CN106870661B - Oval-not rounded three-wheel toothed belt transmission design method of circle- - Google Patents
Oval-not rounded three-wheel toothed belt transmission design method of circle- Download PDFInfo
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- CN106870661B CN106870661B CN201710191389.6A CN201710191389A CN106870661B CN 106870661 B CN106870661 B CN 106870661B CN 201710191389 A CN201710191389 A CN 201710191389A CN 106870661 B CN106870661 B CN 106870661B
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- F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
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- 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
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Abstract
The invention discloses the not rounded three-wheel toothed belt transmission design methods of circle-ellipse-.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 of synchronous belt is calculated, changed according to synchronous belt perimeter slack and the free pitch curve of not rounded tensioning synchronous pulley is calculated by iterative algorithm, tensioning wheel is the not rounded synchronous pulley of free pitch curve, the problem of not rounded 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 synchronization belt sag variable quantity generated in real-time compensation transmission process；The long axis of the radius of circle active synchronization belt wheel pitch curve and oval driven synchronous pulley pitch curve, eccentricity are controlled variable, the adjusting measured by three changes the shape of circle driving wheel pitch curve and oval driven wheel pitch curve, and the specific non-at the uniform velocity transmission for meeting big centre-to-centre spacing requires.
Description
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
Oval-not rounded three-wheel toothed belt transmission design method of circle-.
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 the self-compensating not rounded three-wheel of circle-ellipse-of one kind amount of becoming slack
Toothed belt transmission design method provides a whole set of perfect design theory basis in practical applications for not rounded synchronous pulley,
Realize the non-at the uniform velocity directly accurate transmission between big centre-to-centre spacing.The design method initially sets up the pitch curve of synchronous belt principal and subordinate wheel
Equation, and move synchronous belt pulley transmission ratio using polar coordinates theoretical calculation principal and subordinate is cut；Then the perimeter for calculating synchronous belt, according to synchronization
The parameters of the not rounded tensioning free pitch curve of synchronous pulley are calculated by alternative manner with the variation of perimeter slack.
In order to solve the above technical problems, the technical scheme is that：
The present invention is as follows：
Step 1: determining round active synchronization belt wheel pitch curve and oval driven synchronous pulley pitch curve according to transmission rule
Equation；
Round active synchronization belt wheel is the input link of uniform rotation,For the dynamic seat of round active synchronization belt wheel pitch curve
Mark system x1o1y1Middle x1Axis is to quiet coordinate system xo1The corner of x-axis, θ in y1For p1To moving coordinate system x1o1y1Middle x1The corner cut of axis, it is round
Active synchronization belt wheel cuts polar equation：
p1=r1 (1)
S=2 π × r1 (2)
In formula, p1Diameter, r are cut for round active synchronization belt wheel pitch curve1It is the half of round active synchronization belt wheel pitch curve
Diameter, s are the perimeter of round active synchronization belt wheel pitch curve.
Oval driven synchronous pulley is output link, cuts polar equation：
p2=a × (1-e2)-a×e2×cos(θ2) (3)
In formula, p2Diameter, θ are cut for oval driven synchronous pulley pitch curve2For p2To moving coordinate system x2o2y2Middle x2Axis is cut
Angle, e2For the eccentricity of oval driven synchronous pulley pitch curve, a is the long axis of oval driven synchronous pulley pitch curve.
In formula, b is the short axle of oval driven synchronous pulley pitch curve, and c is the focal length of oval driven synchronous pulley pitch curve.
