Generally, machine efficiency has been an important issue in energy saving, and thus gear efficiency has been of interest to mechanical engineers. Since gear efficiency is better than the efficiency of an internal combustion engine, electric motor and pump, no special interest in analyzing gear efficiency had been taken in mechanical engineering. At present, from the point of view of energy saving, all losses of machine are going to be surveyed. On this technological tendency, gear efficiency becomes to be a problematic issue. In designing gears, engineers must often use a gear efficiency equation that includes the friction coefficient. However, the basic behavior of this friction coefficient has not yet been characterized. Therefore, it is reasonable that the interest of study in gear efficiency is focused on tribological phenomena.
Fifty years ago, Merritt and Buckingham carried out theoretical analyses of gear efficiency. They presented gear efficiency equations of the same calculation form, although the processes they used to obtain their results differed. Merritt introduced his equation in “Gears”⁽¹⁾ under an assumption regarding the input power in meshing gear, whereas Buckingham presented his equation without Merritt's assumption in his famous works “Analytical Mechanics of Gears”⁽²⁾ which was published later than Merritt’s above- mentioned work. In many works^(3),(4) published since 1962, Buckingham’s efficiency equation has been used without Merritt’s assumption and has become well known among gear designers. As a result, the misunderstanding that Buckingham's equation is a correct analytical result is wide-spread. Through theoretical analysis of gear dynamics we will clarify the differences between Merritt's and Buckingham's equations and describe a problem which arises when using Merritt's description in which the tangential force acting at the pitch point force is considered.
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The circular velocity of the driving or the driven gear at meshing point M on the tooth surface is obtained as follows, where point A or B is the insatantaneous revolution center.

_.. (1)

The velocity of the driving tooth relative to that of the driven tooth is calculated as
(2)

The moving velocity of the meshing point M along the line of action is
. (3)
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2.2 Loss of gear and Buckingham's equation
Energy loss dL due to friction, within a small time interval dt, is calculated as

　　　　　　　 (4)

where.

Fig.2 Tooth Surface Velocity and Force at Meching Point

Therefore total energy loss L over the entire meshing region (a-f in Fig. 2) is obtained by integration of Eq. (4) along the line of action for the approach region and recess region as follows:

(5)

Buckingham calculated the input energy W_in as the product of normal force F_n at the contact point and displacement from point f to point a along the line of action. Therefore, the rate of loss can be explained as

(6)

and gear efficiency is

(7)

This is Buckingham's result.
Niemann and Winter^(5),(6) derived the following the rate of loss for the case of contact ratio larger than 1:

(8)

where e is the contact ratio.
The manner of introducing Eq.(8) from the meshing condition is the same as that of Buckingham above.
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3.1 Direction of friction force
In the approach region, the surface velocity of the driving tooth v₁ is smaller than that of the driven tooth v₂. The existence of relative velocity between the two teeth induces the driving tooth to be pulled by friction. The direction of the friction force coincides with the pulled direction. Thus, the vector sum of the reaction force on the driving tooth at the contact point in the approach region has a friction angle r that is inclined to the line of action. Moreover, action line A'B' intersects centerline O₁O₂ at p_e' ( Fig. 3(a) ). Point p_e' is located inside the driven pitch circle.

Fig.3 Action Line of vector Sum of Force

In contrast, in the recess region, the driven tooth is pulled, since the surface velocity of the driven tooth is smaller than that of the driving tooth. Therefore the action line A"B" of the vector sum of the reaction force on the contact point M" in this case passes through point p_e'' which is also located inside the driven pitch circle ( Fig. 3(b) ).
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3.2 Relationship between moment and efficiency

As mentioned above, under frictional conditions, the position of point p_e, at which the action line of force intersects centerline O₁O₂, differs from the position of pitch point p. Considering the distance between the base circle center and p_e, Merritt presented the relationship between driving moment M₁ and driven moment M₂ as

(9)
Since the transmission ratio i of gears is given by the ratio of the two distances between the pitch point and the gear center, efficiency is obtained as

(10)

Representing the fixed distance between pitch point p and contact point M' or M" along the line of action as x, we find that the distance between p and p_e' along the centerline in the approach region ( Fig. 4(a) ) is larger than the distance between p and p"_e in the recess region ( Fig. 4(b) ). The difference between the lengths pp_e' and pp"_e can be confirmed by considering that the two lengths pm of DpmM' and pn of DpnM'' are the same because the two triangles are in congruence. Thus, efficiency ht at contact point M" located at a distance x from pitch point p in the recess region is greater than that at M' of the approach region, but the difference between these two efficiencies is very small.

