Abbreviations and symbols
Contents
Material properties
| \( E \) | Modulus of elasticity (Young's modulus) | Pa |
| \( G \) | Shear modulus (modulus of rigidity) | Pa |
| \( \nu \) [nu] | Poisson's ratio | ——— |
| \( \rho \) [rho] | Density | kg/m3 |
| \( \alpha \) [alpha] | Coefficient of thermal expansion | 1/°C |
| \( HRC \), \( R_c \) | Rockwell hardness (C scale) | ——— |
| \( HV \) | Vickers hardness | ——— |
| \( c_{tw} \) | Thin wire wave speed | m/sec |
| \( c_d \) | Dilatational wave speed | m/sec |
| \( c_s \) | Shear wave speed | m/sec |
| \( c_{tp} \) | Thin-plate wave speed | m/sec |
| \( Q_{material} \) | Q of the specified material at a specified frequency and strain | ——— |
Electrical
| \( L \) | Inductance | H |
| \( C \) | Capacitance | F |
| \( R \) | Resistance | ohm |
| \( Y \) | Admittance | ohm |
| \( Z \) | Impedance | ohm |
| \( v \), \( V \) | Voltage | volt |
| \( i \), \( I \) | Current | ampere or amp |
| \( q \) | Charge | coulomb |
| \( E \) | Electric field strength | volt/m |
| \( \varepsilon \) [epsilon] | Permittivity | F/m |
| \( \varepsilon_o \) | Permittivity of free space | F/m |
| \( K \) | Dielectric constant | ——— |
Stress & strain
| \( \sigma \) [sigma] | Tensile or compressive stress | Pa |
| \( \epsilon \) [epsilon] | Tensile or compressive strain | ——— |
| \( \tau \) [tau] | Shear stress | Pa |
| \( \gamma \) [gamma] | Shear strain | ——— |
| \( \sigma_{ij} \) [sigma] | Stress on i plane acting in j direction (see note) | Pa |
| \( \epsilon_{ij} \) [epsilon] | Strain on i plane acting in j direction (see note) | ——— |
| \( \sigma_v \) [sigma] | von Mises stress | Pa |
| \( k_t \) | Stress concentration factor | ——— |
Piezoelectric ceramics & transducer
| \( T_c \) | Curie temperature | °C |
| \( d \) | Charge constant or strain constant | m/volt |
| \( g \) | Voltage constant | m2/coulomb |
| \( \kappa \) [kappa] | Electromechanical coupling coefficient | ——— |
| \( \kappa_{eff} \) [kappa] | Electromechanical coupling coefficient (effective) | ——— |
| \( C_o \) | Piezoelectric clamped (blocked) capacitance | F |
| \( tan\delta \) [tan delta] | Dielectric loss tangent | ——— |
| \( T \) | Stress (piezoelectric analysis only) | Pa |
| \( S \) | Strain (piezoelectric analysis only) | ——— |
| \( E \) | Electric field strength | volt/m |
| \( D \) | Dielectric displacement | coulomb |
| \( Y \) | Young's modulus (piezoelectric analysis only) | Pa |
| \( s \) | Compliance (piezoelectric analysis only) | 1/Pa |
| \( f_s \) | Series resonance frequency | Hz, kHz |
| \( f_p \) | Parallel resonance frequency | Hz, kHz |
| \( f_{sc} \) | Short circuit resonance frequency | Hz, kHz |
| \( f_{oc} \) | Open circuit resonance frequency | Hz, kHz |
| \( f_m \) | Frequency of minimum (absolute value) impedance | Hz, kHz |
| \( f_n \) | Frequency of maximum (absolute value) impedance | Hz, kHz |
Performance
| \( f \) | Frequency | Hz, kHz |
| \( f_r \) | Resonant frequency (generic) | Hz, kHz |
| \( u \), \( U \) | Amplitude (displacement) | µ [peak], m |
| \( \overline{U} \) [u bar] | Average amplitude | µ [peak], m |
| \( \dot{u} \), \( \dot{U} \) [u dot] | Velocity | m/sec |
| \( \ddot{u} \), \( \ddot{U} \) [u double-dot] | Acceleration | m/sec2 |
| \( \widehat{U} \) [U hat] | Uniformity of amplitude | ——— |
| \( \widehat{A} \) [A hat] | Asymmetry of amplitude | ——— |
| \( G \) | Gain | ——— |
| \( p \), \( P \) | Power | W or kW |
| \( I \) | Acoustic power intensity | W/m2 |
| \( \overline{I} \) [I bar] | Average acoustic power intensity | W/m2 |
| \( \varphi \) | tuning rate | Hz/mm |
| \( Q_{stack} \) | Q of the stack | ——— |
| \( \delta \) | Log decrement | ——— |
Waves
| \( \lambda \) [lambda] | Wavelength | m |
| \( \Gamma \) [gamma] | Half wavelength | m |
