control of vibrations in multi-mass systems with locally controlled absorbers

qThis paper was not presented at any IFAC meeting. This paperwas recommended for publication in revised form by Associate EditorJ. Z. Sasiadek under the direction of Editors K. Furuta and M. Araki.

*Corresponding author. Tel.: #49-89-1489-6713; fax: #49-89-1489-6383.

E-mail address: damir."[email protected] (D. FilipovicH ).

Automatica 37 (2001) 213}220

Brief Paper

Control of vibrations in multi-mass systems with locally controlledabsorbersq

Damir FilipovicH *, Dierk SchroK derInstitute for Electric Drives, Technical University Munich, Arcisstr. 21, 80333 Munich, Germany

Received 24 June 1997; revised 21 April 2000; received in "nal form 20 June 2000

A solution for mass displacements in multi-mass systems with a general locally controlled absorberwhich uses a single local feedback is derived and used for vibration control. A strategy for absorberplacement in the system as well as the concept of remote vibration control has been proposed.

Abstract

The main goal of this paper is the presentation of a local control concept to suppress vibrations in multi-mass systems. Thecontroller needs only restricted information from the multi-mass system, thus surmounting the need for perfect knowledge of thesystem under consideration, and reducing the sensor cost. A general solution for mass displacements is derived, and a strategy forplacement of a locally controlled absorber in the multi-mass system is suggested. Besides, the possibilities of remote suppression andsimultaneous suppression from di!erent system points with a single absorber are presented. ( 2000 Elsevier Science Ltd. All rightsreserved.

Keywords: Vibration dampers; Multivariable systems; Decentralized control; Local control

1. Introduction

As technology advances the excited system modesoften represent a limitation for further performance im-provements of the controlled system behaviour. There-fore, the controller in the broader sense should be fairly`awarea of the system complexity in order to be able toundertake an additional task of vibration control.

With a completely known and linear system, the state-space control can be the acceptable method. Neverthe-less, the state-space control requires many sensors orhigh-order observers, the latter always being relativelyslow.

If nonlinearities are present, the direct control is evenmore problematic. These nonlinearities should also be

fully known so that the system becomes observable. Thecontrol of systems with substantial nonlinearities is un-der research world-wide (Engell, 1995; Gupta & Sinha,1996; Freeman & KokotovicH , 1996; Isidori, 1995).

The local control should be preferred for vibrationcontrol for several reasons: If locally controlled, thewhole system does not have to be observable; the numberof sensors is reduced; the spillover can be avoided; thesystem robustness in the presence of parameter perturba-tions is larger.

In our case the vibration controller will be a locallycontrolled absorber (LCA) which comprises the passiveabsorber and an internal actively controlled dynamicfeedback. By means of a feedback compensator of theLCA, vibration suppression characteristics should beon-line tuneable. The design of the passive part will notbe treated here and one of the methods for the design ofpassive absorbers (Korenev & Reznikov, 1993; Hunt,1979) will be assumed.

As an introduction to this paper the following canbe used: (MuK ller & Schiehlen, 1976, 1985; Klotter, 1981)where the similar analysis has been carried out for pas-sive multi-mass systems, and general considerations

0005-1098/01/$ - see front matter ( 2000 Elsevier Science Ltd. All rights reserved.PII: S 0 0 0 5 - 1 0 9 8 ( 0 0 ) 0 0 1 4 0 - 0

about vibration suppression conditions have beendiscussed.

The main question that we want to answer in thispaper, independently of the type of LCA, and with anytype of external excitation force, is the following: If a cer-tain LCA is attached at the pth mass of the n-massprimary system, and the qth mass is acted upon by anexternal force, what is the behaviour of the particular rthsystem mass (see Fig. 1)? The contribution of this paper isto give a general solution of this problem in a compactform, and to apply it to show the possibility of remotevibration control and of the simultaneous suppression atdi!erent system points with a single LCA. The stabilityproperties of this solution will not be treated here inorder to keep the paper within the publishable limits. Aninterested reader can "nd it in FilipovicH (1998).

The LCA are introduced in Section 2. The solution forthe primary system alone, without the LCA, is derived inSection 3.1. The general solution for the whole combinedsystem, i.e. primary system together with the LCA, isderived in Section 3.2. Further, the e!ects of the solutionwill be discussed in detail and applied for the vibrationcontrol. This includes also a proposal for the generalstrategy for the LCA placement in the system.

