Lifetime and meshing-teeth temperature of a crossed ...

Bulletin of the JSME

Journal of Advanced Mechanical Design, Systems, and ManufacturingVol.11, No.6, 2017

Paper No.17-00385© 2017 The Japan Society of Mechanical Engineers[DOI: 10.1299/jamdsm.2017jamdsm0081]

Lifetime and meshing-teeth temperature of a crossed helical gear consisting of a plastic gear and a metal gear:

In case of no-lubrication

Mikio TAKAHASHI*, Takayoshi ITAGAKI*, Hideo TAKAHASHI*, Takao KOIDE** and Yuki KOBORI***

*Department of Mechanical Engineering, National Institute of Technology, Kisarazu College 2-11-1 Kiyomidai-higashi, Kisarazu, Chiba 292-0041 Japan

E-mail: [email protected] ** Department of Mechanical and Aerospace Engineering, Tottori University

4-101 Minami, Koyama-cho, Tottori, 680-8552, JAPAN *** Advanced course of Mechanical and Electrical Engineering, National Institute of Technology, Kisarazu College

Kiyomidai-higashi, Kisarazu, Chiba 292-0041, JAPAN

Abstract The strength of plastic greatly varies with temperature; that is, the strength increases as the temperature decreases. Therefore, the lifetime of a plastic gear can be increased by reducing the meshing-teeth temperature of the gear. The meshing-teeth temperature can be reduced by changing the material of the mating gear from plastic to metal considering that a metal gear will have high thermal conductivity. In this study, we investigated the load-carrying characteristics of an unlubricated crossed helical gear consisting of a plastic gear meshed with a metal gear. We performed gear lifetime experiments and meshing-teeth temperature survey experiments. The results showed that the failure mode of the plastic crossed helical gear was caused by the breakage of the tooth and the rim. Then, we confirmed that the stress ratio could be an index for the lifetime evaluation of the plastic crossed helical gear that could fail by tooth breakage. Additionally, by testing gear pairs with several dimensions, we confirmed that the mean flash temperature could be an index for the meshing-teeth temperature evaluation of the plastic crossed helical gear. Furthermore, we proposed a lifetime estimation method for the plastic crossed helical gear that could fail by tooth breakage and verified the validity of the proposed method based on the experimental results.

Keywords : Gear, Plastic, Crossed helical gear, Lifetime estimation, Temperature

1. Introduction

Crossed helical gears have certain advantages such as the ability to achieve precise meshing even when assembly

errors occur, simple production methods, and the possibility to set an arbitrary skew condition between shafts. However, the load-carrying capacity of crossed helical gears is considered to be lower than that of parallel-axis cylindrical gears because crossed helical gears involve point contact meshing. The load-carrying capacity of the crossed helical gears can be improved by using plastic as a material for the gears. This improvement can be achieved because the contact areas of plastic gears are larger than those of metal gears owing to the greater elastic deformability of plastic. Furthermore, plastic gears have additional advantages over metal gears, such as lower noise, weight, and cost.

In this study, we investigated the load-carrying characteristics of an unlubricated crossed helical gear consisting of a plastic gear meshed with a metal gear. The use of the combination of a plastic gear and a metal gear in a crossed helical gear can reduce the meshing-teeth temperature owing to the high thermal conductivity of the metal gear. The strength of plastic greatly varies with temperature; that is, the strength increases as the temperature decreases.

