kinematic design of a seven-bar linkage with optimized

Research ArticleKinematic Design of a Seven-Bar Linkage with OptimizedCentrodes for Pure-Rolling Cutting

Biao Liu1 Yali Ma1 DelunWang1 Shaoping Bai2 Yangyang Li1 and Kangkang Li1

1School of Mechanical Engineering Dalian University of Technology Dalian China2Department of Mechanical and Manufacturing Engineering Aalborg University Aalborg Denmark

Correspondence should be addressed to Delun Wang dlunwangdluteducn

Received 9 February 2017 Revised 20 March 2017 Accepted 13 April 2017 Published 9 May 2017

Academic Editor Anna M Gil-Lafuente

Copyright copy 2017 Biao Liu et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A novel method for designing a seven-bar linkage based on the optimization of centrodes is presented in this paper The proposedmethod is applied to the design of a pure-rolling cutting mechanism wherein close interrelation between the contacting linesand centrodes of two pure-rolling bodies is formulated and the genetic optimization algorithm is adopted for the dimensionalsynthesis of the mechanismThe optimization is conducted to minimize the error between mechanism centrodes and the expectedtrajectories subject to the design requirements of the opening distance the maximum amount of overlap error and peak value ofshearing force An optimal solution is obtained and the analysis results show that the horizontal slipping and standard deviationof the lowest moving points of the upper shear blade have been reduced by 780 and 801 and the peak value of shear stressdecreases by 29 which indicate better cutting performance and long service life

1 Introduction

A seven-bar linkage has two degrees of freedom whichcan be used in many machines with variable trajectoriesOf all associated machines a typical example is the seven-bar pure-rolling cutting mechanism which generates pure-rolling motion between two contacting bodies [1ndash4]

The design of pure-rolling mechanism is essentially aproblem of trajectory synthesis of linkages for which manysynthesized methods are available The synthesis can becarried out either for a set of given points or for a con-tinuous trajectory [5 6] The synthesis results are eitherexact or approximate Normally exact synthesis is difficultto implement in practice and approximate methods are usedto approximate the given points or continuous trajectory asmuch as possible [7] To evaluate the trajectory deviation ofapproximate synthesis some trajectory deviation measure-ment functions are introduced including deterministic error[8] Fourier deviation [9] shape similarity [10] ambiguityfunction [11] and shape feature matching deviation [12]

Generally there are two different ways to accomplishapproximate synthesis namely direct and indirect synthesis

methods The direct synthesis method generates a mecha-nism directly according to the given points or continuoustrajectory Nelson Larsen [13] used an atlas of coupler curvesto analyze the four-bar linkage but the computation accuracywas unsatisfactory Kramer [14] extended the selective pre-cision synthesis method to generate four-bar motion mech-anism with prescribed input crank rotations which usedthe Hooke-and-Jeeves search method to handle the equalityconstraints during the synthesis process Subbian andFlugrad[15] implemented the continuation method to deal with thesets of polynomial equations in the four-bar path generationsynthesis which was proved to be more effective Never-theless even with these numerical methods the nonlinearsynthesis equations of high order are still difficult to solveCabrera et al [8] used the genetic optimization algorithm tooptimize the position error between the given target pointsand the points reached by the resulting mechanism duringthe synthesis of four-bar planar mechanisms In order toobtain both effectiveness and high accuracy many otheroptimization algorithms are also adopted in the trajectorysynthesis of the mechanism such as simulated annealing [16]and stochastic method [17]

HindawiMathematical Problems in EngineeringVolume 2017 Article ID 1586375 11 pageshttpsdoiorg10115520171586375

2 Mathematical Problems in Engineering

The indirect synthesis method [10 12] is used to searchfor the matching trajectory from the predefined trajectoryatlas instead of directly generating a mechanism schemewhich is done by analyzing the expected trajectory and thenexporting the corresponding mechanism types and sizes Ifthere is a similar scheme the minimum trajectory deviationwill be obtainedThe indirect synthesis method mainly relieson the mass data-storage capacity and rapid retrieval abilityof a computer Although the rapid improvement of computerspromotes the application and development of the indirectsynthesis the difficulties of the establishment of a trajectoryatlas the mass data-storage capacity of a computer and theapproach to effectively search for the bestmatching trajectoryare still challenging problems to be solved

For the problem here the design of seven-bar linkagesfor pure rolling needs to meet both the trajectory and alsoother requirements formachining that is steel plates cuttingThe shear motion of a rolling shear mechanism is generallyrealized by means of the relative motion between the uppershear blade and the lower shear blade The expected shearmotion should be a pure-rolling motion without slipping Inthis regard Wang and Huang [18] developed an optimizedmodel for rolling shear mechanism with single shaft anddouble eccentricity choosing four motion positions as accesspoints to acquire the expected motions while the phasedifference was set to be identical Yang et al [19] used theconstraints of equal radius of crank and equal length oflinkage to set up an optimization model of rolling shearmechanism with roll guide groove Sun et al [20] designeda rolling shear mechanism by optimizing the trajectory ofthe lowest moving point of the upper shear blade but theupper shear blade could not perform pure-rolling motionrelative to the lower blade due to the horizontal slide In [21]a synthesis-optimized model was built to design a rollingshearmechanism using a guiding rod as an additional designvariable while identical phase difference and identical lengthbetween the designed guide rod and the expected guiderod are adopted for four positions In order to improveshear quality decrease blade wear and prolong blade life ofthe cutting machine [22] generally the pure-rolling motionbetween the shear blades can be transformed into a series ofmoving positions and phase angles of the seven-bar linkageswith which an optimizedmethod is adopted to obtain properlinkage sizes In certain situations the synthesis can onlysatisfy some key points the motion accuracy of the designedpure-rolling cutting mechanism is thus low It is difficultto realize the pure-rolling motion during the whole cuttingprocess due to the fact that the cutting performance was notconsidered or embodied in the synthesis

This paper proposes a method for the kinematic design ofa seven-bar linkage to generate pure-rolling motion by opti-mizing the centrodes The introduced method is developedbased upon the interrelation between the centrodes and con-tacting lines of pure-rolling motion A case study of seven-bar rolling shear mechanism is included to demonstrate themethod to accomplish the pure-rolling motion A geneticoptimization algorithm is used to obtain mechanism sizeswith the metric function of minimum approximation errorbetween mechanism centrodes and expected trajectories of

A F

BE

HG

C D

MN

Upper arc blade

Steel plate

Lower straight blade

휔1 휔2

Figure 1 Sketch of a seven-bar rolling shear mechanism

shear blade The constraints of the formulated optimizationproblem for the pure-rolling mechanism include the designrequirements of the opening distance the maximum amountof overlap error and peak value of shearing force Moreoverthe performance of the newly designed rolling shear mech-anism is investigated and compared with the original onewhich shows the advantages of the new method

2 Design Model

21 Design Issue and Problem Formulation The rolling shearmechanism is a typical pure-rolling cutting mechanismwhich is usually used for medium plate rolling shear Theseven-bar linkagemechanism is a common application of therolling shear mechanism [23] as shown in Figure 1

The seven-bar linkage has 2 degrees of freedom cor-responding to cranks 119860119861 and 119864119865 as driving links whichrotate by the same angular velocity and with a constant phasedifference sharing a power input Link 119862119863119866 to which theupper blade is attached outputs motion Generally the lowershear blade is fixed on the frame while the upper shear blademoves relative to the lower shear blade to cut the steel platebetween them as shown in Figure 2

The horizontal sliding of the upper shear blade should beas little as possible to reduce thewear of the bladeMeanwhilethe cutting depth of the upper shear blade should be the sameto reduce the bending deformation of the steel plate ensuringa stable cutting quality of the steel plate Thus the idealmotion of the upper blade should be pure-rolling cuttingrelative to the steel plate during the shearing process to makesure there is no horizontal sliding between the blade and thesteel plate at the cutting contact point

