Mit2 72s09 lec05
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This document provides lecture materials on structural design and analysis from the MIT OpenCourseWare course 2.72 Elements of Mechanical Design. The materials cover topics such as matrix operations, transformation matrices, structural modeling using stick figures, and the use of homogeneous transformation matrices to analyze structural systems and determine deflections and forces. Examples are provided on using HTMs to analyze a simple beam and a drill press structure.
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Mit2 72s09 lec05
- 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.
- 2. 2.72 Elements of Mechanical Design Lecture 05: Structures
- 3. Schedule and reading assignment © Martin Culpepper, All rights reserved Quizzes ‰ Quiz – None Topics ‰ Finish fatigue ‰ Finish HTMs in structures Reading assignment ‰ None ‰ Quiz next time on HTMs 2
- 5. What is a Matrix? bA matrix is an easy way to 1 ⎤ ⎥ ⎦ ⎤ ⎥ ⎦ ⎡ represent a system of linear ⎢bequations ⎣ 2 Linear algebra is the set of “Vector”rules that governs matrix ⎡ ⎢ ⎣ and vector operations a1 a2 a3 a4 “Matrix” © Martin Culpepper, All rights reserved 4
- 6. Matrix Addition/Subtraction You can only add or subtract matrices of the same dimension Operations are carried out entry by entry b b b b ⎤+a a a a1 2 1 2 1 1 2 2 ⎥+ ⎦ (2 x 2) (2 x 2) (2 x 2) − ⎤ ⎥− ⎦ + = + b b b b+a a a a3 4 3 4 3 3 4 4 − − ⎡ ⎢ ⎣ ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ b b b b⎤a a a a1 2 1 2 1 1 2 2 ⎥b b b ba a a a⎦3 4 3 4 3 3 4 4 (2 x 2) (2 x 2) (2 x 2) − = ⎡ ⎢ ⎣ ⎡ ⎢ ⎣ © Martin Culpepper, All rights reserved ⎤ ⎥ ⎦ ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ ⎡ ⎢ ⎣ 5
- 7. Matrix Multiplication ⎤ ⎥ ⎦ An matrix times an matrix produces an matrixm x n n x p m x p b b b b b b++a a a a a a1 2 1 2 1 1 2 3 1 2 2 4 b b b b b b++a a a a a a3 4 3 4 3 1 4 3 3 2 4 4 (2 x 2) (2 x 2) (2 x 2) = ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ © Martin Culpepper, All rights reserved ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ 6
- 8. Matrix Properties © Martin Culpepper, All rights reserved Notation: A, B, C = matrix , c = scalar Cumulative Law: A + B = B + A Distributive Law: c(A + B) = cA + cB C(A + B) = CA + CB Associative Law: A + (B – C) = (A + B) – C A(BC) = (AB)C NOTE that AB does not equal BA !!!!!!! 7
- 9. Matrix Division © Martin Culpepper, All rights reserved To divide in linear algebra we multiply each side by an inverse matrix: AB = C A-1AB = A-1C B = A-1C Inverse matrix properties: A-1A = AA-1 = I (The identity matrix) (AB)-1 = B-1A-1 8
- 10. Structures
- 11. Machines structures Structure = backbone = affects everything Satisfies a multiplicity of needs ‰ Enforcing geometric relationships (position/orientation) ‰ Material flow and access ‰ Reference frame Requires first consideration and serves to link modules: ‰ Joints (bolted/welded/etc…) ‰ Bearings ‰ Shafts Image removed due to copyright restrictions. Please see http://www.clarkmachinetools.com/2003_1.jpg ‰ Parts ‰ Tools ‰ Sensors ‰ Actuators © Martin Culpepper, All rights reserved 10
- 12. Key issues with structural design Machine concepts ‰ Topology Image removed due to copyright restrictions. Please see ‰ Material properties http://www.fortune-cnc.com/uploads/images/1600ge_series.jpg Principles ‰ Thermomechanical ‰ Elastomechanics ‰ Kinematics ‰ Vibration Key tools that help ‰ Stick figures ‰ Parametric system/part error model Visualization of the: Load path Vibration modes Thermal growth © Martin Culpepper, All rights reserved 11
