kinematic design method for six-bar linkage sley drive...

Indian Journal of Fibre & Textile Research Vol. 30, September 2005, pp. 243-25 1

Kinematic design method for six-bar linkage sley drive mechanisms in weaving

Recep Erena & Ahmet Aydemir

Department of Textile Engineering, Faculty of Engineering and Architecture, University of Uludag, GbrUkle Campus, GcrUkle 1 6059, B ursa, Turkey

Received 27 May 2004; revised received 15 October 2004; accepted 28 December 2004

A kinematic design method has been developed for calculating the l ink lengths of a s ix-bar l inkage sley drive mechan.ism satisfying the required sley motion. I nitially, a six-bar mechanism is divided into a crank rocker mechanism and a double rocker mechanism and then the equations are i ntroduced for the kinematic design and analysis. The effect of mechanism parameters on sley motion is also investigated. It is observed that the 6.<p', which is a freely chosen design parameter of the double rocker mechanism, has the most significant effect on sley motion and that the effect of crank rocker mechanism on sley motion is also significant. Other mechanism parameters have no significant effect on sley motion. It i s also observed in the design trials that the practically applicable l i nk lengths can be obtained while the sley motion requirements are being satisfied.

Keywords: Six-bar l i nkage sley drive mechanism, Sley motion, Weaving IPC Code: 1 nt.C1 7 D03C1 9/00

1 Introduction Six-bar l inkages have been used to drive the sley i n

various types of weaving machines such a s shuttle looms, early types of rapier looms where weft insertion system moves together with sley and some air-jet looms. Six-bar mechanisms are preferred i n sley drive in the cases where a four bar mechanism does not satisfy the sley motion requirements for an optimum shed geometry and where a cam mechanism is not preferred because of i ts higher manufacturing cost and a high precision required for its manufacturi ng.

Fig. ! shows a sley motion driven by a six-bar linkage. The six-bar l inkage consists of 2 four-bar mechanisms connected in series. The first mechanism (AoABBo) i s called a crank rocker mechanism (link 2 rotates continuously whereas link 4 swings) and the second mechanism (BoCDDo) is called double rocker mechanism (both l inks 4 and 6 swing). The working principle of the mechanism is as follows: the crank (link 2) rotates continuously at the main shaft speed of the loom. The motion of the crank is transmitted to the rocker (link 4) by the coupler link (link 3). During one revolution of the crank, the rocker swings between its two extreme positions. The extreme positions of the rocker are reached when the crank

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and the coupler l ie in a straight line (Fig. 2) . The rocker executes the angle «l40 between i ts extreme positions. The link 4 is also the drive arm for the double rocker mechanism and its motion is transmitted to the link 6 by the link 5. As the l ink 4 swings, the l ink 6 also swings between its extreme positions by the angle «l60. Because the sley and the l ink 6 are the same rigid body, «l6o is also the sley swing angle. The most forward position of the sley (beat up point) is attained when the l inks 2 and 3 are extended and the rearmost position is attained when the links 2 and 3 are folded on top of each other. Currently, thiS s ix-bar linkage sley drive mechanism is used with especially wide air-jet looms of speed exceeding 500 rpm (ref. I ) .

While reviewing the l iterature, only one research paper2 regarding the design of s ix-bar l inkage sley dri ve mechanisms could be retrieved. In this research paper2, a six-bar l inkage sley drive mechanism was designed so as to minimize the sley angular acceleration and the forces affecting the joints of mechanism. The mechanism l ink lengths were generated randomly within their allowed maximum and minimum values. No mathematical formulation was i ntroduced to calculate the mechanism link lengths. The other available l iterature i s mainly limited to the qualitative assessments3-5 .

It i s well known from the previous studies6- 1 3 that the beat up force depends on cloth fel l position which

244 INDIAN J. FIBRE TEXT. RES .. SEPTEMBER 2005

warp

h 1. _ _ _ _ I

B

Fig. l -Six-bar l inkage sley drive mechanism [Ao. Bo and Do -the joints between the machine frame and the l inks 2. 4 and 6 respectively; A - joint between l inks 2 and 3; B - joint between l inks 3 and 4; C - joint between l inks 4 and 5; and D - joint between l inks 5 and 6. The numbers 2 - 6 are the l ink numbers as indicated in the figure]

Bo

BI!

