Cantilever

Abstract: A method to determine the spring constant of a rectangular atomic force microscope cantilever is proposed that relies solely on the measurement of the resonant frequency and quality factor of the cantilever in fluid (typically air), and knowledge of its plan view dimensions. This method gives very good accuracy and improves upon the previous formulation by Sader et al. [Rev. Sci. Instrum. 66, 3789 (1995)] which, unlike the present method, requires knowledge of both the cantilever density and thickness.

Abstract: About the Authors. Preface. 1. Introduction to Piezoelectric Energy Harvesting. 1.1 Vibration-Based Energy Harvesting Using Piezoelectric Transduction. 1.2 An Examples of a Piezoelectric Energy Harvesting System. 1.3 Mathematical Modeling of Piezoelectric Energy Harvesters. 1.4 Summary of the Theory of Linear Piezoelectricity. 1.5 Outline of the Book. 2. Base Excitation Problem for Cantilevered Structures and Correction of the Lumped-Parameter Electromechanical Model. 2.1 Base Excitation Problem for the Transverse Vibrations. 2.2 Correction of the Lumped-Parameter Base Excitation Model for Transverse Vibrations. 2.3 Experimental Case Studies for Validation of the Correction Factor. 2.4 Base Excitation Problem for Longitudinal Vibrations and Correction of its Lumped-Parameter Model. 2.5 Correction Factor in the Electromechanically Coupled Lumped-Parameter Equations and a Theoretical Case Study. 2.6 Summary. 2.7 Chapter Notes. 3. Analytical Distributed-Parameter Electromechanical Modeling of Cantilevered Piezoelectric Energy Harvesters. 3.1 Fundamentals of the Electromechanically Coupled Distributed-Parameter Model. 3.2 Series Connection of the Piezoceramic Layers. 3.3 Parallel Connection of Piezoceramic Layers. 3.4 Equivalent Representation of the Series and the Parallel Connection Cases. 3.5 Single-Mode Electromechanical Equations for Modal Excitations. 3.6 Multi-mode and Single-Mode Electromechanical FRFs. 3.7 Theoretical Case Study. 3.8 Summary. 3.9 Chapter Notes. 4. Experimental Validation of the Analytical Solution for Bimorph Configurations. 4.1 PZT-5H Bimorph Cantilever without a Tip Mass. 4.2 PZT-5H Bimorph Cantilever with a Tip Mass. 4.3 PZT-5A Bimorph Cantilever. 4.4 Summary. 4.5 Chapter Notes. 5. Dimensionless Equations, Asymptotic Analyses, and Closed-Form Relations for Parameter Identification and Optimization. 5.1 Dimensionless Representation of the Single-Mode Electromechanical FRFs. 5.2 Asymptotic Analyses and Resonance Frequencies. 5.3 Identification of Mechanical Damping. 5.4 Identification of the Optimum Electrical Load for Resonance Excitation. 5.5 Intersection of the Voltage Asymptotes and a Simple Technique for the Experimental Identification of the Optimum Load Resistance. 5.6 Vibration Attenuation Amplification from the Short-Circuit to Open-Circuit Conditions. 5.7 Experimental Validation for a PZT-5H Bimorph Cantilever. 5.8 Summary. 5.9 Chapter Notes. 6. Approximate Analytical Distributed-Parameter Electromechanical Modeling of Cantilevered Piezoelectric Energy Harvesters. 6.1 Unimorph Piezoelectric Energy Harvester Configuration. 6.2 Electromechanical Euler-Bernoulli Model with Axial Deformations. 6.3 Electromechanical Rayleigh Model with Axial Deformations. 6.4 Electromechanical Timoshenko Model with Axial Deformations. 6.5 Modeling of Symmetric Configurations. 6.6 Presence of a Tip Mass in the Euler-Bernoulli, Rayleigh, and Timoshenko Models. 6.7 Comments on the Kinematically Admissible Trial Functions. 6.8 Experimental Validation of the Assumed-Modes Solution for a Bimorph Cantilever. 6.9 Experimental Validation for a Two-Segment Cantilever. 6.10 Summary. 6.11 Chapter Notes. 7. Modeling of Piezoelectric Energy Harvesting for Various Forms of Dynamic Loading. 7.1 Governing Electromechanical Equations. 7.2 Periodic Excitation. 7.3 White Noise Excitation. 7.4 Excitation Due to Moving Loads. 7.5 Local Strain Fluctuations on Large Structures. 7.6 Numerical Solution for General Transient Excitation. 7.7 Case Studies. 7.8 Summary. 7.9 Chapter Notes. 8. Modeling and Exploiting Mechanical Nonlinearities in Piezoelectric Energy Harvesting. 8.1 Perturbation Solution of the Piezoelectric Energy Harvesting Problem: the Method of Multiple Scales. 8.2 Monostable Duffing Oscillator with Piezoelectric Coupling. 8.3 Bistable Duffing Oscillator with Piezoelectric Coupling: the Piezomagnetoelastic Energy Harvester. 8.4 Experimental Performance Results of the Bistable Peizomagnetoelastic Energy Harvester. 8.5 A Bistable Plate for Piezoelectric Energy Harvesting. 8.6 Summary. 8.7 Chapter Notes. 9. Piezoelectric Energy Harvesting from Aeroelastic Vibrations. 9.1 A Lumped-Parameter Piezoaeroelastic Energy Harvester Model for Harmonic Response. 9.2 Experimental Validations of the Lumped-Parameter Model at the Flutter Boundary. 9.3 Utilization of System Nonlinearities in Piezoaeroelastic Energy Harvesting. 9.4 A Distributed-Parameter Piezoaeroelastic Model for Harmonic Response: Assumed-Modes Formulation. 9.5 Time-Domain and Frequency-Domain Piezoaeroelastic Formulations with Finite-Element Modeling. 9.6 Theoretical Case Study for Airflow Excitation of a Cantilevered Plate. 9.7 Summary. 9.8 Chapter Notes. 10. Effects of Material Constants and Mechanical Damping on Power Generation. 10.1 Effective Parameters of Various Soft Ceramics and Single Crystals. 10.2 Theoretical Case Study for Performance Comparison of Soft Ceramics and Single Crystals. 10.3 Effective Parameters of Typical Soft and Hard Ceramics and Single Crystals. 10.4 Theoretical Case Study for Performance Comparison of Soft and Hard Ceramics and Single Crystals. 10.5 Experimental Demonstration for PZT-5A and PZT-5H Cantilevers. 10.6 Summary. 10.7 Chapter Notes. 11. A Brief Review of the Literature of Piezoelectric Energy Harvesting Circuits. 11.1 AC-DC Rectification and Analysis of the Rectified Output. 11.2 Two-Stage Energy Harvesting Circuits: DC-DC Conversion for Impedance Matching. 11.3 Synchronized Switching on Inductor for Piezoelectric Energy Harvesting. 11.4 Summary. 11.5 Chapter Notes. Appendix A. Piezoelectric Constitutive Equations. Appendix B. Modeling of the Excitation Force in Support Motion Problems of Beams and Bars. Appendix C. Modal Analysis of a Uniform Cantilever with a Tip Mass. Appendix D. Strain Nodes of a Uniform Thin Beam for Cantilevered and Other Boundary Conditions. Appendix E. Numerical Data for PZT-5A and PZT-5H Piezoceramics. Appendix F. Constitutive Equations for an Isotropic Substructure. Appendix G. Essential Boundary Conditions for Cantilevered Beams. Appendix H. Electromechanical Lagrange Equations Based on the Extended Hamilton s Principle. Index.

