CN106870661A - Circle-ellipse-not rounded three-wheel toothed belt transmission method for designing - Google Patents

Abstract

The invention discloses circle-ellipse-not rounded three-wheel toothed belt transmission method for designing.The present invention initially sets up the pitch curve equation of Timing Belt principal and subordinate wheel, and polar coordinates theoretical calculation principal and subordinate wheel gearratio is cut in utilization；Then the girth of Timing Belt is calculated, not rounded is calculated by iterative algorithm according to the change of Timing Belt girth slack and is tensioned the free pitch curve of synchronous pulley, tensioning wheel is the not rounded synchronous pulley of free pitch curve, can be with the synchronous belt sag variable quantity of generation in real-Time Compensation transmission process, overcoming traditional two-wheeled not rounded V belt translation can not be while meets non-at the uniform velocity transmission and the in real time problem of tensioning；The radius of circle active synchronization belt wheel pitch curve and the major axis of oval driven synchronous pulley pitch curve, eccentricity are controlled variable, change the shape of circle driving wheel pitch curve and oval driven pulley pitch curve by the regulation of three amounts, the specific non-at the uniform velocity transmission for meeting big centre-to-centre spacing is required.

Description

Circle-ellipse-not rounded three-wheel toothed belt transmission method for designing

Technical field

The present invention relates to a kind of method for designing of not rounded toothed belt transmission, and in particular to one kind amount of becoming slack is self-compensating Circle-ellipse-not rounded three-wheel toothed belt transmission method for designing.

Background technology

Transmission mechanism changes the forms of motion and speed of input and output component, to meet different operating environmental requirement, its In non-uniform transmission mechanism occupy extremely important status, common are linkage, cam mechanism, non-circular gear mechanism etc..Phase For linkage and cam mechanism, non-circular gear mechanism has compact conformation, stable drive, power is larger for transmission, easy reality The advantages of existing dynamic balancing, therefore it has been successfully applied to machining tool, automation, transport, instrument and meter, pump class, flowmeter, spinning On loom tool and agricultural machinery.But non-circular gear drive is only suitable for, and centre-to-centre spacing is smaller, lubrication is convenient non-is at the uniform velocity driven Occasion, is thus suitable for big centre-to-centre spacing, the not rounded flexible element (band/chain) of the inconvenient and low manufacturing cost occasion of lubrication and is driven and meets the tendency of And give birth to.Wherein the chaindriven polygon effect of not rounded is obvious, therefore when having strict demand to non-at the uniform velocity gearratio Changing Pattern Just it is restricted；Simultaneously common frictional V belt translation is due to Elastic Sliding it cannot be guaranteed that accurate gearratio rule.

Current non-round belt (chain) transmission, all only 2 band of not rounded (chain) wheels --- driving wheel and driven pulley, in transmission During due to its pitch curve be not rounded, the slack of band (chain) is real-time change, therefore cannot simultaneously ensure work institute It is required that non-at the uniform velocity gearratio Changing Pattern and band (chain) real-time tensioning.In order to compensate the band (chain) in transmission in practical application Slack change, by additional springs with realize tensioning, due in a period of motion its tensile force be change, and As the amplitude of variation of the aggravation tensile force of non-at the uniform velocity characteristic is bigger, the non-precision being at the uniform velocity driven can be so influenceed in turn, and And dynamics is deteriorated；Therefore in Practical Project, non-round belt (chain) transmission is rarely applied to accurately load high-speed drive Occasion.

The content of the invention

The purpose of the present invention is directed to problem above, proposes the self-compensating circle-ellipse-not rounded three-wheel of one kind amount of becoming slack Toothed belt transmission method for designing, is that not rounded synchronous pulley provides a whole set of perfect design theory basis in actual applications, Realize the non-at the uniform velocity directly accurate transmission between big centre-to-centre spacing.The method for designing initially sets up the pitch curve of Timing Belt principal and subordinate wheel Equation, and move synchronous belt pulley transmission ratio using polar coordinates theoretical calculation principal and subordinate is cut；Then the girth of Timing Belt is calculated, according to synchronization The parameters that not rounded is tensioned the free pitch curve of synchronous pulley are calculated by alternative manner with the change of girth slack.

In order to solve the above technical problems, the technical scheme is that：

It is of the invention to comprise the following steps that：

Step one, according to transmission rule determine round active synchronization belt wheel pitch curve with oval driven synchronous pulley pitch curve Equation；

Round active synchronization belt wheel is the input link of uniform rotation,It is the dynamic seat of round active synchronization belt wheel pitch curve Mark system x₁o₁y₁Middle x₁Axle is to quiet coordinate system xo₁The corner of x-axis, θ in y₁It is p₁To moving coordinate system x₁o₁y₁Middle x₁The corner cut of axle, it is round Active synchronization belt wheel cuts polar equation：

p₁=r₁ (1)

S=2 π × r₁ (2)

In formula, p₁Footpath, r are cut for round active synchronization belt wheel pitch curve₁It is the half of round active synchronization belt wheel pitch curve Footpath, s is the girth of round active synchronization belt wheel pitch curve.

Oval driven synchronous pulley is output link, cuts polar equation：

p₂=a × (1-e₂)-a×e₂×cos(θ₂) (3)

In formula, p₂Footpath, θ are cut for oval driven synchronous pulley pitch curve₂It is p₂To moving coordinate system x₂o₂y₂Middle x₂Axle is cut Angle, e₂It is the eccentricity of oval driven synchronous pulley pitch curve, a is the major axis of oval driven synchronous pulley pitch curve.

In formula, b is the short axle of oval driven synchronous pulley pitch curve, and c is the focal length of oval driven synchronous pulley pitch curve.

