Gears are extensively used to transmit a rotation from a driving shaft to a driven shaft. Depending on the gear application, various configurations of the gear-shaft axes are distinguished. In the general case, gear-shaft axes cross one another; in one particular application, the gear shaft axes intersect one another; finally, in another particular application, the gear shaft axes are parallel to one another.

Gears used to transmit a rotation from a driving shaft to a driven shaft parallel to one another are commonly referred to as “parallel-axes gearing” (or just “P_{a} — axes gearing,” for simplicity). Spur and helical gearing, double-helical and herring-bone gearing, as well as a few more kinds of gearing, are perfect examples of “parallel-axes gearing.” Figure 1 illustrates a trivial application of “P_{a} — axes gearing.” Gearing of only this particular kind, that is P_{a} — axes gearing, is considered in this article.

The root cause for noise excitation in gearboxes and in gear transmissions is discussed in our earlier paper [1]. It is shown that the deviation of the angular base pitch of a gear (and that of a mating pinion) from operating base pitch of the gear pair, is the root cause for an excessive gear-noise excitation and vibration generation. It is also shown that only in geometrically-accurate gearing^{1} (or, in other words, in perfect gearing), the angular base pitch of a gear (and that of a mating pinion) both are equal to the operating base pitch of the gear pair. Does it mean that perfect gear pairs are noiseless, and do not generate noise and vibrations? No, it doesn’t. So, what is the root cause for the noise excitation in geometrically accurate gearing when the gear mesh produces no noise?

To answer this question, let’s begin with the analysis of the interaction of a gear and a mating pinion tooth flank.

### Interaction of a gear and a mating pinion tooth flank

In a parallel-axes gear pair, an input rotation and an input torque are transmitted from a driving shaft to a driven shaft by means of forces acting within the plane of action, PA.

The transmission of rotation in geometrically-accurate P_{a} — gearing is illustrated in Figure 2. Here, the input rotation, ω_{input}, and the input torque, T_{input}, along with the output rotation, ω_{output}, and the output torque, T_{output} are shown.

In Figure 2, the input rotation, *ω*_{p}, is transformed to the corresponding output rotation, *ω*_{g}. The correlation between the rotations can be expressed by the following formula:

where the gear ratio is designated as u.

Magnitude, F_{t} = |F_{t}|, of the acting tangential force, F_{t}, is equal:

where base diameter of the driving pinion is designated as d_{b.p}.

The acting tangential force, F_{t}, is evenly distributed, f_{t.pa}, along the face width, F_{pa}, of the gear pair. An equation:

can be used for the calculation of the distributed load, f_{t.pa}.

When one pair of the teeth is engaged in mesh as illustrated in Figure 3a, the applied load is evenly distributed along a single line of contact, LC, the resultant (or, the equivalent) force, F_{t}, is applied at point, c_{g}, at the middle of a single line of contact, LC. The force F_{t} is along a line of action, LA, through the point c_{g}.

Equation 2 is valid for the calculation of a distributed load in spur parallel-axes gear pairs. In a case of helical parallel-axes gearing the distributed load, f_{t}, equals:

where the base helix angle is designated as *ψ*_{b}.

Both, the normal distributed load, f_{t.n}, and the axial component, f_{t.a}, of the distributed load can be expressed in terms of the distributed load, f_{t}, (see Equation 4).

Equation 3 is valid in cases when a gear and a mating pinion tooth flanks, G and P, contact each other along a single line of contact, LC. As the total contact ratio, m_{t}, in a gear pair is always greater than one (m_{t} > 1), in reality two or even more lines of contact, LC_{i}, are observed. Let’s consider the load transmission for the cases when the plane of action, PA, is shaped in the form of a rectangle.

When two (or more) pairs of the teeth are engaged in mesh as illustrated in Figure 3b, the applied load is evenly distributed along the corresponding number of the lines of contact, LC_{i}. The distributed load, f_{t}, is shared among n*φ* pairs of teeth engaged in mesh at that same instant of time. The distributed load per a pair of teeth, f_{t.n}*φ*, equals:

The resultant force, F_{t}, is equally shared among all the lines of contact, LC_{i}. The resultant force per a pair of teeth, F_{t.n}*φ*, equals:

All the forces F_{t.n}*φ* are along a line of action, LA, through the middle of the lines of contact, LC_{i}.

The resultant (or the equivalent) force, F_{t}, is applied at point c_{g}. Point c_{g} corresponds to the middle of the lines of contact, LC_{i}, and between the first, LC_{i}, and the last, LC_{j}, lines of simultaneous contact. When two lines of contact are observed, point c_{g} is located in the middle in between the lines of contact. When three lines of contact are observed, point c_{g} is located at the center of the second line of contact, and so forth.