C=a × e2 (6)
Step 2: calculating the transmission ratio of round active synchronization belt wheel and oval driven synchronous pulley initial position；
Initial position, the moving coordinate system x of round active synchronization belt wheel pitch curve1o1y1Middle x1Axis is to quiet coordinate system xo1X in y
The corner of axisThe moving coordinate system x of oval driven synchronous pulley pitch curve2o2y2Middle x2Axis is to quiet coordinate system xo1X-axis in y
CornerAccording to cutting, polar coordinates are theoretical to be obtained：
In formula, p1(θ12) and p2(θ21) it is respectively that round active synchronization belt wheel pitch curve and oval driven synchronous pulley section are bent
Line common tangent incision superius C1、C2It is corresponding to cut diameter, p1(θ13) and p3(θ31) it is respectively round active synchronization 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) it is respectively oval driven synchronous pulley
Pitch curve and tensioning synchronous pulley pitch curve common tangent incision superius C3、C4It is corresponding to cut diameter, θ120For round active synchronization belt wheel section
Curve cuts diameter p1(θ12) with oval driven synchronous pulley pitch curve cut diameter p2(θ21) to respective moving coordinate system trunnion axis corner at the beginning of
Value, θ130Diameter p is cut for round active synchronization belt wheel pitch curve1(θ13) with tensioning synchronous pulley pitch curve cut diameter p3(θ31) to respectively
The corner initial value of moving coordinate system trunnion axis, θ230Diameter p is cut for oval driven synchronous pulley pitch curve2(θ23) and tensioning synchronous pulley
Pitch curve cuts diameter p3(θ32) arrive respective moving coordinate system trunnion axis corner initial value, θ12、θ13Respectively round active synchronization belt wheel section
Curve incision superius C1、C6Correspondence cuts diameter to moving coordinate system x1o1y1Middle x1The corner cut of axis, θ21、θ23Respectively oval driven synchronous belt
Take turns pitch curve incision superius C2、C3Correspondence cuts diameter to moving coordinate system x2o2y2Middle x2The corner cut of axis, θ31、θ32Respectively it is tensioned synchronous belt
Take turns pitch curve incision superius C4、C5Correspondence cuts diameter to moving coordinate system x3o3y3Middle x3The corner cut of axis, L1For round active synchronization belt wheel with
Oval driven synchronous pulley centre-to-centre spacing, L2For oval driven synchronous pulley and tensioning synchronous pulley centre-to-centre spacing, L3It is same for round active
Walk belt wheel and tensioning synchronous pulley centre-to-centre spacing；
The round active synchronization belt wheel of initial position is with oval driven synchronous belt pulley transmission ratio：
Step 3: between calculating round active synchronization belt wheel, oval driven synchronous pulley and tensioning synchronous pulley per two-wheeled
Common tangent segment length.
Initial time sets circle of the tensioning synchronous pulley pitch curve to give radius, round active synchronization belt wheel and ellipse
Common tangent segment length T between two point of contact of driven synchronous pulley0, oval driven synchronous pulley and two point of contact of tensioning synchronous pulley it
Between common tangent segment length T1, common tangent segment length T between two point of contact of round active synchronization belt wheel and tensioning synchronous pulley2Point
It is not：
In formula, p '1(θ120)、p′1(θ130) it is respectively p1(θ120)、p1(θ130) first differential, p'2(θ120)、p'2(θ230)
Respectively p2(θ120)、p2(θ230) first differential, p'3(θ130)、p'3(θ230) it is respectively p3(θ130)、p3(θ230) single order it is micro-
Point.
