Abstract
Through research and development Power Engineering and Manufacturing, Ltd(PEM) has developed gear teeth that have a unique involute geometry that can carry a higher horsepower with a longer surface and bending fatigue life. PEM’s trademark gear MEGAGEAR® has symmetric flanks that can carry approximately 33% more horsepower with three to ten times longer surface and bending fatigue life.
PEM’s other gear is trademarked UNIMEGAGEAR® which has non symmetric flanks, and can carry up to 45% more horsepower, also with a three to ten times longer surface and bending fatigue life.
Gears with involute profiles are used because they are capable of transferring forces between the teeth as the gears rotate at a constant angular velocity; this is a very important feature because no vibrations are generated due to the geometry of the load transfer members. Vibrations are due to many other reasons, but they are small in comparison to what could happen if the involute profile curve was never used. The involute curve generated off the base circle diameter can be a very long curve, much longer than the portion that is being used to make the gear tooth. For that reason it is up to the engineer to determine which section of this large involute curve can be used more advantageously. This is commonly known as the generating pressure angle and/or the working pressure angle. The higher the pressure angle is, the more powerful the tooth is, which enables it to carry more horsepower as well as have a longer surface and bending fatigue life.
The Involute Curve
Austrian scientist Leonhard Euler (1707-1783) derived the involute mathematics that resulted in constant angular velocity. Because of the involute property of generating constant angular velocity, nearly all the gears manufactured around the world are made with an involute profile. Any section of the involute curve can be used to form the gear tooth profile. The further away the gear tooth is from the base circle diameter, the bigger the radius of the involute curvature is. This larger involute radius yields a larger area of contact between the teeth, thus reducing the surface compressive stress and increasing the surface fatigue life.
This process is done within limits because the further away from the base circle the involute curve is used, the closer it becomes to being parallel with the base circle. This can make the teeth shorter and have an insufficient profile contact ratio.
At this point it is appropriate to define the gear tooth profile. Figure Number 1 shows how the involute curve is developed. If a pencil is tied to a string that is fixed to a drum whose diameter is the same as the base circle of a gear, when unwinding the string, the pencil scribes a line that is the involute curve. Note that the further away the pencil moves from the base circle the bigger the involute radius becomes.
MEGAGEARS®
Power Engineering & Manufacturing, Ltd has continuously conducted research and development during its 31 years of existence to increase the power density, surface fatigue life and the bending fatigue life.
This slow and cautious advancement enables PEM to reach a point on the involute curve that allows us to make stronger teeth that can carry up to 33 percent more horsepower on symmetric teeth as well as a three to ten times longer surface and bending fatigue life. MEGAGEARS® are generated with 25 degrees pressure angle cutters, but have a working pressure angle of up to 36 degrees and depending on the particular design, sometimes an even higher working pressure angle.
Picture Number 1
Standard gear, 23 teeth, 4.233 NDP,20° PA, 30° HA 1264 HP at 3000 RPM, 50 million load cycles SCS@PCR of 1.0=231,000 PSI, 6.753 O.D.
Picture Number 2
MEGAGEAR®, 19 teeth, 4 NDP, 35° PA, 30° HA 1264 HP at 3000 RPM, 50 million load cycles SCS@SAP=186,000 PSI, 6.813 O.D.
Picture Number 3
MEGAGEAR®, 19 teeth, 4 NDP, 35° PA, 30° HA 1686 HP at 3000 RPM, 50 million load cycles SCS@SAP=215,000 PSI, 6.813 O.D.
Picture Numbers 1, 2, and 3 are of gears that were tested at the University of Newcastle upon Tyne in the UK under the direction of Professor Dieter Hofmann. 6Picture Number 1 is of a conventional 20° pressure angle gear after it was tested at 1,264 horsepower at 3,000 rpm for 50 million load cycles.
Picture Number 2 shows a MEGAGEAR® after it was tested at, 1,264 horsepower at 3,000 rpm for 50 million load cycles identically to the gear in Picture Number 1.
Picture Number 3 shows a MEGAGEAR® after it was tested at a higher horsepower of 1,686 at 3,000 rpm for 50 million load cycles.
Note, that when compared to the conventional 20° pressure angle gear shown on Picture Number 1 the MEGAGEAR® shown in Picture Number 2 that was tested at 1,264 horsepower did not show many fatigue micropits, except at the start of active profile.
