The Electro-Hydraulic Bending Moment Servo Vibration Damper

Copyright © IFAC Control Science and Technology (8th Triennial World Congress) Kyoto, Japan, 1981

THE ELECTRO-HYDRAULIC BENDING MOMENT SERVO VIBRATION DAMPER M. Mochizuki* and Y. Okada** -Digital EqUIpment Corporation Internationaljapan, Bunkyo-ku, Tokyo, japan "Faculty of Mechanical Engineering, Ibaragi University, Hitachi, Ibaragi-ken, japan

Abstract. In the metal cutting process, selfexcited chatter sometimes may take place and it may cause bad effects on the workpiece surface, on machine life and on the cutter. It is generated, first of all, by the extremely resilient dynamic stiffness in the structure of the machine tool; this is especially true when a borine bar is long and slender. As the result, an additional damping unit is highly desirable to improve the cutting stability. This paper introduces the idea and theoretical analysis of a boring bar provided with a bending moment servo vibration damper. Chang's optimum control theory is applied to the controller because the damping force must be most effectively used. Then, the damper is applied to stablize the cutting chatter with good results. Keywords. Machine tools; damping; vibration control; feedback; hydraulic systems; optimal control. INTRODUCTION Selfexcited chatter (1) is widely accepted as one of the major obstacles in machine processes. These phenomena limit the rate of material removal, the machine life and the tool life. Several factors are conceivable as the cause of chattering; but the most significant is the extremely resilient dynamic stiffness in the structure of the machine tool. Numerous research projects have been conducted and reported (6), (7) on this difficulty. In particular, a boring bar operation provides inherent problems because of its low stiffness and high aptitude to cause chatter. l-le had developed a bending moment servo vibration damper (3) to improve the dynamic stiffness of large-size ram structures. We applied this concept to improve the dynamic stiffness of a boring bar. Glaser and Hachitgal ,(4) report the same concept. However, the main problem is how to provide additional damping. To overcome this problem, we applied Chang's optimum control theory (2) to design the transfer function of the controller. As the result, the differential oil pressure is efficiently converted into damping force and may efficiently decrease the system vibration. Then the controlled boring bar system is tested to correct the cutting instability. The results obtained are discussed in detail. DEVELOPMENT AND ANALYSIS OF A BENDING-MOMENT SERVO DAMPER Basic Concept The concept of an active vibration damper is

eST . _ J

2123

characterized by providing an additional damping of the exterior force generated by a suitable force motor to machine tool structures. On the other hand, a bending moment damper is based upon the '.tilization of the motion of cantilever structure to control the hydraulic differential pressure in two chambers in a boring bar. Let us consider the basic operation of a bending moment damper acted on by a beam by a static exterior force and a damping force. As shown in Fig. 1, we assume the external force Fin and the damping force Fc are identical in magnitude and opposite in direction. The bending moments of these forces Min and Mc on the cross-sections mn and m'n' also cancel each other. An alternative means of applying Fc , the bending moment damper serves to correct exterior force Fin by applying Mc directly on the beam. The primary moment generator, as shown schematically in Fig. 2, is based on this idea, the moment is generated by a pressurized chamber with the cross-section area A placed perpendicular to the beam. When a hydraulic pressure P is applied to this chamber, a force F=AP acts in the direction of the beam axis. Since the chamber is located at a distance of i from the nuetral axis NN', it produces a moment of magnitude M=Fi. A pair of bending moment generators are mounted on both sides of the beam as shown in Fig. 3. Vibration is absorbed through the generation of positive and negative moments. But, the boring bar machine cannot utilize this moment generator because there is an installation limitation. Hence, we considered a new idea: the cantilever with two axial hole chambers shown in Fig. 4. The idea is to make the axial hole itself a pressure chamber and to

The Electro-Hydraulic Bending Moment Servo Vibration Damper (3)

There is a small transfer delay in the oil chamber of a moment damper depending on the oil compressibility. This transfer dealy is approximated by a first order lag element. P

For theoretical convenience, the system differential equations (4), (5) and (6) are translated to the block diagram as shown in Fig. 8. From the diagram, we yield the following relations: ~ F(S)

G2(S)

Fd(S) _

(7)

