Methods featured are primarily conformal symplectic exponential time differencing methods, with a focus on families of methods, the construction of methods, and the features and advantages of methods, such as order, stability, and symmetry. Methods are applied to the problem of the damped harmonic oscillator. Construction of both exponential time differencing and integrating factor methods are discussed and contrasted. It is shown how to determine if a system of equations or a method is conformal symplectic with flow maps, how to determine if a method is symmetric by taking adjoints, and how to find the stability region of a method. Exponential time differencing Stormer-Verlet is derived and is shown as the example for how to find the order of a method using Taylor series. Runge-Kutta methods, partitioned exponential Runge-Kutta methods, and their associated tables are introduced, with versions of Euler's method serving as examples. Lobatto IIIA and IIIB methods also play a key role, as a new exponential trapezoid rule is derived. A new fourth order exponential time differencing method is derived using composition techniques. It is shown how to implement this method numerically, and thus it is analyzed for properties such as error, order of accuracy, and structure preservation.

The dynamic properties of the elastic coupling are not readily known; therefore testing has to be performed in order to determine these properties. This is the primary objective for this thesis. The dynamic properties in question are the stiffness and damping. An attempt to determine the dynamic properties was also be carried out through the use of finite element analysis. There are two different configurations of couplings. One configuration forms the coupling from several elastic elements, referred to as HRC elements, which are manufactured in three sizes: A, B, and C. The second configuration, referred to as the HEMD coupling, has a single elastic member in the form of a hollow rubber/fabric ring connecting the input to the output. The couplings have cords made of either polyester or nylon. These cords will affect the dynamic properties of the coupling.

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Ferroelectric and piezoelectric ceramic reinforced metal matrix composites are new materials being explored for vibration damping purposes. The high damping ability of ferroelectric and piezoelectric ceramics such as barium titanate (BaTiO3) and lead zirconate titanate (PZT) is due to the anelastic response of ferroelastic domain walls to applied external stress. In piezoelectric ceramics, vibration energy can also be dissipated through the direct piezoelectric effect if the appropriate electric circuit is connected across the ceramic. In this work we have examined the vibration damping behavior of BaTiO3, nickel-barium titanate (Ni-BaTiO3) composites and nickel-lead zirconate titanate (Ni-PZT) composites. BaTiO3 ceramics were fabricated by a combination of uniaxial pressing and cold isostatic pressing followed by sintering in air. Low frequency (0.1Hz-10Hz) damping capacity of BaTiO3, tanδ has been measured in three-point bend configuration on a dynamic mechanical analyzer. Tanδ has been found to increase with temperature up to the Curie temperature (Tc) of BaTiO3, after which there was a drop in damping capacity values due to the disappearance of ferroelectric domains above Tc. Furthermore within the frequency range tested, tanδ has been found to decrease with increasing vibration frequency. We also observed that tanδ decays with the number of vibration cycles (N). The decrease in tanδ with N, however, is fully recovered if BaTiO3 is heated above the Tc. Ni-BaTiO3 composite composed of a layer of BaTiO3 ceramic sandwiched between two layers of Ni were fabricated using a combination of electroless plating and electroforming. The damping behavior of the composite was analyzed in terms of the damping mechanisms below Tc and the damping mechanisms above Tc of BaTiO3. Below Tc, vibration damping ability of the composite was highly influenced by ferroelastic damping in the BaTiO3 component. Above the Curie temperature, the damping capacity was influence more by the inherent damping mechanisms in the nickel matrix. The damping mechanisms in Ni-PZT composites were evaluated at a low vibration frequency of 1Hz. In these composites we identified ferroelastic domain wall motion as the main damping mechanism active below the Tc of PZT. Using a poled PZT ceramic enhanced the damping capacity of the composite because of favorable ferroelastic domain orientation in the direction of applied stress. Based on our experimental results, we found no evidence of a direct piezoelectric damping mechanism in the Ni-PZT composites. / Ph. D.

This dissertation concentrates on the three principal problems facing experimentalists during their attempts to identify and control lightly damped, large space structures (LSS). The problems are low damping, high modal density, and low natural frequencies of oscillation. They present a blend of difficulties which lead the experimentalist to turn to multiple-input multiple-output (MIMO) identification techniques and high performance compensators. Presented here are two MIMO modal identification techniques: Polyreference, and the Eigensystem Realization Algorithm, as well as two types of compensator-based controllers: Linear Quadratic Gaussian, and Independent Modal Space Control. The various techniques are described in the context of controlling lightly damped LSS. Because the research in this dissertation is primarily applications oriented, problems which experimentalists encounter in the laboratory are addressed as well as the performance of the different identification and control techniques on the test articles. Polyreference and ERA are both shown to perform very well in identifying modal frequencies while overestimating model damping ratios. Simulations show that high modal density combined with noisy data results in standard deviations that increase linearly with respect to mode separation. Pseudo- Inverse IMSC is shown to be robust with respect to system uncertainties. Block Independent Control is shown to possess minor coupling between blocks and provides a powerful control approach to overcome actuator bandwidth limitations. / Ph. D.

The performance of fiber-reinforced composites is often controlled by the properties of the fiber-matrix interface. Good interfacial bonding (or adhesion), to ensure load transfer from matrix to reinforcement, is a primary requirement for effective use of reinforcement properties. Thus, a fundamental understanding of interfacial properties and a quantitative characterization of interfacial adhesion strength can help in evaluating the mechanical behavior and capabilities of composite materials. A large number of analytical techniques have been developed for understanding interfacial adhesion of glass fiber reinforced polymers. Common adhesion tests include contact angle measurements, tension or compression of specially shaped blocks of polymer containing a single fiber, the single fiber pull-out test, single-fiber fragmentation test, short beam shear and transverse tensile tests, and the vibration damping test. Among these techniques, the vibration damping technique has the advantage of being nondestructive as well as highly sensitive for evaluating the interfacial region, and it can allow the materials industry to rapidly determine the mechanical properties of composites. In this work, we contributed a simple optical system for measuring the damping factor of uniaxial fiber-reinforced-polymer composites in the shape of cantilever beams. A single glass fiber- and three single metallic wire-reinforced epoxy resin composites were tested with the optical system. The fiber- (wire-) matrix interfacial adhesion strength measurements were made by microbond test. A reasonable agreement was found between the measured interfacial adhesion strength and micromechanics calculations using results from vibration damping experiments. The study was also extended to multi-fiber composites. The interfacial damping factors in glass-fiber reinforced epoxy-resin composites were correlated with transverse tensile strength, which is a qualitative measurement of adhesion at the fiber-matrix interface. Four different composite systems were tested. For each system, glass fibers with three different surface treatments were used at three different volume fractions. The experimental results also showed an inverse relationship between damping contributed by the interface and composite transverse tensile strength for all of the multi-fiber composites. / Ph. D.