22.322 Mechanical Design II Spring 2013

Summary Lectures 1-4 A cam converts one motion into another form A follower follows the cam profile 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

Summary Lectures 1-4 Follower displacement curves: Graphical Design 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 q intervals according to displacement curve, plot the corresponding points, make tangent lines and draw cam contour tangent to those lines

Summary HW #1 Solution

Summary Lectures 1-4 SVAJ Diagrams (HW #2) SCCA Family of Functions 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 Don’t 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 Kloomok and Muffley method of combining displacement curves

Summary C-1 H-2 P-2

Lecture 5 The general form of a polynomial function is: Polynomial Functions The general form of a polynomial function is: s = Co + C1x + C2x2 + C3x3 + C4x4 + … + Cnxn where s is the follower displacement, x is the independent variable (q/b 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.

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 q to get V and a: Now we can apply the BC’s to solve for C0, C1, C2, C3, C4, C5

Lecture 5 We now have 3 equations and 3 unknowns and can solve for the unknown coefficients!

Lecture 5 𝑠=ℎ 10 𝜃 𝛽 3 −15 𝜃 𝛽 4 +6 𝜃 𝛽 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, C6 and C7, and use two more BCs:

Lecture 5 𝑠=ℎ 35 𝜃 𝛽 4 −84 𝜃 𝛽 5 +70 𝜃 𝛽 6 −20 𝜃 𝛽 7 If we repeat the same analysis but add the two extra BCs and coefficients, we will result in the following 4-5-6-7 polynomial for the displacement function: 𝑠=ℎ 35 𝜃 𝛽 4 −84 𝜃 𝛽 5 +70 𝜃 𝛽 6 −20 𝜃 𝛽 7 This 4-5-6-7 polynomial will have a smoother jerk but a higher acceleration than the 3-4-5 polynomial.

Lecture 5 Single-Dwell Cams 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) Acceleration is negative in rise and fall regions…why go to 0? Subsequently leads to discontinuous jerk

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

Lecture 5 Single-Dwell Cams 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 BC’s are: With 7 BC’s, we need 7 coefficients. If we solve as we have previously, the function turns out to be a 3-4-5-6 polynomial: 𝑠=ℎ 64 𝜃 𝛽 3 −192 𝜃 𝛽 4 +192 𝜃 𝛽 5 −64 𝜃 𝛽 6

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