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Received: 1 August 2017; Revised: 8 September 2017; Accepted: 5 November 2017

2© 2017 The Japan Society of Mechanical Engineers[DOI: 10.1299/jamdsm.2017jamdsm0081]

Mikio Takahashi, Itagaki, Hideo Takahashi, Koide and Kobori,Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.11, No.6 (2017)

Table 1 Dimensions of test gears Material Gear notation m1β20 m1β30 m1β45 m05β45 m2β45

All

Normal module mn [mm] 1 0.5 2 Normal pressure angle αn [°] 20 Number of teeth z 37 34 28 56 14 Helix angle β [°] 20 30.3 44.7 Addendum modification coefficient

x 0

Face width b [mm] 10 Reference diameter d [mm] 39.37 39.38 39.39

POM-C Total profile deviation Fα Grade 5 Grade 7 Grade 6 Grade 6 Grade 8 Total helix deviation Fβ Grade 7 Grade 4 Grade 4 Grade 6 Grade 5

Aluminum alloy

Total profile deviation Fα Grade 5 Grade 6 Grade 6 Grade 8 Grade 7 Total helix deviation Fβ Grade 4 Grade 4 Grade 6 Grade 8 Grade 8

Steel Total profile deviation Fα Grade 5 - - - - Total helix deviation Fβ Grade 4 - - - -

Therefore, a reduction in the meshing-teeth temperature can increase the lifetime of the gear. We performed endurance tests using the crossed helical gear with an axial angle of 40°. We evaluated the lifetime of the gear and experimentally determined the effects of the mating gear material, operating conditions, and gear dimensions on the meshing-teeth temperature. Additionally, we examined a method for estimating the lifetime of the crossed helical gear.

2. Experimental procedure 2.1 Experimental equipment

Figure 1 shows a schematic representation of the experimental equipment. We used an open-type testing machine because a power rotation-type machine cannot maintain a fixed normal load owing to tooth surface wear during the experiment. The center distance and the angle between the gear shafts could be set arbitrarily. The synchronous speed of the motor was 1500 min-1 (50 Hz). A transistor inverter was used to change the frequency of the electrical power source. The frequency was changed in the range of 10–60 Hz. Therefore, the synchronous speed of the motor could be set at random within the range of 300–1800 min-1. The rotational speed of the test gear was monitored using a tachometer. To set the testing torque, a torque meter and a powder brake were employed. During the experiment, the meshing-teeth temperature was measured using an infrared thermometer.

Fig. 1 Schematic of experimental equipment used for testing plastic crossed helical gear.

2.2 Test gears

Table 1 lists the dimensions of the test gears. The plastic m1β20 gear was processed by injection molding, and the other gears were processed by hobbing. In this study, the direction of rotation for both the driving and driven gears was the same. Therefore, in the case of gears with a helix angle of 20°, the axial angle was 40°. The test gears were composed of polyacetal copolymer (POM-C), aluminum alloy, and steel.

2



The notations for the gear pairs are obtained using a combination of abbreviations for the driving gear material (P for POM-C, A for aluminum alloy, and S for steel), driven gear material (P, A, and S), module (m1, m2, … mn), and helix angle. For example, APm1β20 indicates a gear pair with an aluminum alloy driving gear, a POM-C driven gear, module 1, and a helix angle of 20°.

2.3 Experimental conditions

Two types of experiments were performed in this study: a gear lifetime experiment and a meshing-teeth temperature survey experiment. In the gear lifetime experiment, the rotational speed of the test gear was set between 500 and 1000 min-1. The peripheral velocity was approximately in the range of 1–2 m/s at the pitch circle. The applied torque was arbitrarily set between 1.25 and 2.5 Nm to determine the relationship between the applied torque and the lifetime of the gears. The test gear pair used in the gear lifetime experiment was m1β20. The gear lifetime experiment was performed until gear transmission no longer occurred, and at this point, the number of load cycles was counted as the lifetime of the gear. In the meshing-teeth temperature survey experiment, the rotational speed of the test gear was set as 500 min-1. However, for the m1β20, it was set between 500 and 1500 min-1 to confirm the influence of the rotational speed on the meshing-teeth temperature. The applied torque was arbitrarily set so that the relationships between the calculated and experimental values were easy to determine. In both experiments, the backlash was set to 0.2 mm; this value was based on the difference in the adjustment of the center distance. The atmospheric temperature (room temperature) during the experiment was 25 ± 2 °C. The meshing-teeth temperature was measured in the middle of a tooth during operation via an infrared thermometer. Additionally, there was no lubrication applied under this condition. 3. Results and discussion 3.1 Gear failure mode