C=a × e₂ (6)

Step 2, round active synchronization belt wheel is calculated with the gearratio of oval driven synchronous pulley initial position；

Initial position, the moving coordinate system x of round active synchronization belt wheel pitch curve₁o₁y₁Middle x₁Axle is to quiet coordinate system xo₁X in y The corner of axleThe moving coordinate system x of oval driven synchronous pulley pitch curve₂o₂y₂Middle x₂Axle is to quiet coordinate system xo₁X-axis in y CornerAccording to cutting, polar coordinates are theoretical to be obtained：

In formula, p₁(θ₁₂) and p₂(θ₂₁) round active synchronization belt wheel pitch curve is respectively with oval driven synchronous pulley section song Line common tangent incision superius C₁、C₂It is corresponding to cut footpath, p₁(θ₁₃) and p₃(θ₃₁) round active synchronization belt wheel pitch curve is respectively with tensioning Synchronous pulley pitch curve common tangent incision superius C₆、C₅It is corresponding to cut footpath, p₂(θ₂₃) and p₃(θ₃₂) it is respectively oval driven synchronous pulley Pitch curve and tensioning synchronous pulley pitch curve common tangent incision superius C₃、C₄It is corresponding to cut footpath, θ₁₂₀It is round active synchronization belt wheel section Curve cuts footpath p₁(θ₁₂) cut footpath p with oval driven synchronous pulley pitch curve₂(θ₂₁) to respective moving coordinate system trunnion axis corner at the beginning of Value, θ₁₃₀For round active synchronization belt wheel pitch curve cuts footpath p₁(θ₁₃) cut footpath p with tensioning synchronous pulley pitch curve₃(θ₃₁) to each The corner initial value of moving coordinate system trunnion axis, θ₂₃₀For oval driven synchronous pulley pitch curve cuts footpath p₂(θ₂₃) and tensioning synchronous pulley Pitch curve cuts footpath p₃(θ₃₂) to respective moving coordinate system trunnion axis corner initial value, θ₁₂、θ₁₃Respectively round active synchronization belt wheel section Curve incision superius C₁、C₆Correspondence cuts footpath to moving coordinate system x₁o₁y₁Middle x₁The corner cut of axle, θ₂₁、θ₂₃Respectively oval driven Timing Belt Wheel pitch curve incision superius C₂、C₃Correspondence cuts footpath to moving coordinate system x₂o₂y₂Middle x₂The corner cut of axle, θ₃₁、θ₃₂Respectively it is tensioned Timing Belt Wheel pitch curve incision superius C₄、C₅Correspondence cuts footpath to moving coordinate system x₃o₃y₃Middle x₃The corner cut of axle, L₁For round active synchronization belt wheel with Oval driven synchronous pulley centre-to-centre spacing, L₂It is oval driven synchronous pulley and tensioning synchronous pulley centre-to-centre spacing, L₃For round active is same Step belt wheel and tensioning synchronous pulley centre-to-centre spacing；

The round active synchronization belt wheel of initial position is with oval driven synchronous belt pulley transmission ratio：

Between step 3, the round active synchronization belt wheel of calculating, oval driven synchronous pulley and tensioning synchronous pulley are per two-wheeled Common tangent segment length.

Initial time, sets tensioning synchronous pulley pitch curve to give the circle of radius, round active synchronization belt wheel and ellipse Common tangent segment length T between the point of contact of driven synchronous pulley two₀, oval driven synchronous pulley and tensioning synchronous pulley two point of contact it Between common tangent segment length T₁, common tangent segment length T between round active synchronization belt wheel and tensioning synchronous pulley two point of contact₂Point It is not：

In formula, p '₁(θ₁₂₀)、p′₁(θ₁₃₀) it is respectively p₁(θ₁₂₀)、p₁(θ₁₃₀) first differential, p'₂(θ₁₂₀)、p'₂(θ₂₃₀) Respectively p₂(θ₁₂₀)、p₂(θ₂₃₀) first differential, p'₃(θ₁₃₀)、p'₃(θ₂₃₀) it is respectively p₃(θ₁₃₀)、p₃(θ₂₃₀) single order it is micro- Point.

When round active synchronization belt wheel turns over angleOval driven synchronous pulley accordingly turns over angleRound active Synchronous pulley pitch curve incision superius C₁、C₆Corresponding arc length variable quantity is s₁、s₆, oval driven synchronous pulley pitch curve incision superius C₂、C₃Corresponding arc length variable quantity is s₂、s₃, tensioning synchronous pulley pitch curve incision superius C₄、C₅Corresponding arc length variable quantity is s₄、 s₅.Then have：

In formula, p "₁(θ₁) it is p₁(θ₁) second-order differential, p "₂(θ₂) it is p₂(θ₂) second-order differential, p "₃(θ₃) it is p₃(θ₃) Second-order differential, θ₃It is tensioning Timing Belt round cut footpath p₃To moving coordinate system x₃o₃y₃Middle x₃The corner cut of axle.

Any time, the common tangent segment length between round active synchronization belt wheel and the oval point of contact of driven synchronous pulley two T₁₂, common tangent segment length T between oval driven synchronous pulley and tensioning synchronous pulley two point of contact₂₃, round active synchronization belt wheel With the common tangent segment length T between tensioning synchronous pulley two point of contact₁₃Respectively：

In formula, p '₁(θ₁₂)、p′₁(θ₁₃) it is respectively p₁(θ₁₂)、p₁(θ₁₃) first differential, p'₂(θ₂₁)、p'₂(θ₂₃) respectively It is p₂(θ₂₁)、p₂(θ₂₃) first differential, p'₃(θ₃₂)、p'₃(θ₃₁) it is respectively p₃(θ₃₂)、p₃(θ₃₁) first differential,To open Tight synchronous pulley pitch curve moving coordinate system x₃o₃y₃Middle x₃Axle is to quiet coordinate system xo₁The corner of x-axis in y.

