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A gear ratio states the relation between the rotational motions of connected gears. Because conventions differ, specify the input, output and direction. Here the positive reduction ratio isso $i>1$ means that the output has lower angular speed. For a single external pair on fixed axes, tooth-passage counting givesThe minus sign assumes the same positive rotation convention on both parallel axes. KHK uses the driven-to-driver tooth-count ratio for its speed-reduction examples.^[ref]

Ratios multiply through successive stages of a gear train. Consider a 12-tooth input driving a 36-tooth gear, with a 15-tooth gear rigidly attached to that intermediate rotating body and driving a 45-tooth output. Each stage reduces angular speed by three, so the combined reduction ratio is nine. An input of $1800$ revolutions per minute yields an output of $200$ revolutions per minute. Two external meshes reverse direction twice, so input and output rotate in the same direction.

An intermediate gear used only to pass rotation onwards gives a different result. If a 12-tooth input drives a 36-tooth intermediate gear that itself drives a 48-tooth output, the ratio is $(36/12)(48/36)=4$: the intermediate tooth count cancels. The previous ninefold reduction required two distinct tooth counts rigidly sharing the intermediate rotation.

For steady ideal transmission, conservation of energy gives $|{\tau }_{\mathrm{out}}|=i|{\tau }_{\mathrm{in}}|$. If gear efficiency is $\eta =P_{\mathrm{out}}/P_{\mathrm{in}}$, the corresponding relation is $|{\tau }_{\mathrm{out}}|=\eta i|{\tau }_{\mathrm{in}}|$. With $i=9$, $\eta =0.90$ and input torque $2\mathrm{N}\mathrm{m}$, the output torque is $16.2\mathrm{N}\mathrm{m}$. This power balance assumes steady operation: accelerating rotating components can temporarily store kinetic energy. A measured gear ratio establishes kinematics, while torque capacity and losses require additional mechanical information.

A gear reference circle is the imaginary circle used to specify an individual gear's tooth geometry. For a spur gear, its diameter $d$ is defined by gear module $m$ and tooth count $N$ through $d=mN$. It supplies a common datum for tooth spacing and other dimensions, whether or not that gear is currently meshing with another one. The operating pitch circle, in contrast, describes the mounted pair's motion.^[ref]

For a 30-tooth spur gear with gear module $2\mathrm{mm}$, the reference diameter is $60\mathrm{mm}$. Thus successive teeth repeat at an angular interval of $2\pi /30$ and a gear circular pitch of $2\pi \mathrm{mm}$ on that circle. In the standard full-depth, unshifted geometry specified by KHK, the tooth-tip diameter is $d_a=d+2m$, giving $64\mathrm{mm}$ in this example. Measuring the outside diameter and calling it the reference diameter would therefore produce the wrong gear module.^[ref]

The gear base circle is another distinct geometric datum. For a tooth profile constructed as the involute of a circle, the gear base circle generates the curved tooth flank, whereas the gear reference circle fixes its nominal sizing. Their diameters obey $d_b=d\cos \alpha$, with reference gear pressure angle $\alpha$. With $d=60\mathrm{mm}$ and $\alpha =20{}^{\circ }$, the base diameter is approximately $56.382\mathrm{mm}$. These three diameters—base, reference and tip—answer different questions about the same physical gear.^[ref]

A gear train is an interconnected arrangement of gears that transmits rotation through one or more meshes. Its behavior depends on which gears mesh and which are rigidly attached to the same rotating body. A simple gear train passes motion through successive individual gears; a compound gear train uses at least one intermediate rotating body carrying different gears rigidly together.^[ref]

For a chain of external meshes on fixed parallel axes, each mesh satisfies ${\omega }_{j+1}/{\omega }_j=-N_j/N_{j+1}$. Multiplication gives the overall signed gear ratio; two rigidly joined gears instead share exactly the same angular velocity. These are different connection rules and must not be interchanged.

In a simple gear train with tooth counts 16, 40 and 64, the overall relation isThe intermediate count cancels, and the two reversals give equal input/output rotation directions. That intermediate gear can still change spacing and mounting geometry even though it does not change this reduction ratio.

