Screw rotor tooth profile

A screw rotor tooth profile which comprises a pair of male and female rotor tooth profiles engaging each other. The screw rotor tooth profile is configured such that at least one of the male and female rotor tooth profiles has a point of minimum pressure angle or an engaging tooth surface concurring with those of a theoretical tooth profile and is deviated from the theoretical tooth profile such that the amount of deviation increases as going from the point of minimum pressure angle or the engaging tooth surface toward the tooth top side and the tooth bottom side. The screw rotor tooth profile, even in case of the presence of machining and/or assembling errors in the profiles, are capable of assuring excellent mutual tooth engagement therebetween and almost not susceptible to that or those errors, that is, insensitive to manufacturing precision.

BACKGROUND OF THE INVENTION 
The present invention relates to relatively large diameter screw rotor 
tooth profiles for hydraulic systems such as, for example, screw 
compressors, expanding machines or the like and, more particularly, to 
screw rotor tooth profiles which can hardly be influenced by machining 
and/or assembling errors, etc. thereof, that is, which have an 
insensitivity to manufacturing precision thereof. 
Generally, in the field of screw rotor tooth profiles, a pair of male and 
female rotor tooth profiles engaging each other with no clearance provided 
therebetween are referred to as theoretical tooth profiles. 
A pair of rotor tooth profiles with theoretical tooth profiles, however, 
fail to provide smooth rotational movements because they tend to be 
subjected to influences of machining and/or assembling errors in the tooth 
profiles, thermal expansion during their operation, etc. 
In for example, Japanese Patent Unexamined Publication No. 39508-53, it has 
been proposed to modify the rotor tooth profiles in such a manner that the 
male or female rotor tooth profile thereof has a trailing flank deviated 
in a direction causing reduction in its dimension relative to that of the 
theoretical tooth profile thereof, so that a clearance may be defined 
between a pair of rotor tooth profiles to obtain a smooth rotation during 
operation. In the case of large diameter rotors of, for example, 200 mm or 
more, the hob milling process is difficult because of technological 
limitations in the manufacture of proper tools thereof, so that the rotors 
are manufactured by a single cutter. However, such tooth cutting process 
is often accompanied by large machining errors. More particularly, in the 
single cutter machining, the rotor grooves are usually cut one by one and 
therefore, the working conditions differ, so that errors arise in the 
indexing precision becomes. Besides, the manufacturing of high precision 
tools is relatively difficult. 
The above-mentioned modified type of rotor tooth profiles however, are 
disadvantageous in that it becomes difficult to maintain a proper tooth 
engagement on the leading flanks of the male and female rotor tooth 
profiles due to machining errors or errors of distance between the shafts 
during an assembly of the system. As is known, the best force transmitting 
tooth surface (engaging tooth surface) on the leading flank is found at or 
near a point of minimum pressure angle. However, in case where the 
distance between the shafts is shorter than its prescribed distance due to 
the errors during the manufacturing, the bottom part of the tooth of the 
female rotor contacts the top part of the tooth of the male rotor. On the 
contrary, in case where the actual distance between the shafts is longer 
than the prescribed distance, the part of the tooth of the female rotor 
existing on the top side beyond the point of minimum pressure angle 
contact the part of the tooth of the male rotor existing on the bottom 
side away from the point of minimum pressure angle. Thus, in either case 
of the above errors, it is difficult for the engaging tooth surfaces of 
the leading flanks obtain a proper tooth engagement at or near the point 
of minimum pressure angle. The afore-mentioned modified type of screw 
rotor tooth profiles, thus, has a problem of which the rotors sensitively 
suffer influences of manufacturing errors. 
In, for example, U.S. Pat. No. 4,140,445, another type of rotor tooth 
profile is proposed wherein the leading and trailing flanks of the female 
rotor are modified to deviate in a direction diminishing dimensions 
thereof relative to those of the theoretical tooth profiles. 
Since, in the above-described proposed rotor tooth profiles, the amount of 
the deviation near the pitch circle of the tooth flanks is small, these 
rotor tooth profiles cause no particular problem as long as they are 
manufactured exactly according to the theoretical tooth profiles thereof 
or they have no errors noted above. 
In case where the clearance between both rotor tooth profiles has rendered 
extremely narrow due to influences of manufacturing or assembly errors or 
the like, however, a proper tooth engagement at engaging tooth surfaces 
can not be obtained and, in the worst case, there is a possibility that a 
pair of rotor tooth profiles completely fail to engage each other 
profiles. 
SUMMARY OF THE INVENTION 
Accordingly, the present invention aims to provide screw rotor tooth 
profiles which are almost not susceptible to machining and/or assembling 
errors etc. of the rotor tooth profiles, that is, having an insensitivity 
to manufacturing precision. 
