Distributed parameter systems (DPS) are systems defined on infinite dimensional spaces. This includes problems governed by partial differential equations (PDEs) and delay differential equations. In order to numerically implement a controller for a physical system we often first approximate the PDE and the PDE controller using some finite dimensional scheme. However, control design at this level will typically give rise to controllers that are inherently large-scale. This presents a challenge since we are interested in the design of robust, real-time controllers for physical systems. Therefore, a reduction in the size of the model and/or controller must take place at some point. Traditional methods to obtain lower order controllers involve reducing the model from that for the PDE, and then applying a standard control design technique. One such model reduction technique is balanced truncation. However, it has been argued that this type of method may have an inherent weakness since there is a loss of physical information from the high order, PDE approximating model prior to control design. In an attempt to capture characteristics of the PDE controller before the reduction step, alternative techniques have been introduced that can be thought of as controller reduction methods as opposed to model reduction methods. One such technique is LQG balanced truncation. Only recently has theory for LQG balanced truncation been developed in the infinite dimensional setting. In this work, we numerically investigate the viability of LQG balanced truncation as a suitable means for designing low order, robust controllers for distributed parameter systems. We accomplish this by applying both balanced reduction techniques, coupled with LQG, MinMax and central control designs for the low order controllers, to the cable mass, Klein-Gordon, and Euler-Bernoulli beam PDE systems. All numerical results include a comparison of controller performance and robustness properties of the closed loop systems. / Ph. D.

This thesis develops an immersed finite element (IFE) space for numerical simulations arising from beam design with multiple materials. This IFE space is based upon meshes that can be independent of interface of the materials used to form a beam. Both the forward and inverse problems associated with the beam equation are considered. The order of accuracy of this IFE space is numerically investigated from the point of view of both the interpolation and finite element solution of the interface boundary value problems. Both single and multiple interfaces are considered in our numerical simulation. The results demonstrate that this IFE space has the optimal order of approximation capability. / Master of Science

In this thesis, transverse vibration of a cracked beam on an elastic foundation and the effect of crack and foundation parameters on transverse vibration natural frequencies are studied. Analytical formulations are derived for a beam with rectangular cross section. The crack is an open type edge crack placed in the medium of the beam and it is uniform along the width of the beam. The cracked beam rests on an elastic foundation. The beam is modeled by two different beam theories, which are Euler-Bernoulli beam theory and Timoshenko beam theory. The effect of the crack is considered by representing the crack by rotational springs. The compliance of the spring that represents the crack is obtained by using fracture mechanics theories. Different foundation models are discussed / these models are Winkler Foundation, Pasternak Foundation, and generalized foundation. The equations of motion are derived by applying Newton&#039 / s 2nd law on an infinitesimal beam element. Non-dimensional parameters are introduced into equations of motion. The beam is separated into pieces at the crack location. By applying the compatibility conditions at the crack location and boundary conditions, characteristic equation whose roots give the non-dimensional natural frequencies is obtained. Numerical solutions are done for a beam with square cross sectional area. The effects of crack ratio, crack location and foundation parameters on transverse vibration natural frequencies are presented. It is observed that existence of crack reduces the natural frequencies. Also the elastic foundation increases the stiffness of the system thus the natural frequencies. The natural frequencies are also affected by the location of the crack.

abstract: In order to verify the dispersive nature of transverse displacement in a beam, a deep understanding of the governing partial differential equation is developed. Using the finite element method and Newmark’s method, along with Fourier transforms and other methods, the aim is to obtain consistent results across each numerical technique. An analytical solution is also analyzed for the Euler-Bernoulli beam in order to gain confidence in the numerical techniques when used for more advance beam theories that do not have a known analytical solution. Three different beam theories are analyzed in this report: The Euler-Bernoulli beam theory, Rayleigh beam theory and Timoshenko beam theory. A comparison of the results show the difference between each theory and the advantages of using a more advanced beam theory for higher frequency vibrations. / Dissertation/Thesis / Masters Thesis Civil Engineering 2016

