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
\( f_{sc} \)	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 —

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. 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".
4. Units generally conform to the MKS system, particularly when used in equations. However, other units may conveniently be used (e.g., mm in drawings).
5. 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.
6. 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.)