# Browsing by Author "Virgin, Lawrence N"

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Item Open Access A Study of Non-Smooth Impacting Behaviors(2015) George, Christopher MichaelThe dynamics of impacting components is of particular interest to engineers due to concerns about noise and wear, but is particularly difficult to study due to impact's non-linear nature. To begin transferring concepts studied purely analytically to the world of physical mechanisms, four experiments are outlined, and important non-linear concepts highlighted with these systems. A linear oscillator with a kicked impact, an impacting forced pendulum, two impacting forced pendulums, and a cam follower pair are studied experimentally, with complementary numerical results.

Some important ideas highlighted are limit cycles, basins of attraction with many wells, grazing, various forms of coexistence, super-persistent chaotic transients, and liftoff. These concepts are explored using a variety of non-linear tools such as time lag embedding and stochastic interrogation, and discussions of their intricacies when used in non-smooth systems yield important observations for the experimentalist studying impacting systems.

The focus is on experimental results with numerical validation, and spends much time discussing identification of these concepts from an experiment-first mindset, rather than the more traditional analytical-first approach. As such a large volume of experimentally important information on topics such as transducers and forcing mechanism construction are included in the appendices.

Item Open Access An Investigation of Sensitivity to Initial Conditions in an Experimental Structural System(2013) Waite, Joshua JosephThis thesis characterizes the nonlinear behavior of an experimental system that exhibits snap-through buckling behavior. A single-degree-of-freedom snap-through link model is harmonically forced using a Scotch yoke mechanism. In order to establish the sensitivity to initial conditions, experimental basins of attraction are constructed using the stochastic interrogation method. After, frequency sweeps are performed on the system to identify regions of interesting behavior. Then, time series data is collected at specific frequencies of interest to highlight the broad phenomenological behavior of the structural system.

A useful tool when modeling structural systems is numerical analysis. An equation of motion is developed to numerically simulate all experimentally observed results. The numerical results include snap-through boundaries, bifurcation diagrams, full initial condition grid basins of attraction, time-lag embedded basins of attraction, frequency sweeps, and time series of regions of pathological behavior.

Item Open Access Analysis of the Elastica with Applications to Vibration Isolation(2007-05-02T17:38:28Z) Santillan, Sophia TeresaLinear theory is useful in determining small static and dynamic deflections. However, to characterize large static and dynamic deflections, it is no longer useful or accurate, and more sophisticated analysis methods are necessary. In the case of beam deflections, linear beam theory makes use of an approximate curvature expression. Here, the exact curvature expression is used to derive the governing partial differential equations that describe the in-plane equilibrium and dynamics of a long, thin, inextensible beam, where the self-weight of the beam is included in the analysis. These beam equations are expressed in terms of arclength, and the resulting equilibrium shape is called the elastica. The analysis gives solutions that are accurate for any deflection size, and the method can be used to characterize the behavior of many structural systems. Numerical and analytical methods are used to solve or to approximate solutions to the governing equations. Both a shooting method and a finite difference, time-stepping algorithm are developed and implemented to find numerical solutions and these solutions are compared with some analytical approximation method results. The elastica equations are first used to determine both linear and nonlinear equilibrium configurations for a number of boundary conditions and loading types. In the case of a beam with a significant self-weight, the system can exhibit nonlinear static behavior even in the absence of external loading, and the elastica equations are used to determine the weight corresponding to the onset of instability (or self-weight buckling). The equations are also used to characterize linear and nonlinear vibrations of some structural systems, and experimental tests are conducted to verify the numerical results. The linear vibration analysis is applied to a vibration isolator system, where a postbuckled clamped-clamped beam or otherwise highly-deformed structure is used (in place of a conventional spring) to reduce system motion. The method is also used to characterize nonlinear dynamic behavior, and the resulting frequency-response curves are compared with those in the literature. Finally, the method is used to investigate the dynamics of subsea risers, where the effects of gravity, buoyancy, and the current velocity are considered.Item Open Access Assessing the Role of Initial Imperfections in Cylinder Buckling Using 3D Printing(2019) Yang, HaochengThis thesis assesses the role of several types of designed initial imperfection on the buckling behavior of 3D printed cylindrical shells through experimental method. The buckling loads and paths are given by the axial compression tests while the lateral force-displacement relationships are given by the lateral poking tests at first. Then the influences of designed imperfections on them are discussed. The test results show that the effect of designed initial imperfections in this thesis might be overridden by other unpredictable imperfections. Poking tests can provide some information on the buckling loads and paths that is difficult to obtain through compression tests. The designed buckling mode shapes imperfections may generate bigger influences than the designed imperfections as shown in this thesis.

