The application of robotic manipulators in machining systems has gained a great interest in manufacturing because of their lower prices, higher kinematic flexibility and larger workspace compared to conventional CNC machine tools. However, their performance is limited due to the much lower structural rigidity which makes them more susceptible to excessive and unstable vibrations, known as chatter, during the machining process. Highly effective chatter modeling and avoidance methods that have been developed for CNC machining in the past decades are now being used by the industry to design high-performance chatter-free machining operations. The available methods, however, face major difficulties when applied to robotic machining, mainly due to the high flexibility and pose-dependency of the vibration response in robots. High flexibility leads to high-amplitude vibrations which affect the process dynamics and excite structural nonlinearities. The existing approaches to modeling machining vibrations assume linearity of the structural dynamics of the robotic manipulator. This assumption, considering the inherent nonlinearities in the robot’s revolute joints, may cause considerable inaccuracies in predicting the stability of vibrations during the process. This thesis studies the high flexibility and nonlinearity of the robot’s structural dynamics and their effects on chatter vibrations. The research starts with investigating the effects of high flexibility of robot's structure in the process dyamics by considering the modulation of cutting forces by axial vibrations, which is normally ignored in CNC milling due to high rigidity of the machine in this direction. The results of chatter prediction considering this effect are shown and discussed. The rest of the thesis focuses on the structural nonlinearity. Firstly, an experimental study is presented to investigate the extent of nonlinearity in structural dynamics of the robot. The results confirm that structural nonlinearities in robotic machining systems can be effectively excited in the presence of high-amplitude vibrations due to milling forces, such that they cause remarkable differences in chatter prediction. The following step is modeling the structural nonlinearities. For this purpose, the variation of restoring forces with the dynamic response (displacement and velocity) are observed when the robot is subjected to harmonic excitation. Based on the experimental observations, the nonlinear effects are modeled by cubic stiffness and damping characteristics. Parameters of the nonlinear model are then identified using Higher-order Frequency Response Functions (HFRF) extracted from measurements. The identified model can predict the vibration behavior of the robotic machining system when subjected to periodic loads such as milling forces. The developed model of nonlinear structural dynamics is then coupled with the chatter model. Consequently, the system is described by nonlinear Delay Differential Equations (DDE) with periodic coefficients. Bifurcation diagrams for the forced vibrations in the described system are developed using the numerical continuation method. The effects of cutting parameters such as feedrate as well as the nonlinear parameters are studied. The thesis is concluded by proposing the use of in-process FRF in the linear model of chatter stability for quick prediction of stability limits. In this approach, the exact characteristics of the nonlinear mechanisms are not studied; instead, the measured FRF during the milling process are used, which are assumed to represent the nonlinear structural dynamics that are linearized about the applied operational conditions. Two methods of measuring in-process FRF are proposed and employed in the robotic milling system. The measured FRF are then used in the linear chatter model to develop the Stability Lobes Diagram (SLD) which shows the combination of cutting parameters that lead to stable or unstable vibrations. Experimental chatter tests show that better agreement with predictions can be achieved by using in-process FRF instead of FRF measured at the idle state of the system. The results of this thesis contribute to better characterization of vibrations in robotic machining with high-amplitude forces and selecting suitable strategies to enhance productivity of the operation. / Graduate

For a successful design of an engineering system it is essential to pay careful attention to its dynamic response. This is particularly true, in the case of nonlinear systems, since they can exhibit very complex dynamic behaviour, including multiple co-existing stable solutions and chaotic motions, characterized by large sensitivity to initial conditions. In some systems nonlinear characteristics are desired and designed for, but in other cases they are unwanted and can cause fatigue and failure. A type of dynamical system which is highly nonlinear is discontinuous or non-smooth systems. In this work, systems with impacts are primarily investigated, and this is a typical example of a discontinuous system. To enhance or optimize the performance of dynamical systems, some kind of control can be implemented. This thesis concerns implementation of low-cost control strategies for discontinuous systems. Low-cost control means that a minimum amount of energy is used when performing the control actions, which is a desirable situation regardless of the application. The disadvantage of such a method is that the performance might be limited as compared with a control strategy with no restrictions on energy consumption. In this work, the control objective is to enforce a continuous or discontinuous grazing bifurcation of the system, whichever is desirable. In Paper A, the dynamic response and bifurcation behaviour of an impactoscillator with dry friction is investigated. For a one-degree-of-freedom model of the system, analytical solutions are found in separate regions of state space. These are then used to perform a perturbation analysis around a grazing trajectory. Through the analysis, a condition on the parameters of the system is derived, which assures a continuous grazing bifurcation. It is also shown that the result has bearing on the dynamic response of a two-degree-of-freedom model of the system. A low-cost active control strategy for a class of impact oscillators is proposed in Paper B. The idea of the control method is to introduce small adjustments in the position of the impact surface, at discrete moments in time, to assure a continuous bifurcation. A proof is given for what control parameters assures the stabilization. In Paper C, the proposed low-cost control method is implemented in a quarter-car model of a vehicle suspension, in order to minimize impact velocities with the bumpstop in case of high amplitude excitation. It is shown that the control method is effective for harmonic road excitation. / QC 20101118

