2.72 Elements of Mechanical Design

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1 MIT OpenCourseWare Elements of Mechanical Design Spring 2009 For information about citing these materials or our Terms of Use, visit:

2 2.72 Elements of Mechanical Design Lecture 02: Review

3 Intent High-level review of undergrad material as applied to engineering decision making NOT an ME redo or a how to recitation

4 Import Main goal of 2.72 is to teach you how to integrate past knowledge to engineer a system Given this, how do I engineer a mechanical system? modular simple complex system 2.001, , , , 2.008

6 Impact Understand why & how we will use parts of ME core knowledge Problem set Engineering

7 Future help We can t use lecture time to redo the early curriculum BUT We are HAPPY to help outside of lecture IF you ve tried

8 Schedule and reading assignment Reading quiz Changing from sponge to active mode Lecture Mechanics Dynamics Heat transfer Matrix math Hands-on Mechanics Dynamics Heat transfer Matrix math Reading assignment Shigley/Mischke Sections : 08ish pages & Sections : 11ish pages Pay special attention to examples 4.1, 4.4, 5.3, and 5.4 Martin Culpepper, All rights reserved 7

9 Mechanics

10 Free body diagrams Useful for: v v ΣF = 0 = ma Equilibrium Stress, deflection, vibration, etc ΣM = 0 = v Iα Why do we ALWAYS use free body diagrams? Communication Thought process Documentation How will we use free body diagrams? We are dealing with complex systems We will break problem into modules We will model, simulate and analyze mechanical behavior Integrate individual contributions to ascertain system behavior Martin Culpepper, All rights reserved 9

15 Example 1: 0 < x < a Cantilever Forces, moments, & torques y a F b l Why do we care?: Stress Shear & normal Static failure Fatigue failure Why do we care?: Stiffness Displacement Rotation Vibration ( k / m ) ½ But, ends aren t all that matters y a b F M(0) V(0) v Σ F = 0 = F + V ( 0) V ( 0) = V ( x) = F v Σ M = 0 = + F l + M 0 M 0 = F ( ) ( ) a a b x M(x) F V(x) v Σ F = 0 = F + V 0 V 0 = V x = v Σ M = 0 = + F ( ) ( ) ( ) F ( a x) + M ( x) M ( x) = F ( a x) Martin Culpepper, All rights reserved 15

16 Example 1: 0 < x < a But, ends aren t all that matters Relating V(x) & M(x) V ( x )= F M ( x )= F ( x a) v Σ F = 0 a x M(x) V(x) F = F +V ( 0) V (0) = V (x) = F v ΣM = 0 = + F (a x)+ M (x) M (x) = V (x) (a x) b V ( ) ( ) x = d M x dx Shear moment diagrams Solve statics equation Put point of import on plots Use V=dM/dx to generate M plot Master before spindle materials F V M -F a V (x) = F x x M (x) = V (x) (a x) = F (a x) Martin Culpepper, All rights reserved 16

17 Example 1: 0 < x < a Stress h 2 y M(x) σ max M(x+dx) h a b x M(x) F V(x) v Σ F = 0 = F + V 0 V 0 = V x = v Σ M = 0 = + V ( ) ( ) ( ) F ( x) ( a x) M ( x) M ( x) = V ( x) ( a x) I ( x ) = 1 b( ) ( ) 12 x [h x ] 3 σ ( y )= M y σ max = M c I x b I ( x ) ( ) h x = F (a x) ( ) 12 2 h x σ max b( x ) [ ( ) ] 3 F (a x) = 6 b x [ h x ] 2 σ max ( ) ( ) Martin Culpepper, All rights reserved M -F a 6F l bh 2 σ -6F l bh 2 M ( x) = V ( x) ( x a ) = F ( a x) F (a x) σ = 6 bh 2 F (a x) σ = 6 bh 2 17 x x

19 Dynamics

20 Vibration Vibration principles Exchange potential-kinetic energy 2nd order system model c x Blocks and squiggles m F(t) What do they really mean? Why are they important? k How will we apply this? Multi-degree-of-freedom system Mode shape Resonant frequency Gain ω = n k m Estimate ω n (watch units) for: A car suspension system ω ω n Martin Culpepper, All rights reserved 20

, Design of High-Speed, Meso-Scale Nanopositioners Driven by

23 Vibrations: Meso-scale device behavior Underdamped response Noise Free vibration Golda, D. S., Design of High-Speed, Meso-Scale Nanopositioners Driven by Electromagnetic Actuators, Ph.D. Thesis, Massachusetts Institute of Technology, Martin Culpepper, All rights reserved 23

28 Heat transfer

29 Thermal growth errors For uniform temperature ΔL = α L ΔT o STEEL :12L14 ΔL = m m o C L o ΔT ALUMINUM :6061 T 6 ΔL = m m o C L ΔT o POLYMER : Delrin ΔL = m m o C L o ΔT Martin Culpepper, All rights reserved 29

30 Convection and conduction Convection: q& = h A surface (T T ) Why do we care? Heat removal from cutting zone Heat generation in bearings Thermal growth errors Conduction: dt q& = k A cross dx Why do we care? Heat removal from cutting zone Heat generation in bearings Thermal growth errors Common k values to remember Air W /(m o C) 12L W /(m o C) 6061 T6 167 W /(m o C) Martin Culpepper, All rights reserved 30

31 Thermal resistance Thermal resistance ΔT q& = RT Convection q& = (h ( T T ) ) 1 a R 1 T = h A A surface surface Conduction k A q& = dt cross a R T = dx dx k A cross Martin Culpepper, All rights reserved 31

32 Biot (Bi) number Ratio of convective to conductive heat transfer q& convection a (h AΔT ) convection q& conduction k A ΔT L conduction a Bi = hl c k Why do we care? Low Bi High Bi Martin Culpepper, All rights reserved 32

Example of thermal errors For: For: h = 0.

35 Types of errors

36 Machine system perspective System-level approach C 1 C 2 C 3 Linking inputs and outputs 4 C5 C6 Measurement quality C 7 C 8 C 9 [ Outputs]= C [ Inputs] Power Material Temperature changes Torques Machine Desired outputs - Motion, location, rotation - Cutting forces - Speeds Perceived Error Measured outputs - Motion, location, rotation - Cutting forces Real - Speeds error Actual outputs - Motion, location, rotation - Cutting forces - Speeds Fabrication Forces Vibration Speeds errors Martin Culpepper, All rights reserved 36

37 Errors. Accuracy The ability to tell the truth Repeatability Both Ability to do the same thing over & over 1 st make repeatable, then make accurate Calibrate Determinism Machines obey physics! Model understand relationships C 1 C 2 C 3 [Outputs] = C4 C 5 C 6 [Inputs] Range C C C Understand sensitivity [ΔOutputs] = J [ΔInputs] Furthest extents of motion Resolution Smallest, reliable position change Martin Culpepper, All rights reserved 37

38 Categorizing error types Systematic errors Inherent to the system, repeatable and may be calibrated out. Non-systematic errors Errors that are perceived and/or modeled to have a statistical nature Machines are not random, there is no such thing as a random error Consider the error for each set below Link behavior with systematic and non-systematic errors. A B C D Martin Culpepper, All rights reserved 38

39 Exercise

40 Exercise Due Tuesday, start of class: Lathe components Rough sketch(es) of lathe Annotate main components 1 page bullet point summary of where need to use: 2.001, , , , Rules: You may not re-use examples from lecture! You are encouraged to ask any question! You may work in groups, but must submit your own work Martin Culpepper, All rights reserved 40

2.72 Elements of Mechanical Design

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