2.72 Elements of Mechanical Design

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

2 2.72 Elements of Mechanical Design Lecture 03: Shafts

3 Schedule and reading assignment Reading quiz Hand forward lathe exercise quiz Topics inish matrices, errors Shaft displacements Stiffness exercise Reading assignment Shigley/Mischke Sections : 10ish pages & Sections : 21ish pages Pay special attention to example 6.12 (modified Goodman portion) Martin Culpepper, All rights reserved 2

4 Deflection within springs and shafts

5 Shafts, axles and rails Shafts Rotating, supported by bearings/bushings Dynamic/fluctuating analysis Axles Non-rotating, supported by bearings/bushings Static analysis Rails Non-rotating, supports bearings/bushings Static analysis Martin Culpepper, All rights reserved 4

8 In practice, we are concerned with Deflection Stiffness Bearings and stiffness of connectivity points unction of global shaft geometry, sometimes adjacent components Stress Catastrophic failure: Ductile Brittle atigue unction of local shaft geometry Martin Culpepper, All rights reserved 7

9 What is of concern? Deflection and stiffness Beam bending models Superposition Load and stress analysis Bending, shear & principle stresses Endurance limit atigue strength Endurance modifiers Stress concentration luctuating stresses ailure theories Von Mises stress Maximum shear stress Martin Culpepper, All rights reserved 8

10 Materials Steel vs. other materials Aluminum Brass Cast iron Important properties Modulus Yield stress Is density important? atigue life CTE Material treatment Hardening What does hardening do the material properties It is expensive Affects final dimensions You can usually design without this Martin Culpepper, All rights reserved 9

12 Modeling: General forms of equations Lateral bending deflection (middle) L 3 L 3 δ = Const 48E I E I Axial deflection L δ = A E R L R R Lateral bending angles (at ends) L 2 L 2 M L δ = = or 6 E I Const E I Const E I Martin Culpepper, All rights reserved 11

13 Modeling: Stiffness Lateral bending stiffness at middle k b = 48 (E 3 L I ) = Const (E I ) L n Axial stiffness A E k A = L R L R R Torsional stiffness Stepped shafts? k θ = J G L Martin Culpepper, All rights reserved 12

14 Modeling: Stiffness These pop up in many places, memorize them Square cross section I = Circular cross sections bh I = π [(d ) 4 (d ) 4 ] 64 outer inner J = π [(d ) 4 (d ) 4 ] 32 outer inner Martin Culpepper, All rights reserved 13

15 Principles of stiffness: Ratios Everything deforms Impractical to model the stiffness of everything Mechanical devices modeled as high, medium & low stiffness elements Stiffness ratios show what to model as high-, medium, or low stiffness Stiffness ratio R k = k 1st k 2nd AE AE 3EI ½ l = k = R k = l = 4 l2 k axial lateral l l 3 3 EI h 2 Building intuition for stiffness You can t memorize/calculate everything Engineers must be reasonable instruments Car suspension is easy, but flexed muscle vs. bone? l 3 Martin Culpepper, All rights reserved 14

17 Superposition You must be careful, following assumptions are needed Cause and effect are linearly related No coupling between loads, they are independent Geometry of beam does not change too much during loading Orientation of loads does not change too much during loading Use your head, when M = 0, what is going on Superposition is not plug and chug You must visualize You must think Martin Culpepper, All rights reserved 16

5 k (x b 2 -x a 2 ) Non-linear force spring k = function of x ΔE b-a = (x) dx orce-displacement Curve (x)=k x 1

18 Types of springs and behaviors Springs and stiffness k = d(x) / dx Constant force spring k = d(x) / dx = 0 ΔE b-a = (x b -x a ) orce-displacement Curve a (x)= x b Constant stiffness spring k = Constant ΔE b-a = 0.5 k (x b 2 -x a 2 ) Non-linear force spring k = function of x ΔE b-a = (x) dx orce-displacement Curve (x)=k x 1 orce-displacement Curve a x k b Martin Culpepper, All rights reserved Images from Wikimedia Commons, x a b 17

19 Non-conformal contact ball on flat Non-conformal contacts often non-linear Example: bearings, belleville washers, structural connections Is anything ever perfectly conformal? Specific case: Hertzian contact k [N/micron] In-plane stiffness Preload [N] k ( n )= Constant R 3 E 3 3 Martin Culpepper, All rights reserved 18

20 Linearization of non-linear springs If you can linearize over the appropriate range then you can use superposition So how would, and when could, you do this? R = ball radius E = modulus of both materials (both steel) = contact load d = k ( n )= Constant R 3 E 3 3 dδ Martin Culpepper, All rights reserved 19

Practical application to the lathe problem y x Case 11

22 Practical application to the lathe problem y x 1 y( x ) A B = 96EI x2 (11x 9l) y( l 2 )= 1 96EI 7 8 l3 768 E I k = Beam 7 l 3 But, is this really what is going on? Martin Culpepper, All rights reserved 21

27 Group work Obtain an equation for δ total in terms of, k and l Estimate when k is important / should be considered? What issue/scenario would cause k not to be infinite? Look at these causes, if a stiffness is involved, would linearity, and therefore superposition apply? Martin Culpepper, All rights reserved 26

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