A damping force opposes motion in proportion to velocity, adding a y′ term: my″ + cy′ + ky = 0.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The discriminant decides
The characteristic equation is a quadratic, and c² − 4mk decides everything about the behaviour.
Underdamped
A negative discriminant gives complex roots. The system oscillates inside a shrinking exponential envelope.
Overdamped
A positive discriminant gives two real negative roots. The system crawls back with no oscillation at all.
Critically damped
A zero discriminant is the boundary. It returns to rest fastest without overshooting.
Why engineers care
Car suspensions and door closers aim near critical damping: settle quickly, without bouncing.
Friction adds a velocity term
A resistance proportional to speed gives mx″ + cx′ + kx = 0. The new middle term is what removes energy, and its size determines which of three behaviours results.
Three regimes
Underdamped systems oscillate with decaying amplitude, overdamped ones return slowly without oscillating, and critically damped ones return as fast as possible without overshooting.
The discriminant decides
Which regime occurs depends on the sign of c² − 4mk, the discriminant of the characteristic equation. The algebra and the physics correspond exactly, case by case.
Critical damping is usually the design target
Car suspensions and door closers are tuned near critical damping, since it settles fastest without overshoot. It is a design choice made by adjusting c relative to m and k.
Step 2: Try It Yourself
Tap and try it out.
- Point(2, 0.03)
Step 3: Watch an Example
One step at a time.
Watch Yara Classify a System
Yara has m = 1, c = 4 and k = 3.
- Step 1
She computes c², which is 16.
Step 4: Your Turn
Practice makes it stick.
The Discriminant
Problem 1 of 2
m = 1, c = 6, k = 5. What is c² − 4mk?
The Boundary
Problem 2 of 2
m = 1 and k = 9. What value of c gives critical damping?
How Does It Settle
1 of 8
m = 1, c = 2, k = 5. What is c² − 4mk?
2 of 8
A discriminant of −16. Enter 1 underdamped, 2 critically damped, 3 overdamped.
3 of 8
A discriminant of 0. Enter 1 underdamped, 2 critically damped, 3 overdamped.
4 of 8
A discriminant of 25. Enter 1 underdamped, 2 critically damped, 3 overdamped.
5 of 8
m = 1 and k = 16. What value of c gives critical damping?
6 of 8
Which case returns to rest fastest without overshooting? Enter 1 under, 2 critical, 3 over.
7 of 8
Match each case to how the system behaves.
Tap a card on the left to start.
8 of 8
c = 0, meaning no damping at all. Does the system oscillate forever? 1 yes, 0 no.
Step 5: Quick Check
Show what you know.
Question 1 of 2
m = 1, c = 4, k = 4. What is c² − 4mk?
Question 2 of 2
Which damping case oscillates?
What You Learned
- Damping adds a term proportional to velocity, and the discriminant c² − 4mk decides everything.
- Negative means underdamped and oscillating; positive means overdamped and sluggish.
- Zero is critical damping, which returns to rest fastest without overshooting.