When the managers at the University Test Laboratory saw the absence of wide area micropits they increased the horsepower to 1,686 and repeated the test. Surface micropits can now be seen on Picture Number 3.
UNIMEGAGEARS®
A new and unique tooth form development at PEM is an involute based profile but with different pressure angles on each flank. UNIMEGAGEARS® have the advantage of an even higher power density than MEGAGEARS® which allows them to carry up to 45 percent more horsepower with a surface fatigue life and bending fatigue life that is three to ten times longer than that of conventional 20 degrees pressure angle gears.
UNIMEGAGEAR® teeth are generated with cutters that have 15 degrees on the low pressure angle side and a 35 degrees angle on the high pressure angle side. The working pressure angles of UNIMEGAGEARS® are about 42 degrees on the high pressure angle flank and about 28 degrees on the low pressure angle flank.
The theoretical pressure angle for maximum power density is 45 degrees. The power density increases up to 45 degrees working pressure angle. As the working pressure angle increases past 45 degrees, the power density decreases.
UNIMEGAGEARS® usually have a pinion with a long addendum and a gear with a short addendum. We design the gears with a working pressure angle of up to 42 degrees rather than 45 degrees to maintain a balanced working contact angle at the point of mesh on the high, as well as the low side of the 45 degrees operating pressure angle point over the entire tooth contact radial length.
Because the high and low pressure angle flanks are calculated simultaneously, the low working pressure angle flank that is generated with a 15 degrees pressure angle cutter grows to about 28 degrees working pressure angle when the high working pressure angle is at about 42 degrees.
(Terms are defined at the end of the paper)
Equation Number 1 shows that if it is possible to reduce the surface compressive stress by 21 percent when replacing conventional gears with MEGAGEARS® and UNIMEGAGEARS®, the surface fatigue life doubles. The majority of the gear boxes that are made by other companies that we overhaul the reduction in the surface compressive stress is lowered by more than 21 percent and the surface fatigue life on average is tripled. At the same time, the bending fatigue failures are virtually eliminated because the bending stress is so low.
This makes the low pressure angle flank able to carry more load than a conventional gear with a 20 degrees working pressure angle and likewise with a three to ten times longer surface fatigue and bending fatigue life. UNIMEGAGEARS® are suitable for applications where the load is predominantly in one direction. The high working pressure angle flank has the high load carrying capacity. The low working pressure angle flank has about 50 percent or less load carrying capacity for adequate long fatigue life.
Surface Compressive Stress
The entire concept of MEGAGEARS® and UNIMEGAGEARS® is based on maximizing the area of contact during load transfer between the teeth. This is accomplished by maximizing the length of the pinion involute radius at the start of active profile and also at the point of profile contact ratio of 1.0. This involute radius is the shortest at the start of active profile of the gear with the smallest number of teeth.
Heinrich Hertz (1857-1894) the German physicist, born in Hamburg, derived the equation for surface compressive stress between two cylinders which is as follows:
Where B can be calculated as follows:
The variables r1 and r2 in Equation Number 31 designates the involute radius at the point of contact at load transfer between the teeth anywhere on the tooth height. It is often taken at the working pitch radius because it is easier to calculate but is not representative because it can be much smaller at the start of active profile of the gear with the smallest number of teeth. This results in a shorter surface fatigue life.
Equation Number 4 is used to calculate the surface compressive stress from which the surface fatigue life can be estimated. Equation Number 4 shows two very important variables r1 and r2. These are the involute radii at the point of load transfer between the teeth as shown on Figure Number 21 r1 and r2 are not the pitch radii.
Figure Number 2 shows the involute radii of a pair of MEGAGEARS® that were also tested at the University of Newcastle upon Tyne. See Picture Numbers 1, 2, and 5.
Figure Number 3 shows the involute radius of a conventional 20 degrees pressure angle gear tested at the University of Newcastle upon Tyne in the UK.
Figure Number 4 shows the involute radius of a 35 degrees MEGAGEAR®. Note that the involute radius of the MEGAGEAR® in Figure Number 4 is considerably longer than the involute radius of the conventional gear shown in Figure Number 3.
