Plastic Spur Gear Pair Failure

Chapter 3.36: Plastic Spur Gear Pair Failure

3.36 Plastic Spur Gear Pair Failure 

Summary

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Chapter Overview



Gear Geometry



Material Modeling



Contact



Failure Criteria



Experimental Test Machine



Results & Conclusions

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Modeling Tips



Input Files



References



Animation

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5 5

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13 13 14

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3.36-2 Marc User’s Guide Summary

Summary Title Problem features

Plastic Spur Gear Pair Failure • Acetal copolymer gears in contact • UACTIVE user subroutine deactivates failed elements

Geometry

2.026 in

Material Properties

Elastic-plastic

Analysis type

Quasi-static analysis

Boundary conditions

Rigid bodies inside shaft holes hold one gear fixed and rotate the other.

Element type

4-node plane strain element type 11 with variable thickness

FE results

Predicted torque versus twist compared to experimental values. 120

Equivalent Von Mises Stress 4500.00 4044.65 3589.30

100

3133.95 2678.60

1

Inc: 128

Torque (in-lbf)

2223.26 1767.91

80

1312.56 857.21 401.86

1

Inc: 112

-53.49

60 1

Inc: 140

1

40

Inc: 79 0

Experimental Prediction

20 1

Inc: 57

Twist (Radians) 0 0.00

0.02

0.04

0.06

0.08

0.10

CHAPTER 3.36 3.36-3 Plastic Spur Gear Pair Failure

Chapter Overview An elastic-plastic finite element analysis of the quasi-static loading of two acetal copolymer gears in contact is preformed. Torque verses twist of the gear set is compared to actual experimental results. The gear geometry is modeled by plane strain elements with variable thickness between the rim and web. Gear tooth failure is modeled by deactivating elements when the plastic strain of 0.15 is exceeded in the tensile regions.

Gear Geometry Two acetal copolymer spur gears were selected as test specimens. The geometry of the gear teeth was based on the American Gear Manufacturers Association (AGMA) standard: Tooth proportions for Plastic Gears (Reference 36-1). The entire gear pair is modeled to capture the correct torsional stiffness of the gear pair. The specifications for the test gears used are provided in the table below. Basic Specification Data Number of Teeth

40

Diametric pitch

20

Standard pressure angle (degrees)

20

Tooth form

AGMA PT1

Standard addendum (inch)

.0500

Standard whole depth (inch)

.1120

Circular thickness on standard pitch circle (inch)

.250

Basic Rack Data Flank angle (degrees)

20

Tip to reference line (inch)

.0665

Tooth thickness at reference line (inch)

.250

Tip radius (inch)

.0214

The test gears were assembled at a center distance of 2.0620 inches. This gave a nominal backlash of 0.0320 inches. This relative large backlash permitted the test gears to reach relatively high torque levels without having the gear teeth roll back on each other, thereby making contact on the backside of the adjacent tooth. An illustration of the gear model (mesh lines included) assembly is shown in Figure 3.36-1. The rim of the gear teeth is 0.25 inch (geom1) in thickness and the web thickness (geom2 and geom3) is 0.123.

3.36-4 Marc User’s Guide Material Modeling

geom1 Y geom2 Z

X

geom3

Figure 3.36-1 Geometry and Mesh

Material Modeling The material is modeled as elastic-plastic with Young’s modulus of 3.0x105 psi with an initial yield strength of 2500 psi. The Cauchy stress versus true plastic strain curve is shown in Figure 3.36-2. 5000

Cauchy Stress (psi)

4000

3000

True Plastic Strain 2000 0.00 Figure 3.36-2 Material Behavior

0.05

0.10

0.15

0.20

CHAPTER 3.36 3.36-5 Plastic Spur Gear Pair Failure

Contact The contact bodies are shown in Figure 3.36-3 and two circular rigid bodies, drive1 and drive2, are glued to each gear, gear1 and gear2, respectively. Contact body drive1 rotates about the center of the gear while drive2 remains stationary. Two other rigid bodies (drive1out and drive2out) move just like drive1 and drive2, but are non-contacting rigid bodies via contact table. They appear on the post file to visualize where the teeth would be if they were rigid. Since kinematics for the design of a gear set assumes the gears to be rigid; it is convenient to see where the teeth would be if the gear material was rigid. drive1

gear1 gear1

gear2 drive1 drive2

drive1out

drive2out

drive1out drive2out gear2

Y Z

X

drive2

Figure 3.36-3 Contact bodies

Failure Criteria Two user routines are used, PLOTV, captures the total equivalent plastic strain and the mean stress and determines the elements to be deactivated when the mean stress is tensile (> 1000) and the plastic strain exceeds 15%. Subroutine ACTIVE, uses the information from PLOTV to actually deactivate the elements selected. The deactivated elements no longer participate in the analysis. The routines are listed below. subroutine plotv(v,s,sp,etot,eplas,ecreep,t,m,nn,layer,ndi, * nshear,jpltcd) c* * * * * * c define a variable for contour plotting (user subroutine). c v variable to be put onto the post file c s (idss) stress array c sp stresses in preferred direction c etot total strain (generalized) c eplas total plastic strain

