This dissertation introduces a novel three-dimensional pseudo-rigid-body model (3-D PRBM) for straight cantilever beams with rectangular cross sections. The model is capable of capturing the behavior of the neutral axis of a beam loaded with an arbitrary force end-load. Numerical integration of a system of differential equations yields approximate displacement and orientation of the beam's neutral axis at the free end, and curvatures of the neutral axis at the fixed end. This data was used to develop the 3-D PRBM which consists of two torsional springs connecting two rigid links for a total of 2 degrees of freedom (DOF). The 3-D PRBM parameters that are comparable with existing 2-D model parameters are characteristic radius factor (mean: γ = 0.8322), bending stiffness coefficient (mean: KΘ = 2.5167) and parametric angle coefficient (mean: cΘ = 1.2501). New parameters are introduced in the model in order to capture the spatial behavior of the deflected beam, including two parametric angle coefficients (means: cΨ = 1.0714; cΦ = 1.0087). The model is verified in a few locations using ANSYSTM and its use in the design of compliant mechanisms is illustrated through spatial compliant versions of crank slider and double slider mechanisms.

The objective of the dissertation is to develop and describe kinematic models (Pseudo-Rigid-Body Models) for approximating large-deflection of spatial (3D) cantilever beams that undergo multiple bending motions thru end-moment loading. Those models enable efficient design of compliant mechanisms, because they simply and accurately represent the bending and stiffness of compliant beams. To accomplish this goal, the approach can be divided into three stages: development of the governing equations of a flexible cantilever beam, development of a PRBM for axisymmetric cantilever beams and the development of spatial PRBMs for rectangular cross-section beam with multiple end moments. The governing equations of a cantilever beam that undergoes large deflection due to force and moment loading, contains the curvature, location and rotation of the beam. The results where validated with Ansys, which showed to have a Pearson's correlation factor higher than 0.91. The resulting deflections, curvatures and angles were used to develop a spatial pseudo-rigid-body model for the cantilever beam. The spatial pseudo-rigid-body model consists of two links connected thru a spherical joint. For an axisymmetric beam, the PRB parameters are comparable with existing planar PRBMs. For the rectangular PRBM, the parameters depend on the aspect ratio of the beam (the ratio of the beam width over the height of the cross-section). Tables with the parameters as a function of the aspect ratio are included in this work.

This dissertation describes the development and effectiveness of a mathematical model used to predict the behavior of cantilever beams whose loading conditions include parallel combinations of evenly distributed loads and endpoint forces. The large deflection of cantilever beams has been widely studied. A number of models and mathematical techniques have been utilized in predicting the endpoint path coordinates and load-deflection relationships of such beams. The Pseudo-Rigid-Body Model (PRBM) is one such method which replaces the elastic beam with rigid links of a parameterized pivot location and torsional spring stiffness. In this paper, the PRBM method is extended to include cases of a constant distributed load combined with a parallel endpoint force. The phase space of the governing differential equations is used to store information relevant to the characterization of the PRBM parameters. Correction factors are also given to decrease the error in the load-deflection relationship and extend the angular range of the model, thereby further aiding compliant mechanism design. The calculations suggest a simple way of representing the effective torque caused by a distributed load in a PRBM as a function of easily calculated model parameters.

In this thesis the Ramberg-Osgood nonlinear model for describing the behavior of many different materials is investigated. A brief overview of the model as it is currently used in the literature is undertaken and several misunderstandings and possible pitfalls in its application is pointed out, especially as it pertains to more recent approaches to finding solutions involving the model. There is an investigation of the displacement of a cantilever beam under a combined loading consisting of a distributed load across the entire length of the beam and a point load at its end and new solutions to this problem are provided with a mixture of numerical techniques, which suggest strong mathematical consistency within the model for all theoretical assumptions made. A physical experiment was undertaken and the results prove to be inaccurate when using parameters derived from tensile tests, but when back calculating parameters from the beam test the model has a 14.40% error at its extreme against the experimental data suggesting the necessity for further testing.

The aim of the presented thesis is to create a compact publication which deals with properties, solution and examination of behavior of dynamical systems as models of mechanical structures. The opening portion of the theoretical part leads the reader through the subject of description of dynamical systems, offers solution methods and investigates solution stability. As the introduction proceeds, possible forms of structure loading, damping and response are presented. Following chapters discuss extensively the possible approaches to system behavior observation and identification of nonlinear and chaotic phenomena. The attention is also paid to displaying methods and color spaces as these are essential for the examination of complex and sensitive systems. The theoretical part of the thesis ends with an introduction to fractal geometry. As the theoretical background is laid down, the thesis proceeds with an application of the knowledge and shows the approach to numerical simulation and study of models of real structures. First, the reader is introduced to the single pendulum model, as the simplest model to exhibit chaotic behavior. The following double pendulum model shows the obstacles of observing systems with more state variables. The models of free rod and cantilever serve as examples of real structure models with many degrees of freedom. These models show even more that a definite or at least sufficiently relevant monitoring of behavior of such deterministic systems is a challenging task which requires sophisticated approach.