)
S
where
Then,
the
and
defined
manner
the
i/3-octave
absorption
The
is
this
for
Effects
and
atmospheric
in
dy
= Ys
_(f'Ys
) Ys
-
by
equations
(3) and
with
the
atmospheric
is
5.1-7
Yo
Yo
_(f'Yo
)
(4),
respectively.
absorption
effect
(13)
• -[2_(f,Ys,Yo)f/Cr]
<p2>; ,abs = <p2>j e
The subscript abs indicates
mean-square pressure.
(r-rs)
(14)
that the absorption has been included in the
GroundEffects
Similarly, the ground effects are applied to the subband data. The
ground effects factor G is a function of path-length difference
kAr,
cosine of the incidence angle cos @, dimensionless frequency D, and
source-to-image distance kr 2, as discussed in the Ground Reflection and
Attenuation Module (GRA). Figure 3 shows the source-to-observer geometry
for the ground effects computation. Referring to the figure, the
quantities
r, h, and y are obtained from the geometry table. Then,
from the Law of Cosines,
the
source-to-image
distance
is given
by
2
r2 =
r
2
2
r2 =
r
2
+
(2h)
-
+
4h 2 +
4rh
(2r)(2h)
cos
(900
+ y)
(15)
or
Then
the
path-length
Ar
and
the
2
=
cos
difference
r2 -
cosine
of
Q
=
sin
Ar
(16)
is
r
the
r
y
(17)
incidence
sin
y
+
angle
is
2h
(18)
r2
The
dimensionless
frequency
n
and
the
wave
number
k
are
given
by
2_Paf
n
=
(19)
k
2nf
= -ca
(20)
and
5.1-8
where Pa and ca are evaluated from the atmospheric properties table
at the observer altitude
Yo" The value of G is now computed using the
method presented for the Ground Reflection and Attenuation Module. The
mean-squarepressure with ground effects is then given by
,gr = G
(21)
The subscript gr indicates
the mean-square pressure.
that the ground effect
has been included in
SubbandCombination
The
final
step
The
mean-square
relation
is
to
pressure
recombine
is
the
summed
over
subbands
each
into
I/3-octave
i/3-octave
band
bands.
by
the
Nb
*
=
>j
2
*
(22)
h=l
where
j = m(i
has
been
the
received
-
The
Module
of
i.
No
2.
Atmospheric
3.
Ground
4.
Both
appropriate
problem.
output
the
user
attenuation
or
mean-square
and
pressure
observer
spherical
has
four
and
ground
ground
effects
attenuation
option
is
output
selected
form
mean-square
the
the
time
index,
spreading
options
and
concerning
effects:
effects
only
only
atmospheric
of
performs
The
attenuation
effects
of
reception
complete.
always
attenuation
standard
dimensionless
is
atmospheric
processing
of
change.
atmospheric
the
the
values
table
impedance
The
printed
Once
all
data
Propagation
application
The
+ h.
for
noise
characteristic
the
l)
completed
for
pressure
sound
pressure
5.1-9
and
by
the
ground
effects
user
depending
the
noise
<p2>
level
.
data
at
The
user
SPL
in
the
may
on
the
observer
also
decibels,
nature
is
of
the
request
defined
as
(23)
, (Ca
,)2 + 197
SPL = i0 lOgl0 <p2>* + 20 lOgl0 Qa
where Qa and ca are determined from the atmospheric properties
at the observer altitude
Yo"
table
REFERENCE
i.
Montegani,
Sound
Francis
for
J.:
Computation
Fractional-Octave
Bands.
of
Atmospheric
NASA
5.1-10
TP-1412,
Attenuation
1979.
of
TABLEI.-
RANGE
ANDDEFAULT
VALUESOF INPUTPARAMETERS
Input
parameter
Minimum
Default
1
N
b
•
rs , m
O,
.
•
•
•
.
•
3
....
5
9
,
0.i
......
kg/s-m
Maximum
5
×
5.1-11
104
1
2.5
x
105
i00
1
x
106
TABLE
II.-
ANSI
WITH
Octave
1/3
octave
5O
63
63
8O
i00
125
125
160
1/15
STANDARD
OCTAVE
1/15-OCTAVE-BAND
octave
288
47.9
302
Octave
1/15
octave
1 820
1 910
2
000
2
090
55.0
347
2
190
57.5
363
2
290
60.3
380
2
400
63.0
400
315
1/3
octave
331
315
2000
400
2
2
000
500
2 500
66.1
417
2
69.2
437
2 750
72.4
457
2 880
75.9
479
3 020
3
150
525
3
310
87.1
55O
3
470
91.2
575
3
630
95.5
603
3
800
4
000
5OO
i00
500
630
83.2
80.0
630
5OO
630
3
4000
4
150
000
105
661
4
170
ii0
692
4
370
115
724
4
570
120
759
4
790
5
000
125
8OO
5 000
800
132
832
5
250
138
871
5
500
145
912
5
750
151
955
6
030
6
300
160
i000
i000
6
i000
300
166
1050
6
610
174
ii00
6
920
182
1150
7
240
7
590
8
000
2OO
1200
1250
1250
8OOO
8
000
209
1320
8
320
219
1380
8
710
1450
9
120
1510
9
550
1600
i0
000
240
250
BANDS
52.5
50.0
229
25O
1/15
octave
45.7
191
200
1/3-OCTAVE
FREQUENCIES
1/3
Octave
octave
AND
25O
263
275
1600
i0
000
1660
i0
500
1740
ii
000
5.1-12
!
