A vibrating cantilever beam near one of its modes is modelled as a Duffing oscillator…

PHYSICS · Multi-part · Fermi University Physics · Computational Physics · Research software practice, verification and validation · Research software practice, verification and validation

Scientific Computing for Physicists, Ch Are they the right equations? Solution verification, benchmarks and validation

Problem

A vibrating cantilever beam near one of its modes is modelled as a Duffing oscillator

\ddot{x} + 2\zeta\omega_0\dot{x} + \omega_0^2 x + c_3 x^3 = \frac{F_0}{m}\cos\Omega t,

with

\omega_0 = 314\,\mathrm{rad\,s^{-1}}

, damping ratio

\zeta = 0.008

and cubic stiffness

c_3 = 4.5\times10^{7}\,\mathrm{m^{-2}\,s^{-2}}

, the last obtained by fitting a small-amplitude frequency sweep. A verified RK4 code time-steps the equation to its steady state and, for a forcing amplitude

F_0/m = 5.05\,\mathrm{m\,s^{-2}}

, reads off the peak displacement at resonance:

S = 3.20\,\mathrm{mm}

. A step-halving study over three step sizes (in the asymptotic range) gives a numerical uncertainty

u_{\rm num} = 0.02\,\mathrm{mm}

. Refitting

c_3

within the scatter of the calibration data and re-running the simulation gives an input uncertainty

u_{\rm input} = 0.15\,\mathrm{mm}

. A laser-vibrometer measurement of the same beam under the same forcing, averaged over 20 sweeps, gives

D = 3.05 \pm 0.10\,\mathrm{mm}

. ((a)) Compute the comparison error

E = S - D

. ((b)) Combine

u_{\rm num}

u_{\rm input}

and

u_D

(independent sources) into

u_{\rm val}

. ((c)) Using

k = 2

, give the interval for

\delta_{\rm model}

and state whether the comparison provides evidence of model-form error at this forcing level. ((d)) The same

\omega_0

\zeta

and

c_3

, unchanged, are then used to predict the peak amplitude when the forcing is raised to

F_0/m = 202\,\mathrm{m\,s^{-2}}

(forty times larger), a regime in which the Duffing backbone curve is known to bend enough to produce amplitude jumps and bistability. Is that prediction inside the validation domain established in part (c)? Justify.

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