This article is about noise excitation and vibration generation in gear drives when the gear drives operate. It is shown that transmission error is not the root cause for the noise excitation as it is commonly meant. Instead, transmission error is a consequence of deviation of base pitches of a gear and a mating pinion from the operating base pitch in the gear pair. Therefore, to reduce noise excitation in gear drives, the deviation of the actual values of base pitches of a gear and a mating pinion from the operating base pitch in the gear pair must be minimized. This gives an opportunity to get the gear noise problem properly engineered, and to eliminate the necessity of lapping process in production of precision bevel and hypoid gears. This discussion is focused mainly on parallel-axes gearings, but it is also valid for gear pairs with intersected-axes of rotation, as well as with crossed-axes of rotation of the gears.

### Introduction

An excessive noise excitation and vibration generation is a challenging problem for gear drives of modern designs, especially for those operating at high rotations. Parallel-axes gearings (or just P_{a}— gearings, for simplicity), intersected-axes gearings (or just I_{a}— gearings, for simplicity), and crossed-axes gearings (or just C_{a}— gearings, for simplicity) are subject to noise excitation and vibration generation. The noise problem in gears made out of steel and alloys, plastic gears, and so forth is under consideration for a long time. The results of the research can be found out in a plenty of papers published in the top-notch scientific journals, and available in the public domain. There is no reason to list all the papers on gear noise excitation as there are hundreds and hundreds of them. To the best possible extent, the problem under consideration is summarized in the well-known book by Smith [1]. Unfortunately, the output of the research carried out on the subject is more than doubtful. None of the gear experts can even roughly predict when the problem of the gear noise excitation will be solved. Poor understanding of the root cause for gear noise excitation and vibration generation is the chief reason for that.

Commonly, transmission error is considered as the predominant reason for the noise excitation in gear drives. This is not correct, as the transmission error is a consequence of the deviation of the base pitches of gears in the gear drive, and not the cause of the excessive noise excitation in a gear drive. Moreover, transmission error can be expressed in terms of the base pitch deviations. Therefore, the deviation of base pitches of mating gears from the operating base pitch in the gear drive is the root cause for an excessive noise excitation in gear drives, and not the transmission error. Once the importance of the deviation of the base pitches of a gear and a mating pinion from operating base pitch of the gear pair is understood, then it will become clear how the gear noise problem can be solved for gears of all kinds, that is, in case of P_{a}—, I_{a}—, and for C_{a}— gearings.

### Basics

To better understand the root cause of gear noise excitation and vibration generation in gear drives, it is helpful to recall some basics of the gear kinematics and geometry.

**Gear (transverse) base pitch** is defined as a distance, p_{b}, between tooth flanks of two adjacent teeth of the basic rack, R , measured in a transverse section perpendicular to the tooth flanks (Figure 1a). When a gear teeth are generated by means of the basic rack, R , the gear base pitch, p_{b.g}, is measured between tooth flanks of two adjacent teeth of the gear, G ^{i} and G ^{i+1}, measured in a transverse section perpendicular to the tooth flanks (Figure 1b).]

**Pinion (transverse) base pitch** is defined identically to that in a gear. Note: A gear, p_{b.g}, and a mating pinion, p_{b.p}, base pitches can be directly measured in the gear and in the pinion. These two fundamental design parameters can be also calculated.

**Operating base pitch** in a gear pair is defined as a distance, p_{b.op}, between two neighboring desirable lines of contact, LC^{i}_{des} and LC^{i+}^{1}_{des }, of the tooth flanks of the gear, G, and of the mating pinion, P [2]. The operating base pitch in a gear pair is measured along a line of intersection of the plane of action, PA, by a transverse plane as shown in Figure 2.

In intersected-axes gearing, the angular operating base pitch, ϕ_{b.op}_{ }, is measured within the plane of action, PA, as the angular distance between two adjacent desirable lines of contact, LC^{i}_{des} and LC^{i+}^{1}_{des }**,** and is illustrated in Figure 3a [2]. In crossed-axes gearing, the angular operating base pitch, ϕ_{b.op}, is measured within the plane of action, PA, as the angular distance between two adjacent desirable lines of contact, LC^{i}_{des} and LC^{i+}^{1}_{des }**,** and is illustrated in Figure 3b [2].