, _ II <P20 I / /

I I

Fig. 2-Extreme positions of crank rocker mechanism [A' and A" - positions of joint A at the extreme positions of the mechanism: B' and B" - positions of joint B at the extreme positions of mechanism; <1>40 - swing angle of link 4; and <1>20 -angular displacement of l ink 2 between its extreme positions]

is determined by fabric construction and some of the machine settings. The effect of sley motion curve and front dwel l period of sley on beat up force has been studied using a computer controlled hydraulic sley drive mechanism 14 . It is found that the effect of sley motion curve on maximum value of beat up force i s not significant. But, the front dwell of the sley (dwell of the sley at its most forward position) significantly affects the beat up force and an i ncrease in the dwell period reduces the beat up force. Some special sley

drive mechanisms having double beating action serve the same purpose. A sley drive mechanism with a front dwell or double beating action can be suitable for weaving especial ly heavy or dense fabrics at slower speeds. At shuttleless looms with high running speeds, this i s not a common practice. Moreover. it is not possible to obtain a sley front dwell with a four bar mechanism and a six bar mechanism (Fig. I ) . Therefore, i t wil l be sufficient, in many cases, to design a sley mechanism to obtai n a sley motion for an optimum shed geometry without taking into account the beat up force . This paper reports an analytical method to calculate l i nk lengths of a six-bar sley drive mechanism for a required sley motion. Link lengths of a l inkage are defined as the distance between the centers of its joints. The notation for the l ink lengths of this six-bar l inkage sley drive mechanism, which is used in the following sections for the design and analysis of the mechanism, is given below:

'i = [AoBo l , r2 = IAoA I , r) = IAB I , r4 = IBoB I ' r; = [BoC I , rs = ICD I , r6 = 10001 , and r7 = IBoDo l

2 Materials and Methods

2.1 Design Problem

It i s required that the kinematic design of this sixbar sley drive mechanism satisfies the fol lowing weavi ng technology and mechanical criteria:

• Each of forward and return movements of the sley takes place during 1 80° crank rotation.

• A sley swing angle, «J60. • A sley motion suitable for the shed geometry of a

weaving machine. • Transmission angle. It i s a parameter whose value

is l imited to a minimum deviation from 90° in the kinematic design of mechanisms. As the deviation i ncreases, the force component affect ing bearings increases . Although the amount of deviation allowed in practice depends on appl ication, up to 30° deviation i s accepted i n general .

2.2 Design Equations

Design of a s ix-bar sley drive mechanism is carried out by dividing i t i nto 2 four-bar mechanisms. namely the crank rocker mechanism and the double rocker mechanism, and then by developing design equations seperately for each mechanism.

EREN & A Y DEMIR: SIX-BAR LINKAGE SLEY DRIVE MECHANISMS IN W EAVING 245

2.2.1 Crank Rocker Mechanism

The sley movement between its most forward and rearmost positions is required to correspond to 1 800 crank rotation. To satisfy this condition, the crank rocker mechanism must be designed with the time ratio of I , which means that the swing of l ink 4 in both the directions corresponds to 1 800 crank rotation. The extreme positions of this crank rocker mechanism is shown in Fig. 2, where <j)20 is equal to 1 800• This type of crank rocker mechanism is cal led a centric crank rocker mechanism.

For this crank rocker mechanism, link ratios can be calculated using the fol lowing equations (derivation of these equations is given in Appendix 1 ) :

sin <j)40 r) = 2 Ij cos /lmin

l -r� y � I - sin 2 <j)40

2

. . . ( 1 )

. . . (2)

. . . (3)

where <j)40 is the swing angle of link 4 between its extreme positions as shown in Fig. 2; /lmilh the minimum value of transmission angle (defined and expressed mathematically in Appendix 1 ); rJ, the independent design parameter and hence can be selected freely ; and r2, r3 and r4, the link lengths calculated respectively from Eqs (3), ( 1 ) and (2) for given values of <j)40 and /lmin' The value of rl does not affect the motion characteristic of link 4 as long as the link ratios remain constant. Once <j)40 is chosen according to the required motion amplitute, the crank rocker mechanism can be designed w ith different values of /lmin ' In each case, different l ink ratios and hence different motion curves for l ink 4 are obtained. This affects s\ey motion, as discussed in section 3.