Figure 2 shows an example of a test gear after failure. In this study, slight wear of the tooth surface was confirmed as a progressive failure. However, the failure mode associated with gear transmission termination was crack propagation to the rim, with the cracks originating at the tooth root. As a result of the excessive wear, the load-carrying characteristics of the gear may change. As an example, it has been reported that breakage may occur about the pitch point as a consequence of excessive wear (Chen et al., 2017). However, because the degree of wear on the plastic gear, which was implemented in combination with the metal gear, was minimal, we considered that the influence of wear on the stress applied to the tooth root was minimal.

(a) Gear (b) Teeth and rim Fig. 2 Failed plastic gear. The failure of the unlubricated crossed helical gear consisting of a plastic gear and a metal gear was caused by crack propagation from the tooth root to the rim. 3.2 Gear lifetime experiment

First, we performed the gear lifetime experiment using APm1β20 gear pairs. However, after the experiment was repeated several times, failure was confirmed to occur in not only the plastic gear but also the aluminum alloy gear (Fig. 3). The failure was observed in the form of a scratch that occurred about the tooth tip. This aluminum alloy gear

3



was repeatedly used for extended periods of time in the meshing-teeth temperature survey experiment described in Section 3.3. Note that we did not periodically observe the condition of the metal gear because we did not consider that the metal gear-plastic gear combination would fail. Therefore, it is not clear when the failure occurred to the aluminum alloy gear, and it is not possible to clearly explain the reason for the failure. Additionally, because of such gear failure, adequate results were not obtained for several gear lifetime experiments in which APm1β20 gear pairs were used. Thus, we changed the material of the metal gear to steel that was harder than the aluminum alloy; then, the subsequent experiments were performed using SPm1β20 gear pairs. In the case of the steel/plastic gear combination, failure of the steel gear was not confirmed.

Figure 4 shows the relationship between the applied torque and the lifetime of gears. In this figure, PPm1β20(N) indicates the results for the unlubricated POM-C gear pair (Takahashi et al., 2015), and PPm1β20(G) indicates the results for the POM-C gear pair lubricated with grease (Takahashi et al., 2011). The load-carrying capacities of the PPm1β20(N) gear pairs are low because wear increases drastically. In contrast, the load-carrying capacities of the PPm1β20(G) gear pairs are high because the wear and the meshing-teeth temperature are reduced owing to grease lubrication. The SPm1β20 gear pairs are not lubricated. However, their load-carrying capacities are high because the meshing-teeth temperature is reduced by the high thermal conductivity of the metal and the wear does not increase. Thus, when grease lubrication cannot be realized, the load-carrying capacity can be improved drastically by changing mating gear material from plastic to metal.

Figure 5 shows the relationship between the stress ratio, Cu (= σb/σ0), and the lifetime, NL. Here, the stress ratio is the ratio of the maximum bending strength of the plastic at the average meshing-teeth temperature during operation, σ0, to the tooth root stress, σb. A straight line seen in this figure is best-fit line obtained using the experimental results for the plastic gear pair lubricated with grease (PPm1β20(G)); the line indicates that the gear failure mode is the same in this study.

The results in this study and best-fit line show very good correlation. Therefore, the lifetime of the unlubricated crossed helical gear consisting of a plastic gear and a metal gear can be estimated by using the same equation used for estimating the lifetime of the plastic gear pair lubricated with grease. The relationship between the stress ratio and the lifetime of the gears is given by the following equation (Takahashi et al., 2011):

(1)

Here, the maximum bending strength of POM-C, σ0, can be obtained by the following equation for the average meshing-teeth temperature during operation, θm (Takahashi et al., 2009):

(2)

No simple formula is available to calculate the tooth root stress of the crossed helical gear. Therefore, in this study,

the tooth root stress was calculated using the finite element method. The following equation is obtained from equations (1) and (2). Lifetime estimation can be performed by inputting the values of the average meshing-teeth temperature, θm, and the tooth root stress, σb, into this equation (Takahashi et al., 2011).