No description available.

Tunneling is a fundamental effect of quantum mechanics, which allows waves to penetrate into regions that are inaccessible by classical dynamics. We study this phenomenon for generic non-integrable systems with a mixed phase space, where tunneling occurs between the classically separated phase-space regions of regular and chaotic motion. We derive a semiclassical prediction for the corresponding tunneling rates from the regular region to the chaotic sea. This prediction is based on paths which connect the regular and the chaotic region in complexified phase space. We show that these complex paths can be constructed despite the obstacle of natural boundaries. For the standard map we demonstrate that tunneling rates can be predicted with high accuracy, by using only a few dominant complex paths. This gives the semiclassical foundation for the long-conjectured and often-observed exponential scaling with Planck's constant of regular-to-chaotic tunneling rates.

Nonlinear dispersive partial differential equations occur in a variety of areas within mathematical physics and engineering. We study several classes of such equations, including scalar complex partial differential equations, vector partial differential equations, and finally non-local integro-differential equations. For physically interesting families of these equations, we demonstrate the existence (and, when possible, stability) of specific solutions which are relevant for applications. While multiple application areas are considered, the primary application that runs through the work would be the nonlinear dynamics of vortex filaments under a variety of physical models. For instance, we are able to determine the structure and time evolution of several physical solutions, including the planar, helical, self-similar and soliton vortex filament solutions in a quantum fluid. Properties of such solutions are determined analytically and numerically through a variety of approaches. Starting with complex scalar equations (often useful for studying two-dimensional motion), we progress through more complicated models involving vector partial differential equations and non-local equations (which permit motion in three dimensions). In many of the examples considered, the qualitative analytical results are used to verify behaviors previously observed only numerically or experimentally.

Converting vibrations to a usable form of energy has been the topic of many recent investigations. The ultimate goal is to convert ambient or aeroelastic vibrations to operate low-power consumption devices, such as microelectromechanical systems, heath monitoring sensors, wireless sensors or replacing small batteries that have a nite life span or would require hard and expensive maintenance. The transduction mechanisms used for transforming vibrations to electric power include: electromagnetic, electrostatic, and piezoelectric mechanisms. Because it can be used to harvest energy over a wide range of frequencies and because of its ease of application, the piezoelectric option has attracted significant interest. In this work, we investigate the performance of different types of piezoelectric energy harvesters. The objective is to design and enhance the performance of these harvesters. To this end, distributed-parameter and phenomenological models of these harvesters are developed. Global analysis of these models is then performed using modern methods of nonlinear dynamics. In the first part of this Dissertation, global nonlinear distributed-parameter models for piezoelectric energy harvesters under direct and parametric excitations are developed. The method of multiple scales is then used to derive nonlinear forms of the governing equations and associated boundary conditions, which are used to evaluate their performance and determine the effects of the nonlinear piezoelectric coefficients on their behavior in terms of softening or hardening. In the second part, we assess the influence of the linear and nonlinear parameters on the dynamic behavior of a wing-based piezoaeroelastic energy harvester. The system is composed of a rigid airfoil that is constrained to pitch and plunge and supported by linear and nonlinear torsional and flexural springs with a piezoelectric coupling attached to the plunge degree of freedom. Linear analysis is performed to determine the effects of the linear spring coefficients and electrical load resistance on the flutter speed. Then, the normal form of the Hopf bifurcation (flutter) is derived to characterize the type of instability and determine the effects of the aerodynamic nonlinearities and the nonlinear coefficients of the springs on the system's stability near the bifurcation. This is useful to characterize the effects of different parameters on the system's output and ensure that subcritical or "catastrophic" bifurcation does not take place. Both linear and nonlinear analyses are then used to design and enhance the performance of these harvesters. In the last part, the concept of energy harvesting from vortex-induced vibrations of a circular cylinder is investigated. The power levels that can be generated from these vibrations and the variations of these levels with the freestream velocity are determined. A mathematical model that accounts for the coupled lift force, cylinder motion and generated voltage is presented. Linear analysis of the electromechanical model is performed to determine the effects of the electrical load resistance on the natural frequency of the rigid cylinder and the onset of the synchronization region. The impacts of the nonlinearities on the cylinder's response and energy harvesting are then investigated. / Ph. D.

