In order to understand the dynamic characteristics of an ultrasonic motor stator, we proposed a modified two- dimensional axisymmetric finite element model to analyze the three-dimensional vibrational problem of piezoelectric annular and circular plates. In this work, displacement fields are properly assumed and the electric effect is included. Following the finite element method, analyses of axisymmetric and nonaxisymmetric vibration of circular and annular plate, and also the stator of ultrasonic motors can be conducted in a convenient way. Natural frequency, location of contact point and elliptic locus of the stator are then calculated. Effects by different geometry design and selected circumferential wave number are discussed. Comparisons of some typical examples are made between the present work and those available in the literature.

In the present study, effect of satisfying stress boundary conditions, in addition to displacement boundary conditions, in the axisymmetric vibration analysis of circular and annular plates is investigated. A new axisymmetric finite element, which is based on a combination of the conventional displacement-type variational principle and the Reissner¡¦s principle, is proposed. With this formulation, stresses, like displacements, are primary variables, and both displacement and stress boundary conditions can be easily and exactly imposed. Axisymmetric vibration frequencies of some typical circular and annular plates are then obtained with the present approach and are compared with those by the displacement-type axisymmetric finite element. It is found that the conventional finite element, though not satisfying stress boundary conditions, can still obtain sufficiently accurate vibration frequencies of circular and annular plates.

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In the past 10 years there has been a significant amount of research into two-phase particle transport. The terrorist events of September 11, 2001 sparked a series of studies analyzing particle entrainment and deposition in turbulent airflows. One area of research needing further attention has been the study of particles entrained in axisymmetric air jets. An experimental rig was designed and built to study entrainment properties and deposition of Newtonian particles, after injection into a turbulent axisymmetric free air jet. Newtonian spherical particles, ranging from 1mm to 6mm in diameter, were injected into a turbulent airstream and blown through a nozzle into a large, open space. As the particles fell out of the jet stream, their linear distances, from nozzle to initial-ground-contact, were recorded and analyzed. The experiments conducted indicated particle size and density to be significant factors when considering Newtonian particle entrainment. Additionally, particle deposition distribution revealed a consistent positive skewness, as opposed to an expected Gaussian form. The data presented in this paper provide a starting point for understanding entrainment of Newtonian spherical particles in jets. The simple experimental rig geometry and results also provide an opportunity for computational fluid dynamics models to be validated, answering a call from the 2006 Annual Review of Fluid Mechanics. / Master of Science

Pas de résumé / Axisymmetric turbulence is investigated using direct numerical simulations. A fully spectral method is implemented using Chandrasekhar-Kendall eigenfunctions of the curl-operator. The numerical domain is a periodic cylinder with no-penetration and partial slip conditions at the wall. Numerical simulations are first carried out for freely decaying axisymmetric turbulence, starting from a variety of initial conditions. The simulations indicate that the global angular momentum is the most robust invariant of the system. It is further observed that large-scale coherent structures emerge, as in 2D isotropic turbulence. Energy decays more slowly than helicity, and the toroidal kinetic energy decays faster than its poloidal part. In the case where the toroidal kinetic energy becomes negligible, a quasi-two dimensional turbulence in the poloidal plane is obtained, with a behavior compatible with predictions of statistical mechanics theories. Forced and decaying simulations are then carried out to assess the cascade-behavior of the different invariants. The existence of an inverse cascade is shown to explain the robustness of the angular momentum and the possible ‘spontaneous generation’ of this quantity and of circulation in the flow. In helical flows, the existence of a dual cascade is confirmed, with a scenario compatible with the existence of an inverse energy cascade towards the large scales, and a direct cascade of helicity towards the small scales. The inverse energy cascade seems to be mainly associated with the poloidal velocity field. Using a helical decomposition of the flow, it is shown that the direct cascade of helicity seems to subsist even in the absence of net helicity, when the ‘cascade’ of the helicity contained in oppositely polarized modes is considered individually. The scaling of the energy spectra associated with the energy cascade is compatible with elementary dimensional arguments, whereas the scaling of the inverse (presumably helicity) cascade yields an anomalously steep slope. It is shown that this slope adjusts to the value predicted by dimensional analysis when the spectra are computed from a filtered velocity field in which strong intermittent regions of velocity are not accounted for. Finally, a preliminary (but unfortunately unfruitful) attempt is presented to apply a variational principle to the description of turbulent scalar mixing in three-dimensional turbulence.

