1. Modeling in an Engineering Mathematics Class -- Tuned Mass Dampers --
Dr Keith A. Landry, PE, F.ASCE
Assistant Professor, Georgia Southern University, Statesboro, GA
Dr Brian Winkel
Professor Emeritus, United States Military Academy, West Point, NY

2. Agenda Introduction & Background: Engineering Mathematics
Civil Infrastructure: Motion Under Load
Modeling Approach: Simple -> Complex
Free & Undamped
Forced & Undamped
Forced & Damped
Observations & Discussion

3. Background: Engineering Mathematics
2003 -> West Point
Civil & Mechanical Engineering Majors (ABET)
Team Teaching Approach
Integrated Engineering Scenarios into Course Content
Mathematica
Classroom Note-taking & Mathematica
Well-received by students
Show difficulties students have in going from FBD to equations – need practice with “signs” and forces. They often come to DE course, having lost that skill from elementary physics course, If they ever had it.
Be sure to state Newton’s Second Law of Motion which is what permits us to go from FBD to equations, e.g.
m y’’(t) = Sum of external forces
Start with
my’’ = - k y
purchased from “ideal” spring/building store, then drive it to get “trouble in River City” with resonance.
my’’ = - k y + f(t).
How to mitigate resonance, recognize internal resistance/friction of the building (if k y(t) represents stiffness, then c y’(t) represents resistance to motion) hence.
my’’ = - k y – c y’(t) + f(t).

4. Civil Infrastructure: Motion Under Load - Resonance
Tohuku Earthquake (Tokyo 2011)
Clifton Bridge (Bristol, UK 2013)
Show here maximum frequency response – which still can be dangerous.
What can engineer do next?

5. Civil Infrastructure: Motion Under Load - TMDs
TMD: Taipei 101
TMD: Schwedter Strasse, Berlin
TMD: Stockbridge Damper
TMD: Grand Canyon
TMD: Citicorp Building
Introduce “counter” mass which will sway “the other way: - quotes because students’ intuition tells them this could be possible

6. Modeling Approach: Multi-DOF System
Physical observation
Harmonic Motion
Free & Undamped (SDOFS)
Forced & Undamped (SDOFS)
Forced & Damped (SDOFS)
Forced & Damped (MDOFS)
Key Concepts:
Resonance
Damping ratio (ζ)
Then go to 2DOF or two mass system – not necessarily tuned yet. Build system, perhaps converted to linear system of four DE’s in y1(t), y1’(t), y2(t), and y2’(t). Eigenvalue options and what they can tell us, or analytic solution, but not by hand, always using Maple or Mathematica – computer algebra system with analyitic solution capabilities so students can see the form of the solution and the role the coefficients might play, certainly biy plotdting solutions. Or jump right to numerical solution say in MatLab or EXCEL (small step sizes) – but ALWAYS get plots, ALWAYS!!!!!!
Go back to pure oscillator when introducing second mass to prevent resonance and THEN with resistance to drop height of frequency response curve.
Actually, show plot to show motion is “TOTALLY” deadened in pure oscillator case AT the natural frequency when second tuned mass damper to THAT frequency is applied, and near that frequency good things happen, but away from that natural frequency could be trouble again.

7. Observations: Resonance (Undamped)
Mass 1
Mass 1
Mass 2

8. Observations: Resonance (Damped)
Then move to issues such as relative size of mass, new mass m2, original m1,
If m2/m1 increases then the band (in frequency of driver) at which the motion of m2 is controlled widens about the natural frequency, but we cannot have m2/m1 being too high – too expensive, e.g., Empire State bldg. on top of Empire State bldg. to TMD the original Empire State bldg. problems.
Which frequency dominates displacement?
Do we want to mitigate a range of frequencies?
Active vs Passive damping?
This is all passive, what does an active TMD do? This could be closer for it shows engineering mentality of trying to intervene and control a process for better, but there are costs, e.g., more sophisticated mathematics, energy input, more construction mass, etc.

9. Discussion Student Motivation Class Organization Teaching Resources
Student Feedback
Getting Started

10. TMD Modeling Scenarios: SIMIODE
(5-40-S) Tuned Mass Dampers – Part I (Student)
(5-40-S) Tuned Mass Dampers – Part I (Teacher)
(5-40-T) Tuned Mass Dampers – Part II (Student)
(5-40-T) Tuned Mass Dampers – Part II (Teacher)

11. Questions or Comments?