One of the rigid bodies is usually chosen to be fixed andanother moves relative to the chosen one for convenienceduring the motion analysis of two rigid bodies as shown inFigure 3 Rigid body II is fixed in coordinate system 119874119891 minus119909119891119910119891 Rigid body I on which a moving coordinate system119874119898 minus 119909119898119910119898 is built moves in the fixed coordinate system119874119891minus119909119891119910119891 A point119874119898 of body I has a velocity V119874

119898

and body Irotates about point119874119898 with angular velocity120596119874

119898

Themotionstate of body I at any moment is either (a) entire translation

Mathematical Problems in Engineering 3

Position 1

Position 3

Position 2

Position 4

Figure 2 Motion cycle of the rolling shear mechanism

II

I

P

yf

Of xf

ym

Om xm

휔0Γ1

Γ2

O푚

휔O푚

Figure 3 Fixed and moving centrodes of two rigid bodies

or (b) rotation about a specific point 119875 on body I of whichthe velocity in the fixed coordinate system is zero The point119875 is called the instantaneous velocity center and the entiretranslation can be regarded as the point119875 being at infinity Sothe motion of body I can be treated as a pure rotation about119875 at any moment As rigid body I moves the instantaneousvelocity center 119875 traces a trajectory in the fixed coordinatesystem119874119891minus119909119891119910119891 which is called the fixed centrode Γ1 and atrajectory in themoving coordinate system119874119898minus119909119898119910119898 whichis called the moving centrode Γ2Themotion of body I can beregarded as the pure-rolling motion of the moving centrodealong with the fixed centrode with no sliding

As the thickness of the steel plate is far less than the widthof the steel plate and the length of the blade the contactline of the shear blade and steel plate is usually treated as acontact point in practice Thus the ideal cutting motion canbe regarded as the pure-rolling motion between the uppershear blade and the lower shear blade with no sliding atthe contact point The objective is to synthesize the linkagefor pure-rolling shear motion so that the profiles of theupper and lower blades coincide with the moving and fixedcentrodes of the output link 119862119863119866 respectively during theshearing process

22 Kinematic Design Model A seven-bar linkage is chosento establish the kinematic design model of which the topo-logical structure can be obtained as shown in Figure 4

Links 119860119861 and 119864119865 are assigned as the driving links whilethe ternary link 119862119863119866 is assigned as the output coupler towhich the upper shear blade is attached Link 119860119865119867 which isalso a ternary link assigned number 7 is chosen as the frameSo there are 6 movable links corresponding to 6 angles120579119894 119894 = 1 2 6 as shown in Figure 5 Vector equations ofclosed-loopHGDEFH andHGCBAH are listed as follows

HG + GD +DE = HF + FEHG + GC + CB = HA + AB (1)

Usually joints119860 and 119865 are designed with the same heightfor convenience of structure design and power transmissionFor the convenience of the modeling vectorsHK andKF areintroduced to take the place of vectorsHA andHF as shownin Figure 5 The lengths of vectors HK and KF represent thevertical and horizontal distances of joint 119865 relative to joint119867respectively

So (2) can be obtained from the new closed-loopHGDE-FKH andHGCBAFKH as

HG + GD +DE = HK + KF + FEHG + GC + CB = HK + KF + FA + AB (2)

A fixed coordinate system 119867 minus 119909119910 and a moving coor-dinate system 119871 minus 119909119898119910119898 are established at the hinged point119867 and the center of the driven link 119862119863 respectively Besidesthe basic length parameters 119871 119894 119894 = 1 2 6 of the 6movable links shown in Figure 5 1198717 1198718 and120572 are introducedto determine the dimensions of link 119862119863119866 and 1198719 11987110 and11987111 are introduced for vectors AF HK and KF Thus thelength parameters of the linkage are 119871 119894 119894 = 1 2 11Expanding (2) yields

1198716 cos 1205796 + 1198715 cos 1205795 + 1198714 cos 1205794 + 11987111minus 1198712 cos 1205792 = 0

1198716 sin 1205796 + 1198715 sin 1205795 + 1198714 sin 1205794 minus 11987110 minus 1198712 sin 1205792 = 01198716 cos 1205796 + 1198718 cos (1205795 + 120572) + 1198713 cos 1205793 + 1198719 + 11987111minus 1198711 cos 1205791 = 0

1198716 sin 1205796 + 1198718 sin (1205795 + 120572) + 1198713 sin 1205793 minus 11987110minus 1198711 sin 1205791 = 0

(3)

The driving links 119860119861 and 119864119865 have the same angularvelocity with a constant phase difference sharing a power


A

B E

F

Upper arc blade Lower straight blade

1

2

3

4

5

67

HD

G

C

(a) Schematic diagram of the mechanism

A

B E

F

1

2

3

4

5

6

7

H

D

G

C

(b) Topological structure of the mechanism

Figure 4 Schematic diagram and topological structure of the seven-bar linkage

B

A F

E

C

G

P

D

H

y

x

K

L

yG

ym

xm

xG

Γ1

Γ2

L1 L2

L3 L4

L5 L6

L7

L8

L9

L10

L11휃1 휃2

휃3

휃4

휃5

휃6

훼훽1

훽2

Figure 5 Kinematic model of the seven-bar linkage

input That means 1205791 minus 1205792 = 120579119862 The differentiation withrespect to time of (3) yields

minus 11987161205966 sin 1205796 minus 11987151205965 sin 1205795 minus 11987141205964 sin 1205794+ 11987121205962 sin 1205792 = 011987161205966 cos 1205796 + 11987151205965 cos 1205795 + 11987141205964 cos 1205794minus 11987121205962 cos 1205792 = 0minus 11987161205966 sin 1205796 minus 11987181205965 sin (1205795 + 120572) minus 11987131205963 sin 1205793+ 11987111205961 sin 1205791 = 011987161205966 cos 1205796 + 11987181205965 cos (1205795 + 120572) + 11987131205963 cos 1205793minus 11987111205961 cos 1205791 = 0

(4)

where 1205961 = 1205962 which are given quantities denoting theangular velocity of links 119860119861 and 119864119865 Thus 1205795 1205796 1205965 and 1205966can be obtained by solving (3) to (4)

Thedriven link119862119863119866 should generate pure-rolling cuttingmotion between the upper and lower blades During thecutting process the instantaneous center 119875 forms themoving

centrode relative to the driven link 119862119863119866 and the fixed cen-trode relative to the fixed frame represented by curves Γ1 andΓ2 as displayed in Figure 5 In order to derive the kinematicequations of centrodes coordinate transformation matrixis used to transform the points from moving coordinatesystem to fixed coordinate system which is related to rotationangle and translation distance [24] Let the coordinates ofinstantaneous center 119875 be (119909 119910) in the fixed coordinatesystem119867minus119909119910 and (119909119898 119910119898) in the moving coordinate system119871 minus 119909119898119910119898 An additional coordinate system 119866 minus 119909119866119910119866which is established at hinged point 119866 as shown in Figure 5is introduced to implement coordinate transform betweenthe fixed and moving coordinate systems The two sets ofcoordinates (119909 119910) and (119909119898 119910119898) are related by

1199091199101= M119867119866M119866119871

1199091198981199101198981 (5)

where matrix M119867119866 is the homogeneous transformationmatrix from the fixed coordinate system119867minus 119909119910 to 119866 minus 119909119901119910119901andM119866119871 is the one from 119866 minus 119909119901119910119901 to the moving coordinatesystem 119871 minus 119909119898119910119898 They are given as

M119867119866 = [[[cos 1205795 minus sin 1205795 1198716 cos 1205796sin 1205795 cos 1205795 1198716 sin 12057960 0 1

