Module 04
Free vibration and damping
Displace a structure, let go, and watch what happens — the source of the natural frequency, the three damping regimes, and the measurement that gets damping out of a real building.
What this module covers
- Derive ωn = √(k/m) from the free-vibration equation
- Derive the critical damping coefficient and explain what makes it critical
- Write the free-vibration response for underdamped, critical and overdamped cases
- Explain why the damped frequency is below the undamped one, and by how little
- Derive the logarithmic decrement and use it to estimate damping from a decay trace
- Explain why damping is the least reliable number in a dynamic model
Lessons
Where ωn = √(k/m) comes from, what critical damping means, and why the three regimes are one solution rather than three.
Start lesson →The logarithmic decrement, derived and then used — and an honest account of why the number it gives is the least reliable input to any dynamic model.
Start lesson →
Module checkpoint
Check what you have taken in
4 questions
Question 1
A structure has m = 20 000 kg and k = 8.0 MN/m. What is the critical damping coefficient, in kN·s/m?
Question 2
Peaks in a decay trace are 25.0 mm and 9.2 mm, eight cycles apart. What is the damping ratio as a percentage?
Question 3
What does critical damping mean physically?
Question 4
A structure with 5% damping is set vibrating. What fraction of its initial ENERGY remains after 3 cycles? Give the answer as a percentage.