A mathematical model is developed to investigate the grazing dynamics of tapping mode atomic force microscopes (AFM) subjected to a base harmonic excitation. The nonlinear dynamics of the AFM microcantilever are studied in both of the monostable and bistable phases with the microcantilever tip being, respectively, located in the monostable and bistable regions of the static bifurcation diagram in the reference configuration. Free-vibration responses of the AFM probes, including the microcantilever natural frequencies and mode shapes, are determined. It is found that, for the parameters used in a practical operation of an AFM, the natural frequencies and mode shapes of the AFM microcantilever are almost the same as those of a free-end microcantilever with the same geometry and made of an identical material. A multimode Galerkin approximation is utilized to discretize the nonlinear partial-differential equation of motion and associated boundary conditions governing the cantilever response and obtain a set of nonlinearly coupled ordinary-differential equations (ODE) governing the time evolution of the system dynamics. The corresponding nonlinear ODE set is then solved using numerical integration schemes. A comprehensive numerical analysis is performed for a wide range of the excitation amplitude and frequency. The tip oscillations are examined using nonlinear dynamic tools through several examples. The non-smoothness in the tip/sample interaction model is treated rigorously. A higher-mode Galerkin analysis indicates that period doubling bifurcations and chaotic vibrations are possible in tapping mode microscopy for certain operating parameters. It is also found that a single-mode Galerkin approximation, which accurately predicts the tip nonlinear responses far from the sample, is not adequate for predicting all of the nonlinear phenomena exhibited by an AFM, such as grazing bifurcations, and leads to both quantitative and qualitative errors. A point-mass model is also developed based on the single-mode Galerkin procedure to compare with the present distributed-parameter model. In addition, a reduced-order model based on a differential quadrature method (DQM) is employed to explore the dynamics of the AFM probe in the bistable phase where the multimode Galerkin procedure is computationally expensive. We found that the DQM with a few grid points accurately predicts the static bifurcation diagram. Moreover, we found that the DQM is capable of precise prediction of the lowest natural frequencies of the microcantilever with only a few grid points. For the higher natural frequencies, however, a large number of grid points is required. We also found that the natural frequencies and mode shapes of the microcantilever about non-contact equilibrium positions are almost the same as those of the free-end microcantilever. On the other hand, free-vibration responses of the microcantilever about contact equilibrium positions are quite different from those of the free-end microcantilever. Moreover, we used the DQM to discretize the partial-differential equation governing the microcantilever motion and a finite-difference method (FDM) to calculate limit-cycle responses of the AFM tip. It is shown that a combination of the DQM and FDM applied, respectively, to discretize the spatial and temporal derivatives provides an efficient, accurate procedure to address the complicated dynamic behavior exhibited by the AFM probe. The procedure was, therefore, utilized to study the response of the microcantilever to a base harmonic excitation through several numerical examples. We found that the dynamics of the AFM probe in the bistable region is totally different from those in the monostable region. / Ph. D.

