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# AGMA surface endurance strength and AGMA bending strength6

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## AGMA surface endurance strength and AGMA bending strength

(OP)
Does anyone knows how to calculate the AGMA surface endurance strength and the AGMA bending strength?

I'm looking for a general formula (if there is, the ones i know are for steel), if there is? If not, does anyone knows these values for aluminum 2024-t4 and 303 stainless steel ( i need these values to calculate the safety factors).

For example, Sf(safety factor for bending) = (St x Yn/ Kt x Kr)/sigma (at this moment i have all values less the St).
Best regards

### RE: AGMA surface endurance strength and AGMA bending strength

The general formulas are in AGMA 2001 Fundamental Rating Factors and Calculation Methods for Involute Spur and Helical Gear Teeth.

AGMA reviews materials in multiple standards, including AGMA 2004 Gear Materials and Heat Treatment Manual. Neither aluminum alloys nor austenitic stainless steels are discussed beyond a single sentence saying that they are non-standard and require special analyses.

Your two specific alloys are going to be poor gear materials with respect to wear and resistance to contact stress.

Good luck with your analysis.

### RE: AGMA surface endurance strength and AGMA bending strength

(OP)
Thanks for your reply CoryPad. That was what i was seing, that's why i came to the forum to know if anyone knows this values.

About this materials being poor gear materials, they are not, it depends on your application. For example, sdp-si (www.sdp-si.com), a big gear's manufacturer, uses 303 stainless steel and aluminum 2024-t4 (i'm trying to choose which gears to use, that's why i need this values).

Hope someone knows this values.

Best regards

### RE: AGMA surface endurance strength and AGMA bending strength

Are you transmitting power or just motion? I don't think aluminum would be a good choice if you were transmitting power.

### RE: AGMA surface endurance strength and AGMA bending strength

(OP)
Both power and motion, but are small values (~0.03 hp). From matweb, they seem to have a yield and ultimate strength close to other steel metals (hardened ones). Also, it is in shingley's mechanical engineering design book has a good material for gears.

Best regards

### RE: AGMA surface endurance strength and AGMA bending strength

As CoryPad noted, the AGMA gear design approach you are asking about that uses various factors is intended to be used with the material properties recommended by AGMA for this approach. There are other AGMA specs that provide every equation you will need to calculate the precise tooth bending and surface contact stresses of your gear design. You can find some fatigue properties for 2024-T4 aluminum in MIL-HDBK-5. But you probably won't find fatigue data for 303 cres, since its poor fracture characteristics make it unsuitable for structural applications.

### RE: AGMA surface endurance strength and AGMA bending strength

(OP)
Hi, tbuelna, just saw the MIL-HDBK-5 but i coulnd't find what i'm looking for (used ctrl+f and searched for 2024), only could found the ultimate and yield strength, not the allowable contact strength and allowable bending strength.

Best regards

### RE: AGMA surface endurance strength and AGMA bending strength

Tooth bending is based on tensile stress in the root fillet area. For tooth bending you need to consider all of the factors you would with any fatigue analysis, such as load cycles, load conditions, surface finish, mechanical working of the material, fillet shape, etc.

### RE: AGMA surface endurance strength and AGMA bending strength

(OP)
Thanks again tbuelna, but again, the values showned here are for the ultimate strength and yield strength. Maybe i didn't understood the values from the book. For example, (this will be kind of a newbie question) agma gives this equation, based on hardness, for grade 1 steel: <<Contact-fatigue strength number, Sac = 2.22HB + 200 MPa>> and <<Allowable bending stress number, Sat =0.533HB +88.3 MPa>>. St and Sc are used in the used in the bending factor of safety and wear factor of safety equations, respectavily, and are this values i'm looking for, or some sort of calculating them or estimating (both two sources you have posted talk about ultimate strength and yield strength.

Hope you could reply me.

Best regards

EDIT:

After reading the book again, i found that they have changed the designations of what i'm looking for, so title is wrong. What i'm looking for is the allowable stresses numbers (Sat, Sac). Does anyone know this values or a way of calculating them (sorry for confusion tbuelna, since you were always posting info about ultimate strength and yield, hope you can help me in this too :D)?

### RE: AGMA surface endurance strength and AGMA bending strength

Rather than using the simplified AGMA design approach that uses various factors, you need to calculate the actual root fillet tensile stress due to tooth bending. You can find all the equations needed in the AGMA specs. For a fatigue case where there might be varying loads and frequencies, you need to establish a single composite load and number of cycles value to use. Take the root fillet tensile stress value you calculate and compare it to the appropriate curve in the chart above. If the gear tooth is subject to reversing bending loads (this is common with idler gears), make sure to take that into account when using the chart above.

