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A compound gear train has an intermediate rotating body carrying at least two rigidly joined gears. Those gears share angular velocity but can have different tooth counts, allowing successive reductions to multiply without the cancellation found in a simple gear train. For two external stages with the second and third gears rigidly joined,The two external meshes give equal input/output rotation directions.^[ref]

For an original example, use tooth counts $N_1=18$, $N_2=54$, $N_3=12$ and $N_4=60$. The reduction is $3\times 5=15$. Input speed $1800$ revolutions per minute gives intermediate speed $600$ and output speed $120$ revolutions per minute.

This example also permits a reverted gear train, with input and output axes coaxial. If both stages use the same gear module $m$ and standard unshifted spur gears, the two axis separations must agree:Here both tooth-count sums equal72, so $m=2\mathrm{mm}$ gives a common separation $72\mathrm{mm}$. Ratio multiplication alone would not establish this packaging condition.

In steady ideal operation, output torque is fifteen times input torque. With stage gear efficiencies ${\eta }_1,{\eta }_2$, it becomes $15{\eta }_1{\eta }_2$ times input torque, provided the stage power boundaries include the stated losses. A high reduction can therefore trade compact stages against added losses, loads and backlash. Axis separations and support loads remain separate requirements from the desired numerical gear ratio.

A gear is a toothed mechanical element that transmits motion and torque through contact with another gear or a rack and pinion. The teeth maintain a prescribed relation between the motions: unlike a smooth friction drive, the ideal gear ratio is set by geometry and tooth counts. The gear pitch circle supplies an imaginary rolling geometry for describing that relation; it is not the circle through the tooth tips.^[ref]

For two external gears on fixed parallel axes, let $N_1,N_2$ be their tooth counts and let signed angular velocities ${\omega }_1,{\omega }_2$ use the same orientation. Each tooth on one gear must pass a tooth space on the other, so counting tooth passages givesThus a 20-tooth input driving a 40-tooth output makes the output turn half as fast in the opposite direction. An input of $1200$ revolutions per minute gives an output of $600$ revolutions per minute in magnitude. These signs describe an external pair; a rack and pinion instead converts rotation into translation.

The useful trade is between angular speed and torque. In an ideal steady transmission, conservation of energy equates input and output power, $|{\tau }_1{\omega }_1|=|{\tau }_2{\omega }_2|$. The same 20-tooth/40-tooth pair therefore turns an input torque of $2\mathrm{N}\mathrm{m}$ into an output torque of $4\mathrm{N}\mathrm{m}$ while halving angular speed. This does not create energy: more torque accompanies less angular displacement per input revolution. Real gears dissipate energy through friction, and backlash permits a small lost motion on reversal; tooth strength and bearings, rather than the gear ratio alone, determine allowable load.

Gear axial pitch $p_x$ is the axial repeat distance between corresponding flanks of successive teeth of a cylindrical helical gear. The measurement follows a line parallel to the rotation axis on a concentric cylinder. It is not the axial advance of one tooth helix through an entire revolution; that larger distance is gear lead $L$. For $N$ equally spaced teeth, $L=N{p}_x$.^[ref]

On the developed cylinder, neighboring tooth traces separated transversely by gear transverse pitch $p_t$ intersect a fixed circumferential coordinate at axial positions separated by $p_x$. Their slope givesFor $d=60\mathrm{mm}$, $N=30$ and gear helix angle $|\beta |=30{}^{\circ }$, $p_t=2\pi \mathrm{mm}$ and $p_x=2\pi \sqrt{3}\mathrm{mm}\approx 10.8828\mathrm{mm}$. The corresponding gear normal pitch is only $5.4414\mathrm{mm}$. These two distances differ by a factor of two in this example because $\sin 30{}^{\circ }=1/2$.

For an engaged gear face width $b=20\mathrm{mm}$, the ideal gear overlap ratio is $b/p_x\approx 1.8378$: successive tooth-contact traces span about1.84 axial pitches across the face. Holding $p_t$ fixed while letting $\beta$ tend to zero makes $p_x$ diverge. Straight spur gear tooth traces do not cross an axial generator successively, so a finite axial-pitch interpretation does not survive that limit. Helix hand determines orientation; the positive pitch magnitude alone does not specify hand.

