mathematical model of cam profile based on heald frame...

Research ArticleMathematical Model of Cam Profile Based on Heald FrameMotion Characteristics

Honghuan Yin 1 Hongbin Yu 1 Junqiang Peng 1 and Hongyu Shao 2

1School of Mechanical Engineering Tianjin Polytechnic University No 399 Binshuixi Road Xiqing District Tianjin 300387China2School of Mechanical Engineering Tianjin University No 92 Weijin Road Nankai District Tianjin 300072 China

Correspondence should be addressed to Hongbin Yu yuhongbintjpueducn

Received 12 October 2019 Revised 7 January 2020 Accepted 28 January 2020 Published 19 February 2020

Academic Editor Andras Szekrenyes

Copyright copy 2020HonghuanYin et al+is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In this paper the transmission process of the heald frame driven by the dobby is analyzed +e equivalent motion model of thedobby modulator the eccentric mechanism and the motion transmission unit are constructed +en based on the givenmovement characteristics of the heald frame the mathematical model is built to achieve the cam pitch curve and the cam profile ofthe modulator+e numerical solutionmethod for this is developed+e preparation of a mathematical model for the new conceptof the solving cam profile based on the motion characteristics of heald frames is explained in this study By setting a 11th

polynomial motion law of the heald frame due to the inconsistency between the outward and return motion laws of the crank-rocker mechanism an asymmetrical cam profile is obtained under the premise of ensuring that the heald framersquos ascending anddescending motions are consistent +rough the kinematics simulation analysis the correctness of the reverse process is verified

1 Introduction

At present the weaving technology is focused towardshigher speeds higher efficiency and easier control andoperation [1 2] Eren [3ndash6] introduced kinematic modelsfor the motion of rotary dobbies cranks and camshedding mechanisms and equations governing healdframe motion were derived In addition the obtainedheald frame motion curves were compared with eachother while the heald frame motion characteristics weremainly determined the design of the modulator mecha-nism Nowadays in modern high-speed weaving ma-chines rotary dobbies have been developed for theshedding operations Rotary dobbies convert the rota-tional motion of the main shaft of the weaving machineinto an up-down frame motion by means of various gearsarms and eccentrics (cams) [7 8]

In optimal cam profile design particularly focused onreducing vibration Wiederrich [9ndash11] has been the subjectof numerous investigations Stoddart [12ndash15] continuedDudleyrsquos work on polydyne cams for valve trains and

developed a characteristic relationship between vibrationamplitude and cam speed Mermelstein and Acar [16] usedpiecewise polynomials and the complete cam profile can bedesigned as a combined linear system Qiu et al [17]generated optimized motion curves using B-splines andsimultaneously handled many design objectives includingthe control of residual vibrations Mosier [18ndash21] startedwith a candidate acceleration function that was adjusted tosatisfy the necessary constraints In that method no can-didate profile was needed and all of the desired accelerationproperties were defined from the outset with no adjustmentsneeded to produce the final form +e result was bettercontrol over all important acceleration events during thecam cycle Once the acceleration was defined the velocitywas obtained by integrating the acceleration while enforcingboundary conditions and continuity between segments [22]

However no other publications found the research ofrelationship between heald framemovement and cam profilein the literature In particular the cam profile is calculatedbased on the motion characteristics of the heald frame +ispaper conducts a research study on the construction of the

HindawiMathematical Problems in EngineeringVolume 2020 Article ID 2106373 9 pageshttpsdoiorg10115520202106373

cam profile of the modulator through mathematical modelsbased on the heald frame motion characteristics

2 Motion Principle

+e rotary dobby and heald frame shedding controlmechanism consist of a modulator an eccentric mechanisma motion transmission unit (MTU) and a heald frame +emotion principle is presented in Figure 1 +e modulator isused to convert the continuous rotation of the loom mainshaft into an intermittent movement of the shaft +eschematic view of themodulator is shown in Figure 1(a)+econjugate cam (5 and 10) is fixed to the dobby body+e camfollower arms (2 and 8) that are used as an engagementelement are pivoted on gear 1 and can rotate around its pivotaxis O1 transmitting the motion of the loom to the camrollers (3 6 9 and 12) +e rotation of the follower rollersmakes links 4 and 7 to move and drive the main shaft 11+erefore the modulator transforms the uniform rotarymotion of the weaving machine into a nonuniform rotarymotion of the main shaft

Figure 1(b) illustrates the schematic views of the ec-centric mechanism MTU and heald frame+e link 13 is aneccentric disc cam as its center of rotation O1 is differentfrom its geometrical center D Due to this characteristicwhen the gear of the modulator rotates for exampleclockwise the motion of the main shaft is transmitted to theeccentric disc cam and its motion is transmitted to the liftingarm 15 by the ring link 14 +e motion of link 15 istransmitted to the heald frame by the MTU (Figure 1(b))+e MTU consists of a four-bar linkage mechanism(O2FGO3 MTU-I) and a crank slider mechanism (O3HIMTU-II) +e forwardmost and rearmost positions of thelifting arm 15 correspond to the lower and higher positionsof the heald frame 21 respectively Due to this constructionthe eccentric mechanism generates a heald frame motiononly for plain weaving In order to convert it to a rotarydobby mechanism it is necessary to include the necessarymeans to get the lifting arm 15 (hence the heald frame)dwelt at its forwardmost and rearmost positions as manyloom revolutions as required by the weave [3]

3 Mathematical Model of the Shedding Motion

31 MTU and Heald Frame Modeling +e schematic dia-gram ofMTU is shown in Figure 2 MTU-II is equivalent to aslider-crank linkage that changes at each position and thekinematic characteristics of the heald frame are transformedto those of the crank 14 and the link 13 +e link 11 is rigidlycoupled to the link 13 +e angle between them is φ9 aroundthe fixed point O3 +e positive angle between the link 13and the x axis is φ10

Stoddart [12] demonstrated how to develop double-dwell polynomials suitable for polydyne applications DefineL(u) S(u) + S0 as the kinematic characteristic function ofthe heald frame (15) where e is the offset of the slider 15 S0 isthe distance between the initial point I and the x axisl13 O3H and l14 HI e S0 l13 and l14 are known pa-rameters in the slider-crank mechanism (O3HI) S(u) is an

11th-degree polynomial [23] motion law and is described asfollows

S(u) h 336u5

minus 1890u6

minus 4740u7

minus 6615u8

+ 5320u9

1113872

minus 2310u10

+ 420u11

1113873

(1)

+e straight-line motion of the heald frame is transformedfrom the rotational motion of the link 13 +e displacementequation relating the parameters e l14 and l13 to the input andoutput variables L(u) and φ10 may be written as [24]

2l13 L(u)sinφ10 minus e cosφ101113858 1113859 + l214 minus l

213 + L

2(u) + e

21113960 1113961 0

(2)

φ10 arc sinl213 + L2(u) + e2 minus l214( 1113857

2l13L2(u) + e2

11139681113890 1113891 + arc tane

L(u)1113888 1113889

forφ10 isin minus 90deg 90deg( 1113857

(3)

Equation (3) takes a derivative of time and obtainsangular velocity of φ10

ω10 minus2L(u) middot [dL(u)du] + 2l15L(u)sinφ10

2l13 L(u)cos φ10 + e sinφ101113858 1113859 (4)

MTU-I is also equivalent to a four-bar linkage mecha-nism that changes its geometry at each position (Figure 2)Angle φ10 of the link 13 can be obtained from equations (1)to (4) Angle φ8 of the link 11 can be obtained by thegeometric relationship φ8 φ9 + φ10 where φ5 and φ9 arethe known parameters

In this study l12 represents the distance from the pointO3 to the pivot point O2 l11 represents the input link lengthof O3G l10 represents the link length of FG and l9 representsthe output link length of O2F If a Cartesian coordinatesystem (X Y) is centered on the pivot point O3 then angleφ6 can be expressed in terms of φ8

+e complex vector form of the closed vector equation isdefined as

l9eiφ6 + l10e

iφ7 l11eiφ8 + l12 (5)

+e Euler formula is applied

eiφ

cosφ + i sinφ (6)

+e real part is separated from the imaginary part ofequation (5) yielding

l9 cosφ6 + l10 cosφ7 l11 cosφ8 + l12

l9 sinφ6 + l10 sinφ7 l11 sinφ81113896 (7)

Angle φ7 is eliminated to get

l210 l

211 + l

212 + l

29 + 2l11l12 cosφ8 minus 2l9l12 cosφ6

minus 2l9l11 cos φ8 minus φ6( 1113857(8)