]]]

M119866119871 = [[[cos1205732 sin1205732 119871119866119871 cos1205731minus sin1205732 cos1205732 119871119866119871 sin12057310 0 1

]]]

(6)

where 1205795 and 1205796 are orientation angles of links 119863119866 and 1198661198671205731 is the angle between vectors GD and GL and 1205732 is theorientation angle of the 119909-axis of 119871 minus 119909119898119910119898 in coordinatesystem119866minus119909119866119910119866119871119866119871 represents the length of119866119871 Substitutingthe above equation into (5) and upon differentiation withrespect to time one has


1199100= [[[

minus1205965 sin 1205795 minus1205965 cos 1205795 minus11987161205966 sin 12057961205965 cos 1205795 minus1205965 sin 1205795 11987161205966 cos 12057960 0 0]]][[[

cos1205732 sin1205732 119871119866119871 cos1205731minus sin1205732 cos1205732 119871119866119871 sin12057310 0 1]]]1199091198981199101198981 (7)

where and 119910 are the velocities of instantaneous center 119875 1205965and 1205966 are the angular velocities of links 1198715 and 1198716 As thevelocity of instantaneous velocity center 119875 at any moment inthe fixed coordinate system119867minus119909119910 is zero namely = 119910 = 0by arranging and rewriting the above equation the movingcentrode of link 119862119863119866 is expressed as

119909119898 = 1198711198661198711205965 cos (1205731 + 1205732) + 11987161205966 cos (1205732 + 1205796 minus 1205795) 119910119898 = 1198711198661198711205965 sin (1205731 + 1205732) + 11987161205966 sin (1205732 + 1205796 minus 1205795) (8)

Substituting (8) into (5) the fixed centrode is obtained as

119909 = 119871119866119871 (1205965 + 1) cos (1205731 + 1205795) + 1198716 (1205966 + 1) cos 1205796119910 = 119871119866119871 (1205965 + 1) sin (1205731 + 1205795) + 1198716 (1205966 + 1) sin 1205796 (9)

So far both the moving and the fixed centrodes havebeen obtained upon which optimal sizes and positions of themechanism can be searched to ensure that the trajectories ofmoving and fixed centrodes cooperate with each other in theway of pure rolling

3 Optimization Design Case

In this section a design case of seven-bar rolling shear mech-anism as a kind of common pure-rolling cuttingmechanismis considered The ideal shear motion of a rolling shearmechanism should be pure-rollingmotion between the uppershear blade and the lower shear blade With the generatedmoving centrodes and fixed centrodes coinciding with themotion contact lines of the upper shear blade and lowershear blade respectively the pure-rolling motion can beobtained Hence the optimization objective function and theconstraints could be determined by pure-rolling motion andcutting performance requirements A genetic optimizationmethod [25] is employed to determine the proper linkagesizes of rolling shear mechanism thanks to its effectivenessand convenience

31 Design Parameters The design parameters of a rollingshear mechanism are given by the cutting process [26 27]These design parameters include the width of sheared plate119861the maximum shearing thickness ℎmax the shearing overlap119878 and the shearing angle 120572 as shown in Figure 6Thewidth 119861determines the horizontal width of the lower shear blade andthe shearing overlap 119878 gives the overlapping amount betweenthe upper and the lower shear blades in the shearing processThe shear angle 120572 refers to the contact point between thelower shear blade and the tangent of arc upper shear blade

32 Optimization Model Based on the design parametersthe expected trajectories (or profiles) of upper and lowershear blades can be obtained The purpose of genetic opti-mization model is to seek a set of optimal mechanism sizesto minimize the deviation between centrodes and expectedtrajectories of upper and lower shear blades subject tosome specific design requirementsThe detailed optimizationmodel is as follows

321 Optimization Variables The design variables of arolling shearmechanism are generally the lengths of links andpivoting joint positionsThese design variables are defined asoptimization variables expressed by a vector t

t = [1199051 1199052 119905119899]119879 t isin 119877119899 (10)

in which each variable 119905119894 (119894 = 1 2 119899) represents thesize parameter of a mechanism scheme such as the lengthsof links 119871 119894 119894 = 1 2 11 and phase angles 120579119894 119894 =1 2 6 Each optimal scheme can be expressed by vectortlowast called optimal point

322 Optimization Objective Function The objective ofthe design optimization is to make the moving centrodeapproach the profile of the upper shear blade and the fixedcentrode approach the profile of the lower shear blade asmuch as possible Accordingly the objective function of theoptimization design can be defined as the sumof approachingerrors including the approaching error for moving centrodeand upper shear blade together with the approaching errorfor fixed centrode and lower shear blade which will beminimized as follows

min 119880 (t)119880 (t) = 1198801 (t) + 1198802 (t) (11)

where 1198801(t) and 1198802(t) are the curve approximation errorsbetween moving centrode and upper shear blade and fixedcentrode and lower shear blade respectively The errorsshould be evaluated in the moving coordinate system 119874119898 minus119909119898119910119898 and the fixed coordinate system 119874 minus 119909119910 on upper andlower shear blade as shown in Figure 7

The geometric equations of the moving centrode andprofile of the upper shear blade in the moving coordinatesystem 119874119898 minus 119909119898119910119898 can be written as

119910119898 = 119892119898119905 (119909119898) 119910119898 = 119891119898 (119909119898) (1199091198981 le 119909119898 le 119909119898119899) (12)


VW

S

B

R

Upper arc blade

Steel plate


UC

D G H

훼

훽

ℎmax

Figure 6 Design parameters of the rolling shear mechanism

x

y

O

CP

Moving centrode ym = gmt(xm)

Fixed centrode y = gt(x)

Lower straight shear blade y = f(x) = C

Upper arc shear blade ym = fm(xm)

xm1 x

m2 xmj

Om

xmq

xm

ym

xi xpx2x1

Figure 7 Centrodes and profiles of the shear blades

Also equations of the fixed centrode and profile of the lowershear blade in the fixed coordinate system 119874 minus 119909119910 can bewritten as

119910 = 119892119905 (119909) 119910 = 119862 (1199091 le 119909 le 119909119899) (13)

where 119909119899minus1199091 = 119861 and119862 is a constant describing the positionof the sheared plate

The errors 1198801(t) and 1198802(t) can be determined by

1198801 (t) = radic 1119899119899sum119894=1

(119892119905 (119909119894) minus 119862)2

1198802 (t) = radic 1119899119899sum119894=1

(119892119898119905 (119909119898119894) minus 119891119898 (119909119898119894))2(14)

Hence the objective function of optimization for thepure-rolling cutting mechanism design can be expressed as

119880 (t) = radic 1119899119899sum119894=1

(119892119905 (119909119894) minus 119891 (119909119894))2

+ radic 1119899119899sum119894=1

(119892119898119905 (119909119898119894) minus 119891119898 (119909119898119894))2(15)

323 Constraints The constraints of a rolling shear mech-anism mainly include some motion parameters and perfor-mance parameters such as the opening distance the shearingoverlap error of the upper and lower shear blade and the peakvalue of shearing force

(1) Opening Distance Constraint In order tomake the shearedplate get through smoothly between the two shear bladesthe clearance between the upper and lower shear blades aftershearing also known as the opening distance (119867) which isthe function of design variable 119905 should be greater than thedesigned value 119870 associated with the thickness of shearedplate

119867(t) ge 119870 (16)

(2) Overlap Error ConstraintThe overlap error in direction ofplate width should be limited to a given amount The overlapamount is the distance from the lowestmoving point of uppershear blade to the lower shear blade The coordinates of thelowest moving point119882 in the fixed coordinate system can beobtained by a geometrical relationship as shown in Figure 6It can be written as