Step 4, the gearratio for calculating any time round active synchronization belt wheel and oval driven synchronous pulley；

Round active synchronization belt wheel uniform rotation, according to formula (1), (3) solve p₁, p₂, then instantaneous transmission ratio be：

Step 5, calculating any time Timing Belt girth；

Round active synchronization belt wheel pitch curve is designated as C with tensioning synchronous pulley pitch curve common tangent incision superius₆, any time C₁With C₆Between arc length be c₁₁, oval driven synchronous pulley pitch curve with tensioning synchronous pulley pitch curve common tangent incision superius be designated as C₃, any time C₂With C₃Between arc length be c₂₂, it is tensioned synchronous pulley pitch curve and is designated as with driven pulley pitch curve common tangent incision superius C₄, it is tensioned synchronous pulley pitch curve and is designated as C with round active synchronization belt wheel pitch curve common tangent incision superius₅, any time C₄With C₅ Between arc length be c₃₃。

Any time, Timing Belt Zhou Changwei：

C=T₁₂+T₁₃+T₂₃+c₁₁+c₂₂+c₃₃ (16)

Step 6, the free pitch curve of tensioning synchronous pulley are calculated；

Iterative algorithm is as follows：

A () setting tensioning synchronous pulley center of rotation, the radius for being tensioned synchronous pulley is set to variable, is tensioned synchronous pulley Radius initial value gives, and is designated as r_3-0, belt length initial value is calculated according to formula (16) and is designated as C₀。

B () round active synchronization belt wheel turns over 1 °, calculating oval driven synchronous pulley according to gearratio requirement turns over accordingly Angle, be tensioned synchronous pulley corner it is identical with round active synchronization belt wheel.On the premise of ensureing that C is constant, according to formula (16) the round active synchronization belt wheel of reverse turns over corresponding tensioning synchronous pulley radius r at 1 °_3-1, that is, correspond to the p at moment₃。

C () repeats (b) 358 times, obtain round active synchronization belt wheel and turn over 2 °, and 3 ° ..., corresponding tensioning is synchronous at 359 ° Belt wheel radius is respectively r_3-2, r_3-3... ..., r_3-359。

D () so far obtains 360 concentric circles, by the tensioning synchronous pulley radius in (a), (b) and (c), one is taken every 1 ° The radius of individual circle, sequentially takes 360 radiuses, is the center of circle to set tensioning synchronous pulley center of rotation, will take 360 radiuses The outer end point is sequentially connected with, and constitutes a not rounded for closing.

E not rounded that () will obtain in (d) is tensioned being scaled up to footpath or diminution for each moment of synchronous pulley so that new The girth of the not rounded tensioning synchronous pulley for obtaining is equal with the girth of round active synchronization belt wheel and oval driven synchronous pulley.

The radius value at f each moment that () is tried to achieve (e) substitutes into the belt length that formula (16) calculates each moment.

If g the absolute value of the difference of the belt length at () each moment and initial belt length is respectively less than preset value, step (k) is carried out, Otherwise carry out step (h).

H () reduces not rounded tensioning synchronous pulley is each worth to footpath 15 ° before and after belt length maximum position correspondence moment point ~5%, belt length minimum position correspondence moment point before and after 5 °, increase not rounded tensioning synchronous pulley each to footpath value 1~ 5%, then it is fitted with B-spline and obtains new not rounded tensioning synchronous pulley.

I () will be scaled to footpath through the not rounded tensioning synchronous pulley each moment after (h) or reduce so that newly obtain Not rounded tensioning synchronous pulley girth it is equal with the girth of round active synchronization belt wheel and oval driven synchronous pulley.

J () will substitute into formula (16) and be calculated each moment correspondence Timing Belt through the not rounded tensioning synchronous pulley after (i) to footpath Belt length, if the absolute value of the difference of correspondence Timing Belt belt length of each moment and Timing Belt girth initial value is respectively less than preset value, is walked Suddenly (k), otherwise (h) is returned to.

(k) set up not rounded tensioning synchronous pulley each moment to footpath and corresponding cornerRelation is tensioning synchronous pulley Pitch curve equation.

The device have the advantages that：

1st, it is of the invention for the self-compensating circle-ellipse of the amount of becoming slack-not rounded three-wheel toothed belt transmission is carried in actual applications A whole set of perfect design theory basis is supplied, all circle-ellipses-not rounded three-wheel synchronous belt drive mechanism can be applied to, Promote promoting the use of for circle-ellipse-not rounded three-wheel toothed belt transmission.

2nd, driving wheel pitch curve is circle in the present invention, and driven pulley pitch curve is ellipse, and gearratio design is simple, round active The radius of synchronous pulley pitch curve, the major axis of oval driven synchronous pulley pitch curve, eccentricity are controlled variable, by three amounts Regulation change the shape of round active synchronization belt wheel pitch curve and oval driven synchronous pulley pitch curve, meet it is specific it is non-at the uniform velocity It is required that transmission.

3rd, the present invention is using the exact value for cutting polar coordinates theoretical calculation gearratio, it is easy to which programming realization, solving precision is high, side Just it is quick.

4th, the not rounded tensioning synchronous pulley in the present invention is the not rounded synchronous pulley of free pitch curve, can be justified with real-Time Compensation The belt sag variable quantity produced during type active synchronization belt wheel and the driven synchronous belt pulley transmission of free not rounded, realizes big centre-to-centre spacing Between non-at the uniform velocity directly accurate transmission.

