CN105432246A - Synchronous pull type carrot harvesting device driven by non-circular gears - Google Patents

Abstract

The invention discloses a synchronous pull type carrot harvesting device driven by non-circular gears. A disc cutter type harvesting part easily damages carrots in cutting. The synchronism of two hydraulic motors of a hydraulically driven pull type harvesting part is difficult to ensure. N round holes are formed in the inner ends of driving discs and the inner ends of driven discs of the device. Two ends of each pull rod are connected in a pair of round holes of the driving discs and the driven discs through bearings. Outer end covers of the driving discs are in key connection with driving shafts. Outer end covers of the driven discs are in key connection with driven shafts. The hydraulic motors transfer power to a power shaft through the driving non-circular gear and the driven non-circular gear. Two driving bevel gears arranged reversely are fixed to the power shaft, and each driving bevel gear is meshed with one driven bevel gear. Two driven bevel gears are fixed to the driving shafts for two pull rod assemblies. The synchronous pull type carrot harvesting device synchronously inputs power to two groups of pull rods through matching between the non-circular gears and the bevel gears, meets the requirement for the synchronous effect of the two groups of pull rods and is high in reliability.

Description

The synchronous pulling formula carrot harvesting apparatus that non-circular gear drives

Technical field

The invention belongs to agricultural mechanical field, relate to carrot reaping machine, be specifically related to the synchronous pulling formula carrot harvesting apparatus that a kind of non-circular gear drives.

Background technology

Carrot is a kind of agricultural product that China resident eats, the annual production of China carrot accounts for 1/3rd of world's annual production, but its mechanized harvest level is very low, most area also mainly relies on artificial results, the carrot reaping machine technology that minority area uses is not also very ripe, still there is various problem.At present, the results parts of the self-propelled carrot reaping machine at home and abroad market sold are the pulling formula results parts that drive of wheel cutter formula results parts and hydraulic drive mainly.Wheel cutter formula results parts are by wheel cutter excision carrot rice shoot, easy cut wound carrot in results process, results poor quality, and damage ratio is higher; It is drive two groups to pull bar respectively by two hydraulic motors to realize the separation really of carrot seedling that hydraulic-driven pulls formula results parts, in results process, the synchronism of two hydraulic motors is difficult to ensure, reliability is low, and the rate curve pulling bar and carrot effect is undesirable, the results effect of mechanism does not reach best.

Summary of the invention

The object of the invention is for the deficiencies in the prior art, the synchronous pulling formula harvesting apparatus providing a kind of non-circular gear to drive, this device adopts the mode of non-circular gear drive, designs the pitch curve of non-circular gear as required and guarantees to pull the requirement of output speed curve needed for bar; This device adopts non-circular gear and bevel gear mating reaction, pulls the synchronous input power of bar to two groups, and can meet the requirement that two groups pull bar synchronous effect, reliability is high.

In order to achieve the above object, the technical solution used in the present invention is:

The present invention includes and be contained in pulling device on installing rack and non-circular gear drive device.Described pulling device is arranged on below conveyer belt; Described non-circular gear drive device drives pulling device.

Described pulling device comprises two pulling bar assemblies; Described pulling bar assembly comprises driving shaft, initiatively disk, pulls bar, driven disk and driven shaft.Described active disk and driven disk the inner all offer n the circular hole be uniformly distributed along the circumference, n >=5; Initiatively each circular hole of disk and driven disk is connected with the outer ring of the pair of bearings oppositely arranged; By axle sleeve axial location between often pair of bearing; Initiatively the inner ring of the inner often pair of bearing of disk is connected with the one end fits of a pulling bar, and every root pulls the other end of bar and the pair of bearings inner ring of driven disk the inner is connected; The end, two ends pulling bar is all connected with round nut, realizes axial restraint; Active disk and the driven disk axis of two pulling bar assemblies all be arranged in parallel; Be positioned at inside two pulling bar assemblies a pair or two to the interlaced setting of pulling bar.The axis hole that the active disk outer end cap of two pulling bar assemblies is offered is connected by key with a driving shaft respectively, and described driving shaft end and hubcap are bolted; Two driving shafts are bearing on two bearing blocks by bearing, and these two bearing blocks are bolted on installing rack sidepiece; The axis hole that the driven disk outer end cap of two pulling bar assemblies is offered is connected by key with a driven shaft respectively, and described driven shaft end and hubcap are bolted; Two driven shafts are bearing on two bearing blocks by bearing, and these two bearing blocks are bolted on the conveyer belt installing plate at installing rack top.