Vibration is an inherent phenomenon in dynamic mechanical systems. The work undertaken in this thesis is to identify natural frequencies of a variable length blade. Therefore designers can ensure that natural frequencies will not be close to the frequency (or integer multiples) of the main excitation forces in order to avoid resonance. For a wind turbine blade, the frequency range between 0.5 Hz and 30 Hz is relevant. The turbine blade is approximated by a cantilever, therefore, it is fully constrained where attached to a turbine shaft/hub. Flap-wise, edge-wise and torsional natural frequencies are calculated. The MATLAB program “BEAMANALYSIS.m” has been developed for the finite element analysis of a one dimensional model of the beam. Similarly, a three dimensional model of the beam has been developed in a finite element program Unigraphics NX5. The results found using the MATLAB program are compared with those found with NX5. Satisfactory agreement between the results is found for frequencies up to almost 500 Hz. Additionally, the frequencies one might expect in an experiment are identified. Experimental modal analysis has been performed on a uniform and stepped beam made of mild steel to extract the first five flap-wise natural frequencies. The results found have been compared to numerical results and the exact solution of an Euler-Bernoulli beam. Concurrence is found for the frequency range of interest. Although, some discrepancies exist at higher frequencies (above 500 Hz), finite element analysis proves to be reliable for calculating natural frequencies. Finally, the fixed portion and moveable portion of the variable length blade are approximated respectively by a hollow and a solid beam which can be slid in and out. Ten different configurations of the variable length blade, representing ten different positions of the moveable portion are investigated. A MATLAB program named VARIBLADEANALYSIS.m was developed to predict natural frequencies. Similarly three dimensional models of the variable length blade have been developed in the finite element program Unigraphics NX5. / This work was supported by the Research office of CPUT.

The highly optimized performance of nature’s creations and biological assemblies has inspired the development of their engineered counter parts that can potentially outperform conventional systems. In particular, bat wings are populated with air flow hair receptors which feedback the information about airflow over their surfaces for enhanced stability and maneuverability during their flight. The hairs in the bat wing membrane play a role in the maneuverability tasks, especially during low-speed flight. The developments of artificial hair sensors (AHS) are inspired by biological hair cells in aerodynamic feedback control designs. Current mathematical models for hair receptors are limited by strict simplifying assumptions of creeping flow hair Reynolds number on AHS fluid-structure interaction (FSI), which may be violated for hair structures integrated on small-scaled Unmanned Aerial Vehicles (UAVs). This study motivates by an outstanding need to understand the dynamic response of hair receptors in flow regimes relevant to bat-scaled UAVs. The dynamic response of the hair receptor within the creeping flow environment is investigated at distinct freestream velocities to extend the applicability of AHS to a wider range of low Reynolds number platforms. Therefore, a threedimensional FSI model coupled with a finite element model using the computational fluid dynamics (CFD) is developed for a hair-structure and multiple hair-structures in the airflow. The Navier-Stokes equations including continuity equation are solved numerically for the CFD model. The grid independence of the FSI solution is studied from the simulations of the hairstructure mesh and flow mesh around the hair sensor. To describe the dynamic response of the hair receptors, the natural frequencies and mode shapes of the hair receptors, computed from the finite element model, are compared with the excitation frequencies in vacuum. This model is described with both the boundary layer effects and effects of inertial forces due to fluid-structure xiv interaction of the hair receptors. For supporting the FSI model, the dynamic response of the hair receptor is also validated considering the Euler-Bernoulli beam theory including the steady and unsteady airflow.

This study focuses on assessing the accuracy of the Euler-Bernoulli beam theory as computational bases to calculate strain and deflection of composite sandwich beam subjected to three-point and four-point bending. Two groups of composite sandwich beams tests results will be used for comparison purposes. Mechanical properties for the laminated skin are provided by researchers from University of Mississippi (Ellen Lackey et al., 2000). Mechanical properties for the balsa wood core are provided by Alcan Baltek Corporation. Appropriate material properties and test geometries are then used in the Euler-Bernoulli-based algorithm in order to generate analytical data for comparison to experimental data provided by researchers from University of New Orleans (UNO, 2005). The resulting single material cross section is then analyzed in the traditional manner using the Euler-Bernoulli beam theory. In general, the Euler-Bernoulli beam theory provides an appropriate analytical approach in predicting flexural behavior of composite sandwich beams.

In this work, several beam finite element formulations are proposed for failure analysis of planar reinforced concrete beams and frames under monotonic static loading. The localized failure of material is modeled by the embedded strong discontinuity concept, which enhances standard interpolation of displacement (or rotation) with a discontinuous function, associated with an additional kinematic parameter representing jump in displacement (or rotation). The new parameters are local and are condensed on the element level. One stress resultant and two multi-layer beam finite elements are derived. The stress resultant Euler-Bernoulli beam element has embedded discontinuity in rotation. Bending response of the bulk of the element is described by elasto-plastic stress resultant material model. The cohesive relation between the moment and the rotational jump at the softening hinge is described by rigid-plastic model. Axial response is elastic. In the multi-layer beam finite elements, each layer is treated as a bar, made of either concrete or steel. Regular axial strain in a layer is computed according to Euler-Bernoulli or Timoshenko beam theory. Additional axial strain is produced by embedded discontinuity in axial displacement, introduced individually in each layer. Behavior of concrete bars is described by elastodamage model, while elasto-plasticity model is used for steel bars. The cohesive relation between the stress at the discontinuity and the axial displacement jump is described by rigid-damage softening model in concrete bars and by rigid-plastic softening model in steel bars. Shear response in the Timoshenko element is elastic. Finally, the multi-layer Timoshenko beam finite element is upgraded by including viscosity in the softening model. Computer code implementation is presented in detail for the derived elements. An operator split computational procedure is presented for each formulation. The expressions, required for the local computation of inelastic internal variables and for the global computation of the degrees of freedom, are provided. Performance of the derived elements is illustrated on a set of numerical examples, which show that the multi-layer Euler-Bernoulli beam finite element is not reliable, while the stress-resultant Euler-Bernoulli beam and the multi-layer Timoshenko beam finite elements deliver satisfying results.