Similar to the case of a single line of contact, in cases of multiple lines of contact, the force F_{t} is along a line of action, LA, through point c_{g}.

When the gear rotate, *ω*_{g}, the plane of action, PA, travels straight, V_{pa} (Figure 4). The arm, R_{cg}, of the resultant (or the equivalent) force, F_{t}, with respect to the bearing support remains the same and equals:

where:

r_{b.g }— is the base radius of the gear.

a — is the distance of the gear from the bearing support (Figure 4).

Point, c_{g}, dose not migrate in the axial direction of the gear. Therefore, the force, F_{t}, creates a bending moment of a constant value with respect to the bearing support. Thus, no variation of the deflection of the shafts, of the gear housing, and so forth, are observed in the case under consideration. Spur gears create no additional source for noise excitation.

A more general case of contact between tooth flanks, G and P, of a gear and a mating pinion is observed in geometrically-accurate helical P_{a }— axes gearing.

When the gears rotate, commonly either one or two to three lines of contact, LC_{i}, are observed in geometrically-accurate helical P_{a} — axes gearing. In two different cases, it is convenient to distinguish in helical gearing, namely:

(a) when all the lines of contact, LC_{i}, are of a full length, and

(b) when one or more line(s) of contact is of a reduced length.

When the only pair of gear teeth is engaged in mesh, the entire applied load is evenly distributed along a single line of contact, LC. This case is illustrated in Figure 5a.

The distributed load, f_{t}, is calculated from the equation:

The resultant (the equivalent) force, F_{t}, is applied at point c_{g} exactly at the middle of a single line of contact, LC. The force F_{t} is along a line of action, LA, through the point c_{g}.

Point, c_{g}, where the load, F_{t}, is applied does not migrate in the axial direction when a gear rotates. Therefore, bending moment of the force, F_{t}, with respect to the bearing support, is of a constant value.

Normal force, F_{t.n}, is perpendicular to the line of contact, LC. The component F_{t.n} of the resultant (the equivalent) force, F_{t}, is applied at that same point, c_{g}, as the load F_{t} is applied. The magnitude, F_{t.n}, of the component F_{t.n} of the resultant (the equivalent) force, F_{t}, equals to:

Normal distributed load, f_{t.n}, can be calculated from the expression:

When two (or more) pairs of the teeth are engaged in mesh, the applied load is shared among a few number of lines of contact, LC_{i}. This case is illustrated in Figure 5b. The distributed load, f_{t}, is shared among n*φ* pairs of teeth engaged in mesh at that same time. The distributed load per a pair of teeth, f_{t.n}*φ*, equals:

The resultant force, F_{t}, is shared among all the lines of contact, LC_{i}. The resultant force per a line of contact, F_{t.n}*φ*, equals:

The resultant (the equivalent) force, F_{t} [as well as the resultant (the equivalent) normal force, F_{t.n}], is applied at point c_{g}. Point c_{g} corresponds to the middle of the lines of contact, LC_{i}, and between the first, LC_{i}, and the last, LC_{j}, lines of simultaneous contact. When two lines of contact are observed, point c_{g} is located in the middle in between the lines of contact. When three lines of contact are observed, point c_{g} coincides with the center of the second line of contact, and so forth.

The normal distributed load, f_{t.n}, can be calculated from the expression:

Point, c_{g}, does not migrate in the axial direction of the gear. Therefore, the force, F_{t}, creates a bending moment of a constant value with respect to the bearing support. Thus, no variation of the deflection of the shafts, of the gear housing, and so forth, is observed in the case under consideration. No additional source for noise excitation is created in the case under consideration.

Ultimately, one or more lines of contact, LC_{i}, can be not of a full length.

As illustrated in Figure 6a, the line of contact, LC_{i}, is not of a full length, while the next line of contact, LC_{i+1}, is of a full length (that is, the inequality LC_{i} < LC_{i+1} is observed). In the example under consideration, tangential load, F_{t}, is shared between the lines of contact, LC_{i} and LC_{i+1}. The load (|F_{t.1}| + |F_{t.2}|) equals (see Figure 6a):

where:

The loads F_{t.1} and F_{t.2} are applied at points 1 and 2 within the lines of contact. Points 1 and 2 correspond to points within the lines of contact, LC_{i} and LC_{i+1}, at the middle of the portion F_{pa.1} of the face width F_{pa}.

