Analysis of the vibratory excitation of gear systems: Basic
theory
William
D. Mark
Bolt Beranek and NewmanIncorporated,Cambridge,Massachusetts
02138
(Received22 August 1977;revised16 November 1977)
Formulation of the equationsof motion of a generic gear systemin the frequencydomain is shown to
requirethe Fourier-seriescoefficientsof the componentsof vibration excitation;thesecomponentsare the
static transmissionerrors of the individualpairs of meshinggearsin the system.A generalexpressionfor
the static transmissionerror is derived and decomposedinto componentsattributable to elastic tooth
deformationsand to deviationsof tooth facesfrom perfect involute surfaceswith uniform lead and spacing.
The componentdue to tooth-facedeviationsis further decomposedinto appropriatelydefinedmean and
random components.The harmonic componentsof the static transmissionerror that occur at integral
multiplesof the tootli-meshingfrequencyare shown to be causedby tooth deformationsand mean
deviationsof the tooth faces from perfect involute surfaces.Harmonic componentsthat occur at the
remainingmultiplesof gear-rotationfrequenciesare shownto be causedby the random componentsof the
tooth-face deviations.Expressionsfor the Fourier-seriescoefficientsof all componentsof the static
transmissionerror are derivedin terms of two-dimensionalFourier transformsof local tooth-pair stiffnesses
and stiffness-weighted
weighteddeviationsof tooth facesfrom perfectinvolute surfaces.Resultsare valid
for arbitrary, specifiedtooth-facecontact regions and include spur gears as the special case of helical
gearswith zero helix angle.
PACS
numbers:
43.40.At
INTRODUCTION
The principal source of vibratory excitation of gear
systems is the unsteady component of the relative angular motion of pairs of meshing gears. The static
transmission error describes this displacement-type
vibratory excitation. In this paper, expressions are
derived
for
the
Fourier-series
coefficients
of the mean
and random components of the static transmission
er-
ror; in addition, expressions are derived for the power
spectrum of the random component.
are
written
in terms
These expressions
of the two-dimensional
Fourier
transforms of the local tooth-pair stiffness per unit
length of line of contact and the deviations of the tooth
faces from perfect involute surfaces with uniform lead
and spacing. The derivations are carried out for helical gears of nominal involute design. Results for spur
gears are included as the special case of the helical
results for zero helix angle.
The
results
contained
herein
should
be useful
for
a
number of purposes: (i) predicting the vibration excitation of specific gear systems, (ii} understandingthe effects of machining errors on vibration excitation so that
gear manufacturing specifications can include vibration
and noise considerations, (iii) understandingthe effects
of helical design parameters
on vibration excitation
(e.g., pitch, base, and addendacircle radii, pressure
angle, tooth spacing, face width, helix angle, and face
contact region), (iv) designingtooth-face deviationsfrom
perfect involute surfaces to minimize vibration excitation under design constraints of strength, wear, lubrica-
tion, weight, size, or other considerations,and (v} interpreting observed vibration spectra for diagnosing
machinery.
surfaces, and they also contain machining errors that
may vary tooth to tooth. Furthermore, when a pair
of meshing rotating gears is transmitting torque, the
teeth elastically deform, giving rise to an unsteady
component in the relative angular motion of the two
gears. This unsteady component is caused by the periodic variation in the stiffness of the gear mesh that is
attributable to the periodic variation in the numbers of
teeth
in contact
and the variation
in the stiffness
of the
individual tooth pairs as the location of their mutual
line of contact changes during rotation.
The intentional
tooth-face modifications, machining errors and wear,
and tooth deformations all provide non-negligible contributions to the deviation from exactly uniform relative
angular motion of pairs of meshing rotating gears. The
composite effect of the above contributions is described
bythe statictransmission
error,•'-• whichmay bedefined as the deviation •0 from linearity of the angular
position 0 of a gear measured as a function of the angular position of the gear it meshes with when the gear
pair is transmitting a constant torque at low enough
speed so that inertial effects are negligible. In this paper, we shall deal with the lineal transmission error •
=Rb•O, where Rb is the base circle rad'ms of the gear,
whose angular transmission error is •50. It is now
generally recognized that the static transmission error
describesthe principal Sourceof vibratory excitation
of gear systems.•.-6
When a gear system is in operation at normal speed,
the static transmission error of each pair of its meshing gears causes an unsteady component in the total
force transmitted between the meshing teeth of that pair
angularmotion.• However,thetoothfacesof real gears
(in a direction normal to the tooth surfaces); this unsteady force is a consequenceof the rotational inertia of
the gear system. Since, to a first approximation, the
motion of the gears is pure (unsteady)rotation, this
unsteady component of force must have reactions in the
are designed to deviate slightly from perfect involute
shafting bearings that support the two meshing gears.
A pair of meshing gears with rigid, perfect, uniformly
spaced involute teeth would transmit exactly uniform
1409
J. Acoust.Soc.Am. 63(5), May 1978
0001-4966/78/6305-1409500.80
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1409
1410
William D. Mark: Vibratory excitation of gear systems
1410
rotational positions of the two gears as indicated earlier.
Let Cr denote the time average compliance and let
OCr =•Cr - Cr
BASE CYLINDER
denote the instantaneous variation
in compliance.
Further-
more, let us define a loading-dependent component of
transmission error by
PLANE
OF CONTACT
• • Wocr ,
(•)
which arises from variations in the comp,liance of the
BASE
mesh. Thus, we may rewrite the above expression involving the unsteady gear rotations and translations as
CYLINDER
R•>O(•>+v(•>-R?O (•'>-v © =CrW+•'
where
FIG. 1.
we have
(2)
defined
Pair of meshing gears.
Thus, the static transmission errors of pairs of meshing
gears cause unsteady forces on gear casings, which, in
turn, vibrate andradiate sound.7
To illustrate
how the static
transmission
error
is
used in writing the equations of motion of a gear system, we consider the pair of meshing gears shown in
Fig. 1. The radii of the base cylinders of the two gears
are denoted
by R(•t>andR}•'>. All contactbetween
pairs
of meshing teeth takes place in the plane of contact,
which is the plane tangent to the base cylinders of the
two gears. By involute construction, tooth surfaces at
points of tooth contact always are normal to the plane
of contact; thus, forces W transmitted by the teeth always lie in the plane of contact as shown.
Let v(t>andv(•'>denotetheunsteady
components
of
gear translational
displacement measured in the direc-
tion parallel to the plane of contactande.(•) ande(•')the
components of unsteady rotational displacement. Let
Cr denote the (instantaneous) compliance of the mesh
and•(•) and•(•') the components
of statictransmission
error of the teeth on gears (1) and (2) due to any deviations of the tooth faces from perfect, uniformly spaced
involute surfaces when no loading is present. "Errors"
•(•>and•(•') are measuredin a directiondefinedby the
intersection of the plane of contact and a plane normal
to the axesof rotation;•(t> and•(•'>are takenas positive when they are equivalent to removal of material
from perfect involute surfaces.
If we compare the positions of a pair of real gears under loading W with the positions of a pair of fictitious,
unloaded, perfect involute gears, it is evident that for
the teeth of the real gears to be in contact under loading W, the gears must be in positions such that the teeth
havecometogetherby a displacement
CrW+•(t)+•(•')
relative to the fictitious gears.
The quantity • =CrW
+•(•>+•(•') is the transmissionerror. Sinceperfectinvolute gears transmit uniform motion, it is evident from
Fig. 1 that the unsteady instantaneous rotations and
translations of the two gears shown are related to the
transmissionerror • =CrW+•;(t) +•(•'>by
The differential equations of motion of a gear system
of interest may be written in straightforward fashion in
terms of the dynamic forces W acting at the individual
meshes as illustrated in Fig. 1. These equations of
motion are supplemented by an equation which is analo-
gous to Eq. (2) and which is written for each pair of
meshing gears in the system. The resulting combined
set of equations can be solved approximately as follows. Each equationanalogousto Eq. (2) is solvedalgebraically for W and combined with the original differential equations of motion in such a manner that explicit
dependence of the new equations on the time dependent
forces W is eliminated, except for the dependence of
each •½ on W. The new set of simultaneous differential
equations thus obtained has for its right-hand side a
transmission-error excitation vector that is composed
of components
•'= • + •(•>+ •(2) from eachpair of meshing gears in the system.
A solution to this set of equations can be obtained by
setting the value of W in each •[ equal to the mean force
W0 transmitted by the gear pair associated with •.
Except for de components, CrW o not included, the resulting
components
•' =•½+•(t> +•(•') in the excitationvector are
the static transmission errors of the gear pairs in the
system. The set of time-invariant linear differential
equations thus obtained is most conveniently solved in
the frequency domain. To carry out the solution in the
frequency
domain,we requirethe Fourier-seriescoefficients of each static transmission-error
component
•' =•[+•(• +•(•'•, withW setequalto theappropriate
constant W0. Expressions for the Fourier-series
efficients
of the static
transmission
error
are
co-
derived
in the main text of this paper.
In some applications, it may be practical to use an
iterative procedure to arrive at an improved solution.
The above described solution, which is obtained by setting W equal to the appropriate value of W0 in each •,
yields a set of time-dependent rotational and transla-
tional displacementsof the system(or their transforms).
These solutions may be used to evaluate the rotational
and translational displacements (or their transforms) in
each of the equationsanalogousto Eq. (2), and the time-
dependent
iorees W (or their transforms)thenmaybe
The instantaneous mesh compliance Cr varies with the
solved for algebraically
from the resulting expressions.
J. Acoust. Soc. Am., Vol. 63, No. 5, May 1978
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1411
William
D. Mark:Vibratory
excitation
of gearsystems
/
/
/
I
/
o
1411
/
'/
/
I
I
I
/
FIG. 2. Lower figure is end view of a perfeet pair of meshinginvolutehelical gears.
Upper figure shows lines of contact and zone
of contact in the plane of contact,
DRIVER
fl %
PLANE OF
CONTACT
BASE
CYLINDERS
PITCH
CYLINDERS
OF
CONTACT
The set of differentialequations
obtained
by eliminating
the W's are thenresolved;however,to obtainthis secondsolution,onemustuse the appropriatetime-dependentforces W in the force-dependent
components
•v of
the transmission-errorexcitationfunctions•' =• +•,(1)
+•(•'). Thissecond
solution
will yielda second
setof
rotational and translational displacements or their
transforms. Usingthis secondset of displacements,
one may repeat the procedure to obtain a third set of
displacements--and
so on. Alternatively, if oneoperates entirely in the frequency domain without elimina-
tion of the W's, the entire set of simultaneous
equations can be solved "exactly."
I. MECHANICS OF HELICAL GEAR ACTION
A. Contactgeometry
Figure2 illustratesrelevantfeaturesof thegeometry
of an unloaded
perfect pair of meshinginvolutehelical
gears with uniform axial lead.9 Toothcontactis limited
to the portion of the plane of contact that lies within the
addendum
cylindersof the twogearsas indicatedby the
heavyline in the lowerhalf of thefigure. Theupper
half of the figure illustrates the zone of contact where
the planeof contactis theplaneof the paper. The dimensionsof the zoneof contactare determinedby the
lengthL of the activeportionof the path of contactand
theactivefacewidthF of thegears. Thebasepitch,
A is the arc lengthmeasuredonthebasecylindersbetween correspondingpoints on successive teeth. The
basehelixangle•b is thehelixangleonthebasecylinders. The angleqbshownin the lowerhalf of Fig. 2 is
called the pressure angle. The pitch cylinder andbase
cylinder radii are R andRb, respectively, as shown.
Relationshipsamongthesequantities, required later in
the paper, are derived in Appendix A.
As the two gears shownin Fig. 2 rotate about their
respectiveaxes, the zoneof contactshownin the upper
half of the figure remains fixed and the lines of contact
movefrom right {o left throughthe zoneof contact. We
can imagine that the lines of contact are attached to a
belt riding on the twobasecylindersandpassingthrough
the zone of contact.
The lines on this belt are identical
with the lines onthe imaginary"bel•" usedto define
the involutetoothsurfacesonthetwomeshinggears;
hence, at any giveninstantof time, contactbetween
meshing pairs of teeth occurs on the lines of contact
in the plane of contact.
Let us attachto this belt a pair• of rectangularcoordinates x,y as shownin Fig. 2. The origin of this coordinate system is placed midface on the line of contact of
a pair of teeth arbitrarily designatedas j =0. The value
of the x coordinate designatesthe position of the center
line of the zoneof contactrelative to the midiaceposition of the line of contactof toothpair j =0, as shown
in Fig. 2. If we denotethe angular position of one of the
gears with base circle radius Rbby 8, thenx and 8 are
related by
x =R•,
(3)
where the origin 0 =0 has been chosen to coincide with
zoneof contactpositionx =0. The valueof y designates
the axial locationof a genericpointp locatedon a line
of contactas shownin the upper figure.
