| Laboratory
7 Cams: Design |
MECH324 Dynamics of Machines |
OBJECTIVES:
NOTE: there is homework required for this week's Lab (see below). As with all of the Cam Lab work over the next few weeks, you need to work together with your Lab group (usually 3 or 4 people) and submit only one clean copy of your work for the entire group (including the names of all of your Lab group members).
PROBLEM
STATEMENT:
Design a rise-fall-dwell (RFD) cam for a roller-follower and
one for a flat-faced follower based on critical extreme positions (CEP).
The lift profile, s(θ), must be identical for both
cams; that is, both will have the same output motion. Use a polynomial function
to the create the output motion. The boundary conditions and constraints
are as follows:
Here are design constraints on your cam:
Your goal is to design cams that are as small as possible while satisfying all of the constraints above.
After designing your cams, you will be cutting them out of Plexiglas on a CNC mill (second week of cam lab, during times scheduled with your TA). Finally, you will test your cams and compare their actual output to the theoretical (third week of cam lab). Shortly thereafter, you will need to turn in a report summarizing the design (counts for three homework grades).
With your final report (see the course syllabus for the due date), include your design model (e.g., print your Mathcad file). Be sure this includes:
See the Homework 7 report guidelines for more information.
BACKGROUND:
It is highly recommended that you read chapter 8 from the text; everything
you need to know can be found in there. With that said...
Cams are extremely useful for generating a desired output motion. As has been seen in previous homeworks, cams can be modeled as linkages where the link lengths are dependent on the radius of curvature of the cam at the contact point. Because the radius of curvature may be continually changing as you travel around the perimeter, the effective link lengths are constantly changing. The result is a mechanism that allows much more flexible and complex motion than traditional linkages. Because the output motion is what dictates cam geometry, the first step in cam design is specification of the output motion.
OUTPUT
MOTION:
The output motion, or motion of the follower, is often a combination
of rising, falling, and dwelling. Each of these phases can be thought of
as separate motion segments with (relatively) independent equations of motion.
For a critical extreme position (CEP) output, we don't really care about the path
the follower takes (within reason) as long long as it meets certain boundary conditions
that we specify. The boundary conditions may state that the motion must
go through key positions, such as a given lift at a given angle (in our problem
statement, we say that h must be 0.75" at 95°).
Or, the boundary conditions may specify velocity, acceleration, jerk, snap, crackle,
pop (proposed names for 1st, 2nd and 3rd derivatives of jerk) at key points.
In order to generate a polynomial that fits these boundary conditions, the polynomial
must have at least as many variables as boundary conditions. For example,
if we want a curve that goes through three points, the polynomial describing that
curve, P(u), must have three variables (i.e., P(u) = C0 + C1u
+ C2u2). Or, if we want a curve that goes through
two points and has specified slopes at those two points, four variables are needed--one
for each term in the polynomial. These variables, or coefficients, can be
solved with the given the boundary conditions.
As was mentioned in the last paragraph, each segment of motion can be described by independent polynomials--well, not entirely independent. We want the equation of motion for one segment to transition smoothly into the equation of motion for the next segment. During the dwell phase, the follower has no motion (zero output, zero velocity, zero acceleration...). So it would make sense that, as we go from the dwell to the rise, there is no sudden jump in position, velocity or acceleration (a somewhat arbitrary "fundamental law of cam design"). Of course, we'd like for there to be no jump in jerk, snap, crackle, pop, etc. either, but we can only be so picky (you have to choose a reasonable order of continuity). Once each motion segment is generated, the overall output motion of the follower can be described by a piecewise function (or spline) that is relatively smooth over the entire range.
Did I hear you say you wanted an example? Let's generate the s-v-a-j polynomials describing the output motion during the rise/fall period of an RFD profile. Although it is perfectly legal to make two sets of equations, one for the rise and one for the fall, it is possible (and easier) to combine them into one. Because the rise portion immediately follows the dwell and the fall portion terminates in the dwell, we want the position, velocity, and acceleration to be zero at the beginning and end. For the sake of simplicity, however, we will ignore acceleration constraints. In addition, we wish to specify that the follower has a height of 2" halfway through the rise/fall period (at β=1/2). The total rise/fall period, β, is 100°. Thus, we have the following five BCs:
| s(θ/β) | v(θ/β) | |
| @ θ = 0 (θ/β = 0) | s(0) = 0 | v(0) = 0 |
| @ θ = β/2 (θ/β = 0.5) | s(0.5) = 2 | |
| @ θ = β (θ/β = 1) | s(1) = 0 | v(1) = 0 |
Because we have five boundary conditions, we need a fourth-order polynomial describing the position:
Taking the derivative with respect to θ gives the velocity:
With
the five BCs, we have five equations and five unknowns (the coefficients):
| s(0) = C0 = 0 | v(0) = (1/100)[C1] = 0 |
| s(.5) = C0 + C1 (.5) + C2 (.5)2 + C3 (.5)3 + C4 (.5)4 = 2 | |
| s(1) = C0 + C1 + C2 + C3 + C4 = 0 | v(1) = (1/100)[C1 + 2C2 + 3C3 + 4C4] = 0 |
There are a number of ways to solve this system of equations, but putting it in matrix form is probably the easiest method. The coefficients in front of the Ci's (coefficients of the coefficients?) are represented in one matrix, while the boundary conditions are in another. Inverting the coefficient matrix and multiplying by the boundary conditions gives the Ci matrix.
