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CN106838160B - Pascal non-circular-sine non-circular-non-circular three-wheel synchronous belt transmission design method - Google Patents

Pascal non-circular-sine non-circular-non-circular three-wheel synchronous belt transmission design method Download PDF

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CN106838160B
CN106838160B CN201710191507.3A CN201710191507A CN106838160B CN 106838160 B CN106838160 B CN 106838160B CN 201710191507 A CN201710191507 A CN 201710191507A CN 106838160 B CN106838160 B CN 106838160B
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circular
synchronous pulley
pitch curve
pascal
tensioning
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CN106838160A (en
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夏旭东
陈建能
童林
汪应萍
徐继垦
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Anhui Huajingdao E Commerce Co Ltd
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Zhejiang University of Technology ZJUT
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    • F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
    • F16—ENGINEERING ELEMENTS AND UNITS; GENERAL MEASURES FOR PRODUCING AND MAINTAINING EFFECTIVE FUNCTIONING OF MACHINES OR INSTALLATIONS; THERMAL INSULATION IN GENERAL
    • F16H—GEARING
    • F16H7/00—Gearings for conveying rotary motion by endless flexible members
    • F16H7/02—Gearings for conveying rotary motion by endless flexible members with belts; with V-belts
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/10—Geometric CAD
    • G06F30/17—Mechanical parametric or variational design

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Abstract

The invention discloses non-circular-sinusoidal non-circular-non-circular three-wheel toothed belt transmission design methods of Bath main officer of Tibet.The present invention initially sets up the pitch curve equation of synchronous belt principal and subordinate wheel, and utilizes and cut polar coordinates theoretical calculation principal and subordinate wheel transmission ratio；Then the perimeter for calculating synchronous belt, changes according to synchronous belt perimeter slack and calculates non-circular tensioning synchronous pulley pitch curve by iterative algorithm.Tensioning wheel is the non-circular synchronous pulley of free pitch curve, the problem of non-circular V belt translation of traditional two-wheeled cannot meet non-at the uniform velocity transmission and tensioning in real time simultaneously can be overcome with the synchronous belt sag variable quantity generated in the non-circular driving wheel of real-time compensation Bath main officer of Tibet and sinusoidal non-circular driven wheel transmission process；Circular diameter, length, deformation coefficient and order occur for the non-circular driving wheel pitch curve of Bath main officer of Tibet, four parameters such as the amplitude of sinusoidal non-circular driven wheel pitch curve are controlled variable, by adjusting these parameters, change driving wheel and driven wheel pitch curve shape, meet big center away from it is specific it is non-at the uniform velocity transmission require.

Description

Non-circular-sinusoidal non-circular-non-circular three-wheel toothed belt transmission design method of Bath main officer of Tibet
Technical field
The present invention relates to a kind of design methods of non-circular toothed belt transmission, and in particular to self-compensating bar of one kind amount of becoming slack Non-circular-sinusoidal non-circular-non-circular three-wheel toothed belt transmission design method of this main officer of Tibet.
Background technique
Transmission mechanism changes the forms of motion and speed of input and output component, to meet different operating environmental requirement, In non-uniform transmission mechanism occupy extremely important status, common are link mechanism, cam mechanism, non-circular gear mechanism etc..Phase For link mechanism and cam mechanism, non-circular gear mechanism has compact-sized, stable drive, transmitting power larger, easy to be real The advantages that 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, which is only suitable for center, non-to be at the uniform velocity driven away from smaller, lubrication are convenient Occasion, therefore be suitable for big center and be driven away from the non-circular flexible element (band/chain) of, the inconvenient and low manufacturing cost occasion of lubrication to meet the tendency of And it gives birth to.Wherein non-circular chaindriven polygon effect is obvious, therefore when having strict demand to non-at the uniform velocity transmission ratio changing rule Just it is restricted；Frictional V belt translation common simultaneously cannot be guaranteed accurate transmission ratio rule due to Elastic Sliding.
Current non-round belt (chain) transmission, all only 2 non-circular bands (chain) are taken turns --- and driving wheel and driven wheel are being driven In the process due to its pitch curve be it is non-circular, the slack of band (chain) is real-time change, therefore cannot guarantee work institute simultaneously It is required that non-at the uniform velocity transmission ratio changing rule and band (chain) real-time tensioning.In order to compensate for the band (chain) in transmission in practical application Slack variation, by additional springs with realize tensioning, due in a period of motion its tensile force be variation, 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 will affect in turn in this way, and And kinetic characteristics are deteriorated；Therefore in practical projects, non-round belt (chain) transmission is rarely applied to accurately load high-speed drive Occasion.
Summary of the invention
The purpose of the present invention is in view of the above problems, proposing that the self-compensating Bath main officer of Tibet of one kind amount of becoming slack is non-circular-sinusoidal non- Circle-non-circular three-wheel toothed belt transmission design method provides a whole set of for non-circular synchronous pulley in practical applications and perfect sets Theoretical basis is counted, realizes non-at the uniform velocity directly accurate transmission of the big center away between.The design method initially sets up synchronous belt principal and subordinate The pitch curve equation of driving wheel, and synchronous belt pulley transmission ratio is moved using polar coordinates theoretical calculation principal and subordinate is cut；Then synchronous belt is calculated Perimeter calculates non-circular the every of tensioning synchronous pulley pitch curve by alternative manner according to the variation of synchronous belt perimeter slack and joins Number.
In order to solve the above technical problems, the technical scheme is that
The specific steps of the present invention are as follows:
Step 1: non-circular driven synchronous according to the transmission rule calculating non-circular active synchronization belt wheel pitch curve of Bath main officer of Tibet and sine Belt wheel pitch curve equation；
The non-circular active synchronization belt wheel of Bath main officer of Tibet is the input link of uniform rotation, cuts polar equation:
In formula, r11、r12The respectively non-circular active synchronization belt wheel pitch curve first segment curve of Bath main officer of Tibet and second segment curve It is denaturalized polar diameter, b1For the generation circular diameter of the non-circular active synchronization belt wheel pitch curve of Bath main officer of Tibet, n1For the non-circular active synchronization band of Bath main officer of Tibet Take turns the order of pitch curve, m11、m12For the change of Bath main officer of Tibet non-circular active synchronization belt wheel pitch curve first segment curve and second segment curve Shape coefficient,For the non-circular active synchronization belt wheel pitch curve of Bath main officer of Tibet polar diameter to moving coordinate system x1o1y1Middle x1The corner of axis, l are The length of the non-circular pitch curve of Bath main officer of Tibet.
In formula, p1Diameter, θ are cut for the non-circular active synchronization belt wheel pitch curve of Bath main officer of Tibet1For p1To moving coordinate system x1o1y1Middle x1 The corner cut of axis.
Sinusoidal non-circular driven synchronous pulley is the output wheel of synchronous belt, and pitch curve cuts polar equation are as follows:
In formula, r21For the polar diameter of sinusoidal non-circular driven synchronous pulley pitch curve, α is that sinusoidal non-circular driven synchronous pulley section is bent Line parameter, A are sinusoidal amplitude,γ is positive chord joint parameter of curve, and taking γ=π/6, b is proportionality coefficient, a It is the center of the non-circular active synchronization belt wheel pitch curve of Bath main officer of Tibet and sinusoidal non-circular driven synchronous pulley pitch curve away from β is Bath main officer of Tibet The relative rotation of non-circular active synchronization belt wheel pitch curve and sinusoidal non-circular driven synchronous pulley pitch curve, β=[0~2 π].p2For Sinusoidal non-circular driven synchronous pulley pitch curve cuts diameter, θ2For p2To moving coordinate system x2o2y2Middle x2The corner cut of axis,For moving axes It is x2o2y2Middle x2Axis is to quiet coordinate system xo1The corner of x-axis in y.