When round active synchronization belt wheel turns over angleOval driven synchronous pulley accordingly turns over angleRound active
Synchronous pulley pitch curve incision superius C1、C6Corresponding arc length variable quantity is s1、s6, oval driven synchronous pulley pitch curve incision superius
C2、C3Corresponding arc length variable quantity is s2、s3, tensioning synchronous pulley pitch curve incision superius C4、C5Corresponding arc length variable quantity is s4、
s5.Then have：
In formula, p "1(θ1) it is p1(θ1) second-order differential, p "2(θ2) it is p2(θ2) second-order differential, p "3(θ3) it is p3(θ3)
Second-order differential, 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 round active synchronization belt wheel and oval driven two point of contact of synchronous pulley
T12, common tangent segment length T between oval driven synchronous pulley and two point of contact of tensioning synchronous pulley23, round active synchronization belt wheel
With the common tangent segment length T between two point of contact of tensioning 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 round active synchronization belt wheel and oval driven synchronous pulley；
Round active synchronization belt wheel uniform rotation, according to formula (1), (3) solve p1, p2, then instantaneous transmission ratio be：
Step 5: calculating any time synchronous belt perimeter；
Round active synchronization belt wheel pitch curve is denoted as C with tensioning synchronous pulley pitch curve common tangent incision superius6, any time
C1With C6Between arc length be c11, oval driven synchronous pulley pitch curve is denoted as with synchronous pulley pitch curve common tangent incision superius is tensioned
C3, any time C2With C3Between arc length be c22, it is tensioned synchronous pulley pitch curve and is denoted as with driven wheel pitch curve common tangent incision superius
C4, tensioning synchronous pulley pitch curve and round active synchronization belt wheel pitch curve common tangent incision superius are denoted as C5, any time C4With C5
Between arc length be c33。
Any time, synchronous belt Zhou Changwei：
C=T12+T13+T23+c11+c22+c33 (16)
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 (16) and is denoted as C0。
(b) round active synchronization belt wheel turns over 1 °, requires the oval driven synchronous pulley of calculating to turn over accordingly according to transmission ratio
Angle, the corner for being tensioned synchronous pulley is identical as round active synchronization belt wheel.Under the premise of ensureing that C is constant, according to formula
(16) the round active synchronization belt wheel of reverse turns over corresponding tensioning synchronous pulley radius r at 1 °3-1, that is, correspond to the p at moment3。
(c) it repeats (b) 358 times, obtains round active synchronization belt wheel and turn over corresponding tensioning synchronization at 2 °, 3 ° ..., 359 °
Belt wheel 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), 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 the perimeter of round active synchronization belt wheel and oval driven synchronous pulley are 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 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 round active synchronization belt wheel and oval driven synchronous pulley be equal.
(j) it the not rounded tensioning synchronous pulley after (i) is substituted into formula (16) 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 that the not rounded three-wheel toothed belt transmission of the self-compensating circle-ellipse-of the amount of becoming slack carries in practical applications
A whole set of perfect design theory basis has been supplied, the not rounded three-wheel synchronous belt drive mechanism of all circle-ellipses-is can be applied to,
Promote promoting the use of for circle-ellipse-not rounded three-wheel toothed belt transmission.
2, driving wheel pitch curve is circle in the present invention, and driven wheel pitch curve is ellipse, and transmission ratio design is simple, round active
The radius of synchronous pulley pitch curve, the long axis of oval driven synchronous pulley pitch curve, eccentricity are controlled variable, are measured by three
Adjusting change the shape of round active synchronization belt wheel pitch curve and oval driven synchronous pulley pitch curve, meet it is specific it is non-at the uniform velocity
It is required that transmission.
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 not rounded tensioning synchronous pulley in the present invention is the not rounded synchronous pulley of free pitch curve, can be justified with real-time compensation
The belt sag variable quantity generated during type active synchronization belt wheel and freely not rounded driven synchronous belt pulley transmission, realizes big centre-to-centre spacing
Between non-at the uniform velocity directly accurate transmission.
Description of the drawings
Fig. 1 is the transmission schematic diagram of the present invention；
Fig. 2 is transmission ratio and round active synchronization of the round active synchronization belt wheel with oval driven synchronous pulley in the present invention
Belt wheel angle relation curve graph；
Synchronous belt belt length change curve when Fig. 3 is the pitch curve using the not rounded tensioning synchronous pulley in the present invention；
Fig. 4 is oval driven synchronous pulley pitch curve figure in the present invention；
Fig. 5 is the tensioning free pitch curve fitted figure of synchronous pulley in the present invention.
Specific implementation mode
Below in conjunction with the accompanying drawings and case study on implementation the invention will be further described.
