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Learning with Purpose February 1, 2013 Learning with Purpose February 1, 2013 22.322 Mechanical Design II Spring 2013

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Learning with Purpose February 1, 2013 A cam converts one motion into another form A follower follows the cam profile Summary Lectures 1-4 Think of a cam-follower system as a linkage with variable-length effective links Become familiar with basic cam terminology: Translating vs. rotating followers Follower types: Flat-faced Roller Knife-edge Curved

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Learning with Purpose February 1, 2013 Follower displacement curves: Rise, dwell, return/fall Graphical Design Based on a displacement curve, generate the cam contour (HW #1) Understand what these are: Base Circle Pressure Angle Pitch Point Throw/stroke/travel Start with your base circle, divide it into intervals according to displacement curve, plot the corresponding points, make tangent lines and draw cam contour tangent to those lines Summary Lectures 1-4

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Learning with Purpose February 1, 2013 Summary HW #1 Solution

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Learning with Purpose February 1, 2013 SVAJ Diagrams (HW #2) When designing a cam-follower system, consider the higher derivatives of displacement Satisfy the Fundamental Law of Cam Design s, v, a must be continuous j must be finite SCCA Family of Functions Acceleration curves that will change the peak magnitude of velocity, acceleration, jerk Curves are defined by the same set of equations but by varying parameters and coefficients Dont worry about the derivation of those equations/parameters programs like DYNACAM do that for you! Different values lead to different curves depending on the design specifications e.g. modified sine acceleration function leads to a low peak velocity If the follower mass is large, you want to minimize its velocity Summary Lectures 1-4

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Learning with Purpose February 1, 2013 Kloomok and Muffley method of combining displacement curves Summary Lectures 1-4

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Learning with Purpose February 1, 2013 Summary C-1 H-2 P-2

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Learning with Purpose February 1, 2013 Lecture 5 Polynomial Functions The general form of a polynomial function is: s = C o + C 1 x + C 2 x 2 + C 3 x 3 + C 4 x 4 + + C n x n where s is the follower displacement, x is the independent variable (/ or time t) C coefficients are unknown and depend on design specification The number of terms in the polynomial will be equal to the number of boundary conditions on the s v a j diagrams.

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Learning with Purpose February 1, 2013 Lecture 5 For the double-dwell problem, we can define six BCs: Since we have 6 BCs, we need a polynomial with 6 coefficients: Differentiate with respect to to get V and a: Now we can apply the BCs to solve for C 0, C 1, C 2, C 3, C 4, C 5

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Learning with Purpose February 1, 2013 Lecture 5 We now have 3 equations and 3 unknowns and can solve for the unknown coefficients!

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Learning with Purpose February 1, 2013 Lecture 5 This is called a 3-4-5 polynomial because of the exponents Velocity and acceleration are continuous but jerk is not it was left unconstrained. In order to constrain the jerk, we need to add two additional coefficients, C 6 and C 7, and use two more BCs:

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Learning with Purpose February 1, 2013 Lecture 5 This 4-5-6-7 polynomial will have a smoother jerk but a higher acceleration than the 3-4-5 polynomial.

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Learning with Purpose February 1, 2013 Many applications require a single dwell cam profile (rise-fall-dwell) Example: the cam that opens valves in an engine The valve opens on the rise, closes on the return, and remains shut while combustion and compression take place. The cam profiles for the double dwell case may work but will not be optimal (e.g. cycloidal displacement) Lecture 5 Single-Dwell Cams Acceleration is negative in rise and fall regionswhy go to 0? Subsequently leads to discontinuous jerk

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Learning with Purpose February 1, 2013 Although the jerk is finite, it is discontinuous. A better approach is to use the double harmonic function. Lecture 5 Single-Dwell Cams Double harmonic should never be used for double- dwell case non- zero acceleration at one end of interval.

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Learning with Purpose February 1, 2013 If we used a polynomial function to design the single dwell cam, we could use only one function to describe the rise and the return. The BCs are: With 7 BCs, we need 7 coefficients. If we solve as we have previously, the function turns out to be a 3-4-5-6 polynomial: Lecture 5 Single-Dwell Cams

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Learning with Purpose February 1, 2013 Note that the acceleration is reduced compared to the double harmonic, but the jerk is discontinuous at the ends. We could have also setbut that would increase the order of the polynomial by two. Lecture 5 Single-Dwell Cams A high-degree function may have undesirable oscillations between its BCs!