The fraction of critical damping (denoted as ξ, zeta) is a dimensionless measure that indicates how much damping is present in a system compared to the critical damping required to prevent oscillations. The formula for the fraction of critical damping is:

$\xi = \frac{c}{2 \sqrt{km}}$

Where:

• c = damping coefficient (in Ns/m)
• k = stiffness of the spring (in N/m)
• m = mass (in kg)

Given values:

• m = 22.8 kg
• k = 8,142 N/m
• c = 63.3 Ns/m

Now, let's substitute the given values into the formula:

$\xi = \frac{63.3}{2 \sqrt{8142 \times 22.8}}$

First, we calculate the term inside the square root:

$8142 \times 22.8 = 185,634.6$

Next, we find the square root:

$\sqrt{185,634.6} \approx 430.76$

Now, calculate ξ:

$\xi = \frac{63.3}{2 \times 430.76} = \frac{63.3}{861.52} \approx 0.0735$

Conclusion:

The fraction of critical damping is approximately 0.0735.

Would you like further clarification on this topic or any other details? Here are some follow-up questions:

1. What are the consequences of different damping values in mechanical systems?
2. How do you calculate the natural frequency of this system?
3. What is the effect of increasing the damping on the system's oscillations?
4. How do you interpret underdamped, critically damped, and overdamped systems?
5. What role does damping play in the design of suspension systems for vehicles?

Tip: To optimize damping in practical systems, it's often a balance—too much damping can lead to slow response times, while too little can lead to excessive oscillations.