Step 2: calculating the transmission of the non-circular active synchronization belt wheel of Bath main officer of Tibet and sinusoidal non-circular driven synchronous pulley initial position Than:
Initial position, the moving coordinate system x of the non-circular active synchronization belt wheel pitch curve of Bath main officer of Tibet1o1y1Middle x1Axis is to quiet coordinate system xo1The corner of x-axis in yThe moving coordinate system x of sinusoidal non-circular driven synchronous pulley pitch curve2o2y2Middle x2Axis is marked to sitting quietly It is xo1The corner of x-axis in yAccording to cutting, polar coordinates are theoretical to be obtained:
In formula, p1(θ12) and p2(θ21) it is respectively that the non-circular active synchronization belt wheel pitch curve of Bath main officer of Tibet and sine are non-circular driven same Walk belt wheel pitch curve common tangent incision superius C1、C2It is corresponding to cut diameter, p1(θ13) and p3(θ31) it is respectively the non-circular active synchronization of Bath main officer of Tibet Belt wheel pitch curve and tensioning synchronous pulley pitch curve common tangent incision superius C6、C5It is corresponding to cut diameter, p2(θ23) and p3(θ32) be respectively Sinusoidal non-circular driven synchronous pulley pitch curve and tensioning synchronous pulley pitch curve common tangent incision superius C3、C4It is corresponding to cut diameter, θ120 Diameter p is cut for the non-circular active synchronization belt wheel pitch curve of Bath main officer of Tibet1(θ12) with sinusoidal non-circular driven synchronous pulley pitch curve cut diameter p2 (θ21) arrive respective moving coordinate system trunnion axis corner initial value, θ130Diameter p is cut for the non-circular active synchronization belt wheel pitch curve of Bath main officer of Tibet1 (θ13) with tensioning synchronous pulley pitch curve cut diameter p3(θ31) arrive respective moving coordinate system trunnion axis corner initial value, θ230It is sinusoidal non- The driven synchronous pulley pitch curve of circle cuts diameter p2(θ23) with tensioning synchronous pulley pitch curve cut diameter p3(θ32) arrive respective moving coordinate system water The corner initial value of flat axis, θ12、θ13The respectively non-circular active synchronization belt wheel pitch curve incision superius C of Bath main officer of Tibet1、C6Corresponding diameter of cutting is to moving Coordinate system x1o1y1Middle x1The corner of axis, θ21、θ23Respectively sinusoidal non-circular driven synchronous pulley pitch curve incision superius C2、C3It is corresponding Diameter is cut to moving coordinate system x2o2y2Middle x2The corner of axis, θ31、θ32Respectively it is tensioned synchronous pulley pitch curve incision superius C4、C5It is corresponding Diameter is cut to moving coordinate system x3o3y3Middle x3The corner of axis, L1For the non-circular active synchronization belt wheel of Bath main officer of Tibet and sinusoidal non-circular driven synchronous belt Wheel center is away from L2It is sinusoidal non-circular driven synchronous pulley and tensioning synchronous pulley center away from L3For the non-circular active synchronization band of Bath main officer of Tibet Wheel with tensioning synchronous pulley center away from；
The non-circular active synchronization belt wheel of initial position Bath main officer of Tibet and sinusoidal non-circular driven synchronous pulley instantaneous transmission ratio are as follows:
Step 3: calculating the non-circular active synchronization belt wheel of Bath main officer of Tibet, sinusoidal non-circular driven synchronous pulley and tensioning synchronous pulley Common tangent segment length between every two-wheeled.
Initial time sets circle of the tensioning synchronous pulley pitch curve to give radius, the non-circular active synchronization belt wheel of Bath main officer of Tibet With the common tangent segment length T between sinusoidal non-circular driven two point of contact of synchronous pulley0, sinusoidal non-circular driven synchronous pulley and tensioning it is same Walk the common tangent segment length T between two point of contact of belt wheel1And the non-circular active synchronization belt wheel of Bath main officer of Tibet is cut with tensioning synchronous pulley two Common tangent segment length T between point2It is respectively as follows:
In formula, p1'(θ120)、p1'(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 the non-circular active synchronization belt wheel of Bath main officer of Tibet turns over angleSinusoidal non-circular driven synchronous pulley accordingly turns over angleThe non-circular active synchronization belt wheel pitch curve incision superius C of Bath main officer of Tibet1、C6Corresponding arc length variable quantity is s1、s6, sinusoidal non-circular driven Synchronous pulley pitch curve incision superius C2、C3Corresponding arc length variable quantity is s2、s3, it is tensioned synchronous pulley pitch curve incision superius C4、C5 Corresponding arc length variable quantity is s4、s5.Then have:
In formula, p1″(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, θ3To be tensioned synchronous belt round cut diameter p3To moving coordinate system x3o3y3Middle x3The corner of axis.
Any time, the public affairs between the non-circular active synchronization belt wheel of Bath main officer of Tibet and sinusoidal non-circular driven two point of contact of synchronous pulley are cut Line segment length T12, common tangent segment length T between sinusoidal non-circular driven synchronous pulley and tensioning two point of contact of synchronous pulley23And Common tangent segment length T between the non-circular active synchronization belt wheel of Bath main officer of Tibet and tensioning two point of contact of synchronous pulley13It is respectively as follows:
In formula, p1'(θ12)、p1'(θ13) it is respectively p1(θ12)、p1(θ13) first differential, p'2(θ21)、p'2(θ23) respectively For 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 x3Axis is to quiet coordinate system xo1The corner of x-axis in y.