### RE: AGMA surface endurance strength and AGMA bending strength

(OP)
Thanks tbuelna, i'll try to find that equation...Will reply as soon as i make the calculations.

Best regards

EDIT

After searching around, i couldn't find the equation, could you please point it to me or post it here (this was better if forum had private messages, since you are only one helping tbuelna)

Only could find/had these ones (i think that they are not the equation for the root fillet, this are for strength/stress):

bending
1

2

contact stress
3

4

Imagine this (what will be the first stage): 8W motor, 7000 rpm pinion, 13 teeth pinion, 41 teeth gear, 80 diametral pitch (0,3175 module), 0,1 inches face width (2,5 mm), designed for 10^8 cycles, SF > 1,3, sigmalim= 42.5 :

From equation 1, i can transmite a maximum of 60500 psi, and from Ft = (1.95*10^6*P)/dg*n (dg=working pitch diameter, n= rotational speed of pinion, P=power transmitted), Ft=800psi, so Ft<<Ftlim
Now from equation 2, Ft = 9527 psi, so, what is the correct value?

Best regards

### RE: AGMA surface endurance strength and AGMA bending strength

noobita-

I would recommend reading thru AGMA 917-B97 for a step-by-step approach to designing a spur gear. Sections 6.9.5 & 5.6.1 discuss methods for determining tooth bending stress other than the AGMA geometry factor method you are using. Section 6.9.6 discusses contact stress. And sections 10.2 & 10.3 briefly discuss using 300 series cres and wrought aluminum alloys for gears.

### RE: AGMA surface endurance strength and AGMA bending strength

(OP)
Hi tbuelna, again, thanks for your reply, but again this file couldn't clear my doubts(it doesn't have the contact stress for 2024-t4). But from i read from the file, they say that gear designers are based in experience, this is, making the system and testing them, i think that i will try it (from my calculations, for the final stage, the gear will experience a contact stress of 200k psi and a bending stress of 44k psi, which material would you recomend - 4130 or 4340?).

Best regards

### RE: AGMA surface endurance strength and AGMA bending strength

Here's a couple comments based on the description of your gear set provided above. First, it will require quite a bit of effort to produce an 80DP 13T pinion with the precision you seem to be looking for. Second, a hertzian contact stress of 200ksi at 10^8 load cycles is a bit too high. You should keep the contact stresses below around 175ksi for 10^8 cycles, and even then you'll need to use a case hardened alloy steel. For an 80DP gear tooth the most practical case hardening option is nitriding. I'd suggest using vacuum melt Nitralloy N (AMS 6475) or vacuum melt 4340 (AMS 6414). Bending stress of 44ksi at 10^8 cycles (unidirectional) is OK for either material. Third, due to the very small size gear teeth you are using it can be difficult to accurately predict tooth stresses due to the difficulty in controlling geometry errors when manufacturing such small gear teeth. So you need to be very conservative with your stress limits.

### RE: AGMA surface endurance strength and AGMA bending strength

noobita

Can you be more specific for the gears are intended for?
Cycle number?
Etc.,

We have used such gears for specific application that include both for power and motion but for limited cycle numbers.

### RE: AGMA surface endurance strength and AGMA bending strength

(OP)
tbuelna first of all thanks for your reply, again. And sorry for the misunderstand, the system with 80DP is the first stage of gearbox, the one i talk in this last point is last stage (gearbox is composed by 5 stages), which has 48DP (and values presented for stresses are bidirectional values - 1/0.7 = 1.43 x value of unidirectional - hope i have used the right multiplying factor - this was what was stated in the book). The first stage has a bending stress of 10kpsi and contact stress of 40kpsi, so there is no problem there (for 4130/4340 or even astm no.50 there isn't any problem). Only problem is really last stage, which will withstand a higher value of force (last stage is 14 pinion teeth and 39 gear teeth).

israelkk, trying to reply to your questions, the gears are intented to use in a speed-reducer gearbox, in which the motor has 0.009 hp (approximately 7W, so this will be the transmitted power), the motor speed (pinion speed) is about 10600 rpm (no load speed - i'm using this value, but don't know if i should use another value, like max continuous speed, or even a lower value - because force will increase if speed reduces), and is intended to reduce speed to about 55 rpm (so, the ratio is about 193, for that i'm designing a 5 stage gearbox). Would you like that i post my preliminary design of the gearbox? Oh, and there is a space constraint, gears should fit in a space of 1x2 inches (25.4x50.8mm), height isn't a constraint, but should be less than 30mm.