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1412
WilliamD. Mark:Vibratoryexcitation
of gearsystems
1412
tates that the two gears must move together relative
to
rigid perfect involute gears by the distance
=.,•
- (x,y)+
(x,y)+
=ui(x,y)+',li
where
y)
(x,y)+,t•
(x,y),
we have found it convenient
to combine
(4)
the elastic
deformations of both members of the gear pair into the
single deformation
u•(x,y) a=u•("(x,
,"'(x, y) ß
(5)
Equation (4) is not sufficient to determine •(x). For
g to be independentof y and toothnumberj, u•(x, y)must
FIG.
3.
rotation.
Pair
of teeth in contact.
Plane cut normal
Rigid involute teeth; ....
to axes of
with elastic defor-
mationsU(1), u(z);- '-with deformations
U(1), U(2) andinvolutedeviations•(1), •½2)at contactpoint.
vary with y and j so that the force-deformation relationship governed by the tooth stiffness is satisfied. SoIving
Eq. (4) for uy(x,y) yields
u•(x,y)= •(x) -- *l}l)(x,y) --tly-(•')(x, y) ,
(6)
which always is assumed to be greater than or equal to
Consider a plane cut normal to the gear axes through
a pair of teeth in contact. By definition, all points on
this plane lie at the same value of coordinate y. For
every angular position of the gears during their rota-
tion (and, therefore, every value of x), the pair of teeth
is in contact at only a single point within the plane de-
fined by coordinate y. It follows that for a generic
tooth pair j, specification of the values x, y designates
a unique point of contact on the faces of the teeth denoted by j.
B. Expressionfor transmissionerror
zero. Let Kr•(x ,y) denotethe local stiffnessof tooth
pair j per unit length of line of contact as defined in
AppendixB. Let W•(x) denotethe total force transmitted by tooth pair j measured in the plane of contact in
a direction normal to the gear axes, and let dl denote
the differential length of line of contact; then,
dl = sec½
bdY,
which follows directly from the upper half of Fig. 2.
From the definitionof Kv•(x, y), we nowhave
%(x)
=f
Figure 3 illustrates tooth contact in a plane cut normal to the gear axes at a location designated by coordi-
loaded gears deviate from perfect involute faces with
uniform lead and spacing. These unloaded deviations
are designated
by •y
r•(1)(x, y) and r• (x, y), as illustrated
in Fig. 3. The elastic deformations u•(')(x, y) andthe
geometricdeviations
,/(•')(x,y) are measured
in a direc-
tion defined by the intersection of the plane of contact
and a plane normal to the axes of rotation located at co-
ordinatey. Quantitiesu•(')(x , y) and • (x, y) are defined as positive when they are "equivalent" to removal
of material from the faces of unloaded perfect involute
teeth as illustrated in Fig. 3.
Let g(x) denote the transmission error as defined in
the Introduction. Here, the transmission error is expressed as a function of zone of contact position x, which
may be related to the angular position 0 of either of the
real 'meshinggears by Eq. (3). In the treatment to follow, •(x) is assumedto be independentof y. This assumption is essentially equivalent to hypothesizing that
imaginary lines on gear rims drawn parallel to the gear
axes remain parallel to the axes when the gears are
transmitting torque. According to Fig. 3 and the above
comments, the requirement that gear teeth remain in
contact at every value of y within the zone of contact dic-
Kv•(x,y)u•(x,y)dy
(8a)
(8b)
A
= seeCb
Kv•(x, y)
A
wheresuperscripts
(1) and (2) designate gears of the pair and j designates
tooth-pair number. In addition, actual tooth faces of un-
y)u(x,
y)at
=see½
b
nate y. The solid teeth shown in Fig. 3 illustrate rigid
perfect involute teeth. When the gears transmit torque,
the two teeth elastically deform at the point of contact
(1)(.x, y) andu•')(x,y),
byamounts
uy
(7)
x[•(x) -(t)(x y)-*l•')(x,y)]dy
(8c)
where Eqs. (6) and (7) have been used. Limits of integration YA and ys are functions of tooth number j and
zone of contact position x; these limits designate the
endpoints of the line of contact of tooth pair j. For the
idealized situation depicted in Fig. 2, the limits are
determined by the zone of contact shown in the figure.
A truly exact analysis would require that we consider
yA and ys to be dependent on tooth-to-tooth differences
in thegeometric
deviations
•')(x, y), because
suchdifferences cause tooth-to-tooth variations in the regions
of contact on individual tooth faces. Vibratory motions
of the gears cause further variations in the regions of
contact. Inclusion of these effects would horrendously
complicate the analysis. In the present paper, we shall
assume that the zone of contact is independent of gear
vibrations
andtooth-to-tooth
differences
in r/}''(x, y).
Let
us define
l•v•(x)•=sec½• Kv•(x,y)dy,
(9)
A
which is the stiffness of the portion of tooth pair j within
the zone of contact. For either gear of the pair (1), (2),
let us further
define
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1413
WilliamD. Mark: Vibratory excitationof gearsystems
%rj(x)a__
sec•b
1413
•-LINE
OF
CONTACT
ATs=0
Kvj(x,y)•I}')(x,y)dy,
A
z
which is the integral over the line of contact of the stiffness weighted geometric deviation of the face of tooth j.
Then, the total force transmitted by the mesh can be expressed as
0\
---V--?PITCH
PLANEI
=
-
(x) +
•
ß(x)],
I z
',•
I
I I
'• I I - • . I CONS•CT
AT
s'=
O
where we have used gqs. (8)-(10) and where the sum,
mations are taken over all tooth pairs j within the zone
of contact sho•
in Fig. 2. Solving gq. (11) for •(x)
yieIds the desired e•ression
for the transmission
•
ror:
•(x)
W
•
Y•
xX
_x•
(12)
'-
,
• F/Z••
I
F/Z
-'
LINE
OF
CONTACT
OF
TOOTHNO.j
BAS IC RACK
(DRIVER)
where the to•l force W transmitted by the mesh is measured in the plane of contact in a direction normal to •e
gear
axes.
10
FIG. 4. Lower figure is zone of contact projected onto a plane
parallel with pitch plane. Upper figure illustrates position of
line of contact
on tooth face as a function
of zone of contact
position.
S•ce Kr•(x) is the stiffnessof toothpair j, the denom•ator
of both terms in the r•ht-hand
side of Eq.
(12) is the total mesh stiffness, which is a function of
zone of contact position x. Therefore, the first term
in the right-hand side represents the direct contribution
to the transmission
error
from
the elastic
mation of the mesh caused by the load W.
defor-
According to
Eqs. (9) and (10), the secondterm in the right-•nd side
of Eq. (12) represents a weighted average of the sum of
thegeometric
deviations
B(/)and•}z>fromperfectinvolute tooth faces, where this average is •ken over the
to•l length of lines of contact of all mesh•
teeth • the
zone of con•ct
and where the weighting function in this
those of the original gear pair.
tact of the basic
rack
are
Since the lines of con-
identical
with
those
of the
original gear pair, we may establish our tooth coordinates on the basic rack and transform
with complete
generality between these coordinates and points on the
teeth of the corresponding gear of finite diameter.
Let us concentrate on a generic tooth j and define the
coordinate
s•x-ja.
(13)
averageis the local tooth-pairstiffnessKr•(x , y).
When s =0, we have x =fiX; thus, at s =0 the center line
C. Transformation to tooth coordinate system
y =0 with the line of contact of the jth tooth. This line
of the zone of contact illustrated
of contact
Tooth-pairstiffness
Kv•(x, y) anddeviations
r/•')(x,y)
from perfect involute teeth are both most conveniently
expressed in a coordinate system that directly describes
positions on tooth faces.
The coordinates x,y illustrated
in Fig. 2 do not satisfy this requirement. We ghall now
derive
a coordinate
transformation
that
will
enable
us to
express
•:v•(x)and•[))(x)intermsoftooth-pair
stiffnesses and deviations from involute profiles that are de-
scribed in, such a "tooth-face" coordinate system.
Since we are dealing with helical gears, it would be
convenientif we could roll the outer portion (which contains the teeth) of a generic gear onto a plane before defining the new coordinate system. This can be done
with complete rigor by defining our tooth coordinates on
thebasic-rackform of the gear.9 Figure 4 illustrates
a basic-rack
form
and other
material
relevant
to the
is illustrated
in the
in Fig.
sketch
2 coincides at
located
in the
lower-right-hand corner of Fig. 4, which is a projection of the plane of contact onto a plane parallel with the
pitch plane. Positions of the zone of contact shown in
Fig, 4 are denoted by the variable s', which is related
to s by
s' = s cosg)
(14)
because the zone of contact shown in Fig. 4 is a projection onto the pitch plane of, the true zone of contact,
which is located in the base plane.
Two locations of the zone of contact are shown in Fig.
4. In the location described by dashed lines, the line
of contact passes through the center line of the zone of
contact at y =0; hence, for this position we have s' =0.
The position of the zone of contact shown by solid lines
illustrates a situation where the gears have advanced
sure angie ½, the base pitch zX, the base helix angie
the plane of contact, the lines of contact, and the zone
somewhat beyond the position corresponding to s'=0.
The position of the gears shown in the lower-left-hand
corner of the figure corresponds to this latter zone of
of contact
contact.
present discussion.
It is easily shown that the pres-
of the basic-rack
form
are
all identical
with
J. Acoust. Soc. Am., Vol. 63, No. 5, May 1978
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1414
WilliamD. Mark: Vibratoryexcitationof gearsystems
1414
Y
In the upper-right-hand corner of Fig. 4, lines of
contact on the jth tooth of the basic rack are illustrated
for the two positions of the zone of contact shown in the
lower figure. The lines of contact in the upper figure
have been drawn dashed and solid to correspond with the
marking of the two zones of contact. It may be seen
from the upper figure that as x or s increases, the line
of contact moves parallel to itself along the tooth face.
•
F/;:'
siS)/•
The Cartesian y, z coordinates shownin the upper
portion of Fig. 4 illustrate the coordinate system we
shall use to locate a point on the face of a tooth. Let D
denote the height measured perpendicular
(F TAN •b-L)/2
I
(FTAN
•b+L)/2
to the pitch
plane of the path of contact projected onto the tooth face
as illustrated in the lower-left-hand
corner of Fig. 4.
The origin of our tooth coordinate z iS taken at the mid-
point of the range D as shown in the upper-right-hand
corner of the figure. Thus, at zone of contact position
s =0, the line of contact in our y,z coordinate system
passes through the origin of coordinates y =0, z =0 as
TAN
FIG. 5.
,½,i•i••_F/2
Pb-L)/2
The lower and upper curves give the limits of inte-
shown in the upper figure.
gration of the integral in Eq. (20b) as functions of zone of contact position s. Situation depicted is for rectangular zone of
It was pointed out in Sec. I A that every pair of values
x, y of our original coordinate system defines a unique
point of contact on the faces of a pair of teeth in contact. Once j has been specified, it follows from Eq.
contact illustrated in Figs. 2 and 4.
(20a)
(13) that the same must be true for every pair of values
s, y, and, according to Eq. (14), for every pair of values s• y. The position s' of the solid zone of contact
illustrated in the lower portion of Fig. 4, together with
the value of y locating point p in the lower figure, must
therefore define a unique point of contact p on the face
of the jth tooth as shown in the upper figure. For a
general pair of values s, y it is shown in Appendix C that
the corresponding point on the line of contact in our new
tooth coordinate system is
z =l•s+yy,
(16)
where the constants • and 7 may be evaluated from the
pressure angle • and pitch cylinder helix angl•e• by
• = sin•,
(17)
A
= sec½•
fc j( Y, ½s+yy)dy,
YA($)
(20b)
where the first line has the same form as Eqs. (9) and
(10), where we have used the identity of Eq. (19) in
going to the second line, and where we have expressed
the end points YA and Ys of the line of contact of the jth
tooth in terms
of the coordinate
s.
For
situations
where
contact takes place over the rectangular region of tooth
face illustrated in Fig. 4, it is shown in Appendix D that
the limits of integration in Eq. (20b) are determined by
the lower and upper curves denotedby yA(s) and ys(s)
in Fig. 5. A and L also are related to fundamental gear
parameters in Appendix D.
It is evident from Figs. 4 and 5 that we should require
the domain of integration in Eq. (20b) to be truncated
for l yl > • F. This truncation can be accomplishedby
multiplying the integrand of Eq. (20b) by the rectangular
1
7 =- sin• cos• tan•.
