Browsing by Author "Gavin, Henri"

Linear vibration isolation systems, used to reduce the transmissibility of vertical vibration, requires a vertical static displacement that increases with the square of the natural period of the isolation system. The static displacement of a vertical isolation system with a one second natural period is 0.25 m. The nonlinear stiffness of buckled beams loaded in the transverse direction can be designed to reduce the vertical static displacement requirement of vertical systems. This study presents an analysis of large displacement mechanics of slender beams that buckle against a constraint, and extracts the transverse constraint force via the Lagrange multiplier enforcing the constraint. The constraint prescribes a maximum allowable lateral displacement along the length of the beam and a specified longitudinal displacement at the mid span of the beam. No small curvature assumption is involved. Lateral and longitudinal displacements are parameterized in terms of Fourier coefficients. Coefficient values for constrained equilibria are found by minimizing the bending strain energy such that lateral and longitudinal constraints are satisfied. Because the full expression for curvature is used, this is a nonlinear constrained optimization problem.

Edge and mid-point horizontal constraint positions are varied to gain a better understanding of the constraint forces at each position. This modeling approach is then used to design a system of post-buckled leaf springs in order to meet vibration isolation requirements without over-stressing the springs. This process is discussed in detail along with the process and challenges associated with the physical model. Theoretical predictions are compared to laboratory scale measurements. Experimental results from the physical model are compared to the theoretical and numerical simulation results. The potential for rocking responses of the vertical isolation system are quantified via the modeling of the nonlinear dynamics of a platform supported by a system of springs and carrying a mass concentrated above the platform.

In this dissertation, we develop a novel methodology for characterizing and simulating nonstationary, full-field, stochastic turbulent wind fields.

In this new method, nonstationarity is characterized and modeled via temporal coherence, which is quantified in the discrete frequency domain by probability distributions of the differences in phase between adjacent Fourier components.

The empirical distributions of the phase differences can also be extracted from measured data, and the resulting temporal coherence parameters can quantify the occurrence of nonstationarity in empirical wind data.

This dissertation (1) implements temporal coherence in a desktop turbulence simulator, (2) calibrates empirical temporal coherence models for four wind datasets, and (3) quantifies the increase in lifetime wind turbine loads caused by temporal coherence.

The four wind datasets were intentionally chosen from locations around the world so that they had significantly different ambient atmospheric conditions.

The prevalence of temporal coherence and its relationship to other standard wind parameters was modeled through empirical joint distributions (EJDs), which involved fitting marginal distributions and calculating correlations.

EJDs have the added benefit of being able to generate samples of wind parameters that reflect the characteristics of a particular site.

Lastly, to characterize the effect of temporal coherence on design loads, we created four models in the open-source wind turbine simulator FAST based on the \windpact turbines, fit response surfaces to them, and used the response surfaces to calculate lifetime turbine responses to wind fields simulated with and without temporal coherence.

The training data for the response surfaces was generated from exhaustive FAST simulations that were run on the high-performance computing (HPC) facilities at the National Renewable Energy Laboratory.

This process was repeated for wind field parameters drawn from the empirical distributions and for wind samples drawn using the recommended procedure in the wind turbine design standard \iec.

The effect of temporal coherence was calculated as a percent increase in the lifetime load over the base value with no temporal coherence.

This thesis proposes an integrated procedure for identifying the nominal models of the deterministic part and the stochastic part of a system, as well as their model error bounds in different uncertainty structures (e.g. $\mathcal{H}_2$-norm and $\mathcal{H}_{\infty}$-norm) based on the measurement data. In particular, the deterministic part of a system is firstly identified by closed-loop instrumental variable method in which a known external signal sequence uncorrelated with the system noises is injected in the control input for the identifiability of the system in closed loop. By exploiting the second-order statistics of the noise-driven output components, the stochastic part of a system is identified by the improved subspace approach in which a new and straightforward linear-matrix-inequality-based optimization is proposed to obtain a valid model even under insufficient measurement data.