Step 4: calculating the transmission of the non-circular active synchronization belt wheel of any time Bath main officer of Tibet and sinusoidal non-circular driven synchronous pulley Than；
The non-circular active synchronization belt wheel uniform rotation of Bath main officer of Tibet, according to formula (2), (4) solve p1, p2, then instantaneous transmission ratio are as follows:
Step 5: calculating any time synchronous belt perimeter；
The non-circular active synchronization belt wheel pitch curve of Bath main officer of Tibet and tensioning synchronous pulley pitch curve common tangent incision superius are denoted as C6, appoint Anticipate moment C1With C6Between arc length be c11, sinusoidal non-circular driven synchronous pulley pitch curve and tensioning synchronous pulley pitch curve common tangent Incision superius is denoted as C3, any time C2With C3Between arc length be c22, it is tensioned synchronous pulley pitch curve and sinusoidal non-circular driven synchronous belt Wheel pitch curve common tangent incision superius is denoted as C4, tensioning synchronous pulley pitch curve and the non-circular active synchronization belt wheel pitch curve of Bath main officer of Tibet are public Tangent line incision superius is denoted as C5, any time C4With C5Between arc length be c33。
Any time, synchronous belt perimeter are as follows:
C=T12+T13+T23+c11+c22+c33 (14)
Step 6: the free pitch curve of tensioning synchronous pulley calculates；
Iterative algorithm is as follows:
(a) setting tensioning synchronous pulley center of rotation, the radius for being tensioned synchronous pulley are set as variable, are tensioned synchronous pulley Radius initial value is given, is denoted as r3-0, belt length initial value, which is calculated, according to formula (14) is denoted as C0。
(b) the non-circular active synchronization belt wheel of Bath main officer of Tibet turns over 1 °, is required to calculate sinusoidal non-circular driven synchronous belt according to transmission ratio Wheel turns over corresponding angle, and the corner for being tensioned synchronous pulley is identical as the non-circular active synchronization belt wheel of Bath main officer of Tibet.Guaranteeing that C is constant Under the premise of, corresponding tensioning synchronous pulley radius r when turning over 1 ° according to the non-circular active synchronization belt wheel of formula (14) reverse Bath main officer of Tibet3-1, Correspond to the p at moment3。
(c) it repeats (b) 358 times, obtains when the non-circular active synchronization belt wheel of Bath main officer of Tibet turns over 2 °, 3 ° ..., 359 ° corresponding Tight synchronous pulley radius is respectively r3-2, r3-3... ..., r3-359。
(d) 360 concentric circles are so far obtained, by the tensioning synchronous pulley radius in (a), (b) and (c), take one every 1 ° The radius of a circle sequentially takes 360 radiuses, to set tensioning synchronous pulley center of rotation as the center of circle, will take 360 radiuses The outer end point is sequentially connected with, and composition one is closed non-circular.
(e) by obtained in (d) it is non-circular tensioning synchronous pulley each moment to diameter scale up or reduce so that newly The perimeter of obtained non-circular tensioning synchronous pulley and the non-circular active synchronization belt wheel of Bath main officer of Tibet and sinusoidal non-circular driven synchronous pulley Perimeter is equal.
(f) radius value at (e) obtained each moment is substituted into the belt length that formula (14) calculate each moment.
If (g) 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 step (h) is carried out.
(h) 5 ° before and after belt length maximum position corresponds to moment point, reduce non-circular tensioning synchronous pulley respectively to the 1 of diameter value ~5%, 5 ° before and after belt length minimum position corresponds to moment point, increase it is non-circular tensioning synchronous pulley respectively to diameter value 1~ 5%, it is then fitted to obtain new non-circular tensioning synchronous pulley with B-spline.
(i) non-circular tensioning synchronous pulley each moment after (h) is scaled up or is reduced to diameter, so that newly obtaining Non-circular tensioning synchronous pulley perimeter and the non-circular active synchronization belt wheel of Bath main officer of Tibet and sinusoidal non-circular driven synchronous pulley perimeter It is equal.
(j) it the non-circular tensioning synchronous pulley after (i) is substituted into formula (14) to diameter is calculated each moment and correspond to synchronous belt Belt length is walked if each moment corresponds to synchronous belt belt length and the absolute value of the difference of synchronous belt perimeter initial value is respectively less than preset value Suddenly (k), otherwise (h) is returned to.
(k) establish each moment of non-circular tensioning synchronous pulley to diameter and corresponding cornerRelationship is to be tensioned synchronous pulley Pitch curve equation.
The invention has the benefit that
1, the present invention is that non-circular-sinusoidal non-circular-non-circular three-wheel toothed belt transmission of the self-compensating Bath main officer of Tibet of the amount of becoming slack exists A whole set of perfect design theory basis is provided in practical application, can be applied to all Bath main officer of Tibets it is non-circular-sinusoidal it is non-circular- Non-circular three-wheel synchronous belt drive mechanism, the popularization for promoting non-circular-sinusoidal non-circular-non-circular three-wheel toothed belt transmission of Bath main officer of Tibet make With.
2, driving wheel pitch curve is Bath main officer of Tibet curve in the present invention, and driven wheel pitch curve is sinusoidal pattern curve, and Bath main officer of Tibet is bent Four generation circular diameter, length, deformation coefficient and order variables of line, the amplitude of sinusoidal non-circular driven synchronous pulley pitch curve and α, b and β are controlled variable, can change the shape of driving wheel and driven wheel pitch curve by the adjusting of this tittle, are met more Specific non-be at the uniform velocity driven.
3, the present invention is easily programmed realization using the exact value for cutting polar coordinates theoretical calculation transmission ratio, and solving precision is high, side Just quick.
4, the tensioning synchronous pulley in the present invention is the non-circular synchronous pulley of free pitch curve, can be with real-time compensation Bath main officer of Tibet The belt sag variable quantity generated during non-circular active synchronization belt wheel and sinusoidal non-circular driven synchronous belt pulley transmission, realizes big center Non- at the uniform velocity directly accurate transmission away between.
Detailed description of the invention
Fig. 1 is transmission schematic diagram of the invention；
Fig. 2 is the transmission of the non-circular active synchronization belt wheel of Bath main officer of Tibet and sinusoidal non-circular driven synchronous pulley in the embodiment of the present invention Than with the non-circular active synchronization belt wheel angle relation curve graph of Bath main officer of Tibet；
Synchronous belt belt length change curve when Fig. 3 is the pitch curve using the non-circular tensioning synchronous pulley in the embodiment of the present invention Figure；
Fig. 4 is the non-circular active synchronization belt wheel pitch curve figure of Bath main officer of Tibet in the embodiment of the present invention；
Fig. 5 is sinusoidal non-circular driven synchronous pulley pitch curve figure in the embodiment of the present invention；
Fig. 6 is the tensioning free pitch curve figure of synchronous pulley in the embodiment of the present invention.
Specific embodiment
With reference to the accompanying drawing and case study on implementation the invention will be further described.
Non-circular-sinusoidal non-circular-non-circular three-wheel toothed belt transmission design method of Bath main officer of Tibet, the specific steps are as follows:
Step 1: such as Fig. 1, the non-circular active synchronization belt wheel pitch curve base circle diameter (BCD) b of Bath main officer of Tibet1=10mm, length l= 40mm, three-wheel center is away from L1=L2=L3=100mm, the order n of the non-circular active synchronization belt wheel pitch curve of Bath main officer of Tibet1=1, Bath The deformation coefficient m of main officer of Tibet non-circular active synchronization belt wheel pitch curve first segment curve and second segment curve11=m12=1, three-wheel is isoperimetric Long closing convex curve, calculates the non-circular active synchronization belt wheel pitch curve polar diameter of Bath main officer of Tibet according to formula (1):
In formula,For the non-circular active synchronization belt wheel pitch curve moving coordinate system x of Bath main officer of Tibet1o1y1Middle x1Axis is to quiet coordinate system xo1y The corner of middle x-axis.
The non-circular active synchronization belt wheel pitch curve of Bath main officer of Tibet cuts polar equation are as follows:
The non-circular active synchronization belt wheel pitch curve of Bath main officer of Tibet is as shown in Figure 4.
