CN106838158B - Not rounded-not rounded three-wheel toothed belt transmission design method of off-centre operation-Fourier - Google Patents

Not rounded-not rounded three-wheel toothed belt transmission design method of off-centre operation-Fourier Download PDF

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CN106838158B
CN106838158B CN201710191356.1A CN201710191356A CN106838158B CN 106838158 B CN106838158 B CN 106838158B CN 201710191356 A CN201710191356 A CN 201710191356A CN 106838158 B CN106838158 B CN 106838158B
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synchronous pulley
rounded
fourier
pitch curve
belt wheel
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CN106838158A (en
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陈建能
叶军
陈之威
马培峰
周赟
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Ningbo Fulong Synchronous Belt 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 a kind of not rounded-not rounded three-wheel toothed belt transmission design methods of off-centre operation-Fourier.The driving wheel of the present invention is off-centre operation synchronous pulley；Driven wheel is the not rounded synchronous pulley of Fourier；Tensioning wheel is not rounded synchronous pulley, it plays the synchronous belt slack variation brought in real-time compensation driving wheel, driven wheel transmission process.The present invention initially sets up the pitch curve equation of synchronous belt principal and subordinate wheel, and moves synchronous belt pulley transmission ratio using polar coordinates theoretical calculation principal and subordinate is cut；Then the perimeter for calculating synchronous belt calculates the parameters of not rounded tensioning synchronous pulley pitch curve according to the variation of synchronous belt perimeter slack by alternative manner.The present invention meets the non-requirement for being at the uniform velocity driven and being tensioned in real time to realize that the non-at the uniform velocity directly accurate transmission of big centre-to-centre spacing provides a kind of new transmission mechanism and new method.

Description

Not rounded-not rounded three-wheel toothed belt transmission design method of off-centre operation-Fourier
Technical field
The present invention relates to a kind of design methods of not rounded toothed belt transmission, and in particular to one kind amount of becoming slack is self-compensating partially Not rounded-not rounded three-wheel toothed belt transmission design method of heart circle-Fourier.
Background technology
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, transmission 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, that centre-to-centre spacing is smaller, lubrication is convenient non-is at the uniform velocity driven Occasion, therefore not rounded flexible element (band/chain) transmission for being suitable for the inconvenient and low manufacturing cost occasion of big centre-to-centre spacing, lubrication is met the tendency of And it gives birth to.Wherein not rounded chaindriven polygon effect is apparent, 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 not rounded bands (chain) are taken turns --- and driving wheel and driven wheel are being driven In the process since its pitch curve is not rounded, the slack of band (chain) is real-time change, therefore cannot ensure 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 can be influenced in turn in this way, and And kinetic characteristics are deteriorated；Therefore in Practical Project, non-round belt (chain) transmission is rarely applied to accurately load high-speed drive Occasion.
Invention content
The purpose of the present invention is in view of the above problems, proposing that the self-compensating off-centre operation-Fourier of one kind amount of becoming slack is non- Circle-not rounded three-wheel toothed belt transmission design method provides a whole set of for not rounded synchronous pulley and perfect sets in practical applications Theoretical foundation is counted, realizes the non-at the uniform velocity directly accurate transmission between big centre-to-centre spacing.The design method initially sets up synchronous belt principal and subordinate The pitch curve equation of driving wheel, and move synchronous belt pulley transmission ratio using polar coordinates theoretical calculation principal and subordinate is cut；Then synchronous belt is calculated Perimeter calculates not rounded 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 present invention is as follows：
Step 1: determining off-centre operation active synchronization belt wheel pitch curve and the not rounded driven synchronous belt of Fourier according to transmission rule Take turns pitch curve equation；
Off-centre operation active synchronization belt wheel wheel is the input link of uniform rotation, cuts polar equation：
p1=r1+e1×cos(θ1) (1)
s1=2 π × r1 (2)
In formula, p1Diameter, θ are cut for off-centre operation active synchronization belt wheel pitch curve1For p1To moving coordinate system x1o1y1Middle x1Axis Corner cut, e1For the eccentricity of off-centre operation active synchronization belt wheel pitch curve, r1For off-centre operation active synchronization belt wheel pitch curve radius, s1 For off-centre operation active synchronization belt wheel pitch curve perimeter.
The not rounded driven synchronous pulley of Fourier is output wheel, and the not rounded driven synchronous pulley pitch curve of Fourier cuts polar coordinates side Cheng Wei：
In formula, r21,r22The respectively not rounded driven synchronous pulley pitch curve first segment curve of Fourier and second segment curve Polar diameter, n21For the exponent number of the not rounded driven synchronous pulley pitch curve of Fourier, m21For the not rounded driven synchronous pulley pitch curve of Fourier The deformation coefficient of first segment curve, a0,a1,a2,b1,b2For the parameter of the not rounded driven synchronous pulley pitch curve of Fourier,For Fu In the not rounded driven synchronous pulley pitch curve moving coordinate system x of leaf2o2y2Middle x2Axis is to quiet coordinate system xo1The corner of x-axis in y.