Further, the present invention aims to provide screw rotor tooth profiles 
which assure an optimum tooth engagement even in case that various errors 
noted above should occur in the machining and/or assembling etc. 
To this end, according to the invention, there are provided screw rotor 
tooth profile comprising a pair of engageable male and female rotor tooth 
profiles rotatable around a pair of parallel shafts, with tooth profiles 
having leading and trailing flanks. At least one of the male and female 
rotor tooth profiles has a point of minimum pressure angle or an engaging 
tooth surface concurring with those of a theoretical tooth profile and is 
deviated from the theoretical tooth profile thereof such that the amount 
of deviation increases as going from the point of minimum pressure angle 
or the engaging tooth surface toward the tooth top side and the tooth 
bottom side.

DETAILED DESCRIPTION 
Referring now to the drawings wherein like reference numerals are used 
throughout the various views to designate like parts and, more 
particularly to FIG. 1, according to this figure, a pair of screw rotor 
tooth profiles according to the present invention are illustrated with the 
solid lines depicting theoretical tooth profiles comprising a female rotor 
tooth profile 1 and a male rotor tooth profile 2 which engage each other 
with no clearance provided therebetween and rotate around respective 
shafts arranged in parallel. 
In FIGS. 1 and 2, the female rotor tooth profile 1 and a male rotor tooth 
profile 2 are shown on a plane perpendicular to rotating axes of rotors. 
The female rotor is driven by the male rotor. The rotor tooth profiles 1 
and 2 are arranged to rotate respectively around center points 3 and 4 
within a casing (not shown) so as to function as a compressor. 
The female rotor tooth profile 1 comprises, as its main parts, a leading 
flank 5 composed of a first leading flank 7 and second leading flank 8 and 
a trailing flank 6 composed of a first trailing flank 9 and second 
trailing flank 10. These main parts are located inside a pitch circle 11. 
On the other hand, the male rotor tooth profile 2 comprises, as its main 
parts, a leading flank 12 composed of a first leading flank 14 and second 
leading flank 15 and a trailing flank 13 composed of a first trailing 
flank 17 and second trailing flank 16. These main parts are located 
outside a pitch circle 18. 
The first leading flank 7 of the female rotor tooth profile 1 is formed 
between points 19 and 20. The form of the first leading flank 7 between 
the points 19 and 20 is defined by a parabolic curve which is expressed by 
an equation Y.sup.2 =4a (X-E), where, in a rectangular coordinate system 
of axes X, Y with its origin located at the rotary shaft center point 3, E 
represents the distance between the center point 3 and the point 19 and a 
represents the distance between the point 19 and a focal point 21 inside 
the pitch circle 11 on a line connecting the rotary shaft center points 3 
and 4. 
On the other hand, the second leading flank 8, defined between the points 
20 and 22, is formed by an arc of a circle having radius R.sub.2 with its 
center located at a point 23 inside the pitch circle 11 and the first 
trailing flank 9, defined between the points 19 and 24, is created by an 
arc of a circle having radius R.sub.5 of the first trailing flank 17 of 
the male rotor 2. The second leading flank 10, defined between points 24 
and 25, is formed by an arc of a circle having radius R.sub.3 with its 
center located at a point 26 inside the pitch circle 11. 
The form of its first leading flank 14, defined between points 27 and 28, 
is created by the parabola of the first leading flank (between the points 
19 and 20) of the female rotor tooth profile 1. The forms of the second 
leading flank 15, defined between the points 28 and 29, and the second 
trailing flank 16, defined between points 30 and 31, are created 
respectively by the arc of the circle with the radius R.sub.2 of the 
second leading flank 8 (between the points 20 and 22) of the female rotor 
tooth profile 1 and by the arc of the circle with the radius R.sub.3 of 
said second trailing flank 10 (between the points 24 and 25) of the female 
rotor tooth profile 1. The form of the first trailing flank 17 between the 
points 27 and 30 is provided by an arc of a circle having radius R.sub.5 
with its center located at a point 32 on a line connecting the rotary 
shaft center points 3 and 4 of the female and male rotor tooth profiles 1 
and 2. 
In both the rotor tooth profiles 1 and 2 (shown by the solid lines and 
referred to as the theoretical profiles) respectively configured as 
explained above, the first leading flank 5 of the female rotor tooth 
profile 1 has its point of minimum pressure angle (approximately 
30.degree.) disposed at a point 20 connecting the first leading flank 7 to 
the second leading flank 8, while the first leading flank 12 of the male 
rotor tooth profile 2 has its point of minimum pressure angle disposed at 
a point 28 connecting the first leading flank 14 to the second leading 
flank 15. It has been found that, at the time of transmitting a rotating 
force from the male rotor tooth profile 2 to the female rotor tooth 
profile 1, the most efficient transmission of the force is obtained when 
tooth engagement is conducted by the tooth surfaces on the opposing 
leading flanks at or near the point of minimum pressure angle as already 
described and it is not preferable to have tooth surfaces other than those 
concerned in the transmitting of the force by reason of mechanical loss as 
will be explained more fully hereinbelow. 
