Ren-ChungSoong Department ofMechanical …tcsme.org/Papers/Vol32/Vol32No3-4Paper10.pdf · AN...

AN ADJUSTABLE SIX-BAR MECHANISM WITH VARIABLE INPUT SPEED

FOR MECHANICAL FORMING PRESSES

Ren-Chung Soong

Department of Mechanical and Automation Engineering, Kao Yuan University

Kaohsiung, Taiwan, R.O.C.

Contact: [email protected]

Received April 2008, Accepted November 2008

No. 08-CSME-12, E.Le. Accession 3050

ABSTRACT

An adjustable six-bar mechanism mechanical press, in which one of its link length can be adjusted

and its driving crank also can be varied according to different forming processes, is proved to be feasible

in this paper. By properly designing the speed trajectory of the driving crank and the adjusting

magnitudes of the adjustable link in length, the desired kinamatic characteristics of the ram can be

obtained. The examples are given to verify its feasibility and effectiveness in practical applications.

Keywords: Adjustable mechanisms; Variable input speed; Mechanical Presses

UN MECANISME ASIX BARRES AJUSTABLE AVITESSE VARIABLE POUR

PRESSES MECANIQUES DE FORMAGE

RESUME

Cet article demontre la possibilite de concevoir un mecanisme asix barres ajustable, dont la longueur

d'un des maillons peut-etre ajustee et Ie bras de levier modifie pour s'adapter aux differents procedes de

formage. En evaluant convenablement la vitesse de trajectoire du bras de levier et l'amplitude de

l'ajustement de la longueur du maillon ajustable, on peut obtenir les caracteristiques cinematiques

souhaitees. Des exemples sont donnes pour verifier en pratique la faisabilite et l'efficacite de

I'application.

Mots-cles : mecanismes ajustables; vitesse variable; presses mecaniques

Transactions ofthe CSME Ide fa SCGM Vol. 32, No. 3-4,2008 453

1. INTRODUCTION

The metal forming press is one of the most commonly used manufacturing machineries today. In

general, there are two major types of press that have been developed for practical industrial applications,

the one is the mechanical presses the other one is hydraulic presses. The former is fast and energy

efficient, but lacks flexibility. The latter is flexible, but is expensive to build and to operate. There are four

types of metal forming process such as cutting, bending, deep drawing and forging. Among these

processes, the different kinematic requirements of ram have to be satisfied such as trajectories of position,

velocity and acceleration in a cycle. An existing mechanical press usually only is designed for the one of

the four processes mentioned above. Moreover, the kinematic characteristics of the ram are functions of

the link lengths and the kinematic characteristics of driving link of the presses. Therefore, if we can

design a press in which one of its link length and trajectories of position, velocity and acceleration of the

driving link can be adjusted according to different forming processes, the higher flexibility of applications

will be obtained. This is a reasonable choice instead of redesigning the new presses when an existing

press has to satisfy different types of forming processes.

Many researches have been conducted to study mechanical forming presses. Some works focus on

either Finite Element Analysis (FEA) or structure improvement of the presses. For example, computer

simulation and dynamic analysis are performed for a single-point-drive eccentric press [1]. A Lagrange

multiplier method is proposed to synthesize the dimension of a drag-link drive mechanical press for

drawing [2]. A design procedure, which combines the linkage synthesis and the trial and error method to

optimize the dimensions, is developed for the nine-bar linkage press [3]. Also a two phase optimization

technique is proposed to reduce the shaking force and shaking moment of the drag-link mechanical

presses [4]. Some researchers are devoted to improve the performance and to raise flexibility of practical

applications by varying speed trajectory of the input link for mechanical presses. Such as, Yossifon and

shivpuri [5-6] discussed the design, analysis and construction of a servo-motor controlled mechanical

press for precision forming. Doege and Hindersmann [7] designed the non-circular gears to drive

mechanical presses for optimizing kinematics. Van and Chen [8-9] proposed a novel approach by varying

the input speed of the crank to make the ram's motion suitable for both deep-drawing and

precision-cutting processes. Recently, the concept of the hybrid mechanism, also call controllable

mechanism or hybrid machine, is applied to design the mechanical presses. Du and Gue [10] designed a