In formula, p2Diameter, θ are cut for the not rounded driven synchronous pulley pitch curve of Fourier2For p2To moving coordinate system x2o2y2Middle x2 The corner cut of axis.
Step 2: calculating the transmission of off-centre operation active synchronization belt wheel and the not rounded driven synchronous pulley initial position of Fourier Than；
Initial position, the moving coordinate system x of off-centre operation active synchronization belt wheel pitch curve1o1y1Middle x1Axis is to quiet coordinate system xo1In y The corner of x-axisThe moving coordinate system x of the not rounded driven synchronous pulley pitch curve of Fourier2o2y2Middle x2Axis is to quiet coordinate system 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 off-centre operation active synchronization belt wheel pitch curve is not rounded driven synchronous with Fourier Belt wheel pitch curve common tangent incision superius C1、C2It is corresponding to cut diameter, p1(θ13) and p3(θ31) it is respectively off-centre operation active synchronization belt wheel section Curve and tensioning synchronous pulley pitch curve common tangent incision superius C6、C5It is corresponding to cut diameter, p2(θ23) and p3(θ32) it is respectively Fourier Not rounded driven synchronous pulley pitch curve and tensioning synchronous pulley pitch curve common tangent incision superius C3、C4It is corresponding to cut diameter, θ120It is inclined Heart circle active synchronization belt wheel pitch curve cuts diameter p1(θ12) with the not rounded driven synchronous pulley pitch curve of Fourier cut diameter p2(θ21) to respectively From the corner initial value of moving coordinate system trunnion axis, θ130Diameter p is cut for off-centre operation active synchronization belt wheel pitch curve1(θ13) synchronous with tensioning Belt wheel pitch curve cuts diameter p3(θ31) arrive respective moving coordinate system trunnion axis corner initial value, θ230For the not rounded driven synchronous belt of Fourier Wheel pitch curve cuts diameter p2(θ23) with tensioning synchronous pulley pitch curve cut diameter p3(θ32) to respective moving coordinate system trunnion axis corner at the beginning of Value, θ12、θ13Respectively off-centre operation active synchronization belt wheel pitch curve incision superius C1、C6Correspondence cuts diameter to moving coordinate system x1o1y1Middle x1 The corner cut of axis, θ21、θ23The respectively not rounded driven synchronous pulley pitch curve incision superius C of Fourier2、C3Correspondence cuts diameter to moving coordinate system x2o2y2Middle x2The corner cut of axis, θ31、θ32To be tensioned synchronous pulley pitch curve incision superius C4、C5Correspondence cuts diameter to moving coordinate system x3o3y3 Middle x3The corner cut of axis, L1For off-centre operation active synchronization belt wheel and the not rounded driven synchronous pulley centre-to-centre spacing of Fourier, L2It is non-for Fourier The driven synchronous pulley of circle and tensioning synchronous pulley centre-to-centre spacing, L3For tensioning synchronous pulley and off-centre operation active synchronization belt wheel center Away from；
Initial position off-centre operation active synchronization belt wheel is with the not rounded driven synchronous belt pulley transmission ratio of Fourier：
Step 3: calculating in off-centre operation active synchronization belt wheel, the not rounded driven synchronous pulley of Fourier and tensioning synchronous pulley Common tangent segment length between per two-wheeled.