2.2.2 Double Rocker Mechanism

From the design of crank rocker mechan.i.sm, the input motion for the double rocker mechanism (<j)40) i s determined. The output is the s\ey swing angle (<j)60). The design problem for the double rocker mechanism

is the determination of link lengths r'4, rs and r6 satisfying angular displacements of the links 4 (<j)40) and 6 (<j)60). r7 is the freely selected l ink length of double rocker mechanism. Freudenstein ' s equation, used in the design of double rocker mechanism considering Fig. 3 (ref. I S), is given below:

where <j)'4 and <j)6 are the angular positions of l inks 4 and 6 respectively; and KJ, K2 and K3, the three constants defined in terms of double rocker mechanism l ink lengths which are given below:

K, = r7 /r'4 ' K2 = r7 /r6 '

K) = (r412 - r/ + r62 + r/ )/(2r: r6 )

As Freudenstein ' s equation is valid for all positions of a four-bar linkage, following equations can be written for the two positions of double rocker mechanism considering Fig. 4:

where <j)'41 and <j)61 are the angular positions of l inks 4 and 6 for the first position; and <j)'42 and <j)62, the angular positions of l inks 4 and 6 for the second position of the double rocker mechanism.

On the other hand, a six-bar l inkage s\ey drive mechanism produces an approximate dwell around its

5

C

4

Bo

./ ./

D '1'6

Fig. 3-Angular positons of the l inks of double rocker mechanism [<P'4 - angular position of l ink 4; <p, - angular position of l ink 5 ;

and <P6 - angular position of l i nk 6 J

246 INDIAN J. FIBRE TEXT. RES., SEPTEMBER 2005

./ ./ ./

./ ./ ./

Fig. 4-0ead center positIOn of double rocker mechanism [<P'41 - first angular position of l ink 4; <P'42 - second angular position of l ink 4; <P61 - firsl angular position of link 6; q>62 -second angular position of l ink 6; C,q>'4 - angular displacement of l ink 4 beyond its dead center position during which the l i nk 6 is rquired (0 dwel l; C 1 and C2 - first and second positions of joint C ; and 01 and O2 - first and second positions of joint 0]

rearmost position so that a long weft insertion interval is obtained. As shown in Fig. 4, this occurs at the dead center position of the double rocker mechanism at which the angular velocity of l ink 6 (the sley) is zero. A third equation is obtained for this condition by taking the partial derivative of Freudenstein' s equation with respect to <P'4 and equating O<pJO<P'4=0, <P'4=<P'42-�<P'4 and <P6=<P62 (ref. 1 6), where �<p' 4 is the angle of rotation of link 4 after the dead center position is reached during which the link 6 is required to dwel l .

Then, the equation becomes,

From the Eqs (5)-(7), three unknown terms (K" K2 and K3) are observed as shown below:

. . . (8)

K - K2 (coSq>:2 -COSq>: I ) + COS(q>62 -<P:2 ) - COS(q>61 -q>: I ) 1 -COSq>62 -COSq>61

. . . (9)

. . . ( 1 0)

After calculating K" K2 and K3 ' from Eqs (9), (8) and ( 1 0) respectively, the l ink lengths are determined using the fol lowing equations :

, r7 r = -4 K I

r7 r. = -6 K 2

. . . ( I 1 )

. . . ( 1 2)

. . . ( 1 3)

where r7 = � '12 + 1t2 ; I'j and It are chosen freely and g

in Fig. 4 is equal to rl . To complete the design of double rocker

mechanism, it is necessary to derive the mathematical expression for the transmission angle as its value is l imited to a minimum deviation from 90°. I t is defined as the angle (/ldr) between links 5 and 6. Using the law of cosines, it is expressed as fol lows considering Fig .5 :

. . . ( 1 4)

. . . ( 1 5)

where m i s the distance between the centers of joints C and Do.