Described non-circular gear drive device comprises line shaft, initiatively non-circular gear, driven non-circular gear, hydraulic motor, drive bevel gear and driven wheel of differential; Described active non-circular gear is connected by key with power transmission shaft, and power transmission shaft is connected by shaft coupling with the output shaft of hydraulic motor; Described driven non-circular gear is connected by key with line shaft, and engages with active non-circular gear; The two ends of line shaft are bearing on bearing block by bearing, and this bearing block is bolted on installing rack sidepiece; Described line shaft is fixed with two drive bevel gear oppositely arranged, and each drive bevel gear engages with a driven wheel of differential; Two driven wheels of differential are fixed on the driving shaft of two pulling bar assemblies; Described drive bevel gear and the gearratio of driven wheel of differential are 1.

Initiatively the pitch curve design process of non-circular gear and driven non-circular gear is as follows:

(1) carry out kinematics analysis, draw the kinematical equation pulling bar.

Pull the output speed of bar in t horizontal direction:

v＝ωRsin(ωt)

In formula, R is the distance of the initiatively inner center of circular hole axis of disk and active circle disk center, and ω is the angular speed of initiatively disk.

The horizontal range of two pulling bars relative inside two pulling bar assemblies:

X＝L+Rcos(1.5π-φ-θ)-Rcos(1.5π+θ)-2l

In formula, L is the centre-to-centre spacing of initiatively disk and driven disk, and l pulls the offset distance that bar actuating rod part is mounted opposite the installation center of circular hole axis of bar part, and φ is the deflection angle of two relative pulling bar mounting rod parts, and θ=ω t is the angular displacement pulling bar.

(2) according to carrot rice shoot mechanical characteristic, the rate request derivation pulling bar vertical direction is pulled to the curve requirement of speed v with the output shaft variation in angular displacement of hydraulic motor of bar horizontal direction, and it is as follows to set up the rate curve equation pulling bar horizontal direction:

$v=\left\{\begin{array}{cc}\frac{{\mathrm{mx}}^3}{3}-3600mx& 0\&le;x<\frac{\mathrm{\&pi;}}{3}\\ 40h& \frac{\mathrm{\&pi;}}{3}\&le;x<\frac{2\mathrm{\&pi;}}{3}\\ \frac{-{\mathrm{mx}}^3}{3}+180x^2-(h+{180}^{2}m)x+c_2& \frac{2\mathrm{\&pi;}}{3}\&le;x<\frac{4\mathrm{\&pi;}}{3}\\ -40h& \frac{4\mathrm{\&pi;}}{3}\&le;x<\frac{5\mathrm{\&pi;}}{3}\\ \frac{{\mathrm{mx}}^3}{3}-360{\mathrm{mx}}^2+(h+{360}^{2}m)x+c_3& \frac{5\mathrm{\&pi;}}{3}\&le;x<2\mathrm{\&pi;}\end{array}\right.$

In formula, parameter $c_2=\frac{{180}^{3}m}{3}+180h,c_3=\frac{{360}^{3}m}{3}-360h,$ H=-3600m; X is the corner of initiatively non-circular gear;

By $\begin{array}{cc}2{\&Integral;}_0^{\frac{T}{6}}(\frac{{\mathrm{mx}}^3}{3}-3600mx)& dx\end{array}+\frac{hT}{6}={\&Integral;}_0^{\frac{T}{2}}\mathrm{\&omega;}sin\left(\mathrm{\&omega;}t\right)dt$ Obtain parameter m, thus obtain the parameter c in rate curve equation ₂, c ₃and h; In formula, T is the rotation period of initiatively non-circular gear.

(3) according to the pulling bar horizontal direction rate curve equation set up and the pitch curve pulling bar equation of motion reverse noncircular gear pair.

Obtained by v=ω sin ω t=θ ' sin θ obtained the curvilinear equation of θ again by integration, and then obtain the angular speed equation of driven non-circular gear, be i.e. the equation of the angular velocity omega of active disk.

