learning with purpose january 30, 2013 learning with purpose january 30, 2013 22.322 mechanical...

Learning with Purpose January 30, 2013Learning with Purpose January 30, 2013

22.322 Mechanical Design II

Spring 2013

Learning with Purpose January 30, 2013

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.

Lecture 4

SCCA


Normalize 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/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:

Lecture 4

SCCA


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.

Values of b, c, d define the shape of the curve

Normalized velocity

Normalized acceleration

Normalized jerk


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

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

In Zone 1


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.


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)

Lecture 4

SCCA


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

Lecture 4


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

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

Lecture 4


Lecture 4

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


Different 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 functions

• Note how similar the displacement curves look for the double-dwell problem:

Lecture 4


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 applicationsCan be tailored to many design specificationsThe 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

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.

Lecture 4



3-4-5 polynomial:• Equation of cam design’s displacement becomes

Lecture 4


Jerk is unconstrained


4-5-6-7 polynomial:• Equation of cam design’s displacement becomes

Lecture 4


Jerk is constrained

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


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.

Lecture 4

Kloomok and Muffley


Lecture 4

learning with purpose january 30, 2013 learning with purpose january 30, 2013 22.322 mechanical...

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