The load |F_{t.3}| equals (see Figure 6a):

The load |F_{t.3}| is applied at point 3 within the line of contact, LC_{i+1}. Point 3 corresponds to point within the line of contact, LC_{i+1}, at the middle of the portion F_{pa.2} of the face width F_{pa}.

The resultant load, F_{t}, is applied at point c_{g}. The coordinates of point c_{g} can be determined using the principle of the center of gravity, that is, point c_{g} is the “center of gravity” of the points 1, 2, and 3 having the “weights” F_{t.1}, F_{t.2}, and F_{t.3} correspondingly.

As the gears rotate, the lines of contact, LC_{i} and LC_{i+1}, travel together with the plane of action, PA. Points 1, 2, and 3, as well as point c_{g} migrate within the plane of action, PA. The migration of point c_{g} with a linear velocity, V_{cg}, in the axial direction of the gear pair results in a corresponding alteration of the arm, R_{cg}, of the resultant force, F_{t}, in relation to the bearing support of the gear shaft as schematically illustrated in Figure 7. When point c_{g} migrates between points c_{g}* and c_{g}**, the arm |R_{cg}| of the resultant force, F_{t}, changes in the range of:

A variation of magnitude, |ΔR_{cg}|, of the arm, R_{cg}, results in a variable in time torque that bends the gear shaft, and deforms the gear housing, and can cause an extensive unfavorable vibration of the gear housing, and an extensive noise excitation.

Another configuration of the lines of contact, LC_{i} and LC_{i+1}, is illustrated in Figure 6b,

The lines of contact, LC_{i} and LC_{i+1}, shown in Figure 6b , are not of full length (that is, the inequality LC_{i} ≠ LC_{i+1} is observed). In the example under consideration, the tangential load, F_{t}, is shared between the lines of contact, LC_{i} and LC_{i+1}. The load F_{t.1} equals (see Figure 6b):

Point 1 is located in the middle of the line of contact, LC_{i}. The load F_{t.1} is applied at point 1.

The load |F_{t.2}| equals (see Figure 6b):

Point 2 is located in the middle of the line of contact, LC_{i+1}. The load F_{t.2} is applied at point 2.

No load is transmitted by the portion F_{pa.3} of the face width, F_{pa}, as no lines of contact are located there. Therefore, this portion of the gear face is excluded from the analysis.

When calculating contact stress in a parallel-axes gear pair, use of the so-called “Total length of lines of contact” (or just TLLC, for simplicity) is often helpful. “Total length of lines of contact” means a summa of the length of all the lines of contact that occur at a specified instant of time^{2}. When two or more lines of contact do not overlap one another, the importance of the total length of lines of contact is vital: In such a case, an applied load is equally distributed along the TLLC.

The resultant load, F_{t}, is applied at point c_{g}. The coordinates of point c_{g} can be determined using the principle of the center of gravity, that is, point c_{g} is the center of “gravity” of points 1 and 2, having the “weights” F_{t.1}, and F_{t.2}, correspondingly.

As the gears rotate, the lines of contact, LC_{i} and LC_{i+1}, travel together with the plane of action, PA. Points 1 and 2, as well as point c_{g} migrate within the plane of action, PA. The migration of point c_{g} with a linear velocity V_{cg} in the axial direction of the gear pair results in a corresponding alteration of the arm, R_{cg}, of the resultant force, F_{t}, in relation to the bearing support of the gear shaft as schematically illustrated in Figure 6.

When point c_{g} migrates between points c_{g}* and c_{g}**, the arm |R_{cg}| of the resultant force, F_{t}, alters in the range of |R*_{cg}| ≤ |R_{cg}| ≤ R**_{cg} (Figure 7).

A variation of magnitude, |ΔR_{cg}|, of the arm, R_{cg}, results in a variable in time torque that bends the gear shaft and deforms the gear housing and can cause an extensive unfavorable vibration of the gear housing and an extensive noise excitation.

The disclosed approach for the determining of loads acting in every pair of teeth in geometrically-accurate spur and helical parallel-axes gearing can be enhanced to gear pairs with lines of contact of an arbitrary geometry, that is, to gear pairs with lines of contact in the form of arcs of a circle, of a spiral curve, and so forth. Generally speaking, for this purpose the plane-of-action face width, F_{pa}, is subdivided onto several segments within each of them, either zero, or one, or two and so forth, lines of contact occur. In a case of line of contact in a form of a planar curve, the corresponding portion of plane-of-action face width, F_{pa}, is sliced on infinite number of infinitesimally narrow slices. A portion of a line(s) of contact within each infinitesimally narrow slice is considered a straight line segment. In such a manner, loads in geometrically-accurate parallel-axes gear pairs with the lines of contact of an arbitrary geometry can be determined.