This thesis work provides an insight into the design of Force-amplifying Compliant Mechanisms (FaCMs) that are integrated with micromachined resonant accelerometers to increase their sensitivity. An FaCM, by mechanically amplifying the inertial force, enhances the shift in the resonance frequency of the beams used for sensing the acceleration whose effect causes an axial force on the beams. An extensive study on different configurations of resonators namely, single beam resonator, single-ended tuning fork (SETF), and double-ended tuning fork (DETF), is carried out to gain insights about their resonant behavior. The influence of the boundary conditions on the sensor’s sensitivity emerged from the study. We found that not only the force-amplification factor but also the multi-axial stiffness of the FaCM and proof-mass influence the resonance frequency of the resonator as well as the bandwidth of the modified sensor for certain configurations but not all. Thus, four lumped parameters were identified to quantify the effectiveness of an FaCM. These parameters determine the boundary condition of the sensing beams and also the forces and the moment transmitted to them. Also presented in this work is a computationally efficient model, called the Lumped Parameter Model (LPM) for evaluation of the sensitivity. An analytical expression for the frequency-shift of the sensing resonator beams is obtained by considering the FaCM stiffness parameters as well as the lumped stiffness of the suspension of the inertial mass. Various FaCMs are evaluated and compared to understand how the four lumped parameters influence the sensor’s sensitivity. The FaCMs are synthesized using topology optimization to maximize the net amplification factor with the volume constraint. One of the FaCMs outperforms the lever by a factor of six. Microfabrication of resonant accelerometer coupled with FaCM and comb-drive actuator is carried out using a silicon-on-insulator process. Finally, the selection map technique, a compliant mechanism redesign methodology is used for enhancing the amplification of FaCMs. This technique provides scope for further design improvement in FaCMs for given sensor specifications.

A computational framework for linear static aeroelastic analyses is presented. The overall aeroelasticity model is applicable to conceptual aircraft design studies and other low-fidelity aero-structural analyses. A partitioned approach is used, i. e. separate solvers for aerodynamics and structure analyses are coupled in a suitable way, together forming a model for aeroelastic simulations. Aerodynamics are modelled using the vortexlattice method (VLM), a simple computational fluid dynamics (CFD) model based on potential flow. The structure is represented by a three-dimensional (3D) Euler-Bernoulli beam model in a finite element method (FEM) formulation. A particular focus was put on the modularity and loose coupling of aforementioned models. The core of the aeroelastic framework was abstracted, such that it does not depend on any specific details of the underlying aerodynamics and structure modules. The final aeroelasticity model constitutes independent software tools for the VLM and the beam FEM, as well as a framework enabling the aeroelastic coupling. These different tools have been developed as part of this thesis work. A wind tunnel experiment with a simple wing model is presented as a validation test case. An aero-structural analysis of a fully elastic unmanned aerial vehicle (UAV) (OptiMale) is described and results are compared with an existing higherfidelity study. / Rapporten beskriver en beräkningsmodell för linjära, statisk aeroelastiska analyser. Modellen kan användas för konceptuella designstudier av flygplan. En partitionerad metod används, d v s separata lösare för aerodynamik- och strukturanalyser kopplas på ett lämpligt sätt, och bildar tillsammans en modell för aeroelastiska simulationer. Aerodynamik modelleras med hjälp av en så kallad vortex-lattice method (VLM), en enkel modell för beräkningsströmningsdynamik (CFD) som är baserad på friktionsfri strömning. Strukturen representeras av en tredimensionell (3D) Euler-Bernoulli-balkmodell implementerad med hjälp av en finita elementmetod (FEM). Ovannämnda modeller har utvecklats med fokus på modularitet och lös koppling. Kärnan i den aeroelastiska modellen har abstraherats så att den inte beror på specifika detaljer i de underliggande aerodynamik- och strukturmodulerna. Aeroelasticitetsmodellen i sin helhet består av separata mjukvaruprogram för VLM och balk-FEM, såväl som ett ramverk som möjliggör den aeroelastiska kopplingen. Dessa olika program har utvecklats som en del av examensarbetet. Ett vindtunnelförsök med en enkel vingmodell presenteras som ett valideringstest. Dessutom beskrivs en analys av ett elastiskt obemannad flygplan (OptiMale) och resultaten jämförs med en befintlig studie som har genomförts med modeller av högre trovärdighet.