| Preface | 6 |
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| Contents | 8 |
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| 1 Resonance Frequency | 10 |
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| 1.1 Eigenmodes of Ideal Continuum Mechanical Structures | 11 |
| 1.1.1 One-Dimensional Bending Vibrations | 15 |
| 1.1.1.1 Free Bending Vibration of Beams | 16 |
| 1.1.1.2 Free Bending Vibration of Beams Under Tensile Stress (Strings) | 23 |
| 1.1.2 One-Dimensional Bulk Vibrations | 26 |
| 1.1.3 Two-Dimensional Bending Vibrations | 29 |
| 1.1.3.1 Free Bending Vibration of Plates | 30 |
| 1.1.3.2 Free Bending Vibration of Plates Under Tensile Stress (Membranes) | 34 |
| 1.1.4 Torsional Vibration of Thin Beams | 36 |
| 1.2 Lumped-Element Model Resonator | 38 |
| 1.2.1 Damped Linear Resonator | 38 |
| 1.2.1.1 Free Undamped Vibration | 39 |
| 1.2.1.2 Free Damped Vibration | 40 |
| 1.2.1.3 Driven Damped Vibration | 41 |
| 1.2.1.4 Quality Factor | 44 |
| 1.2.1.5 Effective Parameters | 47 |
| 1.2.1.6 Torsional Paddle Resonator | 49 |
| 1.2.2 Coupled Linear Resonators | 51 |
| 1.2.3 Damped Nonlinear Resonators | 54 |
| 1.2.3.1 Sources of Nonlinearity | 55 |
| 1.2.3.2 Solving the Nonlinear Equation of Motion | 59 |
| References | 63 |
| 2 Quality Factor | 66 |
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| 2.1 Medium Interaction Losses | 67 |
| 2.1.1 Liquid Damping | 67 |
| 2.1.1.1 Resonator Immersed in Liquid | 67 |
| 2.1.1.2 Liquid Inside the Resonator | 68 |
| 2.1.2 Gas Damping | 70 |
| 2.1.2.1 Fluidic Regime (Kn | 70 |
| 72 | 70 |
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| 2.1.2.2 Ballistic Regime (Kn | 70 |
| 74 | 70 |
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| 2.2 Clamping Loss | 75 |
| 2.2.1 Cantilever Beams | 76 |
| 2.2.2 Membranes | 77 |
| 2.3 Intrinsic Damping | 78 |
| 2.3.1 Intrinsic Damping Mechanisms | 78 |
| 2.3.1.1 Friction Losses | 79 |
| 2.3.1.2 Fundamental Losses | 86 |
| 2.3.2 Damping Dilution in Strings and Membranes | 90 |
| 2.3.2.1 Damping Dilution in Strings | 91 |
| 2.3.2.2 Damping Dilution in Membranes | 94 |
| References | 97 |
| 3 Responsivity | 100 |
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| 3.1 Frequency Response to Mass | 101 |
| 3.1.1 Point Mass | 102 |
| 3.1.1.1 Strings | 104 |
| 3.1.1.2 Beams | 106 |
| 3.1.2 Distributed Mass | 108 |
| 3.2 Amplitude and Frequency Response to Force | 110 |
| 3.2.1 Amplitude Response to a Force | 110 |
| 3.2.1.1 Quasi-Static Force Sensing (??) | 111 |
| 3.2.1.2 Resonant Force Sensing (?=?) | 111 |
| 3.2.2 Frequency Response to a Force Gradient | 111 |
| 3.2.2.1 Frequency Response to an Electrostatic Potential | 112 |
| 3.3 Frequency Response to Ambient Temperature and Local Heating | 114 |
| 3.3.1 Stress Released Resonators | 115 |
| 3.3.2 Resonators Under Tensile Stress (Strings) | 117 |
| 3.3.2.1 Ambient Temperature | 117 |
| 3.3.2.2 Local Heating at String Center | 120 |
| References | 122 |
| 4 Transduction | 124 |
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| 4.1 Electrodynamic (Actuation and Detection) | 125 |
| 4.1.1 Lorentz Force on a Straight Wire | 126 |
| 4.1.2 Electrodynamically Induced Voltage (Electromotive Force) | 127 |
| 4.2 Electrostatic (Actuation and Detection) | 128 |
| 4.2.1 Electrostatic Forces | 129 |
| 4.2.1.1 Forces Between Electrodes | 130 |
| 4.2.1.2 Dielectric Polarization Force | 132 |
| 4.2.2 Capacitively Induced Current | 135 |
| 4.2.3 Other Capacitive Detection Schemes | 140 |
| 4.3 Thermoelastic (Actuation) | 140 |
| 4.4 Piezoresistive (Detection) | 141 |
| 4.5 Piezoelectric (Actuation and Detection) | 143 |
| 4.5.1 Piezoelectric Actuation | 145 |
| 4.5.2 Piezoelectric Detection | 146 |
| 4.6 Optic (Actuation and Detection) | 147 |
| 4.6.1 Optical Forces | 147 |
| 4.6.2 Interferometric Detection | 148 |
| 4.6.3 Beam Deflection Detection | 150 |
| 4.6.3.1 Optical Leverage | 150 |
| 4.6.3.2 End-Coupled Optical Waveguide | 151 |
| 4.6.4 Plasmonic Detection | 151 |
| References | 152 |
| 5 Measurement and Noise | 157 |
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| 5.1 Amplitude Noise | 157 |
| 5.1.1 Fundamentals | 158 |
| 5.1.1.1 Transduction Chain Noise Transfer | 158 |
| 5.1.1.2 Noise Referred to Input (RTI) | 158 |
| 5.1.2 Thermomechanical Fluctuations | 159 |
| 5.1.2.1 Amplitude Calibration | 162 |
| 5.1.3 Transduction Related Noise | 163 |
| 5.1.3.1 Johnson–Nyquist Thermal Noise | 163 |
| 5.1.3.2 Shot Noise | 165 |
| 5.1.3.3 Hooge (1/f) ``Flicker'' Noise | 166 |
| 5.1.3.4 Noise Equivalent Circuit | 167 |
| 5.1.4 Amplifier Noise | 167 |
| 5.1.4.1 Noise Figure and Noise Temperature | 170 |
| 5.2 Frequency Noise | 171 |
| 5.2.1 Phase-Locked Loop | 171 |
| 5.2.2 Self-Sustained Oscillator | 173 |
| 5.2.3 Allan Variance | 176 |
| References | 178 |
| Index | 181 |
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