On the role of nonsynchronous rotating damping in rotordynamics

Nonsynchronous rotating damping, i.e. energy dissipations occurring in elements rotating at a speed different from the spin speed of a rotor, can have substantial effects on the dynamic behaviour and above all on the stability of rotating systems. The free whirling and unbalance response for systems with nonsynchronous damping are studied using Jeffcott rotor model. The system parameters affecting stability are identified and the threshold of instability is computed. A general model for a multi-degrees of freedom model for a general isotropic machine is then presented. The possibility of synthesizing non-synchronous rotating and nonrotating damping using rotor-and stator-fixed active dampers is then discussed for the general case of rotors with many degrees of freedom.


INTRODUCTION
Damping plays different roles in rotordynamics.It is well known that nonrotating damping is always stabilizing, while the damping of the rotating parts of the machine has a destabilizing effect on whirl- ing motions occurring in supercritical conditions (Dimentberg, 1961; Lalanne and Ferraris, 1990;   Genta, 1998).The effect of damping depends on whether the energy dissipation occurs in stationary or rotating parts of the machine: an eddy current damper with a stationary permanent magnet and a rotating conducting disc can cause severe instability in the supercritical regime (Genta et al., 1996).
Apart from dampers which are stationary or rotate at the same spin speed of the rotor, there can be cases in which energy is dissipated in an element rotating at a different speed, both in the same direction of the rotor (co-rotating damping) or in the .oppositedirection (counter-rotating damping).
A very well known case is the heuristic model of lubricated bearings (Crandall, 1956; 1982; Lund,  1987).The oil film is assumed to rotate at half the spin speed, with the result of giving way to an 468 G. GENTA AND E. BRUSA unstable behaviour, usually referred to as oil whip.The threshold of instability is located at a speed identified on the Campbell diagram by the inter- section of the curve related to the first forward mode with the line of equation A =co/2.In many cases its value is close to twice the first critical speed.
This model has been generalized (Muszynska,  1986; 1990; .1991;1995; Muszynska and Bently,  1989; Muszynska and Grant, 1991) to include the case of fluid rotating with an average velocity dif- ferent from A =co/2.The fluid is so modelled as a damper rotating at a speed COd rco (in the men- tioned papers, the symbol A is used for the ratio cod/co; here a different symbol has been introduced to avoid confusion with the whirl speed A), obtaining a deeper insight in the fluid-induced instabilities of rotating machinery.In Muszynska (1989) the antiswirl effect obtained by injecting a fluid in the rotor-stator clearance in a direction opposite to rotation is studied in terms of non- synchronous rotating damping, eventually counter- rotating.This model can be used for fluid-rotor interactions of all types, as those occurring in bearings, seals, blade-tip gaps, etc., and show that the threshold of instability is very much affected by ratio r, strongly increasing with decreasing r.
The results regarding nonsynchronous rotating damping found in the literature are usually limited to the case of co-rotation of the damping or slow counter-rotation, with speeds lower than the spin speed.
The availability of active dampers has recently broadened the application of nonsynchronous damping to cases in which no fluid-rotor interac- tion is involved.In rotors supported in active mag- netic bearings (AMB), the damping effect is due to the derivative action of the control loop which keeps the rotor in position.Active damlers can also be added to rotors supported in conventional bearings, using techniques derived from AMB technology.
If an active device is present, nonsynchronous damping can be synthesized without the need of resorting to a material device which actually rotates at the required speed: the damper can be stator-fixed or even rotor-fixed and the nonsynchronous damping action can be obtained by choosing a suitable control law.In particular, if digital techniques are used, the choice of the speed ratio r can be defined by the program of the control microprocessor.
In the case of statorless rotors, e.g.spinning spacecrafts, a damping action rotating at a speed different from that of rotation can be easily obtained using active dampers.Stability can thus be achieved even in cases for which passive stabil- ization is difficult (Genta and Brusa, 1997; Nobili  et al., 1998).
The aim of the present paper is studying the effects of nonsynchronous damping on the dynamic behaviour of rotors, firstly using the Jeffcott rotor model and then a general multi-degrees of freedom isotropic model.The control laws which allow one to synthesize such damping action will then be dealt with in detail.

