Solve $($b$)$ For a constant resitive torque T$,$ determine the arm and foot forces, F$1$ and F$2,$ respectively, that must be applied to the linkage to balance this load as the linkage undergoes one complete cycle

Problem $1$:

Elliptical trainers are typically composed of linkages that transfer the user's arm and leg motion to a rotating component with a resistive load. Consider the elliptical mechanism shown to the right. A resistive torque $-$That$(k)$ is applied to the flywheel is pinned at $O_2,$ while a force $F_1$hat$({)}_4$ is applied to the upper link and a force $-F_6$hat$({)}_6$ is applied to the lower, both perpendicular to their respective linkages.

The linkage dimensions are

$||r_{{?}_{?}}(\frac{A}{O_2})||=l_2=10cm,||r_{{?}_{?}}(\frac{B}{A})||=l_3=50cm$

$||r_{{?}_{?}}(\frac{O_4}{B})||=l_4=35cm,$

$r_{{?}_{?}}(\frac{O_4}{O_2})=-d_x$hat$()+d_y$hat$()=-(40cm)hat()+(20cm)hat()$

$||r_{{?}_{?}}(\frac{C}{A})||=l_5=70cm,||r_{\frac{D}{C}}||=l_6=40cm$

$||r_{{?}_{?}}(\frac{D}{B})||=l_7=60cm,$

$||r_{{?}_{?}}(\frac{Q}{O_4})||=l_8=50cm,||r_{{?}_{?}}(\frac{P}{D})||=l_9=50cm.$

For this mechanism

a$)$ Plot the path of the foot point $P$ as the linkage undergoes one complete cycle.

b$)$ For a constant resistive torque $T,$ determine the arm and foot forces, $F_1$ and $F_2,$ respectively, that must be applied to the linkage to balance this load as the linkage undergoes one complete cycle.