Lawrie Virgin

Duke University | DU · Department of Mechanical Engineering and Materials Science (MEMS)

Given a constrained slender structure and subject to thermal loading, that is, relative to ambient conditions and any support structure, a reduction in stiffness not only leads to buckling, but also has a strong effect on the natural frequencies of the system. One of the simplest structural forms is a thin circular panel, clamped around its perimet...

The results of an experimental study on the buckling of a vertically cantilevered plate under corner twisting forces are reported. In this configuration, an interesting and somewhat counter-intuitive behavior is observed in which a laterally loaded slender panel exhibits a subtle instability characterized by nonlinear out-of-plane corner deflection...

Cylinder buckling is notoriously sensitive to small geometric imperfections. This is an underlying motivation for the use of knock-down factors in the design process, especially in circumstances in which minimum weight is a key design goal, an approach well-established at NASA, for example. Not only does this provide challenges in the practical des...

A bistable structural component possesses more than one stable equilibrium configuration. In terms of the strain energy stored in bending, this can be thought of as a system with not only an initial equilibrium configuration represented by an isolated minimum, but also a remote minimum that might be accessed given a sufficient disturbance. Whether...

It is well established that axial loads tend to influence lateral stiffness and hence natural frequencies of slender structural components. For very slender structures, the axial loading can be caused by self-weight (in a gravitational field), and these effects influence equilibrium configurations and dynamic behavior. In some applications, for exa...

This paper presents a new approach to predicting an incipient critical speed in a rotating shaft. Based on the classical governing equations of motion for an eccentric mass on a flexible shaft (the Jeffcott rotor model), the approach is centered on examining the behavior of small perturbations or random disturbances to infer the approach of a criti...

Locating the shear, or flexural, center of non-symmetric cross-sectional beams is a key element in the teaching of structural mechanics. That is, establishing the point on the plane of the cross-section where an applied load, generating a bending moment about a principal axis, results in uni-directional deflection, and no twisting. For example, in...

This brief note examines the lumped dynamic modeling of two simple structural systems: 3D-printed cantilevers, and circular aluminum panels. In each case a set of stiffness and natural frequency measurements are made based on simple experiments, with a view to examining effective mass, and a comparison is made with the basic theory.

This paper considers the case of a relatively large number of parallel columns that buckle simultaneously. The close proximity between columns results in the possibility of contact between adjacent columns as buckling proceeds, and this brings with it some interesting observations on load-carrying capacity. Some experimental results verify the theo...

Folded cantilevers have been utilized in MEMS devices, particularly for suspension. The structures consist of a horizontal beam segment fixed at its left end, a short downward connector (joint) at the right end, and a lower horizontal segment under the upper one. Here, the left end of the lower segment is free and a downward concentrated load is ap...

The teaching of structural stiffness is one of the keystones of the undergraduate curriculum in mechanics and the strength of materials. Standard linear theory, going back to Hooke's law, has proven to be very successful in predicting the performance of elastic structures under load. Many courses in basic mechanics have a conventional laboratory co...

This paper considers the load–deflection behavior of a pyramid-like, shallow lattice structure. It consists of four beams that join at a central apex and when subject to a lateral load, it exhibits a propensity to snap-through: a classical buckling phenomenon. Whether this structural inversion occurs, and the routes by which it happens, depends sen...

Perhaps the two most common forms of potential energy are those associated with gravitational and elastic forces. In an experimental setting, if we can measure the force required to maintain equilibrium, then the extraction of potential energy is relatively straightforward, since the force is the negative vector gradient of the potential. In this p...

Unstable equilibria play an important organizing role in nonlinear dynamic systems in a global sense. However, it is difficult to measure them directly in a physical experiment. In this study, a digital image correlation (DIC) system is used to capture the transient behavior of a post-buckled beam in which trajectories are generated by repeated imp...

This paper describes a primarily experimental study in which a nonlinear structural component (a slender, mechanically-buckled panel) is subject to probing. That is, equilibrium configurations are explored when a specific location on the panel is subject to the application of a (variable) displacement constraint, and characterized by a correspondin...

