Variations in the modal characteristics of a telescopically deploying beam

The equations of motion for a two-segment deploying telescopic beam are derived through application of Lagrange's equation. The outer tube of the beam is fixed at one end and the inner tube slides freely relative to the fixed segment. The resulting nonlinear, non-autonomous set of equations is linearized and simplified to the standard Euler-Bernoulli partial differential equations for an elastic beam by freezing the deployment process at various stages of deployment, and examining the small amplitude and natural modes of vibration of the resulting configuration. Application of the natural boundary conditions and compatibility of motion relations for the two segments in their common region of overlap leads to a transcendental characteristic equation in the frequency parameter Beta(L). Numerical solution of the equation for the characteristic roots determines the modal frequencies, and the corresponding mode shapes are obtained from the general solution of the Euler-Bernoulli equation tailored to the natural boundary conditions. Sample results of modal frequencies and shapes are presented for various stages of deployment and discussed. It is shown that for all intermediate stages of deployment (between 0 and 100 percent) the spectral distribution is drastically altered by the appearance of regions of very closely spaced modal frequencies. The sources of this modal agglomeration are explored.