Equations of Motion
Consider a damped Jeffcott rotor running at con- stant speed (Genta, 1998).Assume that a damper, with damping coefficient Cd, rotating at a speed COd--rCO, is present together with the usual non- rotating and rotating damping Cn and Cr.
By resorting to the complex coordinate z x / iy (Dimentberg, 1961; Genta, 1998), the equation of motion can be written in the form m5 + c: + (k-iCOCr-iCOdCd)Z Fre it + Fn, (1)   where m, k and c are the mass, stiffness and total damping (c Cn + Cr + Cd), respectively.Force Fr rotates at the spin speed (Fr meCO 2 if it is due to the eccentricity e of.the centre of mass) and Fn is fixed in the inertial frame, although possibly being time-dependent.A general nonsynchronous force, FnseiCo*t, rotating at a speed CO* different from CO, can be added.
Note that the same equation can be written using real coordinates, obtaining rn 0 + Fny (2) Equation (2) shows clearly that a circulatory matrix is present due to the rotating (synchronous and nonsynchronous) damping.The damping ma- trix has a stabilizing effect while the circulatory one reduces the stability of the system.

Free Whirling
The homogeneous equation associated with Eq. ( 1) allows one to study the stability of the system.By assuming a solution of the type z-z0 ei'xt, the fol- lowing characteristic equation is obtained" -m, 2 + ic3 + k-icocr icodCd 0. (3) By introducing the nondimensional whirl and spin speeds X-A/cocr, cot_ CO/cocr and co cod/cocr, where cocr V/-/rn is the critical speed of the undamped system, and the damping ratios (z, defined in the usual way (Genta, 1998), the solu- tions of the characteristic equation can be shown to be where 1-" (1 2)/2 and E co'r + cod.
The expression of the imaginary part of the complex whirl speed, i.e. the decay rate, depends on the sign of the nondimensional parameter =" :: where the upper and lower signs indicate forward and backward whirling, respectively.
Stability when E > 0 From Eq. ( 6) it follows immediately that the backward mode is always stable.The condition for stability of the forward mode is < By introducing the ratio r-co/col between the speed of rotation of the damper Cd and the spin speed, the condition for stability becomes < (8) r if-rd A particular case is that of counter-rotating damping with r=-l.The nondimensional thres- hold of instability, which for synchronous damping has a value (/r, is now equal to (/(r-d)-Stability when < 0 In this case the forward mode is always stable.The condition for stability of the backward mode is Conclusions on stability A stability chart is shown in Fig. 1.Forward modes are always stable for counter-rotating damping with Icol >co'r/d, while for counter-rotating damping with a speed lower (in absolute value) than the mentioned value and for co-rotating damping the condition for sta- bility is expressed by Eq. ( 7).
Backward modes are always stable for co- rotating or for counter-rotating damping with IcoI < cotr/d' while for counter-rotating damping with a speed higher (in absolute value) than the mentioned value the condition for stability is ex- pressed by Eq. ( 9).
It is possible to find a value of r which assures a stable working of the machine at any speed for each value of rotating damping.A fairly large zone about rd/r--1, where the value of the threshold of stability is very high, can be found.No threshold of instability exists if rd/r 1.Note that nonsynchronous damping can make the system unstable even at standstill (i.e. for co 0), the condition for stability being Icocl < n + fir + d.
(11) Cd In the case of nonrotating systems the sign of cod has no meaning as the direction of rotation of the damper is immaterial.

Unbalance Response
The unbalance response can be computed by intro- ducing a solution of the type z-z0 exert into Eq.( 1).
The relevant algebraic equation yielding the amplitude of the response is then [1 -con + 2iCnco' + 2i(co'-co)d]Z0 eco '2.(12)   The real and imaginary parts and the absolute value of the nondimensional response are co,z(1 co,z) z0 {(1-co'2) 2 q-4co'2 I('n-+-( 1Systems with Structural Damping In many cases both rotating and nonrotating damp- ing can best be modelled as hysteretic damping. The nonsynchronous damper will be assumed to be of the viscous type and to have a damping coeffi- cient equal to Cd.By resorting to the usual small damping assumptions for hysteretic damping (Ramanujam and Bert, 1983;Genta, 1998), the characteristic equation for free whirling can be shown to be --m, 2 -+k + i[-+-r/k -+-r/rkr -+-Cd(/ COd)] O, (14) where the first double sign is (+) for forward whirling and (-) for backward whirling, while the second one is (+) in subcritical conditions and (-)   in supercritical conditions and backward whirling.
The stability assessment is in this case less straightforward, as the expression for . .depends whether the rotor is working in subcritical forward whirling, supercritical forward whirling or backward whirling conditions.
In the first case (subcritical forward whirling), the expression for E is r/nkn q-r/rkr Cdcod 2m It can he readily shown that if E > 0 the motion is always stable.
(1 6) Cd Condition (16) states that speed cod must be either negative (counter-rotating damping) or small enough in case of a co-rotating damper.
In the second case (supercritical forward whirl- ing), the expression for E is /nkn -/Trkr Cdcod 2m If E > 0 the motion is always stable.If E < 0 the condition for stability is nkn f/rkr CCOd "( -1-cocr.

