ABSTRACT
There is a trend in the industry to make gears with a profile contact ratio greater than 2.0. The main reason is that these gears would run quieter by constantly having two or more teeth sharing the load. In order to obtain a profile contact ratio greater than 2.0, it is necessary to meet two requirements:
- Have a fairly large number of teeth on each gear.
- Have a low generating pressure angle of about 20° or lower.
Use of such gears, perform quieter than gears with a profile contact ratio that is greater than 1. 0 but close to 1.0, where two teeth share the load only part of the time. These gears will be referred to as silent gears or low pressure angle gears. The disadvantage of these gears is that they have low horsepower capacity.
INTRODUCTION
When designing gears, the lower the pressure angle, the higher the surface compressive and bending stresses become. This lowers the surface and bending fatigue lives. Even if we take into account the load division between the two teeth that are carrying the load, you will see that each tooth carries less than a full load.
The amount of load that is being shared between the two teeth in contact depends on the tooth stiffness. The cross sectional area of the tooth is an important factor in load sharing. The gear with the smallest number of teeth will have thicker teeth towards the root diameter because they are made on a smaller base diameter. For this reason, the teeth are made on a smaller cylinder. The involute radii are smaller over the entire flank than on gears that have a much large number of teeth. As the number of gear teeth increases the involute curvature approaches a straight line to the extreme where the gear with an infinite number of teeth will have a tooth profile where the flanks are nearly parallel, see Figure Number 1.
Tooth deflection of low pressure angle gears is also substantial and can have a detrimental effect on the bending fatigue life. The gear tooth can be viewed as a short cantilever beam. The thinner the tooth is, the more it can deflect and have a higher bending stress. The bending stresses are proportional to the deflection, the higher the bending stress, the lower the bending fatigue life becomes.
At Power Engineering and Manufacturing, Ltd. we manufacture only high pressure angle gears between 35° and 43°. One advantage of the higher pressure angle gears is that they are triangular and wide near the root diameter. The design of the
MEGAGEARS® and UNIMEGAGEARS® is such that the fatigue due to bending stress is completely eliminated5.
GEAR DESIGN PRACTICES
We design spur gears to have a profile contact ratio slightly over 1.0. This will insure that there will be no interruption when transferring the load from one tooth to another. The length of time that two teeth carry the load is proportional to the amount of profile contact ratio in an excess of 1.0. Spur gears, which have less than a 1.0 profile contact ratio, have a small amount of rotation when the teeth do not carry any load. This allows the driving shaft to accelerate, and the driven shaft to decelerate. When the teeth engage again to carry the torque, there will be a light impact that eventually may cause premature surface damage and generate noise and vibrations.
The line of load on spur gears is in the axial direction at the same radial position on the full face width of the tooth as shown on Figure Number 2. Because the line of load is parallel with the gear centerline at high speeds the load is transferred nearly instantaneously from one tooth to another over the entire face width. The load that is rapidly applied at the tip of the tooth is like an impacting load, when the line of load exits at the tip of the tooth, and is suddenly released and straightens out from whatever deflection it may have and generates noise like a musician’s tuning fork. Tooth deflection, profile and lead accuracy are the main contributors to noise.
On helical gears that have a helical contact ratio greater than 1.0, the line ofload is diagonal on two teeth, as shown on Figure Number 3. This allows for a smooth load transfer from one tooth to another. When one tooth has the entire helical line of load that represents a 1.0 helical contact ratio the short line of load on the adjacent tooth has a length that is proportional to the amount of the helical contact ratio that is in excess of 1.0. As the gear rotates, the line of load moves in the axial direction until it exits the tooth and enters another. The advantage of the helical gear is that since the line of load is on a diagonal it makes a more placid load transfer from one tooth to another. The helical contact ratio is additive to the profile contact ratio. When the profile contact ratio is smaller than 1.0 it can be compensated for that by making the helical contact ratio sufficiently large so that together the profile contact ratio plus the helical contact ratio are greater than 2.0.
Figure Number 4 shows that the tooth deflection under load is smaller on a helical tooth than the equivalent tooth deflection on a spur gear tooth because there is additional resistance to deflection adjacent to the load in the axial direction.
It is a good practice to make the sum of the profile contact ratio and the helical contact ratio greater than 2.0. This will enable the gear to always have two teeth carrying the load simultaneously.