Step 2: sinusoidal non-circular driven synchronous pulley is output link, output rule determines sinusoidal non-circular driven same Step belt wheel pitch curve cuts polar equation, gives sinusoidal non-circular driven synchronous pulley pitch curve parameter alpha=π/4, A is sine curve Amplitude,γ is positive chord joint parameter of curve, takes γ=π/6, and proportionality coefficient b=cos α is then sinusoidal non-circular driven same Walk belt wheel pitch curve equation are as follows:
r21=50 × [1+0.5773 × cos (0.7071 × β)] (3)
In formula, r21For the polar diameter of sinusoidal non-circular driven synchronous pulley pitch curve, β be the non-circular active synchronization belt wheel of Bath main officer of Tibet with The relative rotation of sinusoidal non-circular driven synchronous pulley, β=[0~2 π].
In formula, p2Diameter, θ are cut for sinusoidal non-circular driven synchronous pulley pitch curve2To cut diameter p2To moving coordinate system x2o2y2Middle x2 The corner of axis,For sinusoidal non-circular driven synchronous pulley pitch curve moving coordinate system x2o2y2Middle x2Axis is to quiet coordinate system xo1X-axis in y Corner.Sinusoidal non-circular driven synchronous pulley pitch curve such as Fig. 5.
Step 3: calculating the transmission of the non-circular active synchronization belt wheel of Bath main officer of Tibet and sinusoidal non-circular driven synchronous pulley initial position Than:
Initial position, the moving coordinate system x of the non-circular active synchronization belt wheel pitch curve of Bath main officer of Tibet1o1y1Middle x1Axis is to quiet coordinate system xo1The corner of x-axis in yThe moving coordinate system x of sinusoidal non-circular driven synchronous pulley pitch curve2o2y2Middle x2Axis is marked to sitting quietly It is xo1The corner of x-axis in yAccording to cutting, polar coordinates are theoretical to be obtained:
In formula, p1(θ12) and p2(θ21) it is respectively that the non-circular active synchronization belt wheel pitch curve of Bath main officer of Tibet and sine are non-circular driven same Walk belt wheel pitch curve common tangent incision superius C1、C2It is corresponding to cut diameter, p1(θ13) and p3(θ31) it is respectively the non-circular active synchronization of Bath main officer of Tibet Belt wheel pitch curve and tensioning synchronous pulley pitch curve common tangent incision superius C6、C5It is corresponding to cut diameter, p2(θ23) and p3(θ32) be respectively Sinusoidal non-circular driven synchronous pulley pitch curve and tensioning synchronous pulley pitch curve common tangent incision superius C3、C4It is corresponding to cut diameter, θ120 Diameter p is cut for the non-circular active synchronization belt wheel pitch curve of Bath main officer of Tibet1(θ12) with sinusoidal non-circular driven synchronous pulley pitch curve cut diameter p2 (θ21) arrive respective moving coordinate system trunnion axis corner initial value, θ130Diameter p is cut for the non-circular active synchronization belt wheel pitch curve of Bath main officer of Tibet1 (θ13) with tensioning synchronous pulley pitch curve cut diameter p3(θ31) arrive respective moving coordinate system trunnion axis corner initial value, θ230It is sinusoidal non- The driven synchronous pulley pitch curve of circle cuts diameter p2(θ23) with tensioning synchronous pulley pitch curve cut diameter p3(θ32) arrive respective moving coordinate system water The corner initial value of flat axis, θ12、θ13The respectively non-circular active synchronization belt wheel pitch curve incision superius C of Bath main officer of Tibet1、C6Corresponding diameter of cutting is to moving Coordinate system x1o1y1Middle x1The corner of axis, θ21、θ23Respectively sinusoidal non-circular driven synchronous pulley pitch curve incision superius C2、C3It is corresponding Diameter is cut to moving coordinate system x2o2y2Middle x2The corner of axis, θ31、θ32Respectively it is tensioned synchronous pulley pitch curve incision superius C4、C5It is corresponding Diameter is cut to moving coordinate system x3o3y3Middle x3The corner of axis, L1For the non-circular active synchronization belt wheel of Bath main officer of Tibet and sinusoidal non-circular driven synchronous belt Wheel center is away from L2It is sinusoidal non-circular driven synchronous pulley and tensioning synchronous pulley center away from L3For the non-circular active synchronization band of Bath main officer of Tibet Wheel with tensioning synchronous pulley center away from；
It is i according to the instantaneous transmission ratio that formula (6) calculate initial position120=0.6340:
Step 4: calculating the non-circular active synchronization belt wheel of Bath main officer of Tibet, sinusoidal non-circular driven synchronous pulley and tensioning synchronous pulley Common tangent segment length between every two-wheeled.
Initial time sets circle of the tensioning synchronous pulley pitch curve to give radius, the non-circular active synchronization belt wheel of Bath main officer of Tibet With the common tangent segment length T between sinusoidal non-circular driven two point of contact of synchronous pulley0, sinusoidal non-circular driven synchronous pulley and tensioning it is same Walk the common tangent segment length T between two point of contact of belt wheel1And the non-circular active synchronization belt wheel of Bath main officer of Tibet is cut with tensioning synchronous pulley two Common tangent segment length T between point is respectively as follows:
In formula, p1'(θ120)、p1'(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.
T is calculated to obtain according to formula (7)0=102.6370mm, T1=105.4651mm, T2=101.5862mm.
When the non-circular active synchronization belt wheel of Bath main officer of Tibet turns over angleSinusoidal non-circular driven synchronous pulley accordingly turns over angleThe non-circular active synchronization belt wheel pitch curve incision superius C of Bath main officer of Tibet1、C6Corresponding arc length variable quantity is s1、s6, sinusoidal non-circular driven Synchronous pulley pitch curve incision superius C2、C3Corresponding arc length variable quantity is s2、s3, it is tensioned synchronous pulley pitch curve incision superius C4、C5 Corresponding arc length variable quantity is s4、s5.Then have:
In formula, p1″(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, θ3To be tensioned synchronous belt round cut diameter p3To moving coordinate system x3o3y3Middle x3The corner cut of axis.
Any time, the public affairs between the non-circular active synchronization belt wheel of Bath main officer of Tibet and sinusoidal non-circular driven two point of contact of synchronous pulley are cut Line segment length T12, common tangent segment length T between sinusoidal non-circular driven synchronous pulley and tensioning two point of contact of synchronous pulley23And Common tangent segment length T between the non-circular active synchronization belt wheel of Bath main officer of Tibet and tensioning two point of contact of synchronous pulley13It is respectively as follows:
In formula, p1'(θ12)、p1'(θ13) it is respectively p1(θ12)、p1(θ13) first differential, p'2(θ21)、p'2(θ23) respectively For p2(θ21)、p2(θ23) first differential, p'3(θ32)、p'3(θ31) it is respectively p3(θ32)、p3(θ31) first differential,For It is tensioned synchronous pulley pitch curve moving coordinate system x3o3y3Middle x3Axis is to quiet coordinate system xo1The corner of x-axis in y.
Step 5: calculating the transmission of the non-circular active synchronization belt wheel of any time Bath main officer of Tibet and sinusoidal non-circular driven synchronous pulley Than；
The non-circular active synchronization belt wheel of Bath main officer of Tibet be uniform rotation, calculate the non-circular active synchronization belt wheel of Bath main officer of Tibet and sine it is non- The driven synchronous pulley instantaneous transmission ratio of circle:
According to formula (10), (11), (12), when the calculating non-circular active synchronization belt wheel of Bath main officer of Tibet rotates a circle, instantaneous transmission ratio It is as shown in Figure 2 with the non-circular active synchronization belt wheel angle relation curve of Bath main officer of Tibet.
