One somewhat surprising result in the vibration of beams is that the natural frequencies of a free-free beam are identical to those of a clamped-clamped beam. Our intuition may tell us that the fixed-fixed beam is stiffer and should have higher natural frequencies but while the fixed beam is indeed statically stiffer, the natural frequencies are identical. As we show below, fixed and free boundary conditions result in the equivalent characteristic equations to find the natural frequency because they differ simply by two differentiations. Similarly, pinned and sliding boundary conditions also have the same equivalence. As a result many pairs of boundary conditions can result in identical sets of natural frequencies but with clearly different mode shapes.

In this article, we compare the performance of a tuned-mass damper mounted at the end of a cantilever beam to the Lanchester damper which was shown in the previous article. The classic single-degree-of-freedom (SDOF) tuned-mass damper is sketched in the figure below. The design approach is to find the equivalent SDOF system for the cantilever beam’s mode of interest and then use the design formulas for an optimal SDOF TMD to determine the stiffness and damping of the absorber.

In this article, we show the robust and broadband performance of a Lanchester damper applied to a cantilever beam and how it achieves good performance without tuning and good performance over a number of modes, not just the primary mode.

Beams are often used in precision engineering applications. One common question is “what are the optimal support locations for a beam?” The answer depends on the desired objective. Below we describe some of the most common support locations: Airy points, Bessel points, minimum deflection, and nodal points. It turns out that these points are relatively close to each other for the uniform beam. The basic problem is sketched in the figure below. A uniform beam is supported on two points and the objective is the determine the placement of the supports in the presence of gravity.