Appendix A

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Kinematic_Coupling_3Groove_Design.xlsTo design three groove kinematic couplingsWritten by Alex Slocum long ago. Last modified 2012/10/28 by Alex Slocum

Material properties

1User defined material aluminumYield stress

System geometry (XY plane is assumed to contain the ball centers) plastic 3.45E+07Dbeq (mm) = 35 Equivalent diameter ball to contact the groove at the same points RC 62 Steel 1.72E+09Rbminor (mm) = 500 "Ball" minor radius CARBIDE 2.76E+09Rbmajor (mm) = 500 "Ball" major radius user defined 2.76E+08Rgroove (mm) = 1.00E+06 Groove radius (negative for a trough) Elastic modulusCostheta = 1 Is ball major radius along groove axis? plastic 2.07E+09Dcoupling (mm) = 625 Coupling diameter RC 62 Steel 2.04E+11Fpreload (N) = -15000 Preload force over each ball (negative pushes ball into groove) CARBIDE 3.10E+11Xerr (mm) = 0.0 X location of error reporting user defined 6.80E+10Yerr (mm) = 0.0 Y location of error reporting Poisson ratioZerr (mm) = 1320.8 Z location of error reporting plastic 0.20

RC 62 Steel 0.29

Matlabball = 2 CARBIDE 0.30Matlabgroove = 2 user defined 0.29Min. yield strength (Pa, psi) 1.72E+09 249,855Largest contact ellipse major diameter (mm) 8.778Smallest contact ellipse major diameter (mm) 8.761Largest contact stress ratio 0.22RMS applied force (N) 30000RMS stiffness (N/micron) 755Z displacement caused by preload (mm) 0.015Applied Z load at zero inclination (N) -30000inclination angle (degrees) 10Applied force's Z,Y,Z values and coordinates Coupling centroid locationFLx (N) = 0.00 XL (mm) = 0 xc (mm) 0.000 XL (m) =FLy (N) = -5209 YL (mm) = 0 yc (mm) 0.000 YL (m) =FLz (N) = -29544 ZL (mm) = 1320.8 zc (mm) 0.000 ZL (m) =Results: Hertz stresses and deformationsError displacements at the point of interest due to applied load (preload displacement subtracted off) (micron)DeltaX (mm) 0.0000 DeltaY (mm) 0.0393 DeltaZ (mm) 0.0056resulting rotation (degrees) wrt cent 0.0000 0.0072Vector displacement (mm) 0.0397

Groove normal forces (preload + applied load) (N) Contact stress (preload + applied load) (Pa)Deflection (+into ball) (m)

from: Applied Load Preload onlyBall-Groove 1 Ball-Groove 1 Ball-Groove 1

Fbnone 7190.8 sigone 4.09E+08 0.142 delone -4.94E-06 2.17E-05Fbntwo 7190.8 sigtwo 4.09E+08 0.142 deltwo -4.94E-06 2.17E-05

Ball-Groove 2 Ball-Groove 2 Ball-Groove 2Fbnthree 20633.2 sigthree 5.81E+08 0.202 delthree 1.21E-05 2.17E-05Fbnfour 24886.7 sigfour 6.18E+08 0.215 delfour 1.66E-05 2.17E-05

Ball-Groove 3 Ball-Groove 3Fbnfive 24886.7 sigfive 6.18E+08 0.215 delfive 1.66E-05 2.17E-05

Fbnsix 20633.2 sigsix 5.81E+08 0.202 delsix 1.21E-05 2.17E-05

Results: Error motions Xerr Yerr ZerrError motions are at X,Y,Z coordinates (m) 0.000 0.000 1.321

Metric units only! Enters numbers in BOLD, Results in RED

Standard 120 degree equal size groove coupling? (contact forces are inclined at 45 to the XY plane. For non standard designs, enter geometry after results section)

Auto select material values (enter other_4 to the right)

Enter 1 for plastic, 2 for steel, 3 for carbide, 4 for user defined, 5 where each ball and groove is defined individually