Step 6: calculating synchronous belt perimeter；
The non-circular active synchronization belt wheel pitch curve of Bath main officer of Tibet and tensioning synchronous pulley pitch curve common tangent incision superius are denoted as C6, appoint Anticipate moment C1With C6Between arc length be c11, the non-circular active synchronization belt wheel of Bath main officer of Tibet and sinusoidal non-circular driven synchronous pulley pitch curve are public Tangent line incision superius is denoted as C2, sinusoidal non-circular driven synchronous pulley pitch curve and tensioning synchronous pulley pitch curve common tangent incision superius are remembered For C3, any time C2With C3Between arc length be c22, it is tensioned synchronous pulley pitch curve and sinusoidal non-circular driven synchronous pulley pitch curve Common tangent incision superius is denoted as C4, it is tensioned on synchronous pulley pitch curve and the non-circular active synchronization belt wheel pitch curve common tangent of Bath main officer of Tibet and cuts Point is denoted as C5, any time C4With C5Between arc length be c33。
Any time, synchronous belt perimeter are as follows:
C=T12+T13+T23+c11+c22+c33 (14)
Initial time is C according to the original perimeter that formula (14) calculate synchronous belt0=682.4570mm；
Each timing synchronization band belt length, each timing synchronization when driving wheel rotates 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 calculates；
Iterative algorithm is as follows:
(a) known to be tensioned synchronous pulley center of rotation, the radius for being tensioned synchronous pulley is set as variable r3, it is tensioned synchronous belt Wheel radius initial value is denoted as r3-0=30mm, synchronous belt original perimeter C0=682.4570mm.
(b) the non-circular active synchronization belt wheel of Bath main officer of Tibet turns overSuch as Fig. 2, according to the non-circular active synchronization belt wheel of Bath main officer of Tibet with Sinusoidal non-circular driven synchronous belt pulley transmission calculates sinusoidal non-circular driven synchronous pulley pitch curve and turns over corresponding angle than relationshipThe corner for being tensioned synchronous pulley pitch curve is identical as the non-circular active synchronization belt wheel pitch curve of Bath main officer of TibetUnder the premise of guaranteeing that synchronous belt perimeter C is constant, r is calculated3-1=40.1564mm.
(c) it repeats (b) 358 times, obtains r3-2, r3-3... ..., r3-359。
(d) 360 concentric circles are so far obtained, by the tensioning synchronous pulley radius in (a), (b) and (c), take one every 1 ° The radius of a circle sequentially takes 360 radiuses, to set tensioning synchronous pulley center of rotation as the center of circle, will take 360 radiuses The outer end point is sequentially connected with, and composition one is closed non-circular.
(e) scaling up each point of non-circular tensioning synchronous pulley obtained in (d) or reduce to diameter, so that new The week of the perimeter and the non-circular active synchronization belt wheel of Bath main officer of Tibet of the non-circular tensioning synchronous pulley arrived and sinusoidal non-circular driven synchronous pulley Length is equal.
(f) radius value at (e) obtained each moment is substituted into formula (14) and calculates corresponding belt length of each moment.
If (g) 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 step (h) is carried out.
(h) 5 ° before and after belt length maximum position corresponds to moment point, reduce non-circular tensioning synchronous pulley respectively to diameter value 3%, 5 ° before and after belt length minimum position corresponds to moment point, increase non-circular tensioning synchronous pulley respectively to the 3% of diameter value, then It is fitted to obtain new non-circular tensioning synchronous pulley with B-spline.
(i) non-circular tensioning synchronous pulley each point after (h) is scaled up or is reduced to diameter, so that newly obtain The perimeter of non-circular tensioning synchronous pulley and the perimeter of the non-circular active synchronization belt wheel of Bath main officer of Tibet and sinusoidal non-circular driven synchronous pulley are equal It is equal.
(j) the non-circular tensioning synchronous pulley that (i) is obtained is substituted into formula (14) to diameter and each moment belt length is calculated, if respectively The absolute value of the difference of the belt length at a moment and initial belt length is respectively less than preset value, carries out step (k), otherwise returns to (h).
(k) establish each moment of non-circular tensioning synchronous pulley to diameter and corresponding cornerRelationship is to be tensioned synchronous pulley Pitch curve equation.Three pitch curves taken turns and phase angle, center of rotation all determine, calculate tensioning synchronous pulley and Bath main officer of Tibet is non-circular Active synchronization belt wheel, the corresponding angle relation of sinusoidal non-circular driven synchronous pulley.
Tensioning synchronous pulley pitch curve such as Fig. 6 after calculating.
Synchronous belt theory belt length variable quantity is 11.86mm in the embodiment, is the 2.23% of synchronous belt total length, because of band It needs to be tensioned, can satisfy actual operation requirements.

Claims (1)