2 l12 l11 cosφ8 minus l9 cosφ6( 1113857 minus l11l9 φ8 minus φ6( 11138571113858 1113859

+ l212 + l

211 minus l

210 + l

29 0

(9)

2 Mathematical Problems in Engineering

1

234

5

6

7

8

9

10

11

12

A

B

C

O1

(a)

1314

15

16 17

18

19

20

21

D

EF

G

H

J

K

LI

O1 O2 O3O4

(b)

Figure 1 Heald frame motion principle (a) Modulator (1) gear (2 8) cam swing arms (3 6 9 12) cam rollers (4 7) modulatorlinks (5 10) conjugate cams and (11) main shaft (b) Eccentric mechanism MTU and heald frame (13) Eccentric disc cam (14) ringlink (15) lifting arm (16) lifting arm link (17) swivel arm (18) rotor link (19) heald frame link (20) support link and (21) healdframe

MTU-II MTU-I

F

E

GH

I

7

9

10

1112

13

14

15

φ4

φ7

φ10

φ9φ8

S 0

φ5 φ6

X

Y

e

O2O3

L(u)

S(u)

Figure 2 Schematic diagram of the MTU

Mathematical Problems in Engineering 3

After some manipulation and substitution the final result is

tanφ62

1113874 1113875 B minus

A2 + B2 minus C2

radic( 1113857

(A minus C) (10)

where A l12 + l11 cosφ8 B l11 sinφ8 and C (A2 +

B2 + l29 minus l210)2l9

φ6 2 arc tanB minus

A2 + B2 minus C2

radic( 1113857

(A minus C)1113890 1113891 forφ6 isin 0deg 90deg( 1113857

(11)

Equation (7) is differentiated by time and gives

l9ω6 sinφ6 + l10ω7 sinφ7 l11ω8 sinφ8

l9ω6 cosφ6 + l10ω7 cosφ7 l11ω8 cosφ81113896 (12)

Equation (12) is written as a matrix

l9 sinφ6 l10 sinφ7

l9 cosφ6 l10 cosφ71113890 1113891

ω6

ω71113890 1113891 ω8

l11 sinφ8

l11 cosφ81113890 1113891 (13)

From equation (13) the ω6 and ω7 angular velocityfunctions can be derived

ω6 ω8l11 sin φ7 minus φ8( 1113857

l9 sin φ7 minus φ6( 1113857 (14)

ω7 ω8l11 sin φ8 minus φ6( 1113857

l10 sin φ6 minus φ7( 1113857 (15)



l9 cosφ6 l10 cosφ71113890 1113891

α6α7

1113890 1113891

minusl9 cosφ6 l10 cosφ7

minus l9 sinφ6 minus l10 sinφ71113890 1113891

ω6

ω71113890 1113891

+ α8l11 sinφ8

l11 cosφ81113890 1113891 + ω8

l11 cosφ8

minus l11 sinφ81113890 1113891

(16)

+e α6 and α7 theoretical anger acceleration functions ofthe follower respectively are

α6 ω26l9 cos φ6 minus φ7( 1113857 + ω2

7l10 minus ω28l11 cos φ8 minus φ7( 1113857

l11 sin φ8 minus φ7( 1113857

(17)

α7 minus ω2

6l9 cos φ6 minus φ8( 1113857 minus ω27l10 cos φ7 minus φ8( 1113857 minus ω2

8l11

l11 sin φ7 minus φ8( 1113857

(18)

32 Eccentric Mechanism Modeling Figure 3 illustrates theeccentric mechanism which is equivalent to a crank-rockermechanism From equations (5) to (18) the φ6 angularmotion characteristic functions can be obtained φ4 isconsidered to affect the output of the φ2 According toFigure 3 φ4 φ6 + φ5 where φ5 is a known parameter

Moreover l8 represents the distance from the pivot pointO2 to the pivot point O1 l7 represents the lifting arm lengthof O2E l6 represents the link length of E D and l5 representsthe output link length of O1D During one revolution of thelink 5 the lifting arm 7 swings between its limit positionsand the swing angle isempty When points O1 D0 and E are onthe same line and O1D0 and D0E are extended the liftingarm 7 reaches its forwardmost position When O1D0 andD0E are folded on top of each other the link 7 reaches itsrearmost position +e eccentric mechanism has no quick-return characteristics

If a Cartesian coordinate system (X Y) is centered onthe pivot point O2 the output angle φ2 can be expressed interms of φ4



+en

l26 l

27 + l

28 + l



+en

2 l8 l7 cosφ4 minus l5 cosφ2( 1113857 minus l7l5 cos φ4 minus φ2( 11138571113858 1113859

+ l28 + l

27 minus l

26 + l

25 0

(21)

After some manipulation and substitution the finalresult is

tanφ2

21113874 1113875

B plusmnA2 + B2 minus C2

radic( 1113857

(A minus C) (22)

where A l8 + l7 cosφ4 B l7 sinφ4 and C A2 + B2 +

l25 minus l262l5

φ2 2 arc tanB plusmn

A2 + B2 minus C2

radic( 1113857

(A minus C)1113890 1113891

forφ2 isin minus 180deg 180deg1113858 1113857

(23)





l5 sinφ2 l6 sinφ3

l5 cosφ2 l6 cosφ31113890 1113891

ω2

ω31113890 1113891 ω4

l7 sinφ4

l7 cosφ41113890 1113891 (25)


ω2 ω4l7 sin φ3 minus φ4( 1113857

l5 sin φ3 minus φ2( 1113857 (26)

ω3 minusω4l7 sin φ4 minus φ2( 1113857

l6 sin φ2 minus φ3( 1113857 (27)



l5 sinφ2 l6 sinφ3

l5 cosφ2 l6 cosφ31113890 1113891

α2α3

1113890 1113891 α4l7 sinφ4

l7 cosφ41113890 1113891

+ ω4l7 cosφ4

minus l7 sinφ41113890 1113891 minus

l5 cosφ2 l6 cosφ3


ω2

ω31113890 1113891

(28)

+e α2 and α3 angular acceleration functions are

α2 ω22l5 cos φ2 minus φ3( 1113857 + ω2


l7 sin φ4 minus φ3( 1113857 (29)

α3 minus ω2


4l7

l7 sin φ3 minus φ4( 1113857 (30)

33 Modulator Modeling Figure 4 shows a schematic dia-gram of the cam-link modulator Angle φ2 of the link O1D

can be obtained from equations (19) to (30) Angle φ2 isequal to angle φ1 φ1 φ2 where φ1 is the main shaft angle ofthe modulator

In Figure 4(a) ω is the angular velocity of the gear +estarting position of the cam swing arm coincides with thehinge point C of the gear A Cartesian coordinate system(X Y) is centered on the cam point O1+e rotation angle ofthe gear in the Cartesian coordinate system is defined as θand it is expressed by θ ωt (Figure 4(b))

From the geometric relationship

xA minus xB( 11138572

+ yA minus yB( 11138572

l22

xA minus xC( 11138572

+ yA minus yC( 11138572

l21

⎧⎨

⎩ (31)

+en

xA minus B1 plusmn

B21 minus 4A1C1

1113969

1113874 1113875

2A1

yA (C minus Ax)

B

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(32)

where A xC minus xB B yC minus yB C (l4 minus l3 minus x2B minus

y2B + x2

C + y2C)2 A1 A2 + B2 B1 2AByB minus 2AC minus 2B2xB

C1 minus 2BCyB + C2 minus B2(l24 minus y2B minus x2

B) xB l3 cosφ1 yB

l3 sinφ1 xc l4 cos θ and yc l4 sin θ+e distance between the cam actual profile and the cam

pitch curve is equal to the roller radius r +e cam profile

along a direction normal to any point A is taken as a distancer in order to obtain the coordinates of the correspondingpoint T(xT yT) on the actual cam profile

+e slope of the normal line n minus n at the cam pitch curveA is

tan c dxA

dyA

minusdxAdθ( 1113857

dyAdθ( 1113857sin c

cos c (33)

where c is the angle between the normal line n minus n and the x

axis and (xA yA) are the coordinates of any point A on thecam pitch curve

Equation (33) can be written as

sin c dxAdθ( 1113857

dxAdθ( 11138572

+ dyAdθ( 11138572

1113969

cos c minusdyAdθ( 1113857

dxAdθ( 11138572

+ dyAdθ( 11138572

1113969⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(34)

According to equations (31) to (34) the coordinates ofthe point T(xT yT) of the actual cam profile can be workedout as follows

xT xA plusmn r cos c

yT yA plusmn r cos c1113896 (35)

