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 05: Structures 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 Matrix Review What is a Matrix? b A matrix is an easy way to 1 ⎤ ⎥ ⎦ ⎤ ⎥ ⎦ ⎡ represent a system of linear ⎢ b equations ⎣ 2 Linear algebra is the set of “Vector” rules that governs matrix ⎡ ⎢ ⎣ and vector operations a 1 a 2 a 3 a 4 “Matrix” © Martin Culpepper, All rights reserved 4 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 a 1 2 1 2 1 1 2 2 ⎥ + ⎦ (2 x 2) (2 x 2) (2 x 2) − ⎤ ⎥ − ⎦ + = + b b b b + a a a a 3 4 3 4 3 3 4 4 − − ⎡ ⎢ ⎣ ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ b b b b ⎤ a a a a 1 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 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 a 1 2 1 2 1 1 2 3 1 2 2 4 b b b b b b ++ a a a a a a 3 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 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 Matrix Division © Martin Culpepper, All rights reserved To divide in linear algebra we multiply each side by an inverse matrix: AB = C A -1 AB = A -1 C B = A -1 C Inverse matrix properties: A -1 A = AA -1 = I (The identity matrix) (AB) -1 = B -1 A -1 8 Structures [...]... Cross-Sectional Area of large sections = Cross-Sectional Area of Drill Bit = Young’s Modulus of Material = © Martin Culpepper, All rights reserved 1 e E Ad A 0 0 0 1 -(D-δ) 0 0 1 δ= FD EAd 31 Drill Press Example Simple Beam Example: Find the HTM from b to c: L c b H a D c b F F cosӨ L e Cross-Sectional Area of large sections = Cross-Sectional Area of Drill Bit = Young’s Modulus of Material = © Martin... Cross-Sectional Area of large sections = Cross-Sectional Area of Drill Bit = Young’s Modulus of Material = © Martin Culpepper, All rights reserved E Ad c Hb = sinӨ cosӨ δ 0 d -sinӨ L 0 1 A FL3 δ= 3EI FL2 Ө= 2EI 34 Drill Press Example Simple Beam Example: Find the HTM from c to d: L c c b H a F FL D F d L e d cosӨ Cross-Sectional Area of large sections = Cross-Sectional Area of Drill Bit = Young’s Modulus of Material... 3 1 2 © Martin Culpepper, All rights reserved 12 Modeling: stick figures 1 Stick figures These types of models are idealizations of the physical behavior The designer must KNOW: 2 Beam bending 3 System bend (a) if beam bending assumptions are valid (b) how to interpret and use the results o this type of these models 3 1 2 © Martin Culpepper, All rights reserved y z x 13 Modeling: stick figures F © Martin... with structural design Machine concepts ‰ ‰ Topology Material properties Image removed due to copyright restrictions Please see 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 © Martin Culpepper, All rights reserved Visualization of the: Load path... -sinӨ L 0 cosӨ D 0 0 1 1 FL3 D= 3EI FL2 Ө= 2EI 28 Drill Press Example Simple Beam Example: Find the HTM from a to b: L b c b H a D a F F d L Ha = b Cross-Sectional Area of large sections = Cross-Sectional Area of Drill Bit = Young’s Modulus of Material = © Martin Culpepper, All rights reserved 1 e A 0 0 0 1 -(D-δ) 0 0 1 Ad E 29 Useful Force-deflection Equations Force Deflection Equations d d= F d Ө FL EA...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…) . 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. 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. Principles  Thermomechanical  Elastomechanics  Kinematics  Vibration Key tools that help  Stick figures  Parametric system/part error model Visualization of the: Load path Vibration