A gear base circle is the fixed circle from which an involute tooth flank is generated. Imagining a taut string unwinding from that circle, the free end traces the involute. It is a geometric construction for the tooth shape, distinct from the gear reference circle used for sizing and the operating pitch circle describing a particular meshing pair.^[ref]

For base radius $r_b$ and unwinding parameter $u\ge 0$, one orientation of the involute isSquaring and adding gives $r=\sqrt{x^2+y^2}=r_b\sqrt{1+u^2}$. The local pressure-angle relation $\tan \alpha =u$ therefore gives $r_b=r\cos \alpha$. At the gear reference circle, this becomesFor reference diameter $d=60\mathrm{mm}$ and reference gear pressure angle $20{}^{\circ }$, the base diameter is $56.382\mathrm{mm}$, approximately. A change in mounting separation does not change this base diameter or the cut tooth profile; it changes which operating geometry describes the mesh.

In a compatible external pair, a common tangent to the two gear base circles is the gear line of action. Their fixed geometry gives $r_{b1}+r_{b2}=a\cos {\alpha }_w$, explaining why the operating pressure angle changes when $a$ changes. Repeating a tooth profile every $2\pi /N$ also gives gear base pitch $p_b=\pi d_b/N$ along the base circumference. Finally, an ideal involute generated this way lies on or outside its gear base circle; a real tooth's lower connecting portion can have a different shape. The gear base circle is not a physical boundary of the gear material.

Gear base pitch is the spacing of successive involute tooth profiles referred to the gear base circle. For base diameter $d_b$, tooth count $N$, gear module $m$ and reference gear pressure angle $\alpha$,It also gives the spacing between successive corresponding tooth contacts measured along the gear line of action. The gear circular pitch uses the reference circle instead, giving $p=\pi m$ and $p_b=p\cos \alpha$.^[ref]

For $m=2\mathrm{mm}$ and $\alpha =20{}^{\circ }$, $p=6.2832\mathrm{mm}$ but $p_b\approx 5.9043\mathrm{mm}$. A second gear with the same module but $25{}^{\circ }$ reference angle has $p_b\approx 5.6945\mathrm{mm}$. Matching reference tooth spacing alone therefore does not make these the same specified mating geometry.

Changing the mounting separation of a compatible involute pair does not change its cut profiles, base circles or gear base pitch. In operating-circle notation, $p_w=2\pi r_w/N$ changes with mounted radius, whilestays fixed. This links the mounted operating pressure angle to an invariant tooth-profile spacing.

For a tooth rotating at angular speed $\omega$, base-circle tangential speed is $r_b|\omega |$. Dividing by gear base pitch gives the gear mesh frequency,A 30-tooth gear at $600$ revolutions per minute gives $f_m=300\mathrm{Hz}$. Finally, the gear contact ratio divides usable gear path of contact length by this base pitch, so profile spacing and available contact length jointly determine overlap between successive tooth engagements.

Gear base-roll speed $u_b$ is the rate at which base-circle rolling advances an ideal involute contact along the gear line of action. For a fixed-axis external pair at constant speed,The equality follows from the tooth-count speed ratio and compatible base pitches. It describes the rate of change of contact position along a geometric line, rather than the gear sliding velocity between material points on the flanks. Base circles and their common tangent define this geometry.^[ref]

For standard module2 gears with20 and40 teeth at20degree reference angle, base radii are $r_{b1}=20\cos 20{}^{\circ }\mathrm{mm}$ and $r_{b2}=40\cos 20{}^{\circ }\mathrm{mm}$. Signed angular velocities ${\omega }_1=50rad/s$ and ${\omega }_2=-25rad/s$ give the same $u_b\approx 0.93969m/s$. The first gear's reference-circle tangential speed is1m/s, so base-roll and pitch-circle speeds are distinct.

In one tooth-entry interval, the base circle advances one gear base pitch:With $f_m\approx 159.155\mathrm{Hz}$, the example advances5.90426mm per entry. An active contact length9.65457mm therefore gives gear contact duration $L/u_b\approx 10.2742\mathrm{ms}$.

For compatible involute teeth, changing center distance changes the operating pressure angle and operating pitch circles, while leaving the manufactured base circles unchanged. At the same shaft speeds, base-roll speed is therefore unchanged even though the usable contact length may change. This separates a kinematic rate from the active interval needed to calculate contact time. If speed varies, contact displacement follows $\int u_b(t)dt$; a constant-speed quotient is then only an approximation.