119909119882 = 119909119881 minus 119877 sin120573119910119882 = 119910119881 minus 119877 (1 minus cos120573) (17)

where 119877 and 120573 are the arc radius and the dip angle of uppershear blade respectively and (119909119881 119910119881) is the coordinate ofmiddle point 119881 on upper shear blade in the fixed coordinatesystem Thus the overlap error constraint is expressed as

|Δ119878| = 1003816100381610038161003816119862 minus 119910119882 minus 1198781003816100381610038161003816 le 01 sdot 119878 (18)

(3) Peak Value of Shearing Force Constraint Generally theforces applied on upper shear blade refer to both shearforce and other forces such as friction force The peakvalue of shear force constraint can be introduced by limitingthe maximum shearing force that usually appears in theinitial shear stage The shearing force [28] of a rolling shearmechanism is expressed as

119875 = 06120590119887120575sdot ℎ2tan120572 (1 + 119885 tan12057206120575 + 11 + 1001205751205901198871198842119883)

(19)


Table 1 Design parameters of the rolling shear mechanism

Platewidth 119861

Maximumthicknessℎmax

Overlapamount 119878 Shear

angle 120572 Openingdistance119867

3500mm 50mm 5mm 224∘ ge200mm

where 120590119861 and 120575 are the ultimate strength and percentageelongation of material for sheared plate 119885 represents theconversion coefficient 119884 is the ratio of shear blade gap withthe thickness of steel plate and 119883 is the ratio of the distancebetween shear blade edge and steel plate with the thicknessof steel plateThe shearing force constraint may be limited byshear angle 120572 because the peak value of shearing force canbe highly correlated to the shear angle Therefore it may begiven by means of specified shear angle 1205720 which is writtenas

120572st ge 1205720 (20)

where 120572st is the initial shear angle of upper shear bladeAccording to the above discussion for determining the sizesof rolling shearmechanism the final design vectormarked astlowast where themechanism sizes achieve pure-rollingmotion ofthe upper shear blade can be obtained bymeans of the geneticoptimization algorithm

4 Results and Analysis

According to the optimization functions and given shearingrequirements the seven-bar mechanism for pure-rollingcutting will be synthesized and the kinematic performancewill be analyzed and compared to the original one

41 Optimization Results The seven-bar mechanism forpure-rolling cutting is shown in Figure 1 The actual designparameters of sheared plate are used as the design parametersof rolling shear mechanism as shown in Table 1

The length of each link and initial phase angles of twocranks are used as optimization variables Given that theconstant 119862 should be set as minus400mm in (13) the constraintof initial shearing angle is selected as follows 120572st ge 15∘Meanwhile the optimization model of rolling shear men-tioned above can be established together with the geneticoptimization algorithm employed Therefore the lengths oflinkages the coordinate of fixed hinge point 119865 and the initialphase angle of crank 119860119861 of the new mechanism can beobtained as shown in Table 2

42 Kinematic Performance Analysis Amajor kinematic per-formance concerned for this design is a pure-rolling motionbetween two blades and is described by deviations betweenfixed and moving centrodes and contacting lines which isintuitively exhibited through the trajectory of the lowestmoving point and arc middle point of upper shear blade Thecutting performance is illustrated by the comparison of theshear angle and shear stress between the original design andthe new design in this paper

Table 2 The design result of the new rolling shear mechanism

Result parameters ValueLength of link 119860119861 1149mmLength of link 119864119865 1149mmLength of link 119861119862 8641mmLength of link 119864119863 8641mmLength of 119860119865 24000mmLength of link 119862119863 24000mmLength of link119863119866 8567mmLength of link119867119866 8072mmCoordinates of fixed hinge point 119865 (minus16381 10330)Initial phase angle of crank 119860119861 1140∘

Figure 8 Motion simulation of rolling shear mechanism

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0minus4000 minus3000 minus2000 minus1000minus5000minus6000

Original designNew designLower straight blade

Figure 9 Fixed centrode of upper shear blade and lower horizontalshear blade

Simulation and performance analysis of the rolling shearmechanism based on ProE and MATLAB software wereconducted Figure 8 shows the motion simulation model ofthe rolling shear mechanism

The comparison of the fixed centrode of the upper shearblade and the lower shear blade between the original andoptimal results is shown in Figure 9 The designed fixed


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Original designNew designUpper arc blade

Figure 10 Moving centrode of upper shear blade and arc profile of upper shear blade

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(b) Trajectory in the shearing process

Figure 11 Trajectory of arc middle point on the upper shear blade

centrode has better straightness in the segment which canapproximate the horizontal contact line in a better way andis in accordance with the objective function Notice that theaxes are not isometric for clear demonstration

Figure 10 shows that the designed moving centrodeapproximates the symmetrical arc perfectly which meansthat it approximates the moving contact arc perfectly whichis in accordance with the objective function Notice that theaxes are not isometric for clear demonstration

Figure 11 shows the trajectory of arc middle point on theupper shear blade which presents the cutting process part

Notice that the axes are not isometric for clear demonstrationThe results demonstrate that the horizontal slipping of thedesigned upper shear blade is confirmed as 097mm com-pared to the original result of 488mm reduced by 801which illustrates that the designed upper shear blade profileis better in the realization of pure-rolling motion and alsoindirectly proves the validity of the method of designingrolling shear mechanism sizes

Figure 12 shows that the trajectory of the lowest movingpoint of upper shear blade is approximately a straight lineand its straightness reflects overlapping evenness of upper


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Original design

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(a) Overall view

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minus401

minus400



Figure 12 Trajectory of the lowest moving point119882 on upper shear blade

Angle of crank AB (∘)310300290280270260250240230220


08

1

12

14

16

18

2

22

24

26

Shea

r ang

le (∘

)

(a) Shear angle

times107

Angle of crank AB (∘)310300290280270260250240230220

07

08

09

1

11

12

13

14

15

16

17

Shea

r for

ce (N

)


(b) Shear force

Figure 13 Comparison of shear angle and stress before and after the design

and lower shear blade Notice that the axes are not isometricfor clear demonstration The standard deviation of optimalresult in trajectory sets of upper arc lowest moving pointduring shearing process is confirmed as 0415mm comparedto the original result of 1890mm being reduced by 780indicating more uniform overlap between upper and lowershear blade

The changes of shear angle and stress before and afterthe design are shown in Figure 13 which indicates that theinitial angle of the designed rolling shear mechanism at the

beginning of the cutting process is roughly 15∘ while theoriginal initial angle is 09∘This improvement will be of greatinterest to improving the initial peak value of shear forceTheshear angle increases to about 22∘ when the shearing processcomes to the stable rolling stage no matter in the originaldesign or in the new design The peak value of shear stress ofthe designed rolling shear mechanism is roughly 12 times 107Ndecreasing by 29 in comparison with the original shearstress peak of 17 times 107N Moreover the above figures alongwith shear angle changing curve show that the shear stress


and shear angle change oppositely Therefore it is beneficialto improve the initial shear angle in order to reduce the initialshear stress

5 Conclusions

A new approach to design a seven-bar linkage for pure-rolling cutting by optimizing centrodes is presented in thispaper Using the genetic optimization algorithm the pro-posed method allows the designer to obtain an optimumlinkage which minimizes the error between the centrodes ofmechanisms and profiles of pure rolling With the proposedmethod a seven-bar rolling shear mechanism is designedwhich has better performance compared to the original onein the following aspects

(1) The horizontal slipping of the designed rolling shearmechanism has been reduced by 780 whichincreases the cutting efficiency and reduces the wearof the shearing blade

(2) The standard deviation of the lowest moving pointon the upper shear blade has been reduced by 801which indicates better quality of steel plates

(3) The peak value of shear stress which indicates thepower performance of rolling shear mechanism isdecreased by 29 for long service life

