The direct and adjoint methods are to linearize the time-averaged solution of bounded dynamical systems about one or more design parameters. Hence, such methods are one way to obtain the gradient necessary in locally optimizing a dynamical system’s time-averaged behavior over those design parameters. However, when analyzing nonlinear systems whose solutions exhibit chaos, standard direct and adjoint sensitivity methods yield meaningless results due to time-local instability of the system. The present work proposes a new method of solving the direct and adjoint linear systems in time, then tests that method’s ability to solve instances of the Lorenz system that exhibit chaotic behavior. Promising results emerge and are presented in the form of a regression analysis across a parametric study of the Lorenz system.

Land surfaces have changed as a result of human and natural processes, such asdeforestation, urbanization, desertification and natural disasters like wildfires. Land use and landcover change impacts local and regional climates through various bio geophysical processes acrossmany time scales. More realistic representation of land surface parameters within the land surfacemodels are essential to for climate models to accurately simulate the effects of past, current andfuture land surface processes. In this study, we evaluated the sensitivity and accuracy of theWeather Research and Forecasting (WRF) model though the default MODIS land cover data andannually updated land cover data over southeast of United States. Findings of this study indicatedthat the land surface fluxes, and moisture simulations are more sensitive to the surfacecharacteristics over the southeast US. Consequently, we evaluated the WRF temperature andprecipitation simulations with more accurate observations of land surface parameters over thestudy area. We evaluate the model performance for the default and updated land cover simulationsagainst observational datasets. Results of the study showed that updating land cover resulted insubstantial variations in surface heat fluxes and moisture balances. Despite updated land use andland cover data provided more representative land surface characteristics, the WRF simulated 2- m temperature and precipitation did not improved due to use of updated land cover data. Further,we conducted machine learning experiments to post-process the Noah-MP land surface modelsimulations to determine if post processing the model outputs can improve the land surfaceparameters. The results indicate that the Noah-MP simulations using machine learning remarkablyimproved simulation accuracy and gradient boosting, and random forest model had smaller meanerror bias values and larger coefficient of determination over the majority of stations. Moreover,the findings of the current study showed that the accuracy of surface heat flux simulations byNoah-MP are influenced by land cover and vegetation type.

The Generation IV International Forum (GIF) has initiated an international collaboration for the research and development of the Generation IV future nuclear energy systems. The Canadian PT-SCWR is Canada’s contribution to the GIF as a GEN-IV advanced energy system. The PT-SCWR is a pressure tube reactor type and considered as an evolution of the conventional CANDU reactor. The PT-SCWR is characterized by bi-directional coolant flow through the High Efficiency Re-entrant Channel (HERC). The Canadian SCWR is a unique design involving high pressure and temperature coolant, a light water moderator, and a thorium-plutonium fuel, and is unlike any operating or conceptual reactor at this time. The SCWR does share some features in common with the BWR configuration (direct cycle, control blades etc…), CANDU (separate low temperature moderator), and the HTGR/HTR (coolant with high propensity to up-scatter), and so it represents a hybrid of many concepts. Because of its hybrid nature there have been subtle feedback effects reported in the literature which have not been fully analyzed and are highly dependent on these unique characteristics in the core. Also given the significant isotopic changes in the fuel it is necessary to understand how the feedback mechanisms evolve with fuel depletion. Finally, given the spectral differences from both CANDU and HTR reactors further study on the few-energy group homogenization is needed. The three papers in this thesis address each one of these issues identified in literature. Models were created using the SCALE (Standardized Computer Analysis for Licensing Evaluation) code package. Through this work, it was found that the lattice is affected by more than one large individual phenomenon but that these phenomena cancel one another to have a small net final change. These phenomena are highly affected by the coolant properties which have major roles in neutron thermalization process since the PT-SCWR is characterized by a tight lattice pitch. It was observed that fresh and depleted fuel have almost similar behaviour with small differences due to the Pu depletion and the production of minor actinides, 233U and xenon. It was also found that a higher thermal energy barrier is recommended for the two-energy-group structure since the PT-SCWR is characterized by a large coolant temperature compared to the conventional water thermal reactors. Two, three and four optimum energy group structure homogenizations were determined based on the behaviour of the neutron multiplication factor and other reactivity feedback coefficients. Robust numerical computations and experience in the physics of the problem were used in the few-energy group optimization methodology. The results show that the accuracy of the expected solution becomes highly independent of the number of energy groups with more than four energy groups used. / Thesis / Doctor of Philosophy (PhD)