For a compound gear train, let a 16-tooth input drive a 48-tooth gear, rigidly joined to a 12-tooth gear that drives a 60-tooth output. ThenAn input of $1500$ revolutions per minute produces $100$ revolutions per minute at the output. The two intermediate tooth counts do not cancel because they belong to different gears sharing rotation, rather than one gear engaging both neighbors.

In steady operation, stage gear efficiencies multiply when all intervening losses are included in their stated power boundaries. If the two stages each have efficiency $0.96$, the total is $0.9216$; at input torque $1\mathrm{N}\mathrm{m}$, the output torque is $15(0.9216)=13.824\mathrm{N}\mathrm{m}$. These fixed-axis relations do not automatically describe arrangements whose gear axes move. In all cases, measured speeds test kinematics, while measured power is needed to establish losses.

Gear transverse module $m_t$ specifies tooth spacing in the plane perpendicular to a cylindrical gear's axis. If $N$ is tooth count and $d$ is reference diameter, $m_t=d/N$, so reference-circle gear transverse pitch is $p_t=\pi m_t$. For a helical gear, this transverse section differs from the section normal to the tooth direction.^[ref]

Unwrapping the reference cylinder makes the tooth direction a slanted line. Projecting its normal spacing into the circumferential direction giveswhere $m_n$ is gear normal module and $\beta$ is reference gear helix angle. At zero helix angle the normal and transverse modules coincide, recovering the spur gear case.

For an original example, take $m_n=2.5\mathrm{mm}$, $\beta =30{}^{\circ }$ and $N=24$. Then $m_t\approx 2.88675\mathrm{mm}$ and $d\approx 69.282\mathrm{mm}$. A compatible 48-tooth gear has reference diameter $138.564\mathrm{mm}$, giving standard separation $103.923\mathrm{mm}$. Treating2.5 as the transverse module instead would incorrectly give diameters60 and120mm and separation90mm.

The transverse specification is convenient for reference diameters and axis separations, while the normal specification is convenient for the tooth-normal geometry. A drawing must identify which module and pressure-angle planes it uses; a numerical module value alone is incomplete for a helical gear. These are reference dimensions: a changed operating separation must additionally be interpreted through the mounted geometry, rather than redefining the gear's original module.

A helical gear is a cylindrical gear whose teeth follow a helix around the reference cylinder. The gear helix angle $\beta$ measures their inclination to the axis. In a conventional external pair on parallel axes, mating gears have opposite hands with matching helix-angle magnitudes. Inclination makes engagement progress across the face rather than beginning everywhere along a straight tooth at once, but also creates a gear axial force that the supports must carry.^[ref]

A helical gear must specify its reference plane. The gear normal module $m_n$ is based on a section normal to the tooth direction, whereas gear transverse module $m_t$ is based on the plane perpendicular to the gear axis. Their geometric relation isSimilarly, reference pressure angles satisfy $\tan {\alpha }_t=\tan {\alpha }_n/\cos \beta$. Equal numbers labeled “module” in different planes therefore need not specify interchangeable gears.^[ref]

For an original example, take $m_n=2\mathrm{mm}$, $N=20$ and $\beta =30{}^{\circ }$. Reference diameter is $46.188\mathrm{mm}$, rather than $40\mathrm{mm}$. With ${\alpha }_n=20{}^{\circ }$, the transverse angle is about $22.796{}^{\circ }$.

At standard mounting, a transmitted torque $2\mathrm{N}\mathrm{m}$ gives gear tangential force $F_t\approx 86.60\mathrm{N}$. Ideal contact-force resolution gives axial magnitudeIts direction depends on helix hand and transmitted-torque direction. The corresponding radial component uses the transverse gear pressure angle. This illustrates the design trade: distributed engagement can improve smoothness, but the same transmission requires axial-load support. The example concerns parallel-axis cylindrical gearing; it should not be applied unchanged to crossed-axis arrangements.

An idler gear is an intermediate gear that passes rotation between other gears without using a separately sized, rigidly coupled gear to form another reduction stage. In a simple gear train, its tooth count cancels from the ideal endpoint ratio, but it changes placement, intermediate speed and the number of direction reversals.