'
(18)
Let fcj( Y,z) denotea real functionof the newcoordi-
functionrect(y/F), whichis definedastx
nates y,z defined on toothS. Here, subscript C denotes
that the functionfct is describedin the new Cartesian
coordinates y,z illustrated in the upper-right-hand corner of Fig. 4. In practice, this function will represent
local tooth-pair stillnesses, deviations from perfect in-
volutefaces, or their product. Let f•(x, y) describethe
samephysicalquantityas fcj(y,z), but in the former
x, y coordinate system illustrated in Fig. 2. It follows
immediately from Eqs. (13), (15), and (16) and the
abovediscussionthat fj(x, y) maybe expressedin terms
of fcj(y,z) by
f•(x, y) =f •(s+/A, y) -=fc•(Y, l•s+7y).
(19)
Instead of dealing separately with Eqs. (9) and (10),
1
rectx •
l, Ixl<•
•
(21)
= O, Ixl>•.
If we explicitly include this requirement in our formulation, we can show--from the slopes and y intercepts
of the limits of integration displayed in Fig. 5--that Eq.
(20b) may be rewritten as
A s+A
-
fj(x) =sec½•
ßrect
=A_
sec•.rect
L
-'s-•
fc•(y,
fc• ,13s
+
let us definethe integral of f•(x, y) over the line of contact of tooth j as
where the changeof variable • =(L/A)y was usedin
J. Acoust. Soc. Am., Vol. 63, No. 5, May 1978
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d•,
1415
1415
William D. Mark: Vibratory excitationof gearsystems
Consequently, for general contact regions defined by
Eq. (25), Eq. (24) can be replaced by
-
s-•
s-•
(27)
FIG. 6. Elliptical contact region 'Qc(Y,z) definedby Eqs. (25)
and (39). l]c(y, z) is unity inside the elliptical region and zero
We may verify this last result by noting that for the
rectangular contact region illustrated in the upper-righthand corner of Fig. 4, we have
•c( Y, z) = rect(y/F) rect(z/D).
elsewhere.
(28)
Substitutionof Eq. (28) into Eq. (27) yields Eq. (24).
For caseswherefc•(y,z) is identically,zerooutside
going to the second line in Eqo (22). Using again the
the face contactregion, Eqs. (26) and (27) reduce to
above definition of the rectangular function, we can
•j(x)=seeCb
f. fc•(y,•s
+yy)dy
show that Eq. (22) may be expressed as
(29)
and
xf.irect(sL'-•)rect(•'L)fc'{-•"
ils+
Y-•)d•. In Sec. II, we shall haveto deal rep_eatedlywith
(23)
From Eqs. (C4), (17), (18), (A4), and (C8), it may be
seen that Eq. (23) can be further simplified to
f
the Fourier transformof the functionf•(x).
_f•(s)
• sec½•
f.i9c(y,•s
+7y)fc,(y,•s
+7y)dy.
(31)
=csc
oo
Let us
denote the right-hand side of Eqo (26) by
Hence, from Eqs. (13), (26), and (31), we have
S-'•
xf.•rect(F2/Alrect(•)fc•(l•••i,
L/D-)d•,f•(x) =ll(s)=_f•(x- jA);
(24)
where.L, A/L, andD/L all maybe expressedin terms
of fundamental gear parameters by relationships derived
in AppendixesC and D. Equation (24) is an expression
for the integral of fc•(y,z) over the line of contactof the
jth tooth for zone of contact position x = s + jA.
i.e., f_•(x)is identicalwith]j(x) exceptfor a shiftof the
origin of the independent variable by fix units.
Let us
definethe Fourier transformsof f_j(s),fc•(Y, z), and
9c( y,z) by
The
{•(g)
af."_{,(s)
exp(-i2•gs)ds,
functionfc •(y, z) is expressedin the Cartesian tooth
coordinates y , z illustrated in the upper-right-hand
(32)
(33)
cor-
ner of Fig. 4. Equation (24) is applicable to situations
where contact takes place over the rectarwular region
f cj(gi,gz) •
of tooth face illustrated in Fig. 4.
f c•(y, z)
Xexp[-i2rt(giy +gzz)]dydz,
(34)
and
D. Generalizationto arbitrary contact regions
The two rectangular functions in Eq. (24) are a direct
consequenceof the finite limits of integration yA(s) and
yB(s) in Eq. (20b) for the case of a rectangular toothface contactregion. Since the limits yA(s) and yB(s)
represent the end points of the line of contact shown in
9c (g•, gz)•
9c( y, z)
x exp[-i27r(g•y+gzz)]dydz,
(35)
where the caret denotes Fourier transform.
In Appendix E, we show that for arbitrary face contact regions
the upper-right-hand corner of Fig. 4, the result of
Eq. (24) can be generalized as follows. Let us define
definedby Eq. (25), the Fourier transform of f•(s) can
a "face contactfunction"9c(y,z), in terms of our
be expressed in terms of the two-dimensiorml trans-
forms of fc•(y,z) and •c(y,z) by
tooth-face coordinates y, z as
1,inside
contact
region
, O, outside contact region,
•(g) =(L/D)see½•
(25)
which is illustrated in Fig,. 6 for an elliptically shaped
contact region. If we multiply our general function
fcj(y,z) by 9c(y,z), we may replace the limits of integration in Eq. (20b)by ya =-co andys =oo; i.e.,
fj(x) =secCb nc(y,•s+yy)fcj(y,•s+yy)dy.
(26)
,
(36)
For the case of the rectangular contact region illus-
trated in Fig. 4, we may obtainthe function•c(g•,ge)
by forming the two-dimensional
Fourier
transform
Eq. (28):.
J. Acoust.Soc. Am., Vol. 63, No. 5, May 1978
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of
1416
WilliamD. Mark: Vibratory excitationof gearsystems
g2
1416
lipse.
The area of the gx, g•. plane inside this ellipse is
ALL POINTS91=(L/A)g,
17%larger than the rectangular area inside the loci of
g:•= (L/D)g LIE ONTHIS LINE
zeros illustrated
in Fig. 7.
For the cases representedby Eqs. (29) and (30),
where fcj(y,z) is identically zero outsideth_etrue face
contact region, we may take 12c(y,z) equal to unity
everywhere in the y,z plane, as can be seen by com-
paringEqs. (26)and(27)with Eqs. (29)^and
(30). Hence
Eq. (36) will yield the correct value of _fj(g)for this
case,if we evaluate•c(gz, g•.)fromEq. (37)withF-oo
g2 = (L/D) g
and D-oo.
However, using the known integral
f.•sinaxdx=7r,
it follows from Eqso (37) and (38) that for any values of
.._
F and D we have
1__
f.if.••c(g,,g•.)dg,dg•.:l. (41)
Thus, when F-oo and D-oo, the right-hand side of Eq.
(37) approaches a two-dimensional delta function with
gl = (L/A) g
,
FIG. 7. Theprojections
of thefunction
'•c[(L/A)g-g
l,
(L/D)g-g 2] in Eq. (36)are shownon the gl andg2 axesfor the
casewhere•½(gi,g2)is defined
by Eq. (37).
•c(gx,g•.)=FD sinc(Fgz)sinc(Dg•.)
,
(37)
where sincx is defined aszz
(38)
For the case of the elliptical contact region illustrated
in Fig. 6 with boundary defined by
[ y•'/(F/2)•']+[z•'/(D/2)•']= 1,
(39)
we can showthat the function •c(gz, ge) takes on the
form z•.
%(g"g")
=2
........ D•'?')•/
everywhere
outside
thetruefacecontact
region,•i(g)
maybeevaluated
fromfc•(gz,g•.)
by
.•(g): (L/D)sec½ofcj[(L/A)g
, (L/D)g].
(42)
Equation (42) also can be derived directly from Eq. (29)
sinex a_simrx/rrx.
^
unitarea concentrated
at gx=0, ga=0. It followsf•om
the abovecommentsandEq. (36)that whenfc•(gz,g•.)
is evaluatedfrom a functionfcj(y,z) that is zero
or (30)by first substituting_•(s)
for theleft-handsides
according to Eq. (32) and Fourier transforming either of
the resulting expressions.
II.
HARMONIC
ANALYSIS
OF STATIC
TRANSMISSION
ERROR
A. Decomposition into mean and random components
Let us return to the general expression for the trans-
,
(40)
mission error provided by Eq. (12). Let us also denote
the stiffness-weighted deviation from a perfect involute
where Jz(') is a Bessel function of the first kind• of order unity. According to p. 409 of Ref. 13• the first
five zeros of Jz(x)/x occur at x =3. 832, 7. 016, 10.173•
tooth, as used in Eq. (10), by
13. 324• and 16.471.
and the same quantity described in the Cartesian tooth
The maximum value of the right-
handsideof Eq. (40)occursat the origingz=g•.=0. z•
Figure7 illus[ratesthe generalresultof Eq. (36)for
'qtr'•(x, y)•-Kvj(x, y)rlt/)(x, y) ,
(43)
coordinatesystemy,z by r/rcj(y,z);
(')
i.e.,
•{r'•(x,y)=•{r'](s+jA
, y)=•l(g•(y,t3s+yy),
(44)
the case where •c(gz•g•.) represents the two-dLmensiona]
as in Eq. (19), where we have used the transformation
transform, Eq. (37), of the rectangular contact region
of Eqs. (15) and (16). According to Eq. (43), the Car-
defined
byEq. (28). For•any
valueofg, .•(g) is gener-
atedby an integrationof fc•(gz,g•.) weightedheavily in
the vicinity of gz = (L/A)g andg•.= (L/D)g by the function
•c[(L/A)g-gz, (L/D)g-gz]. It is evidentfromEq. (37)
and Fig. 7 that the behavior of the function •c(gz•g•.) is
governed by the dimensions F and D of the tooth-face
contact region. In the case of the rectangular contact
region defined by Eq. (28), the locus of the zeros of the
function
•c[(L/A)g-gz, (L/D)g-gz] closesttogz=(L/
A)g, gz = (L/D)g forms the rectangular pattern shownin
Fig. 7.
In the case of the inscribed elliptical contact
region illustrated in Fig. 6• Eq. (40) showsthat the
locusof thezerosof thefunction
•c[(L/A)g-gz, (L/
D)g-g•.] closestto gz = (L/A)g, g•.=(L/D)g forms an el-
tesian representation•rc•(y,z)
('•
maybe expressedas
•lgc•(Y,z) =Kvc(y,z)•l(c']
( y, z) ,
(45)
where Krc(y,z) is the local stiffness of a pair of teeth
per unitlengthof lineof contact
andB•c'•>(y,z)
is theunloaded deviation of the face of tooth j from a perfect involute surface, both expressed in Cartesian tooth coor-
dinates. The Cartesian representation Krc(y,z) of local tooth-pair stiffness is independentof tooth number j.
(-)
We shall nowdecomposeBrc•(y,z) into meanand randomcomponents,
(')
(')
mrc(y,z)
and½rc•(y,z),
respectively,
where the meancomponentis the averageof 'l•c•(
(') y,z)
over all teeth on the same gear; i.e.,
J. Acoust. Soc. Am., Vol. 63, No. 5, May 1978
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1417
William
D. Mark:Vibratory
excitation
of gearsystems
1 N(')'I
"'(y,•)A
=•r•,•
•/•I•
I
•K•C•(y,z)
,
y=0
(46)
where N (') denotes the number of teeth on the gear under
consideration.
By definition•m•c (Y, z) is independent
of tooth number j. The stiffness-weighted random com-
ponent
(•(y,z)
c}.,(•)• • •;;,(•)/x•(•),
(s9)
we can decompose•(x) into mean and random compo-
nents, •,•(x)and •(x),
respectively. Thus,
•(x) =•(x) +•,(x),
(60)
is thedifference
between
thestiffness- where, when W is a constantWo, the mean component
weighted original and mean deviations'
(•)
(.)
(.)
=
(•c•(y,z) •c•(y
, z)- m•c( y, z) .
(47)
(,)
From Eqs. (45) and (46), it followsthat mrc(y,z) may
be e•ressed
1417
as
g,•(x) is the sumof •e load-dependent
component
gv(x)0
andthe mean-deviation
components
r {• (x) andg{Z)(x)
from gears (1) •nd (2),
•
r {•)(x) ,
•.(x) =C•(x)0+•{•(x)
+-.
(61)
and the random componentg,(x) is the sum of the random componentsfrom gears (1) and (2),
•.• =K•c(y,z)
mgc(y,z)
N'" '
•[•(y,z)
5=0
=x•(y, z)m•'•( y, z),
(48)
where we have defined the unloaded mean deviation
of
the tooth faces on gear (.) from perfect involute sur-
••( y,•).
(49)
Therandomcomponent
of •[• (y, z) maybe definedas
•'•
' •'•
- m['•( y,z) .
(•0)
•c•(y,z)•
%•(Y,Z)
(02)
Subscript zero on •v(x)0 denotes situations where W is
assumed to be a consent value W0.
