The demand for strength ratings for plastic gears has been consistently increasing. However, there are no international standards such as ISO, DIN, or AGMA, but only domestic level or in-house guidelines are available. This situation has been a big obstacle in the plastic-gear industry. It makes it difficult for engineers to exchange knowledge on design and production, which are often done on a global level such as in the electronic and automotive industries.

The only widely accepted strength rating method in western countries had been the German guideline VDI 2545, which was withdrawn in 1996. VDI published a new guideline VDI 2736 in 2014 as the successor to the old guideline. On the other hand, a Japanese standard, JIS B 1759, was newly published in 2013 for the calculation of bending load capacity of plastic gears. Both standards are similar in the sense that VDI 2736 is based on DIN 3990 and JIS B 1759 on ISO 6336, which is essentially equivalent to DIN 3990. However, both standards made various adaptations to consider the special characteristics of plastic gears and show differences in several ways.

The main objective of this article is to clarify the differences of VDI 2736 and JIS B 1759. The comparison will be done only for the bending load capacity since JIS B 1759 does not provide other failure modes such as pitting or wear resistance. Hopefully, the article gives an opportunity to initiate a discussion to establish global consensus on the calculation method for plastic gears and to build a well-accepted international standard in the near future.

### 1 Introduction

The applications of plastic gears are greatly expanding in modern industry as alternatives to metal gears. Plastic gears have various benefits in terms of weight, noise, vibration, lubrication, and in design and production when they are injection molded. At the same time, there are several drawbacks such as lower accuracy, lower strength, and higher sensitivity to the operation environment such as temperature and humidity. Both the benefits and the drawbacks mainly stem from their unique material properties. Thus, the strength rating considering the special material properties of the plastics is necessary for the reliable design of plastic gears. Still, no international standard is available for the calculation. Every major plastic-gear supplier has its own calculation method. This situation is a critical issue for the plastic-gear industry because it hinders the exchange of product knowledge and information. It is common for the design and the production of plastic-gear drives to be made on a global level. For instance, an automotive company may have multiple suppliers from different countries for the plastic-gear drives used in their cars. How can the engineers guarantee that all the plastic-gear drives have the same level of safety factors and life expectancies if they are designed by different calculation methods?

In western countries, the only widely accepted strength rating method for plastic gears had been the German guideline VDI 2545 [1], which was withdrawn in 1996. After almost 20 years of inactivity, a new guideline VDI 2736 (abbreviated as VDI) was published in 2014 [2] as the successor to the old guideline. On the other hand, a Japanese standard JIS B 1759 (abbreviated as JIS) was published in 2013 [3] for the calculation of bending load capacity of plastic gears. Both standards are quite similar in the sense that VDI is based on DIN 3990-3 (abbreviated as DIN) [4] and JIS on ISO 6336-3 (abbreviated as ISO) [5], which is essentially equivalent to DIN. However, both DIN and ISO apply only for metal gears, and thus several adaptations have been made in VDI and JIS to consider the special characteristics of plastic-gear geometry and material. In addition, VDI is based on method C of DIN and JIS on method B of ISO. Consequently, the two standards show differences in several ways.

We will clarify the differences of the standards in detail in the following sections:

### 2 Comparisons of VDI 2736 and JIS B 1759

#### 2.1 Comparisons of nominal bending stress calculation

The comparisons of nominal bending stress calculation by VDI and JIS are listed in Table 1.

First, VDI applies the load influence factors (K factors) while JIS does not. Moriwaki [6] explains that JIS didn’t introduce them because dynamic loads would be small and the effect of running-in could be large in plastic gears. However, the applications of plastic gears in high-speed and high-torque conditions are increasing with the development of high-performance plastics. It is questionable if we can ignore dynamic loads for those critical applications. At the very least, engineers should be able to make this decision themselves.

Another important difference is the definition of the nominal load. VDI uses nominal tangential load F_{t} applied on the reference circle while JIS assumes the nominal load F_{wt} is applied on the operating pitch circle.