These general considerations are veri"ed in Section 4;"rstly by means of an example showing the possibility ofsimultaneous local and remote suppression with a singleLCA; secondly, by means of experimental results. Thesummary with areas of application is addressed in thelast section.

2. Locally controlled absorbers (LCA)

The general locally controlled absorber (LCA) com-prises a mass}spring}damper trio and an active localfeedback path. The absorber should be attached to a pri-mary structure, i.e. a multi-body system that is acted uponby external forces. For the feedback design, no informa-tion outside the absorber itself is needed, thus makingthe control completely decoupled from the primary be-haviour. The signal used for the feedback is either theabsolute, x

a, or the relative motion signal, x

a!x

p, and

can be the displacement, velocity or acceleration of theabsorber mass.

The dynamics of the absorber is described by thefollowing equation of motion:

maxKa#c

a(x5

a!x5

p)#k

a(x

a!x

p)#f

a"0, (1)

where fais the feedback force. The corresponding transfer

function is

xa(s)

xp(s)

"

cas#k

a#dF

a(s)

mas2#c

as#k

a#F

a(s)

"

Ca(s)#dF

a(s)

Ma(s)#F

a(s)

. (2)

In#uence of the passive part is collected in Ma(s) and that

of the elastic coupling in Ca(s). The possibility of using

the feedback that is dependent upon the signal relative tothe point of attachment is included by the symbol d. It isd"0 for the absolute signal for the feedback input, e.g.xa, and d"1 for the relative signal for the feedback

input, e.g. xa!x

p.

The active part is

Fa(s)"saG

a(s), (3)

where a"0 for position feedback, a"1 for velocityfeedback, and a"2 for acceleration feedback. G

a(s) is

a transfer function of the feedback compensator with theactuator dynamics included (to simplify further analysis,the actuator dynamics will be neglected). At this pointit is not of crucial importance, how the transfer func-tion G

a(s) is de"ned. One should know that it is

actually dependent on some compensator parametersh"[h

1 2 h8] and we shall therefore sometimes write

Fa(s, h).If G

a(s) is a linear transfer function, such as

Ga(s)"g(1#s¹

1)/(1#s¹

2) where h"[g ¹

1] and

¹2

be a free parameter, the LCA is called the single-frequency linear active resonator (LAR) which is intro-duced by FilipovicH and SchroK der (1996, 1999).

The objective of any resonant absorber is to choosethose parameters h for a given structure F

a(s, h) such that

the absorber becomes marginally stable, i.e. an idealresonator, for a predetermined resonant frequency u.This can be achieved by solving the characteristicequation

Ea(s),M

a(s)#F

a(s)"0 (4)

for the marginally stable pole s"ju. It is a complex-valued equation that should be solved separately for thereal and the imaginary part

RMEa( ju)N"0, IME

a( ju)N"0. (5)

These two real-valued equations have a unique solu-tion if the feedback compensator has two independentparameters h"[h

1h2].

3. Control of vibration in multi-mass systems

A one-dimensional n-degree-of-freedom (n-DOF) sys-tem with the locally controlled absorber is depicted inFig. 1. Elastic couplings C

ibetween masses m

icomprise

a spring and a damper. It is assumed, that the absorber isattached to the mass m

pand that the source of harmonic

vibrations is at the mass mq.

3.1. The primary system alone * without absorber

The analysis of the primary system alone is restrictedto notions needed for later analysis of the combinedsystem with the LCA.

214 D. Filipovic& , D. Schro( der / Automatica 37 (2001) 213}220

Fig. 1. Multi-degree-of-freedom system with the LCA.

The equation of motion of the n-DOF system alonewithout an absorber can be written as

Mx( (t)#CxR (t)#Kx(t)"f(t), (6)

where M, C, K are (n]n) mass, damping, and sti!nessmatrices and x, f are (n]1) position and disturbancevectors (Meirovitch, 1990). These equations will be repre-sented in the Laplace domain as follows:

B(s)x(s)"f(s), (7)

where (the dependence upon the complex variable somitted)

B"CM

1!C

10

!C1

M2

!C2

!C2

} }

} Mn~1

!Cn~1

0 !Cn~1

Mn

D,x"C

x1

x2F

xnD, f"C

0

F

fqF

0 DQrow q (8)

and

Ci"c

is#k

i, i"0,1,2, n,

Mi"m

is2#C

i#C

i~1, i"1,2,2, n.