(3)

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Fig. 3 Failed aluminum alloy gear. Even when the mating gear was made of plastic, failure of the aluminum gear was confirmed after the endurance test was repeated several times. Therefore, the subsequent experiments were performed using a combination of steel and plastic gears. In the case of this combination, failure of the steel gear was not confirmed.

Fig. 4 Relationship between applied torque and lifetime of gears. Even with no-lubrication, the load-carrying capacity of the crossed helical gear consisting of a plastic gear and a metal gear was high.

Fig. 5 Relationship between stress ratio and lifetime of gears. The lifetime of the unlubricated crossed helical gear consisting of a plastic gear and a metal gear can be estimated by considering the stress ratio as an index using the same numerical formula used for estimating the lifetime of the plastic gear pair lubricated with grease.

App

lied

torq

ue

Nm

T

Lifetime of gears NL

Rotational speed min-1

SPm1 20PPm1 20(N)PPm1 20(G)

500 750 1000n

βββ

105 106 107 1080

1

2

3

4

5


Best-fit line for PP(G)

Stre

ss ra

tioσ

σ

b

= (

/

) 0

C u

Lifetime of gears NL

n500 750 1000

SPm1 20β

106 107

1

0.5

0.6

0.7

0.8

0.9

2

5



3.3 Meshing-teeth temperature survey experiment As mentioned, the lifetime of gears that are expected to fail by tooth breakage can be estimated from the stress

ratio. Therefore, to estimate the gear lifetime, it is necessary to determine the average meshing-teeth temperature during operation. In this section, we present a method for estimating the increase in the meshing-teeth temperature. This increase can be determined by subtracting the atmospheric temperature from the meshing-teeth temperature. Three major factors are involved in the increase in the meshing-teeth temperature of the plastic gear: frictional heat, hysteresis heat (heat generated from hysteresis, which is caused by iterative deformation), and heat radiation. For crossed helical gears, the tooth surface temperature distribution is complicated because the contact area changes with the load. Therefore, it is difficult to estimate the heat radiation during operation. Further, it is not possible to calculate the hysteresis heat for crossed helical gears with an elliptical contact area because a method for such a calculation has not yet been developed. However, the frictional heat in crossed helical gears with a large sliding velocity is greater than that in other gear types such as spur gears. Furthermore, because the degree of deformation of the crossed helical gear is smaller than those of other gear types such as spur gears, the amount of hysteresis heat is also small. Therefore, we investigated whether the increase in the meshing-teeth temperature of the crossed helical gear could be estimated directly from the frictional heat.

3.3.1 Mean flash temperature

The increase in the maximum surface temperature in the elliptical contact area (Figure 6(a)) was derived in terms of the flash temperature according to Coleman’s equation (Coleman, 1967) as follows:

(4)

Here, pHmax is the maximum Hertzian pressure [Pa], µ is the coefficient of friction, Vg is the sliding velocity [m/s], λi is the thermal conductivity [W/(mK)], γi is the density [kg/m3], and ci is the specific heat capacity [J/(kgK)]. Vi and di are shown in Figure 6(a).

The coefficient of friction, µ, was calculated from the following equation, which was obtained from Takanashi and Shoji’s equation (Takanashi and Shoji, 1981). Takanashi and Shoji’s equation is intended for two parallel cylinders. Therefore, we replaced the load with the Hertzian pressure and calculated the coefficient of friction.

(5)

Here, C is the coefficient modified from the Takanashi and Shoji equation coefficient C0, which is determined according to the material type and lubricating condition (Table 2). However, the value of the friction coefficient obtained via Eq. (5) is an index of friction coefficient estimation that changes according to the operating condition. Because experiments on the coefficient of friction at crossed cylinders have not been conducted, it was not possible to obtain an accurate friction coefficient. In the future, we plan to develop and propose the friction coefficient estimation formula for two crossed cylinders. Moreover, it is particularly necessary to examine a value for the coefficient C that changes according to the combination of materials.