3.36-6 Marc User’s Guide Failure Criteria

c ecreep total creep strain c t array of state variable (temperature first) c m(1) user element number c m(2) internal element number c m(3) material id c m(4) internal material id c nn integration point number c layer(1) layer number c layer(2) internal layer number c ndi number of direct stress components c nshear number of shear stress components c jpltcd the absolute value of the user's entered post code c* * * * * * implicit real*8 (a-h,o-z) common /mydata/ ielem(30000) dimension s(*),etot(*),eplas(*),ecreep(*),sp(*) dimension m(2),layer(2),t(2) kc=1 call elmvar(18,m(1),nn,kc,v) call elmvar( 7,m(1),nn,kc,ve) if(nn.eq.1.and.ielem(m(1)).ne.1) ielem(m(1)) = 0 if(v.ge.1.0d3.and.ve.ge.0.15d0 ) ielem(m(1)) = 1 return end subroutine uactive(m,n,mode,irststr,irststn,inc,time,timinc) c* * * * * * c user routine to activate or deactivate an element c c m(1) - user element number c m(2) - master element number for local adaptivity c n - internal elsto number c mode(1)=-1 - deactivate element, remove element from post file c mode(1)=-11 - deactivate element, keep element on post file c mode(1)=2 - leave in current status c mode(1)=1 - activate element and add element to post file c mode(1)=11 - activate element and keep status on post file c mode(2)=1 - only activate/deactivate mechanical of coupled c mode(2)=2 - only activate/deactivate thermal part of coupled c mode(3)=0 - activation/deactivation at the end of increment c mode(3)=1 - activation/deactivation at the beg. of increment c irststr - reset stresses to zero c irststn - reset strains to zero c inc - increment number c time - time at beginning of increment c timinc - incremental time

CHAPTER 3.36 3.36-7 Plastic Spur Gear Pair Failure

c* * * * * * implicit real*8 (a-h,o-z) common /mydata/ ielem(30000) dimension m(2),mode(3) ie=m(1) if(ielem(ie).eq.1.and.mode(1).ne.-1) then mode(1)=-1 write(96,*) 'deactivating element ', ie, ' increment ', inc else mode(1)=2 end if return end

3.36-8 Marc User’s Guide Model Review

Model Review The model is complete and ready to run, however, we shall review the contact table option used to glue the rigid bodies drive1 and drive2 onto gear1 and gear2 respectively, while making rigid bodies drive1out and drive2out non-contacting. Then we shall submit the results and check the results as they are generated. FILES OPEN gearpair.mud OK MAIN CONTACT CONTACT TABLES PROPERTIES

MAIN JOBS RUN SUBMIT OPEN POST FILE (RESULTS MENU) DEF ONLY SKIP TO INC 57 SCALAR (Equivalent von Mises Stress) CONTOUR BANDS

CHAPTER 3.36 3.36-9 Plastic Spur Gear Pair Failure

As expected the gears become engaged and deform as shown in Figure 3.36-4. The non-contacting rigid bodies, drive1out and drive2out, are shown as green lines representing rigid gear motion making tooth deformation easy to visualize. Inc: 57 Time: 2.467e+000

4500.00 4044.65 3589.30 3133.95 2678.60 2223.26 1767.91 1312.56 857.21 401.86 -53.49

Y Z

X

1 lcase2 Equivalent Von Mises Stress

Figure 3.36-4 Contour Equivalent von Mises Stress at Increment 57

Another important plot is the torque versus twist which can be generated by using the history plot feature as: HISTORY PLOT COLLECT GLOBAL DATA NODES/VARABLES ADD GLOBAL CURVE Angle Pos drive1 Moment Z drive1

job1

Moment Z drive1 (x100) 0 0

10

20

The first load case brings the gears into contact at the end of increment 1 and this is seen here. Using the copy to clipboard the history data can be exported to say Excel and the data manipulated and compared to experimental results as see in Figure 3.36-7.

30 40 50 60 70 80 90 100 -1.048

0

Angle Pos drive1 (x.01)

110 120 130 8.333

3.36-10 Marc User’s Guide Experimental Test Machine

Experimental Test Machine A parallel axis gear-testing machine developed by Ticona (www.ticona.com) was used to load and record the load-displacement response of the gears (Figure 3.36-5).