To derive an explicit model error bound on the identification model, we investigate a complete asymptotic analysis for identification of the stochastic part of the system. We first derive the asymptotically normal distributions of the empirical sample covariance and block-Hankel matrix of the outputs. Thanks to these asymptotic distributions and the perturbation analysis of singular value decomposition and discrete algebraic Riccati equation, several central limit theorems for the identified controllability matrix, observability matrix, and the state-space matrices in the associated covariance model are derived, as well as the norm bounds of Kalman gain and the innovations covariance matrix in the innovations model. By combining these asymptotic results, the explicit $\mathcal{H}_2$-norm and $\mathcal{H}_{\infty}$-norm bounds of the model error are identified with a given confidence level.

Practical applicability of the proposed combined deterministic-stochastic identification procedure is illustrated by the application to indirect adaptive control of a multi-generator wave energy harvesting system.

Over the past decade, a significant amount of research activity has been devoted to developing electromechanical systems that can convert ambient mechanical vibrations into usable electric power. Such systems, referred to as vibratory energy harvesters, have a number of useful of applications, ranging in scale from self-powered wireless sensors for structural health monitoring in bridges and buildings to energy harvesting from ocean waves. One of the most challenging aspects of this technology concerns the efficient extraction and transmission of power from transducer to storage. Maximizing the rate of power extraction from vibratory energy harvesters is further complicated by the stochastic nature of the disturbance. The primary purpose of this dissertation is to develop feedback control algorithms which optimize the average power generated from stochastically-excited vibratory energy harvesters.

This dissertation will illustrate the performance of various controllers using two vibratory energy harvesting systems: an electromagnetic transducer embedded within a flexible structure, and a piezoelectric bimorph cantilever beam. Compared with piezoelectric systems, large-scale electromagnetic systems have received much less attention in the literature despite their ability to generate power at the watt--kilowatt scale. Motivated by this observation, the first part of this dissertation focuses on developing an experimentally validated predictive model of an actively controlled electromagnetic transducer. Following this experimental analysis, linear-quadratic-Gaussian control theory is used to compute unconstrained state feedback controllers for two ideal vibratory energy harvesting systems. This theory is then augmented to account for competing objectives, nonlinearities in the harvester dynamics, and non-quadratic transmission loss models in the electronics.

In many vibratory energy harvesting applications, employing a bi-directional power electronic drive to actively control the harvester is infeasible due to the high levels of parasitic power required to operate the drive. For the case where a single-directional drive is used, a constraint on the directionality of power-flow is imposed on the system, which necessitates the use of nonlinear feedback. As such, a sub-optimal controller for power-flow-constrained vibratory energy harvesters is presented, which is analytically guaranteed to outperform the optimal static admittance controller. Finally, the last section of this dissertation explores a numerical approach to compute optimal discretized control manifolds for systems with power-flow constraints. Unlike the sub-optimal nonlinear controller, the numerical controller satisfies the necessary conditions for optimality by solving the stochastic Hamilton-Jacobi equation.

Recently multi-agent navigation robots have been gaining increasing popularity indiverse applications such as agriculture, package delivery, exploration, search and rescue due to their maneuverability and collaborativity. The control algorithm is the nucleus of such intelligent and autonomous robots performing tasks. In real-world applications, the robots are subjected to nonlinear dynamics, external disturbances, actuator saturation/dynamics, and modeling, estimation, and measurement errors. Furthermore, teams of robots are needed to perform collaboratively while ensuring inter-robot and robot-obstacle collision avoidance.

To address these needs, a novel control paradigm has been developed for multiagent navigation robots that possesses safety, robustness, resilience, scalability, and computation efficiency. The control rule is fully defined by the current active subset of a superset of inequality constraints, which contrasts methods of minimizing a weighted cost function subject to stability constraints. The advantages of this method are that

• the constraints (equality, inequality, holonomic, nonholonomic, scleronomic, rheonomic, etc.) can be handled without trying to \look ahead" to a finite time horizon;• the nonlinear control actions are specified by instantly solving a linear matrix equation; • it does not involve a cost function; • it does not involve any dynamics linearization; • the control parameters are physically interpretable; • actuator saturation and actuator dynamics are readily incorporated; and • it is applicable to fully nonlinear, time-varying, and/or arbitrary-order dynamical systems; and • it can simultaneously control the position and orientation of mechanical systems in one unified step.

