Why the Spring Model Doesn’t Work The Abelian Group Edward Lee, Seonhee Kim, Adrian So

Model Equation of Motion  Our model equation of motion for the beam is based on the spring equation: where y is the displacement, C is the (normalized) damping constant, and K is the (normalized) spring constant.  Initial conditions: y(0) = y 0 and y’(0) = 0.

Methodology  Data for the vibrating beam were collected (total 9 runs)  Subsequent analysis of data was performed in Matlab using statistical methods

Assumptions  Physical assumptions Initial conditions were precisely as described Density, size of beam were not important (mass is not well-defined)  Mathematical assumptions Errors were identically, independently distributed Gaussian N(0,σ 2 ) Errors independent of measured displacements (homoscedasticity) and of time

Matlab Analysis  Parameters (C, K) for model equation allowed to vary in order to fit the observed data  Fit was determined using a residual-least squares cost function  Set of optimal (C, K) determined for each run

Observed Values of C and K  Data for all 9 runs fit according to the procedure before  Mean C: s –1, σ = s –1  Mean K: 1539 s –2, σ = 4.749s –2

Residuals  Residuals clearly time dependent (structure in residual-time plot) Expected no dependence on time – variance in errors should be same for each measurement Expected scatter with no patterns  Homoscedasticity assumption fails Expected no dependence on measurement – variance in errors should be same for each measurement Expected scatter with no patterns about the x-axis (fitted value)

QQ-Plot  Data do not lie on the expected straight line  There are more extreme values than in a Gaussian Residual distribution decays slower than in a normal distribution

Conclusions and Remarks  All assumptions on the errors were violated  Model decayed much faster than observed  The spring model does not adequately describe the problem of the vibrating beam  To less than 95% confidence, C = ± s –1 and K = 1539 ± 9.3 s –2

Improvements  Tried using least-fourths fitting Penalizes large deviations from experimental data much more

Other Improvements  Use beam equation and model to fit data Drawbacks: more parameters required for fitting  Fourier analysis Model frequency spectrum of observed data and obtain parameters  Use time- and space-dependent error terms ε = ε(x,t)