2 Lecture 4SCCAIn lecture 3, we introduced several acceleration curves:Constant accelerationSimple harmonicModified trapezoidalModified sineCycloidalThese 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 4SCCANormalize the independent variable, cam angle q, by dividing it by the interval period, b: x = q / bThis normalized value, x, then runs from 0 to 1 over any interval.The normalized follower displacement is: y=s/hs = instantaneous follower displacementh = total follower lift/riseThe 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 4Interval 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 velocityNormalized accelerationNormalized jerkValues of b, c, d define the shape of the curve
5 Lecture 4For each zone, there will be a set of equations for s, v, a, and j that is defined by parameters and coefficientsZone 0 all functions are zeroIn Zone 1 Equations for zones 2 through 6 can be found in the text (pages )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 4SCCATo 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, hActual duration, b (radians)Cam velocity, w (rad/sec)
8 Lecture 4Comparing 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 4Modified sine jerk is somewhat less ragged than modified trapezoid but not as smooth as cycloid (which is a full-period cosine)
10 Lecture 4Peak velocities of cycloidal and modified trapezoid functions are sameEach will store the same peak kinetic energy in the follower trainPeak velocity of modified sine is the lowest of the functions shownPrincipal 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 4Peak values of acceleration, velocity, and jerk in terms of total rise, h, and period, b.
12 Lecture 4Different acceleration functions will provide different dynamic characteristics.For low acceleration modified trapezoidalFor low velocity modified sineThe 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 functionsNote how similar the displacement curves look for the double-dwell problem:
13 Lecture 4Polynomial FunctionsThe 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 applicationsCan be tailored to many design specificationsThe general form of a polynomial function is:s = Co + C1x + C2x2 + C3x3 + C4x4 + … + Cnxnwhere 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 4Polynomial FunctionsWe 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.
16 Lecture 4 4-5-6-7 polynomial: Polynomial Functions Equation of cam design’s displacement becomes𝑠=ℎ 35 𝜃 𝛽 4 −84 𝜃 𝛽 𝜃 𝛽 6 −20 𝜃 𝛽 7polynomial has smoother jerk for better vibration control compared to polynomial, cycloidal, and all other functionsHowever, higher peak acceleration is observedJerk is constrained
17 Lecture 4 Kloomok and Muffley Developed a system of CAM design that uses three analytical functionsCycloidHarmonicEighth power polynomialThe 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.