interference and undercutting in involute gears

INTERFERENCE AND UNDERCUTTING IN INVOLUTE GEARS

1. Phenomenon of Interference

• As we discussed in Section 4.8, the constant angular velocity ratio for a pair of gears can be maintained only when the mating tooth profiles are conjugate (or involute) with respect to one another.

• When surfaces of mating gear teeth contact over a non-conjugate portion, interference takes place. In other words, the mating of two non-conjugate (on non-involute) tooth profiles is known as interference.

• The interference is undesirable. Because if any portion of the tooth profile in contact is not involute, the surfaces in contact will not touch each other tangentially causing the improper transmission of motion and power. In such a case, the relative motion between tooth surfaces will not have proper sliding motion and the transmission of motion will be non-uniform motion.

• The phenomenon when the tip of the tooth of gear will dig out or interfere with the flank portion of the tooth portion of the mating gear is known as interference.

2. Concept of Undercutting

• When the tip of the gear tooth undercuts the root (flank) of the mating gear tooth, some portion of the flank will be removed. This process of removal of material due to interference phenomenon is called undercutting.

• The undercutting in meshing gears is illustrated in Fig.4.25.

• The undercutting results in weaker gear teeth. In practice, the teeth are so designed to avoid interference and hence the undercutting.

Discussion:

Consider a pinion driving the wheel as shown in Fig.4.26. When the pinion rotates in clockwise direction, the contact between a pair of involute teeth begins at K and ends at L. So KL is the path of contact between two mating teeth. MN is the common tangent to the base circles.

Now, if the radius of the addendum circle of the pinion is increased to O1N, the point of contact L will shift from L to N. Any further increase in the value of this radius, will result in shifting the point of contact inside the base circle of the wheel. Since an involute exist only outside the base circle, therefore, any profile of teeth inside the base circle will be of non-involute type. In such a case, the tip of tooth on the pinion will then undercut the tooth on the wheel at the root and remove part of the involute profile of tooth on the wheel. This effect is known as interference. In other words, the phenomenon when the tip of tooth undercuts the root on its mating gear is known as interference.

Similarly, if the addendum radius of the wheel increases beyond O̟¿M, then the tip of tooth on wheel will cause interference with the tooth on pinion. The points M and N are called interference points. The conclusion is that to avoid interference, the path of contact should not extend beyond the interference points M and N.

3. Condition to Avoid Interference

• From the above discussion, the condition to avoid interference is that the length of path of contact (KL) should always be less than or equal to the maximum length of path of contact (MN).

• Mathematically, the condition to avoid interference is given by

Length of path of contact ≤ Maximum length of path of contact

where KL = Length of path of contact = KP + PL, and

MN = Maximum length of path of contact = MP + PN

From the geometry of Fig.4.26, we get

Maximum length of path of approach, MP = r sin ϕ

maximum length of path of recess, PN = R sin ϕ

Note 

In addition to equation (4.17), the following conditions can also be used to check whether interference occurs or not:

(i) Length of path of approach ≤ Maximum length of path of approach

(ii) Length of path of recess ≤ Maximum length of path of recess

4. Methods to Avoid Interference

In order to avoid or reduce interference, the following methods may be employed:

1. By modifying addendum of gear teeth. That is, to avoid interference, the interfering portion of the face of the gear tooth can be cut-off; the resulting tooth becomes stub tooth instead of full depth tooth.

2. By increasing the pressure angle.

3. By modifying tooth profile or profile shifting. That is, the non-involute portion of the tooth profile can be made cycloidal; so the profile of teeth is partly involute and partly cycloidal.

4. By increasing the centre distance. That is, the centre distance between mating involute gears can be increased within limits, without affecting the correctness of gearing. This method will increase the pressure angle and will prevent the tip of gear tooth mating with non-involute flank portion of the pinion.

5. By increasing the number of teeth on the mating pinion.

6. By undercutting the radial flank of the pinion. This method is undesirable as it weakens the tooth.

Example 4.9 

Two mating gears have 20 and 40 involute teeth of module 10 mm and 20° pressure angle. The addendum on each wheel is to be made of such a length that the ine of contact on each side of the pitch point has half the maximum possible length. Determine:

(i) the addendum height for each gear wheel,

(ii) the length of path of contact,

(iii) the length of arc of contact, and

(iv) the contact ratio.

Given data: 

TP = 20; TG = 40; m = 10 mm; ϕ = 20°

Solution: Pitch circle radii of pinion and gear are given by

(i) Addendum height for both the wheels:

Given that the addendum on each wheel is to be made of such a length that the line of contact on each side of the pitch point (i.e., the path of approach and the path of recess) has half the maximum possible length.

Therefore the given conditions can be written mathematically as

Using condition (i), we can write

Substituting the value of R and r,  we get

Squaring on both sides, we get

⸫ Addendum height (or simply addendum) of gear wheel,

Similarly, using condition (ii), we get

Substituting the value of R and r, we get

Squaring on both sides, we get

⸫ Addendum height of pinion is given by

(ii) Length of path of contact:

(iii) Length of arc contact:

(iv) Contact ratio: