Inverse dynamics of underactuated flexible mechanical systems governed by quasi-linear hyperbolic partial differential equations

Diese Arbeit befasst sich mit der inversen Dynamik unteraktuierter, flexibler, mechanischer Systeme, welche durch quasi-lineare hyperbolische partielle Differentialgleichungen beschrieben werden können. Diese Gleichungnen, sind zeitlich veränderlichen Dirichlet-Randbedingungen unterworfen, welche durch unbekannte, räumlich disjunkte, also nicht kollokierte Neumann-Randbedingungen erzwungen werden. Die zugrundeliegenden Gleichungen werden zunächst abstrakt hergeleitet, bevor verschiedene mechanische Systeme vorgestellt werden können, die mit der eingangs postulierten Formulierung übereinstimmen. Hierzu werden geometrisch exakte Theorien hergeleitet, welche in der Lage sind große Bewegungen schlanker Strukturen wie Seile und Balken, aber auch ganz allgemein, dreidimensionaler Festkörper zu beschreiben. In der Regel werden Anfangs-Randwertprobleme, die in der nichtlinearen Strukturdynamik auftreten, durch Anwendung einer sequentiellen Diskretisierung in Raum und Zeit gelöst. Diese Verfahren basieren für gewöhnlich auf einer räumlichen Diskretisierung mit finiten Elementen, gefolgt von einer geeigneten zeitlichen Diskretisierung, welche meist auf finiten Differenzen beruht. Ein kurzer Überblick über derartige sequentielle Integrationsverfahren für das vorliegende Anfangs-Randwertproblem wird zunächst anhand der direkten Formulierung des Problems gegeben werden. D.h. es wird zunächst das reine Neumann-Randproblem betrachtet, bevor anschließend ganz allgemein, verschiedene Möglichkeiten zur Einbindung etwaiger Dirichlet-Randbedingungen diskutiert werden. Darauf aufbauend wird das Problem der inversen Dynamik im Kontext räumlich diskreter mechanischer Systeme, welche rheonom-holonomen Servo-Bindungen unterliegen, eingeführt. Eine ausführliche Untersuchung dieser Art von gebundenen Systemen soll die grundlegenden Unterschiede zwischen Servo-Bindungen und klassischen Kontakt-Bindungen herausarbeiten. Die daraus resultierenden Folgen für die Entwicklung geeigneter numerisch stabiler Integrationsverfahren können dabei ebenfalls angesprochen werden, bevor zahlreich ausgewählte Beispiele vorgestellt werden können. Aufgrund der sehr eingeschränkten Anwendbarkeit der sequentiellen Lösung der inversen Dynamik in Raum und Zeit, wird eine eingehende Analyse des vorliegenden Anfangs-Randwertproblems unternommen. Vor allem durch die Freilegung der hyperbolischen Struktur der zugrundeliegenden partiellen Differentialgleichungen werden sich weitere Einblicke in das vorliegende Problem erhofft. Die Erforschung der daraus resultierenden Mechanismen der Wellenausbreitung in kontinuierlichen Strukturen öffnet die Tür zur Entwicklung numerisch stabiler Integrationsverfahren für die inverse Dynamik. So kann unter anderem eine Methode vorgestellt werden, die auf der Integration der partiellen Differentialgleichungen entlang charakteristischer Mannigfaltigkeiten beruht. Dies regt zu der Entwicklung neuartiger Galerkinverfahren an, die ebenfalls in dieser Arbeit vorgestellt werden können. Diese neu entwickelten Methoden können anschlie\ss end auf die Steuerung verschiedener mechanischer Systeme angewendet werden. Darüber hinaus können die neuartigen Integrationsverfahren auch auf flexible Mehrkörpersysteme übertragen werden. Angeführt seien hier beispielsweise die kooperative Steuerung eines an mehreren flexiblen Seilen aufgehängten starren Körpers oder die Steuerung des Endeffektors eines flexiblen mehrgliedrigen Schwenkarms. Ausgewählte numerische Beispiele verdeutlichen die Relevanz der hier vorgeschlagenen, in Raum und Zeit simultanen Integration des vorliegenden Anfangs-Randwertproblems

The Inertial Range of Turbulence in the Inner Heliosheath and in the Local Interstellar Medium

