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

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

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.

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

Checking the assumptions Homoscedasticity Assumption:

Normality assumption

Independence Assumption

Beam Model

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

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.

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

The data

The parameters found  gamma air damping  YI_beam ** beam -- Young's modulus  CI_beam beam -- internal damping  Kp Kp for beam  rho_patch ** linear density of patch  YI_patch ** patch -- Young's modulus  CI_patch patch -- internal damping

Thank You Any Questions?