An Ackerman mechanism was designed with the dimensions $\backslash ($ a$=0\mathrm{.}3\backslash$mathrm$\{$~m$\}\backslash )$ and $\backslash ($ b$=1\backslash$mathrm$\{$~m$\}\backslash ).$ This mechanism presents friction in each of its joints, which can be considered proportional to the speed of rotation between the bodies with a factor of $\backslash (1\mathrm{.}0\backslash$mathrm$\{$Nms$\}/\backslash$mathrm$\{$deg$\}\backslash ).$ The rudder is connected to the left rod so that the input torque is applied directly on it$.$

You propose adding torsion springs of equal stiffness $($k$)$ in each of the joints, so that, in its natural length, the system is in the neutral position $($as seen in the image$).$

$1.$ Propose realistic values for the mass and inertia properties of each of the bodies in the mechanism. Present a table with the proposed values.

$2.$ For the springless mechanism, graph the input torque required to move the tire $\backslash (10^\{\backslash$circ$\}\backslash )$ to the left, with constant angular velocity, in $0\mathrm{.}5$ s or less.

$3.$ Define a value for $\backslash ($ k $\backslash )$ of the springs such that, when the tire is at $\backslash (10^\{\backslash$circ$\}\backslash )$ and no torque is applied to the steering wheel, a moment is produced that returns the tire to $\backslash (0^\{\backslash$circ$\}\backslash )$ and has the same order of magnitude as that found in $2.$

$4.$ Assuming small angles, find the equivalent second$-$order model of your system and calculate the corresponding wn and zetas values.

$5.$ Calculate and plot the magnification factor of your mechanism as a function of the input torque frequency. Comment on the influence of the input torque frequencies on the rudder with respect to the effect on the rotation of the wheels.