5.1-13
0
,.Q
0
0
.._
o
o
0
0
g
I..l
0
t_
.,.4
o
o
o
!
o
0
5.1-14
o
0
o_
0
0
-_1
0
o
0
>,
0
m
0
I
0
4..1
I
Q.I
o
0
!
m
-M
t_
0
5.1-15
5.2
GENERAL
SUPPRESSION
MODULE
INTRODUCTION
The
a noise
General
table
pression
Suppression
produced
factor
directivity
is
noise
Module
any
supplied
angle,
suppressed
by
and
is
by
the
a
noise
the
azimuthal
in
applies
ANOPP
user
as
a
format
suppression
module.
function
directivity
same
noise
source
of
angle.
as
the
The
input
factor
The
noise
noise
frequency,
output
to
suppolar
table
of
table.
SYMBOLS
c
ambient
f
frequency,
<p2/
speed
of
sound,
m/s
(ft/s)
Hz
mean-square
Pref
reference
S
suppression
e
polar
P_
ambient
¢
azimuthal
acoustic
pressure,
pressure,
2
×
10 -5
re
Pa
2
pc
4
(4.177
×
10 -7
ib/ft
2)
factor
directivity
density,
angle,
kg/m
3
deg
(slugs/ft
directivity
angle,
3)
deg
INPUT
A
table
of
must
be provided.
tation
of sound
mean-square
acoustic
In addition,
pressure
levels.
pressure
ambient
Noise
f
frequency,
e
polar
<p2 (f,e,¢)>
angle,
directivity
o0
Po0
suppression
are
Table
mean-square
ambient
speed
ambient
density,
deg
angle,
acoustic
Ambient
c
a
Hz
directivity
azimuthal
and
conditions
of
sound,
kg/m
3
5.2-1
deg
pressure,
Conditions
m/s
(ft/s)
(slugs/ft
3)
re
2 4
pc
required
factor
for
compu-
Suppression
f
frequency,
8
polar
Table
Hz
directivity
azimuthal
S (f,8,_)
Factor
angle,
directivity
suppression
deg
angle,
deg
factor
OUTPUT
The
output
is
the
suppressed
noise
Suppressed
f
frequency,
@
polar
table.
Noise
Table
Hz
directivity
azimuthal
angle,
directivity
suppressed
deg
angle,
deg
mean-square
acoustic
pressure,
re
Q
2 4
c
METHOD
The
suppression
factor
is
defined
as
<p2>s
S
-
(i)
,
<p2>
where
is
the
each
is
<p2>_
the
unsuppressed
element
suppression
of
the
factor
printed
output
defined
as
is
suppressed
mean-square
mean-square
acoustic
input
table
to
noise
yield
available
the
of
acoustic
pressure.
by
the
suppressed
the
pressure
The
appropriate
noise
suppressed
table.
sound
and
module
value
In
pressure
<p2>
*
multiplies
of
the
addition,
level
SPL
2
2 *
pc
(2)
SPL
=
I0
lOgl0
<p
>s
+
20
lOgl0
Pref
5.2-2
6.
RECEIVED
NOISE
6.1
NOISE
LEVELS
MODULE
INTRODUCTION
Previous
data;
that
modules
is,
independent
data
data
to
noise
are
There
are
The
pressure
adds
a weighting
or
PNLT
the
more
of
Noise
been
concerned
time,
Levels
to
time,
produce
and
Noise
as
a
factor
that
of
data.
and
III
source
these
as
Level
Level
are
of
the
Effective
level.
This
to
a
(L A)
function
have
can
produce
time
be
level
The
which
Module
can
prior
is
and
III
II
referred
Noise
module
by
of
tables
the
One,
user.
noise
levels.
c
speed
AD i
second
P
tone
f
frequency,
i
standard
LA
A-weighted
sound
pressure
level,
dB(A)
D-weighted
sound
pressure
level,
dB(D)
difference
correction
noisiness
(ft/s)
of
band
sound
pressure
difference
Hz
i/3-octave-band
index,
noys
6.1-1
for
noise
of
and
OASPL,
The
PNLT
number
rela(PNL)
(PNLT)
LA,
or
the
mean-square
level
some,
the
level
characteristics.
correction
m/s
account
pure
tone
levels.
of
D-weighted
noise
discrete
sound,
the
frequency
constants
of
to
Module
(EFF)
to compute
can also
sum the
noise
computation
frequency
and
perceived
of
noise
integration
perceived
observer.
selected
describe
simple
factor
bands.
tone-corrected
sounds
Levels
pressure
to
a
C
N
II
source
used
is
a weighting
frequency
scales
by
noise
add
are
(OASPL)
sound
The
that
level
(LD)
function
sources
k
Level
and
(LEV),
Level
SYMBOLS
Ak,B
with
observer,
Module
observer,
scales
the
impact
level
required
ceived
varied
pressure
pressure.
The
noise
many
level
annoyance
PNL
of
A-weighted
tive
to
the
frequency
function
sound
spectrum.
acoustic
have
frequency,
levels.
overall
sound
over
a
ANOPP
have
In
integrated
noise
The
that
variables.
are
as
within
data
all
LD,
PNL,
of
table
effective
data
from
adds
these
is
pertwo
or
Nt
total
OASPL
overall
O
observer
PNL
perceived
PNLT
tone-corrected
<p2>
mean-square
SPL
sound
pressure
t
time,
s
W
weighting
P
density,
noys
value
sound
pressure
level,
dB
index
noise
level,
PNdB
perceived
acoustic
noise
level,
pressure,
level,
Pa 2
PNdB
(ib2/ft
4)
dB
function
kg/m
3
(slugs/ft
sea
level
3)
Subscripts:
a
ambient
r
standard
value
Superscript:
*
dimensionless
quantity
INPUT
The
pared
by
f
o
input
the
is
a
table
Propagation
frequency,
Hz
reception
time,
observer
c
(o)
speed
p
(o)
density
of
the
mean-square
Module
(PRO).