Best regards

PS: i use SI system, but has i think this is an american forum, i'm posting the american units. If you prefer (and is better for me) i can only post in the SI system..

### RE: AGMA surface endurance strength and AGMA bending strength

What is the maximum torque the gearbox outputs?
Is it rotate multiple rotations (continuous rotation) or just limited angular output against a torque load?
How many cycles the gearbox will see against the maximum torque?
If it is for limited angular output, how fast it needs to reach the maximum angle against the maximum torque?

### RE: AGMA surface endurance strength and AGMA bending strength

(OP)
So,the maximum torque that the gearbox will have to output will be 5.0Nm(approximately 780 oz-in). Multiple rotations, without limits. Against the maximum torque, since it must handle 10^8 cycles in maximum continous torque, it should handle at least 10^4 cycles at the stall torque (which is the 5Nm).

Would you like me to post the calculations that i already have with materials, etc, and each stage of the gearbox?

Thanks for your help and best regards (eng-tips doesn't have pm right?)

### RE: AGMA surface endurance strength and AGMA bending strength

Remember, that to show that the system for sure will last 10^4 cycles at the stall torque you should prove by testing that it can live 10^5 cycles (factor of safety of 10 for fatigue analysis some will argue factor of safety of 4). Same apply for the maximum continuous torque, you need to show by tests between 4x10^8 to 10^9 cycle before failure. The criteria for failure should be established too. It may be a fracture of the gear tooth or position accuracy loss, depends on the application. I doubt that you can achieve those requirements in the space you defined using SDP-SI, PIC or BERG type gears. You probably need to use custom designed case hardened and grinded gears, if any.

### RE: AGMA surface endurance strength and AGMA bending strength

(OP)
Thanks for your answer israelkk, i was looking already for custom spur gear, the suppliear gave me the fallowing materials (the ones they use):

1) Carbon Steel
SC45 (1045)
40Cr

2) Alloy Steel
42CrMo (SAE 4140, SCM 440)
20CrNiMo (AISI 8620)
40CrNiMo (SAE 4340)
20CrMnTi (This is most used for spur gear and helical gear in China)

17CrNiMo6 or 18CrNiMo7 for spiral bevel gear

3) Aluminum
6061-T6 for some gears and timing pulley.

For now, my problem is really find the specific materials for each gear, so it can last all cycles

PS: This afternoon i will try to post an already made design which lasts highest torques that my gears are supposed to handle - hope this can help

EDIT

So, here is the details of the other gearbox:

(13/54) -> 80dp
(15/53) -> 80dp
(15/35) -> 64dp
(13/27) -> 48dp
(14/39) -> 48dp

Ratio is, approximately, 198. In this design, the maximum torque (stall torque) is 7.3 N.m, power transmitted 8W, and maximum continuous torque is 2N.m. Which material do you think is used in this gearbox? PS: this is a tested gearbox, data was taken from internet, don't know any other specs.

Best Regards

### RE: AGMA surface endurance strength and AGMA bending strength

Each gear width is necessary in order to estimate their ability to live the stall torque and the maximum torque for the expected life cycle.
Are those gears use rack/profile shift or are as is?

### RE: AGMA surface endurance strength and AGMA bending strength

Is your fatigue life of 10^8 load cycles based on the 13T input pinion?

The knock down factor of 0.70 for fully reverse tooth bending fatigue stress is fine for vacuum melt quality steel alloys.

The allowable fatigue stress limit for a given material will vary based on the reliability rate used. For example, a material at L2 (98% reliability) will have a lower fatigue stress limit than it does at L10 (90% reliability). The statistical reliability rate required in your gear drive is something you must consider when calculating fatigue life. And it will depend on how critical your application is.

### RE: AGMA surface endurance strength and AGMA bending strength

(OP)
First of all, thanks again. DO you know if eng-tips has private message? Since would do this proccess much easier.

@israelkk, modifying what i posted about the gears:

(13/54) -> 80dp, face width(F) 2.5mm (0.1inches), x1=0.34 (pinion), x2=-0.34 (gear)
(15/53) -> 80dp, F 3mm (0.12inches), x1=0.34 (pinion), x2=-0.34 (gear)
(15/35) -> 64dp, F 3.5mm (0.16inches), x1=0.34 (pinion), x2=-0.34 (gear)
(13/27) -> 48dp, F 5mm (0.2inches), x1=0.3 (pinion), x2=-0.14 (gear)
(14/39) -> 48dp, F 8mm (0.32inches), x1=0.34 (pinion), x2=-0.34 (gear)

PS: I'm not sure about the profile shift, so these values are only estimates

I'm really trying to find a suitable material (this gears that i have posted are for the dynamixel mx-64, so the design works and material will last).