Max shear stress/(ult. tensile/2)

A B C D E F G H1234

5

6789101112131415161718

19

2021222324252627282930313233343536373839404142434445464748495051525354

Appendix A

Page 2

deltaX 0.00E+00 RMS 3.97E-05deltaY 3.93E-05 Homogenous Transformation Matrix:deltaZ 5.59E-06 1.00E+00 -0.00E+00 0.00E+00 0.00E+00EpsX -2.91E-05 -0.00166666015031778 0.00E+00 1.00E+00 2.91E-05 9.17E-07EpsY 0.00E+00 -0.00E+00 -2.91E-05 1.00E+00 5.59E-06EpsZ 0.00E+00 0.00E+00 0.00E+00 0.00E+00 1.00E+00Generic data entry for non-120 degree couplingsNOTE! For calculation of angular errors, the coupling is assumed to lie in the XY plane. Ball 1 must lie in quadrants 1 0r 2, and Balls 2 & 3 must lie in quadrants 3 and 4Enter X,Y,Z coordinates and alpha, beta, gamma direction cosines for Ball 1Dball (m) 0.0350Dcoupling (m) 0.625

Contact point 1 Contact point 2Xba = 0.012374 Xbb = -0.012374Yba = 0.312500 Ybb = 0.312500Zba = -0.012374 Zbb = -0.012374Aba = -0.707107 Abb = 0.707107Bba = 0.000000 Bbb = 0.000000Gba = 0.707107 Gbb = 0.707107

User values if matlabs = 5Enter characteristics for groove 1 and ball 1Egone = 2.04E+11 Groove material elastic modulus, or enter value (mm):vgone = 0.29000 Groove material Poisson ratio, or enter value (mm):Rgone = 1000 Groove radius of curvatureEbone = 2.04E+11 Ball material elastic modulus, or enter value (Pa):vbone = 0.29 Ball material Poisson ratio, or enter value (Pa):Eeone = 1.11E+11 Equivelent modulusDbone = 3.50E-02 Equivelent diameter ball that would contact the groove at the same pointsRpone = 0.5 "Ball" major radius of curvature, or enter value (mm):Raone = 0.5 "Ball" minor radius of curvture, or enter value (mm):Reone = 0.24993751562109 Equivelent radiusctone = 0.00024993751562 Cos(theta)theta_1 = 1.57054638927667alpha_1 = 1.00317603045331beta_1 = 1.00125019938518lambda_1 = 0.74903902176841Sone = 1.72E+09 Allowable Hertz stress, or enter value:Enter X,Y,Z coordinates and alpha, beta, gamma direction cosines for Ball 2

Contact point 3 Contact point 4Xbc = -0.276820 Xbd = -0.264446Ybc = -0.145533 Ybd = -0.166967Zbc = -0.012374 Zbd = -0.012374Abc = 0.353553 Abd = -0.353553Bbc = -0.612372 Bbd = 0.612372Gbc = 0.707107 Gbd = 0.707107

User values if matlabs = 5Enter characteristics for groove 2 and pin 2Egtwo = 2.04E+11 Groove material elastic modulus, or enter value (mm):vgtwo = 0.29000 Groove material Poisson ratio, or enter value (mm):Rgtwo = 1000 Groove radius of curvatureEbtwo = 2.04E+11 Ball material elastic modulus, or enter value (Pa):vbtwo = 0.29 Ball material Poisson ratio, or enter value (Pa):Eetwo = 1.11E+11 Equivelent modulusDbtwo = 3.50E-02 Equivelent diameter ball that would contact the groove at the same pointsRptwo = 0.5 "Ball" major radius of curvature, or enter value (mm):Ratwo = 0.5 "Ball" minor radius of curvture, or enter value (mm):Retwo = 0.24993751562109 Equivelent radius

A B C D E F G H555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899100101102103104105106107108109

Appendix A

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cttwo = 0.00024993751562 Cos(theta)theta_2 = 1.57054638927667alpha_2 = 1.00317603045331beta_2 = 1.00125019938518lambda_2 = 0.74903902176841Stwo = 1.72E+09 Allowable Hertz stressEnter X,Y,Z coordinates and alpha, beta, gamma direction cosines for Ball 3