The governing mechanisms of magnetic field annihilation in the outer heliosphere is an intriguing topic. It is currently believed that the turbulent fluctuations pervade the inner heliosheath (IHS) and the Local Interstellar Medium (LISM). Turbulence, magnetic reconnection, or their reciprocal link may be responsible for magnetic energy conversion in the IHS.   As 1-day averaged data are typically used, the present literature mainly concerns large-scale analysis and does not describe inertial-cascade dynamics of turbulence in the IHS. Moreover, lack of spectral analysis make IHS dynamics remain critically understudied. Our group showed that 48-s MAG data from the Voyager mission are appropriate for a power spectral analysis over a frequency range of five decades, from 5e-8 Hz to 1e-2 Hz [Gallana et al., JGR 121 (2016)]. Special spectral estimation techniques are used to deal with the large amount of missing data (70%). We provide the first clear evidence of an inertial-cascade range of turbulence (spectral index is between -2 and -1.5). A spectral break at about 1e-5 Hz is found to separate the inertial range from the enegy-injection range (1/f energy decay). Instrumental noise bounds our investigation to frequencies lower than 5e-4 Hz. By considering several consecutive periods after 2009 at both V1 and V2, we show that the extension and the spectral energy decay of these two regimes may be indicators of IHS regions governed by different physical processes. We describe fluctuations’ regimes in terms of spectral energy density, anisotropy, compressibility, and statistical analysis of intermittency.   In the LISM, it was theorized that pristine interstellar turbulence may coexist with waves from the IHS, however this is still a debated topic. We observe that the fluctuating magnetic energy cascades as a power law with spectral index in the range [-1.35, -1.65] in the whole range of frequencies unaffected by noise. No spectral break is observed, nor decaying turbulence

A simultaneous space-time discretization approach to the inverse dynamics of geometrically exact strings

A continuous space-time Galerkin method is newly proposed for the numerical solution of inverse dynamics problems. The proposed space-time finite element method is combined with servo-constraints to partially prescribe the motion of the underlying mechanical system. The new approach to the feedforward control of infinite-dimensional mechanical systems is motivated by the classical method of characteristics. In particular, it is shown that the simultaneous space-time discretization is much better suited to solve the inverse dynamics problem than the semi-discretization approach commonly applied in structural dynamics. Representative numerical examples dealing with elastic strings undergoing large deformations demonstrate the capabilities of the newly devised space-time finite element method

A knee brace design to reduce the energy consumption of walking

Thesis (S.B.)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, 2005.Includes bibliographical references (leaves 51-52).Recent research into the mechanics of walking indicates that a quasi passive wearable device could be created which dramatically reduces the metabolic energy used in walking especially when the wearer is carrying additional torso weight. Target population groups include military personnel who must carry heavy battle packs and body armor, hikers, letter carriers, and the quasi disabled. This latter group includes a significant fraction of the elderly who have reduced leg strength and/or higher weight torsos. The device is called PUUMA, an acronym for Personal Unpowered Universal Mobility Assistant. Though walking has been studied extensively, there appears to be a limited understanding of the interplay between the kinetic and potential energy of the torso when driven by legs that can store and release energy. This thesis introduces a simplified model which enables simulation of the entire walking process including the epoch following heel strike. One simulation goal was to explore the knee spring properties which enable lossless walking. Simulations show that there are two knee spring configurations which allow for lossless walking. It is also shown that the percentage of kinetic energy transferred to a knee spring can be a significant fraction of the torso kinetic energy.(cont.) PIJUMA's basic idea is the incorporation of torsion springs at the knee joints which absorb torso kinetic energy following heel strike and then release that stored energy later in the step. An application of the capstan effect is introduced which enables a practical implementation of two knee spring configurations. In particular, the design allows the thigh and shank to be dynamically coupled to a microprocessor controlled knee spring thereby allowing both unimpeded leg swing and kinetic energy transfer to the knee spring. Another use of the capstan effect is introduced which allows for a microprocessor controlled brake that can freeze the knee at its maximum torsion and then release it later in the walking cycle. A design is shown which embodies the architectural ideas created. Several of the key components were designed, prototyped and tested.by Matthew R. Carvey.S.B

Coupled Kinetic-Fluid Simulations of Ganymede's Magnetosphere and Hybrid Parallelization of the Magnetohydrodynamics Model