The potential of hydrogen as a clean energy carrier is hotly debated, but it promises to significantly contribute to a sustainable energy future. Hydrogen can replace fossil fuels in carbon-intensive industries, heavy transport and aviation, and support a renewable energysystem by acting as energy storage to balance intermittent supply. With increasing investment, high demand projections and the promise of hydrogen to reduce carbon emissions, it is becoming increasingly important to include hydrogen in energy models. While some studies include hydrogen in their energy models, they don’t comprehensively analyse the effects of uncertainty, which is significant in the hydrogen sector. This thesis addresses this gap by developing a Nordic energy model that includes hydrogen supply and storage using OSeMOSYS, based on the European OSeMBE model. In addition to a scenario analysis, a global sensitivity analysis is performed to identify the most influential uncertainties and key interactions between the hydrogen and electricity sectors. The study identifies hydrogen demand and carbon pricing as key uncertain drivers of change, affecting system costs and emissions levels. Uncertainty about the efficiency of carbon capture and the potential for biomass technology with carbon capture and storage also significantly impact emissions. While the share of renewables is projected to be robust, the technologies used for hydrogen production are susceptible to uncertainties. Steam reforming dominates in the absence of a strong carbon price. Electricity and hydrogen from biomass can provide negative emissions and have the potential to play an important role in decarbonisation. However, biomass availability is limited and policy support like carbon pricing is needed to make these technologies competitive in the market. A key link between the electricity and the hydrogen system is electrolysers. However, while cheaper electricity makes electrolysers more attractive, the cost and performance of hydrogen production technologies, such as steam reforming or biomass gasification, are more relevant in determining which hydrogen technologies will dominate. Hydrogen storage and fuel cells aren’t used in the study, except in small amounts for some of the runs in the sensitivity analysis. However, this may change with a specified time-dependent hydrogen demand or a finer time representation in the model. The thesis shows that uncertainties around hydrogen have a much larger impact on emissions than uncertainties around the electricity system. Hydrogen technologies are in close competition, with steam reforming difficult to displace. While in the Nordic countries, the advance of renewables in electricity generation seems unstoppable, the hydrogen sector needs public policy support to become an ally in decarbonisation rather than a burden. / Vätgasens potential som en ren energibärare är omdiskuterad, men den kan bidra avsevärt till en hållbar energiframtid. Vätgas kan ersätta fossila bränslen i koldioxidintensiva industrier, tunga transporter och luftfart, och stödja ett förnybart energisystem genom att fungera som energilagring för att balansera intermittenta leveranser. Med ökande investeringar, prognoser om hög efterfrågan och löftet om att vätgas kan minska koldioxidutsläppen blir det allt viktigare att inkludera vätgas i energimodeller. Vissa studier inkluderar vätgas i sina energimodeller, men de analyserar inte effekterna av osäkerhet på ett heltäckande sätt, vilket är betydande inom vätgassektorn. Den här avhandlingen adresserar detta gap genom att utveckla en nordisk energimodell som inkluderar vätgasförsörjning och lagring med hjälp av OSeMOSYS, baserat på den europeiska OSeMBE-modellen. Förutom en scenarioanalys utförs en global känslighetsanalys för att identifiera de mest inflytelserika osäkerheterna och viktiga interaktioner mellan vätgas- och elsektorerna. Studien identifierar efterfrågan på vätgas och prissättningen på koldioxid som viktiga osäkra drivkrafter för förändring, vilket påverkar systemkostnader och utsläppsnivåer. Osäkerheten kring koldioxidavskiljningens effektivitet och potentialen för biomassateknik med koldioxidavskiljning och lagring påverkar också utsläppen avsevärt. Även om andelen förnybara energikällor förväntas vara robust, är de tekniker som används för vätgasproduktion känsliga för osäkerheter. Ångreformering dominerar i avsaknad av ett starkt koldioxidpris. Elektricitet och vätgas från biomassa kan ge negativa utsläpp och har potential att spela enviktig roll i utfasningen av fossila bränslen. Tillgången på biomassa är dock begränsad och politiskt stöd i form av t.ex. koldioxidpriser behövs för att göra dessa tekniker konkurrenskraftiga på marknaden. En viktig länk mellan el- och vätgassystemet är elektrolysörer. Men även om billigare el gör elektrolysörer mer attraktiva, är kostnaden och prestandan för vätgasproduktionstekniker såsom ångreformering eller förgasning av biomassa mer relevanta för att avgöra vilka vätgastekniker som kommer att dominera. Vätgaslagring och bränsleceller används inte i studien, förutom i små mängder för några av körningarna i känslighetsanalysen. Detta kan dock förändras med en specificerad tidsberoende vätgasefterfrågan eller en finare tidsrepresentation i modellen. Avhandlingen visar att osäkerheter kring vätgas har en mycket större inverkan på utsläppen än osäkerheter kring elsystemet. Vätgasteknikerna konkurrerar nära varandra, men ångreformering är svår att ersätta. I de nordiska länderna verkar framstegen för förnybara energikällor inom elproduktion vara ostoppbara, men vätgassektorn behöver offentligt politiskt stöd för att bli en allierad i utfasningen av fossila bränslen snarare än en börda.

