delayed resonator with speed feedback – design and performance analysis

Delayed resonator with speed feedback – designand performance analysis

Damir Filipovi�cc a,1, Nejat Olgac b,*

a Institute for Electrical Drives, Technical University of Munich, Arcisstr. 21, 80333 Munich, Germanyb Department of Mechanical Engineering, University of Connecticut, 191 Auditorium Road, U-139, Storrs,

CT 06269-3139, USA

Received 8 July 1999; accepted 8 August 2000

Abstract

In this work a tunable torsional vibration absorption mechanism, the delayed resonator

(DR) is studied. The tuning feedback used is time delayed proportional control on the angular

velocity of the absorber. Dynamic analysis of the absorber and its tuning features are pre-

sented. Single- and dual-frequency resonance characteristics are introduced, both of which are

achieved owing to the added delay in control. Combined system stability, for such time de-

layed dynamics and relevant topics of relative stability and dominant pole placement are

discussed. A design tool is suggested based on the property called, the degree-of-stability.

Experimental results are also presented, using a torsional vibration setup involving electric

motor drives. They support the theoretical findings strongly. � 2002 Elsevier Science Ltd. All

rights reserved.

Keywords: Torsional vibrations; Vibration absorption; Resonator; Delayed resonator; Stability; Time-

delay

1. Introduction

The paper elaborates on a recently introduced real-time tunable vibration ab-sorption strategy, the delayed resonator (DR) [1–3]. The underlying control strategyis a partial state feedback with time delay. It has been shown that this controlstructure can convert the absorber into a resonator, with a tunable resonance

Mechatronics 12 (2002) 393–413

*Corresponding author. Tel.: +1-860-486-2382; fax: +1-860-486-5088.

E-mail addresses: [email protected] (D. Filipovi�cc), [email protected]

(N. Olgac).1 Fax: +49-89-289-28336.

0957-4158/02/$ - see front matter � 2002 Elsevier Science Ltd. All rights reserved.

PII: S0957-4158 (00 )00065-9

frequency. This resonator functions as a perfect vibration absorber against a tonalexcitation even when its frequency is varying with time.

The DR absorber is studied for transverse vibration cases in the earlier efforts[1–4,7,12]. Its usage for torsional oscillations was first introduced by Filipovi�cc andOlgac [3]. This paper presents two contributions. First is on the analysis of single-and dual-frequency resonances for delayed speed feedback. It is shown that thetorsional DR can be tuned to resonate at not only one but two distinct frequencies.At these settings the DR can absorb bi-tonal oscillations. These two frequencies,however, are not tunable, i.e., they are fixed for a given passive absorber. Both thedirect problem (i.e., the analysis) and the indirect problem (i.e., the synthesis) of theabsorber for a given pair of frequencies are presented.

Second contribution is on the stability measure of the operation. When the DR iseffectively suppressing the tonal vibrations, the combined system (the primary systemand the DR absorber) should remain asymptotically stable. This feature is studiedfrom various points-of-view. The characteristic equation representing the combinedsystem exhibits the form of a quasi-polynomial, i.e. a polynomial with exponential,e�ss delay terms where s represents the time delay. This class of systems are not easyto analyze [5,6,8,9]. They possess infinitely many finite characteristic roots due to thetranscendentality of the equation. The dominant (i.e., the rightmost) root dictates

Nomenclature\ anglej j absolute value (module)d e the ceiling functionR the real part of complex numberI the imaginary part of complex numberc; ca torsional stiffnessd; da torsional dampingfL; fa the load torque and the feedback torqueg; gc; g� gain of the feedbackJ1; J2; Ja moments of inertial root locus branch identifier (counter)n1; n2; na speed of rotation (r/min)q squared frequency ratio of two DFFDR frequenciess the complex variablet timeTd sampling timev the complex variables; sc; s� time delay of the feedbacku1;u2;ua angle of rotationfa damping coefficientxa natural frequency of passive absorberxc;xc1;xc2 resonant frequencies of DR

394 D. Filipovi�cc, N. Olgac / Mechatronics 12 (2002) 393–413

the system behavior. The stability, tuning speed (i.e., settling) characteristics arestudied using this information to impart a better design of the absorber.