Theoretical ly , the infinite number of solutions for the link lengths of double rocker mechanism can be found by changing the values of <P'41 . <P40, <P6 1 , <p6Q and �<P'4' Among these, the solution that satisfies the weaving technology and mechanical criteria can be accepted as the suitable solution .

2.3 Kinematic Analysis of Mechanism

The equations are derived to calculate the angular displacement, angular velocity and angul ar acceleration of the s ley based on Freudenstein' s equation.

EREN & A YDEMIR: SIX-BAR LINKAGE SLEY DRIVE M ECHANISMS IN WEAVING 247

D 6

J.Ldr 5 _____ /' ----- -----/' ----- /'

Do

C ----- >< /' /' I /' /' l h

Bo

/' /'

/'

-;-�--------------�--------j 9

Fig. 5-Transmission angle in double rocker mechanism [qJ'4 - angular position of link 4; /ldr - transmission angle; g and h - horizontal and vertical distances between joints Bo and Dol

2.3.1 Crank Rocker Mechanism

The angular displacement, angular velocity and angular acceleration of the link 4 are obtained with respect to the angle of crank shaft (2) by using the following equations according to Fig.6 (ref. l S) :

[ -b ± �b2 - 4ac

1 4 = 2 arctan 2a

where

a=( l -k2)cosr(kJ-k3) , b=-2sin2 c=kJ+k3-( 1 +k2)coS2.

and

[ -b ± � b2 - 4de

1 3 = 2 arctan

2d

where

. . . ( 1 6)

. . . ( 1 7)

d = (k4 + 1) cos 2 + ks - ki ' e = (k4 - 1) cos 2 + kJ + k,

and

. . . ( 1 8)

. . . ( 1 9)

4

3 B

Fig. 6-Angular positions of l inks of the crank rocker mechanism [qJ2 - angular position of l ink 2; qJ3 - angular position of l ink 3 ;

and qJ4 - angular position of link 4 1

where C.th, W3 and W4 are the angular velocities of links 2-4 respectively.

t r2 2 '3 2 r2 . U4 = -W2 COS(2 - 3 ) + -W3 + -u2 sm(2 - j ) r4 r4 r4

-W/ cos( 3 - 4 ) ) fine 4 - 3 ) . . . (20)

where U2 and U4 are the angular acceleration of the l inks 2 and 4 respectively .

2.3.2 Double Rocker Mechanism

According to Fig. 3 , the kinematic analysis equations can be written for the double rocker mechanism in the same way as written for the crank rocker mechanism. The fol lowing Eqs (2 1 )-(23) show the relationships between the output motion of crank rocker mechanism and the input motion of double rocker mechanism:

. . . (2 1 )

4! = 4 when the sley is at beat up point.

. . . (22)

. . . (23)

The angular displacement of l inks S and 6, angular velocities of l inks S and 6, and angular acceleration of l ink 6 were calculated using the following equations:

Angular displacement of link 6 (sley)

[ -B ±�B2 -4AC 1 6 = 2 arctan

2A . . . (24)

248 INDIAN J. FIBRE TEXT. RES., SEPTEMBER 2005

where

A = (l - KJ cos<p: - (K, - K3 )

S = -2 sin <p: C = K, + Kj - ( 1 + K2 ) cos<p: and

Angular displacement of link 5

( -S ± .J S2 - 4DE 1 <P5 = 2 arctan

2 D

where

D = (K4 + l ) cos <p: + K5 - K, ' E = (K4 - l) cos<p: + K, + Ks

and

Angular velocities of links 6 (sley) and 5

. . . (2S)

. . . (26)

. . . (27)

where Ws and W6 are the angular velocities of l inks S and 6 respectively .

Angular acceleration of link 6 (sley)

where U6 i s the angular acceleration of l ink 6. The sley angular displacement, angular velocity

and angular acceleration were calculated with respect to the loom main shaft angle (<P2) by using Eqs ( 1 6)-(28) .

