of Vertic Weight a

A mo of tab dis the pre pre typ ma I fun H ma com mo acc ori etc ma po hig the hor eff ma stru loa cas ma mo Mx tra to sys cap M Min Gye 82-D Eng (e-m & M pch

Abstract—In 3 otion is worse th

structure differ ble weight and scussed. Then a e horizontal t edicting motio ediction is intro pes in order t achine structure

Index Terms— nction, counterw

High precision anufactures b

mponents and ost important curacy of ma iginated from c. [1,2]. In o achine tool g

sitioning have gh effective ma e accuracy of v rizontal motion fect of table w ainly discussed

The first diffe ucture. Table w ad on ball scre

se. Misalignm achine head an oving table. M

x, My and Mz

ansfer function vertical case. The second on stem is assem pacity and rel

Manuscript receiv nistry of Knowled Su-Jin Kim is eongsang Nationa 55-751-6075; fax: Duy Le, Hyun-K gineering, Gyeong mail: duyle2003@ Gyung-Ho Kim, C Materials, Daejon, h657@kimm.re.kr)

The A

3-axis machine han the other tw rences. These m

d vibration o additional comp transfer funct n error of v oduced to evalu to help engin es.

—Vertical guid weight vibration

I. INTROD

n machine cen because the d consistency o

factor of the achine tools. geometric, cut rder to impro geometric err

been being ch achine design vertical motion

ns because of weight and vib d in the current erence is the e weight in verti ew which is la ment of connec nd counterwei Moreover addit

z - are propos

[4,5] of linear ne is about cou mbled with cou

liability, the p

ved April 7, 201 ge Economy of So with the Depart al University, Jin : 82-55-762-0227; Kwang Cho are gsang National Un @gmail.com; elgar3 Chun-Hong Park a , 305-343 South K ).

Assessmen

in 3-axi

Duy Le, H

center, accur wo horizontal m main problems,

f counterweigh ponents are pro

tion for the ertical guide. uate the effect o

eering designe

e, horizontal n.

DUCTION

nters are requi demand of of quality are precision com Mainly, posit tting force, dy ove machine rors as well haracterized an

[3]. In 3-axis m n is worse than

structure diffe bration of cou

research. effect of grav ical motion cau argely differen cting chain or ight causes m tional moment sed to modify

guide so that it unterweight. Al unterweight to precision is r

10. This work is outh Korea. tment of Mechan nju, 660-701 Sou e-mail: sujinkim@

with Department niversity, Jinju, 66 328@nate.com). are with Korea Inst Korea (e-mail: gyu

nt on Dif

is Machin

Hyun-Kwang

racy of vertica motions becaus gravity effect o ht, are mainl posed to modif application on

And vibration of counterweigh ers to optimiz

guide, transfe

ired in most o high accurat e growing. Th mponents is th tion errors ar ynamic loading tool accuracy as precision nd predicted fo

machine center n the other two

rences. Gravity unterweight ar vity to machin uses large axia nt to horizonta r wire between moments around t components

the horizonta t can be applied lthough vertica o increase thei educed due to

supported by th

nical Engineering uth Korea (Phone @gnu.ac.kr).

t of Mechanica 0-701 South Kore

titute of Machiner ungho@kimm.re.kr

fferences

ne Tool:

g Cho, Su-Jin

al se of ly fy n n ht ze er of te he he re g, y, n or r, o y re ne al al n d - al d al ir o he g, e: al ea ry r; vibration decelerate marks on during m counterwe counterwe models. E ratio of m vibration counterwe optimize m

II. TH

Rail d separated made with rigid. The the more pressure ( ball slides NSK roll shows the Fig. 1 (b) the maxim negligible

of Vertic

Weight a

Kim, Gyung [6,7]. Specif es until stops n the workpiec machine proc eight: mecha eight, which Estimating and machine parts a prediction is eight types in machine struct

HE DIFFERENCE

M deformation:

rails which are h the machine b en bolted rail t

general and la (q) - from spind

s is 25.103 _Pa.

ing guides [8] e deformation m

gives the analy mum deforma e or can be ove

Fig. 1 (b) Nastran a

cal and H

and Coun

-Ho Kim, Ch

fically, when at the desired ce, which redu cess. There anical, hydrau

can be divide characterizing are suggested i proposed to other to help ture.