Received
Noise
acoustic
Data
Table
s
index
of
sound
at
at
observer,
observer,
re
re
cr
Pr
2
*
<p2(f,t,o)>
pressure
mean-square
acoustic
pressure,
6.1-2
re
_a
4
C
a
as
pre-
OUTPUT
The output of this module is one or more of the following
depending on the desires of the user:
tables,
Overall SoundPressure Level Table
t
reception time, s
o
observer index
OASPL
(t,o)
overall
sound pressure level,
dB
A-Weighted Sound Pressure Level Table
t
reception time, s
o
observer index
LA(t,o)
A-weighted sound pressure level,
dB(A)
D-Weighted Sound Pressure Level Table
t
reception time, s
o
observer index
LD(t,o)
D-weighted sound pressure level,
dB(D)
Perceived Noise Level Table
t
reception time, s
o
observer index
PNL(t, o)
perceived noise level,
PNdB
Tone-Corrected Perceived Noise Level Table
t
reception time, s
o
observer index
PNLT(t,o)
tone-corrected perceived noise level,
PNdB
METHOD
The method for each noise level scale is presented as a separate
section. Further details and a comparison
of the
noise
level
scales
are
6.1-3
presented in reference i.
input,
they
are
summed
If two
element
or
by
more
received
element
prior
noise
to
data
the
noise
tables
are
level
computation.
Overall
The
quency
overall
spectra.
Sound
Pressure
sound
pressure
level
It
expressed
in
is
Level
is
the
a
(OASPL)
simple
units
integration
of
decibels
of
in
the
the
fre-
form
n
OASPL
=
i0
Z
lOgl0
>[
<p2
+
20
*<a>
lOgl0
Pa
c
+
(i)
197
i=l
where
n
is
dimensionless
the
is
the
number
of
frequency
mean-square
A-Weighted
The
A-weighted
i/3-octave
band
each
frequency.
The
table
I and
Sound
sound
each
is
values
in
in
for
Pressure
pressure
that
plotted
terms
pressure
level
figure
the
i.
input
the
ith
Level
applies
representative
of
the
<p2>
i
(LA)
the
factor
degree
factors
equation
and
band.
a weighting
of
A-weighting
The
table
frequency
for
of
are
LA
to
annoyance
given
of
in
is
n
LA
=
l0
lOgl0
p2>
W(i,A
+
20
lOgl0
Pa_Ca)
+
(2)
197
i=l
where
W(i,A)
is
the
A-weighting
D-Weighted
The
ferent
D-weighted
weighting
D-weighting
equation
factors
for
LD
Sound
sound
factor
Pressure
pressure
is
are
factor.
level
applied
given
in
Level
is
to
the
table
II
(LD)
similar
data.
to
The
and
LA
values
plotted
except
a
of
the
in
figure
2.
0a
ca
difThe
is
n
LD
=
i0
7[<>[
lOgl0
p2
W(i,D
+
20
lOgl0
+
(3)
197
i=l
where
W(i,D)
is
the
D-weighting
factor.
Perceived
The
a
function
acoustic
perceived
of
both
pressure.
noise
level
frequency
This
Noise
scale
Level
applies
and
sound
weighting
is
(PNL)
a
intensity,
accomplished
6.1-4
weighting
to
the
with
factor,
which
mean-square
the
use
of
a
is
noisiness rating measured in noys. Figure 3 shows the values of the
noisiness rating as a function of frequency and sound pressure level
(SPL). The SPL is defined as
SPL =
The
noisiness
lOgl0
rating
computation.
sound
i0
The
pressure
*
<p2>
has
been
20
.( Ca.)2 + 197
lOgl0
rating
in
for
a
functional
given
form
value
of
10BI
(SPL-A
I) -I
10B2
(SPL-A
3)
10B3
(SPL-A
3 )
10B5
(SPL-A5)
The
(A 1
_
The
coefficients
table
III.
for
ease
frequency
of
and
is
_0
N
(4)
Pa
expressed
noisiness
level
+
A k
and
computation
of
Bk
PNL
I.
Determine
band
the
value
2.
Determine
the
maximum
3.
Compute
the
total
are
uses
values
of
N,
noys
noys
functions
in
of
N.
noys,
value
value
of
from
(SPL
< A I)
SPL
-< A 2)
<
(A 2 -< SPL
<
(A 3 <
<- A 4)
SPL
(A 4 -< SPL
<
frequency
as
The
from
process
equation
(5)
A 3)
150)
given
in
is
(5)
for
each
Nma x
the
equation
(6)
Nt
for
4.
Compute
the
=
Nmax
0"15
I/3-octave
the
PNL
+
40
+
N_-
Nmaxl
bands
perceived
=
II_i=l
noise
33.22
6.1-5
lOgl0
level
N t
in
units
of
PNdB
from
(7)
Tone-Corrected Perceived Noise Level (PNLT)
The tone-corrected perceived noise level is perceived noise level
modified for the impact of pure tone content of the noise spectra. Pure
tones provide an additional irritation
not found in broadband noise. The
methodprovides for detection of pure tone content in i/3-octave-band
spectra and correction of the PNL for the impact of the pure tones.