From equations (31) to (35) the analytical solution of thecam profile equation can be determined computationally

4 Results and Discussion

41 Calculation A mathematical model was used to as-certain the relationship between heald frame motioncharacteristics and cam profile In the calculation processthe program was written in Visual Studio 2012 and ran on apersonal computer

Figure 5 shows the kinematic curves of the heald framefor displacement velocity acceleration and jerk [20] for the11th-degree polynomial functions +ese polynomial func-tions are taken as the motion characteristics (Displac-11thVelocity-11th Accel-11th and Jerk-11th) of the heald framein combination with equation (1) +e program sets180deg2000 as the rotation step of the gear and calculates from0deg to 180deg

According to the actual size of the test bench as shown inFigure 6 the parameter h is 1068mm the length of l14 is375mm e is 200mm the fixed angle φ9 is 9765deg the lengthof l13 is 200mm and the distance S0 is 325mm

+is program employed equations (1) to (4) and used theabove parameter values as input data and the φ10 motioncharacteristics (Angular displac-φ10 Angular velocity-φ10Angular accel-φ10 and Jerk-φ10) can be obtained and arepresented in Figure 7 normalized with respect to the cal-culated maximum angular displacement value of 31deg Resultsare shown for the half gear cycle only after which a steadystate is achieved

Based on the test bench the length of link l11 is 150mmthe length of l12 is 695mm the length of l10 is 550mm the

D

E

F

X

Y

Φ5

67

8D0

D1

E0 F0

F1

E1

O1 O2

φ2

φ3 φ4

φ6

φ5

Figure 3 Schematic diagram of the eccentric mechanism


fixed angle φ5 is 100deg and the length of link l9 is 185mmAccording to equations (5) to (18) and the above parametervalues the φ6 motion characteristics (Angular displac-φ6Angular velocity-φ6 Angular accel-φ6 and Jerk-φ6) can beobtained and the results are presented in Figure 8 nor-malized with respect to the maximum angular displacementvalue of 364deg

+e length of link l5 is 30mm the length of l6 is 170mmand the length of link l7 is 96mm In order to ensure noquick-return motion of eccentric mechanism the length of

l8 is calculated to be 19294mm According to equations (19)to (30) and the above parameter values when D0 movescounterclockwise and clockwise to D1 the φ2 motioncharacteristics (Angular displac-φ2 minus 1 Angular velocity-φ2 minus 1 Angular accel-φ2 minus 1 and Angular jerk-φ2 minus 1) and(Angular displac-φ2 minus 2 Angular velocity-φ2 minus 2 Angularaccel-φ2 minus 2 and Angular jerk-φ2 minus 2) are obtained re-spectively which are presented in Figure 9 normalized withrespect to the maximum angular displacement value of 180deg

+e length of l3 is 425mm the length of l4 is 111mm thelength of l2 is 81mm and the length of l1 is 56mm In Figure 4the rotation angle θ of the gear is ωt and the step is 180deg2000θ isin [0

deg 180deg) According to equations (31) to (35) and the

above parameter values the 2000 points are obtained throughthe program +e final cam profiles 1 and 2 and cam pitchcurves 1 and 2 fitted through nonuniform rational B-splines(NURBS) can be established +ey are presented in Figure 10

42 Simulation In order to verify the correctness of theproposed mathematical model a virtual prototype of therotary dobby and the heald frame shedding control

ω

A

B

C12

3 4

O1

(a)

A

B

C

Y

X

1

2

3

4T

n

nγ θ

ωrRoller

Cam profile

Cam pitchcurve

O1φ1

(b)

Figure 4 Cam-link modulator

0 30 60 90 120 150 180ndash12ndash10ndash08ndash06ndash04ndash02

0002040608101214

ndash24ndash20ndash16ndash12ndash08ndash04

0004081216202428


Dim

ensio

nles

s acc

el


Dim

ensio

nles

s jer

k

Dim

ensio

nles

s velo

city

Dim

ensio

nles

s disp

lac

Gear angle θ (deg)

Displac-11th Velocity-11th

Jerk-11thAccel-11th

Figure 5 11th-degree polynomial curves

Figure 6 +e test bench


Dim

ensio

nles

s acc

el

Dim

ensio

nles

s jer

k



0004081216202428

Dim

ensio

nles

s ang

ular

vel

ocity


0002040608101214

Dim

ensio

nles

s ang

ular

disp

lac

0 30 60 90 120 150 180Gear angle θ (deg)

Angular displac-φ10

Angular velocity-φ10Angular jerk-φ10

Angular accel-φ10

Figure 7 Motion characteristics of angular φ10


mechanism is developed using SolidWorks 2016 +enoncentral symmetrical cam defined by the cam profilecurve 1 and 2 is employed on the modulator +e motionsimulation is performed on the heald frame driven by therotary dobby with this modulator +e obtained simulationcharacteristics (Displac-SHF Velocity-SHF Accel-SHF andJerk-SHF) of the heald frame are compared with the

characteristics (Displac-11th Velocity-11th Accel-11th andJerk-11th) (Figure 11)

As it can be seen in Figure 12 the period of the gear is180deg and the modulator shedding motion characteristics(Displac-SHF Velocity-SHF Accel-SHF and Jerk-SHF) ofthe heald frame are very close to the 11th polynomial curveswhich proves the correctness of the proposed mathematicalmodel At the same time the main shaft of the rotary dobbyis rotated for one cycle and the same heald motion char-acteristics as motion characteristics (Displac-SHF Velocity-SHF Accel-SHF and Jerk-SHF) can be formed meaningthat regardless the forward or reverse rotation of the gearthe heald frame rise laws are consistent with the fall laws

+e cam profile 3 and 4 are formed by copying androtating the cam profile 1 and 2 by 180deg and combining them(Figure 12) whereas the cam profile 3 is shown as the dashdot line and the cam profile 4 is shown as the solid line +ecam pitch curve 3 and 4 are shown as the dash double-dotline and the dashed line respectively

+e two cams formed by cam profile 3 and 4 areemployed in the modulator +e two cams are mounted onthe rotary dobby transmission mechanism in order to obtainthe movement law of the heald frame Motion characteristics


Dim

ensio

nles

s ang

ular

acce

l


Dim

ensio

nles

s jer

k


0004081216202428

Dim

ensio

nles

s ang

ular

velo

city


0002040608101214

Dim

ensio

nles

s ang

ular

disp

lac

0 30 60 90 120 150 180Gear angle θ (deg)

Angular displac-φ6


Angular accel-φ6



Dim

ensio

nles

s ang

ular

acce

l


Dim

ensio

nles

s jer

k


0004081216202428

Dim

ensio

nles

s ang

ular

vel

ocity


0002040608101214

Dim

ensio

nles

s ang

ular

disp

lac

Angular displac-φ2-1Angular displac-φ2-2Angular accel-φ2-1

Angular velocity-φ2-1Angular velocity-φ2-2Angular jerk-φ2-1Angular jerk-φ2-2

Angular accel-φ2-2

0 30 60 90 120 150 180Gear angle θ (deg)


A

C

T

B

ω

Tprime

Aprime

1

2

3

Roller

Cam profile 1Cam profile 2

Cam pitchcurve 2

Cam pitchcurve 1

O3

Figure 10 Cam profile and cam pitch curve 1 and 2


Dim

ensio

nles

s acc

el


Dim

ensio

nles

s jer

k


0004081216202428

Dim

ensio

nles

s velo

city


0002040608101214

Dim

ensio

nles

s disp

lac

Gear angle θ (deg)

Displac-11th

Displac-SHFVelocity-11th

Velocity-SHFJerk-11th

Jerk-SHFAccel-11th

Accel-SHF

Figure 11 11th-degree polynomial curves and simulation char-acteristics of the heald frame


(Displac-CP3 Velocity-CP3 Accel-CP3 and Jerk-CP3Displac-CP4-1 Velocity-CP4-1 Accel-CP4-1 and Jerk-CP4-1) are obtained from the simulation of the heald framemotion driven by the rotary dobby with the aforementionedmodulators θ isin [0

deg 360deg) +e results are compared with

each other and presented in Figure 13It can be seen that the motion characteristics (Displac-

CP3 Velocity-CP3 Accel-CP3 and Jerk-CP3 Displac-CP4-1 Velocity-CP4-1 Accel-CP4-1 and Jerk-CP4-1) ofthe heald frame formed by cam profiles 3 and 4 respec-tively are very different Figure 14 shows the deviationcurves +e motion of the heald frame by the modulatorwith cam profile 3 or 4 resulted in different characteristicsin the upward and downward sections More specificallythe forward and reverse rotation of the dobby modulatormain shaft resulted in different heald frame motioncharacteristics