Brief description of the drawings

Fig. 1 is transmission schematic diagram of the invention；

Fig. 2 is the gearratio and round active synchronization of round active synchronization belt wheel and oval driven synchronous pulley in the present invention Belt wheel angle relation curve map；

Fig. 3 is Timing Belt belt length change curve when being tensioned the pitch curve of synchronous pulley using the not rounded in the present invention；

Fig. 4 is oval driven synchronous pulley pitch curve figure in the present invention；

Fig. 5 is the tensioning free pitch curve fitted figure of synchronous pulley in the present invention.

Specific embodiment

Below in conjunction with the accompanying drawings and case study on implementation the invention will be further described.

Circle-ellipse-not rounded three-wheel toothed belt transmission method for designing, comprises the following steps that：

Step one, such as Fig. 1, give round active synchronization belt wheel pitch curve radius r₁=30mm, round active synchronization belt wheel Pitch curve cuts footpath p₁=r₁, tensioning synchronous pulley is the not rounded belt wheel according to the change fitting of Timing Belt girth slack；Three-wheel center Away from L₁=L₂=L₃=100mm, three-wheel calculates round active synchronization belt wheel section to wait girth closing convex curve according to formula below The girth s=188.4956mm of curve：

S=2 π × r₁ (1)

Step 2, the eccentricity e for giving oval driven synchronous pulley pitch curve₂=0.8, according to round active synchronization belt wheel The pitch curve principle equal with oval driven synchronous pulley pitch curve girth, calculates oval driven synchronous pulley pitch curve long respectively Axle a=35.1022mm, short axle b=21.0613mm.

It is determined that the polar equation of cutting of oval driven synchronous pulley pitch curve is：

p₂=7.0204-28.0818 × cos (θ₂) (4)

In formula, θ₂For oval driven synchronous pulley pitch curve cuts footpath p₂To moving coordinate system x₂o₂y₂Middle x₂The corner cut of axle.

Oval driven synchronous pulley pitch curve is as shown in Figure 4.

Step 3, round active synchronization belt wheel is calculated with the gearratio of oval driven synchronous pulley initial position：

Initial position, the moving coordinate system x of round active synchronization belt wheel pitch curve₁o₁y₁Middle x₁Axle is to quiet coordinate system xo₁X in y The corner of axleThe moving coordinate system x of oval driven synchronous pulley pitch curve₂o₂y₂Middle x₂Axle is to quiet coordinate system xo₁X-axis in y CornerAccording to cutting, polar coordinates are theoretical to be obtained：

In formula, p₁(θ₁₂) and p₂(θ₂₁) round active synchronization belt wheel pitch curve is respectively with oval driven synchronous pulley section song Line common tangent incision superius C₁、C₂It is corresponding to cut footpath, p₁(θ₁₃) and p₃(θ₃₁) round active synchronization belt wheel pitch curve is respectively with tensioning Synchronous pulley pitch curve common tangent incision superius C₆、C₅It is corresponding to cut footpath, p₂(θ₂₃) and p₃(θ₃₂) it is respectively oval driven synchronous pulley Pitch curve and tensioning synchronous pulley pitch curve common tangent incision superius C₃、C₄It is corresponding to cut footpath, θ₁₂₀It is round active synchronization belt wheel section Curve cuts footpath p₁(θ₁₂) cut footpath p with oval driven synchronous pulley pitch curve₂(θ₂₁) to respective moving coordinate system trunnion axis corner at the beginning of Value, θ₁₃₀For round active synchronization belt wheel pitch curve cuts footpath p₁(θ₁₃) cut footpath p with tensioning synchronous pulley pitch curve₃(θ₃₁) to each The corner initial value of moving coordinate system trunnion axis, θ₂₃₀For oval driven synchronous pulley pitch curve cuts footpath p₂(θ₂₃) and tensioning synchronous pulley Pitch curve cuts footpath p₃(θ₃₂) to respective moving coordinate system trunnion axis corner initial value, θ₁₂、θ₁₃Respectively round active synchronization belt wheel section Curve incision superius C₁、C₆Correspondence cuts footpath to moving coordinate system x₁o₁y₁Middle x₁The corner cut of axle, θ₂₁、θ₂₃Respectively oval driven Timing Belt Wheel pitch curve incision superius C₂、C₃Correspondence cuts footpath to moving coordinate system x₂o₂y₂Middle x₂The corner cut of axle, θ₃₁、θ₃₂Respectively it is tensioned Timing Belt Wheel pitch curve incision superius C₄、C₅Correspondence cuts footpath to moving coordinate system x₃o₃y₃Middle x₃The corner cut of axle, L₁For round active synchronization belt wheel with Oval driven synchronous pulley centre-to-centre spacing, L₂It is oval driven synchronous pulley and tensioning synchronous pulley centre-to-centre spacing, L₃For round active is same Step belt wheel and tensioning synchronous pulley centre-to-centre spacing；

The gearratio for calculating initial position according to formula (6) is i₁₂₀=1.1667：

Between step 4, the round active synchronization belt wheel of calculating, oval driven synchronous pulley and tensioning synchronous pulley are per two-wheeled Common tangent segment length.

Initial time, sets tensioning synchronous pulley pitch curve to give the circle of radius, round active synchronization belt wheel and ellipse Common tangent segment length T between the point of contact of driven synchronous pulley two₀, oval driven synchronous pulley and tensioning synchronous pulley two point of contact it Between common tangent segment length T₁, common tangent segment length T between round active synchronization belt wheel and tensioning synchronous pulley two point of contact₂Point It is not:

T is calculated according to formula (7)₀=103.3370mm, T₁=100.7259mm, T₂=102.9287mm.