The proposed method in this paper is used for theseven-bar pure-rolling cutting mechanism It can also beapplicable to the other types of pure-rolling mechanismIn this paper the genetic optimization algorithm methodis adopted for obtaining an optimal solution and otheroptimization methods can be considered to search for betteroptimal solutions in future study

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The authors acknowledge with appreciation the financialsupport from the National Natural Science Foundation ofChina (Grant no 51275067)

References

[1] G Figliolini and J Angeles ldquoSynthesis of conjugate Genevamechanisms with curved slotsrdquo Mechanism and Machine The-ory vol 37 no 10 pp 1043ndash1061 2002

[2] L-MWu J Ren and T-B Yin ldquoOptimal design of rolling shearmechanism based on improved genetic algorithmsrdquo in Proceed-ings of the International Conference on Digital Manufacturingand Automation (ICDMA rsquo10) pp 68ndash71 chn December 2010

[3] J Wang T Hashimoto H Nishikawa and M Kaneta ldquoPurerolling elastohydrodynamic lubrication of short stroke recip-rocating motionrdquo Tribology International vol 38 no 11-12 pp1013ndash1021 2005

[4] P H Pan and R M Qian ldquoDesign of the new pure rollingcircular arc gearrdquo Jiangsu Machine Building Automation no 1pp 20ndash22 2003

[5] Y S ShenTheory of machines and mechanisms Tsinghua Uni-versity Press Beijing China 2nd edition 2005

[6] S Bai D Wang and H Dong ldquoA unified formulation fordimensional synthesis of Stephenson linkagesrdquo Journal ofMech-anisms and Robotics vol 8 no 4 Article ID 041009 2016

[7] Y Liu and W B Xiao ldquoProgress in research on the theory ofmechanisms for path generationrdquo Journal of Computer-AidedDesign and Computer Graphics vol 17 no 4 pp 627ndash636 2005

[8] J A Cabrera A Simon and M Prado ldquoOptimal synthesis ofmechanisms with genetic algorithmsrdquoMechanism andMachineTheory vol 37 no 10 pp 1165ndash1177 2002

[9] I Uiiah and S Kota ldquoOptimal synthesis of mechanisms forpath generation using fourier descriptors and global searchmethodsrdquo Journal ofMechanical Design vol 119 no 10 pp 504ndash510 1997

[10] Z Li and C L Gui ldquoMorphologically Analyzingmethod for thecharacteristic of coupler curves of linkage mechanismrdquo ChineseJournal ofMechanical Engineering vol 38 no 2 pp 27ndash30 2002

[11] W Q Cao ldquoThe identification of coupler curve by usingfuzzy mathematicsrdquo Journal of Shanxi Institute of MechanicalEngineering no 2 pp 21ndash28 1992

[12] X B Sun R B Xiao and XWang ldquoAn approach tomechanicalpath generation and retrieve based on quantification andextraction of the geometrical features of curverdquo Chinese Journalof Mechanical Engineering vol 36 no 11 pp 98ndash105 2000

[13] G Nelson Larsen Analysis of the four-bar linkage Publishedjointly by the Technology Press of the Massachusetts Instituteof Technology and Wiley New York NY USA 1951

[14] S N Kramer ldquoSelective precision synthesis of the fourmdashbarmotion generator with prescribed input timingrdquo Journal ofMechanical Design vol 101 no 11 pp 614ndash618 1979

[15] T Subbian and D R Flugrad ldquoFour-bar path generationsynthesis by a continuation methodrdquo Journal of MechanicalDesign vol 113 no 1 pp 63ndash69 1991

[16] D S Johnson C R Aragon L A McGeoch and CSchevon ldquoOptimization by simulated annealing an experimen-tal evaluationmdashPart II graph coloring and number partition-ingrdquo Operations Research vol 39 no 3 pp 378ndash406 1991

[17] S Kirkpatrick C D Gelatt Jr and M P Vecchi ldquoOptimizationby simulated annealingrdquo American Association for the Advance-ment of Science vol 220 no 4598 pp 671ndash680 1983

[18] J R Wang and X Q Huang ldquoOptimization of the shaft of arolling shear with single shaft and double eccentricityrdquo HeavyMachinery no 1 2005

[19] J G Yang Z W Yue and H C Zhang ldquoOptimization of shearmechanism in plate shear with roll guide grooverdquo Journal ofTaiyuan University of Science amp Technology vol 24 no 4 pp297ndash300 2006

[20] S F Sun X H Yang Q X Yan and D Y Yang ldquoSimulationresearch of mechanism parameter on cut-to-length rollingshearrdquoMetallurgical Equipment no 170 2008

[21] M A Li-Feng Q X Huang L I Ying et al ldquoEstablishment andapplication of mathematical model on spatial shear mechanismoptimization of new-type steel rolling shearrdquo Journal of SichuanUniversity no 2 pp 170ndash174 2008

[22] Q Huang L Ma J Li J Wang Q Zhang and C ZhaoldquoPrinciple of asymmetric crank mechanism of new-type rollingshearrdquo Chinese Journal of Mechanical Engineering vol 44 no 5pp 119ndash123 2008

[23] L J Meng and Y W Chen ldquoApplication of offset mechanism ofcrankshaft in rolling-type heavy steel shearrdquo Journal of TaiyuanUniversity of Science amp Technology 2009


[24] D L Wang and Y Gao Mechanical Principle China MachinePress Beijing China 2011

[25] H X Lin W Huang T X Lin et al ldquoStudy of syntheticallygenetics annealing optimization for the luffing mechanismtrajectory of a plane linkrdquo Journal of Computer-Aided Design ampComputer Graphics vol 13 no 8 pp 724ndash729 2001

[26] X P Zhang G J Yang and J F Wang ldquoAnalysis of factorsinfluencing cutting quality and strength parameters of rollingcut shearsrdquo China Mechanical Engineering no 7 pp 780ndash7842008

[27] S L Li X L Qin and G B Zhao ldquoRolling shear characteristicsand analysis of key parametersrdquo Industrial Technology no 242008

[28] J M Wen and F Wang ldquoShear force calculation of the rollingshear mechanismrdquoMetallurgical Equipment no 1 2011

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Journal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


The indirect synthesis method [10 12] is used to searchfor the matching trajectory from the predefined trajectoryatlas instead of directly generating a mechanism schemewhich is done by analyzing the expected trajectory and thenexporting the corresponding mechanism types and sizes Ifthere is a similar scheme the minimum trajectory deviationwill be obtainedThe indirect synthesis method mainly relieson the mass data-storage capacity and rapid retrieval abilityof a computer Although the rapid improvement of computerspromotes the application and development of the indirectsynthesis the difficulties of the establishment of a trajectoryatlas the mass data-storage capacity of a computer and theapproach to effectively search for the bestmatching trajectoryare still challenging problems to be solved

For the problem here the design of seven-bar linkagesfor pure rolling needs to meet both the trajectory and alsoother requirements formachining that is steel plates cuttingThe shear motion of a rolling shear mechanism is generallyrealized by means of the relative motion between the uppershear blade and the lower shear blade The expected shearmotion should be a pure-rolling motion without slipping Inthis regard Wang and Huang [18] developed an optimizedmodel for rolling shear mechanism with single shaft anddouble eccentricity choosing four motion positions as accesspoints to acquire the expected motions while the phasedifference was set to be identical Yang et al [19] used theconstraints of equal radius of crank and equal length oflinkage to set up an optimization model of rolling shearmechanism with roll guide groove Sun et al [20] designeda rolling shear mechanism by optimizing the trajectory ofthe lowest moving point of the upper shear blade but theupper shear blade could not perform pure-rolling motionrelative to the lower blade due to the horizontal slide In [21]a synthesis-optimized model was built to design a rollingshearmechanism using a guiding rod as an additional designvariable while identical phase difference and identical lengthbetween the designed guide rod and the expected guiderod are adopted for four positions In order to improveshear quality decrease blade wear and prolong blade life ofthe cutting machine [22] generally the pure-rolling motionbetween the shear blades can be transformed into a series ofmoving positions and phase angles of the seven-bar linkageswith which an optimizedmethod is adopted to obtain properlinkage sizes In certain situations the synthesis can onlysatisfy some key points the motion accuracy of the designedpure-rolling cutting mechanism is thus low It is difficultto realize the pure-rolling motion during the whole cuttingprocess due to the fact that the cutting performance was notconsidered or embodied in the synthesis