Note: Operating base pitch, p_{b.op}, in a gear pair is determined before the tooth flanks, G and P, are constructed. This means that the tooth flanks, G and P, are not constructed yet, but the operating base pitch, p_{b.op}, can be already calculated). This fundamental design parameter can be only calculated, and it cannot be directly measured in a gear pair [2].

**Transmission error: **in a gear pair is defined as the difference between the angular position that the output shaft of a drive would occupy if the drive were perfect and the actual position of the output.

### Noise Excitation and Vibration Generation

In perfect (noiseless) gear pairs:

The actual value of the base pitch of a gear^{1}, ϕ_{b.g}, equals to operating base pitch, ϕ_{b.op}, of the gear pair, that is, an equality ϕ_{b.g }= ϕ_{b.op} is valid;

The actual value of the base pitch of a mating pinion, ϕ_{b.p}, also equals to operating base pitch, ϕ_{b.op}, of the gear pair, that is, an equality ϕ_{b.p }= ϕ_{b.op }is valid;

Ultimately, in perfect (noiseless) gear pairs, a following set of two equalities is observed.

Both fulfillment and violation of the condition of Equation 1 is explained as such: For simplicity but without loss of generality, transmission error in a gear pair may be illustrated by means of a wheel that rolls over a surface, as shown in Figure 4. When a perfect wheel rolls over a perfect flat surface (Figure 4a) with no slippage, the relative motion of the wheel may be viewed as a superposition of a rotation, ω, and a translation, V. This case corresponds to meshing of two perfect gears. No sources for noise excitation are observed under such a scenario, as this is a trivial case of rolling of a circle over a plane.

In another scenario, a perfect wheel rolls over a rough surface as schematically shown in Figure 4b. The relative motion of the wheel may be viewed as a superposition of a rotation, ω, and a translation, V, and a certain additional motion. This case corresponds to meshing of an imperfect gear with a perfect pinion. As the gear is imperfect, an additional motion, V_{noise}, that causes the noise excitation, becomes inevitable. The transmission error is designated here as Δϕ_{g} because it is caused by the imperfect gear. It is evident that the gear (and not the pinion) needs to be improved in order to reduce/eliminate the transmission error, and to reduce noise excitation in the gear pair.

In another scenario, a rough wheel rolls over a perfect plane surface as schematically shown in Figure 4c. The relative motion of the wheel may be viewed as a superposition of a rotation, ω, and a translation, V, and a certain additional motion. This case corresponds to meshing of a perfect gear with an imperfect pinion. As the pinion is imperfect, an additional motion, V_{noise}, that causes the noise excitation, becomes inevitable. The transmission error is designated here as Δϕ, because it is caused by the imperfect pinion. It is evident that the pinion (and not the gear) needs to be improved in order to reduce/eliminate the transmission error, and to reduce noise excitation in the gear pair.

Ultimately, a rough wheel rolls over a rough surface as schematically shown in Figure 4d. The relative motion of the wheel may be viewed as a superposition of a rotation, ω, and a translation, V, and a certain additional motion. This case corresponds to meshing of an imperfect gear with an imperfect pinion. As both, the gear and the pinion, are imperfect, an additional motion, V_{noise}, that causes the noise excitation, becomes inevitable. The transmission error is designated here as Δϕ, because it is caused by the imperfect gear and an imperfect pinion together. The transmission error, Δϕ, in this case is a function of the components Δϕ_{g} and Δϕ_{p}, that is, Δϕ = Δϕ(Δϕ_{g},Δϕ_{p}). In order to reduce/eliminate the noise excitation when the gear pair operates, it is necessary to know how much each of the components, Δϕ_{g} and Δϕ_{p}, contribute to the transmission error, Δϕ. Unfortunately, the actual values of the components, Δϕ_{g} and Δϕ_{p}, commonly are not known. Therefore, the problem of reduction of gear noise in gear pairs of this kind turns to an indefinite one. Thus, the measured values of the transmission error cannot be used for the improvement neither of the gear, nor of the pinion.

Fortunately, the actual values of the components, Δϕ_{g} and Δϕ_{p}, as well as of the transmission error, Δϕ, can be expressed in terms of variation of base pitches of both, of a gear and a mating pinion.

