91Ó°ÊÓ

Natural frequency is an essential concept when studying damped harmonic oscillators. It refers to the frequency at which a system oscillates when it is not subjected to any damping forces. For a spring-mass system, the natural frequency can be calculated using the formula \( \omega_0 = \sqrt{\frac{k}{m}} \), where \( k \) is the spring constant, and \( m \) is the mass of the object attached to the spring.
In our problem, we determined the natural frequency by substituting the given values:

• \( k = 41.0 \, \mathrm{N/m} \)
• \( m = 0.835 \, \mathrm{kg} \)

Calculating this gives:\[\omega_0 = \sqrt{\frac{41.0}{0.835}} \approx 7.01 \, \mathrm{rad/s}\]
Understanding natural frequency helps us comprehend how quickly an undamped system vibrates. This parameter provides a foundational understanding for any subsequent analysis of the system when damping is introduced.

The damping ratio, denoted as \( \zeta \), is a dimensionless measurement that characterizes how oscillations in a system diminish. It compares the damping forces present in the system to its natural frequency.
In simpler terms, the damping ratio tells us how fast the oscillations fade out. For a spring-mass-damper system, the formula for the damping ratio is:\[\zeta = \frac{b}{2\sqrt{km}}\]
Here, \( b \) is the damping coefficient, \( k \) is the spring constant, and \( m \) is the mass.Using the values from the exercise:

• \( b = 0.662 \, \mathrm{Ns/m} \)
• \( k = 41.0 \, \mathrm{N/m} \)
• \( m = 0.835 \, \mathrm{kg} \)

The damping ratio is calculated as:\[\zeta \approx 0.0576\]
This value suggests a lightly damped system, meaning that the amplitude will gradually decrease over time but not very rapidly.

When a damped harmonic oscillator is subject to a damping force, the frequency of oscillation changes from its natural frequency. This modified frequency is known as the damped frequency.
The damped angular frequency \( \omega_d \) can be calculated using the formula:\[\omega_d = \omega_0 \sqrt{1 - \zeta^2}\]
This frequency takes into account the damping factor \( \zeta \) and the natural frequency \( \omega_0 \).Substituting the known values:

• \( \omega_0 = 7.01 \, \mathrm{rad/s} \)
• \( \zeta = 0.0576 \)

We find that:\[\omega_d \approx 7.00 \, \mathrm{rad/s}\]
The damped frequency is slightly lower than the natural frequency, indicating that the presence of damping slows down the system’s oscillation, albeit slightly in this case. Understanding the damped frequency is key for predicting and analyzing the behavior of a system under real-world conditions.

The amplitude decrease per cycle is an important aspect of a damped oscillator and is determined by how much the maximum amplitude reduces after each complete swing of the system.
The reduction in amplitude per cycle can be calculated using the expression:\[\text{Fractional Decrease} = 1 - e^{-2\pi\zeta/\sqrt{1-\zeta^2}}\]
This formula takes into account the damping ratio \( \zeta \) to quantify the exponential drop in amplitude.For our exercise, using the calculated damping ratio:

• \( \zeta = 0.0576 \)

The fractional decrease per cycle is:\[\text{Fractional Decrease} \approx 0.356\]
Therefore, the amplitude decreases by approximately 35.6% with each cycle. Knowing the rate of amplitude decrease is crucial for understanding how quickly a system loses energy and approaches rest.