However in Figure 15 the heald frame motion char-acteristics (Displac-CP3 Velocity-CP3 Accel-CP3 andJerk-CP3) generated by the dobby modulator when it rotatesforward with cam profile 3 and themovement characteristics(Displac-CP4-2 Velocity-CP4-2 Accel-CP4-2 and Jerk-CP4-2) generated by the dobby modulator when it rotatesreversely with cam profile 4 are compared As it can beobserved the values of motion characteristics (Displac-CP3Velocity-CP3 Accel-CP3 and Jerk-CP3 Displac-CP4-2Velocity-CP4-2 Accel-CP4-2 and Jerk-CP4-2) have thesame variation trend +is implies that modulators with

different cam profiles can produce exactly the same healdframe motion characteristics

+e above findings form the foundation for the mod-ulator design and further theoretical analysis of the rotarydobby structure

5 Conclusions

When the geometric dimensions of the modulator ec-centric mechanism and MTU as well as the motion curveof the heald frame are given the analytical mathematicalmodel of the cam profile and the cam pitch curve can bedetermined In this mathematical model it is possible toascertain the relationship between heald frame motioncharacteristics and cam profile By means of this model2000 points of cam profile is obtained and the error of thismodel can be reduced by increasing the number of cal-culation points +e model proposed in this article enablesthe analysis of the heald frame motion characteristics andthe cam profile design Given different heald frame dis-placement curves and parameter values the cam profileand motion characteristics of each motion transfer pro-cess can be obtained based on the proposed mathematicalmodel

A

CT

B

ω1

2

3

Cam profile 4

Cam profile 3Cam pitchcurve 4

Cam pitchcurve 3

Roller

O3



Dim

ensio

nles

s acc

el


Dim

ensio

nles

s jer

k


0004081216202428

Dim

ensio

nles

s velo

city


0002040608101214

Dim

ensio

nles

s disp

lac

0 60 120 180 240 300 360Gear angle θ (deg)

Displac-CP3Displac-CP4-1

Velocity-CP3Velocity-CP4-1Jerk-CP3Jerk-CP4-1

Accel-CP3Accel-CP4-1

Figure 13 +e motion characteristics of the heald frame with thesame rotation direction

Dim

ensio

nles

s jer

k


000002004006008010012

ndash08ndash06ndash04ndash02

0002040608


Dim

ensio

nale

ss ac

cel


Dim

ensio

nles

s velo

city

Dim

ensio

nles

s disp

lac

3600 60 120 180 240 300Gear angle θ (deg)

Displac-D-CP3ampCP4-1Accel-D-CP3ampCP4-1Velocity-D-CP3ampCP4-1Jerk-D-CP3ampCP4-1

Figure 14 +e deviations of motion characteristics


Dim

ensio

nles

s acc

el


Dim

ensio

nles

s jer

k


0004081216202428

Dim

ensio

nles

s velo

city


0002040608101214

Dim

ensio

nles

s disp

lac

3600 60 120 180 240 300Gear angle θ (deg)




Figure 15 +e motion characteristics of the heald frame withdifferent rotation directions


Based on the particularity of the rotary dobby structurea cam profile is obtained and the motion law of the healdframe is solved in the forward direction to verify the cor-rectness of the inverse model At the same time regardless ofwhether the dobby is rotating forward or backward themotion of the heald frame is the same while the cam isasymmetrical

Two cam profiles are obtained the heald frame motioncharacteristics are solved in the forward direction and theasymmetric motion characteristics of the heald frame areobtained +e asymmetric deviation revealed the cam pro-files the eccentric mechanism and the motion transmissionmechanism When the roller moves clockwise along one ofthe cam profiles and counterclockwise along the other camprofile the exact same heald frame motion characteristicsare produced

A good correlation is found between the simulation andcalculation results of the heald frame displacement velocityacceleration and jerk Future work will include the devel-opment of a virtual prototype which will verify the math-ematical model and may give a wealth of dynamicinformation about the system as well as similar systems yetto be built

Data Availability

All datarsquos used to support the findings of this study areincluded within the article

Disclosure

Honghuan Yin and Hongbin Yu are co-first authors

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+is research was supported by the National Key RampDProgram of China (2017YFB1104202)

References

[1] Anon ldquoRotary dobby-the efficient shed forming mechanismfor modern weaving machinesrdquo International Textile BulletinFabric Forming vol 66 pp 45-46 1983

[2] R Marks and A T C Robinson Principles of Weaving +eTextile Institute London UK 1976

[3] R Eren G O enA and Y Turhan ldquoKinematics of rotarydobby and analysis of heald frame motion in weaving pro-cessrdquo Textile Research Journal vol 78 no 12 pp 1070ndash10792008

[4] Staubli Dobby machines catalog 2006[5] P A Korolev and V N Lohmanov ldquoKinematics of con-

nections of the shedding mechanism of a circular loom TKP-110-Urdquo Lzvestiya Vysshikh Uchebnykh Zavedenii SeriyaTeknologiya vol 4 pp 116ndash119 2011

[6] K Sadettin D M Taylan and K Ali ldquoCam motion tuning ofshedding mechanism for vibration reduction of heald framerdquo

Gazi University Journal of Science vol 23 no 2 pp 227ndash2322010

[7] G Abdulla and O Can ldquoDesign of a new rotary dobbymechanismrdquo Industria Textile vol 69 no 6 pp 429ndash4332018

[8] D Mundo G A Danieli and H S Yan ldquoKinematic opti-mization of mechanical presses by optimal synthesis of cam-integrated linkagesrdquo Transactions of the Canadian Society forMechanical Engineering vol 30 no 4 pp 519ndash532 2006

[9] J L Wiederrich Design of cam profiles for systems with highinertial loading PhD thesis Stanford University StanfordCA USA 1973

[10] M Chew and C H Chuang ldquoMinimizing residual vibrationsin high-speed cam-follower systems over a range of speedsrdquoJournal of Mechanical Design vol 117 no 1 pp 166ndash1721995

[11] K C Gupta and J L Wiederrich ldquoDevelopment of camprofiles using the convolution operatorrdquo Journal of Mecha-nisms Transmissions and Automation in Design vol 105no 4 pp 654ndash657 1983

[12] D A Stoddart ldquoPolydyne cam design-Irdquo Machine Designvol 1 pp 121ndash135 1953

[13] D A Stoddart ldquoPolydyne cam design-IIrdquo Machine Designvol 2 pp 146ndash154 1953

[14] D A Stoddart ldquoPolydyne cam design-IIIrdquo Machine Designvol 3 pp 149ndash164 1953

[15] J E Shigley and J J Uicker Aeory of Machines andMechanisms McGraw-Hill New York NY USA 1980

[16] S P Mermelstein and M Acar ldquoOptimising cam motionusing piecewise polynomialsrdquo Engineering with Computersvol 19 no 4 pp 241ndash254 2004

[17] H Qiu C-J Lin Z-Y Li H Ozaki J Wang and Y Yue ldquoAuniversal optimal approach to cam curve design and its ap-plicationsrdquo Mechanism and Machine Aeory vol 40 no 6pp 669ndash692 2005

[18] R G Mosier ldquoModern cam designrdquo International Journal ofVehicle Design vol 23 no 12 pp 38ndash55 2000

[19] R L Norton Machine Design An Integrated Approachpp 111ndash116 Prentice-Hall Upper Saddle River NJ USA2000

[20] R L Norton Cam Design and Manufacturing HandbookIndustrial Press New York NY USA 2009

[21] A Gabil and H Baris ldquoSynthesis work about drivingmechanism of a novel rotary dobby mechanismrdquo Tekstil VeKonfeksiyon vol 3 pp 218ndash224 2010

[22] F W Flocker ldquoA versatile cam profile for controlling in-terface force in multiple-dwell cam-follower systemsrdquo Journalof Mechanical Design vol 134 no 9 pp 1ndash6 2012

[23] E E Peisekah ldquoImproving the polydyne cam design methodrdquoRussian Engineering Journal vol 12 pp 25ndash27 1966

[24] M P Koster ldquoVibrations of cam mechanismsrdquo PhillipsTechnical Library Series Macmillan Press London UK 1974


cam profile of the modulator through mathematical modelsbased on the heald frame motion characteristics

