A general theory for the determination of natural frequencies and mode shapes for a set of elastically connected axially loaded Euler–Bernoulli beams is developed. A normal-mode solution is applied to a set of non-dimensional coupled partial differential equations. The natural frequencies are the eigenvalues of a matrix of differential operators. The matrix operator is shown to be self-adjoint leading to an orthogonality condition for the mode shapes.

In the special case of identical beams, it is shown that the natural frequencies are organized into sets of intramodal frequencies in which each mode shape is a product of a spatial mode and a discrete mode. An exact solution is available for the general case. However the natural frequencies and mode shapes are then determined using a complicated numerical method. A Rayleigh–Ritz method using mode shapes of the corresponding unstretched beams is developed as an alternative.