Add a periodic push and the equation becomes my″ + cy′ + ky = F cos(ωt). The system now has two frequencies in play.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Transient and steady state
The homogeneous part fades away with damping. What remains is the steady state, oscillating at the forcing frequency.
Resonance
When the forcing frequency matches the natural frequency, the response grows enormous. Small pushes at the right rhythm add up.
Without damping
With no damping and exact resonance, the amplitude grows without bound. The solution contains a factor of t.
With damping
Damping caps the peak, but the response near resonance can still be many times the input.
Where it shows up
A pushed swing, a wine glass breaking to a sung note, a radio tuning to one station, and bridges that had to be redesigned.
An external driving force
A periodic force makes the equation nonhomogeneous. The solution has a transient part that decays and a steady-state part that persists at the driving frequency.
Resonance
Driving at the system's natural frequency produces amplitude growing without bound in the undamped case. Pushing in rhythm with the natural motion adds energy every cycle.
Damping bounds the response
With damping the amplitude stays finite but peaks sharply near the natural frequency. How sharp that peak is depends on how light the damping is, which is what quality factor measures.
Both useful and dangerous
Radio tuning and musical instruments exploit resonance; bridges and buildings are designed to avoid it. It is one of the clearest cases of a differential equation result having direct engineering consequences.
Step 2: Try It Yourself
Tap and try it out.
- Point(0, 0)
Step 3: Watch an Example
One step at a time.
Watch Sam Spot a Resonance
Sam has y″ + 9y = 2cos(3t) and wants to know what happens.
- Step 1
He reads the natural frequency from ω² = 9, so ω = 3.
Step 4: Your Turn
Practice makes it stick.
The Match
Problem 1 of 2
y″ + 16y = cos(4t). What is the natural frequency?
The Swing
Problem 2 of 2
A swing with a natural frequency of 2. At what forcing frequency does resonance occur?
Match the Rhythm
1 of 8
y″ + 25y = cos(5t). What is the natural frequency?
2 of 8
y″ + 25y = cos(5t). Is this resonance? 1 yes, 0 no.
3 of 8
y″ + 25y = cos(2t). Is this resonance? 1 yes, 0 no.
4 of 8
A damped system driven at resonance. Does the amplitude stay bounded? 1 yes, 0 no.
5 of 8
An undamped system at exact resonance. Does the amplitude stay bounded? 1 yes, 0 no.
6 of 8
The transient part of a damped forced solution. What does it approach in the long run?
7 of 8
Sort each example by whether it is resonance.
Tap something to move it.
- Empty
- Empty
8 of 8
y″ + 4y = cos(2t). What is the natural frequency?
Step 5: Quick Check
Show what you know.
Question 1 of 2
y″ + 36y = cos(6t). What is the natural frequency?
Question 2 of 2
When does resonance occur?
What You Learned
- A forced system oscillates at the forcing frequency once the transient has faded.
- Resonance happens when the forcing frequency matches the natural frequency.
- Without damping, resonance grows without bound; with damping, the peak is capped but still large.