(56)

$\frac{{z}_{j}{J}_{0}({z}_{j})}{{J}_{1}({z}_{j})}=(1-\nu ),j=1,2,3,\dots $ (57)

(57)

$z=(\gamma a)=\mathrm{2}.0\mathrm{67364},5.\mathrm{395106},8.575401,11.734360\dots $ (58)

(58)

The modeshapes of Eq. (55) satisfy the orthonormality relation

${\int}_{0}^{a}{U}_{p}(r){U}_{q}(r)rdr=\frac{{a}^{2}}{2}{\delta}_{pq}$ (59)

(59)

The modal excitation ${f}_{j}$ is given ...

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