Gear Design Symbol Reference (GB/T 2821 / ISO 701)

Overview

Gear design involves a large number of symbols covering geometric dimensions, meshing forces, material properties, and strength safety factors. This reference compiles the complete notation used across all gear calculation standards, based on GB/T 2821-2003 (Gear Terminology) and ISO 701:1998 (International Gear Notation).

Symbols are rendered in LaTeX throughout this article. Subscripted letter combinations (e.g. $K_{H\alpha}$ , $\sigma_{FP}$ ) use inline math notation; simple numeric sub/superscripts use Unicode characters (e.g. $d_1$ , N/mm²).

Chord dimensions — symbols $\hat{h}$ and $\hat{s}$ denote values measured along a chord rather than an arc (chord tooth height and chord tooth thickness respectively).

Geometric Parameters

Symbol	Meaning	Unit
$A$	Bevel gear mounting distance	mm
$A_k$	Outer cone height	mm
$A_a$	Crown-to-back distance	mm
$a$	Centre distance (standard and addendum-modified gears)	mm
$a_w$ ( $a'$ )	Centre distance for angular-modified gears	mm
$b$	Face width	mm
$b_{cal}$	Calculated face width	mm
$b_{eH}$	Effective face width for bevel gear contact strength	mm
$b_{eF}$	Effective face width for bevel gear bending strength	mm
$c$	Bottom clearance	mm
$c^*$	Bottom clearance coefficient	—
$d$	Diameter; reference circle diameter	mm
$d_1$ , $d_2$	Reference circle diameter of pinion, wheel	mm
$d_{a1}$ , $d_{a2}$	Tip circle diameter of pinion, wheel	mm
$d_{b1}$ , $d_{b2}$	Base circle diameter of pinion, wheel	mm
$d_{f1}$ , $d_{f2}$	Root circle diameter of pinion, wheel	mm
$d_w$ ( $d'$ )	Pitch circle diameter	mm
$D_M$ ( $d_p$ )	Pin (ball) diameter for span measurement	mm
$h$	Tooth height	mm
$h_w$ ( $h'$ )	Working tooth depth	mm
$h_a$	Addendum	mm
$h_f$	Dedendum	mm
$h_a^*$	Addendum coefficient	—
$h_{an}^*$	Normal addendum coefficient	—
$h_{at}^*$	Transverse addendum coefficient	—
$h_{aP}$ , $h_{fP}$	Tool tip height and root height (basic rack)	mm
$h_{a0}$	Tool addendum	mm
$h_{a0}^*$	Tool addendum coefficient	—
$h_f$	Dedendum	mm
$h_{f0}$	Tool dedendum	mm
$h_{Fa}$	Bending moment arm — load at tooth tip	mm
$h_{Fe}$	Bending moment arm — load at outer boundary of single-pair contact	mm
$\hat{h}_{amn}$	Bevel gear midpoint normal chord addendum	mm
$\hat{h}_{ca}$	Helical gear constant chord addendum	mm
$\hat{h}_a$	Helical gear pitch circle chord addendum	mm
$\hat{h}$	Pitch circle chord tooth height	mm
$\bar{h}$	Constant chord tooth height	mm
$h'_a$	Bevel gear pitch cone addendum	mm
$h'_f$	Bevel gear pitch cone dedendum	mm
$L$	Length	mm
$m$	Module; equivalent mass	mm; kg/mm
$m_n$	Normal module	mm
$m_t$	Transverse module	mm
$m_{nm}$	Bevel gear midpoint normal module	mm
$p$	Pitch; reference circle pitch	mm
$p_b$	Base circle pitch	mm
$p_{ba}$	Normal base pitch	mm
$p_{bt}$	Transverse base pitch	mm
$p_n$	Normal pitch	mm
$p_t$	Transverse pitch	mm
$q$	Auxiliary coefficient; worm diameter coefficient	—
$q_s$	Root fillet parameter	—
$R$	Cone distance	mm
$R_a$	Arithmetic mean surface roughness	μm
$R'$	Pitch cone distance	mm
$R_i$	Small-end cone distance	mm
$R_m$	Midpoint cone distance	mm
$R_z$	Ten-point surface roughness height	μm
$r$	Radius; reference circle radius	mm
$r_a$	Tip circle radius	mm
$r_b$	Base circle radius	mm
$r_f$	Root circle radius	mm
$s$	Tooth thickness; reference circle tooth thickness	mm
$s_a$	Tip tooth thickness	mm
$s_f$	Root tooth thickness	mm
$s_n$	Normal tooth thickness	mm
$s_t$	Transverse tooth thickness	mm
$s_{Fn}$	Tooth thickness at critical section	mm
$s_0$	Tool tooth thickness	mm
$\hat{s}_n$	Helical gear pitch circle chord tooth thickness	mm
$\hat{s}_{amn}$	Bevel gear midpoint normal chord tooth thickness	mm
$\bar{s}$	Helical gear constant chord tooth thickness	mm
$\hat{s}$	Pitch circle chord tooth thickness	mm
$W$ , $W_k$	Span (base tangent length) over $k$ teeth	mm
$W^*$	Span for module $m = 1$	mm
$x$	Profile shift coefficient (radial)	—
$x_1$ , $x_2$	Profile shift coefficients of pinion, wheel	—
$x_\Sigma$	Sum of profile shift coefficients	—
$x_t$	Tooth thickness variation coefficient; transverse shift coefficient	—
$x_n$	Normal profile shift coefficient	—
$y$	Centre distance modification coefficient	—
$y_0$	Centre distance modification at cutting	—
$\Delta y$	Addendum modification coefficient	—
$z$	Number of teeth	—
$z_1$ , $z_2$	Number of teeth of pinion, wheel	—
$z_n$ , $z_v$	Equivalent number of teeth of helical gear	—
$z_{vm}$	Mean equivalent number of teeth of bevel gear pair	—
$z_p$	Number of teeth of planar gear	—
$z_0$	Number of teeth of tool	—

Angles and Helix Parameters

Symbol	Meaning	Unit
$\alpha$	Pressure angle; tooth profile angle	°, rad
$\alpha_n$	Normal reference-circle pressure angle	°, rad
$\alpha_t$	Transverse reference-circle pressure angle	°, rad
$\alpha'$	Working pressure angle	°, rad
$\alpha'_t$	Transverse working pressure angle	°, rad
$\alpha_A$	Tip circle pressure angle	°, rad
$\alpha_{an}$	Normal tip pressure angle	°, rad
$\alpha_{at}$	Transverse tip pressure angle	°, rad
$\alpha_{Fan}$	Normal load angle at tooth tip	°, rad
$\alpha_{Fat}$	Transverse load angle at tooth tip	°, rad
$\alpha_{Fen}$	Normal load angle at outer boundary of single-pair contact	°, rad
$\alpha_{Fet}$	Transverse load angle at outer boundary of single-pair contact	°, rad
$\alpha_M$	Pressure angle at pin (ball) centre on involute	°, rad
$\alpha_m$	Bevel gear midpoint equivalent-gear reference pressure angle	°, rad
$\alpha'_m$	Bevel gear midpoint equivalent-gear working pressure angle	°, rad
$\alpha_y$	Pressure angle at any point $y$	°
$\alpha_0$	Tool tooth profile angle; bevel gear profile angle	°
$\alpha'_0$	Working pressure angle during cutting	°, rad
$\beta$	Reference-circle helix angle; transverse tooth profile angle	°, rad
$\beta_b$	Base helix angle	°, rad
$\beta_e$	Helix angle at outer boundary of single-pair contact	°, rad
$\gamma$	Auxiliary angle	°, rad
$\delta$	Pitch (reference) cone angle	°, rad
$\delta_a$	Tip cone angle	°, rad
$\delta_f$	Root cone angle	°, rad
$\theta_a$	Addendum angle	°, rad
$\theta_f$	Dedendum angle	°, rad
$\Sigma$	Shaft angle	—

Contact Ratio and Involute Function

Symbol	Meaning	Unit
$\varepsilon_\alpha$	Transverse contact ratio	—
$\varepsilon_\beta$	Overlap (longitudinal) contact ratio	—
$\varepsilon_\gamma$	Total contact ratio	—
$\text{inv}\,\alpha$	Involute function of angle $\alpha$	—
$j$	Backlash	mm

Forces and Loads

Symbol	Meaning	Unit
$F_t$	Nominal tangential force on reference circle, transverse plane	N
$F_{tm}$	Tangential force at midpoint of bevel gear	N
$F_{tH}$	Tangential force for $K_{H\alpha}$ calculation	N
$F_{bt}$	Nominal tangential force on base circle, transverse plane	N
$F_{bn}$	Nominal tangential force on base circle, normal plane	N
$F_n$	Normal force	N
$F_r$	Radial force	N
$F_x$	Axial force	N
$w_m$	Mean load per unit face width	N/mm
$w_{max}$	Maximum load per unit face width	N/mm
$P$	Power	kW
$T_1$ , $T_2$	Nominal torque of pinion, wheel	N·m
$M$	Bending moment; pin-measurement distance	N·m