Because the high pressure angle gear teeth are more triangular in shape than the silent gears, it makes them stiffer than the silent low pressure angle gear teeth that can deflect more under load. For that reason, the high pressure angle helical gears are not as silent as the low pressure angle gears. See Figure Number 5.
Tooth deflection under load is very small due to the triangular shape of the high pressure angle gears. Because the teeth are stiffer, they do not deflect and when the load is released they do not act like a musical tuning fork vibrating to generate sound. Because the teeth are stiffer, some noise is generated when the load is transferred from one tooth to another upon impact. Note that the High Pressure Angle Gear shown in Figure Number 5 has two base circle diameters. The smaller base circle is for the super high pressure angle flank, while the bigger base circle near the root diameter is for the low pressure angle flank. UNIMEGAGEARS® have a non-symmetric tooth, and are manufactured with a non-symmetric teeth hobs or shaper cutters. The power density, durability, and efficiency of high pressure angle gears are substantially greater than those of low pressure angle gears because the involute curve at the gear teeth is further away from the base circle diameter, and the tooth whole depth is shorter. The sliding at the pitch diameter is zero on all gear teeth and grows in both directions the further away a point is from the pitch diameter. Force times distance is work. This is how the sliding forces at the gear mesh are converted into heat, which is the gear mesh efficiency.
Note that in Figure Number 6 the further away the point is on the involute curve; the larger the radius of curvature is R1 < R2 < R3. The larger the radii of the involute curve becomes, the larger the area of contact between the teeth becomes. It is desirable to maximize the area of contact between the gear teeth to reduce the surface compressive stress and increase the surface fatigue life. This method of maximizing the involute radius is a result of increasing the operating pressure angle.
Figure Number 7 shows that the pressure angle at the working pitch diameter is designated by a straight line that is tangent with the involute curve at the working pitch diameter and a line that is passing thought the gear center and a mid transverse point on a tooth, hence the name “High Pressure Angle Gears”. The increased area ofload transfer between the gear teeth also increases the oil film thickness to approach a hydrostatic load transfer and to reduce the mechanical concentrated load transfer that can initiate premature surface fatigue. A hydrostatic load transfer is desirable because it reduces or eliminates mechanical load transfer that has undesirable concentrated points of the load resulting in a reduced surface fatigue life.
Because of the inherent high power density Power Engineering and Manufacturing, Ltd only manufactures high pressure angle gears. With the advent of computer controlled gear tooth grinding equipment we are able to do crowning, tip relief, root relief, leading helical relief, and compensate for the torsional shaft deflection. This tooth profile and lead refinements help improve the load distribution along the full tooth radial and axial lengths to maximize the surface and bending fatigue lives, and to operate quieter.
Some applications require that the gear noise be extremely low, even much lower than the legal requirement. Low pressure angle helical gears with a profile contact ratio slightly gn:aler lliau 2.0, were found to satisfy the low level noise requirements. For high horsepower applications, the low pressure angle silent gears become very big because they have a relatively low power density. Large gears generate a high pitch line velocity that causes a high oil turbulence that in turn generates heat and loss of horsepower.
High pressure angle gears inherently have a higher power density and can be used for high horsepower transmissions. For comparison Table Number 1 shows the features of low pressure angle gears as compared to high pressure angle gears. To make the comparison valid the center distance between the pinion and the gear, the horsepower, the input speed, and the reduction ratio were kept the same for all three types of gears. The low pressure angle gears have a generating pressure angle of 17°, a working pressure angle of 20°, and a helical angle of 12°.
SILENT GEARS
The pinion has 20 teeth and the gear has 81 teeth. It is designed to have a reduction ratio slightly greater than 4: 1 to avoid vibrations and to have every tooth in mesh with every tooth for a longer wear life. Input speed was selected to be 2,000 rpm. The profile contact ratio is 2.05. Surface compressive stress at the start of active profile is 225,000 psi. The stress at the start of active profile was selected as 255,000 psi because it is higher than the stress at the pitch diameter. This higher surface compressive stress causes the tooth to have a lower surface fatigue life. Because there is a 2.05: 1 profile contact ratio, two pairs of gear teeth are constantly in contact. The load was assumed to be evenly divided between the two pairs of teeth so that at any given time each gear mesh will only carry 200 horsepower not 400, but together the low pressure angle gears carry 400 horsepower just like the high pressure angle gears5.
MEGAGEARS®
MEGAGEAR® is defined as a high pressure angle gear because of the high operating pressure angle, which is shown on Table Number 1 to be 31.4°, (It can be up to 40° depending on the individual gear mesh design), and the generating pressure angle is 25°.