7)
Cd Again condition (17) states a maximum value for cod for achieving stability.Nonsynchronous damping enhances stability when cod < cocr and acts to reduce stability if cod > COcr.
In the third case (backward whirling), the expres- sion for E is , -flnkn ]rkr Cdcod If E < 0 the motion is always stable if COd > 0, i.e. if the damper is co-rotating.In case of counter- rotating damping the condition for stability is rlnkn -t-r]rkr ICOd] < (1 8) Cd If E > 0 the condition for stability is 77nkn q-]rkr Cd If COd is positive or negative with ]COd] < cocr non- synchronous damping is stabilizing, while its effect is opposite and if [cOal > cocr.

ROTOR WITH MANY DEGREES OF FREEDOM
The general equation of an isotropic, multi-degrees of freedom rotor with nonsynchronous damping can be written, with reference to the complex coordinates q, in the form described in Genta (1998)" Mi + (C-iwG)l + (K + Kw 2 -iwfr-idCd)q Fn + co2Fr eict. (20) Matrix K has been introduced to take into account centrifugal stiffening and C Cn / C; / Cd.The rotating forcing function has been assumed to be due to unbalance and factor co 2 has consequently been extracted from vector Fr to state it explicitly.
Equation ( 20) can be used for the study of free and forced behaviour and the stability of the system.Although little can be said in general, the conclusions drawn from the Jeffcott rotor, i.e. that counter-rotating damping generally increases the stability of forward modes while decreasing that of backward ones, still holds in a qualitative way.

SYNTHESIZING ROTATING AND NONROTATING DAMPING
Stator-fixed Active Damper As already stated, a material device which rotates at the nonsynchronous speed Wd is not needed, if ac- tive dampers are used.Consider a rotor with many degrees of freedom with both rotating and non- rotating damping, provided of an active damper supplying the control force Fc.Equation (20) modi- fies as Mii + (C -iG)l + (K + K -iCr)q Fn + co2Fr eit + Fc.

Fcy
--Cd(/l) + codCd(q) The equation of motion ( 21) can be written with reference to the state space in the usual way Az + Bnun / BrurCo 2 / Bcuc, (24 where the real state vector is z-[R(/I) 3: (/1) 3: (q)X (q)X] x, ( 25) and the dynamic matrix is If the nonrotating and rotating inputs Un and Ur reduce to the corresponding force vectors Fn and Fr, the input gain matrices are simply The simplest way to synthesize nonsynchronous rotating damping is by using a number of actuators and sensors positioned at given locations of the rotor.In the following it will be assumed that each actuator is able to supply two control forces in directions x and y and that the force due to the ith actuator is assumed as the ith control input The corresponding input gain matrix is then -Ii Bc. 0 where 5i is a matrix with n rows (n is the number of the real degrees of frecdom of the system) and 2 columns with the two nonvanishing entries in the rth and sth columns, corresponding to the degrees of freedom for the displacements at the actuator location in x and y directions respectively.
Each sensor is assumed to read the velocities and the displacements in directions x and y at a given location, which in general is not the same as the corresponding actuator location (non-co-located actuators and sensors).
The ith output vector is then Si-[% y; xi The output-gain matrix is then a matrix with 4 rows and 2n columns having the structure Ci 0 ;T where matrix is the same as matrix i with the only difference that the nonzero elements are located at the rth and s'th rows corresponding to the sensor positions.The two matrices are exactly the same in the case of co-located sensors .andactuators.
In the simple case of decentralized control, in which the ith actuator is driven by the ith sensor, an ideal proportional control law can be stated by simply closing the loop with the constant-gain matrix Pi UGi---eiYi, (28) where 0 0 -a] (29) P---Cdio 0.) d 0 This control-gain matrix allows one to synthesize exactly a nonsynchronous damper with constant damping coeffcient Cdi rotating at the speed Wd.It is clearly a highly idealized result, as no sensor, actuator, or controller dynamics has been taken into account and the velocity at the sensor location has been assumed to be readily available.
In any practical application, the actual transfer function of the various elements in the control loop must be carefully accounted for and a more complex control law, at least of the PID type, must be implemented.The standard technology devel- oped for active magnetic bearings can however be used.