This paper proposes a method for the kinematic design ofa seven-bar linkage to generate pure-rolling motion by opti-mizing the centrodes The introduced method is developedbased upon the interrelation between the centrodes and con-tacting lines of pure-rolling motion A case study of seven-bar rolling shear mechanism is included to demonstrate themethod to accomplish the pure-rolling motion A geneticoptimization algorithm is used to obtain mechanism sizeswith the metric function of minimum approximation errorbetween mechanism centrodes and expected trajectories of

A F

BE

HG

C D

MN

Upper arc blade

Steel plate


휔1 휔2

Figure 1 Sketch of a seven-bar rolling shear mechanism

shear blade The constraints of the formulated optimizationproblem for the pure-rolling mechanism include the designrequirements of the opening distance the maximum amountof overlap error and peak value of shearing force Moreoverthe performance of the newly designed rolling shear mech-anism is investigated and compared with the original onewhich shows the advantages of the new method

2 Design Model

21 Design Issue and Problem Formulation The rolling shearmechanism is a typical pure-rolling cutting mechanismwhich is usually used for medium plate rolling shear Theseven-bar linkagemechanism is a common application of therolling shear mechanism [23] as shown in Figure 1

The seven-bar linkage has 2 degrees of freedom cor-responding to cranks 119860119861 and 119864119865 as driving links whichrotate by the same angular velocity and with a constant phasedifference sharing a power input Link 119862119863119866 to which theupper blade is attached outputs motion Generally the lowershear blade is fixed on the frame while the upper shear blademoves relative to the lower shear blade to cut the steel platebetween them as shown in Figure 2

The horizontal sliding of the upper shear blade should beas little as possible to reduce thewear of the bladeMeanwhilethe cutting depth of the upper shear blade should be the sameto reduce the bending deformation of the steel plate ensuringa stable cutting quality of the steel plate Thus the idealmotion of the upper blade should be pure-rolling cuttingrelative to the steel plate during the shearing process to makesure there is no horizontal sliding between the blade and thesteel plate at the cutting contact point

One of the rigid bodies is usually chosen to be fixed andanother moves relative to the chosen one for convenienceduring the motion analysis of two rigid bodies as shown inFigure 3 Rigid body II is fixed in coordinate system 119874119891 minus119909119891119910119891 Rigid body I on which a moving coordinate system119874119898 minus 119909119898119910119898 is built moves in the fixed coordinate system119874119891minus119909119891119910119891 A point119874119898 of body I has a velocity V119874

119898

and body Irotates about point119874119898 with angular velocity120596119874

119898

Themotionstate of body I at any moment is either (a) entire translation


Position 1

Position 3

Position 2

Position 4


II

I

P

yf

Of xf

ym

Om xm

휔0Γ1

Γ2

O푚

휔O푚














(3)



A

B E

F


1

2

3

4

5

67

HD

G

C


A

B E

F

1

2

3

4

5

6

7

H

D

G

C



B

A F

E

C

G

P

D

H

y

x

K

L

yG

ym

xm

xG

Γ1

Γ2

L1 L2

L3 L4

L5 L6

L7

L8

L9

L10

L11휃1 휃2

휃3

휃4

휃5

휃6

훼훽1

훽2




(4)




1199091199101= M119867119866M119866119871

1199091198981199101198981 (5)



]]]


]]]

(6)



1199100= [[[






119909 = 119871119866119871 (1205965 + 1) cos (1205731 + 1205795) + 1198716 (1205966 + 1) cos 1205796119910 = 119871119866119871 (1205965 + 1) sin (1205731 + 1205795) + 1198716 (1205966 + 1) sin 1205796 (9)







t = [1199051 1199052 119905119899]119879 t isin 119877119899 (10)



min 119880 (t)119880 (t) = 1198801 (t) + 1198802 (t) (11)



119910119898 = 119892119898119905 (119909119898) 119910119898 = 119891119898 (119909119898) (1199091198981 le 119909119898 le 119909119898119899) (12)


VW

S

B

R

Upper arc blade

Steel plate


UC

D G H

훼

훽

ℎmax


x

y

O

CP





xm1 x

m2 xmj

Om

xmq

xm

ym

xi xpx2x1



119910 = 119892119905 (119909) 119910 = 119862 (1199091 le 119909 le 119909119899) (13)



1198801 (t) = radic 1119899119899sum119894=1

(119892119905 (119909119894) minus 119862)2

1198802 (t) = radic 1119899119899sum119894=1

(119892119898119905 (119909119898119894) minus 119891119898 (119909119898119894))2(14)


119880 (t) = radic 1119899119899sum119894=1

(119892119905 (119909119894) minus 119891 (119909119894))2

+ radic 1119899119899sum119894=1

(119892119898119905 (119909119898119894) minus 119891119898 (119909119898119894))2(15)



119867(t) ge 119870 (16)






119875 = 06120590119887120575sdot ℎ2tan120572 (1 + 119885 tan12057206120575 + 11 + 1001205751205901198871198842119883)

(19)



Platewidth 119861




3500mm 50mm 5mm 224∘ ge200mm


120572st ge 1205720 (20)










minus600

minus500

minus400

minus300

minus200

minus100

0

100

200

300

400

500







1500

100050

00

2000

2500

minus10

00

minus15

00

minus20

00

minus50

0

minus25

00

minus600

minus500

minus400

minus300

minus200

minus100

0



minus28

75

minus28

70

minus28

65

minus28

60

minus28

55

minus28

50

minus28

45

minus28

40

minus28

35

minus28

30

minus28

25

minus400

minus350

minus300

minus250

minus200


(a) Overall view

minus28

40

minus28

39

minus28

38

minus28

37

minus28

36

minus28

35

minus28

34

minus28

33

minus28

32

minus28

31

minus28

30

minus410

minus405

minus400

minus395

minus390

minus385

minus380










minus400

minus350

minus300

minus250

minus200


Original design

Shearing process

New design

(a) Overall view


minus408

minus407

minus406

minus405

minus404

minus403

minus402

minus401

minus400




Angle of crank AB (∘)310300290280270260250240230220


08

1

12

14

16

18

2

22

24

26

Shea

r ang

le (∘

)

(a) Shear angle

times107

Angle of crank AB (∘)310300290280270260250240230220

07

08

09

1

11

12

13

14

15

16

17

Shea

r for

ce (N

)


(b) Shear force







5 Conclusions








Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





Position 1

Position 3

Position 2

Position 4


II

I

P

yf

Of xf

ym

Om xm

휔0Γ1

Γ2

O푚

휔O푚














(3)



A

B E

F


1

2

3

4

5

67

HD

G

C


A

B E

F

1

2

3

4

5

6

7

H

D

G

C



B

A F

E

C

G

P

D

H

y

x

K

L

yG

ym

xm

xG

Γ1

Γ2

L1 L2

L3 L4

L5 L6

L7

L8

L9

L10

L11휃1 휃2

휃3

휃4

휃5

휃6

훼훽1

훽2




(4)




1199091199101= M119867119866M119866119871

1199091198981199101198981 (5)