2-degree-of -freedom seven-bar linkage mechanism whose performances are programmable, including

the trajectory and velocity of the ram driven by a large constant speed motor and a small servomotor. A

Genetic Algorithm to optimize the design parameters of the linkage is also included. Meng et al. [11] used

the inverse kinematic analysis and optimal synthesis method to design a hybrid driven a seven-bar linkage

mechanical press. Mundo et al. [12] presented a design method to optimize kinematics of mechanical

Transactions ofthe CSME Ide la SCGM Vol. 32, No. 3-4,2008 454

presses by optimal synthesis of cam-integrated linkages.

This paper proposes a new design concept for mechanical forming presses that the driving link is

driven by a servomotor and the one of its link length can be adjusted. The adjustable link is a screw-nut

link also driven by servomotor corresponding to the driving link. By properly designing the kinematic

trajectories of driving crank such as position, velocity and acceleration and determining the magnitude of

the link length of the adjustable link, the desired forming performance of an existing press can be

obtained for satisfying different type forming processes.

2. THE ADJUSTABLE MECHANICAL FORMING PRESS

An adjustable mechanical forming press defined in this paper is a six-bar mechanism in which there is

a screw-nut link its link length can be adjusted and its speed of driving link also can be varied and driven

by servomotors as shown in Fig. 1.

Fig. 1 An adjustable mechanical forming press

3. SPEED TRAJECTORY OF THE INPUT CRANK

We assume the input link of the adjustable mechanical forming press is a crank. In this paper, theposition trajectory ofthe crank is defined by an nth order Bezier curve [9] ¢J (t) with parameter t as

follows:II

¢J(t) = Le; .R;,II (t);;0

Where

Transactions ofthe CSME Ide la SCGM Vol. 32, No. 3-4, 2008

(1)

455

n'· .B. (t) = . . t' . (1- t)n-II,n ., ( ')'Z.· n -Z .

t E [0,1] (2)

in which¢ (t) is a Bezier curve that represents the angular displacement of the input link defined by

control points e;. Parameter t is regarded as the normalized time from 0 to 1. Since the Bezier

curve is nth order differentiable, this guarantees smoothness of the entire motion. Hence, the angularvelocity wet) and acceleration aCt) of the input link can be derived by continuously differentiating

Eqs. (1) and (2) with respect to the time as follows:

Where

w(t) = d¢(t) = Ie; .dBi,n (t)dt i=O dt

d 2¢(t) n d 2B. ((t)aCt) = ="e . I,n

dt 2 f::t I dt 2

dB. (t) n' . 1 .',n = . . t'- . (1- t)n-Idt (i -1)!-(n - O!

n' . .---'--. t' . (1- ty-'-Ii!-(n - i-I)!

(3)

(4)

(5)

n! .ti-2 .(I-t)n-i _ n! .ti-I.(I_t)n-i-l(i-2)!-(n-i)! (i-l)!-(n-i-l)!

n' n'- . . t i-1• (1- tr-i-l + . .t i .(1- tr-i-2

(i -1)!'(n - i-I)! i!-(n - i - 2)!

(6)

After doing the kinematic analysis of the adjustable mechanical forming press by vector loop approach,

all the kinematic magnitude (positions, velocities and accelerations) of each movable link can be obtained

as function of the driving crank motion.

4. THE ADJUSTING MAGNITUDE OF LINK LENGTH

In order to be corresponding with the angular position of driving link, the adjusting magnitude of the

adjustable link in length is also designed with Bezier curve. The adjusting magnitude of the adjustable

link in length !1r is also defined by an nth order Bezier curve !1r(t) with parameter t the same with

equation (1) as follow:

n

!1r (t) = I A; . B;,n (t);=0


(7)

456

Where !:!r(t) is a Bezier curve that represents the adjusting magnitude of link length of the adjustable

link defined by control points A.i .

