際際滷

際際滷Share a Scribd company logo

Mit2 72s09 lec09

Jasim Almuhandis
Jasim Almuhandis

This document summarizes key points from a lecture on mechanical design alignment methods: - Common alignment methods discussed include elastic averaging, compliant kinematic couplings, quasi-kinematic, active kinematic, and passive kinematic couplings. Accuracy and repeatability are important factors. - Passive kinematic couplings provide fixed alignment through contact forces and preloading. Modeling considers contact forces, stresses, stiffness variations, friction, and thermal effects. - Ball-based couplings can be modeled as equivalent radii and moduli to determine displacements and contact stiffness scaling laws. Proper preloading improves repeatability. - Lubrication significantly improves radial repeatability of kinematic couplings compared to unlubricated couplings

1 of 14
Download to read offline
MIT OpenCourseWare http://ocw.mit.edu 2.72 Elements of Mechanical Design Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
2.72 Elements of Mechanical Design Lecture 09: Alignment
Schedule and reading assignment Quiz Thursday: Hale 6.1 Soon: Bolted joint qualifying quiz Topics Lab notebooks Alignment methods Kinematic coupling grade bump = 遜 grade for use/design Reading assignment Read: 8.2 Examples: All in 8.2 息 Martin Culpepper, All rights reserved 2
Lab notebooks Technical quality/quantity Appropriate equations, codes Units Important results highlighted/boxed/noted/explained Graphical quality/quantity Appropriate sketches/pictures Pasted CAD/etc Archival quality Can this be copied? Understood by others? Best practices Dating and number of pages Permanent pen No blank spaces (X out) 息 Martin Culpepper, All rights reserved 3
Ideal alignment interface Repeatable Accuracy Stiffness (sensitive?) Load capacity Perfect constraint Lowest energy state 6 DOF Metal molds High natural frequency 息 Martin Culpepper, All rights reserved 4
Common alignment methods Elastic averagElastic averagiingng Compliant kinCompliant kineematicmatic QuasiQuasi--kinematickinematic Active kinematicActive kinematic Passive kinematicPassive kinematic AccuracyAccuracy RepeatabilityRepeatability Accuracy & repeatability Accuracy & repeatability Desired Position Elastic BElastic B Passive KCPassive KC Active KCActive KC Elastic CElastic C Elastic AElastic A QuasiQuasi--KCKC ErrorErrorErrorError 息 Martin Culpepper, All rights reserved 5
Pin-hole 6 DOF Metal molds 息 Martin Culpepper, All rights reserved 6
3 2 1 Alignment schemes 息 Martin Culpepper, All rights reserved 7
Exact constraint couplings Exact constraint (EC): Constraints = DOF to be constrained Deterministic saves $ Balls (inexpensive) & grooves (more difficult to make) In KC design the issues are: KNOW what is happening in the system (coupling) MANAGE forces, deflections, stresses and friction There are many types of EC couplings, our time limits us to a semi-focused study on kinematic couplings Balls Tetrahedral groove Maxwell V groove Kelvin 息 Martin Culpepper, All rights reserved 8
