The author proposes the method of the gear tooth surface modification developed during 1986-1992 at MIL Helicopter in Moscow, former Soviet Union. This is a good example how an advanced technology was kept in secret for a long time because its military application. The following innovations were developed and tested on helicopter transmissions:

- helical tooth surface modification on the direction of the contact pass

- parabolic curve of tooth modification

- calculation procedure for sizes of spur and helical modifications.

It was also tested on a computer program for gear mesh simulation enables to predict transmission error and design smooth low noise drive. Significant research work was done on manufacturing and testing parabolic shape of tooth profile modification on spur and helical gears for helicopter main power gear boxes.

Gear tooth modification is a way to fight against not conjugated contact in real gear world. Not conjugated contact may be caused by different things. It can be spacing error, tooth profile, lead, deflections under the load or misalignments in positioning of the gears. Any of these issues can cause the edge contact in the mesh. The edge contact can take place twice during the engagement of a pare of teeth. The first time the tip of the pinion tooth impacts the root of the gear tooth. Then the edge contact takes place at the end of contact of two teeth. In both cases the edge contact causes higher contact stresses on the tooth surface. Modification of the tooth surface is the way to avoid the edge contact.

The tooth profile modification is very widely used on spiral bevel gears. It has bean developed naturally. Tooth profile modification on spiral bevel and hypoid gears relates very much to the contact pattern. The other word is the bearing of contact. Setting a contact pattern on spiral bevel gear tooth automatically means creating a tooth modification. It was done this way on spiral bevel gears because the complicity of them. No other ways were available for a long time for spiral bevel gear tooth surface inspection. Unlike spiral bevel gears helical and spur gears can be measured much easier. Profile, lead and spacing can be inspected on simple fixtures. This is why manufacturing of spur and helical gears was always focused on detailed inspection of each individual tooth parameters. 99.99% of spur and helical gears are not tested for the contact pattern because it does not give any useful manufacturing information. Profile, spacing, lead and run-out give all necessary data for making corrections in manufacturing process. The inspection of spur and helical gears is broken on parts so it does not allow to see an influence of all details together.

Hear is one example of looking at a helical gear tooth from a different point of view. The invention was done in 1987( Russian patents: #1556211, #1555976) and it was developed on helicopter transmissions during 1987-1992. The base idea of this invention was developing the main part of tooth modification in the direction of the contact pass. In spiral bevel world such kind of modification is generated automatically by changing the ration on the gear cutting machine. On cylindrical gears the modification of the tooth surface was divided on two parts: profile modification and lead modification. The other words are tip or root relieve and crowning. It is common opinion that the tooth profile modification is for preventing of the edge contact and crowning is for compensation of misalignments of positioning. It is correct for spur gears. The contact area of spur teeth locates parallel to the tooth tip. During the action it moves from the tip to the root or in opposite direction. The direction of the movement is perpendicular to the tooth tip. It works differently on helical gears. Unlike spur gear the contact area on helical gear locates across the tooth surface. The contact area of helical teeth does not parallel to the tooth tip. A regular tip or root relieve decreases the contact are because the modification does not parallel to the contact area any more. Lower area of contact increases tooth contact stresses. The bearing of the contact becomes smaller. The contact bearing is smaller on modified spur gear teeth and it increases the contact stresses on spur gear teeth as well. This is a price to pay for eliminating of the edge tooth contact during the action. Unfortunately tip and root relieve does not eliminate the edge contact on helical teeth. The edge impact appears on the new place. It is not on the tip or the root anymore. The edge contact takes place on the side edge of the tooth. The edge contact still exists on the helical gear even with profile modification. The profile modification made it even worse because it decreases the size of the contact pattern. There is a large difference between the action of spur and helical teeth. A spur tooth impacts the gear tooth by a line and a helical pinion tooth impacts the helical gear tooth by a point. The spur pinion tooth engages with the mating gear tooth by the tip edge. The helical pinion tooth engages with the mating gear by the corner on the tip. The contact stresses are higher on the corner impact. The contact area on a spur gear tooth has an equal size during the contact action. Unlike that the contact area on a helical gear tooth is changing during the action.

A new shape of the helical tooth surface modification has bean developed. The deepest point of the modification has to be in the point of the corner contact. On the rest of the modified surface the depth modification has to be the same along the tooth contact. The changing of the depth of modification is determined along the contact pass but not along the tooth profile. A number of manufacturing processes were developed and patented for manufacturing of the helical modification. (USSR patent #1555976). One way of manufacturing is grinding on the dish wheel grinding CNC machine. The helical modification can be generated by changing the machine ratio. A dish wheel gear grinding machine with a rolling drum was also used for the helical tooth modification. A straight slots on the rolling drum enable to make helical modification.

The dimensions of the tooth modification can be determined as well as the shape of the modification curvature. The maximum depth of modification can be calculated as:

T= t1+ t2 + t3 + t4,

t1 = s1 + s2,

s1 – pinion spacing error,

s2 - gear spacing error,

t2 = p1 + p2,

p1 – pinion profile error,

p2 – gear profile error,

t3 = L1 +L2,

L1- pinion lead error,

L2- gear lead error,

t4 = Dc +Db,

Contact deformation Dc = Cp + Cg,

Cp – contact deformation of the pinion tooth surface

Cg – contact deformation of the gear tooth surface

Bending deformation Db= Bp + Bg,

Bp – bending deformation of the pinion tooth,

Bg – bending deformation of the gear tooth.