These features are achieved through a novel generalization of Gauss’s Principleof Least Constraint (GPLC). GPLC was originally conceived to incorporate hard equality constraints into second-order dynamical systems. The contribution of this dissertation is to define the control actions from the Lagrange multipliers associated with inequality constraints (e.g., collision avoidance constraints) and to accommodate dynamical systems of any order. Thus, the constrained equations of motion are expressed as a Karush-Kuhn-Tucker (KKT) system (a linear matrix equation), which is solved without iteration at each time step.

This constraint-based control has been applied to the navigation control of multiagent, multi-swarm systems of double integrators, fully nonlinear quadrotor drones, and nonholonomic, differential drive, wheeled mobile robots subjected to actuator saturation, actuator dynamics, and external disturbances. Two types of constraints are considered for the aforementioned three types of systems: path following and collision avoidance constraints. Each constraint can be formulated based on vector norms or vector components and can be in either equality or inequality format. Thevector-component-based collision constraints lead to a natural byproduct of resolving deadlock in navigating swarms. Furthermore, through a partition of collision avoidance constraints among colliding agents, the control architecture for the navigation swarms can be centralized or decentralized. Numerical studies on swarms of double integrators, nonlinear quadrotor drones, and nonholonomic wheeled mobile robots have demonstrated the effectiveness and efficiency of the proposed approach.

In earthquake engineering, the selection of input ground motions for nonlinear structural dynamic analysis is an important element in performance based structural design. Probabilistic seismic hazard analysis (PSHA) provides a basis for determination of ground motion characteristics by incorporating regressions on ground motion metrics from past recorded earthquakes for known seismic sources, propagation paths and local site conditions. Due to aleatoric variability in the processes of fault rupture, seismic wave propagation, and local site response; and, the epistemic uncertainty in models of these phenomena due in part to limited data, there exist uncertainties in seismic hazard assesments. This uncertainty should be interpreted carefully before the resulting ground motion predictions are adopted for dynamic analysis and design of structures. This thesis examines the sensitivity of algorithmic parameters and the choice of earthquake magnitude recurrence relations on the conditional ground motion characteristics. For this purpose, new correlation coefficients between pseudo-spectral acceleration (PSA), peak ground velocity (PGV), and cumulative absolute velocity (CAV) are derived using data from the PEER-NGA earthquake database. Finally, the sensitivity of the conditional mean spectrum to earthquake recurrence models is investigated. It is concluded that the choice of the Gutenberg-Richter or Characteristic Magnitude model can significantly affect the conditional mean spectra.

The primary focus of this thesis is on the modeling and control of a high performance vibration isolation platform used in the protection of mission-critical equipment. The equipment isolation system (EIS) consists of a pair of frames—the bottom-frame contains four concave-up bowls, the top-frame contains four concave-down bowls—between which four rigid steel balls can roll, effectively decoupling the top-frame (and equipment) from the base excitation. In order for the four balls to remain in contact with the dishes during times of large angular rotation, the platform is assumed to conform to a saddle shape. Developing a high-fidelity analytical model to capture the nonlinear nature of the EIS requires (i) enforcement of kinematic rolling constraints to the translational and rotational dynamics of the platform center, (ii) precise measurements of the bowl profiles and an accompanying parameterized model, and (iii) calibrating a linear-viscous damping model to experimental free-responses. To this end, nonholonomic equations of state are found from the d'Alembert-Lagrange principle which incorporates velocity constraints via Lagrange multipliers. The bowl profiles are measured using high-resolution photogrammetry and establish the system's potential energy function. Bowl shape parameters and damping parameters are optimized using a Levenberg-Marquardt least squares algorithm; the damping rate is shown to depend upon the mass of the isolated object.

To demonstrate the capabilities of the proposed model, numerical simulations are performed and compared to laboratory results. For an undamped free-response, energy is conserved while, for the damped free-response, non-zero displacement equilibria exist. It is shown that for the non-quadratic potential energy function of this particular system, free response trajectories are strongly sensitive to slight perturbations in initial conditions and that the system is, in fact, chaotic.

