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The influence of the carriage speed speed on the compliance

of the tool-holder

Citation for published version (APA):

Kals, H. J. J., & Hoogenboom, A. J. (1969). The influence of the carriage speed speed on the compliance of the tool-holder. (TH Eindhoven. Afd. Werktuigbouwkunde, Laboratorium voor mechanische technologie en

werkplaatstechniek : WT rapporten; Vol. WT0227). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1969

Document Version:

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(2)

. technische hogeschool eindhoven

laboratorium voor mec:hanisc:he tec:hnoJogie en werkplaatstec:hniek rapport van de sectie: Verspaningstechnologie

titel:

auteur{s):

sec:tieleider:

hoogleraar: samenvatting

The influence of the carriage speed on the compliance of the tool-holder.

ir. H.J.J. Kala ir. A.J. Hoogenboom

dr.ir. A.C.H. van der Wolf

prof. dr. F.C. Veenstra Summary

The effect of the motion of the carriage at the compliance of the Peter - Vanherck test rig has been measured for several conditions. A qualitative explanation of the

phenomenon is given. _H

Actually this explanation is based on the res~lts of analogue

computer experiments.

prognose

,

,

,

biz. 1 van20 biz • rapport nr. 0227 coderin.g: P.7.c. trefwoord: Stabiliteit Gereedschaps-werktuigen. datum: 1-12-1969 aantal biz. 20 gesc:hikt voor publ ic:atie in:

Note to be presented to C.l.R.P. Group Ma.

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0 10 -15 r20 - 251-30 I- 3H--

50-rapport nr" 0227

biz.

2 . van20

blz.l

Introduction.

Displacement measurement of the test~rig and of the carriage

as well are carried out during harmonic excitation of the test-rig.

With the aid of the results of these measurements some conclusions are made.

These conclusions lead to a simple mathematical model, which has been investigated on analogue computer.

The computer results are compared with the original experimental data.

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o

5 1Q 15 20 25 30 50

rapport nr. 0227

biz.

3 van20

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1. Experiments.

1.1. Program of Measurements

The following measurements are carried out. See Fig. 1.

a) The modulus of the transferfunction x;y (strain gauges

measurement) at v •

a

and v • 200 ~m/s. for three values

of the damping ratio 10:£ the tes trig viz. Case I, Case II

and Case III. See Figs. 2, 3,and 4.

b) The amplitude ratio

I

xl I £ measured at natural frequency

a

of the testrig (~155 Hz) as a funcyion of the carriage

speed also for the three cases mentioned. See Fig. 5. c) The absolute amplitude Ixl measured at natural frequency

for v -

a

and v

=

200 ~m/s. also for the cases I, II, and

III. See Table I.

d) The absolute amplitude Iyl measured at natural frequency

for v -

a

and v • 200 ~m/s. also for the three cases.

See Table I.

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01lQ -15 r30 -rapport nt. 0227 blL 4 van20

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1.2. Remarks

a) A comparison of absolute motion Ixl and the relative motion . x-y shows a same tendency with respect to decrease of·the

amplitude. See Table I and the Figs. 2, 3, and 4.

b) The frequency at which the maximum amplitude of IXpIl occurs (about natural frequency) does not change with a varying carriage speed. See Figs. 2, 3, and 4.

c) The low frequency compliance (~ 40 Hz) is not influenced by

a change in the carriage speed. See Figs. 2, 3, and 4.

d) An increase of carriage speed V results in a decrease of the

test-rig motion x and at the same time an increase of the

carriage motion y. See Table I.

1.3. Notes

a) The in Table I mentioned y data for v

=

0 are measured in

an abSOlute way. As a matter of fact, there is no relative motion between carriage and frame in this case.

b) However, at v - 200 ~m/s. the frame movement is close to

zero. so that the absolute movement y equa~s the relative

displacement between bed and carriage.

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0 -10 r- 151--20 '- 25-30 I--35 r-40 r45 -50

I--rapport nr. 0227 biz. 5 van20 blZ.l

1.4. Conclusions

The movement of the carriage influences the coupling of carriage and frame.

