22.322 Mechanical Design II

1. 22.322 Mechanical Design II Spring 2013

2. Lecture 4 SCCA • In lecture 3, we introduced several acceleration curves: • Constant acceleration • Simple harmonic • Modified trapezoidal • Modified sine • Cycloidal • These very different looking curves can all be defined by the same equation with only a change of numeric parameters. • This family of acceleration functions is referred to as the SCCA (sine-constant-cosine-acceleration) functions and will all have the same general shape. • To reveal this similitude, it is first necessary to normalize the variables in the equations.

3. Lecture 4 SCCA • Normalize the independent variable, cam angle q, by dividing it by the interval period, b: x = q /b • This normalized value, x, then runs from 0 to 1 over any interval. • The normalized follower displacement is: y=s/h • s = instantaneous follower displacement • h = total follower lift/rise • The normalized variable y then runs from 0 to 1 over any follower displacement. • The general shapes of the s v a j functions of the SCCA family are shown:

4. Lecture 4 Interval b divided into five zones  zones 0 and 6 represent the dwells on either side of rise (or fall) Widths of zones 1-5 are defined in terms of b and one of three parameters, b, c, d. Normalized velocity Normalized acceleration Normalized jerk Values of b, c, d define the shape of the curve

5. Lecture 4 For each zone, there will be a set of equations for s, v, a, and j that is defined by parameters and coefficients Zone 0  all functions are zero In Zone 1  Equations for zones 2 through 6 can be found in the text (pages 421-425) Note that Ca, Cv, and Cj are dimensionless factors applied to acceleration, velocity, and jerk, respectively: At the end of the rise in zone 5 when x=1, the expression for displacement must have y=1 to match the dwell in zone 6

6. Lecture 4 • For the five standard members of the SCCA family: Infinite number of family members as b, c, and d can take on any set of values that add to 1.

7. Lecture 4 SCCA • To apply the SCCA functions to an actual cam design problem only requires that they be multiplied or divided by factors appropriate to the particular problem: • Actual rise, h • Actual duration, b (radians) • Cam velocity, w (rad/sec)

8. Lecture 4 • Comparing the shapes and relative magnitudes of cycloidal, modified trapezoidal, and modified sine acceleration curves (acceptable cams): • Cycloidal has theoretical peak acceleration ~1.3 times that of modified trapezoid’s peak value for the same cam specification. • Peak acceleration of modified sine is between those of cycloidal and modified trapezoids

9. Lecture 4 • Modified sine jerk is somewhat less ragged than modified trapezoid but not as smooth as cycloid (which is a full-period cosine)

10. Lecture 4 • Peak velocities of cycloidal and modified trapezoid functions are same • Each will store the same peak kinetic energy in the follower train • Peak velocity of modified sine is the lowest of the functions shown • Principal advantage of the modified sine acceleration curve and why it is often chosen for applications in which the follower mass is very large

11. Lecture 4 Peak values of acceleration, velocity, and jerk in terms of total rise, h, and period, b.

12. Lecture 4 • Different acceleration functions will provide different dynamic characteristics. • For low acceleration  modified trapezoidal • For low velocity  modified sine • The designer must ultimately choose the appropriate function. • Remember, it’s important to consider the higher derivatives of displacement! • Nearly impossible to recognize differences by looking only at displacement functions • Note how similar the displacement curves look for the double-dwell problem:

13. Lecture 4 Polynomial Functions • The class of polynomial functions is one of the more versatile types that can be used for cam design. • Not limited to single- or double-dwell applications • Can be tailored to many design specifications • 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

14. Lecture 4 Polynomial Functions • We structure a polynomial cam design problem by deciding how many boundary conditions we want to specify on the s v a j diagrams. • Number of BCs then determines the degree of the resulting polynomial. • If k represents the number of chosen BCs, there will be k equations in k unknown C coefficients and the degree of the polynomial will be n = k – 1.

15. Lecture 4 Polynomial Functions • 3-4-5 polynomial: • Equation of cam design’s displacement becomes Jerk is unconstrained

16. Lecture 4 Polynomial Functions • 4-5-6-7 polynomial: • Equation of cam design’s displacement becomes 4-5-6-7 polynomial has smoother jerk for better vibration control compared to 3-4-5 polynomial, cycloidal, and all other functions However, higher peak acceleration is observed Jerk is constrained

17. Lecture 4 Kloomok and Muffley • Developed a system of CAM design that uses three analytical functions • Cycloid • Harmonic • Eighth power polynomial • The selection of the profiles to suit particular requirements is made according to the following criteria: • 1) Cycloid provides zero acceleration at both ends. Therefore it can be coupled to a dwell at each end. Because the pressure angle is relatively high and the acceleration returns to zero, two cycloids should not be coupled together. • 2) The harmonic provides the lowest peak acceleration and pressure angle of the three curves. • 3) The eighth-power polynomial has a non-symmetrical acceleration curve and provides a peak acceleration and pressure angle intermediate between the harmonic and the cycloid.

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