In most EISs, additional damping suppresses isolator displacements but may significantly increase the accelerations sustained by the isolated object. To demonstrate the potential benefits of semi-active vibration suppression, control theory is used to compare existing techniques to the optimal solution. This thesis presents a method to optimize control trajectories for systems subject to state and control inequality constraints decentralized in terms of each control. Optimal Lagrange multipliers enforcing the inequality constraints may be found at any time through Pontryagin's minimum principle. In so doing, the set of differential algebraic Euler-Lagrange equations are transformed into a nonlinear two-point boundary-value problem for states and costates whose solution meets the necessary conditions for optimality. The optimal performance of inequality constrained control systems is calculable allowing for comparison to previous, sub-optimal solutions. The method is applied to the control of time-lagged damping forces in a linearized EIS subjected to constraints imposed by the physical implementation of a particular controllable damper. Two cases are investigated: a single-degree-of-freedom isolator with one actuator and a three-degree-of-freedom system actuated by three devices. Potential improvements in terms of acceleration response are shown to be achievable, for which parameterized control laws are proposed over varying input excitation frequencies.

A new model for Rolling Isolation Systems(RIS) is developed using Gauss's Principle of Least Constraint (GPLC). Gauss Principle of Least Constraint is a versatile method that can be used to model systems with constrained dynamics and is agnostic to the type of constraints that need to be enforced. In GPLC models, constraints are enforced in the acceleration level regardless of their classification and the equations of constrained motion are then presented in KKT form. Since, enforcing constraints in differentiated form may lead to numerical integration issues, a novel method, Direct State Correction has been proposed and implemented. Direct State Correction, unlike other constraint stabilization techniques, ensures that the constraints are exactly satisfied at each time step by correcting the states of the system before computing each state derivative. The widely used method of Baumgarte stabilization does not stabilize constraint errors for the RIS system studied in this work and in addition it appears to provide no additional stability or accuracy when implemented along with Direct State Correction Method.

In order to optimize the RIS, a data set of real recorded building motions from the Strong Motion Virtual Data Center is put together with consideration of the type of buildings, earthquakes and locations the RIS is expected to perform in. A total of 62 tri-axial floor motions were selected from 32 buildings and 5 representative earthquakes. It is discovered buildings undergo significant vertical motion and the current isolation system is designed to only be effective for horizontal motion, and therefore a vertical isolation system is added to the current RIS.

The dynamics of the RIS are purely dependent on the dish shape and damping present in the system, so with these design variables in mind, the new RIS with vertical isolation is optimized under a probabilistic framework. The optimization uses Incremental Dynamic Analysis (IDA) and the theorem of total probability in order to arrive at an objective function that minimizes the overall rate of exceedance of a desired threshold probability. The optimization is found to be sensitive to the initial starting guess of the design variables, the input motion used to simulate the RIS and the thresholds set for evaluating the performance of the RIS. The optimal shape of the dishes is found to be a variation of a cubic profile and linearly increasing rolling resistance damping. This new design for the RIS shows significant improvement over the current system. Finally, an implementation of the proposed design of the RIS dish shape, damping and vertical isolation system is presented.

The main objectives of this research are: (i) to elaborate a unified nonlinear viscoelastic model for rubber-like materials, in finite strain, accounting for material softening under deformation, and for damage induced anisotropy, (ii) to conceive, implement and test, simple, robust and efficient frictional rolling and sliding contact algorithms, in steady-state, as alternatives to existing, general purpose, contact solving strategies, (iii) to develop and verify high fidelity and computationally efficient modeling tools for isotropic and anisotropic viscoelastic objects in steady-state motion, (iv) to investigate, numerically and through experimentation, the influence of various material parameters, including material nonlinearities such at the Payne effect and the Mullins effect, as well as geometric parameters and contact surface conditions, on viscoelastic rolling resistance, and (iv) to explore, analytically and through experimentation, the conditions under which favorable mechanical synergies occur between material components and develop novel composites with improved structural performances.

A new constitutive model that unifies the behavioral characterizations of rubber-like materials in a broad range of loading regimes is proposed. The model reflects two fundamental aspects of rubber behavior in finite strain: (i) the Mullins effect, and (ii) hyper-viscoelasticity with multiple time scales, including at high strain rates. Suitable means of identifying the system's parameters from simple uniaxial extension tests are explored. A directional approach extending the model to handle softening induced anisotropy is also discussed.

Novel, simple, and yet robust and efficient algorithms for solving steady-state, frictional, rolling/sliding contact problems, in two and three dimensions are presented. These are alternatives to powerful, well established, but in particular instances, possibly `cumbersome' general-purpose numerical techniques, such as finite-element approaches based on constrained optimization. The proposed algorithms are applied to the rolling resistance of cylinders and spheres.