The system matrix B is tridiagonal: the matrix elementson the main diagonal b

iirepresent SDOF systems* the

mass with its connections to other masses * and otherelements b

ijrepresent only an elastic coupling between

masses miand m

j. If one end of the system is free, then

C0"0.The derivation of the solution for (7) is given in

the appendix. The "nal solution for system mass dis-placements with the factorisation of the numeratorpolynomial we give here

xr"

BrqB

fq, B

rq"Bm6 ~1

1

m6 ~1<m6

Bnm6 `1

, r"1,2, n (9)

where

m6"min(r, q), m6 "max(r, q) (10)

with the de"nitions of symbols given in the appendix. It isobvious that always B

rq"B

qr.

For the interpretation of solution (9) we shall examinethe case r(q, i.e. B

rq(s)"Br~1

1(s)<q~1

r(s)Bn

q`1(s): If the

subsystem comprising masses m1

to mr~1

, and represent-ed by Br~1

1, has low damping, the B

rqwill be close to zero

for the natural frequencies of that subsystem Br~11

, and itwill be a good absorber at those lightly damped "xedfrequencies u

i. Thus, if B

rq+0 then x

r+0 independent-

ly of the disturbance source position q. This is in accord-ance with MuK ller and Schiehlen (1985).

On the other hand, if Bnq`1

(s)+0 for some other fre-quencies u

j, then this subsystem comprising masses

mq`1

to mn

is a good absorber at uj. Therefore, the

disturbances at frequencies ujare absorbed by that sub-

system directly from the source and they are not spreadto the rest of the system. Hence, x

r+0 for all r4q. This

corresponds to and generalises results given in Klotter(1981, Section 5.22) where only purely elastic couplings(without damping) have been examined.

The factor <q~1r

(s) gives only negative real (stable)transmission zeros.

Remark 1. The masses miwithin subsystems Br~1

1(s) and

Bnq`1

(s) are supposed to have connections only to theirneighbouring masses. However, solution (9) holds alsofor more complex interdependencies within the parti-cular subsystem Bv

u, insofar as this system is connected to

the rest of the system with a single coupling.

3.2. Combined system with the absorber

The results to be derived here do not depend either onthe LCA type or the shape of the excitation force.

The combined (global) system with the absorber in-cluded is extended to (n#1)-DOF system

AxH"fH (11)

where

A"

F 0

F F

B#A3 F !Ca!F

a

F F

F 0

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

02!Ca!dF

a20 F M

a#F

a

Qrow p

Qrow a

Ccolumn p

Ccolumn a

(12)

D. Filipovic& , D. Schro( der / Automatica 37 (2001) 213}220 215

Fig. 2. Explanation of vibration control possibilities, n"8, p"3,q"6.

xH"Cx

2

xaD, fH"C

f

2

0 D (13)

and

a8ij"G

Ca#dF

afor i"j"p,

0 otherwise,

Ca"c

as#k

a, M

a"m

as2#C

a.

The form of matrix A re#ects our intention to allow onlythe absorber motion (state) signals to be used for thefeedback.

The new row and column in matrix A do not have tobe added as (n#1)th row and column in the systemmatrix. Therefore we shall call it the row a and thecolumn a.

The "nal solution for system mass displacements isderived in the appendix and is given as follows:

xr"

Arq

Afq, r"1,2,n, (14)

Arq"G

Bm6 ~11

m6 ~1<m6

An,am6 `1

, q(p and r(p, (a)

Brq

(Ma#F

a), r4p4q or q4p4r, (b)

Am6 ~1,a1

m6 ~1<m6

Bnm6 `1

, q'p and r'p. (c)

(15)

The de"nition of Av,au

can be found in the appendix. It isagain A

rq"A

qr.

This solution is valid for any LCA with an absolute orrelative state signal of the absorber only, and for anyshape of excitation forces.

Let us examine the consequences of solution (14) and(15) keeping in mind Fig. 1 where p(q (in this case (15a)does not apply):

(a) Solutions for xr, r4p according to (15b) all contain

factor (Ma(s)#F

a(s, h)) that equals the characteristic

equation of the absorber alone (Eq. (4)). Hence, if thisequation is solved for h, given the frequency u

c,

Ma( ju

c)#F

a( ju

c, h

c)"0, (16)

all vibration energy that comes from force fq

to massm

pwill be absorbed by the LCA at m

pand no energy

at this frequency is transmitted to any of mr, r4p.

