Question

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.

          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.

PART IV: ANALYTICAL LINKAGE SYNTHESIS [10 points]
In class this two-position analytical motion synthesis example
(for which P2 and c2 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 S⃗1
and S⃗2 has the same value as the angle α2 given between
vectors Z⃗1 and Z⃗2.
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 p21.
14. [3 points] Now rewrite the same equation using complex
polar notation (as in ae^jθ). Use the lowercase expressions
for lengths (u is the length of U, s is the length of S, and
p21 is the length of P2) and use angles from the figure.
15. [4 points] Apply the Euler identity e^jθ = cos(θ) + jsin(θ) 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.

Added by Rodney C.

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