= Two-mode separation of tension and bending stiffness
{title2=$f_n^2/n^2=a+bn^2$}
<Two-mode separation of tension and bending stiffness> uses two identified transverse <eigenmodes> to separate coefficients that a single resonance cannot determine independently. For the uniform <simply supported beam> model derived in <vibroscope area estimator>,
$$
\frac{f_n^2}{n^2}=a+bn^2,\qquad
a=\frac{T}{4\mu L^2},\qquad b=\frac{\pi^2B}{4\mu L^4}.
$$
Thus plotting $f_n^2/n^2$ against $n^2$ gives an intercept associated with tension and a slope associated with bending. Frequencies alone identify these ratios; determining dimensional $T$ and $B$ also requires the remaining scale quantities.
An original exact example takes $f_1=800\,\mathrm{Hz}$ and $f_2=\sqrt{3328000}\,\mathrm{Hz}\approx1824.28\,\mathrm{Hz}$. The two transformed values are $640000$ and $832000\,\mathrm{Hz^2}$ at $n^2=1$ and $4$. Their difference gives
$$
b=\frac{832000-640000}{4-1}=64000\,\mathrm{Hz^2},\qquad
a=640000-b=576000\,\mathrm{Hz^2}.
$$
Hence $r=b/a=1/9$. Using the first mode as if $b=0$ would use $640000$ instead of $576000$ for the tension coefficient; the corresponding area correction is a factor $10/9$ under this model.
Two observations fit two free coefficients, so that fit alone does not validate the model. A third identified mode should satisfy $f_3^2=9(a+9b)=10368000\,\mathrm{Hz^2}$, or $f_3\approx3219.94\,\mathrm{Hz}$, in this example. Repeated measurements and another mode can test consistency.
For $T>0$ and $B\geq0$, the same model requires $2\leq f_2/f_1<4$. A ratio outside that range contradicts at least one assumption or the mode assignment. Actual clamped grips require the clamped-end frequency model described in the companion theory.<Vibroscope method for determination of cross-sectional area of glass and carbon fibres - Theory and further development by Bo Madsen and others>[§2.2, clamped-end boundary conditions and frequency equation]{ref} Do not label arbitrary neighbouring resonance peaks as modes one and two without additional evidence.
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