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Why the Spring Model Doesn’t WorkThe Abelian GroupEdward 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
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,σ2)
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: 0.7806 s–1, σ = 0.1021 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 = 0.7806 ± 0.2001 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)