$7\mathrm{.}8-$ Dynamic response of a single$-$wheel trailer. $($Section $7\mathrm{.}1$ and Chentar $7).$

Although it is known that single$-$wheel trailers sometimes behave

poorly, it is not always clear why they do so$.$ One possibility is tire

flexibility and loose wheel mounting give rise to unstable behavior.

To explore this concept, the linearized equation of motion for the

system shown right was formed, and after assuming a solution of the

form $C{e}^{pt},$ the equation governing values of $p$ was expressed in terms

of the constant $v($the forward measure of the hitch$-$point velocity$)$ as:

$p^3+5\mathrm{.}06p^2+(50\mathrm{.}6+3\mathrm{.}4v)p+33\mathrm{.}7v=0$

Classify the equation governing $p$ by circling the relevant qualifiers.

In view of the coefficients of $p,$ it is possible to know certain facts about $p_1,p_2,p_3($the values of

$p$ that satisfy the equation$)$ and to know the solution $C_1{e}^{p_{1}t}+C_2{e}^{p_{2}t}+C_3{e}^{p_{3}t}$ is stable unstable

$($circle one$)$ when $v<mn>0</mn>$ because:

Explain:

Calculate the missing values of $p_1,p_2,p_3$ in the

table below. Sketch the root locus for $0v30$

and extrapolate for $v<mn>0</mn>$ and $v>30.$ Use arrows

to show the direction of increasing $v.$

Complete the following with a value of $v$ or range

of values of $v$ from the previous table.

Your younger sister is driving the car with attached trailer. What do you tell her about safe

speeds$/$direction when driving with the trailer?

Result:

For each graph shown below, circle the associated value of $v.$

$($PLEASE EXPLAIN SUM OF ROOTS & PRODUCT OF ROOTS PLEASE!$)$