Discussi•.
faces with uniform spacing as
•['•(y,•)O
•
;,(x):; •,•'(x)+;7'(x) ß
From gqs. (40), (48), (40), (53), and
(58), it followsthatg{•)(x) andr {a)(x) are thecomponents of transmission error caused by mean deviations of
the unloaded teeth on gears (1) and (2) from perfect
volute surfaces. If imperfections in gear design and
m•ufacturing were to result • tooth-face and flank errors that provide no contribution to the mean error de-
It thenfollows, from gqs. (45), (4•), (48), and (50) that
finedby gq. (40), thenr {•)(x) andr{a)(x)wouldbe the
the stiffness-weighted random component may be expressed as
componentsof transmission error caused by intentional
des•n deviations from perfect involute teeth.
('•
•gc•(y.z)
=Kvc(y. z)•}(,•)
y
(sz)
ß
Accordingto the conventionestablishedby gqs. (10)
and(20), we have, usingthe newnotationof gqs. (43)
and (44),
The decompositionof g(x) into mean and random componentsg•(x) and g,(x) is motivatedalso by frequencydomain considerations.
As the gear pair shown in Fig.
2 rotates, gears (1) and (2) re•rn to their origiml positionsas x traverses dis•nees of N{•)• andN{a)• units,
respectively. From this fact, it is evident that when W
,•(x)=sec½..•a(.•
,•(y.gs+yy)dy (S2a)is a consent
- (,)
-(,)
=m•(x)+•(x)
.
(52b)
-,., (x) and• [.I(x) are definedin a manneraml wheremg•
ogousto gq. (52a)-i. e.,
-•'•(x)•, sec•
tory =
•A(S)
(rc•(Y,
(,) •s +yy)dy
(53)
(54)
--and where the decompositionof Eq. (47) has been
used. Similarly, usingthe conventionof Eqs. (9) and
(20), we •ve
_ =seeCo( •B
(s)
Kv•(x)
Kvc(y, •s +yy)dy.
a•A (s)
(55)
If we substituteEq. (52b), applied• ge•s (1) and (2),
into Eq. (12) and define the instan•neous to•l mesh
stiffness
•,a :M•(• •t•a):Ma(•{a•a),
where Mt and Ma are the smallest positive integers that
•'>
f •B(s)
•['] (x) • seeCo
•
the s•tie transmission error g(x) is a
periodic function of x wi• period
as
satisfy the right-handequalityin gq. (03). Mt and Ma
are the numbers of rotations •at gears (1) and (2) must
makebeforepo•ts common
tobothmeetagain. If N
andN•) are prme to oneanother,thenM• =N{•) andMa
=N {1)'
Since the static transmission error/7(x)
withperiodN•X, it canbe representedby a Fourier
serieswhosqfundamental
component
hasa periodof N•zX
units. Similarly, therandomcomponents
•(X)(x)and
g(•')(x)from gears(1) and(2) are periodicwithperiods
N(x)•xandN(•'>,x,respectively. It followsdirectly then
fromgq. (63)thatthefundamental
components
of g}X)(x)
andr(•')(x) are respectively, the Mlth and M•.thharmonies of •(x).
Kv(x)• •.. I•vy(x),
(56)
is periodic
It also follows from lgq. (63) that ff
N{x)andN (•')are prime to oneanother, the only har-
moniesof g(,1)(x)and•,(2)(x)thatpotentiallyoccurat the
same frequenciesare the tooth-meshingharmonics with
and the following components of transmission error'
c,,(x)• w/zq.(x) ,
y
fundamental period equal to the base pitch •x. However,
when W is a constant, our above definition of {re(x) ineludes all components of the static transmission error
that are periodic with period zX. Thus, as is proved
later in the paper, •r(1)(x)and•r(•')(x)containno nonzero
J. Acoust.Soc. Am., Vol. 63, No. 5, May 1978
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1418
William
D.Mark:
Vibratory
excitation
ofgear
systems
harmonics at integral multiples of the tooth-meshing
fundamental
frequency.Consequently,
when/V
(•) and
N m)are primeto oneanother,•r(•)(x)and , (x) contain no nonzero harmonics at the same frequencies.
In summary, the nonzero harmonic componentsof
•=(x) and •(x) always occur at different frequencies,
and whenN (•) andN ø') are prime to one another, the
nonzeroharmoniccomponents
of •r(•)(x)and m)(x)also
occur at different frequencies.
1418
For helical gears with contactratios of the order of
five or larger, Eq2 (71) with M = 1 provides excellent
accuracy--
K;•(x) :•[2
- •'Kv(x)],
M =1
(72)
--whereas for contact ratios of about 2.5 or larger,
M = 2 provides excellent accuracy--
K;•(x): R•,'[3- 3R•,'Kr(x)
+/•$•'K•(x)],M: 2.
(73)
For spur gears, Eq. (71) with M = 3 providesexcellent
B. Fourier-seriesrepresentationof mean component
accuracy--
WhenW is a constant,the mean componentof •(x )
givenby Eq. (61) is peri6dicwith periodA. Hence, it
/cr•(x)=R•'[4- 6•'KT(x)+4•2K•(x)- •'•aK•(x)],M=3.
may be expressed in terms of the Fourier series
•=(x)=•
a•, exp(i2•rnx/•),
(64)
where the expansion coefficients a•, are given by
I
=•
(•mn
_AI•.
(65)
_(l) and urn,
•(2) are, res•ctively, the
and where a•,, •'m,,
expansioncoefficientsof the load-dependentcomponent
•(x)0 andthemean-deviation
components
•}•)(x)and
•}•)(x) of the teeth of gears (1) and(2) from perfect involute profiles;
i.e.,
•Wn
=•.a/g •(x) 0exp(- i2x•/A•x
provides acceptableresults for any helical or spur
gears.
For anyvalueof M, the partial sumof Eq. (71) expressesK•(x) as a linear combinationof powersof
KT(x). Therefore, accordingto Eqs. (69) and(71), we
J[A/2•m(x)
exp(i2•rnx/A)dx
-(•) + Utmn
-(•')
= OlWn+ OZmn
(74)
For roughcalculations,theapproximation
of Eq. (72)
,
(66)
requiretheFourier-series
coefficients
of[Kr(x)]'for
l = 0, 1, 2,.;..
a•,,•A
1f•'/2[Kr(x)]'
exp(i2•rnx/A)dx,(75)
="•-•/2
wherec•.•-=c•..
1
•}'•(x)ex•- i2xnx/•)ax.
from Eq. (75) we have
OlKn'''
(67)
1. Fourier-series
representations
of reciprocalmesh
Kn, 1 > 1
ot
• =•ot
•:,,
1=1
where6,.0is •onecker's delta. Noticethatfor 1>1
stiffness and load-dependent component
we have
Accordingto Eqs. (57) and (66), the Fourier-series
coefficients of the load-dependentcomponent•w(x)0
may be expressed as
aw.=Woa(•lr).,
From the convolution
theoremfor
Fourier series Eqs. (G7)-(G12) and the associative
property of the convolutionoperation, it follows that
•d
• =
Let us denotethese coefficientsby
(68)
where
i.e., a• maybe computed
by 1- 1'successive
discrete
convolutions,where each may be carried out by forming the convolutionof the precedingresult with
Finally, usingthe partial sum of Eq. (71) to approximateK•(x), we see from Eqs. (69), (71), and(75) that
1-j.:I•'K•(x)
exp(i2•rnx/A)dx
(69)wemayexpresstheFouriercoefficientsofK•(x) as
O/(1/K)n=X/2
are the Fourier-series
mesh stiffness 1/K•,(x).
coefficients of the reciprocal
In Appendix F, it is shown
that K;•(x) may be expressedby the convergentinfinite
(72)-(74).
where the partial sumof the right-handside of Eq. (70)
obtainedby retaining terms throughm =M may be expressed as
(71)
which contains only one term for each power of
[-Kr(x)/•r].
•r is the me= of Kr(x) definedby Eq.
(F2). The numericalcoefficientsin Eqs. (70) •d (71)
the b•omi•
To evaluate the Fourier coefficients a•r, in Eq. (76),
we note first that Eq. (55) is in the form of the notariohal conventionof Eq. (20b), exceptthatKrc(y, •) is
independentof j. Hence, whenthe conventionof Eq.
(32) is appliedto •r•(x), we have
•(x): K;' (f:•) R, / ,
•e
where the values of M required for satisfactory accura-
cy in Eq. (77) are the sameas thoseindicatedby Eqs.
series
coefficients defied by Eq. (F6).
R•(x): K_'
•(s): K_
•(x- j •x),
('•8)
where K_
r(s), consideredas a functionof the variable
s, is independentof tooth-pair numberj, as examination of Eqs. (55) and (78) will show. Notice that
Krc(y,x) , ya(s), andys(s) in Eq. (55) are all indepen-
J. Acoust.Soc.Am., Vol. 63, No. 5, May 1978
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1419
WilliamD. Mark:Vibratoryexcitation
of gearsystems
dentofj. Hence, accordingto Eqs. (56) and(78),
1419
2. Fourier-series
representationof mean deviation
Kv(x) may be expressed as
components
Let us nowturn to the expressionEq. (58) for the
Kv(x
)=.•.•
.K
T(X
-j•X)
=rep•K_T(X)
,
(79) component
of transmission error causedby the mean
whererep•K_T(X
) is the rep function
ll definedby Eq.
deviation from perfect involute surfaces of the teeth on
(G 13); therefore, from Eq. (G 17), the Fourier coefficients of KT(x) can be given by
form of Eq. (20b); hence,applyingthenotationof Eq.
a•,•=(1/•,)•y(n/A),
(80)
where
.I•y(g)
=f_•_Ky(x)exp(i2wgx)dx
(81)
is the Fourier transformof K_y(x)
as definedby Eq.
Recalling the general notational conventionestablished
in Sec. I, Eqs. (36) and (42) provide expressionsfor
one of a pair of meshinggears. Equation(53) is in the
(32) to • •j(x),
(')
we have
-
(,)
(,)
m•(x) =m_•(s) =m_
• )(x-ja) ,
(88)
wherem_
}')(s), considered
asa function
ofs, is independentofj becauseof the independence
of rn}•(y, z),
ya(s), andys(s)onj, asfollowsfromEq. (53). Hence,
using Eq. (88), we have
• •z•(•.)(x)
=•.•m_
;)(x_jA)•
repam_
x(-)(x).(89)
Let am,•
(') denote the Fourier-series
coefficients
of
•y(n/A)foruseinEq.(80);bycomparing
Eqs.(20b)
rep•_m•
(')(x),which,according
toEq. (G17),aregiven
and(55),wehavefor thisapplication
•(g)= •y(g) when by
fcj(g•,gz)=Rvc(gr,
gz),where,
according
to•q. (34),
f.:
x exp[-i2•(g•y +g2z)]dydz
(82)
is the two-dimensional Fourier transform of the local
stiffnessKvc(y,z) of a pair of teethper unit lengthof
line of contact expressed in the tooth coordinates de-
finedby Eqs. (15), (16), and(19). Thus, for cases
whereEq. (42) is applicable,we have, usingEqs. (80)
(o)
am,.=(1/a)•(')(n/a),
(90)
where•')(g) is theFouriertransform
ofm_•')(x
) as
definedby Eq. (G16). Then, usingour general notationa! convention,Eqs. (36) and(42) provideexpressionsfor Fn_
(•')(g) in terms of the two-dimensiona!
Fourier transform mkc(gz,g2)
^
of the stiffness-weighted
mean deviation m•:c(Y,
(')
z) expressedby Eq. (48)'
•Z•'c)(g,,
g•)=
and (82),
x exp[-i2•r(g•y+g•.z)]ayaz.
ag,=(L/Da)sec½•RTc[(nL/Aa),
(nL/Oa)],
(83)
whichare the Fourier-series coefficientsof Kr(x), as
defined by Eq. (75) for l = 1.
According
to Eq. (F2), Eq. (75) for l = 1, andEq.
(80), we havefor the meanstiffness
•r =as0=(1/•X)i•v(0).
(84)
Thus, for caseswhereEq. (83) is applicable,we have
Ka,=(L/Da)
sec•œrc(0,
0).
(85)
Accordingto Eqs. (83) and(85), the terms in the summation of Eq. (77) dependonly on the ratio
• =
m•'c)(y,z)
.•'vc(O,
O)
'
(86)
whosemultiple self-convolutionyields
= "•/"v
(87)
(9[)
For the cases governedby Eq. (42), it follows from
Eqs. (90) and (91) that the Fourier coefficients of
repam_
•' )(x)are givenby
(') =(L/D a) sec½•
ct•
• K
(')rtnL/Aa),
(nL/Da)] ß
(92)
Finally, from Eqs. (58) and(89), •(')(x) may be expressed
as
• •' )(x)=K•(x)rep•rp_
•')(x),
(93)
whereK•(x) andrep•m_
•')(x)bothare periodicwith
period •X. Hence, using Eq. (G12), from the definition
of O/(1/K)n
providedby Eq. (69) andthat for (') provided just below Eq. (89), we can express the Fourier-
series coefficientsof •)(x) by the discrete convolution
(.)
a(•,)=a (•/K)•.am,•,
(94)
whichis definedby Eq. (G12). The sequencesof coefficients a(•/•)• and a m'• are to be evaluated using Eqs.
for use in Eq. (77).