### Forces acting in transverse section of geometrically-accurate parallel-axes involute gear pair

In addition to forces that act in the plane of action, friction forces act in transverse section of the gear pair as illustrated in Figure 8.

At an instant of time when two contact points, K_{1} and K_{2}, are observed, the resultant tangential force, F_{t}, is equally shared between the points, K_{1} and K_{2}, that is, the forces F_{t.1} and F_{t.2} are equal to one another. The forces, F_{t.1} and F_{t.2}, create the input torque, T_{p}.

Friction forces, F_{f.1} and F_{f.2}, act perpendicular to the line of action, LA. As the instant relative motion of the driving gear and the driven pinion is an instant rotation about the pitch point, P, and the contact points, K_{1} and K_{2}, are located from the opposite sides of the pitch point, the friction forces, F_{f.1} and F_{f.2}, are pointed oppositely to each other. The friction forces, F_{f.1} and F_{f.2}, are equal:

Here, friction coefficient is designated as *μ*.

Variation (due to lubrication) of magnitudes of the forces, F_{f.1} and F_{f.2}, can also produce noise.

Friction forces, F_{f.1} and F_{f.2}, create a friction torque, T_{fr}, that is calculated as:

Friction torque, T_{fr}, is opposite to the input torque, T_{p}.

When the gears rotate, the distances, K_{1}P and K_{2}P, alter. However, the summa (K_{1}P + K_{2}P) remains of a constant value. Therefore, in geometrically-accurate parallel-axes involute gearing, the friction torque, T_{fr}, also is of a constant value.

### Conclusion

Noise excitation in perfect parallel-axes gearing is the subject of the article. As it is shown in our earlier published article [1], base pitch variation is the root cause for an excessive noise excitation in P_{a} — gearing. Theoretically, geometrically-accurate gearing that features zero base pitch variation promises to be noiseless, as no noise is produced by the gear mesh. In reality it can be noisy.

Variation of the gear loading along with a variation of point at which the resultant load is applied is the root cause for an excessive noise excitation in geometrically-accurate parallel-axes gearing. This results in a corresponding variation of the deformation of the gear shafts, of the gear housing, and so forth. Ultimately, perfect parallel-axes gearing can produce an excessive noise. Noise of this kind is also referred to as the “gear noise”; however, it is produced not by the gear mesh, but by the other components of the gear mechanism instead: by the shafts, by bearings, as well as by the gear housing, and so forth (all of these components vibrate due to the variation of the resultant force in gear mesh. This source of noise has to be considered as a “gear noise,” as control over the level of the noise can be achieved by the proper selection of design parameters of the gears: profile angle, contact (total) ratio, helix angle, and so forth.

Gears with a low tooth count (that is, LTC — gears, for simplicity) are affected more by the load variation.

A variation of the tangential force (a) in magnitude, and (b) point (and line) of application of the resultant force can also be caused by angular oscillation of the plane of action, PA, around the axis of instant rotation, P_{ln}.

In a multiple-stage gear drive, every shaft (and every gear pair) causes its own source of noise/vibration. All the vibrations superpose each other.

The disclosed approach can be used for intersected-axes gearing (I_{a} — gearing, for simplicity), as well as for crossed-axes gearing (C_{a} — gearing, for simplicity).

### References

- Radzevich, S.P., “The Root Cause for Gear Noise Excitation,”
*Gear Solutions*magazine, December 2018, pp. 24-29. - Radzevich, S.P., Theory of Gearing: Kinematics, Geometry, and Synthesis, 2nd Edition, revised and expanded, CRC Press, Boca Raton, FL, 2018, 934 pages.

### Footnotes

^{1} Geometrically-accurate gears are those that meet three fundamental laws of gearing [2]. Gears only of this particular kind are capable of transmitting a uniform rotation of an input shaft to a uniform rotation of an output shaft, that is, with a constant angular velocity ratio.

^{2} It is important to stress here that the gear teeth loading strongly depends on the total length of the line of contact (*TLLC*), as well as of the contact ratio in the gear pair. The influence of the total length of the line of contact, and of the contact ratio on the gear teeth loading is complex: It could happen that when the pitch helix angle of a gear, *ψ** _{g}*, goes up, the total length of the lines of contact goes down, that is, the contact ratio can be greater in this case, but the total length of the line of contact can be shorter. The latter results in a higher contact stress, and a higher bending stress in the gear teeth.