Gear circular pitch is the arc distance between corresponding points on successive gear teeth, measured on the gear reference circle. If that circle has diameter $d$ and tooth count $N$, its circumference contains $N$ repetitions, givingwhere $m$ is gear module. The length is measured along the reference arc, rather than as a straight chord between tooth positions.^[ref]

For a spur gear with $m=2\mathrm{mm}$ and $N=30$, reference diameter is $60\mathrm{mm}$, gear circular pitch is $2\pi \mathrm{mm}\approx 6.283\mathrm{mm}$, and angular tooth spacing is $2\pi /30$. The corresponding chord on the reference circle isThe small numerical difference matters when a dimensional inspection is intended to measure reference arc spacing. Increasing the tooth count while holding gear module fixed enlarges the gear reference circle but preserves gear circular pitch.

For an involute tooth profile, the gear base circle provides another spacing, the gear base pitch. Since $d_b=d\cos \alpha$, with reference gear pressure angle $\alpha$,At $\alpha =20{}^{\circ }$, the preceding example gives $p_b\approx 5.904\mathrm{mm}$. Compatible contact must preserve the spacing of successive tooth flanks along their gear line of action; this explains why equal gear circular pitch with incompatible gear pressure angles is not enough to specify a mating pair. Tooth thickness and backlash are additional quantities: gear circular pitch includes an entire tooth-and-space repetition.

Gear contact duration is the time for a specified tooth pair to traverse its active gear path of contact. For a fixed-axis external spur gear pair at constant speed, the contact location advances along the gear line of action at the common gear base-roll speedThis is the speed of the contact location along a geometric line, not the relative tooth-surface gear sliding velocity. The latter vanishes at the gear pitch point, while the contact location continues moving. For active path length $L$, duration is $T=L/u_b$. The geometry relates gear contact ratio to active path length divided by gear base pitch.^[ref]

The time between successive tooth-pair entries is $1/f_m$, with gear mesh frequency $f_m=N_1|{\omega }_1|/(2\pi )$. Since $r_{b1}=r_{w1}\cos {\alpha }_w$ for compatible operating geometry, the base-circle distance advanced per entry isConsequentlyA contact ratio above1 means one pair remains engaged after the next pair enters; it does not mean one pair contacts for an entire shaft revolution.

For the standard module2,20/40-tooth example with20degree reference angle, active length is9.65457mm and base pitch is5.90426mm. At driving angular velocity $50rad/s$, $u_b=0.93969m/s$, $f_m\approx 159.155\mathrm{Hz}$ and $T\approx 10.2742\mathrm{ms}$. The entry interval is6.28319ms, giving $T{f}_m\approx 1.63519$. Doubling both shaft speeds halves duration while preserving geometric gear contact ratio. Variable speed requires integrating the changing base-roll speed; a helical gear also needs a stated transverse slice or a whole-face contact convention.

Gear contact ratio measures how much successive tooth engagements overlap. For an external spur gear pair, its transverse value iswhere $L$ is usable gear path of contact length along the gear line of action and $p_b$ is gear base pitch. One tooth pair remains engaged while contact advances through length $L$; successive pairs begin after an advance of $p_b$. Thus ${ϵ}_{\alpha }$ is also the ideal mean number of tooth pairs simultaneously engaged.^[ref]

For a tip-limited, noninterfering external involute pair with tip radii $r_{a1},r_{a2}$, base radii $r_{b1},r_{b2}$, axis separation $a$ and operating pressure angle ${\alpha }_w$, right-triangle geometry givesFor an original example, use module $2\mathrm{mm}$, tooth counts 20 and 40, reference angle $20{}^{\circ }$ and standard separation $60\mathrm{mm}$. With tip radii $22$ and $42\mathrm{mm}$, the formula gives $L\approx 9.6546\mathrm{mm}$ and $p_b\approx 5.9043\mathrm{mm}$, hence ${ϵ}_{\alpha }\approx 1.6352$.