3 Results and Discussion A computer program was developed for the design

and analysis of six-bar sley drive mechanism. Firstly, the required sley swing angle (<P60) was decided and then different sets of link lengths were calculated by changing, <p' 4 1 > <P40, <P6 1 , L1<p' 4 and /J.lllin ' Some sets of link lengths lacked being practical as they were not proportionate and the transmission angle in the double rocker mechanism exceeded the allowed l imits. However, some solutions were obtained with proportionate link lengths and with mIll lmum deviation of the transmission angles from 90°. The sley angular displacement, angular velocity and angular acceleration curves were obtained with respect to crank angle (the angle of main shaft of a weaving machine) with these link lengths using kinematic analysis equations. I t is observed that the effect of <p' 4 1 , <P40 and <P61 on sley motion is of secondary importance and that the parameter L1<P'4 1 and the minimum transmission angle (/J.lllin) in the crank rocker mechanism have a significant effect on sley motion. Therefore <P'4 1 . <P40 and <P6 1 are adapted to produce the practically acceptable l ink lenghts and a minimum deviation of the transmission angle from 90° in the double rocker mechanism while L1<P'4 and /J.min are being changed to obtain a required sley motion curve. The effect of L1<p' 4 and ).Llllin on sley motion are discussed below .

3.1 Effect of t.CP'4 on Sley Motion

The double rocker mechanism was designed with four different values of L1<P'4 which are -So, 0°, +So and +8°. In all cases, the crank rocker mechanism remained the same. The mechanism parameters and calculated l ink ratios are given in Table I . The link

Table I-Design parameters and calculated l ink ratios

CP2o= 1 80° r2/rl=0. 1 756

t.cP� , deg

CP:I ' deg

CP6J , deg r:/ '7 r5 /r7 r6/r7

J-ldrmin ' deg

Ild, max ' deg

Crank rocker mechanism

CP4o=30° �lmin=70° rirl =0.755 1 r4/rl=0.6787

Double rocker mechanism

[cp60=200] -5 0 +5 +8

-8 -8 -8 -8 1 50 140 1 30 1 25 0.4629 0.4 9 1 4 0.5343 0.5800 0.3286 0.3829 0.41 1 4 0.4000 0.4657 0.377 1 0.29 1 4 0.2429 83 84 8 1 77 96 98 1 00 1 03

EREN & A YDEMIR: SIX-BAR LINKAGE SLEY DRIVE MECHANISMS IN WEA VING 249

lengths of the crank rocker and the double rocker mechanisms can be determined by mUltiplying the link ratios given in the table by the freely chosen rl and r7 values.

Figs 7-9 show the angular displacement, angular velocity and angular acceleration curves of the sley respectively. The loom speed was taken as SOO rpm in the calculations. The significant influence of �<P/4 on sley motion can be observed clearly in these curves . As �<P/4 increases from -So to +8°, the sley moves

24 ------------ ---------,

:'-;-'T-'r-l� B i

30 60 90 120 1 50 180 210 240 270 300 330 360

Main shaft angle, deg

Fig. 7-Sley angular d isplacement curves for different L1<P'4 values [A - L1<p'4=-5°; B - L1q>'4=Oo; C - L1<p'4=5°; and D -L1q>'4=8° ]

Main shaft angle, deg

Fig. 8-Sley angular velocity curves for different L1<P'4 values [A - L1q>'4=-5°; B - L1q>'4=Oo; C - L1q>'4=5°; and D - L1<p'4=8°]

� 1200 " r-.;-\--� c :. 900 B � 600 1 � 300 I i -30: l'�\V"" lij -600 >. � -900

-1200

3 0

-1 500 L._. _ _____ __ _ ___________ __ ------' Main shaft angle. deg

Fig. 9-Sley angular acceleration curves for different L1<P'4 values [A - L1q>'4=-5°; B - L1q>'4=00; C - L1q>'4=5°; and D - L1<p'4=8°]

backward faster and stays longer around its back position. Hence, it allows more t ime for weft insertion. In the case of positive values of �<P/4' the double rocker mechanism reaches its extreme position (the links 4 and S are extended) before that of the crank rocker. Therefore, the sley osci l lates around its rearmost position during �<P'4 degree rotation of link 4 in anti clockwise and clockwise directions. As the positive value of �<P/4 increases, the sley moves towards i ts rearmost position even quicker and its osci l lation around the rearmost position covers a larger period of loom main shaft rotation. But, the increase in �<P/4 causes higher sley speed and accelerations. For this reason, an optimum value of �<P'4 should be determined according to shed geometry of a specific weaving machine.