Item Open Access Complex Behavior in the Dynamic Response of a Non-smoothly Forced Mechanical System: Numerical and Experimental Investigations(2013) Kini, Ashwath ThukaramThis thesis describes the theoretical and experimental investigations on a non-smooth impacting system consisting of a forced oscillating mass with an impacting barrier, which has the ability to impart energy into the vibrating system. The system is also forced by the means of a sinusoidal excitation using a scotch yoke mechanism. Experiments are conducted to obtain the time delay state-space, time series and Poincare sections. Bifurcation diagrams are obtained by conducting forward and reverse frequency sweep. The obtained results are compared with expected linear non-impacting behavior and interesting phenomena including hysteresis, multiple-period orbits, transient and sustained chaos are observed. Numerical simulations were conducted and correlations were obtained between the theoretical and experimental results.

Item Open Access Nonlinear Behavior of Systems with Multiple Equilibria(2019) Guan, YueThis study describes the nonlinear behavior of a number of various systems with multiple equilibria, from discrete mechanical systems to high-dimensional continuous structures. All these systems are capable of exhibiting sophisticated potential landscapes, including multiple equilibria with different stability properties, whereas their nonlinearities are somewhat sensitive to the geometric conditions. Similar behavior and equivalent relationships are developed for various systems.

First, numerical and experimental investigations are presented for systems with three mechanical/structural degrees-of-freedom (DOF). Considering moderate complexity between low-order and relatively high-order systems, the three-DOF system is able to exhibit a visible configuration space. Useful insights are provided by observations of the iso-potentials and experimental transient trajectories meandering within them. Hyperboloidal passable tubes are found around index-1 saddles on the iso-potential shapes, enabling possible transitions between stable equilibria. As a result, transient trajectories have a tendency to slow down, temporarily oscillate, and separate from each other in the vicinity of these saddles. These phenomena have been verified by experiments, which imply possible existence of an unstable equilibrium in dynamics. Bifurcation structures and morphing potential landscapes are revealed by varying key geometric parameters of the system. Parametric excitation shows a possibility to stabilize unstable equilibria in dynamics under the right amplitudes and frequencies - this is exposed both theoretically and experimentally. A practical four-member pyramidal lattice frame is used as an example of a complex system, adequately modeled by three-DOF. Though significantly different from the mass-spring system, the pyramidal frame presents similar universal features and behavior as the discrete system, with its three dominant modes.

Static nonlinear behavior of higher-order structures is then investigated. For geodesic lattice domes with rigid joints, the complete load-displacement relationship and multiple equilibrium configurations are exhibited both in experiment and in simulation. Multiple snaps are observed, when the system discontinuously pops from one stable equilibrium configuration to another. Symmetry breaks as a result of the equilibrium path bifurcation. Experimental result shows the sensitivity of the structure due to minor perturbations. Geometric parameters have a qualitative influence on the system’s nonlinearity. Furthermore, for a shallow arch structure, the geometric conditions to maintain a stable snapped-though equilibrium position (in addition to the nominally unloaded configuration) are studied. Critical stability boundaries are generated in the parameter space. When the boundary is crossed, the stable inverted equilibrium disappears, and as a result, the structure will snap back to its initial configuration spontaneously. A set of 3D-printed arches on both sides of the critical boundary are produced for verification purposes. The results have also been extended to thermal conditions. Finally, as a real high-dimensional system, the buckling and post-buckling behavior of a cylindrical shell is taken into consideration. Various initial imperfections are tested. While the structure is sensitive to some initial imperfection shapes (e.g., a post-buckled deformation, a dimple imperfection, etc.), some other initial imperfection shapes (e.g., an axisymmetric half ‘sine’ wave) hardly have any influence on the buckling behavior. Therefore, the sensitivity of the system can be reduced by applying prescribed initial imperfections in certain `sensitive’ shapes. Lateral probing tests under varying axial loads exhibit a view of the underlying potential landscape and implies an upper limit of the critical buckling force.

In contrast to previous studies on known systems, in the last part, a quadratic regression method is developed to locate unstable equilibria for an unknown system from its transient responses. The method shows great accuracy and efficiency for various systems with two or three mechanical/structural DOFs, especially in locating the `most important’ index-1 saddle point. It is also beneficial in identifying systems with adjacent equilibria or saddle point ghosts. The method also shows a robustness against noise, if a proper zero-phase filter is applied.

