Forced vibrations and resonance：練習問題

What is the primary characteristic of a forced vibration?

For a damped, forced oscillator, how does heavy damping affect the frequency (\(f_{res}\)) at which the maximum amplitude occurs, compared to the natural frequency (\(f_0\)) of the undamped system?

A mechanical system is subjected to forced vibrations. If the damping on the system is increased, how does this affect the amplitude of oscillation at the resonant frequency?

A car suspension system includes dampers (shock absorbers). The primary physical role of these dampers in the context of forced oscillations caused by road bumps is to:

Resonance occurs in a forced oscillatory system when:

Describe the effect of introducing light damping versus heavy damping on the resonance peak observed when plotting the amplitude of a forced oscillator against the driving frequency.

When a vibrating system is heavily damped, the frequency at which the maximum amplitude occurs is slightly different from the natural frequency (\(f_0\)) of the free oscillation. State whether this resonance frequency is higher or lower than \(f_0\) and justify why.

A mechanical system undergoing oscillations has a natural frequency $f_0 = 15 \text{ Hz}$. It is subjected to a periodic driving force of variable frequency $f$.

(a) Describe the phenomenon of resonance in this system, referring to the energy transfer and the amplitude of oscillation.

(b) The system is initially lightly damped. An engineer then increases the damping significantly. Sketch a graph showing the variation of oscillation amplitude with driving frequency for both the original system (low damping) and the modified system (high damping). Clearly label the natural frequency $f_0$ on the frequency axis.

(c) Explain two key effects that increasing the damping has on the resonance curve, specifically addressing the peak amplitude and the resonance frequency.

A structure, such as a tall chimney, can be modelled as a simple mechanical oscillator. The structure has a natural frequency $f_0$. It is subject to forced vibrations due to wind passing over it.

(a) Define the term Quality factor ($Q$) in the context of forced oscillations, and explain how its value relates to the sharpness of the resonance curve.

(b) The structure has a mass $m = 8.0 \times 10^4 \text{ kg}$ and an effective spring constant $k = 3.2 \times 10^7 \text{ N m}^{-1}$. Calculate the natural frequency $f_0$ of the structure.

(c) During high wind conditions, the structure undergoes heavy damping. Describe and explain the need for heavy damping in large civil engineering structures like bridges or tall buildings when they are exposed to a wide range of potential driving frequencies (e.g., wind or traffic).

(d) If the Quality factor of the structure is $Q=2.0$, and the maximum displacement under resonance is $A_{res}$. When the structure is driven at a frequency far below $f_0$, its amplitude is $A_{static}$. Explain whether $A_{res}$ is exactly twice $A_{static}$ for this heavily damped system, and why, relating your answer to the shift in resonance frequency $f_r$.