The largest moon in the solar system, Ganymede, is the only moon known to possess a strong intrinsic magnetic field. The interaction between the Jovian plasma and Ganymede's magnetic field creates a mini-magnetosphere with periodically varying upstream conditions, which creates a perfect laboratory in nature for studying magnetic reconnection and magnetospheric physics. Using the latest version of Space Weather Modeling Framework (SWMF), we study the upstream plasma interactions and dynamics in this subsonic, sub-Alfvénic system. We have developed a coupled fluid-kinetic Hall Magnetohydrodynamics with embedded Particle-in-Cell (MHD-EPIC) model for Ganymede's magnetosphere, with a self-consistently coupled resistive body representing the electrical properties of the moon's interior, improved inner boundary conditions, and high resolution charge and energy conserved PIC scheme. I reimplemented the boundary condition setup in SWMF for more versatile control and functionalities, and developed a new user module for Ganymede's simulation. Results from the models are validated with Galileo magnetometer data of all close encounters and compared with Plasma Subsystem (PLS) data. The energy fluxes associated with the upstream reconnection in the model is estimated to be about 10^-7 W/cm^2, which accounts for about 40% to the total peak auroral emissions observed by the Hubble Space Telescope. We find that under steady upstream conditions, magnetopause reconnection in our fluid-kinetic simulations occurs in a non-steady manner. Flux ropes with length of Ganymede's radius form on the magnetopause at a rate about 3/minute and create spatiotemporal variations in plasma and field properties. Upon reaching proper grid resolutions, the MHD-EPIC model can resolve both electron and ion kinetics at the magnetopause and show localized crescent shape distribution in both ion and electron phase space, non-gyrotropic and non-isotropic behavior inside the diffusion regions. The estimated global reconnection rate from the models is about 80 kV with 60% efficiency. There is weak evidence of $sim 1$ minute periodicity in the temporal variations of the reconnection rate due to the dynamic reconnection process. The requirement of high fidelity results promotes the development of hybrid parallelized numerical model strategy and faster data processing techniques. The state-of-the-art finite volume/difference MHD code Block Adaptive Tree Solarwind Roe Upwind Scheme (BATS-R-US) was originally designed with pure MPI parallelization. The maximum problem size achievable was limited by the storage requirements of the block tree structure. To mitigate this limitation, we have added multithreaded OpenMP parallelization to the previous pure MPI implementation. We opt to use a coarse-grained approach by making the loops over grid blocks multithreaded and have succeeded in making BATS-R-US an efficient hybrid parallel code with modest changes in the source code while preserving the performance. Good weak scalings up to 50,0000 and 25,0000 of cores are achieved for the explicit and implicit time stepping schemes, respectively. This parallelization strategy greatly extends the possible simulation scale by an order of magnitude, and paves the way for future GPU-portable code development. To improve visualization and data processing, I have developed a whole new data processing workflow with the Julia programming language for efficient data analysis and visualization. As a summary, 1. I build a single fluid Hall MHD-EPIC model of Ganymede's magnetosphere; 2. I did detailed analysis of the upstream reconnection; 3. I developed a MPI+OpenMP parallel MHD model with BATSRUS; 4. I wrote a package for data analysis and visualization.PHDClimate and Space Sciences and EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/163032/1/hyzhou_1.pd

Flat systems, equivalence and trajectory generation

Flat systems, an important subclass of nonlinear control systems introduced via differential-algebraic methods, are defined in a differential geometric framework. We utilize the infinite dimensional geometry developed by Vinogradov and coworkers: a control system is a diffiety, or more precisely, an ordinary diffiety, i.e. a smooth infinite-dimensional manifold equipped with a privileged vector field. After recalling the definition of a Lie-Backlund mapping, we say that two systems are equivalent if they are related by a Lie-Backlund isomorphism. Flat systems are those systems which are equivalent to a controllable linear one. The interest of such an abstract setting relies mainly on the fact that the above system equivalence is interpreted in terms of endogenous dynamic feedback. The presentation is as elementary as possible and illustrated by the VTOL aircraft

Sub-Seafloor Characterization and Stability of Submarine Slope Sediments using Dynamic and Static Piezocone Penetrometers

The exploitation of natural resources (e.g. oil, gas and gas hydrate), development of cable and pipeline routes, land reclamation activities in coastal and nearshore environments, and protection of coastal communities require a sound knowledge of the geotechnical properties and characteristics of sub-seafloor soils, especially when dealing with the stability of submarine slopes. In situ dynamic and static piezocone penetration tests (CPTU) are powerful cost- and time-efficient techniques for measuring geotechnical and stratigraphical properties. The penetration process of a dynamic-CPTU device results in a non-linear decreasing penetration rate from an initial penetration rate of up to 10 m/s. In contrast, static-CPTU cones penetrate the soil with a constant penetration rate of usually 2 cm/s. This difference, known as strain-rate effect, causes elevated dynamic-CPTU parameters for fine-grained soils. In this thesis, different strain-rate correction solutions are compared (e.g. logaritmic, inverse sin-hyperbolic, power-law, velocity ratio) in order to find the optimal solution and improve it by considering a modified non-dimensional velocity ratio. Visual observations and a simple statistical methods (e.g. two-sample Kolmogorov-Smirnov test), resulting in the coefficients of determination up to 0.82, demonstrate that the modified inverse sin-hyperbolic equation is best suited to correct for the strain-rate effect in dynamic-CPTU tests. A multi-disciplinary approach described by the combination of bathymetrical, geophysical, sedimentological and geotechnical datasets are used to develop an area-wide sub-seafloor model near the Nice international airport (southeastern France). The sub-seafloor model consists of a geometrical/sedimentological model using bathymetrical map, several core profiles and geophysical chirp transects, and a geotechnical/strength model using results from vane shear and fall cone penetration experiments, high number of dynamic- and several static-CPTU tests. Based on the sub-seafloor model, the role of free gas in the soil and different failure geometries are discussed and demonstrated using 2D numerical slope stability assessments