The curse of dimensionality looms over many studies in science and engineering. Low-order systems provide conceptual clarity but often fail to reveal the extent of possible complexity, whereas high-order systems present a host of daunting challenges to the analyst, not least the classification and visualization of typical behavior. In this paper, w...

Nonlinear bifurcations and instabilities of autonomous nonconservative systems, mainly involving the fluid loading of a solid or structure, are reviewed and described in this accessible, pictorial overview. In contrast to the earlier papers in this series (focusing on the instability of elastic deformable systems, and low-order periodically-forced...

This paper details a new method to estimate the location of unstable equilibria, specifically saddle-points, based on transient trajectories from experiments. 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 mode...

Introduction to Experimental Nonlinear Dynamics - by Lawrence N. Virgin March 2000

The pin-ended, slender, Euler strut has been used as the archetypal buckling problem for many years (Euler in Additamentum I de curvis elasticis, methodus inveniendi lineas curvas maximi minimivi proprietate gaudentes, Bousquet, Lausanne, 1744). Even though it is not conventionally imperfection-sensitive (i.e., in which the magnitude of the bucklin...

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...

The equilibrium configuration of an engineering structure, able to withstand a certain loading condition, is usually associated with a local minimum of the underlying potential energy. However, in the nonlinear context, there may be other equilibria present, and this brings with it the possibility of a transition to an alternative (remote) minimum....

A phase space boundary between transition and non-transition, similar to those observed in chemical reaction dynamics, is shown experimentally in a macroscopic system. We present a validation of the phase space flux across rank one saddles connecting adjacent potential wells and confirm the underlying phase space conduits that mediate the transitio...

This is the third part in a trilogy of papers examining ways in which additive manufacturing can be used to facilitate the introduction of basic principles in structural analysis. Each paper uses 3D-printing and simple, but non-trivial, slender geometric forms, to provide a hands-on aspect to structural behavior in which flexure plays a dominant ro...

Vertical circular rings and a system of three nested rings are tested and analyzed. The rings are clamped to a flat rigid base and are loaded vertically at the top by either a concentrated load or a rigid plate. The tests involve rings made by 3D printing. In the analysis, each ring is modeled as an inextensible elastica. For downward quasi-static...

This paper reviews some examples of bifurcation in low-order, periodically driven dynamical systems. The generic loss of stability is a key component in dynamical systems theory, and provides a central pillar in assessing qualitative changes in system dynamics. Although bifurcation tends to be thought of in rather abstract, theoretical terms, we sh...

This research revisits the analysis of roll motion and the possible capsize of floating vessels in beam seas. Many analytical investigations of this topic have adopted the softening Duffing equation, which is similar to the ship roll equation of motion. Here we focus on the loss of stability of periodic oscillations and its relevance to ship capsiz...

The suppression of expansion in thin clamped panels subjected to elevated thermal loading often results in buckling. However, a number of possible post-buckled equilibrium configurations typically exist, and which shape ensues depends on a number of factors including the role of symmetry, boundary conditions, aspect ratio, and the effect of small g...

Energy dissipation is often the most challenging component of system identification in the modeling of dynamical behavior in mechanical systems. Even for a relatively simple single-degree-of-freedom system such as the rigid-arm pendulum, it can be difficult to choose the form of the best damping model, as well as the subsequent challenge of estimat...

This paper describes experiments and analysis of the complete post-buckling behavior of shallow geodesic lattice domes. Although individual members are straight, their geometric arrangement approximates a curved surface and typical behavior is highly nonlinear, including the possibility of sudden jumps in which there may be multiple discontinuous p...

This paper exploits the accuracy and versatility of additive manufacturing to display interesting buckling behavior in slender elastic columns. A set of parallel columns were printed to relatively high precision, and then subjected to axial loading. The load-deflection behavior is influenced by the post-buckled mutual contact between adjacent colum...