Three different approaches can be found in the literature for the analysis of suchtime delayed systems:

(i) root locus analysis [2,7],(ii) modified Nyquist method [2],(iii) stability chart method [2,7].

This text revisits (iii) with the aim of introducing a new interpretation for the degree-of-stability. It is organized as follows: In Section 2 single frequency DR with speedfeedback is presented. Section 3 contains direct and indirect problem of dual fre-quency fixed DR (DFFDR). Section 4 addresses the stability issues including newresults of degree-of-stability analysis for determining the limitations of the meth-odology. In Section 5 the description of the laboratory set-up for torsional DR isgiven. Experimental results are presented in Section 6.

2. DR with speed feedback

Vibration in rotational systems, such as electrical drives, turbines, ship drives, canbe suppressed substantially at critical points of the structures by torsional dynamicabsorbers. A recent technique is taken into account in this work, the DR vibrationabsorber [1,4]. DR can enforce complete vibration absorption against time varyingsingle frequency excitation as it is demonstrated both analytically and experimentallyin the earlier work. The implementation of DR methodology for torsional oscilla-tions is considered in this paper.

A typical torsional DR with speed feedback is shown in Fig. 1. The symbols usedin this figure and throughout the text are defined in the nomenclature. The feedbackshown uses either the absolute speed na of the absorber mass (solid lines in Fig. 1b)or its relative speed with respect to the primary system (the dashed line). Taking thenominal speed of the primary as constant (for simplicity, n ¼ 0), the governingequation of motion of DR is derived as

Ja €uuaðtÞ þ da _uuaðtÞ þ cauaðtÞ � g _uuaðt � sÞ ¼ 0 ð1Þ

Fig. 1. The structure (a) and the dynamic model (b) of the rotational DR.

D. Filipovi�cc, N. Olgac / Mechatronics 12 (2002) 393–413 395

of which the characteristic equation is

CEðsÞ Jas2 þ dasþ ca � ge�sss ¼ 0: ð2ÞThis equation is transcendental and is often referred to as the quasi-polynomial in theliterature [8].

Without feedback (g ¼ 0) the DR represents a stable passive absorber with two

poles s1;2 ¼ �faxa � jxa

ffiffiffiffiffiffiffiffiffiffiffiffiffi1� f2a

qwhere xa ¼

ffiffiffiffiffiffiffiffiffiffiffica=Ja

pis the undamped natural fre-

quency and fa ¼ daxa=2ca is the damping ratio.When g takes very small positive (g ¼ 0þ) or negative values (g ¼ 0�), (Eq. 2) has

two finite and infinitely many unbounded roots. This equation is a retarded func-tional differential equation (RFDE) [8,9] and is proven to be stable for small g values(obviously so for g ¼ 0).

When g increases maintaining the delay, s, constant, the roots of CEðsÞ get closerto the imaginary axis, Fig. 2. For a certain critical gain gc one pair of poles reach theimaginary axis. For this operating point the two imaginary poles induce the domi-nant behavior, implying that the DR is a true resonator. The root s ¼ �jxc dependson both the gain gc and the time delay sc. For g larger than gc the DR structure isunstable, i.e., the dominant imaginary roots move to the unstable right half complexplane. As these roots are on the imaginary axis, all other roots should remain in thestable left half-plane of s.

The characteristic equation CEðsÞ should be satisfied for the marginally stableroot s ¼ jxc. Imposing this condition on (2) and separating the real and imaginaryparts gives the solution

Fig. 2. A root-locus plot for varying gain and constant delay.


gc ¼ ð�1Þl Jaxc

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2faxaxcÞ2 þ ðx2

a � x2cÞ

2q

; ð3Þ

sc ¼1

xc

lp�

þ arctanx2

a � x2c

2faxaxc

�; l ¼ 0; 1; 2; . . . ð4Þ

There are multiple solutions for the time delay sc but gc is single valued for a givenxc. The multiplicity of sc causes the branching of root-locus plot, as shown in Fig. 2[1].

3. Dual frequency fixed DR (DFFDR)

In [1,2] it is shown that for some particular delays, s, two pairs of poles can beplaced on the imaginary axis for some corresponding gains g. The DR exhibits tworesonant frequencies for such cases despite the fact that the underlying mechanicalstructure is a single degree-of-freedom (SDOF) system.