Initial time sets circle of the tensioning synchronous pulley pitch curve to give radius, off-centre operation active synchronization belt wheel and Fu In common tangent segment length T between not rounded two point of contact of driven synchronous pulley of leaf0, the not rounded driven synchronous pulley of Fourier and tensioning it is same Walk the common tangent segment length T between two point of contact of belt wheel1, between off-centre operation active synchronization belt wheel and tensioning synchronous pulley two point of contact Common tangent segment length T2Respectively：
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 off-centre operation active synchronization belt wheel turns over angleThe not rounded driven synchronous pulley of Fourier accordingly turns over angle Off-centre operation active synchronization belt wheel pitch curve incision superius C1、C6Corresponding arc length variable quantity is s1、s6, the not rounded driven synchronization of Fourier Belt wheel pitch curve incision superius C2、C3Corresponding arc length variable quantity is s2、s3, tensioning synchronous pulley pitch curve incision superius C4、C5It is corresponding Arc length variable quantity be s4、s5.Then have：
In formula, p1"(θ1) it is p1(θ1) second-order differential, p2"(θ2) it is p2(θ2) second-order differential, p3"(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 common tangent between off-centre operation active synchronization belt wheel and not rounded two point of contact of driven synchronous pulley of Fourier Segment length T12, common tangent segment length T between the not rounded driven synchronous pulley of Fourier and tensioning synchronous pulley two point of contact23, it is eccentric Circle active synchronization belt wheel and the common tangent segment length T being tensioned between two point of contact of synchronous pulley13Respectively：
In formula, p '1(θ12)、p’1(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 any time off-centre operation active synchronization belt wheel and the not rounded driven synchronous pulley of Fourier Than；
Off-centre operation active synchronization belt wheel uniform rotation, according to formula (1), (4) solve p1, p2, then instantaneous transmission ratio be：
Step 5: calculating any time synchronous belt perimeter；
Off-centre operation active synchronization belt wheel pitch curve is denoted as C with tensioning synchronous pulley pitch curve common tangent incision superius6, when arbitrary Carve C1With C6Between arc length be c11, the not rounded driven synchronous pulley pitch curve of Fourier with tensioning synchronous pulley pitch curve common tangent on Point of contact is denoted as C3, any time C2With C3Between arc length be c22, it is tensioned on synchronous pulley pitch curve and driven wheel pitch curve common tangent Point of contact is denoted as C4, tensioning synchronous pulley pitch curve and driving wheel pitch curve common tangent incision superius are denoted as C5, any time C4With C5Between Arc length be c33。
Any time, synchronous belt Zhou Changwei：
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 is calculated according to formula (14) and is denoted as C0。
(b) off-centre operation active synchronization belt wheel turns over 1 °, is required to calculate the not rounded driven synchronous pulley of Fourier according to transmission ratio Corresponding angle is turned over, the corner for being tensioned synchronous pulley is identical as off-centre operation active synchronization belt wheel.Ensureing the constant premises of C Under, corresponding tensioning synchronous pulley radius r when turning over 1 ° according to formula (14) reverse off-centre operation active synchronization belt wheel3-1, i.e., to it is corresponding when The p at quarter3。
(c) it repeats (b) 358 times, obtains off-centre operation active synchronization belt wheel and turn over corresponding at 2 °, 3 ° ..., 359 ° be tensioned together It is respectively r to walk belt wheel radius3-2, r3-3... ..., r3-359。
(d) 360 concentric circles are so far obtained, by the tensioning synchronous pulley radius in (a), (b) and (c), one is taken 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 not rounded.
(e) each moment of the not rounded tensioning synchronous pulley obtained in (d) is scaled up or is reduced to diameter so that is new The perimeter of obtained not rounded tensioning synchronous pulley and off-centre operation active synchronization belt wheel and the week of the not rounded driven synchronous pulley of Fourier Length is equal.
(f) radius value at (e) obtained each moment is substituted into the belt length that formula (14) calculates each moment.
If (g) belt length at each moment and the absolute value of the difference of initial belt length are 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 not rounded 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 not rounded tensioning synchronous pulley respectively to diameter value 1~ 5%, it is then fitted to obtain new not rounded tensioning synchronous pulley with B-spline.
(i) not rounded tensioning synchronous pulley each moment that will be after (h) scales up or is reduced to diameter so that newly obtains Not rounded tensioning synchronous pulley perimeter and the perimeter of off-centre operation active synchronization belt wheel and the not rounded driven wheel of Bath main officer of Tibet be equal.
(j) it the not rounded 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 not rounded tensioning synchronous pulley to diameter and corresponding cornerRelationship is to be tensioned synchronous pulley Pitch curve equation.
The device have the advantages that：
1, the present invention is not rounded-not rounded three-wheel toothed belt transmission of the self-compensating off-centre operation-Fourier of the amount of becoming slack in reality A whole set of perfect design theory basis is provided in the application of border, and it is not rounded-not rounded to can be applied to all off-centre operation-Fourier Three-wheel synchronous belt drive mechanism promotes promoting the use of for not rounded-not rounded three-wheel toothed belt transmission of off-centre operation-Fourier.
2, driving wheel pitch curve is off-centre operation in the present invention, and driven wheel pitch curve is Fourier's curve, and off-centre operation is actively same The radius of step belt wheel pitch curve, eccentricity are controlled variables, and the not rounded driven synchronous pulley pitch curve adjustable parameter of Fourier is more, has a0,a1,a2,b1,b2And become property coefficient and exponent number, this seven parameters are adjusted, more specific non-at the uniform velocity require can be met Transmission.
3, the tensioning synchronous pulley in the present invention is the not rounded synchronous pulley of free pitch curve, can be with real-time compensation off-centre operation The belt sag variable quantity generated during active synchronization belt wheel and the not rounded driven synchronous belt pulley transmission of Fourier, realizes big centre-to-centre spacing Between non-at the uniform velocity directly accurate transmission.