Fig.4 Distance between p and p"_e

After studying forces under the above frictional condition, Merritt noted that
“ignoring this difference, which is quite negligible, the mean normal tooth reaction for a tangential load F will be F seca”, where F represents the tangential force of the pitch circle. He also discussed why the normal force is F seca.
For small displacement du on the arc of the pitch circles, the angular displacement is du/r₁ for the driving gear and -du/r₂ for driven gear. The amount of sliding between the tooth surfaces at a distance x from the pitch point is

. (11)

Instantaneous loss dL of the meshing gear induced by friction is obtained as

, (12)

where m=m1=m2
Input energy is

. (13)

Here, representing the relation between displacement dx along the line of action and du on the pitch circle as du=dxcosa, we can obtain the efficiency equation by the method described in section 2.2:

(14)

This equation is Merritt’s result obtained under his assumption that the normal force is Fseca and is equivalent to Buckingham's equation (7) because r_b1=r₁cosa and m=m1=m2
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At first, in the approach region, the direction of friction force F_f acting on the driving tooth surface is downward, as shown in Fig. 5. Normal force F_n, which generates friction force F_f at the contact point, acts on the line of action. Thus, the total force F₁ at contact point M on the tooth surface is the vector sum of these two forces. The force F₁ and component of force of moment M₁ must be in equilibrium.
The crossing angle between the total force F₁ and normal force F_n is equal to friction angle r(=arctan m ). It is shown that the direction of input force at contact point M is inclined with respect to the line of action by friction angle r. Therefore the arms of driving moment M₁ and driven moment M₂ are given as

(15)

Representing the force balance between acting force F₁ and reaction F’₂ at the contact point as (M1/r_m1=M2/r_m2) ,we obtain the moment ratio as

(16)

where.

Equation (16) shows that the moment ratio varies with the position of the contact point. This is the basis of Merritt's analysis. Therefore, if the driving moment acts uniformly, the driven moment varies with the movement of the contact point.

Fig. 6 Velocity and Force in Recess Region under frictional Condition

Similarly, in the recess region (Fig. 6), the two arms of the moment and moment ratio are obtained as
(17)

(18)

where

Fig.7 Moment Ratio vs Meshing Position

Figure 7 shows the relationship of the moment ratios of the approach and recess regions to the location of the contact point on the line of action. The vertical axis, labeled M₂/(i M₁), corresponds to efficiency, as described by Merritt. Here, i=r_b2/r_b1. That is to say, Fig.7 shows the variation of instantaneous efficiency h_t at the contact point. The mean efficiency of the entire region of contact is calculated as

(19)

Expanding the logarithmic terms of Eq. (19) into power series and ignoring series higher than the third power, we can rewrite the above equation as

(20)　

Moreover, ignoring m²tana(g₁²-g₂²) of the numerator and m²tana of the denominator in Eq. (20), since these terms are sufficiently small, Buckingham's equation (7) can be obtained. It is clear from the fact that third-power and higher series were ignored in Eq. (19) and (20) that Buckingham's equation (7) is a second-order approximation.
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In section 4 we compared Buckingham's and Merritt’s equations with an analytical result for gear efficiency, and examined an approximation method in which input power is calculated as the product of displacement and force along the line of action. In this section we will study a problem arising in the treatment of tangential force at the pitch point.

5.1 Force in base circle system
The efficiency is normally considered to be the ratio of transmission power to input during the entire operation time rather than at a given instant of meshing. The efficiency is given not as an instantaneous value, but as a mean ratio of output power to input power. Therefore, the relation of efficiency h to the input power and output power is

. (21)

Considering that the transmission ratio is independent of moment, the moment ratio is given as

(22)

Fig.8 Forces of Base Circle System
Figure 8 shows the relation of Eq. (22). If the diameter of the driving base circle is constant, that of the driven base circle is h -fold less than the real diameter. Here the line of action of force A'B' does not coincide with the line of action AB and passes through a point deflected slightly forwards pitch point p. The line of action of force intersects centerline O₁O₂ at point p' located inside the pitch circle of the driven gear. In section 3.1 it was shown that the line of action of force deflects from the line of action under frictional conditions. In such a case the gear tooth meshing condition is satisfied because the line of action of force passes through the contact point. In the case of Fig. 8 there is no contact point anywhere on the line of action of force and the meshing condition is not satisfied on this line.

This seems to be an unusual condition at first glance, but considering that efficiency h is a mean value for the entire meshing region, this is not unreasonable. Because the reduced radius of the base circle is not an instantaneous one, but an average one for the operation time, the actual force does not act along the line of action of force A'B'.
Moments in this system are given as

, (23)

where M₃ is the moment acting on the two bearings located at the centers O₁ and O₂. Thus the following relation among these moments is obtained:

. (24)

This is moment balance and is valid regardless of whether friction is occurring. Also, this is reasonable in terms of the dynamics.
However by Buckingham’s method, the above moment balance cannot be obtained because the driven moment M₂ cannot be expressed as a function of h, since in Buckingham’s method input force is considered to act along the line of action passing through the pitch point.

Fig.9 Virtual Pitch Circle System

On the other hand, we can consider that there is a virtual pitch point p' located at a point different from the actual pitch point p. For the sake of transferring the pitch point, the radius of the pitch circle is shown in Fig. 9, and indicates that

. (25)

In this case, the radii of the virtual pitch circles are given as

(26)

Thus the three forces that act at O₁, O₂ and p' are the same, and the force and moment balance in this system is also valid.
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5.2 Forces in pitch circle system
　Expressing the moment ratio as a function of pitch circle radius as

(27)

we can obtain, using the previous method described in section 5.1, Fig. 10(a), where the radius of the driven pitch circle is reduced to r_p2 . In this case, since the distance between centers of two circles is fixed, the two pitch circles cannot come into contact. Although this case practically can not exist, dynamic force balance is satisfied throughout the system.