Item Open Access Nonlinear Dynamics of Discrete and Continuous Mechanical Systems with Snap-through Instabilities(2012) Wiebe, RichardThe primary focus of this dissertation is the characterization of snap-through buckling of discrete and continuous systems. Snap-through buckling occurs as the consequence of two factors, first the destabilization, or more often the disappearance of, an equilibrium position under the change of a system parameter, and second the existence of another stable equilibrium configuration at a remote location in state space. In this sense snap-through buckling is a global dynamic transition as the result of a local static instability.

In order to better understand the static instabilities that lead to snap-through buckling, the behavior of mechanical systems in the vicinity of various local bifurcations is first investigated. Oscillators with saddle-node, pitchfork, and transcritical bifurcations are shown analytically to exhibit several interesting characteristics, particularly in relation to the system damping ratio. A simple mechanical oscillator with a transcritical bifurcation is used to experimentally verify the analytical results. The transcritical bifurcation was selected since it may be used to represent generic bifurcation behavior. It is shown that the damping ratio may be used to predict changes in stability with respect to changing system parameters.

Another useful indicator of snap-through is the presence of chaos in the dynamic response of a system. Chaos is usually associated snap-through, as in many systems large amplitude responses are typically necessary to sufficiently engage the nonlinearities that induce chaos. Thus, a pragmatic approach for identifying chaos in experimental (and hence noisy) systems is also developed. The method is applied to multiple experimental systems showing good agreement with identification via Lyapunov exponents.

Under dynamic loading, systems with the requisite condition for snap-through buckling, that is co-existing equilibria, typically exhibit either small amplitude response about a single equilibrium configuration, or large amplitude response that transits between the static equilibria. Dynamic snap-through is the name given to the large amplitude response, which, in the context of structural systems, is obviously undesirable. This phenomenon is investigated using experimental, numerical, and analytical means and the boundaries separating safe (non-snap-through) from unsafe (snap-through) dynamic response in forcing parameter space are obtained for both a discrete and a continuous arch. Arches present an ideal avenue for the investigation of snap-through as they typically have multiple, often tunable, stable and unstable equilibria. They also have many direct applications in both civil engineering, where arches are a canonical structural element, and mechanical engineering, where arches may be used to approximate the behavior of curved plates and panels such as those used on aircraft.

Item Open Access Nonsmooth Dynamics in Two Interacting, Impacting Pendula(2012) George, Christopher MichaelThis thesis reviews the experimental investigation of a non-smooth dynamical system consisting of two pendula; a large pendulum attached to a frame with an impact wall, and a small pendulum, which shares its axis of rotation with the large pendulum and can impact against the large pendulum. The system is forced with a sinusoidal horizontal motion, and due to the nonlinearities present in pendula as well as the discontinuous forcing from impacts, exhibits a wide range of behavior. Periodic, quasi-periodic, and chaotic responses all are possible, hysteresis is present, and grazing bifurcations allow for spontaneous change of behavior and the appearance of chaotic responses without following a traditional route to chaos. This thesis follows from existing non-linear dynamics research on forced pendula, impacting systems (such as a bouncing ball) and doubly impacting systems (ball bouncing on top of a bouncing ball).

Item Open Access On Locating Unstable Equilibria and Probing Potential Energy Fields in Nonlinear Systems Using Experimental Data(2020) Xu, YawenThis study focuses on a series of data-driven methods to study nonlinear dynamic systems. First, a new method to estimate the location of unstable equilibria, specifically saddle-points, based on transient trajectories from experiments is proposed. We describe a system in which saddle-points (not easily observed in a direct sense) influence the behavior of trajectories that pass `close-by' them. This influence is used to construct a model and thus identify a more accurate estimate of the location using a number of refinements associated with linearization and regression. The method is verified on a rolling-ball model. Both simulations and experiments were conducted. The experiments consists of a small ball rolling on a relatively shallow curved surface under the influence of gravity: a potential energy surface in two dimensions. Tracking the motion of the ball with a digital camera provides data that compares closely with the output of numerical simulation. The experimental results suggest that this method can effectively locate the saddle equilibria in a system, and the robustness of the approach is assessed relative to the effect of noise, size of the local neighborhood, etc., in addition to providing information on the local dynamics. Given the relative simplicity of the experiment system used and a-priori knowledge of the saddle-points, it is a useful testing environment for system identification in a nonlinear context. Furthermore, a post-buckled beam model is used to test this method. Because in real world applications, continuous elastic structures are more common. The experiment results successfully capture both the stable and unstable configurations. However, the natural frequency provided by this regression method underestimates the natural frequency of the second mode. This is the result of low sampling rate in the experiment which leads to inaccurate estimation of velocity and acceleration from numerical differentiation. Simulation results from finite element method with higher sampling rate do not have this issue.