In the ideal periodic timing model, two tooth pairs are engaged for about $63.52\%$ of a tooth-passage period and one pair for about $36.48\%$. This does not mean that force is divided equally between engaged pairs: tooth flexibility and contact geometry affect load sharing. A value below one would leave gaps in ideal engagement, while a value exactly one leaves no overlap margin for tolerances and deformation.

The plot shows contact timing, rather than physical tooth shapes or force sharing. For helical gears, the transverse measure must additionally be distinguished from gear overlap ratio caused by the inclination of teeth across the gear face. Finally, real usable contact can be shortened by modified or unavailable flank portions; the tip-circle formula is conditional on its stated geometric assumptions.

Gear face width $b$ describes the width of the toothed portion. For a cylindrical spur gear or helical gear, it is measured parallel to the rotation axis, rather than along the slanting tooth trace. A projecting hub need not be part of the toothed width. Manufacturer drawings distinguish face width, hub width and total length as separate dimensions.^[ref]

The available width of one gear is not necessarily the gear effective face width of a pair. For two aligned cylindrical gears with equal width $b$ and relative axial offset $e$, a simple interval-overlap model givesAn original $b=20\mathrm{mm}$ pair offset by $4\mathrm{mm}$ has only $16\mathrm{mm}$ of common axial extent. If a hypothetical normal load $1000\mathrm{N}$ were uniformly spread across that extent, mean load per axial width would rise from $50$ to $62.5N/mm$. This is bookkeeping for an ideal uniform load, not a tooth-strength prediction. Actual gear face load distribution also depends on alignment and deformation.

For helical gears, width also enters gear overlap ratio:With gear normal module $m_n=2\mathrm{mm}$ and gear helix angle $|\beta |=30{}^{\circ }$, reducing engaged width from $20$ to $16\mathrm{mm}$ reduces ideal overlap from $1.5915$ to $1.2732$. The overall gear ratio is unchanged. Width therefore affects engagement and loading even when tooth counts and shaft speeds remain the same.^[ref]

The gear helix angle $\beta$ measures a cylindrical gear tooth's inclination to the rotation axis, normally on the reference cylinder. A straight spur gear has $\beta =0$; a helical gear has a nonzero inclination and a specified hand. The cylinder and reference direction matter because another measurement plane or cylinder radius can give another angle.^[ref]

Let $L$ be the positive gear lead, the axial advance of one tooth helix in a complete turn, and let the reference diameter be $d$. Unwrapping that cylinder produces a right triangle with circumferential side $\pi d$ and axial side $L$:For $d=60\mathrm{mm}$ and $|\beta |=30{}^{\circ }$, $L\approx 326.484\mathrm{mm}$. Doubling the lead on that same cylinder gives angle approximately $16.102{}^{\circ }$, rather than15degrees: the relation involves a tangent, not a linear angle scaling.

With $N$ equally spaced teeth, gear axial pitch is $p_x=L/N=p_t/\tan |\beta |$, where $p_t$ is gear transverse pitch. For nonzero $\beta$, this also gives $p_x=\pi m_n/\sin |\beta |$. The corresponding axial-pitch quantity diverges in the straight-tooth limit; it is not a finite replacement for the transverse tooth spacing of a spur gear.

Larger helix angle changes reference diameter at fixed gear normal module and tooth count, increases axial thrust for a fixed tangential load, and increases face-overlap geometry at fixed face width. These consequences must be evaluated together with gear contact ratio and support loads. A magnitude such as30degrees does not by itself specify right versus left hand, nor the thrust direction under a particular applied torque.

Gear lead $L$ is the axial advance of one tooth helix over a complete turn around a cylindrical helical gear. It differs from gear axial pitch $p_x$, which steps between successive teeth at one circumferential position. With $N$ equally spaced tooth helices, $L=N{p}_x$. The lead magnitude does not specify whether the helix is right-handed or left-handed.^[ref]

Unwrapping a reference cylinder of diameter $d$ gives a right triangle with axial side $L$ and circumferential side $\pi d$. Since gear helix angle $\beta$ is measured from the axis,For $d=60\mathrm{mm}$ and $|\beta |=30{}^{\circ }$, $L=60\pi \sqrt{3}\mathrm{mm}\approx 326.484\mathrm{mm}$. If $N=30$, then gear axial pitch is $10.8828\mathrm{mm}$; confusing the two distances would introduce a factor30.

