A moment-equation-copula-closure method for nonlinear vibrational systems subjected to correlated noise

We develop a moment equation closure minimization method for the inexpensive approximation of the steady state statistical structure of nonlinear systems whose potential functions have bimodal shapes and which are subjected to correlated excitations. Our approach relies on the derivation of moment equations that describe the dynamics governing the two-time statistics. These are combined with a non-Gaussian pdf representation for the joint response-excitation statistics that has i) single time statistical structure consistent with the analytical solutions of the Fokker-Planck equation, and ii) two-time statistical structure with Gaussian characteristics. Through the adopted pdf representation, we derive a closure scheme which we formulate in terms of a consistency condition involving the second order statistics of the response, the closure constraint. A similar condition, the dynamics constraint, is also derived directly through the moment equations. These two constraints are formulated as a low-dimensional minimization problem with respect to unknown parameters of the representation, the minimization of which imposes an interplay between the dynamics and the adopted closure. The new method allows for the semi-analytical representation of the two-time, non-Gaussian structure of the solution as well as the joint statistical structure of the response-excitation over different time instants. We demonstrate its effectiveness through the application on bistable nonlinear single-degree-of-freedom energy harvesters with mechanical and electromagnetic damping, and we show that the results compare favorably with direct Monte-Carlo Simulations

Large Angle Transient Dynamics (LATDYN) user's manual

A computer code for modeling the large angle transient dynamics (LATDYN) of structures was developed to investigate techniques for analyzing flexible deformation and control/structure interaction problems associated with large angular motions of spacecraft. This type of analysis is beyond the routine capability of conventional analytical tools without simplifying assumptions. In some instances, the motion may be sufficiently slow and the spacecraft (or component) sufficiently rigid to simplify analyses of dynamics and controls by making pseudo-static and/or rigid body assumptions. The LATDYN introduces a new approach to the problem by combining finite element structural analysis, multi-body dynamics, and control system analysis in a single tool. It includes a type of finite element that can deform and rotate through large angles at the same time, and which can be connected to other finite elements either rigidly or through mechanical joints. The LATDYN also provides symbolic capabilities for modeling control systems which are interfaced directly with the finite element structural model. Thus, the nonlinear equations representing the structural model are integrated along with the equations representing sensors, processing, and controls as a coupled system

A Metastable Modular Structure Approach for Shape Morphing, Property Tuning and Wave Propagation Tailoring

The emerging concept of reconfigurable mechanical metamaterials has received increasing attention for realizing future advanced multifunctional adaptive structural systems partially due to their advantages over conventional bulk materials that are beneficial and desirable in many engineering applications. However, some of the critical challenges remain unaddressed before the concept can effectively and efficiently achieve real-world impacts. For instance, in the state-of-art, modules of mechanical metamaterials only reconfigure collectively to achieve global topology adaptation. As a result, the structure merely exhibits limited number of configurations that are discretely different from each other, which greatly undermines the benefits and impact of the reconfiguration effect. Additionally, most of the metamaterials investigations are focusing on the “materials” characteristics assuming infinite domain without considering the “structure” aspect of the systems. The effects of having finite domains and boundary conditions will generate new research issues and phenomena that are critical to real-world systems. To address the challenges and fundamentally advance the state of the art of multifunctional adaptive structures, this dissertation seeks to create a paradigm shift by exploiting and harnessing metastable modular mechanics and dynamics. Through developing new analysis and synthesis methodologies and conducting rigorous analytical, numerical, and experimental investigations, this research creates a new class of reconfigurable metastructure that can achieve mechanical property and topology adaptation as well as adaptive non-reciprocal vibration/wave transmission. The intellectual merit of this dissertation lies in introducing metastable modules that can be synergistically assembled and individually tuned to realize near continuous topology and mechanical property adaptation and elucidating the intricate nonlinear dynamics afforded by the metastructure. This research reveals different kinds of nonlinear instabilities that are able to facilitate the onset of supratransmission, a bandgap transmission phenomenon pertained to nonlinear periodic metastructure. In addition, utilizing this novel phenomenon, supratransmission, together with inherent spatial asymmetry of strategically configured constituents, the proposed metastructure is shown to be able to facilitate unprecedented broadband non-reciprocal vibration and wave transmission and on-demand adaptation. Since the proposed approach depends primarily on scale-independent principles, the broader impact of this dissertation is that the proposed metastructure could foster a new generation of reconfigurable structural and material systems with unprecedented adaptation and unconventional vibration control and wave transmission characteristics that are applicable to vastly different length scales for a wide spectrum of applications.PHDMechanical EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/147525/1/wuzhen_1.pd