This short paper describes a useful teaching tool, ideal for demonstration purposes within the classroom or lab setting. It is based on the simple dynamic response of flexible cantilevers and evolves naturally from the underlying principles of a vibrating reed tachometer. Utilizing a 3D-printer, these ideas conveniently encompass the phenomenon of...

A phase space boundary between transition and non-transition, similar to those observed in chemical reaction dynamics, is shown experimentally in a macroscopic system. We present a validation of the phase space flux across rank one saddles connecting adjacent potential wells and confirm the underlying phase space conduits that mediate the transitio...

The objective of the present paper is to provide experimental evidence of isolated resonances in the frequency response of nonlinear mechanical systems. More specifically, this work explores the presence of isolas, which are periodic solutions detached from the main frequency response, in the case of a nonlinear set-up consisting of two masses slid...

The material contained in this paper focuses on using 3D printing of relatively simple, flexible structural components and plane frames. The relatively high resolution of modern 3D printers facilitates the production of slender structures, and thus provides an opportunity to exploit geometric parameter variations to enhance a practical understandin...

It is well established that the lateral bending stiffness of thin panels is considerably enhanced by judicious use of ribs or stiffeners. This increase in stiffness is primarily due to a disproportionate increase in the second moment of area, and because relatively little mass is added, stiffened panels are especially appealing in an aerospace engi...

This paper provides a companion study to a previous paper by the same author (Virgin, 2017). In that paper, 3D printing was used to provide a hands-on experience for students of (linear) structural analysis based on the lateral stiffness of plane frames. In this paper, a related set of structural plane frames is investigated in terms of their natur...

In dissipative dynamical systems, equilibrium (stationary) points have a dominant organizing effect on transient motion in phase space, especially in nonlinear systems. These time-independent solutions are readily defined in the context of ordinary differential equations, that is, they occur when all the time derivatives are simultaneously zero...

Structural analysis forms a key component in many courses in civil, mechanical and aerospace engineering. Conventionally, the matrix stiffness method, a subset of finite element analysis, tends to occupy a central position in a typical syllabus, with a special focus on plane frames providing a bridge between basic structural components with pedagog...

A uniform elastic cantilever is subjected to a uniformly distributed load or a concentrated load at its tip. The angle of the fixed end with the horizontal is varied until the maximum horizontal distance (projection) from the fixed end to the horizontal location of the tip is attained. The beam is modeled as an inextensible elastica, and numerical...

The equilibrium configuration of an engineering structure, able to withstand a certain loading condition, is usually associated with a local minimum of the underlying potential energy. However, in the nonlinear context, there may be other equilibria present, and this brings with it the possibility of a transition to an alternative (remote) minimum....

Curved structures, such as beams, arches, and panels are capable of exhibiting snap-through buckling behavior when loaded laterally, that is they can exhibit multiple stable equilibria, sometimes after any external loading is removed. This is a consequence of highly nonlinear force-deflection relations with perhaps multiple crossings of the zero-fo...

When the side of a beverage can or the domed lid of a jar is pushed inward, all or part of the structure may suddenly snap into an inverted configuration. The velocity of the pushing motion affects this instability. Most previous analyses of snap-through have considered force control (increasing the pushing force, e.g., a weight). Snap-through unde...

This paper describes the process of estimating Young’s modulus for the thermoplastic material commonly used in a type of 3D printer. Its twin goals are to compare and contrast a number of simple techniques from elementary structural analysis and to assess the influence of the printer density settings and print orientation (effective material anisot...

In rotordynamics, it is often important to be able to predict critical speeds. The passage through resonance is generally difficult to model. Rotating shafts with a disk are analyzed in this study, and experiments are conducted with one and two disks on a shaft. The approach presented here involves the use of a relatively simple prediction techniqu...

This research considers the structural behavior of slender, mechanically buckled beams and panels of the type commonly found in aerospace structures. The specimens were deflected and then clamped in a rigid frame in order to exhibit snap-through. That is, the initial equilibrium and the buckled (snapped-through) equilibrium configurations both co-e...

Rotating shafts often experience undesirable large-amplitude whirling oscillations associated with resonance at critical speeds. This paper further develops a nondestructive technique in which measured information about the growing nature of the response is used to predict an incipient critical speed. A number of models of varying degrees of sophis...