A useful geometric check keeps the lead fixed while changing the chosen concentric cylinder. The helix's axial advance per revolution remains $L$, but its angle satisfies $\tan |\beta (r)|=2\pi r/L$. For the same lead on diameter $40\mathrm{mm}$, the angle is approximately $21.052{}^{\circ }$, rather than $30{}^{\circ }$. Thus a helix-angle specification needs its cylinder, normally the reference cylinder. Lead can remain finite when the physical tooth is much shorter than one complete turn: it describes the continued helix, not a requirement that the manufactured tooth cover a full revolution.

Gear mesh frequency is the rate at which successive teeth pass through a mesh. For a fixed-axis gear with $N$ teeth rotating at $f_{\mathrm{rot}}$ revolutions per second,where $n$ is speed in revolutions per minute and $\omega$ is angular velocity in radians per second. The formula follows by counting $N$ tooth passages per complete turn. Both gears of one steady compatible mesh give the same result because their tooth-passage rates agree.^[ref]

For an original example, a24-tooth input rotating at1500 revolutions per minute gives $f_m=600\mathrm{Hz}$. Its60-tooth mate turns at600 revolutions per minute, also giving $60(600)/60=600\mathrm{Hz}$. The shaft rotation frequencies differ even though the mesh frequency agrees.

A compound gear train can have different mesh frequencies at successive stages. If that60-tooth gear is rigidly joined to a20-tooth gear driving an80-tooth output, the second mesh frequency is $20(600)/60=200\mathrm{Hz}$. The final output turns at150 revolutions per minute. By contrast, every mesh of an ideal fixed-axis simple gear train shares the same tooth-passage frequency: its intermediate gear engages both neighbors with the same tooth count and rotation rate.

A gear vibration spectrum may contain components at the mesh frequency and its integer multiples, together with other mechanical excitations. Measuring rotation with a rotary encoder lets a test compare observed frequency components with tooth-passage predictions. A spectral peak near $f_m$ does not by itself establish a structural natural frequency or identify a particular defect; the forcing and the system's frequency response both shape measured vibration.^[ref] For trains with moving gear axes, the simple fixed-axis formula requires the appropriate relative rotational rate rather than an unqualified absolute shaft speed.

The gear module specifies the size of metric gear teeth through the gear reference circle. For a spur gear with reference diameter $d$ and $N$ teeth,The tooth count is dimensionless, so gear module has units of length, conventionally millimetres. A larger gear module means a larger reference diameter at fixed tooth count. It does not mean a larger gear ratio, which depends on the tooth counts of the meshing pair.^[ref]

The gear circular pitch $p$ is the arc length between equivalent points on adjacent teeth, measured along the gear reference circle. Dividing the circumference by the tooth count givesFor example, a spur gear with $m=1.5\mathrm{mm}$ and $N=24$ has reference diameter $36\mathrm{mm}$ and gear circular pitch $1.5\pi \mathrm{mm}\approx 4.712\mathrm{mm}$. A 48-tooth gear of the same gear module has diameter $72\mathrm{mm}$ but the same tooth spacing. Their standard axis separation is $(36+72)/2=54\mathrm{mm}$, and their reduction gear ratio is $48/24=2$.

Conversely, two gears can have the same diameter without compatible tooth spacing: $d=36\mathrm{mm}$ can describe 24 teeth of gear module $1.5\mathrm{mm}$ or 18 teeth of gear module $2\mathrm{mm}$. Such teeth do not form the specified standard pair. For ordinary compatible spur gears, matching gear module is necessary but does not by itself specify the complete tooth geometry: the gear pressure angle and other tooth details must also agree. The definition uses reference diameter, not the diameter through the tooth tips or the operating pitch circle of a particular mounting.

Gear normal module $m_n$ is the gear normal pitch divided by $\pi$, measured normal to the tooth direction of a helical gear. It describes a different section from gear transverse module $m_t$, which uses the circumferential spacing in a plane perpendicular to the rotation axis.^[ref]

If $\beta$ is reference gear helix angle, geometric projection givesFor an original inference example, reference diameter $d=60\mathrm{mm}$, tooth count $N=30$ and $\beta =30{}^{\circ }$ imply $m_t=2\mathrm{mm}$ but $m_n=\sqrt{3}\mathrm{mm}\approx 1.73205\mathrm{mm}$. The normal pitch is approximately $5.4414\mathrm{mm}$, compared with transverse pitch $6.2832\mathrm{mm}$. This is a geometric example, not a claim that the inferred normal module belongs to a particular manufacturer's preferred series.