Large Angle Transient Dynamics (LATDYN) demonstration problem manual

LATDYN is a computer code for modeling the Large Angle Transient Dynamics of structures. The objective in developing the code was to investigate new techniques for analyzing flexible deformation and control/structure interaction problems associated with large angular motions of spacecraft. Such motions may consist of pointing the entire spacecraft or articulation of individual components, events which occur frequently during construction, operation, and maintenance of large spacecraft. This type of analysis is beyond the routine capability of conventional analytical tools without simplifying assumptions. In some instances, the motion may be sufficiently slow and the spacecraft (or component) sufficiently rigid to simplify the analysis of dynamics and controls by making pseudo-static and/or rigid body assumptions. LATDYN introduces a new approach to the problem by combining finite element structural analysis, multi-body dynamics, and control system analysis in a single tool. It includes a new type of finite element that can deform and rotate through large angles at the same time, and which can be connected to other finite elements either rigidly or through mechanical joints. LATDYN also provides symbolic capabilities for modeling control systems which are interfaced directly with the finite element structural model. Thus, the nonlinear equations representing the structural model are integrated along with the equations representing sensors, processing, and controls as a coupled system

Towards High-Quality Black-Box Chemical Reaction Rates with System-Specific Potential Energy Surfaces

The calculation of highly-accurate reaction rate constants (k(T)) is one of the central topics in theoretical chemical kinetics. Two approaches for doing this are dominant in literature: application of heuristic corrections of the transition state theory (TST) and wave packet propagation with the aim to represent exact quantum mechanical dynamics. While the first approach is easy to handle but suffers from intrinsic approximations and limited accuracy, the second approach enables convergence towards the exact result, but at the expense of a complex handling and massive costs. This limits its application to a small circle of highly-specialized theoreticians. A new method that might be able to bridge the gap between easy application and convergence towards the exact result is the ring polymer molecular dynamics (RPMD) method. It is based on the isomorphism between quantum statistical mechanics and classical statistical mechanics of a fictitious ring polymer. With this, configurational state sums and free energy surfaces can be obtained from probabilistic samplings of the system's accessible phase space with classical MD of ring polymers. Based on these free energy surfaces reaction rate constants can be obtained that converge towards the results of wave packet propagations, if the size of the ring polymer is adequate. In order to conduct RPMD calculations, a sufficiently accurate representation of the thermally accessible potential energy surface (PES) of the system on which the ring polymers are propagated is needed. In principle, this surface could be represented by ad hoc calculations of energies and gradients based on quantum chemical methods like density functional theory (DFT) or second order Moller-Plesset perturbation theory (MP2). However, since many millions of single gradient calculations are needed to converge a free energy surface and the associated k(T) value, this approach is impractical. Instead, analytical representations of PESs that are fitted to DFT or MP2 results are commonly used. The parametrization of these representations is quite demanding, though, thus being a task for experts. The present thesis deals with new methods for the automated parametrization of analytical PES representations of reactive systems and the successive k(T) calculations based on RPMD. These representations are built on a combination of the quantum mechanical derived force field (QMDFF) method by \Grimme and the empirical valence bond method (EVB) by Warshel, being plugged together recently by Hartke and Grimme (EVB-QMDFF). In line with this thesis a crucial improvement of this combination of methods was done, complementing it with newly developed EVB concepts. For practical usage a new program package was developed, which enables the automated generation of an EVB-QMDFF-PES representation and calculations of RPMD-free-energy surfaces, recrossing corrections as well as k(T) values and Arrhenius parameters for comparison with experimental data, based on the preoptimized reaction path of an arbitrary thermal ground state system. The abilities of the new methods and the associated implementation were thoroughly benchmarked in different kinds of applications. These are calculations of k(T) values and Arrhenius parameters of arbitrary systems from a reaction data base and their comparison to literature values, theoretical molecular force experiments with quantitative investigations of force-dependent reactivities for different systems, a thorough study of urethane synthesis being part of our cooperation with Covestro AG and finally a combination of calculated rate constants of several elementary reactions for describing the dynamics of larger systems based on the kinetic Monte Carlo (KMC) method

A Schr\"odinger Equation for Evolutionary Dynamics

We establish an analogy between the Fokker-Planck equation describing evolutionary landscape dynamics and the Schr\"{o}dinger equation which characterizes quantum mechanical particles, showing how a population with multiple genetic traits evolves analogously to a wavefunction under a multi-dimensional energy potential in imaginary time. Furthermore, we discover within this analogy that the stationary population distribution on the landscape corresponds exactly to the ground-state wavefunction. This mathematical equivalence grants entry to a wide range of analytical tools developed by the quantum mechanics community, such as the Rayleigh-Ritz variational method and the Rayleigh-Schr\"{o}dinger perturbation theory, allowing us to not only make reasonable quantitative assessments but also explore fundamental biological inquiries. We demonstrate the effectiveness of these tools by estimating the population success on landscapes where precise answers are elusive, and unveiling the ecological consequences of stress-induced mutagenesis -- a prevalent evolutionary mechanism in pathogenic and neoplastic systems. We show that, even in a unchanging environment, a sharp mutational burst resulting from stress can always be advantageous, while a gradual increase only enhances population size when the number of relevant evolving traits is limited. Our interdisciplinary approach offers novel insights, opening up new avenues for deeper understanding and predictive capability regarding the complex dynamics of evolving populations