The area of the tooth surface that has to be modified depends on contact ratio. If the contact ratio less than 2 the only portion of the tooth surface can be modified. The tooth with contact ratio less then 2 has to have a not modified area in the middle of the tooth surface. On spur gears the place on the tooth where the tip modification has to start can be calculated by the following formulas:

Rt = R_low_active +Np

Rt – radius of curvature of involute in the start point of the tip modification,

R_low_active – radius of involute in the lowest point of the active profile,

Np – normal pitch.

The root modification starts on the root and ends on the calculated point of the active tooth profile. The point where the root modification ends can be calculated by the following formula:

Rr = R_high_active -Np

Rt – radius of curvature of involute in the start point of the tip modification,

R_high_active – radius of involute in the highest point of the active profile,

Np – normal pitch.

The helical gear contact ratio is determined also by overlapping of the tooth surfaces along the gear width. The ending modification root point can be located even higher than the starting point of the tip modification on the helical gear tooth. However it does not mean overlapping of modified areas of the tooth. The start point of the tip modification and the end point of the root modification on the helical gear can be calculated from the following formulas:

Rt = R_low_active +Np – B*TAN(Lead_angle)

Rt – radius of curvature of involute in the start point of the tip modification,

R_low_active – radius of involute in the lowest point of the active profile,

Np – normal pitch.

Rr = R_high_active –Np + B*TAN(Lead_angle)

Rt – radius of curvature of involute in the start point of the tip modification,

R_high_active – radius of involute in the highest point of the active profile,

Np – normal pitch.

B – gear face width

As it can be seen from the formulas helical modification provides much longer relieve. The difference is the length of an additional relieve which is:

B*TAN(Lead_angle)

This addition represents an additional distance on the base circle while the modified areas of mating teeth are in the contact. The additional distance provides additional period of time for transferring the contact from one pare of teeth to the next pare of teeth. Stretching of the transferring time allows to decrease picks of the impact stresses and prevent distortion of lubrication film.

The area of modification in ideal helical gear has to be a ruled surface as the helical involute surface of the not modified tooth area. The surface of modification can be represented by a surface built on the pass of parallel lines. The lines are parallel to each other and in the same time they parallel to the main axis of the contact ellipse. The location of the lines is defined by the pass curve which has the shape to parabolic function of the transmission error.

Predesigned parabolic shape of transmission error was proposed by the author in 1986 (USSR patent #1593354). Initially it was patented an applied for modification of spur gear teeth. Then the idea was applied to helical gears. Predesigned parabolic shape of transmission error was used also for crowning of not involute point contact gears such as Novikov gears. Unfortunately most of research work was frozen during financial crises in 1994.

The parabolic shape of transmission error is the most desirable shape because it provides the lowest pick of dynamic load when the contact goes from one pare of teeth to the next pare of teeth. The reason why the parabola is the best can be strictly explained mathematically with derivatives and integrals. Here is an engineering explanation that might be interesting not only for University professors. Transmission error means that the speed of gear rotation is not constant. It is well-known that it has to be acceleration if the speed is not constant. The acceleration always creates a dynamic force. A good example is a car. When it takes off the dynamic force pulls the driver back and it pushes the driver forward when the car stops. The same forces exist in a gear mesh. The gear speed increases when it goes from point 1 to point 2 and it decreases when it goes from point 2 to point 3. The question is what is the best way to get from point 1 to point 3. How would you drive a car to get to the next traffic light and don’t spill your coffee? You will probably constantly accelerate the half of the distance and then constantly brake another half of the way. The constant acceleration means the constant first derivative of the speed or the constant second derivative of the distance. The constant second derivative from the distance means that the distance is changing by the law of parabola. It has to be taken in consideration that it was a life explanation of the parabolic concept. The transmission error is different under different loads and different profile modifications have to be used.

The shape of transmission error curve for a spur gear set depends on the tooth profile. Spur gear tooth profile has to have modification that provides parabolic shape of transmission error. It is a well established practice to measure spur gear tooth profile by taking deviations from the theoretical involute profile. The parabolic curve of the tooth profile deviation provides parabolic curve of transmission error. The picture shows root and tip modification on one tooth. Each modification area has two parabolas. One parabola is concave and another parabola is convex. The concave shape of the graph does not mean a concave shape of the tooth profile. It is a deviation from the theoretical involute and it is concave on the graph because the large scale.

The shape of transmission error of a helical gear depends on the tooth profile and it depends on the lead error as well. As it was explained above helical tooth modification has to be build parallel to the contact ellipse. The height and the shape of helical tooth modification have to be measured on the sides of the tooth. In the case of measuring on CMM the all helical tooth surface has to be digitized.

Conclusion.

(1) Modification of helical tooth differs from modification of spur tooth. Applying spur modification to a helical tooth increases surface contact stress. Helical modification decreases dynamic stresses because it has higher length of modified area.

(2) The calculation procedure for sizes of spur and helical modification has bean developed.

(3) The concave-convex parabolic curve of tooth surface modification provides lowest dynamic stress.