]]]


]]]

(6)



1199100= [[[






119909 = 119871119866119871 (1205965 + 1) cos (1205731 + 1205795) + 1198716 (1205966 + 1) cos 1205796119910 = 119871119866119871 (1205965 + 1) sin (1205731 + 1205795) + 1198716 (1205966 + 1) sin 1205796 (9)







t = [1199051 1199052 119905119899]119879 t isin 119877119899 (10)



min 119880 (t)119880 (t) = 1198801 (t) + 1198802 (t) (11)



119910119898 = 119892119898119905 (119909119898) 119910119898 = 119891119898 (119909119898) (1199091198981 le 119909119898 le 119909119898119899) (12)


VW

S

B

R

Upper arc blade

Steel plate


UC

D G H

훼

훽

ℎmax


x

y

O

CP





xm1 x

m2 xmj

Om

xmq

xm

ym

xi xpx2x1



119910 = 119892119905 (119909) 119910 = 119862 (1199091 le 119909 le 119909119899) (13)



1198801 (t) = radic 1119899119899sum119894=1

(119892119905 (119909119894) minus 119862)2

1198802 (t) = radic 1119899119899sum119894=1

(119892119898119905 (119909119898119894) minus 119891119898 (119909119898119894))2(14)


119880 (t) = radic 1119899119899sum119894=1

(119892119905 (119909119894) minus 119891 (119909119894))2

+ radic 1119899119899sum119894=1

(119892119898119905 (119909119898119894) minus 119891119898 (119909119898119894))2(15)



119867(t) ge 119870 (16)






119875 = 06120590119887120575sdot ℎ2tan120572 (1 + 119885 tan12057206120575 + 11 + 1001205751205901198871198842119883)

(19)



Platewidth 119861




3500mm 50mm 5mm 224∘ ge200mm


120572st ge 1205720 (20)










minus600

minus500

minus400

minus300

minus200

minus100

0

100

200

300

400

500







1500

100050

00

2000

2500

minus10

00

minus15

00

minus20

00

minus50

0

minus25

00

minus600

minus500

minus400

minus300

minus200

minus100

0



minus28

75

minus28

70

minus28

65

minus28

60

minus28

55

minus28

50

minus28

45

minus28

40

minus28

35

minus28

30

minus28

25

minus400

minus350

minus300

minus250

minus200


(a) Overall view

minus28

40

minus28

39

minus28

38

minus28

37

minus28

36

minus28

35

minus28

34

minus28

33

minus28

32

minus28

31

minus28

30

minus410

minus405

minus400

minus395

minus390

minus385

minus380










minus400

minus350

minus300

minus250

minus200


Original design

Shearing process

New design

(a) Overall view


minus408

minus407

minus406

minus405

minus404

minus403

minus402

minus401

minus400




Angle of crank AB (∘)310300290280270260250240230220


08

1

12

14

16

18

2

22

24

26

Shea

r ang

le (∘

)

(a) Shear angle

times107

Angle of crank AB (∘)310300290280270260250240230220

07

08

09

1

11

12

13

14

15

16

17

Shea

r for

ce (N

)


(b) Shear force







5 Conclusions








Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





A

B E

F


1

2

3

4

5

67

HD

G

C


A

B E

F

1

2

3

4

5

6

7

H

D

G

C



B

A F

E

C

G

P

D

H

y

x

K

L

yG

ym

xm

xG

Γ1

Γ2

L1 L2

L3 L4

L5 L6

L7

L8

L9

L10

L11휃1 휃2

휃3

휃4

휃5

휃6

훼훽1

훽2




(4)




1199091199101= M119867119866M119866119871

1199091198981199101198981 (5)



]]]


]]]

(6)



1199100= [[[






119909 = 119871119866119871 (1205965 + 1) cos (1205731 + 1205795) + 1198716 (1205966 + 1) cos 1205796119910 = 119871119866119871 (1205965 + 1) sin (1205731 + 1205795) + 1198716 (1205966 + 1) sin 1205796 (9)







t = [1199051 1199052 119905119899]119879 t isin 119877119899 (10)



min 119880 (t)119880 (t) = 1198801 (t) + 1198802 (t) (11)



119910119898 = 119892119898119905 (119909119898) 119910119898 = 119891119898 (119909119898) (1199091198981 le 119909119898 le 119909119898119899) (12)


VW

S

B

R

Upper arc blade

Steel plate


UC

D G H

훼

훽

ℎmax


x

y

O

CP





xm1 x

m2 xmj

Om

xmq

xm

ym

xi xpx2x1



119910 = 119892119905 (119909) 119910 = 119862 (1199091 le 119909 le 119909119899) (13)



1198801 (t) = radic 1119899119899sum119894=1

(119892119905 (119909119894) minus 119862)2

1198802 (t) = radic 1119899119899sum119894=1

(119892119898119905 (119909119898119894) minus 119891119898 (119909119898119894))2(14)


119880 (t) = radic 1119899119899sum119894=1

(119892119905 (119909119894) minus 119891 (119909119894))2

+ radic 1119899119899sum119894=1

(119892119898119905 (119909119898119894) minus 119891119898 (119909119898119894))2(15)



119867(t) ge 119870 (16)






119875 = 06120590119887120575sdot ℎ2tan120572 (1 + 119885 tan12057206120575 + 11 + 1001205751205901198871198842119883)

(19)



Platewidth 119861




3500mm 50mm 5mm 224∘ ge200mm


120572st ge 1205720 (20)










minus600

minus500

minus400

minus300

minus200

minus100

0

100

200

300

400

500







1500

100050

00

2000

2500

minus10

00

minus15

00

minus20

00

minus50

0

minus25

00

minus600

minus500

minus400

minus300

minus200

minus100

0



minus28

75

minus28

70

minus28

65

minus28

60

minus28

55

minus28

50

minus28

45

minus28

40

minus28

35

minus28

30

minus28

25

minus400

minus350

minus300

minus250

minus200


(a) Overall view

minus28

40

minus28

39

minus28

38

minus28

37

minus28

36

minus28

35

minus28

34

minus28

33

minus28

32

minus28

31

minus28

30

minus410

minus405

minus400

minus395

minus390

minus385

minus380










minus400

minus350

minus300

minus250

minus200


Original design

Shearing process

New design

(a) Overall view


minus408

minus407

minus406

minus405

minus404

minus403

minus402

minus401

minus400




Angle of crank AB (∘)310300290280270260250240230220


08

1

12

14

16

18

2

22

24

26

Shea

r ang

le (∘

)

(a) Shear angle

times107

Angle of crank AB (∘)310300290280270260250240230220

07

08

09

1

11

12

13

14

15

16

17

Shea

r for

ce (N

)


(b) Shear force







5 Conclusions








Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





1199100= [[[






119909 = 119871119866119871 (1205965 + 1) cos (1205731 + 1205795) + 1198716 (1205966 + 1) cos 1205796119910 = 119871119866119871 (1205965 + 1) sin (1205731 + 1205795) + 1198716 (1205966 + 1) sin 1205796 (9)







t = [1199051 1199052 119905119899]119879 t isin 119877119899 (10)



min 119880 (t)119880 (t) = 1198801 (t) + 1198802 (t) (11)



119910119898 = 119892119898119905 (119909119898) 119910119898 = 119891119898 (119909119898) (1199091198981 le 119909119898 le 119909119898119899) (12)


VW

S

B

R

Upper arc blade

Steel plate


UC

D G H

훼

훽

ℎmax


x

y

O

CP





xm1 x

m2 xmj

Om

xmq

xm

ym

xi xpx2x1



119910 = 119892119905 (119909) 119910 = 119862 (1199091 le 119909 le 119909119899) (13)