The biological characteristics of fisheries stocks that are assessed for management considerations are rarely homogeneous over time or space. However, stock assessment scientists largely ignore this heterogeneity in their models. This thesis addresses the effects of spatial and temporal heterogeneity on stock assessment models using Atlantic Weakfish (Cynoscion regalis) as a case study. First, spatial and temporal variation was incorporated into length-, weight-, and maturity-at-age estimates using mixed-effects models (Chapter Two). The resulting heterogeneous weight and maturity parameters then were applied to per-recruit analyses to examine the sensitivity of biological reference points to spatial and temporal variation in life history attributes (Chapter Three). Mixed-effects life history models incorporating spatial and temporal variation revealed distinct regional and annual trends that were not visible from standard homogeneous models. In several instances, the homogeneous modelling approach produced life history estimates that varied significantly from the spatial and temporal means produced by the heterogeneous models. In some cases, this difference was so great that the homogeneous means were much higher or lower than the heterogeneous means for all regions or years. Minimized AIC statistics revealed that spatially and temporally integrated mixed-effects models were more robust and descriptive of Atlantic Weakfish life history than the standard homogeneous models. Per-recruit and biological reference points derived from these life history estimates in Chapter Three were found to be highly sensitive to spatial and temporal variations in weight parameters. In several cases, reference points used as management targets were so significantly different that ignoring spatial and temporal heterogeneity in Atlantic Weakfish life history would likely cause overfishing and decline of Weakfish in certain regions and years. / Master of Science / Many stocks of commercially and recreationally harvested marine fish have displayed a declining trend in recent years. Marine fisheries are a vital component of the global economy, and, as such, sophisticated management measures have been developed to reduce and reverse this trend. These management strategies are based on regular reports from fisheries stock assessment scientists, who evaluate the status of fish stocks by modelling life history and productivity trends. One of the greatest challenges to stock assessments is the identification and incorporation of variability in fish populations. There is an inherent variation in fish growth, maturity, and productivity among geographical locations and over time. To produce the most effective management strategies, stock assessments must incorporate this spatial (regional) and temporal (annual) variation. In this thesis, I used mixed effects models to integrate spatial and temporal variation in life history and productivity using Atlantic Weakfish (Cynoscion regalis) as a case study. Distinct trends were observed in fishery-independent data for this species that were reflected in spatially and temporally incorporated models. However, these trends were masked in the standard models which incorporated neither spatial nor temporal variation. This oversight could cause weakfish to be overfished in certain regions and years and underfished in others. To maximize the effectiveness of management and the sustainable fisheries yield in all regions and years for Atlantic Weakfish and other harvested species, I highly recommend using spatially and temporally incorporated life history and productivity models such as the ones developed in this thesis.