This situation is depicted in Fig. 2a where the `boldamasses represent the masses brought to a standstill.Such an absorber forms a `screena that protects allmasses `behinda it from disturbances coming fromthe opposite side. Furthermore, once the feedbackparameters h in F

a(s, h) are set according to (16) the

solutions for other mi, i'p are given by Eq. (15c).

These solutions di!er from zero, Ai~1,a1

( juc, h

c)O0,

once (16) is satis"ed. Therefore, these masses con-tinue to vibrate, however, with smaller amplitudesince the resonator absorbs a part of vibrating en-ergy.

(b) If we want to bring any mass between mass mp

andmass m

qto a standstill, i.e. any mass m

r, p(r(q,

Eq. (15c) has to be used and solved for

Ar~1,a1

( juc, h

c)"0. (17)

This solution includes all masses mi, i"1,2, r!1

and the absorber mass ma. It can be said that

this absorber comprises all these masses mi,

i"1,2, r!1, a and the r-mass resonator that is soformed were attached to mass m

r, as shown in

Fig. 2b. From the point of view of the elementarysingle-mass LCA, it absorbs the vibrational energynot from the point of attachment, m

p, but from

a remote point, and thus the remote absorption isintroduced. In this case, provided Eq. (17) holds, themass m

ris the only one motionless system mass in

the steady state since Aiq( ju

c, h

c)O0 for all iOr.

(c) Solutions for xr, r5q all contain the same factor

Aq~1,a1

, Eq. (15c), which does not depend upon themass m

r. Solving

Aq~1,a1

( juc, h

c)"0, (18)


makes the remote absorber that includes massesm

i, i"1,2, q!1, a, attached at the mass m

q, and

the vibrational energy is, therefore, absorbed at thesource. Thus, all masses m

i, i5q are brought to

a standstill (Fig. 2c). Other masses mr, r"1,2,

q!1, a serve as parts of the remote absorber.(d) If by any chance q"p, the elementary LCA sup-

presses vibrations directly at the source and Eq. (15b)holds for each mass in the primary system. If condi-tion (16) is satis"ed, all vibrational energy at fre-quency u

cis taken over by the elementary LCA, and

all primary masses mr, r"1,2, n are motionless in

the steady state. This is the most desirable case.

Remark 2. The structure of the solution (14) and (15)reveals that the subsystems, generally represented by Bv

u,

<vu

and Av,au

, do not have to be one-dimensionally built,and that they can have a more complex structure. Theyonly have to be separable in factors, i.e. only the connec-tions between these subsystems should be represented bya single coupling C

i. As a matter of fact, these couplings

do not have to be built strictly as Ci"c

is#k

i; they only

have to be linear such that expressions (14) and (15) hold.

Taking into consideration the discussion given in a}d,a general strategy for the positioning of the absorber ina one-dimensional multi-mass system can be sum-marized:

(i) Attach the vibration absorber to the source of vibra-tions, if possible. Vibrations are then not spread toother parts of the system and all the vibration energyis absorbed directly at the source.

(ii) If an absorber cannot be attached to the source ofvibrations, it should be attached as close as possibleto the source, in order to localize the in#uence ofvibrations. The part of the system `behinda the ab-sorber, looking from the source of vibrations, will befreed from harmonic disturbances.

(iii) When the source of vibrations cannot be localized,or there is more than one source, the absorber shouldbe attached to the point that is very important fora particular case and that should stay vibrationlessfor all operating conditions.

(iv) The last and the most demanding solution is theremote absorption, which can be applied when thepoint to be freed from vibrations cannot be reached(the absorber cannot be attached to that point) andthat point is closer to the source of vibrations thanthe absorber is. In that case the absorber comprisesa part of the system including the absorber mass. Theexpression for formation of an ideal resonator fromthe multi-mass subsystem can be rather complex andtherefore a narrower stability range should be ex-pected.

4. Veri5cation of results

A simple simulation model is built to verify simulta-neous local and remote suppression at di!erent systempoints using a single locally controlled absorber. Experi-mental control of torsional vibrations in a three-massdrive system is presented afterwards. A harmonic excita-tion is applied in all cases.

4.1. Example of remote vibration control

Example 4.1. Results given in previous sections will beveri"ed on the two-mass primary system with the absorb-er attached at the mass m

1. For this system, we have

n"2, p"1, (q, r)3M1,2N.