Conversely, specifying $m_n=2\mathrm{mm}$ for the same30 teeth and30degree helix angle gives reference diameter $69.282\mathrm{mm}$. Thus two drawings both labeled “module2” can describe different gears when one means normal module and the other transverse module.

Pressure angles need the same care: for reference tooth geometry, $\tan {\alpha }_t=\tan {\alpha }_n/\cos \beta$. Matching only a module number, without its plane, pressure-angle convention and helix geometry, does not establish a compatible pair. In a conventional parallel-axis external mesh, the two helical gears also require appropriate opposite hands; the normal-module definition alone does not specify that connection.

Gear normal pitch $p_n$ is the repeat spacing of corresponding helical gear tooth flanks normal to their tooth direction on the developed reference cylinder. It equals $\pi$ times gear normal module $m_n$. It is a different geometric spacing from gear transverse pitch $p_t$, measured circumferentially, and from gear axial pitch $p_x$, measured parallel to the rotation axis.^[ref]

Unwrap the reference cylinder into a plane. The equally spaced tooth traces become parallel straight lines making gear helix angle $\beta$ with the axis. Projecting the transverse spacing onto the normal to those traces givesFor reference diameter $d=60\mathrm{mm}$, $N=30$ teeth and $\beta =30{}^{\circ }$, $p_t=\pi d/N=2\pi \mathrm{mm}$, whereas $p_n=\sqrt{3}\pi \mathrm{mm}\approx 5.4414\mathrm{mm}$. Multiplying $p_n$ by $N$ gives about $163.24\mathrm{mm}$, not the $188.50\mathrm{mm}$ circumference: normal pitch is not the circumferential step.

At $\beta =0$, normal and transverse pitches agree, as expected for a spur gear. For nonzero gear helix angle, using one in place of the other introduces a factor $\cos \beta$. Normal pitch is also distinct from gear base pitch, which refers to the base-circle geometry of an involute. Matching a pitch value without its measurement convention, pressure angle and tooth geometry is insufficient to establish a compatible mesh.

The gear path of approach is the part of the active gear path of contact followed before the contact reaches the gear pitch point. For a conventional external involute pair in the stated driving direction, contact starts where the driven gear's tip circle intersects the usable gear line of action, and then moves toward the pitch point. This is a straight-line distance along the common normal, not an arc around a pitch circle.^[ref]

Let gear1 drive gear2. Write $r_{a2}$ for gear2's tip radius, $r_{b2}$ for its base radius, $r_{w2}$ for its operating radius and ${\alpha }_w$ for operating pressure angle. A right triangle from the base-circle tangency gives tip-to-tangency distance $\sqrt{r_{a2}^2-r_{b2}^2}$. Subtracting the pitch-point-to-tangency distance givesThis formula assumes the full tip-limited interval is usable without interference or profile truncation.

For an original standard module2,20/40-tooth pair at60mm center distance and20degree reference angle, gear2 has tip radius42mm, operating radius40mm and base radius $40\cos 20{}^{\circ }\mathrm{mm}$. Consequently $L_a\approx 5.0586\mathrm{mm}$. If gear1 rotates at $50rad/s$, contact advances along the line at $u_b=|{\omega }_1|r_{b1}=0.93969m/s$, giving an approach contribution to gear contact duration of about $5.383\mathrm{ms}$. This contact-location speed differs from gear sliding velocity, which tends to zero on arriving at the pitch point.

The total active length adds gear path of recess. Reversing the followed engagement direction exchanges which segment is called approach and which is recess. If altered usable flanks exclude the nominal pitch point, an assumed positive tip-limited formula must not replace an inspection of the actual active interval.

The gear path of contact is the locus of actual tooth-contact points during an engagement. For a conventional external involute pair, its active part is a finite straight segment of the gear line of action. Its endpoints are set by the usable tooth flanks, rather than by the infinite geometric line alone. The part before the gear pitch point is the gear path of approach; the part after it is the gear path of recess in the stated direction of transmission.^[ref]

For a standard noninterfering external pair driven by gear1, let $r_{ai}$ be tip radii, $r_{bi}$ base radii, $r_{wi}$ operating radii and ${\alpha }_w$ the operating pressure angle. When tip circles bound the full usable contact,Adding gives $L=L_{\mathrm{app}}+L_{\mathrm{rec}}$. The two lengths need not be equal even when the gears use the same module and reference angle.