ES OF VERTICAL

MOTION REVIEW

There are tw e bolted to mac body itself whi type should be arger deformati dle weight - act According to ], NSK LY35 model of bolte ysis result solv ation is about ercomed by usi

(a) Bolted rai analysis of bol

Horizonta

nterweig

hun-Hong Par

the machine point, it caus uce the surface

are three ki ulic and pn ed into two v g stiffness and d n each model. evaluate the e machine desi

L AND HORIZO W

wo rail types: chine body; the ich is assumed considered to ion. A study c ting on the rail the technical is chosen. Fi ed rail of rollin ed by Nastran i 0.2µm. This ng stronger rai

(a)

l model ted rail model

al Motion

ght

rk e's head es wavy e quality inds of neumatic vibration damping Finally, effect of igners to NTAL one is e other is

Fig. 2 Different stiffness in vertical sliding guide

Rail Stiffness: Different from horizontal guides, vertical guides deal with big moment from spindle weight. It causes tension on top side and compression on bottom side of the contact area between guide and rail. In case of rolling guide, tensile stiffness and compressive stiffness are the same because special structure of the rails helps them to take the pressure with same area on both sides. But in sliding guide, tensile stiffness is smaller than compressive stiffness because of different areas as shown in Fig. 2.

Ball screw deformation: Ball screw in vertical case normally loads large axial force - from the spindle - on the ball nut. It causes axial displacement on the screw during the motion. Fig. 3 (a) gives an example of screw deformation in case of fix-fix assembly with 300kg load, screw length 1m, screw diameter 25mm, E = 207GPa. And Fig. 3 (b) shows the results of displacement when the ball nut moves from top to bottom of the guide. This error can be avoided by adding counterweight support or compensating on machine controller.

Backlash: In case of no counterweight, there is no backlash because the load pushes down on the nut keeping it in constant contact with the screw. If counterweight is present, backlash depends on the percentage of the counterweight as shown in Fig. 4. But nowadays these problems can be solved by using anti backlash ball screw [9] or compensating for backlash on the machine controller [10]. So, in vertical motion, backlash is not necessary to be considered as an issue. Accuracy is maintained whether the load is being raised or lowered. Another advantage of vertical motion applications is that the torque needed to lower the load is less than that required to raise it. This means there are possible opportunities for downsizing the motor. However it is always necessary to brake the screw shaft with the motor to prevent any backdriving.

(a) 

  

(b) 

[image:2.612.113.258.49.140.2]

Fig. 3 (a) Fix-fix ball screw displacement model (b) Displacement and ball nut position

Fig. 4 Backlash in horizontal and vertical cases (a) Horizontal case

(b) Vertical without counterweight (c) 100% counterweight

(d) 70% counterweight

III. VERTICAL MODEL A. Current Transfer Function

Equilibrium equations from transfer function of [4].

0

F=

∑

G G (1)

(

)

(

)

, , ,

, , ,

0

0

v v

h h n m

z ij z ij z r z e j i

n m

y ij y ij y r y e j i

f K z f f

f K y f f

− − + =

− − + =

∑∑

∑∑

0 M =

∑

G G (2)

(

)

(

)

{

}

(

)

(

)

{

}

(

)(

)

{

}

(

)

,

,

, ,

0

0

0 v v

h h

v v h h

n m

z ij z ij x vci ry j i

n m

y ij y ij x hci rz j i

n m n m

z ij z ij y cj z y ij y ij

j i j i

f K z R X M

f K y R X M

f K z R Y R f K y

− − − =

− − − =

− − − − =

∑∑

∑∑

∑∑

∑∑

They work almost perfectly in case of horizontal motion. But in the case of vertical motion they might need to be modified. That is the reason to build the separated model of forces and moments in case of vertical motion.

B. Force and Moment Modeling

Assuming ball nut is the center of system coordinates as shown in Fig. 5 (a). Counterweight force and head weight will be considered generally as C(Cx, Cy, Cz) and W(Wx, Wy,

Wz). By projecting them onto each plane of the coordinates,

moments and forces components are expressed:

[image:2.612.314.539.170.572.2]

(a)   

[image:3.612.80.296.46.519.2]

(b) 

Fig. 5 Force and moment model (a) General model (b) Projected components

(

)

(

)

(

) (

)

(

)

(

)

x y W y C z W z C

y x W x C z W z C

z x W x C y W y C

M W z C z W y C y M W z C z W x C x

M W y C y W x C x

⎧ = − + + +

⎪⎪ _{= + − +}

⎨ ⎪

= − + + +

⎪⎩

(3)

(

)

(

)

y y y

z z z

F W C

F W C

⎧ = − + ⎪

⎨

= − + ⎪⎩

(4) Those forces and moments are different to components in

Eqns. (1) and (2). It is suggested to modify and generalize the components within those equations to improve the application of transfer function.