Oval-not rounded three-wheel toothed belt transmission design method of circle-, is as follows：
Step 1: such as Fig. 1, round active synchronization belt wheel pitch curve radius r is given1=30mm, round active synchronization belt wheel
Pitch curve cuts diameter p1=r1, tensioning synchronous pulley is the not rounded belt wheel according to the variation fitting of synchronous belt perimeter slack；Three-wheel center
Away from L1=L2=L3=100mm, three-wheel are that perimeters is waited to close convex curve, and round active synchronization belt wheel section is calculated according to following formula
The perimeter s=188.4956mm of curve：
S=2 π × r1 (1)
Step 2: the eccentricity e of given oval driven synchronous pulley pitch curve2=0.8, according to round active synchronization belt wheel
The pitch curve principle equal with oval driven synchronous pulley pitch curve perimeter, it is long to calculate separately oval driven synchronous pulley pitch curve
Axis a=35.1022mm, short axle b=21.0613mm.
Determine that the polar equation of cutting of oval driven synchronous pulley pitch curve is：
p2=7.0204-28.0818 × cos (θ2) (4)
In formula, θ2Diameter p is cut for oval driven synchronous pulley pitch curve2To moving coordinate system x2o2y2Middle x2The corner cut of axis.
Oval driven synchronous pulley pitch curve is as shown in Figure 4.
Step 3: calculating the transmission ratio of round active synchronization belt wheel and oval driven synchronous pulley initial position：
Initial position, the moving coordinate system x of round active synchronization belt wheel pitch curve1o1y1Middle x1Axis is to quiet coordinate system xo1X in y
The corner of axisThe moving coordinate system x of oval driven synchronous pulley pitch curve2o2y2Middle x2Axis is to quiet coordinate system xo1X-axis in y
CornerAccording to cutting, polar coordinates are theoretical to be obtained：
In formula, p1(θ12) and p2(θ21) it is respectively that round active synchronization belt wheel pitch curve and oval driven synchronous pulley section are bent
Line common tangent incision superius C1、C2It is corresponding to cut diameter, p1(θ13) and p3(θ31) it is respectively round active synchronization 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) it is respectively oval driven synchronous pulley
Pitch curve and tensioning synchronous pulley pitch curve common tangent incision superius C3、C4It is corresponding to cut diameter, θ120For round active synchronization belt wheel section
Curve cuts diameter p1(θ12) with oval driven synchronous pulley pitch curve cut diameter p2(θ21) to respective moving coordinate system trunnion axis corner at the beginning of
Value, θ130Diameter p is cut for round active synchronization belt wheel pitch curve1(θ13) with tensioning synchronous pulley pitch curve cut diameter p3(θ31) to respectively
The corner initial value of moving coordinate system trunnion axis, θ230Diameter p is cut for oval driven synchronous pulley pitch curve2(θ23) and tensioning synchronous pulley
Pitch curve cuts diameter p3(θ32) arrive respective moving coordinate system trunnion axis corner initial value, θ12、θ13Respectively round active synchronization belt wheel section
Curve incision superius C1、C6Correspondence cuts diameter to moving coordinate system x1o1y1Middle x1The corner cut of axis, θ21、θ23Respectively oval driven synchronous belt
Take turns pitch curve incision superius C2、C3Correspondence cuts diameter to moving coordinate system x2o2y2Middle x2The corner cut of axis, θ31、θ32Respectively it is tensioned synchronous belt
Take turns pitch curve incision superius C4、C5Correspondence cuts diameter to moving coordinate system x3o3y3Middle x3The corner cut of axis, L1For round active synchronization belt wheel with
Oval driven synchronous pulley centre-to-centre spacing, L2For oval driven synchronous pulley and tensioning synchronous pulley centre-to-centre spacing, L3It is same for round active
Walk belt wheel and tensioning synchronous pulley centre-to-centre spacing；
The transmission ratio that initial position is calculated according to formula (6) is i120=1.1667：
Step 4: between calculating round active synchronization belt wheel, oval driven synchronous pulley and tensioning synchronous pulley per two-wheeled
Common tangent segment length.
Initial time sets circle of the tensioning synchronous pulley pitch curve to give radius, round active synchronization belt wheel and ellipse
Common tangent segment length T between two point of contact of driven synchronous pulley0, oval driven synchronous pulley and two point of contact of tensioning synchronous pulley it
Between common tangent segment length T1, common tangent segment length T between two point of contact of round active synchronization belt wheel and tensioning synchronous pulley2Point
It is not:
T is calculated to obtain according to formula (7)0=103.3370mm, T1=100.7259mm, T2=102.9287mm.