5. DESIGN OF OPTIMIZATION

The optimization procedure is applied to determine the design variables to obtain the desired forming

performance for an existing adjustable six-bar mechanical forming press. For the purpose to fmd the

proper speed trajectory of the driving crank and the adjusting magnitude of the adjustable link in length

the general optimization equations can be defined as follows:

n/

Minimizing f(eoe, ..,en_l,en,A.o,A.l,. ..,A·n_I,A.n) = Lobi;;=1

Subject to

i=l, ... ,nc

i = 1,. oo,ng

(8)

(9)

(10)

Where obi; is the desired performance objective function, n; denotes the number of the desired

performance objective function, nc and ng denote the number of equality and inequality constrained

equations. Note that the equality and inequality constraints are defined to obtain the desired

performance.

Up to here, all information for optimization is derived. Any optimization methods can be used to

determine the design variables. A sequential quadratic programming subroutine [13] is applied to solve

design variables in this approach.

6. EXAMPLES AND DISCUSSIONSHere examples will be demonstrated to prove the feasibility of this proposed approach. A 10lb order

Bezier curve (with 11 control points) is used to represent the trajectory of the input crank and the

adjusting magnitude of the adjustable link in length. It is clear that eo, ,.1.0' en and A.n are the

boundary conditions for speed trajectory of the crank and the adjusting magnitude of the adjustable link in

length in a consecutive cycle. Therefore, eo = eudc ' ,.1.0 = 0, en = eudc + 27r and A.n = 0 must be

Transactions ofthe CSME Ide fa SCGM Vol. 32, No. 3-4, 2008 457

specified, and elide is the corresponding crank angle when the ram is at upper dead center. And the

average speed of input crank is set to be 60 rpm in all examples. An existing press in which the link

length of link 4 is adjustable and its link lengths are shown in Fig. 2 and Table 1, respectively. In these

two examples, the goal is to design the proper input speed trajectory of an existing six-bar mechanism

forming press shown in Fig. 2 to have the desired ram performance: smooth pressing to avoid large

transient force and vibration, long dwelling time with lower forming speed during its work stroke over a

relatively large rotational angle of the driving link to ensure uniform metal deformation and to minimize

spring back and driving torque.

Table I Dimensions of an existing press

fl (mm) f2 (mm) f3 (mm) f4 (mm) rs (mm)

80 42.16 82.35 46.39 59.8 251.84 150 148 20

d

Fig. 2 An existing adjustable mechanical forming press

Example 1

In this example, in order to prolong the life of dies, to raise the quality of products, to lower the

vibrations and noises of the presses and to decrease the loading of the brake of the presses in the return

stroke, the work is designing the speed trajectory of the driving link and the adjusting magnitudes of

adjustable link in length for an existing six-bar mechanism mechanical press to keep approximately

constant speed of ram over the specific period before and after forming and to minimize the peak of

acceleration of the ram.

The objective function can be defined as follows:


Minimize f(B» ... ,B9 A, ,....,~) = peak of aram

Subject to

cZ (BI'".,B9 ) = az(O)-az(l) = 0

C3(el' ... ,e9,~,.... ,A9) =s(ta) =Samax =Semax

(11)

(12)

(13)

(14)

(15)

Where a ram is the linear acceleration of the ram, S denotes the displacement of the ram for the

adjustable press, samax and semax are the maximum linear displacement of the ram for the adjustable

press and the existing press, respectively, ta represents the normalize time corresponding to Sa max and

