10.1 Experiments with the three-spectral inverse problem for a  (Page 2/2)

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$\frac{\sigma }{{\stackrel{˜}{\ell }}_{k}}\left({\stackrel{˜}{v}}_{k}\left(t\right)-{\stackrel{˜}{v}}_{k+1}\left(t\right)\right)+\frac{\sigma }{{\stackrel{˜}{\ell }}_{k-1}}\left({\stackrel{˜}{v}}_{k}\left(t\right)-{\stackrel{˜}{v}}_{k-1}\left(t\right)\right)+{\stackrel{˜}{m}}_{k}{\stackrel{˜}{v}}_{k}^{\text{'}\text{'}}=0,\phantom{\rule{1.em}{0ex}}k=1,2,\cdots ,{n}_{2}$

By assuming the displacement of bead ${m}_{k}$ behaves as ${u}_{k}{e}^{\lambda t}$ , the equations of motion for a beaded string give recurrence relations for the amplitudes of each individual bead:

${u}_{k}={R}_{2k-2}\left({\lambda }^{2}\right){u}_{1},\phantom{\rule{1.em}{0ex}}{\stackrel{˜}{u}}_{k}={\stackrel{˜}{R}}_{2k-2}\left({\lambda }^{2}\right){\stackrel{˜}{u}}_{1}$

Boyko and Pivovarchik show that the characteristic polynomial of the system of equations governing the motion of the beads is given by

$\phi \left({\lambda }^{2}\right)={R}_{2{n}_{1}}\left({\lambda }^{2}\right){\stackrel{˜}{R}}_{2{n}_{2}-1}\left({\lambda }^{2}\right)+{R}_{2{n}_{1}-1}\left({\lambda }^{2}\right){\stackrel{˜}{R}}_{2{n}_{2}}\left({\lambda }^{2}\right).$

The zeros of $\phi \left({\lambda }^{2}\right)$ , ${\lambda }_{k}^{2}$ { $k=1,...,{n}_{1}+{n}_{2}$ }, are the eigenvalues of the system of ordinary differential equations that describe the motion of the string and are the frequencies at which the whole string vibrates. The polynomials ${R}_{2{n}_{1}}\left({\lambda }^{2}\right)$ and ${\stackrel{˜}{R}}_{2{n}_{2}}\left({\lambda }^{2}\right)$ have roots that are the eigenvalues of the fixed-fixed boundary value problem for the left and right strings formed by clamping the string at the point where the left and right parts meet. The polynomials ${R}_{2{n}_{1}-1}\left({\lambda }^{2}\right)$ and ${\stackrel{˜}{R}}_{2{n}_{2}-1}\left({\lambda }^{2}\right)$ have roots that are the corresponding eigenvalues of the fixed-flat boundary value problem. Although for the forward problem it is usually simpler to solve the eigenvalue problem generated directly from the ODE's given by equations and and the assumption that the displacements are complex exponentials, it is important to consider these $R$ polynomials for the inverse problem.

The three-spectral inverse problem

Denote the spectra of the unclamped string by ${\lambda }_{k}$ , and the spectra of the left and right parts by ${\nu }_{k,\ell }$ and ${\nu }_{k,r}$ . $L$ is the length of the whole string and ${L}_{\ell }$ and ${L}_{r}$ are the lengths of the separate parts. From this information we immediately construct three polynomials:

${p}_{w}\left({\lambda }^{2}\right)=L\prod _{k=1}^{{n}_{1}+{n}_{2}}\left(1,-,\frac{{\lambda }^{2}}{{\lambda }_{k}^{2}}\right),\phantom{\rule{10.0pt}{0ex}}{p}_{\ell }\left({\lambda }^{2}\right)={L}_{\ell }\prod _{k=1}^{{n}_{1}}\left(1,-,\frac{{\lambda }^{2}}{{\nu }_{k,\ell }^{2}}\right),\phantom{\rule{10.0pt}{0ex}}{p}_{r}\left({\lambda }^{2}\right)={L}_{r}\prod _{k=1}^{{n}_{2}}\left(1,-,\frac{{\lambda }^{2}}{{\nu }_{k,r}^{2}}\right)$

Note that ${p}_{w}\left({\lambda }^{2}\right)$ is proportional to $\phi \left({\lambda }^{2}\right)$ , ${p}_{\ell }\left({\lambda }^{2}\right)$ is proportional to ${R}_{2{n}_{1}}\left({\lambda }^{2}\right)$ , and ${p}_{r}$ is proportional to ${\stackrel{˜}{R}}_{2{n}_{2}}\left({\lambda }^{2}\right)$ . It is known that the ratio of polynomials ${R}_{2{n}_{1}}\left(z\right)/{R}_{2{n}_{1}-1}\left(z\right)$ has the continued fraction expansion:

