The modal analysis and seismic response of a vibrating cantilever, with or without
tip mass and rotary inertia, are investigated in this study using a shear
deformable beam model and including the effect of vertical load. Based on existing
approaches, an original method is proposed that does not use fourth-order
uncoupled equations to determine modal deflection and rotation. In fact, the
approach presented herein transforms the second-order coupled system into a
first-order system which can be solved more easily using matrix algebra and
Laplace transform. Furthermore, the proposed form allows a straightforward
demonstration of orthogonality conditions, that is, the problem is self-adjoint,
and the solution in the case of forced response using modal superposition. In
addition, even if the solution presented herein is applicable only to the cantilever
with a tip mass and rotary inertia, the scope is general, and the approach
can be applied to shear deformable beams with other boundary conditions.
Finally, the seismic response by modal superposition is shown, and some examples
are proposed and discussed for the case of uniform or continuously varying
cross-sectional properties.

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