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The natural frequency of a system is a key concept in understanding how systems vibrate or oscillate without external force. It is essentially the frequency at which a system naturally tends to oscillate in the absence of damping or external forcing. For a system with a mass (\(m\)) and stiffness (\(k\)), the natural frequency (\(ω_n\)) can be calculated using the formula:

• \(ω_n = \sqrt{\frac{k}{m}}\)

This formula shows that the natural frequency is dependent on the stiffness and the mass of the system. In essence, a stiffer or lighter system will have a higher natural frequency. In our example, with \(k = 1\) and \(m = 4\), the natural frequency is \(ω_n = 0.5\).Understanding the natural frequency helps in predicting the behavior of the system at different frequencies.

The damping ratio is a dimensionless measure describing how oscillations in a system decay after a disturbance. A crucial aspect to study, the damping ratio (\(ζ\)) informs us about the nature of the system's response:

• \(ζ = \frac{b}{2\sqrt{mk}}\), where \(b\) is the damping coefficient.

In simple terms, it tells us how quickly the vibrations decrease in amplitude. A damping ratio less than 1 indicates underdamping, which is a condition where the system oscillates but the amplitude steadily decreases to zero. For our exercise, with \(b = 2\), the damping ratio is calculated as \(ζ = 0.5\). This means the system is underdamped, resulting in oscillations that diminish over time, but not instantly.

The frequency response function (\(H(ω)\)) is essential for understanding how a system reacts to different input frequencies. It helps in plotting the frequency response curves, showing how the amplitude of the system's output varies with frequency. The general expression for the frequency response is:

• \(H(ω) = \sqrt{\left(1-\left(\frac{ω^2}{ω_n^2}\right)\right)^2 + \left(\frac{2ζω}{ω_n}\right)^2}\)

This function captures the dynamic behavior of the system, reflecting how amplitudes change when exposed to varying frequencies. For the system with \(ω_n = 0.5\) and \(ζ = 0.5\), sketching this curve helps visualize resonant peaks and how the system amplitude approaches zero at higher frequencies.

Light damping refers to a condition where the system's damping ratio is less than 1, suggesting that the system is not critically damped. This condition allows the system to oscillate with decreasing amplitude. Presence of light damping means that while the system will experience some reduction in oscillatory response, it does not return to equilibrium immediately. In our system, with a damping ratio \(ζ < 1\), the system is referred to as lightly damped.

Notably, light damping impacts the frequency response curve by creating a resonance peak near the natural frequency. This peak represents the system's maximum response to a stimulus, after which the amplitude reduces as frequency increases beyond the natural frequency. Practically, this attribute is significant in scenarios like vehicle suspensions or seismograph designs where oscillations need to be reduced effectively while maintaining a desired level of responsiveness.