When round active synchronization belt wheel turns over angleOval driven synchronous pulley accordingly turns over angleRound active
Synchronous pulley pitch curve incision superius C1、C6Corresponding arc length variable quantity is s1、s6, oval driven synchronous pulley pitch curve incision superius
C2、C3Corresponding arc length variable quantity is s2、s3, tensioning synchronous pulley pitch curve incision superius C4、C5Corresponding arc length variable quantity is s4、
s5.Then have：
In formula, p "1(θ1) it is p1(θ1) second-order differential, p "2(θ2) it is p2(θ2) second-order differential, p "3(θ3) it is p3(θ3)
Second-order differential, θ3To be tensioned synchronous belt round cut diameter p3To moving coordinate system x3o3y3Middle x3The corner of axis.
Any time, the common tangent segment length between round active synchronization belt wheel and oval driven two point of contact of synchronous pulley
T12, common tangent segment length T between oval driven synchronous pulley and two point of contact of tensioning synchronous pulley23, round active synchronization belt wheel
With the common tangent segment length T between two point of contact of tensioning 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,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 ratio of any time round active synchronization belt wheel and oval driven synchronous pulley；
Round active synchronization belt wheel uniform rotation, p1=r1, p is solved according to formula (4)2, then round active synchronization band is calculated to obtain
Wheel and oval driven synchronous pulley instantaneous transmission ratio：
According to formula (10), (11), (12), when calculating round active synchronization belt wheel and rotating a circle, round active synchronization belt wheel
With the transmission ratio variation such as Fig. 2 of oval driven synchronous pulley.
Step 6: calculating synchronous belt perimeter；
Round active synchronization belt wheel pitch curve is denoted as C with tensioning synchronous pulley pitch curve common tangent incision superius6, any time
C1With C6Between arc length be c11, round active synchronization belt wheel and oval driven synchronous pulley pitch curve common tangent incision superius are denoted as C2,
Oval driven synchronous pulley pitch curve is denoted as C with tensioning synchronous pulley pitch curve common tangent incision superius3, any time C2With C3Between
Arc length be c22, tensioning synchronous pulley pitch curve and oval driven synchronous pulley pitch curve common tangent incision superius are denoted as C4, tensioning
Synchronous pulley pitch curve is denoted as C with round active synchronization belt wheel pitch curve common tangent incision superius5, any time C4With C5Between arc
A length of c33。
Any time, synchronous belt Zhou Changwei：
C=T12+T13+T23+c11+c22+c33 (14)
Initial time calculates synchronous belt original perimeter C according to formula (14)0=515mm；
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 belt length initial value are denoted as C0=515mm.
(b) round active synchronization belt wheel turns over 1 °, requires the oval driven synchronous pulley of calculating to turn over accordingly according to transmission ratio
Angle, the corner for being tensioned synchronous pulley is identical as round active synchronization belt wheel.Under the premise of ensureing that C is constant, according to formula
(14) when the round active synchronization belt wheel of reverse turns over 1 °, corresponding tensioning synchronous pulley radius 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
To not rounded tensioning synchronous pulley perimeter and the perimeter of round active synchronization belt wheel and oval driven synchronous pulley be 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 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 and the perimeter of round active synchronization belt wheel and oval driven synchronous pulley of not rounded tensioning synchronous pulley are equal.
(j) each point is calculated to diameter substitution formula (14) 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 round active is same
Walk belt wheel angle relation corresponding with the driven synchronous pulley of ellipse.
The free pitch curve of tensioning synchronous pulley such as Fig. 5 after calculating.
Synchronous belt theory belt length variable quantity is 12mm in the embodiment, is the 2.3% of synchronous belt total length, because with needs
Tensioning, can meet actual operation requirements.