For instance, when a deviation is given to the female rotor tooth profile 
alone, the point 20 of minimum pressure angle on the first leading flank 5 
of the female rotor tooth profile 1, as shown in FIGS. 1 through 3, is 
located on the theoretical tooth profile indicated by the solid line and, 
as preceeding from the point 20 toward the top side and bottom side of the 
tooth, the deviation is continuously increased as illustrated by a dotted 
line. That is, the deviation from the theoretical profile is given in a 
direction causing tooth thickness to diminish. The modified form of the 
female rotor tooth profile 1 is defined as follows. Reference numeral 8' 
indicates the second leading flank deviated from the second leading flank 
8 of the theoretical tooth profile shown by the solid line. The second 
leading flank 8' is defined between the points 20 and 22' and formed by an 
arc of a circle having radius R.sub.2 ' with its center located at a point 
23' on a line connecting the point of minimum pressure angle 20 and the 
center 23 of the radius R.sub.2. The tip point 22' of the tooth indicates 
a point of intersection of an arc of the radius R.sub.2 ' with an 
extension of a line connecting the rotation center 3 of the female tooth 
profile 1 to the center point 23' of the circular arc. Here, it is 
determined that the amount of deviation between the radii R.sub. 2 and 
R.sub.2 ' is .delta..sub.2 (0.1-0.15 mm) and the radius R.sub.2 ' equals 
to the value R.sub.2 -.delta..sub.2. 
On the other hand, the first leading flank 7' deviated from the first 
leading flank 7, of the theoretical tooth profile, is formed by a 
hyperbola extending through a point 19' inwardly deviated by .delta..sub.1 
(0.1-0.15 mm) from the lowest tooth bottom of the theoretical tooth 
profile and the point of minimum pressure angle 20. The amount of the 
deviation continuously increases as the deviation proceeds from the point 
20 toward the point 19' of the tooth bottom. The reason for using the 
hyperbola on the first leading flank of the modified tooth profile instead 
of the parabola forming that of the theoretical tooth profile resides in 
the fact that the normal line at the point 20 of the first leading flank 7 
of the theoretical tooth profile can be the same as the normal line at the 
point 20 of the deviated first leading flank 7'; that is, both the first 
leading flanks can have a common normal line. Additionally, by forming the 
first leading flank 7' with the hyperbola and the second leading flank 8' 
with the circular arc respectively, the amount of the deviation from the 
theoretical tooth profile can continuously increase as the deviation 
proceeds from the point of minimum pressure angle 20 toward the tooth top 
and the tooth bottom thereof. Thus, by the transmitting of rotative force 
at the point of minimum pressure angle 20 where the hyperbola is connected 
to the circular arc, the forces acting in the normal line direction and 
the radial direction can be diminished to thereby make it possible to 
reduce mechanical loss and also extend the life-time of bearings which 
support the rotor. Especially, in oil-cooled screw compressors, because 
the transmission torque between tooth surfaces is rather small, it is 
sufficient only to ensure the narrow tooth engagement at or near the point 
of minimum pressure angle. 
The trailing flank 6 connected to the leading flank 5 is formed by a first 
trailing flank 9' and a second trailing flank 10' which are deviated in 
the normal line direction by a constant amount identical to that of the 
deviation .delta..sub.1 of the first leading flank 7'. More particularly, 
the first trailing flank 9' between the points 19' and 24' is deviated 
from the first trailing flank 9 of the theoretical tooth profile, shown by 
the solid line, by the amount .delta..sub.1 in the tooth thickness 
decreasing direction. And, the second trailing flank 10' between the 
points 24' and 25' is formed by an arc of a circle with radius R.sub.3 ' 
having its center located at a point 26. Here, it is determined that the 
value of the radius R.sub.3 ' equals to R.sub.3 -.delta..sub.1. 
Of note, said amounts of the deviation .delta..sub.1 and .delta..sub.2 may 
be identical or different. 
When the parabola forming the first leading flank 7 of the theoretical 
tooth profile and the amount of the deviation .delta..sub.1 are given, the 
hyperbola forming the first leading flank 7' is obtained as follows. 
In FIG. 2, the parabola forming the first leading flank 7 is expressed by 
an equation (1): 
EQU Y.sup.2 =4.multidot.a (X-.delta..sub.1) (1) 
And, the hyperbola forming the deviated first leading flank 7' is expressed 
by an equation (2): 
##EQU1## 
Since two normal lines at the point 20 of minimum pressure angle on the 
those two kinds of curves must agree with each other, in other words, the 
gradients of the tangent lines at the point 20 thereof must agree with 
each other, from the equation (1), 
##EQU2## 
and from the equation (2), 
##EQU3## 
The equation (3)=the equation (4), then 
##EQU4## 
Thus, constituent dimensions for the hyperbola are obtained from the x and 
y coordinate values at the point 20 and the value of .phi.. 
