Problem I

In class we saw that for our $1^{st}$ attempt of asymmetrical rise$-$fall$-$dwell $($RFD$)$ cam design $($Example $8-9)$ the position value overshot $($see figure $8-31)$ the desired maximum value of $S=1\mathrm{.}0in.$ We reasoned on our $2^{nd}$ attempt, that adding a boundary condition $V=0\mathrm{.}0i\frac{n}{r}ad$ would fix the overshoot. It does, but causes other problems $($see figure $8-32).$ In our $3^{rd}$ attempt, we decided to use three segments for the cam design, with one polynomial for the rise and one for the fall as shown in the first part of Example $8-10.$ We solved for the rise polynomial first and the fall polynomial second. We calculated the acceleration at the end of the rise and used it as the acceleration boundary condition at the start of the fall. Sadly, this $3^{rd}$ result was also not desirable; it had significant undershoot and a very large acceleration magnitude. Part two of Example $8-10$ in the book is a $4^{th}$ attempt at asymmetrical RFD cam design. The example is described in some detail, but the work done to solve for the coefficients is not shown.

For this HW you will set up the boundary condition tables for the rise and fall segments for the $4$ th attempt using the following steps: setup and solve for the fall boundary conditions first $($since it has the lowiest a$$eleration$),$ setup and solve for the rise boundary conditions second while using the acceleration at the start of the fall as the acceleration boundary condition at the end of the rise.

Complete the following steps:

Setup and solve for the fall polynomial coefficients $($since it has the lowest acceleration$).$

Complete the $S,V,$ and A fall equations. Make sure the position $($ S $)$ equation is equal to the fall equation shown below.

Setup and solve for the rise polynomial coefficients using the acceleration at the start of the fall as the acceleration boundary condition at the end of the rise.

Complete the $S,V,$ and A rise equations. Make sure the position $($ S $)$ equation is equal to the rise equation shown below.

Use MATLAB, plot the S$.$V$,$ and A curves as a function of cam angles in degrees.

How do your SVA curves from this $4$ th attempt compare to those of the $3$rd attempt shown in figure $8-33?$

Fall

Rise

$s=h[1-6(\frac{}{{}_2}{)}^2+8(\frac{}{{}_1}{)}^3-3(\frac{}{{}_2}{)}^4]$

$s=h[9\mathrm{.}333(\frac{}{{}_1}{)}^3-13\mathrm{.}667(\frac{}{{}_2}{)}^4+5\mathrm{.}333(\frac{}{{}_1}{)}^5]$