Research
Introduction
Below are some brief descriptions of various recent and on-going projects being carried out in the Nonlinear Dynamics Lab at Duke. They involve various aspects of nonlinear behavior in vibration and buckling situations encountered in a number of practical application. For more specific details the list of journal publications can be consulted.
Global Behavior of Nonlinear Systems
This research considers some global behavior aspects of nonlinear systems, and in particular, the dynamics associated with an indeterminate saddle-node bifurcation. Certain nonlinear oscillators display the well-known jump to resonance under various excitation parameter combinations; such jumps lead to a region of hysteresis in which stable, competing attractors exist. At certain levels of excitation, the catchment regions of these attractors are non-fractal, but as parameters are varied, it has been shown that completely determinate basins give way to fractal basin boundaries and ultimately an erosion of the entire basin such that jumps to resonance become indeterminate, since initial conditions can never be known accurately enough to overcome the fractal nature of the basin.
Experimentally, this study is considering two different nonlinear systems, one governed by a quadratic nonlinearity (the "escape" equation) and another governed by a cubic nonlinearity (Duffing's equation). Taking advantage of the "ball-rolling-on-a-hill" concept, tracks in the shape of potential energy surfaces have been constructed upon which a cart can roll, and the entire system can be excited over a range of forcing frequencies and amplitudes. Initial condition maps of the cart are generated in attempts to experimentally demonstrate the indeterminate bifurcation; these maps are constructed through slow parameter sweeps and through the relatively new method of stochastic interrogation, which allows an entire basin of attraction to be visited from one time series that can be perturbed.
Nonlinear Aeroelasticity and Control
The nonlinear behavior and flutter characteristics of an aeroelastic system with control surface freeplay are being studied. A low-order, state-space numerical model of the aeroelastic typical section has been created which allows for the inclusion of structural freeplay and preload in the control surface degree of freedom. A physical model of the system has also been constructed for use in the Duke University low-speed wind tunnel
One of the most interesting aspects of the behavior of the airfoil is that hysteresis (in the type of limit cycle behavior) is present. That is, the type of (flutter) response and the flow speeds at which they occur depends on whether the flow rate is increasing or decreasing. This is nicely captured by the spectrograms shown below.
Previous work included the design of a linear control law to significantly raise the flutter speed of the linear system. Current work focuses on the nonlinear behavior induced by the freeplay in the control surface. A variable structure (sliding mode) control is also being developed.
Controlling Chaos
A signature of chaos that can be found in a nonlinear system is the system's sensitive dependence on its initial conditions. These systems can also display a sensitive dependence on small parameter changes. Sensitive dependence, among other things, makes the analysis of nonlinear systems exceedingly complicated and has convinced many to restrict their study to linear systems. Recently, however, researchers have capitalized on this intriguing feature of a nonlinear system's dynamics in order to stabilize chaotic behavior.
At Duke we are applying a controlling scheme originally created by Ott, Grebogi and Yorke to a model of the forced Duffing Oscillator. The goal is to stabilize a selected unstable chaotic response using only very small parameter perturbations. Part of the attraction to this method of control is that there are theoretically an infinite amount of chaotic orbits embedded within the systems global response. This suggests that there is some flexibility in choosing the desired behavior of the system. The validity of the theoretical concepts as well as their practical limitations will be determined when the control scheme is applied to our experimental apparatus.
Alternative Experimental Techniques and New Phenomena in an Impacting System
This project encompasses two main foci: new non-invasive data acquisition techniques and new dynamic phenomena, namely grazing bifurcations. The system under study is an impacting pendulum under sinusoidal base excitation. Both numerical and experimental approaches are used to study the system.
In the past, a potentiometer or similar device has been used experimentally to provide continuous monitoring of the displacement of the impacting pendulum. This techniques has the disadvantage that it significantly affects the damping (and thereby the dynamic behavior) of the system under study. We investigate two non-invasive methods of data acquisition. The first involves monitoring the occurrence of impact with a piezoceramic material, and the second involves recording the sound of the impacts. A nearly one-dimensional map can be constructed from the inter-spike intervals, and information about the behavior of the system can be extracted.