When round active synchronization belt wheel turns over angleOval driven synchronous pulley accordingly turns over angleRound active Synchronous pulley pitch curve incision superius C₁、C₆Corresponding arc length variable quantity is s₁、s₆, oval driven synchronous pulley pitch curve incision superius C₂、C₃Corresponding arc length variable quantity is s₂、s₃, tensioning synchronous pulley pitch curve incision superius C₄、C₅Corresponding arc length variable quantity is s₄、 s₅.Then have：

In formula, p "₁(θ₁) it is p₁(θ₁) second-order differential, p "₂(θ₂) it is p₂(θ₂) second-order differential, p "₃(θ₃) it is p₃(θ₃) Second-order differential, θ₃It is tensioning Timing Belt round cut footpath p₃To moving coordinate system x₃o₃y₃Middle x₃The corner of axle.

Any time, the common tangent segment length between round active synchronization belt wheel and the oval point of contact of driven synchronous pulley two T₁₂, common tangent segment length T between oval driven synchronous pulley and tensioning synchronous pulley two point of contact₂₃, round active synchronization belt wheel With the common tangent segment length T between tensioning synchronous pulley two point of contact₁₃Respectively：

In formula, p '₁(θ₁₂)、p′₁(θ₁₃) it is respectively p₁(θ₁₂)、p₁(θ₁₃) first differential, p'₂(θ₂₁)、p'₂(θ₂₃) respectively It is p₂(θ₂₁)、p₂(θ₂₃) first differential, p'₃(θ₃₂)、p'₃(θ₃₁) it is respectively p₃(θ₃₂)、p₃(θ₃₁) first differential,To open Tight synchronous pulley pitch curve moving coordinate system x₃o₃y₃Middle x₃Axle is to quiet coordinate system xo₁The corner of x-axis in y.

Step 5, the gearratio for calculating any time round active synchronization belt wheel and oval driven synchronous pulley；

Round active synchronization belt wheel uniform rotation, p₁=r₁, p is solved according to formula (4)₂, then round active synchronization band is calculated Wheel and oval driven synchronous pulley instantaneous transmission ratio：

According to formula (10), (11), (12), when the round active synchronization belt wheel of calculating rotates a circle, round active synchronization belt wheel With the gearratio change such as Fig. 2 of oval driven synchronous pulley.

Step 6, calculating Timing Belt girth；

Round active synchronization belt wheel pitch curve is designated as C with tensioning synchronous pulley pitch curve common tangent incision superius₆, any time C₁With C₆Between arc length be c₁₁, round active synchronization belt wheel is designated as C with oval driven synchronous pulley pitch curve common tangent incision superius₂, Oval driven synchronous pulley pitch curve is designated as C with tensioning synchronous pulley pitch curve common tangent incision superius₃, any time C₂With C₃Between Arc length be c₂₂, it is tensioned synchronous pulley pitch curve and is designated as C with oval driven synchronous pulley pitch curve common tangent incision superius₄, tensioning Synchronous pulley pitch curve is designated as C with round active synchronization belt wheel pitch curve common tangent incision superius₅, any time C₄With C₅Between arc A length of c₃₃。

Any time, Timing Belt Zhou Changwei：

C=T₁₂+T₁₃+T₂₃+c₁₁+c₂₂+c₃₃ (14)

Initial time, Timing Belt original perimeter C is calculated according to formula (14)₀=515mm；

Each timing synchronization band belt length, each timing synchronization when driving wheel is rotated one week are sequentially calculated according to above method Band belt length change curve such as Fig. 3.

Step 7, the free pitch curve of tensioning synchronous pulley are calculated.

Iterative algorithm is as follows：

A () known tensioning synchronous pulley center of rotation, the radius for being tensioned synchronous pulley is set to variable r₃, it is tensioned Timing Belt Wheel radius initial value is designated as r_3-0=30mm, Timing Belt belt length initial value is designated as C₀=515mm.

B () round active synchronization belt wheel turns over 1 °, calculating oval driven synchronous pulley according to gearratio requirement turns over accordingly Angle, be tensioned synchronous pulley corner it is identical with round active synchronization belt wheel.On the premise of ensureing that C is constant, according to formula (14) when the round active synchronization belt wheel of reverse turns over 1 °, corresponding tensioning synchronous pulley radius r_3-1=30.0562mm.

C () repeats (b) 358 times, obtains r_3-2, r_3-3... ..., r_3-359。

D () so far obtains 360 concentric circles, by the tensioning synchronous pulley radius in (a), (b) and (c), one is taken every 1 ° The radius of individual circle, sequentially takes 360 radiuses, is the center of circle to set tensioning synchronous pulley center of rotation, will take 360 radiuses The outer end point is sequentially connected with, and constitutes a not rounded for closing.

E not rounded that () will obtain in (d) is tensioned being scaled up to footpath or diminution for each point of synchronous pulley so that new The girth of the not rounded tensioning synchronous pulley for arriving is equal with the girth of round active synchronization belt wheel and oval driven synchronous pulley.

The radius value at f each moment that () is tried to achieve (e) substitutes into the belt length that formula (14) calculates each moment.

If g the absolute value of the difference of the belt length at () each moment and initial belt length is respectively less than preset value, step (k) is carried out, Otherwise carry out step (h).

H () reduces what not rounded tensioning synchronous pulley was each worth to footpath 5 ° before and after belt length maximum position correspondence moment point 3%, 5 ° before and after belt length minimum position correspondence moment point, increase not rounded tensioning synchronous pulley is each worth to footpath 3%, then It is fitted with B-spline and obtains new not rounded tensioning synchronous pulley.