2 Motion Principle

+e rotary dobby and heald frame shedding controlmechanism consist of a modulator an eccentric mechanisma motion transmission unit (MTU) and a heald frame +emotion principle is presented in Figure 1 +e modulator isused to convert the continuous rotation of the loom mainshaft into an intermittent movement of the shaft +eschematic view of themodulator is shown in Figure 1(a)+econjugate cam (5 and 10) is fixed to the dobby body+e camfollower arms (2 and 8) that are used as an engagementelement are pivoted on gear 1 and can rotate around its pivotaxis O1 transmitting the motion of the loom to the camrollers (3 6 9 and 12) +e rotation of the follower rollersmakes links 4 and 7 to move and drive the main shaft 11+erefore the modulator transforms the uniform rotarymotion of the weaving machine into a nonuniform rotarymotion of the main shaft

Figure 1(b) illustrates the schematic views of the ec-centric mechanism MTU and heald frame+e link 13 is aneccentric disc cam as its center of rotation O1 is differentfrom its geometrical center D Due to this characteristicwhen the gear of the modulator rotates for exampleclockwise the motion of the main shaft is transmitted to theeccentric disc cam and its motion is transmitted to the liftingarm 15 by the ring link 14 +e motion of link 15 istransmitted to the heald frame by the MTU (Figure 1(b))+e MTU consists of a four-bar linkage mechanism(O2FGO3 MTU-I) and a crank slider mechanism (O3HIMTU-II) +e forwardmost and rearmost positions of thelifting arm 15 correspond to the lower and higher positionsof the heald frame 21 respectively Due to this constructionthe eccentric mechanism generates a heald frame motiononly for plain weaving In order to convert it to a rotarydobby mechanism it is necessary to include the necessarymeans to get the lifting arm 15 (hence the heald frame)dwelt at its forwardmost and rearmost positions as manyloom revolutions as required by the weave [3]

3 Mathematical Model of the Shedding Motion

31 MTU and Heald Frame Modeling +e schematic dia-gram ofMTU is shown in Figure 2 MTU-II is equivalent to aslider-crank linkage that changes at each position and thekinematic characteristics of the heald frame are transformedto those of the crank 14 and the link 13 +e link 11 is rigidlycoupled to the link 13 +e angle between them is φ9 aroundthe fixed point O3 +e positive angle between the link 13and the x axis is φ10

Stoddart [12] demonstrated how to develop double-dwell polynomials suitable for polydyne applications DefineL(u) S(u) + S0 as the kinematic characteristic function ofthe heald frame (15) where e is the offset of the slider 15 S0 isthe distance between the initial point I and the x axisl13 O3H and l14 HI e S0 l13 and l14 are known pa-rameters in the slider-crank mechanism (O3HI) S(u) is an

11th-degree polynomial [23] motion law and is described asfollows

S(u) h 336u5

minus 1890u6

minus 4740u7

minus 6615u8

+ 5320u9

1113872

minus 2310u10

+ 420u11

1113873

(1)

+e straight-line motion of the heald frame is transformedfrom the rotational motion of the link 13 +e displacementequation relating the parameters e l14 and l13 to the input andoutput variables L(u) and φ10 may be written as [24]

2l13 L(u)sinφ10 minus e cosφ101113858 1113859 + l214 minus l

213 + L

2(u) + e

21113960 1113961 0

(2)

φ10 arc sinl213 + L2(u) + e2 minus l214( 1113857

2l13L2(u) + e2

11139681113890 1113891 + arc tane

L(u)1113888 1113889

forφ10 isin minus 90deg 90deg( 1113857

(3)

Equation (3) takes a derivative of time and obtainsangular velocity of φ10

ω10 minus2L(u) middot [dL(u)du] + 2l15L(u)sinφ10

2l13 L(u)cos φ10 + e sinφ101113858 1113859 (4)

MTU-I is also equivalent to a four-bar linkage mecha-nism that changes its geometry at each position (Figure 2)Angle φ10 of the link 13 can be obtained from equations (1)to (4) Angle φ8 of the link 11 can be obtained by thegeometric relationship φ8 φ9 + φ10 where φ5 and φ9 arethe known parameters

In this study l12 represents the distance from the pointO3 to the pivot point O2 l11 represents the input link lengthof O3G l10 represents the link length of FG and l9 representsthe output link length of O2F If a Cartesian coordinatesystem (X Y) is centered on the pivot point O3 then angleφ6 can be expressed in terms of φ8

+e complex vector form of the closed vector equation isdefined as

l9eiφ6 + l10e

iφ7 l11eiφ8 + l12 (5)

+e Euler formula is applied

eiφ

cosφ + i sinφ (6)

+e real part is separated from the imaginary part ofequation (5) yielding



Angle φ7 is eliminated to get

l210 l

211 + l

212 + l



2 l12 l11 cosφ8 minus l9 cosφ6( 1113857 minus l11l9 φ8 minus φ6( 11138571113858 1113859

+ l212 + l

211 minus l

210 + l

29 0

(9)


1

234

5

6

7

8

9

10

11

12

A

B

C

O1

(a)

1314

15

16 17

18

19

20

21

D

EF

G

H

J

K

LI

O1 O2 O3O4

(b)


MTU-II MTU-I

F

E

GH

I

7

9

10

1112

13

14

15

φ4

φ7

φ10

φ9φ8

S 0

φ5 φ6

X

Y

e

O2O3

L(u)

S(u)




tanφ62

1113874 1113875 B minus

A2 + B2 minus C2

radic( 1113857

(A minus C) (10)


B2 + l29 minus l210)2l9


A2 + B2 minus C2

radic( 1113857


(11)






l9 cosφ6 l10 cosφ71113890 1113891

ω6

ω71113890 1113891 ω8

l11 sinφ8

l11 cosφ81113890 1113891 (13)


ω6 ω8l11 sin φ7 minus φ8( 1113857

l9 sin φ7 minus φ6( 1113857 (14)

ω7 ω8l11 sin φ8 minus φ6( 1113857

l10 sin φ6 minus φ7( 1113857 (15)



l9 cosφ6 l10 cosφ71113890 1113891

α6α7

1113890 1113891



ω6

ω71113890 1113891

+ α8l11 sinφ8

l11 cosφ81113890 1113891 + ω8

l11 cosφ8

minus l11 sinφ81113890 1113891

(16)


α6 ω26l9 cos φ6 minus φ7( 1113857 + ω2


l11 sin φ8 minus φ7( 1113857

(17)

α7 minus ω2


8l11

l11 sin φ7 minus φ8( 1113857

(18)






+en

l26 l

27 + l

28 + l



+en


+ l28 + l

27 minus l

26 + l

25 0

(21)


tanφ2

21113874 1113875


radic( 1113857

(A minus C) (22)


l25 minus l262l5


A2 + B2 minus C2

radic( 1113857

(A minus C)1113890 1113891


(23)





l5 sinφ2 l6 sinφ3

l5 cosφ2 l6 cosφ31113890 1113891

ω2

ω31113890 1113891 ω4

l7 sinφ4

l7 cosφ41113890 1113891 (25)



l5 sin φ3 minus φ2( 1113857 (26)


l6 sin φ2 minus φ3( 1113857 (27)



l5 sinφ2 l6 sinφ3

l5 cosφ2 l6 cosφ31113890 1113891

α2α3

1113890 1113891 α4l7 sinφ4

l7 cosφ41113890 1113891

+ ω4l7 cosφ4


l5 cosφ2 l6 cosφ3


ω2

ω31113890 1113891

(28)


α2 ω22l5 cos φ2 minus φ3( 1113857 + ω2


l7 sin φ4 minus φ3( 1113857 (29)

α3 minus ω2


4l7

l7 sin φ3 minus φ4( 1113857 (30)






+ yA minus yB( 11138572

l22


+ yA minus yC( 11138572

l21

⎧⎨

⎩ (31)

+en

xA minus B1 plusmn

B21 minus 4A1C1

1113969

1113874 1113875

2A1

yA (C minus Ax)

B

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(32)


y2B + x2



B) xB l3 cosφ1 yB





tan c dxA

dyA



cos c (33)





dxAdθ( 11138572

+ dyAdθ( 11138572

1113969


dxAdθ( 11138572

+ dyAdθ( 11138572

1113969⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(34)











D

E

F

X

Y

Φ5

67

8D0

D1

E0 F0

F1

E1

O1 O2

φ2

φ3 φ4

φ6

φ5










ω

A

B

C12

3 4

O1

(a)

A

B

C

Y

X

1

2

3

4T

n

nγ θ

ωrRoller

Cam profile

Cam pitchcurve

O1φ1

(b)