In the crossed helical gears, the effect of the tooth trace direction component on the sliding speed is very large. Therefore, we used the sliding velocity in the tooth trace direction (VF) as the value for the sliding velocity (Vi). The maximum Hertzian pressure and the shape of the contact ellipse (Figure 6(b)) were determined by Nemoto’s method (Nemoto et al., 1994).

The velocity of the tooth surface, Vi, and the diameter of the contact ellipse in the direction of tooth surface movement, di (Figure 6(b)), were calculated from the following equations (Takahashi et al., 2011):

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(6)

(7)

(8)

Here, V01 and V02 are the peripheral velocities of the gears [m/s], δ is the axial angle [rad], a is the radius of the major axis of the contact ellipse [m], b is the radius of the minor axis of the contact ellipse [m], and ψi is the angle between the tooth trace direction and the major axis of the contact ellipse [rad].

The mean flash temperature was calculated by averaging the flash temperatures along the action line (Takahashi et al., 2015).

Table 2 Coefficient C for different materials and lubricating conditions Materials Lubrication Coefficient C

POM–POM No-lubrication 3.19 Grease 1.27

POM–metal No-lubrication 6.83 Grease 3.80

3.3.2 Mean flash temperature and increase in meshing-teeth temperature

Figure 7 shows the effects of the gear material on the increase in the meshing-teeth temperature (Takahashi et al., 2014). In this figure, the vertical axis represents the increase in the meshing-teeth temperature obtained experimentally, and the horizontal axis represents the applied torque. At the same applied torque, the contact ellipse of the metal and plastic gear pair (APm1β45) was smaller than that of the plastic gear pair (PPm1β45). In addition, the maximum Hertzian pressure increased. Therefore, the increase in the meshing-teeth temperature of the metal and plastic gear pair could possibly be higher than that of the plastic gear pair. However, the experimental results indicated that the increase in the meshing-teeth temperature of the metal and plastic gear pair was lower than that of the plastic gear pair. This was because the thermal conductivity of the metal was much higher than that of the plastic.

Figure 8 shows the relationship between the increase in the meshing-teeth temperature and the mean flash

(a) Elliptical contact area and vectors of object movement (b) Dimensions of contact ellipse

Fig. 6 Contact ellipse of crossed helical gear.

d2

V1

V2 d1

VF

a b

ψ’i

di

Tooth trace direction

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temperature for all the gear pairs used in this study. In this figure, the vertical axis represents the experimental increase in the meshing-teeth temperature, and the horizontal axis represents the calculated mean flash temperature. The results of the SPm1β20 gear pair were the average value of increase in meshing-teeth temperature during the gear lifetime experiment. The remaining results demonstrated the increase in meshing-teeth temperature at the equilibrium in the meshing-teeth temperature survey experiment. This figure, shows that well the relationship between the increase in the meshing-teeth temperature and the mean flash temperature can be well approximated by a straight line with good correlation. This result indicates that the effects of the gear dimensions, operating conditions, and material of the mating metal gear on the increase in the meshing-teeth temperature can be evaluated from the mean flash temperature. The equation relating the increase in the meshing-teeth temperature and the mean flash temperature for the metal and plastic gear pair obtained in this study is as follows:

(9)

Fig. 7 Effect of gear material on increase in meshing-teeth temperature. Even in the case of a plastic crossed helical gear, the meshing-teeth temperature can be reduced by changing the mating gear material from plastic to metal (Takahashi et al., 2014).

Fig. 8 Relationship between increase in meshing-teeth temperature and mean flash temperature. This figure indicates that the effects of the gear dimensions, operating conditions, and material of the mating metal gear on the increase in the meshing-teeth temperature (experimental values) can be evaluated from the mean flash temperature (calculated values).