Figure 3.36-5 Parallel axis gear-testing machine The test gears were lubricated with oil prior to loading to eliminate any shearing forces acting on the tooth flanks that were in contact. Torque was measured on the stationary side and load was applied on the motor side. Two high precision encoders were used to measure the angular displacement of both gears. These encoders have a positional accuracy of 57600 counts per revolution. The rate of loading was set by the time for encoder position on the motor side. The stationary was not totally rigid. It required some angular displacement for the torquemeter to record data. To obtain the true angular displacement, the relative displacement between both gears was recorded. This gave a rate for the relative angular displacement between the motor gear and stationary gear to be about 0.002 radians per minute. Five tests were made per gear set at ambient conditions. A plot of applied torques verses relative displacement was recorded. The results are shown in Figure 3.36-6. Test 2 and Test 4 did not reach tooth failure. This is due to that Test 4 was not taken up to the breaking torque and Test 2 reached

CHAPTER 3.36 3.36-11 Plastic Spur Gear Pair Failure

the set limited encoder position before breaking. Static Mesh Bending on Acetal Copolymer Gears 140

120

Torque (in -lb)

100

Te st 1

80

Te st 2 Te st 3 Te st 4

60

Te st 5

40

20

0 0

0.01

0.02

0.03

Figure 3.36-6 Plot of Experimental Results

0.04 Radians

0.05

0.06

0.07

0.08

3.36-12 Marc User’s Guide Results & Conclusions

Results & Conclusions A plot of applied torque verses twist was made and gives excellent representation of the experimental results (Figure 3.36-7). At the beginning, a two teeth pair (on each gear) come into contact, then as these teeth bend, the tooth leading this pair begins to come into contact (Figure 3.36-7 Inc: 79). Later (Figure 3.36-7 Inc: 112) there are four teeth on each gear in contact with their counterparts. At increment 112, the first element is deactivated (leading tooth on top moving gear) followed by several more shown in increment 128. After increment 128, elements begin to fail in the stationary gear and the torque drops off dramatically. Based on the results of this analysis, the mechanical behavior and prediction of copolymer acetal gears is very complex. The results indicate that to optimize a gear set, a non-linear analysis is required to be performed. Only under low loads and deformation can a linear elastic approach be suitable. Clearly combining computer simulations with material and component testing has led to a far better understanding of copolymer acetal gear design; this understanding could not be achieved by either simulation or testing alone. It is envisioned that with a few more material tests, the torque-displacement response of the gear pair can be simulated with confidence thus advancing the technology of copolymer acetal gear applications.

120

Equivalent Von Mises Stress 4500.00 4044.65 3589.30

100

3133.95 2678.60

1

Inc: 128

Torque (in-lbf)

2223.26 1767.91

80

1312.56 857.21 401.86

1

Inc: 112

-53.49

60 1

Inc: 140

1

40

Inc: 79 0

Experimental Prediction

20 1

Inc: 57

Twist (Radians) 0 0.00

0.02

0.04

0.06

0.08

0.10

Figure 3.36-7 Predictions versus experimental results of torque versus twist of the gear pair

CHAPTER 3.36 3.36-13 Plastic Spur Gear Pair Failure

Modeling Tips

becomes,  = s  1 + e  and the true strain becomes,  = ln  1 + e  . The work hardening plot (Figure 3.36-2) then becomes the Cauchy stress versus the total plastic strain,  p =  –   E .

Fig 3.1 · Celcon acetal copolymer stress-strain properties (ISO 527) 120 100 Stress, MPa

The material used was Celcon grade M90 (Toughened; Impact Modified) which is the red curve taken from Reference 36-2, Figure 3.1 duplicated herein. It was assumed that this stress strain data was in engineering measures of stress and strain (s, e) and they needed to be converted to true values, (   ) where the Cauchy stress

25% Glass Coupled

80 Unfilled, 9.0 Melt Flow

60 40

Toughened; Impact Modified

20 0

0

2

4

6

8

10

12

Strain, %

Input Files File

Description

gearpair.mud

Mentat model file

gearpair_job1.dat

Marc input file

gearpair.f

User subroutine to define invoke failure criterion

References 36-1. American National Standard/AGMA Standard, Tooth Proportions for Plastic Gears, ANSI/AGMA 1006-A97, 1997. 36-2. Designing with Celcon http://www.kmsbearings.com/pdf/Celcon_Design%20Guide_3.9.07.pdf

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3.36-14 Marc User’s Guide Animation

Animation Click on the figure below to play the animation.