(77) and(92)--or for arbitrary contactregions,Eq.
(36) insteadof Eq. (92). Also see Eqs. (85)-(87) in
Usingthe results of Eqs. (85)-(87), we may computethe coefficientsa(•/r),of the reciprocal mesh
connectionwith the evaluation of Eq. (77). The
Fourier-series coefficients of the load-dependentand
stiffnessusingEq. (77). Finally, accordingto Eq.
mean-deviationcomponents
are to be combinedusing
(68), the Fourier-series coefficients of the load-de-
Eq. (65) to provide the coefficients of the total mean
pendentcomponent
•(x)0 are obtainedby multiplying
the a (•!r)• by the meanloadW0 transmittedbythemesh.
odic with period A.
The results of Eqs. (85)-(87) are valid for situations
whereKvc(o, -) is computedfrom the local tooth-pair
stiffnessKvc(y, z) definedas zero everywhereoutside
the true face contactregion as in Eq. (42).
componentof static transmission error which is peri-
C. Fourier-series
representation
of randomcomponent
Therandom
component
•(x) ofthetransmission
error of a pair of meshing gears is the sum of the ran-
J. Acoust.Soc.Am., Vol. 63, No. 5, May 1978
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1420
WilliamD. Mark:Vibratoryexcitation
of gearsystems
dom components
•;•'•(x) from the individualgears as
described by Eq. (62).
The period of each individual
randomcomponent
is N(')A; thus, the harmonicsof
1420
at. ,-N(,)/X
.__
xexp•N,.) ).
occur at generally different frequencies as described
The random component of the transmission error of
each gear may be expressed by the Fourier series
s)g}?(s);
N(.)/x
/- i2•rnj\
each of the random components of the meshing pair
earlier.
•
(103)
However, from the convolution theorem for Fourier
transforms, we have
• (•);•] =
('>
exp[i2•rnx/(N('>A)],(95)
•(•)•7(•
- •)•
where the expansion coefficients are given by
(.) 1
i("
=
x exp[- i2•rnx/(N(')A)]dx.
(96)
It followsfrom the fact that •r(')(x)is periodicwith period N (')/x andfrom Eq. (50) that we may express
•("(x) as
(o•)
where index j denotes tooth number and each advance
of index l represents one rotation of the gear under
consideration. Using the notational convention of Eq.
(32) appliedto •(•?(x), we have
(.)
• (•7(x)=! • (s)=•_(•7(x-•),
where s =x-j/x
as before.
•(•?(x lN('•)=•('•(x-j•
(08)
Hence,
lN('•)
Using Eq. (00) •d the fact that Kr(x) is peri•ie
•riod &, we may replace Eq. (0•) by
(OO)
• (1/K)n* KJ
(04)
--
where we h•ve used the notation o• •q. (G4) •d
•q.
(G6) •s it •ppHes to •q. (69). •y comb[n[n••qs. ([03)
•d ([04), we obtainthe desired expression•o• the
Yourlea-seriescoefficientso• •'>(x)'
=
•rnN(')A
Y=O
• ([/K)n'_K• •(.)•
xexp[N(.) ],
(105)
where• (,)(g) denotes
theFouriertransformof (•?(s)
as defined by Eq. (E3).
Wemayevaluate
•?[(n- N("n')/(N("&)]
byusing
the
general results establishedin Sec. I--i. e., Eq. (36)
or (42). Comparing Eqs. (20b) and (54), we see that
for the case governed by Eq. (42), we have
wi•
•(.)[L(n
-N(')n')
L(n
DN(.)&
•(')-1
X(•c• • AN(.)& ,
,
•=0
•(,)
N (ø).1
=•
rep•,.,a[K'/(x
--jA)•.•(?(X
_jA)],
(•00)
where the rep function is defined by Eq. (G13) and
where the period of the rep function in Eq. (100) is
N (')A. From Eqs. (96), (100), and (G13)-(G17), the
Fourier-series
coefficients of •r(')(x) may be expressed
as
(.)
1
Y--0
•
(x-jA)•_(•7(x-j•);•(.,,
where (•c•(g•,gs) is the two-dimensionalFourier
['
tr•sform of ((')
•c•z,
• (.)
t
• ((2)
•cAY,
(rc•(g•,g•)
=
x exp[- i 2•(g•y +g;• ) dyd•],
(107)
(.)
where (•c•(Y, •) is the stiffness-weightederror expressed by Eq. (51). The coefficients a {•/r). are to be
evaluated using Eq. (77) •d Eqs. (85)-(87).
D. Power-spectrumrepresentationof random component
(•0•)
where we have used the notation of Eq. (E3) and where
the Fourier transform indicated in Eq. (101) is taken
with respect to the variable x. For a generic function
f(x), we have
If, in Eq. (95), we write
•0("= •/•("a,
(•08)
then we may express the power-spectral density of
(.)
(.,
_ •"),
•[f(x-j/x);g]
=f.:f(x
_j/x)exp(-i2;•gx)dx
where 6(. ) is the Dirac delta function. As a conse=exp(i2•rgj/x)
f.:f(s)
exp(i2•rgs)ds
quenceof Parseval'sformula, $•(')(k)satisfies
= exp(- i2•rgj/x)ff•[f(s);g].
(102)
5
•r•'>(x)in termsof thesquared
magnitudes
of cer.byx
•
1
•t/(')a/2
Hence, Eq. (101) is equivalent to
J. Acoust.Soc. Am., Vol. 63, No. 5, May 1978
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'
1421
William D. Mark: Vibratory excitationof gearsystems
1421
where
we have
defined
=N(.)
Fn*(j')F,(j'+j).
(118)
Equation (109) expresses the power-spectral density
N(')_,•
of •(')(x) in terms of the squaredmagnitudesof its
Fourier-series coefficientsa (') In Eq. (117), the
,,•,':
N(')_ 1
squared magnitudes l
are expressed in terms of
the finite discrete autocorrelationfunctionR(rø.)(j)of
j"
the sequence F.(j)
From
1
the form
defined by Eq. (111).
of the
Fourier-series
coefficients
(.)
o
c/r, provided by Eqs. (111) and (112), it is shown in
,
AppendixH that a(')=-0 at values of n=pN (') for p
j'
=
FIG. 8. Shifting the terms associated with Eqs. (113)-(117).
. , -2,-1,0,1,2,...
, which, according toEqs.
(•i) and
(95),aretheintegral
multiples
ofthetoothmeshing fundamental frequency.
which justifies using the term power-spectral "density"
to describe the discrete spectrum of Eq. (109).
Let us now expandthe functionfcs(y,z) introduced in
Sec. I in a genericset of functions$•cc=(Y,
z); i.e. 10
Referring to Eq. (105), let us define
= 1L
F.(j)
A
,
•(',(.
n-N(')n')
a(•l•:).,.•s N(.)A
(111)
f cs(y,z)
=• as.=+/•c,•(y,z)
,
(1i9)
where subscripts K and C on •rc=(Y, z) denote stiffness
weighting and the Cartesian tooth coordinate system,
which when substituted into Eq. (105) yields
(.)1N
(')-•
(-N(.)
i2,rnj•
at"-N(')
E
Fnj)exp
];
E. Results using expansionsin tooth coordinates
(112)
and where the same set of expansion functions is as-
sumedto be usedfor •very toothj onthe gear under
consideration. Following the notational convention of
Eqs. (20b) and (32), let us define
hence,
,..
y"=0
x exp
•c=(s) •= secCb
j'=0
-
----i
,
(113)
where the asterisk denotes the complex conjugate.
From the fact that the gear under consideration has
N (') teethand, therefore, that the period in indexj of
f•B(s)
rs(x)=_f
s(s)=secCb
E= as.=•A (s) +rc=(Y
, •s+yy)dy
=• as,=•=(s),
that
Moreover,
(114)
according to Eq. (120).
(121)
Let us now define
dp=(s
) __A
A-•K•(s)•_•:=(s
).
since harmonic index n is always an integer,
we have
(120)
which is the integral of •c=(Y, z) over the line of contact. Combining Eqs. (20b), (32) and (119) yields
•(')[(n
•:s - N(')n')/(N (')a)] is N('), it followsfrom Eq. (111)
Fn(j") =Fn(j" + N(')).
+•cc=(Y,•s + 7y)dy ,
•B
(s)
A(s)
(122)
The Fourier transform of Eq. (122), as defined by
Eq. (E3), is
exp(- i2;mj"/N ('•) = exp[- i2;rn(j " + N(") /N("] .
•)=(g)=a" •[K•J(s)•/•=(s);g]
From Eqs. (114) and (115), we see that all terms in
the doublesummationin Eq. (113) for whichj" < j'
may be shifted abovethe diagonalj"=j' in the manner
illustrated in Fig. 8.
Such a shift is permissible,
be-
cause each shifted term has the value of j" increased
by N (') units, whereasthe value ofj' remains the
same.
=a'•,. • (•/r),,•/•=[g
-(n'/a)], (123)
where the second line was obtained by substituting
(ø) •n Eq. (104)andwhere•=(g)
ß_•=(s)fore_•s(s)
•s the
Fourier transform of •=(s).
If, in Eq. (113), we let
ß •c=(y,z)
For exp•s•on functions
that are zero everywhere outside the true
facecontactregion,wemayevaluate•=(g) usingthe
j =j" -j',
then it follows from Fig. 8 that Eq. (113) may be re-
generalresult of Eq. (42); i.e. xs
• •=(g) =(L/D) sec•c=[(L/A)g,
placed by
(.) (N1
(''.•[•t•.•
)]exp
(-i2•rnj)
la.
Fn*(j')F,(j'+j
N(.)
('))•'•t
•
(L/D)g],
(124)
where, from Eq. (34), we see that
=
1 •(')'•
=•
[-i2•nj\
• •(r'2(j)
expk
N"' }'
Y=0
(11•)
•xc=(gz,g•)
=
x exp[- i2•(g• y +g•.z)]dydz
J. Acoust. Soc. Am., Vol. 63, No. 5, May 1978
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(25)
1422
WilliamD. Mark:Vibratory
excitation
of ge•ar
systems
is the two-dimensional
Fourier
pansion function •KCm(Y,Z).
transform
1422
(.)
of the ex-
½•o•(Y,
z)=•
Equation (124) is to be
usedin the evaluationof •m(g) by Eq. (123). For expansion functions •fcm(Y, Z) that are not zero every-
(.)
m
by,m•Kcm(Y,
z)ß
(131)
Here,according
toEqs.(11•)and(131),weletfc•(y,z
)
be evaluatedusingEq. (36). •6 The sequenceof coef-
take on the role of ½•c•(Y,z). Consequently,according
to Eqs. (20b)and(54), f•(x) takesontherole of [ •'•
Thus, it follows from Eq. (121) that the Fourier trans-
ficients c•(•! •),, in Eq. (123) are to be evaluated using
formo• _(•(s) maybe expressed
as
whereoutsidethetruefacecontactregion,•_Kin(g)
may
Eq. (77) and Eqs. (85) through (87).
• •(g)
(') =y'. b•,m
(')_•m(g)
,
1. Fourier-series
coefficients
of mean-deviation
(132)
where the b•,m
(') are the expansioncoefficientsusedin
components
Using Eqs. (119)-(123) , we can express the Fourier(.)
series coefficients c•m, of Eq. (67) in terms of the
Eq. (131). Settingg=(n-N('•n')/N('•A andsubstituting
Eq. (132) into Eq. (111) yields
functions
•m(g), whenthestiffness-weighted
meande-
(.)
m•cc(Y,
z)=• a•"%ccm(Y,
z),
(.)
(53) and (88)with Eqs. (20b), (32), and (119)-(121), we
see that the Fourier transform m_s
A (. )(g) ofm_
•(')(s)may
be expressed as
• •nN(.)n
,.)
--
=• bJ;•(•m(n/N"'A)
,
(126)
where in this situation the expansion coefficients am
are independent of tooth number •. Comparing Eqs.
.o
(.) A-1
(.)
viations m•c(• , z) given by Eq. (48) are expanded in
the functions •cm(•,z)
as in Eq. (119); i.e.,
where the second line is a consequence of Eq. (123).
Therefore, according to Eqs. (112)and(133) , the
Fourier-series
coefficients of •r(')(x) may be expressed
as
"' • B•("(n)$m(n/N(')A),
Otrn =
m
where
we have
where the coefficientsa(•'• are thoseusedin Eq. (126).
