Tapered Beams in MEMS: A Symbolic Modeling Framework with Applications to Energy Harvesting

This book addresses important aspects of MEMS designs that are well established in engineering practice but rarely discussed in the standard textbooks. One such aspect is the ubiquitous use of tapered beams in the sensing and actuation elements of MEMS designs. As explained in this book, the tapered beam has distinct advantages over the standard rectangular beam but these advantages are often left unarticulated due to the blind trust in the finite-element models of MEMS devices. In this monograph, the authors take a fundamental, physics-based approach to the modeling of tapered beams in MEMS that is based on a rigorous perturbation analysis of the traditional Euler-Bernoulli beam. The authors demonstrate how perturbation methods combined with symbolic modeling and the tools of computer algebra enable the development of semi-analytical models for tapered-beam MEMS elements. They pay particular attention to the application of these novel models to piezoelectric MEMS energy harvesters with tapered-beam elements, including the development of lumped-parameter circuit models that can be readily used for fast electro-mechanical simulations. Another important aspect of MEMS designs that is extensively addressed in the book is the uncertainty quantification (UQ) of tapered-beam MEMS elements using both Monte Carlo and polynomial chaos expansion methods. These UQ methods are applied to the design of variation-aware piezoelectric energy harvesters. With consistent focus on MEMS devices with tapered beam elements, this up-to-date monograph Foreword Preface References Acknowledgements Contents Nomenclature 1 Introduction to Piezoelectric MEMS Energy Harvesting 1.1 MEMS Piezoelectric Vibrational Energy Harvesting 1.1.1 Microelectromechanical Systems 1.1.2 Energy Harvesting 1.1.3 Vibrations as a Source of Energy 1.1.4 Piezoelectric Transduction 1.1.5 Figures of Merit of Vibration Energy Harvesters 1.1.6 Survey of Piezoelectric Energy Harvesters 1.2 Variability-Aware CAD of MEMS 1.2.1 Variability in MEMS 1.2.1.1 Types of Variability 1.2.2 Uncertainty Quantification 1.2.2.1 Survey of Forward Uncertainty Propagation Methods 1.2.3 Uncertainty Propagation in Microfabrication References 2 Review of Euler–Bernoulli Rectangular Beam Theory 2.1 Static Beam Equation 2.1.1 Deriving the Bending Equation 2.1.2 The Dynamic Equation of a Beam 2.2 Euler–Bernoulli Boundary Value Problem Formulation 2.3 Separation of Variables 2.4 Differential Eigenanalysis 2.5 Mass Normalization of the Mode Shapes 2.6 Piezoelectric Energy Harvester Models 2.6.1 Lumped Parameter Models of MEMS Devices 2.6.2 Piezoelectric Constitutive Equations 2.6.3 System Model of a Rectangular Beam PEH 2.6.3.1 Governing Equations 2.6.3.2 Mode Shapes 2.6.3.3 Electromechanical Coupled Modal Response ODE 2.6.3.4 Equivalent Circuit Model (ECM) 2.6.4 Survey of Piezoelectric Energy Harvester Models References 3 Tapered-Beam Piezoelectric MEMS Energy Harvesters 3.1 Tapered Beams 3.1.1 Tapered Beams in Nature 3.1.2 Tapered Beams in Engineering 3.1.3 Tapered Beams in MEMS 3.2 Tapered Beams in Piezoelectric Energy Harvesters 3.2.1 Strain Distribution 3.3 Tapered Cantilever Model 3.3.1 Survey of Tapered Cantilever Models 3.3.1.1 Approximate Solutions 3.4 Finite Element Modeling and Simulation 3.4.1 CoventorWare Modeling and Simulation 3.4.1.1 3D Model Construction 3.4.1.2 Simulation and Analyses 3.4.2 MEMS+ Modeling and Simulation 3.4.2.1 3D Model Construction 3.4.3 Output Power Computation 3.4.4 ECM from FEM 3.4.4.1 Identification of Cr, Lr, Rr, and