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Mechanical Design II Spring 2013

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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.

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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:

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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

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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 ) 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

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**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.

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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)

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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

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

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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

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

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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:

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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

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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.

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**Lecture 4 3-4-5 polynomial: Polynomial Functions**

Equation of cam design’s displacement becomes 𝑠=ℎ 10 𝜃 𝛽 3 −15 𝜃 𝛽 𝜃 𝛽 5 Jerk is unconstrained

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**Lecture 4 4-5-6-7 polynomial: Polynomial Functions**

Equation of cam design’s displacement becomes 𝑠=ℎ 35 𝜃 𝛽 4 −84 𝜃 𝛽 𝜃 𝛽 6 −20 𝜃 𝛽 7 polynomial has smoother jerk for better vibration control compared to polynomial, cycloidal, and all other functions However, higher peak acceleration is observed Jerk is constrained

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**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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