1. Vibrating Beam Inverse Problem Team K.E.Y Scott Clark ● Asya Monds ● Hanh Pham SAMSI Undergraduate Workshop 2007

2. Outline First Model (spring) Potential Problems How to improve Second Model (beam) Results

3. The first model: Spring Model We have observations : (t 1 ; y 1 ); … ; (t m ; y m ). The goal is to estimate the unknown parameters C and K.

4. Our cost function: Now we need to minimize the cost function. After running the script, we get: C=0.7284; K=1537.8

6. Checking the assumptions Homoscedasticity Assumption:

7. Normality assumption

8. Independence Assumption

9. Beam Model

10. Cost Function  Minimize it. How?  7 parameters, YI, CI, ρ, etc  Extremum may be dense in parameter space  Find “reasonable” values  Set beam to same as patch, search near given data, try to minimize a new cost function

11. A new cost function  Needs to take into account spatial variations as well as frequency variations from the model and the data  So we use a weighted least squares cost function minimized a simplex method (fminsearch).  This doesn’t work. Phase change too much to overcome.

12. What now then?  Limit the search. Fewer parameters, smaller variations.  And then, it works! (kind of)

13. The data

14. The parameters found  gamma 0.17916724273807 air damping  YI_beam ** 0.00000669769791 beam -- Young's modulus  CI_beam 0.01013780051727 beam -- internal damping  Kp 0.00036441688104 Kp for beam  rho_patch ** 0.08291057427864 linear density of patch  YI_patch ** 0.20912090251615 patch -- Young's modulus  CI_patch 0.00009783931434 patch -- internal damping

15. Thank You Any Questions?