Claims (1)
1. the not rounded three-wheel toothed belt transmission design method of circle-ellipse-, it is characterised in that：This method is specific as follows：
Step 1: determining round active synchronization belt wheel pitch curve and oval driven synchronous pulley pitch curve side according to transmission rule
Journey；
Round active synchronization belt wheel is the input link of uniform rotation,For the moving coordinate system of round active synchronization belt wheel pitch curve
x1o1y1Middle x1Axis is to quiet coordinate system xo1The corner of x-axis, θ in y1For p1To moving coordinate system x1o1y1Middle x1The corner cut of axis, quiet coordinate system
xo1The origin of y is arranged at the center of rotation of round active synchronization belt wheel, moving coordinate system x1o1y1Origin be arranged in round master
At the center of rotation of dynamic synchronous pulley, round active synchronization belt wheel cuts polar equation：
p1=r1 (1)
S=2 π × r1 (2)
In formula, p1Diameter, r are cut for round active synchronization belt wheel pitch curve1For the radius of round active synchronization belt wheel pitch curve, s is
The perimeter of round active synchronization belt wheel pitch curve；
Oval driven synchronous pulley is output link, cuts polar equation：
p2=a × (1-e2)-a×e2×cos(θ2) (3)
In formula, p2Diameter, θ are cut for oval driven synchronous pulley pitch curve2For p2To moving coordinate system x2o2y2Middle x2The corner cut of axis is moved
Coordinate system x2o2y2Origin be arranged at the center of rotation of oval driven synchronous pulley, e2It is bent for oval driven synchronous pulley section
The eccentricity of line, a are the long axis of oval driven synchronous pulley pitch curve；
In formula, b is the short axle of oval driven synchronous pulley pitch curve, and c is the focal length of oval driven synchronous pulley pitch curve；
C=a × e2 (6)
Step 2: calculating the transmission ratio of round active synchronization belt wheel and oval driven synchronous pulley initial position；
Initial position, the moving coordinate system x of round active synchronization belt wheel pitch curve1o1y1Middle x1Axis is to quiet coordinate system xo1X-axis in y
CornerThe moving coordinate system x of oval driven synchronous pulley pitch curve2o2y2Middle x2Axis is to quiet coordinate system xo1X-axis turns in y
AngleAccording to cutting, polar coordinates are theoretical to be obtained：
In formula, p1(θ12) and p2(θ21) it is respectively that round active synchronization belt wheel pitch curve and oval driven synchronous pulley pitch curve are public
Tangent line incision superius C1、C2It is corresponding to cut diameter, p1(θ13) and p3(θ31) it is respectively that round active synchronization belt wheel pitch curve is synchronous with tensioning
Belt wheel pitch curve common tangent incision superius C6、C5It is corresponding to cut diameter, p2(θ23) and p3(θ32) it is respectively that oval driven synchronous pulley section is bent
Line and tensioning synchronous pulley pitch curve common tangent incision superius C3、C4It is corresponding to cut diameter, θ120For round active synchronization belt wheel pitch curve
Cut diameter p1(θ12) with oval driven synchronous pulley pitch curve cut diameter p2(θ21) arrive respective moving coordinate system trunnion axis corner cut initial value,
θ130Diameter p is cut for round active synchronization belt wheel pitch curve1(θ13) with tensioning synchronous pulley pitch curve cut diameter p3(θ31) arrive each automatic seat
The corner cut initial value of mark system trunnion axis, θ230Diameter p is cut for oval driven synchronous pulley pitch curve2(θ23) bent with tensioning synchronous pulley section
Line cuts diameter p3(θ32) arrive respective moving coordinate system trunnion axis corner cut initial value, θ12、θ13Respectively round active synchronization belt wheel pitch curve
Incision superius C1、C6Correspondence cuts diameter to moving coordinate system x1o1y1Middle x1The corner cut of axis, θ21、θ23Respectively oval driven synchronous pulley section
Curve incision superius C2、C3Correspondence cuts diameter to moving coordinate system x2o2y2Middle x2The corner cut of axis, θ31、θ32Respectively it is tensioned synchronous pulley section
Curve incision superius C4、C5Correspondence cuts diameter to moving coordinate system x3o3y3Middle x3The corner cut of axis, L1For round active synchronization belt wheel and ellipse
Driven synchronous pulley centre-to-centre spacing, L2For oval driven synchronous pulley and tensioning synchronous pulley centre-to-centre spacing, L3For round active synchronization band
