Two-mode separation of tension and bending stiffness ($f_n^2/n^2=a+b{n}^2$)

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,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 $f_2=\sqrt{3328000}\mathrm{Hz}\approx 1824.28\mathrm{Hz}$. The two transformed values are $640000$ and $832000\mathrm{H}{\mathrm{z}}^2$ at $n^2=1$ and $4$. Their difference givesHence $r=b/a=1/9$. 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 $f_3^2=9(a+9b)=10368000\mathrm{H}{\mathrm{z}}^2$, or $f_3\approx 3219.94\mathrm{Hz}$, in this example. Repeated measurements and another mode can test consistency.

For $T>0$ and $B\ge 0$, the same model requires $2\le f_2/f_1<4$. 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.^[ref] Do not label arbitrary neighbouring resonance peaks as modes one and two without additional evidence.