Semax' la is the link length of the adjustable link, lu and II are, respectively, the maximum and the

minimum link length of the adjustable link to satisfy the Grashof low for keeping the driving link to be a

crank, v denotes the linear velocity of the ram of the adjustable press, tds and tde represent the time

of the beginning and the end in two specific periods, e v and ea are small numbers and set to be 0.5 in

two examples. The first two constrained equations of equality are for having continuous angular velocity

and acceleration of the input crank in two consecutive cycles. The third constrained equation of equality

guarantees to have the same stroke of the adjustable press with the existing press. The first two

constrained equations of inequality keep the adjustable press to be a crank rocker six-bar press. The last

two constrained equations of inequality are in order to force the velocity of the ram to be an approximate

constant speed during periods from tds1 to tdel and from tdsz to tdeZ

The optimal control points of the speed trajectory of the driving crank and the adjusting magnitude of

link length of adjustable link are, respectively, shown in Table 2 and Table 3 in this example. The

magnitude and fluctuations of the angular displacement, velocity and acceleration of the driving crank are

shown in Fig. 3, respectively. The magnitude and fluctuations of the linear displacement, velocity and

acceleration of the ram are shown in Fig. 4, respectively. The adjusting magnitude of link length of

adjustable link is shown in Fig. 5.


Table 2 The control points of the speed trajectory of the driving crank for Example 1

76.2 129.5 237.5 o 185.5 360 162.5 184.4 283.7

Table 3 The control points of the adjusting magnitude of the adjustable link for Example 1

65.23 0 0 0 0 0 0 0 o

·200r--~-~--,---,---,----r---,-----.------.---,

2 ------;

-_._---_.,,-----

, . .."'------'.----_._----· . .· . ., . .· . ., . .

, . ., . .· . .

-...... ' .:,. .....-->'\-_:_-----_!_-----~------~-- ---,------~----)<~~: -----

, , ' I '

'\ : ----- existing press : /' :

'\ : -- adjustable press :,' :------:- \,:--:------r-----1- -----~- -----y-----:---, \' , , , I' I

..:_----~-:------~-----_.:__ ._--~----};------;, \' , I I " ,, \ , , I I'

'\ • , 'I'

··\:,·1·····:/·····" , ,I" , .,~~';~--T-----y---

------0._---- .._--

, . ,------'- 0. -_._-----300

-320 '--_-'--_-'--_...L.._--'--_-'-_--'__'---_-'--_-'-_-'

o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9Nonnalized lime

'0 -260

~E

~ -280g-o

E.s -240

~":;

I • , I. __ . ., ----- .. ----.-.--._-, I • ,

, I • ,I I , ,

I I , I

, ""------~-----,--" -- -~!'<. -.:-. --_..:- -_. --~- ----- ~------~---- _.; -----i l,,~,"~ ~ : ~ ~ ~ 1

--- +-/~:-------:----- :------i------i------i-----',' " .."

,'/1 ': ~] ~ ~"--_-'--_-'--_-'-_-1__-'--_-'-_--'-__'---_-'--_-'

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9Normalized lime

5 -- ----~------:- -- ----:- - -.: ----- existing press ,

~ : -- adjuslable press :4 ------~------, - ----,-- -- - --,-- - - - - -:--- --

~u :::'0 ~ __ . __ .~_. .~._. _..... _.

~E

~g.iJ

'":;g>«

(a) (a)

i

,',• I • : , :: \ , , :• ----','-' ----,-------,-------,--- ----,--- ----,-, {A,. , ., _

, , 'I \' , ,

: ----- existing press :: \: : :

-- adjuslable press: :: \ ' :-._._-~. - ----~ -- --- -~------~."----~ - ... -~-- .. _.;~\-----

: :: 1 '\: .' \, I'

.. ----~. - -- - -~ -- --- -~ - -----~. --- --~ _. -_. :~_ .. -_.:, , I , , J' ,

': :::, ,I

--- ---~- --_. -~ - -- - - -~ - -- - - -~ - - - -- -~ :-~ - - ----~ - - - -- -~ - - - - -.:--- - - ~"'''', : " : : : '

' ..: /~ ~ : : :