I () will be tensioned being scaled up to footpath or diminution for synchronous pulley each point through the not rounded after (h) so that newly obtain The girth of not rounded tensioning synchronous pulley is equal with the girth of round active synchronization belt wheel and oval driven synchronous pulley.

J () will substitute into formula (14) and be calculated each point correspondence Timing Belt band through the not rounded tensioning synchronous pulley after (i) to footpath It is long, if the absolute value of the difference of each point correspondence Timing Belt belt length and Timing Belt girth initial value is respectively less than preset value, carry out step K (), otherwise returns to (h).

(k) set up not rounded tensioning synchronous pulley each moment to footpath and corresponding cornerRelation is tensioning synchronous pulley Pitch curve equation.Three pitch curves taken turns and phase angle, center of rotation all determine, calculate tensioning synchronous pulley and round active is same Step belt wheel angle relation corresponding with oval driven synchronous pulley.

The free pitch curve of tensioning synchronous pulley such as Fig. 5 after calculating.

Timing Belt theory belt length variable quantity is 12mm in the embodiment, is the 2.3% of Timing Belt total length, because band needs Tensioning, can meet actual operation requirements.

Claims (1)

1. circle-ellipse-not rounded three-wheel toothed belt transmission method for designing, it is characterised in that：The method is specific as follows：

Step one, according to transmission rule determine round active synchronization belt wheel pitch curve with oval driven synchronous pulley pitch curve side Journey；

Round active synchronization belt wheel is the input link of uniform rotation,It is the moving coordinate system of round active synchronization belt wheel pitch curve x₁o₁y₁Middle x₁Axle is to quiet coordinate system xo₁The corner of x-axis, θ in y₁It is p₁To moving coordinate system x₁o₁y₁Middle x₁The corner cut of axle, round active Synchronous pulley cuts polar equation：

p₁=r₁ (1)

S=2 π × r₁ (2)

In formula, p₁Footpath, r are cut for round active synchronization belt wheel pitch curve₁It is the radius of round active synchronization belt wheel pitch curve, s is The girth of round active synchronization belt wheel pitch curve；

Oval driven synchronous pulley is output link, cuts polar equation：

p₂=a × (1-e₂)-a×e₂×cos(θ₂) (3)

In formula, p₂Footpath, θ are cut for oval driven synchronous pulley pitch curve₂It is p₂To moving coordinate system x₂o₂y₂Middle x₂The corner cut of axle, e₂ It is the eccentricity of oval driven synchronous pulley pitch curve, a is the major axis of oval driven synchronous pulley pitch curve；

$a=\frac{s}{\[(2\mathrm{\π}-4)\×\sqrt{1-e_2^2}+4\]}---\left(4\right)$

$b=\sqrt{a^2-c^2}---\left(5\right)$

In formula, b is the short axle of oval driven synchronous pulley pitch curve, and c is the focal length of oval driven synchronous pulley pitch curve；

C=a × e₂ (6)

Step 2, round active synchronization belt wheel is calculated with the gearratio of oval driven synchronous pulley initial position；

Initial position, the moving coordinate system x of round active synchronization belt wheel pitch curve₁o₁y₁Middle x₁Axle is to quiet coordinate system xo₁X-axis in y CornerThe moving coordinate system x of oval driven synchronous pulley pitch curve₂o₂y₂Middle x₂Axle is to quiet coordinate system xo₁X-axis turns in y AngleAccording to cutting, polar coordinates are theoretical to be obtained：

$\left\{\begin{array}{c}{\mathrm{\θ}}_{12}={\mathrm{\θ}}_{21}={\mathrm{\θ}}_{120}\\ {\mathrm{\θ}}_{13}={\mathrm{\θ}}_{31}={\mathrm{\θ}}_{130}\\ {\mathrm{\θ}}_{23}={\mathrm{\θ}}_{32}={\mathrm{\θ}}_{230}\\ p_2\left({\mathrm{\θ}}_{21}\right)-p_1\left({\mathrm{\θ}}_{12}\right)=-L_1\×cos\left({\mathrm{\θ}}_{120}\right)\\ p_3\left({\mathrm{\θ}}_{31}\right)-p_1\left({\mathrm{\θ}}_{13}\right)=-L_3\×cos\left({\mathrm{\θ}}_{130}\right)\\ p_3\left({\mathrm{\θ}}_{32}\right)-p_2\left({\mathrm{\θ}}_{23}\right)=-L_2\×cos\left({\mathrm{\θ}}_{230}\right)\end{array}\right.---\left(7\right)$

In formula, p₁(θ₁₂) and p₂(θ₂₁) round active synchronization belt wheel pitch curve is respectively with oval driven synchronous pulley pitch curve public affairs Tangent line incision superius C₁、C₂It is corresponding to cut footpath, p₁(θ₁₃) and p₃(θ₃₁) to be respectively round active synchronization belt wheel pitch curve synchronous with tensioning Belt wheel pitch curve common tangent incision superius C₆、C₅It is corresponding to cut footpath, p₂(θ₂₃) and p₃(θ₃₂) it is respectively oval driven synchronous pulley section song Line and tensioning synchronous pulley pitch curve common tangent incision superius C₃、C₄It is corresponding to cut footpath, θ₁₂₀It is round active synchronization belt wheel pitch curve Cut footpath p₁(θ₁₂) cut footpath p with oval driven synchronous pulley pitch curve₂(θ₂₁) to the corner cut initial value of respective moving coordinate system trunnion axis, θ₁₃₀For round active synchronization belt wheel pitch curve cuts footpath p₁(θ₁₃) cut footpath p with tensioning synchronous pulley pitch curve₃(θ₃₁) arrive each automatic seat The corner cut initial value of mark system trunnion axis, θ₂₃₀For oval driven synchronous pulley pitch curve cuts footpath p₂(θ₂₃) bent with tensioning synchronous pulley section Line cuts footpath p₃(θ₃₂) to respective moving coordinate system trunnion axis corner cut initial value, θ₁₂、θ₁₃Respectively round active synchronization belt wheel pitch curve Incision superius C₁、C₆Correspondence cuts footpath to moving coordinate system x₁o₁y₁Middle x₁The corner cut of axle, θ₂₁、θ₂₃Respectively oval driven synchronous pulley section Curve incision superius C₂、C₃Correspondence cuts footpath to moving coordinate system x₂o₂y₂Middle x₂The corner cut of axle, θ₃₁、θ₃₂Respectively it is tensioned synchronous pulley section Curve incision superius C₄、C₅Correspondence cuts footpath to moving coordinate system x₃o₃y₃Middle x₃The corner cut of axle, L₁It is round active synchronization belt wheel and ellipse Driven synchronous pulley centre-to-centre spacing, L₂It is oval driven synchronous pulley and tensioning synchronous pulley centre-to-centre spacing, L₃It is round active synchronization band Wheel and tensioning synchronous pulley centre-to-centre spacing；