JIS explains that this is because the load capacity of a gear should be determined in terms of the strength of a gear pair, not a single gear. According to this change, JIS also modified the tooth form factor Y_{F} to use the transverse pressure angle at the pitch circle *α*_{wt} instead of the normal pressure angle *α*_{n}. JIS explains that this change is first made on the pressure angle from normal to transverse after validating the formula in ISO, and then from reference to operating according to the usage of F_{wt}. However, as the operating pitch diameter is defined as d_{wt}= d(cos*α*_{t} /cos*α*_{wt}) , and F_{t}⁄cos*α*_{t} and F_{wt}⁄cos*α*_{wt} are the same as shown in Table 2. Thus, the change to use F_{wt} and *α*_{wt} makes no difference in the calculation result.

There is another difference in the tooth form factor Y_{F} and in the stress correction factor Y_{S}. In calculating both factors, VDI assumes that the load is applied at the tooth tip when calculating the geometry factors while JIS takes the load applied at the highest point of single tooth contact. The VDI’s approach follows the method C in DIN and gives a more conservative result (lower safety) to consider lower quality and high-dimensional variation of plastic gears. However, this approach is questionable as new materials with better mechanical properties have been developed in recent years and the advances in design and manufacturing technologies have shown high-quality plastic gears can be achieved. As a compromise, it is preferable to allow the engineer to choose the load application point. The tooth form factor Y_{F} for internal gears are approximated as 2 in VDI while JIS follows so-called 60° tangent method per ISO. Clearly the approach of VDI might be regarded as too simplified.

Both VDI and JIS use the helix angle factor as ISO to convert the tooth root stress of a virtual spur gear to that of the corresponding helical gear.

VDI uses the contact ratio factor Y_{ε} according to method C in DIN. The factor is used to convert the stress calculated by the tooth form factor and the stress correction factor for application of load at the tooth tip to a value approximating the condition where determinant position of load is at the outer point of single pair tooth contact. JIS is based on ISO method B, and there is no need to include the contact ratio factor.

JIS newly introduced the tooth fillet factor Y_{f} that was not included in VDI and ISO. This is to consider the change in root stress if the root fillet is not defined by the standard basic rack. The introduction of the factor might be regarded as the proper approach since injection-molded plastic gears can have various fillet shape. JIS defines Y_{f}>1 if the root fillet is not based on the standard basic rack such as radii. If the root fillet is optimized, then Y_{f}<1. However, the calculation formulas are not yet given. Only empirical formulas shown in the annex based on FEM for the cases of arc shaped fillet giving Y_{f}>1 are available. It is common practice for plastic gears to optimize root fillet shape such as elliptical curves to have a bigger radius than the fillet cut by the basic rack. It should be possible to provide the general calculation formula considering arbitrary fillet shape.

JIS applies the rim thickness factor Y_{B} by using the modified formula from ISO as shown in Table 1. Moriwaki [6] explains that the modification has been made by using the results from operating tests and FEM to consider the lower stiffness of plastic gears relative to the metal gears. Figure 1 shows a graphical comparison of the factor according to the backup ratio for external and internal gears. As the rim thickness factor for internal gear is defined as the ratio of normal module in ISO, we converted the factor as the ratio of the tooth height assuming h_{t}=2.25m_{n} of the standard basic rack. The figure shows the effect of the rim thickness is considerably smaller in both external and internal gears. VDI doesn’t apply the rim thickness factor, the same as DIN method C.

Neither VDI nor JIS apply the deep tooth factor used in ISO and DIN. This is because the deep tooth factor is only meaningful for high-precision gears with accuracy grade equal or less than 4, which is generally difficult to achieve in plastic gears.

#### 2.2 Comparisons of permissible bending stress calculation

The comparisons of the permissible bending stress calculation are listed in Table 3.

Both VDI and JIS define the calculation method for permissible bending stress *σ*_{FG} and *σ*_{FP} based on the allowable bending stress *σ*_{Flim} measured from the gear test rig. Note VDI assumes the failure probability of 10% for the assessment of the measured data for the allowable stress while JIS assumes 1%. JIS does not provide any data for allowable bending stress while VDI provides for four different materials (POM, PA 66, PET, PE).