Therefore,
=
according to Eqs. (90) and (127), we have
•
(')=A'1• a.(.)•'
•,,
.
(128)
However,
writingout•hediscrete
convolution
in Eq.
(94), we have
(133)
(134)
defined
y,mexp(-i2;rnj•
.
• •v(•".,
(135)
Accordingto Eq. (la5), the coefficientsB•'•(n) in Eq.
(.)
(134) are determinedby the expansioncoefficients
of Eq. (131). Thefunctions
$•(n/N('•a) are to be eval-
uated using Eqs. (123)-(125) and •e aeeomp•ying discussion.
mn=
(I/K
3. Power spectra of random components
.
.
-
a
'
(120)
where the second line is a consequence of Eq. (128).
Comparing Eq. (123) with Eq. (129) yields
(') I•' directly in terms of the expansion
To express I c•r,
coefficients by,m
(') of Eq. (131), we shall use Eq. (117).
Accordingto Eqs. (118)and(133), R•(•)(j)maybe expressed as
(130)
which is the desired expression for the Fourier-series
coefficientsof •m
(ø)(x)in terms of the coefficients
of the expansionof m•(•(•, z) providedby Eq. (126).
Toevaluate
•m(•/A)for usein Eq. (13•),weseefrom
Eq. (123) that we require the function •_•m(g) only at
those values ofg =p/•
where p is an integer. For ex-
pansion functions •Kcm(•, z) that are zero everywhere
outside
thetruefacecontact
region,_•m(g)is tobe
evaluated using Eqs. (124) and (125).
Otherwise, Eq.
(36) may be used.is
Let us definethe averageof (b•,m
•'•)z over all teeth on
the gear under consideration as
• 1• • (b•'•)•
the discrete cross-correlation
bj,m,
•'• and bj,
•'•m,, as
()2
t
2. Fourier-seriescoefficientsof random components
To arrive
at an expression for the Fourier-series
coefficientsc•(') of the randomcomponents
•(')(x) of the
transmission error,
we expand the stiffness-weighted
random errors of Eq. (51) in the .functions
of Eq. (119):
coefficient p.,.,,(j)
•'•
of
]'
1 •(')'[
(la8)
Y' =0
andthe discrete cross spectrumS •') ,(n) of b (') and
J. Acoust. Soc. Am., Vol. 63, No. 5, May 1978
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1423
WilliamD. Mark:Vibratoryexcitationof gearsystems
S,.,,.,,(n)
=N•'• •
p•,•
•'• ,,(j) exp
\
/
.
(139)
Combining Eqs. (136)-(139) with Eq. (117) yields the
desired expression for la,.,
.
=
uated using Eqs. (123)-(125) and the accompanying discussion. The other quantities in Eq. (140) are deter-
(') of Eqß (131),
minedby the expansioncoefficientsbs.m
as can be seen from Eqs. (137)-(139).
SUMMARY
AND
DISCUSSION
Expressions for the Fourier-series
coefficients of
the static transmission error of a pair of meshing helical gears have been given here in terms of the local
stiffness of a pair of teeth per unit length of line of contact and the deviations of the tooth faces from perfect
involute surfaces with uniform lead and spacing.
ß
Results for spur gears are included as the special
case of helical gears with zero helix angle. For application of the results to spur gears, we note from Eq.
(C4) that for zero helix angles we have (L/A)=0,
a fact
that also is evident from the definition of A implied by
Fig. 4. In particular,
it was observed that Eq. (42)
is applicable in the evaluation of several quantities.
For spu•'gea•'s, Eq. (42) reduces to
_••(g)=(L/D)fcs[O
, (L/D)g],
(141)
x exp[- i2•T(L/i)gz]dz.
(142)
The first major result derived in the paper was the
error as a function of zone of contact position x, which
may be related to the angular position of either one of
the meshing gears by x =Rb0, where Rb is the base
circle radius of the gear under consideration and 0 is
its angular position measured in radians. Equation
(12) expresses the transmission error as the sum of a
load-dependent component and of components from
each of the meshing gears that arise from deviations
of the tooth faces from perfect involute surfaces posiintervals.
A general transformation, Eq. (20b), was then dewhich
enabled
us to describe
The static transmission
error
was then decomposed
into meanandrandomcomponents,Eq. (60), where
tions, Eq. (57), plus components arising from the
mean deviation of the tooth faces of each of the meshing
gears from perfect involute surfaces, Eq. (58). The
random component of the transmission error, Eq. (62),
consists of contributions, Eq. (59), from each of the
meshing gears. The random contributions from each
meshing gear arise from differences between the mean
tooth-face surface for each gear and the actual surfaces of the individual teeth of that gear. It was shown
that, when the force W transmitted by the mesh is
assumed to be constant, the mean component of static
transmission error contributes nonzero harmonics only
at integral multiples of the tooth-meshing frequency,
whereas the random components from each gear may
contribute to all rotational harmonics of that gear except the tooth-meshing harmonics.
Expressions for the Fourier-series
coefficients of
the mean and random components of static transmission
error are provided in Secs. IIB and IIC, respectively.
An expression for the power spectrum of the random
component of the transmission error from each gear is
providedin Sec. IID. In Sec. IIE, the stiffnessweighted
mean and random deviations from perfect involute tooth
faces are expanded in series,
the Fourier-series
and new expressions for
coefficients
of the mean
as a function
error
and random
are derived.
(134) and (135).
general expression, Eq. (12), for the transmission
rived
interest.
These expressions are provided by Eq. (130), and Eqs.
/cs[O,
(L/D)g]=
f.•[f_;fcs(Y,
z,dy]
at uniform
cases of practical
components of static transmission
where, from Eq. (34), we have
tioned
of the quantities in-
volved by Eq. (36), which reduces to Eq. (42) for many
dependent component caused by elastic tooth deforma-
x
(4o)
Thefunctions
•(nN <')A)in Eq. (140)are tobeeval-
III.
dimensional Fourier transforms
the mean component, Eq. (61), consists of the load-
<ø>
<"
1423
of zone
contact position x integrals over the line of contact of
local tooth-pair stillnesses and stiffness-weighted deviations from perfect involute teeth, both described in
a Cartesian tooth-face coordinate system. For rectangular zones of contact, this transformation takes the
form of Eq. (24). For arbitrary contact regions defined in the tooth-face coordinate system by Eq. (25),
the transformation takes the form of Eq. (26) or (27),
or Eq. (29) or (30); it is expressed in terms of the two-
It is particularly
instructive
of these expressions.
to compare the forms
Equations (130) and (134) both
are series expansions over the same index m as the
original representations of the tooth-surface devia-
tions, Eqs. (126) and (131).
Each term in Eqs. (130)
and (134) is proportional to the Fourier transform
•m(g) describedby Eq. (123). In the caseof the
(') givenbyEq . (130) for
Fourier-series
coefficients c•m,
the components of transmission error caused by the
mean deviations from perfect involute surfaces, the
Fourier transforms•,,(g) are evaluatedat the values
g =n/A, which correspond to the tooth-meshing harmonics, as expected. In the case of the Fourier-series
coefficients at,(') givenby Eq. (134) for the random
components of the transmission error, the Fourier
transforms•m(g)are evaluated
at thevaluesg =n/
(N(')A), whichcorrespondto the rotationalharmonics
of the gear under consideration,
also as expected.
(') andB•(')(n)
However, the expansion coefficients am
in Eqs. (130) and (134) have essential differences. In
(o)
Eq. (130), the expansion coefficients am are identical
to theoriginalexpansion
coefficients
usedin Eq. (1,26)
to represent the stiffness-weighted mean deviation
from a perfect involute surface.
On the other hand, in
Eq. (134) the expansioncoefficientsB(m')(n)
are, accord-
J. Acoust. Soc. Am., Vol. 63, No. 5, May 1978
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1424
WilliamD. Mark: Vibratoryexcitationof gearsystems
ing to Eq. (135), finite discreteFourier transforms•
of the original coefficients used in Eq. (131) to expand
the stiffness-weighted random deviations of the individual tooth
surfaces.
The discrete
nature
of the Fourier
transform of Eq. (135) is not an artifact of an approximate sampling procedure, as is usually the case;
rather, the discrete nature arises from the finite num-
ber of N (') teeth on the gear underconsideration.
Equations (134) and (135) predict the so-called sidebands that are observed in experimental vibration spectra in the neighborhoods of the tooth-meshing harmon-
ics. (For an experimentally obtained spectrum showing
such sidebands, see, for example, Ref. 5.) To show
that Eqs. (134) and (135) predict these sidebands, we
notefirst that Bm
(')(n) is a periodicfunctionof indexn
with periodin n equalto N ('). This periodicityis a
well-known property of the discrete Fourier trans-
form,• andfollowsfrom the fact that, for integerp, we
have
exp[- i2•r(n+pN(' ))j/N(' )] = exp(- i2•rnj/N(')) .
(143)
The periodicity of B('>(n)followsfrom Eqs. (135) and
(143):
By'(n +pN(ø>)
=B•("(n), P= . . ., - 2,- 1, O,1, 2, ....
(144)
As a consequence
of Eq. (144), if Bm(')(n)
is unusually
large for a particular valueof n, sayn =n', thenBm(')(n)
will be unusuallylarge at the periodic intervals n = n'
+pN ('). It is an experimentalfact that the first few
rotational harmonics are usually the largest,
and be-
1424
deviations, in detail, after manufacture. Hence, a
statistical approach to the description of the harmonic
content of the random component of transmission error
is in order. With this requirement in mind, the power
spectrum representation of the random componentof'
the transmission error provided in Sec. IID was developed. Further development of the statistical treatment
required the series expansion of the stiffness-weighted
random errors described by Eq. (131). This expansion
permitted us to develop the expression for the squared
magnitudesof the Fourier-series coefficients
given by Eq. (140). Once the expansion functions
•I,Kc,•(•, z) used in Eq. (131) have been chosen, evaluation of the right-hand side of Eq. (140) requires, in
addition to Krc(•,z),
only the rms values of the expansion coefficients
and their
discrete
functions defined by Eq. (138).
cross-correlation
It is possible to develop
simple statistical models of these correlation functions
which will permit prediction of the main features of
the power spectra of the random components of trans-
mission error from tooth lag "correlation interval parameters" and estimates of the rms expansion coefficients obtained either by measurement or from manu-
facturing specifications. These methods, and application of other aspects of the present paper, will be described in a second paper.
APPENDIX
A:
PARAMETERS
RELATIONSHIPS
AMONG
GEAR
Let •band •bb denote, respectively, pitch and base
cylinder helix angles, R and Rb pitch and base cylinder
radii, L h axial lead of helix, qbpressure angle, and Ac
causeof this fact andEq. (144), B•(')(n)will be unusually
and A circular andbase pitches.9 It may be shown
largefor valuesofn nearpN•'), p .... , - 2, - 1, O,1, 2, ....
However, the frequencies associated with values of the
rotational harmonic numbers that are exactly equal to
from Fig. 2 that
n=pN(') are the tooth-meshing
harmonics. Thus, if
valuesof B•(')(n)are unusuallylargefor small n, then
tan•b= 2•rR/Lh,
(A1)
tan•b•= 27rR•/L• ,
(A2)
cos•b=R•/R.
(A3)
Eq. (144) predicts that the rotational harmonics will
be unusually large in the neighborhoods of the toothmeshing harmonics.
These rotational harmonics are
the sidebands observed in experimental vibration spec-
and
tra.
expressioninto Eq. (A2) yields the fundamentalrelaAccording to the results presented in Sec. IIB, pre-
diction
of the Fourier-series
coefficients
of the mean
component of the static transmission error requires
both the local stiffness KTc(y , z) of a pair of teeth per
unit length of line of contact and the mean deviation
m(c')(Y,
z) of the toothfacesof eachgear from perfect
involute surfaces. The local stiffness KTc(y , z) can be
computed, with effort, from design drawings by the
method outlined in Appendix B.
Solving Eq. (A3) for R• and substituting the resulting
tionship
tan•b = cos•btan•.
Furthermore,
(A4)
from the lower half of Fig. 2, we have
=nln ,
(A5)
which yields, when combined with Eq. (A3),
zX= zXccosqb.
(A6)
The mean deviation
m(c')(Y,
z) also is a quantitythat canbe estimatedfrom
APPENDIX
gear-tooth design drawings.
TOOTH-PAIR
STIFFNESS
Krj(X,y)
In contrast, prediction of the Fourier-series
coefficients of the random component of transmission error
requires, in addition to Krc(j•,z), the differences
the line of contact of tooth pair j due to a unit force
(.)