The procedure may be illustrated
with reference to figure 4, which
showsan example spectrum, and table IV, which gives the spectrum in
tabular form. The following steps are performed:
i. Computethe second difference
2.
If
ADi
=
<
-5,
_D i
SPLi+
1 -
check
2SPL i
to
see
LiDi
+
SPLi_
if
of the SPL, which is
(8)
1
SPL i
is
a
local
maximum,
that
is,
if
SPL i
3.
If
SPL i
noise
>
is
SPLi_
a
level
These
250
dashed
4.
The
maximum,
defined
values
2500
Hz.
The
in
are
the
(9)
average,
is
=
SPL.
-
background,
1
given
in
table
noise
IV
at
levels
are
shown
of
as
4.
the
local
maximum
and
the
background
noise
(ii)
1
in
correction
the
range
C(f,F)
500
<
fi
can
<
now
be
determined.
5000,
"0
=
frequencies
SPL.
frequency
frequencies
C (f,F)
or
as
background
figure
between
F
discrete
For
compute
1
(I0)
The
l
5.
SPLi_
> SPLi+
2
difference
F
1 +
=
lines
level
SPL i
SPL i
average
and
and
local
SPLi+
SPLi
1
(F <
F/3
(3
6.7
< F
(20
6.1-6
<
3)
20)
-< F)
(12)
and for frequencies in the range fi _
500
or
0
C(f,F) =
F/6
(3 <
3.3
6.
The
tone
correction
quency
with
a
level
PNLT
PNLT
is
=
2.
the
Cma x
Then
computed
PNL
_
5000,
(F
<
3)
F
<
(20
is
correction
value
of
fi
maximum
value
of
the
the
PNL
(13)
-< F)
discrete
which
in the example
occurs
the tone-corrected
perceived
from
20)
freat 2500
noise
as
(14)
+ Cma x
REFERENCE
i.
Edge,
Philip
M.,
Quantification
TN
D-7977,
Jr.;
of
and
Cawthorn,
Community
1976.
6.1-7
Hz
Exposure
Jimmy
to
M.:
Selected
Aircraft
Noise.
Methods
NASA
for
TABLEI.-
I/3-octaveband center
frequency
5O
63
8O
I00
125
160
200
250
315
400
5OO
630
8OO
WEIGHTING
FUNCTION
FORA-WEIGHTED
SOUND
PRESSURE
LEVEL
W(i,A)
dB
i/3-octaveband
center
correction
dB
W(i,A)
correction
frequency
0.00096
.0024
.0056
.0123
.0245
.0457
.0813
.138
.219
.331
.479
.646
.832
-30.2
1 000
1.0
-26.2
1
250
1.148
-22.5
1 600
1.259
1.0
-19.1
2 000
1.318
1.2
-16.1
2
500
i. 349
1.3
-13.4
3
150
1.318
1.2
-i0.9
4
000
i. 259
1.0
1.112
0
.6
.5
-8.6
5
000
-6.6
6
300
.977
-4.8
8
000
.776
-i.I
-3.2
i0
000
.562
-2.5
-i .9
12
500
.372
-4.3
--.8
6.1-8
--.i
TABLEII.-
i/3-octaveband center
frequency
5O
63
8O
I00
125
160
200
250
315
400
5OO
630
8OO
WEIGHTING
FUNCTION
FORD-WEIGHTED
SOUND
PRESSURE
LEVEL
W(i,D)
dB
correction
i/3-octaveband
center
dB
W(i,D)
correction
frequency
0.0525
.0813
.126
.191
.282
.398
.55O
.692
.832
.912
.933
.891
.871
-12.8
-i0.9
-9.0
-7.2
-5.5
-4.0
-2.6
-1.6
1
000
1.0
0
1
250
1.585
2.0
1
600
3.090
4.9
2
000
6.166
2
500
ii .482
10.6
3
150
14.125
11.5
4
000
12.882
ii.i
5
000
9.120
9.6
--.8
6
300
5.754
7.6
--.4
8
000
3.548
5.5
--.3
i0
000
2.188
--.5
12
500
--.6
6.1-9
.724
7.9
3.4
-1.4
TABLE
III.-
CONSTANTS
OF
REQUIRED
PERCEIVED
FOR
NOISE
COMPUTATION
LEVEL
i/3-octaveband
center
A 1
B1
A 2
B 2
A 3
B3
A4
B5
A 5
frequency,
Hz
5O
49
55'0.058098
64 i
.043478
91.01
63
44
.068160
51
.058098
60
.040570
85.88
.030103
51
8O
39
.068160
46
.052288
56 _
.036831
87.32
.030103
49
i00
34
.059640
42
.047534
53
.036831
79.85
.030103
47
125
30
.053013
39
.043573
51
.035336
79.76
.030103
46
160
27
.053013
36
.043573
48
.033333
75.96
.030103
45
200
24
.053013
33
.040221i46
.033333
73.96
.030103
43
250
21
.053013
30
.037349
44
.032051
74.91
.030103
42
315
18
.053013
27
.034859
42
.030675
94.63
.030103
41
400
16
.053013
25
.034859
40
.030103
i00 .00
.03010340
5OO
16
.053013
25
.034859
40
.030103
i00 .00
.030103
630
16
.053013
25
.034859
40
.030103
i00 .00
.030103140
0.079520
0.030103
52
40
8OO
16
.053013
25
.034859
40
.030103
i00 .00
.030103140
1
000
16
.053013
25
.034859
40
.03O103
i00 .00
.030103
40
1
250
15
.059640
23
.034859
38
.030103
i00 .00
.030103
38
1
600
12
.053013
21
.040221
34
.029960
i00 .00
.029960
34
2 000
9
.053013
18
.037349
32
.029960
i00 .00
.029960
32
2
500
5
.047712
15
.034859
30
.029960
i00 .00
.029960J30
3
150
4
.047712
14
.034859
29
.029960
i00 .00
.029960
29
4
000
5
.053013
14
.034859
29
.029960
I00 .00
.029960
29
5 000
6
.053013
15
.034859
30
.029960
i00 .00
.02996030
6
300
i0
.068160
17
.037349
31
.029960
i00 .00
.029960!31
8
000
17
.079520
23
.037349
37
.042285
44 .29
.029960
i0
000
21
.0596401
29
.043573
41
.042285
5O .72
.029960137
6.1-10
34
TABLE
IV.-
OF
Band
EXAMPLE
DISCRETE
i
fi
PROBLEM
FREQUENCY
SPLi
FOR
CORRECTION
AD i
19
8O
70
20
i00
62
21
125
70
2
22
160
80
-8
23
200
82
-i
250
83
-8
25
315
76
ii
26
400
80
-4
27
5OO
80
-i
28
630
79
C (f,F)
79
2/3
79
2
0
8OO
78
3O
1
000
80
31
1
250
78
0
32
1
600
76
5
33
2
000
79
34
2
500
85
35
3
150
79
36
4
000
78
-6
37
5
000
71
-4
38
6
300
60
-5
39
8
000
54
-3
40
I0
000
45
6.1-11
SPL i
16
24
29
DETERMINATION
3
-4
3
-12
5
10--
.