The service life factor Y_{NT} is applied to the data for *σ*_{Flim} to obtain the allowable bending stress at the required number of load cycles in the limited life region. Neither VDI nor JIS provides a general formula for the factor Y_{NT}. Instead, VDI provides the data and respective equations of *σ*_{FlimN} directly including the number of load cycles for PA 66 and POM considering temperature given in Equations 1 and 2.

On the other hand, JIS calculates the permissible bending stress based on the allowable bending stress *σ*_{Flim} with the temperature factor Y_{θ}, the temperature rise factor Y_{Δθ}, the lubrication factor Y_{L}, and the material factor Y_{M} as shown in Table 3. JIS does not provide any data for the allowable bending stress *σ*_{Flim} and the calculation formulas for the factors except general comments on the decision criteria. Instead, it provides calculation examples for the factors based on test results for POM test gears meshing with steel gears in its Annex. For instance, the Annex shows the formula for the allowable bending stress, the service life factor, and the temperature factor as shown in Equations 3-5. It also shows specific values for the temperature rise factor, the lubrication factor, and the material factor, but more extensive work shall be made to obtain an estimation formula.

Essentially the calculation of the permissible bending stress in both standards has the same concept that the stress is represented as a function of temperature, torque, and load cycles, but JIS might be said to have more proper structure for further investigation of each operating parameters.

VDI applies the stress correction factor Y_{ST} from the reference test gears to obtain the permissible bending stress *σ*_{FG} while JIS doesn’t apply the factor. VDI sets Y_{ST}=2.0, the same as DIN and ISO.

The setup and test condition for the gear test rig is shown in Table 4. Both standards allow different types of test rigs but prefer mechanically non-closed loop type (power absorption type) test rigs. JIS defines the standard test condition more specifically. Considering the large number of plastic materials and cost for the test, it is almost impossible to include a complete set of data into the standards. However, the formal definition of the test procedure makes it inevitable to gain a reliable material database.

For the test gears, VDI shows three different types (Size 1, Size 2, Size 3) based on the work from respective sources while JIS specifies only one dimension. Table 5 shows the comparison of the dimensions of the test gears from VDI and JIS. For VDI, only the type Size 1 is shown since the normal module of it is the same as that from JIS (m_{n}=1). The biggest difference is the number of teeth. The test gear in VDI has the number of teeth of 17 while JIS specifies a relatively large number of teeth (z_{1}=50). It is difficult to assess which test gear is more suitable for the test, but at least the test gear in VDI has a benefit to reduce the testing time. Note that the test gear in VDI has positive profile shift coefficient (x_{1}=0.259), presumably to prevent undercut. The allowable stress data of plastic-gear materials is most important in strength rating. The standardization of the test gears together with the test setup will surely accelerate the process to obtain reliable data.

### 3 Conclusion

This article clarified the differences between VDI 2736 and JIS B 1759 for the bending load capacity of plastic gears. Both standards have their own merits, and it is not easy to state which standard is superior to the other. Based on the comparison in this article, however, the authors are hoping to initiate a discussion to build a global consensus on the strength-rating method for plastic gears. It cannot be emphasized enough that a well-established international standard is most important for the rapid evolution of plastic-gear technology.

### References

- VDI 2545, Zahnräder aus thermoplastischen Kunststoffen, 1981.
- VDI 2736 Blatt 2, Thermoplastische Zahnräder – Stirnradgetriebe Tragfähigkeits-berechnung, 2014.
- JIS B 1759, Estimation of tooth bending strength of cylindrical plastic gears, 2013.
- DIN 3990 Teil 3, Tragfähigkeitsberechnung von Stirnrädern – Berechnung der Zahnfußfähigkeit, 1987.
- ISO 6336-3, Calculation of load capacity of spur and helical gears – Part 3, 2007.
- Moriwaki, I., et al., New Japanese Standard JIS B 1759 on load capacity of plastic gears, Proceedings of International Gear Conference 2014, pp. 1172 – 1178, Lyon, France, 2014.

*This paper was first presented at International Conference on Gears 2019, 3rd International Conference on High Performance Plastic Gears 2019, 3rd International Conference on Gear Production 2019, Garching/Munich (VDI-Berichte 2355, 2019, VDI Verlag GmbH, Page 1267-1278).*