For the original module2,20/40-tooth example at20degree reference angle and60mm standard separation, tip radii22 and42mm give $L_{\mathrm{app}}\approx 5.0586\mathrm{mm}$ and $L_{\mathrm{rec}}\approx 4.5960\mathrm{mm}$. Their sum is $9.6546\mathrm{mm}$; dividing by gear base pitch $5.9043\mathrm{mm}$ gives transverse gear contact ratio about1.6352.

These are distances along the line of contact, not arc lengths traveled around a pitch circle. A shortened or altered flank can reduce the active interval, and interference can invalidate the full tip-limited construction. The labels approach and recess also depend on which direction is being followed. A practical contact inspection therefore needs the profile geometry and transmission direction, not just a sketch of two nominal pitch circles.

The gear path of recess is the part of the active gear path of contact followed after passing the gear pitch point, in the stated driving direction. For a standard external involute pair driven by gear1, contact ends at the driving gear's tip circle when that tip bounds the usable profile. The recess length lies on the gear line of action; it is not an arc of the gear pitch circle.^[ref]

Use driving-gear tip radius $r_{a1}$, base radius $r_{b1}$, operating radius $r_{w1}$ and operating pressure angle ${\alpha }_w$. The base-circle tangent triangle givesFor a standard module2,20/40-tooth pair at60mm separation and20degree reference angle, $r_{a1}=22\mathrm{mm}$, $r_{w1}=20\mathrm{mm}$ and $r_{b1}=20\cos 20{}^{\circ }\mathrm{mm}$ yield $L_r\approx 4.5960\mathrm{mm}$. The corresponding gear path of approach is about5.0586mm, so the pitch point does not bisect the active contact interval.

At driving angular velocity $50rad/s$, the contact-location speed is $u_b=|{\omega }_1|r_{b1}\approx 0.93969m/s$. Recess therefore lasts about4.891ms. Adding approach gives gear contact duration $T\approx 10.274\mathrm{ms}$. Tooth-passage frequency is $20(50)/(2\pi )\approx 159.155\mathrm{Hz}$, so $T{f}_m\approx 1.6352$, agreeing with the transverse gear contact ratio calculated as active length divided by gear base pitch.

A shorter usable driving flank reduces recess and can change gear sliding-friction loss, because sliding grows with distance from the pitch point. The tip-circle formula alone cannot describe profile modifications or an interfering contact. The geometry and direction must be specified before applying the approach/recess labels.

A gear pitch circle is an imaginary circle used to represent the motion or reference geometry of a gear. A useful distinction is between the gear reference circle, specified for an individual gear, and the operating pitch circle, determined for a meshing pair. The terminology entry in KHK Gear Nomenclature calls the latter the intermeshing pitch circle and also uses the name working pitch circle.^[ref]

For an external pair with fixed axes and constant gear ratio, imagine two operating pitch circles in contact and rolling without slipping. If their radii are $r_1,r_2$ and their axis separation is $a$, their geometry and signed angular velocities obeyThe last equality follows by matching the gear ratio to tooth counts $N_1,N_2$. It is an ideal kinematic construction: the imaginary circles themselves are not physical rims carrying the contact force.

For standard unshifted straight-tooth gears at their standard axis separation, the gear reference circles coincide with the operating pitch circles. The gear module $m$ specifies reference diameter $d=mN$. With $m=2\mathrm{mm}$ and tooth counts 20 and 40, the reference diameters are $40\mathrm{mm}$ and $80\mathrm{mm}$, giving reference radii $20\mathrm{mm}$ and $40\mathrm{mm}$ and standard separation $60\mathrm{mm}$. The reference diameters differ from the larger tooth-tip diameters. If a compatible pair is operated at another axis separation, its gear reference circles remain properties of the individual gears, while its operating pitch circles must fit the new separation and the same tooth-count ratio. This is why an unspecified “pitch diameter” should not silently mix reference geometry with operating geometry.^[ref]