Asymptotic eigenfrequency distributions for the \(N\)-beam Euler-Bernoulli coupled beam equation with dissipative joints (Q1176784)

The following theorem is proved: If the \(n\) beams have the same length, then there will be at most \(n\) streams of eigenfrequencies, each lying asymptotically on a vertical line. More generally, if the beams have different lengths, but these lengths have ratios \(\ell_ 1:\ell_ 2:\ell_ 3:\dots:\ell_ n=p_ 1:p_ 2:\dots:p_ n\), with all \(p_ i\) integers, then there will be at most \(p_ 1+p_ 2+\dots+p_ n\) streams of vertical eigenfrequencies. However, if the ratios of the lengths are irrational, then no distinct streams occur, and the eigenfrequencies form a chaotic pattern.