@tbuelna, the 10^8 cycles are only for the first pinion. The objective is that the entire system lasts at least 20000 hours (if not possible, 10000 hours) in the max continuous torque (so, first pinion should last something like 3x10^8cyles and last stage gear only approximately 3x10^6 cycles (again, this for maximum continuous operation, not operation at stall torque).

### RE: AGMA surface endurance strength and AGMA bending strength

As far as I know the SDP, PIC, BERG gears are not corrected (profile/rack shifted) therefore, they will last much less than corrected gears.

10000 hours at 7000 rpm at 8W is 4.2 x 10^9, it is more than 10^8. \

You need to specify the gearbox output load at continuous torque and at pick torque and the number of gearbox output RPM in both cases. From this data you go back and calculate the RPM and torque at each gearbox stage and check if the gears can live for the desired cycles 10000/20000 hours definition is not clear unless it continuously rotate at a specific rpm and torque for the whole 10000/20000 hours).

I did a quick check for the pinion made of material at 120 Brinnel hardness (2024 T4) therefore, the gears from SDP, PIC and BERG will not live more than 3000 rotations for the pinion and 22000 for the gear in bending loads. For surface loads they will fail instantly (I used 8W, 7000 rpm).

No matter what material you choose all gears must be custom designed and manufactured. You may get away with 2024-T4 for the pinion but it must have profile shift/rack shift to avoid undercut and excessive sliding velocity. From by experience the pinion is usually not the most loaded gear, generally a middle stage and/or the output stage are the most critical.

### RE: AGMA surface endurance strength and AGMA bending strength

Sorry for my previous calculations I used incorrect HP (10 times larger 0.11HP instead of 0.011HP). The pinion and gear from SDP (120 Brinnel hardness (2024 T4)) may be OK for the 8W, 7000 rpm case. The program I used didn't allow the 13 tooth pinion without correction to avoid undercut and with the correction it's OK. However, the gears from the catalog have undercut to my best knowledge. Therefore, they need to be carefully checked.

### RE: AGMA surface endurance strength and AGMA bending strength

(OP)
Hi israelkk, once again thanks for your help. The sdp can make the profile shift of the gears by request (from what i read from their website), but at this stage the gears would be custom made. Could you tell me what program did you used (this because i searched online for a lot of programs and couldn't find a suitable one, kissoft seemed the best, but with the tutorial version you can't do much)? I made the calculations in a self made excel worksheet, and according to my calculations i would be needing a material with a hardness of at least 240 (this for last stage), for first stage the 120HB would be ok

Best regards

### RE: AGMA surface endurance strength and AGMA bending strength

I use an old program from Fairfield Mfg. They used to sell it 25 years ago and then made it free on their website. Lately they removed it and it is no longer available. Just to clarify, all gear calculation formulas for surface stresses in the literature and those used by AGMA, DIN, ISO etc., assumes proper lubrication between mating teeth where always an oil film exists between the mating gear teeth. If the oil film breaks from any reason and a metal to metal contact exists, the formulas are no longer valid and the gear considered ruined. Therefore, use the formulas with care if you use only grease.

### RE: AGMA surface endurance strength and AGMA bending strength

noobita-

As israelkk noted above, each tooth of your 13T pinion operating at 7000rpm for 10k hours will be subject to 10^9 load cycles. This is quite a large number of load cycles, and the stress limits used when designing for this case basically means you'll end up with unlimited fatigue life.

The most important thing to remember is to design all of your drive system components (gears, bearings, shafts, etc) so that they have balanced fatigue life. It usually does no good to have some components in the system that have much greater fatigue life than others.

### RE: AGMA surface endurance strength and AGMA bending strength

(OP)
Thanks again israelkk and tbuelna.

Responding first to israelkk, yeah, i know all the formulas are based in proper oiled systems.

tbuelna, that's why i'm asking here, don't know if it is asking too much, help on finding suitable materials. I already posted all the system design, i can post the calculations (need to find the proper material to resist the stall torque too, not only the maximum continuous ratings - those i already decipher - but for the stall torque i don't think the system can handle it). How do you normally choose the material for this?

Best regards

### RE: AGMA surface endurance strength and AGMA bending strength

When analyzing your drivetrain components such as gears and bearings, you need to run a couple different cases. There is the 10K hour composite lifecycle fatigue case you describe above. There might also be a max torque case for a limited number of hours. And there is the stalled condition you describe.