Passive kinematic couplings Fabricate and forget 村 micrometer with best practices, 10s of nm recently What is important? Contact forces Contact stress Stiffness vs. geometry Stiffness vs. preload Friction & settling Thermal loading Preload repeatability Preload (nesting load) is the force applied to keep the coupling components engaged and prevent tipping Preload 息 Martin Culpepper, All rights reserved 9
Ball motions: Displacements iABall iABall ee n iABalln n ER F _ _ 3 1 2 2 __ 16 9 =裡隆 v iBBall iBBall ee n iBBalln n ER F _ _ 3 1 2 2 __ 16 9 =裡隆 v Hertz 1857-1894 v v v 裡隆Ball _i = 隆Ball _iA +隆Ball _iB This assumes that the ball-ball stiffness is > ~10x ball-groove stiffness Ball far-field point B1 B3 B2 A B iABall _隆 v iBBall _隆 v iBall _隆 v Groove far-field points 息 Martin Culpepper, All rights reserved 10
Load balance: Force and moment Preload Force balance (3 equations) Error Contact 裡F v relative = 0 = ( preloadF v + ErrorF v )+ ( Moment balance (3 equations) M r relative = ( iBalli M _1 6 r = )裡 + ( preloadM r )+ ( errorM r )= ( preloadpreload Fr vv + Errorerror Fr vv )+ 裡 iBalliBall Fr __ vv Goal: 1. Solve 6 equations for contact forces 2. Solve normal displacements 3. Solve relative displacements/rotations Ball far-field point Given geometry, materials, preload force, error force, solve for local distance of approach A B Groove far-field points )6_5_4_3_2_1_ BallBallBallBallBallBall FFFFFF vvvvvv +++++ 息 Martin Culpepper, All rights reserved 11
Modeling round interfaces Equivalent radius 1 R =e 1 1 1 1 + + + R1major R1minor R2major R2minor Equivalent modulus 1 Ee = 2 2 Poissons ratio1侶1 1侶2 + E1 E2 Youngs modulus 1 9 F 2 3 n 隆 = n 16 Re Ee 2 Important scaling law Scaling with Matl properties and geometry Contact stiffness 0 05 10 20 30 0 250 500 750 1000 Fn [N] k[N/micron] Preload should be repeatable in magnitude & direction Degree of nonlinearity is reduced as preload is increased 3 E 3 3k 隆 = 2 R E 隆( ) ( 0.5 ) 0.5 k ( )F = Constant (R 1 2 ) F 1 n n e e n n n e e n 息 Martin Culpepper, All rights reserved 12
Friction and lubrication The trend of the data is important Wear in vs. snow balling Magnitude depends on coupling design and test conditions Slocum, A. H., Precision Engineering, 1988: Kinematic couplings for precision fixturing Experimental determination of repeatability and stiffness Number of Trials Radial Repeatability (Unlubricated) 2 0 Displacement,亮m Radial Repeatability (Lubricated) Displacement,亮m 2 0 Number of Trials 息 Martin Culpepper, All rights reserved Courtesy of Elsevier, Inc., http://www.sciencedirect.com. Used with permission. 13

