On the eigenfrequencies of a cantilevered beam with a tip mass and in-span support

The present study addresses essentially to the investigation of the bending eigenfrequencies of a cantilevered Bernoulli-Euler beam with a tip mass which has an in-span support. First, an exact frequency equation is established via a boundary value formulation and then an approximate frequency equation is derived using the Lagrange's multipliers method. For a design engineer who is interested in a quick but accurate determination of the fundamental eigenfrequency of the system, an approximate formula is also established by using the Dunkerley's procedure. Finally, a formula for the sensitivity of the eigenfrequencies with respect to small changes in the location of the in-span support is derived. Numerical results on a digital computer justify the usefulness of the obtained analytical results.