Abbreviations and symbols

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
\( f_m \) 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
\( \overline{I} \) 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
\( S_n \) 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

Machining

Ra Roughness, average
TIR Total indicator reading

Other

\( m \) Mass kg
\( k \) Stiffness N/m
[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:

  1. 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.
  2. 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.
  3. 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).
  4. 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.)