### Variation of Base Pitches in a Gear Pair

A steady rotation of the driving pinion, cannot be transmitted by the gear pair smoothly to the driven gear if the total deviation of the base pitch, Δp_{b.}_{Σ}, is not of a zero value (Δp_{b.}_{Σ} ≠ 0). The base pitch error, Δp_{b.}_{Σ}, causes a transmission error, Δϕ. The transmission error, Δϕ, can be expressed in terms of the total deviation of the base pitch, Δp_{b.}_{Σ}.

When a base pitch error is observed, theoretically, the tooth flanks, G and P, are separated from one another. In reality, they contact each another, as both, the driving pinion and the driven gear, are loaded by an operating torque. As the gears are loaded, there is no gap between the gear line of contact, lc_{g}, and the pinion line of contact, lc_{p}. The gear line of contact, lc_{g}, travels toward the pinion line of contact, lc_{p}. The distance covered by the pinion line of contact, lc_{p}, to get in contact with the gear line of contact, lc_{g}, depends on the actual values of the deviations, Δ_{pb.g} and Δ_{pb.p}. This distance, in nature, is the operating base pitch error. In a simple case, the total deviation of the base pitches, Δp_{b.}_{Σ}, is equal to the sum of the instant deviations^{2}, Δ_{pb.g} and Δ_{pb.p}, and can be calculated as:

In a more general case, the total deviation of the base pitches, Δp_{b.}_{Σ}, is calculated as:

Here, in Equation 3, the multipliers, a(ϕ_{p}) and b(ϕ_{p}), are functions of the angular configuration of the input shaft, that is, of the angular configuration of the pinion. The instant value of the total the total deviation of the base pitch, Δp_{b.}_{Σ}, is within the interval:

Once the deviations of the base pitch are measured in both, in the gear, and in the mating pinion, then each of the components of the total deviation of the base pitch, Δp_{b.}_{Σ}, can be corrected so as to keep the base pitches, p_{b.g} and p_{b.p}, equal to the operating base pitch, p_{b.op}. This is important as it makes clear how much deviations in the tooth flank geometry in a gear and a mating pinion contribute to the total the total deviation in the base pitch. Moreover, it becomes clear of the components, either the gear, or the pinion, needs in more improvement to reduce the operating base pitch variation.

Refer to Figure 5 to express an instant value of the transmission error, Δϕ, in terms of the total deviation of the base pitch, Δp_{b.}_{Σ}.

In a case of perfect parallel-axes gear pair, contact point, K, occupies certain position within the line of action, LA. For a specified value of the angle of rotation of the gear, ϕ_{g}, the distance, PK, traveled by the contact point together with the plane of action,PA, can be calculated from the ΔPKO_{g}:

where:

r_{g }— is the radius of the pitch circle of the gear

*φ*_{t} — is the transverse pressure angle in the gear pair

To derive the Equation 5, sine rule was applied to the ΔPKO_{g}.

In reality, because of the base pitch error is not equal to zero, the gears make contact not at point, K, but they contact one another at point, K*, instead. The distance between the points, K and K*, equals to the total deviation of the base pitch, Δp_{b.}_{Σ}, that is, an equality KK* = Δp_{b.}_{Σ} is valid. The deviation, Δp_{b.}_{Σ}, is calculated from Equation 3.

An additional rotation through an angle, Δϕ, is caused by the deviation of Δp_{b.}_{Σ}. In nature, this additional rotation is equal to the instant value of the transmission error. In order to determine the angle, Δϕ, it is necessary to determine the distance, K*O_{g}. This distance can be determined from the ΔPK*O_{g}:

To derive the Equation 6, cosine rule was applied to the ΔPK*O_{g}.

Then, the rule of sine is applied to the ΔPK*O_{g}:

Ultimately, the angle, Δϕ, equals to:

For the determination of the distance, KO_{g}, another approach can be used. It can be written from the ΔAPO_{g} that:

and then from the ΔAKO_{g}:]

Equations 9 through 11 make possible an expression for the calculation of the distance, KO_{g}:

The determined distance, KO_{g} (see Equation 12), is used for the derivation of Equation 8. The approach, applied for the derivation of Equation 8, can also be used for the calculation of the components, Δϕ_{g} and Δϕ_{p}, of the transmission error, Δϕ. The components, Δϕ_{g} and Δϕ_{p}, are contributed by the gear and the pinion, correspondingly.

Another approach can be used for the derivation of Equation 8. The transmission error, Δϕ, can be expressed in terms of the total deviation of the base pitch, Δp_{b.}_{Σ}, and of base diameters of a gear and a mating pinion in a gear pair.