Again due to the angular velocity omega of active non-circular gear ₁known, so the gearratio i of noncircular gear pair ₁₂can ask.The centre-to-centre spacing a of given noncircular gear pair again, obtains the pitch curve equation of noncircular gear pair by following formula:

$\left\{\begin{array}{c}{r}_1=\frac{a}{i_{12}}\\ r_2=a-r_1=\frac{{\mathrm{ai}}_{12}}{1+i_{12}}\end{array}\right.$

In formula, r ₁for the radius vector of active non-circular gear, r ₂for the radius vector of driven non-circular gear.

The benefit effect that the present invention has is:

The mode input power that the present invention adopts hydraulic motor to be coordinated with two pairs of bevel gears by non-circular gear drive, reduce by power set, and ensure that two groups pull the running of bar component synchronization, improve the reliability of results parts, the more important thing is the pitch curve that designs non-circular gear as required and guarantee to pull the requirement of output speed curve needed for bar, make operating efficiency optimum, reduce the damage ratio of carrot, improve fruit seedling separating effect.The carrot of the present invention's results also leaves some seedling leaves, and being convenient to carrot has long shelf life.

Accompanying drawing explanation

Fig. 1 is operation principle schematic diagram of the present invention;

Fig. 2 is the stereogram of pulling device in the present invention;

Fig. 3 is driven disk and the sectional arrangement drawing pulling bar and driven shaft in the present invention;

Fig. 4 is the mechanism principle figure of non-circular gear drive device in the present invention;

Fig. 5 is the motion analysis figure of pulling device in the present invention;

Fig. 6 is the output speed curve map pulling bar in the present invention;

Fig. 7 is the pitch curve mesh schematic representation of initiatively non-circular gear and driven non-circular gear in the present invention.

In figure: 1, carrot, 2, conveyer belt, 3, pulling device, 4, installing rack, 5, non-circular gear drive device, 6, hydraulic motor, 7, driving shaft, 8, initiatively disk, 9, pull bar, 10, driven disk, 11, driven shaft, 12, line shaft, 13-1, initiatively non-circular gear, 13-2, driven non-circular gear.

Embodiment

Below in conjunction with drawings and Examples, the invention will be further described.

As shown in Figure 1, the synchronous pulling formula carrot harvesting apparatus that non-circular gear drives, comprises and is contained in pulling device 3 on installing rack 4 and non-circular gear drive device 5.Pulling device 3 is arranged on below conveyer belt 2; Non-circular gear drive device 5 drives pulling device 3.

As shown in Figures 2 and 3, pulling device 3 comprises two pulling bar assemblies; Pull bar assembly to comprise driving shaft 7, initiatively disk 8, pull bar 9, driven disk 10 and driven shaft 11;

Initiatively disk 8 and driven disk 10 the inner all offer five circular holes be uniformly distributed along the circumference; Initiatively each circular hole of disk 8 and driven disk 10 is connected with the outer ring of the pair of bearings oppositely arranged; By axle sleeve axial location between often pair of bearing; Initiatively the inner ring of the often pair of bearing in disk 8 the inner is connected with the one end fits of a pulling bar 9, and every root pulls the other end of bar 9 and the pair of bearings inner ring of driven disk 10 the inner is connected; The end, two ends pulling bar 9 is all connected with round nut, realizes axial restraint; Active disk 8 and driven disk 10 axis of two pulling bar assemblies all be arranged in parallel; Be positioned at inside two pulling bar assemblies a pair or two to the interlaced setting of pulling bar 9.The axis hole that active disk 8 outer end cap of two pulling bar assemblies is offered is connected by key with a driving shaft 7 respectively, and driving shaft 7 end and hubcap are bolted; Two driving shafts 7 are bearing on two bearing blocks by bearing, and these two bearing blocks are bolted on installing rack 4 sidepiece; The axis hole that driven disk 10 outer end cap of two pulling bar assemblies is offered is connected by key with a driven shaft 11 respectively, and driven shaft 11 end and hubcap are bolted; Two driven shafts 11 are bearing on two bearing blocks by bearing, and these two bearing blocks are bolted on the conveyer belt installing plate at installing rack 4 top.