Nr 3.5 Tapered Beam EH Design and Fabrication 3.5.1 Energy Harvester Process Platform 3.5.2 Design Targets 3.5.2.1 Design Philosophy 3.5.3 Key Designs 3.5.3.1 First-Generation Devices 3.5.3.2 Second-Generation Devices 3.5.3.3 Third-Generation Devices 3.6 Energy Harvester Characterization 3.6.1 Experimental Setup for Characterization 3.6.2 Characterization Methodology 3.6.3 Visual Inspection of the Device 3.6.4 Residual Stress Analysis Methodology 3.6.5 Device Output Characterization Methodology 3.7 Characterization Results of the Tapered Beam EH 3.7.1 Visual Inspection of the Device 3.7.2 Residual Stress Analysis 3.7.2.1 First-Generation Devices 3.7.2.2 Second-Generation Devices 3.7.3 Device Output Characterization 3.7.3.1 First-Generation Devices 3.7.3.2 Second-Generation Devices 3.8 Conclusion References 4 Modal Analysis 4.1 Tapered Beam EH Model Formulation Workflow 4.2 Governing Equation of Motion 4.2.1 Internal Moment 4.2.2 Applying Equilibrium Condition 4.3 Spatial Modal Equation Formulation 4.3.1 Separation of Variables 4.3.2 Spatial Modal Equation of the Free Undamped Beam 4.3.2.1 Modal Solution in Terms of Special Functions 4.4 Perturbation Expansion of the Modal Solution 4.4.1 Iterative Perturbed Tapered System (ITPS) Formulation 4.4.1.1 Boundary Conditions of the ITPS 4.4.2 Iterative Solution to the ITPS 4.5 Normalization Conditions 4.5.1 Normalization Conditions for the Straight Cantilever Beam 4.5.2 Normalization Conditions for the TaperedBeam Response 4.5.3 Normalization Conditions for Perturbation Components 4.5.3.1 Companion Form Normalization Conditions for Tapered Beam 4.6 Finding Eigenvalues of the Perturbed System 4.6.1 Finding the 0th Perturbation Component of theEigenvalue 4.6.2 Eigenvalues of the Higher Perturbation Components 4.6.3 Evaluating Eigenvalue Perturbation Components Using Fn(x) Orthogonality Property 4.7 Green's Function of the Spatial Modal Equation 4.7.1 Discovering Green's Function of the System 4.7.1.1 Green's Function Boundary Conditions 4.7.2 Proof That the System Has a Zero Eigenvalue 4.7.3 Application of Fredholm Alternative Theorem 4.7.4 Response to the Null-Space Deflation G0,r(x,xi) 4.7.5 Impulse Response Gdelta,r(x,xi) 4.7.5.1 Boundary Conditions 4.7.6 Solution for Gdelta(x,xi) 4.8 Solution of the Perturbed Spatial Modal Equation 4.8.1 Nominal Component psi0,r(x) 4.8.2 Applying Green's Function to Solve Higher Perturbation Components 4.8.2.1 Evaluating c1 Using Normalization Conditions 4.9 Tapered Beam Modal Analysis Algorithm 4.10 Conclusion References 5 Symbolic Model Order Reduction Using Perturbation Analysis 5.1 Introduction 5.2 Background 5.3 Symbolic Perturbation Expansion 5.4 Algorithm for Symbolic Modal Analysis 5.4.1 Iteration Initialization 5.4.2 Symbolic Eigenvalue Computation 5.4.3 Symbolic Green's Function Derivation 5.4.4 Symbolic Mode-Shape Computation 5.4.5 Symbolic Normalization 5.4.6 Symbolic Implementation 5.5 Model Evaluation Modules 5.5.1 Numerical Evaluation of Perturbation Expansions 5.5.2 Computation of Temporal Response 5.6 Modal Analysis Validation Against MEMS+ Results 5.6.1 Resonance Frequency Estimation Results 5.6.2 Mode-Shape Estimation Results 5.6.3 Timing Comparison with FEM 5.7 Conclusions References 6 Lumped-Parameter Modeling 6.1 Full Electromechanical Model Formulation 6.1.1 The Governing System 6.1.1.1 Internal Moment Evaluation 6.1.1.2 Applying Equilibrium Condition 6.1.2 Modal Response with Electrical Coupling 6.1.3 Electrical Circuit Equation with Mechanical Coupling 6.2 Steady-State Analysis 6.2.1 Steady-State Solution with Harmonic Inputs 6.2.2 Numerical Evaluation of the