A suspension system for a vehicle requires an initial design evaluation. For the purposes of this

evaluation, the suspension system can be modelled as a simple single$-$degree$-$of$-$freedom system as

shown in Figure Q$1.$ It has been proposed that the combined spring stiffness $($k$)$ in this system is

$50$kN$/$m$,$ the combined damping $($c$)$ is $3$kNs$/$m and the vehicle body mass $($m$)$ is $260$kg$.$ This

suspension system must be evaluated at four common road speed limits: $30$km$/$h$,50$km$/$h$,100$km$/$h

and $120$km$/$h$.$ It must also be evaluated for the condition when the vehicle is fully loaded, which simply

involves adding $80$kg to the vehicle body mass.

A specification for this suspension system has been devised and stipulates that when travelling on a

specific undulating road, which can be simulated as a pure sinusoidal motion at the wheel axis, with

an amplitude A of $50$mm and a peak$-$to$-$peak distance S of $4$m $($see Figure Q$1)$:

$$ At speeds above $90$km$/$h the maximum vibration amplitude of the vehicle body should not exceed

$15$mm$.$

$$ At speeds up to $50$km$/$h the maximum acceleration amplitude of the vehicle body should not

exceed $2$g$.$

$$ The maximum travel in the suspension cannot exceed $70$mm$.$

$$ The maximum force transmitted to the vehicle body should not exceed $8\mathrm{.}6$kN$.$

$($a$)$ Evaluate this suspension system by analysing both the unloaded and loaded scenarios at the four

common road speed limits given above to identify where it does not comply with the specification.

$[25]$

$($b$)$ To fully comply with the specification, it is proposed to change the combined spring stiffness $($k$)$

in the system to either $40$kN$/$m or $60$kN$/$m$.$ Assess which is the preferred option. To justify your

answer, you only need to focus on the specific areas where the original system failed to meet the

specification. $[10]$

$($c$)$ As part of this analysis, it is also necessary to identify the vehicle speed at which the relative

amplitude values $($Z$)$ in the system exceed the absolute amplitude values $($X$).$ Derive an

expression for this vehicle speed and use it to calculate the respective speeds for each of the

three design option scenarios $($loaded and unloaded$).$