We analyze the transverse vibrations of a thin homogeneous beam which is symmetric with respect to the x-y and x-z planes. The cross section of the beam at x is assumed to have the form
$D(x)={(x,y,z)|x∈[0,1],$

$y=xαy1,$

$z=xβz1,$

$(y1,z1)∈D1}$
where D1 is the cross section at x = 1. Expressions are obtained from which the eigenvalues and eigenfunctions can be easily found for 0 ≤ α < 2 and all combinations of clamped, hinged, guided, and free boundary conditions at both ends of the beam.
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