The second part of this project investigates the case when the impact does not occur at the equilibrium position of the system. For various forcing parameters, this means that the pendulum may never impact, may always impact, or may undergo grazing or low-velocity impact. This third case produces a new class of bifurcations which occur at the transition between the 'never-impacting' range and the 'always-impacting' range of parameters.
The plots above show the combinations of forcing parameters that lead to the overturning of a rocking block (and the colors correpsond to how quickly the block overturns, if it does).
Solar Sails
Solar sails are based on the idea of using the sun's photons for propulsion for deep space exploration. These spacecraft are basically huge lightweight kites. The support structures tend to be very slender and liable to buckling and vibration problems. Solar sail booms may take the form of thin cantilvers with cables attached.
New Projects
A rolling ball (see also video section). A conventional paradigm of linear oscillations is the motion of a small ball as it rolls on a parabolic curve. The unique minimum provides a compelling demonstration of stable equilibrium. If the curve is then shaken at close to the natural frequency of the ball undergoing free oscillations then a large amplitude response occurs due to resonance.
If we take this concept a couple of steps further we can envsion a small ball rolling along a two-dimensional surface, and furthermore, we can design the surface to exhibit various maxima and minima. Now, given some uncertainty in the initial conditions it can be very difficult to predict which long-term behavior (in this case a surface minimum) the ball will eventually end up in (given a little energy dissipation). And if we shake the surface harmonically (say) then chaos is a typical response.
The pictures above show a contour plot of a typical surface together with its physical realization (taking advantage of a computer-controlled milling machine to exactly machine a specified surface from a solid block of Lexan)
The above diagrams show the transient time prior to settling in one of the four potential wells (a), and the corresponding initial condition map (b), generated using numerical simulation. Below is an experimental equivalent, in which a digital camera tracked the destinations of many initial conditions.
If we force the surface harmonically (using the scotch-yoke shaker described elsewhere on this site) we can easily illustrate chaos. The phase projections shows two trajectories started from essentially the same position and under the same forcing conditions. They rapidly become uncorrelated.
Non-Smooth Systems
Discretely-forced systems are an extension of piecewise linear systems but where the discontinuity is in the excitation rather than a characteristic of the mechanical system (e.g., bilinear stiffness). This project involves both experimental and theoretical research and is sponsored by the NSF. There is currently an open position for a research assistant.
A typical application of this type of work is systems restrained by cables. The possibility of alternating taut/slack cables results in a harsh dynamic environment. This can be related to a certain type of billiard problem.
The Elastica
Although the elastica occupies a well-established place in classical mechanics, there has been much less work done on the dynamic behavior of the elastica. Below are shown some typical elastica-type structures, generally becoming unstable to due the effect of gravity (self-weight). This project seeks to compute and measure natural frequencies and their corresponding mode shapes regardless of the extent of the underlying equilibrium configuration.
In the example on the left, a cable pulls on the tip of a thin cantilever. In the middle and on the right, a cable's length is increased such that buckling out of plane occurs.
Below is shown a thin polycarbonate beam with its ends pushed together, and inclined at an angle:
Large Amplitude Vibrations
When the amplitude of vibrations are not restricted to small values, the range of possible behavior is greatly enhanced. This is especially true for forced systems. The picture below is a spectrogram of the response of a cantilever initially undergoing very large amplitude oscilaltions but then settling down to equilibrium via linear oscillations due to damping.
Snap-through and Continuation
This project is sponsored by the US Air Force (AFOSR) and involves the complex nonlinear dynamic behavior of curved beams and panels. The work is being undertaken in collaboration with the Air Force Research Lab (AFRL) in Dayton, Ohio. A research assitant is also being sought to work on this project.
Vibration and buckling in rocket engine nozzles
The next generation of rockets to replace the space shuttle involves the development of the ARES system. A number of vibration problems have been identified with the various stages, and this project will involve a number of modeling developments to assess the response of slender structural components in extreme and hostile environments.