Extrusion is a metal forming process used extensively in industry to produce different structural, mechanical, electrical, architectural, automotive and aerospace application parts. Currently after extrusion, the rod is subjected to environmental wear due to long storage time and hence requires an additional cleaning process before further use. This cleaning process can be eliminated by extruding a polymer coated metal rod workpiece such that the polymer coating is sustained on the final product after the extrusion process. In the present research study a new upper bound analytical model is developed to predict the forces required to conduct extrusion of a polymer coated metal rod successfully. The search for the lower upper bound power functional is modeled as a non linear optimization problem. Optimizing the functional also determines the set of constraints defining the shape of rigid plastic deformation boundaries and the final coating thickness. Also an upper bound analytical model was developed to predict forces for failure of the polymer coating during the extrusion. Both the analytical models for successful and failed extrusion are compared to obtain critical die angle which can provide tooling and process design guidelines. Finite element analysis simulations were modeled using commercially available software package, ABAQUS. Predictions of FEA simulations were in good agreement with published results and with the predictions of analytical model developed in this study.

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An experimental scale model study was conducted to investigate the onset of fluidelastic instability in a tube array subjected to axisymmetric jet flow. A tube array was constructed using aluminum tubes with 44.45 mm outer diameter, $D$, which were arranged in a square pattern with 88 mm pitch, $P$. The pitch to diameter ratio, $P/D$, was approximately 2.0. The tubes were flexibly mounted using threaded rod and tuned to a first mode natural frequency, $f_n$, of 9 Hz. Auxiliary damping devices were added to each tube, and tuned, to achieve a damping ratio, $\zeta$, of 1 % of critical. The mass damping parameter, $m(2 \pi \zeta)/(\rho D^{2})$, of the tube array was 27.9. The tube array was tested under uniform flow conditions in McMaster University's 2 ft wind tunnel to establish the critical reduced velocity, $V_{cr}/(f_n D)$, of 30.0 at the onset of fluidelastic instability. The uniform flow test established a basis for comparing the results with the existing literature and evaluating the validity of the proposed partial admission calculation. The tube array was also tested in open air using an axisymmetric jet, with two different physical arrangements, the first with the jet aimed between tubes and perpendicular to the tube spans and the second with the jet aimed at a tube face and perpendicular to the tube spans. In each case the jet flow velocity was incrementally increased to characterize the onset of fluidelastic instability. To characterize the flow dispersion through the tube array a series of velocity profile measurements were also collected. The measured velocity profiles were used to estimate the spanwise function of transverse average gap velocity, $\bar{V}(x)$, which was used to predict the equivalent critical uniform gap flow velocity, $V_{cr}$, using the concept of partial admission. The predicted $V_{cr}$ values showed reasonable agreement with the experimental results. However, the prediction method did indicate instabilities in tube rows where instability was not actually observed. A simplified prediction approach was developed which was based on using a predicted three dimensional velocity profile, $V(x,y)$, at the $z$ location of the first row tube gap, under the assumption of free field conditions, to calculate an estimate of the spanwise function of transverse average gap velocity, $\bar{V}(x)$. Although the predictions of $V_{cr}$ agreed reasonably well with the experimental results, first row instabilities were not observed in any of axisymmetric jet flow experiments. Therefore, this method can be used to estimate the the critical uniform gap velocity, $V_{cr}$, but not the spatial location of the instability. Based on the results of the experiments and calculations, adoption of the modified partial admission formula is recommended and possible avenues for further investigation and verification are suggested. / Thesis / Master of Applied Science (MASc)

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