Figure Number 3, and Picture Number 4 show the profile of these conventional gear teeth. These are equal size gears and the involute radii at the start of active profile r1 = r2 = .753 inches
Picture Number 5 shows the profile of a MEGAGEAR®. Note that the high pressure angle makes the tooth shape triangular. These gears are similar in size to the conventional gears discussed above and fit on the same center distance. Their involute radius of curvature is r1 = r2 = 1.642 inches.
Also note that in Figure Number 4 the base circle is smaller than the base circle of the conventional gear shown in Figure Number 3. The smaller the base circle becomes, the larger the involute radius is.
Figure Number 5 shows the involute radii of a UNIMEGAGEAR®. Note that this gear has two base circles; one for the high pressure angle and one for the low pressure angle. The high fatigue load capacity is on the high pressure angle. The involute radius of curvature on the high pressure angle is r1 = r2 = 1.843 inches. The involute radius on the low pressure angle is r1 = r2 = 1.026 inches. The profile of these teeth can be seen on Picture Number 6.
These gears were not tested at the University of Newcastle upon Tyne, however, PEM has manufactured thousands of these gears and they are working extremely well. They are designed with the same performance and fatigue parameters as the MEGAGEARS® and they have proven themselves to be very powerful and durable.
The American Gear Manufacturers Association(AGMA) calculates the surface fatigue life based on the surface compressive stress at the pitch diameter. Figure Number 6 shows cylinders that are developed from the Hertz Equations that represent the involute radii at the pitch diameter at the start of active profile, with a profile contact ratio of 1.0 marked as single tooth contact. The axial length of the cylinders is equal to the face width of the gears that they represent. The calculations are made with equal size gears and equal torque for the conventional 25 degrees pressure angle gears and 33.7 degrees working pressure angle for MEGAGEARS®.
If we use the involute radius in Equation Number 4 of the conventional 25 degrees gear of .700 from Figure Number 6 and later we substitute it with the involute radius of 1.625 of a MEGAGEAR® also from Figure Number 6 we can see that since it is in the denominator, the entire quantity under the square root will become smaller. This is how the surface compressive stress magnitude is being reduced and the surface fatigue life is being increased.
Table Number 1
Contact Width at Different Position on the Teeth
| Surface Compressive Stress (PSI) | Contact Width (IN) | Involute Radii | ||||||
| Pinion | Gear | |||||||
| 25° | 33.7° | 25° | 33.7° | 25° | 33.7° | 25° | 33.7° | |
| Pitch Diameter | 129,000 | 118,000 | .035 | .042 | .676 | 1.625 | 5.751 | 6.942 |
| Contact Ratio = 1.0 | 143,000 | 123,000 | .032 | .041 | .989 | 1.670 | 5.463 | 6.898 |
| Start of Active Profile | 172,000 | 125,000 | .026 | .040 | .700 | 1.625 | 5.751 | 5.909 |
It can be seen on Table Number 1 that in all three cases the surface compressive stresses are higher on the conventional 25 degrees pressure angle gears, while at the same time the area of contact between the teeth is smaller than those of the MEGAGEAR®. The reason for that is simply the involute radii of the conventional 25 degree pressure angle gears is smaller than the involute radii of the MEGAGEARS® even though the gears are of similar diameter and face widths.
If this comparison would have been made by using a 20 degrees conventional design in place of the 25 degrees, the surface compressive stresses would have been higher because all the involute radii of the 20 degrees gears are smaller than those of the 25 degrees gears resulting in a shorter fatigue life. maintenance engineering manager
Helical vs. Spur Gears
When spur gears rotate, the load is transferred instantaneously from one tooth to another along the full face width of the tooth on a line that is parallel with the gear centerline. The instant that the forces change from load sharing to no load sharing, the force magnitude increases again along the full length of the tooth. The tooth deflection has been rapidly relieved on the unloaded tooth creates a slight impact on the tooth that now carries the full load. On accurate ground gears that operate at fairly low speeds, the position of transition from load sharing to no load sharing can be identified as a line on the gear teeth after some time of usage. Spur gears that are heavily loaded make a high frequency whining noise that is similar to the sound of a tuning fork when the pinched fork is suddenly released. Because of the sudden load release of one tooth and sudden increase in force on the tooth that now carries the load at the transition point, it may be stated that a mild shock load occurs. In most cases, because of the oil film thickness, no damage occurs. However, inevitably the force is somewhat higher than the force that corresponds to the full load.