Steady-State Solution 6.3 Transient Analysis 6.3.1 Implicit Transient Solution 6.3.2 State-Space Model of the Electromechanical System 6.3.2.1 Single-Mode System 6.3.2.2 Multiple-Mode System 6.4 Equivalent Circuit Model Extraction 6.5 Equivalent Circuit Simulation 6.5.1 Analytical Solution for AC Analysis 6.5.2 AC Analysis Using CppSim 6.5.3 Device Simulation Workflow Using ECM 6.6 Simulation Results 6.6.1 Model Validation for AC Analysis 6.6.2 Beam Deflection AC Analysis 6.6.3 Transient Analysis 6.6.4 Timing Comparison with FEM 6.7 Conclusion References 7 Uncertainty Quantification Using Monte Carlo Methods 7.1 Uncertainty Model of MEMS Process Variability 7.1.1 Sources of Process Variability 7.1.2 Developing Variability Models of Parameters with Aleatoric Uncertainty 7.2 Uncertainty Quantification Strategies for MEMS 7.2.1 Identifying Uncertainty in the Model 7.2.2 Nonintrusive and Intrusive Uncertainty Quantification 7.3 Nonintrusive Uncertainty Quantification Strategies 7.3.1 Monte Carlo Simulation 7.3.2 Stochastic Collocation Simulation 7.3.2.1 Stochastic Collocation Algorithm 7.4 Monte Carlo-Based UQ Results for Tapered PEH 7.4.1 Modal Analysis Uncertainty Measures 7.4.1.1 Resonance Frequency 7.4.1.2 Average Curvature of the Mode Shape 7.4.2 Steady-State Analysis Uncertainty Measures 7.4.2.1 Maximum Output Power Results at Optimal Load 7.5 ECM and ECM Response UQ Results 7.6 Conclusion References 8 Uncertainty Quantification Using Polynomial-Chaos Expansion 8.1 The Stochastic Governing System 8.1.1 General MEMS Stochastic Equation 8.1.2 Stochastic Tapered Beam Piezoelectric EH Governing System 8.2 Polynomial Chaos Expansion of the MEMS Stochastic Equation 8.2.1 Finding Orthogonal Polynomials 8.2.1.1 Calculating Expansion Modes 8.2.1.2 Statistics of the Solution 8.3 Galerkin Projection on PCE 8.3.1 General Procedure to Galerkin Projection PCE 8.3.2 PCE Applied to Stochastic MEMS Equation 8.3.3 PCE with Galerkin Projection Applied to the Tapered PEH Model 8.3.3.1 Augmented Model Interpretation 8.4 Stochastic Testing 8.4.1 Setting Up the DAE 8.4.2 Testing Nodes 8.5 Uncertainty Propagation of the Tapered Beam Model 8.5.1 PCE of the Tapered Beam Coupled Electromechanical Model 8.5.2 Probability Transformation 8.5.3 Applying Probability Transformation to Uncertainty Propagation 8.5.4 Uncertainty Dependency Graph 8.5.5 Uncertainty Propagation Using ProbabilityTransformation 8.5.5.1 PDF Transformation Results 8.6 Hierarchical Uncertainty Quantification of Tapered PEH 8.6.1 Estimating the PDFs of ECM Parameters 8.6.2 Determining the gPC Basis Functions 8.6.3 Determining the Quadrature Points 8.7 Conclusion References 9 Variation-Aware Design of Reliable Piezoelectric MEMS Energy Harvesters 9.1 Process-Aware MEMS Device Design 9.1.1 Design Flow Using MEMS+ 9.1.1.1 Device Geometry Optimization 9.1.2 Design Validation Using CoventorWare 9.1.3 Process-Aware Analytical Model 9.1.3.1 Adapting ECM Construction to the Device Taper 9.2 Motivation of Variability-Aware MEMS Design 9.3 Variability-Aware MEMS Device Design Flow 9.3.1 Design Flow with Variability Constraints 9.3.2 Design Flow with Variability Tests Decoupled References Epilogue A Mathematical Properties of Perturbation Solutions of Tapered-Beam Systems A.1 Self-Adjoint Operator Helps Evaluate Eigenvalues A.2 Proof that the Tapered Beam System is Self-Adjoint A.3 Self-Adjointedness Implication on Fn(x) B Some Properties of the Green's Function for Perturbation Solutions of Tapered-Beam Systems B.1 Eigenvalue Expansion View of the Green's Function B.2 Fredholm Alternative Theorem for a Boundary Value Problem References Index