As shown in Figure 4, non-circular gear drive device 5 comprises line shaft 12, initiatively non-circular gear 13-1, driven non-circular gear 13-2, hydraulic motor 6, drive bevel gear and driven wheel of differential; Initiatively non-circular gear 13-1 is connected by key with power transmission shaft, and power transmission shaft is connected by shaft coupling with the output shaft of hydraulic motor 6; Driven non-circular gear 13-2 is connected by key with line shaft 12, and engages with active non-circular gear 13-1; The two ends of line shaft 12 are bearing on bearing block by bearing, and this bearing block is bolted on installing rack 4 sidepiece; Line shaft 12 is fixed with two drive bevel gear oppositely arranged, and each drive bevel gear engages with a driven wheel of differential; Two driven wheels of differential are fixed on the driving shaft 7 of two pulling bar assemblies; The gearratio of drive bevel gear and driven wheel of differential is 1.

Initiatively the pitch curve design process of non-circular gear 13-1 and driven non-circular gear 13-2 is as follows:

(1) as shown in Figure 5, carry out kinematics analysis, draw the kinematical equation pulling bar.

Pull the output speed of bar in t horizontal direction:

v＝ωRsin(ωt)

In formula, R is the distance in the initiatively inner center of circular hole axis of disk 8 and active disk 8 center of circle, and ω is the angular speed of initiatively disk 8.

The horizontal range of two pulling bars relative inside two pulling bar assemblies:

X＝L+Rcos(1.5π-φ-θ)-Rcos(1.5π+θ)-2l

In formula, L is the centre-to-centre spacing of initiatively disk 8 and driven disk 10, l pulls the offset distance that bar actuating rod part is mounted opposite the installation center of circular hole axis of bar part, and φ is the deflection angle of two relative pulling bar mounting rod parts, and θ=ω t is the angular displacement pulling bar.

(2) as shown in Figure 6, according to carrot rice shoot mechanical characteristic, the rate request pulling bar vertical direction is derived the curve requirement that the speed v that pulls bar horizontal direction change with the output shaft angular displacement beta of hydraulic motor 6, and the rate curve equation setting up pulling bar horizontal direction is as follows:

$v=\left\{\begin{array}{cc}\frac{{\mathrm{mx}}^3}{3}-3600mx& 0\&le;x<\frac{\mathrm{\&pi;}}{3}\\ 40h& \frac{\mathrm{\&pi;}}{3}\&le;x<\frac{2\mathrm{\&pi;}}{3}\\ \frac{-{\mathrm{mx}}^3}{3}+180x^2-(h+{180}^{2}m)x+c_2& \frac{2\mathrm{\&pi;}}{3}\&le;x<\frac{4\mathrm{\&pi;}}{3}\\ -40h& \frac{4\mathrm{\&pi;}}{3}\&le;x<\frac{5\mathrm{\&pi;}}{3}\\ \frac{{\mathrm{mx}}^3}{3}-360{\mathrm{mx}}^2+(h+{360}^{2}m)x+c_3& \frac{5\mathrm{\&pi;}}{3}\&le;x<2\mathrm{\&pi;}\end{array}\right.$

In formula, parameter $c_2=\frac{{180}^{3}m}{3}+180h,c_3=\frac{{360}^{3}m}{3}-360h,$ H=-3600m; X is the corner of initiatively non-circular gear;

By $\begin{array}{cc}2{\&Integral;}_0^{\frac{T}{6}}(\frac{{\mathrm{mx}}^3}{3}-3600mx)& dx+\frac{hT}{6}={\&Integral;}_0^{\frac{T}{2}}\mathrm{\&omega;}sin\left(\mathrm{\&omega;}t\right)dt\end{array}$ Obtain parameter m, thus obtain the parameter c in rate curve equation ₂, c ₃and h; In formula, T is the rotation period of initiatively non-circular gear.

(3) according to the pulling bar horizontal direction rate curve equation set up and the pitch curve pulling bar equation of motion reverse noncircular gear pair.

Obtained by v=ω sin ω t=θ ' sin θ obtained the curvilinear equation of θ again by integration, and then obtain the angular speed equation of driven non-circular gear, be i.e. the equation (because the gearratio of drive bevel gear and driven wheel of differential is 1) of the angular velocity omega of active disk 8.

Again due to the angular velocity omega of active non-circular gear ₁known, so the gearratio i of noncircular gear pair ₁₂can ask.The centre-to-centre spacing a of given noncircular gear pair again, obtains the pitch curve equation of noncircular gear pair by following formula:

$\left\{\begin{array}{c}{r}_1=\frac{a}{i_{12}}\\ r_2=a-r_1=\frac{{\mathrm{ai}}_{12}}{1+i_{12}}\end{array}\right.$

In formula, r ₁for the radius vector of active non-circular gear, r ₂for the radius vector of driven non-circular gear.The pitch curve of noncircular gear pair as shown in Figure 7.