A Schrödinger Equation for Evolutionary Dynamics

We establish an analogy between the Fokker–Planck equation describing evolutionary landscape dynamics and the Schrödinger equation which characterizes quantum mechanical particles, showing that a population with multiple genetic traits evolves analogously to a wavefunction under a multi-dimensional energy potential in imaginary time. Furthermore, we discover within this analogy that the stationary population distribution on the landscape corresponds exactly to the ground-state wavefunction. This mathematical equivalence grants entry to a wide range of analytical tools developed by the quantum mechanics community, such as the Rayleigh–Ritz variational method and the Rayleigh–Schrödinger perturbation theory, allowing us not only the conduct of reasonable quantitative assessments but also exploration of fundamental biological inquiries. We demonstrate the effectiveness of these tools by estimating the population success on landscapes where precise answers are elusive, and unveiling the ecological consequences of stress-induced mutagenesis—a prevalent evolutionary mechanism in pathogenic and neoplastic systems. We show that, even in an unchanging environment, a sharp mutational burst resulting from stress can always be advantageous, while a gradual increase only enhances population size when the number of relevant evolving traits is limited. Our interdisciplinary approach offers novel insights, opening up new avenues for deeper understanding and predictive capability regarding the complex dynamics of evolving populations

Pulsed Laser Cooling for Cavity-Optomechanical Resonators

A pulsed cooling scheme for optomechanical systems is presented that is capable of cooling at much faster rates, shorter overall cooling times, and for a wider set of experimental scenarios than is possible by conventional methods. The proposed scheme can be implemented for both strongly and weakly coupled optomechanical systems in both weakly and highly dissipative cavities. We study analytically its underlying working mechanism, which is based on interferometric control of optomechanical interactions, and we demonstrate its efficiency with pulse sequences that are obtained by using methods from optimal control. The short time in which our scheme approaches the optomechanical ground state allows for a significant relaxation of current experimental constraints. Finally, the framework presented here can be used to create a rich variety of optomechanical interactions and hence offers a novel, readily available toolbox for fast optomechanical quantum control.Comment: 6 pages, 4 figure

Limit cycle behavior of smart fluid dampers under closed loop control

Semiactive vibration dampers offer an attractive compromise between the simplicity and fail safety of passive devices, and the weight, cost, and complexity of fully active systems. In addition, the dissipative nature of semiactive dampers ensures they always remain stable under closed loop control, unlike their fully active counterparts, However undesirable limit cycle behavior remains a possibility, which is not always property considered during the controller design. Smart fluids provide an elegant means to produce semiactive damping, since their resistance to flow can be directly controlled by the application of an electric or magnetic field. However the nonlinear behavior of smart fluid dampers makes it difficult to design effective controllers, and so a wide variety of control strategies has been proposed in the literature. In general, this work has overlooked the possibility of undesirable limit cycle behavior under closed loop conditions. The aim of the present study is to demonstrate how the experimentally observed limit cycle behavior of smart dampers can be predicted and explained by appropriate nonlinear models. The study is based upon a previously developed feedback control strategy, but the techniques described are relevant to other forms of smart damper control

Laser Theory for Optomechanics: Limit Cycles in the Quantum Regime

Optomechanical systems can exhibit self-sustained limit cycles where the quantum state of the mechanical resonator possesses nonclassical characteristics such as a strongly negative Wigner density, as was shown recently in a numerical study by Qian et al. [Physical Review Letters, 109, 253601 (2012)]. Here we derive a Fokker-Planck equation describing mechanical limit cycles in the quantum regime which correctly reproduces the numerically observed nonclassical features. The derivation starts from the standard optomechanical master equation, and is based on techniques borrowed from the laser theory due to Haake's and Lewenstein. We compare our analytical model with numerical solutions of the master equation based on Monte-Carlo simulations, and find very good agreement over a wide and so far unexplored regime of system parameters. As one main conclusion, we predict negative Wigner functions to be observable even for surprisingly classical parameters, i.e. outside the single-photon strong coupling regime, for strong cavity drive, and rather large limit cycle amplitudes. The general approach taken here provides a natural starting point for further studies of quantum effects in optomechanics.Comment: 17 pages, 7 figure