Browsing by Author "Vogl, Gregory William"

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    Comprehensive Theory of Heat Transfer in Heterogeneous Materials
    Vogl, Gregory William (Virginia Tech, 2003-01-06)
    For over forty years, researchers have attempted to refine the Fourier heat equation to model heat transfer in engineering materials. The equation cannot accurately predict temperatures in some applications, such as during transients in microscale (< 10^-12 s) situations. However, even in situations where the time duration is relatively large, the Fourier heat equation might fail to predict observed non-Fourier behavior. Therefore, non-Fourier models must be created for certain engineering applications, in which accurate temperature modeling is necessary for design purposes. In this thesis, we use the Fourier heat equation to create a general non-Fourier, but diffusive, equation that governs the matrix temperature in a composite material. The composite is composed of a matrix with embedded particles. We let the composite materials be governed by Fourier's law and let the heat transfer between the matrix and particles be governed by contact conductance. After we make certain assumptions, we derive a general integro-differential equation governing the matrix temperature. We then non-dimensionalize the general equation and show that our model reduces to that used by other researchers under a special limit of a non-dimensional parameter. We formulate an initial-boundary-value problem in order to study the behavior of the general matrix temperature equation. We show that the thermalization time governs the transition of the general equation from its small-time limit to its large-time limit, which are both Fourier heat equations. We also conclude that our general model cannot accurately describe temperature changes in an experimental sand composite.
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    Nonlinear Dynamics of Circular Plates under Electrical Loadings for Capacitive Micromachined Ultrasonic Transducers (CMUTs)
    Vogl, Gregory William (Virginia Tech, 2006-12-05)
    We created an analytical reduced-order model (macromodel) for an electrically actuated circular plate with an in-plane residual stress for applications in capacitive micromachined ultrasonic transducers (CMUTs). After establishing the equations governing the plate, we discretized the system by using a Galerkin approach. The distributed-parameter equations were then reduced to a finite system of ordinary-differential equations in time. We solved these equations for the equilibrium states due to a general electric potential and determined the natural frequencies of the axisymmetric modes for the stable deflected position. As expected, the fundamental natural frequency generally decreases as the electric forcing increases, reaching a value of zero at pull-in. However, strain-hardening effects can cause the frequencies to increase with voltage. The macromodel was validated by using data from experiments and simulations performed on silicon-based microelectromechanical systems (MEMS). For example, the pull-in voltages differed by about 1% from values produced by full 3-D MEMS simulations. The macromodel was then used to investigate the response of an electrostatically actuated clamped circular plate to a primary resonance excitation of its first axisymmetric mode. The method of multiple scales was used to derive a semi-analytical expression for the equilibrium amplitude of vibration. The plate was found to always transition from a hardening-type to a softening-type behavior as the DC voltage increases towards pull-in. Because the response of CMUTs is highly influenced by the boundary conditions, an updated reduced-order model was created to account for more realistic boundary conditions. The electrode was still considered to be infinitesimally thin, but the electrode was allowed to have general inner and outer radii. The updated reduced-order model was used to show how sensitive the pull-in voltage is with respect to the boundary conditions. The boundary parameters were extracted by matching the pull-in voltages from the macromodel to those from finite element method (FEM) simulations for CMUTs with varying outer and inner radii. The static behavior of the updated macromodel was validated because the pull-in voltages for the macromodel and FEM simulations were very close to each other and the extracted boundary parameters were physically realistic. A macromodel for CMUTs was then created that includes both the boundary effects and an electrode of finite thickness. Matching conditions ensured the continuity of displacements, slopes, forces, and moments from the composite to the non-composite regime of the CMUT. We attempted to validate this model with results from FEM simulations. In general, the center deflections from the macromodel fell below those from the FEM simulation, especially for relatively high residual stresses, but the first natural frequencies that accompany the deflections were very close to those from the FEM simulations. Furthermore, the forced vibration characteristics also compared well with the macromodel predictions for an experimental case in which the primary resonance curve bends to the right because the CMUT is a hardening-type system. The reduced-order model accounts for geometric nonlinear hardening, residual stresses, and boundary conditions related to the CMUT post, allows for general design variables, and is robust up to the pull-in instability. However, even more general boundary conditions need to be incorporated into the model for it to be a more effective design tool for capacitive micromachined ultrasonic transducers.