Damping Factor Calculation for Spring-Mass-Damper System

This problem asks us to determine the damping factor of a spring-mass-damper system. The damping factor, also known as the damping ratio, is a crucial dimensionless parameter that describes how oscillations in a system decay after a disturbance. It helps in understanding the system's dynamic behavior, particularly its stability and response to external forces.

To calculate the damping factor ($\zeta$), we need to know the actual damping coefficient ($c$) and the critical damping coefficient ($c_c$). The formula for the damping factor is:

$$\zeta = \frac{c}{c_c}$$

First, let's list the given parameters for the spring-mass-damper system:

• Spring strength (stiffness), \(k = 25 \, \text{kN/m} = 25 \times 10^3 \, \text{N/m}\)
• Mass, \(m = 0.1 \, \text{kg}\)
• Coefficient of damping, \(c = 40 \, \text{N-s/m}\)

Natural Frequency Determination

Before we can find the critical damping coefficient, we must first calculate the undamped natural frequency ($\omega_n$) of the system. The natural frequency is the frequency at which the system would oscillate if there were no damping and no external forces.

The formula for the undamped natural frequency is:

$$\omega_n = \sqrt{\frac{k}{m}}$$

Now, let's substitute the given values:

$$\omega_n = \sqrt{\frac{25 \times 10^3 \, \text{N/m}}{0.1 \, \text{kg}}}$$

$$\omega_n = \sqrt{250 \times 10^3 \, \text{s}^{-2}}$$

$$\omega_n = \sqrt{25 \times 10^4 \, \text{s}^{-2}}$$

$$\omega_n = (5 \times 10^2) \, \text{rad/s}$$

$$\omega_n = 500 \, \text{rad/s}$$

Critical Damping Coefficient Calculation

Next, we need to calculate the critical damping coefficient ($c_c$). Critical damping is the minimum amount of damping required to prevent oscillation in a system when it returns to its equilibrium position.

The formula for the critical damping coefficient is:

$$c_c = 2m\omega_n$$

Substitute the values of mass ($m$) and natural frequency ($\omega_n$) into this formula:

$$c_c = 2 \times 0.1 \, \text{kg} \times 500 \, \text{rad/s}$$

$$c_c = 2 \times 50 \, \text{N-s/m}$$

$$c_c = 100 \, \text{N-s/m}$$

Damping Factor Determination

Finally, we can calculate the damping factor ($\zeta$) using the actual damping coefficient ($c$) and the critical damping coefficient ($c_c$).

The formula is:

$$\zeta = \frac{c}{c_c}$$

Substitute the values for $c$ and $c_c$:

$$\zeta = \frac{40 \, \text{N-s/m}}{100 \, \text{N-s/m}}$$

$$\zeta = 0.4$$

Summary of Calculation Steps

Here's a quick overview of the values and steps:

The calculated damping factor for the spring-mass-damper system is 0.4. This value indicates that the system is underdamped, meaning it will oscillate with decreasing amplitude when disturbed.