0
-10
_=
-2o
-30
I
100
50
I
200
I
I
500
1000
FREQUENCY,
Figure
I.-
Decibel
correction
for
A-weighted
6.1-12
I
2000
i
5000
Hz
sound
pressure
level.
10000
10--
0
-10
,
-20
-3O
I
5O
100
I
I
200
500
I
1000
FREQUENCY,
Figure
2.-
Decibel
correction
6.1-13
for
I
D-weighted
I
2000
5000
10000
Hz
sound
pressure
I
level.
600
500
400
Z
m,
,-.,_300
SPL:120
d
200
IO0
110
100
0
50
1
,
100
200
500
I000
FREQUENCY,
Figure
3.- Perceived
noisiness
2000
Hz
of sound pressure
6 .i-14
5000
levels.
I0000
go
80
m
(U
(U
..J
70
(U
L
(/}
t-
60
e0
I/3
BACKGROUND
NOISE
ESTIMATE
50
,,liiI,ll,JlJlJ,llJ
40
21
24
Standard
60
125
27
30
One-Third
Octave
500
1000
250
Frequency,
Figure
4.-
33
Example
spectrum
tone-corrected
perceived
6.1-15
Band
36
Number
2000
4000
Hz
for
computation
noise
level.
of
800O
6.2
EFFECTIVE
NOISE
MODULE
INTRODUCTION
The
which
the
a
impact
sure
to
(EFF)
of
noise
The
a
data
due
is
important
an
variety
of
LEV
a
indexes
noise
which
take-offs
module
from
of
The
noise
exposure
is
I and
table
of
is
cumulative
the
into
indexes;
of
as
of
expo-
Noise
Module
effective
this
per-
module.
however,
Multiple
EPNL
tone-corrected
scales
consideration
exposure.
index
II
level
For
Effective
incorporated
landings.
Levels
a
of
noise
time.
operations,
exposure
and
in
computes
table
terminal
cumulative
considered
various
and
consideration.
used
other
computes
location
airport
(EPNL),
multiple
are
This
to
commonly
level
tion,
(LEV)
observer
time-averaged
most
from
Module
of
noise
noise
problems
the
Levels
function
computes
ceived
are
Noise
are
There
they
require
and
landing
take-off
ANOPP.
a
perceived
function
noise
of
level
observer
posi-
produced
by
module.
SYMBOLS
EPNL
effective
n
number
o
observer
PNLT
tone-corrected
<pnlt2>
mean-square
t
time,
_t
reception
perceived
of
time
noise
level,
EPNdB
segments
index
perceived
equivalent
noise
of
level,
PNdB
PNLT
s
time
increment,
s
INPUT
The
produced
input
by
the
is
a
LEV
table
of
the
Input
At
reception
tone-corrected
module.
time
increment,
6.2-1
Constant
s
perceived
noise
level
Tone-Corrected
t
time,
O
observer
Perceived
Level
Table
s
index
tone-corrected
PNLT(t,o)
Noise
perceived
noise
level,
PNdB
OUTPUT
The
output
initial,
is
a
maximum,
table
and
of
final
Effective
O
observer
n(o)
number
of
(o)
effective
PNLT
i (o)
initial
PNLTf
time
PNLT
each
Noise
noise
level
and
observer.
Level
Table
segments
noise
value,
PNLT
final
(o)
perceived
at
Perceived
perceived
maximum
x (o)
effective
values
index
EPNL
PNLTma
the
PNLT
PNLT
level,
EPNdB
PNdB
value,
value,
PNdB
PNdB
METHOD
The
First,
computation
the
PNLT
of
is
EPNL
converted
requires
back
the
to
time
mean-square
average
of
pressure
the
by
PNLT.
the
relation
<pnlt2>
The
EPNL
can
=
be
(1)
10PNLT/I0
computed
in
integral
form
as
(2)
EPNL
=
l0
lOgl0
_tl f <pnlt2>
dt
where
ti
and
the PNLT
data.
tf
The
are
the
segments
initial
and
final
times
for
are
determined
by comparing
time
At
used
the
increment
tabulated
then
a
initial,
allow
time.
new
If
segment
maximum,
assessment
the
is
and
of
in
started.
final
the
Geometry
tabulated
time
In
PNLT
quality
addition
values
of
Module
entries
the
are
EPNL
6.2-2
to
to
the
each
the
differences
are
more
the
value
of
for
each
printed
predictions.
segment
reception
than
At
the
of
in
apart,
EPNL,
observer
the
to
7. UTILITIES
7.1
THERMODYNAMIC
UTILITIES
INTRODUCTION
The
of
the
prediction
mixture
are
engine
to
of
thermodynamics
needed
processes
convert
modules.
