Problem #$1$: $(34$ points $...32$ points from questions, $2$ points for consistent units$).$ The group of students then

moved on to studying springs in a more practical setting, which is with suspension systems in cars. The students

study the suspension system of the $2023$ Porsche Taycan Cross Turismo, which has a mass of approximately

$2,250kg($on the order of the weight of some trucks$).$ Suppose the suspension spring has a spring constant $k$

equal to $20,000\frac{N}{m},$ and the damping coefficient $b$ of the shock absorber is $1500N*\frac{s}{m}.$ In a testing

environment of its suspension system, the car is lifted $0\mathrm{.}1$ m above the equilibrium position and released from

rest to simulate a bump.

a$.$ Determine the natural angular frequency of the Porsche's suspension system without damping. Then,

determine the damping parameter $($or $2).$ What kind of damping is this system? Justify your answer. $(6$

points$)$

b$.$ Determine the quality factor $Q$ of this system. Car manufacturers typically desire a quality factor of around

$3$ for a comfortable driving experience. If your calculated $Q$ was not near $3,$ discuss a reasonable modification

that could be made to achieve this quality factor. $(6$ points$)$

c$.$ Determine the angular frequency of the damped system. Then, write the equation for the position as a

function of time for the damped system using your conclusion from part $"a"-$ be sure to include relevant

numbers into the equation. $(6$ points$)$

d$.$ On a test drive, the Porsche hits a bump on the road that causes it to oscillate. How long does it take for the

amplitude of oscillation to decrease to $10\%$ of its initial value? Suppose you wanted this to happen much more

quickly, what kind of damping would have to be used, and what would the natural angular frequency now have

to be$?$ Explain your answers. $(7$ points$)$

e$.$ Suppose the vehicle is driving over a series of small, periodic bumps on a road that act like a driving force at

a frequency of $1\mathrm{.}2$ Hz $.$ Determine whether the driving angular frequency is close to the system's natural angular

frequency and discuss the implications for the vehicle's ride quality. To answer the implications to the ride

quality, consider the following: What happens as the driving frequency approaches the natural frequency? Why

might this impact the ride quality? Be sure to explain your responses. $(7$ points$)$