4, 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.
Description of the drawings
Fig. 1 is the transmission principle figure of the present invention；
Fig. 2 is that off-centre operation active synchronization belt wheel becomes with the not rounded driven synchronous belt pulley transmission ratio of Fourier in the embodiment of the present invention Change curve graph；
Synchronous belt belt length change curve when Fig. 3 is the pitch curve using the not rounded tensioning synchronous pulley in the embodiment of the present invention Figure；
Fig. 4 is the not rounded driven synchronous pulley pitch curve figure of Fourier in the embodiment of the present invention；
Fig. 5 is not rounded tensioning synchronous pulley pitch curve figure in the embodiment of the present invention.
Specific implementation mode
The invention will be further described with reference to the accompanying drawings and embodiments.
Not rounded-not rounded three-wheel toothed belt transmission design method of off-centre operation-Fourier, is as follows：
Step 1: such as Fig. 1, off-centre operation active synchronization belt wheel pitch curve radius r is given1=30mm, eccentric distance e1=15, root The perimeter s of off-centre operation active synchronization belt wheel is calculated according to following formula1=188.4956mm.
s1=2 π × r1 (1)
The polar equation of cutting of off-centre operation active synchronization belt wheel pitch curve is：
p1=r1+e1×cos(θ1) (2)
The not rounded driven wheel pitch curve equation coefficient of Fourier is a1=3.05, a2=1.32, b1=-3.78, b2=2.65, According to the off-centre operation active synchronization belt wheel pitch curve principle equal with the not rounded driven wheel pitch curve perimeter of Fourier, Fourier is calculated The parameter a of not rounded driven synchronous pulley pitch curve0=22.55；The exponent number n of the not rounded driven synchronous pulley pitch curve of Fourier21= 1, it is denaturalized Coefficient m21=1, the above parameter is substituted into the polar diameter that formula (3) calculates the not rounded driven synchronous pulley pitch curve of Fourier r21、r22：
?By numerical solution in section, r is obtained21Numerical solution.
In formula, u2Justify tangent line and polar diameter r on active synchronization belt wheel pitch curve for sinusoidal non-eccentricity21Angle；
CornerCorner cut θ2, angle u2Between in the presence of relation of plane：
Determine that the polar equation of cutting of the not rounded driven synchronous pulley pitch curve of Fourier is：
The not rounded driven synchronous pulley pitch curve such as Fig. 4 of Fourier.
Step 2: calculating the transmission of off-centre operation active synchronization belt wheel and the not rounded driven synchronous pulley initial position of Fourier Than；
Initial position, the moving coordinate system x of off-centre operation active synchronization belt wheel pitch curve1o1y1Middle x1Axis is to quiet coordinate system xo1In y The corner of x-axisThe moving coordinate system x of the not rounded driven synchronous pulley pitch curve of Fourier2o2y2Middle x2Axis is to quiet coordinate system 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 off-centre operation active synchronization belt wheel pitch curve is not rounded driven synchronous with Fourier Belt wheel pitch curve common tangent incision superius C1、C2It is corresponding to cut diameter, p1(θ13) and p3(θ31) it is respectively off-centre operation active synchronization belt wheel section Curve and tensioning synchronous pulley pitch curve common tangent incision superius C6、C5It is corresponding to cut diameter, p2(θ23) and p3(θ32) it is respectively Fourier Not rounded driven synchronous pulley pitch curve and tensioning synchronous pulley pitch curve common tangent incision superius C3、C4It is corresponding to cut diameter, θ120It is inclined Heart circle active synchronization belt wheel pitch curve cuts diameter p1(θ12) with the not rounded driven synchronous pulley pitch curve of Fourier cut diameter p2(θ21) to respectively From the corner initial value of moving coordinate system trunnion axis, θ130Diameter p is cut for off-centre operation active synchronization belt wheel pitch curve1(θ13) synchronous with tensioning Belt wheel pitch curve cuts diameter p3(θ31) arrive respective moving coordinate system trunnion axis corner initial value, θ230For the not rounded driven synchronous belt of Fourier Wheel pitch curve cuts diameter p2(θ23) with tensioning synchronous pulley pitch curve cut diameter p3(θ32) to respective moving coordinate system trunnion axis corner at the beginning of Value, θ12、θ13Respectively off-centre operation active synchronization belt wheel pitch curve incision superius C1、C6Correspondence cuts diameter to moving coordinate system x1o1y1Middle x1 The corner cut of axis, θ21、θ23The respectively not rounded driven synchronous pulley pitch curve incision superius C of Fourier2、C3Correspondence cuts diameter to moving coordinate system x2o2y2Middle x2The corner cut of axis, θ31、θ32To be tensioned synchronous pulley pitch curve incision superius C4、C5Correspondence cuts diameter to moving coordinate system x3o3y3 Middle x3The corner cut of axis, L1For off-centre operation active synchronization belt wheel and the not rounded driven synchronous pulley centre-to-centre spacing of Fourier, L2It is non-for Fourier The driven synchronous pulley of circle and tensioning synchronous pulley centre-to-centre spacing, L3For tensioning synchronous pulley and off-centre operation active synchronization belt wheel center Away from；
The transmission ratio that initial position is calculated according to formula (8) is i120=0.6730.