The complete solution for primary mass displacementsusing (15) is given by

Cx1

x2D"

1

ACA

11A

12A

21A

22DC

f1f2D, (19)

where

CA

11A

12A

21A

22D"

CM

2(M

a#F

a) C

1(M

a#F

a)

C1(M

a#F

a) M

1(M

a#F

a)#(M

a!C

a)(C

a#dF

a)D

and A"(M1M

2!C2

1)(M

a#F

a)#M

2(M

a!C

a)(C

a#

dFa). Thus, putting M

a( ju)#F

a( ju,h)"0 gives A

11( ju)"

A12

( ju)"A21

( ju)"0 and the mass m1

is motionless,independently of where the source of harmonic distur-bance is. Also, if the disturbance acts on the mass m

1then

both masses are brought to a standstill. In order tosuppress vibrations at the source when the mass m

2is

acted upon by the harmonic disturbance f2, the remote

absorber that consists of Ma

and M1

is to be built bysolving A

22( ju,h)"0.

Assume the following primary system parameters:m

1"4 kg, c

1"5 N s/m, k

1"100 N/m, and m

2"2kg,

c2"5N s/m, k

2"50N/m with the absorber para-

meters ma"1kg, c

a"0.3N s/m, k

a"10 N/m.

Suppose that the mass m1

is acted upon by a harmonicforce f

1with frequency u

1and the mass m

2by another

harmonic force f2

with frequency u2. A solution is

searched for, such that both disturbances are absorbeddirectly at the particular source.

Consequently, two complex-valued equations shouldbe solved simultaneously, namely A

11( ju

1, h)"0 and

A22

( ju2, h)"0. This leads to four real-valued equations

and, therefore, the feedback compensator should havefour independent parameters. Let us use the velocityfeedback and a compensator with the structure

Ga(s)"g

1#s¹1#s2¹2

21#s¹

3#s2¹2

4

, (20)


Fig. 3. Vibration suppression with the dual-frequency resonating ab-sorber * the LAR and the remote LAR operate simultaneously.

where h"[g,¹1,¹

2,¹

4] and ¹

3is an additional free

parameter that is set in this example to ¹3"0.1 s.

For given frequencies u1"6 rad/s and u

2"8 rad/s

the solution is

Ga(s)"!0.6

1#1.095s#0.136s2

1#0.1s#0.00365s2. (21)

The response of the combined system with the feedbackcompensator (21) is shown in Fig. 3. For the frequencyu

1the elementary single mass LAR suppresses all vibra-

tional energy at u1

from the mass m1. Thus, the rest of

the system is not in#uenced by the disturbance f1

at all.The absorber together with m

1, c

1,k

1is at the same time

an ideal resonator at frequency u2

and the harmonicdisturbance f

2is completely absorbed from the mass m

2.

The mass m1

as a part of the remote absorber vibrateswith frequency u

2(the energy from the force f

1has been

completely absorbed by the absorber mass ma). The ab-

sorber mass performs biharmonic oscillations.If the system would have more resiliently coupled

masses between m2

and the ground, and provided thisnew system is stable, all these new masses would bemotionless in steady state, because both harmonic forceshave been suppressed directly at their respective source.

4.2. Experimental vibration control

In order to verify the features of LCAs, laboratoryexperiments are conducted at the Institute for Electric

Drives, Technical University of Munich. It is chosen tobuild a torsional electric drive system with a torquecontrolled ac servo-machine as the actuator.

The primary system comprises a dc machine with anadditional load mass. The electric motor is either speedor current (torque) controlled by a line converter. Thecontroller is built of an induction servomotor* with aninternal resolver* controlled by a frequency converter.The velocity is measured using incremental encoders.

An elastic coupling between the absorber and the loadmass is designed and built at the Institute with its equiva-lent sti!ness k

aand damping c

a.

The absorber feedback is designed in the Matlab-Simulink environment. The generated C code is down-loaded into a DSP card whose primary CPU is a TexasInstruments TMS320C31 microcontroller operating at40MHz. The control sampling time is set to ¹

d"1ms.

The D/A converters generate the output control torque.All stabilizable parameter pairs g,¹ for our laboratory

system are calculated for the feedback compensator withthe transfer function g/(1#s¹). The stability analysis inFilipovicH (1998) showed that the operating points arestable for suppression frequencies u(50 rad/s.