Wheel and tensioning synchronous pulley centre-to-centre spacing；Moving coordinate system x3o3y3Origin be arranged tensioning synchronous pulley center of rotation at；
The round active synchronization belt wheel of initial position is with oval driven synchronous belt pulley transmission ratio：
Step 3: the public affairs between calculating round active synchronization belt wheel, oval driven synchronous pulley and tensioning synchronous pulley per two-wheeled
Tangent line segment length；
Initial time sets circle of the tensioning synchronous pulley pitch curve to give radius, and round active synchronization belt wheel and ellipse are driven
Common tangent segment length T between two point of contact of synchronous pulley0, between oval driven synchronous pulley and two point of contact of tensioning synchronous pulley
Common tangent segment length T1, common tangent segment length T between two point of contact of round active synchronization belt wheel and tensioning synchronous pulley2Respectively：
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 round active synchronization belt wheel turns over angleOval driven synchronous pulley accordingly turns over angleRound active synchronization band
Take turns pitch curve incision superius C1、C6Corresponding arc length variable quantity is s1、s6, oval driven synchronous pulley pitch curve incision superius C2、C3It is right
The arc length variable quantity answered is s2、s3, tensioning synchronous pulley pitch curve incision superius C4、C5Corresponding arc length variable quantity is s4、s5；Then
Have：
In formula, p "1(θ1) it is p1(θ1) second-order differential, p "2(θ2) it is p2(θ2) second-order differential, p "3(θ3) it is p3(θ3) 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 T between round active synchronization belt wheel and oval driven two point of contact of synchronous pulley12, it is ellipse
Common tangent segment length T between two point of contact of the driven synchronous pulley of circle and tensioning synchronous pulley23, round active synchronization belt wheel and tensioning
Common tangent segment length T 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 round active synchronization belt wheel and oval driven synchronous pulley；
Round active synchronization belt wheel uniform rotation, according to formula (1), (3) solve p1, p2, then instantaneous transmission ratio be：
Step 5: calculating any time synchronous belt perimeter；
Round active synchronization belt wheel pitch curve is denoted as C with tensioning synchronous pulley pitch curve common tangent incision superius6, any time C1With
C6Between arc length be c11, oval driven synchronous pulley pitch curve and tensioning synchronous pulley pitch curve common tangent incision superius are denoted as C3,
Any time C2With C3Between arc length be c22, tensioning synchronous pulley pitch curve and driven wheel pitch curve common tangent incision superius are denoted as C4,
Tensioning synchronous pulley pitch curve is denoted as C with round active synchronization belt wheel pitch curve common tangent incision superius5, any time C4With C5Between
Arc length be c33；
Any time, synchronous belt Zhou Changwei：
C=T12+T13+T23+c11+c22+c33 (16)
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 (16) and is denoted as C0；
(b) round active synchronization belt wheel turns over 1 °, requires the oval driven synchronous pulley of calculating to turn over corresponding angle according to transmission ratio
Degree, the corner for being tensioned synchronous pulley are identical as round active synchronization belt wheel；It is anti-according to formula (16) under the premise of ensureing that C is constant
Round active synchronization belt wheel is asked to turn over corresponding tensioning synchronous pulley radius r at 1 °3-1, that is, correspond to the p at moment3；
(c) it repeats (b) 358 times, obtains round active synchronization belt wheel and turn over corresponding tensioning synchronous pulley at 2 °, 3 ° ..., 359 °
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), 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 perimeter of round active synchronization belt wheel and oval driven synchronous pulley be 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
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 and the perimeter of round active synchronization belt wheel and oval driven synchronous pulley of circle tensioning synchronous pulley are equal；
(j) it the not rounded tensioning synchronous pulley after (i) is substituted into formula (16) 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.
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