KAHIBARO
Table of Contents
When resonance happens
Resonance is the strong response of an oscillating system when it is driven at a frequency close to its natural frequency. A small repeated push can produce a very large motion if the timing matches the system well.
A simple way to think about resonance is a child on a swing. If the pushes come at the right rhythm, the swing goes higher and higher. If the pushes come too fast or too slowly, the effect is much weaker. The size of the driving force may stay the same, but the amplitude of the motion changes greatly depending on the driving frequency.
Resonance in a driven oscillator
Consider a mass attached to a spring that is also being pushed by an external periodic force. The driving force can be written as
$$
F(t) = F_0 \cos(\omega t)
$$
where $F_0$ is the force amplitude and $\omega$ is the driving angular frequency.
The system itself has a natural angular frequency, for a mass spring system,
$$
\omega_0 = \sqrt{\frac{k}{m}}
$$
If there were no damping, the response becomes especially large when
$$
\omega = \omega_0
$$
This is the resonance condition in the ideal case.
Resonance occurs when the driving frequency is equal to, or very close to, the natural frequency of the system.
For an ideal undamped oscillator,
$$
\omega_{\text{res}} = \omega_0
$$
Why the amplitude becomes large
In forced oscillation, energy is supplied by the external force. At resonance, the force acts in step with the motion in such a way that energy is added very efficiently during each cycle. Because of this, the oscillator gains energy faster than at other driving frequencies.
If damping is present, some energy is continuously lost. Then the amplitude does not grow without limit. Instead, it reaches a steady value where energy gained from the driving force equals energy lost to damping.
Role of damping
Real systems always have some damping. Because of damping, the resonance peak is not infinitely high. It becomes lower and broader as damping increases.
With light damping, resonance is sharp and the amplitude near the resonant frequency is very large. With heavy damping, resonance is less noticeable.
The steady state amplitude of a driven damped oscillator is
$$
A(\omega) = \frac{F_0/m}{\sqrt{(\omega_0^2 - \omega^2)^2 + (b\omega/m)^2}}
$$
where $b$ is the damping constant.
This formula shows that the amplitude depends strongly on the driving frequency $\omega$.
In real systems, damping limits the resonant amplitude.
Less damping, higher and sharper resonance peak.
More damping, lower and broader resonance peak.
Resonance curve
If we plot amplitude versus driving frequency, we get a resonance curve. The curve has a peak near the natural frequency.
The exact peak position depends on damping. For small damping, it lies very close to $\omega_0$.
Phase and resonance
A driven oscillator does not always move exactly in step with the driving force. The phase difference changes with driving frequency. Far below resonance, the motion is nearly in phase with the force. Far above resonance, the motion is nearly opposite in phase. Near resonance, the phase difference is about $\pi/2$.
This change in phase is one of the signs that the system is passing through resonance.
Examples of resonance
Resonance appears in many areas of physics and engineering. A swing is one familiar example. Musical instruments use resonance to strengthen sound. A tuning fork resonates strongly at its own natural frequency. Buildings and bridges can also resonate, which is important for safety when exposed to wind, earthquakes, or repeated forces.
An electrical circuit can also show resonance, though that belongs to a later topic. The main idea is the same, a system responds most strongly when driven near its natural frequency.
| System | Natural oscillation | Driving source | Resonance effect |
|---|---|---|---|
| Swing | Back and forth motion | Periodic pushes | Large swing amplitude |
| Tuning fork | Mechanical vibration | Sound or удар, corrected below | |
| Guitar string | String vibration | Plucking or sound coupling | Strong sound at certain frequencies |
| Bridge | Structural vibration | Wind or repeated loading | Dangerous large oscillations |
Energy viewpoint
Resonance can also be understood through energy transfer. The driver supplies energy every cycle. If the timing is poor, some pushes help little or may partly oppose the motion. If the timing is right, almost every push adds energy effectively. That is why resonance produces the largest steady motion.
Important ideas to remember
Resonance is not just large motion. It is specifically large response caused by matching the driving frequency to the system's natural frequency. The sharper the resonance, the more sensitive the system is to frequency. Damping reduces this sensitivity.
Key facts about resonance:
Resonance is the maximum response of a driven oscillator.
It occurs when the driving frequency is near the natural frequency.
Damping prevents unlimited growth of amplitude.
Near resonance, energy transfer from the driver to the oscillator is most efficient.
A simple comparison
| Driving frequency | Response |
|---|---|
| Much less than natural frequency | Small to moderate amplitude |
| Near natural frequency | Maximum amplitude |
| Much greater than natural frequency | Small amplitude |
Resonance is one of the most important ideas in oscillations because it explains why repeated small forces can produce very large effects when their timing matches the system correctly.