Helical gears that have a combined profile and helical contact ratio sum that is greater than two are superior to spur gears because the load is transferred gently from one tooth to another due to the fact that two teeth will always share the load. An incoming tooth begins to carry the load before the outgoing tooth is relieved of load. The tooth deflection of the helical gears is more complex because the line of load is on a diagonal; therefore, the tooth deflection occurs instantaneously along the entire radial length of the tooth. The line of load moves axially as the gears rotate. The deflection of a helical gear tooth is less than that of an equivalent spur gear tooth because the line of load in the diagonal direction, allows the metal adjacent to the line of load in the axial direction to resist deflection as shown on Figure Number 8.
If the deflection is lower, the bending stress is also lower. Because spur gears have an axial line of load along the tooth, they operate with tooth deflection. This spur gear deflection generates noise but it can also initiate vibrations, particularly in a mechanism where a natural frequency of vibration is near the gear tooth meshing frequency.
Helical gears with a helical contact ratio that is greater than 1.0 transfer the load to the next tooth before the load is removed from the adjacent tooth. This feature makes the helical gear quieter and less likely to induce vibrations. For these reasons, helical gears are superior to spur gears.
Long Addendum and Short Addendum
Figure Number 9 shows the gear teeth of a conventional 20 degree pair of gears. Note that the base circle at the pinion is at the root diameter. The involute radii is shown at the pinion start of active profile which is the most critical point on the tooth. This is the point where the involute radius is the smallest which generates the highest surface compressive stress and consequently the shortest surface fatigue life.
Figure Number 10 shows the gear teeth of a 33. 7 degrees MEGA GEAR®. It can be seen that the pinion base circle is substantially smaller than the pinion root diameter. This enables the involute radii to be longer than the conventional 20 degrees gears. This leads to a lower surface compressive stress and a higher surface fatigue life.
The practice of making the pinion with a long addendum and the gear with a short addendum further helps the pinion involute radius become longer. This in tum helps to increase the surface fatigue life. See Figure Number 6.
The bending stress and fatigue life is fairly simple to calculate if it is assumed that the gear tooth is a short cantilever beam. To calculate the bending stress for helical gears it is virtually impossible. For that reason a finite element analysis program was integrated with the gear design program that instantly calculates to tooth bending stress from which the bending fatigue life can be determined. Figure Number 11 shows the tooth profiles of a conventional gear, a MEGAGEAR®, and a UNIMEGAGEAR®.
Note that Figure Number 11, B and C of the MEGAGEAR® and UNIMEGAGEAR® teeth have a wider base at the root diameter than the teeth of a conventional gear shown on Figure Number 11A. This makes the tooth thicker which reduces the bending stress and increases the bending fatigue life. The most damaging bending stress force is the normal force at the tip of the tooth. The amount of material available to absorb the steady operating force or an impact force is indicated by the dotted line as a continuation of the normal force. Note that the MEGAGEAR® and the UNIMEGAGEAR® have a substantially larger amount of material to absorb forces and the forces are in compression. There is no shearing force. Figure Number 11A has a smaller amount of material to absorb forces, but it also has shearing stress because the normal force exits the tooth. An extremely high force from a shock load can shear the tip of a conventional gear tooth.
The tip of the MEGAGEARS® and the UNIMEGAGEARS® are narrower, but because of the absence of a shearing stress and the fact that the teeth are wider, they have a substantially lower bending stress. Also the higher angle at the tooth tip makes the normal force have a longer radial force component that provides a compressive stress that actually reduces the tensile stress generated by the tangential force component of the normal force. This is why even a thinner tip carries the load without damage.
Surface Fatigue Life
The surface fatigue life is calculated by using the surface compressive stress as shown on Equation Number 4. Durability tests conducted by the University of Newcastle upon Tyne as well as many organizations around the world have shown that a good steel such as 8620 carburized and case hardened to RC 58 to 62 has a surface fatigue life of 50 million load cycles at 225,000 psi as shown on Picture Numbers 1, and 3.