The synchronous pulling formula carrot harvesting apparatus that this non-circular gear drives, operation principle is as follows:

The hydraulic motor 6 of non-circular gear drive device 5 drives initiatively non-circular gear 13-1, driven non-circular gear 13-2 and active non-circular gear 13-1 engaged transmission, and power is passed to two drive bevel gear oppositely arranged through line shaft 12; Two drive bevel gear engage transferring power separately to the active disk 8 of two pulling device 3 with driven wheel of differential, the active disk 8 of two pulling bar assemblies is rotated backward, and drives respective pulling bar 9.Carrot 1 under the clamping of conveyer belt 2 along with conveyer belt moves obliquely, when carrot enters the inner side of two pulling bar assemblies, two pulling bar assemblies relative a pair or two just in time start interlaced to be clamped by carrot to pulling bar and down pull, now the rice shoot of carrot completes carrot seedling fruit and is separated under pulling bar and the acting in conjunction of conveyer belt, reaches the object gathering in the crops carrot.

Can be optimized by the pitch curve of design active non-circular gear 13-1 and driven non-circular gear 13-2 and pull bar output speed (pulling the component of rotating speed at vertical direction of bar), improve the operating efficiency of the synchronous pulling formula carrot harvesting apparatus that this non-circular gear drives, reduce the damage ratio of carrot, improve fruit seedling separating effect.Initiatively the pitch curve that engages of non-circular gear 13-1 and driven non-circular gear 13-2 as shown in Figure 7 time, the curve that the output speed pulling bar horizontal direction changes with the output shaft angular displacement beta of hydraulic motor 6 is as shown in Figure 6.In Fig. 7, initiatively non-circular gear 13-1 is single order non-circular gear, and driven non-circular gear 13-2 is five rank non-circular gears.Visible, when pulling bar and carrot effect, the speed pulling bar horizontal direction is increased to maximum from zero when the output shaft variation in angular displacement of hydraulic motor 6 is less, then keep maximal rate steadily to pull carrot, meet the feature that carrot rice shoot is easily pulled off under steady high speed pull effect.

Claims (2)

1. the synchronous pulling formula carrot harvesting apparatus that drives of non-circular gear, comprises and is contained in pulling device on installing rack and transmission device, it is characterized in that: described transmission device is non-circular gear drive device; Described pulling device is arranged on below conveyer belt; Described non-circular gear drive device drives pulling device;

Described pulling device comprises two pulling bar assemblies; Described pulling bar assembly comprises driving shaft, initiatively disk, pulls bar, driven disk and driven shaft; Described active disk and driven disk the inner all offer n the circular hole be uniformly distributed along the circumference, n >=5; Initiatively each circular hole of disk and driven disk is connected with the outer ring of the pair of bearings oppositely arranged; By axle sleeve axial location between often pair of bearing; Initiatively the inner ring of the inner often pair of bearing of disk is connected with the one end fits of a pulling bar, and every root pulls the other end of bar and the pair of bearings inner ring of driven disk the inner is connected; The end, two ends pulling bar is all connected with round nut, realizes axial restraint; Active disk and the driven disk axis of two pulling bar assemblies all be arranged in parallel; Be positioned at inside two pulling bar assemblies a pair or two to the interlaced setting of pulling bar; The axis hole that the active disk outer end cap of two pulling bar assemblies is offered is connected by key with a driving shaft respectively, and described driving shaft end and hubcap are bolted; Two driving shafts are bearing on two bearing blocks by bearing, and these two bearing blocks are bolted on installing rack sidepiece; The axis hole that the driven disk outer end cap of two pulling bar assemblies is offered is connected by key with a driven shaft respectively, and described driven shaft end and hubcap are bolted; Two driven shafts are bearing on two bearing blocks by bearing, and these two bearing blocks are bolted on the conveyer belt installing plate at installing rack top;

Described non-circular gear drive device comprises line shaft, initiatively non-circular gear, driven non-circular gear, hydraulic motor, drive bevel gear and driven wheel of differential; Described active non-circular gear is connected by key with power transmission shaft, and power transmission shaft is connected by shaft coupling with the output shaft of hydraulic motor; Described driven non-circular gear is connected by key with line shaft, and engages with active non-circular gear; The two ends of line shaft are bearing on bearing block by bearing, and this bearing block is bolted on installing rack sidepiece; Described line shaft is fixed with two drive bevel gear oppositely arranged, and each drive bevel gear engages with a driven wheel of differential; Two driven wheels of differential are fixed on the driving shaft of two pulling bar assemblies; Described drive bevel gear and the gearratio of driven wheel of differential are 1.