 |
 |
 |
C0 = 0 |
Plugging these values back into the position polynomial:
Easy enough? For the cam you are to design for lab, it's slightly trickier since there are 9 boundary conditions, so 9 coefficients. Solve for these coefficients to obtain your s-v-a-j polynomials. Your TA will be verifying that you have these at the beginning of this week's Lab period. This is your grade for the day! You should turn in your polynomial coefficient values along with the work used to arrive at the values. Also, include a printout of plots of your s, v, a, and j diagrams based on the polynomial.
SURFACE & CUTTING
PROFILES:
Now that the output motion is defined, it is just a matter of
translating it into an equation for the cam surface. This is basically just
geometry.
ROLLER
FOLLOWER: There's good news and bad news. The good news is that
there are equations in the book that give equations for the x and y coordinates
of the pitch curve, the path the center of the roller follower makes. Because
we will be using a cutter equal in diameter to the follower diameter, this is
also the path of the cutter. Now for the bad news; I don't find the equations
in the book all that intuitive which makes it difficult to generate the cam surface
plot from the given pitch curve equations. I found it easier to ignore their
equations and come up with my own. Stop your cryin'; this isn't as tough
as it sounds! Using simple geometry (Pythagorean's Theorem, Law of Cosines,
etc.), you can come up with an equation for the distance from the center of the
cam, O2, to the center of the roller, A. Once you find this,
the x and y components of the pitch curve is this equation times cosine and sine
of theta. Similarly, you can find an equation for the distance from O2
to the cam/follower contact point. This is slightly trickier since you have
to take into account the fact that the contact point is NOT at θ
degrees (like point A is). All of this geometry is made simpler, however,
by the fact that ε=0 in our case. Feel free,
of course, to use the equations given in the book, but I didn't find them helpful
(except for the fact that the distance b is numerically equal to the velocity--that's
key). See the modified
version of the above figure that may help.
It is up
to you to come up with the radius of the prime circle (the main design element
of the roller follower). You do this by keeping the pressure angle less
than 30 degree and minimize the size of the cam. You must also beware of
undercutting. This will probably involve iterating Rp until these
conditions are met.
FLAT-FACED
FOLLOWER: Again, there are equations in the book. Again, feel
free to use them, but I didn't. I recommend checking out the modified
version of the above figure that may make the geometry easier. This time,
however, they provide the surface profile (there is no such thing as a
pitch curve for flat-faced followers). It is up to you to generate the cutting
path equation.
Like for the roller follower, you must minimize
the cam size yet keep from undercutting. This will involve defining a minimum
radius of curvature.
NEXT WEEK:
ALL of your cam design has to be done
before you cut next week. You need to sign up for a cutting time (during your
scheduled Lab time) on the schedule to be posted by your TA. You must bring a
printout with the following to lab next week:
1) Values for coefficients
(C0, C1, C2...C8) along with s, v, a, and j plots
2) Chosen value for the roller
follower cam prime circle radius (Rp) and plots of the pitch curve
radius of curvature and pressure angle over an entire revolution to show that
your design meets the specified requirements.
3) Chosen value for the flat-flaced
follower cam base circle radius (Rb) along with work to show that the
choice meets the design requirement for the minimum cam radius of curvature.
4)
Cam surface AND cutter path plots (on the same graph) for the roller follower
cam
5) Cam surface AND cutter path plots (on the same graph) for the flat-faced
follower cam
6) NC files (named "roller.nc" and "flat.nc")
on a flash drive or CD-ROM to cut both the roller follower and flat-faced follower
cams (see Lab 8 for details)
Your TA will check each of these off before letting you cut your cams to make sure everything looks hunky dory. If you don't have everything listed above prepared properly by the beginning of your scheduled time, there will be deductions in your Lab grade for next week (e.g., -10 for each item above). If you need to reschedule your cutting time due to problems with your design or data, there will be further deductions (e.g., -20 for a reschedule).
ADDITIONAL RESOURCES:
