Gear Base Pitch
Calculate the base pitch of an involute gear, p_b = π·m·cos(φ), from the module m (mm) and the pressure angle φ (degrees). The base pitch is the distance between two homologous flanks of consecutive teeth, measured along the base circle (or, equivalently, along the line of action) — different from the circular pitch (π·m), measured on the pitch circle. The base pitch is a FUNDAMENTAL property of involute meshing for an elegant reason: for two meshes to transmit motion correctly, they must have the SAME base pitch — it is the conjugacy condition of involute profiles. Moreover, the base pitch appears directly in the CONTACT RATIO (the average number of teeth in simultaneous contact, found by dividing the line-of-action length by the base pitch): a contact ratio above 1 (ideally above 1.4) ensures there is always at least one tooth pair meshed, transmitting motion continuously and smoothly, without impacts. The base pitch is also the basis of checking gears 'over two pins' or by span measurement (W over teeth), classic dimensional-control methods. It is an essential parameter in gear geometry and metrology. Enter the module and the pressure angle.
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Gear base pitch
The base pitch of an involute gear is p_b = π·m·cos(φ), built from the module m and the pressure angle φ. It is the distance between two corresponding flanks of consecutive teeth, measured along the base circle (or, equivalently, along the line of action) — unlike circular pitch (π·m), which gets measured on the pitch circle. Base pitch stands as a fundamental property of involute gearing for an elegant reason: two meshing gears transmit motion correctly only if they share the same base pitch — the conjugate-action condition for involute profiles. Base pitch further enters directly into the contact ratio (the average number of teeth in mesh at once, obtained by dividing the length of the line of action by the base pitch): a contact ratio greater than 1, and ideally above 1.4, guarantees that at least one pair of teeth stays engaged at all times, transmitting motion continuously and smoothly, with no impact at each tooth handover. Base pitch also underlies gear inspection over two pins and the W measurement across teeth (span measurement with a disc micrometer), classic dimensional control methods. It ranks as an essential parameter in gear geometry and metrology. Enter the module and the pressure angle.
Related Tools
Gear Base Diameter
Calculate the base circle diameter of an involute gear, d_b = d·cos(φ), from the pitch diameter d (mm) and the pressure angle φ (degrees). The base circle is the circle from which the INVOLUTE tooth profile is generated — the standard profile of modern gears. The involute is the curve traced by the tip of a string unwinding from a cylinder: that cylinder is exactly the base circle. The entire active tooth profile (the part that actually transmits force) is ABOVE the base circle; below it there is no involute profile. The base diameter is fundamental in gear geometry because it defines the involute profile and, with it, key properties: the LINE OF ACTION (the line tangent to both base circles of the mesh, along which tooth contact travels, always in the same direction — why involute gears transmit uniform motion), the base pitch and the contact ratio. The relation d_b = d·cos(φ) shows that the pressure angle is the angle between the line of action and the tangent to the pitch circles. It is an essential parameter in designing and manufacturing (generating) involute gears. Enter the pitch diameter and the pressure angle.
Minimum Pinion Teeth
Calculate the minimum number of pinion teeth to avoid interference, z_min = 2 ÷ sin²(φ), from the pressure angle φ (degrees). Interference is a geometric problem occurring when gears with FEW teeth mesh: the pinion tooth flank (the part below the base circle, where the involute profile does not exist) collides with the larger gear's tooth tip, causing vibration, noise, rapid wear or jamming. To avoid it, the pinion needs a minimum tooth count depending on the pressure angle: LARGER pressure angles ('fatter' teeth at the root) allow pinions with FEWER teeth without interference. For the standard 20° pressure angle, the theoretical minimum is about 17-18 teeth; for 14.5° (old standard), about 32; for 25°, about 12. When a pinion with fewer than the minimum is needed (for a high gear ratio in little space), profile CORRECTION (profile shift, corrected teeth) or undercut (root relief) is used, avoiding interference at the cost of weakening the tooth. This calculation is fundamental in designing a gear pair's geometry. Enter the pressure angle.
Tooth Thickness at Pitch Circle
Calculate the tooth thickness measured at the pitch circle of a standard gear, s = (π·m) ÷ 2, from the module m (mm). In a standard (uncorrected) gear, the circular pitch (the distance from one tooth to the next, along the pitch circle) is p = π·m, and it splits equally between the TOOTH (the solid part) and the SPACE (the gap between teeth): half each, hence s = π·m/2. This equality between tooth thickness and space width is what lets two standard gears of the same module mesh perfectly, with one's tooth fitting the other's space with proper clearance. Tooth thickness is a fundamental parameter: it sets the tooth STRENGTH (thicker teeth resist bending more) and the mesh backlash. In CORRECTED gears (with profile shift, used to avoid interference in small pinions, adjust center distance or balance pinion-gear strength), the pitch-circle tooth thickness DIFFERS from π·m/2 — it increases in a positively corrected pinion (strengthening it) and decreases in the gear. Measuring tooth thickness (by the chordal method, with a gear-tooth caliper, or over pins) is a classic gear quality-control check. Enter the module.
Gear Contact Stress (Hertz)
Calculate the Hertzian contact stress on gear tooth surfaces, σ_H = C_p·√(F_t ÷ (b·d·I)), from the elastic coefficient C_p (√MPa, a function of the pair's elastic moduli), the tangential force F_t (N), the face width b (mm), the pinion pitch diameter d (mm) and the geometry factor I (dimensionless). This is the basis of SURFACE FATIGUE (pitting) design — the second fundamental gear failure mode, distinct from bending breakage. When two teeth touch, the contact is practically a LINE, and even moderate loads create very high contact stresses (hundreds of MPa) in the tiny contact area, per Hertz theory. Under repeated cycles, these stresses cause sub-surface fatigue that flakes off small bits of material, forming craters (pitting) that progress, destroy the tooth profile, generate noise and vibration and lead to failure. The elastic coefficient C_p gathers the materials' elastic properties (steel-steel, steel-bronze, etc.), and the geometry factor I, the curvature and contact ratio. Contact stress is compared with the material's pitting resistance (which depends strongly on surface HARDNESS — so gears are often case-hardened). It is one of the two central AGMA criteria. Enter the elastic coefficient, tangential force, face width, diameter and geometry factor.
Moisture: Dry / Wet Basis
Convert a food's moisture from wet basis to dry basis, Xdb = Xwb/(1 − Xwb), where Xwb is the water fraction relative to total mass (wet basis) and Xdb relative to dry mass. The dry basis is preferred in drying calculations because the denominator (dry mass) does not change during the process, unlike the total mass. Confusing the two bases is a common and serious error. Enter the wet-basis moisture (fraction).
Bend Allowance (Flat Length)
Calculate the material length consumed in a bend (bend allowance), BA = (π/180)·θ·(R + K·t), from the bend angle θ (degrees), inner bend radius R (mm), thickness t (mm) and K factor (neutral-line factor, typically 0.33-0.5). This is one of the most important — and subtlest — calculations in sheet metal work: to make a bent part to correct dimensions, you must know the FLAT sheet (blank) size before bending. The catch is that, on bending, the outer face STRETCHES and the inner face COMPRESSES, and there is an intermediate line — the neutral line — that does not change length. The K factor locates that neutral line within the thickness (not exactly in the middle, but shifted inward, so K < 0.5). The total developed length is the sum of the straight flanges plus each bend's allowance. Getting this wrong makes out-of-size parts — a costly production error. So blank development (with K factors calibrated by material and process) is a critical step in sheet-part design, now automated in sheet-metal CAD. Enter the angle, inner radius, thickness and K factor.
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