GEARX Logo Study on Interference and Undercutting of Double Involute Gears

The smaller number of teeth can achieve a more reliable gear transmission when the center distance is fixed.But small number of teeth will produce the problem of undercutting.Undercutting is an important problem in designing and manufacturing gears with small numbers of teeth.Conditions of undercutting for gears vary with the profile and manufacturing methods When undercutting occurs,the generated gear tooth is comparatively weak,and the transmission capacity of gear set is substantially reduced.The conjugate area of the tooth profile is decreased and the tooth contact
stress is increased rapidly at the discontinuity of the tooth surface.The service life Of the gear set is thus reduced.Beside,undercutting may cause dulling wear of the cutter on its corners.For all these reasons,undercutting is undesirable and should be avoided.
Many gears used in aircraft and other transmissions have size limitations based on the minimum number of teeth that can be cut on a pinion without undercutting the teeth.If the number of teeth is made fewer than this minimum,a weaker tooth will be produced because of the undercutting.Undercutting occurs naturally and increases as the number of teeth cut

Gear designers currently have two methods available for reducing undercut.These methods can be used separately or in combination with one another.The first method is to increase the pressure angle.An increase in the pressure angle of the gear will allow a decrease in the.number of teeth without undercutting.The second gear design method is the use of long-
short addendum design.This method is accomplished by increasing the addendum of the pinion and the decreasing the addendum of the gear by an equal .,amount But the two methods will bring the tip pointing without the undercutting while the number of teeth is smaller.In other words,the tip pointing and the undercut of root of involute gear are difficult to eliminate simultaneously while the number of the teeth is smaller.
Combining the advantages of the involute gear and the double circular gear, professor Zhang advanced a new type of involute gear, which was called“the double:involute gear," The double involute gear has the advan tages:in case of special requirements or the number of the teeth less than 14,a gear with the unequal radii of the base circles of the addendum and the dedendum or unequal correction of the addendum thickness and the dedendum be used,the tip pointing and the undercut of root of the gear can be avoided at the same time.

Successful designing of a pair of gears involutesgeometrical requirements necessary to secure smooth and continuous action.Importance of correct geometryincrease as the nember of teeth in a gear is reduced.
Interference,the non-conjugate contact between the-teeth of meshing gears, takes place between the tooth tips of one gear and the tooth fillets of the other.The actual effect is the involute tip or face of the driven gear tends to dig out the non involute flank of the driver.
When a gear tooth is produced by a generating process,interference is automatically eliminated because the cutting tool removes the interfering portion of the flank.In the production of a spur or a helical gear cut by hob or rank cutter, the teeth will have aportion of the involute profile removed if the number of teeth to be cut in the blank is less than some mini
mum which is dictated by the rack-curer or hob tooth geometry.This action is the result of involute interference between the tip of the rack cutter tooth and the flank of gear tooth and is known as undercutting.When undercutting occurs.the thickness near the gear fillets will be decreased.Therefore,the load capacity of the tooth is reduced and the length of the line of action may reduce.
Mathematically,the problem of preventing undercutting is the problem of avoiding the appearance of singular points on the generated tooth shape.The general conditions of non-undercutting have been proposed by Litvin.
The tooth profile of basic profile of dong between the gear blank and rack Cutter is usually applied to determine the limit cutter.We shall consider under what circumstances a singular point appears on the working surface of gears generated by the rack cutter.

The gear tooth is not undercut when the gear fillet of root and the involute curve is in tangency.At that minimum number of teeth Zmin,the point of tangency lies on the base circle.
The double involute gear must be manufactured as helical gear,so it must be noted that the transverse profile is different from the normal profile for the rack cutter of the double involute gear.Based on gearing theory, the study about the
minimum number of teeth of the double involute helical gear will be completed in the transverse plane.
The process of generating the involute flank profile on an external helical gear tooth may be visualized and treated as the pure rolling of a rack with.the straight flank rack-curer profile on the pitch surface of the gear blank in the transverse plane.
Helical gears have two significant planes,normal and transverse.The involute features.however, are contained in the transverse plane.The cutter tip is circular-arc in the normal plane,but it is elliptical-are in the transverse plane.In the generating process,the tooth profile and tooth fillet are simultaneously produced by the rack-cutter(or hob).The elliptical arcs generate the fillet surfaces of the double involute gear, while the straight generates the working tooth surfaces
of tIle double involute gear.

In general,the involute gears of larger pressure angle and longer working than the standard gears are used in aviation and shipping,but tip pointing of tooth often appears when the number of teeth is small or for x-gear.
It can be seen from the formula(9)and as stated above,that increase the pressure angle,shorten the height coefficient of the addendum and x-gear can decrease the minimum number of teeth without undercutting,but it will also lead to the new problem of the tip pointing and insufficient of contact ratio.
Usually, the tooth thickness at the tip circle sais demanded in order to ensure the toe strength of tooth of gear.It is about 0.25 mn for soft flank gears and about 0.4mn for hard flank gears.
Comparing the involute gear with the double involute gear can notice that the tooth thickness at the tip circle of the double involute gear is weaker when tte pressure angles of the addendum involute profile and the dedendum involute profile are equal.
But it can make up for a loss to use the special featuremat a gear with the unequal radius of the base circles of the addendum and the dedendum or unequal correction of the addendum thickness and the dedendum thickness be used,the pointing and the undercut of rootof the gear can be avoided at the sametime.
It can be seen that demand ofthe tooth thickness is met for double involute gear with the minimum number of the teeth at the tip circle.

Based on the theory of gearing,undercutting of involute gears has been studied.The meshing interference and cutter interference is analyzed.A method is presented for the determination of the minimum number of teeth that can be cut in a double involute helical gear without undercutting by a rounded-tooth tip rack-cutter.The study is completed according to
the cutting engagement of the rack cutter and a gear in transverse plane.The analysis is carried out combining the tip pointing with the minimum number of teeth to avoid undercutting.
Fan Zhimin Zhong Yunqing
(Qingdao University of Science and Technology,China)

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