Step 3: calculating in off-centre operation active synchronization belt wheel, the not rounded driven synchronous pulley of Fourier and tensioning synchronous pulley Common tangent segment length between per two-wheeled.
Initial time sets circle of the tensioning synchronous pulley pitch curve to give radius, off-centre operation active synchronization belt wheel and Fu In common tangent segment length T between not rounded two point of contact of driven synchronous pulley of leaf0, the not rounded driven synchronous pulley of Fourier and tensioning it is same Walk the common tangent segment length T between two point of contact of belt wheel1, between off-centre operation active synchronization belt wheel and tensioning synchronous pulley two point of contact Common tangent segment length T2Respectively：
T is calculated to obtain according to formula (9)0=108.3370mm, T1=110.6045mm, T2=107.3296mm.
When off-centre operation active synchronization belt wheel turns over angleThe not rounded driven synchronous pulley of Fourier accordingly turns over angle Off-centre operation active synchronization belt wheel pitch curve incision superius C1、C6Corresponding arc length variable quantity is s1、s6, the not rounded driven synchronization of Fourier Belt wheel pitch curve incision superius C2、C3Corresponding arc length variable quantity is s2、s3, tensioning synchronous pulley pitch curve incision superius C4、C5It is corresponding Arc length variable quantity be s4、s5.Then have：
In formula, p1"(θ1) it is p1(θ1) second-order differential, p2"(θ2) it is p2(θ2) second-order differential, p3"(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 common tangent between off-centre operation active synchronization belt wheel and not rounded two point of contact of driven synchronous pulley of Fourier Segment length T12, common tangent segment length T between the not rounded driven synchronous pulley of Fourier and tensioning synchronous pulley two point of contact23, it is eccentric Circle active synchronization belt wheel and the common tangent segment length T being tensioned between two point of contact of synchronous pulley13Respectively：
In formula, p '1(θ12)、p’1(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 ratio of any time off-centre operation active synchronization belt wheel and the not rounded driven wheel of Fourier；
Off-centre operation active synchronization belt wheel is uniform rotation, and then according to formula (2), (6) solve p1, p2, then off-centre operation is calculated to obtain Active synchronization belt wheel and the not rounded driven wheel instantaneous transmission ratio of Fourier：
According to formula (12), (13), (14), when calculating circle driving wheel rotates a circle, in off-centre operation active synchronization belt wheel and Fu The not rounded driven synchronous belt pulley transmission of leaf than relationship such as Fig. 2.
Step 5: calculating synchronous belt perimeter；
Off-centre operation active synchronization belt wheel pitch curve is denoted as C with tensioning synchronous pulley pitch curve common tangent incision superius6, when arbitrary Carve C1With C6Between arc length be c11, on off-centre operation active synchronization belt wheel and the not rounded driven synchronous pulley pitch curve common tangent of Fourier Point of contact is denoted as C2, the not rounded driven synchronous pulley pitch curve of Fourier and tensioning synchronous pulley pitch curve common tangent incision superius are denoted as C3, Any time C2With C3Between arc length be c22, tensioning synchronous pulley pitch curve and the not rounded driven synchronous pulley pitch curve of Fourier are public Tangent line incision superius is denoted as C4, it is tensioned synchronous pulley pitch curve and is denoted as with off-centre operation active synchronization belt wheel pitch curve common tangent incision superius C5, any time C4With C5Between arc length be c33。
Any time, synchronous belt Zhou Changwei：
C=T12+T13+T23+c11+c22+c33 (16)
It carves at the beginning, the original perimeter that synchronous belt is calculated according to formula (16) is C0=563.98mm；
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 6: 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 are denoted as C0=563.98mm.
(b) off-centre operation active synchronization belt wheel turns overIt is not rounded driven with Fourier according to off-centre operation active synchronization belt wheel Synchronous belt pulley transmission calculates the not rounded driven synchronous pulley of Fourier and turns over corresponding angle than relationship such as Fig. 2 The corner for being tensioned synchronous pulley is identical as off-centre operation active synchronization belt wheelEnsureing that synchronous belt perimeter C is constant Under the premise of, calculate r3-1=30.0562mm.
(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), one is taken 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 not rounded.