The round active synchronization belt wheel of initial position is with oval driven synchronous belt pulley transmission ratio：

$i_{120}=\frac{p_2\left({\mathrm{\θ}}_{120}\right)}{p_1\left({\mathrm{\θ}}_{120}\right)}---\left(8\right)$

Step 3, the public affairs for calculating round active synchronization belt wheel, oval driven synchronous pulley and being tensioned between the every two-wheeled of synchronous pulley Tangent line segment length；

Initial time, sets tensioning synchronous pulley pitch curve driven with ellipse to give the circle of radius, round active synchronization belt wheel Common tangent segment length T between the point of contact of synchronous pulley two₀, between oval driven synchronous pulley and tensioning synchronous pulley two point of contact Common tangent segment length T₁, common tangent segment length T between round active synchronization belt wheel and tensioning synchronous pulley two point of contact₂Respectively：

In formula, p '₁(θ₁₂₀)、p′₁(θ₁₃₀) it is respectively p₁(θ₁₂₀)、p₁(θ₁₃₀) first differential, p '₂(θ₁₂₀)、p′₂(θ₂₃₀) respectively It is p₂(θ₁₂₀)、p₂(θ₂₃₀) first differential, p '₃(θ₁₃₀)、p′₃(θ₂₃₀) it is respectively p₃(θ₁₃₀)、p₃(θ₂₃₀) first differential；

When round active synchronization belt wheel turns over angleOval driven synchronous pulley accordingly turns over angleRound active synchronization Belt wheel pitch curve incision superius C₁、C₆Corresponding arc length variable quantity is s₁、s₆, oval driven synchronous pulley pitch curve incision superius C₂、C₃ Corresponding arc length variable quantity is s₂、s₃, tensioning synchronous pulley pitch curve incision superius C₄、C₅Corresponding arc length variable quantity is s₄、s₅；Then Have：

$\left\{\begin{array}{c}{s}_1={\∫}_{{\mathrm{\θ}}_{120}}^{{\mathrm{\θ}}_{12}}\left(p_1\right({\mathrm{\θ}}_1)+p_1^{\′\′}({\mathrm{\θ}}_1\left)\right){\mathrm{d\θ}}_1\\ s_2={\∫}_{{\mathrm{\θ}}_{120}}^{{\mathrm{\θ}}_{21}}\left(p_2\right({\mathrm{\θ}}_2)+p_2^{\′\′}({\mathrm{\θ}}_2\left)\right){\mathrm{d\θ}}_2\\ s_3={\∫}_{{\mathrm{\θ}}_{230}}^{{\mathrm{\θ}}_{21}}\left(p_2\right({\mathrm{\θ}}_2)+p_2^{\′\′}({\mathrm{\θ}}_2\left)\right){\mathrm{d\θ}}_2\\ s_4={\∫}_{{\mathrm{\θ}}_{230}}^{{\mathrm{\θ}}_{32}}\left(p_3\right({\mathrm{\θ}}_3)+p_3^{\′\′}({\mathrm{\θ}}_3\left)\right){\mathrm{d\θ}}_3\\ s_5={\∫}_{{\mathrm{\θ}}_{130}}^{{\mathrm{\θ}}_{31}}\left(p_3\right({\mathrm{\θ}}_3)+p_3^{\′\′}({\mathrm{\θ}}_3\left)\right){\mathrm{d\θ}}_3\\ s_6={\∫}_{{\mathrm{\θ}}_{130}}^{{\mathrm{\θ}}_{13}}\left(p_1\right({\mathrm{\θ}}_1)+p_1^{\′\′}({\mathrm{\θ}}_1\left)\right){\mathrm{d\θ}}_1\end{array}\right.---\left(10\right)$

In formula, p "₁(θ₁) it is p₁(θ₁) second-order differential, p "₂(θ₂) it is p₂(θ₂) second-order differential, p "₃(θ₃) it is p₃(θ₃) two Rank differential, θ₃It is tensioning Timing Belt round cut footpath p₃To moving coordinate system x₃o₃y₃Middle x₃The corner cut of axle；

Any time, the common tangent segment length T between round active synchronization belt wheel and the oval point of contact of driven synchronous pulley two₁₂, it is ellipse Common tangent segment length T between the driven synchronous pulley of circle and tensioning synchronous pulley two point of contact₂₃, round active synchronization belt wheel with tensioning Common tangent segment length T between the point of contact of synchronous pulley two₁₃Respectively：