2. the synchronous pulling formula carrot harvesting apparatus of non-circular gear driving according to claim 1, is characterized in that: initiatively the pitch curve design process of non-circular gear and driven non-circular gear is as follows:

(1) carry out kinematics analysis, draw the kinematical equation pulling bar;

Pull the output speed of bar in t horizontal direction:

v＝ωRsin(ωt)

In formula, R is the distance of the initiatively inner center of circular hole axis of disk and active circle disk center, and ω is the angular speed of initiatively disk;

The horizontal range of two pulling bars relative inside two pulling bar assemblies:

X＝L+Rcos(1.5π-φ-θ)-Rcos(1.5π+θ)-2l

In formula, L is the centre-to-centre spacing of initiatively disk and driven disk, and l pulls the offset distance that bar actuating rod part is mounted opposite the installation center of circular hole axis of bar part, and φ is the deflection angle of two relative pulling bar mounting rod parts, and θ=ω t is the angular displacement pulling bar;

(2) according to carrot rice shoot mechanical characteristic, the rate request derivation pulling bar vertical direction is pulled to the curve requirement of speed v with the output shaft variation in angular displacement of hydraulic motor of bar horizontal direction, and it is as follows to set up the rate curve equation pulling bar horizontal direction:

$v=\left\{\begin{array}{cc}\frac{{\mathrm{mx}}^3}{3}-3600mx& 0\&le;x<\frac{\mathrm{\&pi;}}{3}\\ 40h& \frac{\mathrm{\&pi;}}{3}\&le;x<\frac{2\mathrm{\&pi;}}{3}\\ \frac{-{\mathrm{mx}}^3}{3}+180x^2-\left(h+{180}^{2}m\right)x+c_2& \frac{2\mathrm{\&pi;}}{3}\&le;x<\frac{4\mathrm{\&pi;}}{3}\\ -40h& \frac{4\mathrm{\&pi;}}{3}\&le;x<\frac{5\mathrm{\&pi;}}{3}\\ \frac{{\mathrm{mx}}^3}{3}-360{\mathrm{mx}}^2+\left(h+{360}^{2}m\right)x+c_3& \frac{5\mathrm{\&pi;}}{3}\&le;x<2\mathrm{\&pi;}\end{array}\right.$

In formula, parameter $c_2=\frac{{180}^{3}m}{3}+180h,c_3=\frac{{360}^{3}m}{3}-360h,$ H=-3600m; X is the corner of initiatively non-circular gear;

By $2{\&Integral;}_0^{\frac{T}{6}}(\frac{{\mathrm{mx}}^3}{3}-3600mx)dx+\frac{hT}{6}={\&Integral;}_0^{\frac{T}{2}}\mathrm{\&omega;}sin\left(\mathrm{\&omega;}t\right)dt$ Obtain parameter m, thus obtain the parameter c in rate curve equation ₂, c ₃and h; In formula, T is the rotation period of initiatively non-circular gear;

(3) according to the pulling bar horizontal direction rate curve equation set up and the pitch curve pulling bar equation of motion reverse noncircular gear pair;

Obtain-∫ vdt=cos θ by v=ω sin ω t=θ ' sin θ, then obtained the curvilinear equation of θ by integration, and then obtain the angular speed equation of driven non-circular gear, be i.e. the equation of the angular velocity omega of active disk;

Again due to the angular velocity omega of active non-circular gear ₁known, so the gearratio i of noncircular gear pair ₁₂can ask; The centre-to-centre spacing a of given noncircular gear pair again, obtains the pitch curve equation of noncircular gear pair by following formula:

$\left\{\begin{array}{c}{r}_1=\frac{a}{i_{12}}\\ r_2=a-r_1=\frac{{\mathrm{ai}}_{12}}{1+i_{12}}\end{array}\right.$

In formula, r ₁for the radius vector of active non-circular gear, r ₂for the radius vector of driven non-circular gear.