Mit2 72s09 lec06
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This document provides a summary of a lecture on mechanical constraints. The key points covered include: - Exact constraint means applying the minimum number of constraints to constrain each degree of freedom. Under or over constraint can cause problems. - Bearing layouts must be designed to provide the desired motion while avoiding over constraint. Thermal expansion must also be considered. - Examples of bearing layouts for constraining a spindle shaft are analyzed for whether they provide good or bad constraint. Back-to-back bearing configurations are generally better to allow for thermal expansion. - Students complete an exercise to analyze the constraint layout in their lathe spindle and propose an alternative layout.
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Mit2 72s09 lec06
- 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 06: Constraints
- 3. Schedule and reading assignment Quiz ‰ Quiz – HTMs Topics ‰ Principles of exact constraint ‰ Bearing layout exercises ‰ Spindle shaft constraint/bearing layout Reading assignment ‰ Chapter 11 in Shigley and Mischke • Read sections 11.1 – 11.6, 11.9 • Skim sections 11.7 – 11.8, 11.10 – 11.12 ‰ This is 40ish pages, but most of it is pictures/graphs/examples © Martin Culpepper, All rights reserved 2
- 5. Under, exact and over constraint Constraints are fundamental to mechanical design ‰ A mechanical designers goal is to control, i.e. to constrain, parts so that they are where they are supposed to be. Exact constraint: ‰ There should be one constraint for each degree-of-freedom that is constrained. Under constraint ‰ Too few constraints, part is not held in all the directions it needs to be Over constraint ‰ Too many constraints, some constraints may fight each other when trying to do the same job. © Martin Culpepper, All rights reserved 4
- 6. Mechanical constraints We want to learn how to model and design each We first need to know: ‰ How they should be used ‰ What their functional requirements are How they should be used and their FRs depend upon: ‰ How they are laid out ‰ Dos and don’ts Learn to lay them out right ‰ Use this to obtain their Functional Requirements ‰ Then do the detailed design of each © Martin Culpepper, All rights reserved 5
- 7. Constraints and Degrees-of-freedom Rigid components have 6 degrees-of-freedom We will represent an ideal constraint as a line 6 – C = R C = # of Non-Redundant Constraints R = # of Independent Degrees of Freedom Example: 6 - 2constraints = 4DoFs Rz Courtesy John Hopkins, MIT © Martin Culpepper, All rights reserved 6
- 8. Constraints, compliance and motion Exact constraint: Achieve desired motion ‰ By applying minimum number of constraints y ‰ Arranging constraints in correct constraint topology ‰ Adding constraints only when necessary z For now: ‰ Start with ideal constraints x Stage Ground ‰ Considering small motions ‰ Constraints = lines Focus on rigid stage attached to ground ‰ What do we mean by rigid? ‰ What do we mean by constraint? - Stiffness ratios © Martin Culpepper, All rights reserved 7
- 9. The benefits of exact constraint Penalties for over constraint © Martin Culpepper, All rights reserved Parts of machines are now always the same strength and stiffness. Large, stiff components have a tendency to “kill” their smaller counterparts when they are connected so that they are forced to fight. Exact constraint design helps to prevent fights, therfore all your parts live in harmony. 8 X Y A B A B Loose Binding Figure by MIT OpenCourseWare.
