PlanetaryGearTrains.nb

Homework #10

Getting Geared Up

Problem #1 - Analysis of a simple gear set

Problem Statement

A 78-tooth spur gear is in mesh with a 27-tooth pinion. The pitch is 6 and the pressure angle is 20°. The pinion transmits 33 kW at 1600 rpm. The = 11, the pinion is made of 4340 steel and the gear is made of A-7-d nodular-iron.
Determine the following:
What is the contact ratio for this gearset?
Find the torque on each shaft.
Size these spur gears (both the pinion and the gear) for a surface safety factor of 1.2, assuming steady torque.

Next time use a 72 tooth Gear and a 24 tooth Pinion(see Boston Gear)

Given

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Remove ["Global`*"] ;  Off[General :: "spell"] ; Off[General :: &q ... 1600 * 2 π) Radian/Minute ; Qv = 11 ; pinionMaterial = 4340 ; gearMaterial = A7nodularIron ;

Assumptions

We assume 100% efficiency in the transmission of power and torque in the gearset.
Realistically it would be more like 97-99% efficiency.

Find

contact ratio, torque on each shaft, and size (face width) of gears for surface safety factor of 1.2

Solution

Compute Contact Ratio

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pc [d_, N_] := (π d)/N ; (** circular pitch **)pb [ pc_, ϕ_] := pc Cos[ϕ ... (** the base pitch**)contactRatio[lenAction_, basePitch_] := lenAction/basePitch ;

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dgear = Ngear / pitch RowBox[{RowBox[{agear, , =, , RowBox[{1., , /, , pitch}]}], }] dpinion = Npinion / pitch apinion = agear

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As we would expect, the contact ratio is the same for both gears!

Find the torque on each gear

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Note the torque is the not same for both the pinion and the gear (we ignore the 1-3% losses in the transmission), but the tangential force is! The torque ratio is the same as the gear ratio.

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Determine the size of the gears for a surface safety factor of 1.2

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Cp[Ep_, νp_, Eg_, νg_] := 1/(π (((1 - νp^2)/Ep) + ((1 - νg^2)/Eg)))^(1/2)

We get the tangential force from the torque

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Surface Geometry Factor, see page 724.

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$Clear[sgf] ;  sgf[rp_, xp_, pitch_, ϕ_, center2centerDistance_, dp_] := Module[{& ... istance Sin[ϕ] - ρp ; sgf = Cos[ϕ]/((1/ρp + 1/ρg) dp)]$

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$surfGeoFactor = FullSimplify[sgf[dpinion/2, 0, pitch, pressureAngle, (dpinion + dgear)/2, dpinion]//N ] /.{( Inch)/Inch^2^(1/2) -> 1}$

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From Figure 11-22, page 719,

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Or we can use the equations developed by Norton, on pages 718-719.

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$KvCvus[Qv_, Vt_] := Module[{A, B}, B = (12 - Qv)^(2/3)/4 ; A = 50 + 56 (1 - B) ; Kv = (A/(A + Vt^(1/2)))^B]/.Foot/Minute->1$

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Very close agreement!

From Table 11-16, page 720, we'll try Face < 2"

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From Table 11-17, page 721, we the uniform value

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From page 721, we use the default size factor.

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From page 726, we assume a reasonable surface finish so

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$RowBox[{elasticCoe, , =, , RowBox[{Cp, [, RowBox[{3 10^7PoundForce/Inch^2, ,, , 0.28, , ,, RowBox[{2.5, , 10^7, PoundForce/Inch^2}], ,, 0.3}], ]}]}]$

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$RowBox[{2177.71, , PoundForce/Inch^2^(1/2)}]$

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$RowBox[{114472., , PoundForce/Inch^2^(1/2), , PoundForce/(Face Inch)^(1/2)}]$

Compute the Surface Fatigue Strength

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The pinion is made of 4340 and is nitrided. The range of ultimate strengths is 150-175 kpsi. We will use the minimum value, i.e., 150,000 psi. Also note that CH is the relative hardness measure and represents the relative hardness of the gear to the pinion (the pinion is given the value 1).

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CHpinion = 1 ; (** see page 734**)CL = 1 ; CT = 1 ; CR = 1 ; SfbprimePinion = 150 10^3 PoundForce / Inch^2 ;

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So the face should be say 0.875 Inch wide. We can check the produce of face width and pitch to see if its in the recommended range.

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And it is not. The recommended range is 8-16. I don't know the basis for this.

We should check the gear side as well.

Compute the minimum required face width for the nodular iron gear required to meet the surface safety factor of 1.2.

Note: nitriding is a case (surface) hardening process and as such the second approach to establishing CH should be used. No value for Rq was given, but Figure 7-1, on page 446 has an example of a cam that has been ground polished and in the notes Rq is given as 0.175 (I assume μin), so I will use that (I expect the surface of the cam to be compariable to that of a well manufactured gear).

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$RowBox[{RowBox[{Rq, , =, , 0.175}], ;}] RowBox[{RowBox[{CHgear, , =, , RowBox[{1, , +, , ... , (** see page 734**)}] CL = 1 ; CT = 1 ; CR = 1 ; SfbprimeGear = 92 10^3 PoundForce / Inch^2 ;$

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So we get a small effect on the gear from the case hardened pinion.

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$RowBox[{RowBox[{(, RowBox[{92068.9, , PoundForce}], )}], /, Inch^2}]$

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$RowBox[{67349.1, , PoundForce/Inch^2^(1/2), , PoundForce/(Face Inch)^(1/2)}]$

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So we see that the demands on the pinion require a wider face than for the gear, so we will specify the face requirement based upon the Pinion, and will round it up to the next standard width.

After reviewing the catalog at Boston Gear, I find that they only provide gears of this type in 2" face width, so that would be what I would specify. :-)

Planetarty Gear Train

Problem Statement

We have a planetary gear train with 8 tooth sun, and 40 tooth planets.
The module is 1 mm. The sun is the input and the carrier is the output.
Answer the following questions about this planetary gear train.
- what is the train ratio for this planetary gear train?
- what is the gear ratio for this gear train?
- what is the torque ratio for this gear train?
- how many planets can this planetary have?

Given

Remove ["Global`*"] ;  <<Miscellaneous`Units` ; mm = Milli Meter ; Nsun = 8 ; Nplanet = 40 ; module = 1 mm ; input = sun ; output = arm ;

Assumptions

none

Solution

First lets determine the number of teeth in the Ring gear

what is the train ratio for this planetary gear train?

what is the gear ratio for this gear train?

That is the gear ratio is 12:1 or 12 to 1.

what is the torque ratio for this gear train?

the torque ratio is the same as the gear ratio, i.e. we get twelve times as much torque out of the planetary as we put in, (but at one twelveth the speed).

how many planets can this planetary have?

Since Nring is an even number we can have an even number of planets.
In addition because the planets are quite large, three of them will not fit within the ring gear without interference, so the answer is THIS PLANETARY CAN HAVE ONLY ONE OR TWO PLANETS.

Created by Mathematica (April 20, 2004)