H(S)

G2(S) . GcCS)

F1S) - ---,l:---:+----=G-l(7:S")-.-G-=-c-("'S")-

G2(S) - H(S) Gl(S) (8)

Where H(S) is the optimum control ratio. The functions Gl(S) and G2(S) are the following. G (S) = 1

V(S) G2(S) = Z(S)

~ Z(S)

(1 + TS)(mS 2 +

Z (S)

cs

+ K)

1

(19)

+ TS

H(S)

=

·i~·W!Fo d, 2 w.W +

R(S)

2

vz f3 2 z .z +f3

(20)

Taking into account eqs. (10) and (11), the denominator of eq . (20) is comprehended to be a sixth order polynominal, and can be represented as follows: d,2W.W+ f32 z . z = A1 2z o"Zo = A12(-S2 + r12)(-S2 + r22)(-S2 + r3 2 )

(10)

(12)

The damping force must reduce the vibration of the structure most efficiently. The design criterion is to minimize the following performance index 1 0 • (13)

Where a and S are Lagrange multipliers and
~tX = H·H· ~ff

Where, H (S) mus t satisfy Chang' s t11r.::~e conditions and R(S) denotes an arbitrary function which does not possess any poles in the lefthalf plane. Assuming the external force to be white noise,g?~S) = Fo(constant). From eqs. (14), (15) and (19), solving H(S), we have

(9)

(11)

W(S) V(S)

Applying the optimization theory by S.S.L. Chang(2)to eq. (18), we obtain the condition that To is to be stationary.

(6)

1 + TS

i

2125

(14)

(21)

Since the function H(S) must be stable, all the poles -rI' -r 2 and -r3 are to be in the left-hand plane. Thus, we have (22)

H(S)

Where Vo(S) denotes an arbitrary polynominal, and all the coefficients of this polynominal must have arbitrary positive values. When d. ~ 0, Vo(S) must be equal to V(S). Assuming Vo (S) = AS+B, we can determine A and B, and the optimum compensating transfer function Go(S) is determined as follows: Gc(S) = G2(S) - H(S) Gl(S)H(S)

(23)

The numerator of eq . (23) can be represented as follows: ZoV-ZVo = (1+TS)\ (1-Am)S3 + (rl+ r2+ r3-AC- mB )S2

(15) Where ~l = Gl(-S), G2 = G2(-S) and H = H(-S). The relations between mean square values and power spectra are given as follows:

j t f- ."" ~XI( X
"TTJ -J'"

I =--. (2

2TTj

fj,..

_j".

fJ'"

To obtain the simplest realizable optimum compensating transfer function, A and Bare decided as follows: A

f{'{d{S)ds

1 . (ex2 Pxx CS ) +p2 PMiS»)ds 10=-'

(24)

(16) (17)

Introducing eqs . (16) and (17) into eq. (13) yields. .

211J .J"

+ (r l r 2+ r 2r 3+ r 3 rC AK- BC )S + rlr2rrBKl

(18)

(25)

llm

(rl + r2 + r3)m - C B

m2

(26)

We obtain Vo(S) as follows: Vo(S) =~+ ill

(rr + r2 + r3)m - C

m2

(27)

2126

M. Mochizuki and Y. Okada

As a result of the above, the optimum control ratio H(S) is obtained from eqs. (27) and (20) and the optimum compensating transfer function is obtained from eqs. (27) and (23). Calculation of H(S) and Gc(S)

Cutting Behavior of This Boring Bar with Homent Servo Damper The effect of this controlled boring bar to the metal cutting stabilization is tested using an exclusive turning-type cutting apparatus. Cutting materials are JIS-20ll and JIS-S45C. Cutting conditions are shown as follows:

The values of the coefficients of this experimental device are shown in the nomenclature (1) In the case of the workmaterial JIS-20ll table. The optimum controller is designed Cut speed 110 m/min. with the aid of a computer. The roots of the Feed 0.026 mm/rev. 2 system characteristic equation !l( W·W + f32 z· = 0 and the mean square values of
z

The Electro-Hydraulic Bending Moment Servo Vibration Damper

2127

( 6)

Okada, Y. Analysis and experiments of the electro-magnetic servo -vibration damper. Bull. of the JSME, Vol. 20, No. 144, 1977, pp. 696.