The vibration control in the whole combined system isdepicted in Fig. 4. The primary system is driven by thespeed reference n

13%&"n

$##n

!#sinut, with u"30 rad/s.

The exercise is to suppress the undesired dither n!#

sinut.The base speed n

$#"380 rpm is set. The absorber and

primary responses (na

and n1) as well as the control

torque fa

are displayed in Fig. 4. At t"0.55 s the feed-back control is activated and the absorber is tuned to thedither frequency u. The steady state is reached afterapprox. 0.4 s and the vibration is suppressed by about20dB.

5. Conclusions

The paper presents a general solution in the compactform for the system behaviour (mass displacements) ina one-dimensional multi-mass elastic linear system whichcomprises one locally controlled absorber (LCA). A strat-egy for placement of LCA in the multi-mass system hasbeen suggested in order to achieve the desired vibrationcontrol.

The given solution also reveals the possibility of theremote vibration control: LCA feedback parameters canbe controlled such that any point of special concern inthe system, even the one further away from the point ofattachment, can be freed from vibrations. In the case ofremote suppression, the controller design depends on thereduced set of system parameters.

These theoretical considerations have been proved ex-perimentally using a torsional three-mass drive system.The presented vibration control is robust in the presenceof primary system perturbations. However, it is essential


Fig. 4. Experimental results of vibration suppression for n$#+380 rpm.

that the LCA structure and its parameters are known,since the variations in passive parameters of the LCAwould in#uence the suppression ratio, if not followed byappropriate adjustment of the feedback controller.

Possible applications of such a local vibration controlis in multi-mass systems which cannot be e$ciently dir-ectly controlled, be it because of inability of the motioncontroller to appropriately in#uence the system behav-iour, or because the source of vibrations is too far awayfrom the present actuator. Another possibility for ap-plication of LCAs is the replacement of already presentpassive absorbers, or the improvement of vibrationsuppression by adding the locally controlled actuatorto the passive absorber. This would be the case for e.g.structural control, car suspension control, control insteel and paper mills.

Appendix A

A.1. Derivation of the solution for the primary systemwithout an absorber

We are looking for the solution for system mass dis-placements x

i, i"1,2, n which will be appropriately

factorized to make the concept of LCA applicable.The solution for the displacement of the particular

mass mr

can be found using the Cramer's rule

xr"

detBr

detB"

BrqB

fq, (A.1)

where Brq

is a cofactor of the element brq

of the matrix B,obtained by suppressing all elements in the row r andcolumn q. This suppression brings elements C

ionto the

main diagonal of Brq

between elements brr

and bqq

. Aftercarrying out elementary row and column transforma-tions, the determinant B

rqcan be brought to the form of

a lower triangular determinant of determinants

Brq"(!1)r`qDB

(m6 1~1)D D!C

m6D2D!C

m6 ~1DDB( nm6 `1

) D

(A.2)

for 1((r, q)(n, where m6

and m6 are de"ned with (10),and the annotation B(v

u) stands for the determinant of

the principal quadratic submatrix

B(vu)"K

buu

2 buv

F } F

bvu

2 bvvK. (A.3)

According to the Laplace expansion for determinants(Blanus\ a, 1963), it follows from (A.2):

Brq"B

(m6 1~1)C

m62C

m6 ~1B( nm6 `1

) (A.4)

In order to achieve an elegant solution for Eq. (7) thefollowing annotation is introduced as the extension of thede"nition domain:

v<u

"G1, u'v,v

<i/u

Ci, u4v,


Bvu"G1, u'v,

B(vu) u4v. (A.5)

Then, we have the solution for (7) given with (9).

A.2. Derivation of the solution for the combined systemwith the absorber

The solution for the displacement of the particularmass m

rcan be found using the same methodology as

before for the primary system

xr"

detAr

detA"

Arq

Afq

(A.6)

The similar derivation procedure holds for Arq

as beforefor B

rq. With a preliminary assumption that 1(r(p, as

shown in Fig. 1 (due to symmetry, the case p(r(n canbe examined by analogy) is

Arq"A(m6 !1)1

Cm62C

m6 ~1A( n,am6 `1

) for 1((r, p)(n,

(A.7)

where A(v,au )

is the extension of A(vu)

by ath row and ath

column

A(v, au )

"Kauu

2 auv

aua

F } F F

avu

2 avv

ava

aau

2 aav

aaaK. (A.8)