Contact point 5 Contact point 6Xbe = 0.264446 Xbf = 0.276820Ybe = -0.166967 Ybf = -0.145533Zbe = -0.012374 Zbf = -0.012374Abe = 0.353553 Abf = -0.353553Bbe = 0.612372 Bbf = -0.612372Gbe = 0.707107 Gbf = 0.707107

User values if matlabs = 5Enter characteristics for groove 3 and ball 3Egthree = 2.04E+11 Groove material elastic modulus, or enter value (mm):vgthree = 0.29000 Groove material Poisson ratio, or enter value (mm):Rgthree = 1000 Groove radius of curvatureEbthree = 2.04E+11 Ball material elastic modulus, or enter value (Pa):vbthree = 0.29 Ball material Poisson ratio, or enter value (Pa):Eethree = 1.11E+11 Equivelent modulusDbthree = 3.50E-02 Equivelent diameter ball that would contact the groove at the same pointsRpthree = 0.5 "Ball" major radius of curvature, or enter value (mm):Rathree = 0.5 "Ball" minor radius of curvture, or enter value (mm):Rethree = 0.24993751562109 Equivelent radiusctthree = 0.00024993751562 Cos(theta)theta_3 = 1.57054638927667alpha_3 = 1.00317603045331beta_3 = 1.00125019938518lambda_3 = 0.74903902176841Sthree = 1.72E+09 Allowable Hertz stressPreload forces' X,Y,Z components and coordinatesFpxone = 0 Fpxtwo = 0 Fpxthree = 0Fpyone = 0 Fpytwo = 0 Fpythree = 0Fpzone = -15000 Fpztwo = -15000 Fpzthree = -15000Xpone = 0 Xptwo = -0.270632938683 Xpthree = 0.270632938682637Ypone = 0.3125 Yptwo = -0.15625 Ypthree = -0.15625Zpone = 0.07 Zptwo = 0.07 Zpthree = 0.07Calculations:Build Force Moment equilibrium matrices: AF = B (Equations 1-6)Matrix A Matrix F B with loadsFbn1 Fbn2 Fbn3 Fbn4 Fbn5 Fbn6

-7.07E-01 7.07E-01 3.54E-01 -3.54E-01 3.54E-01 -3.54E-01 Fbn1 -0.00E+000.00E+00 0.00E+00 -6.12E-01 6.12E-01 6.12E-01 -6.12E-01 Fbn2 5.21E+037.07E-01 7.07E-01 7.07E-01 7.07E-01 7.07E-01 7.07E-01 Fbn3 7.45E+042.21E-01 2.21E-01 -1.10E-01 -1.10E-01 -1.10E-01 -1.10E-01 Fbn4 -6.88E+030.00E+00 0.00E+00 1.91E-01 1.91E-01 -1.91E-01 -1.91E-01 Fbn5 -0.00E+002.21E-01 -2.21E-01 2.21E-01 -2.21E-01 2.21E-01 -2.21E-01 Fbn6 -0.00E+00

Res. Forces with applied loads Res forces with preload onlyfbnone 7190.84 fone 10606.60fbntwo 7190.84 ftwo 10606.60fbnthree 20633.20 fthree 10606.60fbnfour 24886.69 ffour 10606.60fbnfive 24886.69 ffive 10606.60fbnsix 20633.20 fsix 10606.60

A B C D E F G H110111112113114115116117118119120121122123

124

125126127128129130131132133134135136137138139140141142143144145146147148149150151152153154155156157158159160161162163164