... L,·~,_._T>/:···-·····~-···-····

800

600

~

~ 400

.sEl!! 200":;'0.2:-gQj>

----- existing press

-- adjuslable press

· , ,_____ L. '. ,__

· , ,· , ,

14,--,..--.,.---r---,----,---,--,---.,.---,---=",",

12 -- ---~------~------~ -r-'--.=..--'-'O".=..--",-.=..--",''''"-"'--==-=;

.>f.Cl!!o'0

~ 6 -·-:::-·-·t':':'"·-:· ~-_"':::--":~:-.:-.-::~.-:::-:-.-:~:-.-:-::~_-:::-----:!':':":::::'~.~ " , ,

ffii 4 ····_-t------:--- ------.'.-- ... , ------:------«

0'~ 10 ..

l

0 -4000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Normalized lime Normalized lime

(b) (b)


0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9Normalized time

20000 r-........,.--,---,--..,---r--r----,----r--,-,--,

15000 ----- ----- ----- ----- ------;-----:l------~----- -----,------~ : :~ :~ ----- existing press I 1'1 '

E -- adJ'ustable press : : :1 :E ' ,'I '

E 10000 - ----:------~·_----~------~-·-·--~----1-~:-----~----- ------,------

~ ~'~ j j i ! \ ;

'0 :,:::: : ~ :.9. 5000 --- _..:- -- _. - .:- -- - - - -:- --- --.; - - - --- +. ---l-r -} --- -:. ---_. ----- ---- ---... :::::: : ~ :~ ."" I' I I

~ ,:://(---~\ i : ~ ~ , _o ~~~-:_-~~~~~;;;;!- -rOOM_Or ~I--r--\! ,i~ __--~----

: : : : ' : \ : ,,: :

: : : \ :,/: I

, I I \" I

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9Normalized time

(c)

Fig 3 The angular displacement, velocity and

acceleration of the driving crank for Example 1

(c)

Fig 4 The displacement, velocity and acceleration of

the ram for Example 1

E 30 ,---,---,---,-----,----,----,.----,.----,.----,.---,.§.

0.90.80.70.4 0.5 0.6Normalized time

0.30.20.1

, ,

~Vl- 5 ----.-~-------:-. ----~-- ---~-------:-------~------~-------~.----.~---- .., , , : : ; , , ,

"'6'IIIQ).<:: 0 ~--'-_...J-. _~_.i_.._ _=::::...oio.__....._ ......_ .....__J

f- a

, .! 25 -- - : - - - -- - ~- - --- --~~~~~- ~;i~~;~~-~;~~~-~--- __ oj ------

i ,." --adjustable press :

.E 20 - - - - -! ---- --:-- -----t-- --- -:-------:-- -----:-- ----t- ----~--- --- (- ---f; : :::::::C) , ""'"c: .".".,.Q.) ., I , , , • • •

:;;; 15 - ----~------~- -----~------~-------~------~------~-------~------~------

:§ :':::::':'0 : : : :: ::G : : : :: :'0 10 -----:------,----- -~------~-------~------~------,-------~------~------.2 :: :

'"'"E

Fig 5 The adjusting magnitude oflink length of the adjustable link for Example 1

Example 2

ill this example, in order to ensure uniform metal deformation and to minimize spring back and driving

torque, the work is designing the speed trajectory of the driving link and the adjusting magnitudes of the

adjustable link in length for an existing six-bar mechanism mechanical press to keep approximately zero

speed of the ram over a specific period before and after forming and to minimize the peak of acceleration

of the ram.

The objective function is the same with Example 1. The constrained equations are also the same with

Example 1 from Equation 10 to Equation 13 except Equation 15 as follow:

Transactions ofthe CSME Ide La SCGM VoL. 32, No. 3-4, 2008 461

(16)

where a denotes the linear acceleration of the ram for the adjustable press, Sv and sa are small

numbers and set to be the same value with the example 1. The purposes of the three constrained equations

of equality and the first two constrained equations of inequality are the same with example 1. The last

constrained equations of inequality are in order to force the ram to have an approximate dwelling forming

period from tds to tde .