3.2 Effect of Transmission Angle on Sley Motion

To show how the minimum transmission angle in the crank rocker mechanism affects the sley motion, the crank rocker mechanism was designed with minimum transmission angle values of 74°, 70° and 60° for �<P'4=0° and +So in the double rocker mechanism. Table 2 shows the design data for the crank rocker mechanism. 60° was taken as the allowed lowest value for the transmission angle in the crank rocker mechanism. No solution was obtained for transmisson angles above 7So for <p4o=30°. The link lengths calculated with �min=7So were not proportionate and unsuitable for practical use. Therefore, 74° was taken as the upper value for the transmission angle. The design data for the double rocker mechanism when �<P/4 =0° and �<P'4 =+So are given i n Table 1 .

The sley angular displacement curves are shown in Figs 1 0 and 1 1 . The lower the minimum value of transmission angle, the quicker the sley moves towards its rearmost position and the slower it goes towards its most forward position. A more space is obtained in the shed for weft insertion with the lower values of transmission angle as the sley remains closer to its rearmost position for a longer period of

Table 2-Design data for crank rocker mechanism

[q>4o=400] Parameter i!min=60° i!min= 70° i!min=74°

r2/'i 0.2293 0. 1 756 0.0940

r1/'i 0.5 1 69 0.75 5 1 0.9364

r4/'i 0.8862 0.6787 0.3634

250 INDIAN J . FIBRE TEXT. RES., SEPTEMBER 2005

30 60 90 120 150 1 60 210 240 270 300 330 360 Main shaft angle. deg

Fig. I O-Sley angular displacement curves (ll.<p'4 =0°)

fA - �min=74° ; B - �min=700; and C -�min=600J

: --" -- A cl o . . . -•. -�--,--.--�---�--�-��-�-""-I 30 60 90 120 150 180 210 240 270 300 330 360

Main shaft angle. deg Fig. I I -Sley angular displacement curves (ll.<p'4=50)

[A - �min=74°; B - �min=700: and C - �min=600J

loom main shaft angle. Compared with �CP'4' however, the effect of minimum transmission angle i n the crank rocker mechanism o n sley motion i s less significiant. Moreover, reducing the minimum value of transmission angle worsens the force transmiss ion which IS not desired from machine dynamic viewpoint.

4 Conclusions 4.1 A s ix-bar l inkage sley drive mechanism can be

designed with practically acceptable l ink lengths whi le satisfying required sley motion specifications and minimum deviation of transmission angle from 90°.

4.2 �CP'4 i s found to be the most influencial parameter to shape the sley motion. I1min of the crank rocker mechanism also has some effect on sley motion. The effect of other mechanism parameters on sley motion is of secondary importance, but their values should be adjusted to obtain the practically acceptable l ink lengths and minimum deviation of the transmission angle (l1dr) from 90°.

References I Tsudakoma ZAX209i air jet loom cata/oge (Tsudakoma

Corp., Nomachi Kanazawa 92 1 -8650, Japan) . 2 Tomas J. 1 Mechanisms, 5 ( 1 970) 495.

3 Alpay H R, Weaving Machines (TMMOB Publication, Bursa. Turkey), 1 985.

4 Talavasek 0 & Svaty V , ShulIleless Weaving Machines (Elsevier Scientific Publ ishing Company, Amsterdam/Oxford/Newyork), 1 98 / .

5 Ormerod A & Sondhelm W S , Weaving: Technology and

Operations (The Textile Institute, Manchester), 1 998.