- v

=

O. The harmonic forces between carriage and frame are

not large enough to exceed Coulomb-friction forces in order to cause a relative displacement y between carriage and frame.

The coupling between the carriage and frame is. rigid. See Note 1.3.a.

- v

=

large. Now, we have to deal with a relative velocity

between carriage and frame. The Coulomb-friction transformes into a viscous friction. The coupling between carriage and

z) frame is viscous and almost independed on the carriage speed. See Note 1.3.b.

- v

=

small. The velocity amplitude of the carriage is equal

or larger than the nominal speed V.

In this case the carriage will periodically stand still. The coupling between carriage and frame is periodically rigid and viscous, and depends upon the carriage speed.

So we can divide the velocity range in two parts. See Fig. 5.

In the following the situation of a very loose and a rigid

coupling of carriage and frame is simulated by varying the

quantity p • This quantity represents the viscous friction

y

between carriage and frame.·

Z) The viscous coupling between.carriage and frame is also independed

on the preload. That is the re~son why the dynamic behaviour will

not be influenced by locking the carriage at a certain speed.

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01-lO f- 11- 2Of- 25- 30- 351-

5Ot-rapport nr. 0227 blL 6 van 20 biLl

2. Mathematical Model.

2.1. Differential equations

In the following part a very simplified model of the combination test rig-carriage-frame is formuh.ted.

There are two simultaneous differential equations which describe the

displacements x and y.

The solution of these equations is formed with the aid of an analogue computer.

mx

x

+ p x

(x-y)

+ c x (x-y) -

r

Px

(x-y)

+

c x (x-y) - m . y

y+

p y

y

where

m mass of the testrig

x

m y mass of the carriage

Px damping cons tant:b£ the ::ttis;t rig

P

y clamping constari£"between cardage and bed

c stiffness of the testrig.

x See Fig. 1. Substitute of 2 c x 2S c x x 2S c y x Wx

.-

m Px

...

W Py

-

w x x x and m x k where _ l : ; , ; m m y

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. rapport nr. 0227 0 -1Q r - lSi- 3Ot- 35-I 110-I I bt-I

I

I

re-werkplaatstechnfek

biz.

7 van 20

blz.l

Sx damping ratio of the test rig,

reduced damping ratio of the carriage,

gives ] .. 26 xC') p + (x-y) and

2

x + -til x-y

--

C til X X x 26 1 28 x

(x-y)

+ (x-y) ...

y

+ ---1:.

Y

til til 2k til x x x m

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01- 51lQ - 15120 - 251- 301- 351-

50-rapportnr. 0227 biz. 8 van 20 blL

1

2.2. Dimensionless Computer Eguations f

The quantities, x, i.

X,

y,

Y

and yare related to their maximum

values, xm'

x

m'

x

m• Ym'

Y

m and

Y

m respectively.

In case of an harmonic excitation, with the maximum circular

frequency W t the following relation between these maximum

max values will exist:

.

x -m W max m x •• 2 x .. W x m max m Let us assume x - y • m m

Now the eqs modify into:

W 2 W

..

P

[(~

)

-

(~

)]

(~)

..

= = - (_x_)

-'

{ 213 (_x_) c x W x W x m xm ,max max x Ym m 2 +

(_:x_)'

[(~

>_(}-)]}

max m m and~

The analogue computer circuit is given in Fig. 6.

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r---~---, 0 1Q - 151-2S I -30 I - 451-

SOl-rapport nr. 0227 biz. 9 vall20 biz.

I

2.3. Analogue Experiment

Most of the parameters of both eqs are known except the value of B • y W x We assume ~---. w 0.5, k - 0,16, S - 10 max m y Prosram of Measurements

The following measurements are carried out: and

of the transferfunction The modulus

x-Y

p for B

III 10 and By - 3 for three

y

values of Bx viz. Case I, Case II ,and Case III.

a -

y 3.

See and compare Fig. 7, 8, and 9 with Fig. 2, 3, and

4 respectively. ~,.