The optimal control points of the speed trajectory of the driving crank and the adjusting magnitude of

link length of adjustable link are, respectively, shown in Table 4 and Table 5 for this example. The

magnitude and fluctuations of the angular displacement, velocity and acceleration of the driving crank is

shown in Fig. 6, respectively. The magnitude and fluctuations of the linear displacement, velocity and

acceleration of the ram is shown in Fig. 7, respectively. The magnitude and fluctuation of link length for

adjustable link is shown in Fig. 8.

Table 4 The control points of the speed trajectory of the driving crank for Example 2

68.2 55.6 260.3 230.6 74.4 242.6 261.9 142.8

Table 5 The control points of the adjusting magnitude of the adjustable link for Example 2

70 5.32 0 0 0 0 0 0

291.8

24.87

From the design results as shown in Fig. 4, the ram of the adjustable press keeping an approximate

constant speed before and after forming can be verified in Example 1. Therefore, the goal of Example 1,

prolonging the life of dies, raising the quality of products, lowering the vibrations and noises of the

presses and decreasing the loading of the brake of the presses in the return stroke, can be certainly

obtained. According to the design results as shown in Fig. 7, the ram of the adjustable press having an

approximate specific dwelling period including before and after forming is proved in Example 2. The

purpose of Example 2, to ensure uniform metal deformation and to minimize spring back and driving

torque, can be surely reached in Example 2.

Moreover, the peak of acceleration of the ram of the adjustable press is substantially minimized in two

examples as results shown in Fig. 4 and Fig. 7. The trajectories of angular velocity and acceleration of

driving crank are continuous in two consecutive cycles coincide with our design constraints as shown in

Fig. 3 and Fig. 6.

Transactions ofthe CSME Ide la SCGM Vol. 32, No. 3-4, 2008 462

From the results, apparently, it is not necessary to redesign the link dimensions when the design

requirements have been changed. We just have to find out a new set of control points of speed trajectory

of the driving crank and the adjusting magnitude of link length of the adjustable link by optimization

procedure for an existing adjustable press.

7. CONCLUSION

An adjustable six-bar mechanism press, in which one of its link length can be adjusted and its

driving crank also can be varied according to different forming processes, is proved to be feasible in this

paper. It can be a new choice for metal forming processes. By properly designing the speed trajectory of

the driving crank and the adjusting magnitude of link length of the adjustable link with Bezier curve can

satisfy the kinematic requirements of the ram for different forming processes. That makes the adjustable

six-bar mechanism presses programmable and adjustable and increases the flexibility of practical

applications. Moreover, the linear acceleration peak of the ram can substantially minimized, it is excellent

for dynamic forming performance to the adjustable forming presses. But the cost has to pay is to

overcome the problem of the control technology for the driving crank and the adjustable link

simultaneously.

, , ,, , ,

5 --- - -- ~ -- -- - - ~ - -- - - .: -- -- - _: -- - - - _: - -- - - - ~-- - - - - ~ - - - -- -; - - -- - -; --/: : ----- existing press : : : ,,~"

j :: -- adjustable press: : : / :i 4 .. _- .-~-._---~.-----:--_. --;------; - -- .. _~-_ ... ·r-·~;;,F~---<-----~ :::::: ,~/ : :~ 3 ------[------[------~------r------:--~),~:-'--f--- r----r-----~ "".,' "

~ 2 --. - -- ~ - -- -. - ~ _. - -- - ~ -- --: --;;,~~_. - . - - ~ - - -- - - ~ -- - . - - ~ - . - - -- t-----% : ;*,',:::'

i:J>__<CJi I ' I',,," , , , . , , ,

,,/! ~ : ' : ~ : l ~_1l<-----'-_-'--_-'------'_--'-__-'-_-'-----'-_-'----'

o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9Nonnalized time

(a)