Then, a method to identify potential energy through probing a force field is presented. A small ball resting on a curve in a gravitational field offers a simple and compelling example of potential energy. The force required to move the ball, or to maintain it in a given position on a slope, is the negative of the vector gradient of the potential field: the steeper the curve, the greater the force required to push the ball up the hill (or keep it from rolling down). We thus observe the turning points (horizontal tangency) of the potential energy shape as positions of equilibrium (in which case the 'restoring force' drops to zero). We appeal directly to this type of system using both one and two-dimensional shapes: curves and surfaces. The shapes are produced to a desired mathematical form generally using additive manufacturing, and we use a combination of load cells to measure the forces acting on a small steel ball-bearing subject to gravity. The measured forces, as a function of location, are then subject to integration to recover the potential energy function. The utility of this approach, in addition to pedagogical clarity, concerns extension and applications to more complex systems in which the potential energy would not be typically known {\it a priori}, for example, in nonlinear structural mechanics in which the potential energy changes under the influence of a control parameter, but there is the possibility of force {\it probing} the configuration space. A brief example of applying this approach to a 1-D simple elastic structure is also presented. For multi-dimensional continuous elastic systems, it would be hard to derive the whole potential energy field. However, it is possible to learn the potential energy difference between different equilibria. This information could help us learn the global stability of the stable equilibria, \textit{i.e.}, how much energy is required to escape from the stable equilibria.

Finally, a case study using the two above-mentioned methods on short square box columns is presented. This case study relies on simulation from the finite element method. The buckling of short square box column is dominated by the local buckling of the panel on each side of the column. Hence, the buckling of short box columns shares strong similarities with the buckling of a rectangular panel under uni-axial load. The primary, secondary and tertiary

bifurcation of a series of square box columns with different height-to-width ratio is presented. Then, we focus on the column with height-to-width ratio of 1.4142, in which the primary and second bifurcation would happen almost simultaneously. And thus, the differences in the energy level between different stable equilibria are important. The simulation results show that after the secondary bifurcation, the energy `well' depth for these stable equilibria are similar initially. With the further increase of buckling load, the energy well for the second mode is deeper and the second mode becomes the more stable configuration. We also study the dynamic snap-through of the post-buckled column. The regression method is used to estimate the equilibria configuration and the natural frequencies with great accuracy. We notice an interesting phenomenon, there can be an energy exchange between different sides of the box column and hence, the real parts of the eigenvalue of the Jacobian matrix are positive if we only take the shape of one surface into account, whereas, if we take two next surfaces into the regression method, the real parts become negative.

Item Open Access Post-Buckled Stability and Modal Behavior of Plates and Shells(2012) Lyman, Theodore ClarenceIn modern engineering there is a considerable interest in predicting the behavior of post-buckled structures. With current lightweight, aerospace, and high performance applications, structural elements frequently operate beyond their buckled load. This is especially true of plates, which are capable of maintaining stability at loads several times their critical buckling load. Additionally, even structures such as cylindrical shells may be pushed into a post-buckled range in these extreme applications.

Because of the nature of these problems, continuation methods are particularly well suited as a solution method. Continuation methods have been extensively applied to a range of problems in mathematics and physics but have been used to a lesser extent in engineering problems. In the present work, continuation methods are used to solve a variety of buckling and stability problems of discrete dynamical systems, plates and cylinders. The continuation methods, when applied to dynamic mechanical systems, also provide very useful information regarding the modal behavior of the structure, including linearized natural frequencies and mode shapes as a by-product of the solution method.

To verify the results of the continuation calculations, the commercial finite element code ANSYS is used as an independent check. To confirm previously unseen stable equilibrium shapes for square plates, a set of experiments on polycarbonate plates is also presented.

Item Open Access Stiffness and frequency of slender structures: An experimental study utilizing 3D printing(2018) Giliberto, Joseph VincentThis study analyzes the effect of geometric changes to the stiffness and frequency of slender structures. Geometric changes were made by altering the width and length of the structure as well as adding structural components. 3D printing was utilized to create the slender structures which were tested experimentally. Stiffness was determined by finding the slope of the linear region of the structure's force vs deflection plot. The frequency of the structure was obtained by putting a time series of the structure's oscillations through a Fast Fourier transform which provides a peak signifying the structures in plane frequency. Additionally, several structures were combined to create a springs in parallel system. Results of analysis show that for a structure with constant material properties that increasing/decreasing the length will lead to an decrease/increase in stiffness and frequency while altering the width of the structure will increase stiffness, but have no effect on frequency. It is also shown that additional structural components added to a simple structure increases its stiffness and frequency. Analysis of the springs in parallel system will give a non-linear force vs deflection plot which is made up of linear regions. The slope of the curve changes when the deflection is equal to the spacing between structures. These results are useful for designing structures to fulfill their requirements in the overall system.