Calculations of oil film thickness under load 3,4,5 show that MEGAGEARS® as well as UNIMEGAGEARS® have a thicker oil film than conventional 20 or 25 degrees pressure angle gears. The thicker oil film, together with a good surface finish in the order of 15 micro inches, and accurate profile and lead will improve the load distribution to be uniform and not create localized high surface compressive stresses that will lead to premature surface fatigue.
The number of Surface Fatigue Load Cycles (SFLC) of 8620 steel carburized and case hardened to RC 56-62 may be calculated per Equation Number 6.
Because of improvements in the steel quality and manufacturing processes, a slightly higher surface compressive stress than 225,000 psi may be used.
Table Number 2
Surface Fatigue Life
| Surface Compressive Stress (PSI) | Surface Fatigue Life (Percent) | |||
| 25° | 33.7° | 25° | 33.7° | |
| Pitch Diameter | 129,000 | 118,000 | 100% | 181% |
| Contact Ratio = 1.0 | 143,000 | 123,000 | 100% | 273% |
| Start of Active Profile | 172,000 | 125,000 | 100% | 839% |
Table Number 2 shows that there is a big spread in the surface compressive stress (33 percent) and surface fatigue life of the conventional gears. At the same time Table Number 2 shows that the same spread of the MEGAGEARS® is much smaller (6 percent). It is desirable to have uniform stress to have a uniform surface fatigue life over the entire tooth area.
If AGMA surface fatigue life is calculated to be the design criteria for establishing the surface fatigue life then it can be shown that the gear will have a substantial shorter life at the point of contact ratio of 1.0 or at the start of active profile.
We can calculate the percent reduction in the surface fatigue life at the profile contact ratio of 1.0 by using Equation Number 6.
This shows that in this example the conventional gears will have half as much surface fatigue life at the point of profile contact ratio of 1.0 as was calculated at the pitch diameter. At the same time the MEGAGEARS® surface fatigue life at the point of profile contact ratio of 1.0 in relation to the pitch diameter is;
The MEGAGEARS® are losing only 11 percent surface fatigue life. This shows the advantage of the MEGAGEARS® over conventional gears. To properly determine the surface fatigue life it should be calculated at the profile contact ratio of 1.0 to achieve the desired service life of a gear box.
Conclusion
The mathematical derivation of the MEGAGEARS® was tested and proven to be able to carry 33 percent more horsepower by durability tests conducted at the University of Newcastle upon Tyne in England.
Both MEGAGEARS® and UNIMEGAGEARS® have been in use at Power Engineering and Manufacturing, Ltd for 31 years and perform excellent according to the theoretical calculation expectations.
References:
1. Machine Design. Shigley, J.E. McGraw-Hill. 1956.
2. AGMA Surface Durability (Pitting) of Helical and Herringbone Gear Teeth. 211.02, 1974.
3. 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
4. Moyer, C. A. The Use of Elastohydrodynamic Lubrication in Understanding Bearing Performance. SAE Technical Paper Series. Vol 80. 1971.
5. Moyer, C. A. Using the Modified Lambda Ratio to Advance Bearing and Gear Performance. SAE Technical Paper Series. September 10-13 1990.
6. Design Unit – Gear Technology Centre. “The Surface Durability of PEM “MEGAGEARS”. University of Newcastle upon Tyne. DU2876. July 2002.
Definitions: (ABC order)
| B | Contact Width |
| BL | B Cylindrical Tangential Contact Width
L Cylinder Length (Gear Face Width) |
| BR | Base Radius |
| Cos | Cosine |
| F | Force |
| E | Modules of Elasticity |
| FW | Gear Face Width |
| GPA | Generating Pressure Angle |
| HA | Helical Angle |
| L | Gear Face Width |
| N | Number of Teeth |
| NDP | Normal Diametral Pitch |
| OD | Outside Diameter |
| PA | Pressure Angle |
| PCR | Profile Contact Ratio |
| R1 | Pinion involute radius length |
| R2 | Gear involute radius length |
| RPM | Rotations Per Minute |
| SAP | Start of Active Profile |
| SCS | Surface Compressive Stress |
| SFLC | Surface Fatigue Load Cycles |
| T | Torque |
| µ | Poison Ratio |
| PD | Pitch Diameter |
| PSI | Pounds per Square Inch |
| VF | Velocity Factor |