½cs(•, z) between each tooth face of each gear and the
meandeviation,/(c')(•,z) for the gear underconsidera(')(•,, z) cannotbe pretion. These random deviations½cs
dicted before a gear is manufactured; moreover, it
would be a very time-consuming task to measure such
B:
INTEGRAL
EQUATION
FOR LOCAL
Let ks(r, r'; x) denotethe deformationat a pointr on
applied at point r', where force and deformationare
both measured in a direction defined by the intersection of the plane of contact and a plane normal to the
axes of gear rotation and where x denotes the position
of the zone of contact as illustrated in Fig. 2. The
influencefunctionks(r, r';x) is assumedto includebend-
J. Acoust Soc. Am., Vol. 63, No. 5, May 1978
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1425
William
D. Mark:Vibratory
excitation
of gearsystems
ing, shear, and Hertzian deformation components. The
unitforceusedtodefineks(r, r";x) is assumed
tobe
applied uniformly to a circular area of diameter equal
to the width of the contactline betweenthe two gear
teeth. Since the Hertzian deformation componentof
real teeth is a (weakly)nonlinearfunctionof loading
because of dependence of the width of the contact line
on loading, specification of the area of applied force
implies that our influencefunctionis definedfor a given
nominalloadingcondition. The deformationuj(v;x) at
pointv causedby a lineal force densityp•(r';x), where
• and•' are onthe line of contact,maybe expressed
in terms of kj(•, v';x) by
!
u•(•;x)=
/
!
!
kj(•, • ;x)pj(• ;x)dv ,
(B1)
whered•' denotesdifferentialdistancemeasuredalong
the line of contact and where the integral is taken over
the portion of the line of contact contained in the zone
of contact. Deformationsu•(x,y) in Eqs. (Sa) and (Sb)
anduj(•;x) in Eq. (B1) are related by
!
!
u•(x,y) = u•(ysec½•;x),
(B2)
where we note from Fig. 2 that distance • along the
(B3)
Equation (B1) may be regarded as a Fredholm integral equation of the first kind for the lineal force den-
sityp•(v';x). WemayformallyinvertEq. (B1)to yield
k•(•, • !;x)u•(•
! !;x)dv!,
p•(•;x)=
inversek}•(v,v';x) oftheinfluence
function
ks(v
, v';x).
However,k•(v, v';x) neednotbe evaluatedin order to
determineKr•(x,y). Let •(v';x) be the solutionto Eq.
(B1) for the deformationu•(v;x)=•, whichis a constant
independent of v; i.e.,
•=f k•(v,
v';x)•(v';x)dv'.
(B4)
wherek•z(•,•';x) is theinverseofk•(•, •';x). Integration of p•(v;x) over the line of contactyields the total
force W• transmitted by tooth pair j:
From Eq. (B4) and the fact that • is independent
of v,
we have
•(v;x)=•f k•(v,
v';x)dv'.
Furthermore, the symmetry of the influence function
t
kj•(v,v ;x) possesses
thesamesymmetry;i.e.,
t
t
k•'(v, v ;x) =k•'(v , v;x).
Using Eq. (Bll),
p•(v;x)/•=
=f •fk••(•,
•';x)d4u•(•';x)d•'
'
',
(B5)
where, in going to the secondline, we have used Eq.
(B4) and, in goingto the last line, we have defined
Kv•(x,
= k•(u, • ,;x) du.
, • ,)zxf
(B6)
To put Eq. (B5) into the form of Eq. (8b), we transform
v' to y usingr'=y secCb,as in Eq. (B3), therebyob-
(Bll)
we obtainfrom Eq. (B10)
kj•(v , v;x)dv .
(Big)
K;•(x, v)=•(v; x)/• ,
(B13)
Kv•(x,y) =p•(y see½•;x)/•.
(B14)
Notice that both sides of Eq. (B14) have the dimension
of force dividedby lengthsquared.It followsfrom Eqs.
(BO)and
(B14)thatthelinealforce
density
•(v;x) that
yields a constantdeformation•. along the line of contact
providesthe local tooth-pairstiffnessKv•(x,y) per
unit length of line of contact. Equation (B0) is the integral equationwhosesolution•(v'; x) yields the local
tooth-pair stiffness with the aid of Eq. (B14).
TO TOOTH
SYSTEM
Figure 4 showsthe zone of contact projected onto a
plane parallel with the pitch plane, whereas Fig. 2
shows the zone of contact in the plane of contact. Since
the angle between these two planes is the pressure
angle •b, it follows from the sketch in the lower-left-
hand corner of Fig. 4 that
L'=L cos•.
Kv•(x,
YB
t y sec½,)u•(y
t sec½,;x)dy.
(B7)
A
From Eqs. (8b), (B2), and (B7), we have
Kv•(x,y) =K[•(x,y see½•)
Fig. 4, is measured in a plane parallel with the pitch
plane, we also have
=f k•(v,
ysec½b;x)dv,
(B8b)
where Eq. (B6) was used in going to the secondline.
Equation(B8b) expressesKv•(x,y) in terms of the
(C2)
From the sketch in the lower-right-hand corner of
Fig. 4, we have
tan½'=L '/A =(L/A)cos•),
(BSa)
(C1)
Moreover, since distances, definedby Eq. (13), is
measuredin the plane of contact, whereass', shownin
s' =s cos•.
taining
W•= sec½,
18
Comparing Eqs. (B6) and (B12) shows that
COORDINATE
f Kv•(x,
' v
(B10)
ks(v,v';x) (Maxwell'sreciprocaltheorem)implies that
APPENDIX C: TRANSFORMATION
=
(B0)
and combiningthis result with Eq. (B8a) yields, finally,
line of contact is related to y by
• = Y sec•bb
ß
1425
(C3)
accordingto Eq. (C1). Moreover, from Fig. 2 and
the definition of A shownin Fig. 4, it follows that:
L/A = tan½•;
(C4)
hence,
tan½'= eos• tan½•
J. Acoust.Soc.Am., Vol. 63, No. 5, May 1978
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(C5)
1426
William D. Mark: Vibratoryexcitationof gearsystems
1426
or, using Eq. (A4), we have
GEAR(1)
tan•' = cos2qb
tan½.
(C6)
From the tooth shown in the upper-right-hand
of Fig. 4, we have
9oø+•
corner
tan0 = D/A.
/
R(1)
ß
q-..
(C 7)
However, from the sketch in the lower-left-hand
ner of Fig. 4, we have
DRIVER
,,
..
cor-
D =L' tan(•
L(•)
= L sin•,
(C8)
according to Eq. (C1). Therefore, by combining Eqs.
(C7), (C8), (C4), and(A4), we have
GEAR (2)
tan0 = sin(• eos(• tan½.
(C9)
Equations (C1), (C2), (C6), and (C9) relate quantities
FIG. D-1. Gear geometry required to relate path of contact
length œ to fundamental gear parameters.
shown in Fig. 4 to basic gear parameters.
Equations
(C6) and (C9) are consistent with results given on pp.
166-167 of Ref. 9 (different notation).
Thus, two corners of the range of integration occur at
To establish the validity of the transformation
Eqs. (15)-(18),
s'= ñ «(F -A) tan½',
of
we first note from the tooth shown in
whereas
the upper-right-hand corner of Fig. 4 that the coordinate z of the point p may be expressed as
z = 6z - 6z0.
from
We must now transform
(C10)
the definitions
6z = s ' tan•
of 6z and s
!
6z0= y tan0.
By combining Eqs. (C9)-(C12),
OF CONTACT
The limits of integration shown in Fig. 5 are easily
see
fine the y coordinate end points of the line of contact illustrated in Fig. 4, it follows from considerations of the
of contact
relative
to the line
of con-
tact in the lower portion of Fig. 4 that two corners of
the range of integration shownin Fig. 5 occur at s'=
ña , whereas the other two corners occur at s = ñ(L'
+a') .
!
!
From the lower-right-hand
a'= «(F -A) tan½'.
(D2)
(D8)
whereL (t) andL (•')are the segments
of thepathof
contacton the two sidesof the pitchpoint. /{•(t)and
/{ (t) are, respectively,the radius of the addendum
(D1)
hence,
that
L =L (•)+L (•)
sketch in Fig. 4, we have
tan½'
=a/(•-•);
(D7)
Figure D-1 showsthe gear geometry required to relate L to fundamental parameters. The gear pair
shown in Fig. D-1 is the same pair illustrated in Fig.
2; the plane of the paper of Fig. D-1 is normal to the
gear axes. By comparing Figs. 2 and D-1, one may
AND
determined from Fig. 4. Recalling that yA(s) and yB(s)de-
of the zone
s = ñ «(F tan½•+ L ).
5 was determined with the aid of Eq. (D5).
which is the transformation given by Eqs. (15)-(18).
motion
(D6)
values of the variable y. The slopeAlL shownin Fig.
(C13)
INTEGRATION
s = ñ «(F tanCb- L)
The corners of the range of integration defined by
these values of s are shown in Fig. 5 at the appropriate
we have
z =s sin(• -y sin(• cos• tan½,
OF
(D5)
and
(C12)
OF PATH
of s' to the
s' in Eqs. (DS) and (D4) are
left-hand corner of Fig. 4 and from Eq. (C2). From
the gear tooth shown in the upper-right-hand corner of
Fig. 4, we also have
LENGTH
the above,values
(D4)
According to Eqs. (C1), (C2), (C5), and (D5), it follows that the values of s corresponding to the values of
(Cll)
LIMITS
at
A = L / tanCb.
.
as may be easily seen from the sketch in the lower-
D:
occur
corresponding values of s. First, we need an expression for A in terms of fundamental gear parameters,
which may be obtained from Eq. (C4):
that
= s sin•,
APPENDIX
two corners
s'= ñ [L '+ «(F -A)tan½'].
Since 6z is measured in a plane perpendicular to the
gear axes in a direction normal to the pitch plane, it
follows
the other
(D3)
circle and the radius of the pitch circle of gear (1)
shown in Fig. D-1.
We may,solvefor L a) in terms of •, R•a), andR (•)
using the law of cosines
az=bz+ cz - 2bccosa,
(D9)
where a, b, and c are sides of a triangle with angle a
opposite side a. Solving Eq. (D9) for b, we have
J. Acoust.
Soc.Am.,Vol.63,No.5, May1978
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1427
WilliamD. Mark:Vibratoryexcitationof gearsystems
b -c cosc•
ñ(c•'cos•'•+a•'-c•')•/•' .
(D10)
1427
thecoordinate
transformation
of Eqs. (15)and(16).
We shall denote this region of integration in the •, z
Applying
Eq. (D10)to solvefor La), we identifyL a) with coordinatesby f•½. From Eqs. (15) and (16), we have
b, R•a) witha, R a) withc, and•b+90øwitha. Recognizing the fact that cos(90ø+ •b)= - sinqb,it follows from Eq.
(D10) that
y =y, s = z/•3 - yy/13,
from
L (•): - R (•)simb
which
we obtain
ayas= Io(y,s)/o(y,z)laya•
ñ [(R(•))•'sin2qb
+(RJ•))2_ (R(•))2]•/•.
=g-•ayaz.
=R (•){- sinqb
+ [sin"'qb
+(R•(•)/Rn)),._ 1]•/•.}.
_
(E6)
Thus, when we express the inner double integration of
(Dll)
The negative sign was rejected before the radical in
Eq. (Dll), becauseusing the negative sign would al-
waysleadto a negativevaluefor L (1). Simplechecks
showthat Eq. (Dll) gives correct results for qb=0 ø
and qb= 90ø. Denotingthe pitch radius and addendum
radiusof gear(2) byR(•')andR•(•'),we havefrom Fig.
D-1 in a completely analogous fashion •
L (a):R(•.){_sinqb
+ [sin•'qb
+(R•(•')/R
,a))a
_ 1]'/a}.
(D12)
Equations(D8), (Dll),
(g5)
and (D12) determine L in terms
of the pressure angle and the pitch and addendum circle
radii of gears (1) and (2).
Eq. (E4) in the y,z coordinates, we obtain
•(g)=•'•sec½•
' fff•c•(g•,ga)
ff exp
{-i2•
c
X[(g
-•g•)(•½)-(g•+yg•)y]}dydzdg•
=•'•sec•,•
•]cs(g•,g•'
•f <cexp{-igx[
-(•+gO
Y+(•-g2)4
}Odzdg•d
(g•)
Moreover, from Eqs. (17) and (C8), we have
• =O/L,
(E8)
•d from Eqs. (17), (18), (A4), and (C4), we also have
APPENDIXE' FOURIERTRANSFORM
OF .f/is)
7/• = • L/A.
To establish the result of Eq. (36), we first note
(E9)
that we may expressfcj(Y, z) in terms of its two-dimen-
SubstitutingEqs. (ES) •d (E9) into Eq, (E7) yields
sional Fourier transform by using the Fourier mate to
Eq. (34)'
f•(g)=• sec•b
x exp[i2•(g•y +g,.z)]dgxdg,..