C. Setting error of chain or wire

Fig. 6 Misalignment of connecting chain/wire to spindle head

In assembly process of connecting the chain or wire to the machine head, misalignment usually happens. Fig. 6 is an example that visualizes the changes of counterweight components Cx during the movement of the head, where

[image:3.612.329.531.265.698.2]

h x h L≤ ≤ + , h is initial height, L is travelling height. Fig. 7 demonstrates the moment errors through setting error following the case:

(

)

(

)

1000,0, 0 , 0, 800 0.4 1

x z

C C

N

h m

L m

= = = W C C

0.3 0.2 0.1

0.1 0.2

0.4 C

W

C

W

C

W

x m

x m

y m

y m

z m

z m

= = − =

= − =

=

At a quite large setting error of 50mm the error of Mx is

less than 10N/m, My and Mx are less than 3%. These errors

can be overcomed by increasing the stiffness of the vertical rail.

(a) 

(b) 

[image:3.612.75.254.568.718.2]

(c) 

Fig. 7 Moment error and setting error (a) Mx

(b) My

IV. COUNTERWEIGHT AND VIBRATION A. Vibration Model

Machine tool with counterweight has higher load capacity and more safety during operation. But adding one more component to the machine structure means dynamic problem needs to be reconsidered. And vibration analysis is coming up. By simplifying machine center as Lin's work [6], vibration model can be carried out. Ball nut m3 is considered

as a source giving motion to machine head m2. Machine head

is connected to counterweight system m1 by chain or steel

wire with stiffness k1 and damping c1. Connection between

the spindle and ball nut has stiffness k2 and damping c2. Fig. 8

(a) shows the mechanical counterweight model and vibration equation is:

{ } { }

{ } {

( )

}

M x +C x +K x = F t (5)

(

)

(

)

1 1 1

1 1 1 1 1

2 2 2

2 1 1 2 2 1 1 2 2

3 3 3

0 0 0 0 0

0 0 0

0 0 1 0 0 0 0 0 0 ( )

x x x

m C C K K

x x x

m C C C C K K K K

x x x f t

− −

⎡ ⎤ ⎡ ⎤ ⎡ ⎤

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤

⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎢ ⎥ ₊⎢ _{− +} ⎥ ₊⎢ _{− +} ⎥ ₌⎢ ⎥

⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥_{⎣ ⎦}

⎣ ⎦⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦⎣ ⎦

Fig. 8 (b) presents the hydraulic or pneumatic counterweight model. Vibration equation in this case can be expressed as Eqn. (5) with m1 << m2.

[image:4.612.312.541.210.612.2]

B. Vibration analysis

Fig. 9 introduces a schematic model built in MATLAB Simulink. Initial parameters of this study case are:

m1 = 28 (kg)

m2 = 38 (kg)

k1 = 1.15x104 (N/m)

c1 = 121 (Ns/m)

k2 =9.86x106 (N/m)

c2 = 9730 (Ns/m)

Fig. 8 Vibration model (a) Mechanical counterweight (b) Hydraulic and Pneumatic counterweight

One complete move of machine head to the target is normally divided in 3 stages: acceleration, constant and deceleration. Fig. 10 expresses velocity and acceleration through time during one move of the ball nut, meanwhile f(t) in Eqn. (5) should be:

if 0 0.5

( ) 0 if 0.5 1.5

if 1.5

a t

f t t

a t

≤ ≤ ⎧

⎪

=⎨ ≤ ≤

⎪− ≤

⎩

(6)

From that input condition displacements or position errors between spindle head and ball nut positions are plotted in Fig. 11. Fig. 11 (b) gives better results compare to Fig. 11 (a), which mean hydraulic and pneumatic counterweight and non-counterweight are more reliable than mechanical counterweight.

Fig. 9 Matlab simulink

(a) 

 (b) 

Fig. 10 Ball nut input (a) Velocity (b) Acceleration

[image:4.612.168.531.217.685.2] [image:4.612.75.287.328.719.2]

A poi bef err the and c2,

[image:5.612.76.299.46.351.2]

stu

Fig. 11 Displa (a (b) Hydraulic

As mentioned int, it keeps m fore considere rors are expres e relation betw

d m2, stiffness

respectively. udy case above

(a

(b acement of spin a) Mechanical

and pneumatic counterw

above, after t moving up and d as permanen sed in Fig. 11. ween position e

ratio of k1 and

The vertical d .

