PART IV: ANALYTICAL LINKAGE SYNTHESIS [10 points]
In class this two-position analytical motion synthesis example
(for which $P_2$ and $c_2$ are specified) was partially solved for the
left side only. This part of the exam asks you to repeat the
methodology for the RIGHT side. Shown here is a figure that
identifies the labeling of vectors and angles. Triangle ABP
represents characteristic points on a rigid coupler.
12. [1 point] TRUE or FALSE: The angle between vectors $\vec{S}_1$
and $\vec{S}_2$ has the same value as the angle $\alpha_2$ given between
vectors $\vec{Z}_1$ and $\vec{Z}_2$.
13. [2 points] Write the vector loop equation for the RIGHT
side of the mechanism, in a way that would enable the
determination of unknown lengths $u$ and $s$, while also
achieving the required displacement $p_{21}$.
14. [3 points] Now rewrite the same equation using complex
polar notation (as in $ae^{j\theta}$). Use the lowercase expressions
for lengths ($u$ is the length of $U$, $s$ is the length of $S$, and
$p_{21}$ is the length of $P_2$) and use angles from the figure.
15. [4 points] Apply the Euler identity $e^{j\theta} = cos(\theta) + jsin(\theta)$ to extract the real and imaginary scalar equations from your vector
equation. For the limited time of this exam, you are not being asked to solve for any variables of interest, only to extract
independent scalar equations. Remember to write equations set equal to zero and also to eliminate $j$ wherever appropriate.
Need to simplify with factoring, as long as each scalar equation is legitimate and correct.