Appendix A

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Original ball corrdinatesxboneO 0.0000000 xbtwoO -0.2706329 xbthreeO 0.2706329yboneO 0.3125000 ybtwoO -0.1562500 ybthreeO -0.1562500zboneO 0.0000000 zbtwoO 0.0000000 zbthreeO 0.0000000New ball coordinates (=original + ball deflection*direction cosines)xboneN 0.00000000000 xbtwoN -0.2706337 xbthreeN 0.2706337yboneN 0.3125000 ybtwoN -0.1562486 ybthreeN -0.1562486zboneN -0.0000035 zbtwoN 0.0000101 zbthreeN 0.0000101Ball centers' deflectionsdxone 0.0000E+00 dxtwo -7.9414E-07 dxthree 7.9414E-07dyone 0.0000E+00 dytwo 1.3755E-06 dythree 1.3755E-06dzone -3.4960E-06 dztwo 1.0139E-05 dzthree 1.0139E-05Theory applicability check:Initiial dist. between balls Final dist. between balls DifferenceLotI 5.4127E-01 LotN 5.4127E-01 DLotI 7.9397E-07LttI 5.4127E-01 LttN 5.4127E-01 DLttI -1.5883E-06LtoI 5.4127E-01 LtoN 5.4127E-01 DLtoI 7.9397E-07Change in length/distance between balls Deflection/ball radius Ratio (should be >5)

1.47E-06 2.83E-04 1.93E+022.93E-06 9.48E-04 3.23E+021.47E-06 9.48E-04 6.46E+02

Coupling centroid is assumed to be at intersection of coupling triangle's angle bisectors but if only apply load over balls, gives erroroneous valueInitial centroid Distance from ball to centroid Error motion at centroid from weighted ball motion initial theory was dzone*(LotI-Dcone)/LotI+dztwo*(LttI-Dctwo)/LttI+dzthree*(LtoI-Dcthree)/LtoIxci 0.000000000 Dcone 0.312500000 dxc 0.00E+00 0yci 0.000000000 Dctwo 0.312500000 dyc 9.17E-07zci 0.000000000 Dcthree 0.312500000 dzc 5.59E-06Original angles between balls Original altitude lengthsAngone ### angle at ball 1 Aone 0.4687 Ball 1 to side 2 3Angtwo ### angle at ball 2 Atwo 0.4688 Ball 2 to side 1 3Angthree ### angle at ball 3 Athree 0.4688 Ball 3 to side 2 1New angles between balls Original sides' angle with X axisAngoneN 60.0003 angle at ball 1 Aot 60 Side opposite ball 3AngtwoN 59.9999 angle at ball 2 Att 0 Side opposite ball 1AngthreeN 59.9999 angle at ball 3 Ato 120 Side opposite ball 2New sides' angle with X axisAotN 59.99985439712 Side opposite ball 3 0AttN 0 Side opposite ball 1 0AtoN 120.00014560288 Side opposite ball 2Original altitudes' slope angles and Y interceptsAmtwoO 30 AbtwoO 0AmthreeO 150 AbthreeO 0Rotation about opposite side (radians)Ttt -7.46E-06 rotation about side 23 due to Z motion at ball 1Tto 2.16E-05 rotation about side 13 due to Z motion at ball 2Tot 2.16E-05 rotation about side 12 due to Z motion at ball 3Coupling error rotationsEpsX -2.91E-05 EpsZ1 0.00E+00 Z rot from ball 1EpsY 0.00E+00 EpsZ2 -5.08E-06 Z rot from ball 2EpsZ 0.00E+00 EpsZ3 5.08E-06 Z rot from ball 3Coupling HTM Point of interest

1.00E+00 -0.00E+00 0.00E+00 0.00E+00 Xerr 00.00E+00 1.00E+00 2.91E-05 9.17E-07 Yerr 0

-0.00E+00 -2.91E-05 1.00E+00 5.59E-06 Zerr 1.32080.00E+00 0.00E+00 0.00E+00 1.00E+00 1 1

A B C D E F G H165166167168169170171172173174175176177178179180181182183184185186187188189190191192193194195196197198199200201202203204205206207208209210211212213214215216217218219

Appendix A

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Error displacements at the point of interestDeltaX 0.00E+00DeltaY 3.93E-05DeltaZ 5.59E-06

0

A B C D E F G H220221222223224