More Related Content

Mit2 72s09 lec09

  • 1. MIT OpenCourseWare http://ocw.mit.edu 2.72 Elements of Mechanical Design Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
  • 3. Schedule and reading assignment Quiz Thursday: Hale 6.1 Soon: Bolted joint qualifying quiz Topics Lab notebooks Alignment methods Kinematic coupling grade bump = 遜 grade for use/design Reading assignment Read: 8.2 Examples: All in 8.2 息 Martin Culpepper, All rights reserved 2
  • 4. Lab notebooks Technical quality/quantity Appropriate equations, codes Units Important results highlighted/boxed/noted/explained Graphical quality/quantity Appropriate sketches/pictures Pasted CAD/etc Archival quality Can this be copied? Understood by others? Best practices Dating and number of pages Permanent pen No blank spaces (X out) 息 Martin Culpepper, All rights reserved 3
  • 5. Ideal alignment interface Repeatable Accuracy Stiffness (sensitive?) Load capacity Perfect constraint Lowest energy state 6 DOF Metal molds High natural frequency 息 Martin Culpepper, All rights reserved 4
  • 6. Common alignment methods Elastic averagElastic averagiingng Compliant kinCompliant kineematicmatic QuasiQuasi--kinematickinematic Active kinematicActive kinematic Passive kinematicPassive kinematic AccuracyAccuracy RepeatabilityRepeatability Accuracy & repeatability Accuracy & repeatability Desired Position Elastic BElastic B Passive KCPassive KC Active KCActive KC Elastic CElastic C Elastic AElastic A QuasiQuasi--KCKC ErrorErrorErrorError 息 Martin Culpepper, All rights reserved 5
  • 7. Pin-hole 6 DOF Metal molds 息 Martin Culpepper, All rights reserved 6
  • 8. 3 2 1 Alignment schemes 息 Martin Culpepper, All rights reserved 7
  • 9. Exact constraint couplings Exact constraint (EC): Constraints = DOF to be constrained Deterministic saves $ Balls (inexpensive) & grooves (more difficult to make) In KC design the issues are: KNOW what is happening in the system (coupling) MANAGE forces, deflections, stresses and friction There are many types of EC couplings, our time limits us to a semi-focused study on kinematic couplings Balls Tetrahedral groove Maxwell V groove Kelvin 息 Martin Culpepper, All rights reserved 8
  • 10. Passive kinematic couplings Fabricate and forget 村 micrometer with best practices, 10s of nm recently What is important? Contact forces Contact stress Stiffness vs. geometry Stiffness vs. preload Friction & settling Thermal loading Preload repeatability Preload (nesting load) is the force applied to keep the coupling components engaged and prevent tipping Preload 息 Martin Culpepper, All rights reserved 9
  • 11. Ball motions: Displacements iABall iABall ee n iABalln n ER F _ _ 3 1 2 2 __ 16 9 =裡隆 v iBBall iBBall ee n iBBalln n ER F _ _ 3 1 2 2 __ 16 9 =裡隆 v Hertz 1857-1894 v v v 裡隆Ball _i = 隆Ball _iA +隆Ball _iB This assumes that the ball-ball stiffness is > ~10x ball-groove stiffness Ball far-field point B1 B3 B2 A B iABall _隆 v iBBall _隆 v iBall _隆 v Groove far-field points 息 Martin Culpepper, All rights reserved 10
  • 12. Load balance: Force and moment Preload Force balance (3 equations) Error Contact 裡F v relative = 0 = ( preloadF v + ErrorF v )+ ( Moment balance (3 equations) M r relative = ( iBalli M _1 6 r = )裡 + ( preloadM r )+ ( errorM r )= ( preloadpreload Fr vv + Errorerror Fr vv )+ 裡 iBalliBall Fr __ vv Goal: 1. Solve 6 equations for contact forces 2. Solve normal displacements 3. Solve relative displacements/rotations Ball far-field point Given geometry, materials, preload force, error force, solve for local distance of approach A B Groove far-field points )6_5_4_3_2_1_ BallBallBallBallBallBall FFFFFF vvvvvv +++++ 息 Martin Culpepper, All rights reserved 11
  • 13. Modeling round interfaces Equivalent radius 1 R =e 1 1 1 1 + + + R1major R1minor R2major R2minor Equivalent modulus 1 Ee = 2 2 Poissons ratio1侶1 1侶2 + E1 E2 Youngs modulus 1 9 F 2 3 n 隆 = n 16 Re Ee 2 Important scaling law Scaling with Matl properties and geometry Contact stiffness 0 05 10 20 30 0 250 500 750 1000 Fn [N] k[N/micron] Preload should be repeatable in magnitude & direction Degree of nonlinearity is reduced as preload is increased 3 E 3 3k 隆 = 2 R E 隆( ) ( 0.5 ) 0.5 k ( )F = Constant (R 1 2 ) F 1 n n e e n n n e e n 息 Martin Culpepper, All rights reserved 12
  • 14. Friction and lubrication The trend of the data is important Wear in vs. snow balling Magnitude depends on coupling design and test conditions Slocum, A. H., Precision Engineering, 1988: Kinematic couplings for precision fixturing Experimental determination of repeatability and stiffness Number of Trials Radial Repeatability (Unlubricated) 2 0 Displacement,亮m Radial Repeatability (Lubricated) Displacement,亮m 2 0 Number of Trials 息 Martin Culpepper, All rights reserved Courtesy of Elsevier, Inc., http://www.sciencedirect.com. Used with permission. 13