(w.1)
Using Eqs. (32) and (El), we may rewrite Eq. (20b) as
A (s)
Xexp{i2•[g•y +ga(•s+ •y)•dg•dg•.dy,
(E2)
where we shall now consider the limits of integration
ya(s) and ys(s) as defining an arbitrary contact region.
This arbitrary region may be different from the specific region delineated in Fig. 5 that yields the rectangular contact region defined by Eq. (28) when the
transformation to tooth coordinates y, z is made. Let
us denote the Fourier transform of a generic function
f(s) by
](g)=•:•[
f(s);g]
--f.•f(s)
exp(i2•rgs)ds
. (E3)
Using the notation of Eq. (E3), the Fourier transform
of Eq. (E2) with respect to s may be expressed as
. =o
(El0)
Equation(36) is a direct consequence
of Eqs. (El0),
(25), and (35).
YB
(s)
j•-.:
-
]c•(g•,g•)
APPENDIX
F' SERIES REPRESENTATION
RECIPROCAL
MESH STIFFNESS
OF
The total instantaneous mesh stiffness KT(x) defined
by Eq. (56) can be represented as
Kv(x) = Kv + •5Kv(x)
=•[
• + •n•(x)/•],
where
(F1)
'
=X
(F2)
•v(x)ax
is the me• stiffness of the •riodic function Kv(x).
The reciprocal mesh stiffness is, from Eq. (F1),
KT•(x)=•g[ 1+ 5Kv•v] '•
fYB
(S)
x [(g - •g•.)s- (g, + yg•.)y
]}dydsdg
figs..
Let us now transform the two-dimensional
(E4)
region of
integration in Eq. (E4) in the variables y, s to the comparable region in tooth-face coordinates y, z by using
=K$• •-•
' I •-
<1,
(F3)
where, in going to the second line, we have used the
J. Acoust. Soc. Am., Vol. 63, No. 5, May 1978
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1428
WilliamD. Mark: Vibratoryexcitationof gearsystems
1428
usual power-series representationof (1 + x)'x and, in
where the expansion coefficients in Eq. (62) are given
going to the last line, we have used Eq. (F1). The condition for convergence of the infinite series in Eq. (F3)
by
G(x) exp(- i27•nx/A)dx.
1;•/•.
•/•'
=
Otn
•
Using the binomial
will always be satisfied in practice.
theorem, we have
Let •(g) denotetheFouriertransformofG(x)defined
- •r =,=•(•)[-•'r
J'
(F4)
Combining Eqs. (F3) and (F4) yields
_ •
m=O
G(g) =
[_•(•)],•(• Ky_-R•
l =0
(63)
<1,
whose
Fourier
mate
(64)
is
G(x)
=/• •(g)exp(i27•gx)
dg.
(F5)
where
G(x) exp(- i27•gx)dx ,
(65)
From Eqs. (62) and (65), it is evident that
(D =
• !(• - •i!
are the binomial
(•B)
coefficients.
Generally,
a few terms
of the outer sum in Eq. (F5) will provide excellent
accuracy.
Let us define the partial
sum
5(g)=• c•.6(g
- n/A),
(G6)
which c• be verified by substituting Eq. (66) into Eq.
(65) and comparing the result with Eq. (62).
Fourier series of product. Let G'(x) and G•(x) be
=
=
R•
'
(FV)
For any sum, we have
M
m
M
two periodic functions possessing the same period A
as in Eq. (61). -Let G'(x) andG"(x) possessthe complex Fourier-series
representations
M
G'(x) =
Thus, Eq. (F?) may be written as
=
L•J
'
• exp(i2•kx/A)
•• a ntW
k
(67)
and
However, it is evident from Eq. (F$) that
(D = (:-,);
(F•0)
•"(•) =•
•w
where
consequently, we have
w • exp(i27•x/A).
(69)
Denotethe productof G'(x) andG"(x) by G(x), which
also is periodic with period
C(x)• • '(x)• "(x)
where, in going to the second line, we have used a result on p. 822 of Ref. 13 and, in going to the third line,
we have used gq. (F10). The partial sum of gq. (F9),
therefore, may be written as
=• a,w",
usi• the convention of Eq. (69).
(68), and (610), we have
• Eq. (F12), each power of [-Kr(x)/Ky ] appearsonly
once; the original form of K•(x) givenby Eq. (F7)
G:
THEOREMS
FOR
FOURIER
•=-
G(x+n A)=G(x),
i.e.,
n=...,-2,-1,0,1,2,
•(•• ' "•
SERIES
Fourier series andtransform. Let G(xi be a periodic function with period •;
....
Then, G(x) possesses the complex Fourier-series
(G1)
rep-
resentation
where, in going to the second line, we have made the
substitution n = k + m. Comparing both sides of Eq.
(611) yields
•. =
which
G(x)=E •"exp(i2•Tnx/A)
,
(62)
From Eqs. (67),
--
does not have this property.
APPENDIX
(G10)
•
••
t It • •.*•.t It,
is the convolution
theorem
(G12)
for
Fourier
series.
Fourier series of repfumtion. •fine
J. Acoust. Soc. Am., Vol. 63, No. 5, May 1978
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1429
William D. Mark: Vibratory excitationof gearsystems
= •
rep•v(x)/x
v(x ,jA),
1429
the exponential function in Eq. (H4) always is unity;
(G13)
hence,
+ja); (p - n')/a] =
which is periodic with period A. According to Eqs.
(G2) and (G13), we may represent rep•v(x) by
•.. v(x-j/•)=•. a,exp(i2•nx/A),
j=. eo
where, here, the a,are
the Fourier-series
(H5)
(G14)
r•--- oo
(L'(s);(p - n')/a].
coefficients
of rep•v(x). Accordingto the Poissonsumformula,19
we have
Combining Eqs. (H3) and (H5), and reversing the orders
of summation and Fourier
1
N(')
1
j=o
,
a
•-•(.) -n
transformation,
=•'•
j=o
yields
• )(x ,op-n
a '
(H6)
Z v(x-jA)=-•E 9(n/A)
exp(i2•nx/A), (G15)
But, from Eq. (54), we have
where
9(g)a_
- • v(x)exp(-i2•gx)dx,
is the Fourier transform of v(x).
(G16)
From Eqs. (G14)
•d (G15), we can see that the Fourier-series coefficients of rep•v(x) may be expressed in terms of the
Fourier tr•sform
of v(x) by
a•= (1/•)9(n/•)
.
(H7)
However, from Eqs. (46) and (47), we see that
N('). 1
(G1
.)
APPENDIX
H: PROOF THAT CONTRIBUTION
TO
TOOTH-MESHING
HARMONICS
FROM RANDOM
COMPONENTS
OF STATIC TRANSMISSION
ERROR
j--O
,.,
e•:c•(Y = ;
(H8)
,
hence, from Eqs. (H2) and (H6)-(H8),
it follows that
at the tooth-meshing
harmonicsn =pN(') we have
(')=0 ß
O•rr•
IS ZERO
ACKNOWLEDGMENTS
According to Eqs. (64) and (95), the tooth-meshing
harmonicsoccur at the harmonicsn =pN(') of the randomcomponents
•(')(x) of the static transmission
error,
where p is any integer or zero.
I am indebted
For values of
n =pN ('), we have
exp(- i2•nj/N (')) = exp(- i2•pj) -=1,
since p and j are integers.
(H1)
Consequently, at the tooth-
Brown
for
his encour-
agement in the work reported herein and for providing
time for the writing of the paper. Discussions with
Dr. Fred R. Kern, Jr. have been helpful particularly
in the interpretations of gear machining errors.
2j. Zeman, "DynamischeZusatzkrafterin Zahnradgetrieben,"
1 •(')-I
Z. Ver. Dtsch. Zucker Ing. 99, 244 (1957).
3S. L. Harris, "DynamicLoadson the Teeth of Spur Gears,"
•0
_•
N(')'1 •
•
=
Proc. Inst. Mech. Eng. 172, 87-100 (1958).
,
,
(H2)
where the second line is a consequence of Eq. (111).
Usi• the notation of Eq. (E3), we have
.
A.
(Macmillan, New York, 1941; republished by Dover, New
York, 1966), pp. 163-167.
Eq. (112),
•
Neal
1See,for example, A. Sloane,EngineeringKinematics
meshingharmonicsn=pN(') of •r(')(x), we havefrom
rn - N(')
to Dr.
N(').i
N('). 1
•o
•=o
4R. W. Gregory, S. L. Harris, andR. G. Munro, "Dynamic
Behavior of Spur Gears,"
2i8 (i963-i964).
Proc. Inst. Mech. Eng. 178, 207-
5H. K. Kohler, A. Pratt, and A.M.
and Noise of Parallel-Axis
Thompson, "Dynamics
Gearing," Proc. Inst. Mech.
Eng. 184, 111-121 (1969-1970).
6D. B. Welbourn, "Gear Errors and Their Resultant Noise
Spectra," Proc. Inst. Mech. Eng. 184, 131-139 (1969-1970).
?H. Opitz, "Noise of Gears," Phil. Trans. R. Soc. London
Ser. A 263, 369-380
N(.)-I
_
•
+ja);g],
(H3)
(1968-1969).
8Inthe main text of the paper, we deal with meshstiffness
KT'(x)=1/C• insteadof mesh compliance. See Eq. (õ6). •
is obtained by setting n =0 in Eq. (77).
9E. Buckhxgham,
AnalyticalMechanicsof Gears(McGraw--Hill,
where, in going to the second line, we have used Eq.
(98) with•=s +jA and where the Fourier transform in
Eq. (H3) is to be taken with respect to the variable s.
Applying Eq. (102) to the present situation, using a
negative value of A, we find
•[• x•(s
(') +ja);g]= exp(i2•gja)V•[[•(3•(s);g]. (H4)
For val•es of g =(p- n')/•X, wherep andn' are integers,
New York, 1949; republishedby Dover, New York, 1963),
Chaps. 1, 4, 7, and 8.
WOurderivationof Eq. (12) followsalongthe general lines of
a derivation carried out by P. R. Nayak in Report TIR No. 86
(Bolt Beranek and Newman, Inc., Cambridge, MA, 1973)
which applies to a more restricted class of situations than
Eq. (12).
lip. M. Woodward,ProbabilityandInformationTheorywith
Applicationsto Radar, 2nded. (Pergamon,Oxford, 1964).
J. Acoust. Soc. Am., Vol. 63, No. 5, May 1978
Downloaded 20 Jun 2012 to 195.220.21.230. Redistribution subject to ASA license or copyright; see http://asadl.org/journals/doc/ASALIB-home/info/terms.jsp
1430
William D. Mark: Vibratory excitationof gearsystems
12A.Papoulis,Systems
andTransformswithApplications
in
Optics (McGraw-Hill,
New York, 1968), p. 95.
13M.AbramowitzandI. Ao Stegun,Handbook
of Mathematical
Functions with Formulas, Graphs, and Mathematical Tables
(U.S. GPO, Washington, DC, 1964).
14R.Bracewell,TheFourier TransformandIts Applications
(McGraw-Hill,
New York, 1965), p. 367.
•SW.B. Davenport,Jr., andW. L. Root,An Introductionto
the Theory of Random Signals and Noise (McGraw-Hill,
York, 1958), pp. 89, 90.
New
•The expansion
of Eq. (119)maybe interpretedin twoways.
1430
to be used, as appropriate,
to evaluate the Fourier transforms
_•Km(g)
asindicated
in themaintextofthepaper.In the
secondinterpretation of Eq. (119), we expandthe product
•c(Y, z)fcj(y,z). In this case, the limits yA(s)andyB(s)are
determined by the domain of definition of the expansion func-
tions •Kcm(Y,z) and the application of Eq. (42) provided by
Eq. (124) is to be used to compute the Fourier transforms
•Km
(g).
.
17j. W. Cooley,P. A. W. Lewis, andP. D. Welch, "The
Finite Fourier Transform,"
IEEE
acoust. AU-17, 77-85 (1969).
Trans.
Audio Electro-
In the first interpretation, we expandthe functionfcj(y, z)
1By.C. Fung,Foundations
of SolidMechanics(l•rentice-Hall,
over the domain of its definition irrespective of whether this
domain coincides with or is larger than the true face contact
region. In this interpretation, either of Eqs. (36) or (42) are
19A.Papculls,TheFourierIntegralandIts Applications
Englewood Cliffs, NJ, 1965), p. 9.
(McGraw-Hill,
New York, 1962).
J. Acoust. Soc. Am., Vol. 63, No. 5, May 1978
Downloaded 20 Jun 2012 to 195.220.21.230. Redistribution subject to ASA license or copyright; see http://asadl.org/journals/doc/ASALIB-home/info/terms.jsp