The DFFDR design problem is solved here, for the speed feedback case. Similarsteps are taken as in [12], where only the position feedback is considered. Twoproblems are posed here:

(a) Direct problem: Given a passive absorber – inertia, stiffness and damping – findall double frequency resonators, i.e., all pairs gc; sc for DFFDR and correspond-ing resonant frequencies xc1;xc2.(b) Inverse problem: Given two resonant frequencies, design a DFFDR that willresonate with these two frequencies, i.e., find Ja, da, ca, gc, sc.

3.1. Direct problem

It will be shown in the next section that, DFFDR feature appears only on twoadjacent branches of the locus (i.e., l ¼ 0; 2 or 2,4 etc. for g > 0). The expressions(3,4) are reconsidered for the two imaginary pairs of poles with resonance fre-quencies xc1, and xc2. They satisfy the following:

gcðxc1Þ ¼ gcðxc2Þ; ð5Þ

scðxc1; lÞ ¼ scðxc2; lþ 2Þ; l ¼0; 2; 4; . . . for gc > 0;

1; 3; 5; . . . for gc < 0;

�ð6Þ

Condition (5) gives a simple relationship

xc1xc2 ¼ x2a: ð7Þ

It is trivial to show analitically that xc1jl < xc2jlþ2. In other words, for a given shigher numbered branch crosses the imaginary axis at a higher frequency. This canbe observed from the root locus plot of Fig. 2.

A dimensionless quantity q is introduced as

q ¼ xc1

xa

¼ xa

xc2

¼ffiffiffiffiffiffiffixc1

xc2

r< 1: ð8Þ


Using this notation Eq. (6) becomes

Eðq; lÞ q2 � 1ð Þlþ 2q2

1þ q2p � arctan

1� q2

2faq

� �¼ 0: ð9Þ

Eðq; lÞ parametrically depends only on the damping ratio fa. This expression isshown to increase monotonously in q within the interval of interest q 2 ð0; 1Þ, seeFig. 3.

The parameter q corresponding to a particular damping ratio fa can be foundnumerically for different branch identifiers l. This concludes the direct problem, i.e.,we find the resonant frequencies using (8) and the gain gc and the delay sc using (3)and (4).

By analyzing the expression Eðq; lÞ we can conclude more about the frequencyratio q. The second term in Eðq; lÞ is always smaller than p=2, because only the coremeaning of arctan is taken into account here. The first term in (9) should also besmaller than p=2. This condition results in a limitation for q as

q2 ¼ xc1

xc2

61þ 2l3þ 2l

ð10Þ

or for different l values

A few observations are made from this table:• The frequency ratio has an upper limit, depending on the branches selected to car-

ry the resonant poles. This feature of DFFDR frequencies appears in this simpleform only for speed feedback utilization for the absorber. For position and accel-eration feedback cases similar conditions to (7)–(9) arise only in much more com-plex forms [4,12].

• For increasing values of q 2 ð0; 1Þ the number of viable branches decreases, e.g.,for q ¼ 0:5, l ¼ 0; 1; 2; . . ., however for q ¼ 0:75, l ¼ 1; 2; . . ..

l 0 1 2 3 ...

q2 6 1/3 3/5 5/7 7/9 ...

Fig. 3. The function Eðq; lÞ for fa ¼ 0:98.


• The two frequencies corresponding to DFFDR form a nested set, i.e.,

xc1jl¼0 < xc1jl¼1 < xc1jl¼2 < ;

xc2jl¼0 > xc2jl¼1 > xc2jl¼2 > :

This is also observed from Fig. 3. Notice that the solutions of (9) are

qðl ¼ 0Þ < qðl ¼ 1Þ < qðl ¼ 2Þ < ;

depending on the branch of interest. Using the definition of q this translates into

xc1jl¼0 < xc1jl¼1 < ;

xc2jl¼0 > xc2jl¼1 > ;

which concurs with the nesting feature above. This point was not observed in theearlier investigations, i.e., for position and acceleration feedback studies. It mayform a good basis for the design of the DFFDR, and for the identification of theoperating branches.