- 13. Modeling: stick figures Image removed due to copyright restrictions. Please see http://americanmachinetools.com/images/diagram-lathe.jpg 1 2 3 © Martin Culpepper, All rights reserved 12
- 14. Modeling: stick figures 1. Stick figures 2. Beam bending 3. System bend. 1 2 3 These types of models are idealizations of the physical behavior. The designer must KNOW: (a) if beam bending assumptions are valid (b) how to interpret and use the results o this type of these models y z © Martin Culpepper, All rights reserved 13 x
- 15. Modeling: stick figures F © Martin Culpepper, All rights reserved 14
- 17. Translational Transformation MatrixTranslational Transformation Matrix © Martin Culpepper, All rights reserved P=(A+L1, B+L2)=(A’, B’) X Y P=(A, B) A B X’ Y’ L1 L2 A B 1 1 1 L1 L2 0 0 0 0 1 = A’ B’ 1 16
- 18. Translational Transformation Matrix General 2D transformation matrix 1 0 0 1 0 0 ΔX 1 ΔY 1 0 0 1 0 0 L1 L2 1 © Martin Culpepper, All rights reserved 17
- 19. Rotational Transformation MatrixRotational Transformation Matrix Y Y’ B’ B X X’ Ө Ө P=(A, B)P=(A’, B’) A A’ Ө © Martin Culpepper, All rights reserved 18
- 20. Rotational Transformation MatrixRotational Transformation Matrix Y P=(A’, B’)Y’ X’ Ө Ө Ө B X A A’ = A cosӨ + B sin Ө © Martin Culpepper, All rights reserved 19
- 21. Rotational Transformation Matrix Y P=(A’, B’)Y’ X’ Ө Ө Ө B X A A’ = A cosӨ + B sin Ө B’ = -A sinӨ + B cos Ө © Martin Culpepper, All rights reserved 20
- 22. Rotational Transformation MatrixRotational Transformation Matrix Y Y’ P=(A’, B’) X’ Ө Ө A’ A A’ B’ B X cosӨ sinӨ 0 A = BB’ -sinӨ cosӨ 0 A’ = A cosӨ + B sin Ө 1 10 0 1 B’ = -A sinӨ + B cos Ө © Martin Culpepper, All rights reserved 21
- 23. Rotational Transformation MatrixRotational Transformation Matrix Y Y’ (+Ө) Counter Clockwise X’ cosӨ sinӨ 0 General 2D rotational matrix: Ө -sinӨ cosӨ 0 0 0 1Ө X Y Y’ (-Ө) Clockwise cosӨ -sinӨ 0 cosӨsinӨsinӨcosӨ0 0 0 0 Ө 1-sinӨ cosӨ 0 X Ө 0 0 1 X’ © Martin Culpepper, All rights reserved 22
- 24. Homogeneous Transformation Matrix General 2D HTM translational and rotational matrix: cosӨ sinӨ Δx -sinӨ cosӨ Δy 0 0 1 © Martin Culpepper, All rights reserved 23
- 25. HTM Applications
- 26. Simple Beam Example:Simple Beam Example © Martin Culpepper, All rights reserved X’ B Y’ L F X A Y 25
- 27. Simple Beam Example:Simple Beam Example X’ Y’ B X Y A L F D Ө © Martin Culpepper, All rights reserved 26
- 28. Force Deflection EquationsUseful Force-deflection Equations © Martin Culpepper, All rights reserved F F M d d Ө d Ө d = FL EA d = FL3 3EI Ө = Ө =d = ML2 2EI FL2 2EI ML EI 27
- 29. Simple Beam Example:Simple Beam Example A Y X Ө © Martin Culpepper, All rights reserved X’ B Y’ L cosӨ 0 0 -sinӨ sinӨ cosӨ= L D 1 B BHA = L D 1 0 0 1 A D = Ө = FL3 3EI FL2 2EI 28 D
- 30. Simple Beam Example:Drill Press Example Find the HTM from a to b: c d L L H b D a F e Cross-Sectional Area of large sections = A Cross-Sectional Area of Drill Bit = Ad Young’s Modulus of Material = E b a F 1 0 0 bHa = 0 1 -(D-δ) 0 0 1 © Martin Culpepper, All rights reserved 29
- 31. Force Deflection EquationsUseful Force-deflection Equations © Martin Culpepper, All rights reserved F F M d d Ө d Ө d = FL EA d = FL3 3EI Ө = Ө =d = ML2 2EI FL2 2EI ML EI 30