(e) each point of the not rounded tensioning synchronous pulley obtained in (d) is scaled up or is reduced to diameter so that new The perimeter of the not rounded tensioning synchronous pulley arrived and off-centre operation active synchronization belt wheel and the perimeter of the not rounded driven synchronous pulley of Fourier It is equal.
(f) radius value at (e) obtained each moment is substituted into the belt length that formula (16) calculates each moment.
If (g) belt length at each moment and the absolute value of the difference of initial belt length are 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 not rounded tensioning synchronous pulley respectively to diameter value 3%, 5 ° before and after belt length minimum position corresponds to moment point, increase not rounded tensioning synchronous pulley respectively to the 3% of diameter value, then It is fitted to obtain new not rounded tensioning synchronous pulley with B-spline.
(i) not rounded tensioning synchronous pulley each point that will be after (h) is scaled up or is reduced to diameter so that is newly obtained The perimeter of the perimeter and the not rounded driven synchronous pulley of off-centre operation active synchronization belt wheel and Fourier of not rounded tensioning synchronous pulley is homogeneous Deng.
(j) each point is calculated to diameter substitution formula (16) in the not rounded tensioning synchronous pulley after (i) and corresponds to synchronous belt band It is long, if each point 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, carry out step (k), otherwise (h) is returned to.
(k) establish each moment of not rounded 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 off-centre operation actively The corresponding angle relation of the not rounded driven synchronous pulley of synchronous pulley and Fourier.
Tensioning synchronous pulley pitch curve such as Fig. 5 after calculating.
Synchronous belt theory belt length variable quantity is 25.78mm in the embodiment, is the 4.8% of synchronous belt total length, because of band It needs to be tensioned, actual operation requirements can be met.

Claims (1)

1. not rounded-not rounded three-wheel toothed belt transmission design method of off-centre operation-Fourier, it is characterised in that：This method is specifically such as Under：
Step 1: determining off-centre operation active synchronization belt wheel pitch curve and the not rounded driven synchronous pulley section of Fourier according to transmission rule Curvilinear equation；
Off-centre operation active synchronization belt wheel is the input link of uniform rotation, cuts polar equation：
p1=r1+e1×cos(θ1) (1)
s1=2 π × r1 (2)
In formula, p1Diameter, θ are cut for off-centre operation active synchronization belt wheel pitch curve1For p1To moving coordinate system x1o1y1Middle x1The corner cut of axis, e1For the eccentricity of off-centre operation active synchronization belt wheel pitch curve, r1For off-centre operation active synchronization belt wheel pitch curve radius, s1For bias Circle active synchronization belt wheel pitch curve perimeter；Moving coordinate system x1o1y1Origin be arranged in the rotation of off-centre operation active synchronization belt wheel At the heart；
The not rounded driven synchronous pulley of Fourier is output wheel, and the not rounded driven synchronous pulley pitch curve of Fourier cuts polar equation For：
In formula, r21,r22The respectively polar diameter of Fourier's not rounded driven synchronous pulley pitch curve first segment curve and second segment curve, n21For the exponent number of the not rounded driven synchronous pulley pitch curve of Fourier, m21For the not rounded driven synchronous pulley pitch curve of Fourier first The deformation coefficient of section curve, a0,a1,a2,b1,b2For the parameter of the not rounded driven synchronous pulley pitch curve of Fourier,For Fourier Not rounded driven synchronous pulley pitch curve moving coordinate system x2o2y2Middle x2Axis is to quiet coordinate system xo1The corner of x-axis in y；Moving coordinate system x2o2y2Origin be arranged at the center of rotation of the not rounded driven synchronous pulley of Fourier；Quiet coordinate system xo1The origin setting of y exists At the center of rotation of off-centre operation active synchronization belt wheel；
In formula, p2Diameter, θ are cut for the not rounded driven synchronous pulley pitch curve of Fourier2For p2To moving coordinate system x2o2y2Middle x2Axis is cut Angle；
Step 2: calculating the transmission ratio of off-centre operation active synchronization belt wheel and the not rounded driven synchronous pulley initial position of Fourier；
Initial position, the moving coordinate system x of off-centre operation active synchronization belt wheel pitch curve1o1y1Middle x1Axis is to quiet coordinate system xo1X-axis in y CornerThe moving coordinate system x of the not rounded driven synchronous pulley pitch curve of Fourier2o2y2Middle x2Axis is to quiet coordinate system xo1y The corner of middle x-axisAccording to cutting, polar coordinates are theoretical to be obtained：