3.2. Indirect problem

For the indirect problem, two complex valued equations CEðjxc1Þ ¼ 0 andCEðjxc2Þ ¼ 0 should be solved for five unknowns ðJa; ca; da; gc; scÞ. The real andimaginary parts of the two equations yield four real equations. Thus, there is one freeparameter which should be selected possibly based on some desirable design features.For example the inertia Ja can be selected as a free parameter in order to limit themass ratio between the absorber and the primary. In most practical applicationshigher mass ratios than 10% would be unacceptable. Once Ja is prescribed, the so-lution for the remaining unknowns is given in the following expressions (the deri-vation of which are in Appendix A):

xa ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffixc1xc2

p; ð11Þ

q ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffixc1=xc2

p; ð12Þ

l ¼ 3q2 � 1

2ð1� q2Þ

� ; ð13Þ

sc ¼2p

xc1 þ xc2

ðlþ 1Þ; ð14Þ

fa ¼1� q2

2q tanxc1sc: ð15Þ

The symbol d e designates the ceiling function – the smallest integer number which islarger than the argument. From (11) and (15) da and ca can be evaluated for a givenJa. This concludes the design of DR.


4. Stability analysis

Desired operation of DR is achieved by using the gain g and the time delay s asper Eqs. (3) and (4). By definition these values render the DR structure marginalstability. When such an absorber is installed on a vibrating primary, however, theglobal system – the primary and the DR together – must be asymptotically stable.Since the only determining factor for g and s is the excitation frequency x, thestability of the global system should be guaranteed a priori for the full operatingfrequency range.

Several methodologies have been used in the earlier studies in order to resolvethese stability concerns [1,2,4,7]. Root locus analysis and a modified Nyquist im-plementation have been pursued in [1,2,7]. A more practical scheme, stability chartmethod, is presented in [4,7] to state the nature of the ‘combined system’ stability ing; s-plane. That is, once a DR is combined with a primary, and a certain excitationfrequency, x, is declared, this chart states the stability of the operation for therespective gðxÞ, sðxÞ feedback parameters. Indeed, these parametric plots revealthe desired stability (and instability) zone in x, as explained in the followingparagraphs.

The stability chart method follows the D-subdivision concept [8,10,11]. It emanatesfrom the continuity property of the roots of quasi-polynomials for varying param-eters (say s and g). This property indicates that for two sets of parameters½s1 g1�; ½s2 g2� there is at least one continuous path beginning at the correspondingroots of one polynomial p1ðs; s1; g1Þ and ending at another p2ðs; s2; g2Þ. If that pathdoes not include roots on the imaginary axis jx (i.e., marginal stability) bothpolynomials have the same stability property: they are either stable or unstable.

The stability is then defined in the boundary crossing theorem [11] that states: If theparameters of a stable polynomial are changing in an interval and its roots do notcross the imaginary axis, this polynomial is stable within the parameter interval ofconcern. Then the obvious goal of the D-subdivision method is to find all marginallystable roots in the parameter plane. The method consists of two steps. First, theparameters ½s�; g�� are found in order to enforce purely imaginary roots. The solu-tions divide the parameter plane into regions with the same stability property, ormore accurately, with the same number of unstable roots, say m. Crossing the linebetween two regions means that one real pole (or one complex pair of poles) crossesthe imaginary axis. In the second step the stable region is found. If in one region theparameter set ½s0 g0� represents stable characteristic equation, then the stability ispreserved under all continuous variations of ½s g� within that region. As statedearlier, it is guaranteed that the RFDE is always stable for g ¼ 0. Then the startingpoint ½s0 g0� is selected for this subdivision. Usually the stability regions are denotedby DðmÞ where m is the number of unstable poles, so that Dð0Þ indicates the stableregion.

In the first step we find the parameters ½s� g�� to enforce purely imaginary rootsfor

pðjx; s�; g�Þ ¼ Rfpðjx; s�; g�Þg þ jIfpðjx; s�; g�Þg ¼ 0:


The equations Rfpðjx; s�; g�Þg¼ 0, Ifpðjx; s�; g�Þg¼ 0 should be solved simulta-neously for two parameters s� and g�. The parametric curves s�ðxÞ, g�ðxÞ are thenplotted in the s; g-plane.