0002040608101214


0004081216202428


Dim

ensio

nles

s acc

el


Dim

ensio

nles

s jer

k

Dim

ensio

nles

s velo

city

Dim

ensio

nles

s disp

lac

Gear angle θ (deg)


Jerk-11thAccel-11th




Dim

ensio

nles

s acc

el

Dim

ensio

nles

s jer

k



0004081216202428

Dim

ensio

nles

s ang

ular

vel

ocity


0002040608101214

Dim

ensio

nles

s ang

ular

disp

lac

0 30 60 90 120 150 180Gear angle θ (deg)



Angular accel-φ10









Dim

ensio

nles

s ang

ular

acce

l


Dim

ensio

nles

s jer

k


0004081216202428

Dim

ensio

nles

s ang

ular

velo

city


0002040608101214

Dim

ensio

nles

s ang

ular

disp

lac

0 30 60 90 120 150 180Gear angle θ (deg)

Angular displac-φ6


Angular accel-φ6



Dim

ensio

nles

s ang

ular

acce

l


Dim

ensio

nles

s jer

k


0004081216202428

Dim

ensio

nles

s ang

ular

vel

ocity


0002040608101214

Dim

ensio

nles

s ang

ular

disp

lac



Angular accel-φ2-2

0 30 60 90 120 150 180Gear angle θ (deg)


A

C

T

B

ω

Tprime

Aprime

1

2

3

Roller


Cam pitchcurve 2

Cam pitchcurve 1

O3



Dim

ensio

nles

s acc

el


Dim

ensio

nles

s jer

k


0004081216202428

Dim

ensio

nles

s velo

city


0002040608101214

Dim

ensio

nles

s disp

lac

Gear angle θ (deg)

Displac-11th



Jerk-SHFAccel-11th

Accel-SHF










5 Conclusions


A

CT

B

ω1

2

3

Cam profile 4


Cam pitchcurve 3

Roller

O3



Dim

ensio

nles

s acc

el


Dim

ensio

nles

s jer

k


0004081216202428

Dim

ensio

nles

s velo

city


0002040608101214

Dim

ensio

nles

s disp

lac

0 60 120 180 240 300 360Gear angle θ (deg)





Dim

ensio

nles

s jer

k


000002004006008010012


0002040608


Dim

ensio

nale

ss ac

cel


Dim

ensio

nles

s velo

city

Dim

ensio

nles

s disp

lac

3600 60 120 180 240 300Gear angle θ (deg)




Dim

ensio

nles

s acc

el


Dim

ensio

nles

s jer

k


0004081216202428

Dim

ensio

nles

s velo

city


0002040608101214

Dim

ensio

nles

s disp

lac

3600 60 120 180 240 300Gear angle θ (deg)









Data Availability


Disclosure




Acknowledgments


References



























1

234

5

6

7

8

9

10

11

12

A

B

C

O1

(a)

1314

15

16 17

18

19

20

21

D

EF

G

H

J

K

LI

O1 O2 O3O4

(b)


MTU-II MTU-I

F

E

GH

I

7

9

10

1112

13

14

15

φ4

φ7

φ10

φ9φ8

S 0

φ5 φ6

X

Y

e

O2O3

L(u)

S(u)




tanφ62

1113874 1113875 B minus

A2 + B2 minus C2

radic( 1113857

(A minus C) (10)


B2 + l29 minus l210)2l9


A2 + B2 minus C2

radic( 1113857


(11)






l9 cosφ6 l10 cosφ71113890 1113891

ω6

ω71113890 1113891 ω8

l11 sinφ8

l11 cosφ81113890 1113891 (13)


ω6 ω8l11 sin φ7 minus φ8( 1113857

l9 sin φ7 minus φ6( 1113857 (14)

ω7 ω8l11 sin φ8 minus φ6( 1113857

l10 sin φ6 minus φ7( 1113857 (15)



l9 cosφ6 l10 cosφ71113890 1113891

α6α7

1113890 1113891



ω6

ω71113890 1113891

+ α8l11 sinφ8

l11 cosφ81113890 1113891 + ω8

l11 cosφ8

minus l11 sinφ81113890 1113891

(16)


α6 ω26l9 cos φ6 minus φ7( 1113857 + ω2


l11 sin φ8 minus φ7( 1113857

(17)

α7 minus ω2


8l11

l11 sin φ7 minus φ8( 1113857

(18)






+en

l26 l

27 + l

28 + l



+en


+ l28 + l

27 minus l

26 + l

25 0

(21)


tanφ2

21113874 1113875


radic( 1113857

(A minus C) (22)


l25 minus l262l5


A2 + B2 minus C2

radic( 1113857

(A minus C)1113890 1113891


(23)





l5 sinφ2 l6 sinφ3

l5 cosφ2 l6 cosφ31113890 1113891

ω2

ω31113890 1113891 ω4

l7 sinφ4

l7 cosφ41113890 1113891 (25)



l5 sin φ3 minus φ2( 1113857 (26)


l6 sin φ2 minus φ3( 1113857 (27)



l5 sinφ2 l6 sinφ3

l5 cosφ2 l6 cosφ31113890 1113891

α2α3

1113890 1113891 α4l7 sinφ4

l7 cosφ41113890 1113891

+ ω4l7 cosφ4


l5 cosφ2 l6 cosφ3


ω2

ω31113890 1113891

(28)


α2 ω22l5 cos φ2 minus φ3( 1113857 + ω2


l7 sin φ4 minus φ3( 1113857 (29)

α3 minus ω2


4l7

l7 sin φ3 minus φ4( 1113857 (30)






+ yA minus yB( 11138572

l22


+ yA minus yC( 11138572

l21

⎧⎨

⎩ (31)

+en

xA minus B1 plusmn

B21 minus 4A1C1

1113969

1113874 1113875

2A1

yA (C minus Ax)

B

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(32)


y2B + x2



B) xB l3 cosφ1 yB





tan c dxA

dyA



cos c (33)





dxAdθ( 11138572

+ dyAdθ( 11138572

1113969


dxAdθ( 11138572

+ dyAdθ( 11138572

1113969⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(34)











D

E

F

X

Y

Φ5

67

8D0

D1

E0 F0

F1

E1

O1 O2

φ2

φ3 φ4

φ6

φ5










ω

A

B

C12

3 4

O1

(a)

A

B

C

Y

X

1

2

3

4T

n

nγ θ

ωrRoller

Cam profile

Cam pitchcurve

O1φ1

(b)



0002040608101214


0004081216202428


Dim

ensio

nles

s acc

el


Dim

ensio

nles

s jer

k

Dim

ensio

nles

s velo

city

Dim

ensio

nles

s disp

lac

Gear angle θ (deg)


Jerk-11thAccel-11th




Dim

ensio

nles

s acc

el

Dim

ensio

nles

s jer

k



0004081216202428

Dim

ensio

nles

s ang

ular

vel

ocity


0002040608101214

Dim

ensio

nles

s ang

ular

disp

lac

0 30 60 90 120 150 180Gear angle θ (deg)



Angular accel-φ10









Dim

ensio

nles

s ang

ular

acce

l


Dim

ensio

nles

s jer

k


0004081216202428

Dim

ensio

nles

s ang

ular

velo

city


0002040608101214

Dim

ensio

nles

s ang

ular

disp

lac

0 30 60 90 120 150 180Gear angle θ (deg)

Angular displac-φ6


Angular accel-φ6



Dim

ensio

nles

s ang

ular

acce

l


Dim

ensio

nles

s jer

k


0004081216202428

Dim

ensio

nles

s ang

ular

vel

ocity


0002040608101214

Dim

ensio

nles

s ang

ular

disp

lac



Angular accel-φ2-2

0 30 60 90 120 150 180Gear angle θ (deg)


A

C

T

B

ω

Tprime

Aprime

1

2

3

Roller


Cam pitchcurve 2

Cam pitchcurve 1

O3



Dim

ensio

nles

s acc

el


Dim

ensio

nles

s jer

k


0004081216202428

Dim

ensio

nles

s velo

city


0002040608101214

Dim

ensio

nles

s disp

lac

Gear angle θ (deg)

Displac-11th



Jerk-SHFAccel-11th

Accel-SHF










5 Conclusions


A

CT

B

ω1

2

3

Cam profile 4


Cam pitchcurve 3

Roller

O3



Dim

ensio

nles

s acc

el


Dim

ensio

nles

s jer

k


0004081216202428

Dim

ensio

nles

s velo

city


0002040608101214

Dim

ensio

nles

s disp

lac

0 60 120 180 240 300 360Gear angle θ (deg)