| \( \Gamma_{tw} \) [gamma] | Thin-wire half-wavelength | m |
| \( k_w \) | Wave number | 1/m |
| \( c_{eff} \) | Wave speed (effective) | m/sec |
Fatigue
| \( N \) | Number of cycles to failure | Cycles |
| \( S \) | Stress | Pa |
| \( R \) | Stress ratio | Pa |
| \( S_n \) | Endurance limit (fatigue limit) | Pa |
| \( {S'}_n \) | Endurance limit for R.R. Moore low frequency rotating bending tests | Pa |
| \( K_f \) | Fatigue notch factor | ——— |
Math
| ≈ | Approximately equal |
| α | Proportional to |
| Δ [delta] | Small increment or change |
| ∑ [sigma] | Summation |
| ∫ | Integral |
| \( \ln \) | Natural logarithm |
| \( e \) (Napierian base) | 2.71828183... |
| \( \overline{Z} \) [character overlined] | 1. Average 2. Magnitude of a complex number |
| \( Z' \) [character prime] | Real part of a complex number |
| \( Z'' \) [character doubleprime] | Imaginary part of a complex number |
Statistics
| \( s \) | Standard deviation |
| \( R^2 \) | Coefficient of determination |
Geometry
| \( r \), \( R \) | Radius | mm |
| \( d \), \( D \) | Diameter | mm |
| Ø | Diameter | mm |
| ØOD, O.D. | Outside diameter | mm |
| ØID, I.D. | Inside diameter | mm |
| \( h \) | Height (thickness) | m, mm |
| \( \tilde {A} \) [A tilde] | Area | m2 |
| \( \tilde {V} \) [V tilde] | Volume | m3 |
Machining
| Ra | Roughness, average |
| TIR | Total indicator reading |
Other
| \( m \) | Mass | kg |
| \( k \) | Stiffness | N/m |
| Ṕ [P accent] | Pressure | Pa |
| \( W \) | Energy | N m = joule |
| \( \widehat{W} \) [W hat] | Energy density | N m/m2 = joule/m2 |
| \( KE \) | Kinetic energy | N m = joule |
| \( PE \) | Potential energy | N m = joule |
Units
| Hz | Hertz |
| kHz | kiloHertz |
| µ [mu] | Micron |
| °C | Degrees Celcius (centigrade) |
| N | newton |
| Pa | pascal |
| MPa | Mega-pascal (=106 Pa) |
| GPa | Giga-pascal (=109 Pa) |
| Ω | ohm |
| C | coulomb |
| F | farad [coulomb/volt] |
| H | henry [volt-second/ampere] |
Acronyms
| ASTM | American Society for Testing and Materials |
| AISI | American Iron and Steel Institute |
| CARD | Computer Aided Resonator Design |
| FEA | Finite Element Analysis |
| FEM | Finite Element Method |
| PZT | Lead Zirconate Titinate peizoelectric ceramic |
| RMS | Root mean squared |
Notes —
- These abbreviations and symbols generally conform to those in the literature. However, some have been changed for clarity or because of conflicts with other abbreviations and symbols.
- In some cases the choice of abbreviation or symbol will depend on the topic. For example, discussions of mechanics use \( \sigma \) for stress whereas discussions of piezoelectrics use \( S \) for stress. In fatigue both \( \sigma \) and \( S \) are used. These conflicts occur because of established conventions.
- In some cases the same abbreviation or symbol may have different meanings depending on the context. For example, \( E \) is used for both "modulus of elasticity" and "electric field strength".
- Units generally conform to the MKS system, particularly when used in equations. However, other units may conveniently be used (e.g., mm in drawings).
- When both lowercase and uppercase characters are shown, the lowercase character indicates an instantaneous value whereas the uppercase character indicates a magnitude (e.g., RMS, peak, peak‑to‑peak). For example, \( u \) is the instantaneous amplitude (displacement) whereas \( U \) is the peak amplitude.
- When a stress or strain symbol is followed by double subscripts, the first subscript indicates the associated plane (where the plane is indicated by the axis that is perpendicular to the plane). The second subscript indicates the direction of stress or strain. For example, \( \sigma_{XY} \) indicates a stress acting on the X plane in the Y direction. (See Juvinall, p. 21.)