In formula, p '₁(θ₁₂)、p′₁(θ₁₃) it is respectively p₁(θ₁₂)、p₁(θ₁₃) first differential, p '₂(θ₂₁)、p′₂(θ₂₃) it is respectively p₂ (θ₂₁)、p₂(θ₂₃) first differential, p '₃(θ₃₂)、p′₃(θ₃₁) it is respectively p₃(θ₃₂)、p₃(θ₃₁) first differential,It is tensioning Synchronous pulley pitch curve moving coordinate system x₃o₃y₃Middle x₃Axle is to quiet coordinate system xo₁The corner of x-axis in y；

Step 4, the gearratio for calculating any time round active synchronization belt wheel and oval driven synchronous pulley；

Round active synchronization belt wheel uniform rotation, according to formula (1), (3) solve p₁, p₂, then instantaneous transmission ratio be：

$i_{12}=\frac{p_2}{p_1}---\left(14\right)$

Step 5, calculating any time Timing Belt girth；

Round active synchronization belt wheel pitch curve is designated as C with tensioning synchronous pulley pitch curve common tangent incision superius₆, any time C₁With C₆Between arc length be c₁₁, oval driven synchronous pulley pitch curve and tensioning synchronous pulley pitch curve common tangent incision superius are designated as C₃, Any time C₂With C₃Between arc length be c₂₂, tensioning synchronous pulley pitch curve and driven pulley pitch curve common tangent incision superius are designated as C₄, Tensioning synchronous pulley pitch curve is designated as C with round active synchronization belt wheel pitch curve common tangent incision superius₅, any time C₄With C₅Between Arc length be c₃₃；

$\left.\begin{array}{c}{c}_{11}=\underset{{\mathrm{\θ}}_{12}}{\overset{{\mathrm{\θ}}_{13}}{\∫}}\[p_1\left({\mathrm{\θ}}_1\right)+p_1^{\′\′}\left({\mathrm{\θ}}_1\right)\]{\mathrm{d\θ}}_1\\ c_{22}=\underset{{\mathrm{\θ}}_{23}}{\overset{{\mathrm{\θ}}_{21}}{\∫}}\[p_2\left({\mathrm{\θ}}_2\right)+p_2^{\′\′}\left({\mathrm{\θ}}_2\right)\]{\mathrm{d\θ}}_2\\ c_{33}=\underset{{\mathrm{\θ}}_{31}}{\overset{{\mathrm{\θ}}_{32}}{\∫}}\[p_3\left({\mathrm{\θ}}_3\right)+p_3^{\′\′}\left({\mathrm{\θ}}_3\right)\]{\mathrm{d\θ}}_3\end{array}\right\}---\left(15\right)$

Any time, Timing Belt Zhou Changwei：

C=T₁₂+T₁₃+T₂₃+c₁₁+c₂₂+c₃₃ (16)

Step 6, the free pitch curve of tensioning synchronous pulley are calculated；

Iterative algorithm is as follows：

A () setting tensioning synchronous pulley center of rotation, the radius for being tensioned synchronous pulley is set to variable, is tensioned synchronous pulley radius Initial value gives, and is designated as r_3-0, belt length initial value is calculated according to formula (16) and is designated as C₀；

B () round active synchronization belt wheel turns over 1 °, calculating oval driven synchronous pulley according to gearratio requirement turns over corresponding angle Degree, the corner for being tensioned synchronous pulley is identical with round active synchronization belt wheel；It is anti-according to formula (16) on the premise of ensureing that C is constant Round active synchronization belt wheel is asked to turn over corresponding tensioning synchronous pulley radius r at 1 °_3-1, that is, correspond to the p at moment₃；

C () repeats (b) 358 times, obtains round active synchronization belt wheel and turn over 2 °, 3 ° ..., corresponding tensioning synchronous pulley at 359 ° Radius is respectively r_3-2, r_3-3... ..., r_3-359；

D () so far obtains 360 concentric circles, by the tensioning synchronous pulley radius in (a), (b) and (c), a circle is taken every 1 ° Radius, sequentially take 360 radiuses, with set tensioning synchronous pulley center of rotation be the center of circle, 360 outer ends of radius will be taken Point is sequentially connected with, and constitutes a not rounded for closing；

E not rounded that () will obtain in (d) is tensioned being scaled up to footpath or diminution for each moment of synchronous pulley so that newly obtain Not rounded tensioning synchronous pulley girth it is equal with the girth of round active synchronization belt wheel and oval driven synchronous pulley；

The radius value at f each moment that () is tried to achieve (e) substitutes into the belt length that formula (16) calculates each moment；

If g the absolute value of the difference of the belt length at () each moment and initial belt length is respectively less than preset value, step (k) is carried out, otherwise Carry out step (h)；

(h) belt length maximum position correspondence moment point before and after 5 °, reduce not rounded tensioning synchronous pulley each to footpath value 1~ 5%, 5 ° before and after belt length minimum position correspondence moment point, increase not rounded tensioning synchronous pulley is each worth to footpath 1~5%, Then it is fitted with B-spline and obtains new not rounded tensioning synchronous pulley；

I () will be scaled to footpath through the not rounded tensioning synchronous pulley each moment after (h) or reduce so that what is newly obtained is non- The girth of circle tensioning synchronous pulley is equal with the girth of round active synchronization belt wheel and oval driven synchronous pulley；

J () will substitute into formula (16) and be calculated each moment correspondence Timing Belt belt length through the not rounded tensioning synchronous pulley after (i) to footpath, If the absolute value of the difference of each moment correspondence Timing Belt belt length and Timing Belt girth initial value is respectively less than preset value, step (k) is carried out, Otherwise return to (h)；

(k) set up not rounded tensioning synchronous pulley each moment to footpath and corresponding cornerIt is bent that relation is tensioning synchronous pulley section Line equation.