Rotor-fixed Active Damper
The active damper considered in the previous section is stator-fixed, i.e. is physically not rotating.The same effect can however be obtained by using an active damper which rotates together with the rotor, as it is mandatory, for instance, in the case of statorless .(orfree in Crandall, 1995) rotors.In such a case the equation of motion is better written in a rotor-fixed reference frame, obtaining where r=qe-it.Equation ( 30) is equivalent to Eq. ( 20).
Operating as in the previous case, the following expression for the control-gain matrix yielding an ideal proportional controller is obtained: Similarly, it is possible to synthesize nonrotating damping using a rotor-fixed active damper by intro- ducing the following control-gain matrix:

CONCLUSIONS
The role of nonsynchronous damping in rotordynamics, has been studied.In particular, its effect on the stability and the unbalance response of a Jeffcott rotor has been dealt with in detail.
Co-rotating damping has been shown to decrease the stability of the system.By increasing the rota- tional speed of the damper, the stability of forward modes is reduced.
Moderate counter-rotating damping, i.e. a counter-rotating damping with low damping coeffi- cient or characterized by a speed with low absolute value, increases the stability of forward modes while reducing that of backward modes, which however remain stable.A value of the product of the damping coefficient and the speed causing both forward and backward modes to be always stable was found.
Backward modes can become unstable, while forward ones show no threshold of instability, if a stronger counter-rotating damping is present.
A general model for an axi-symmetric rotating machine with many degrees of freedom was de- scribed.Even if no general result can be obtained in this case, the conclusions seen for the Jeffcott rotor still hold in a qualitative way.
Nonsynchronous rotating dampers require the presence of an element rotating at a speed which is different from that of the rotor and thus have limited practical applications.However, as active dampers can synthesize nonsynchronous damping without the need of a material object rotating at the required speed, there is no difficulty in incorpo- rating this form of damping in systems running on magnetic bearings.
If active dampers are used, the devices synthe- sizing nonrotating damping or damping rotating at any arbitrary speed can all be located within the rotor.This is particularly interesting in statorless rotors as spinning spacecraft, in which the only conventional way of applying nonrotating damping is to introduce de-spun devices.
FIGUREStability chart for the forward and backward whirl modes.Nondimensional parameter cO'(r/( as a function of rCd/ Economic and environmental factors are creating ever greater pressures for the efficient generation, transmission and use of energy.Materials developments are crucial to progress in all these areas: to innovation in design; to extending lifetime and maintenance intervals; and to successful operation in more demanding environments.Drawing together the broad community with interests in these areas, Energy Materials addresses materials needs in future energy generation, transmission, utilisation, conservation and storage.The journal covers thermal generation and gas turbines; renewable power (wind, wave, tidal, hydro, solar and geothermal); fuel cells (low and high temperature); materials issues relevant to biomass and biotechnology; nuclear power generation (fission and fusion); hydrogen generation and storage in the context of the 'hydrogen economy'; and the transmission and storage of the energy produced.As well as publishing high-quality peer-reviewed research, Energy Materials promotes discussion of issues common to all sectors, through commissioned reviews and commentaries.The journal includes coverage of energy economics and policy, and broader social issues, since the political and legislative context influence research and investment decisions.S SU UB BS SC CR RI IP PT TI IO ON N I IN NF FO OR RM MA AT TI IO ON N Volume 1 (2006), 4 issues per year Print ISSN: 1748-9237 Online ISSN: 1748-9245 Individual rate: £76.00/US$141.00Institutional rate: £235.00/US$435.00Online-only institutional rate: £199.00/US$367.00For special IOM 3 member rates please email s su ub bs sc cr ri ip pt ti io on ns s@ @m ma an ne ey y. .cco o. .uuk k E ED DI IT TO OR RS S D Dr r F Fu uj ji io o A Ab be e NIMS, Japan D Dr r J Jo oh hn n H Ha al ld d, IPL-MPT, Technical University of Denmark, Denmark D Dr r R R V Vi is sw wa an na at th ha an n, EPRI, USA F Fo or r f fu ur rt th he er r i in nf fo or rm ma at ti io on n p pl le ea as se e c co on nt ta ac ct t: : Maney Publishing UK Tel: +44 (0)113 249 7481 Fax: +44 (0)113 248 6983 Email: subscriptions@maney.co.uk or Maney Publishing North America Tel (toll free): 866 297 5154 Fax: 617 354 6875 Email: maney@maneyusa.comFor further information or to subscribe online please visit w ww ww w. .mma an ne ey y. .cco o. .uuk k C CA AL LL L F FO OR R P PA AP PE ER RS S Contributions to the journal should be submitted online at http://ema.edmgr.comTo view the Notes for Contributors please visit: www.maney.co.uk/journals/notes/emaUpon publication in 2006, this journal will be available via the Ingenta Connect journals service.To view free sample content online visit: w ww ww w. .i in ng ge en nt ta ac co on nn ne ec ct t. .cco om m/ /c co on nt te en nt t/ /m ma an ne ey y