On the leading flank 12 of the male rotor tooth profile 2, a point of 
minimum pressure angle 28 is located on the theoretical tooth profile 
shown by a solid line. 
Reference numeral 14' indicates a first leading flank deviated from the 
first leading flank 14 of the theoretical tooth profile shown by the solid 
line to the tooth thickness decreasing direction. The amount of this 
deviation on the first leading flank 14' continuously increases from the 
point 28 of minimum pressure angle. And, numeral 15' indicates a second 
leading flank deviated from the second leading flank 15 of the theoretical 
tooth profile shown by the solid line to the tooth thickness decreasing 
direction. The amount of deviation on said second leading flank 15' 
continuously increases from the point 28 of minimum pressure angle, and 
further a first and second trailing flanks 17' and 16' are given a 
constant amount of deviation. Although not shown in the drawing, it is 
possible, if desired, to select the point of minimum pressure angle on the 
trailing flank 13 of the theoretical tooth profile. 
In FIG. 5, on the trailing flank 6 of the female rotor tooth profile 1, the 
point 24 of minimum pressure angle (about 15.degree.) is located on the 
theoretical tooth profile shown by a solid line. 
Numeral 9' indicates a first trailing flank deviated from the first 
trailing flank 9 of the theoretical tooth profile to the same direction as 
those already mentioned in the previous drawings and the amount of 
deviation on the first leading flank 9' also continuously increases from 
the point of minimum pressure angle 24. 
Numeral 10' indicates a second trailing flank deviated from the second 
trailing flank 10 and the amount of deviation on the second following 
flank 10' also continuously increases from the point of minimum pressure 
angle 24. 
As will be understood from the foregoing description, by forming the 
leading and trailing flanks of the female rotor tooth profile 1 or the 
leading flank of the male rotor tooth profile 2 such that the amount of 
deviation thereof from the theoretical tooth profiles continuously 
increase as going from the respective points of minimum pressure angles 
toward the tooth top direction and the tooth bottom direction as well, a 
pair of screw rotor tooth profiles can be obtained which are almost not 
susceptible to machining and/or assembling errors, etc. involved in the 
rotor tooth profiles, that is, which have an insensitivity to 
manufacturing precision thereof. 
In other words, even in case of the presence of various errors it is 
possible to have tooth engagement in a narrow area at or close to a point 
of minimum pressure angle which assuredly carries out the transmission of 
rotative torque. 
Further, in case of rotor diameter being large, the single cutter is used 
for tooth cutting because the manufacture of hob for tooth cutting tool is 
difficult, but such cutting cannot cause precise cutting. The modified 
rotor tooth profiles according to the present invention, however, make it 
possible to provide excellent tooth engagement required for force 
transmission even in case of the errors of cutting being large. 
Accordingly, the modified tooth profiles are very effective when applied 
to a large-sized screw compressor adopting large diameter rotor tooth 
profiles. Besides, said effect of the invention can also be provided by 
using the afore-described modified type of rotor tooth profiles in job 
cutting operations or the like where there is a possibility of occurrence 
of large errors. 
Although, in the disclosed embodiments of the invention, explanation has 
been made with respect to the cases in which the point of minimum pressure 
angle coincides with the engaging tooth surface position (the driving 
force position), as apparent, such coincidence with the point of minimum 
pressure angle is not necessarily required as long as the tooth engaging 
position is set on the theoretical tooth profile. 
Further, while in the embodiments described above, the cases of giving 
deviation only to one of the female and male rotor tooth profiles have 
been described, it is to be understood that both of the rotor tooth 
profiles may be given deviations respectively if desired. 
Besides, although, in one embodiment, a parabola has been adopted on the 
first leading flank of the theoretical tooth profile, it is also possible 
to use a circular arc or other curvilinear lines: that is, in the 
application of the present invention, the theoretical profile undergoes no 
particular limitation as regards the configuration thereof. 
As noted above, according to the invention screw rotor tooth profiles, at 
least one of the female and male rotor tooth profiles constructed as a 
pair of theoretical tooth profiles engaging each other with no clearance 
provided therebetween has a point of minimum pressure angle or an engaging 
tooth surface selected on the theoretical tooth profile to thereby obtain 
tooth engagement required for force transmission only at or close to said 
point of minimum pressure angle or an engaging tooth surface position, so 
that a pair of screw rotor tooth profiles can be provided which can hardly 
be influenced by machining and/or assembling errors, etc., that is, which 
have an insensitivity to manufacturing precision thereof.