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

NameDate

Figure — use these to answer the problems

Set the frequency to match the natural one and imagine each push landing in time with the motion. That accumulation is resonance.

y = 1 sin(1x) + 0

the response amplitude1

the forcing frequency1

the time shown0

Figure 1

Straightforward practice. Get the method working first.

5 problems

y″ + 36y = cos(6t). What is the natural frequency?
Answer
2
When does resonance occur?
- When the forcing frequency matches the natural frequency.
- Whenever the forcing is large enough.
y″ + 25y = cos(5t). What is the natural frequency?
Answer
y″ + 25y = cos(5t). Is this resonance? 1 yes, 0 no.
Answer
y″ + 25y = cos(2t). Is this resonance? 1 yes, 0 no.
Answer

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
An undamped system at exact resonance. Does the amplitude stay bounded? 1 yes, 0 no.
Answer
The transient part of a damped forced solution. What does it approach in the long run?
Answer

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

4 problems

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