Advanced Space Plasma Simulations Using a High-Order Accurate Method and the Magnetohydrodynamics with Embedded Particle-in-Cell Model

Global magnetohydrodynamics (MHD) models have been important tools for space physics research in recent decades. In order to improve the numerical accuracy and the physics capability of an MHD mode, a fifth-order accurate finite difference scheme for hyperbolic equations on block-adaptive curvilinear grids is developed to improve the accuracy of the Michigan MHD model BATS-R-US. To model kinetic phenomena, like magnetic reconnection, BATS-R-US is two-way coupled with a particle-in-cell (PIC) code iPIC3D to incorporate kinetic physics into a global model. The two-way coupled model is called magnetohydrodynamics with embedded particle-in-cell (MHD-EPIC) model. This dissertation research focuses on the development of the fifth-order scheme and the applications of the MHD-EPIC model. The fifth-order finite-difference scheme constructs the face fluxes with a monotonicity preserving limiter MP5, and achieves high-order spatial derivatives by a flux correction step. This scheme is generalized to curvilinear grids with a free-streaming discretization. For the locally refined mesh, high-order accuracy is also achieved by careful interpolation of ghost cells near the grid resolution changes. Numerical tests are presented to demonstrate the accuracy and robustness of the algorithm. The MHD-EPIC model is applied to study Earth's dayside magnetopause reconnection and Mercury's magnetotail reconnection. From the Earth simulation, the generation and evolution of flux transfer events (FTEs) are studied. It is found the magnetic field signature of FTEs at their early formation stage is similar to a `crater FTE'. After the FTE core field grows to a significant value, it becomes an FTE with typical flux rope structure. Kinetic phenomena, such as the crescent electron phase space distribution, the Larmor electric field, and the lower hybrid drift instability are identified from the global simulation. The Mercury simulations apply MHD-EPIC to study the magnetotail reconnection. The properties of the magnetotail flux ropes agree well with the MESSENGER observations. The reconnection dawn-dusk asymmetry also arises from the simulations; the reconnection jets are stronger on the dawn side, which agrees with the MESSENGER observations.PHDAtmospheric, Oceanic & Space ScienceUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/138711/1/yuxichen_1.pd

Towards a solution of the closure problem for convective atmospheric boundary-layer turbulence

We consider the closure problem for turbulence in the dry convective atmospheric boundary layer (CBL). Transport in the CBL is carried by small scale eddies near the surface and large plumes in the well mixed middle part up to the inversion that separates the CBL from the stably stratified air above. An analytically tractable model based on a multivariate Delta-PDF approach is developed. It is an extension of the model of Gryanik and Hartmann [1] (GH02) that additionally includes a term for background turbulence. Thus an exact solution is derived and all higher order moments (HOMs) are explained by second order moments, correlation coefficients and the skewness. The solution provides a proof of the extended universality hypothesis of GH02 which is the refinement of the Millionshchikov hypothesis (quasi- normality of FOM). This refined hypothesis states that CBL turbulence can be considered as result of a linear interpolation between the Gaussian and the very skewed turbulence regimes. Although the extended universality hypothesis was confirmed by results of field measurements, LES and DNS simulations (see e.g. [2-4]), several questions remained unexplained. These are now answered by the new model including the reasons of the universality of the functional form of the HOMs, the significant scatter of the values of the coefficients and the source of the magic of the linear interpolation. Finally, the closures 61 predicted by the model are tested against measurements and LES data. Some of the other issues of CBL turbulence, e.g. familiar kurtosis-skewness relationships and relation of area coverage parameters of plumes (so called filling factors) with HOM will be discussed also

Multibody Systems with Flexible Elements

Multibody systems with flexible elements represent mechanical systems composed of many elastic (and rigid) interconnected bodies meeting a functional, technical, or biological assembly. The displacement of each or some of the elements of the system is generally large and cannot be neglected in mechanical modeling. The study of these multibody systems covers many industrial fields, but also has applications in medicine, sports, and art. The systematic treatment of the dynamic behavior of interconnected bodies has led to an important number of formalisms for multibody systems within mechanics. At present, this formalism is used in large engineering fields, especially robotics and vehicle dynamics. The formalism of multibody systems offers a means of algorithmic analysis, assisted by computers, and a means of simulating and optimizing an arbitrary movement of a possibly high number of elastic bodies in the connection. The domain where researchers apply these methods are robotics, simulations of the dynamics of vehicles, biomechanics, aerospace engineering (helicopters and the behavior of cars in a gravitational field), internal combustion engines, gearboxes, transmissions, mechanisms, the cellulose industry, simulation of particle behavior (granulated particles and molecules), dynamic simulation, military applications, computer games, medicine, and rehabilitation