Incr

ease

in m

eshi

ng-te

eth

tem

pera

ture

Applied torque Nm

℃b

θ

Rotational speed = 500 min-1n

T

Axial angle = 90°δβPPm1 45βAPm1 45

0 0.2 0.4 0.6 0.8 1 1.2

10

20

30

40

50

60

Incr

ease

in m

eshi

ng-te

eth

tem

pera

ture

Mean flash temperature fm θ

℃b

θ

℃

SPm1 20APm1 20APm1 30APm1 45APm05 45Apm2 45


500 750 1000 1500 n

βββββ

β

0 20 40 60 80 100

20

40

60

80

100

8



4. Lifetime estimation method Figure 9 shows the flowchart of a lifetime estimation method for plastic crossed helical gears that are expected to

fail by tooth breakage. If the average meshing-teeth temperature during operation, θm, and the tooth root stress, σb, are obtained, the lifetime of the gears, NL, can be determined by equation (3). If θm is unknown, the increase in the meshing-teeth temperature, θb, can be estimated using the mean flash temperature, θfm, which is calculated from the gear dimensions and the operating conditions. By adding the atmospheric temperature, θa, to θb, it is possible to estimate θm. Therefore, the lifetime can be estimated under any operating condition and for any gear dimensions, material of the mating metal gear, and atmospheric temperature.

Figure 10 shows the comparison of the estimated lifetime curves and the experimental results. The estimated lifetime curves are calculated according to the flowchart shown in Figure 10. Most of the experimental results at an atmospheric temperature of 25 ± 2 °C are located between the estimated lifetime curves for atmospheric temperatures of 20 and 30 °C. Therefore, the experimental results and the estimated lifetime curves are in good agreement. However, this verification is a comparison between the estimated lifetime and the experimental result only at the atmospheric of 25 ± 2 ° C. In the future, it is necessary to verify the relationship between the estimated lifetime and the experimental result when the atmospheric temperature is different.

The equation relating the stress ratio, Cu, and the lifetime, NL, is applicable only to the gear pair SPm1β20 and has not yet been confirmed for other gear dimensions. Additionally, the relationship between the mean flash temperature, θfm, and the increase in the meshing-teeth temperature, θb, changes with the mating gear material (combination of a plastic gear pair or combination of a plastic gear and a metal gear), manufacturing method of gear, etc. Therefore, to obtain these equations (Eq. (1) and (9)) for other gear dimensions, it is necessary to perform additional experiments for confirmation. However, using the proposed method, the lifetime of a plastic crossed helical gear that is expected to fail by tooth breakage over a wide range of conditions can be estimated using a small number of experiments.

Fig. 9 Flowchart of lifetime estimation method for plastic crossed helical gears that are expected to fail by tooth (or rim) breakage.

Applied torque: T

Atmospheric temperature: θa

Increase in meshing-teeth temperature: θb

Meshing-teeth temperature: θm

Tooth root stress: σb Bending strength: σ0

Rotational speed: n

Dimensions of gears

Mean flash temperature: θfm

Stress ratio: Cu = (σb/σ0)

Lifetime of gears: NL

9



(a) Rotational speed n = 500 min-1

(b) Rotational speed n = 750 min-1

(c) Rotational speed n = 1000 min-1

Fig. 10 Comparison of estimated lifetime curves and experimental results. The estimated lifetime curves are calculated using the flowchart shown in Figure 9, and the color of the curves indicates the atmospheric (room) temperature. The data points are experimental results. Most of the experimental results at an atmospheric temperature of 25 ± 2 °C are located between the estimated lifetime curves for atmospheric temperatures of 20 °C (green) and 30 °C (orange).

Rotational speed Experimental resultsAtmospheric temp.

25 ± 2 °C

App

lied

torq

ue

Nm

T

Lifetime of gears N L

n500 min-1

Lifetime estimationAtmospheric temp.

