Question

You are hired by an automotive company to choose a spring to use on each of the four wheels of a 2000 kg racing vehicle designed to go 180 miles per hour. When you set the car on the tires the entire car is supposed to sag not more than 10 cm. The car also needs shocks to damp out road vibrations. The damping constant should be chosen such that the amplitude of oscillation reduces to 1/3 the original undamped amplitude after one full period of oscillation. Give values for k, the spring constant, and b, the damping constant. State any other assumptions you might need to make.

Answer 1

From the text I understood that's it's an "under dumped" system. In this case we have a so-called "amplitude reduction factor". Its formula:

Consider 2 oscillations, one after n periods than the first one; their eqs are:

The amplitudes are the factors in front of cosine function. Let's divide them:

;   

=damping ratio. This has the formula:

c = damping coefficient (this has to be determined)

c_c = critical damping coefficient

(angular frequency for "natural" oscillations)

Replace all these formulas in eq(1) and get for the exponential:

Make the logarithm of eq(1) and get:

In this case:

n=1

x1/x2=3--> ln(3)~1.1

k=already calculated (first line)

m=mass of the vehicle (in the text)

Extract from eq(3), replace in eq(2) and get c.