As given in [1] the characteristic equation of the global system can be written as

pðs; g; sÞ AðsÞ þ ge�ssPðsÞ ¼ 0; ð16Þwhere AðsÞ and P ðsÞ are polynomials in s, and the condition degAðsÞ > deg P ðsÞtypically holds. Therefore the global structure is a retarded differential equation(RFDE).

The solutions for s ¼ jx are

s�ðxÞ ¼ 1

xpl½ þ \P ðjxÞ � \AðjxÞ�; ð17Þ

g�ðxÞ ¼ ð�1Þl AðjxÞP ðjxÞ

; l ¼ 0; 1; 2; . . . ð18Þ

By plotting the parametric graph s�; g�, the stable regions are easy to determine. Thiswill be explained in more details using the model of the experimental setup.

Remark. By definition, expressions (17) and (18) can also be directly used to find theoperating points of the DR as well, if the characteristic Eq. (2) of the DR is usedinstead of the characteristic equation of the global system, (16). In other wordsAðsÞ ¼ mas2 þ dasþ ca and P ðsÞ ¼ s, yield sc ¼ s�, gc ¼ g� for DR alone, isolatedfrom the primary. The comparison of these two sets of stability charts yield theoperable and inoperable frequency regions as explained later in Section 5.

4.1. Degree of stability

For practical implementations the degree-of-stability (DOS) of the system is im-portant. It implies the distance of the dominant characteristic roots of (16) from theimaginary axis. DOS can also be considered as a stability margin, as well as the‘tuning speed’ indicator of the absorber. It is clear that, if the excitation frequencyshows a step variation, this dominant pole will dictate the settling time of the ef-fective absorption. The largest real part of all roots of the system characteristicequation (16) is of interest

maxi

ðRfsigÞ ¼ �r0: ð19Þ

This number determines DOS of the system. It can be interpreted as the C-stability[11] where the C-region is the half-plane Rfsg < �r0.

For the characteristic equation (16) we require a stability margin in which the realpart �r0 is smaller than a certain �r�. This can be achieved by mapping the halfplane Rfsg < �r� via transformation s ¼ v� r� onto the left half plane of the newcomplex variable v. The necessary and sufficient condition for DOS to be better than�r0 is that the new ‘shifted’ equation in v is stable, which can be assessed again byusing D-subdivision method.


If the shifted equation

C0ðvÞ Aðv� r�Þ þ ge�ðv�r�ÞsPðv� r�Þ ¼ 0

A0ðv; r�Þ þ ge�ðv�r�ÞsP 0ðv; r�Þ ¼ 0 ð20Þ

has poles on the imaginary axis, then the original characteristic Eq. (16) has poles onthe s ¼ �r� vertical line.

A similar procedure for the D-subdivision follows. Solving RfC0ðjxÞg ¼ 0 andIfC0ðjxÞg ¼ 0 for g�

0and s�

0gives

s�0 ðxÞ ¼ 1

xpl0�

þ \P 0ðjx; r�Þ � \A0ðjx; r�Þ�; ð21Þ

g�0 ðxÞ ¼ ð�1Þl

0e�r�s�

0 A0ðjx;r�ÞP 0ðjx; r�Þ

; l ¼ 0; 1; 2; . . . ð22Þ

For r� ¼ 0 the solution is the same as Eqs. (17,18). With r� > 0 the expressionfor s�

0is still of the same ‘shape’ as (17), but the expression for g�

0has an ad-

ditional factor e�r�s�0compared to expression (18). This additional factor has an

important meaning. Without it the solutions would be of the similar shape as forthe stability margin for the characteristic Eq. (16). Here, the gain is weighted bythe factor e�r�s�

0so that g�

0decreases exponentially as the delay s�

0increases. This

means that the stability reserve of the system decreases exponentially with thetime delay s. Hence, to move the dominant poles to the same distance r, smallergains g� should be selected. Furthermore with larger delays the absorber settlingtime (also known as the tuning speed) increases which is not desirable. Thereforethe lowest possible branch number l should be used with smallest positive timedelay s.

An expected consequence of enforcing a higher DOS is a reduction in the oper-ating frequency range. That is, if dominant poles are allowed to be closer to theimaginary axis the operable frequency range becomes more relaxed. This feature willbe better visualized in the examples of the following section.