1198801 (t) = radic 1119899119899sum119894=1

(119892119905 (119909119894) minus 119862)2

1198802 (t) = radic 1119899119899sum119894=1

(119892119898119905 (119909119898119894) minus 119891119898 (119909119898119894))2(14)


119880 (t) = radic 1119899119899sum119894=1

(119892119905 (119909119894) minus 119891 (119909119894))2

+ radic 1119899119899sum119894=1

(119892119898119905 (119909119898119894) minus 119891119898 (119909119898119894))2(15)



119867(t) ge 119870 (16)






119875 = 06120590119887120575sdot ℎ2tan120572 (1 + 119885 tan12057206120575 + 11 + 1001205751205901198871198842119883)

(19)



Platewidth 119861




3500mm 50mm 5mm 224∘ ge200mm


120572st ge 1205720 (20)










minus600

minus500

minus400

minus300

minus200

minus100

0

100

200

300

400

500







1500

100050

00

2000

2500

minus10

00

minus15

00

minus20

00

minus50

0

minus25

00

minus600

minus500

minus400

minus300

minus200

minus100

0



minus28

75

minus28

70

minus28

65

minus28

60

minus28

55

minus28

50

minus28

45

minus28

40

minus28

35

minus28

30

minus28

25

minus400

minus350

minus300

minus250

minus200


(a) Overall view

minus28

40

minus28

39

minus28

38

minus28

37

minus28

36

minus28

35

minus28

34

minus28

33

minus28

32

minus28

31

minus28

30

minus410

minus405

minus400

minus395

minus390

minus385

minus380










minus400

minus350

minus300

minus250

minus200


Original design

Shearing process

New design

(a) Overall view


minus408

minus407

minus406

minus405

minus404

minus403

minus402

minus401

minus400




Angle of crank AB (∘)310300290280270260250240230220


08

1

12

14

16

18

2

22

24

26

Shea

r ang

le (∘

)

(a) Shear angle

times107

Angle of crank AB (∘)310300290280270260250240230220

07

08

09

1

11

12

13

14

15

16

17

Shea

r for

ce (N

)


(b) Shear force







5 Conclusions








Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





VW

S

B

R

Upper arc blade

Steel plate


UC

D G H

훼

훽

ℎmax


x

y

O

CP





xm1 x

m2 xmj

Om

xmq

xm

ym

xi xpx2x1



119910 = 119892119905 (119909) 119910 = 119862 (1199091 le 119909 le 119909119899) (13)



1198801 (t) = radic 1119899119899sum119894=1

(119892119905 (119909119894) minus 119862)2

1198802 (t) = radic 1119899119899sum119894=1

(119892119898119905 (119909119898119894) minus 119891119898 (119909119898119894))2(14)


119880 (t) = radic 1119899119899sum119894=1

(119892119905 (119909119894) minus 119891 (119909119894))2

+ radic 1119899119899sum119894=1

(119892119898119905 (119909119898119894) minus 119891119898 (119909119898119894))2(15)



119867(t) ge 119870 (16)






119875 = 06120590119887120575sdot ℎ2tan120572 (1 + 119885 tan12057206120575 + 11 + 1001205751205901198871198842119883)

(19)



Platewidth 119861




3500mm 50mm 5mm 224∘ ge200mm


120572st ge 1205720 (20)










minus600

minus500

minus400

minus300

minus200

minus100

0

100

200

300

400

500







1500

100050

00

2000

2500

minus10

00

minus15

00

minus20

00

minus50

0

minus25

00

minus600

minus500

minus400

minus300

minus200

minus100

0



minus28

75

minus28

70

minus28

65

minus28

60

minus28

55

minus28

50

minus28

45

minus28

40

minus28

35

minus28

30

minus28

25

minus400

minus350

minus300

minus250

minus200


(a) Overall view

minus28

40

minus28

39

minus28

38

minus28

37

minus28

36

minus28

35

minus28

34

minus28

33

minus28

32

minus28

31

minus28

30

minus410

minus405

minus400

minus395

minus390

minus385

minus380










minus400

minus350

minus300

minus250

minus200


Original design

Shearing process

New design

(a) Overall view


minus408

minus407

minus406

minus405

minus404

minus403

minus402

minus401

minus400




Angle of crank AB (∘)310300290280270260250240230220


08

1

12

14

16

18

2

22

24

26

Shea

r ang

le (∘

)

(a) Shear angle

times107

Angle of crank AB (∘)310300290280270260250240230220

07

08

09

1

11

12

13

14

15

16

17

Shea

r for

ce (N

)


(b) Shear force







5 Conclusions








Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






Platewidth 119861




3500mm 50mm 5mm 224∘ ge200mm


120572st ge 1205720 (20)










minus600

minus500

minus400

minus300

minus200

minus100

0

100

200

300

400

500







1500

100050

00

2000

2500

minus10

00

minus15

00

minus20

00

minus50

0

minus25

00

minus600

minus500

minus400

minus300

minus200

minus100

0



minus28

75

minus28

70

minus28

65

minus28

60

minus28

55

minus28

50

minus28

45

minus28

40

minus28

35

minus28

30

minus28

25

minus400

minus350

minus300

minus250

minus200


(a) Overall view

minus28

40

minus28

39

minus28

38

minus28

37

minus28

36

minus28

35

minus28

34

minus28

33

minus28

32

minus28

31

minus28

30

minus410

minus405

minus400

minus395

minus390

minus385

minus380










minus400

minus350

minus300

minus250

minus200


Original design

Shearing process

New design

(a) Overall view


minus408

minus407

minus406

minus405

minus404

minus403

minus402

minus401

minus400




Angle of crank AB (∘)310300290280270260250240230220


08

1

12

14

16

18

2

22

24

26

Shea

r ang

le (∘

)

(a) Shear angle

times107

Angle of crank AB (∘)310300290280270260250240230220

07

08

09

1

11

12

13

14

15

16

17

Shea

r for

ce (N

)


(b) Shear force







5 Conclusions








Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





1500

100050

00

2000

2500

minus10

00

minus15

00

minus20

00

minus50

0

minus25

00

minus600

minus500

minus400

minus300

minus200

minus100

0



minus28

75

minus28

70

minus28

65

minus28

60

minus28

55

minus28

50

minus28

45

minus28

40

minus28

35

minus28

30

minus28

25

minus400

minus350

minus300

minus250

minus200


(a) Overall view

minus28

40

minus28

39

minus28

38

minus28

37

minus28

36

minus28

35

minus28

34

minus28

33

minus28

32

minus28

31

minus28

30

minus410

minus405

minus400

minus395

minus390

minus385

minus380










minus400

minus350

minus300

minus250

minus200


Original design

Shearing process

New design

(a) Overall view


minus408

minus407

minus406

minus405

minus404

minus403

minus402

minus401

minus400




Angle of crank AB (∘)310300290280270260250240230220


08

1

12

14

16

18

2

22

24

26

Shea

r ang

le (∘

)

(a) Shear angle

times107

Angle of crank AB (∘)310300290280270260250240230220

07

08

09

1

11

12

13

14

15

16

17

Shea

r for

ce (N

)


(b) Shear force







5 Conclusions








Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





minus400

minus350

minus300

minus250

minus200


Original design

Shearing process

New design

(a) Overall view


minus408

minus407

minus406

minus405

minus404

minus403

minus402

minus401

minus400




Angle of crank AB (∘)310300290280270260250240230220


08

1

12

14

16

18

2

22

24

26

Shea

r ang

le (∘

)

(a) Shear angle

times107

Angle of crank AB (∘)310300290280270260250240230220

07

08

09

1

11

12

13

14

15

16

17

Shea

r for

ce (N

)


(b) Shear force







5 Conclusions








Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






5 Conclusions








Acknowledgments


References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of

















Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




kinematic design of a seven-bar linkage with optimized

Documents