Note

It can be proved that the observed effects exist is a wide range of values Band k •

Y m

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0 -1Q r-15 I25 -30 I50

-rapport nr. 0227 biz. 1

a

van 20 blZ.l

3. Conclusions.

After comparing the experimental and~I'computed data, the mathematical

model seems to be fairly good.

The influence of the carriage speed on the transferfunction may be decreased by adding mass to the carriage, adjusting backlash

between carri~ge and frame. or using a lubricate with a high

viscosity.

We have to notice that the system is nonlinear. Therefore the dynamic behaviour will depend on the amplitude of the exciting force.

Using the test rig characteristics for computing the stability limit-values we must keep in mind either this nonlinearity as the velocity influence.

(12)

o

5 1Q 15 20 25 30 35 4S 50 rapport nr. 0227 p testrig

v

frame m x m y

biz.

11 van20

b,z.l

1----"""

X y

Fig. 1. Testrig mounted on the carriage.

(13)

:I

...

CIt t tJ: ~ I'! t:= c;: ;; CIt CIt I I I I I I I I I I. I :E

O,4~m

...

~ N "CI 0 D

....

..

....

n :r :I ;-~ 4 -___

fx;

y

l--~~ ----+---+---~---~~---~

...

~----~~--

...

~~~

...

~--~---_+---

CD n :::r' :::I

ft'

:::r' CD :::r' 0 CD CD fI) n :::r' 0 9. CD

-,

:::I a. ::r ~ 200 Hz «I> :::I 0 •

Fig. 2. Transfer function, case I,

a

"CI

1

::. :I

...

• 0 N N

'"

< DO ::0 N o 5!:

,..

L-~ __________________________________________________________________________________________ --__ ~ ______________ -L-~

(14)

I I I I I

0.4

ll:

..

'111:" "Q Q a

..

..

f

:r :s

.-

'111:"

Ix ;

y

I

I

I

it

n :r ::I 0' n :r

/.

:r

/-'

0 CO

~

CD

~

tn n ~ :r 0 2-CD

:i*

a.. :r ~ 100 CD :J 0 Fig_) • Transferfunction, I I I

/ \

V

=

0 .,

If

'

~

J

/",..---.,

p"

V

~

200"m/sec\

~

case II.

I

I

I

I

I

I

!

-o I ~ ColI I. ~

"

~

~

200 Hz CIt I 3 "Q

1.

:s

..

.

0 N N "'-J

I

I

I

S!:: ~

...

f'"

--

w < I» re 0 S!:: f'"

(15)

t t It to: :;: N ~

I; \

-CIt 0

'"

1:1 I I I I I .1 I I

1

I I. I :I

T

."

a

!It lJlU

!

71:" 0.4

N

. " ir ::II D V = 0

..

-

..

-•

n ;;r r'. 0 ::II N ;-

..

\

N 71:" 4

..

"-J

-

..

. •

\

I

n J x ; Y f f

\'

.

I

-

\\

..

II

pi

!

~

v = 2001lm/sec:

~

VI

\

/;

~

;-I'

\

n ::r ::a ~~

~

::r CI:I •..

:f?

!"

::r 0 ~ CD

~

CI:I

n

---

~ ::r 0

e.

~ !2: 1:'1 CI:I

--:i .I:>-CL c ::r 100 200 Hz I» 0 :::0 < I N CD

,

0 ::a )

Fig. 4 • Transferfunction, case III.· !2:

1:'1

(16)

~ !C :/01:" "'0 D D

-..

ft n :r ::II ;-:/01:"

;-.n

=-::s i' n

=-CD

=-

0 CIQ CD CIt n ::r 0 2-cD

Er

Q. ::s-~ CD ::s I 1 I I

~

to--.

----r---

---IX - XI

. f - , P f 0

.

In this part:

r---

~ .. ~

/ '

. /

/

~/ /

V~