The most difficult case to analyze accurately is the stalled condition. Since the stall torque is almost 4X the max operating torque, this is likely the case that will drive your design. You would need to go through each mesh and determine the precise point during a single tooth passing thru mesh where the surface contact stresses are highest. This would be affected by factors like contact ratio, pressure angle, profile shift, geometry errors, etc. The max static contact stress should be well below what would produce a permanent deformation in the tooth flank surface. What you'll find is that when you account for the combined worst case condition at each of your five gear meshes the result will be rather discouraging. But if this is a requirement which you must be able to demonstrate by analysis that your gear drive is capable of meeting, then that's what you must do. Also, remember to check your bearings are good at this stalled condition.

### RE: AGMA surface endurance strength and AGMA bending strength

To design for stall torque you must specify a number of loading to stall torque (7.3N-m) that the gear will/may see. Even if in real life this should not happen, during the development process it will surely happen and a lot, as a result of mistakes, improper behavior of unfinished control system, human errors etc,. To this you need to take a safety of factor of x4 as tbuelna stated. When you will have this number then you can make calculations and select gear sizes and materials. In aerospace many systems are designed to live the extensive testings during the development even though in real life they work only one time.

If the system is needed to be as accurate as in optical sight systems then not only a breakage of the gear teeth is an issue but deformation of the teeth surface will be an issue. For some systems even a permanent deformation of the teeth can be allowed as the torque is transmitted without a breakage of the teeth.

### RE: AGMA surface endurance strength and AGMA bending strength

What is important is what you can show by analysis. Testing only serves to validate your analysis work. A permanent deformation of a gear tooth or bearing race surface would only be acceptable in a failure condition, where the device would subsequently be removed from service.

### RE: AGMA surface endurance strength and AGMA bending strength

(OP)
Let me see if i understood what you said. Imagine i do calculations for a specific diametral pitch, face width, profile shift, etc...it gives me an value of 20kpsi (this is only for the maximum continous torque). Then, i multiply by the cycles factor and the Safety factor, which you said i should use 4, and have to find a material that should withstand 80kpsi, for example. Is this correct?

Now in the design, for first stage pinion i have this value for bending force: 8698,649473 psi (maximum continuous), for 20000 hours, n = 3,57x10^8cycles, which gives yn pinion 0,955, temperature factor 1, reliability factor of 0,85. So, if you say safetic factor should be 4, material should be abble to withstand at least 31kpsi, is this?

Best regards

### RE: AGMA surface endurance strength and AGMA bending strength

When I said x4 safety factor it was on the "desired life cycle" at stall, maximum continuous, etc,. Not on the stress level. For example, if you expect the gearbox to see stall torque 100 times during development, testings, and application phases you need to show by calculations and tests that after 400 cycles to stall that the gearbox did the desired job. The reason for this factor is that fatigue is a wide spread statistical phenomena. Ultimate and yield stress of materials are more strict and controlled statistical phenomena than the fatigue life of same material. While the "stress level safety factor of safety" in aerospace can go from 1.25 to 4 depends how critical and risky is a failure to human life, in fatigue the safety factor on the life cycle to failure can go from x4 to x10. For example, even though in tests your gear box lived the 400 cycles to stall torque, you can guaranty only 100 cycles.

### RE: AGMA surface endurance strength and AGMA bending strength

I would suggest that you assemble a cumulative damage model. Miner's Rule is the simplest to use, the Inverse Power Law-Weibull model is more complicated but the results are more accurate.

Remember that, no matter which technique you use, the more load stages that you can assemble into your model, the more accurate the results will be. Start with a minimum of 3.

Have a look at this page to get you started - http://www.weibull.com/hotwire/issue116/hottopics1...

### RE: AGMA surface endurance strength and AGMA bending strength

(OP)
According to calculations, i would need a material with a bending strength of 50kpsi+, that is even possible? Exists any material like this (hardened of course - i know nitrided, like 4140 have 42kpsi - and this is for maximum continous, not even stall, at stall i would need 4x10^8, what seems impossible)?

### RE: AGMA surface endurance strength and AGMA bending strength

gearcutter makes a very good point about how and when a few discrete elevated load cycles get applied can produce different results on fatigue life. If these elevated load cycles occur early in the component's life their effect on fatigue life is different than if they occur later in the component's operating life. For example, if your component is good for say 1000 load cycles at the stall condition when new, after operating for 90% of its design life (18K hours) at normal conditions, it would only be able to handle a tiny fraction of the number of stall condition load cycles as when new. Also consider the opposite case, where a large percentage of the stall condition load cycles occur within the initial 10% of design life (2K hours) at normal conditions. There might not be sufficient fatigue life remaining in the component to last 20K hours under normal conditions.

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