Two and three-dimensional boundary element formulations of isotropic, transversely isotropic, and fully orthotropic, compressible and incompressible, viscoelastic layers of finite thickness are presented, in a moving frame of reference. The proposed formulations are based on two-dimensional Fourier series expansions of relevant mechanical fields in the continuum of the layers and support any linear viscoelastic material model characterized by general frequency-domain master-curves. These modeling techniques result in a compliance matrix for the upper boundary of the layers, including the effects of steady-state motion. Such characterizations may be used as components in various problem settings to generate sequences of high fidelity solutions for varying parameters. These are applied, in combination with appropriate contact solvers, to the rolling resistance of rigid cylinders and spheres.

The problem of a viscoelastic sphere moving across a rigid surface is significantly more complicated than that of a rigid indenter on a viscoelastic plane. The additional difficulties raised by the former may explain why previous work on this topic is so sparse. A new boundary element formulation for the multi-layered viscoelastic coating of a rigid sphere is developed. The model relies on the assumption of a relatively small contact surface in order to decouple equilibrium equations in the frequency domain. It is applied in combination with an adapted rolling contact solving strategy to the rolling resistance of a coated sphere.

New modeling approaches yielding rolling resistance estimates for rigid spheres (and cylinders) on viscoelastic layers of finite thicknesses are also introduced, as lower-cost alternatives to more comprehensive solution-finding strategies, including those proposed in this work. Application examples illustrate the capabilities of the different approaches over their respective ranges of validity.

The computational tools proposed in this dissertation are verified by comparison to dynamic finite element simulations and to existing solutions in limiting cases. The dependencies of rolling resistance on problem parameters are explored. It is for instance shown that, on orthotropic layers, the dissipated power varies with the direction of motion, which suggests new ways of optimizing the level of damping in various engineering applications of very high impact. Interesting lateral viscoelastic effects resulting from material asymmetry are unveiled. These phenomena could be harnessed to achieve smooth and `invisible' guides across three-dimensional viscoelastic surfaces, and hence suggest new ways of controlling trajectories, with a broad range of potential applications.

A new experimental apparatus is designed and assembled to measure viscoelastic rolling resistance. Experiments are conducted by rolling steel balls between sheets of rubber. Principal sources of measurement error, specific to the device, are discussed. Rolling resistance predictions are obtained using the computational tools presented in this dissertation, and compared to the measurements. Interesting conclusions are drawn regarding the fundamental influence of the Payne effect on viscoelastic rolling friction.

The work presented in this dissertations finally touches on the mechanical behavior of casing-infill composite tubes, as potential new lightweight structural elements. The axial behavior of composite circular tubes is addressed analytically. The influence of material parameters and geometry on structural performances are revealed and presented in original graphical forms. It is for instance shown that significantly improved overall stiffness and capacity at yield can be obtained using a moderately soft and highly auxetic infill, which further highlights the need to develop new lightweight auxetic materials, without compromising their stiffness. It is furthermore concluded that limited mechanical synergies can be expected in metal-polymer composite tubes, within the linear range of the materials involved. This prediction is confirmed by a bending experiment conducted on an Aluminum-Urethane composite tube. The experiment however reveals unexpected and quite promising mechanical synergies under large deformations. This novel composite has a potential influence on the design and performance of lightweight protecting structures against shocks and accelerations due to impacts, which justifies that it be characterized further.

Rolling Isolation Systems provide a simple and effective means for protecting components from horizontal floor vibrations. In these systems a platform rolls on four steel balls which, in turn, rest within shallow bowls. The trajectories of the balls is uniquely determined by the horizontal and rotational velocity components of the rolling platform, and thus provides nonholonomic constraints. In general, the bowls are not parabolic, so the potential energy function of this system is not quadratic. This thesis presents the application of Gauss's Principle of Least Constraint to the modeling of rolling isolation platforms. The equations of motion are described in terms of a redundant set of constrained coordinates. Coordinate accelerations are uniquely determined at any point in time via Gauss's Principle by solving a linearly constrained quadratic minimization. In the absence of any modeled damping, the equations of motion conserve energy. This mathematical model is then used to find the bowl profile that minimizes response acceleration subject to displacement constraint.