-200,-~-~-~---,---,---~-.-----,-~-------,...........~

",+\+-----+···--~------~------~------f---)<-- ~-----

, \: : ----- existing press : ,': :E ,\,: -- adjustable press : ,': :i -240 c_ ----c\;\~,-f------r------r------r-----y----:- ----:------ "\""'0 •

i:••••••I••••••i••••J\\'\<.i•••••V.'·•••••I••••••I•••••. , , , .. , , . ,• , I • ,

-320 '----'---'----'------'---'---'---'------'----'---'o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Nonnalized time

(a)


-4oo0'---:"'-:----''?::--:'0.-=-3--:0:'-.4,...--=0':.S:--0:'.-=-6--:0':.7::----:'0.'::e--:0~.g=--:

Nonnalized time


-- adjustable press, , ,, , , ,... ----_._ ..... '_ .. - ...._----, , , ,

: : : '

----~-----~----~-----:-----7

°0'---0"-.1---0L.2--0-'-.3--0-'-.4--0--'-.S----'0.-6-~0.'-7--oL.e---:o-'-.g--'

Nonnalized time

600 ._--_.'.----_. __ . __ .

: ----- existing press

, __.~ a~~~::~~le_~~~s.

(b) (b)

20000,.--,---,--....-,---,-.,.-....-...,---,-,

, , ,_...--_._ .. ------,.-----.

, , , ,_···-·,-··--·r------,·--·--l-·, , , ,

: ----- existing press

: -- adjustable press

, , . , ,-._~ _.. _.~. _.... ~ .. _ .. -:... _ .. ~ ...

, 1,1 , , ,, ,,1

"I,"':1I ,I ,- -. - -... • t ~ .. - .

1'1 '

: ::" I" ,

.-j.. -.~ J.\I' ,

: : ~, ' I, ' II' ,

0'-"';".lo-+'--r~.. -L. -~,,' \:, '

\ ~ /'. '-50000'----,0-'-.1--0--'-.2---..,.0."-3---:0-'-.4--0,....-=-5-.,.0"-.6--"0"'=.7,---0,....-=-e-"'O."-g---'

Nonnalized time

15000

".c155 5000

~§«

1:~.sE ooסס1 ..~

, , ,_'•• J ~, , ,

, , ,_._'•• __ ._J l

, , ,, ., ., ., ,, ,, ,, ., ", , , , ,.,-- .. ' ,-"---' ····-r-·----,---, , ,

, , ,, , ,, , ,, , ,


-- adjustable press

-150 '--_'--_-'-_-J'__,__...L...._---'---_-L_--'__'----..J

o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 o.e O.gNormalized time

(c) (c)

Fig 6 The angular displacement, velocity and

acceleration of the driving crank for Example 2

Fig 7 The displacement, velocity and acceleration

of the ram for Example 2


-----:------: -_ .. ;-- --:··_-":'-·--1'", , , , , ,, , " ,: :' :, ,

I , , ,. _.. -,.. .. ~ .. - .. - - - -.. - _. - -.- - - - - .- -_., ,

~~ 15

15

" ,, .,.,.,~ 25 •. _.-;•••.• - "-- -- _w_: w J __ ••• '._. _ .'--._ •• -:_ •• - ~-

.2!: : ----- existing press : •~ : - adjustable press :

~ 20 • _.~. __ •• " •.• - -:"":'- :". --:- .• _-~---_.~-_.

, : : : :, , ,

, ,, , , , I , , , •

.~ 5 .. ---:- •• -~ .-- ~--- -;... ---;---_.:-_. -~_. _.:-._ .• ~_.-

~ j ~ ! ' i ' : ~ ~nl " I • , ,

~ O-~-_:_i ; ; ~~.... 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Normalized time

Fig 8 The adjusting magnitude oflink length for adjustable link for Example 2

8. REFERENCES

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[13] Tseng, C. Ro, Liao, W. Co, and Yang, T. C. Most Users' Manual, 2001, Department of MechanicalEngineering, National Chiao Tung University, Rsinchu, Taiwan, R.Ooc.

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