Question

You are riding in an automobile of mass $3000kg$. Assuming that you are examining the oscillation characteristics of its suspension system. The suspension sags $15cm$ when the entire automobile is placed on it. Also, the amplitude of oscillation decreases by $50\%$ during one complete oscillation. Estimate the value of :


$A$: the spring constant $k$.

$B$: the damping constant $b$ for the spring and shock absorber system of one wheel, assuming that each wheel supports $750kg$

Answer 1

$A$:

Displacement in the suspension system $x=15cm=0.15m$

As there are $4$ springs in parallel to support the mass of the automobile.
The equation of restoring force for the system

$F=-4kx=Mg$

$k$ is spring constant for suspension system.

$k=\frac{Mg}{4x}$

$k=\frac{3000\times 10}{4\times 0.15}=5\times {10}^{4}N/m$

$B$:

For damping factor 𝑏, the equation for displacement is given by,
$x=x_0{e}^{-bt/2m}$
Now, the amplitude of oscillation decreases by $50\%$,
i.e.,$x=x_0/2,\text{when}t=T$

Time period, $T=2\pi \sqrt{\frac{m}{k}}=2\pi \sqrt{\frac{750}{5\times {10}^4}}=0.769s$

Therefore,

$\frac{x_0}{2}=x_0{e}^{-\frac{bT}{2m}}$

$b=\frac{2m\times ln2}{T}$
$b=\frac{2\times 750\times 0.693}{0.769}=1351.58kg/s$