Novel wind turbine designs are deemed acceptable through a simulation-based certification process that involves generating a synthetic wind record and using it as an input to a computer model of the turbine. Naturally, whether the simulation loads reflect the loads that the turbine would actually experience depends on the accuracy of the wind turbine model and, more importantly, on the accuracy of the method used to generate the synthetic wind record. The simulation methods that are commonly used for this purpose are spectral-based and produce Gaussian, stationary random fields. These methods prescribe a power spectral density (PSD) of the wind velocity, which fixes the magnitudes of the Fourier components, then assumes that the Fourier phase angles are independent and uniformly distributed. An inverse Fast Fourier Transform (IFFT) is then used to transform the wind velocity field to the time domain.

This thesis applies the concept of phase coherence---i.e., Fourier phase angles that are not independent---to the stochastic modeling and simulation of wind velocity fields. Using a large dataset available from the National Wind Technology Center (NWTC), a joint distribution is characterized for the mean wind speed U, turbulence σ_u, Kaimal length scale L, and a metric for the degree of phase coherence in wind data, R. The correlations between these four parameters, the vertical height, and another phase coherence parameter are presented; only U, σ_u, and L have any significant degree of correlation. The joint distribution is used to generate synthetic wind records, which are then compared with measured data that have the same parameter values. For data with low to medium coherence values, the synthetic records have a similar qualitative appearance to the data. For high levels of phase coherence, the records simulated with the proposed model were qualitatively different from records with the same parameter values due to the variation of the phase difference spread in the spectral domain. Lastly, the importance of correctly modeling phase coherence is demonstrated by using the data and the synthetic records as inputs to a single-degree-of-freedom (SDOF) oscillator and comparing the peak response statistics and damage equivalent loads (DELs).

The rolling isolation system (RIS) studied in this dissertation functions on the principle of a rolling pendulum; an isolated object rests on a steel frame that is supported at its corners by ball-bearings that roll between shallow steel bowls, dynamically decoupling the floor motion from the response of the object. The primary focus of this dissertation is to develop predictive models that can capture experimentally-observed phenomena and to advance the state-of-the-art by proposing new isolation technologies to surmount current performance limitations. To wit, a double RIS increases the system's displacement capacity, and semi-active and passive damped RISs suppress the system's displacement response.

This dissertation illustrates the performance of various high-performance isolation strategies using experimentally-validated predictive models. Effective modeling of RISs is complicated by the nonholonomic and chaotic nature of these systems which to date has not received much attention. Motivated by this observation, the first part of this dissertation addresses the high-fidelity modeling of a single, undamped RIS, and later this theory is augmented to account for the double (or stacked) configuration and the supplemental damping via rubber-coated bowl surfaces. The system's potential energy function (i.e. conical bowl shape) and energy dissipation model are calibrated to free-response experiments. Forced-response experiments successfully validate the models by comparing measured and predicted peak displacement and acceleration responses over a range of operating conditions.

Following the experimental analyses, numerical simulations demonstrate the potential benefits of the proposed technologies. This dissertation presents a method to optimize damping force trajectories subject to constraints imposed by the physical implementation of a particular controllable damper. Potential improvements in terms of acceleration response are shown to be achievable with the semi-active RIS. Finally, extensive time-history analyses establish how the undamped and damped RISs perform when located inside biaxial, hysteretic, multi-story structures under recorded earthquake ground motions. General design recommendations, supported by critical-disturbance spectra and peak-response distributions, are prescribed so as to ensure the uninterrupted operation of vital equipment.

The full-scale base-isolated structure studied in this dissertation is the only base-isolated building in South Island of New Zealand. It sustained hundreds of earthquake ground motions from September 2010 and well into 2012. Several large earthquake responses were recorded in December 2011 by NEES@UCLA and by GeoNet recording station nearby Christchurch Women's Hospital. The primary focus of this dissertation is to advance the state-of-the art of the methods to evaluate performance of seismic-isolated structures and the effects of soil-structure interaction by developing new data processing methodologies to overcome current limitations and by implementing advanced numerical modeling in OpenSees for direct analysis of soil-structure interaction.