The
ANOPP.
to
are
these
the
to
analyze
the
as
static
The
variables
detailed
properties
processes.
in
ANOPP.
is
for
the
needed
both
efficiency
of
these
in
the
in
ANOPP
knowledge
of
variables
perform
increased
requires
gas
thermodynamic
total
utilities
allows
noise
engine.
stored
engine
This
aircraft-engine
of
the
fuel-air
All
data
The
noise-prediction
functions
design
on
capability
within
of
the
functional
given
in
modules.
A
brief
description
of
each
utility
is
the
following:
i.
Gas-Properties
specific
enthalpy,
ture
given
for
ambient
2.
are
and
Computes
the
entropy
absolute
ratio
function
humidity,
inputs,
outputs,
in
The
are
any
available
- Computes
number
temperature,
herein.
methods
Utility
Mach
total
The
mented
of
Flow-Variables
pressure,
All
values
-
specific
of
as
a
specific
heats,
function
fuel-to-air
of
ratio,
and
ranges
standard
for
from
gas
use
in
given
and
gas
method
of
extracted
for
the
for
each
pressure,
mass
flow
1
text.
in
utility
variables
references
module
static
of
static
rate,
total
properties.
input
dynamics
any
the
values
to
All
are
are
3;
given
however,
presented
flow
cp
f
cross-sectional
specific
heat
fuel-to-air
H
at
area,
table
I.
they
are
docu-
thermodynamic
utilities
ANOPP.
constant
m 2
(ft 2)
pressure,
m2/K-s
ratio
absolute
humidity,
percent
h
specific
enthalpy,
m2/s
M
Mach
m
mass,
kg
(slugs)
mass
flow
rate,
mole
fraction
a
2
(ft2/s
number
kg/s
(slugs/s)
7.1-1
2)
2
in
in
SYMBOLS
A
tempera-
and
temperature.
temperature,
detail
Utility
and
(ft2/°R-s
2)
P
pressure,
R
dry-air
Pa
gas
universal
(ib/ft
constant,
gas
gas
T
temperature,
U
velocity,
m/s
molecular
weights
constant,
Xi
mass
Y
ratio
P
density,
m2/K-s
K
entropy
(ft2/°R-s
kg-m2/K-s
2
(ft2/°R-s
2
of
of
3
gas
gas
2)
constituents
constituents
heats
(slugs/ft
function,
3)
m2/K-s
2
(ft2/°R-s
2)
Subscripts:
a
air
r
reference
s
static
t
total
co
ambient
value
(see
table
III)
Superscript:
*
dimensionless
quantity
GAS-PROPERTIES
UTILITY
Input
fuel-to-air
H a
T
T
co
ratio
absolute
humidity,
ambient
temperature,
local
temperature,
percent
re
mole
re
fraction
T r
T co
7.1-2
2)
(slugs-ft2/°R-s
(ft/s)
specific
kg/m
2
(OR)
fractions
of
m2/K-s
constant,
R
Wi
2)
2)
Output
The
gas-properties
properties
data
table
when
needed
can
be
within
interpolated
ANOPP.
to
The
gas
provide
constant
the
is
gas-
also
provided•
R*
gas
constant,
re
R
Gas-Properties
T
temperature,
h* (T*)
specific
enthalpy,
_* (T*)
specific
entropy
¥ (T*)
ratio
re
of
Table
T
re
RT
function,
specific
re
R
heats
Method
The
constituents
(nitrogen,
dioxide
and
assumed
to
their
of
oxygen,
and
water).
The
be
negligible.
molecular
weights.
n
the
gas
argon)
within
the
the
products
and
amounts
of
the
II
gives
Table
The
mass
of
flow
other
of
dry
gas
air
(carbon
constituents
five
gas
those
combustion
gas
the
each
are
of
are
constituents
constituent
is
and
given
by
_
0.75558
m1
0.23154
m 2
m 3
-
3.43185f
(i)
3.13753f
= m a
m4
0.00622H
a
+
1.28432f
0.01289
m 5
m
The
amount
of
fuel
the
air.
oxygen
in
cannot
exceed
marized
gas
is
in
that
0.06767•
table
can
Therefore,
The
III.
X 1
Then,
be
burned
the
constants
the
is
value
mass
used
limited
of
the
in
fraction
to
the
availability
fuel-to-air
equations
X i
of
ratio
(i)
each
are
of
f
sum-
constituent
m1
X 2
m 2
1
(2)
x5
i
i m51
7.1-3
where the total
mass m is
5
(3)
m = _ mi
i=l
The dimensionless gas constant
computed
from
the
universal
gas
R
for
constant
each
R
constituent
gas
can
be
as
R/WII
n _
R/W21
1
(4)
IRi
_R/w5J
where
R
is the
dry-air
the
ith gas
constituent
gas
given
constant
in table
and
II.
stant
then
given
the
of
the
mixture
is
by
W i
is the molecular
The
dimensionless
following
matrix
weight
gas
con-
of
product:
D
R1
R*
=
Ix I,
x 2 .....