ACKNOWLEDGEMENTS The authors wish to express sincere appreciation to Dr. T. Nakada, Dr. N. Tominari, Mr. K. Saito and Mr. H. Takahashi for their cooperation.

(7)

Cowly, A., and Boyle, A. Active dampers for machine tools. C. I. R. P., Vol . 17, 1969, pp. 103.

Nomenclature Table F~n

equivalent damping constant

(5.0 N·sec/m)

E:

Young's modulus

(2.06xlO l l N/m 2 )

F:

exterior force

(N) (N)

C:

fd:

damping force transfer function of controller

H(S) :

optimum control ratio

i:

servo valve input current

(A)

I:

moment of inertia of cross-section

(1. 24xlO- 9m4 )

K:

stiffness of main structure length of beam

(7xl0 4 N/m)

M*:

moment generated throughout the beam

(N·m)

m:

mass of main structure

(0.3 Kg)

P:

supply pressure

(1.4xl0 6 Kg/m 2 )

R(S):

Chang's arbitrary function

S:

Laplace operator

(l/sec)

T:

time constant

(2.27xlO- 3 l/sec)

V:

shearing force

(N)

X:

displacement

(m)

4> :

angle of deflection

(rad. )

).J :

m

----- -- --- -r---¥-----.- -M-;~-------

GC
.e:

m

N-{- tt-M,n-t-_

t--t+-ff--N' .

____I!c;. ____L..-_,..--_...J n

Fig.

Mc

nO- - - - - - - - - -

Stastic Aspect

1

pressure

chamber

(0.222 m)

Fi g.

Dimension of a praimaxy moment generator

2

F

pressure amplification degree

REFERENCES (1) 11erritt, H. E. Theory of self-excited l-lachine tool chatter. ASME Journal of Engineering for Industy, Nov. 1965. (2 )

Chang, S. S. L. Synthesis of Optimum Control System. McGraw-Hill, 1961, pp. 11-86. (3 )

Mochizuki, M., Takahashi and Tominari. Development of a quasi-moment damper. Bull. of the JSME, Vol. 20, No. 148, October, 1977. (4 )

Glaser and Nachligal. Development of a hydraulic chambered activity controlled boring bar. Trans. ASME (B),1977, pp. 362. (5 )

Pestel, E. C., and Leckie, F. A. Matrix Methods In Elastomechanics. McGraw-Hill, 1961.

~ . ~. c Fig. 4 Actuation Concept of Pressurized Chamber

Fi'1. 3

Dynamic

Aspect

dB 30

~

20

l-~

10

o•10

• ..-:1

20

5~

-10

·

.

o

· ·

-100

~-<>-~\

100~ 20,0 Hr

",-

-~"ii-'
1'"0...

-200

Fig. 11

Frequency Response

M. Mochizuki and Y. Okada

2128

Point

beam I

I,

Point 1 BB -section 1. main 2.pressure pip e co nne c ting part 3.exhaust pip e connecting part 4. shank parts 5. oil c hamber s 6. cu tter 7. c utt e r clamper 8. servo valve 9. co il of velocity detector

Fig. 5

beam

Point 2

Fig. 6 Equivalent Model

Experimental scheme of Boring Bar to.

jW

I

~=O~ - 3.22+j4630 -;

'1

ve lo city feedback

Xl0' jW 12

~./

-3. 25+j4000

-----; posi t i nal feedback Kt+ oo

Kt=O

I

4

-1.86+j393

IJ'

-1.86-j393

I

Kt +

CXl<

( X')

-3. 25 -j4000 -3.22-j4630

K =0 t

o

I

20

40

(a)

Fig. 7

Root

60

80

100

-12

(1.!

Y

(b)

The mean square values

X(S)

F(S)

Fig.S

Block

Fig.

9

with o ut

Diagram

Feedback

Fi g. 10

-0

Mean square values and root locus

(a)

(a)

The root Locus of

cx'w +(I'zz

Locus

(b)

Step

with Feedback

Respon se

S45C

Fig. 12

(b)

.l1S2011

Work pieces surface after cutting