This dissertation presents a novel method for recovering force-displacement relations within the isolators of building structures with unknown nonlinearities from sparse seismic-response measurements of floor accelerations. The method requires only direct matrix calculations (factorizations and multiplications); no iterative trial-and-error methods are required. The method requires a mass matrix, or at least an estimate of the floor masses. A stiffness matrix may be used, but is not necessary. Essentially, the method operates on a matrix of incomplete measurements of floor accelerations. In the special case of complete floor measurements of systems with linear dynamics, real modes, and equal floor masses, the principal components of this matrix are the modal responses. In the more general case of partial measurements and nonlinear dynamics, the method extracts a number of linearly-dependent components from Hankel matrices of measured horizontal response accelerations, assembles these components row-wise and extracts principal components from the singular value decomposition of this large matrix of linearly-dependent components. These principal components are then interpolated between floors in a way that minimizes the curvature energy of the interpolation. This interpolation step can make use of a reduced-order stiffness matrix, a backward difference matrix or a central difference matrix. The measured and interpolated floor acceleration components at all floors are then assembled and multiplied by a mass matrix. The recovered in-service force-displacement relations are then incorporated into the OpenSees soil structure interaction model.

Numerical simulations of soil-structure interaction involving non-uniform soil behavior are conducted following the development of the complete soil-structure interaction model of Christchurch Women's Hospital in OpenSees. In these 2D OpenSees models, the superstructure is modeled as two-dimensional frames in short span and long span respectively. The lead rubber bearings are modeled as elastomeric bearing (Bouc Wen) elements. The soil underlying the concrete raft foundation is modeled with linear elastic plane strain quadrilateral element. The non-uniformity of the soil profile is incorporated by extraction and interpolation of shear wave velocity profile from the Canterbury Geotechnical Database. The validity of the complete two-dimensional soil-structure interaction OpenSees model for the hospital is checked by comparing the results of peak floor responses and force-displacement relations within the isolation system achieved from OpenSees simulations to the recorded measurements. General explanations and implications, supported by displacement drifts, floor acceleration and displacement responses, force-displacement relations are described to address the effects of soil-structure interaction.

Low frequency noise in measured sensor data is amplified when integrated. In the integration of measured acceleration data to displacement, such low frequency noise can lead to significant drift errors. In non-real-time applications, time domain and frequency domain detrending methods can be employed to remove bias and drift errors. For real-time applications, recursive high-pass digital filters, such as Butterworth filters, are computationally simple to implement. This research focuses on developing a discrete time state-space model to simultaneously filter out low frequency noise, deconvolve, and integrate voltage measurements from a geophone sensor. A circuit model for the sensor was chosen. Forward and inverse dynamical systems describing the circuit were derived utilizing the theory of linear time-invariant systems. The stability and accuracy of Butterworth filter design using the bilinear transformation method can be affected by the filter order and cut-off frequency. This research reveals the root cause of the numerical instability of high-pass digital Butterworth filters having low cutoff frequencies (less than half a percent of the sampling frequency) and high filter orders (greater than 6). These instabilities arise when filter coefficients are computed from discrete time poles and can be avoided by converting a continuous-time state-space model for the filter to discrete time via a matrix exponential. The method is demonstrated using measured geophone data.

Abstract

This master thesis investigates the effect of soil structure interaction on the behavior of base isolated structures. A linear soil model coupled with a nonlinear model of a base-isolated structure is assembled as a nonlinear system in state-space. Transient responses to earthquake ground motions are computed by integrating the nonlinear equations with a fixed-time step.Earthquake ground motions representative of near fault and far-field conditions are generated and tested for this system. The effects of soil compliance and friction in the base isolation system are then evaluated. An eigenvalue analysis of the dynamics matrix of the couple system gives the evidence that the soil structure interaction increases the periods of the structure positively. It is shown that the

base isolation system with large hysteretic frictional force is less effective in achieving the goals of seismic isolation; large isolation-level friction couples the structure above it too tightly, which results in less deformation in base isolator and larger roof acceleration than intended for a seismically-isolated structure. In addition, the base isolation system with large hysteretic frictional force produces larger residual displacements for both base isolator and the above floors. The impact factor of shear wave velocity to the stiffness of soil is more difficult to assess. Therefore, nonlinear liquefied soil model is

recommended to be calibrated for soil structure interaction evaluation as well as to be compared with empirical testing. To this end, nonlinear models for liquefying soil are investigated.

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