Load Factors

Symbol	Meaning	Unit
$K$	Overall load factor	—
$K_A$	Application factor	—
$K_V$	Dynamic factor	—
$K_{H\alpha}$	Transverse load distribution factor (contact)	—
$K_{H\beta}$	Helix load distribution factor (contact)	—
$K_{F\alpha}$	Transverse load distribution factor (bending)	—
$K_{F\beta}$	Helix load distribution factor (bending)	—
$K_m$	Wear factor for open gear drives	—

Tooth Stiffness

Symbol	Meaning	Unit
$c_\gamma$	Mean total tooth stiffness per unit face width (mesh stiffness)	N/(mm·μm)
$c'$	Maximum stiffness of a single tooth pair per unit face width	N/(mm·μm)
$q$	Single tooth-pair flexibility per unit face width	μm·mm/N
$g_{va}$	Equivalent meshing line length of bevel gear	—

Tooth Form Factors (Y-Factors)

Symbol	Meaning	Unit
$Y_F$	Tooth form factor — load at outer boundary of single-pair contact	—
$Y_{Fa}$	Tooth form factor — load at tooth tip	—
$Y_{Fs}$	Combined tooth form factor	—
$Y_S$	Stress correction factor — load at outer boundary	—
$Y_{Sa}$	Stress correction factor — load at tooth tip	—
$Y_{ST}$	Stress correction factor of test gear	—
$Y_\varepsilon$	Contact ratio factor (bending)	—
$Y_\beta$	Helix angle factor (bending)	—
$Y_{NT}$	Life factor (bending)	—
$Y_X$	Size factor (bending)	—
$Y_K$	Bevel gear factor (bending)	—
$Y_{RrelT}$	Relative surface condition factor	—
$Y_{\delta relT}$	Relative root fillet sensitivity factor	—

Zone and Elastic Factors (Z-Factors)

Symbol	Meaning	Unit
$Z_H$	Zone factor	—
$Z_E$	Elasticity factor	$\sqrt{\mathrm{N/mm^2}}$
$Z_\varepsilon$	Contact ratio factor (contact)	—
$Z_\beta$	Helix angle factor (contact)	—
$Z_B$ , $Z_D$	Single-pair contact factor of pinion, wheel	—
$Z_L$	Lubricant factor	—
$Z_v$	Velocity factor	—
$Z_R$	Roughness factor	—
$Z_W$	Work hardening factor	—
$Z_{NT}$	Life factor (contact)	—
$Z_X$	Size factor (contact)	—
$Z_K$	Bevel gear factor (contact)	—

Material and Strength Properties

Symbol	Meaning	Unit
$E$	Elastic modulus (Young's modulus)	N/mm²
$G$	Shear modulus	N/mm²
$\nu$	Poisson's ratio	—
$\rho$	Density; radius of curvature	kg/mm³, mm
$\rho_{fp}$	Root fillet radius of basic rack	mm
$\rho_F$	Root fillet radius at critical section	mm
$\rho_f$	Root fillet radius	mm
HB	Brinell hardness	—
HRC	Rockwell hardness C-scale	—
HV1	Vickers hardness at $F = 9.8\,\text{N}$	—
HV10	Vickers hardness at $F = 98.1\,\text{N}$	—
$\sigma_b$	Ultimate tensile strength	N/mm²
$\sigma_F$	Calculated tooth-root bending stress	N/mm²
$\sigma_{F0}$	Basic value of tooth-root stress	N/mm²
$\sigma_{FE}$	Reference bending stress (material property)	N/mm²
$\sigma_{FG}$	Permissible peak bending stress	N/mm²
$\sigma_{FP}$	Allowable tooth-root stress	N/mm²
$\sigma_{Flim}$	Endurance bending stress limit of test gear	N/mm²
$\sigma_H$	Calculated contact stress	N/mm²
$\sigma_{H0}$	Basic value of contact stress	N/mm²
$\sigma_{HG}$	Permissible peak contact stress	N/mm²
$\sigma_{Hp}$	Allowable contact stress	N/mm²
$\sigma_{Hlim}$	Endurance contact stress limit of test gear	N/mm²

Safety Factors and Life

Symbol	Meaning	Unit
$S_F$	Calculated safety factor for bending	—
$S_{Fmin}$	Minimum required safety factor for bending	—
$S_H$	Calculated safety factor for contact	—
$S_{Hmin}$	Minimum required safety factor for contact	—
$N_L$	Number of stress cycles	—
$N_c$	Number of cycles at endurance limit	—
$N_0$	Maximum cycle count for static strength	—

Kinematic Parameters

Symbol	Meaning	Unit
$i$	Gear ratio (speed ratio)	—
$u$	Tooth ratio $u = z_2/z_1 \geq 1$	—
$n$	Rotational speed	r/min
$n_1$ , $n_2$	Rotational speed of pinion, wheel	r/min
$n_E$	Critical speed	r/min
$n_{E1}$	Critical speed of pinion	r/min
$n_p$	Number of planet gears	—
$N$	Critical speed ratio; exponent	—
$\upsilon$	Peripheral velocity; pitch-line velocity	m/s
$\eta$	Sliding ratio; efficiency	—
$\nu$	Kinematic viscosity of lubricant	mm²/s (cSt)

Planetary Gear Quantities

Symbol	Meaning	Unit
$m_p$	Equivalent mass of planet gear	kg/mm
$m_s$	Equivalent mass of sun gear	kg/mm
$m_{red}$	Reduced (induced) mass	kg/mm
$\Theta_1$ , $\Theta_2$	Moment of inertia of pinion, wheel	kg·mm²

Tooth Run-in Allowances

Symbol	Meaning	Unit
$C_a$	Tip relief amount	μm
$C_{ay}$	Tip relief amount from run-in	μm
$y_\alpha$	Profile run-in allowance	μm
$y_\beta$	Helix run-in allowance	μm
$x_\beta$	Helix run-in coefficient	—

Gear Error and Deviation

Symbol	Meaning	Unit
$F_\beta$	Total helix deviation allowance	μm
$F_{\beta x}$	Total initial helix deviation allowance	μm
$F_{\beta y}$	Total helix deviation allowance after run-in	μm
$f_{f\alpha}$	Tooth profile form deviation allowance	μm
$f_{ma}$	Helix deviation component from manufacturing/assembly error	—
$f_{pb}$	Base pitch limit deviation (may use $f_{pt}$ from GB/T 10095.1)	μm

Miscellaneous Coefficients

Symbol	Meaning	Unit
$C$	Pitch point; transmission accuracy factor; coefficient	—
$C_B$	Basic rack coefficient	—
$C_Q$	Blank structure coefficient	—
$e$	Auxiliary quantity	—
$k$	Number of teeth spanned; number of spaces (internal gear)	—
$\psi$	Geometric pressure coefficient; tooth thickness ratio	—
$\psi_a$	Face width coefficient relative to centre distance	—
$\psi_d$	Face width coefficient relative to reference circle diameter	—

Subscripts

Subscript	Meaning
A	Of sun gear (planetary)
B	Of ring gear (internal)
C	Of planet gear
$v$ , $n$	Equivalent
X	Of planet carrier
0	Of tool / cutter
1	Of pinion; of worm
2	Of wheel; of worm gear
Ⅰ	High-speed stage
Ⅱ	Low-speed stage

Notes

Geometric element symbols follow GB/T 2821-2003 and ISO 701:1998.
Gear accuracy deviation symbols are defined separately in GB/T 10095-2008 and are not listed here.
Symbols for worm drives, pin-tooth drives, and flexible-tooth drives are not included.
Chord dimension symbols $\hat{h}$ and $\hat{s}$ denote values measured along a chord; $\bar{h}$ and $\bar{s}$ denote constant-chord values.

References

GB/T 2821-2003 — Gear Terminology
ISO 701:1998 — International Gear Notation: Symbols for Geometrical Data
GB/T 10095-2008 — Cylindrical Gears: Accuracy Requirements
GB/T 3480-1997 — Calculation of Load Capacity of Spur and Helical Gears

Try the Calculators

These symbols appear throughout the gear calculators on this site:

Gear Ratio Calculator — $i$ , $z_1$ , $z_2$ , $n$ , $T$ , $\eta$
Gear Module Calculator — $m$ , $z$ , $d$ , $a$ , $h_a$ , $h_f$
Bevel Gear Calculator — $\delta$ , $R$ , $m_{nm}$ and bevel-specific symbols
Gear Undercut Check — $x$ , $z$ , $\alpha$ , $h_a^*$
Base Tangent Length — $W_k$ , $k$ , $\text{inv}\,\alpha$ , $m$ , $z$