Fig.10 Forces of Pitch Circle System

There is another consideration that the two pitch circles come into contact at the pitch point, as shown in Fig. 10(b). In this case the driving force at the pitch point is F and the driven force or the reaction force should be hF, because r_p2 is not changed. Then force balance at the pitch point is not satisfied. Moreover, the two forces acting at centers O₁ and O₂ are different and the momentmust disappear from this system to satisfy moment balance. Therefore the treatment of tangential force at the pitch point is not suitable for considering force and moment balance.
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In this section an example of the misunderstanding of the behavior of a planetary gear mechanism due to treatment of tangential force on the pitch circle, even in the case of no friction, is given. Figure 11(a) shows a planetary gear mechanism that is composed of two external sun gears. In this case the driving gear is the large sun gear z₃ and the carrier H is driven. Moreover, the small sun gear z₂ is stationary. Kutzbach's diagram is shown in Fig. 11(b), which indicates that the carrier rotates faster than the large sun gear z₃. Figure 11(c) shows the relation among forces acting on the planetary gear. From these diagrams we can deduce that planetary gear z₁₁ rotates clockwise around the pitch point of z₂ as an instantaneous rotational center. This does not occur regardless of what is indicated on Kutzbach’s diagram when the carrier is loaded. In actuality, the system is locked and cannot drive the carrier from the large sun gear. To explain this phenomenon, we consider the rack and pinion system shown in Fig. 12.

Fig.11 Planetary Gear System Generating Lock

Fig.12 Simulation System of Planetary Gear Mechanism shown in Fig.11

In the system, rack 2 is fixed and corresponds to the small sun gear z₂, rack 3 is the driver element and corresponds to the large sun gear z₃. T11 and T12 are teeth of one gear body corresponding to planetary gears z₁₁ and z₁₂, at the center of which load is applied.
When rack 3 moves rightward and reaches the position shown by the broken line ( Fig. 12(a) ), according to Kutzbach’s diagram ( Fig. 11(b) ), the pinion rotates clockwise and the center of the pinion moves from c to the point c’ located forward to the contact point q'₁. Since the pinion is forced to rotate clockwise due to the movement of rack 3, the instantaneous rotational center must be the contact point of rack 2 with T12.
According to the force balance shown in Fig. 11(c), normal force Fp acts at point p1 on rack 2 in the reverse direction against the movement of the gear center. If p1 is a fixed fulcrum, the existence of force Fp is easily understandable. However, point p1 on T12 can move rightward freely. Therefore, for the case when the pinion is pushed rightward by rack 3 under load Fc at its center, it is difficult to understand how contact between rack 2 and T12 can be maintained. Rotational movement around p1 as an instantaneous rotational center is not practical in this case.
There is an other case regarding contact of T12 with rack 2. Figure 12(b) shows that the contact point of T12 with the stationary rack 2 is on the left side of rack 2. In this case rightward movement of T12 is impossible because of obstruction due to the stationary rack 2. Therefore, even if rack 3 pushes T11 under load at its center, rack 3 cannot move. This is a locking condition. Then we can say, regarding Fig. 11(a), that, when the loaded carrier is driven from the large sun gear z3, the planetary gear system should be locked. However, this phenomenon does not occur if the carrier is not loaded.
In Figs. 12(a) and (b) the directions of the horizontal components of normal forces Fp and Fpb are the same, and these directions coincide with the direction of tangential force acting at the pitch point of the planetary gear. Then there is no difference between the two cases regarding the direction of the horizontal component of force. This means that different behavior can be explained if we look at the tangential force at the pitch point. From previous study, it has become clear that Kutzbach's diagram includes an unverified case.
If the driving and driven elements are exchanged in the system shown in Fig. 11(a), locking does not occur. There are reports(7),(8) in which this phenomenon is explained as self-locking, but this is not the case; it is generated from only geometrical conditions of the planetary gear mechanism and is not generated from friction as in the case of self-locking.
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References
(1 ) Merritt H.E., Gears, Sir Isaac Pitman & Sons (1946), p.238-246.
(2) Buckingham, E., Analytical Mechanics of Gears, McGraw Hill (1949), p.395-406.
(3) Dudley, D., Gear Handbook, 1st edition, McGraw Hill, New York (1962),p.14.2-14.4.
(4) Townsend, D. P., Gear Handbook, 2nd edition, McGraw Hill, New York (1992),p.12.4-12.13.
(5) Niemann, G., Maschinenelemente, Bd.2, Springer, Berlin (1965), p53-57.
(6) Niemann, G. and Winter, H., Maschinenelemente, Bd. 2, Springer, Berlin (1983),p220.
(7) Jakobsson B., Efficiency of epicyclic gears considering the influence of the number of teeth. Transaction of Chalmers University of Technology. Gotenburg. No. 309 (1960),p.33-36.
(8) Loomann, J., Zahnradgetribe, 3 Auflage, Springer-Verlag, (1996),p.46-51.