Forced Oscillation and Resonance

Pushing at the right rhythm, and what it costs.

12 problems · about 23 minutes · F-TF.B.5, F-IF.B.4

What this lesson teaches

The student explains resonance and identifies when it occurs.

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.

Straightforward practice. Get the method working first.

5 problems

y″ + 36y = cos(6t). What is the natural frequency?
Answer 6
Why 6.
When does resonance occur?
Answer When the forcing frequency matches the natural frequency.
Why When the frequencies match.
y″ + 25y = cos(5t). What is the natural frequency?
Answer 5
Why The square root of 25.
y″ + 25y = cos(5t). Is this resonance? 1 yes, 0 no.
Answer 1
Why The frequencies match.
y″ + 25y = cos(2t). Is this resonance? 1 yes, 0 no.
Answer 0
Why 2 does not match 5.

The same ideas with more to keep track of.

3 problems

A damped system driven at resonance. Does the amplitude stay bounded? 1 yes, 0 no.
Answer 1
Why Damping caps the peak.
An undamped system at exact resonance. Does the amplitude stay bounded? 1 yes, 0 no.
Answer 0
Why Nothing removes the energy being added.
The transient part of a damped forced solution. What does it approach in the long run?
Answer 0
Why It fades away.

Mixed problems. Work out what kind of question it is before you start.

4 problems

Sort each example by whether it is resonance.
Answer Resonance: Pushing a swing exactly in time with its motion, A sung note matching a glass and breaking it · Not resonance: Pushing a swing at random moments, A steady constant force with no oscillation
Why The push must match the natural rhythm.
y″ + 4y = cos(2t). What is the natural frequency?
Answer 2
Why The square root of 4.
The Match: y″ + 16y = cos(4t). What is the natural frequency?
Answer 4
Why 4.
The Swing: A swing with a natural frequency of 2. At what forcing frequency does resonance occur?
Answer 2
Why 2.