Three models are presented which predict frequencies and mode shapes of transverse vibration for a continuous prismatic Bernoulli-Euler beam on elastic supports, subjected to a compressive axial load. The first model, which approximates support stiffnesses by an equivalent elastic foundation, is found to be inaccurate for wave lengths close to the support spacing. A discrete mass model is formulated which accounts for axial load by stability functions which modify the element stiffness matrices. A continuous model is formulated which yields an exact solution for Bernoulli-Euler beam theory. The frequencies predicted by the discrete mass model and continuous model are in excellent agreement. A method of predicting axial compressive load and support stiffness based on measured frequency and phase data is presented which can be used for either the discrete mass model or the continuous model. A frequency reduction factor is derived which accounts for the effects of shear deformation and rotatory inertia. Tests are performed on an eight span beam with compressive axial load. Test results show that the models accurately predict frequencies and mode shapes of vibration. Results indicate that the method formulated can be used to determine compressive axial load and support stiffness. / Master of Science

<div>This body of work examines analytical and numerical models to simulate the response of structures in shock absorption applications. Specifically, the work examines the prediction of cushion curves of polymer foams, and a topological examination of a $\chi$ shape unit cell found in architected mechanical elastomeric metamaterials. The $\chi$ unit cell exhibits the same effective stress-strain relationship as a closed cell polymer foam. Polymer foams are commonly used in the protective packaging of fragile products. Cushion curves are used within the packaging industry to characterize a foam's impact performance. These curves are two-dimensional representations of the deceleration of an impacting mass versus static stress. The main drawback with cushion curves is that they are currently generated from an exhaustive set of experimental test data. This work examines modeling the shock response using a continuous rod approximation with a given impact velocity in order to generate cushion curves without the need of extensive testing. In examining the $\chi$ unit cell, this work focuses on the effects of topological changes on constitutive behavior and shock absorbing performance. Particular emphasis is placed on developing models to predict the onset of regions of quasi-zero-modulus (QZM), the length of the QZM region and the cushion curve produced by impacting the unit cell. The unit cell's topology is reduced to examining a characteristic angle, defining the internal geometry with the cell, and examining the effects of changing this angle.</div><div>However, the characteristic angle cannot be increased without tradeoffs; the cell's effective constitutive behavior evolves from long regions to shortened regions of quasi-zero modulus. Finally, this work shows that the basic $\chi$ unit cell can be tessellated to produce a nearly equivalent force deflection relationship in two directions. The analysis and results in this work can be viewed as new framework in analyzing programmable elastomeric metamaterials that exhibit this type of nonlinear behavior for shock absorption.</div>