(a ) 

) 

ndle due to inpu counterweight c counterweigh weight

the head move down for a ve nt stop. The tw . Figs. 11 (a), ( errors and wei d k2, damping r

dash lines are

) 

ut of ball nut t

h, or without

es to the targe ery short period wo first upward (b) and (c) giv ight ratio of m ration of c1 and

marked for th et

d d ve m1

[image:5.612.316.543.50.355.2]

d he

Fig. 11 Po

C. Cha Accord difference counterwe velocity r

2 m

1 ( 4r −

and sho One ex counterwe

v3 = 20

r = 100 Chain L

( )

1

v t =

( )

1

a t = 1( )

F t =

From E maximum

osition errors a (a) W (b) W (c) W ain Vibration ding to chain

e between v eight system, a atio and chain

2

3

2 2

1 min

)

p v

p ≤v ≤ +

own by Fig. 13 ample is carrie eight system.

(m/min) 0 (mm)

LH0822, p = 1

0.001341 cos×

(

0.22sin 165

= −

(

66sin 165t

= −

Eqn. (8) Fig. 1 m amplitude is l

(b) 

(c) 

after the spindl With weight ra With stiffness ra With damping ra

n and sprocke velocity of as described in

model is given

2 2 min

1 4

p r +

3 (a).

ed out to explor

2.7 [11]

(

165

)

1 m/s

3

t +

)

2

5 m/st

)

+ 3000 N

t

3 (b) shows th less than 3%.

le head reach th atio

atio atio

et assembly machine he Fig. 12. The re n by

re the force vib

he force vibrat

 

 

he target

there is ad and elation of

(7)

ration of

[image:5.612.72.295.497.630.2]

[image:6.612.81.292.53.205.2]

Fig. 12 Velocity difference by chain assembly

(a) 

 

(b) 

Fig. 13 (a) Velocity ratio and chain model (b) Force fluctuation with chain LH0822

V. CONCLUSION

This assessment reviews the differences between vertical and horizontal motion in machine center. From the review two problems are raised and solutions are also proposed. Specific forces and moment model for vertical guide are built. But for the purpose of applying modifications to current transfer function, the model needs to be generalized. Vibration of mechanical counterweight and hydraulic or air counterweight system are predicted and evaluated.

Further works and experiments need to be solved for generalize the results into practical applications.

ACKNOWLEDGMENT

The authors wish to acknowledge the Ministry of Knowledge Economy of South Korea and Korea Institute of Machinery & Materials for supporting this work.

REFERENCES

[1] Alexander H.Slocum, PRECISION MACHINE DESIGN. Prentice-Hall International, Inc, 1992.

[2] Chana Raksiri, Manukid Panichkun, "Geometric and force errors compensation in a 3-axis CNC milling machine", Int. J. Mach Tools Manuf, vol. 44, 2004, pp. 1283-1291.

[3] H. Vermeulen, M. Vermeulen, S. Wetzels, W. Pril, "Design for Precision: Current Status and Trends", Keynote paper, Section Precision Engineering, Eindhoven University of Technology, The Netherlands, Annals of the CIRP, vol. 47, Feb. 1998.

[4] Eiji Shamoto, Chun-Hong Park, Toshimichi Moriwaki, "Analysis and Improvement of Motion Accuracy of Hydro Static Feed Table", CIRP Annals - Manufacturing Technology, vol. 50, 2001, pp. 2285-290. [5] Chun-Hong Park, Yoon-Jin Oh, Joon-Hee Hong, Chan-Hong Lee,

"Theoretical Verification on the Motion Error Analysis Method of Hydrostatic bearing Table Using a Transfer Function", Int. J. Prec Eng and Manuf, vol. 4, 2003, pp. 64-70.

[6] S. Y. Lin, Y. C. Fang, C. W. Huang, "Improvement strategy for machine tool vibration induced from the movement of a counterweight during machining process", Int. J. Mach Tools Manuf, vol. 48, 2008, pp. 870-877.

[7] Ming-Hung Tsai, Ming-Chang Shih, "A Study of the Pneumatic Counterweight of Machine Tools Conventional and Active Pressure Control Method", JSME Int. Journal, Series C, vol. 49, 2006, pp. 890-896.

[8] NSK LINEAR ROLLING GUIDES. NSK Technical Journal, pp. A14-A78. Available:

http://pdf.directindustry.com/pdf/nsk-precision-america/linear-rolling-guides-17324-1238.html

[9] Thomson BSA Lead and Ball Screw. Available:

http://www.thomsonbsa.com/pdf/ThomsonBSA-Lead-and-Ball-Screw -Catalog.pdf

[10] CNC Linear Controller, P/N 8800. Available: http://www.sherline.com/8800inst.pdf [11] Renold Lifting Chain Catalogue. Available:

http://www.renold.com/nmsruntime/saveasdialog.asp?lID=463&sID= 1860

[image:6.612.90.278.226.462.2]