- 32. Simple Beam Example:Drill Press Example Find the HTM from a to b: c d L L H b D a F e Cross-Sectional Area of large sections = A Cross-Sectional Area of Drill Bit = Ad Young’s Modulus of Material = E b a F 1 0 0 bHa = 0 1 -(D-δ) 0 0 1 FD δ = EAd © Martin Culpepper, All rights reserved 31
- 33. Simple Beam Example:Drill Press Example Find the HTM from b to c: c d L L H b c b D a F F cosӨ -sinӨ L e cHb = sinӨ cosӨ δ 0 0 1 Cross-Sectional Area of large sections = A Cross-Sectional Area of Drill Bit = Ad Young’s Modulus of Material = E © Martin Culpepper, All rights reserved 32
- 34. Force Deflection EquationsUseful Force-deflection Equations © Martin Culpepper, All rights reserved F F M d d Ө d Ө d = FL EA d = FL3 3EI Ө = Ө =d = ML2 2EI FL2 2EI ML EI 33
- 35. Simple Beam Example:Drill Press Example Find the HTM from b to c: c d L L H b c b D a F F cosӨ -sinӨ L e cHb = sinӨ cosӨ δ 0 0 1 Cross-Sectional Area of large sections = A Cross-Sectional Area of Drill Bit = Ad Young’s Modulus of Material = E δ = FL3 3EI Ө = FL2 2EI © Martin Culpepper, All rights reserved 34
- 36. Simple Beam Example:Drill Press Example c d L L H a F e Cross-Sectional Area of large sections = A Cross-Sectional Area of Drill Bit = Ad Young’s Modulus of Material = E Find the HTM from c to d: F b FLD c d cosӨ -sinӨ -δ1 dHc = sinӨ cosӨ H+δ2 0 0 1 © Martin Culpepper, All rights reserved 35
- 37. Force Deflection EquationsUseful Force-deflection Equations © Martin Culpepper, All rights reserved F F M d d Ө d Ө d = FL EA d = FL3 3EI Ө = Ө =d = ML2 2EI FL2 2EI ML EI 36
- 38. Simple Beam Example:Drill Press Example c d L L H Find the HTM from c to d: c F b FLD a F e d cosӨ -sinӨ -δ1 dHc = sinӨ cosӨ H+δ2 Cross-Sectional Area of large sections = A 0 0 1 Cross-Sectional Area of Drill Bit = Ad Young’s Modulus of Material = E FLH FLH2 FH Ө = EI δ1 = 2EI δ2 = EA 37© Martin Culpepper, All rights reserved
- 39. Simple Beam Example:Drill Press Example Find the HTM from d to e: c d L L H b D a F e Cross-Sectional Area of large sections = A Cross-Sectional Area of Drill Bit = Ad Young’s Modulus of Material = E ed F FL 1 0 -L eHd = 0 1 0 0 0 1 © Martin Culpepper, All rights reserved 38
- 40. Simple Beam Example:Drill Press Example Find the HTM from a to e: c d L L H b D a F e eHa= eHd dHc cHb bHa Find the vector aV from e to a:e 0 0 aV = eHae 1 Cross-Sectional Area of large sections = A Cross-Sectional Area of Drill Bit = Ad Young’s Modulus of Material = E © Martin Culpepper, All rights reserved 39
- 41. Simple Beam Example:Method for building system’s HTM • Identify key nodes around the system’s structural loop • Create HTMs for each member between each node • Multiply the member’s HTMs in the correct order © Martin Culpepper, All rights reserved 40
- 42. More on HTMs
- 43. 3D HTMs 1 0 0 X 0 sin cos Z ⎤ ⎥For x-axis Y0 sincos ⎥x x rotation ⎥ ⎥ ⎦ cos(Ө)~1 & sin(Ө)~ Ө X0 sin ⎤cos ⎥For y-axis ⎥ ⎥ ⎥ ⎦ ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ rotation sin 0 cos Z θ θx x θy y θy y 0 0 0 1 z zθ θ θ θ z z − θ θ θ θ rotation Z0 0 1 ⎡ ⎢ ⎢ ⎢ ⎢ 0 0 0 1⎣ For small Ө: −⎡ ⎢ Y0 1 0⎢ ⎢ ⎢ ⎣ Xsin 0⎡ cos ⎢ Ysin 0−For z-axis cos⎢ ⎢ ⎢ 0 0 0 1⎣ © Martin Culpepper, All rights reserved 42
- 44. HTM Rotation •Remember order of multiplication matters: 180 deg 90 deg rightflip 180 deg flip 90 deg right •To combine a translation and rotation, again multiply the HTM matrices together •Note that the order of the rotation and translation matrices does matter, so makes sure the answer makes sense!!! © Martin Culpepper, All rights reserved 43