Gear Ratio and Involute Gear Calculator

For each external mesh, the reduction convention is $r=z_{driven}/z_{driver}=n_{in}/n_{out}$. A compound train multiplies stage ratios and efficiencies; $T_{out}=T_{in}r\eta$ and $P=T\omega$. Speeds and torques are magnitudes; each external mesh reverses rotation. This follows KHK Gear Technical Reference, attached page 5, Table 2.1 (printed page 600).

The involute is $\operatorname{inv}\alpha=\tan\alpha-\alpha$, with the subtracted angle in radians. Solve $\operatorname{inv}\alpha_w=\operatorname{inv}\alpha+2\tan\alpha(x_1+x_2)/(z_1+z_2)$ by bracketed bisection. Then $a_w=a_0\cos\alpha/\cos\alpha_w$ and $y=(a_w-a_0)/m$. Reference and base radii are $mz/2$ and $(mz/2)\cos\alpha$. KHK attached pages 6–9 provide the involute, undercut and shifted-pair equations.

With $k=x_1+x_2-y$, the KHK option uses $h_{a,j}=m(1+x_j-k)$; the unshortened option uses $h_{a,j}=m(1+x_j)$. Both use $r_{f,j}=m(z_j/2-1.25+x_j)$. KHK Table 4.3 preserves radial clearance $0.25m$ by shortening both tips. KHK Table 3.5 describes the unshortened individual-gear convention; KG Table 15's published contact ratio is consistent with that convention.

The contact path is $L=\sqrt{r_{a1}^2-r_{b1}^2}+\sqrt{r_{a2}^2-r_{b2}^2}-a_w\sin\alpha_w$ and $\varepsilon_\alpha=L/(\pi m\cos\alpha)$. Radii, rather than diameters, belong inside these square roots (KG attached page 2). Top-land arc thickness follows the involute tooth half-angle. The full-depth rack undercut screen is $x\ge1-z\sin^2\alpha/2$. Mesh force is the ideal input torque divided by working pitch radius; separating force is $F_t\tan\alpha_w$.

Vehicle speed follows wheel rotations times rolling circumference: wheel RPM = engine RPM divided by transmission and final-drive ratios. The cruise calculation reverses this relation. Explicit unit conversions handle RPM, angular speed, power and road speed. The vehicle model follows the TREMEC Toolbox instructions.

• External involute spur gears share module and reference pressure angle; parallel, rigid shafts and a standard full-depth rack with addendum coefficient 1 and dedendum coefficient 1.25.
• Working center distance is the theoretical zero-backlash distance implied by the chosen shifts. Existing gears with independently fixed center distance or tooth thinning need a different backlash calculation.
• Profile shifts and efficiency values are user inputs, not material or catalogue recommendations. The optional second stage contributes ratio and losses only; its tooth geometry is not checked.
• The drawing samples involute flanks with ten points per side and joins them to root-circle arcs using radial connectors. It omits trochoidal fillets, cutter-tip radius, undercut removal and face width.
• The play input advances one tooth pitch per cycle at an illustrative rate independent of entered RPM. Only the drawing transforms depend on it; outlines and engineering calculations remain cached.
• Steady torque, constant estimated efficiency and no slip. Tire diameter is an effective rolling diameter; six default car ratios are illustrative.

Use for preliminary geometry and kinematics, not gear strength certification or manufacturing export. Ratios remain valid when warnings flag undercut, but the reported ideal involute contact ratio may then overestimate actual usable engagement. Increase tooth count or shift and check a cutter-generated profile before selecting such gears.

Errors suppress results for a non-real working angle, pointed teeth, an invalid tip/root arrangement, below-base-circle or root interference, inadequate root clearance, or contact ratio at or below one. A top land below 0.25 module and a contact ratio below 1.2 are advisory screening thresholds chosen here, not compliance criteria. Positive top-land thickness alone does not guarantee manufacturability.

The model excludes elastic deflection, dynamic loads, material strength, heat treatment, lubrication, wear, actual backlash, bearing loads beyond ideal mesh forces and thermal growth. It implements manufacturer geometry equations and makes no claim of ISO, DIN, AGMA or JIS design compliance. Check against the standard in force when specifying a production gear.

Vehicle results assume locked mechanical drive, no tire slip or torque-converter slip and constant rolling diameter. Drag, engine power, rev limits, road loads and tire growth determine attainable speed separately.