**1. Introduction**

High-rate dynamics are defined as events having amplitudes greater than 100 *g* over durations less than 100 ms [1]. Some examples of high-rate systems may include civil structures exposed to blast, passenger vehicles experiencing collisions, and aerial or spacecraft vehicles subjected to ballistic impacts. Such systems have the potential to experience rapid changes in mechanical configuration through damage. Economic investments and lives could be saved if fast detection of parameter changes can be accurately quantified [2]. A variable input observer has been studied by the authors as a potential solution to increasing convergence times through richer inputs [3]. However, there is a need to validate high-rate structural health monitoring (SHM) methods [4].

An experimental testbed has been designed and built to test and validate SHM methods systems experiencing high-rate dynamics. The development of an experimental testbed is critical, because the experimentation on real-life high-rate systems would be complex, difficulty to verify, and potentially very costly. This testbed design incorporates a cantilever beam with a roller that restrains the displacement in the vertical direction and is allowed to move freely along the length of the beam. Additionally, the mass at the free end of the beam can be dropped through the de-energizing of the electromagnet that detaches the mass from the beam. The roller is a moving cart that provides a changing boundary condition while the mass drop provides a sudden change in mechanical configuration. This system is easily controllable and repeatable in a laboratory setting.

To develop analytical solutions for this beam structure, the Euler-Bernoulli beam theory is applied. The system is modeled as clamped-pinned-free with a point mass at free end. To the best knowledge of the authors, this configuration has not been previously studied analytically nor experimentally. There is no mention of this beam configuration in the book authored by Blevins [5]. The clamped-pinned-free without mass [6] and the clamped-free with mass at free end without pin [7] have been studied. In more recent years, researchers have investigated the free vibration of multi-span beams with flexible constraints [8], axial vibrations of multi-span beams with concentrated masses [9], multi-span beams with moving masses [10], and multi-span beams carrying spring-mass systems [11,12].

Using beam theory, Section 2 derives the transcendental equation. A generalized form is presented that is applicable to any pin location and any mass value. Derived results are verified through comparison between well known cases in literature. Section 3 calculates the eigenvalues for normalized pinned location for various mass ratios. Section 4 calculates the mode shapes for several different pinned locations, while Section 5 compares the results from the theoretical calculations to experimental data.