In formula, p1(θ12) and p2(θ21) it is respectively off-centre operation active synchronization belt wheel pitch curve and the not rounded driven synchronous pulley of Fourier Pitch curve common tangent incision superius C1、C2It is corresponding to cut diameter, p1(θ13) and p3(θ31) it is respectively off-centre operation active synchronization belt wheel pitch curve With tensioning synchronous pulley pitch curve common tangent incision superius C6、C5It is corresponding to cut diameter, p2(θ23) and p3(θ32) it is respectively that Fourier is not rounded Driven synchronous pulley pitch curve and tensioning synchronous pulley pitch curve common tangent incision superius C3、C4It is corresponding to cut diameter, θ120For off-centre operation Active synchronization belt wheel pitch curve cuts diameter p1(θ12) with the not rounded driven synchronous pulley pitch curve of Fourier cut diameter p2(θ21) to each automatic The corner initial value of coordinate system trunnion axis, θ130Diameter p is cut for off-centre operation active synchronization belt wheel pitch curve1(θ13) and tensioning synchronous pulley Pitch curve cuts diameter p3(θ31) arrive respective moving coordinate system trunnion axis corner initial value, θ230For the not rounded driven synchronous pulley section of Fourier Curve cuts diameter p2(θ23) with tensioning synchronous pulley pitch curve cut diameter p3(θ32) arrive respective moving coordinate system trunnion axis corner initial value, θ12、θ13Respectively off-centre operation active synchronization belt wheel pitch curve incision superius C1、C6Correspondence cuts diameter to moving coordinate system x1o1y1Middle x1Axis Corner cut, θ21、θ23The respectively not rounded driven synchronous pulley pitch curve incision superius C of Fourier2、C3Correspondence cuts diameter to moving coordinate system x2o2y2Middle x2The corner cut of axis, θ31、θ32To be tensioned synchronous pulley pitch curve incision superius C4、C5Correspondence cuts diameter to moving coordinate system x3o3y3 Middle x3The corner cut of axis, L1For off-centre operation active synchronization belt wheel and the not rounded driven synchronous pulley centre-to-centre spacing of Fourier, L2It is non-for Fourier The driven synchronous pulley of circle and tensioning synchronous pulley centre-to-centre spacing, L3For tensioning synchronous pulley and off-centre operation active synchronization belt wheel center Away from；Moving coordinate system x3o3y3Origin be arranged tensioning synchronous pulley center of rotation at；
Initial position off-centre operation active synchronization belt wheel is with the not rounded driven synchronous belt pulley transmission ratio of Fourier：
Step 3: calculating every two in off-centre operation active synchronization belt wheel, the not rounded driven synchronous pulley of Fourier and tensioning synchronous pulley Common tangent segment length between wheel；
Initial time sets circle of the tensioning synchronous pulley pitch curve to give radius, off-centre operation active synchronization belt wheel and Fourier Common tangent segment length T between not rounded two point of contact of driven synchronous pulley0, the not rounded driven synchronous pulley of Fourier with tensioning synchronous belt Take turns the common tangent segment length T between two point of contacts1, the public affairs between off-centre operation active synchronization belt wheel and tensioning synchronous pulley two point of contact cut Line segment length T2Respectively：
In formula, p '1(θ120)、p′1(θ130) it is respectively p1(θ120)、p1(θ130) first differential, p'2(θ120)、p'2(θ230) respectively For p2(θ120)、p2(θ230) first differential, p'3(θ130)、p'3(θ230) it is respectively p3(θ130)、p3(θ230) first differential；
When off-centre operation active synchronization belt wheel turns over angleThe not rounded driven synchronous pulley of Fourier accordingly turns over angleIt is eccentric Circle active synchronization belt wheel pitch curve incision superius C1、C6Corresponding arc length variable quantity is s1、s6, the not rounded driven synchronous pulley of Fourier Pitch curve incision superius C2、C3Corresponding arc length variable quantity is s2、s3, tensioning synchronous pulley pitch curve incision superius C4、C5Corresponding arc Long 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) two Rank differential, θ3To be tensioned synchronous belt round cut diameter p3To moving coordinate system x3o3y3Middle x3The corner cut of axis；
Any time, the common tangent segment length between off-centre operation active synchronization belt wheel and not rounded two point of contact of driven synchronous pulley of Fourier Spend T12, common tangent segment length T between the not rounded driven synchronous pulley of Fourier and tensioning synchronous pulley two point of contact23, off-centre operation master Dynamic synchronous pulley and the common tangent segment length T being tensioned between two point of contact of synchronous pulley13Respectively：
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,For tensioning 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 ratio of any time off-centre operation active synchronization belt wheel and the not rounded driven synchronous pulley of Fourier；
Off-centre operation active synchronization belt wheel uniform rotation, according to formula (1), (4) solve p1, p2, then instantaneous transmission ratio be：
Step 5: calculating any time synchronous belt perimeter；