This paper shows how the presence of unstable equilibrium configurations of elastic continua is reflected in the behaviour of transients induced by large perturbations. A beam that is axially loaded beyond its critical state typically exhibits two buckled stable equilibrium configurations, separated by one or more unstable equilibria. If the beam i...

A buckled beam with immovable pinned ends is considered. Attached to the beam are either one concentrated mass, two concentrated masses, a spring–mass system (that could model a human, robot, or passive vibration absorber), or a horizontal rigid bar with two vertical end springs (a “bounce–pitch” system that could model an animal or a vehicle). In...

The deformation and vibration of vertical, highly flexible loops are investigated primarily from an experimental perspective. Both upright and hanging loops are considered with a small clamped length. The effects of self-weight on the loop static deformation and lowest frequency for in-plane symmetric vibrations are determined. Good agreement is at...

In the present paper, the observation and characterization of isolated response curves (IRCs) are experimentally reported in the case of a nonlinear system consisting of two masses sliding on an horizontal guide. Transverse springs are attached to one mass to provide the nonlinear restoring force, and a harmonic motion of the complete system is imp...

This paper focuses on thoroughly exploring the finite-time transient behaviors occurring in a periodically driven non-smooth dynamical system. Prior to settling down into a long-term behavior, such as a periodic forced oscillation, or a chaotic attractor, responses may exhibit a variety of transient behaviors involving regular dynamics, co-existing...

This paper presents the snap-through phenomenon and effect of self-contact of the spatial elastica subjected to mid-length torque. One end of the elastica is clamped while the other end is placed in a sleeve joint. The total arc-length of the elastica can be varied by sliding the end through the sleeve joint. At a certain value of total arc-length,...

The prediction of critical speeds of a rotating shaft is a crucial issue in a variety of industrial applications ranging from turbomachinery to disk storage systems. The modeling and analysis of rotordynamic systems is subject to a number of complications, but perhaps the most important characteristic is to pass through a critical speed under spin-...

This paper describes some typical behavior encountered in the response of a harmonically-excited mechanical system in which a severe nonlinearity occurs due to an impact. Although such systems have received considerable recent attention (most of it from a theoretical viewpoint), the system scrutinized in this paper also involves a discrete input of...

In the present paper, isolated response curves in a nonlinear system consisting of two masses sliding on a horizontal guide are examined. Transverse springs are attached to one mass to provide the nonlinear restoring force, and a harmonic motion of the complete system is imposed by prescribing the displacement of their supports. Numerical simulatio...

For very slender structural components, self-weight may compete with elastic flexural stiffness in determining equilibrium configurations. In cases where the inherent elastic stiffness is low (relative to self-weight) we observe a variety of types of highly nonlinear behavior in the equilibrium shapes, together with changes in the natural frequenci...

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The equilibria and stability of a shallow prestressed arch (beam–column) are investigated theoretically and experimentally. The deflection of the arch is unilaterally constrained by a displacement-control device. Both snap-through and remote coexisting equilibria are observed. Force–deflection curves for primary and secondary equilibrium branches a...

Crassulacean acid metabolism (CAM) photosynthesis functions as an endogenous circadian rhythm coupled to external environmental forcings of energy and water availability. This paper explores the nonlinear dynamics of a new CAM photosynthesis model (Bartlett et al., 2014) and investigates the responses of CAM plant carbon assimilation to different c...

The deformation and vibration of vertical flexible loops are investigated theoretically and experimentally. Both upright and hanging loops are considered. Potential applications include nanorings and carbon nanotubes as force sensors or structural components. The upright tubes rest on a rigid or linearly elastic (Winkler) foundation, and cases with...

In this paper, we examine the potential for coexisting responses in a harmonically forced buckled beam. It is shown experimentally that such structures may present many more responses than might be observed using frequency sweep-up and sweep-down testing, with some responses being observed only very infrequently. The primary contribution of this wo...