(5)
x5_
R5
B
The
dimensionless
ratio
and
The
computed
gas
absolute
dimensionless
from
c__
constant
humidity
=
R
in
can
is
table
specific
cp/R_,
9
heat
be
given
as
a
function
of
fuel-to-air
IV.
at
expressed
constant
as
pressure
(which
is
follows:
*, 1 (T/Tr)
,
*
Cp
°
c
_ I _ p, 2 (T/Tr)
(6)
!
LCp,
7.1-4
5 (T/Tr)
and R_ are the componentmass fractions
where Xi
=
from
equations
specific
the
ratio
and gas constants
l
(2)
and
(4)
and
heats
at constant
of specific
heats
T/T r
T
T
•
pressure
y
is
The
_.
are
values
given
of
in
table
the
component
V.
Finally,
Cp, i
c
P
y
=
Cp
The
ratio
ratio
is
The
of
(7)
.
-
1
specific
plotted
heats
in
enthalpy
as
figure
per
a
function
of
temperature
and
fuel-to-air
i.
unit
mass
of
a
fluid
is
defined
as
(8)
h
=
Cp(T/T
T
r)
dT
o0
Expressed
in
dimensionless
h* (T*)
Equation
(9)
can
form,
= fl T* Cp_T
*/
be
=
expressed
_i
T
becomes
the
following:
(9)
• T ,_) dT •
in
T/Tr
h*(T*)
this
terms
of
an
absolute
scale
T/T r
as
,
Cp
d
- _
T
_i
oO
Cp
(I0)
d
oo
or
(ii)
h*(T*)
In
general,
and
the
in
terms
the
absolute
of
the
=
h*r(T , T_)
,
specific
h.rlfT*--)_/
_ IT_*
enthalpy
is
Then
the
humidity.
component
-
specific
a
function
specific
enthalpies
of
the
fuel-to-air
enthalpy
of
table
h r
V
is
ratio
expressed
as
h*
r,ll
h 21
h r
=
Xl,
R
X 2,
...,
7.1-5
R
X
•
(12)
The
specific
air
ratio
enthalpy
for
The
zero
entropy
is
plotted
absolute
function
=
Cp
T
_T T
as
a
humidity
_
per
function
in
of
figure
unit
mass
temperature
and
fuel-to-
2.
of
a
fluid
is
given
by
dT
(13)
oO
which
may
be
expressed
in
T*
_*
Equation
_.(T*T*)
can
be
_*(T*)
=
_* (T*)
=
form
as
*
= fl
(14)
dimensionless
(14)
dT*
expressed
in
terms
of
T/T r
as
Cp
ITZTr
(T/Tr)
d(T/T
r)
* r)
-
d(T/T
(15)
d(T/T
r)
or
In
general,
the
ratio
and
expressed
_r(T*TI)
specific
absolute
in terms
-
entropy
humidity.
of the
(16)
_r(T*)
function
Then
component
is
a
the
specific
specific
function
entropy
entropy
of
the
function
functions
fuel-to-air
of
_r
is
table
V
as
m
_r, i
_r* =
ER*iXl , R2X2,
*
R 5 x 5_
...,
(17)
_r'2
.
_r,5
m
The
specific
entropy
fuel-to-air
ratio
The
(17) .
value
a
given
gas
It
of
can
T*.
value
function
for
zero
properties
be
is
table
interpolated
Alternately,
of
h*
or
plotted
absolute
as
humidity
is
formed
to
determine
the
d
from
a
function
in
temperature
T*
7.1-6
h*,
can
temperature
and
3.
equations
y,
_*.
of
figure
(7),
or
be
_*
(12),
for
determined
and
a given
for
FLOW-VARIABLES
UTILITY
Input
The inputs to this utility
and the total flow variables.
are the gas-properties
table
Total Flow Variables
mt
total
mass flow rate, re APt/R_ t
Pt
total
pressure, re p_
Tt
total
temperature, re T
0o
Gas-Properties
T
temperature,
h*(T*)
specific
enthalpy,
_*(T*)
specific
entropy
yCT*)
ratio
The
of
outputs
re
T
re
the
RT
function,
specific
are
Mach
Ps
T
re
R
heats
static
flow
Static
M
Table
variables.
Flow
Variables
number
static
pressure,
re
static
temperature,
p_
re
T
S
Method
The
specific
computation
heats
or
can
be
variable
Constant
As
to
the
derived
in
Mach
number
reference
as
follows:
performed
ratio
of
Ratio
2,
7.1-7
the
with
either
specific
of
ratio
of
heats.
Specific
total
constant
mass
Heats
flow
rate
mt
is
related
l_[y+l_
mt
where
the
=
2
ratio
of
gas-properties
be
has
subsonic
nique
root
is
4)
the
constant
used
from
here,
energy
equation
1
the
total
heats,
2)
+X-
IM
one
Mach
the
(18).
interest
find
from
rate,
equation
of
dimensionless
_
determined
flow
and
to
(ref.
is
subsonic
the
ratio
Tt
*
T
mass
for
one
specific
temperature
X
the
form
one
the
is
heats
Given
closed
roots,
versus
From
with
in
two
(ref.
plotted
specific
table.
determined
tion
(18)
so
mass
the
the
expression
addition,
the
cannot
the
equa-
flow.