The drug discovery process is challenging, time-consuming, and costly, with drug target identification being an essential step in developing effective therapies. Drug repurposing offers a strategy for identifying new uses for existing drugs, aiming to simplify the process. Machine learning models and network analysis methods have demonstrated promise in both drug target identification and repurposing, providing powerful tools for analyzing complex biological data. This thesis will explore the applications of neural networks and multilayer biological networks for drug repurposing opportunities and network inference problems applied to signaling pathways. A novel machine learning and network-based workflow is presented for identifying drug targets for cystinosis, a rare disease that causes progressive kidney disease, currently lacking effective therapies to prevent the kidney failure. This approach permits to recapitulate the disease mechanisms in the context of renal tubular physiology and identify candidate drug targets for further validation using a cross-species workflow and disease-relevant screening technologies. While machine learning approaches have shown promise, they often need more mechanistic understanding, which is necessary for robust drug target identification and repurposing strategies. Mechanistic models provide crucial insights into the underlying biological mechanisms, complementing machine learning techniques. However, inferring mechanistic signaling networks from omics data poses challenges due to non-identifiability, resulting in multiple valid solutions consistent with the data. After that, the focus shifts towards quantifying signaling model diversity through solver-agnostic solution sampling with CORNETO, an ongoing effort that aims to unify network inference problems via constrained optimization. Mechanistic signaling networks can be inferred from omics data and prior knowledge using combinatorial optimization and mathematical solvers to find the optimal network. However, this problem is in general, non-identifiable, and several solutions may be equally valid. Ignoring the existence of these alternative solutions leads to an incomplete picture of the hypothesis space of consistent mechanistic signaling networks. To alleviate this issue, an algorithm to explore the space of alternative solutions and to conduct sensitivity analysis on the optimal solution is implemented and presented. These algorithms are applied to data from pancreatic cancer cell lines treated with kinase inhibitors to study cellular responses to drug perturbations by inferring mechanistic signaling networks from omics data.

Clinching is a joining process that is becoming more and more important in industry due to the increasing use of multimaterial designs. Despite the already widespread use of the process, there is still a need for research to understand the mechanisms and design of clinched joints. In contrast to the tool parameters, process and material disturbances have not yet been investigated to a relatively large extent. However, these also have a great influence on the properties and applicability of clinching. The effect of process disturbances on the clinched joint are investigated with numerical and experimental methods. The investigated process variations are the history of the sheets using the pre-hardening of the material, different sheet thicknesses, sheet arrangements and punch strokes. For the consideration of the material history, a specimen geometry for pre-stretching specimens in uniaxial tension is used, from which the pre-stretched secondary specimens are taken. A finite element model is set up for the numerical investigations. Suitable clinching tools are selected. With the simulation, selected process influences can be examined. The effort of the numerical investigations is considerably reduced with the help of a statistical experimental design according to Taguchi. To confirm the simulation results, experimental investigations of the clinch point geometry by using micrographs and the shear strength of the clinched joint are performed. The analysis of the influence of difference disturbance factors on the clinching process demonstrate the importance of the holistic view of the clinching process.

In recent years, climate change has intensified the frequency of severe rainfall events, raising concerns, particularly in urban areas where impervious surfaces dominate. The resultant reliance on man-made drainage increases pluvial flooding risks, threatening infrastructure and urban resilience. As the global population increasingly shifts to urban living, the vulnerability to flooding grows. Understanding how areas respond to rainfall is crucial for proactive flood risk mitigation. Available hydrological models offer insights and predictions, but are often linked with long simulation times and high computational cost. Semi-distributed models, like the Spatially Distributed Travel Time (SDTT) approach, offer simplified model formulations suitable for screening applications. This thesis extends Ekeroth's (2022) SDTT model for watershed delineation and travel time formulations, focusing on ensemble runs of multi-input rainfall/infiltration scenarios. As there is often many uncertain factors in hydrological modeling, there is a need for faster models capable of generating a distribution of scenarios to represent the uncertainty of real systems. Even a quick and simple model should account for the multifaceted aspects of urban flooding, including rainfall-infiltration dynamics and the variations in rainfall intensity. Script modules were developed to analyze rainfall severity, peak discharge distribution, and parameter impact efficiently. In three urban watersheds with an average size of 0.45 km2, 120 scenarios distinguished by intensity distribution, rainfall duration, soil composition of pervious areas, and antecedent moisture conditions, were simulated