A key feature of many nonlinear dynamical systems is the presence of co-existing solutions, i.e, nonlinear systems are often sensitive to initial conditions. While there have been many studies to explore this behavior from a numerical perspective, in which case it is trivial to prescribe initial conditions (for example using a regular grid), this i...

This paper studies a system composed of two pendulums attached to a common base that is oscillated horizontally. The pendulums share a common pivot line, but move independently and are only coupled together through collisions. Impact dynamics for the collisions of the pendulums with each other and with fixed barriers yield complex nonlinear behavio...

It is well established that certain structural buckling problems are extremely sensitive to small changes in configuration: geometric imperfections, load application, symmetry, boundary conditions, etc. This paper considers the behavior of a very shallow arch under lateral point loading, and specifically under the influence of changes in the therma...

This short paper takes a close look at a relatively simple harmonically-excited mechanical oscillator. Throughout the range of forcing frequencies the basins of attraction are investigated by applying strong perturbations to steady-state behavior. In this way, a more general solution space is mapped out. Numerical simulation of the equation of moti...

Slender curved structures can often be found as components of complex structures in civil, mechanical, and aerospace systems. Under extreme loadings, a curved structure might undergo snap-through buckling, i.e., the structure is forced to its inverted configuration, inducing fatigue. Therefore, it is important to identify the stability boundaries o...

Steady-state motions of a woman's ponytail during level, straight, walking and running are examined. Based on reported data, formulas have been developed for the relationship of the forward speed to the frequencies of vertical and sideways motion of the head, and of the form of that motion. The ponytail is modeled as a compound pendulum or a multi-...

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 resp...

The effect of damping on the behaviour of oscillations in the vicinity of bifurcations of nonlinear dynamical systems is investigated. Here, our primary focus is single degree-of-freedom conservative systems to which a small linear viscous energy dissipation has been added. Oscillators with saddle-node, pitchfork and transcritical bifurcations are...

Slender curved structures may experience a loss of stability called snap-through, causing the curvature on part or all of the structure to invert inducing fatigue damage. This paper presents a framework for analyzing the transient responses of slender curved structures. A numerical study of snap-through in a shallow arch-like model under periodic e...

Continuation methods are used to examine the static and dynamic postbuckled behavior of a uniaxially loaded, simply supported plate. Continuation methods have been extensively used to study problems in mathematics and physics; however, they have not been as widely applied to problems in engineering. When paired with a Galerkin approximation, contin...

Geometrically nonlinear structures often possess multiple equilibrium configurations. Under extreme conditions of excitation, it is possible for these structures to exhibit oscillations about and between these co-existing configurations. This behavior may have serious implications for fatigue in the context of aircraft surface panels. Snap-through...

A method for estimating model parameters based on chaotic system response data is described. This estimation problem is made challenging by sensitive dependence to initial conditions. The standard maximum likelihood estimation method is practically infeasible due to the non-smooth nature of the likelihood function. We bypass the problem by introduc...

A nonlinear Duffing-type dynamical system, in which the stability of equilibria is modulated in a time-dependent manner, is investigated both experimentally and numerically. This is a low-order dynamical system with some interesting available choices in the coordinate system. The system is found to exhibit a variety of interesting nonlinear behavio...

This paper considers the behavior of a spatial elastica in a gravitational field. The slenderness of the system considered is such that the weight becomes an important consideration in determining elastic equilibrium configurations. Both ends of the elastica are clamped in an initially (planar) horizontal orientation at a fixed distance apart. Howe...

The sign of the largest Lyapunov exponent is the fundamental indicator of chaos in a dynamical system. However, although the extraction of Lyapunov exponents can be accomplished with (necessarily noisy) the experimental data, this is still a relatively data-intensive and sensitive endeavor. This paper presents an alternative pragmatic approach to i...

Continuation and path following methods have been applied to many nonlinear problems in mathematics and physics. There is less widespread application of these methods, however, to structural systems. Since structural buckling and stability problems are primarily concerned with system behavior as a control parameter (most often the load) varies, the...