1.巴斯噶非圆—正弦非圆—非圆三轮同步带传动设计方法，其特征在于：该方法具体如下：1. Pascal non-circular-sine non-circular-non-circular three-wheel synchronous belt drive design method, is characterized in that: the method is specifically as follows: 步骤一、根据传动规律计算巴斯噶非圆主动同步带轮节曲线和正弦非圆从动同步带轮节曲线方程；Step 1: Calculate the pitch curve equation of the Pascal non-circular active synchronous pulley and the sinusoidal non-circular driven synchronous pulley pitch curve according to the transmission law; 巴斯噶非圆主动同步带轮为匀速转动的输入构件，其切极坐标方程：The Pascal non-circular active synchronous pulley is an input member rotating at a constant speed, and its tangential polar coordinate equation: 式中，r11、r12分别为巴斯噶非圆主动同步带轮节曲线第一段曲线与第二段曲线的变性极径，b1为巴斯噶非圆主动同步带轮节曲线的发生圆直径，n1为巴斯噶非圆主动同步带轮节曲线的阶数，m11、m12为巴斯噶非圆主动同步带轮节曲线第一段曲线与第二段曲线的变形系数，为巴斯噶非圆主动同步带轮节曲线的极径到动坐标系x1o1y1中x1轴的转角，动坐标系x1o1y1的原点设置在巴斯噶非圆主动同步带轮的转动中心处，l为巴斯噶非圆节曲线的展长；In the formula, r 11 and r 12 are the modified polar diameters of the first segment curve and the second segment curve of the Pascal non-circular active synchronous pulley pitch curve, respectively, and b 1 is the difference of the Pascal non-circular active synchronous pulley pitch curve. The diameter of the occurrence circle, n 1 is the order of the pitch curve of the Paskar non-circular active synchronous pulley, m 11 and m 12 are the deformation of the first and second curves of the Paskar non-circular active synchronous pulley pitch curve coefficient, is the rotation angle from the polar diameter of the pitch curve of the Pascal non-circular active synchronous pulley to the x 1 axis in the moving coordinate system x 1 o 1 y 1 , and the origin of the moving coordinate system x 1 o 1 y 1 is set at the Pascal non-circular At the rotation center of the active synchronous pulley, l is the extension of the Pascal non-circular pitch curve; 式中，p1为巴斯噶非圆主动同步带轮节曲线的切径，θ1为p1到动坐标系x1o1y1中x1轴的切角；In the formula, p 1 is the cut diameter of the pitch curve of the Pascal non-circular active synchronous pulley, and θ 1 is the cut angle from p 1 to the x 1 axis in the moving coordinate system x 1 o 1 y 1 ; 正弦非圆从动同步带轮为同步带的输出轮，其节曲线的切极坐标方程为：The sinusoidal non-circular driven synchronous pulley is the output pulley of the synchronous belt, and the tangential coordinate equation of its pitch curve is: 式中，r21为正弦非圆从动同步带轮节曲线的极径，α为正弦非圆从动同步带轮节曲线参数，A为正弦曲线的振幅，γ为正弦节曲线参数，取γ＝π/6，b为比例系数，a为巴斯噶非圆主动同步带轮节曲线与正弦非圆从动同步带轮节曲线的中心距，β为巴斯噶非圆主动同步带轮节曲线与正弦非圆从动同步带轮节曲线的相对转角，β＝[0～2π]；p2为正弦非圆从动同步带轮节曲线的切径，θ2为p2到动坐标系x2o2y2中x2轴的切角，为动坐标系x2o2y2中x2轴到静坐标系xo1y中x轴的转角，动坐标系x2o2y2的原点设置在正弦非圆从动同步带轮的转动中心处，静坐标系xo1y的原点设置在巴斯噶非圆主动同步带轮的转动中心处；In the formula, r 21 is the pole diameter of the sinusoidal non-circular driven synchronous pulley pitch curve, α is the sinusoidal non-circular driven timing pulley pitch curve parameter, A is the amplitude of the sinusoidal curve, γ is the sine pitch curve parameter, take γ=π/6, b is the proportional coefficient, a is the center distance between the pitch curve of the Pascal non-circular driving synchronous pulley and the pitch curve of the sine non-circular driven synchronous pulley, and β is the pitch of bar The relative rotation angle between the pitch curve of the sine non-circular driving synchronous pulley and the pitch curve of the sine non-circular driven synchronous pulley, β=[0～2π]; p 2 is the tangent of the pitch curve of the sine non-circular driven synchronous pulley, θ 2 is the chamfering angle from p 2 to the x 2 axis in the moving coordinate system x 2 o 2 y 2 , is the rotation angle from the x 2 axis in the moving coordinate system x 2 o 2 y 2 to the x axis in the static coordinate system xo 1 y, the origin of the moving coordinate system x 2 o 2 y 2 is set at the rotation of the sinusoidal non-circular driven synchronous pulley At the center, the origin of the static coordinate system xo 1 y is set at the rotation center of the Pascal non-circular driving synchronous pulley; 步骤二、计算巴斯噶非圆主动同步带轮与正弦非圆从动同步带轮初始位置的传动比：Step 2. Calculate the transmission ratio of the initial position of the Pascal non-circular driving synchronous pulley and the sinusoidal non-circular driven synchronous pulley: 初始位置，巴斯噶非圆主动同步带轮节曲线的动坐标系x1o1y1中x1轴到静坐标系xo1y中x轴的转角正弦非圆从动同步带轮节曲线的动坐标系x2o2y2中x2轴到静坐标系xo1y中x轴的转角根据切极坐标理论得：The initial position, the rotation angle from the x 1 axis in the moving coordinate system x 1 o 1 y 1 to the x axis in the static coordinate system xo 1 y of the Pascal non-circular driving synchronous pulley pitch curve The rotation angle from the x 2 axis in the moving coordinate system x 2 o 2 y 2 to the x axis in the static coordinate system xo 1 y in the sinusoidal non-circular driven synchronous pulley pitch curve According to the tangential coordinate theory, we get: 式中，p1(θ12)和p2(θ21)分别为巴斯噶非圆主动同步带轮节曲线与正弦非圆从动同步带轮节曲线公切线上切点C1、C2对应的切径，p1(θ13)和p3(θ31)分别为巴斯噶非圆主动同步带轮节曲线与张紧同步带轮节曲线公切线上切点C6、C5对应的切径，p2(θ23)和p3(θ32)分别为正弦非圆从动同步带轮节曲线与张紧同步带轮节曲线公切线上切点C3、C4对应的切径，θ120为巴斯噶非圆主动同步带轮节曲线切径p1(θ12)与正弦非圆从动同步带轮节曲线切径p2(θ21)到各自动坐标系水平轴的转角初值，θ130为巴斯噶非圆主动同步带轮节曲线切径p1(θ13)与张紧同步带轮节曲线切径p3(θ31)到各自动坐标系水平轴的转角初值，θ230为正弦非圆从动同步带轮节曲线切径p2(θ23)与张紧同步带轮节曲线切径p3(θ32)到各自动坐标系水平轴的转角初值，θ12、θ13分别为巴斯噶非圆主动同步带轮节曲线上切点C1、C6对应切径到动坐标系x1o1y1中x1轴的转角，θ21、θ23分别为正弦非圆从动同步带轮节曲线上切点C2、C3对应切径到动坐标系x2o2y2中x2轴的转角，θ31、θ32分别为张紧同步带轮节曲线上切点C4、C5对应切径到动坐标系x3o3y3中x3轴的转角，动坐标系x3o3y3的原点设置在张紧同步带轮的转动中心处，L1为巴斯噶非圆主动同步带轮与正弦非圆从动同步带轮中心距，L2为正弦非圆从动同步带轮与张紧同步带轮中心距，L3为巴斯噶非圆主动同步带轮与张紧同步带轮中心距；In the formula, p 1 (θ 12 ) and p 2 (θ 21 ) are the tangent points C 1 , C 2 on the common tangent line of the Pascal non-circular driving timing pulley pitch curve and the sinusoidal non-circular driven timing pulley pitch curve, respectively. Corresponding tangent diameters, p 1 (θ 13 ) and p 3 (θ 31 ) are respectively corresponding to the tangent points C 6 and C 5 on the common tangent of the Pascal non-circular active timing pulley pitch curve and the tension timing pulley pitch curve. The tangent diameter of , p 2 (θ 23 ) and p 3 (θ 32 ) are the tangent points corresponding to the tangent points C 3 and C 4 on the common tangent of the sinusoidal non-circular driven synchronous pulley pitch curve and the tensioning synchronous pulley pitch curve. Diameter, θ 120 is the tangent diameter p 1 (θ 12 ) of the pitch curve of the Pascal non-circular driving synchronous pulley and the tangent diameter p 2 (θ 21 ) of the sine non-circular driven synchronous pulley pitch curve to the horizontal axis of each automatic coordinate system The initial value of the rotation angle, θ 130 is the pitch curve cut diameter p 1 (θ 13 ) of the non-circular active synchronous pulley of Pascal and the pitch cut diameter p 3 (θ 31 ) of the tensioning timing pulley to the horizontal axis of each automatic coordinate system The initial value of the rotation angle, θ 230 is the difference between the sine non-circular