Part $2$: Assignment

The suspension system for one wheel of an old$-$fashioned pickup truck can be illustrated as shown in Figure $3$ a$.$ The mass of the vehicle is $M$ and the mass of the wheel is $m.$ The suspension spring has a constant $k_1$ and the tire has a spring constant $k_2$ The damping constant of the shock absorber is $b.$

$($a$)$ Car wheel suspension system

$($b$)$ shock absorber

Figure $3$ Car suspension model

Based on the description above suggest a suitable model for this system.

Draw the free body diagram of the suggested model.

Write the equations of motion of the system.

For zero initial conditions, take Laplace transform of both sides of the equations obtained in $(3).$ Obtain the transfer functions

$G_1(s)=\frac{x_1(s)}{Y(s)}=\frac{?}{\text{d}\text{e}\text{n}}$

$u{m}_{-}1,$ and $,G_2(s)=\frac{x_2(s)}{Y(s)}=\frac{?}{\text{d}\text{e}\text{n}}$

$u{m}_{-}2$