Dim

ensio

nles

s jer

k


000002004006008010012


0002040608


Dim

ensio

nale

ss ac

cel


Dim

ensio

nles

s velo

city

Dim

ensio

nles

s disp

lac

3600 60 120 180 240 300Gear angle θ (deg)




Dim

ensio

nles

s acc

el


Dim

ensio

nles

s jer

k


0004081216202428

Dim

ensio

nles

s velo

city


0002040608101214

Dim

ensio

nles

s disp

lac

3600 60 120 180 240 300Gear angle θ (deg)









Data Availability


Disclosure




Acknowledgments


References




























tanφ62

1113874 1113875 B minus

A2 + B2 minus C2

radic( 1113857

(A minus C) (10)


B2 + l29 minus l210)2l9


A2 + B2 minus C2

radic( 1113857


(11)






l9 cosφ6 l10 cosφ71113890 1113891

ω6

ω71113890 1113891 ω8

l11 sinφ8

l11 cosφ81113890 1113891 (13)


ω6 ω8l11 sin φ7 minus φ8( 1113857

l9 sin φ7 minus φ6( 1113857 (14)

ω7 ω8l11 sin φ8 minus φ6( 1113857

l10 sin φ6 minus φ7( 1113857 (15)



l9 cosφ6 l10 cosφ71113890 1113891

α6α7

1113890 1113891



ω6

ω71113890 1113891

+ α8l11 sinφ8

l11 cosφ81113890 1113891 + ω8

l11 cosφ8

minus l11 sinφ81113890 1113891

(16)


α6 ω26l9 cos φ6 minus φ7( 1113857 + ω2


l11 sin φ8 minus φ7( 1113857

(17)

α7 minus ω2


8l11

l11 sin φ7 minus φ8( 1113857

(18)






+en

l26 l

27 + l

28 + l



+en


+ l28 + l

27 minus l

26 + l

25 0

(21)


tanφ2

21113874 1113875


radic( 1113857

(A minus C) (22)


l25 minus l262l5


A2 + B2 minus C2

radic( 1113857

(A minus C)1113890 1113891


(23)





l5 sinφ2 l6 sinφ3

l5 cosφ2 l6 cosφ31113890 1113891

ω2

ω31113890 1113891 ω4

l7 sinφ4

l7 cosφ41113890 1113891 (25)



l5 sin φ3 minus φ2( 1113857 (26)


l6 sin φ2 minus φ3( 1113857 (27)



l5 sinφ2 l6 sinφ3

l5 cosφ2 l6 cosφ31113890 1113891

α2α3

1113890 1113891 α4l7 sinφ4

l7 cosφ41113890 1113891

+ ω4l7 cosφ4


l5 cosφ2 l6 cosφ3


ω2

ω31113890 1113891

(28)


α2 ω22l5 cos φ2 minus φ3( 1113857 + ω2


l7 sin φ4 minus φ3( 1113857 (29)

α3 minus ω2


4l7

l7 sin φ3 minus φ4( 1113857 (30)






+ yA minus yB( 11138572

l22


+ yA minus yC( 11138572

l21

⎧⎨

⎩ (31)

+en

xA minus B1 plusmn

B21 minus 4A1C1

1113969

1113874 1113875

2A1

yA (C minus Ax)

B

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(32)


y2B + x2



B) xB l3 cosφ1 yB





tan c dxA

dyA



cos c (33)





dxAdθ( 11138572

+ dyAdθ( 11138572

1113969


dxAdθ( 11138572

+ dyAdθ( 11138572

1113969⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(34)











D

E

F

X

Y

Φ5

67

8D0

D1

E0 F0

F1

E1

O1 O2

φ2

φ3 φ4

φ6

φ5










ω

A

B

C12

3 4

O1

(a)

A

B

C

Y

X

1

2

3

4T

n

nγ θ

ωrRoller

Cam profile

Cam pitchcurve

O1φ1

(b)



0002040608101214


0004081216202428


Dim

ensio

nles

s acc

el


Dim

ensio

nles

s jer

k

Dim

ensio

nles

s velo

city

Dim

ensio

nles

s disp

lac

Gear angle θ (deg)


Jerk-11thAccel-11th




Dim

ensio

nles

s acc

el

Dim

ensio

nles

s jer

k



0004081216202428

Dim

ensio

nles

s ang

ular

vel

ocity


0002040608101214

Dim

ensio

nles

s ang

ular

disp

lac

0 30 60 90 120 150 180Gear angle θ (deg)



Angular accel-φ10









Dim

ensio

nles

s ang

ular

acce

l


Dim

ensio

nles

s jer

k


0004081216202428

Dim

ensio

nles

s ang

ular

velo

city


0002040608101214

Dim

ensio

nles

s ang

ular

disp

lac

0 30 60 90 120 150 180Gear angle θ (deg)

Angular displac-φ6


Angular accel-φ6



Dim

ensio

nles

s ang

ular

acce

l


Dim

ensio

nles

s jer

k


0004081216202428

Dim

ensio

nles

s ang

ular

vel

ocity


0002040608101214

Dim

ensio

nles

s ang

ular

disp

lac



Angular accel-φ2-2

0 30 60 90 120 150 180Gear angle θ (deg)


A

C

T

B

ω

Tprime

Aprime

1

2

3

Roller


Cam pitchcurve 2

Cam pitchcurve 1

O3



Dim

ensio

nles

s acc

el


Dim

ensio

nles

s jer

k


0004081216202428

Dim

ensio

nles

s velo

city


0002040608101214

Dim

ensio

nles

s disp

lac

Gear angle θ (deg)

Displac-11th



Jerk-SHFAccel-11th

Accel-SHF










5 Conclusions


A

CT

B

ω1

2

3

Cam profile 4


Cam pitchcurve 3

Roller

O3



Dim

ensio

nles

s acc

el


Dim

ensio

nles

s jer

k


0004081216202428

Dim

ensio

nles

s velo

city


0002040608101214

Dim

ensio

nles

s disp

lac

0 60 120 180 240 300 360Gear angle θ (deg)





Dim

ensio

nles

s jer

k


000002004006008010012


0002040608


Dim

ensio

nale

ss ac

cel


Dim

ensio

nles

s velo

city

Dim

ensio

nles

s disp

lac

3600 60 120 180 240 300Gear angle θ (deg)




Dim

ensio

nles

s acc

el


Dim

ensio

nles

s jer

k


0004081216202428

Dim

ensio

nles

s velo

city


0002040608101214

Dim

ensio

nles

s disp

lac

3600 60 120 180 240 300Gear angle θ (deg)









Data Availability


Disclosure




Acknowledgments


References




























l5 sinφ2 l6 sinφ3

l5 cosφ2 l6 cosφ31113890 1113891

α2α3

1113890 1113891 α4l7 sinφ4

l7 cosφ41113890 1113891

+ ω4l7 cosφ4


l5 cosφ2 l6 cosφ3


ω2

ω31113890 1113891

(28)


α2 ω22l5 cos φ2 minus φ3( 1113857 + ω2


l7 sin φ4 minus φ3( 1113857 (29)

α3 minus ω2


4l7

l7 sin φ3 minus φ4( 1113857 (30)






+ yA minus yB( 11138572

l22


+ yA minus yC( 11138572

l21

⎧⎨

⎩ (31)

+en

xA minus B1 plusmn

B21 minus 4A1C1

1113969

1113874 1113875

2A1

yA (C minus Ax)

B

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(32)


y2B + x2



B) xB l3 cosφ1 yB





tan c dxA

dyA



cos c (33)





dxAdθ( 11138572

+ dyAdθ( 11138572

1113969


dxAdθ( 11138572

+ dyAdθ( 11138572

1113969⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(34)











D

E

F

X

Y

Φ5

67

8D0

D1

E0 F0

F1

E1

O1 O2

φ2

φ3 φ4

φ6

φ5










ω

A

B

C12

3 4

O1

(a)

A

B

C

Y

X

1

2

3

4T

n

nγ θ

ωrRoller

Cam profile

Cam pitchcurve

O1φ1

(b)



0002040608101214


0004081216202428


Dim

ensio

nles

s acc

el


Dim

ensio

nles

s jer

k

Dim

ensio

nles

s velo

city

Dim

ensio

nles

s disp

lac

Gear angle θ (deg)