0 °C 10 °C 20 °C 30 °C 40 °C

SPm1 20β

106 107 1081.75

2

2.25

2.5

2.75


25 ± 2 °C

App

lied

torq

ue

Nm

T


n750 min-1


0 °C 10 °C 20 °C 30 °C 40 °C

SPm1 20β

106 107 1081.5

1.75

2

2.25

2.5


25 ± 2 °C

App

lied

torq

ue

Nm

T


n1000 min-1


0 °C 10 °C 20 °C 30 °C 40 °C

SPm1 20β

106 107 1081

1.25

1.5

1.75

2

10



5. Conclusion A gear lifetime experiment and a meshing-teeth temperature survey experiment were performed using an

unlubricated crossed helical gear consisting of a plastic gear and a metal gear, and the experimental results were discussed. The main results can be summarized as follows: (1) The failure mode of the unlubricated crossed helical gear consisting of a plastic gear and a metal gear was caused

by crack propagation from the tooth root to the rim. (2) For the unlubricated crossed helical gear consisting of a plastic gear and a metal gear, a significant correlation was

observed between the lifetime and the stress ratio, which is the ratio of the tooth root stress to the maximum bending stress of the plastic at the meshing-teeth temperature.

(3) The mean flash temperature could be a useful parameter for evaluating the increase in the meshing-teeth temperature of the crossed helical gear consisting of a plastic gear and a metal gear.

(4) The meshing-teeth temperature of the plastic crossed helical gear decreases when the mating gear material is changed from plastic to metal. Therefore, the lifetime of the plastic crossed helical gear can be increased by using a metal mating gear instead of a plastic mating gear.

References Chen, Y., Itagaki, T., Takahashi, H., Takahashi, M. and Iizuka, H., Fatigue life of injection-molded-plastic-helical-gear

added with carbon powder made from rice hull, Proceedings of The JSME International Conference on Motion and Transmissions (MPT2017 - Kyoto) (2017), pp.512-517.

Coleman, W., A Scoring Formula for Bevel and Hypoid Gear Teeth, Transactions of the ASME, Ser. F, 89, 2 (1967), pp.114-126.

Nemoto, R., Naruse, C. and Haizuka, S., Contact Stress State between Tooth Surface of Crossed Helical Gears, Bulletin of the University of Electro-Communications, Vol.7, No.2 (1994), pp.167-173.

Takahashi, M., Takahashi, H., Taki, T. and Koide, T., A Life Estimation Method of the Plastic Helical Gear and Spur Gear Pairs, Transactions of the Japan Society of Mechanical Engineering, C, Vol.75, No.755 (2009), pp.2068-2074 (in Japanese).

Takahashi, M., Takahashi, H. and Koide, T., A Life Estimation Method of Plastic Crossed Helical Gear Pair (Case of Grease Lubrication), Journal of Japan Society for Design Engineering, Vol.46, No.5 (2011), pp.286-294 (in Japanese).

Takahashi, M., Itagaki, T., Takahashi, H. and Koide, T., Lifetime and meshing teeth temperature of plastic crossed helical gear: case of grease lubrication, Proceedings of International Gear Conference 2014 (2014), pp.148-157.

Takahashi, M., Takahashi, H. Koide, T. and Itagaki, T., A Lifetime Estimation Method of Plastic Crossed Helical Gear Pair (Case of No-Lubrication), Journal of Japan Society for Design Engineering, Vol.50, No.8 (2015), pp.409-414 (in Japanese).

Takahashi, M., Itagaki, T., Takahashi, H. and Koide, T., An Estimation Method of Meshing Teeth Temperature of Plastic Crossed Helical Gear Pair, Journal of Japan Society for Design Engineering, Vol.51, No.6 (2016), pp.419-426 (in Japanese).

Takanashi, S. and Shoji, A., Measurement of the Coefficient of Kinetic Friction of Plastic Materials used for Gears, Journal of the Japan Society of Precision Engineering, Vol.47, No.8 (1981), pp.944-948 (in Japanese).

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