The articles in this volume give an overview and introduction to nonlinear phenomena in structural dynamics. Topics treated are approximate methods for analyzing nonlinear systems (where the level of nonlinearity is assumed to be relatively small), vibration isolation, the mitigation of undesirable torsional vibration in rotating systems utilizing...

The ability to isolate a structure or machine from the undesirable effects of applied motion (especially vibration) has wide application. Suspension systems are incorporated into large buildings to protect them from earthquake excitation, mountain bikes and vehicles in general are designed to minimize the transfer of unwanted accelerations from the...

This book is based on a one-week worksop at CISM. In order to fully appreciate the benefits of nonlinearity in certain engineering systems it is important to understand the underlying behavior of linear systems, and this first chapter provides a general overview of linear dynamical systems and then begins to explore the effect of nonlinearities.

Here, we explore the single particle dynamics of superparamagnetic beads exposed to multifrequency ratchets. Through a combination of theory, simulation, and experiment, we determine the important tuning parameters that can be used to implement multiplexed separation of polydisperse colloidal mixtures. In particular, our results demonstrate that th...

This paper models a slender, flexible structure used as a drill string or riser in the offshore oil and gas industry that connects the well-head with a floating control vessel. These systems are used in deep-water drilling applications and present considerable design challenges due to their extreme flexibility and susceptibility to buckling and vib...

The nonlinear dynamic behavior of superparamagnetic beads transported through a two-dimensional potential energy landscape is explored empirically and through numerical simulation. The beads are driven through a periodic array of micromagnets by an external rotating field oriented at an angle θ relative to the magnetization direction of the substra...

Examples of vibro-impact mechanical and structural systems are not uncommon. Such systems are capable of exhibiting an interesting spectrum of non-smooth dynamic behavior when a characteristic changes abruptly. However, they present strong challenges to the analyst and designer trying to predict dynamic behavior. This paper considers an example of...

The optimal distribution of material to minimize the vertical deflection of the free end of a horizontal cantilever is determined. The beam is only subjected to its own weight. Large deflections are considered, and the structure is modeled as an inextensible elastica. A minimum-area constraint is included, and is active in a region near the tip. Af...

This paper considers the free vibration of a plane, rectangular, portal frame consisting of slender members. Natural frequencies and mode shapes are influenced by the addition of mass at the corners of the frame. The members are sufficiently slender that axial effects occur, and may ultimately lead to buckling. The results from both theoretical and...

Snap-through buckling can reduce the life-span of structural systems such as aircraft surface paneling. This is envisioned to be a specific problem in hypersonic vehicles in which severe thermal loading and acoustic excitation conspire to create an especially hostile environment for structural elements. A shallow arch, and two simplified link model...

The characterization of chaos as a random-like response from a deterministic dynamical system with an extreme sensitivity to initial conditions is well-established, and has provided a stimulus to research in nonlinear dynamical systems in general. In a formal sense, the computation of the Lyapunov Exponent spectrum establishes a quantitative measur...

We 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...

This paper models flexible risers and pipelines as slender elastica structures. The theoretical formulation leads to a type of nonlinear boundary value problem that can be solved numerically given appropriate boundary conditions. The offsetting effects of gravity and buoyancy are included in the analysis. These forces can provide considerable axial...

An underlying potential energy function can provide visual and intuitive insight into a system's stability and overall behavior. In particular, the motion of a ball moving along a curve or surface in a gravitational field provides a macroscale demonstration of interesting dynamics. We investigate the motion of a small ball rolling along a smooth tw...

The optimal distribution of material to maximize the critical load of columns has been studied extensively in the past, along with initial postbuckling behavior. Here, large postbuckling deflections are analyzed for optimal columns with pinned ends. Small vibrations of the optimal columns about postbuckled equilibrium shapes are also investigated....

The nonlinear dynamic behavior of superparamagnetic beads exposed to a periodic array of micromagnets and an external rotating field is simulated as a function of the relative size of the bead with respect to the micromagnet size and the strength of the external field relative to the pole density of the substrate. For large bead sizes, it is confir...