driven synchronous pulley pitch curve cut diameter p 2 (θ 23 ) and the tension synchronous pulley pitch curve cut diameter p 3 (θ 32 ) to the horizontal axis of each automatic coordinate system The initial value of the rotation angle, θ 12 and θ 13 are respectively the rotation angles from the tangent points C 1 and C 6 on the pitch curve of the Pascal non-circular active synchronous pulley to the x 1 axis in the moving coordinate system x 1 o 1 y 1 , θ 21 , θ 23 are the rotation angles from the tangent points C 2 , C 3 on the pitch curve of the sine non-circular driven synchronous pulley to the x 2 axis in the moving coordinate system x 2 o 2 y 2 corresponding to the tangent diameter, θ 31 , θ 32 Respectively, the tangent points C 4 and C 5 on the pitch curve of the tensioning synchronous pulley correspond to the corners of the tangent to the x 3 axis in the moving coordinate system x 3 o 3 y 3 , and the origin of the moving coordinate system x 3 o 3 y 3 is set at At the rotation center of the tensioning synchronous pulley, L 1 is the center distance between the Pascal non-circular driving synchronous pulley and the sinusoidal non-circular driven synchronous pulley, and L 2 is the sinusoidal non-circular driven synchronous pulley and the tensioning synchronous belt Wheel center distance, L3 is the center distance between Pascal non - circular active synchronous pulley and tensioning synchronous pulley; 初始位置巴斯噶非圆主动同步带轮与正弦非圆从动同步带轮瞬时传动比为：The instantaneous transmission ratio of the Baskar non-circular driving synchronous pulley and the sinusoidal non-circular driven synchronous pulley at the initial position is: 步骤三、计算巴斯噶非圆主动同步带轮、正弦非圆从动同步带轮和张紧同步带轮每两轮之间的公切线段长度；Step 3: Calculate the length of the common tangent line segment between every two wheels of the Pascal non-circular driving synchronous pulley, the sinusoidal non-circular driven synchronous pulley and the tensioning synchronous pulley; 初始时刻，设定张紧同步带轮节曲线为给定半径的圆，巴斯噶非圆主动同步带轮与正弦非圆从动同步带轮两切点之间的公切线段长度T0、正弦非圆从动同步带轮与张紧同步带轮两切点之间的公切线段长度T1以及巴斯噶非圆主动同步带轮与张紧同步带轮两切点之间的公切线段长度T2分别为：At the initial moment, the pitch curve of the tensioning synchronous pulley is set as a circle with a given radius, and the common tangent line segment length T 0 between the two tangent points of the Pascal non-circular driving synchronous pulley and the sinusoidal non-circular driven synchronous pulley is T 0 , The length of the common tangent line between the two tangent points of the sinusoidal non-circular driven timing pulley and the tensioning timing pulley T 1 and the common tangent between the two tangent points of the Pascal non-circular driving timing pulley and the tensioning timing pulley The segment lengths T2 are respectively: 式中，p′1(θ120)、p′1(θ130)分别为p1(θ120)、p1(θ130)的一阶微分，p'2(θ120)、p'2(θ230)分别为p2(θ120)、p2(θ230)的一阶微分，p'3(θ130)、p'3(θ230)分别为p3(θ130)、p3(θ230)的一阶微分；In the formula, p′ 1 (θ 120 ) and p′ 1 (θ 130 ) are the first-order differentials of p 1 (θ 120 ) and p 1 (θ 130 ), respectively, and p′ 2 (θ 120 ) and p′ 2 ( θ 230 ) are the first-order differentials of p 2 (θ 120 ) and p 2 (θ 230 ), respectively, and p′ 3 (θ 130 ) and p′ 3 (θ 230 ) are respectively p 3 (θ 130 ) and p 3 ( The first derivative of θ 230 ); 当巴斯噶非圆主动同步带轮转过角度正弦非圆从动同步带轮相应转过角度巴斯噶非圆主动同步带轮节曲线上切点C1、C6对应的弧长变化量为s1、s6，正弦非圆从动同步带轮节曲线上切点C2、C3对应的弧长变化量为s2、s3，张紧同步带轮节曲线上切点C4、C5对应的弧长变化量为s4、s5；则有：When the Pascal non-circular active synchronous pulley rotates through the angle The sine non-circular driven synchronous pulley rotates through the corresponding angle The arc length changes corresponding to the tangent points C 1 and C 6 on the pitch curve of the Pascal non-circular driving timing pulley are s 1 and s 6 , and the tangent points C 2 and C 3 on the pitch curve of the sinusoidal non-circular driven timing pulley The corresponding arc length changes are s 2 and s 3 , and the arc length changes corresponding to the tangent points C 4 and C 5 on the pitch curve of the tensioning synchronous pulley are s 4 and s 5 ; then there are: 式中，p″1(θ1)为p1(θ1)的二阶微分，p″2(θ2)为p2(θ2)的二阶微分，p″3(θ3)为p3(θ3)的二阶微分，θ3为张紧同步带轮切径p3到动坐标系x3o3y3中x3轴的转角；In the formula, p″ 1 (θ 1 ) is the second-order differential of p 1 (θ 1 ), p″ 2 (θ 2 ) is the second-order differential of p 2 (θ 2 ), and p″ 3 (θ 3 ) is p 3 (θ 3 ) second-order differential, θ 3 is the rotation angle of the cut diameter p 3 of the tensioning synchronous pulley to the x 3 axis in the moving coordinate system x 3 o 3 y 3 ; 任意时刻，巴斯噶非圆主动同步带轮与正弦非圆从动同步带轮两切点之间的公切线段长度T12、正弦非圆从动同步带轮与张紧同步带轮两切点之间的公切线段长度T23以及巴斯噶非圆主动同步带轮与张紧同步带轮两切点之间的公切线段长度T13分别为：At any time, the common tangent length T 12 between the two tangent points of the Pascal non-circular driving synchronous pulley and the sinusoidal non-circular driven synchronous pulley, and the two tangents between the sinusoidal non-circular driven synchronous pulley and the tensioning synchronous pulley The common tangent line length T 23 between the points and the common tangent line length T 13 between the two tangent points of the Pascal non-circular active synchronous pulley and the tensioning synchronous pulley are respectively: 式中，p′1(θ12)、p′1(θ13)分别为p1(θ12)、p1(θ13)的一阶微分，p'2(θ21)、p'2(θ23)分别为p2(θ21)、p2(θ23)的一阶微分，p'3(θ32)、p'3(θ31)分别为p3(θ32)、p3(θ31)的In the formula, p′ 1 (θ 12 ) and p′ 1 (θ 13 ) are the first-order differentials of p 1 (θ 12 ) and p 1 (θ 13 ), respectively, and p′ 2 (θ 21 ), p′ 2 ( θ 23 ) are the first-order differentials of p 2 (θ 21 ) and p 2 (θ 23 ), respectively, and p′ 3 (θ 32 ) and p′ 3 (θ 31 ) are respectively p 3 (θ 32 ), p 3 ( θ 31 ) of 一阶微分，为张紧同步带轮节曲线动坐标系x3o3y3中x3轴到静坐标系xo1y中x轴的转角；first-order differential, is the rotation angle from the x 3 axis in the dynamic coordinate system x 3 o 3 y 3 to the x axis in the static coordinate system xo 1 y; 步骤四、计算任意时刻巴斯噶非圆主动同步带轮与正弦非圆从动同步带轮的传动比；Step 4: Calculate the transmission ratio of the Pascal non-circular driving synchronous pulley and the sinusoidal non-circular driven synchronous pulley at any time; 巴斯噶非圆主动同步带轮匀速转动，根据式(2)，(4)解得p1，p2，则瞬时传动比为：Pascal non-circular active synchronous pulley rotates at a constant speed, according to formulas (2), (4) to solve p 1 , p 2 , then the instantaneous transmission ratio is: 步骤五、计算任意时刻同步带周长；Step 5. Calculate the perimeter of the synchronous belt at any time; 巴斯噶非圆主动同步带轮节曲线与张紧同步带轮节曲线公切线上切点记为C6，任意时刻C1与C6间的弧长为c11，正弦非圆从动同步带轮节曲线与张紧同步带轮节曲线公切线上切点记为C3，任意时刻C2与C3间的弧长为c22，张紧同步带轮节曲线与正弦非圆从动同步带轮节曲线公切线上切点记为C4，张紧同步带轮节曲线与巴斯噶非圆主动同步带轮节曲线公切线上切点记为C5，任意时刻C4与C5间的弧长为c33；The tangent point on the common tangent of the pitch curve of the Pascal non-circular active synchronous pulley and the pitch curve of the tensioned synchronous pulley is recorded as C 6 , the arc length between C 1 and C 6 at any time is c 11 , and the sine non-circular driven synchronization is The tangent point on the common tangent line between the pitch curve of the pulley and the pitch curve of the tensioning synchronous pulley is recorded as C 3 , the arc length between C 2 and C 3 at any time is c 22 , the pitch curve of the pitch synchronous pulley and the sine non-circular driven The tangent point on the common tangent line of the synchronous pulley pitch curve is recorded as C 4 , and the tangent point on the common tangent line between the tension synchronous pulley pitch curve and the Pascal non-circular active synchronous pulley pitch curve is recorded as C 5 , and C 4 and C at any time The arc length between 5 is c 33 ; 任意时刻，同步带周长为：At any time, the perimeter of the timing belt is: C＝T12+T13+T23+c11+c22+c33 (14)C=T 12 +T 13 +T 23 +c 11 +c 22 +c 33 (14) 步骤六、张紧同步带轮自由节曲线计算；Step 6. Calculation of the free section curve of the tensioning synchronous pulley; 迭代算法如下：The iterative algorithm is as follows: (a)设定张紧同步带轮转动中心，张紧同步带轮的半径设置为变量，张紧同步带轮半径初始值给定，记为r3-0，根据式(14)计算带长初始值记为C0；(a) Set the rotation center of the tensioning synchronous pulley, the radius of the tensioning synchronous pulley is set as a variable, the initial value of the tensioning synchronous pulley radius is given, denoted as r 3-0 , and the belt length is calculated according to the formula (14). The initial value is denoted as C 0 ; (b)巴斯噶非圆主动同步带轮转过1°，根据传动比要求计算正弦非圆从动同步带轮转过相应的角度，张紧同步带轮的转角与巴斯噶非圆主动同步带轮相同；在保证C不变的前提下，根据式(14)反求巴斯噶非圆主动同步带轮转过1°时对应的张紧同步带轮半径r3-1，即对应时刻的p3；(b) The non-circular driving synchronous pulley rotates by 1°, and the sine non-circular driven synchronous pulley rotates by the corresponding angle according to the transmission ratio requirement. The pulleys are the same; on the premise of ensuring that C remains unchanged, according to formula (14), find the corresponding tensioning synchronous pulley radius r 3-1 when the Pascal non-circular active synchronous pulley rotates by 1°, that is, p at the corresponding time 3 ; (c)重复(b)358次，得到巴斯噶非圆主动同步带轮转过2°，3°，…，359°时对应的张紧同步带轮半径分别为r3-2，r3-3，……，r3-359；(c) Repeat (b) 358 times to get the radii of the tensioned synchronous pulley corresponding to the 2°, 3 °, . 3 ,...,r 3-359 ; (d)至此得到360个同心圆，按(a)、(b)和(c)中的张紧同步带轮半径，每隔1°取一个圆的半径，顺次取360个半径，以设定张紧同步带轮转动中心为圆心，将所取360个半径的外端点顺次连接，组成一个封闭的非圆；(d) So far, 360 concentric circles are obtained. According to the radii of the tensioning synchronous pulley in (a), (b) and (c), take the radius of a circle every 1°, and take 360 radii in sequence to set Set the rotation center of the tensioning synchronous pulley as the center of the circle, and connect the outer end points of the 360 radii taken in sequence to form a closed non-circle; (e)将(d)中得到的非圆张紧同步带轮的各时刻的向径按比例放大或缩小，使得新得到的非圆张紧同步带轮的周长与巴斯噶非圆主动同步带轮及正弦非圆从动同步带轮的周长均相等；(e) Enlarging or reducing the radial direction of the non-circular tensioning synchronous pulley obtained in (d) at each moment in proportion, so that the perimeter of the newly obtained non-circular tensioning synchronous pulley is the same as the Pascal non-circular driving pulley. The perimeter of the synchronous pulley and the sinusoidal non-circular driven synchronous pulley are equal; (f)将(e)所求得的各个时刻的半径值代入式(14)计算各个时刻的带长；(f) Substitute the radius value of each moment obtained by (e) into formula (14) to calculate the band length of each moment; (g)若各个时刻的带长与初始带长之差的绝对值均小于预设值，则进行步骤(k)，否则进行步骤(h)；(g) if the absolute value of the difference between the band length at each moment and the initial band length is less than the preset value, then step (k) is performed, otherwise step (h) is performed; (h)在带长最大位置对应时刻点的前后5°，减小非圆张紧同步带轮各自向径值的1～5％，在带长最小位置对应时刻点的前后5°，增加非圆张紧同步带轮各自向径值的1～5％，然后用B样条进行拟合得到新的非圆张紧同步带轮；(h) At 5° before and after the corresponding time point at the maximum belt length position, reduce 1 to 5% of the respective radial values of the non-circular tensioning synchronous pulleys, and at 5° before and after the corresponding time point at the minimum belt length position, increase the 1~5% of the respective radial values of the circular tensioning synchronous pulleys, and then fitting with B-splines to obtain new non-circular tensioning synchronous pulleys; (i)将经(h)后的非圆张紧同步带轮各时刻的向径按比例放大或缩小，使得新得到的非圆张紧同步带轮的周长与巴斯噶非圆主动同步带轮及正弦非圆从动同步带轮的周长均相等；(i) Enlarging or reducing the radial direction of the non-circular tensioning synchronous pulley at each moment after (h) in proportion, so that the perimeter of the newly obtained non-circular tensioning synchronous pulley is automatically synchronized with the Pascal non-circular synchronous pulley. The perimeter of the pulley and the sine non-circular driven synchronous pulley are equal; (j)将经(i)后的非圆张紧同步带轮向径代入式(14)计算得到各时刻对应同步带带长，若各时刻对应同步带带长与同步带周长初始值之差的绝对值均小于预设值，进行步骤(k)，否则回到(h)；(j) Substitute the radial direction of the non-circular tensioned synchronous pulley after (i) into the formula (14) to calculate the corresponding timing belt length at each moment. If the corresponding timing belt length at each moment and the initial value of the timing belt circumference are calculated If the absolute value of the difference is less than the preset value, go to step (k), otherwise return to (h); (k)建立非圆张紧同步带轮的各时刻的向径与对应转角关系即为张紧同步带轮节曲线方程。(k) The radial direction and the corresponding rotation angle of each moment when the non-circular tensioning synchronous pulley is established The relationship is the equation of the pitch curve of the tensioning synchronous pulley.
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