The
interval-halving
value.
flow
rate
adiabatic
using
number
supersonic
the
number
for
In
for
Tt
Mach
flow
for
the
The
in
of
Mach
figure
a perfect
technumber
is
4.
gas
stagnation-
is
2
(19)
2
S
Rearranging
yields
=
T
1
+
Y
1
Ts
which
is
plotted
in
Similarly,
is
related
20)
2
for
to
the
figure
an
5
for
7
isentropic
=
1.4.
process,
stagnation-temperature
the
ratio
stagnation-pressure
by
the
following
ratio
relation:
7-1
(21)
Tt
After
\Pt/
substituting
equation
(19)
and
rearranging,
the
static
pressure
Ps
is
Ps*
which
is
also
=
Pt* /( 1
plotted
+
in
Y
2
1
figure
(22)
M2)Y-I
5
for
7
=
1.4.
7.1-8
Variable Ratio of Specific Heats
Computation of the static temperature and static pressure using a
variable ratio of specific heats requires two relations.
The first is
derived from continuity
and
the
first
law of thermodynamics,
and
the
second
tions
is
are
The
derived
from
relation
law
of
thermodynamics.
The
two
equa-
is
(23)
first
=
Substituting
second
PsUA
the
U
the
simultaneously.
continuity
=
and,
from
solved
law
_ 2 (h t
of
thermodynamics,
(24)
- h s)
equation
(24)
into
(23)
and
applying
the
ideal-gas
law
yields
Ps
=
By rearranging
becomes
•*
mt
From
given
where
by
q2(ht and
=
the
expressing
T*
s
second
in
#(
PS
*
Pt
* 2 h t
law
(25)
hs )
of
dimensionless
h*
s
form,
equation
(25)
)
(26)
thermodynamics,
the
change
in
entropy
As
is
by
_
As
=
is
the
rearranging
in
R
in
+
entropy
yields
Ct
-
function.
the
(27)
Cs
For
an
isentropic
process,
As
=
0,
so
following:
(28)
R
7.1-9
By taking the exponential of both sides and putting in dimensionless form,
equation (28) becomes
Ps
-exp[-(_t
Pt
(29)
- _s)_
Equations (26) and (29) can be solved simultaneously for the static
temperature and static pressure. Two roots exist, one for subsonic flow
and one for supersonic flow. To ensure that the subsonic flow case is
determined, the equations are solved using an interval-halving
technique.
The static-to-total
temperature and pressure ratios computedwith both
constant and variable ratios of specific heats are given in tables VI
and VII.
From the continuity
expressed as
equation (eq. (23)), the Machnumber can be
M=
(30)
DsA_--RT
Applying
the
perfect-gas
mt
Table
VIII
specific
s
law
and
expressing
in
dimensionless
form
yields
Pt
compares
the
Mach
numbers
for
constant
and
variable
ratios
heats.
REFERENCES
i.
2.
McBride,
Heimel,
Involving
the
Shapiro,
Beckett,
First
Ascher
Flow.
Liepmann,
Wiley
4.
J.;
Thermodynamic
Fluid
3.
Bonnie
Sanford:
H.
&
W.;
Sons,
Royce;
Algorithms.
H.:
Volume
and
Inc.,
and
Sheldon;
Ehlers,
Properties
18
The
to
Elements.
Dynamics
I.
Ronald
Roshko,
A.:
NASA
and
Press
Janet
6000 °
K
G.;
for
SP-3001,
210
Elements
Gordon,
Substances
1963.
Thermodynamics
Co.,
and
of
Compressible
c.1953.
of
Gasdynamics.
John
c.1957.
Hurt,
McGraw-Hill
James:
Book
Numerical
Co.,
c.1967.
7.1-10
Calculations
and
of
TABLEI.-
RANGES
OF INPUTPARAMETERS
Input
parameter
Minimum
0
Ha, percent
.
........
Pt
........
Ta
........
t
7.1-11
0.06767
4
0
0
mt
Maximum
0.6847
0.i
i0.0
0.8
1.2
0.5
7.0
TABLEII.-
Index
Constituent
Nitrogen (N2)
28.01340
2
Oxygen (02)
31.99880
3
Carbon
4
Water
(H20)
18.01534
5
Argon
(Ar)
39.94800
Mass
fraction
of
nitrogen
Mass
fraction
of
oxygen
Mass
fraction
of
argon
Mass
of
Mass
Mass
of
water
burned
of
fuel
Mass
oxygen
burned
fuel
carbon
burned
fraction
Standard
Universal
Molecular weight
1
TABLE
fuel
GASCONSTITUENTS
sea
gas
required
dioxide
III.-
STORED
in
in
in
dry
dry
dry
per
PRIMARY
air
air
air
unit
44.00995
(CO2)
CONSTANTS
0.75558
..............
...............
0.23154
...............
0.01289
mass
of
.........................
produced
per
unit
3.42185
mass
of
.........................
dioxide
produced
1.28432
per
unit
of
3.13753
.........................
of
level
water
per
percent
temperature,
constant,
kg-m2/K-s
mole
K
2
fraction
........
0.00622
...............
288.15
...............
8314.32
7.1-12
TABLEIV.- VARIATIONIN DIMENSIONLESS
GASCONSTANTR
R*
percent
mole
for
Ha,
fraction,
of
f
1.2
0 .000
.005
.010
.015
.020
.025
.030
.035
.040
.0_5
.050
.055
.060
.065
1,00016
1,00032
1,00048
1,00064
1,00079
1,00095
,00110
,00125
,00140
,00155
,00170
,0018.
,00199
,00213
7.1-13
1,00763
1,00775
1,00787
1,00799
1.00811
1,00822
1.00834
1,0084_
1,00857
1,00868
1,00879
1,00890
1,00901
1,00912
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