Jerk-11thAccel-11th




Dim

ensio

nles

s acc

el

Dim

ensio

nles

s jer

k



0004081216202428

Dim

ensio

nles

s ang

ular

vel

ocity


0002040608101214

Dim

ensio

nles

s ang

ular

disp

lac

0 30 60 90 120 150 180Gear angle θ (deg)



Angular accel-φ10









Dim

ensio

nles

s ang

ular

acce

l


Dim

ensio

nles

s jer

k


0004081216202428

Dim

ensio

nles

s ang

ular

velo

city


0002040608101214

Dim

ensio

nles

s ang

ular

disp

lac

0 30 60 90 120 150 180Gear angle θ (deg)

Angular displac-φ6


Angular accel-φ6



Dim

ensio

nles

s ang

ular

acce

l


Dim

ensio

nles

s jer

k


0004081216202428

Dim

ensio

nles

s ang

ular

vel

ocity


0002040608101214

Dim

ensio

nles

s ang

ular

disp

lac



Angular accel-φ2-2

0 30 60 90 120 150 180Gear angle θ (deg)


A

C

T

B

ω

Tprime

Aprime

1

2

3

Roller


Cam pitchcurve 2

Cam pitchcurve 1

O3



Dim

ensio

nles

s acc

el


Dim

ensio

nles

s jer

k


0004081216202428

Dim

ensio

nles

s velo

city


0002040608101214

Dim

ensio

nles

s disp

lac

Gear angle θ (deg)

Displac-11th



Jerk-SHFAccel-11th

Accel-SHF










5 Conclusions


A

CT

B

ω1

2

3

Cam profile 4


Cam pitchcurve 3

Roller

O3



Dim

ensio

nles

s acc

el


Dim

ensio

nles

s jer

k


0004081216202428

Dim

ensio

nles

s velo

city


0002040608101214

Dim

ensio

nles

s disp

lac

0 60 120 180 240 300 360Gear angle θ (deg)





Dim

ensio

nles

s jer

k


000002004006008010012


0002040608


Dim

ensio

nale

ss ac

cel


Dim

ensio

nles

s velo

city

Dim

ensio

nles

s disp

lac

3600 60 120 180 240 300Gear angle θ (deg)




Dim

ensio

nles

s acc

el


Dim

ensio

nles

s jer

k


0004081216202428

Dim

ensio

nles

s velo

city


0002040608101214

Dim

ensio

nles

s disp

lac

3600 60 120 180 240 300Gear angle θ (deg)









Data Availability


Disclosure




Acknowledgments


References


































ω

A

B

C12

3 4

O1

(a)

A

B

C

Y

X

1

2

3

4T

n

nγ θ

ωrRoller

Cam profile

Cam pitchcurve

O1φ1

(b)



0002040608101214


0004081216202428


Dim

ensio

nles

s acc

el


Dim

ensio

nles

s jer

k

Dim

ensio

nles

s velo

city

Dim

ensio

nles

s disp

lac

Gear angle θ (deg)


Jerk-11thAccel-11th




Dim

ensio

nles

s acc

el

Dim

ensio

nles

s jer

k



0004081216202428

Dim

ensio

nles

s ang

ular

vel

ocity


0002040608101214

Dim

ensio

nles

s ang

ular

disp

lac

0 30 60 90 120 150 180Gear angle θ (deg)



Angular accel-φ10









Dim

ensio

nles

s ang

ular

acce

l


Dim

ensio

nles

s jer

k


0004081216202428

Dim

ensio

nles

s ang

ular

velo

city


0002040608101214

Dim

ensio

nles

s ang

ular

disp

lac

0 30 60 90 120 150 180Gear angle θ (deg)

Angular displac-φ6


Angular accel-φ6



Dim

ensio

nles

s ang

ular

acce

l


Dim

ensio

nles

s jer

k


0004081216202428

Dim

ensio

nles

s ang

ular

vel

ocity


0002040608101214

Dim

ensio

nles

s ang

ular

disp

lac



Angular accel-φ2-2

0 30 60 90 120 150 180Gear angle θ (deg)


A

C

T

B

ω

Tprime

Aprime

1

2

3

Roller


Cam pitchcurve 2

Cam pitchcurve 1

O3



Dim

ensio

nles

s acc

el


Dim

ensio

nles

s jer

k


0004081216202428

Dim

ensio

nles

s velo

city


0002040608101214

Dim

ensio

nles

s disp

lac

Gear angle θ (deg)

Displac-11th



Jerk-SHFAccel-11th

Accel-SHF










5 Conclusions


A

CT

B

ω1

2

3

Cam profile 4


Cam pitchcurve 3

Roller

O3



Dim

ensio

nles

s acc

el


Dim

ensio

nles

s jer

k


0004081216202428

Dim

ensio

nles

s velo

city


0002040608101214

Dim

ensio

nles

s disp

lac

0 60 120 180 240 300 360Gear angle θ (deg)





Dim

ensio

nles

s jer

k


000002004006008010012


0002040608


Dim

ensio

nale

ss ac

cel


Dim

ensio

nles

s velo

city

Dim

ensio

nles

s disp

lac

3600 60 120 180 240 300Gear angle θ (deg)




Dim

ensio

nles

s acc

el


Dim

ensio

nles

s jer

k


0004081216202428

Dim

ensio

nles

s velo

city


0002040608101214

Dim

ensio

nles

s disp

lac

3600 60 120 180 240 300Gear angle θ (deg)









Data Availability


Disclosure




Acknowledgments


References

































Dim

ensio

nles

s ang

ular

acce

l


Dim

ensio

nles

s jer

k


0004081216202428

Dim

ensio

nles

s ang

ular

velo

city


0002040608101214

Dim

ensio

nles

s ang

ular

disp

lac

0 30 60 90 120 150 180Gear angle θ (deg)

Angular displac-φ6


Angular accel-φ6



Dim

ensio

nles

s ang

ular

acce

l


Dim

ensio

nles

s jer

k


0004081216202428

Dim

ensio

nles

s ang

ular

vel

ocity


0002040608101214

Dim

ensio

nles

s ang

ular

disp

lac



Angular accel-φ2-2

0 30 60 90 120 150 180Gear angle θ (deg)


A

C

T

B

ω

Tprime

Aprime

1

2

3

Roller


Cam pitchcurve 2

Cam pitchcurve 1

O3



Dim

ensio

nles

s acc

el


Dim

ensio

nles

s jer

k


0004081216202428

Dim

ensio

nles

s velo

city


0002040608101214

Dim

ensio

nles

s disp

lac

Gear angle θ (deg)

Displac-11th



Jerk-SHFAccel-11th

Accel-SHF










5 Conclusions


A

CT

B

ω1

2

3

Cam profile 4


Cam pitchcurve 3

Roller

O3



Dim

ensio

nles

s acc

el


Dim

ensio

nles

s jer

k


0004081216202428

Dim

ensio

nles

s velo

city


0002040608101214

Dim

ensio

nles

s disp

lac

0 60 120 180 240 300 360Gear angle θ (deg)





Dim

ensio

nles

s jer

k


000002004006008010012


0002040608


Dim

ensio

nale

ss ac

cel


Dim

ensio

nles

s velo

city

Dim

ensio

nles

s disp

lac

3600 60 120 180 240 300Gear angle θ (deg)




Dim

ensio

nles

s acc

el


Dim

ensio

nles

s jer

k


0004081216202428

Dim

ensio

nles

s velo

city


0002040608101214

Dim

ensio

nles

s disp

lac

3600 60 120 180 240 300Gear angle θ (deg)









Data Availability


Disclosure




Acknowledgments


References


































5 Conclusions


A

CT

B

ω1

2

3

Cam profile 4


Cam pitchcurve 3

Roller

O3



Dim

ensio

nles

s acc

el


Dim

ensio

nles

s jer

k


0004081216202428

Dim

ensio

nles

s velo

city


0002040608101214

Dim

ensio

nles

s disp

lac

0 60 120 180 240 300 360Gear angle θ (deg)





Dim

ensio

nles

s jer

k


000002004006008010012


0002040608


Dim

ensio

nale

ss ac

cel


Dim

ensio

nles

s velo

city

Dim

ensio

nles

s disp

lac

3600 60 120 180 240 300Gear angle θ (deg)




Dim

ensio

nles

s acc

el


Dim

ensio

nles

s jer

k


0004081216202428

Dim

ensio

nles

s velo

city


0002040608101214

Dim

ensio

nles

s disp

lac

3600 60 120 180 240 300Gear angle θ (deg)









Data Availability


Disclosure




Acknowledgments


References






























Data Availability


Disclosure




Acknowledgments


References



























mathematical model of cam profile based on heald frame...

Documents