within approximately 3.5 minutes, enabling comprehensive hydrological analysis. The successful implementation of the new modules implicate a promising tool for hydrological risk-screening analysis in urban environments, although further research should investigate incorporating probability-based scenarios and bigger input rainfall datasets. / Under senare år har klimatförändringarna intensifierat förekomsten av skyfall, något som är särskilt oroväckande i stadsområden där marktäckningen huvudsakligen består av hårdgjorda ytor. Genom att asfaltera och bygga försvinner markens naturliga infiltrationsförmåga. Detta leder till ett ökat beroende av konstgjorda dräneringssystem som sällan är dimensionerade för särskilt intensiva regnhändelser. Urbana översvämningar innebär inte bara ett hot mot infrastruktur och den bebyggda miljön, men den globala befolkningens ökade bosättning i urbana områden medför att sårbarheten vid översvämningar ökar även den. För att kunna hantera översvämningsrisken i ett urbant område är förståelse för avrinningsområdets respons till ett skyfall viktigt. Det finns hydrologiska modeller på marknaden som erbjuder prognoser, men dessa är oftast baserade på komplexa fysiska beskrivningar som medför långa processtider och beräkningskostnader. Samtidigt finns nytänkande modeller som skär ner på processtiderna genom att minska den spatiala upplösningen på beräkningarna, såsom SDTT (Spatially Distributed Travel Time) formuleringen, som erbjuder förenklade analyser lämpliga som screeningverktyg. Denna studie utvidgar Ekeroths (2022) SDTT-modell med fokus på ensemblekörningar av regn- och infiltrationsscenarier. Eftersom det ofta finns flertalet osäkra faktorer i hydrologisk modellering finns ett behov av snabbare modeller som kan genera en fördelning av möjliga utfall givet olika scenarier. Samtidigt behöver även en snabb och enkel modell beakta de mångfacetterade aspekterna av urbana översvämningar, exempelvis gällande dynamiken mellan regn och infiltration och skyfallsegenskaper. Kodmoduler utvecklades för att effektivt analysera utfallen av regnscenarierna och att finna de mest allvarliga händelserna, fördelningen av värden inom de simulerade utfallen, samt inflytandet från parametrarna som definierar scenarierna. I tre urbana avrinningsområden med en genomsnittlig storlek på 0.45 km2 simulerades 120 scenarier inom 3,5 minuter, vilket möjliggör hydrologisk analys på en hanterbar tid. Implementeringen av de nya modulerna pekar mot ett lovande verktyg för hydrologisk risk-screeninganalys i urbana miljöer. Samtidigt bör framtida studier fortsatt undersöka möjligheten att inkludera sannolikhetsbaserade scenarier och körning av större dataset.

Modeling and simulation have become fully ingrained into the set of design and development tools that are broadly used in the field of power electronics. To state simply, they represent the fastest and safest way to study a circuit or system, thus aiding in the research, design, diagnosis, and debugging phases of power converter development. Advances in computing technologies have also enabled the ability to conduct reliability and production yield analyses to ensure that the system performance can meet given requirements despite the presence of inevitable manufacturing variability and variations in the operating conditions. However, the trustworthiness of all the model-based design techniques depends entirely on the accuracy of the simulation models used, which, thus far, has not yet been fully considered. Prior to this research, heuristic safety factors were used to compensate for deviation of real system performance from the predictions made using modeling and simulation. This approach resulted invariably in a more conservative design process. In this research, a modeling and design approach with parametric and model-form uncertainty quantification is formulated to bridge the modeling and simulation accuracy and reliance gaps that have hindered the full exploitation of model-based design techniques. Prior to this research, a few design approaches were developed to account for variability in the design process; these approaches have not shown the capability to be applicable to complex systems. This research, however, demonstrates that the implementation of the proposed modeling approach is able to handle complex power converters and systems. A systematic study for developing a simplified test bed for uncertainty quantification analysis is introduced accordingly. For illustrative purposes, the proposed modeling approach is applied to the switching model of a modular multilevel converter to improve the existing modeling practice and validate the model used in the design of this large-scale power converter. The proposed modeling and design methodology is also extended to design optimization, where a robust multi-objective design and optimization approach with parametric and model form uncertainty quantification is proposed. A sensitivity index is defined accordingly as a quantitative measure of system design robustness, with regards to manufacturing variability and modeling inaccuracies in the design of systems with multiple performance functions. The optimum design solution is realized by exploring the Pareto Front of the enhanced performance space, where the model-form error associated with each design is used to modify the estimated performance measures. The parametric sensitivity of each design point is also considered to discern between cases and help identify the most parametrically-robust of the Pareto-optimal design solutions. To demonstrate the benefits of incorporating uncertainty quantification analysis into the design optimization from a more practical standpoint, a Vienna-type rectifier is used as a case study to compare the theoretical analysis with a comprehensive experimental validation. This research shows that the model-form error and sensitivity of each design point can potentially change the