- 10. Constraint examples 1 2 More examples: y 3 z x 4 5 6 © Martin Culpepper, All rights reserved 9
- 12. Mechanical constraints: Some bearing types Sliding ‰ Bushings ‰ etc… Radial rolling ‰ Radial ‰ Shallow groove ‰ Deep groove (Conrad) ‰ Angular contact ‰ Tapered © Martin Culpepper, All rights reserved 11
- 13. Examples drawn from your lathe © Martin Culpepper, All rights reserved 12
- 14. Example: Carriage constraint x z y Assume these are bushings as in your lathe © Martin Culpepper, All rights reserved 13
- 15. Example cont. y x z Misalignment & assembly errors © Martin Culpepper, All rights reserved 14
- 16. Example cont. x z y Thermally Induced growth © Martin Culpepper, All rights reserved 15
- 17. x Practical embodiment y z Flexure that is stiff in y and z, yet compliant in x © Martin Culpepper, All rights reserved 16
- 18. Group exercise – Carriage constraints Identify the motions that you desire for the carriage and the minimum # of constraints that are needed to yield only these motions. Identify the constraints from each bearing set and determine how they act in concert to yield the desired motions. © Martin Culpepper, All rights reserved 17
- 20. Housing Avoiding over constraint How to deal with the thermal growth issue ‰ The shaft typically gets hotter than the housing because the housing has better ability to carry heat away ‰ Whether the outer or inner race are fixed, matters… Constraining front and rear bearings ‰ One bearing set should be axially and radially restrained ‰ The other bearing set should ONLY have radial restraint 19© Martin Culpepper, All rights reserved Shaft Chuck NutGear Part Tool 8.0 04.010.0 Gear
- 21. Housing Examples: Good or bad Outer race fixed axially, if shaft heats is this bad? ‰ Think about what happens to the preload… Shaft Chuck © Martin Culpepper, All rights reserved 20
- 22. Housing Examples: Good or bad Outer race fixed axially, if shaft heats is this bad? ‰ Think about what happens to the preload… Shaft Chuck This is the back-to-back config. It should be used when the outer race is not rotating © Martin Culpepper, All rights reserved 21
- 23. Housing Examples: Good or bad Assume the outer race does not rotate Shaft Chuck Nut Gear Gear Sliding permitted Spacer 1 © Martin Culpepper, All rights reserved 22
- 24. Housing Examples: Good or bad Assume the outer race does not rotate Shaft Chuck Nut Gear Gear Sliding permitted Spacer 2 © Martin Culpepper, All rights reserved 23
- 25. Housing Examples: Good or bad Shaft Chuck Nut Gear Gear Assume the outer race does not rotate 3 © Martin Culpepper, All rights reserved 24
- 26. Housing Examples: Good or bad Shaft Chuck Nut Gear Gear Spring Assume the outer race does not rotate 4 © Martin Culpepper, All rights reserved 25
- 27. Housing Examples: Good or bad Assume the outer race does not rotate Shaft Chuck Nut Gear Gear 5 © Martin Culpepper, All rights reserved 26
- 28. Housing 27© Martin Culpepper, All rights reserved Gear Gear Examples: Good or bad Assume the outer race does not rotate Shaft Chuck 6
- 29. Housing Examples: Good or bad Assume the outer race does not rotate Good ‰ Front set constrained radially and axially ‰ Rear set constrained only radially Shaft Chuck Nut Gear Gear Sliding permitted Spacer 1 BUT, sliding means some gap must exist and therefore one must precision fabricate if a small gap is desired © Martin Culpepper, All rights reserved 28
- 30. Housing Examples: Good or bad Assume the outer race does not rotate Bad ‰ Front set constrained only radially ‰ Rear set constrained only radially Shaft Chuck Nut Gear Gear Sliding permitted Spacer 2 This design will not work if axial loads are to be applied along both directions © Martin Culpepper, All rights reserved 29
- 31. Housing Examples: Good or bad Assume the outer race does not rotate Good ‰ Front set constrained radially and ½ axially ‰ Rear set constrained radially and ½ axially Shaft Chuck Nut Gear Gear 3 BUT, adding a spring increases part count/cost Axial stiffness values (left vs. right) will be different © Martin Culpepper, All rights reserved 30
- 32. Housing Examples: Good or bad Assume the outer race does not rotate BAD ‰ Front set is constrained only ½ axially ‰ Rear set is constrained only ½ axially Shaft Chuck Nut Gear Gear Spring 4 This design will not work if axial loads are to be applied along both directions © Martin Culpepper, All rights reserved 31
- 33. Housing Examples: Good or bad Assume the outer race does not rotate BAD ‰ Front set is constrained ½ axially and radially ‰ Rear set is constrained ½ axially and radially Shaft Chuck Nut Gear Gear 5 At high speeds/loads, thermal growth may kill the bearing sets ‰ Like a double face-to-face… © Martin Culpepper, All rights reserved 32
- 34. Housing Examples: Good or bad Bad ‰ Front set constrained radially and ½ axially ‰ Rear set constrained radially (if flexure is of proper stiffness) Assume the outer race does not rotate Gear Gear Shaft Chuck 6 Left bearing set is not in the back-to-back configuration Shaft can pop out… © Martin Culpepper, All rights reserved 33
- 35. Group exercise
- 36. Group exercise – Spindle constraints The spindles you have seen use tapered roller bearings. First, sketch a layout from one of the previous lathes and diagnose its layout Second, generate and sketch a different way to constrain the spindle shaft. © Martin Culpepper, All rights reserved 35