Abstract: A new frequency modulation (FM) technique has been demonstrated which ennances the sensitivity of attractive mode force microscopy by an order of magnitude or more. Increased sensitivity is made possible by operating in a moderate vacuum ( < 10 ’ Torr), which increases the Q of the vibrating cantilever. In the FM technique, the cantilever serves as the frequency determining element of an oscillator. Force gradients acting on the cantilever cause instantaneous frequency modulation of the oscillator output, which is demodulated with a FM detector. Unlike conventional “slope detection,” the FM technique offers increased sensitivity through increased Q without restricting system bandwidth. Experimental comparisons of FM detection in vacuum (Q50 000) versus slope detection in air (Q100) demonstrated an improvement of more than 10 times in sensitivity for a fixed bandwidth. This improvement is evident in images of magnetic transitions on a thin-film CoPtCr magnetic disk. In the future, the increased sensitivity offered by this technique should extend the range of problems accessible by force microscopy.

Abstract: The spring constant of microfabricated cantilevers used in scanning force microscopy (SFM) can be determined by measuring their resonant frequencies before and after adding small end masses These masses adhere naturally and can be easily removed before using the cantilever for SFM, making the method nondestructive The observed variability in spring constant—almost an order of magnitude for a single type of cantilever—necessitates calibration of individual cantilevers in work where precise knowledge of forces is required Measurements also revealed that the spring constant scales with the cube of the unloaded resonant frequency, providing a simple way to estimate the spring constant for less precise work