For example, a20-tooth input driving a30-tooth idler gear that drives a50-tooth output givesAt750 input revolutions per minute, the idler turns at500 in the opposite direction and the output at300 in the original direction. Replacing the idler by60 teeth preserves output speed but changes idler speed to250 revolutions per minute.^[ref]

With standard module $2\mathrm{mm}$ and collinear centers, the30-tooth idler gives input/output axis spacingThe60-tooth replacement gives190mm, despite the unchanged output ratio.

An idler support reaction also remains necessary even when its net transmitted torque is zero. For the30-tooth arrangement, suppose input torque is $2\mathrm{N}\mathrm{m}$, so both ideal mesh tangential loads are $100\mathrm{N}$. Their moments about the idler's center cancel, but in this collinear layout the two tangential forces point in the same direction on the idler. Equal, opposite radial mesh components cancel, leaving a mesh-induced shaft-support reaction of $200\mathrm{N}$ opposite the tangential forces.

The drawing shows tangential mesh components and their balancing support reaction, rather than physical tooth profiles. Additional shaft loads are outside this ideal balance. The idler therefore affects geometry, forces and potential losses even though its count cancels from the ideal speed ratio.

The operating pitch circles of two meshing gears are the imaginary circles that reproduce their motion by rolling without slipping. They are properties of the working pair and its mounting, whereas a gear reference circle specifies one gear's nominal geometry. Working pitch circle and intermeshing pitch circle are alternative names.^[ref]

Consider an external pair of spur gears on fixed parallel axes, with axis separation $a$ and tooth counts $N_1,N_2$. Their operating radii $r_{w1},r_{w2}$ must satisfy both contact geometry and the tooth-count gear ratio:Solving givesThus the operating radii scale with the actual separation, while the gear ratio stays fixed by tooth counts. These are kinematic equations, conditional on the teeth actually maintaining compatible contact.

For an original example, choose gear module $m=2\mathrm{mm}$ and tooth counts 20 and 40. Reference radii are $20\mathrm{mm}$ and $40\mathrm{mm}$; their standard separation is $60\mathrm{mm}$. If a compatible mounting uses $a=61\mathrm{mm}$ instead, operating radii become $20.333\dots \mathrm{mm}$ and $40.666\dots \mathrm{mm}$. At input angular velocity $30rad/s$ the output has $-15rad/s$, and both operating tangential speed magnitudes are $0.61m/s$. Using the reference radii in that speed calculation would incorrectly give $0.60m/s$.

For tooth flanks formed as involutes, the fixed gear base circles permit constant-ratio motion over a range of separations. If their radii are $r_{b1},r_{b2}$, the common-normal geometry giveswhere ${\alpha }_w$ is the operating pressure angle. Increasing separation therefore increases that angle, while leaving each gear reference circle and gear base circle unchanged. This geometric possibility does not guarantee adequate tooth contact, clearance or strength at an arbitrary separation: the physical tooth extents and mounting must still be checked.^[ref]

The operating pressure angle ${\alpha }_w$ is the gear pressure angle of an actual meshing pair at its mounted axis separation. For external spur gears with involute profiles, it is the angle between the gear line of action and the common tangent to the operating pitch circles. Changing the separation changes this angle even though the individual gear base circles remain fixed.^[ref]

Let the two base radii be $r_{b1},r_{b2}$ and the mounted axis separation be $a$. The right-triangle geometry of the gear line of action givesIf $a_0=r_1+r_2$ is the reference separation and both reference gear pressure angles equal $\alpha$, then $r_{bi}=r_i\cos \alpha$, henceThe gear reference circles and gear ratio remain unchanged by this mounting adjustment; the operating pitch circles expand or contract to fit $a$.

For example, reference radii $20\mathrm{mm}$ and $40\mathrm{mm}$ with $\alpha =20{}^{\circ }$ give $a_0=60\mathrm{mm}$. Mounting a compatible pair at $a=61\mathrm{mm}$ givesAt an imposed gear tangential force $100\mathrm{N}$, the associated gear separating force is about $41.30\mathrm{N}$, compared with $36.40\mathrm{N}$ at the reference angle. Holding torque instead of tangential load would additionally require using the changed operating radius.

