Q14.21(b):

You are riding in an automobile of mass 3000 kg. Assuming that you are examining the oscillation characteristics of its suspension system. The suspension sags 15 cm when the entire automobile is placed on it. Also, the amplitude of oscillation decreases by 50% during one complete oscillation. Estimate the values of (b) the damping constant $b$ for the spring and shock absorber system of one wheel, assuming that each wheel supports 750 kg.

Step 1: Given Variables & System Parameters

For a single wheel system, we define the following parameters based on the problem statement:

• Mass per wheel, $m = 750\text{ kg}$
• Static sag, $x_0 = 0.15\text{ m}$
• Amplitude decay ratio: $A(T) = 0.5 A(0)$, where $T$ is the period of oscillation.

Step 2: Determining the Spring Constant ($k$)

The spring constant $k$ is derived from the equilibrium condition where the downward force of gravity equals the restoring spring force [Per Hooke's Law: $F = kx$]:

$mg = k x_0$

$k = \frac{mg}{x_0} = \frac{750 \times 9.8}{0.15}$

$k = \frac{7350}{0.15} = 49000\text{ N/m}$

Step 3: Analyzing Damped Harmonic Motion

For a damped oscillator, the displacement is given by $x(t) = A_0 e^{-(b/2m)t} \cos(\omega' t + \phi)$. The amplitude decays as $A(t) = A_0 e^{-(b/2m)t}$. [Per the theory of Damped Oscillations].

Given that after one period ($t = T$), the amplitude is $50\%$ of the initial amplitude:

$0.5 A_0 = A_0 e^{-(b/2m)T}$

$\ln(0.5) = -\frac{bT}{2m} \implies \frac{bT}{2m} = \ln(2) \approx 0.693$

Step 4: Calculating Period ($T$) and Damping Constant ($b$)

Assuming the damping is light ($\omega' \approx \omega_0 = \sqrt{k/m}$):

$\omega_0 = \sqrt{\frac{49000}{750}} \approx \sqrt{65.33} \approx 8.08\text{ rad/s}$

$T = \frac{2\pi}{\omega_0} = \frac{2 \times 3.1416}{8.08} \approx 0.777\text{ s}$

Now, solve for $b$ using the relationship derived in Step 3:

$b = \frac{2m \ln(2)}{T}$

$b = \frac{2 \times 750 \times 0.693}{0.777}$

$b = \frac{1039.5}{0.777} \approx 1337.8\text{ kg/s}$

Step 5: Visual Representation of Damped Decay

Final Solution: The estimated value of the damping constant $b$ is approximately $1338\text{ kg/s}$.