,I case III

r---

r---..

----...

y:>

v

/

case II

. /

..'

/

'"

/

case I 160 1

/

-...

1

J/

I

1

1/

. /

I

I

I

I

Fig. 5. Amplitude ratio

IX;XI

at the natural frequency versus carriage-speed.

I. I " 4

-a

"'0

..

0

...

/

-

~ 0

/

N N "'-J

/

I ~ S!:

,...

-""

< 320 llm/sec. III :=

,

N 0 S!:

,...

---1

(17)

rapport nr. 0227 Or-10 r-15 r -:20 !

-I 125 -30 r-35

r-

5Or-w.rkplaatstechnlek case 1

The signals are to small for accurate measurements. case II ~" V ... 0 V = 200llm/sec.

I

x

I

45 mV 36 mV

'/y /

1,.5 mV 3,2 mV case III V III 0 V • 200tJm/sec. -Ix

I

90 mV 66 mV .

ely I

2,2 mV 3,7 mV y

...

,.. 44 llm

Table 1. Experiment data.

biz.

16 vao20

blz.l

(18)

...

C> ~

..

~ "0 Q

a

...

....

&

':T :I i"

..

.

~ I -

X--

..

+

Y

m Ym

a:-n :r

..

x +

-::s iii" n

..

x m :r et :r

rv

0 IC et tn n :r 0

Fig.6. Analogue computer circuit.

2 2-et

5r

Q. 6 :r 0 < et ::s 7 to.)

-C> 1ft

.

x

x

m Potentiometers 26 x 28 til k 'l x m til max 26x D Ym + til 82-til max til 10 x til max 12 k m 0 x -x m

..

Q "0 "0 ~

....

:::II

..

.

0 N N ..., ..., ~ ::0 N o 5!: J'I ---~--~---~~

(19)

-I I J I I I I 1:

..

'"

I

¥I

"Q iii"" D

-..

if , n ) :r :I

.-'"

~

S'

n :T :s ;;" n 6

=

10 ::r CD y :T

_._.L._._

0 ~ ca

---to

---

roo . - ' - -

B

=

3

_.-'-

y n

.----.'"

:T

.

0 2-CD :f Q. ::r 0 100 < to

-:s

Fig. 7. Analogue computer transferfunction, case I.

-.... I

~

.---.

I I.

I

I

I

I

, I

I

I I

!

!

~

.

...,;;:

~

..

200 Hz

.

co I

a

"Q

!

:I

..

.

0 N N

....

----

!2: !"I

-

C1:> < D> ::0 N CD !2: !"I - - - '

(20)

o I

,

I I .1' I I I I I I :E

a

'tI

...

~ 'tI . "

!

D D ::::I

..

..

'"

..

n ::r Go ::::I

I

N i"

I

N JIt'"

I

...s

I

I

.

I

I I

I

I

I

I

/

~ey=

10 x-y p

V/-·~

,

/

\

i

..

,,"

'.

CD

/;.

.' a

=

3

"

n

V"

y ::r :s ~.

/

::r ,,/ CD

~

.

...--

-~

::r 0

~

CD CD

--

.".,.... CIJ n

.-.-::r

.

-

~

0 2-

...

S!: CD !'"

:r

-

\D Q. ::r < ~ 100 200 Hz 0> :::s

..

.

.

N :s 0

0 Fig. 8. Analogue computer transferfunction, case II. S!:

!'"

(21)

o I I I J I I I I I. I ~ --_._-

..

a

"0

..

"0 ~

!

"0 D a

I

1\

~

-..

"

-•

n

/

\ a

y

=

0 :r ::I N i" 10 N ~

I

..., ! I i !

I

.,/""

..

\

i

I

x-y

I

I I P

I

I

,

i/

lay

=

3 , i I I

~

..

I I I • j

.

I

!

,

\

)

/

~

,I"

S'

~./.'

\

n ::r' :s

fi'

/

/

::r'

/"

,",

CD ::r'

~

V,,·'

~

0

,.

cc

"

CD ~ til

.

..

---n

.-"-::r'

.-

,

0 g,. ~ 5!: CD t"

:r

N 0. 0 ::r' 100 < ~ 200 Hz II> ;:s

.,

I

..

N :s • •

.

0

0 Fig. 9. Analogue computer transferfunction, case III. 5!:

t"

- - l

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