Damping

Friction, and the three ways a system can settle.

12 problems · about 24 minutes · F-IF.B.4, A-REI.B.4

NameDate

Figure — use these to answer the problems

This decaying envelope is what damping imposes. An underdamped system oscillates inside a curve like this one.

y = 3 · -0.5^x + 0

the starting amplitude3

the damping rate-0.5

the time shown2

Figure 1

Straightforward practice. Get the method working first.

5 problems

m = 1, c = 4, k = 4. What is c² − 4mk?
Answer
2
Which damping case oscillates?
- Underdamped, where the roots are complex.
- Overdamped, where the roots are real.
m = 1, c = 2, k = 5. What is c² − 4mk?
Answer
A discriminant of −16. Enter 1 underdamped, 2 critically damped, 3 overdamped.
Answer
A discriminant of 0. Enter 1 underdamped, 2 critically damped, 3 overdamped.
Answer

The same ideas with more to keep track of.

3 problems

A discriminant of 25. Enter 1 underdamped, 2 critically damped, 3 overdamped.
Answer
m = 1 and k = 16. What value of c gives critical damping?
Answer
Which case returns to rest fastest without overshooting? Enter 1 under, 2 critical, 3 over.
Answer

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

4 problems

9
Match each case to how the system behaves.
Draw a line from each item on the left to its match on the right.
- Underdamped
- Critically damped
- Overdamped
- Oscillates inside a shrinking envelope
- Returns to rest fastest without overshooting
- Crawls back slowly with no oscillation
c = 0, meaning no damping at all. Does the system oscillate forever? 1 yes, 0 no.
Answer
The Discriminant
m = 1, c = 6, k = 5. What is c² − 4mk?
Answer
The Boundary
m = 1 and k = 9. What value of c gives critical damping?
Answer