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

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

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Page 1: Why the Spring Model Doesn’t Work The Abelian Group Edward Lee, Seonhee Kim, Adrian So

Why the Spring Model Doesn’t WorkThe Abelian GroupEdward Lee, Seonhee Kim, Adrian So

Page 2: 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) = y0 and y’(0) = 0.

( ) ( ) ( ) 0y t Cy t Ky t

Page 3: Why the Spring Model Doesn’t Work The Abelian Group Edward Lee, Seonhee Kim, Adrian So

Methodology Data for the vibrating beam were collected

(total 9 runs) Subsequent analysis of data was performed in

Matlab using statistical methods

Page 4: Why the Spring Model Doesn’t Work The Abelian Group Edward Lee, Seonhee Kim, Adrian So

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,σ2)

Errors independent of measured displacements (homoscedasticity) and of time

Page 5: Why the Spring Model Doesn’t Work The Abelian Group Edward Lee, Seonhee Kim, Adrian So

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

Page 6: Why the Spring Model Doesn’t Work The Abelian Group Edward Lee, Seonhee Kim, Adrian So

Observed Values of C and K Data for all 9 runs fit according to the

procedure before Mean C: 0.7806 s–1, σ = 0.1021 s–1

Mean K: 1539 s–2, σ = 4.749s–2

Page 7: Why the Spring Model Doesn’t Work The Abelian Group Edward Lee, Seonhee Kim, Adrian So
Page 8: Why the Spring Model Doesn’t Work The Abelian Group Edward Lee, Seonhee Kim, Adrian So
Page 9: Why the Spring Model Doesn’t Work The Abelian Group Edward Lee, Seonhee Kim, Adrian So
Page 10: Why the Spring Model Doesn’t Work The Abelian Group Edward Lee, Seonhee Kim, Adrian So

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)

Page 11: Why the Spring Model Doesn’t Work The Abelian Group Edward Lee, Seonhee Kim, Adrian So
Page 12: Why the Spring Model Doesn’t Work The Abelian Group Edward Lee, Seonhee Kim, Adrian So

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

Page 13: Why the Spring Model Doesn’t Work The Abelian Group Edward Lee, Seonhee Kim, Adrian So

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 = 0.7806 ± 0.2001 s–1 and K = 1539 ± 9.3 s–2

Page 14: Why the Spring Model Doesn’t Work The Abelian Group Edward Lee, Seonhee Kim, Adrian So

Improvements Tried using least-fourths fitting

Penalizes large deviations from experimental data much more

Page 15: Why the Spring Model Doesn’t Work The Abelian Group Edward Lee, Seonhee Kim, Adrian So
Page 16: Why the Spring Model Doesn’t Work The Abelian Group Edward Lee, Seonhee Kim, Adrian So
Page 17: Why the Spring Model Doesn’t Work The Abelian Group Edward Lee, Seonhee Kim, Adrian So

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)