As the driving pinion rotates steadily, and the distance, KK*, is specified, then this distance is equivalent to an additional rotation of the driving pinion through an angle, Δϕ_{p}:

Taking into account that Δϕ_{g} = Δϕ_{p} / u (here u is the gear ratio in the gear pair), Equation 13 immediately returns an expression for the angular deviation, Δϕ:

In this case there is no need to determine the deviation, Δϕ, using the rule of sine for this purpose. The rotation of the gear through an angle, Δϕ, causes an additional rotation of the gear. This additional rotation can be represented by the rotation vector, ω_{bpv}(ϕ_{g}). The rotation vector, ω_{bpv}(ϕ_{g}), is referred to as the “base pitch variation rotation, ω_{bpv}(ϕ_{g})”. The resultant rotation vector of the gear:

Calculations similar to that performed above can be also performed for gears that operate on intersected axes of rotations (bevel gears), as well as on crossing axes of rotations (hypoid gears, worm gears, and so forth).

The transmission error is an integral parameter of accuracy of the gear pair. When only transmission error is specified, it is not clear what has to be done to keep the base pitches equal, that is, whether a gear, or a mating pinion have to be corrected, and to what extent.

### Conclusions

**1. **To transmit a rotation smoothly from a driving shaft to a driven shaft by means of a gear pair, base pitch of the gear, ϕ_{b.g}, and that of the pinion, ϕ_{b.p}, must be equal to the operating base pitch, ϕ_{b.op}, in the gear pair, that is, two equalities: ϕ_{b.g}= ϕ_{b.op} and ϕ_{b.p}= ϕ_{b.op} must be fulfilled at every instant of time.

**2. **A deviation of base pitch of a gear and that of a mating pinion from an operating base pitch of the gear pair, and not the transmission error, is the root cause for the excessive noise excitation and vibration generation in gear drives of all kinds: in parallel-axes, intersected-axes, as well as in crossed-axes gear drives.

**3. **Transmission error is a consequence of the base pitch variations, and it can be expressed in terms of deviations of a base pitch of a gear and that of a mating pinion from an operating base pitch of the gear pair.

**4. **The actual values of base pitches of a gear and a mating pinion can be directly measured, and then the results of the measurements can be compared with calculated values of the operating base pitches in the gear pair. These measurements can be used to improve the quality of both, of the gear and the pinion, as well as of the gear drive. No measurements of the transmission error can be helpful in this regard.

**5. **As the deviations of the base pitches are commonly not known, appropriate tolerances for the accuracy of the base pitches of a gear and a mating pinion can be set. Once, the deviations of the base pitches of a gear and a mating pinion from the operating base pitch of a gear pair are kept within the corresponding tolerance bands, then the noise excitation in the gear drive will not exceed the permissible level.

**6. **The disclosed approach gives an opportunity to get the gear noise problem properly engineered, and kept under control. It also gives an opportunity to eliminate the necessity of lapping process in production of bevel and hypoid gears.

### Endnotes

**1.** In order to not to be confused with the applied designations, it is instructive to note here that, generally (that is, in cases of I_{a} —, and C_{a} — gearings, as well as in perfect gearings with the axes misalignment), the base pitch in a gear pair is an angular parameter, and, thus, it is designated as ϕ_{b}. Only in a reduced case of perfect P_{a} — gearings the base pitch is a linear parameter.

**2. **The instant values of the deviations, Δp_{b.g} and Δp_{b.p}, commonly are not known. However, the calculations can be performed for the tolerances, [Δp_{b.g}]and [Δp_{b.p}], for the corresponding deviations.

### References

- Smith, J.D., Gear Noise and Vibration, 2nd ed., Marcel Dekker Inc., New York, 2003, 318p.
- Radzevich, S.P., Theory of Gearing: Kinematics, Geometry, and Synthesis, 2nd edition, revised and expanded, CRC Press, Boca Raton, Florida, 2018, 898 pages.

^{Printed with permission of the copyright holder, the American Gear Manufacturers Association, 1001 N. Fairfax Street, Suite 500, Alexandria, Virginia 22314. Statements presented in this paper are those of the authors and may not represent the position or opinion of the American Gear Manufacturers Association. (AGMA) This paper was presented October 2017 at the AGMA Fall Technical Meeting in Columbus, Ohio. 17FTM17}