$\frac{{R}_{2{n}_{1}}\left({\lambda }^{2}\right)}{{R}_{2{n}_{1}-1}\left({\lambda }^{2}\right)}={\ell }_{n}+\frac{1}{-\frac{{m}_{n}}{\sigma }{\lambda }^{2}+\frac{1}{{\ell }_{n-1}+\frac{1}{-\frac{{m}_{n-1}}{\sigma }{\lambda }^{2}+...+\frac{1}{{\ell }_{1}+\frac{1}{-\frac{{m}_{1}}{\sigma }{\lambda }^{2}+\frac{1}{{\ell }_{0}}}}}}}$

which reveals the masses of the beads and the lengths between them that we are looking for. Since ${p}_{\ell }\left({\lambda }^{2}\right)$ is proportional to ${R}_{2{n}_{1}}\left({\lambda }^{2}\right)$ , we search for a second polynomial ${q}_{\ell }\left({\lambda }^{2}\right)$ such that ${p}_{\ell }\left({\lambda }^{2}\right)/{q}_{\ell }\left({\lambda }^{2}\right)$ gives the sought after continued fraction expansion. The same reasoning applies to the right part of the string.

${q}_{\ell }\left({\lambda }^{2}\right)=\sum _{k=1}^{{n}_{1}}\frac{{\lambda }^{2}{p}_{w}\left({\nu }_{k,\ell }^{2}\right)}{{\nu }_{k,\ell }^{2}{p}_{r}\left({\nu }_{k,\ell }^{2}\right)}\prod _{j=1,j\ne k}^{{n}_{1}}\frac{\left({\lambda }^{2}-{\nu }_{j,\ell }^{2}\right)}{\left({\nu }_{k,\ell }^{2}-{\nu }_{j,\ell }^{2}\right)}+\prod _{k=1}^{{n}_{1}}\frac{{\nu }_{k,\ell }^{2}-{\lambda }^{2}}{{\nu }_{k,\ell }^{2}}$
${q}_{r}\left({\lambda }^{2}\right)=\sum _{k=1}^{{n}_{2}}\frac{{\lambda }^{2}{p}_{w}\left({\nu }_{k,r}^{2}\right)}{{\nu }_{k,r}^{2}{p}_{\ell }\left({\nu }_{k,r}^{2}\right)}\prod _{j=1,j\ne k}^{{n}_{2}}\frac{\left({\lambda }^{2}-{\nu }_{j,r}^{2}\right)}{\left({\nu }_{k,r}^{2}-{\nu }_{j,r}^{2}\right)}+\prod _{k=1}^{{n}_{2}}\frac{{\nu }_{k,r}^{2}-{\lambda }^{2}}{{\nu }_{k,r}^{2}}$

After constructing these polynomials, we find their roots, which give us the second set of spectra for each of the left and right string parts. We now have all of the necessary information (fixed-fixed and fixed-flat spectra) to consider the problem reduced to two two-spectral problems. We use the algorithm presented in to recover the lengths and masses from the continued fraction expansion using only the roots of the polynomials.

Experimental setup

In the laboratory we have a beaded monochord consisting of steel piano wire held taut between two 5C collet fixtures. Behind one of the collets is a force transducer, and behind the other is a tensioner. The collets are initially open while the string is tensioned. The tension is monitored during this process via the force transducer. The collets are then closed and the system is ready for collecting data. Two phototransistors are positioned at each end of the string in perpendicular axes. In this way we can monitor the transverse vibrations both parallel and perpendicular to the ground at both ends of the string. The clamp is placed somewhere in the middle of the string such that at least one bead is left on each side of it. Both the tension and the phototransistors are monitored via a data acquisition card in a PC, and the information is collected and processed by MATLAB scripts.

Experimental results

Experimental testing has given very positive results. Despite the difficulties in collecting real data, the inverse algorithm just described has produced reasonable bead masses and lengths of string between the beads.

The peaks of the FFT plots shown in Figures , , and , give the observed frequencies for each section of the string. These are the eigenvalues that are used to construct the polynomials used during the inversion procedure.

Two sets of lenghts and masses that we recovered are depicted visually in figures and .

Results for a string with 3 beads

 ${m}_{1}$ ${\stackrel{˜}{m}}_{1}$ ${\stackrel{˜}{m}}_{2}$ ${\ell }_{1}$ ${\ell }_{2}$ ${\stackrel{˜}{\ell }}_{1}$ ${\stackrel{˜}{\ell }}_{2}$ ${\stackrel{˜}{\ell }}_{3}$ Measured 30.8 30.8 17.8 0.203 0.245 0.184 0.203 0.289 Recovered 29.5 29.7 16.1 0.207 0.240 0.192 0.211 0.273

Results for a string with 4 beads

 ${m}_{1}$ ${m}_{2}$ ${\stackrel{˜}{m}}_{1}$ ${\stackrel{˜}{m}}_{2}$ ${\ell }_{1}$ ${\ell }_{2}$ ${\ell }_{3}$ ${\stackrel{˜}{\ell }}_{1}$ ${\stackrel{˜}{\ell }}_{2}$ ${\stackrel{˜}{\ell }}_{3}$ Measured 30.8 17.8 30.8 30.8 0.127 0.229 0.092 0.362 0.152 0.162 Recovered 27.7 17.2 29.6 29.0 0.139 0.226 0.082 0.368 0.150 0.159

Acknowledgements

This Connexions module describes work conducted as part of Rice University's VIGRE program, supported by National Science Foundationgrant DMS–0739420.

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