A numerical procedure is also developed to determine these dominant poles. For agiven operating point ðgc; scÞ we seek the particular one of the infinitely many so-lutions s ¼ r� þ jx of the characteristic Eq.(16) for which r� is the smallest. This stillseems like very difficult task, because it lends itself into solving two nonlinear andtranscendental equations for infinitely many discrete pairs of ðr�;xÞ. There is,however, an additional piece of valuable information we can utilize: For a given scthere can only be one particular root locus branch which carries the root �r� � jx(see Fig. 2) for a given gc. It has been shown earlier that the operable sections of anybranch have frequency limits, say x 2 ½xi;xj�, both of which are known [2,7,12].Then, the mentioned two nonlinear equations are solved numerically forðr�;x 2 ½xi;xj�Þ existence of which can be shown.

This constrained numerical search is performed for any given ðgc; scÞ in orderto determine the DOS (which is characterized by r�). The procedure is demon-strated in the numerical analysis section for the experimental set-up for furtherclarity.


5. Experimental set up and results

In order to demonstrate the above findings a laboratory test is conducted. Theprimary system consists of a 2.7 kW DC motor with the rotor inertiaJ1 ¼ 0:0334 N m s2 and the load inertia J2 ¼ 0:2112 N m s2. It is modeled as a two-mass system, as shown in Fig. 4, where the stiffness and damping between the motorand the load mass are c � 122000 N m=rad and d � 1:27 N m s=rad. The firstelectric motor is either speed or current (torque) controlled by a rectifier (line con-verter). An incremental encoder with 5000 lines per revolution is used for angularmeasurements. A second dc motor controls the actuator for the absorber.

The passive elements of the absorber are selected for operating in the frequencyrange of (10�20) Hz. For this a fictitious DFFDR (dual frequency resonance) issynthesized with the 2 limiting frequencies taken as the two resonances of concern. Itis known that this absorber functions stably in the interval of xc1 < x < xc2 [12].Given xc1 ¼ 62:8 radð10 HzÞ=s, xc2 ¼ 125:4 rad=sð20 HzÞ, expressions (11)–(15)yield xa ¼ 88:9 rad=s, fa ¼ 0:194. With the rotor inertia Ja ¼ J1 as the absorberinertia, ca ¼ 263:7 N m=rad, da ¼ 1:152 N m s=rad are calculated. After the cou-pling is built using standard parts available, the actual parameters are measured asca � 390 N m=rad, da � 0:877 N m s=rad (xa � 108 rad=s, fa � 0:122). The devia-tion in these properties, obviously, introduces different passive absorber featuresthan those desired. The measured values are embedded in the model and used in allthe ensuing analysis.

The DR feedback is programmed simply in Matlab–Simulink environment,Fig. 5, and downloaded into a DSP card with TMS320C31 chip. The sampling timeis set to Td ¼ 1 ms. The line converter on the DR side is used to mimic sinusoidalexcitation torques up to 20 Hz in frequency.

For obvious reasons the feedback using the absolute shaft velocity is not ap-propriate, instead the control torque is formed using the relative velocity between the

Fig. 4. Schematic representation of laboratory set-up.


two inertias (the load and the absorber). A stability analysis shows that with therelative speed feedback the global system becomes very close to the stability margin,as seen in Fig. 6. Clearly the stable operating points (part of the dashed line whichremains on the side of abscissa with respect to the solid line) are very close to themarginal stability.

Another critical implementation detail is to remove the low frequency contentsof the absorber speed (i.e., extracting the dc component or drift). This is achievedby processing the absorber speed signal through a very low cut-off filter, such as 1rad/s (see Fig. 5, block ‘DC speed’). This process removes the undesired low fre-quency bias which appears due to numerical differentiation of the encoder readings.The quadrature encoder unit outputs 20,000 lines per revolution which correspondsto a resolution of 0.018�. This position measurement is transformed into speed withsimple difference quotient Du=Td that gives the smallest measurable speed differenceof �3 r/min. This is the upperbound of the velocity drift mentioned earlier.The whole Simulink program runs in approximately 330 ls of loop speed on theDSP.

