Bending vibrations of wedge beams with any number of point masses

The natural frequencies and mode shapes of beams with constant width and linearly tapered depth (or thickness) carrying any number of point masses at arbitrary positions along the length of the beams were investigated using the Euler–Bernoulli equation. Use of the closed-form (exact) solutions for the natural frequencies and mode shapes of the unconstrained single-tapered beam (without carrying any point masses) and incorporation of the expansion theorem, the equation of motion for the associated constrained beam (carrying any point masses) were derived. Solution of the last equation will yield the desired natural frequencies and mode shapes of the constrained single-tapered beam. The bending vibrations of a single-tapered beam with six kinds of boundary conditions were investigated. Comparison with the existing literature or the traditional finite element method results reveals that the adopted approach has excellent accuracy and simple algorithm.