The angle formula is only a geometric condition. Usable tooth flanks must still reach the contact region and maintain an adequate gear contact ratio; excessive separation can destroy useful contact. Likewise, the existence of a real inverse cosine does not establish that a proposed mounting has suitable backlash or mechanical strength.^[ref]

A reverted gear train returns the output rotation axis to the input axis, usually through a two-stage compound gear train. The input and output gears can rotate independently on coaxial shafts, while the intermediate gears share another rotating body. Coaxial axes are a layout condition in addition to the desired gear ratio.

For standard unshifted spur gears, let gears1 and2 mesh in the first stage and gears3 and4 in the second, with2 and3 rigidly connected. The common intermediate-axis separation requiresIf both stages use the same gear module, this reduces to $N_1+N_2=N_3+N_4$. The reduction ratio independently satisfies $i=(N_2/N_1)(N_4/N_3)$. Both conditions must be satisfied; multiplying arbitrary stage ratios does not ensure a reverted layout.^[ref]

For an original design example, choose counts20,60,16 and64, with module $2\mathrm{mm}$ in both stages. Both sums are80, so both separations are $80\mathrm{mm}$. The reduction is $3\times 4=12$, taking an input of1440 revolutions per minute to an output of120. Two external meshes give the same input/output rotation direction.

The gears occupy different axial planes so that coaxial input and output do not require them to be one rigidly joined gear. A practical design still needs independent shaft support, compatible tooth geometry in each stage and adequate clearances. Operating with shifted tooth geometry or another mounting changes the applicable separation equations; the simple tooth-sum condition belongs to the stated standard-gear model.

A simple gear train transmits rotation through individual intermediate gears, each intermediate gear engaging its neighbors without a separate, differently sized gear rigidly sharing that intermediate rotation. For external meshes on fixed parallel axes, consecutive signed ratios multiply and intermediate tooth counts cancel:where $m$ is the number of external meshes. An intermediate idler gear changes placement and potentially output direction, but not this ideal endpoint tooth-count ratio.^[ref]

For example, a chain with tooth counts 18, 30, 42 and 72 has three meshes andAt input speed $800$ revolutions per minute, output speed is $200$ revolutions per minute in the opposite direction. Removing one intermediate gear while maintaining a compatible two-mesh layout preserves the fourfold reduction but reverses the direction conclusion.

The cancellation concerns kinematics, not every engineering consequence. Intermediate tooth counts affect physical sizes, axis positions and angular speeds of intermediate rotating bodies; additional meshes and supports can change gear efficiency and backlash. Moreover, if two different intermediate gears are rigidly coupled, their tooth counts do not cancel by the same argument: that arrangement is a compound gear train.

A spur gear is a cylindrical gear with straight teeth parallel to its rotation axis. A conventional external pair transmits motion between parallel axes. Unlike the inclined teeth of a helical gear, the ideal straight-tooth geometry produces no axial thrust from the tooth-normal contact force; there are still tangential and separating loads that the mounting must carry.^[ref]

For compatible spur gears, reference diameter is $d=mN$, where $m$ is gear module and $N$ is tooth count. Standard reference axis separation is $(d_1+d_2)/2$. Two external spur gears have signed angular velocity relation ${\omega }_2/{\omega }_1=-N_1/N_2$: they rotate in opposite directions. Matching gear module alone is insufficient; tooth profiles and gear pressure angles must also be compatible.

For an original example, take gear module $3\mathrm{mm}$ and tooth counts 20 and 60. Reference diameters are $60\mathrm{mm}$ and $180\mathrm{mm}$, so standard axis separation is $120\mathrm{mm}$. Driving the smaller spur gear at $600$ revolutions per minute gives the larger one $200$ revolutions per minute in the opposite direction. In a steady ideal transmission, an input torque of $2\mathrm{N}\mathrm{m}$ becomes $6\mathrm{N}\mathrm{m}$ at the output. The corresponding input gear tangential force is $2/0.030\approx 66.7\mathrm{N}$; multiplying this same force by the output reference radius $0.090\mathrm{m}$ recovers the output torque.

A simple bench demonstration marks one tooth on each spur gear and counts complete turns, testing the three-to-one gear ratio. Measuring transmitted torques and rotational speeds additionally tests gear efficiency. Observing the rotation ratio alone cannot establish a tooth's allowable load or the losses caused by friction and backlash.