The stability chart corresponding to this system is depicted in Fig. 6. The pa-rameter plane is divided into many regions by curves ½s�; g�� which are defined by(17,18). As already stated, the only stable region the one sandwiched between thesolid lines for + and )g’s (middle plot).

The operating points sc; gc of the DR are plotted in the same graph with dashedlines using parametric expressions of (4,3). The DFFDR points A1;A2; . . . can befound from (8,9) and (4,3)

Point A1 A2 A3 A4 xc1 (rad/s) 57.6 78.6 86.4 90.7 xc2 (rad/s) 202.5 148.3 135 128.6 gc N m s 4.92 )2.49 1.84 )1.54 sc m s 24.1 55.4 85.2 114.6

Fig. 5. Matlab–Simulink depiction of the test setting.


Fig. 6. Stability chart of the laboratory set-up with stable frequency ranges (upper for g > 0, lower for

g < 0).


Unfortunately, the DFFDRs for this system are in regions Dð2Þ and thereforeunstable. The stable frequency ranges for g > 0 and g < 0 are also shown in Fig. 6.For instance, to suppress the excitation frequency within the ‘stable’ zone of g > 0(upper figure) one can find the appropriate g and s by going to the respective plot inthe middle figure. It is clear that some of these stable operating frequencies may beserviced on one or more branches. For instance x ¼ 100 rad/s is common tol ¼ 0; 1; 2; 3; for both + and )g’s.

Notice that the natural undamped frequency xa of the passive absorber in thiscase can be suppressed without any delay (sc ¼ 0). For this operating point we havegc ¼ da. This property can also be used as a system identification scheme. We canfind the stiffness and damping coefficients of ca; da by detecting the marginally stableresponse of the absorber for sc ¼ 0.

The curves s�0; g�0 of the constant DOS are depicted in the upper part of Fig. 7.They are obtained from (21,22). Observe that the points B1 and B2 have the same

Fig. 7. Stability chart with the degree-of-stability curves (upper) and the real part of the front running

poles (lower) of the laboratory model, for g > 0.


resonant frequency xc. The exponential weighting from (22) indicates that the pointB2 has worse absorption properties – it is closer to the stability margin than thepoint B1 (see lower part of Fig. 7). Therefore the point B1 should be preferred tooperate.

The lower part of Fig. 7 is generated by the numerical procedure for finding thefront running root as explained in Section 4.1. From the relative stability point of

Fig. 9. Double-frequency resonance of the DR for sc ¼ 0.12 s and gc ¼ �1:61 N m s/rad.

Fig. 8. Impulse responses of the absorber for s ¼ 0:01 s and varying g’s.


Fig. 10. Comparison of modelled and measured operating points.

Fig. 11. Experimental results showing vibration absorption for ndc ¼ 0.


view, it is observed that the points near the DFFDRs are the most critical operatingpoints (see the cusps at these regions).

The model of the set up analyzed in earlier sections, does not include sensor andactuator dynamics, nonlinearities in the absorber elastic coupling and modellingdiscrepancies due to the discrete implementation of the feedback. The experimentaleffort presented here addresses issues much more global than the nuances createdby such influences. Therefore, at this stage of experimental demonstration, wefeel omfortable in comparing the present model and the ensuing experimentalresults.

First, only the DR characteristics are studied, blocking the load mass J2 withoutan excitation torque on the primary. Fig. 8 shows the responses of the DR to animpulse torque disturbance for different feedback gains g while the delay s ¼ 0:01 sis kept constant. The passive absorber (g ¼ 0) is stable, and with increasing gthe absorber reaches the stability margin (g ¼ 2:31) and then becomes unstable.Fig. 9 shows the double resonance of the DR achieved for sc ¼ 0:12 s,gc ¼ �1:61 N m s=rad. The respective frequencies are xc1 ¼ 83 rad=s, xc2 ¼131:5 rad=s.

In Fig. 10 experimentally observed operating DR points are compared with theanalytical counterparts. The differences are remarkably small. This is the testimonyof ignorable influences of the non-modelled features as discussed above.

Fig. 12. Experimental results of vibration absorption at base speed ndc ¼ 400 r/min.