performance space and the resultant Pareto Front. As a result, ignoring these main sources of uncertainty in the design will result in incorrect decision-making and the choice of a design that is not an optimum design solution in practice. / Ph. D. / Modeling and simulation have become fully ingrained into the set of design and development tools that are broadly used in the field of power electronics. To state simply, they represent the fastest and safest way to study a circuit or system, thus aiding in the research, design, diagnosis, and debugging phases of power converter development. Advances in computing technologies have also enabled the ability to conduct reliability and production yield analyses to ensure that the system performance can meet given requirements despite the presence of inevitable manufacturing variability and variations in the operating conditions. However, the trustworthiness of all the model-based design techniques depends entirely on the accuracy of the simulation models used, which has not yet been fully considered. In this research, a modeling and design approach with parametric and model-form uncertainty quantification is formulated to bridge the modeling and simulation accuracy and reliance gaps that have hindered the full exploitation of model-based design techniques. The proposed modeling and design methodology is also extended to design optimization, where a robust multi-objective design and optimization approach with parametric and model-form uncertainty quantification is proposed. A sensitivity index is defined accordingly as a quantitative measure of system design robustness, with regards to manufacturing variability and modeling inaccuracy in the design of systems with multiple performance functions. This research shows that the model-form error and sensitivity of each design point can potentially change the performance space and resultant Pareto Front. As a result, ignoring these main sources of uncertainty in the design will result in incorrect decision making and the choice of a design that is not an optimum design solution in practice.

We model the nonlinear dynamics of a cantilever beam with tip mass system subjected to different excitation and exploit the nonlinear behavior to perform sensitivity analysis and propose a parameter identification scheme for nonlinear piezoelectric coefficients. First, the distributed parameter governing equations taking into consideration the nonlinear boundary conditions of a cantilever beam with a tip mass subjected to principal parametric excitation are developed using generalized Hamilton's principle. Using a Galerkin's discretization scheme, the discretized equation for the first mode is developed for simpler representation assuming linear and nonlinear boundary conditions. We solve the distributed parameter and discretized equations separately using the method of multiple scales. We determine that the cantilever beam tip mass system subjected to parametric excitation is highly sensitive to the detuning. Finally, we show that assuming linearized boundary conditions yields the wrong type of bifurcation. Noting the highly sensitive nature of a cantilever beam with tip mass system subjected to parametric excitation to detuning, we perform sensitivity of the response to small variations in elasticity (stiffness), and the tip mass. The governing equation of the first mode is derived, and the method of multiple scales is used to determine the approximate solution based on the order of the expected variations. We demonstrate that the system can be designed so that small variations in either stiffness or tip mass can alter the type of bifurcation. Notably, we show that the response of a system designed for a supercritical bifurcation can change to yield a subcritical bifurcation with small variations in the parameters. Although such a trend is usually undesired, we argue that it can be used to detect small variations induced by fatigue or small mass depositions in sensing applications. Finally, we consider a cantilever beam with tip mass and piezoelectric layer and propose a parameter identification scheme that exploits the vibration response to estimate the nonlinear piezoelectric coefficients. We develop the governing equations of a cantilever beam with tip mass and piezoelectric layer by considering an enthalpy that accounts for quadratic and cubic material nonlinearities. We then use the method of multiple scales to determine the approximate solution of the response to direct excitation. We show that approximate solution and amplitude and phase modulation equations obtained from the method of multiple scales analysis can be matched with numerical simulation of the response to estimate the nonlinear piezoelectric coefficients. / Master of Science / The domain of structural dynamics involves the evaluation of the structures response when subjected to time-varying loads. This field has many applications. For instance, by observing specific variations in the response of a structure such as bridge or a structural element such as a beam, one can diagnose the state of the structure or one of its elements. At much smaller scales, one can use a device to observe small variations in the response of a beam to detect the presence of bio-materials or gas particles in air. Additionally, one can use the response of a structure to harvest energy of ambient vibrations that are freely available. In this thesis, we develop a mathematical framework for evaluating the response of a cantilever beam with a tip mass to small variations in material properties caused by fatigue and to small variations in the tip mass caused by additional mass that gets bound to the structure. We also exploit the response of the beam to evaluate nonlinear material properties of piezoelectric materials that have been suggested for use in charging micro sensors, vibration control, load sensing and for high power energy transfer applications.