For p4q(n the determinant A(n, am6 `1

)decomposes

further into two parts:

A(n, am6 `1

)"A( nm6 `1

) ) (Ma#F

a) for p4q(n. (A.9)

We introduce the following extension of the de"nitiondomain:

Av,au

"G1, u'v,B(v,a

u), u4v. (A.10)

Furthermore, it is obvious that Avu"Bv

usince they both

do not depend on absorber parameters. The determinantAv,a

ucan be represented by the elements of the primary

matrix B using the theorem for the edged determinantgiven in Blanus\ a (1963)

Av,au

(s)"(Ma#F

a)Bv

u#(M

a!C

a)

](Ca#dF

a)Bp~1

uBvp`1

. (A.11)

However, it is not factorizable. Using the same proced-ure, as presented in (A.7) and (A.9), for all possible p's, q's

and r's the complete solution for the problem (11)* ourmain goal * is derived and shown with (14) and (15).

References

Blanus\ a, D. (1963). Vis\ a matematika, I dio, prvi svezak: algebra ialgebarska analiza. Tehnic\ ka knjiga, Zagreb (in Croatian).

Engell, S. (Ed.). 1995. Entwurf nichtlinearer Regelungen. R. OldenbourgVerlag, MuK nchen (in German).

FilipovicH , D. (1998). Resonating and Bandpass Vibration Absorbers withLocal Dynamic Feedback. Ph.D. thesis, Technische UniversitaK tMuK nchen, Munich, Germany.

Freeman, R. A., & KokotovicH , P. V. (1996). Robust nonlinear controldesign. Boston: BirkhaK user.

FilipovicH , D., & SchroK der, D. (1996). Multiple-frequency vibration sup-pression with the linear active absorber. In Proceedings of PEMC '96conference I. Nagg, S. Hala& sz and K. Kusurtz (Eds.), Budapest,September vol. 1 (pp. 58}65) (invited paper).

FilipovicH , D., & SchroK der, D. (1999). Vibration absorption with linearactive resonators * continuous and discrete time design andanalysis. Journal of Vibration and Control 5, 685}708.

Gupta, M. M., & Sinha, N. K. (Eds.). 1996. Intelligent control systems,theory and applications. New York: IEEE Press.

Hunt, J. B. (1979). Dynamic vibration absorbers. London: MechanicalEngineering Publishers Ltd.

Isidori, A. (1995). Nonlinear control systems (3rd ed.). Communicationsand control engineering series. Berlin: Springer.

Klotter, K. (1981). Technische Schwingungslehre, vol. 2 (2nd ed.). Berlin:Springer (in German).

Korenev, B. G., & Reznikov, L. M. (1993). Dynamic vibration absorbers,theory and technical applications. Chichester, UK: Wiley.

Meirovitch, L. (1990). Dynamics and control of structures. New York:Wiley.

MuK ller, P. C., & Schiehlen, W. O. (1976). Lineare schwingungen.Wiesbaden, Germany: Akademische Verlagsgesellschaft (in German).

MuK ller, P. C., & Schiehlen, W. O. (1985). Linear vibrations. Mechanics:Dynamical systems. Dordrecht: Martinus Nijho!. (Englishtranslation).

Damir FilipovicH received Dipl. Ing. andM.Sc. in Electrical Engineering fromUniversity of Zagreb in 1990 and 1993respectively, both in the "eld of control ofelectrical drives. He received Dr.-Ing. fromTechnical University of Munich in 1998 inthe "eld of control of vibrations. He wasa recepient of DAAD research scholarship.Since 1998 he is with MTU Aero Engines,Daimler Chrysler, as system engineer withresponsibilities in software development,control, and safety for jet engine systems.

Dierk SchroK der (M'84}SM'91) was born in1941. He received the Dipl.-Ing. and Dr.-Ing. degrees from the Technical UniversityDarmstadt, Germany, in 1966 and 1969,respectively. He spent 10 years in di!erentpositions with Asea, Brown & Bovery(ABB). In 1979 he was named Professorand Chairman of the Institute of Elec-tronics and Power Electronics at theUniversity Kaiserslautern. In 1983 he wasnamed Professor and Chairman of the In-stitute of Power Electronics and Electrical

Drives at the Technical University Munich, Germany.


control of vibrations in multi-mass systems with locally controlled absorbers

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