The second experiment shows the absorption. The primary system is disturbed asshown in Fig. 4, n1ref ¼ ndc þ nac sinxt, and x ¼ 84 rad=s is used. Fig. 11 shows theabsorption for ndc ¼ 0. At t � 0:2 s the feedback control is activated and the ab-sorber is tuned to the excitation frequency. The steady state is reached after approx.0.4 s and the vibration is suppressed by about 24 dB. The control parametersgc ¼ �1:221 N m s=rad, sc ¼ 41:1 m s are used for this experiment.

Similar absorption characteristics are obtained at rotating speed ndc ¼ 400 r=min,Fig. 12. The best absorption (around 20 dB) is achieved with gc ¼�0:951 N m s=rad, sc ¼ 44:1 ms. Note the differences of gc and sc compared to thecase with ndc ¼ 0, although the excitation frequency remains the same, i.e.,x ¼ 84 rad=s. These differences originate from the drive train irregularities in theline converter effects. Again, they are acceptable at this stage for proving the un-derlying concept. Furthermore, a natural frequency of the primary system at 304 Hzwas experimentally observed which coincided with the current ripple of the lineconverter at 300 Hz. Therefore this mode is inevitably excited, which is obvious in n2trace of Fig. 12. Despite this undesired excitation the absorption quality at frequency13.4 Hz (¼ 84 rad=s) is not noticeably affected.

The simulations given in Figs. 13 and 14 reflect the absorber tuning speeds. Theycorrespond to the points of operation at A and B in Fig. 7. As mentioned earlierthe DOS is directly correlated to these tuning speeds. Apparently point B has a

Fig. 13. Simulation results for the absorption at 110 rad/s, point A.


higher DOS than A, therefore absorption takes place faster (in 1.5 s as opposed to3.5 s).

6. Conclusion

This work presents further elaboration on DR vibration absorption methodologyin particular for torsional vibration cases. Electric drives are taken as the applicationobjectives; both theoretical and experimental results are discussed. Following con-tributions are presented.

Delayed speed feedback is considered as opposed to position or acceleration. Thisnatural proposition for electric drive applications is also shown to offer some simpleranalytical steps for the DFFDR.

As a design tool for selecting the structural parameters of the DR, a DFFDRlinked approach is suggested. It is based on the observation that, for a given fre-quency range a DR can be simply formed by setting these frequency limits as thedual frequencies of DFFDR and using this absorber as the basis for DR imple-mentation. On another front, using the DOS concept it is proved analytically thatthe stability margin decreases exponentially for increasing feedback delays.

Several experimental observations are presented in the text. They show betterthan 20 dB absorption over the passive absorber, although the absorber mechanismat hand and its operating properties (friction etc.) are far from ideal.

Fig. 14. Simulation results for the absorption at 99 rad/s, point B.


Acknowledgements

The authors acknowledge the financial support of DAAD to the first author andof SEW–Eurodrive Foundation to the second author during this work. The studywas carried out at the Institute of Electrical Drives at the Technical University ofMunich. The cooperation of the institute personnel and their director Prof. D.Schr€ooder are also appreciated.

Appendix A. Solving the indirect DFFDR problem

From the direct problem equation (7) xc1xc2 ¼ x2a which yields the natural fre-

quency xa. The only absorber parameter remaining is the damping ratio fa, or al-ternatively the damping constant da. Simple mathematical manipulations over theequations CEðjxc1Þ¼ 0, CEðjxc2Þ¼ 0 result in

cosxc1sc ¼ cosxc2sc; ð23Þ

da ¼ Jax2

a � x2c2

xc2 tanxc2sc; ð24Þ

gc ¼da

cosxc2sc: ð25Þ

Solving Eq. (4) for fa in terms of xc, and setting the condition fajxc¼xc1¼ fajxc¼xc2

yields

tanxc1sc þ tanxc2sc ¼ 0: ð26ÞEqs. (23) and (26) give cosðxc1 þ xc2Þsc ¼ 1, which implies:

sc ¼2p

xc1 þ xc2

ðlþ 1Þ; l ¼ 0; 1; 2; . . . ð27Þ

From (27) the smallest sc is found, which gives positive da (damping constant) in (24).Smaller time delay is preferred from practicality standpoint although the larger scsolutions are also valid mathematically. These findings lead to the DFFDR betweenlth and ðlþ 2Þth branch.

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delayed resonator with speed feedback – design and performance analysis

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