Item Open Access Techniques to Assess Acoustic-Structure Interaction in Liquid Rocket Engines(2008-04-25) Davis, Robert BenjaminAcoustoelasticity is the study of the dynamic interaction between elastic structures and acoustic enclosures. In this dissertation, acoustoelasticity is considered in the context of liquid rocket engine design. The techniques presented here can be used to determine which forcing frequencies are important in acoustoelastic systems. With a knowledge of these frequencies, an analyst can either find ways to attenuate the excitation at these frequencies or alter the system in such a way that the prescribed excitations do result in a resonant condition. The end result is a structural component that is less susceptible to failure. The research scope is divided into three parts. In the first part, the dynamics of cylindrical shells submerged in liquid hydrogen (LH_{2}) and liquid oxygen (LOX) are considered. The shells are bounded by rigid outer cylinders. This configuration gives rise to two fluid-filled cavities: an inner cylindrical cavity and an outer annular cavity. Such geometries are common in rocket engine design. The natural frequencies and modes of the fluid-structure system are computed by combining the rigid wall acoustic cavity modes and the*in vacuo*structural modes into a system of coupled ordinary differential equations. Eigenvalue veering is observed near the intersections of the curves representing natural frequencies of the rigid wall acoustic and the*in vacuo*structural modes. In the case of a shell submerged in LH_{2}, system frequencies near these intersections are as much as 30% lower than the corresponding*in vacuo*structural frequencies. Due to its high density, the frequency reductions in the presence of LOX are even more dramatic. The forced responses of a shell submerged in LH_{2}and LOX while subject to a harmonic point excitation are also presented. The responses in the presence of fluid are found to be quite distinct from those of the structure*in vacuo*. In the second part, coupled mode theory is used to explore the fundamental features of acoustoelastic systems. The result is the development of relatively simple techniques that allow analysts to make informed decisions concerning the importance of acoustic-structure coupling without resorting to more time consuming and complex methods. In this part, a new nondimensional parameter is derived to quantify the fundamental strength of a particular acoustic-structure interaction irrespective of material and fluid properties or cavity size. It is be shown that, in some cases, reasonable approximations of the coupled acoustic-structure frequencies can be calculated without explicit knowledge of the uncoupled component mode shapes. Monte Carlo simulations are performed to determine the parameter values over which the approximate coupled frequency expressions are accurate. General observations concerning the forced response of acoustoelastic systems are then made by investigating the response of a simplified two mode system. The third part of this research discusses the implementation of a component mode synthesis (CMS) technique for use with geometrically complex acoustoelastic systems. The feasibility of conceptually similar techniques was first demonstrated over 30 years ago. Since that time there have been remarkable advancements in computational methods. It is therefore reasonable to question the extent to which CMS remains a computationally advantageous approach for acoustoelastic systems of practical interest. This work demonstrates that relative to the most recent release of the popular finite element software package, ANSYS, CMS techniques have a significant computational advantage when the forced response of an acoustoelastic system is of interest. However, recent improvements to the unsymmetric eigensolver available in ANSYS have rendered CMS a less efficient option when calculating system frequencies and modes. The CMS technique is then used to generate new results related to geometrically complex acoustoelastic systems.Item Open Access The synchronization of superparamagnetic beads driven by a micro-magnetic ratchet.(Lab Chip, 2010-08-21) Gao, Lu; Gottron, Norman J; Virgin, Lawrence N; Yellen, Benjamin BWe present theoretical, numerical, and experimental analyses on the non-linear dynamic behavior of superparamagnetic beads exposed to a periodic array of micro-magnets and an external rotating field. The agreement between theoretical and experimental results revealed that non-linear magnetic forcing dynamics are responsible for transitions between phase-locked orbits, sub-harmonic orbits, and closed orbits, representing different mobility regimes of colloidal beads. These results suggest that the non-linear behavior can be exploited to construct a novel colloidal separation device that can achieve effectively infinite separation resolution for different types of beads, by exploiting minor differences in their bead's properties. We also identify a unique set of initial conditions, which we denote the "devil's gate" which can be used to expeditiously identify the full range of mobility for a given bead type.