These high pressure angle gears have a 33% higher horsepower carrying capacity than the conventional gears (not silent gears, See Table Number 1). The high pressure angle gears have a surface and bending fatigue life that is greater than threefold of conventional gears.
UNIMEGAGEARS®
UNIMEGAGEARS® are a special case of high pressure angle gears that can carry an even higher load with a longer surface and bending fatigue life. The interesting feature is that these gear teeth have non-symmetric flanks. They have a high generating pressure angle of 35° and a low pressure generating pressure angle of 15°. A working pressure angle range up to 43° on the high pressure angle and up to 28° on the low pressure angle can be generated with these hobs. The theoretical pressure angle at the maximum horsepower, which an involute gear tooth can carry, is 45°. These non-symmetric teeth usually have pinions with long addendums; we stop at 43° to maintain a balanced mesh on the high and the low side of 45° operating pressure angle over the entire tooth contact radial length. The advantage of these non-symmetric gears is that, they can carry about 45% more horsepower than the conventional gears on the high pressure angle flank. The low pressure angle flank has a good load carrying capacity; however the surface fatigue life at the low pressure angle flank can be under 50% of the surface fatigue life on the high pressure angle flank. The low pressure angle flank is expanded to approximately 28° from the 15° generating pressure angle. This makes the lower pressure angle flank capable of carrying a slightly higher load than the conventional 20° or 25° pressure angle gears.
These non-symmetric high pressure angle gears are suitable for applications where the load is predominantly in one direction such as in hoists, cars, trucks, tractors, and many other applications. These gears have a 45% to 50% higher horsepower carrying capacity, than conventional gears. They also have a fatigue life that is threefold or greater than that of conventional gears.
CONCLUSION
Silent gears are successful in operating at reduced noise levels primarily because they have a profile contact ratio that is greater than 2.0. The helix angle is also a great help, to reduce noise because the load is transferred to the incoming tooth slowly, not suddenly like in spur gears.
MEGAGEARS® and UNIMEGAGEARS® have a profile contact ratio of about 1.0 and a helical contact ratio that is greater than 1.0. As can be seen from the surface compressive stress comparison between the high pressure angle gears and the silent gear, the high pressure angle gears can carry 16 times as much horsepower with the same amount of metal and for the same surface fatigue life as the silent gears. The high pressure angle non-symmetric gears have an even higher pressure angle and are able to carry 21 times more horsepower than the silent gears do for the same surface compressive stress and the same surface fatigue life.
If we examine the bending stress of all the gears, it can be concluded that bending fatigue life of all high pressure angle gears is very long and it should not be a prnblem unless there are many shock loads of a very high magnitude.
Equal load division between the teeth is difficult to achieve because the pinion and the gear teeth do not have an identical cross sectional area that determines its stiffness. The smaller the number of teeth, the thicker the cross section becomes towards the root diameter when there is no undercut.
The tooth that has a thinner cross section will deflect more and consequently have a higher bending stress. This higher bending stress reduces the bending fatigue life.
Silent gears, if designed properly do not have a bending fatigue problem. High pressure angle gears have an infinite bending fatigue life because of the wide base cross sectional area. Figure Number 5 shows the variation in transverse tooth cross sectional area. Table Number I shows the respective bending stresses and the resulting bending fatigue lives.
REFERENCES
1. Moyer, C. A and Bahney, L.L., “Modifying the Lambda Ratio to Functional Line Contacts”, STLE Trib. Trans. Vol. 33,4, pp 535-542, 1990
2. Moyer, C. A. The Use of Elastohydrodynamic Lubrication in Understanding Bearing Performance. SAE Technical Paper Series. Vol 80. 1971 .
3. Moyer, C. A. Using the Modified Lambda Ratio to Advance Bearing and Gear Performance. SAE Technical Paper Series. September 10-13 1990.
4. Herscovici, Saul., ” Spur and Helical Gear Design”, SAE Technical Paper Series. 1985. 5. Herscovici, Saul., “Increased Power Density, Efficiency, and Durability With MEGAGEARS® and UNIMEGAGEARS®”, SAE Technical Paper Series. 2005.
CONTACT
Saul Herscovici
President
Power Engineering & Manufacturing Ltd.
2635 WCF&N Drive
Waterloo, IA 50704
Phone: 319-232-2311