6 Greenwood K & Cowhig W T, 1 Text Inst. 47 ( 1 956) 24 1 .

7 Greenwood K & Cowhig W T. 1 Text lnst, 47 ( 1 956) 255.

8 Greenwood K & Vaughan G N, 1 Text lIut. 47 ( 1 956) 274.

9 Greenwood K & Vaughan G N, 1 Text lnst, 48 ( 1 957) 39.

10 Zhang Z & Mohamed M H. Text Res 1. 59 ( 1 989) 395.

I I Bul lerwell A C & Mohamed M H, Text Res 1. 6 1 ( 1 99 1 ) 2 1 4.

1 2 Plate E A & Hepworth K, 1 Text Inst. 62 ( 1 97 1 ) 5 1 5 .

1 3 Plate E A & Hepworth K , 1 Text /nst, 64 ( 1 973) 233.

1 4 Sternheim A & Grosberg P, l Text /nst, 8 2 ( 1 99 1 ) 325.

15 Soylemez E, Mechanisms (Middle East Technical Uni versity Publication, Ankara, Turkey), 1 985.

1 6 Tin i � F , Design and Construction of a Rapier Drive System. MSc thesis, M iddle East Technical University, Ankara, Turkey, 1 983.

Appendix I-Derivation of equations calculating l ink lengths of a centric crank rocker mechanism satisfying a required swing angle of the rocker based on minimum transmission angle criterion

Eq. (A- I ) is obtained by expressing S distance in Fig . A- I using the law of cosines for both AoABa and ABaB triangles and then by equating them each other.

. . . (A- I )

Fig. A-2 shows the extreme positions of a centric crank rocker mechanism. As X distance ( x = IBaFI ) belongs to both A"B,l

and BoFB' triangles, it can be expressed for both triangles as follows:

Equating righthand sides of these two equations yields Eq.(A-2):

. . . (A-2)

Eq.(A-3). as given below, i s obtained as the terms 1/ + I} and

rJ2 + r.2 cancel out each other i n Eq. (A- I ) :

. . . (A-3)

s in <P2 = 0 i s obtained by taking first derivative of Eq. (A-3)

and putting � = 0 to determine <pz values at which minimum d<p2

EREN & A YDEMIR: SIX-BAR LINKAGE SLEY DRIVE MECHANISMS·IN WEA VING 25 1

3

B

Fig. A-I-Transmission angle in a crank rocker mechanism [Il - transmission angle; S - distance between joints A and Bo; <1>2 - angular position of link 2]

B'

" "-, A" \ - '- -I

/ ./ / I

Fig. A-2-Dead center positions of a centric crank rocker mechanism [<1>40- swing angle of rocker; A' and A"- positions of joint A at the dead center positions of mechanism; B' and B" -positions of joint B at the dead center positions of mechanism ]

and maximum values of transmission angle are achieved (Fig. A-3). This gives <1>2=0 and <1>2=1[. Eqs (A-4) and (A-5) as given below, can be written for <1>2=0 and <l>2=7t. respectively using Eq. (A-3):

. . . (A-4)

. . . (A-5)

On the other hand, Eq.(A-6) is written from BoFB' triangle in

Fig. A-2.

. . . (A-6)

Fig. A-3-Minimum and maximum values of transmission angle in the crank rocker mechanism [Ilmin- minimum transmission angle; Ilmax- maximum transmission angle; A I and B I - positions of joints A and B at which transmission angle reaches its maximum value; A2 and B2 - positions of joints A and B at which transmission angle reaches its minimum value ]

By substituting sin <1>

40 for !i In Eq. (A-4), the fol lowing 2 r •

expression for .2 is obtained:

s in <1>

40

.2 = __ 2_

'i cosllm;n

'i

. . . (A-7)

By substituting !i.sin <1>

40 for !i I n Eq. (A-2), the fol lowing 'i 2 'i

expression for !i. is obtained:

!i. = 'i

'i

. . . (A-8)

Following expression is written by dividing both r2 and r4 by rl in Eq. (A-6):

!i = !i.sin <1>40

'i 'i 2 . . . (A-9)

Initial ly, rl i s chosen freely and then r2, r3 and r4 are calculated by Eqs (A-9), (A-7) and (A-8) for given values of <1>40 and Ilmin.

kinematic design method for six-bar linkage sley drive...

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