We investigate theoretically and experimentally the nonlinear responses of a clamped-clamped buckled beam to a variety of external harmonic excitations and internal resonances. We assume that the beam geometry is uniform and its material is homogeneous. We initially buckle the beam by an axial force beyond the critical load of the first buckling mode, and then we apply a transverse harmonic excitation that is uniform over its span. The beam is modeled according to the Euler-Bernoulli beam theory and small strains and moderate rotation approximations are assumed. We derive the equation of motion governing the nonlinear transverse planar vibrations and associated boundary conditions using the extended Hamilton's principle. The governing equation is a nonlinear integral-partial-differential equation in space and time that possesses quadratic and cubic nonlinearities. A closed-form solution for such equations is not available and hence we seek approximate solutions. We use perturbation methods to investigate the slow dynamics in the neighborhood of an equilibrium configuration. A Galerkin approximation is used to discretize the nonlinear partial-differential equation governing the beam's response and obtain a set of nonlinearly coupled ordinary-differential equations governing the time evolution of the response. We based our theory on a multi-mode Galerkin discretization. To investigate the large-amplitude dynamics, we use a shooting method to numerically integrate the discretized equations and obtain periodic orbits. The stability and bifurcations of these periodic orbits are investigated using Floquet theory. We solve the nonlinear buckling problem to determine the buckled configurations as a function of the applied axial load. We compare the static buckled configurations obtained from the discretized equations with the exact ones. We find out that the number of modes retained in the discretization has a significant effect on these static configurations. We consider three cases: primary resonance, subharmonic resonance of order one-half of the first vibration mode, and one-to-one internal resonance between the first and second modes. We obtain interesting dynamics, such as phase-locked and quasiperiodic motions, resulting from a Hopf bifurcation, snapthrough motions, and a sequence of period-doubling bifurcations leading to chaos. To validate our theoretical results, we ran an experiment, which is a modified version of the experiment designed by Kreider and Nayfeh. We find that the obtained theoretical results are in good qualitative agreement with the experimental results. In the case of one-to-one internal resonance, we report, theoretically and experimentally, energy transfer between the first mode, which is externally excited, and the second mode. / Ph. D.

The objective of the present research is to investigate unsteady aerodynamic models with state equation representations that are valid up to the high angle of attack regime with the purpose of evaluating them as computationally affordable models that can be used in conjunction with the equations of motion to simulate wing rock. The unsteady aerodynamic models with state equation representations investigated are functional approaches to modeling aerodynamic phenomena, not directly derived from the physical principles of the problem. They are thought to have advantages with respect to the physical modeling methods mainly because of the lower computational cost involved in the calculations. The unsteady aerodynamic multi-axis models with state equation representations investigated in this report assume the decomposition of the airplane into lifting surfaces or panels that have their particular aerodynamic force coefficients modeled as dynamic state-space models. These coefficients are summed up to find the total aircraft force coefficients. The products of the panel force coefficients and their moment arms with reference to a given axis are summed up to find the global aircraft moment coefficients. Two proposed variations of the state space representation of the basic unsteady aerodynamic model are identified using experimental aerodynamic data available in the open literature for slender delta wings, and tested in order to investigate their ability to represent the wing rock phenomenon. The identifications for the second proposed formulation are found to match the experimental data well. The simulations revealed that even though it was constructed with scarce data, the model presented the expected qualitative behavior and that the concept is able to simulate wing rock. / Ph. D.

We tackle the problem of identifying the structural dynamics of the twin tails of the F-15 fighter plane. The objective is to first investigate and identify the different possible attractors that coexist for the same operating parameters. Second is to develop a model that simulates the experimentally determined dynamics. Third is to suppress the high-amplitude vibrations of the tails due to either principal parametric or external excitations. To understand the dynamical characteristics of the twin-tails, the model is excited parametrically. For the same excitation amplitude and frequency, five different responses are observed depending on the initial conditions. The coexisting five responses are the result of the nonlinearities. After the experimental identification of the system, we develop a model to capture the dynamics realized in the experiment. We devise a nonlinear control law based on cubic velocity feedback to suppress the response of the model to a principal parametric excitation. The performance of the control law is studied by comparing the open- and closed-loop responses of the system. Furthermore, we conduct experiments to verify the theoretical analysis. The theoretical and experimental findings indicate that the control law not only leads to effective vibration suppression, but also to effective bifurcation control. We investigate the design of a neural-network-based adaptive control system for active vibration suppression of the model when subjected to a parametric excitation. First, an emulator neural network was trained to represent the structure and thus used to predict the future responses of the model. Second, a neurocontroller is developed to determine the necessary control action. The computer-simulation studies show great promise for artificial neural networks to control the model vibrations caused by parametric excitations. We investigate the use of four different control strategies to suppress high-amplitude responses of the F-15 fighter to a primary resonance excitation. The control strategies are linear velocity feedback, nonlinear velocity feedback, positive position feedback, and saturation-based control. For each case, we conduct bifurcation analyses for the open- and closed-loop responses of the system and investigate theoretically the performance of the different control strategies. We also calculate the instantaneous power requirements of each control law. The experimental results agree with the theoretical findings. / Ph. D.