Off-centre operation active synchronization belt wheel pitch curve is denoted as C with tensioning synchronous pulley pitch curve common tangent incision superius6, any time C1 With C6Between arc length be c11, the not rounded driven synchronous pulley pitch curve of Fourier with tensioning synchronous pulley pitch curve common tangent incision superius It is denoted as C3, any time C2With C3Between arc length be c22, it is tensioned synchronous pulley pitch curve and driven wheel pitch curve common tangent incision superius It is denoted as C4, tensioning synchronous pulley pitch curve and driving wheel pitch curve common tangent incision superius are denoted as C5, any time C4With C5Between arc A length of c33；
Any time, synchronous belt Zhou Changwei：
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 is calculated according to formula (14) and is denoted as C0；
(b) off-centre operation active synchronization belt wheel turns over 1 °, requires the calculating not rounded driven synchronous pulley of Fourier to turn over according to transmission ratio Corresponding angle, the corner for being tensioned synchronous pulley are identical as off-centre operation active synchronization belt wheel；Under the premise of ensureing that C is constant, root Corresponding tensioning synchronous pulley radius r when turning over 1 ° according to formula (14) reverse off-centre operation active synchronization belt wheel3-1, that is, correspond to the moment p3；
(c) it repeats (b) 358 times, obtains off-centre operation active synchronization belt wheel and turn over corresponding tensioning synchronous belt at 2 °, 3 ° ..., 359 ° It is respectively r to take turns radius3-2, r3-3... ..., r3-359；
(d) 360 concentric circles are so far obtained, by the tensioning synchronous pulley radius in (a), (b) and (c), a circle is taken every 1 ° Radius, sequentially take 360 radiuses, using set tensioning synchronous pulley center of rotation as the center of circle, the outer end of 360 radiuses will be taken Point is sequentially connected with, and composition one is closed not rounded；
(e) each moment of the not rounded tensioning synchronous pulley obtained in (d) is scaled up or is reduced to diameter so that newly obtained Not rounded tensioning synchronous pulley perimeter and the not rounded driven synchronous pulley of off-centre operation active synchronization belt wheel and Fourier perimeter it is equal It is equal；
(f) radius value at (e) obtained each moment is substituted into the belt length that formula (14) calculates each moment；
If (g) belt length at each moment and the absolute value of the difference of initial belt length are respectively less than preset value, step (k) is carried out, otherwise Carry out step (h)；
(h) 5 ° before and after belt length maximum position corresponds to moment point, reduce it is not rounded tensioning synchronous pulley respectively to diameter value 1~ 5%, 5 ° before and after belt length minimum position corresponds to moment point, increase not rounded tensioning synchronous pulley respectively to the 1~5% of diameter value, Then it is fitted to obtain new not rounded tensioning synchronous pulley with B-spline；
(i) not rounded tensioning synchronous pulley each moment that will be after (h) scales up or is reduced to diameter so that is newly obtained is non- The perimeter of the perimeter and the not rounded driven synchronous pulley of off-centre operation active synchronization belt wheel and Fourier of circle tensioning synchronous pulley is equal；
(j) it the not rounded tensioning synchronous pulley after (i) is substituted into formula (14) to diameter is calculated each moment and correspond to synchronous belt belt length, 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, step (k) is carried out, Otherwise (h) is returned to；
(k) establish each moment of not rounded tensioning synchronous pulley to diameter and corresponding cornerRelationship is to be tensioned synchronous pulley section song Line equation.
CN201710191356.1A 2017-03-28 2017-03-28 Not rounded-not rounded three-wheel toothed belt transmission design method of off-centre operation-Fourier Active CN106838158B (en)

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CN104462638A (en) * 2014-10-09 2015-03-25 浙江理工大学 Design method of high-order modified Pascal spiral curve non-circular gear pair
CN104455211A (en) * 2014-10-09 2015-03-25 浙江理工大学 Design method of high-order modified Fourier non-circular gear pair
CN104455313A (en) * 2014-10-09 2015-03-25 浙江理工大学 Design method of high-order modified eccentric circular gear and non-circular gear pair conjugated with high-order modified circular gear

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* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US4865577A (en) * 1988-09-08 1989-09-12 Trustees Of Columbia University In The City Of New York Noncircular drive
CN202659811U (en) * 2012-06-04 2013-01-09 浙江理工大学 Non-circular gear pair with Fourier function pitch curve
CN104462638A (en) * 2014-10-09 2015-03-25 浙江理工大学 Design method of high-order modified Pascal spiral curve non-circular gear pair
CN104455211A (en) * 2014-10-09 2015-03-25 浙江理工大学 Design method of high-order modified Fourier non-circular gear pair
CN104455313A (en) * 2014-10-09 2015-03-25 浙江理工大学 Design method of high-order modified eccentric circular gear and non-circular gear pair conjugated with high-order modified circular gear

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